LMFDB - Embedded newform 1680.3.bd.a.769.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,3,Mod(769,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1680.769");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1680.bd (of order $2$, degree $1$, not 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:	$45.7766844125$
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} + 96 x^{14} - 532 x^{13} + 3236 x^{12} - 12864 x^{11} + 49526 x^{10} - 141436 x^{9} + \cdots + 33750 $ x^16 - 8x^15 + 96x^14 - 532x^13 + 3236x^12 - 12864x^11 + 49526x^10 - 141436x^9 + 362298x^8 - 722060x^7 + 1208164x^6 - 1570812x^5 + 1591101x^4 - 1183860x^3 + 619650x^2 - 202500*x + 33750
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.14
Root			$0.500000 - 0.442923i$ of defining polynomial
Character	$\chi$	$=$	1680.769
Dual form			1680.3.bd.a.769.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} +(2.40341 + 4.38447i) q^{5} +(6.94781 - 0.853218i) q^{7} +3.00000 q^{9} +O(q^{10})$	\(q+1.73205 q^{3} +(2.40341 + 4.38447i) q^{5} +(6.94781 - 0.853218i) q^{7} +3.00000 q^{9} -2.88097 q^{11} -13.8145 q^{13} +(4.16283 + 7.59413i) q^{15} -24.1754 q^{17} +6.53403i q^{19} +(12.0340 - 1.47782i) q^{21} +28.8420i q^{23} +(-13.4472 + 21.0754i) q^{25} +5.19615 q^{27} -32.9589 q^{29} -2.43276i q^{31} -4.98998 q^{33} +(20.4394 + 28.4118i) q^{35} +50.9799i q^{37} -23.9275 q^{39} +21.5225i q^{41} +13.5554i q^{43} +(7.21024 + 13.1534i) q^{45} -40.7305 q^{47} +(47.5440 - 11.8560i) q^{49} -41.8730 q^{51} +17.2758i q^{53} +(-6.92415 - 12.6315i) q^{55} +11.3173i q^{57} -1.47488i q^{59} +111.568i q^{61} +(20.8434 - 2.55965i) q^{63} +(-33.2020 - 60.5694i) q^{65} -120.293i q^{67} +49.9558i q^{69} +90.3855 q^{71} +21.4890 q^{73} +(-23.2913 + 36.5037i) q^{75} +(-20.0164 + 2.45809i) q^{77} +66.1324 q^{79} +9.00000 q^{81} -78.5172 q^{83} +(-58.1035 - 105.996i) q^{85} -57.0865 q^{87} +90.9724i q^{89} +(-95.9807 + 11.7868i) q^{91} -4.21367i q^{93} +(-28.6483 + 15.7040i) q^{95} +44.1972 q^{97} -8.64290 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 48 q^{9}+O(q^{10}) $ 16 * q + 48 * q^9	$ 16 q + 48 q^{9} - 96 q^{11} + 24 q^{15} + 24 q^{21} + 24 q^{25} + 64 q^{29} + 8 q^{35} + 144 q^{39} + 224 q^{49} + 48 q^{51} + 368 q^{65} + 384 q^{71} + 608 q^{79} + 144 q^{81} - 440 q^{85} - 224 q^{91} + 560 q^{95} - 288 q^{99}+O(q^{100}) $ 16 * q + 48 * q^9 - 96 * q^11 + 24 * q^15 + 24 * q^21 + 24 * q^25 + 64 * q^29 + 8 * q^35 + 144 * q^39 + 224 * q^49 + 48 * q^51 + 368 * q^65 + 384 * q^71 + 608 * q^79 + 144 * q^81 - 440 * q^85 - 224 * q^91 + 560 * q^95 - 288 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$-1$	$-1$	$1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	1.73205			0.577350
$3$	1.73205			0.577350
$4$		0			0
$5$	2.40341	+	4.38447i	0.480682	+	0.876895i
$5$	2.40341	+	4.38447i	0.480682	+	0.876895i
$6$		0			0
$7$	6.94781	−	0.853218i	0.992544	−	0.121888i
$7$	6.94781	−	0.853218i	0.992544	−	0.121888i
$8$		0			0
$9$	3.00000			0.333333
$10$		0			0
$11$	−2.88097			−0.261906			−0.130953	−	0.991389i	$-0.541804\pi$
$11$	−2.88097			−0.261906			−0.130953	+	0.991389i	$0.541804\pi$
$12$		0			0
$13$	−13.8145			−1.06266			−0.531328	−	0.847166i	$-0.678307\pi$
$13$	−13.8145			−1.06266			−0.531328	+	0.847166i	$0.678307\pi$
$14$		0			0
$15$	4.16283	+	7.59413i	0.277522	+	0.506275i
$16$		0			0
$17$	−24.1754			−1.42208			−0.711041	−	0.703150i	$-0.751776\pi$
$17$	−24.1754			−1.42208			−0.711041	+	0.703150i	$0.751776\pi$
$18$		0			0
$19$			6.53403i			0.343896i	0.985106	+	0.171948i	$0.0550062\pi$
$19$			6.53403i			0.343896i	−0.985106	+	0.171948i	$0.944994\pi$
$20$		0			0
$21$	12.0340	−	1.47782i	0.573045	−	0.0703722i
$22$		0			0
$23$			28.8420i			1.25400i	0.779019	+	0.627000i	$0.215717\pi$
$23$			28.8420i			1.25400i	−0.779019	+	0.627000i	$0.784283\pi$
$24$		0			0
$25$	−13.4472	+	21.0754i	−0.537889	+	0.843016i
$26$		0			0
$27$	5.19615			0.192450
$28$		0			0
$29$	−32.9589			−1.13651			−0.568257	−	0.822851i	$-0.692382\pi$
$29$	−32.9589			−1.13651			−0.568257	+	0.822851i	$0.692382\pi$
$30$		0			0
$31$		−	2.43276i		−	0.0784762i	−0.999230	−	0.0392381i	$-0.987507\pi$
$31$		−	2.43276i		−	0.0784762i	0.999230	−	0.0392381i	$-0.0124931\pi$
$32$		0			0
$33$	−4.98998			−0.151212
$34$		0			0
$35$	20.4394	+	28.4118i	0.583982	+	0.811767i
$36$		0			0
$37$			50.9799i			1.37783i	0.724840	+	0.688917i	$0.241914\pi$
$37$			50.9799i			1.37783i	−0.724840	+	0.688917i	$0.758086\pi$
$38$		0			0
$39$	−23.9275			−0.613525
$40$		0			0
$41$			21.5225i			0.524940i	0.964940	+	0.262470i	$0.0845371\pi$
$41$			21.5225i			0.524940i	−0.964940	+	0.262470i	$0.915463\pi$
$42$		0			0
$43$			13.5554i			0.315243i	0.987500	+	0.157621i	$0.0503826\pi$
$43$			13.5554i			0.315243i	−0.987500	+	0.157621i	$0.949617\pi$
$44$		0			0
$45$	7.21024	+	13.1534i	0.160227	+	0.292298i
$46$		0			0
$47$	−40.7305			−0.866605			−0.433303	−	0.901248i	$-0.642652\pi$
$47$	−40.7305			−0.866605			−0.433303	+	0.901248i	$0.642652\pi$
$48$		0			0
$49$	47.5440	−	11.8560i	0.970287	−	0.241959i
$50$		0			0
$51$	−41.8730			−0.821040
$52$		0			0
$53$			17.2758i			0.325959i	0.986629	+	0.162979i	$0.0521104\pi$
$53$			17.2758i			0.325959i	−0.986629	+	0.162979i	$0.947890\pi$
$54$		0			0
$55$	−6.92415	−	12.6315i	−0.125894	−	0.229664i
$56$		0			0
$57$			11.3173i			0.198549i
$58$		0			0
$59$		−	1.47488i		−	0.0249980i	−0.999922	−	0.0124990i	$-0.996021\pi$
$59$		−	1.47488i		−	0.0249980i	0.999922	−	0.0124990i	$-0.00397867\pi$
$60$		0			0
$61$			111.568i			1.82898i	0.404609	+	0.914490i	$0.367408\pi$
$61$			111.568i			1.82898i	−0.404609	+	0.914490i	$0.632592\pi$
$62$		0			0
$63$	20.8434	−	2.55965i	0.330848	−	0.0406294i
$64$		0			0
$65$	−33.2020	−	60.5694i	−0.510800	−	0.931837i
$66$		0			0
$67$		−	120.293i		−	1.79542i	−0.440590	−	0.897709i	$-0.645231\pi$
$67$		−	120.293i		−	1.79542i	0.440590	−	0.897709i	$-0.354769\pi$
$68$		0			0
$69$			49.9558i			0.723997i
$70$		0			0
$71$	90.3855			1.27304			0.636518	−	0.771262i	$-0.280374\pi$
$71$	90.3855			1.27304			0.636518	+	0.771262i	$0.280374\pi$
$72$		0			0
$73$	21.4890			0.294370			0.147185	−	0.989109i	$-0.452979\pi$
$73$	21.4890			0.294370			0.147185	+	0.989109i	$0.452979\pi$
$74$		0			0
$75$	−23.2913	+	36.5037i	−0.310550	+	0.486715i
$76$		0			0
$77$	−20.0164	+	2.45809i	−0.259953	+	0.0319233i
$78$		0			0
$79$	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
$82$		0			0
$83$	−78.5172			−0.945990			−0.472995	−	0.881065i	$-0.656827\pi$
$83$	−78.5172			−0.945990			−0.472995	+	0.881065i	$0.656827\pi$
$84$		0			0
$85$	−58.1035	−	105.996i	−0.683570	−	1.24702i
$86$		0			0
$87$	−57.0865			−0.656166
$88$		0			0
$89$			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
$92$		0			0
$93$		−	4.21367i		−	0.0453083i
$94$		0			0
$95$	−28.6483	+	15.7040i	−0.301561	+	0.165305i
$96$		0			0
$97$	44.1972			0.455642			0.227821	−	0.973703i	$-0.426840\pi$
$97$	44.1972			0.455642			0.227821	+	0.973703i	$0.426840\pi$
$98$		0			0
$99$	−8.64290			−0.0873020
$100$		0			0
$101$		−	180.052i		−	1.78269i	−0.453326	−	0.891345i	$-0.649763\pi$
$101$		−	180.052i		−	1.78269i	0.453326	−	0.891345i	$-0.350237\pi$
$102$		0			0
$103$	107.208			1.04086			0.520428	−	0.853906i	$-0.325772\pi$
$103$	107.208			1.04086			0.520428	+	0.853906i	$0.325772\pi$
$104$		0			0
$105$	35.4020	+	49.2108i	0.337162	+	0.468674i
$106$		0			0
$107$		−	33.1521i		−	0.309833i	−0.987928	−	0.154917i	$-0.950489\pi$
$107$		−	33.1521i		−	0.309833i	0.987928	−	0.154917i	$-0.0495109\pi$
$108$		0			0
$109$	108.819			0.998338			0.499169	−	0.866505i	$-0.333639\pi$
$109$	108.819			0.998338			0.499169	+	0.866505i	$0.333639\pi$
$110$		0			0
$111$			88.2997i			0.795493i
$112$		0			0
$113$		−	157.658i		−	1.39520i	−0.716488	−	0.697600i	$-0.754251\pi$
$113$		−	157.658i		−	1.39520i	0.716488	−	0.697600i	$-0.245749\pi$
$114$		0			0
$115$	−126.457	+	69.3192i	−1.09963	+	0.602776i
$116$		0			0
$117$	−41.4436			−0.354219
$118$		0			0
$119$	−167.966	+	20.6269i	−1.41148	+	0.173335i
$120$		0			0
$121$	−112.700			−0.931405
$122$		0			0
$123$			37.2781i			0.303074i
$124$		0			0
$125$	−124.724	−	8.30611i	−0.997790	−	0.0664489i
$126$		0			0
$127$		−	28.2047i		−	0.222085i	−0.993816	−	0.111042i	$-0.964581\pi$
$127$		−	28.2047i		−	0.222085i	0.993816	−	0.111042i	$-0.0354189\pi$
$128$		0			0
$129$			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	$0.794529\pi$
$132$		0			0
$133$	5.57495	+	45.3972i	0.0419169	+	0.341332i
$134$		0			0
$135$	12.4885	+	22.7824i	0.0925074	+	0.168758i
$136$		0			0
$137$			244.855i			1.78726i	0.448800	+	0.893632i	$0.351852\pi$
$137$			244.855i			1.78726i	−0.448800	+	0.893632i	$0.648148\pi$
$138$		0			0
$139$			238.245i			1.71399i	0.515325	+	0.856995i	$0.327671\pi$
$139$			238.245i			1.71399i	−0.515325	+	0.856995i	$0.672329\pi$
$140$		0			0
$141$	−70.5472			−0.500335
$142$		0			0
$143$	39.7992			0.278316
$144$		0			0
$145$	−79.2138	−	144.507i	−0.546302	−	0.996603i
$146$		0			0
$147$	82.3487	−	20.5352i	0.560195	−	0.139695i
$148$		0			0
$149$	10.8398			0.0727503			0.0363751	−	0.999338i	$-0.488419\pi$
$149$	10.8398			0.0727503			0.0363751	+	0.999338i	$0.488419\pi$
$150$		0			0
$151$	112.999			0.748340			0.374170	−	0.927360i	$-0.377928\pi$
$151$	112.999			0.748340			0.374170	+	0.927360i	$0.377928\pi$
$152$		0			0
$153$	−72.5262			−0.474027
$154$		0			0
$155$	10.6664	−	5.84693i	0.0688154	−	0.0377222i
$156$		0			0
$157$	−103.693			−0.660462			−0.330231	−	0.943900i	$-0.607127\pi$
$157$	−103.693			−0.660462			−0.330231	+	0.943900i	$0.607127\pi$
$158$		0			0
$159$			29.9226i			0.188193i
$160$		0			0
$161$	24.6085	+	200.389i	0.152848	+	1.24465i
$162$		0			0
$163$		−	199.576i		−	1.22439i	−0.790706	−	0.612196i	$-0.790287\pi$
$163$		−	199.576i		−	1.22439i	0.790706	−	0.612196i	$-0.209713\pi$
$164$		0			0
$165$	−11.9930	−	21.8784i	−0.0726848	−	0.132597i
$166$		0			0
$167$	−87.0777			−0.521424			−0.260712	−	0.965417i	$-0.583957\pi$
$167$	−87.0777			−0.521424			−0.260712	+	0.965417i	$0.583957\pi$
$168$		0			0
$169$	21.8411			0.129237
$170$		0			0
$171$			19.6021i			0.114632i
$172$		0			0
$173$	106.597			0.616166			0.308083	−	0.951359i	$-0.400313\pi$
$173$	106.597			0.616166			0.308083	+	0.951359i	$0.400313\pi$
$174$		0			0
$175$	−75.4468	+	157.901i	−0.431124	+	0.902292i
$176$		0			0
$177$		−	2.55458i		−	0.0144326i
$178$		0			0
$179$	−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	$0.172209\pi$
$181$			186.431i			1.03000i	−0.857189	+	0.515002i	$0.827791\pi$
$182$		0			0
$183$			193.241i			1.05596i
$184$		0			0
$185$	−223.520	+	122.526i	−1.20822	+	0.662301i
$186$		0			0
$187$	69.6485			0.372452
$188$		0			0
$189$	36.1019	−	4.43345i	0.191015	−	0.0234574i
$190$		0			0
$191$	177.076			0.927102			0.463551	−	0.886070i	$-0.346575\pi$
$191$	177.076			0.927102			0.463551	+	0.886070i	$0.346575\pi$
$192$		0			0
$193$		−	153.453i		−	0.795092i	−0.917582	−	0.397546i	$-0.869862\pi$
$193$		−	153.453i		−	0.795092i	0.917582	−	0.397546i	$-0.130138\pi$
$194$		0			0
$195$	−57.5076	−	104.909i	−0.294911	−	0.537996i
$196$		0			0
$197$		−	214.041i		−	1.08650i	−0.839570	−	0.543252i	$-0.817193\pi$
$197$		−	214.041i		−	1.08650i	0.839570	−	0.543252i	$-0.182807\pi$
$198$		0			0
$199$		−	178.176i		−	0.895359i	−0.894194	−	0.447679i	$-0.852251\pi$
$199$		−	178.176i		−	0.895359i	0.894194	−	0.447679i	$-0.147749\pi$
$200$		0			0
$201$		−	208.354i		−	1.03658i
$202$		0			0
$203$	−228.992	+	28.1211i	−1.12804	+	0.138528i
$204$		0			0
$205$	−94.3650	+	51.7275i	−0.460317	+	0.252329i
$206$		0			0
$207$			86.5260i			0.418000i
$208$		0			0
$209$		−	18.8243i		−	0.0900686i
$210$		0			0
$211$	398.012			1.88631			0.943156	−	0.332350i	$-0.107842\pi$
$211$	398.012			1.88631			0.943156	+	0.332350i	$0.107842\pi$
$212$		0			0
$213$	156.552			0.734987
$214$		0			0
$215$	−59.4335	+	32.5793i	−0.276435	+	0.151532i
$216$		0			0
$217$	−2.07568	−	16.9024i	−0.00956533	−	0.0778911i
$218$		0			0
$219$	37.2200			0.169954
$220$		0			0
$221$	333.972			1.51118
$222$		0			0
$223$	6.96741			0.0312440			0.0156220	−	0.999878i	$-0.495027\pi$
$223$	6.96741			0.0312440			0.0156220	+	0.999878i	$0.495027\pi$
$224$		0			0
$225$	−40.3417	+	63.2262i	−0.179296	+	0.281005i
$226$		0			0
$227$	242.118			1.06660			0.533299	−	0.845927i	$-0.320952\pi$
$227$	242.118			1.06660			0.533299	+	0.845927i	$0.320952\pi$
$228$		0			0
$229$			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
$232$		0			0
$233$			435.850i			1.87060i	0.353856	+	0.935300i	$0.384870\pi$
$233$			435.850i			1.87060i	−0.353856	+	0.935300i	$0.615130\pi$
$234$		0			0
$235$	−97.8921	−	178.582i	−0.416562	−	0.759922i
$236$		0			0
$237$	114.545			0.483311
$238$		0			0
$239$	−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	$0.170347\pi$
$241$			245.810i			1.01996i	−0.860186	+	0.509980i	$0.829653\pi$
$242$		0			0
$243$	15.5885			0.0641500
$244$		0			0
$245$	166.250	+	179.961i	0.678572	+	0.734534i
$246$		0			0
$247$		−	90.2645i		−	0.365443i
$248$		0			0
$249$	−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	$0.905388\pi$
$252$		0			0
$253$		−	83.0928i		−	0.328430i
$254$		0			0
$255$	−100.638	−	183.591i	−0.394659	−	0.719965i
$256$		0			0
$257$	421.734			1.64099			0.820495	−	0.571654i	$-0.193698\pi$
$257$	421.734			1.64099			0.820495	+	0.571654i	$0.193698\pi$
$258$		0			0
$259$	43.4969	+	354.198i	0.167942	+	1.36756i
$260$		0			0
$261$	−98.8767			−0.378838
$262$		0			0
$263$			68.6354i			0.260971i	0.991450	+	0.130486i	$0.0416536\pi$
$263$			68.6354i			0.260971i	−0.991450	+	0.130486i	$0.958346\pi$
$264$		0			0
$265$	−75.7454	+	41.5209i	−0.285832	+	0.156683i
$266$		0			0
$267$			157.569i			0.590145i
$268$		0			0
$269$		−	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	$-0.178540\pi$
$272$		0			0
$273$	−166.243	+	20.4153i	−0.608950	+	0.0747814i
$274$		0			0
$275$	38.7410	−	60.7175i	0.140876	−	0.220791i
$276$		0			0
$277$		−	385.440i		−	1.39148i	−0.718294	−	0.695740i	$-0.755077\pi$
$277$		−	385.440i		−	1.39148i	0.718294	−	0.695740i	$-0.244923\pi$
$278$		0			0
$279$		−	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	$0.357070\pi$
$282$		0			0
$283$	−501.435			−1.77185			−0.885927	−	0.463825i	$-0.846477\pi$
$283$	−501.435			−1.77185			−0.885927	+	0.463825i	$0.846477\pi$
$284$		0			0
$285$	−49.6203	+	27.2001i	−0.174106	+	0.0954389i
$286$		0			0
$287$	18.3634	+	149.534i	0.0639840	+	0.521026i
$288$		0			0
$289$	295.450			1.02232
$290$		0			0
$291$	76.5519			0.263065
$292$		0			0
$293$	−337.308			−1.15122			−0.575611	−	0.817724i	$-0.695236\pi$
$293$	−337.308			−1.15122			−0.575611	+	0.817724i	$0.695236\pi$
$294$		0			0
$295$	6.46659	−	3.54476i	0.0219207	−	0.0120161i
$296$		0			0
$297$	−14.9699			−0.0504039
$298$		0			0
$299$		−	398.438i		−	1.33257i
$300$		0			0
$301$	11.5657	+	94.1806i	0.0384244	+	0.312892i
$302$		0			0
$303$		−	311.859i		−	1.02924i
$304$		0			0
$305$	−489.166	+	268.143i	−1.60382	+	0.879159i
$306$		0			0
$307$	−112.285			−0.365751			−0.182875	−	0.983136i	$-0.558540\pi$
$307$	−112.285			−0.365751			−0.182875	+	0.983136i	$0.558540\pi$
$308$		0			0
$309$	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	$0.924329\pi$
$312$		0			0
$313$	−139.927			−0.447052			−0.223526	−	0.974698i	$-0.571757\pi$
$313$	−139.927			−0.447052			−0.223526	+	0.974698i	$0.571757\pi$
$314$		0			0
$315$	61.3181	+	85.2355i	0.194661	+	0.270589i
$316$		0			0
$317$			39.7776i			0.125481i	0.998030	+	0.0627407i	$0.0199841\pi$
$317$			39.7776i			0.125481i	−0.998030	+	0.0627407i	$0.980016\pi$
$318$		0			0
$319$	94.9535			0.297660
$320$		0			0
$321$		−	57.4212i		−	0.178882i
$322$		0			0
$323$		−	157.963i		−	0.489049i
$324$		0			0
$325$	185.767	−	291.147i	0.571591	−	0.895836i
$326$		0			0
$327$	188.480			0.576391
$328$		0			0
$329$	−282.987	+	34.7519i	−0.860144	+	0.105629i
$330$		0			0
$331$	13.6736			0.0413101			0.0206550	−	0.999787i	$-0.493425\pi$
$331$	13.6736			0.0413101			0.0206550	+	0.999787i	$0.493425\pi$
$332$		0			0
$333$			152.940i			0.459278i
$334$		0			0
$335$	527.421	−	289.114i	1.57439	−	0.863026i
$336$		0			0
$337$		−	364.873i		−	1.08271i	−0.840795	−	0.541354i	$-0.817912\pi$
$337$		−	364.873i		−	1.08271i	0.840795	−	0.541354i	$-0.182088\pi$
$338$		0			0
$339$		−	273.071i		−	0.805519i
$340$		0			0
$341$			7.00871i			0.0205534i
$342$		0			0
$343$	320.211	−	122.938i	0.933560	−	0.358421i
$344$		0			0
$345$	−219.030	+	120.064i	−0.634869	+	0.348013i
$346$		0			0
$347$			469.783i			1.35384i	0.736056	+	0.676920i	$0.236686\pi$
$347$			469.783i			1.35384i	−0.736056	+	0.676920i	$0.763314\pi$
$348$		0			0
$349$		−	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
$352$		0			0
$353$	−473.408			−1.34110			−0.670550	−	0.741864i	$-0.733942\pi$
$353$	−473.408			−1.34110			−0.670550	+	0.741864i	$0.733942\pi$
$354$		0			0
$355$	217.234	+	396.293i	0.611926	+	1.11632i
$356$		0			0
$357$	−290.926	+	35.7268i	−0.814918	+	0.100075i
$358$		0			0
$359$	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
$362$		0			0
$363$	−195.202			−0.537747
$364$		0			0
$365$	51.6469	+	94.2179i	0.141498	+	0.258131i
$366$		0			0
$367$	331.526			0.903341			0.451671	−	0.892185i	$-0.350828\pi$
$367$	331.526			0.903341			0.451671	+	0.892185i	$0.350828\pi$
$368$		0			0
$369$			64.5676i			0.174980i
$370$		0			0
$371$	14.7400	+	120.029i	0.0397306	+	0.323529i
$372$		0			0
$373$		−	416.519i		−	1.11667i	−0.829614	−	0.558337i	$-0.811440\pi$
$373$		−	416.519i		−	1.11667i	0.829614	−	0.558337i	$-0.188560\pi$
$374$		0			0
$375$	−216.028	−	14.3866i	−0.576074	−	0.0383643i
$376$		0			0
$377$	455.311			1.20772
$378$		0			0
$379$	−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
$382$		0			0
$383$	135.702			0.354314			0.177157	−	0.984183i	$-0.443310\pi$
$383$	135.702			0.354314			0.177157	+	0.984183i	$0.443310\pi$
$384$		0			0
$385$	−58.8851	−	81.8536i	−0.152948	−	0.212607i
$386$		0			0
$387$			40.6663i			0.105081i
$388$		0			0
$389$	−703.375			−1.80816			−0.904081	−	0.427361i	$-0.859443\pi$
$389$	−703.375			−1.80816			−0.904081	+	0.427361i	$0.859443\pi$
$390$		0			0
$391$		−	697.267i		−	1.78329i
$392$		0			0
$393$			273.006i			0.694672i
$394$		0			0
$395$	158.943	+	289.956i	0.402388	+	0.734065i
$396$		0			0
$397$	−2.16495			−0.00545326			−0.00272663	−	0.999996i	$-0.500868\pi$
$397$	−2.16495			−0.00545326			−0.00272663	+	0.999996i	$0.500868\pi$
$398$		0			0
$399$	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	$0.591719\pi$
$402$		0			0
$403$			33.6075i			0.0833932i
$404$		0			0
$405$	21.6307	+	39.4603i	0.0534092	+	0.0974327i
$406$		0			0
$407$		−	146.871i		−	0.360863i
$408$		0			0
$409$		−	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
$412$		0			0
$413$	−1.25840	−	10.2472i	−0.00304697	−	0.0248117i
$414$		0			0
$415$	−188.709	−	344.257i	−0.454721	−	0.829534i
$416$		0			0
$417$			412.652i			0.989572i
$418$		0			0
$419$			636.723i			1.51963i	0.650141	+	0.759813i	$0.274710\pi$
$419$			636.723i			1.51963i	−0.650141	+	0.759813i	$0.725290\pi$
$420$		0			0
$421$	816.589			1.93964			0.969821	−	0.243819i	$-0.0784003\pi$
$421$	816.589			1.93964			0.969821	+	0.243819i	$0.0784003\pi$
$422$		0			0
$423$	−122.191			−0.288868
$424$		0			0
$425$	325.092	−	509.506i	0.764922	−	1.19884i
$426$		0			0
$427$	95.1916	+	775.151i	0.222931	+	1.81534i
$428$		0			0
$429$	68.9342			0.160686
$430$		0			0
$431$	288.911			0.670328			0.335164	−	0.942160i	$-0.391208\pi$
$431$	288.911			0.670328			0.335164	+	0.942160i	$0.391208\pi$
$432$		0			0
$433$	−114.744			−0.264997			−0.132498	−	0.991183i	$-0.542300\pi$
$433$	−114.744			−0.264997			−0.132498	+	0.991183i	$0.542300\pi$
$434$		0			0
$435$	−137.202	−	250.294i	−0.315408	−	0.575389i
$436$		0			0
$437$	−188.454			−0.431246
$438$		0			0
$439$		−	637.443i		−	1.45203i	−0.687677	−	0.726017i	$-0.741369\pi$
$439$		−	637.443i		−	1.45203i	0.687677	−	0.726017i	$-0.258631\pi$
$440$		0			0
$441$	142.632	−	35.5679i	0.323429	−	0.0806529i
$442$		0			0
$443$			184.901i			0.417384i	0.977981	+	0.208692i	$0.0669207\pi$
$443$			184.901i			0.417384i	−0.977981	+	0.208692i	$0.933079\pi$
$444$		0			0
$445$	−398.866	+	218.644i	−0.896328	+	0.491335i
$446$		0			0
$447$	18.7751			0.0420024
$448$		0			0
$449$	−313.278			−0.697723			−0.348862	−	0.937174i	$-0.613432\pi$
$449$	−313.278			−0.697723			−0.348862	+	0.937174i	$0.613432\pi$
$450$		0			0
$451$		−	62.0057i		−	0.137485i
$452$		0			0
$453$	195.721			0.432054
$454$		0			0
$455$	−282.360	−	392.496i	−0.620571	−	0.862629i
$456$		0			0
$457$		−	174.826i		−	0.382551i	−0.981536	−	0.191275i	$-0.938738\pi$
$457$		−	174.826i		−	0.382551i	0.981536	−	0.191275i	$-0.0612623\pi$
$458$		0			0
$459$	−125.619			−0.273680
$460$		0			0
$461$		−	379.263i		−	0.822695i	−0.911479	−	0.411348i	$-0.865058\pi$
$461$		−	379.263i		−	0.822695i	0.911479	−	0.411348i	$-0.134942\pi$
$462$		0			0
$463$		−	403.499i		−	0.871489i	−0.900071	−	0.435744i	$-0.856485\pi$
$463$		−	403.499i		−	0.871489i	0.900071	−	0.435744i	$-0.143515\pi$
$464$		0			0
$465$	18.4747	−	10.1272i	0.0397306	−	0.0217789i
$466$		0			0
$467$	−807.901			−1.72998			−0.864991	−	0.501788i	$-0.832676\pi$
$467$	−807.901			−1.72998			−0.864991	+	0.501788i	$0.832676\pi$
$468$		0			0
$469$	−102.636	−	835.772i	−0.218840	−	1.78203i
$470$		0			0
$471$	−179.601			−0.381318
$472$		0			0
$473$		−	39.0528i		−	0.0825640i
$474$		0			0
$475$	−137.707	−	87.8645i	−0.289910	−	0.184978i
$476$		0			0
$477$			51.8275i			0.108653i
$478$		0			0
$479$		−	439.564i		−	0.917671i	−0.888521	−	0.458835i	$-0.848267\pi$
$479$		−	439.564i		−	0.917671i	0.888521	−	0.458835i	$-0.151733\pi$
$480$		0			0
$481$		−	704.263i		−	1.46416i
$482$		0			0
$483$	42.6232	+	347.083i	0.0882467	+	0.718599i
$484$		0			0
$485$	106.224	+	193.782i	0.219019	+	0.399550i
$486$		0			0
$487$		−	37.9665i		−	0.0779599i	−0.999240	−	0.0389799i	$-0.987589\pi$
$487$		−	37.9665i		−	0.0779599i	0.999240	−	0.0389799i	$-0.0124108\pi$
$488$		0			0
$489$		−	345.675i		−	0.706903i
$490$		0			0
$491$	7.32753			0.0149237			0.00746184	−	0.999972i	$-0.497625\pi$
$491$	7.32753			0.0149237			0.00746184	+	0.999972i	$0.497625\pi$
$492$		0			0
$493$	796.794			1.61622
$494$		0			0
$495$	−20.7725	−	37.8946i	−0.0419646	−	0.0765547i
$496$		0			0
$497$	627.981	−	77.1185i	1.26354	−	0.155168i
$498$		0			0
$499$	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
$502$		0			0
$503$	70.5081			0.140175			0.0700876	−	0.997541i	$-0.477672\pi$
$503$	70.5081			0.140175			0.0700876	+	0.997541i	$0.477672\pi$
$504$		0			0
$505$	789.432	−	432.738i	1.56323	−	0.856908i
$506$		0			0
$507$	37.8299			0.0746152
$508$		0			0
$509$			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
$512$		0			0
$513$			33.9518i			0.0661829i
$514$		0			0
$515$	257.665	+	470.051i	0.500321	+	0.912721i
$516$		0			0
$517$	117.343			0.226969
$518$		0			0
$519$	184.631			0.355744
$520$		0			0
$521$		−	214.185i		−	0.411104i	−0.978646	−	0.205552i	$-0.934101\pi$
$521$		−	214.185i		−	0.411104i	0.978646	−	0.205552i	$-0.0658990\pi$
$522$		0			0
$523$	121.732			0.232757			0.116379	−	0.993205i	$-0.462871\pi$
$523$	121.732			0.232757			0.116379	+	0.993205i	$0.462871\pi$
$524$		0			0
$525$	−130.678	+	273.493i	−0.248910	+	0.520939i
$526$		0			0
$527$			58.8130i			0.111600i
$528$		0			0
$529$	−302.861			−0.572515
$530$		0			0
$531$		−	4.42465i		−	0.00833268i
$532$		0			0
$533$		−	297.324i		−	0.557830i
$534$		0			0
$535$	145.355	−	79.6783i	0.271691	−	0.148931i
$536$		0			0
$537$	−477.840			−0.889832
$538$		0			0
$539$	−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	$0.312677\pi$
$542$		0			0
$543$			322.908i			0.594674i
$544$		0			0
$545$	261.537	+	477.113i	0.479884	+	0.875437i
$546$		0			0
$547$		−	1045.16i		−	1.91071i	−0.295457	−	0.955356i	$-0.595472\pi$
$547$		−	1045.16i		−	1.91071i	0.295457	−	0.955356i	$-0.404528\pi$
$548$		0			0
$549$			334.703i			0.609660i
$550$		0			0
$551$		−	215.354i		−	0.390843i
$552$		0			0
$553$	459.475	−	56.4253i	0.830877	−	0.102035i
$554$		0			0
$555$	−387.148	+	212.221i	−0.697564	+	0.382380i
$556$		0			0
$557$			601.731i			1.08031i	0.841566	+	0.540154i	$0.181634\pi$
$557$			601.731i			1.08031i	−0.841566	+	0.540154i	$0.818366\pi$
$558$		0			0
$559$		−	187.262i		−	0.334995i
$560$		0			0
$561$	120.635			0.215035
$562$		0			0
$563$	363.871			0.646308			0.323154	−	0.946346i	$-0.395257\pi$
$563$	363.871			0.646308			0.323154	+	0.946346i	$0.395257\pi$
$564$		0			0
$565$	691.245	−	378.916i	1.22344	−	0.670648i
$566$		0			0
$567$	62.5303	−	7.67896i	0.110283	−	0.0135431i
$568$		0			0
$569$	902.900			1.58682			0.793410	−	0.608688i	$-0.208304\pi$
$569$	902.900			1.58682			0.793410	+	0.608688i	$0.208304\pi$
$570$		0			0
$571$	847.207			1.48372			0.741862	−	0.670552i	$-0.233943\pi$
$571$	847.207			1.48372			0.741862	+	0.670552i	$0.233943\pi$
$572$		0			0
$573$	306.705			0.535263
$574$		0			0
$575$	−607.856	−	387.845i	−1.05714	−	0.674512i
$576$		0			0
$577$	306.535			0.531256			0.265628	−	0.964076i	$-0.414421\pi$
$577$	306.535			0.531256			0.265628	+	0.964076i	$0.414421\pi$
$578$		0			0
$579$		−	265.788i		−	0.459047i
$580$		0			0
$581$	−545.522	+	66.9923i	−0.938937	+	0.115305i
$582$		0			0
$583$		−	49.7711i		−	0.0853707i
$584$		0			0
$585$	−99.6060	−	181.708i	−0.170267	−	0.310612i
$586$		0			0
$587$	−467.524			−0.796463			−0.398232	−	0.917285i	$-0.630376\pi$
$587$	−467.524			−0.796463			−0.398232	+	0.917285i	$0.630376\pi$
$588$		0			0
$589$	15.8958			0.0269877
$590$		0			0
$591$		−	370.730i		−	0.627293i
$592$		0			0
$593$	−744.542			−1.25555			−0.627775	−	0.778395i	$-0.716034\pi$
$593$	−744.542			−1.25555			−0.627775	+	0.778395i	$0.716034\pi$
$594$		0			0
$595$	−494.130	−	686.868i	−0.830470	−	1.15440i
$596$		0			0
$597$		−	308.611i		−	0.516936i
$598$		0			0
$599$	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	$0.224641\pi$
$601$			779.606i			1.29718i	−0.761137	+	0.648591i	$0.775359\pi$
$602$		0			0
$603$		−	360.879i		−	0.598472i
$604$		0			0
$605$	−270.865	−	494.130i	−0.447710	−	0.816744i
$606$		0			0
$607$	567.539			0.934990			0.467495	−	0.883996i	$-0.345157\pi$
$607$	567.539			0.934990			0.467495	+	0.883996i	$0.345157\pi$
$608$		0			0
$609$	−396.626	+	48.7072i	−0.651274	+	0.0799790i
$610$		0			0
$611$	562.672			0.920903
$612$		0			0
$613$			801.088i			1.30683i	0.756999	+	0.653416i	$0.226665\pi$
$613$			801.088i			1.30683i	−0.756999	+	0.653416i	$0.773335\pi$
$614$		0			0
$615$	−163.445	+	89.5947i	−0.265764	+	0.145682i
$616$		0			0
$617$			71.2583i			0.115492i	0.998331	+	0.0577458i	$0.0183913\pi$
$617$			71.2583i			0.115492i	−0.998331	+	0.0577458i	$0.981609\pi$
$618$		0			0
$619$			821.793i			1.32761i	0.747904	+	0.663807i	$0.231060\pi$
$619$			821.793i			1.32761i	−0.747904	+	0.663807i	$0.768940\pi$
$620$		0			0
$621$			149.867i			0.241332i
$622$		0			0
$623$	77.6193	+	632.059i	0.124590	+	1.01454i
$624$		0			0
$625$	−263.345	−	566.811i	−0.421351	−	0.906897i
$626$		0			0
$627$		−	32.6047i		−	0.0520011i
$628$		0			0
$629$		−	1232.46i		−	1.95939i
$630$		0			0
$631$	916.907			1.45310			0.726551	−	0.687113i	$-0.241122\pi$
$631$	916.907			1.45310			0.726551	+	0.687113i	$0.241122\pi$
$632$		0			0
$633$	689.377			1.08906
$634$		0			0
$635$	123.663	−	67.7876i	0.194745	−	0.106752i
$636$		0			0
$637$	−656.798	+	163.785i	−1.03108	+	0.257119i
$638$		0			0
$639$	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	$0.304975\pi$
$642$		0			0
$643$	256.297			0.398596			0.199298	−	0.979939i	$-0.436134\pi$
$643$	256.297			0.398596			0.199298	+	0.979939i	$0.436134\pi$
$644$		0			0
$645$	−102.942	+	56.4290i	−0.159600	+	0.0874869i
$646$		0			0
$647$	689.147			1.06514			0.532571	−	0.846385i	$-0.321226\pi$
$647$	689.147			1.06514			0.532571	+	0.846385i	$0.321226\pi$
$648$		0			0
$649$			4.24910i			0.00654714i
$650$		0			0
$651$	−3.59518	−	29.2758i	−0.00552255	−	0.0449705i
$652$		0			0
$653$			395.775i			0.606087i	0.952977	+	0.303044i	$0.0980028\pi$
$653$			395.775i			0.606087i	−0.952977	+	0.303044i	$0.901997\pi$
$654$		0			0
$655$	−691.081	+	378.826i	−1.05509	+	0.578361i
$656$		0			0
$657$	64.4670			0.0981233
$658$		0			0
$659$	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	$-0.978334\pi$
$661$		−	89.9129i		−	0.136026i	0.997684	−	0.0680128i	$-0.0216659\pi$
$662$		0			0
$663$	578.456			0.872482
$664$		0			0
$665$	−185.644	+	133.551i	−0.279164	+	0.200829i
$666$		0			0
$667$		−	950.600i		−	1.42519i
$668$		0			0
$669$	12.0679			0.0180387
$670$		0			0
$671$		−	321.423i		−	0.479021i
$672$		0			0
$673$		−	833.478i		−	1.23845i	−0.785213	−	0.619226i	$-0.787447\pi$
$673$		−	833.478i		−	1.23845i	0.785213	−	0.619226i	$-0.212553\pi$
$674$		0			0
$675$	−69.8738	+	109.511i	−0.103517	+	0.162238i
$676$		0			0
$677$	44.1092			0.0651540			0.0325770	−	0.999469i	$-0.489629\pi$
$677$	44.1092			0.0651540			0.0325770	+	0.999469i	$0.489629\pi$
$678$		0			0
$679$	307.074	−	37.7099i	0.452244	−	0.0555373i
$680$		0			0
$681$	419.360			0.615800
$682$		0			0
$683$			130.146i			0.190550i	0.995451	+	0.0952750i	$0.0303731\pi$
$683$			130.146i			0.190550i	−0.995451	+	0.0952750i	$0.969627\pi$
$684$		0			0
$685$	−1073.56	+	588.488i	−1.56724	+	0.859107i
$686$		0			0
$687$			390.727i			0.568744i
$688$		0			0
$689$		−	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	$0.826987\pi$
$692$		0			0
$693$	−60.0492	+	7.37428i	−0.0866511	+	0.0106411i
$694$		0			0
$695$	−1044.58	+	572.600i	−1.50299	+	0.823885i
$696$		0			0
$697$		−	520.316i		−	0.746508i
$698$		0			0
$699$			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	$0.392664\pi$
$702$		0			0
$703$	−333.104			−0.473832
$704$		0			0
$705$	−169.554	−	309.312i	−0.240502	−	0.438741i
$706$		0			0
$707$	−153.623	−	1250.96i	−0.217289	−	1.76940i
$708$		0			0
$709$	−970.049			−1.36819			−0.684097	−	0.729391i	$-0.739803\pi$
$709$	−970.049			−1.36819			−0.684097	+	0.729391i	$0.739803\pi$
$710$		0			0
$711$	198.397			0.279040
$712$		0			0
$713$	70.1657			0.0984092
$714$		0			0
$715$	95.6539	+	174.499i	0.133782	+	0.244054i
$716$		0			0
$717$	−337.249			−0.470362
$718$		0			0
$719$		−	1369.36i		−	1.90453i	−0.305266	−	0.952267i	$-0.598745\pi$
$719$		−	1369.36i		−	1.90453i	0.305266	−	0.952267i	$-0.401255\pi$
$720$		0			0
$721$	744.861	−	91.4719i	1.03309	−	0.126868i
$722$		0			0
$723$			425.756i			0.588874i
$724$		0			0
$725$	443.205	−	694.622i	0.611318	−	0.958099i
$726$		0			0
$727$	1106.85			1.52249			0.761246	−	0.648463i	$-0.224588\pi$
$727$	1106.85			1.52249			0.761246	+	0.648463i	$0.224588\pi$
$728$		0			0
$729$	27.0000			0.0370370
$730$		0			0
$731$		−	327.708i		−	0.448301i
$732$		0			0
$733$	−297.353			−0.405665			−0.202833	−	0.979213i	$-0.565015\pi$
$733$	−297.353			−0.405665			−0.202833	+	0.979213i	$0.565015\pi$
$734$		0			0
$735$	287.954	+	311.701i	0.391774	+	0.424083i
$736$		0			0
$737$			346.560i			0.470231i
$738$		0			0
$739$	−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
$742$		0			0
$743$		−	733.877i		−	0.987722i	−0.869541	−	0.493861i	$-0.835585\pi$
$743$		−	733.877i		−	0.987722i	0.869541	−	0.493861i	$-0.164415\pi$
$744$		0			0
$745$	26.0525	+	47.5268i	0.0349698	+	0.0637943i
$746$		0			0
$747$	−235.552			−0.315330
$748$		0			0
$749$	−28.2860	−	230.335i	−0.0377650	−	0.307523i
$750$		0			0
$751$	71.0036			0.0945455			0.0472727	−	0.998882i	$-0.484947\pi$
$751$	71.0036			0.0945455			0.0472727	+	0.998882i	$0.484947\pi$
$752$		0			0
$753$			254.651i			0.338182i
$754$		0			0
$755$	271.584	+	495.443i	0.359714	+	0.656216i
$756$		0			0
$757$			708.172i			0.935497i	0.883861	+	0.467749i	$0.154935\pi$
$757$			708.172i			0.935497i	−0.883861	+	0.467749i	$0.845065\pi$
$758$		0			0
$759$		−	143.921i		−	0.189619i
$760$		0			0
$761$		−	797.652i		−	1.04816i	−0.851668	−	0.524081i	$-0.824409\pi$
$761$		−	797.652i		−	1.04816i	0.851668	−	0.524081i	$-0.175591\pi$
$762$		0			0
$763$	756.052	−	92.8462i	0.990894	−	0.121686i
$764$		0			0
$765$	−174.310	−	317.989i	−0.227857	−	0.415672i
$766$		0			0
$767$			20.3748i			0.0265643i
$768$		0			0
$769$			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
$772$		0			0
$773$	643.514			0.832489			0.416245	−	0.909253i	$-0.363346\pi$
$773$	643.514			0.832489			0.416245	+	0.909253i	$0.363346\pi$
$774$		0			0
$775$	51.2715	+	32.7139i	0.0661567	+	0.0422115i
$776$		0			0
$777$	75.3389	+	613.490i	0.0969613	+	0.789562i
$778$		0			0
$779$	−140.629			−0.180525
$780$		0			0
$781$	−260.398			−0.333416
$782$		0			0
$783$	−171.259			−0.218722
$784$		0			0
$785$	−249.216	−	454.637i	−0.317473	−	0.579156i
$786$		0			0
$787$	−919.663			−1.16857			−0.584284	−	0.811549i	$-0.698625\pi$
$787$	−919.663			−1.16857			−0.584284	+	0.811549i	$0.698625\pi$
$788$		0			0
$789$			118.880i			0.150672i
$790$		0			0
$791$	−134.516	−	1095.37i	−0.170058	−	1.38480i
$792$		0			0
$793$		−	1541.26i		−	1.94358i
$794$		0			0
$795$	−131.195	+	71.9164i	−0.165025	+	0.0904608i
$796$		0			0
$797$	−246.244			−0.308964			−0.154482	−	0.987996i	$-0.549371\pi$
$797$	−246.244			−0.308964			−0.154482	+	0.987996i	$0.549371\pi$
$798$		0			0
$799$	984.675			1.23238
$800$		0			0
$801$			272.917i			0.340721i
$802$		0			0
$803$	−61.9091			−0.0770973
$804$		0			0
$805$	−819.454	+	589.512i	−1.01796	+	0.732313i
$806$		0			0
$807$		−	161.599i		−	0.200246i
$808$		0			0
$809$	−609.838			−0.753817			−0.376908	−	0.926251i	$-0.623013\pi$
$809$	−609.838			−0.753817			−0.376908	+	0.926251i	$0.623013\pi$
$810$		0			0
$811$			605.726i			0.746888i	0.927653	+	0.373444i	$0.121823\pi$
$811$			605.726i			0.746888i	−0.927653	+	0.373444i	$0.878177\pi$
$812$		0			0
$813$		−	499.379i		−	0.614242i
$814$		0			0
$815$	875.035	−	479.663i	1.07366	−	0.588543i
$816$		0			0
$817$	−88.5717			−0.108411
$818$		0			0
$819$	−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	$0.317486\pi$
$822$		0			0
$823$			317.029i			0.385211i	0.981276	+	0.192606i	$0.0616938\pi$
$823$			317.029i			0.385211i	−0.981276	+	0.192606i	$0.938306\pi$
$824$		0			0
$825$	67.1014	−	105.166i	0.0813350	−	0.127474i
$826$		0			0
$827$		−	90.7251i		−	0.109704i	−0.998494	−	0.0548520i	$-0.982531\pi$
$827$		−	90.7251i		−	0.109704i	0.998494	−	0.0548520i	$-0.0174687\pi$
$828$		0			0
$829$			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
$832$		0			0
$833$	−1149.40	+	286.623i	−1.37983	+	0.344085i
$834$		0			0
$835$	−209.284	−	381.790i	−0.250639	−	0.457234i
$836$		0			0
$837$		−	12.6410i		−	0.0151028i
$838$		0			0
$839$			73.5993i			0.0877227i	0.999038	+	0.0438613i	$0.0139660\pi$
$839$			73.5993i			0.0877227i	−0.999038	+	0.0438613i	$0.986034\pi$
$840$		0			0
$841$	245.289			0.291663
$842$		0			0
$843$	422.548			0.501244
$844$		0			0
$845$	52.4932	+	95.7618i	0.0621221	+	0.113328i
$846$		0			0
$847$	−783.018	+	96.1576i	−0.924460	+	0.113527i
$848$		0			0
$849$	−868.510			−1.02298
$850$		0			0
$851$	−1470.36			−1.72780
$852$		0			0
$853$	822.046			0.963711			0.481856	−	0.876251i	$-0.339963\pi$
$853$	822.046			0.963711			0.481856	+	0.876251i	$0.339963\pi$
$854$		0			0
$855$	−85.9449	+	47.1119i	−0.100520	+	0.0551017i
$856$		0			0
$857$	625.685			0.730087			0.365044	−	0.930990i	$-0.381054\pi$
$857$	625.685			0.730087			0.365044	+	0.930990i	$0.381054\pi$
$858$		0			0
$859$		−	1396.65i		−	1.62590i	−0.582332	−	0.812951i	$-0.697860\pi$
$859$		−	1396.65i		−	1.62590i	0.582332	−	0.812951i	$-0.302140\pi$
$860$		0			0
$861$	31.8064	+	259.001i	0.0369412	+	0.300814i
$862$		0			0
$863$			601.569i			0.697068i	0.937296	+	0.348534i	$0.113320\pi$
$863$			601.569i			0.697068i	−0.937296	+	0.348534i	$0.886680\pi$
$864$		0			0
$865$	256.196	+	467.371i	0.296180	+	0.540313i
$866$		0			0
$867$	511.734			0.590236
$868$		0			0
$869$	−190.525			−0.219246
$870$		0			0
$871$			1661.79i			1.90791i
$872$		0			0
$873$	132.592			0.151881
$874$		0			0
$875$	−873.643	+	48.7073i	−0.998449	+	0.0556654i
$876$		0			0
$877$			570.762i			0.650812i	0.945574	+	0.325406i	$0.105501\pi$
$877$			570.762i			0.650812i	−0.945574	+	0.325406i	$0.894499\pi$
$878$		0			0
$879$	−584.234			−0.664658
$880$		0			0
$881$			1095.44i			1.24341i	0.783253	+	0.621703i	$0.213559\pi$
$881$			1095.44i			1.24341i	−0.783253	+	0.621703i	$0.786441\pi$
$882$		0			0
$883$		−	305.870i		−	0.346399i	−0.984887	−	0.173199i	$-0.944590\pi$
$883$		−	305.870i		−	0.346399i	0.984887	−	0.173199i	$-0.0554105\pi$
$884$		0			0
$885$	11.2005	−	6.13970i	0.0126559	−	0.00693751i
$886$		0			0
$887$	−637.527			−0.718745			−0.359372	−	0.933194i	$-0.617009\pi$
$887$	−637.527			−0.718745			−0.359372	+	0.933194i	$0.617009\pi$
$888$		0			0
$889$	−24.0648	−	195.961i	−0.0270695	−	0.220429i
$890$		0			0
$891$	−25.9287			−0.0291007
$892$		0			0
$893$		−	266.134i		−	0.298022i
$894$		0			0
$895$	−663.056	−	1209.59i	−0.740845	−	1.35150i
$896$		0			0
$897$		−	690.116i		−	0.769360i
$898$		0			0
$899$			80.1812i			0.0891893i
$900$		0			0
$901$		−	417.650i		−	0.463541i
$902$		0			0
$903$	20.0325	+	163.126i	0.0221843	+	0.180649i
$904$		0			0
$905$	−817.401	+	448.070i	−0.903206	+	0.495105i
$906$		0			0
$907$			785.727i			0.866292i	0.901324	+	0.433146i	$0.142597\pi$
$907$			785.727i			0.866292i	−0.901324	+	0.433146i	$0.857403\pi$
$908$		0			0
$909$		−	540.155i		−	0.594230i
$910$		0			0
$911$	909.376			0.998217			0.499109	−	0.866539i	$-0.333661\pi$
$911$	909.376			0.998217			0.499109	+	0.866539i	$0.333661\pi$
$912$		0			0
$913$	226.205			0.247761
$914$		0			0
$915$	−847.260	+	464.438i	−0.925968	+	0.507582i
$916$		0			0
$917$	134.484	+	1095.11i	0.146657	+	1.19424i
$918$		0			0
$919$	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
$922$		0			0
$923$	−1248.63			−1.35280
$924$		0			0
$925$	−1074.42	−	685.538i	−1.16154	−	0.741122i
$926$		0			0
$927$	321.624			0.346952
$928$		0			0
$929$			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
$932$		0			0
$933$			253.707i			0.271926i
$934$		0			0
$935$	167.394	+	305.372i	0.179031	+	0.326601i
$936$		0			0
$937$	−1012.18			−1.08024			−0.540119	−	0.841589i	$-0.681621\pi$
$937$	−1012.18			−1.08024			−0.540119	+	0.841589i	$0.681621\pi$
$938$		0			0
$939$	−242.361			−0.258106
$940$		0			0
$941$			1384.55i			1.47136i	0.677331	+	0.735679i	$0.263137\pi$
$941$			1384.55i			1.47136i	−0.677331	+	0.735679i	$0.736863\pi$
$942$		0			0
$943$	−620.753			−0.658275
$944$		0			0
$945$	106.206	+	147.632i	0.112387	+	0.156225i
$946$		0			0
$947$			1216.79i			1.28489i	0.766332	+	0.642444i	$0.222080\pi$
$947$			1216.79i			1.28489i	−0.766332	+	0.642444i	$0.777920\pi$
$948$		0			0
$949$	−296.860			−0.312814
$950$		0			0
$951$			68.8968i			0.0724467i
$952$		0			0
$953$			30.5870i			0.0320955i	0.999871	+	0.0160478i	$0.00510838\pi$
$953$			30.5870i			0.0320955i	−0.999871	+	0.0160478i	$0.994892\pi$
$954$		0			0
$955$	425.588	+	776.387i	0.445642	+	0.812971i
$956$		0			0
$957$	164.464			0.171854
$958$		0			0
$959$	208.915	+	1701.21i	0.217847	+	1.77394i
$960$		0			0
$961$	955.082			0.993841
$962$		0			0
$963$		−	99.4564i		−	0.103278i
$964$		0			0
$965$	672.810	−	368.810i	0.697212	−	0.382187i
$966$		0			0
$967$		−	1878.59i		−	1.94270i	−0.237650	−	0.971351i	$-0.576377\pi$
$967$		−	1878.59i		−	1.94270i	0.237650	−	0.971351i	$-0.423623\pi$
$968$		0			0
$969$		−	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	$0.880509\pi$
$972$		0			0
$973$	203.274	+	1655.28i	0.208915	+	1.70121i
$974$		0			0
$975$	321.758	−	504.281i	0.330008	−	0.517211i
$976$		0			0
$977$			750.938i			0.768616i	0.923205	+	0.384308i	$0.125560\pi$
$977$			750.938i			0.768616i	−0.923205	+	0.384308i	$0.874440\pi$
$978$		0			0
$979$		−	262.089i		−	0.267710i
$980$		0			0
$981$	326.457			0.332779
$982$		0			0
$983$	−1525.09			−1.55147			−0.775735	−	0.631059i	$-0.782621\pi$
$983$	−1525.09			−1.55147			−0.775735	+	0.631059i	$0.782621\pi$
$984$		0			0
$985$	938.458	−	514.429i	0.952749	−	0.522263i
$986$		0			0
$987$	−490.148	+	60.1921i	−0.496604	+	0.0609849i
$988$		0			0
$989$	−390.966			−0.395314
$990$		0			0
$991$	−1259.84			−1.27128			−0.635641	−	0.771985i	$-0.719264\pi$
$991$	−1259.84			−1.27128			−0.635641	+	0.771985i	$0.719264\pi$
$992$		0			0
$993$	23.6834			0.0238504
$994$		0			0
$995$	781.210	−	428.231i	0.785135	−	0.430383i
$996$		0			0
$997$	−1406.62			−1.41086			−0.705428	−	0.708781i	$-0.749245\pi$
$997$	−1406.62			−1.41086			−0.705428	+	0.708781i	$0.749245\pi$
$998$		0			0
$999$			264.899i			0.265164i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.3.bd.a.769.14		16
4.3	odd	2		210.3.h.a.139.11	yes	16
5.4	even	2	inner	1680.3.bd.a.769.4		16
7.6	odd	2	inner	1680.3.bd.a.769.3		16
12.11	even	2		630.3.h.e.559.3		16
20.3	even	4		1050.3.f.e.601.11		16
20.7	even	4		1050.3.f.e.601.6		16
20.19	odd	2		210.3.h.a.139.6	yes	16
28.27	even	2		210.3.h.a.139.14	yes	16
35.34	odd	2	inner	1680.3.bd.a.769.13		16
60.59	even	2		630.3.h.e.559.14		16
84.83	odd	2		630.3.h.e.559.6		16
140.27	odd	4		1050.3.f.e.601.2		16
140.83	odd	4		1050.3.f.e.601.15		16
140.139	even	2		210.3.h.a.139.3	✓	16
420.419	odd	2		630.3.h.e.559.11		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.h.a.139.3	✓	16	140.139	even	2
210.3.h.a.139.6	yes	16	20.19	odd	2
210.3.h.a.139.11	yes	16	4.3	odd	2
210.3.h.a.139.14	yes	16	28.27	even	2
630.3.h.e.559.3		16	12.11	even	2
630.3.h.e.559.6		16	84.83	odd	2
630.3.h.e.559.11		16	420.419	odd	2
630.3.h.e.559.14		16	60.59	even	2
1050.3.f.e.601.2		16	140.27	odd	4
1050.3.f.e.601.6		16	20.7	even	4
1050.3.f.e.601.11		16	20.3	even	4
1050.3.f.e.601.15		16	140.83	odd	4
1680.3.bd.a.769.3		16	7.6	odd	2	inner
1680.3.bd.a.769.4		16	5.4	even	2	inner
1680.3.bd.a.769.13		16	35.34	odd	2	inner
1680.3.bd.a.769.14		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1680.bd (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} +(2.40341 + 4.38447i) q^{5} +(6.94781 - 0.853218i) q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+1.73205 q^{3} +(2.40341 + 4.38447i) q^{5} +(6.94781 - 0.853218i) q^{7} +3.00000 q^{9} -2.88097 q^{11} -13.8145 q^{13} +(4.16283 + 7.59413i) q^{15} -24.1754 q^{17} +6.53403i q^{19} +(12.0340 - 1.47782i) q^{21} +28.8420i q^{23} +(-13.4472 + 21.0754i) q^{25} +5.19615 q^{27} -32.9589 q^{29} -2.43276i q^{31} -4.98998 q^{33} +(20.4394 + 28.4118i) q^{35} +50.9799i q^{37} -23.9275 q^{39} +21.5225i q^{41} +13.5554i q^{43} +(7.21024 + 13.1534i) q^{45} -40.7305 q^{47} +(47.5440 - 11.8560i) q^{49} -41.8730 q^{51} +17.2758i q^{53} +(-6.92415 - 12.6315i) q^{55} +11.3173i q^{57} -1.47488i q^{59} +111.568i q^{61} +(20.8434 - 2.55965i) q^{63} +(-33.2020 - 60.5694i) q^{65} -120.293i q^{67} +49.9558i q^{69} +90.3855 q^{71} +21.4890 q^{73} +(-23.2913 + 36.5037i) q^{75} +(-20.0164 + 2.45809i) q^{77} +66.1324 q^{79} +9.00000 q^{81} -78.5172 q^{83} +(-58.1035 - 105.996i) q^{85} -57.0865 q^{87} +90.9724i q^{89} +(-95.9807 + 11.7868i) q^{91} -4.21367i q^{93} +(-28.6483 + 15.7040i) q^{95} +44.1972 q^{97} -8.64290 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 48 q^{9}+O(q^{10}) \) 16 * q + 48 * q^9	\( 16 q + 48 q^{9} - 96 q^{11} + 24 q^{15} + 24 q^{21} + 24 q^{25} + 64 q^{29} + 8 q^{35} + 144 q^{39} + 224 q^{49} + 48 q^{51} + 368 q^{65} + 384 q^{71} + 608 q^{79} + 144 q^{81} - 440 q^{85} - 224 q^{91} + 560 q^{95} - 288 q^{99}+O(q^{100}) \) 16 * q + 48 * q^9 - 96 * q^11 + 24 * q^15 + 24 * q^21 + 24 * q^25 + 64 * q^29 + 8 * q^35 + 144 * q^39 + 224 * q^49 + 48 * q^51 + 368 * q^65 + 384 * q^71 + 608 * q^79 + 144 * q^81 - 440 * q^85 - 224 * q^91 + 560 * q^95 - 288 * q^99

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