LMFDB - Embedded newform 1050.3.f.e.601.16

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			601.16
Root			$1.20711 - 4.25887i$ of defining polynomial
Character	$\chi$	$=$	1050.601
Dual form			1050.3.f.e.601.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} +2.44949i q^{6} +(4.63050 - 5.24961i) q^{7} +2.82843 q^{8} -3.00000 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} +2.44949i q^{6} +(4.63050 - 5.24961i) q^{7} +2.82843 q^{8} -3.00000 q^{9} +11.7671 q^{11} +3.46410i q^{12} -24.8280i q^{13} +(6.54852 - 7.42407i) q^{14} +4.00000 q^{16} +7.26100i q^{17} -4.24264 q^{18} -23.0050i q^{19} +(9.09260 + 8.02027i) q^{21} +16.6412 q^{22} -26.4223 q^{23} +4.89898i q^{24} -35.1122i q^{26} -5.19615i q^{27} +(9.26101 - 10.4992i) q^{28} -57.0861 q^{29} -10.2222i q^{31} +5.65685 q^{32} +20.3812i q^{33} +10.2686i q^{34} -6.00000 q^{36} +14.2196 q^{37} -32.5340i q^{38} +43.0034 q^{39} -16.2271i q^{41} +(12.8589 + 11.3424i) q^{42} +82.3114 q^{43} +23.5342 q^{44} -37.3668 q^{46} -51.6631i q^{47} +6.92820i q^{48} +(-6.11686 - 48.6167i) q^{49} -12.5764 q^{51} -49.6561i q^{52} +64.8273 q^{53} -7.34847i q^{54} +(13.0970 - 14.8481i) q^{56} +39.8459 q^{57} -80.7320 q^{58} +81.7685i q^{59} +13.1240i q^{61} -14.4564i q^{62} +(-13.8915 + 15.7488i) q^{63} +8.00000 q^{64} +28.8233i q^{66} +22.4035 q^{67} +14.5220i q^{68} -45.7648i q^{69} +91.5022 q^{71} -8.48528 q^{72} +71.9256i q^{73} +20.1096 q^{74} -46.0100i q^{76} +(54.4875 - 61.7726i) q^{77} +60.8160 q^{78} -45.2802 q^{79} +9.00000 q^{81} -22.9486i q^{82} -17.7328i q^{83} +(18.1852 + 16.0405i) q^{84} +116.406 q^{86} -98.8761i q^{87} +33.2823 q^{88} +77.5800i q^{89} +(-130.338 - 114.966i) q^{91} -52.8446 q^{92} +17.7054 q^{93} -73.0627i q^{94} +9.79796i q^{96} -6.15741i q^{97} +(-8.65055 - 68.7544i) q^{98} -35.3012 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	4.63050	−	5.24961i	0.661501	−	0.749945i
$7$	4.63050	−	5.24961i	0.661501	−	0.749945i
$8$	2.82843			0.353553
$9$	−3.00000			−0.333333
$10$		0			0
$11$	11.7671			1.06973			0.534867	−	0.844936i	$-0.320362\pi$
$11$	11.7671			1.06973			0.534867	+	0.844936i	$0.320362\pi$
$12$			3.46410i			0.288675i
$13$		−	24.8280i		−	1.90985i	−0.296848	−	0.954925i	$-0.595935\pi$
$13$		−	24.8280i		−	1.90985i	0.296848	−	0.954925i	$-0.404065\pi$
$14$	6.54852	−	7.42407i	0.467752	−	0.530291i
$15$		0			0
$16$	4.00000			0.250000
$17$			7.26100i			0.427118i	0.976930	+	0.213559i	$0.0685055\pi$
$17$			7.26100i			0.427118i	−0.976930	+	0.213559i	$0.931494\pi$
$18$	−4.24264			−0.235702
$19$		−	23.0050i		−	1.21079i	−0.795925	−	0.605395i	$-0.793015\pi$
$19$		−	23.0050i		−	1.21079i	0.795925	−	0.605395i	$-0.206985\pi$
$20$		0			0
$21$	9.09260	+	8.02027i	0.432981	+	0.381918i
$22$	16.6412			0.756417
$23$	−26.4223			−1.14880			−0.574398	−	0.818576i	$-0.694764\pi$
$23$	−26.4223			−1.14880			−0.574398	+	0.818576i	$0.694764\pi$
$24$			4.89898i			0.204124i
$25$		0			0
$26$		−	35.1122i		−	1.35047i
$27$		−	5.19615i		−	0.192450i
$28$	9.26101	−	10.4992i	0.330750	−	0.374972i
$29$	−57.0861			−1.96849			−0.984243	−	0.176819i	$-0.943419\pi$
$29$	−57.0861			−1.96849			−0.984243	+	0.176819i	$0.943419\pi$
$30$		0			0
$31$		−	10.2222i		−	0.329749i	−0.986315	−	0.164874i	$-0.947278\pi$
$31$		−	10.2222i		−	0.329749i	0.986315	−	0.164874i	$-0.0527219\pi$
$32$	5.65685			0.176777
$33$			20.3812i			0.617612i
$34$			10.2686i			0.302018i
$35$		0			0
$36$	−6.00000			−0.166667
$37$	14.2196			0.384314			0.192157	−	0.981364i	$-0.438452\pi$
$37$	14.2196			0.384314			0.192157	+	0.981364i	$0.438452\pi$
$38$		−	32.5340i		−	0.856158i
$39$	43.0034			1.10265
$40$		0			0
$41$		−	16.2271i		−	0.395782i	−0.980224	−	0.197891i	$-0.936591\pi$
$41$		−	16.2271i		−	0.395782i	0.980224	−	0.197891i	$-0.0634093\pi$
$42$	12.8589	+	11.3424i	0.306164	+	0.270056i
$43$	82.3114			1.91422			0.957110	−	0.289725i	$-0.0935639\pi$
$43$	82.3114			1.91422			0.957110	+	0.289725i	$0.0935639\pi$
$44$	23.5342			0.534867
$45$		0			0
$46$	−37.3668			−0.812321
$47$		−	51.6631i		−	1.09922i	−0.835423	−	0.549608i	$-0.814777\pi$
$47$		−	51.6631i		−	1.09922i	0.835423	−	0.549608i	$-0.185223\pi$
$48$			6.92820i			0.144338i
$49$	−6.11686	−	48.6167i	−0.124834	−	0.992178i
$50$		0			0
$51$	−12.5764			−0.246597
$52$		−	49.6561i		−	0.954925i
$53$	64.8273			1.22316			0.611579	−	0.791184i	$-0.290535\pi$
$53$	64.8273			1.22316			0.611579	+	0.791184i	$0.290535\pi$
$54$		−	7.34847i		−	0.136083i
$55$		0			0
$56$	13.0970	−	14.8481i	0.233876	−	0.265145i
$57$	39.8459			0.699050
$58$	−80.7320			−1.39193
$59$			81.7685i			1.38591i	0.720982	+	0.692953i	$0.243691\pi$
$59$			81.7685i			1.38591i	−0.720982	+	0.692953i	$0.756309\pi$
$60$		0			0
$61$			13.1240i			0.215148i	0.994197	+	0.107574i	$0.0343083\pi$
$61$			13.1240i			0.215148i	−0.994197	+	0.107574i	$0.965692\pi$
$62$		−	14.4564i		−	0.233168i
$63$	−13.8915	+	15.7488i	−0.220500	+	0.249982i
$64$	8.00000			0.125000
$65$		0			0
$66$			28.8233i			0.436717i
$67$	22.4035			0.334381			0.167190	−	0.985925i	$-0.446531\pi$
$67$	22.4035			0.334381			0.167190	+	0.985925i	$0.446531\pi$
$68$			14.5220i			0.213559i
$69$		−	45.7648i		−	0.663257i
$70$		0			0
$71$	91.5022			1.28876			0.644382	−	0.764704i	$-0.277115\pi$
$71$	91.5022			1.28876			0.644382	+	0.764704i	$0.277115\pi$
$72$	−8.48528			−0.117851
$73$			71.9256i			0.985282i	0.870233	+	0.492641i	$0.163968\pi$
$73$			71.9256i			0.985282i	−0.870233	+	0.492641i	$0.836032\pi$
$74$	20.1096			0.271751
$75$		0			0
$76$		−	46.0100i		−	0.605395i
$77$	54.4875	−	61.7726i	0.707630	−	0.802242i
$78$	60.8160			0.779693
$79$	−45.2802			−0.573167			−0.286584	−	0.958055i	$-0.592520\pi$
$79$	−45.2802			−0.573167			−0.286584	+	0.958055i	$0.592520\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		−	22.9486i		−	0.279860i
$83$		−	17.7328i		−	0.213648i	−0.994278	−	0.106824i	$-0.965932\pi$
$83$		−	17.7328i		−	0.213648i	0.994278	−	0.106824i	$-0.0340681\pi$
$84$	18.1852	+	16.0405i	0.216490	+	0.190959i
$85$		0			0
$86$	116.406			1.35356
$87$		−	98.8761i		−	1.13651i
$88$	33.2823			0.378208
$89$			77.5800i			0.871686i	0.900023	+	0.435843i	$0.143550\pi$
$89$			77.5800i			0.871686i	−0.900023	+	0.435843i	$0.856450\pi$
$90$		0			0
$91$	−130.338	−	114.966i	−1.43228	−	1.26337i
$92$	−52.8446			−0.574398
$93$	17.7054			0.190381
$94$		−	73.0627i		−	0.777263i
$95$		0			0
$96$			9.79796i			0.102062i
$97$		−	6.15741i		−	0.0634785i	−0.999496	−	0.0317392i	$-0.989895\pi$
$97$		−	6.15741i		−	0.0634785i	0.999496	−	0.0317392i	$-0.0101046\pi$
$98$	−8.65055	−	68.7544i	−0.0882709	−	0.701576i
$99$	−35.3012			−0.356578
$100$		0			0
$101$			142.983i			1.41568i	0.706374	+	0.707839i	$0.250330\pi$
$101$			142.983i			1.41568i	−0.706374	+	0.707839i	$0.749670\pi$
$102$	−17.7858			−0.174370
$103$		−	14.5292i		−	0.141060i	−0.997510	−	0.0705302i	$-0.977531\pi$
$103$		−	14.5292i		−	0.141060i	0.997510	−	0.0705302i	$-0.0224691\pi$
$104$		−	70.2243i		−	0.675234i
$105$		0			0
$106$	91.6797			0.864903
$107$	78.4228			0.732923			0.366462	−	0.930433i	$-0.380569\pi$
$107$	78.4228			0.732923			0.366462	+	0.930433i	$0.380569\pi$
$108$		−	10.3923i		−	0.0962250i
$109$	74.7424			0.685710			0.342855	−	0.939388i	$-0.388606\pi$
$109$	74.7424			0.685710			0.342855	+	0.939388i	$0.388606\pi$
$110$		0			0
$111$			24.6291i			0.221884i
$112$	18.5220	−	20.9985i	0.165375	−	0.187486i
$113$	−85.2417			−0.754351			−0.377175	−	0.926142i	$-0.623105\pi$
$113$	−85.2417			−0.754351			−0.377175	+	0.926142i	$0.623105\pi$
$114$	56.3506			0.494303
$115$		0			0
$116$	−114.172			−0.984243
$117$			74.4841i			0.636617i
$118$			115.638i			0.979984i
$119$	38.1175	+	33.6221i	0.320315	+	0.282539i
$120$		0			0
$121$	17.4642			0.144332
$122$			18.5602i			0.152133i
$123$	28.1061			0.228505
$124$		−	20.4444i		−	0.164874i
$125$		0			0
$126$	−19.6456	+	22.2722i	−0.155917	+	0.176764i
$127$	66.7734			0.525775			0.262887	−	0.964827i	$-0.415325\pi$
$127$	66.7734			0.525775			0.262887	+	0.964827i	$0.415325\pi$
$128$	11.3137			0.0883883
$129$			142.568i			1.10518i
$130$		0			0
$131$		−	28.8330i		−	0.220099i	−0.993926	−	0.110049i	$-0.964899\pi$
$131$		−	28.8330i		−	0.220099i	0.993926	−	0.110049i	$-0.0351009\pi$
$132$			40.7624i			0.308806i
$133$	−120.767	−	106.525i	−0.908026	−	0.800939i
$134$	31.6834			0.236443
$135$		0			0
$136$			20.5372i			0.151009i
$137$	119.450			0.871897			0.435948	−	0.899972i	$-0.356413\pi$
$137$	119.450			0.871897			0.435948	+	0.899972i	$0.356413\pi$
$138$		−	64.7211i		−	0.468994i
$139$			200.948i			1.44567i	0.691021	+	0.722834i	$0.257161\pi$
$139$			200.948i			1.44567i	−0.691021	+	0.722834i	$0.742839\pi$
$140$		0			0
$141$	89.4832			0.634633
$142$	129.404			0.911293
$143$		−	292.154i		−	2.04303i
$144$	−12.0000			−0.0833333
$145$		0			0
$146$			101.718i			0.696700i
$147$	84.2066	−	10.5947i	0.572834	−	0.0720729i
$148$	28.4392			0.192157
$149$	5.70484			0.0382875			0.0191438	−	0.999817i	$-0.493906\pi$
$149$	5.70484			0.0382875			0.0191438	+	0.999817i	$0.493906\pi$
$150$		0			0
$151$	−216.490			−1.43371			−0.716854	−	0.697223i	$-0.754419\pi$
$151$	−216.490			−1.43371			−0.716854	+	0.697223i	$0.754419\pi$
$152$		−	65.0680i		−	0.428079i
$153$		−	21.7830i		−	0.142373i
$154$	77.0570	−	87.3597i	0.500370	−	0.567271i
$155$		0			0
$156$	86.0069			0.551326
$157$		−	108.287i		−	0.689726i	−0.938653	−	0.344863i	$-0.887925\pi$
$157$		−	108.287i		−	0.689726i	0.938653	−	0.344863i	$-0.112075\pi$
$158$	−64.0359			−0.405290
$159$			112.284i			0.706190i
$160$		0			0
$161$	−122.349	+	138.707i	−0.759929	+	0.861533i
$162$	12.7279			0.0785674
$163$	234.395			1.43800			0.719002	−	0.695008i	$-0.244599\pi$
$163$	234.395			1.43800			0.719002	+	0.695008i	$0.244599\pi$
$164$		−	32.4542i		−	0.197891i
$165$		0			0
$166$		−	25.0779i		−	0.151072i
$167$			41.8830i			0.250796i	0.992106	+	0.125398i	$0.0400209\pi$
$167$			41.8830i			0.250796i	−0.992106	+	0.125398i	$0.959979\pi$
$168$	25.7177	+	22.6847i	0.153082	+	0.135028i
$169$	−447.432			−2.64753
$170$		0			0
$171$			69.0151i			0.403597i
$172$	164.623			0.957110
$173$		−	13.0301i		−	0.0753183i	−0.999291	−	0.0376592i	$-0.988010\pi$
$173$		−	13.0301i		−	0.0753183i	0.999291	−	0.0376592i	$-0.0119901\pi$
$174$		−	139.832i		−	0.803631i
$175$		0			0
$176$	47.0683			0.267434
$177$	−141.627			−0.800154
$178$			109.715i			0.616375i
$179$	24.1573			0.134957			0.0674786	−	0.997721i	$-0.478505\pi$
$179$	24.1573			0.134957			0.0674786	+	0.997721i	$0.478505\pi$
$180$		0			0
$181$		−	43.5107i		−	0.240390i	−0.992750	−	0.120195i	$-0.961648\pi$
$181$		−	43.5107i		−	0.240390i	0.992750	−	0.120195i	$-0.0383521\pi$
$182$	−184.325	−	162.587i	−1.01278	−	0.893335i
$183$	−22.7315			−0.124216
$184$	−74.7335			−0.406160
$185$		0			0
$186$	25.0392			0.134619
$187$			85.4408i			0.456903i
$188$		−	103.326i		−	0.549608i
$189$	−27.2778	−	24.0608i	−0.144327	−	0.127306i
$190$		0			0
$191$	−246.572			−1.29095			−0.645477	−	0.763780i	$-0.723341\pi$
$191$	−246.572			−1.29095			−0.645477	+	0.763780i	$0.723341\pi$
$192$			13.8564i			0.0721688i
$193$	161.875			0.838733			0.419366	−	0.907817i	$-0.362252\pi$
$193$	161.875			0.838733			0.419366	+	0.907817i	$0.362252\pi$
$194$		−	8.70789i		−	0.0448860i
$195$		0			0
$196$	−12.2337	−	97.2334i	−0.0624170	−	0.496089i
$197$	−44.8380			−0.227604			−0.113802	−	0.993503i	$-0.536303\pi$
$197$	−44.8380			−0.227604			−0.113802	+	0.993503i	$0.536303\pi$
$198$	−49.9235			−0.252139
$199$		−	114.767i		−	0.576718i	−0.957522	−	0.288359i	$-0.906890\pi$
$199$		−	114.767i		−	0.576718i	0.957522	−	0.288359i	$-0.0931097\pi$
$200$		0			0
$201$			38.8040i			0.193055i
$202$			202.209i			1.00104i
$203$	−264.338	+	299.680i	−1.30216	+	1.47626i
$204$	−25.1529			−0.123298
$205$		0			0
$206$		−	20.5474i		−	0.0997448i
$207$	79.2669			0.382932
$208$		−	99.3122i		−	0.477462i
$209$		−	270.702i		−	1.29522i
$210$		0			0
$211$	−219.602			−1.04077			−0.520384	−	0.853932i	$-0.674211\pi$
$211$	−219.602			−1.04077			−0.520384	+	0.853932i	$0.674211\pi$
$212$	129.655			0.611579
$213$			158.486i			0.744068i
$214$	110.907			0.518255
$215$		0			0
$216$		−	14.6969i		−	0.0680414i
$217$	−53.6626	−	47.3340i	−0.247293	−	0.218129i
$218$	105.702			0.484870
$219$	−124.579			−0.568853
$220$		0			0
$221$	180.277			0.815731
$222$			34.8308i			0.156895i
$223$			3.57102i			0.0160136i	0.999968	+	0.00800678i	$0.00254866\pi$
$223$			3.57102i			0.0160136i	−0.999968	+	0.00800678i	$0.997451\pi$
$224$	26.1941	−	29.6963i	0.116938	−	0.132573i
$225$		0			0
$226$	−120.550			−0.533407
$227$			68.0647i			0.299844i	0.988698	+	0.149922i	$0.0479023\pi$
$227$			68.0647i			0.299844i	−0.988698	+	0.149922i	$0.952098\pi$
$228$	79.6917			0.349525
$229$			180.019i			0.786109i	0.919515	+	0.393054i	$0.128582\pi$
$229$			180.019i			0.786109i	−0.919515	+	0.393054i	$0.871418\pi$
$230$		0			0
$231$	106.993	+	94.3752i	0.463175	+	0.408550i
$232$	−161.464			−0.695965
$233$	115.207			0.494449			0.247225	−	0.968958i	$-0.420481\pi$
$233$	115.207			0.494449			0.247225	+	0.968958i	$0.420481\pi$
$234$			105.336i			0.450156i
$235$		0			0
$236$			163.537i			0.692953i
$237$		−	78.4276i		−	0.330918i
$238$	53.9062	+	47.5488i	0.226497	+	0.199785i
$239$	−15.6873			−0.0656374			−0.0328187	−	0.999461i	$-0.510448\pi$
$239$	−15.6873			−0.0656374			−0.0328187	+	0.999461i	$0.510448\pi$
$240$		0			0
$241$		−	303.442i		−	1.25909i	−0.776962	−	0.629547i	$-0.783240\pi$
$241$		−	303.442i		−	1.25909i	0.776962	−	0.629547i	$-0.216760\pi$
$242$	24.6981			0.102058
$243$			15.5885i			0.0641500i
$244$			26.2481i			0.107574i
$245$		0			0
$246$	39.7481			0.161578
$247$	−571.170			−2.31243
$248$		−	28.9128i		−	0.116584i
$249$	30.7141			0.123350
$250$		0			0
$251$			80.5241i			0.320813i	0.987051	+	0.160407i	$0.0512805\pi$
$251$			80.5241i			0.320813i	−0.987051	+	0.160407i	$0.948719\pi$
$252$	−27.7830	+	31.4977i	−0.110250	+	0.124991i
$253$	−310.913			−1.22891
$254$	94.4318			0.371779
$255$		0			0
$256$	16.0000			0.0625000
$257$			39.1012i			0.152145i	0.997102	+	0.0760723i	$0.0242380\pi$
$257$			39.1012i			0.152145i	−0.997102	+	0.0760723i	$0.975762\pi$
$258$			201.621i			0.781477i
$259$	65.8439	−	74.6474i	0.254224	−	0.288214i
$260$		0			0
$261$	171.258			0.656162
$262$		−	40.7760i		−	0.155633i
$263$	−79.1644			−0.301005			−0.150503	−	0.988610i	$-0.548089\pi$
$263$	−79.1644			−0.301005			−0.150503	+	0.988610i	$0.548089\pi$
$264$			57.6467i			0.218359i
$265$		0			0
$266$	−170.791	−	150.649i	−0.642071	−	0.566349i
$267$	−134.373			−0.503268
$268$	44.8070			0.167190
$269$		−	53.2775i		−	0.198058i	−0.995085	−	0.0990288i	$-0.968426\pi$
$269$		−	53.2775i		−	0.198058i	0.995085	−	0.0990288i	$-0.0315736\pi$
$270$		0			0
$271$			237.638i			0.876892i	0.898757	+	0.438446i	$0.144471\pi$
$271$			237.638i			0.876892i	−0.898757	+	0.438446i	$0.855529\pi$
$272$			29.0440i			0.106779i
$273$	199.128	−	225.751i	0.729405	−	0.826928i
$274$	168.928			0.616524
$275$		0			0
$276$		−	91.5295i		−	0.331629i
$277$	−168.185			−0.607167			−0.303583	−	0.952805i	$-0.598183\pi$
$277$	−168.185			−0.607167			−0.303583	+	0.952805i	$0.598183\pi$
$278$			284.183i			1.02224i
$279$			30.6666i			0.109916i
$280$		0			0
$281$	128.175			0.456140			0.228070	−	0.973645i	$-0.426758\pi$
$281$	128.175			0.456140			0.228070	+	0.973645i	$0.426758\pi$
$282$	126.548			0.448753
$283$		−	72.7293i		−	0.256994i	−0.991710	−	0.128497i	$-0.958985\pi$
$283$		−	72.7293i		−	0.256994i	0.991710	−	0.128497i	$-0.0410153\pi$
$284$	183.004			0.644382
$285$		0			0
$286$		−	413.168i		−	1.44464i
$287$	−85.1859	−	75.1396i	−0.296815	−	0.261810i
$288$	−16.9706			−0.0589256
$289$	236.278			0.817570
$290$		0			0
$291$	10.6649			0.0366493
$292$			143.851i			0.492641i
$293$			273.216i			0.932477i	0.884659	+	0.466238i	$0.154391\pi$
$293$			273.216i			0.932477i	−0.884659	+	0.466238i	$0.845609\pi$
$294$	119.086	−	14.9832i	0.405055	−	0.0509632i
$295$		0			0
$296$	40.2191			0.135875
$297$		−	61.1436i		−	0.205871i
$298$	8.06786			0.0270734
$299$			656.014i			2.19403i
$300$		0			0
$301$	381.143	−	432.103i	1.26626	−	1.43556i
$302$	−306.163			−1.01379
$303$	−247.655			−0.817342
$304$		−	92.0201i		−	0.302698i
$305$		0			0
$306$		−	30.8058i		−	0.100673i
$307$			336.086i			1.09474i	0.836889	+	0.547372i	$0.184372\pi$
$307$			336.086i			1.09474i	−0.836889	+	0.547372i	$0.815628\pi$
$308$	108.975	−	123.545i	0.353815	−	0.401121i
$309$	25.1653			0.0814412
$310$		0			0
$311$			497.276i			1.59896i	0.600693	+	0.799480i	$0.294891\pi$
$311$			497.276i			1.59896i	−0.600693	+	0.799480i	$0.705109\pi$
$312$	121.632			0.389846
$313$			252.024i			0.805188i	0.915379	+	0.402594i	$0.131891\pi$
$313$			252.024i			0.805188i	−0.915379	+	0.402594i	$0.868109\pi$
$314$		−	153.141i		−	0.487710i
$315$		0			0
$316$	−90.5604			−0.286584
$317$	−270.203			−0.852377			−0.426188	−	0.904634i	$-0.640144\pi$
$317$	−270.203			−0.852377			−0.426188	+	0.904634i	$0.640144\pi$
$318$			158.794i			0.499352i
$319$	−671.737			−2.10576
$320$		0			0
$321$			135.832i			0.423153i
$322$	−173.027	+	196.161i	−0.537351	+	0.609196i
$323$	167.040			0.517150
$324$	18.0000			0.0555556
$325$		0			0
$326$	331.484			1.01682
$327$			129.458i			0.395895i
$328$		−	45.8971i		−	0.139930i
$329$	−271.211	−	239.226i	−0.824351	−	0.727132i
$330$		0			0
$331$	−306.813			−0.926926			−0.463463	−	0.886116i	$-0.653393\pi$
$331$	−306.813			−0.926926			−0.463463	+	0.886116i	$0.653393\pi$
$332$		−	35.4656i		−	0.106824i
$333$	−42.6588			−0.128105
$334$			59.2315i			0.177340i
$335$		0			0
$336$	36.3704	+	32.0811i	0.108245	+	0.0954794i
$337$	485.903			1.44185			0.720924	−	0.693014i	$-0.243718\pi$
$337$	485.903			1.44185			0.720924	+	0.693014i	$0.243718\pi$
$338$	−632.764			−1.87208
$339$		−	147.643i		−	0.435525i
$340$		0			0
$341$		−	120.286i		−	0.352744i
$342$			97.6020i			0.285386i
$343$	−283.543	−	193.009i	−0.826656	−	0.562708i
$344$	232.812			0.676779
$345$		0			0
$346$		−	18.4273i		−	0.0532581i
$347$	−8.02712			−0.0231329			−0.0115665	−	0.999933i	$-0.503682\pi$
$347$	−8.02712			−0.0231329			−0.0115665	+	0.999933i	$0.503682\pi$
$348$		−	197.752i		−	0.568253i
$349$		−	649.899i		−	1.86217i	−0.364798	−	0.931087i	$-0.618862\pi$
$349$		−	649.899i		−	1.86217i	0.364798	−	0.931087i	$-0.381138\pi$
$350$		0			0
$351$	−129.010			−0.367551
$352$	66.5647			0.189104
$353$		−	396.496i		−	1.12322i	−0.827403	−	0.561608i	$-0.810183\pi$
$353$		−	396.496i		−	1.12322i	0.827403	−	0.561608i	$-0.189817\pi$
$354$	−200.291			−0.565794
$355$		0			0
$356$			155.160i			0.435843i
$357$	−58.2352	+	66.0214i	−0.163124	+	0.184934i
$358$	34.1636			0.0954291
$359$	−398.981			−1.11137			−0.555684	−	0.831394i	$-0.687543\pi$
$359$	−398.981			−1.11137			−0.555684	+	0.831394i	$0.687543\pi$
$360$		0			0
$361$	−168.231			−0.466014
$362$		−	61.5334i		−	0.169982i
$363$			30.2489i			0.0833303i
$364$	−260.675	−	229.933i	−0.716141	−	0.631683i
$365$		0			0
$366$	−32.1472			−0.0878339
$367$		−	506.200i		−	1.37929i	−0.724147	−	0.689646i	$-0.757766\pi$
$367$		−	506.200i		−	1.37929i	0.724147	−	0.689646i	$-0.242234\pi$
$368$	−105.689			−0.287199
$369$			48.6812i			0.131927i
$370$		0			0
$371$	300.183	−	340.318i	0.809119	−	0.917300i
$372$	35.4108			0.0951903
$373$	−278.751			−0.747322			−0.373661	−	0.927565i	$-0.621898\pi$
$373$	−278.751			−0.747322			−0.373661	+	0.927565i	$0.621898\pi$
$374$			120.832i			0.323079i
$375$		0			0
$376$		−	146.125i		−	0.388631i
$377$			1417.34i			3.75951i
$378$	−38.5766	−	34.0271i	−0.102055	−	0.0900188i
$379$	515.959			1.36137			0.680685	−	0.732576i	$-0.261682\pi$
$379$	515.959			1.36137			0.680685	+	0.732576i	$0.261682\pi$
$380$		0			0
$381$			115.655i			0.303556i
$382$	−348.706			−0.912842
$383$			238.715i			0.623276i	0.950201	+	0.311638i	$0.100878\pi$
$383$			238.715i			0.623276i	−0.950201	+	0.311638i	$0.899122\pi$
$384$			19.5959i			0.0510310i
$385$		0			0
$386$	228.926			0.593073
$387$	−246.934			−0.638073
$388$		−	12.3148i		−	0.0317392i
$389$	170.748			0.438941			0.219471	−	0.975619i	$-0.429567\pi$
$389$	170.748			0.438941			0.219471	+	0.975619i	$0.429567\pi$
$390$		0			0
$391$		−	191.852i		−	0.490671i
$392$	−17.3011	−	137.509i	−0.0441355	−	0.350788i
$393$	49.9402			0.127074
$394$	−63.4105			−0.160940
$395$		0			0
$396$	−70.6025			−0.178289
$397$			80.7184i			0.203321i	0.994819	+	0.101660i	$0.0324155\pi$
$397$			80.7184i			0.203321i	−0.994819	+	0.101660i	$0.967584\pi$
$398$		−	162.305i		−	0.407801i
$399$	184.506	−	209.175i	0.462422	−	0.524249i
$400$		0			0
$401$	469.375			1.17051			0.585255	−	0.810849i	$-0.300994\pi$
$401$	469.375			1.17051			0.585255	+	0.810849i	$0.300994\pi$
$402$			54.8772i			0.136510i
$403$	−253.798			−0.629770
$404$			285.967i			0.707839i
$405$		0			0
$406$	−373.830	+	423.812i	−0.920763	+	1.04387i
$407$	167.323			0.411114
$408$	−35.5715			−0.0871851
$409$			671.238i			1.64117i	0.571525	+	0.820585i	$0.306352\pi$
$409$			671.238i			1.64117i	−0.571525	+	0.820585i	$0.693648\pi$
$410$		0			0
$411$			206.893i			0.503390i
$412$		−	29.0584i		−	0.0705302i
$413$	429.253	+	378.629i	1.03935	+	0.916778i
$414$	112.100			0.270774
$415$		0			0
$416$		−	140.449i		−	0.337617i
$417$	−348.052			−0.834657
$418$		−	382.830i		−	0.915862i
$419$			16.1628i			0.0385747i	0.999814	+	0.0192874i	$0.00613974\pi$
$419$			16.1628i			0.0385747i	−0.999814	+	0.0192874i	$0.993860\pi$
$420$		0			0
$421$	−463.356			−1.10061			−0.550304	−	0.834964i	$-0.685488\pi$
$421$	−463.356			−1.10061			−0.550304	+	0.834964i	$0.685488\pi$
$422$	−310.564			−0.735934
$423$			154.989i			0.366405i
$424$	183.359			0.432451
$425$		0			0
$426$			224.134i			0.526135i
$427$	68.8961	+	60.7709i	0.161349	+	0.142321i
$428$	156.846			0.366462
$429$	506.025			1.17955
$430$		0			0
$431$	425.457			0.987140			0.493570	−	0.869706i	$-0.335692\pi$
$431$	425.457			0.987140			0.493570	+	0.869706i	$0.335692\pi$
$432$		−	20.7846i		−	0.0481125i
$433$			480.947i			1.11073i	0.831606	+	0.555366i	$0.187422\pi$
$433$			480.947i			1.11073i	−0.831606	+	0.555366i	$0.812578\pi$
$434$	−75.8904	−	66.9404i	−0.174863	−	0.154240i
$435$		0			0
$436$	149.485			0.342855
$437$			607.845i			1.39095i
$438$	−176.181			−0.402240
$439$			73.3796i			0.167152i	0.996501	+	0.0835759i	$0.0266341\pi$
$439$			73.3796i			0.167152i	−0.996501	+	0.0835759i	$0.973366\pi$
$440$		0			0
$441$	18.3506	+	145.850i	0.0416113	+	0.330726i
$442$	254.950			0.576809
$443$	−72.4361			−0.163513			−0.0817563	−	0.996652i	$-0.526053\pi$
$443$	−72.4361			−0.163513			−0.0817563	+	0.996652i	$0.526053\pi$
$444$			49.2582i			0.110942i
$445$		0			0
$446$			5.05019i			0.0113233i
$447$			9.88107i			0.0221053i
$448$	37.0440	−	41.9969i	0.0826876	−	0.0937431i
$449$	721.760			1.60748			0.803742	−	0.594978i	$-0.202839\pi$
$449$	721.760			1.60748			0.803742	+	0.594978i	$0.202839\pi$
$450$		0			0
$451$		−	190.945i		−	0.423382i
$452$	−170.483			−0.377175
$453$		−	374.972i		−	0.827752i
$454$			96.2580i			0.212022i
$455$		0			0
$456$	112.701			0.247152
$457$	622.396			1.36192			0.680958	−	0.732322i	$-0.261564\pi$
$457$	622.396			1.36192			0.680958	+	0.732322i	$0.261564\pi$
$458$			254.585i			0.555863i
$459$	37.7293			0.0821989
$460$		0			0
$461$		−	600.196i		−	1.30194i	−0.759102	−	0.650971i	$-0.774362\pi$
$461$		−	600.196i		−	1.30194i	0.759102	−	0.650971i	$-0.225638\pi$
$462$	151.311	+	133.467i	0.327514	+	0.288889i
$463$	127.513			0.275406			0.137703	−	0.990474i	$-0.456028\pi$
$463$	127.513			0.275406			0.137703	+	0.990474i	$0.456028\pi$
$464$	−228.344			−0.492122
$465$		0			0
$466$	162.927			0.349628
$467$			11.4438i			0.0245048i	0.999925	+	0.0122524i	$0.00390016\pi$
$467$			11.4438i			0.0245048i	−0.999925	+	0.0122524i	$0.996100\pi$
$468$			148.968i			0.318308i
$469$	103.740	−	117.610i	0.221193	−	0.250767i
$470$		0			0
$471$	187.559			0.398214
$472$			231.276i			0.489992i
$473$	968.566			2.04771
$474$		−	110.913i		−	0.233995i
$475$		0			0
$476$	76.2349	+	67.2442i	0.160157	+	0.141269i
$477$	−194.482			−0.407719
$478$	−22.1853			−0.0464127
$479$			205.867i			0.429784i	0.976638	+	0.214892i	$0.0689400\pi$
$479$			205.867i			0.429784i	−0.976638	+	0.214892i	$0.931060\pi$
$480$		0			0
$481$		−	353.045i		−	0.733981i
$482$		−	429.131i		−	0.890314i
$483$	−240.247	−	211.914i	−0.497406	−	0.438745i
$484$	34.9284			0.0721662
$485$		0			0
$486$			22.0454i			0.0453609i
$487$	−603.451			−1.23912			−0.619559	−	0.784950i	$-0.712689\pi$
$487$	−603.451			−1.23912			−0.619559	+	0.784950i	$0.712689\pi$
$488$			37.1204i			0.0760664i
$489$			405.983i			0.830232i
$490$		0			0
$491$	−522.576			−1.06431			−0.532154	−	0.846647i	$-0.678617\pi$
$491$	−522.576			−1.06431			−0.532154	+	0.846647i	$0.678617\pi$
$492$	56.2123			0.114253
$493$		−	414.503i		−	0.840776i
$494$	−807.756			−1.63513
$495$		0			0
$496$		−	40.8888i		−	0.0824372i
$497$	423.701	−	480.351i	0.852518	−	0.966501i
$498$	43.4363			0.0872214
$499$	−124.569			−0.249636			−0.124818	−	0.992180i	$-0.539835\pi$
$499$	−124.569			−0.249636			−0.124818	+	0.992180i	$0.539835\pi$
$500$		0			0
$501$	−72.5435			−0.144797
$502$			113.878i			0.226849i
$503$			33.8355i			0.0672673i	0.999434	+	0.0336337i	$0.0107079\pi$
$503$			33.8355i			0.0672673i	−0.999434	+	0.0336337i	$0.989292\pi$
$504$	−39.2911	+	44.5444i	−0.0779586	+	0.0883818i
$505$		0			0
$506$	−439.698			−0.868968
$507$		−	774.975i		−	1.52855i
$508$	133.547			0.262887
$509$			630.938i			1.23956i	0.784774	+	0.619782i	$0.212779\pi$
$509$			630.938i			1.23956i	−0.784774	+	0.619782i	$0.787221\pi$
$510$		0			0
$511$	377.582	+	333.052i	0.738907	+	0.651765i
$512$	22.6274			0.0441942
$513$	−119.538			−0.233017
$514$			55.2974i			0.107582i
$515$		0			0
$516$			285.135i			0.552588i
$517$		−	607.924i		−	1.17587i
$518$	93.1174	−	105.567i	0.179763	−	0.203798i
$519$	22.5687			0.0434850
$520$		0			0
$521$		−	492.821i		−	0.945914i	−0.881086	−	0.472957i	$-0.843187\pi$
$521$		−	492.821i		−	0.945914i	0.881086	−	0.472957i	$-0.156813\pi$
$522$	242.196			0.463977
$523$			331.944i			0.634692i	0.948310	+	0.317346i	$0.102792\pi$
$523$			331.944i			0.634692i	−0.948310	+	0.317346i	$0.897208\pi$
$524$		−	57.6659i		−	0.110049i
$525$		0			0
$526$	−111.955			−0.212843
$527$	74.2235			0.140842
$528$			81.5247i			0.154403i
$529$	169.138			0.319731
$530$		0			0
$531$		−	245.306i		−	0.461969i
$532$	−241.535	−	213.050i	−0.454013	−	0.400469i
$533$	−402.887			−0.755885
$534$	−190.031			−0.355864
$535$		0			0
$536$	63.3667			0.118221
$537$			41.8417i			0.0779175i
$538$		−	75.3457i		−	0.140048i
$539$	−71.9776	−	572.077i	−0.133539	−	1.06137i
$540$		0			0
$541$	−96.8104			−0.178947			−0.0894735	−	0.995989i	$-0.528518\pi$
$541$	−96.8104			−0.178947			−0.0894735	+	0.995989i	$0.528518\pi$
$542$			336.071i			0.620057i
$543$	75.3627			0.138789
$544$			41.0744i			0.0755045i
$545$		0			0
$546$	281.609	−	319.261i	0.515767	−	0.584726i
$547$	−772.034			−1.41140			−0.705699	−	0.708512i	$-0.749367\pi$
$547$	−772.034			−1.41140			−0.705699	+	0.708512i	$0.749367\pi$
$548$	238.900			0.435948
$549$		−	39.3721i		−	0.0717161i
$550$		0			0
$551$			1313.27i			2.38343i
$552$		−	129.442i		−	0.234497i
$553$	−209.670	+	237.704i	−0.379150	+	0.429844i
$554$	−237.850			−0.429332
$555$		0			0
$556$			401.896i			0.722834i
$557$	−42.3682			−0.0760650			−0.0380325	−	0.999277i	$-0.512109\pi$
$557$	−42.3682			−0.0760650			−0.0380325	+	0.999277i	$0.512109\pi$
$558$			43.3692i			0.0777225i
$559$		−	2043.63i		−	3.65587i
$560$		0			0
$561$	−147.988			−0.263793
$562$	181.267			0.322540
$563$		−	370.393i		−	0.657891i	−0.944349	−	0.328946i	$-0.893307\pi$
$563$		−	370.393i		−	0.657891i	0.944349	−	0.328946i	$-0.106693\pi$
$564$	178.966			0.317316
$565$		0			0
$566$		−	102.855i		−	0.181722i
$567$	41.6745	−	47.2465i	0.0735001	−	0.0833272i
$568$	258.807			0.455647
$569$	419.012			0.736401			0.368201	−	0.929746i	$-0.379974\pi$
$569$	419.012			0.736401			0.368201	+	0.929746i	$0.379974\pi$
$570$		0			0
$571$	−113.210			−0.198266			−0.0991329	−	0.995074i	$-0.531607\pi$
$571$	−113.210			−0.198266			−0.0991329	+	0.995074i	$0.531607\pi$
$572$		−	584.307i		−	1.02152i
$573$		−	427.075i		−	0.745332i
$574$	−120.471	−	106.263i	−0.209880	−	0.185128i
$575$		0			0
$576$	−24.0000			−0.0416667
$577$		−	604.749i		−	1.04809i	−0.851690	−	0.524046i	$-0.824422\pi$
$577$		−	604.749i		−	1.04809i	0.851690	−	0.524046i	$-0.175578\pi$
$578$	334.147			0.578109
$579$			280.376i			0.484242i
$580$		0			0
$581$	−93.0902	−	82.1117i	−0.160224	−	0.141328i
$582$	15.0825			0.0259150
$583$	762.829			1.30845
$584$			203.436i			0.348350i
$585$		0			0
$586$			386.385i			0.659360i
$587$		−	458.151i		−	0.780496i	−0.920710	−	0.390248i	$-0.872389\pi$
$587$		−	458.151i		−	0.780496i	0.920710	−	0.390248i	$-0.127611\pi$
$588$	168.413	−	21.1894i	0.286417	−	0.0360365i
$589$	−235.162			−0.399257
$590$		0			0
$591$		−	77.6616i		−	0.131407i
$592$	56.8784			0.0960784
$593$			780.457i			1.31612i	0.752967	+	0.658059i	$0.228622\pi$
$593$			780.457i			1.31612i	−0.752967	+	0.658059i	$0.771378\pi$
$594$		−	86.4700i		−	0.145572i
$595$		0			0
$596$	11.4097			0.0191438
$597$	198.782			0.332968
$598$			927.744i			1.55141i
$599$	11.7391			0.0195977			0.00979887	−	0.999952i	$-0.496881\pi$
$599$	11.7391			0.0195977			0.00979887	+	0.999952i	$0.496881\pi$
$600$		0			0
$601$		−	182.878i		−	0.304290i	−0.988358	−	0.152145i	$-0.951382\pi$
$601$		−	182.878i		−	0.304290i	0.988358	−	0.152145i	$-0.0486181\pi$
$602$	539.018	−	611.086i	0.895379	−	1.01509i
$603$	−67.2106			−0.111460
$604$	−432.980			−0.716854
$605$		0			0
$606$	−350.236			−0.577948
$607$			777.171i			1.28035i	0.768230	+	0.640174i	$0.221138\pi$
$607$			777.171i			1.28035i	−0.768230	+	0.640174i	$0.778862\pi$
$608$		−	130.136i		−	0.214040i
$609$	−519.061	−	457.846i	−0.852317	−	0.751800i
$610$		0			0
$611$	−1282.69			−2.09934
$612$		−	43.5660i		−	0.0711863i
$613$	−106.169			−0.173196			−0.0865978	−	0.996243i	$-0.527600\pi$
$613$	−106.169			−0.173196			−0.0865978	+	0.996243i	$0.527600\pi$
$614$			475.298i			0.774101i
$615$		0			0
$616$	154.114	−	174.719i	0.250185	−	0.283635i
$617$	680.569			1.10303			0.551515	−	0.834165i	$-0.314050\pi$
$617$	680.569			1.10303			0.551515	+	0.834165i	$0.314050\pi$
$618$	35.5892			0.0575877
$619$			861.054i			1.39104i	0.718507	+	0.695520i	$0.244826\pi$
$619$			861.054i			1.39104i	−0.718507	+	0.695520i	$0.755174\pi$
$620$		0			0
$621$			137.294i			0.221086i
$622$			703.255i			1.13063i
$623$	407.265	+	359.235i	0.653716	+	0.576621i
$624$	172.014			0.275663
$625$		0			0
$626$			356.416i			0.569354i
$627$	468.870			0.747798
$628$		−	216.574i		−	0.344863i
$629$			103.249i			0.164147i
$630$		0			0
$631$	−489.247			−0.775352			−0.387676	−	0.921796i	$-0.626722\pi$
$631$	−489.247			−0.775352			−0.387676	+	0.921796i	$0.626722\pi$
$632$	−128.072			−0.202645
$633$		−	380.362i		−	0.600888i
$634$	−382.125			−0.602721
$635$		0			0
$636$			224.569i			0.353095i
$637$	−1207.06	+	151.870i	−1.89491	+	0.238414i
$638$	−949.980			−1.48900
$639$	−274.507			−0.429588
$640$		0			0
$641$	962.813			1.50205			0.751024	−	0.660275i	$-0.229560\pi$
$641$	962.813			1.50205			0.751024	+	0.660275i	$0.229560\pi$
$642$			192.096i			0.299215i
$643$		−	581.069i		−	0.903684i	−0.892098	−	0.451842i	$-0.850767\pi$
$643$		−	581.069i		−	0.903684i	0.892098	−	0.451842i	$-0.149233\pi$
$644$	−244.697	+	277.414i	−0.379964	+	0.430766i
$645$		0			0
$646$	236.230			0.365681
$647$			339.036i			0.524012i	0.965066	+	0.262006i	$0.0843840\pi$
$647$			339.036i			0.524012i	−0.965066	+	0.262006i	$0.915616\pi$
$648$	25.4558			0.0392837
$649$			962.177i			1.48255i
$650$		0			0
$651$	81.9849	−	92.9464i	0.125937	−	0.142775i
$652$	468.789			0.719002
$653$	289.383			0.443159			0.221580	−	0.975142i	$-0.428879\pi$
$653$	289.383			0.443159			0.221580	+	0.975142i	$0.428879\pi$
$654$			183.081i			0.279940i
$655$		0			0
$656$		−	64.9083i		−	0.0989456i
$657$		−	215.777i		−	0.328427i
$658$	−383.551	−	338.317i	−0.582904	−	0.514160i
$659$	−62.6627			−0.0950875			−0.0475438	−	0.998869i	$-0.515139\pi$
$659$	−62.6627			−0.0950875			−0.0475438	+	0.998869i	$0.515139\pi$
$660$		0			0
$661$		−	148.826i		−	0.225153i	−0.993643	−	0.112577i	$-0.964090\pi$
$661$		−	148.826i		−	0.225153i	0.993643	−	0.112577i	$-0.0359104\pi$
$662$	−433.899			−0.655436
$663$			312.248i			0.470962i
$664$		−	50.1559i		−	0.0755360i
$665$		0			0
$666$	−60.3287			−0.0905836
$667$	1508.35			2.26139
$668$			83.7660i			0.125398i
$669$	−6.18519			−0.00924543
$670$		0			0
$671$			154.432i			0.230152i
$672$	51.4355	+	45.3695i	0.0765409	+	0.0675141i
$673$	−833.049			−1.23781			−0.618907	−	0.785464i	$-0.712424\pi$
$673$	−833.049			−1.23781			−0.618907	+	0.785464i	$0.712424\pi$
$674$	687.170			1.01954
$675$		0			0
$676$	−894.864			−1.32376
$677$		−	391.042i		−	0.577611i	−0.957388	−	0.288805i	$-0.906742\pi$
$677$		−	391.042i		−	0.577611i	0.957388	−	0.288805i	$-0.0932581\pi$
$678$		−	208.799i		−	0.307962i
$679$	−32.3240	−	28.5119i	−0.0476053	−	0.0419910i
$680$		0			0
$681$	−117.891			−0.173115
$682$		−	170.110i		−	0.249427i
$683$	−992.775			−1.45355			−0.726775	−	0.686875i	$-0.758982\pi$
$683$	−992.775			−1.45355			−0.726775	+	0.686875i	$0.758982\pi$
$684$			138.030i			0.201798i
$685$		0			0
$686$	−400.990	−	272.956i	−0.584534	−	0.397894i
$687$	−311.802			−0.453860
$688$	329.246			0.478555
$689$		−	1609.54i		−	2.33605i
$690$		0			0
$691$			697.871i			1.00994i	0.863136	+	0.504972i	$0.168497\pi$
$691$			697.871i			1.00994i	−0.863136	+	0.504972i	$0.831503\pi$
$692$		−	26.0601i		−	0.0376592i
$693$	−163.463	+	185.318i	−0.235877	+	0.267414i
$694$	−11.3521			−0.0163574
$695$		0			0
$696$		−	279.664i		−	0.401816i
$697$	117.825			0.169046
$698$		−	919.095i		−	1.31676i
$699$			199.544i			0.285470i
$700$		0			0
$701$	−222.920			−0.318003			−0.159002	−	0.987278i	$-0.550828\pi$
$701$	−222.920			−0.318003			−0.159002	+	0.987278i	$0.550828\pi$
$702$	−182.448			−0.259898
$703$		−	327.122i		−	0.465323i
$704$	94.1367			0.133717
$705$		0			0
$706$		−	560.729i		−	0.794234i
$707$	750.608	+	662.085i	1.06168	+	0.936472i
$708$	−283.254			−0.400077
$709$	267.679			0.377544			0.188772	−	0.982021i	$-0.439549\pi$
$709$	267.679			0.377544			0.188772	+	0.982021i	$0.439549\pi$
$710$		0			0
$711$	135.841			0.191056
$712$			219.429i			0.308187i
$713$			270.094i			0.378814i
$714$	−82.3570	+	93.3683i	−0.115346	+	0.130768i
$715$		0			0
$716$	48.3147			0.0674786
$717$		−	27.1713i		−	0.0378958i
$718$	−564.244			−0.785856
$719$		−	217.258i		−	0.302167i	−0.988521	−	0.151083i	$-0.951724\pi$
$719$		−	217.258i		−	0.302167i	0.988521	−	0.151083i	$-0.0482762\pi$
$720$		0			0
$721$	−76.2728	−	67.2776i	−0.105787	−	0.0933115i
$722$	−237.915			−0.329522
$723$	525.577			0.726939
$724$		−	87.0213i		−	0.120195i
$725$		0			0
$726$			42.7784i			0.0589235i
$727$		−	771.678i		−	1.06146i	−0.847543	−	0.530728i	$-0.821919\pi$
$727$		−	771.678i		−	1.06146i	0.847543	−	0.530728i	$-0.178081\pi$
$728$	−368.650	−	325.174i	−0.506388	−	0.446668i
$729$	−27.0000			−0.0370370
$730$		0			0
$731$			597.664i			0.817598i
$732$	−45.4630			−0.0621080
$733$		−	1103.98i		−	1.50611i	−0.657960	−	0.753053i	$-0.728580\pi$
$733$		−	1103.98i		−	1.50611i	0.657960	−	0.753053i	$-0.271420\pi$
$734$		−	715.875i		−	0.975306i
$735$		0			0
$736$	−149.467			−0.203080
$737$	263.624			0.357699
$738$			68.8457i			0.0932868i
$739$	−264.136			−0.357423			−0.178712	−	0.983901i	$-0.557193\pi$
$739$	−264.136			−0.357423			−0.178712	+	0.983901i	$0.557193\pi$
$740$		0			0
$741$		−	989.295i		−	1.33508i
$742$	424.523	−	481.283i	0.572134	−	0.648629i
$743$	1069.03			1.43880			0.719401	−	0.694595i	$-0.244417\pi$
$743$	1069.03			1.43880			0.719401	+	0.694595i	$0.244417\pi$
$744$	50.0784			0.0673097
$745$		0			0
$746$	−394.213			−0.528436
$747$			53.1983i			0.0712160i
$748$			170.882i			0.228451i
$749$	363.137	−	411.689i	0.484829	−	0.549652i
$750$		0			0
$751$	987.190			1.31450			0.657250	−	0.753673i	$-0.271720\pi$
$751$	987.190			1.31450			0.657250	+	0.753673i	$0.271720\pi$
$752$		−	206.653i		−	0.274804i
$753$	−139.472			−0.185222
$754$			2004.42i			2.65838i
$755$		0			0
$756$	−54.5556	−	48.1216i	−0.0721635	−	0.0636529i
$757$	1173.09			1.54966			0.774830	−	0.632169i	$-0.217835\pi$
$757$	1173.09			1.54966			0.774830	+	0.632169i	$0.217835\pi$
$758$	729.677			0.962634
$759$		−	538.518i		−	0.709509i
$760$		0			0
$761$		−	271.712i		−	0.357045i	−0.983936	−	0.178523i	$-0.942868\pi$
$761$		−	271.712i		−	0.357045i	0.983936	−	0.178523i	$-0.0571318\pi$
$762$			163.561i			0.214647i
$763$	346.095	−	392.369i	0.453598	−	0.514245i
$764$	−493.144			−0.645477
$765$		0			0
$766$			337.594i			0.440723i
$767$	2030.15			2.64687
$768$			27.7128i			0.0360844i
$769$		−	1031.82i		−	1.34177i	−0.741563	−	0.670883i	$-0.765915\pi$
$769$		−	1031.82i		−	1.34177i	0.741563	−	0.670883i	$-0.234085\pi$
$770$		0			0
$771$	−67.7252			−0.0878407
$772$	323.751			0.419366
$773$		−	877.988i		−	1.13582i	−0.823091	−	0.567910i	$-0.807752\pi$
$773$		−	877.988i		−	1.13582i	0.823091	−	0.567910i	$-0.192248\pi$
$774$	−349.218			−0.451186
$775$		0			0
$776$		−	17.4158i		−	0.0224430i
$777$	129.293	+	114.045i	0.166400	+	0.146776i
$778$	241.474			0.310378
$779$	−373.304			−0.479210
$780$		0			0
$781$	1076.71			1.37863
$782$		−	271.320i		−	0.346957i
$783$			296.628i			0.378835i
$784$	−24.4675	−	194.467i	−0.0312085	−	0.248044i
$785$		0			0
$786$	70.6260			0.0898550
$787$		−	142.727i		−	0.181356i	−0.995880	−	0.0906780i	$-0.971097\pi$
$787$		−	142.727i		−	0.181356i	0.995880	−	0.0906780i	$-0.0289034\pi$
$788$	−89.6759			−0.113802
$789$		−	137.117i		−	0.173786i
$790$		0			0
$791$	−394.712	+	447.486i	−0.499004	+	0.565721i
$792$	−99.8470			−0.126069
$793$	325.844			0.410901
$794$			114.153i			0.143770i
$795$		0			0
$796$		−	229.534i		−	0.288359i
$797$		−	317.415i		−	0.398262i	−0.979973	−	0.199131i	$-0.936188\pi$
$797$		−	317.415i		−	0.398262i	0.979973	−	0.199131i	$-0.0638119\pi$
$798$	260.932	−	295.819i	0.326982	−	0.370700i
$799$	375.126			0.469495
$800$		0			0
$801$		−	232.740i		−	0.290562i
$802$	663.796			0.827676
$803$			846.354i			1.05399i
$804$			77.6081i			0.0965274i
$805$		0			0
$806$	−358.924			−0.445315
$807$	92.2793			0.114349
$808$			404.418i			0.500518i
$809$	1392.39			1.72112			0.860560	−	0.509350i	$-0.170114\pi$
$809$	1392.39			1.72112			0.860560	+	0.509350i	$0.170114\pi$
$810$		0			0
$811$			1087.46i			1.34089i	0.741958	+	0.670447i	$0.233898\pi$
$811$			1087.46i			1.34089i	−0.741958	+	0.670447i	$0.766102\pi$
$812$	−528.675	+	599.360i	−0.651078	+	0.738128i
$813$	−411.601			−0.506274
$814$	236.631			0.290701
$815$		0			0
$816$	−50.3057			−0.0616492
$817$		−	1893.58i		−	2.31772i
$818$			949.274i			1.16048i
$819$	391.013	+	344.899i	0.477427	+	0.421122i
$820$		0			0
$821$	172.333			0.209906			0.104953	−	0.994477i	$-0.466531\pi$
$821$	172.333			0.209906			0.104953	+	0.994477i	$0.466531\pi$
$822$			292.591i			0.355950i
$823$	215.439			0.261773			0.130886	−	0.991397i	$-0.458218\pi$
$823$	215.439			0.261773			0.130886	+	0.991397i	$0.458218\pi$
$824$		−	41.0948i		−	0.0498724i
$825$		0			0
$826$	607.055	+	535.463i	0.734934	+	0.648260i
$827$	−797.435			−0.964250			−0.482125	−	0.876102i	$-0.660135\pi$
$827$	−797.435			−0.964250			−0.482125	+	0.876102i	$0.660135\pi$
$828$	158.534			0.191466
$829$		−	389.170i		−	0.469445i	−0.972062	−	0.234723i	$-0.924582\pi$
$829$		−	389.170i		−	0.469445i	0.972062	−	0.234723i	$-0.0754182\pi$
$830$		0			0
$831$		−	291.305i		−	0.350548i
$832$		−	198.624i		−	0.238731i
$833$	353.006	−	44.4146i	0.423777	−	0.0533188i
$834$	−492.220			−0.590192
$835$		0			0
$836$		−	541.404i		−	0.647612i
$837$	−53.1162			−0.0634602
$838$			22.8577i			0.0272764i
$839$		−	280.262i		−	0.334043i	−0.985953	−	0.167021i	$-0.946585\pi$
$839$		−	280.262i		−	0.334043i	0.985953	−	0.167021i	$-0.0534149\pi$
$840$		0			0
$841$	2417.83			2.87494
$842$	−655.285			−0.778248
$843$			222.006i			0.263353i
$844$	−439.204			−0.520384
$845$		0			0
$846$			219.188i			0.259088i
$847$	80.8681	−	91.6804i	0.0954760	−	0.108241i
$848$	259.309			0.305789
$849$	125.971			0.148376
$850$		0			0
$851$	−375.715			−0.441498
$852$			316.973i			0.372034i
$853$		−	245.874i		−	0.288246i	−0.989560	−	0.144123i	$-0.953964\pi$
$853$		−	245.874i		−	0.288246i	0.989560	−	0.144123i	$-0.0460361\pi$
$854$	97.4339	+	85.9431i	0.114091	+	0.100636i
$855$		0			0
$856$	221.813			0.259127
$857$		−	798.850i		−	0.932147i	−0.884746	−	0.466074i	$-0.845668\pi$
$857$		−	798.850i		−	0.932147i	0.884746	−	0.466074i	$-0.154332\pi$
$858$	715.627			0.834064
$859$			1274.93i			1.48420i	0.670287	+	0.742102i	$0.266171\pi$
$859$			1274.93i			1.48420i	−0.670287	+	0.742102i	$0.733829\pi$
$860$		0			0
$861$	130.146	−	147.546i	0.151156	−	0.171366i
$862$	601.688			0.698013
$863$	−736.114			−0.852971			−0.426486	−	0.904494i	$-0.640249\pi$
$863$	−736.114			−0.852971			−0.426486	+	0.904494i	$0.640249\pi$
$864$		−	29.3939i		−	0.0340207i
$865$		0			0
$866$			680.162i			0.785407i
$867$			409.245i			0.472024i
$868$	−107.325	−	94.6680i	−0.123647	−	0.109064i
$869$	−532.816			−0.613137
$870$		0			0
$871$		−	556.236i		−	0.638617i
$872$	211.404			0.242435
$873$			18.4722i			0.0211595i
$874$			859.623i			0.983551i
$875$		0			0
$876$	−249.158			−0.284426
$877$	−1103.15			−1.25787			−0.628936	−	0.777457i	$-0.716509\pi$
$877$	−1103.15			−1.25787			−0.628936	+	0.777457i	$0.716509\pi$
$878$			103.774i			0.118194i
$879$	−473.223			−0.538366
$880$		0			0
$881$		−	269.425i		−	0.305817i	−0.988240	−	0.152909i	$-0.951136\pi$
$881$		−	269.425i		−	0.305817i	0.988240	−	0.152909i	$-0.0488640\pi$
$882$	25.9517	+	206.263i	0.0294236	+	0.233859i
$883$	1077.49			1.22026			0.610129	−	0.792302i	$-0.291118\pi$
$883$	1077.49			1.22026			0.610129	+	0.792302i	$0.291118\pi$
$884$	360.553			0.407865
$885$		0			0
$886$	−102.440			−0.115621
$887$			500.908i			0.564721i	0.959308	+	0.282361i	$0.0911175\pi$
$887$			500.908i			0.564721i	−0.959308	+	0.282361i	$0.908883\pi$
$888$			69.6615i			0.0784477i
$889$	309.194	−	350.534i	0.347800	−	0.394302i
$890$		0			0
$891$	105.904			0.118859
$892$			7.14204i			0.00800678i
$893$	−1188.51			−1.33092
$894$			13.9739i			0.0156308i
$895$		0			0
$896$	52.3882	−	59.3926i	0.0584689	−	0.0662864i
$897$	−1136.25			−1.26672
$898$	1020.72			1.13666
$899$			583.546i			0.649106i
$900$		0			0
$901$			470.712i			0.522433i
$902$		−	270.038i		−	0.299376i
$903$	748.425	+	660.160i	0.828820	+	0.731074i
$904$	−241.100			−0.266703
$905$		0			0
$906$		−	530.290i		−	0.585309i
$907$	−789.727			−0.870702			−0.435351	−	0.900261i	$-0.643376\pi$
$907$	−789.727			−0.870702			−0.435351	+	0.900261i	$0.643376\pi$
$908$			136.129i			0.149922i
$909$		−	428.950i		−	0.471893i
$910$		0			0
$911$	885.912			0.972461			0.486231	−	0.873830i	$-0.338371\pi$
$911$	885.912			0.972461			0.486231	+	0.873830i	$0.338371\pi$
$912$	159.383			0.174763
$913$		−	208.663i		−	0.228547i
$914$	880.200			0.963020
$915$		0			0
$916$			360.038i			0.393054i
$917$	−151.362	−	133.511i	−0.165062	−	0.145596i
$918$	53.3573			0.0581234
$919$	1411.37			1.53577			0.767883	−	0.640590i	$-0.221310\pi$
$919$	1411.37			1.53577			0.767883	+	0.640590i	$0.221310\pi$
$920$		0			0
$921$	−582.119			−0.632051
$922$		−	848.805i		−	0.920613i
$923$		−	2271.82i		−	2.46134i
$924$	213.987	+	188.750i	0.231587	+	0.204275i
$925$		0			0
$926$	180.331			0.194742
$927$			43.5877i			0.0470201i
$928$	−322.928			−0.347983
$929$			1394.66i			1.50125i	0.660727	+	0.750627i	$0.270248\pi$
$929$			1394.66i			1.50125i	−0.660727	+	0.750627i	$0.729752\pi$
$930$		0			0
$931$	−1118.43	+	140.719i	−1.20132	+	0.151148i
$932$	230.413			0.247225
$933$	−861.308			−0.923160
$934$			16.1839i			0.0173275i
$935$		0			0
$936$			210.673i			0.225078i
$937$			448.660i			0.478826i	0.970918	+	0.239413i	$0.0769550\pi$
$937$			448.660i			0.478826i	−0.970918	+	0.239413i	$0.923045\pi$
$938$	146.710	−	166.325i	0.156407	−	0.177319i
$939$	−436.518			−0.464876
$940$		0			0
$941$		−	1061.98i		−	1.12856i	−0.825583	−	0.564281i	$-0.809153\pi$
$941$		−	1061.98i		−	1.12856i	0.825583	−	0.564281i	$-0.190847\pi$
$942$	265.248			0.281580
$943$			428.757i			0.454673i
$944$			327.074i			0.346477i
$945$		0			0
$946$	1369.76			1.44795
$947$	−470.708			−0.497051			−0.248526	−	0.968625i	$-0.579946\pi$
$947$	−470.708			−0.497051			−0.248526	+	0.968625i	$0.579946\pi$
$948$		−	156.855i		−	0.165459i
$949$	1785.77			1.88174
$950$		0			0
$951$		−	468.006i		−	0.492120i
$952$	107.812	+	95.0977i	0.113248	+	0.0998925i
$953$	625.282			0.656119			0.328060	−	0.944657i	$-0.393605\pi$
$953$	625.282			0.656119			0.328060	+	0.944657i	$0.393605\pi$
$954$	−275.039			−0.288301
$955$		0			0
$956$	−31.3747			−0.0328187
$957$		−	1163.48i		−	1.21576i
$958$			291.139i			0.303903i
$959$	553.113	−	627.065i	0.576760	−	0.653874i
$960$		0			0
$961$	856.506			0.891266
$962$		−	499.281i		−	0.519003i
$963$	−235.268			−0.244308
$964$		−	606.884i		−	0.629547i
$965$		0			0
$966$	−339.761	−	299.691i	−0.351719	−	0.310240i
$967$	593.257			0.613503			0.306751	−	0.951790i	$-0.400758\pi$
$967$	593.257			0.613503			0.306751	+	0.951790i	$0.400758\pi$
$968$	49.3963			0.0510292
$969$			289.321i			0.298577i
$970$		0			0
$971$		−	1633.10i		−	1.68187i	−0.541133	−	0.840937i	$-0.682004\pi$
$971$		−	1633.10i		−	1.68187i	0.541133	−	0.840937i	$-0.317996\pi$
$972$			31.1769i			0.0320750i
$973$	1054.90	+	930.490i	1.08417	+	0.956311i
$974$	−853.408			−0.876189
$975$		0			0
$976$			52.4962i			0.0537871i
$977$	−1137.81			−1.16459			−0.582296	−	0.812977i	$-0.697846\pi$
$977$	−1137.81			−1.16459			−0.582296	+	0.812977i	$0.697846\pi$
$978$			574.147i			0.587062i
$979$			912.891i			0.932472i
$980$		0			0
$981$	−224.227			−0.228570
$982$	−739.034			−0.752580
$983$			746.446i			0.759355i	0.925119	+	0.379677i	$0.123965\pi$
$983$			746.446i			0.759355i	−0.925119	+	0.379677i	$0.876035\pi$
$984$	79.4961			0.0807888
$985$		0			0
$986$		−	586.195i		−	0.594518i
$987$	414.352	−	469.752i	0.419810	−	0.475939i
$988$	−1142.34			−1.15621
$989$	−2174.86			−2.19905
$990$		0			0
$991$	−765.218			−0.772168			−0.386084	−	0.922464i	$-0.626172\pi$
$991$	−765.218			−0.772168			−0.386084	+	0.922464i	$0.626172\pi$
$992$		−	57.8256i		−	0.0582919i
$993$		−	531.415i		−	0.535161i
$994$	599.204	−	679.319i	0.602821	−	0.683420i
$995$		0			0
$996$	61.4282			0.0616749
$997$			1799.05i			1.80446i	0.431255	+	0.902230i	$0.358071\pi$
$997$			1799.05i			1.80446i	−0.431255	+	0.902230i	$0.641929\pi$
$998$	−176.167			−0.176520
$999$		−	73.8872i		−	0.0739612i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.3.f.e.601.16		16
5.2	odd	4		210.3.h.a.139.15	yes	16
5.3	odd	4		210.3.h.a.139.2	✓	16
5.4	even	2	inner	1050.3.f.e.601.1		16
7.6	odd	2	inner	1050.3.f.e.601.12		16
15.2	even	4		630.3.h.e.559.5		16
15.8	even	4		630.3.h.e.559.12		16
20.3	even	4		1680.3.bd.a.769.12		16
20.7	even	4		1680.3.bd.a.769.6		16
35.13	even	4		210.3.h.a.139.7	yes	16
35.27	even	4		210.3.h.a.139.10	yes	16
35.34	odd	2	inner	1050.3.f.e.601.5		16
105.62	odd	4		630.3.h.e.559.4		16
105.83	odd	4		630.3.h.e.559.13		16
140.27	odd	4		1680.3.bd.a.769.11		16
140.83	odd	4		1680.3.bd.a.769.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.h.a.139.2	✓	16	5.3	odd	4
210.3.h.a.139.7	yes	16	35.13	even	4
210.3.h.a.139.10	yes	16	35.27	even	4
210.3.h.a.139.15	yes	16	5.2	odd	4
630.3.h.e.559.4		16	105.62	odd	4
630.3.h.e.559.5		16	15.2	even	4
630.3.h.e.559.12		16	15.8	even	4
630.3.h.e.559.13		16	105.83	odd	4
1050.3.f.e.601.1		16	5.4	even	2	inner
1050.3.f.e.601.5		16	35.34	odd	2	inner
1050.3.f.e.601.12		16	7.6	odd	2	inner
1050.3.f.e.601.16		16	1.1	even	1	trivial
1680.3.bd.a.769.5		16	140.83	odd	4
1680.3.bd.a.769.6		16	20.7	even	4
1680.3.bd.a.769.11		16	140.27	odd	4
1680.3.bd.a.769.12		16	20.3	even	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)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} +2.44949i q^{6} +(4.63050 - 5.24961i) q^{7} +2.82843 q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} +2.44949i q^{6} +(4.63050 - 5.24961i) q^{7} +2.82843 q^{8} -3.00000 q^{9} +11.7671 q^{11} +3.46410i q^{12} -24.8280i q^{13} +(6.54852 - 7.42407i) q^{14} +4.00000 q^{16} +7.26100i q^{17} -4.24264 q^{18} -23.0050i q^{19} +(9.09260 + 8.02027i) q^{21} +16.6412 q^{22} -26.4223 q^{23} +4.89898i q^{24} -35.1122i q^{26} -5.19615i q^{27} +(9.26101 - 10.4992i) q^{28} -57.0861 q^{29} -10.2222i q^{31} +5.65685 q^{32} +20.3812i q^{33} +10.2686i q^{34} -6.00000 q^{36} +14.2196 q^{37} -32.5340i q^{38} +43.0034 q^{39} -16.2271i q^{41} +(12.8589 + 11.3424i) q^{42} +82.3114 q^{43} +23.5342 q^{44} -37.3668 q^{46} -51.6631i q^{47} +6.92820i q^{48} +(-6.11686 - 48.6167i) q^{49} -12.5764 q^{51} -49.6561i q^{52} +64.8273 q^{53} -7.34847i q^{54} +(13.0970 - 14.8481i) q^{56} +39.8459 q^{57} -80.7320 q^{58} +81.7685i q^{59} +13.1240i q^{61} -14.4564i q^{62} +(-13.8915 + 15.7488i) q^{63} +8.00000 q^{64} +28.8233i q^{66} +22.4035 q^{67} +14.5220i q^{68} -45.7648i q^{69} +91.5022 q^{71} -8.48528 q^{72} +71.9256i q^{73} +20.1096 q^{74} -46.0100i q^{76} +(54.4875 - 61.7726i) q^{77} +60.8160 q^{78} -45.2802 q^{79} +9.00000 q^{81} -22.9486i q^{82} -17.7328i q^{83} +(18.1852 + 16.0405i) q^{84} +116.406 q^{86} -98.8761i q^{87} +33.2823 q^{88} +77.5800i q^{89} +(-130.338 - 114.966i) q^{91} -52.8446 q^{92} +17.7054 q^{93} -73.0627i q^{94} +9.79796i q^{96} -6.15741i q^{97} +(-8.65055 - 68.7544i) q^{98} -35.3012 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4} - 48 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more