LMFDB - Embedded newform 1050.3.f.e.601.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1050,3,Mod(601,1050)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1050.601");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1050.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$28.6104277578$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 88 x^{14} - 476 x^{13} + 2744 x^{12} - 10640 x^{11} + 39126 x^{10} - 108488 x^{9} + \cdots + 33124 $ x^16 - 8x^15 + 88x^14 - 476x^13 + 2744x^12 - 10640x^11 + 39126x^10 - 108488x^9 + 259514x^8 - 486436x^7 + 690168x^6 - 725188x^5 + 619745x^4 - 430504x^3 + 108130x^2 + 42224*x + 33124
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			601.2
Root			$-0.207107 - 0.264184i$ of defining polynomial
Character	$\chi$	$=$	1050.601
Dual form			1050.3.f.e.601.6

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} +2.44949i q^{6} +(-0.853218 + 6.94781i) q^{7} -2.82843 q^{8} -3.00000 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} +2.44949i q^{6} +(-0.853218 + 6.94781i) q^{7} -2.82843 q^{8} -3.00000 q^{9} +2.88097 q^{11} -3.46410i q^{12} -13.8145i q^{13} +(1.20663 - 9.82568i) q^{14} +4.00000 q^{16} +24.1754i q^{17} +4.24264 q^{18} -6.53403i q^{19} +(12.0340 + 1.47782i) q^{21} -4.07430 q^{22} -28.8420 q^{23} +4.89898i q^{24} +19.5367i q^{26} +5.19615i q^{27} +(-1.70644 + 13.8956i) q^{28} +32.9589 q^{29} -2.43276i q^{31} -5.65685 q^{32} -4.98998i q^{33} -34.1892i q^{34} -6.00000 q^{36} -50.9799 q^{37} +9.24052i q^{38} -23.9275 q^{39} -21.5225i q^{41} +(-17.0186 - 2.08995i) q^{42} -13.5554 q^{43} +5.76193 q^{44} +40.7887 q^{46} -40.7305i q^{47} -6.92820i q^{48} +(-47.5440 - 11.8560i) q^{49} +41.8730 q^{51} -27.6291i q^{52} +17.2758 q^{53} -7.34847i q^{54} +(2.41326 - 19.6514i) q^{56} -11.3173 q^{57} -46.6109 q^{58} +1.47488i q^{59} -111.568i q^{61} +3.44045i q^{62} +(2.55965 - 20.8434i) q^{63} +8.00000 q^{64} +7.05690i q^{66} -120.293 q^{67} +48.3508i q^{68} +49.9558i q^{69} -90.3855 q^{71} +8.48528 q^{72} +21.4890i q^{73} +72.0964 q^{74} -13.0681i q^{76} +(-2.45809 + 20.0164i) q^{77} +33.8385 q^{78} +66.1324 q^{79} +9.00000 q^{81} +30.4375i q^{82} +78.5172i q^{83} +(24.0679 + 2.95563i) q^{84} +19.1703 q^{86} -57.0865i q^{87} -8.14861 q^{88} +90.9724i q^{89} +(95.9807 + 11.7868i) q^{91} -57.6840 q^{92} -4.21367 q^{93} +57.6016i q^{94} +9.79796i q^{96} -44.1972i q^{97} +(67.2374 + 16.7669i) q^{98} -8.64290 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4} - 48 q^{9}+O(q^{10}) $ 16 * q + 32 * q^4 - 48 * q^9	$ 16 q + 32 q^{4} - 48 q^{9} + 96 q^{11} - 16 q^{14} + 64 q^{16} + 24 q^{21} - 64 q^{29} - 96 q^{36} + 144 q^{39} + 192 q^{44} - 176 q^{46} - 224 q^{49} - 48 q^{51} - 32 q^{56} + 128 q^{64} - 384 q^{71} - 224 q^{74} + 608 q^{79} + 144 q^{81} + 48 q^{84} + 416 q^{86} + 224 q^{91} - 288 q^{99}+O(q^{100}) $ 16 * q + 32 * q^4 - 48 * q^9 + 96 * q^11 - 16 * q^14 + 64 * q^16 + 24 * q^21 - 64 * q^29 - 96 * q^36 + 144 * q^39 + 192 * q^44 - 176 * q^46 - 224 * q^49 - 48 * q^51 - 32 * q^56 + 128 * q^64 - 384 * q^71 - 224 * q^74 + 608 * q^79 + 144 * q^81 + 48 * q^84 + 416 * q^86 + 224 * q^91 - 288 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$701$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.41421			−0.707107
$2$	−1.41421			−0.707107
$3$		−	1.73205i		−	0.577350i
$3$		−	1.73205i		−	0.577350i
$4$	2.00000			0.500000
$5$		0			0
$5$		0			0
$6$			2.44949i			0.408248i
$7$	−0.853218	+	6.94781i	−0.121888	+	0.992544i
$7$	−0.853218	+	6.94781i	−0.121888	+	0.992544i
$8$	−2.82843			−0.353553
$9$	−3.00000			−0.333333
$10$		0			0
$11$	2.88097			0.261906			0.130953	−	0.991389i	$-0.458196\pi$
$11$	2.88097			0.261906			0.130953	+	0.991389i	$0.458196\pi$
$12$		−	3.46410i		−	0.288675i
$13$		−	13.8145i		−	1.06266i	−0.847166	−	0.531328i	$-0.821693\pi$
$13$		−	13.8145i		−	1.06266i	0.847166	−	0.531328i	$-0.178307\pi$
$14$	1.20663	−	9.82568i	0.0861880	−	0.701834i
$15$		0			0
$16$	4.00000			0.250000
$17$			24.1754i			1.42208i	0.703150	+	0.711041i	$0.251776\pi$
$17$			24.1754i			1.42208i	−0.703150	+	0.711041i	$0.748224\pi$
$18$	4.24264			0.235702
$19$		−	6.53403i		−	0.343896i	−0.985106	−	0.171948i	$-0.944994\pi$
$19$		−	6.53403i		−	0.343896i	0.985106	−	0.171948i	$-0.0550062\pi$
$20$		0			0
$21$	12.0340	+	1.47782i	0.573045	+	0.0703722i
$22$	−4.07430			−0.185196
$23$	−28.8420			−1.25400			−0.627000	−	0.779019i	$-0.715717\pi$
$23$	−28.8420			−1.25400			−0.627000	+	0.779019i	$0.715717\pi$
$24$			4.89898i			0.204124i
$25$		0			0
$26$			19.5367i			0.751411i
$27$			5.19615i			0.192450i
$28$	−1.70644	+	13.8956i	−0.0609441	+	0.496272i
$29$	32.9589			1.13651			0.568257	−	0.822851i	$-0.307618\pi$
$29$	32.9589			1.13651			0.568257	+	0.822851i	$0.307618\pi$
$30$		0			0
$31$		−	2.43276i		−	0.0784762i	−0.999230	−	0.0392381i	$-0.987507\pi$
$31$		−	2.43276i		−	0.0784762i	0.999230	−	0.0392381i	$-0.0124931\pi$
$32$	−5.65685			−0.176777
$33$		−	4.98998i		−	0.151212i
$34$		−	34.1892i		−	1.00556i
$35$		0			0
$36$	−6.00000			−0.166667
$37$	−50.9799			−1.37783			−0.688917	−	0.724840i	$-0.741914\pi$
$37$	−50.9799			−1.37783			−0.688917	+	0.724840i	$0.741914\pi$
$38$			9.24052i			0.243171i
$39$	−23.9275			−0.613525
$40$		0			0
$41$		−	21.5225i		−	0.524940i	−0.964940	−	0.262470i	$-0.915463\pi$
$41$		−	21.5225i		−	0.524940i	0.964940	−	0.262470i	$-0.0845371\pi$
$42$	−17.0186	−	2.08995i	−0.405204	−	0.0497607i
$43$	−13.5554			−0.315243			−0.157621	−	0.987500i	$-0.550383\pi$
$43$	−13.5554			−0.315243			−0.157621	+	0.987500i	$0.550383\pi$
$44$	5.76193			0.130953
$45$		0			0
$46$	40.7887			0.886712
$47$		−	40.7305i		−	0.866605i	−0.901248	−	0.433303i	$-0.857348\pi$
$47$		−	40.7305i		−	0.866605i	0.901248	−	0.433303i	$-0.142652\pi$
$48$		−	6.92820i		−	0.144338i
$49$	−47.5440	−	11.8560i	−0.970287	−	0.241959i
$50$		0			0
$51$	41.8730			0.821040
$52$		−	27.6291i		−	0.531328i
$53$	17.2758			0.325959			0.162979	−	0.986629i	$-0.447890\pi$
$53$	17.2758			0.325959			0.162979	+	0.986629i	$0.447890\pi$
$54$		−	7.34847i		−	0.136083i
$55$		0			0
$56$	2.41326	−	19.6514i	0.0430940	−	0.350917i
$57$	−11.3173			−0.198549
$58$	−46.6109			−0.803636
$59$			1.47488i			0.0249980i	0.999922	+	0.0124990i	$0.00397867\pi$
$59$			1.47488i			0.0249980i	−0.999922	+	0.0124990i	$0.996021\pi$
$60$		0			0
$61$		−	111.568i		−	1.82898i	−0.404609	−	0.914490i	$-0.632592\pi$
$61$		−	111.568i		−	1.82898i	0.404609	−	0.914490i	$-0.367408\pi$
$62$			3.44045i			0.0554911i
$63$	2.55965	−	20.8434i	0.0406294	−	0.330848i
$64$	8.00000			0.125000
$65$		0			0
$66$			7.05690i			0.106923i
$67$	−120.293			−1.79542			−0.897709	−	0.440590i	$-0.854769\pi$
$67$	−120.293			−1.79542			−0.897709	+	0.440590i	$0.854769\pi$
$68$			48.3508i			0.711041i
$69$			49.9558i			0.723997i
$70$		0			0
$71$	−90.3855			−1.27304			−0.636518	−	0.771262i	$-0.719626\pi$
$71$	−90.3855			−1.27304			−0.636518	+	0.771262i	$0.719626\pi$
$72$	8.48528			0.117851
$73$			21.4890i			0.294370i	0.989109	+	0.147185i	$0.0470212\pi$
$73$			21.4890i			0.294370i	−0.989109	+	0.147185i	$0.952979\pi$
$74$	72.0964			0.974276
$75$		0			0
$76$		−	13.0681i		−	0.171948i
$77$	−2.45809	+	20.0164i	−0.0319233	+	0.259953i
$78$	33.8385			0.433827
$79$	66.1324			0.837119			0.418559	−	0.908189i	$-0.362535\pi$
$79$	66.1324			0.837119			0.418559	+	0.908189i	$0.362535\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$			30.4375i			0.371189i
$83$			78.5172i			0.945990i	0.881065	+	0.472995i	$0.156827\pi$
$83$			78.5172i			0.945990i	−0.881065	+	0.472995i	$0.843173\pi$
$84$	24.0679	+	2.95563i	0.286523	+	0.0351861i
$85$		0			0
$86$	19.1703			0.222910
$87$		−	57.0865i		−	0.656166i
$88$	−8.14861			−0.0925978
$89$			90.9724i			1.02216i	0.859533	+	0.511081i	$0.170755\pi$
$89$			90.9724i			1.02216i	−0.859533	+	0.511081i	$0.829245\pi$
$90$		0			0
$91$	95.9807	+	11.7868i	1.05473	+	0.129525i
$92$	−57.6840			−0.627000
$93$	−4.21367			−0.0453083
$94$			57.6016i			0.612783i
$95$		0			0
$96$			9.79796i			0.102062i
$97$		−	44.1972i		−	0.455642i	−0.973703	−	0.227821i	$-0.926840\pi$
$97$		−	44.1972i		−	0.455642i	0.973703	−	0.227821i	$-0.0731600\pi$
$98$	67.2374	+	16.7669i	0.686096	+	0.171091i
$99$	−8.64290			−0.0873020
$100$		0			0
$101$			180.052i			1.78269i	0.453326	+	0.891345i	$0.350237\pi$
$101$			180.052i			1.78269i	−0.453326	+	0.891345i	$0.649763\pi$
$102$	−59.2174			−0.580563
$103$		−	107.208i		−	1.04086i	−0.853906	−	0.520428i	$-0.825772\pi$
$103$		−	107.208i		−	1.04086i	0.853906	−	0.520428i	$-0.174228\pi$
$104$			39.0734i			0.375706i
$105$		0			0
$106$	−24.4317			−0.230488
$107$	−33.1521			−0.309833			−0.154917	−	0.987928i	$-0.549511\pi$
$107$	−33.1521			−0.309833			−0.154917	+	0.987928i	$0.549511\pi$
$108$			10.3923i			0.0962250i
$109$	−108.819			−0.998338			−0.499169	−	0.866505i	$-0.666361\pi$
$109$	−108.819			−0.998338			−0.499169	+	0.866505i	$0.666361\pi$
$110$		0			0
$111$			88.2997i			0.795493i
$112$	−3.41287	+	27.7912i	−0.0304721	+	0.248136i
$113$	−157.658			−1.39520			−0.697600	−	0.716488i	$-0.745749\pi$
$113$	−157.658			−1.39520			−0.697600	+	0.716488i	$0.745749\pi$
$114$	16.0050			0.140395
$115$		0			0
$116$	65.9178			0.568257
$117$			41.4436i			0.354219i
$118$		−	2.08580i		−	0.0176763i
$119$	−167.966	−	20.6269i	−1.41148	−	0.173335i
$120$		0			0
$121$	−112.700			−0.931405
$122$			157.781i			1.29328i
$123$	−37.2781			−0.303074
$124$		−	4.86553i		−	0.0392381i
$125$		0			0
$126$	−3.61990	+	29.4770i	−0.0287293	+	0.233945i
$127$	−28.2047			−0.222085			−0.111042	−	0.993816i	$-0.535419\pi$
$127$	−28.2047			−0.222085			−0.111042	+	0.993816i	$0.535419\pi$
$128$	−11.3137			−0.0883883
$129$			23.4787i			0.182006i
$130$		0			0
$131$			157.620i			1.20321i	0.798795	+	0.601604i	$0.205471\pi$
$131$			157.620i			1.20321i	−0.798795	+	0.601604i	$0.794529\pi$
$132$		−	9.97996i		−	0.0756058i
$133$	45.3972	+	5.57495i	0.341332	+	0.0419169i
$134$	170.120			1.26955
$135$		0			0
$136$		−	68.3784i		−	0.502782i
$137$	−244.855			−1.78726			−0.893632	−	0.448800i	$-0.851852\pi$
$137$	−244.855			−1.78726			−0.893632	+	0.448800i	$0.851852\pi$
$138$		−	70.6482i		−	0.511943i
$139$		−	238.245i		−	1.71399i	−0.515325	−	0.856995i	$-0.672329\pi$
$139$		−	238.245i		−	1.71399i	0.515325	−	0.856995i	$-0.327671\pi$
$140$		0			0
$141$	−70.5472			−0.500335
$142$	127.824			0.900172
$143$		−	39.7992i		−	0.278316i
$144$	−12.0000			−0.0833333
$145$		0			0
$146$		−	30.3900i		−	0.208151i
$147$	−20.5352	+	82.3487i	−0.139695	+	0.560195i
$148$	−101.960			−0.688917
$149$	−10.8398			−0.0727503			−0.0363751	−	0.999338i	$-0.511581\pi$
$149$	−10.8398			−0.0727503			−0.0363751	+	0.999338i	$0.511581\pi$
$150$		0			0
$151$	−112.999			−0.748340			−0.374170	−	0.927360i	$-0.622072\pi$
$151$	−112.999			−0.748340			−0.374170	+	0.927360i	$0.622072\pi$
$152$			18.4810i			0.121586i
$153$		−	72.5262i		−	0.474027i
$154$	3.47627	−	28.3075i	0.0225732	−	0.183815i
$155$		0			0
$156$	−47.8549			−0.306762
$157$			103.693i			0.660462i	0.943900	+	0.330231i	$0.107127\pi$
$157$			103.693i			0.660462i	−0.943900	+	0.330231i	$0.892873\pi$
$158$	−93.5253			−0.591932
$159$		−	29.9226i		−	0.188193i
$160$		0			0
$161$	24.6085	−	200.389i	0.152848	−	1.24465i
$162$	−12.7279			−0.0785674
$163$	199.576			1.22439			0.612196	−	0.790706i	$-0.290287\pi$
$163$	199.576			1.22439			0.612196	+	0.790706i	$0.290287\pi$
$164$		−	43.0451i		−	0.262470i
$165$		0			0
$166$		−	111.040i		−	0.668916i
$167$		−	87.0777i		−	0.521424i	−0.965417	−	0.260712i	$-0.916043\pi$
$167$		−	87.0777i		−	0.521424i	0.965417	−	0.260712i	$-0.0839572\pi$
$168$	−34.0372	−	4.17990i	−0.202602	−	0.0248803i
$169$	−21.8411			−0.129237
$170$		0			0
$171$			19.6021i			0.114632i
$172$	−27.1109			−0.157621
$173$			106.597i			0.616166i	0.951359	+	0.308083i	$0.0996875\pi$
$173$			106.597i			0.616166i	−0.951359	+	0.308083i	$0.900313\pi$
$174$			80.7325i			0.463980i
$175$		0			0
$176$	11.5239			0.0654765
$177$	2.55458			0.0144326
$178$		−	128.654i		−	0.722778i
$179$	−275.881			−1.54123			−0.770617	−	0.637298i	$-0.780052\pi$
$179$	−275.881			−1.54123			−0.770617	+	0.637298i	$0.780052\pi$
$180$		0			0
$181$		−	186.431i		−	1.03000i	−0.857189	−	0.515002i	$-0.827791\pi$
$181$		−	186.431i		−	1.03000i	0.857189	−	0.515002i	$-0.172209\pi$
$182$	−135.737	−	16.6690i	−0.745808	−	0.0915882i
$183$	−193.241			−1.05596
$184$	81.5775			0.443356
$185$		0			0
$186$	5.95903			0.0320378
$187$			69.6485i			0.372452i
$188$		−	81.4609i		−	0.433303i
$189$	−36.1019	−	4.43345i	−0.191015	−	0.0234574i
$190$		0			0
$191$	−177.076			−0.927102			−0.463551	−	0.886070i	$-0.653425\pi$
$191$	−177.076			−0.927102			−0.463551	+	0.886070i	$0.653425\pi$
$192$		−	13.8564i		−	0.0721688i
$193$	−153.453			−0.795092			−0.397546	−	0.917582i	$-0.630138\pi$
$193$	−153.453			−0.795092			−0.397546	+	0.917582i	$0.630138\pi$
$194$			62.5043i			0.322187i
$195$		0			0
$196$	−95.0881	−	23.7120i	−0.485143	−	0.120979i
$197$	214.041			1.08650			0.543252	−	0.839570i	$-0.317193\pi$
$197$	214.041			1.08650			0.543252	+	0.839570i	$0.317193\pi$
$198$	12.2229			0.0617319
$199$			178.176i			0.895359i	0.894194	+	0.447679i	$0.147749\pi$
$199$			178.176i			0.895359i	−0.894194	+	0.447679i	$0.852251\pi$
$200$		0			0
$201$			208.354i			1.03658i
$202$		−	254.631i		−	1.26055i
$203$	−28.1211	+	228.992i	−0.138528	+	1.12804i
$204$	83.7460			0.410520
$205$		0			0
$206$			151.615i			0.735996i
$207$	86.5260			0.418000
$208$		−	55.2581i		−	0.265664i
$209$		−	18.8243i		−	0.0900686i
$210$		0			0
$211$	−398.012			−1.88631			−0.943156	−	0.332350i	$-0.892158\pi$
$211$	−398.012			−1.88631			−0.943156	+	0.332350i	$0.892158\pi$
$212$	34.5517			0.162979
$213$			156.552i			0.734987i
$214$	46.8842			0.219085
$215$		0			0
$216$		−	14.6969i		−	0.0680414i
$217$	16.9024	+	2.07568i	0.0778911	+	0.00956533i
$218$	153.893			0.705932
$219$	37.2200			0.169954
$220$		0			0
$221$	333.972			1.51118
$222$		−	124.875i		−	0.562499i
$223$		−	6.96741i		−	0.0312440i	−0.999878	−	0.0156220i	$-0.995027\pi$
$223$		−	6.96741i		−	0.0312440i	0.999878	−	0.0156220i	$-0.00497284\pi$
$224$	4.82653	−	39.3027i	0.0215470	−	0.175459i
$225$		0			0
$226$	222.961			0.986555
$227$			242.118i			1.06660i	0.845927	+	0.533299i	$0.179048\pi$
$227$			242.118i			1.06660i	−0.845927	+	0.533299i	$0.820952\pi$
$228$	−22.6345			−0.0992743
$229$			225.586i			0.985093i	0.870286	+	0.492547i	$0.163934\pi$
$229$			225.586i			0.985093i	−0.870286	+	0.492547i	$0.836066\pi$
$230$		0			0
$231$	34.6694	+	4.25754i	0.150084	+	0.0184309i
$232$	−93.2218			−0.401818
$233$	435.850			1.87060			0.935300	−	0.353856i	$-0.115130\pi$
$233$	435.850			1.87060			0.935300	+	0.353856i	$0.115130\pi$
$234$		−	58.6101i		−	0.250470i
$235$		0			0
$236$			2.94977i			0.0124990i
$237$		−	114.545i		−	0.483311i
$238$	237.540	+	29.1708i	0.998066	+	0.122566i
$239$	−194.711			−0.814690			−0.407345	−	0.913274i	$-0.633545\pi$
$239$	−194.711			−0.814690			−0.407345	+	0.913274i	$0.633545\pi$
$240$		0			0
$241$		−	245.810i		−	1.01996i	−0.860186	−	0.509980i	$-0.829653\pi$
$241$		−	245.810i		−	1.01996i	0.860186	−	0.509980i	$-0.170347\pi$
$242$	159.382			0.658603
$243$		−	15.5885i		−	0.0641500i
$244$		−	223.136i		−	0.914490i
$245$		0			0
$246$	52.7192			0.214306
$247$	−90.2645			−0.365443
$248$			6.88089i			0.0277455i
$249$	135.996			0.546168
$250$		0			0
$251$			147.023i			0.585749i	0.956151	+	0.292875i	$0.0946118\pi$
$251$			147.023i			0.585749i	−0.956151	+	0.292875i	$0.905388\pi$
$252$	5.11931	−	41.6868i	0.0203147	−	0.165424i
$253$	−83.0928			−0.328430
$254$	39.8875			0.157038
$255$		0			0
$256$	16.0000			0.0625000
$257$		−	421.734i		−	1.64099i	−0.571654	−	0.820495i	$-0.693698\pi$
$257$		−	421.734i		−	1.64099i	0.571654	−	0.820495i	$-0.306302\pi$
$258$		−	33.2039i		−	0.128697i
$259$	43.4969	−	354.198i	0.167942	−	1.36756i
$260$		0			0
$261$	−98.8767			−0.378838
$262$		−	222.909i		−	0.850796i
$263$	−68.6354			−0.260971			−0.130486	−	0.991450i	$-0.541654\pi$
$263$	−68.6354			−0.260971			−0.130486	+	0.991450i	$0.541654\pi$
$264$			14.1138i			0.0534614i
$265$		0			0
$266$	−64.2013	−	7.88417i	−0.241358	−	0.0296397i
$267$	157.569			0.590145
$268$	−240.586			−0.897709
$269$		−	93.2991i		−	0.346837i	−0.984848	−	0.173418i	$-0.944519\pi$
$269$		−	93.2991i		−	0.346837i	0.984848	−	0.173418i	$-0.0554813\pi$
$270$		0			0
$271$		−	288.317i		−	1.06390i	−0.846776	−	0.531949i	$-0.821460\pi$
$271$		−	288.317i		−	1.06390i	0.846776	−	0.531949i	$-0.178540\pi$
$272$			96.7016i			0.355521i
$273$	20.4153	−	166.243i	0.0747814	−	0.608950i
$274$	346.278			1.26379
$275$		0			0
$276$			99.9116i			0.361999i
$277$	385.440			1.39148			0.695740	−	0.718294i	$-0.255077\pi$
$277$	385.440			1.39148			0.695740	+	0.718294i	$0.255077\pi$
$278$			336.929i			1.21197i
$279$			7.29829i			0.0261587i
$280$		0			0
$281$	243.958			0.868179			0.434090	−	0.900870i	$-0.357070\pi$
$281$	243.958			0.868179			0.434090	+	0.900870i	$0.357070\pi$
$282$	99.7688			0.353790
$283$			501.435i			1.77185i	0.463825	+	0.885927i	$0.346477\pi$
$283$			501.435i			1.77185i	−0.463825	+	0.885927i	$0.653523\pi$
$284$	−180.771			−0.636518
$285$		0			0
$286$			56.2846i			0.196799i
$287$	149.534	+	18.3634i	0.521026	+	0.0639840i
$288$	16.9706			0.0589256
$289$	−295.450			−1.02232
$290$		0			0
$291$	−76.5519			−0.263065
$292$			42.9780i			0.147185i
$293$		−	337.308i		−	1.15122i	−0.817724	−	0.575611i	$-0.804764\pi$
$293$		−	337.308i		−	1.15122i	0.817724	−	0.575611i	$-0.195236\pi$
$294$	29.0411	−	116.459i	0.0987793	−	0.396118i
$295$		0			0
$296$	144.193			0.487138
$297$			14.9699i			0.0504039i
$298$	15.3298			0.0514422
$299$			398.438i			1.33257i
$300$		0			0
$301$	11.5657	−	94.1806i	0.0384244	−	0.312892i
$302$	159.805			0.529156
$303$	311.859			1.02924
$304$		−	26.1361i		−	0.0859741i
$305$		0			0
$306$			102.568i			0.335188i
$307$		−	112.285i		−	0.365751i	−0.983136	−	0.182875i	$-0.941460\pi$
$307$		−	112.285i		−	0.365751i	0.983136	−	0.182875i	$-0.0585405\pi$
$308$	−4.91618	+	40.0328i	−0.0159616	+	0.129977i
$309$	−185.690			−0.600938
$310$		0			0
$311$			146.478i			0.470990i	0.971876	+	0.235495i	$0.0756711\pi$
$311$			146.478i			0.470990i	−0.971876	+	0.235495i	$0.924329\pi$
$312$	67.6771			0.216914
$313$		−	139.927i		−	0.447052i	−0.974698	−	0.223526i	$-0.928243\pi$
$313$		−	139.927i		−	0.447052i	0.974698	−	0.223526i	$-0.0717568\pi$
$314$		−	146.643i		−	0.467017i
$315$		0			0
$316$	132.265			0.418559
$317$	−39.7776			−0.125481			−0.0627407	−	0.998030i	$-0.519984\pi$
$317$	−39.7776			−0.125481			−0.0627407	+	0.998030i	$0.519984\pi$
$318$			42.3170i			0.133072i
$319$	94.9535			0.297660
$320$		0			0
$321$			57.4212i			0.178882i
$322$	−34.8017	+	283.392i	−0.108080	+	0.880100i
$323$	157.963			0.489049
$324$	18.0000			0.0555556
$325$		0			0
$326$	−282.243			−0.865775
$327$			188.480i			0.576391i
$328$			60.8749i			0.185594i
$329$	282.987	+	34.7519i	0.860144	+	0.105629i
$330$		0			0
$331$	−13.6736			−0.0413101			−0.0206550	−	0.999787i	$-0.506575\pi$
$331$	−13.6736			−0.0413101			−0.0206550	+	0.999787i	$0.506575\pi$
$332$			157.034i			0.472995i
$333$	152.940			0.459278
$334$			123.147i			0.368702i
$335$		0			0
$336$	48.1358	+	5.91127i	0.143261	+	0.0175931i
$337$	364.873			1.08271			0.541354	−	0.840795i	$-0.317912\pi$
$337$	364.873			1.08271			0.541354	+	0.840795i	$0.317912\pi$
$338$	30.8880			0.0913846
$339$			273.071i			0.805519i
$340$		0			0
$341$		−	7.00871i		−	0.0205534i
$342$		−	27.7215i		−	0.0810572i
$343$	122.938	−	320.211i	0.358421	−	0.933560i
$344$	38.3406			0.111455
$345$		0			0
$346$		−	150.751i		−	0.435695i
$347$	469.783			1.35384			0.676920	−	0.736056i	$-0.263314\pi$
$347$	469.783			1.35384			0.676920	+	0.736056i	$0.263314\pi$
$348$		−	114.173i		−	0.328083i
$349$		−	26.7504i		−	0.0766487i	−0.999265	−	0.0383244i	$-0.987798\pi$
$349$		−	26.7504i		−	0.0766487i	0.999265	−	0.0383244i	$-0.0122020\pi$
$350$		0			0
$351$	71.7824			0.204508
$352$	−16.2972			−0.0462989
$353$		−	473.408i		−	1.34110i	−0.741864	−	0.670550i	$-0.766058\pi$
$353$		−	473.408i		−	1.34110i	0.741864	−	0.670550i	$-0.233942\pi$
$354$	−3.61272			−0.0102054
$355$		0			0
$356$			181.945i			0.511081i
$357$	−35.7268	+	290.926i	−0.100075	+	0.814918i
$358$	390.155			1.08982
$359$	98.8174			0.275257			0.137629	−	0.990484i	$-0.456052\pi$
$359$	98.8174			0.275257			0.137629	+	0.990484i	$0.456052\pi$
$360$		0			0
$361$	318.306			0.881735
$362$			263.653i			0.728323i
$363$			195.202i			0.537747i
$364$	191.961	+	23.5736i	0.527366	+	0.0647626i
$365$		0			0
$366$	273.284			0.746678
$367$			331.526i			0.903341i	0.892185	+	0.451671i	$0.149172\pi$
$367$			331.526i			0.903341i	−0.892185	+	0.451671i	$0.850828\pi$
$368$	−115.368			−0.313500
$369$			64.5676i			0.174980i
$370$		0			0
$371$	−14.7400	+	120.029i	−0.0397306	+	0.323529i
$372$	−8.42734			−0.0226541
$373$	−416.519			−1.11667			−0.558337	−	0.829614i	$-0.688560\pi$
$373$	−416.519			−1.11667			−0.558337	+	0.829614i	$0.688560\pi$
$374$		−	98.4979i		−	0.263363i
$375$		0			0
$376$			115.203i			0.306391i
$377$		−	455.311i		−	1.20772i
$378$	51.0557	+	6.26984i	0.135068	+	0.0165869i
$379$	−560.942			−1.48006			−0.740028	−	0.672576i	$-0.765188\pi$
$379$	−560.942			−1.48006			−0.740028	+	0.672576i	$0.765188\pi$
$380$		0			0
$381$			48.8520i			0.128221i
$382$	250.424			0.655560
$383$		−	135.702i		−	0.354314i	−0.984183	−	0.177157i	$-0.943310\pi$
$383$		−	135.702i		−	0.354314i	0.984183	−	0.177157i	$-0.0566900\pi$
$384$			19.5959i			0.0510310i
$385$		0			0
$386$	217.015			0.562215
$387$	40.6663			0.105081
$388$		−	88.3945i		−	0.227821i
$389$	703.375			1.80816			0.904081	−	0.427361i	$-0.140557\pi$
$389$	703.375			1.80816			0.904081	+	0.427361i	$0.140557\pi$
$390$		0			0
$391$		−	697.267i		−	1.78329i
$392$	134.475	+	33.5338i	0.343048	+	0.0855454i
$393$	273.006			0.694672
$394$	−302.700			−0.768274
$395$		0			0
$396$	−17.2858			−0.0436510
$397$			2.16495i			0.00545326i	0.999996	+	0.00272663i	$0.000867915\pi$
$397$			2.16495i			0.00545326i	−0.999996	+	0.00272663i	$0.999132\pi$
$398$		−	251.979i		−	0.633114i
$399$	9.65610	−	78.6302i	0.0242007	−	0.197068i
$400$		0			0
$401$	−227.907			−0.568347			−0.284174	−	0.958773i	$-0.591719\pi$
$401$	−227.907			−0.568347			−0.284174	+	0.958773i	$0.591719\pi$
$402$		−	294.656i		−	0.732976i
$403$	−33.6075			−0.0833932
$404$			360.103i			0.891345i
$405$		0			0
$406$	39.7693	−	323.844i	0.0979538	−	0.797644i
$407$	−146.871			−0.360863
$408$	−118.435			−0.290281
$409$		−	219.271i		−	0.536116i	−0.963403	−	0.268058i	$-0.913618\pi$
$409$		−	219.271i		−	0.536116i	0.963403	−	0.268058i	$-0.0863818\pi$
$410$		0			0
$411$			424.102i			1.03188i
$412$		−	214.416i		−	0.520428i
$413$	−10.2472	−	1.25840i	−0.0248117	−	0.00304697i
$414$	−122.366			−0.295571
$415$		0			0
$416$			78.1468i			0.187853i
$417$	−412.652			−0.989572
$418$			26.6216i			0.0636881i
$419$		−	636.723i		−	1.51963i	−0.650141	−	0.759813i	$-0.725290\pi$
$419$		−	636.723i		−	1.51963i	0.650141	−	0.759813i	$-0.274710\pi$
$420$		0			0
$421$	816.589			1.93964			0.969821	−	0.243819i	$-0.0784003\pi$
$421$	816.589			1.93964			0.969821	+	0.243819i	$0.0784003\pi$
$422$	562.874			1.33382
$423$			122.191i			0.288868i
$424$	−48.8634			−0.115244
$425$		0			0
$426$		−	221.398i		−	0.519714i
$427$	775.151	+	95.1916i	1.81534	+	0.222931i
$428$	−66.3043			−0.154917
$429$	−68.9342			−0.160686
$430$		0			0
$431$	−288.911			−0.670328			−0.335164	−	0.942160i	$-0.608792\pi$
$431$	−288.911			−0.670328			−0.335164	+	0.942160i	$0.608792\pi$
$432$			20.7846i			0.0481125i
$433$		−	114.744i		−	0.264997i	−0.991183	−	0.132498i	$-0.957700\pi$
$433$		−	114.744i		−	0.264997i	0.991183	−	0.132498i	$-0.0423000\pi$
$434$	−23.9036	−	2.93545i	−0.0550773	−	0.00676371i
$435$		0			0
$436$	−217.638			−0.499169
$437$			188.454i			0.431246i
$438$	−52.6371			−0.120176
$439$			637.443i			1.45203i	0.687677	+	0.726017i	$0.258631\pi$
$439$			637.443i			1.45203i	−0.687677	+	0.726017i	$0.741369\pi$
$440$		0			0
$441$	142.632	+	35.5679i	0.323429	+	0.0806529i
$442$	−472.307			−1.06857
$443$	−184.901			−0.417384			−0.208692	−	0.977981i	$-0.566921\pi$
$443$	−184.901			−0.417384			−0.208692	+	0.977981i	$0.566921\pi$
$444$			176.599i			0.397747i
$445$		0			0
$446$			9.85341i			0.0220928i
$447$			18.7751i			0.0420024i
$448$	−6.82574	+	55.5825i	−0.0152360	+	0.124068i
$449$	313.278			0.697723			0.348862	−	0.937174i	$-0.386568\pi$
$449$	313.278			0.697723			0.348862	+	0.937174i	$0.386568\pi$
$450$		0			0
$451$		−	62.0057i		−	0.137485i
$452$	−315.315			−0.697600
$453$			195.721i			0.432054i
$454$		−	342.406i		−	0.754198i
$455$		0			0
$456$	32.0101			0.0701976
$457$	174.826			0.382551			0.191275	−	0.981536i	$-0.438738\pi$
$457$	174.826			0.382551			0.191275	+	0.981536i	$0.438738\pi$
$458$		−	319.027i		−	0.696566i
$459$	−125.619			−0.273680
$460$		0			0
$461$			379.263i			0.822695i	0.911479	+	0.411348i	$0.134942\pi$
$461$			379.263i			0.822695i	−0.911479	+	0.411348i	$0.865058\pi$
$462$	−49.0300	−	6.02107i	−0.106125	−	0.0130326i
$463$	403.499			0.871489			0.435744	−	0.900071i	$-0.356485\pi$
$463$	403.499			0.871489			0.435744	+	0.900071i	$0.356485\pi$
$464$	131.836			0.284128
$465$		0			0
$466$	−616.385			−1.32271
$467$		−	807.901i		−	1.72998i	−0.501788	−	0.864991i	$-0.667324\pi$
$467$		−	807.901i		−	1.72998i	0.501788	−	0.864991i	$-0.332676\pi$
$468$			82.8872i			0.177109i
$469$	102.636	−	835.772i	0.218840	−	1.78203i
$470$		0			0
$471$	179.601			0.381318
$472$		−	4.17160i		−	0.00883815i
$473$	−39.0528			−0.0825640
$474$			161.991i			0.341752i
$475$		0			0
$476$	−335.932	−	41.2538i	−0.705740	−	0.0866675i
$477$	−51.8275			−0.108653
$478$	275.363			0.576073
$479$			439.564i			0.917671i	0.888521	+	0.458835i	$0.151733\pi$
$479$			439.564i			0.917671i	−0.888521	+	0.458835i	$0.848267\pi$
$480$		0			0
$481$			704.263i			1.46416i
$482$			347.628i			0.721220i
$483$	−347.083	−	42.6232i	−0.718599	−	0.0882467i
$484$	−225.400			−0.465703
$485$		0			0
$486$			22.0454i			0.0453609i
$487$	−37.9665			−0.0779599			−0.0389799	−	0.999240i	$-0.512411\pi$
$487$	−37.9665			−0.0779599			−0.0389799	+	0.999240i	$0.512411\pi$
$488$			315.561i			0.646642i
$489$		−	345.675i		−	0.706903i
$490$		0			0
$491$	−7.32753			−0.0149237			−0.00746184	−	0.999972i	$-0.502375\pi$
$491$	−7.32753			−0.0149237			−0.00746184	+	0.999972i	$0.502375\pi$
$492$	−74.5563			−0.151537
$493$			796.794i			1.61622i
$494$	127.653			0.258408
$495$		0			0
$496$		−	9.73105i		−	0.0196191i
$497$	77.1185	−	627.981i	0.155168	−	1.26354i
$498$	−192.327			−0.386199
$499$	397.886			0.797366			0.398683	−	0.917089i	$-0.369467\pi$
$499$	397.886			0.797366			0.398683	+	0.917089i	$0.369467\pi$
$500$		0			0
$501$	−150.823			−0.301044
$502$		−	207.922i		−	0.414187i
$503$		−	70.5081i		−	0.140175i	−0.997541	−	0.0700876i	$-0.977672\pi$
$503$		−	70.5081i		−	0.140175i	0.997541	−	0.0700876i	$-0.0223279\pi$
$504$	−7.23979	+	58.9541i	−0.0143647	+	0.116972i
$505$		0			0
$506$	117.511			0.232235
$507$			37.8299i			0.0746152i
$508$	−56.4095			−0.111042
$509$			505.073i			0.992285i	0.868241	+	0.496143i	$0.165251\pi$
$509$			505.073i			0.992285i	−0.868241	+	0.496143i	$0.834749\pi$
$510$		0			0
$511$	−149.301	−	18.3348i	−0.292175	−	0.0358802i
$512$	−22.6274			−0.0441942
$513$	33.9518			0.0661829
$514$			596.422i			1.16035i
$515$		0			0
$516$			46.9574i			0.0910028i
$517$		−	117.343i		−	0.226969i
$518$	−61.5139	+	500.912i	−0.118753	+	0.967012i
$519$	184.631			0.355744
$520$		0			0
$521$			214.185i			0.411104i	0.978646	+	0.205552i	$0.0658990\pi$
$521$			214.185i			0.411104i	−0.978646	+	0.205552i	$0.934101\pi$
$522$	139.833			0.267879
$523$		−	121.732i		−	0.232757i	−0.993205	−	0.116379i	$-0.962871\pi$
$523$		−	121.732i		−	0.232757i	0.993205	−	0.116379i	$-0.0371286\pi$
$524$			315.240i			0.601604i
$525$		0			0
$526$	97.0651			0.184534
$527$	58.8130			0.111600
$528$		−	19.9599i		−	0.0378029i
$529$	302.861			0.572515
$530$		0			0
$531$		−	4.42465i		−	0.00833268i
$532$	90.7944	+	11.1499i	0.170666	+	0.0209585i
$533$	−297.324			−0.557830
$534$	−222.836			−0.417296
$535$		0			0
$536$	340.240			0.634776
$537$			477.840i			0.889832i
$538$			131.945i			0.245251i
$539$	−136.973	−	34.1567i	−0.254124	−	0.0633705i
$540$		0			0
$541$	600.626			1.11021			0.555107	−	0.831779i	$-0.312677\pi$
$541$	600.626			1.11021			0.555107	+	0.831779i	$0.312677\pi$
$542$			407.741i			0.752290i
$543$	−322.908			−0.594674
$544$		−	136.757i		−	0.251391i
$545$		0			0
$546$	−28.8716	+	235.104i	−0.0528785	+	0.430593i
$547$	−1045.16			−1.91071			−0.955356	−	0.295457i	$-0.904528\pi$
$547$	−1045.16			−1.91071			−0.955356	+	0.295457i	$0.904528\pi$
$548$	−489.711			−0.893632
$549$			334.703i			0.609660i
$550$		0			0
$551$		−	215.354i		−	0.390843i
$552$		−	141.296i		−	0.255972i
$553$	−56.4253	+	459.475i	−0.102035	+	0.830877i
$554$	−545.094			−0.983924
$555$		0			0
$556$		−	476.489i		−	0.856995i
$557$	−601.731			−1.08031			−0.540154	−	0.841566i	$-0.681634\pi$
$557$	−601.731			−1.08031			−0.540154	+	0.841566i	$0.681634\pi$
$558$		−	10.3213i		−	0.0184970i
$559$			187.262i			0.334995i
$560$		0			0
$561$	120.635			0.215035
$562$	−345.009			−0.613896
$563$		−	363.871i		−	0.646308i	−0.946346	−	0.323154i	$-0.895257\pi$
$563$		−	363.871i		−	0.646308i	0.946346	−	0.323154i	$-0.104743\pi$
$564$	−141.094			−0.250167
$565$		0			0
$566$		−	709.136i		−	1.25289i
$567$	−7.67896	+	62.5303i	−0.0135431	+	0.110283i
$568$	255.649			0.450086
$569$	−902.900			−1.58682			−0.793410	−	0.608688i	$-0.791696\pi$
$569$	−902.900			−1.58682			−0.793410	+	0.608688i	$0.791696\pi$
$570$		0			0
$571$	−847.207			−1.48372			−0.741862	−	0.670552i	$-0.766057\pi$
$571$	−847.207			−1.48372			−0.741862	+	0.670552i	$0.766057\pi$
$572$		−	79.5984i		−	0.139158i
$573$			306.705i			0.535263i
$574$	−211.474	−	25.9698i	−0.368421	−	0.0452435i
$575$		0			0
$576$	−24.0000			−0.0416667
$577$		−	306.535i		−	0.531256i	−0.964076	−	0.265628i	$-0.914421\pi$
$577$		−	306.535i		−	0.531256i	0.964076	−	0.265628i	$-0.0855793\pi$
$578$	417.829			0.722888
$579$			265.788i			0.459047i
$580$		0			0
$581$	−545.522	−	66.9923i	−0.938937	−	0.115305i
$582$	108.261			0.186015
$583$	49.7711			0.0853707
$584$		−	60.7801i		−	0.104075i
$585$		0			0
$586$			477.025i			0.814037i
$587$		−	467.524i		−	0.796463i	−0.917285	−	0.398232i	$-0.869624\pi$
$587$		−	467.524i		−	0.796463i	0.917285	−	0.398232i	$-0.130376\pi$
$588$	−41.0703	+	164.697i	−0.0698475	+	0.280098i
$589$	−15.8958			−0.0269877
$590$		0			0
$591$		−	370.730i		−	0.627293i
$592$	−203.920			−0.344459
$593$		−	744.542i		−	1.25555i	−0.778395	−	0.627775i	$-0.783966\pi$
$593$		−	744.542i		−	1.25555i	0.778395	−	0.627775i	$-0.216034\pi$
$594$		−	21.1707i		−	0.0356409i
$595$		0			0
$596$	−21.6796			−0.0363751
$597$	308.611			0.516936
$598$		−	563.477i		−	0.942269i
$599$	276.145			0.461011			0.230505	−	0.973071i	$-0.425962\pi$
$599$	276.145			0.461011			0.230505	+	0.973071i	$0.425962\pi$
$600$		0			0
$601$		−	779.606i		−	1.29718i	−0.761137	−	0.648591i	$-0.775359\pi$
$601$		−	779.606i		−	1.29718i	0.761137	−	0.648591i	$-0.224641\pi$
$602$	−16.3564	+	133.191i	−0.0271702	+	0.221248i
$603$	360.879			0.598472
$604$	−225.999			−0.374170
$605$		0			0
$606$	−441.035			−0.727780
$607$			567.539i			0.934990i	0.883996	+	0.467495i	$0.154843\pi$
$607$			567.539i			0.934990i	−0.883996	+	0.467495i	$0.845157\pi$
$608$			36.9621i			0.0607929i
$609$	396.626	+	48.7072i	0.651274	+	0.0799790i
$610$		0			0
$611$	−562.672			−0.920903
$612$		−	145.052i		−	0.237014i
$613$	801.088			1.30683			0.653416	−	0.756999i	$-0.273335\pi$
$613$	801.088			1.30683			0.653416	+	0.756999i	$0.273335\pi$
$614$			158.796i			0.258625i
$615$		0			0
$616$	6.95253	−	56.6149i	0.0112866	−	0.0919074i
$617$	−71.2583			−0.115492			−0.0577458	−	0.998331i	$-0.518391\pi$
$617$	−71.2583			−0.115492			−0.0577458	+	0.998331i	$0.518391\pi$
$618$	262.605			0.424928
$619$		−	821.793i		−	1.32761i	−0.747904	−	0.663807i	$-0.768940\pi$
$619$		−	821.793i		−	1.32761i	0.747904	−	0.663807i	$-0.231060\pi$
$620$		0			0
$621$		−	149.867i		−	0.241332i
$622$		−	207.151i		−	0.333040i
$623$	−632.059	−	77.6193i	−1.01454	−	0.124590i
$624$	−95.7098			−0.153381
$625$		0			0
$626$			197.887i			0.316114i
$627$	−32.6047			−0.0520011
$628$			207.385i			0.330231i
$629$		−	1232.46i		−	1.95939i
$630$		0			0
$631$	−916.907			−1.45310			−0.726551	−	0.687113i	$-0.758878\pi$
$631$	−916.907			−1.45310			−0.726551	+	0.687113i	$0.758878\pi$
$632$	−187.051			−0.295966
$633$			689.377i			1.08906i
$634$	56.2540			0.0887287
$635$		0			0
$636$		−	59.8452i		−	0.0940963i
$637$	−163.785	+	656.798i	−0.257119	+	1.03108i
$638$	−134.285			−0.210477
$639$	271.156			0.424345
$640$		0			0
$641$	737.240			1.15014			0.575070	−	0.818104i	$-0.304975\pi$
$641$	737.240			1.15014			0.575070	+	0.818104i	$0.304975\pi$
$642$		−	81.2058i		−	0.126489i
$643$		−	256.297i		−	0.398596i	−0.979939	−	0.199298i	$-0.936134\pi$
$643$		−	256.297i		−	0.398596i	0.979939	−	0.199298i	$-0.0638661\pi$
$644$	49.2170	−	400.777i	0.0764239	−	0.622325i
$645$		0			0
$646$	−223.393			−0.345810
$647$			689.147i			1.06514i	0.846385	+	0.532571i	$0.178774\pi$
$647$			689.147i			1.06514i	−0.846385	+	0.532571i	$0.821226\pi$
$648$	−25.4558			−0.0392837
$649$			4.24910i			0.00654714i
$650$		0			0
$651$	3.59518	−	29.2758i	0.00552255	−	0.0449705i
$652$	399.152			0.612196
$653$	395.775			0.606087			0.303044	−	0.952977i	$-0.401997\pi$
$653$	395.775			0.606087			0.303044	+	0.952977i	$0.401997\pi$
$654$		−	266.551i		−	0.407570i
$655$		0			0
$656$		−	86.0902i		−	0.131235i
$657$		−	64.4670i		−	0.0981233i
$658$	−400.205	−	49.1467i	−0.608214	−	0.0746910i
$659$	47.8147			0.0725565			0.0362782	−	0.999342i	$-0.488450\pi$
$659$	47.8147			0.0725565			0.0362782	+	0.999342i	$0.488450\pi$
$660$		0			0
$661$			89.9129i			0.136026i	0.997684	+	0.0680128i	$0.0216659\pi$
$661$			89.9129i			0.136026i	−0.997684	+	0.0680128i	$0.978334\pi$
$662$	19.3374			0.0292106
$663$		−	578.456i		−	0.872482i
$664$		−	222.080i		−	0.334458i
$665$		0			0
$666$	−216.289			−0.324759
$667$	−950.600			−1.42519
$668$		−	174.155i		−	0.260712i
$669$	−12.0679			−0.0180387
$670$		0			0
$671$		−	321.423i		−	0.479021i
$672$	−68.0743	−	8.35979i	−0.101301	−	0.0124402i
$673$	−833.478			−1.23845			−0.619226	−	0.785213i	$-0.712553\pi$
$673$	−833.478			−1.23845			−0.619226	+	0.785213i	$0.712553\pi$
$674$	−516.008			−0.765590
$675$		0			0
$676$	−43.6822			−0.0646187
$677$		−	44.1092i		−	0.0651540i	−0.999469	−	0.0325770i	$-0.989629\pi$
$677$		−	44.1092i		−	0.0651540i	0.999469	−	0.0325770i	$-0.0103714\pi$
$678$		−	386.181i		−	0.569588i
$679$	307.074	+	37.7099i	0.452244	+	0.0555373i
$680$		0			0
$681$	419.360			0.615800
$682$			9.91182i			0.0145335i
$683$	−130.146			−0.190550			−0.0952750	−	0.995451i	$-0.530373\pi$
$683$	−130.146			−0.190550			−0.0952750	+	0.995451i	$0.530373\pi$
$684$			39.2042i			0.0573161i
$685$		0			0
$686$	−173.861	+	452.847i	−0.253442	+	0.660127i
$687$	390.727			0.568744
$688$	−54.2218			−0.0788107
$689$		−	238.657i		−	0.346382i
$690$		0			0
$691$			714.724i			1.03433i	0.855885	+	0.517167i	$0.173013\pi$
$691$			714.724i			1.03433i	−0.855885	+	0.517167i	$0.826987\pi$
$692$			213.194i			0.308083i
$693$	7.37428	−	60.0492i	0.0106411	−	0.0866511i
$694$	−664.373			−0.957310
$695$		0			0
$696$			161.465i			0.231990i
$697$	520.316			0.746508
$698$			37.8308i			0.0541988i
$699$		−	754.914i		−	1.07999i
$700$		0			0
$701$	463.854			0.661704			0.330852	−	0.943683i	$-0.392664\pi$
$701$	463.854			0.661704			0.330852	+	0.943683i	$0.392664\pi$
$702$	−101.516			−0.144609
$703$			333.104i			0.473832i
$704$	23.0477			0.0327383
$705$		0			0
$706$			669.500i			0.948301i
$707$	−1250.96	−	153.623i	−1.76940	−	0.217289i
$708$	5.10915			0.00721632
$709$	970.049			1.36819			0.684097	−	0.729391i	$-0.260197\pi$
$709$	970.049			1.36819			0.684097	+	0.729391i	$0.260197\pi$
$710$		0			0
$711$	−198.397			−0.279040
$712$		−	257.309i		−	0.361389i
$713$			70.1657i			0.0984092i
$714$	50.5253	−	411.431i	0.0707638	−	0.576234i
$715$		0			0
$716$	−551.762			−0.770617
$717$			337.249i			0.470362i
$718$	−139.749			−0.194636
$719$			1369.36i			1.90453i	0.305266	+	0.952267i	$0.401255\pi$
$719$			1369.36i			1.90453i	−0.305266	+	0.952267i	$0.598745\pi$
$720$		0			0
$721$	744.861	+	91.4719i	1.03309	+	0.126868i
$722$	−450.153			−0.623481
$723$	−425.756			−0.588874
$724$		−	372.862i		−	0.515002i
$725$		0			0
$726$		−	276.058i		−	0.380245i
$727$			1106.85i			1.52249i	0.648463	+	0.761246i	$0.275412\pi$
$727$			1106.85i			1.52249i	−0.648463	+	0.761246i	$0.724588\pi$
$728$	−271.474	−	33.3381i	−0.372904	−	0.0457941i
$729$	−27.0000			−0.0370370
$730$		0			0
$731$		−	327.708i		−	0.448301i
$732$	−386.482			−0.527981
$733$		−	297.353i		−	0.405665i	−0.979213	−	0.202833i	$-0.934985\pi$
$733$		−	297.353i		−	0.405665i	0.979213	−	0.202833i	$-0.0650147\pi$
$734$		−	468.849i		−	0.638759i
$735$		0			0
$736$	163.155			0.221678
$737$	−346.560			−0.470231
$738$		−	91.3124i		−	0.123730i
$739$	−604.951			−0.818608			−0.409304	−	0.912398i	$-0.634228\pi$
$739$	−604.951			−0.818608			−0.409304	+	0.912398i	$0.634228\pi$
$740$		0			0
$741$			156.343i			0.210989i
$742$	20.8456	−	169.747i	0.0280938	−	0.228769i
$743$	733.877			0.987722			0.493861	−	0.869541i	$-0.335585\pi$
$743$	733.877			0.987722			0.493861	+	0.869541i	$0.335585\pi$
$744$	11.9181			0.0160189
$745$		0			0
$746$	589.047			0.789608
$747$		−	235.552i		−	0.315330i
$748$			139.297i			0.186226i
$749$	28.2860	−	230.335i	0.0377650	−	0.307523i
$750$		0			0
$751$	−71.0036			−0.0945455			−0.0472727	−	0.998882i	$-0.515053\pi$
$751$	−71.0036			−0.0945455			−0.0472727	+	0.998882i	$0.515053\pi$
$752$		−	162.922i		−	0.216651i
$753$	254.651			0.338182
$754$			643.908i			0.853989i
$755$		0			0
$756$	−72.2037	−	8.86690i	−0.0955076	−	0.0117287i
$757$	−708.172			−0.935497			−0.467749	−	0.883861i	$-0.654935\pi$
$757$	−708.172			−0.935497			−0.467749	+	0.883861i	$0.654935\pi$
$758$	793.291			1.04656
$759$			143.921i			0.189619i
$760$		0			0
$761$			797.652i			1.04816i	0.851668	+	0.524081i	$0.175591\pi$
$761$			797.652i			1.04816i	−0.851668	+	0.524081i	$0.824409\pi$
$762$		−	69.0872i		−	0.0906657i
$763$	92.8462	−	756.052i	0.121686	−	0.990894i
$764$	−354.153			−0.463551
$765$		0			0
$766$			191.912i			0.250538i
$767$	20.3748			0.0265643
$768$		−	27.7128i		−	0.0360844i
$769$			261.829i			0.340480i	0.985403	+	0.170240i	$0.0544543\pi$
$769$			261.829i			0.340480i	−0.985403	+	0.170240i	$0.945546\pi$
$770$		0			0
$771$	−730.465			−0.947426
$772$	−306.906			−0.397546
$773$			643.514i			0.832489i	0.909253	+	0.416245i	$0.136654\pi$
$773$			643.514i			0.832489i	−0.909253	+	0.416245i	$0.863346\pi$
$774$	−57.5109			−0.0743035
$775$		0			0
$776$			125.009i			0.161094i
$777$	−613.490	−	75.3389i	−0.789562	−	0.0969613i
$778$	−994.722			−1.27856
$779$	−140.629			−0.180525
$780$		0			0
$781$	−260.398			−0.333416
$782$			986.084i			1.26098i
$783$			171.259i			0.218722i
$784$	−190.176	−	47.4239i	−0.242572	−	0.0604897i
$785$		0			0
$786$	−386.089			−0.491207
$787$		−	919.663i		−	1.16857i	−0.811549	−	0.584284i	$-0.801375\pi$
$787$		−	919.663i		−	1.16857i	0.811549	−	0.584284i	$-0.198625\pi$
$788$	428.082			0.543252
$789$			118.880i			0.150672i
$790$		0			0
$791$	134.516	−	1095.37i	0.170058	−	1.38480i
$792$	24.4458			0.0308659
$793$	−1541.26			−1.94358
$794$		−	3.06170i		−	0.00385604i
$795$		0			0
$796$			356.353i			0.447679i
$797$			246.244i			0.308964i	0.987996	+	0.154482i	$0.0493708\pi$
$797$			246.244i			0.308964i	−0.987996	+	0.154482i	$0.950629\pi$
$798$	−13.6558	+	111.200i	−0.0171125	+	0.139348i
$799$	984.675			1.23238
$800$		0			0
$801$		−	272.917i		−	0.340721i
$802$	322.309			0.401882
$803$			61.9091i			0.0770973i
$804$			416.707i			0.518292i
$805$		0			0
$806$	47.5281			0.0589679
$807$	−161.599			−0.200246
$808$		−	509.263i		−	0.630276i
$809$	609.838			0.753817			0.376908	−	0.926251i	$-0.376987\pi$
$809$	609.838			0.753817			0.376908	+	0.926251i	$0.376987\pi$
$810$		0			0
$811$			605.726i			0.746888i	0.927653	+	0.373444i	$0.121823\pi$
$811$			605.726i			0.746888i	−0.927653	+	0.373444i	$0.878177\pi$
$812$	−56.2422	+	457.984i	−0.0692638	+	0.564020i
$813$	−499.379			−0.614242
$814$	207.707			0.255169
$815$		0			0
$816$	167.492			0.205260
$817$			88.5717i			0.108411i
$818$			310.097i			0.379091i
$819$	−287.942	−	35.3604i	−0.351577	−	0.0431751i
$820$		0			0
$821$	890.751			1.08496			0.542480	−	0.840069i	$-0.317486\pi$
$821$	890.751			1.08496			0.542480	+	0.840069i	$0.317486\pi$
$822$		−	599.770i		−	0.729648i
$823$	−317.029			−0.385211			−0.192606	−	0.981276i	$-0.561694\pi$
$823$	−317.029			−0.385211			−0.192606	+	0.981276i	$0.561694\pi$
$824$			303.230i			0.367998i
$825$		0			0
$826$	14.4918	+	1.77964i	0.0175445	+	0.00215453i
$827$	−90.7251			−0.109704			−0.0548520	−	0.998494i	$-0.517469\pi$
$827$	−90.7251			−0.109704			−0.0548520	+	0.998494i	$0.517469\pi$
$828$	173.052			0.209000
$829$			1366.86i			1.64880i	0.566007	+	0.824400i	$0.308487\pi$
$829$			1366.86i			1.64880i	−0.566007	+	0.824400i	$0.691513\pi$
$830$		0			0
$831$		−	667.601i		−	0.803371i
$832$		−	110.516i		−	0.132832i
$833$	286.623	−	1149.40i	0.344085	−	1.37983i
$834$	583.578			0.699733
$835$		0			0
$836$		−	37.6487i		−	0.0450343i
$837$	12.6410			0.0151028
$838$			900.463i			1.07454i
$839$		−	73.5993i		−	0.0877227i	−0.999038	−	0.0438613i	$-0.986034\pi$
$839$		−	73.5993i		−	0.0877227i	0.999038	−	0.0438613i	$-0.0139660\pi$
$840$		0			0
$841$	245.289			0.291663
$842$	−1154.83			−1.37153
$843$		−	422.548i		−	0.501244i
$844$	−796.024			−0.943156
$845$		0			0
$846$		−	172.805i		−	0.204261i
$847$	96.1576	−	783.018i	0.113527	−	0.924460i
$848$	69.1033			0.0814897
$849$	868.510			1.02298
$850$		0			0
$851$	1470.36			1.72780
$852$			313.105i			0.367494i
$853$			822.046i			0.963711i	0.876251	+	0.481856i	$0.160037\pi$
$853$			822.046i			0.963711i	−0.876251	+	0.481856i	$0.839963\pi$
$854$	−1096.23	−	134.621i	−1.28364	−	0.157636i
$855$		0			0
$856$	93.7684			0.109543
$857$		−	625.685i		−	0.730087i	−0.930990	−	0.365044i	$-0.881054\pi$
$857$		−	625.685i		−	0.730087i	0.930990	−	0.365044i	$-0.118946\pi$
$858$	97.4877			0.113622
$859$			1396.65i			1.62590i	0.582332	+	0.812951i	$0.302140\pi$
$859$			1396.65i			1.62590i	−0.582332	+	0.812951i	$0.697860\pi$
$860$		0			0
$861$	31.8064	−	259.001i	0.0369412	−	0.300814i
$862$	408.582			0.473993
$863$	−601.569			−0.697068			−0.348534	−	0.937296i	$-0.613320\pi$
$863$	−601.569			−0.697068			−0.348534	+	0.937296i	$0.613320\pi$
$864$		−	29.3939i		−	0.0340207i
$865$		0			0
$866$			162.272i			0.187381i
$867$			511.734i			0.590236i
$868$	33.8047	+	4.15135i	0.0389456	+	0.00478267i
$869$	190.525			0.219246
$870$		0			0
$871$			1661.79i			1.90791i
$872$	307.786			0.352966
$873$			132.592i			0.151881i
$874$		−	266.515i		−	0.304937i
$875$		0			0
$876$	74.4401			0.0849772
$877$	−570.762			−0.650812			−0.325406	−	0.945574i	$-0.605501\pi$
$877$	−570.762			−0.650812			−0.325406	+	0.945574i	$0.605501\pi$
$878$		−	901.480i		−	1.02674i
$879$	−584.234			−0.664658
$880$		0			0
$881$		−	1095.44i		−	1.24341i	−0.783253	−	0.621703i	$-0.786441\pi$
$881$		−	1095.44i		−	1.24341i	0.783253	−	0.621703i	$-0.213559\pi$
$882$	−201.712	−	50.3007i	−0.228699	−	0.0570302i
$883$	305.870			0.346399			0.173199	−	0.984887i	$-0.444590\pi$
$883$	305.870			0.346399			0.173199	+	0.984887i	$0.444590\pi$
$884$	667.943			0.755592
$885$		0			0
$886$	261.490			0.295135
$887$		−	637.527i		−	0.718745i	−0.933194	−	0.359372i	$-0.882991\pi$
$887$		−	637.527i		−	0.718745i	0.933194	−	0.359372i	$-0.117009\pi$
$888$		−	249.749i		−	0.281249i
$889$	24.0648	−	195.961i	0.0270695	−	0.220429i
$890$		0			0
$891$	25.9287			0.0291007
$892$		−	13.9348i		−	0.0156220i
$893$	−266.134			−0.298022
$894$		−	26.5520i		−	0.0297002i
$895$		0			0
$896$	9.65306	−	78.6055i	0.0107735	−	0.0877293i
$897$	690.116			0.769360
$898$	−443.042			−0.493365
$899$		−	80.1812i		−	0.0891893i
$900$		0			0
$901$			417.650i			0.463541i
$902$			87.6893i			0.0972166i
$903$	−163.126	−	20.0325i	−0.180649	−	0.0221843i
$904$	445.923			0.493278
$905$		0			0
$906$		−	276.791i		−	0.305509i
$907$	785.727			0.866292			0.433146	−	0.901324i	$-0.357403\pi$
$907$	785.727			0.866292			0.433146	+	0.901324i	$0.357403\pi$
$908$			484.235i			0.533299i
$909$		−	540.155i		−	0.594230i
$910$		0			0
$911$	−909.376			−0.998217			−0.499109	−	0.866539i	$-0.666339\pi$
$911$	−909.376			−0.998217			−0.499109	+	0.866539i	$0.666339\pi$
$912$	−45.2691			−0.0496372
$913$			226.205i			0.247761i
$914$	−247.241			−0.270504
$915$		0			0
$916$			451.173i			0.492547i
$917$	−1095.11	−	134.484i	−1.19424	−	0.146657i
$918$	177.652			0.193521
$919$	270.279			0.294101			0.147050	−	0.989129i	$-0.453022\pi$
$919$	270.279			0.294101			0.147050	+	0.989129i	$0.453022\pi$
$920$		0			0
$921$	−194.484			−0.211166
$922$		−	536.358i		−	0.581733i
$923$			1248.63i			1.35280i
$924$	69.3389	+	8.51508i	0.0750421	+	0.00921546i
$925$		0			0
$926$	−570.634			−0.616235
$927$			321.624i			0.346952i
$928$	−186.444			−0.200909
$929$			1053.28i			1.13378i	0.823795	+	0.566889i	$0.191853\pi$
$929$			1053.28i			1.13378i	−0.823795	+	0.566889i	$0.808147\pi$
$930$		0			0
$931$	−77.4674	+	310.654i	−0.0832088	+	0.333678i
$932$	871.699			0.935300
$933$	253.707			0.271926
$934$			1142.55i			1.22328i
$935$		0			0
$936$		−	117.220i		−	0.125235i
$937$			1012.18i			1.08024i	0.841589	+	0.540119i	$0.181621\pi$
$937$			1012.18i			1.08024i	−0.841589	+	0.540119i	$0.818379\pi$
$938$	−145.149	+	1181.96i	−0.154743	+	1.26009i
$939$	−242.361			−0.258106
$940$		0			0
$941$		−	1384.55i		−	1.47136i	−0.677331	−	0.735679i	$-0.736863\pi$
$941$		−	1384.55i		−	1.47136i	0.677331	−	0.735679i	$-0.263137\pi$
$942$	−253.994			−0.269633
$943$			620.753i			0.658275i
$944$			5.89954i			0.00624951i
$945$		0			0
$946$	55.2290			0.0583816
$947$	1216.79			1.28489			0.642444	−	0.766332i	$-0.277920\pi$
$947$	1216.79			1.28489			0.642444	+	0.766332i	$0.277920\pi$
$948$		−	229.089i		−	0.241655i
$949$	296.860			0.312814
$950$		0			0
$951$			68.8968i			0.0724467i
$952$	475.080	+	58.3416i	0.499033	+	0.0612832i
$953$	30.5870			0.0320955			0.0160478	−	0.999871i	$-0.494892\pi$
$953$	30.5870			0.0320955			0.0160478	+	0.999871i	$0.494892\pi$
$954$	73.2951			0.0768293
$955$		0			0
$956$	−389.422			−0.407345
$957$		−	164.464i		−	0.171854i
$958$		−	621.638i		−	0.648891i
$959$	208.915	−	1701.21i	0.217847	−	1.77394i
$960$		0			0
$961$	955.082			0.993841
$962$		−	995.978i		−	1.03532i
$963$	99.4564			0.103278
$964$		−	491.620i		−	0.509980i
$965$		0			0
$966$	490.850	+	60.2783i	0.508126	+	0.0623999i
$967$	−1878.59			−1.94270			−0.971351	−	0.237650i	$-0.923623\pi$
$967$	−1878.59			−1.94270			−0.971351	+	0.237650i	$0.923623\pi$
$968$	318.764			0.329301
$969$		−	273.600i		−	0.282353i
$970$		0			0
$971$			712.009i			0.733274i	0.930364	+	0.366637i	$0.119491\pi$
$971$			712.009i			0.733274i	−0.930364	+	0.366637i	$0.880509\pi$
$972$		−	31.1769i		−	0.0320750i
$973$	1655.28	+	203.274i	1.70121	+	0.208915i
$974$	53.6927			0.0551260
$975$		0			0
$976$		−	446.271i		−	0.457245i
$977$	−750.938			−0.768616			−0.384308	−	0.923205i	$-0.625560\pi$
$977$	−750.938			−0.768616			−0.384308	+	0.923205i	$0.625560\pi$
$978$			488.859i			0.499856i
$979$			262.089i			0.267710i
$980$		0			0
$981$	326.457			0.332779
$982$	10.3627			0.0105526
$983$			1525.09i			1.55147i	0.631059	+	0.775735i	$0.282621\pi$
$983$			1525.09i			1.55147i	−0.631059	+	0.775735i	$0.717379\pi$
$984$	105.438			0.107153
$985$		0			0
$986$		−	1126.84i		−	1.14284i
$987$	60.1921	−	490.148i	0.0609849	−	0.496604i
$988$	−180.529			−0.182722
$989$	390.966			0.395314
$990$		0			0
$991$	1259.84			1.27128			0.635641	−	0.771985i	$-0.280736\pi$
$991$	1259.84			1.27128			0.635641	+	0.771985i	$0.280736\pi$
$992$			13.7618i			0.0138728i
$993$			23.6834i			0.0238504i
$994$	−109.062	+	888.099i	−0.109720	+	0.893460i
$995$		0			0
$996$	271.992			0.273084
$997$			1406.62i			1.41086i	0.708781	+	0.705428i	$0.249245\pi$
$997$			1406.62i			1.41086i	−0.708781	+	0.705428i	$0.750755\pi$
$998$	−562.695			−0.563823
$999$		−	264.899i		−	0.265164i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.3.f.e.601.2		16
5.2	odd	4		210.3.h.a.139.3	✓	16
5.3	odd	4		210.3.h.a.139.14	yes	16
5.4	even	2	inner	1050.3.f.e.601.15		16
7.6	odd	2	inner	1050.3.f.e.601.6		16
15.2	even	4		630.3.h.e.559.11		16
15.8	even	4		630.3.h.e.559.6		16
20.3	even	4		1680.3.bd.a.769.3		16
20.7	even	4		1680.3.bd.a.769.13		16
35.13	even	4		210.3.h.a.139.11	yes	16
35.27	even	4		210.3.h.a.139.6	yes	16
35.34	odd	2	inner	1050.3.f.e.601.11		16
105.62	odd	4		630.3.h.e.559.14		16
105.83	odd	4		630.3.h.e.559.3		16
140.27	odd	4		1680.3.bd.a.769.4		16
140.83	odd	4		1680.3.bd.a.769.14		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.h.a.139.3	✓	16	5.2	odd	4
210.3.h.a.139.6	yes	16	35.27	even	4
210.3.h.a.139.11	yes	16	35.13	even	4
210.3.h.a.139.14	yes	16	5.3	odd	4
630.3.h.e.559.3		16	105.83	odd	4
630.3.h.e.559.6		16	15.8	even	4
630.3.h.e.559.11		16	15.2	even	4
630.3.h.e.559.14		16	105.62	odd	4
1050.3.f.e.601.2		16	1.1	even	1	trivial
1050.3.f.e.601.6		16	7.6	odd	2	inner
1050.3.f.e.601.11		16	35.34	odd	2	inner
1050.3.f.e.601.15		16	5.4	even	2	inner
1680.3.bd.a.769.3		16	20.3	even	4
1680.3.bd.a.769.4		16	140.27	odd	4
1680.3.bd.a.769.13		16	20.7	even	4
1680.3.bd.a.769.14		16	140.83	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1050.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} +2.44949i q^{6} +(-0.853218 + 6.94781i) q^{7} -2.82843 q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} +2.44949i q^{6} +(-0.853218 + 6.94781i) q^{7} -2.82843 q^{8} -3.00000 q^{9} +2.88097 q^{11} -3.46410i q^{12} -13.8145i q^{13} +(1.20663 - 9.82568i) q^{14} +4.00000 q^{16} +24.1754i q^{17} +4.24264 q^{18} -6.53403i q^{19} +(12.0340 + 1.47782i) q^{21} -4.07430 q^{22} -28.8420 q^{23} +4.89898i q^{24} +19.5367i q^{26} +5.19615i q^{27} +(-1.70644 + 13.8956i) q^{28} +32.9589 q^{29} -2.43276i q^{31} -5.65685 q^{32} -4.98998i q^{33} -34.1892i q^{34} -6.00000 q^{36} -50.9799 q^{37} +9.24052i q^{38} -23.9275 q^{39} -21.5225i q^{41} +(-17.0186 - 2.08995i) q^{42} -13.5554 q^{43} +5.76193 q^{44} +40.7887 q^{46} -40.7305i q^{47} -6.92820i q^{48} +(-47.5440 - 11.8560i) q^{49} +41.8730 q^{51} -27.6291i q^{52} +17.2758 q^{53} -7.34847i q^{54} +(2.41326 - 19.6514i) q^{56} -11.3173 q^{57} -46.6109 q^{58} +1.47488i q^{59} -111.568i q^{61} +3.44045i q^{62} +(2.55965 - 20.8434i) q^{63} +8.00000 q^{64} +7.05690i q^{66} -120.293 q^{67} +48.3508i q^{68} +49.9558i q^{69} -90.3855 q^{71} +8.48528 q^{72} +21.4890i q^{73} +72.0964 q^{74} -13.0681i q^{76} +(-2.45809 + 20.0164i) q^{77} +33.8385 q^{78} +66.1324 q^{79} +9.00000 q^{81} +30.4375i q^{82} +78.5172i q^{83} +(24.0679 + 2.95563i) q^{84} +19.1703 q^{86} -57.0865i q^{87} -8.14861 q^{88} +90.9724i q^{89} +(95.9807 + 11.7868i) q^{91} -57.6840 q^{92} -4.21367 q^{93} +57.6016i q^{94} +9.79796i q^{96} -44.1972i q^{97} +(67.2374 + 16.7669i) q^{98} -8.64290 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4} - 48 q^{9}+O(q^{10}) \) 16 * q + 32 * q^4 - 48 * q^9	\( 16 q + 32 q^{4} - 48 q^{9} + 96 q^{11} - 16 q^{14} + 64 q^{16} + 24 q^{21} - 64 q^{29} - 96 q^{36} + 144 q^{39} + 192 q^{44} - 176 q^{46} - 224 q^{49} - 48 q^{51} - 32 q^{56} + 128 q^{64} - 384 q^{71} - 224 q^{74} + 608 q^{79} + 144 q^{81} + 48 q^{84} + 416 q^{86} + 224 q^{91} - 288 q^{99}+O(q^{100}) \) 16 * q + 32 * q^4 - 48 * q^9 + 96 * q^11 - 16 * q^14 + 64 * q^16 + 24 * q^21 - 64 * q^29 - 96 * q^36 + 144 * q^39 + 192 * q^44 - 176 * q^46 - 224 * q^49 - 48 * q^51 - 32 * q^56 + 128 * q^64 - 384 * q^71 - 224 * q^74 + 608 * q^79 + 144 * q^81 + 48 * q^84 + 416 * q^86 + 224 * q^91 - 288 * q^99

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