LMFDB - Embedded newform 464.3.d.b.175.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [464,3,Mod(175,464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(464, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("464.175");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 464 = 2^{4} \cdot 29 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	464.d (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:	$12.6430842663$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 69 x^{18} + 1795 x^{16} + 24222 x^{14} + 189561 x^{12} + 892623 x^{10} + 2508433 x^{8} + \cdots + 21609 $ x^20 + 69x^18 + 1795x^16 + 24222x^14 + 189561x^12 + 892623x^10 + 2508433x^8 + 3989370x^6 + 3148495x^4 + 862449*x^2 + 21609
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{36}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			175.15
Root			$-2.79820i$ of defining polynomial
Character	$\chi$	$=$	464.175
Dual form			464.3.d.b.175.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.34966i q^{3} -1.82181 q^{5} +0.690645i q^{7} +3.47908 q^{9} +O(q^{10})$	$q+2.34966i q^{3} -1.82181 q^{5} +0.690645i q^{7} +3.47908 q^{9} -14.2821i q^{11} +11.2627 q^{13} -4.28063i q^{15} +28.7655 q^{17} +27.1784i q^{19} -1.62278 q^{21} +24.2420i q^{23} -21.6810 q^{25} +29.3216i q^{27} +5.38516 q^{29} +8.53755i q^{31} +33.5582 q^{33} -1.25822i q^{35} -26.9435 q^{37} +26.4635i q^{39} +16.1674 q^{41} -25.6365i q^{43} -6.33821 q^{45} +73.0506i q^{47} +48.5230 q^{49} +67.5893i q^{51} +74.9103 q^{53} +26.0192i q^{55} -63.8601 q^{57} -73.7015i q^{59} -90.0602 q^{61} +2.40281i q^{63} -20.5184 q^{65} -26.8696i q^{67} -56.9607 q^{69} +124.423i q^{71} +135.584 q^{73} -50.9431i q^{75} +9.86386 q^{77} -81.1222i q^{79} -37.5843 q^{81} +82.5008i q^{83} -52.4052 q^{85} +12.6533i q^{87} +55.4929 q^{89} +7.77851i q^{91} -20.0604 q^{93} -49.5137i q^{95} +151.078 q^{97} -49.6885i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 8 q^{5} - 68 q^{9}+O(q^{10}) $ 20 * q - 8 * q^5 - 68 * q^9	$ 20 q - 8 q^{5} - 68 q^{9} + 16 q^{13} + 40 q^{17} - 48 q^{21} + 188 q^{25} - 120 q^{33} - 80 q^{37} - 72 q^{41} + 72 q^{45} - 28 q^{49} + 96 q^{53} + 104 q^{57} - 96 q^{61} - 80 q^{65} + 352 q^{69} - 312 q^{73} - 192 q^{77} + 60 q^{81} + 336 q^{85} - 152 q^{89} - 472 q^{93} + 56 q^{97}+O(q^{100}) $ 20 * q - 8 * q^5 - 68 * q^9 + 16 * q^13 + 40 * q^17 - 48 * q^21 + 188 * q^25 - 120 * q^33 - 80 * q^37 - 72 * q^41 + 72 * q^45 - 28 * q^49 + 96 * q^53 + 104 * q^57 - 96 * q^61 - 80 * q^65 + 352 * q^69 - 312 * q^73 - 192 * q^77 + 60 * q^81 + 336 * q^85 - 152 * q^89 - 472 * q^93 + 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/464\mathbb{Z}\right)^\times$.

$n$	$117$	$175$	$321$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	2.34966i	0.783221i	0.920131	+	0.391611i	$0.128082\pi$
$3$	2.34966i	0.783221i	−0.920131	+	0.391611i	$0.871918\pi$
$4$	0	0
$5$	−1.82181	−0.364361	−0.182181	−	0.983265i	$-0.558316\pi$
$5$	−1.82181	−0.364361	−0.182181	+	0.983265i	$0.558316\pi$
$6$	0	0
$7$	0.690645i	0.0986635i	0.998782	+	0.0493318i	$0.0157092\pi$
$7$	0.690645i	0.0986635i	−0.998782	+	0.0493318i	$0.984291\pi$
$8$	0	0
$9$	3.47908	0.386564
$10$	0	0
$11$	− 14.2821i	− 1.29837i	−0.760629	−	0.649187i	$-0.775109\pi$
$11$	− 14.2821i	− 1.29837i	0.760629	−	0.649187i	$-0.224891\pi$
$12$	0	0
$13$	11.2627	0.866360	0.433180	−	0.901307i	$-0.357391\pi$
$13$	11.2627	0.866360	0.433180	+	0.901307i	$0.357391\pi$
$14$	0	0
$15$	− 4.28063i	− 0.285376i
$16$	0	0
$17$	28.7655	1.69209	0.846044	−	0.533113i	$-0.178978\pi$
$17$	28.7655	1.69209	0.846044	+	0.533113i	$0.178978\pi$
$18$	0	0
$19$	27.1784i	1.43044i	0.698899	+	0.715220i	$0.253674\pi$
$19$	27.1784i	1.43044i	−0.698899	+	0.715220i	$0.746326\pi$
$20$	0	0
$21$	−1.62278	−0.0772754
$22$	0	0
$23$	24.2420i	1.05400i	0.849865	+	0.527001i	$0.176684\pi$
$23$	24.2420i	1.05400i	−0.849865	+	0.527001i	$0.823316\pi$
$24$	0	0
$25$	−21.6810	−0.867241
$26$	0	0
$27$	29.3216i	1.08599i
$28$	0	0
$29$	5.38516	0.185695
$29$	5.38516	0.185695
$30$	0	0
$31$	8.53755i	0.275405i	0.990474	+	0.137702i	$0.0439718\pi$
$31$	8.53755i	0.275405i	−0.990474	+	0.137702i	$0.956028\pi$
$32$	0	0
$33$	33.5582	1.01691
$34$	0	0
$35$	− 1.25822i	− 0.0359492i
$36$	0	0
$37$	−26.9435	−0.728204	−0.364102	−	0.931359i	$-0.618624\pi$
$37$	−26.9435	−0.728204	−0.364102	+	0.931359i	$0.618624\pi$
$38$	0	0
$39$	26.4635i	0.678552i
$40$	0	0
$41$	16.1674	0.394326	0.197163	−	0.980371i	$-0.436827\pi$
$41$	16.1674	0.394326	0.197163	+	0.980371i	$0.436827\pi$
$42$	0	0
$43$	− 25.6365i	− 0.596198i	−0.954535	−	0.298099i	$-0.903647\pi$
$43$	− 25.6365i	− 0.596198i	0.954535	−	0.298099i	$-0.0963526\pi$
$44$	0	0
$45$	−6.33821	−0.140849
$46$	0	0
$47$	73.0506i	1.55427i	0.629335	+	0.777134i	$0.283328\pi$
$47$	73.0506i	1.55427i	−0.629335	+	0.777134i	$0.716672\pi$
$48$	0	0
$49$	48.5230	0.990266
$50$	0	0
$51$	67.5893i	1.32528i
$52$	0	0
$53$	74.9103	1.41340	0.706701	−	0.707512i	$-0.250183\pi$
$53$	74.9103	1.41340	0.706701	+	0.707512i	$0.250183\pi$
$54$	0	0
$55$	26.0192i	0.473077i
$56$	0	0
$57$	−63.8601	−1.12035
$58$	0	0
$59$	− 73.7015i	− 1.24918i	−0.780954	−	0.624589i	$-0.785267\pi$
$59$	− 73.7015i	− 1.24918i	0.780954	−	0.624589i	$-0.214733\pi$
$60$	0	0
$61$	−90.0602	−1.47640	−0.738199	−	0.674584i	$-0.764323\pi$
$61$	−90.0602	−1.47640	−0.738199	+	0.674584i	$0.764323\pi$
$62$	0	0
$63$	2.40281i	0.0381398i
$64$	0	0
$65$	−20.5184	−0.315668
$66$	0	0
$67$	− 26.8696i	− 0.401039i	−0.979690	−	0.200519i	$-0.935737\pi$
$67$	− 26.8696i	− 0.401039i	0.979690	−	0.200519i	$-0.0642630\pi$
$68$	0	0
$69$	−56.9607	−0.825517
$70$	0	0
$71$	124.423i	1.75244i	0.481908	+	0.876222i	$0.339944\pi$
$71$	124.423i	1.75244i	−0.481908	+	0.876222i	$0.660056\pi$
$72$	0	0
$73$	135.584	1.85732	0.928658	−	0.370937i	$-0.120964\pi$
$73$	135.584	1.85732	0.928658	+	0.370937i	$0.120964\pi$
$74$	0	0
$75$	− 50.9431i	− 0.679242i
$76$	0	0
$77$	9.86386	0.128102
$78$	0	0
$79$	− 81.1222i	− 1.02686i	−0.858131	−	0.513431i	$-0.828374\pi$
$79$	− 81.1222i	− 1.02686i	0.858131	−	0.513431i	$-0.171626\pi$
$80$	0	0
$81$	−37.5843	−0.464004
$82$	0	0
$83$	82.5008i	0.993986i	0.867755	+	0.496993i	$0.165563\pi$
$83$	82.5008i	0.993986i	−0.867755	+	0.496993i	$0.834437\pi$
$84$	0	0
$85$	−52.4052	−0.616532
$86$	0	0
$87$	12.6533i	0.145441i
$88$	0	0
$89$	55.4929	0.623516	0.311758	−	0.950162i	$-0.399082\pi$
$89$	55.4929	0.623516	0.311758	+	0.950162i	$0.399082\pi$
$90$	0	0
$91$	7.77851i	0.0854782i
$92$	0	0
$93$	−20.0604	−0.215703
$94$	0	0
$95$	− 49.5137i	− 0.521197i
$96$	0	0
$97$	151.078	1.55750	0.778750	−	0.627334i	$-0.215854\pi$
$97$	151.078	1.55750	0.778750	+	0.627334i	$0.215854\pi$
$98$	0	0
$99$	− 49.6885i	− 0.501905i
$100$	0	0
$101$	32.6869	0.323633	0.161816	−	0.986821i	$-0.448265\pi$
$101$	32.6869	0.323633	0.161816	+	0.986821i	$0.448265\pi$
$102$	0	0
$103$	− 61.3478i	− 0.595610i	−0.954627	−	0.297805i	$-0.903746\pi$
$103$	− 61.3478i	− 0.595610i	0.954627	−	0.297805i	$-0.0962545\pi$
$104$	0	0
$105$	2.95640	0.0281562
$106$	0	0
$107$	− 32.9750i	− 0.308178i	−0.988057	−	0.154089i	$-0.950756\pi$
$107$	− 32.9750i	− 0.308178i	0.988057	−	0.154089i	$-0.0492442\pi$
$108$	0	0
$109$	6.30640	0.0578569	0.0289284	−	0.999581i	$-0.490791\pi$
$109$	6.30640	0.0578569	0.0289284	+	0.999581i	$0.490791\pi$
$110$	0	0
$111$	− 63.3083i	− 0.570345i
$112$	0	0
$113$	−119.124	−1.05419	−0.527097	−	0.849805i	$-0.676720\pi$
$113$	−119.124	−1.05419	−0.527097	+	0.849805i	$0.676720\pi$
$114$	0	0
$115$	− 44.1643i	− 0.384038i
$116$	0	0
$117$	39.1837	0.334904
$118$	0	0
$119$	19.8667i	0.166947i
$120$	0	0
$121$	−82.9785	−0.685773
$122$	0	0
$123$	37.9879i	0.308845i
$124$	0	0
$125$	85.0438	0.680350
$126$	0	0
$127$	− 133.412i	− 1.05049i	−0.850951	−	0.525246i	$-0.823973\pi$
$127$	− 133.412i	− 1.05049i	0.850951	−	0.525246i	$-0.176027\pi$
$128$	0	0
$129$	60.2372	0.466955
$130$	0	0
$131$	− 121.191i	− 0.925125i	−0.886587	−	0.462563i	$-0.846930\pi$
$131$	− 121.191i	− 0.925125i	0.886587	−	0.462563i	$-0.153070\pi$
$132$	0	0
$133$	−18.7706	−0.141132
$134$	0	0
$135$	− 53.4184i	− 0.395692i
$136$	0	0
$137$	−32.2763	−0.235594	−0.117797	−	0.993038i	$-0.537583\pi$
$137$	−32.2763	−0.235594	−0.117797	+	0.993038i	$0.537583\pi$
$138$	0	0
$139$	78.0620i	0.561597i	0.959767	+	0.280799i	$0.0905993\pi$
$139$	78.0620i	0.561597i	−0.959767	+	0.280799i	$0.909401\pi$
$140$	0	0
$141$	−171.644	−1.21734
$142$	0	0
$143$	− 160.855i	− 1.12486i
$144$	0	0
$145$	−9.81073	−0.0676602
$146$	0	0
$147$	114.013i	0.775597i
$148$	0	0
$149$	−182.590	−1.22544	−0.612720	−	0.790300i	$-0.709925\pi$
$149$	−182.590	−1.22544	−0.612720	+	0.790300i	$0.709925\pi$
$150$	0	0
$151$	− 195.412i	− 1.29412i	−0.762440	−	0.647058i	$-0.775999\pi$
$151$	− 195.412i	− 1.29412i	0.762440	−	0.647058i	$-0.224001\pi$
$152$	0	0
$153$	100.077	0.654101
$154$	0	0
$155$	− 15.5538i	− 0.100347i
$156$	0	0
$157$	−245.914	−1.56633	−0.783164	−	0.621815i	$-0.786396\pi$
$157$	−245.914	−1.56633	−0.783164	+	0.621815i	$0.786396\pi$
$158$	0	0
$159$	176.014i	1.10701i
$160$	0	0
$161$	−16.7426	−0.103992
$162$	0	0
$163$	− 104.743i	− 0.642596i	−0.946978	−	0.321298i	$-0.895881\pi$
$163$	− 104.743i	− 0.642596i	0.946978	−	0.321298i	$-0.104119\pi$
$164$	0	0
$165$	−61.1365	−0.370524
$166$	0	0
$167$	− 21.6850i	− 0.129850i	−0.997890	−	0.0649250i	$-0.979319\pi$
$167$	− 21.6850i	− 0.129850i	0.997890	−	0.0649250i	$-0.0206808\pi$
$168$	0	0
$169$	−42.1520	−0.249420
$170$	0	0
$171$	94.5557i	0.552957i
$172$	0	0
$173$	−203.838	−1.17825	−0.589127	−	0.808040i	$-0.700528\pi$
$173$	−203.838	−1.17825	−0.589127	+	0.808040i	$0.700528\pi$
$174$	0	0
$175$	− 14.9739i	− 0.0855650i
$176$	0	0
$177$	173.174	0.978383
$178$	0	0
$179$	− 67.3420i	− 0.376212i	−0.982149	−	0.188106i	$-0.939765\pi$
$179$	− 67.3420i	− 0.376212i	0.982149	−	0.188106i	$-0.0602349\pi$
$180$	0	0
$181$	32.6670	0.180481	0.0902403	−	0.995920i	$-0.471236\pi$
$181$	32.6670	0.180481	0.0902403	+	0.995920i	$0.471236\pi$
$182$	0	0
$183$	− 211.611i	− 1.15635i
$184$	0	0
$185$	49.0859	0.265329
$186$	0	0
$187$	− 410.832i	− 2.19696i
$188$	0	0
$189$	−20.2508	−0.107147
$190$	0	0
$191$	250.711i	1.31263i	0.754489	+	0.656313i	$0.227885\pi$
$191$	250.711i	1.31263i	−0.754489	+	0.656313i	$0.772115\pi$
$192$	0	0
$193$	−52.5002	−0.272022	−0.136011	−	0.990707i	$-0.543428\pi$
$193$	−52.5002	−0.272022	−0.136011	+	0.990707i	$0.543428\pi$
$194$	0	0
$195$	− 48.2114i	− 0.247238i
$196$	0	0
$197$	−170.910	−0.867562	−0.433781	−	0.901018i	$-0.642821\pi$
$197$	−170.910	−0.867562	−0.433781	+	0.901018i	$0.642821\pi$
$198$	0	0
$199$	58.0168i	0.291542i	0.989318	+	0.145771i	$0.0465662\pi$
$199$	58.0168i	0.291542i	−0.989318	+	0.145771i	$0.953434\pi$
$200$	0	0
$201$	63.1346	0.314102
$202$	0	0
$203$	3.71924i	0.0183214i
$204$	0	0
$205$	−29.4538	−0.143677
$206$	0	0
$207$	84.3399i	0.407439i
$208$	0	0
$209$	388.164	1.85725
$210$	0	0
$211$	224.473i	1.06386i	0.846790	+	0.531928i	$0.178532\pi$
$211$	224.473i	1.06386i	−0.846790	+	0.531928i	$0.821468\pi$
$212$	0	0
$213$	−292.353	−1.37255
$214$	0	0
$215$	46.7048i	0.217232i
$216$	0	0
$217$	−5.89642	−0.0271724
$218$	0	0
$219$	318.577i	1.45469i
$220$	0	0
$221$	323.977	1.46596
$222$	0	0
$223$	− 206.571i	− 0.926325i	−0.886273	−	0.463163i	$-0.846715\pi$
$223$	− 206.571i	− 0.926325i	0.886273	−	0.463163i	$-0.153285\pi$
$224$	0	0
$225$	−75.4299	−0.335244
$226$	0	0
$227$	− 88.9673i	− 0.391926i	−0.980611	−	0.195963i	$-0.937217\pi$
$227$	− 88.9673i	− 0.391926i	0.980611	−	0.195963i	$-0.0627833\pi$
$228$	0	0
$229$	−312.825	−1.36605	−0.683025	−	0.730395i	$-0.739336\pi$
$229$	−312.825	−1.36605	−0.683025	+	0.730395i	$0.739336\pi$
$230$	0	0
$231$	23.1768i	0.100332i
$232$	0	0
$233$	16.0101	0.0687128	0.0343564	−	0.999410i	$-0.489062\pi$
$233$	16.0101	0.0687128	0.0343564	+	0.999410i	$0.489062\pi$
$234$	0	0
$235$	− 133.084i	− 0.566316i
$236$	0	0
$237$	190.610	0.804261
$238$	0	0
$239$	− 79.4928i	− 0.332606i	−0.986075	−	0.166303i	$-0.946817\pi$
$239$	− 79.4928i	− 0.332606i	0.986075	−	0.166303i	$-0.0531830\pi$
$240$	0	0
$241$	−310.931	−1.29017	−0.645084	−	0.764112i	$-0.723178\pi$
$241$	−310.931	−1.29017	−0.645084	+	0.764112i	$0.723178\pi$
$242$	0	0
$243$	175.584i	0.722569i
$244$	0	0
$245$	−88.3996	−0.360815
$246$	0	0
$247$	306.101i	1.23928i
$248$	0	0
$249$	−193.849	−0.778511
$250$	0	0
$251$	95.5427i	0.380648i	0.981721	+	0.190324i	$0.0609539\pi$
$251$	95.5427i	0.380648i	−0.981721	+	0.190324i	$0.939046\pi$
$252$	0	0
$253$	346.227	1.36849
$254$	0	0
$255$	− 123.135i	− 0.482881i
$256$	0	0
$257$	450.968	1.75474	0.877369	−	0.479816i	$-0.159297\pi$
$257$	450.968	1.75474	0.877369	+	0.479816i	$0.159297\pi$
$258$	0	0
$259$	− 18.6084i	− 0.0718471i
$260$	0	0
$261$	18.7354	0.0717832
$262$	0	0
$263$	− 154.826i	− 0.588691i	−0.955699	−	0.294345i	$-0.904898\pi$
$263$	− 154.826i	− 0.588691i	0.955699	−	0.294345i	$-0.0951016\pi$
$264$	0	0
$265$	−136.472	−0.514989
$266$	0	0
$267$	130.390i	0.488351i
$268$	0	0
$269$	344.712	1.28146	0.640729	−	0.767768i	$-0.278632\pi$
$269$	344.712	1.28146	0.640729	+	0.767768i	$0.278632\pi$
$270$	0	0
$271$	− 321.091i	− 1.18484i	−0.805630	−	0.592420i	$-0.798173\pi$
$271$	− 321.091i	− 1.18484i	0.805630	−	0.592420i	$-0.201827\pi$
$272$	0	0
$273$	−18.2769	−0.0669483
$274$	0	0
$275$	309.651i	1.12600i
$276$	0	0
$277$	312.665	1.12875	0.564377	−	0.825517i	$-0.309117\pi$
$277$	312.665	1.12875	0.564377	+	0.825517i	$0.309117\pi$
$278$	0	0
$279$	29.7028i	0.106462i
$280$	0	0
$281$	−29.2825	−0.104208	−0.0521041	−	0.998642i	$-0.516593\pi$
$281$	−29.2825	−0.104208	−0.0521041	+	0.998642i	$0.516593\pi$
$282$	0	0
$283$	− 274.731i	− 0.970782i	−0.874297	−	0.485391i	$-0.838677\pi$
$283$	− 274.731i	− 0.970782i	0.874297	−	0.485391i	$-0.161323\pi$
$284$	0	0
$285$	116.341	0.408213
$286$	0	0
$287$	11.1659i	0.0389056i
$288$	0	0
$289$	538.454	1.86316
$290$	0	0
$291$	354.982i	1.21987i
$292$	0	0
$293$	−197.711	−0.674780	−0.337390	−	0.941365i	$-0.609544\pi$
$293$	−197.711	−0.674780	−0.337390	+	0.941365i	$0.609544\pi$
$294$	0	0
$295$	134.270i	0.455152i
$296$	0	0
$297$	418.775	1.41002
$298$	0	0
$299$	273.030i	0.913145i
$300$	0	0
$301$	17.7057	0.0588230
$302$	0	0
$303$	76.8032i	0.253476i
$304$	0	0
$305$	164.072	0.537942
$306$	0	0
$307$	− 274.770i	− 0.895017i	−0.894280	−	0.447508i	$-0.852311\pi$
$307$	− 274.770i	− 0.895017i	0.894280	−	0.447508i	$-0.147689\pi$
$308$	0	0
$309$	144.147	0.466494
$310$	0	0
$311$	498.598i	1.60321i	0.597854	+	0.801605i	$0.296020\pi$
$311$	498.598i	1.60321i	−0.597854	+	0.801605i	$0.703980\pi$
$312$	0	0
$313$	−374.490	−1.19645	−0.598227	−	0.801326i	$-0.704128\pi$
$313$	−374.490	−1.19645	−0.598227	+	0.801326i	$0.704128\pi$
$314$	0	0
$315$	− 4.37745i	− 0.0138967i
$316$	0	0
$317$	−240.543	−0.758810	−0.379405	−	0.925231i	$-0.623871\pi$
$317$	−240.543	−0.758810	−0.379405	+	0.925231i	$0.623871\pi$
$318$	0	0
$319$	− 76.9115i	− 0.241102i
$320$	0	0
$321$	77.4802	0.241371
$322$	0	0
$323$	781.800i	2.42043i
$324$	0	0
$325$	−244.186	−0.751343
$326$	0	0
$327$	14.8179i	0.0453147i
$328$	0	0
$329$	−50.4520	−0.153350
$330$	0	0
$331$	− 150.863i	− 0.455779i	−0.973687	−	0.227890i	$-0.926817\pi$
$331$	− 150.863i	− 0.455779i	0.973687	−	0.227890i	$-0.0731825\pi$
$332$	0	0
$333$	−93.7386	−0.281497
$334$	0	0
$335$	48.9512i	0.146123i
$336$	0	0
$337$	190.087	0.564055	0.282028	−	0.959406i	$-0.408993\pi$
$337$	190.087	0.564055	0.282028	+	0.959406i	$0.408993\pi$
$338$	0	0
$339$	− 279.901i	− 0.825668i
$340$	0	0
$341$	121.934	0.357578
$342$	0	0
$343$	67.3538i	0.196367i
$344$	0	0
$345$	103.771	0.300786
$346$	0	0
$347$	− 334.477i	− 0.963910i	−0.876196	−	0.481955i	$-0.839927\pi$
$347$	− 334.477i	− 0.963910i	0.876196	−	0.481955i	$-0.160073\pi$
$348$	0	0
$349$	−386.383	−1.10711	−0.553557	−	0.832811i	$-0.686730\pi$
$349$	−386.383	−1.10711	−0.553557	+	0.832811i	$0.686730\pi$
$350$	0	0
$351$	330.240i	0.940856i
$352$	0	0
$353$	−133.391	−0.377878	−0.188939	−	0.981989i	$-0.560505\pi$
$353$	−133.391	−0.377878	−0.188939	+	0.981989i	$0.560505\pi$
$354$	0	0
$355$	− 226.676i	− 0.638523i
$356$	0	0
$357$	−46.6802	−0.130757
$358$	0	0
$359$	− 410.303i	− 1.14290i	−0.820635	−	0.571452i	$-0.806380\pi$
$359$	− 410.303i	− 1.14290i	0.820635	−	0.571452i	$-0.193620\pi$
$360$	0	0
$361$	−377.664	−1.04616
$362$	0	0
$363$	− 194.972i	− 0.537112i
$364$	0	0
$365$	−247.008	−0.676734
$366$	0	0
$367$	544.463i	1.48355i	0.670648	+	0.741776i	$0.266016\pi$
$367$	544.463i	1.48355i	−0.670648	+	0.741776i	$0.733984\pi$
$368$	0	0
$369$	56.2475	0.152432
$370$	0	0
$371$	51.7364i	0.139451i
$372$	0	0
$373$	55.6448	0.149182	0.0745909	−	0.997214i	$-0.476235\pi$
$373$	55.6448	0.149182	0.0745909	+	0.997214i	$0.476235\pi$
$374$	0	0
$375$	199.824i	0.532865i
$376$	0	0
$377$	60.6514	0.160879
$378$	0	0
$379$	512.831i	1.35312i	0.736389	+	0.676558i	$0.236529\pi$
$379$	512.831i	1.35312i	−0.736389	+	0.676558i	$0.763471\pi$
$380$	0	0
$381$	313.474	0.822767
$382$	0	0
$383$	− 138.427i	− 0.361428i	−0.983536	−	0.180714i	$-0.942159\pi$
$383$	− 138.427i	− 0.361428i	0.983536	−	0.180714i	$-0.0578409\pi$
$384$	0	0
$385$	−17.9701	−0.0466755
$386$	0	0
$387$	− 89.1914i	− 0.230469i
$388$	0	0
$389$	132.836	0.341482	0.170741	−	0.985316i	$-0.445384\pi$
$389$	132.836	0.341482	0.170741	+	0.985316i	$0.445384\pi$
$390$	0	0
$391$	697.335i	1.78346i
$392$	0	0
$393$	284.759	0.724578
$394$	0	0
$395$	147.789i	0.374149i
$396$	0	0
$397$	693.912	1.74789	0.873945	−	0.486025i	$-0.161554\pi$
$397$	693.912	1.74789	0.873945	+	0.486025i	$0.161554\pi$
$398$	0	0
$399$	− 44.1046i	− 0.110538i
$400$	0	0
$401$	595.379	1.48474	0.742368	−	0.669992i	$-0.233702\pi$
$401$	595.379	1.48474	0.742368	+	0.669992i	$0.233702\pi$
$402$	0	0
$403$	96.1558i	0.238600i
$404$	0	0
$405$	68.4714	0.169065
$406$	0	0
$407$	384.810i	0.945480i
$408$	0	0
$409$	−401.801	−0.982399	−0.491200	−	0.871047i	$-0.663441\pi$
$409$	−401.801	−0.982399	−0.491200	+	0.871047i	$0.663441\pi$
$410$	0	0
$411$	− 75.8386i	− 0.184522i
$412$	0	0
$413$	50.9015	0.123248
$414$	0	0
$415$	− 150.301i	− 0.362170i
$416$	0	0
$417$	−183.419	−0.439855
$418$	0	0
$419$	− 298.317i	− 0.711975i	−0.934491	−	0.355987i	$-0.884145\pi$
$419$	− 298.317i	− 0.711975i	0.934491	−	0.355987i	$-0.115855\pi$
$420$	0	0
$421$	−540.634	−1.28417	−0.642084	−	0.766635i	$-0.721930\pi$
$421$	−540.634	−1.28417	−0.642084	+	0.766635i	$0.721930\pi$
$422$	0	0
$423$	254.149i	0.600825i
$424$	0	0
$425$	−623.666	−1.46745
$426$	0	0
$427$	− 62.1996i	− 0.145667i
$428$	0	0
$429$	377.955	0.881014
$430$	0	0
$431$	742.588i	1.72294i	0.507807	+	0.861471i	$0.330456\pi$
$431$	742.588i	1.72294i	−0.507807	+	0.861471i	$0.669544\pi$
$432$	0	0
$433$	−336.183	−0.776403	−0.388202	−	0.921574i	$-0.626904\pi$
$433$	−336.183	−0.776403	−0.388202	+	0.921574i	$0.626904\pi$
$434$	0	0
$435$	− 23.0519i	− 0.0529929i
$436$	0	0
$437$	−658.859	−1.50769
$438$	0	0
$439$	− 733.097i	− 1.66993i	−0.550306	−	0.834963i	$-0.685489\pi$
$439$	− 733.097i	− 1.66993i	0.550306	−	0.834963i	$-0.314511\pi$
$440$	0	0
$441$	168.815	0.382801
$442$	0	0
$443$	− 664.117i	− 1.49914i	−0.661928	−	0.749568i	$-0.730262\pi$
$443$	− 664.117i	− 1.49914i	0.661928	−	0.749568i	$-0.269738\pi$
$444$	0	0
$445$	−101.097	−0.227185
$446$	0	0
$447$	− 429.026i	− 0.959791i
$448$	0	0
$449$	−139.219	−0.310065	−0.155032	−	0.987909i	$-0.549548\pi$
$449$	−139.219	−0.310065	−0.155032	+	0.987909i	$0.549548\pi$
$450$	0	0
$451$	− 230.904i	− 0.511982i
$452$	0	0
$453$	459.152	1.01358
$454$	0	0
$455$	− 14.1709i	− 0.0311449i
$456$	0	0
$457$	−644.897	−1.41115	−0.705577	−	0.708634i	$-0.749312\pi$
$457$	−644.897	−1.41115	−0.705577	+	0.708634i	$0.749312\pi$
$458$	0	0
$459$	843.452i	1.83759i
$460$	0	0
$461$	558.356	1.21119	0.605593	−	0.795775i	$-0.292936\pi$
$461$	558.356	1.21119	0.605593	+	0.795775i	$0.292936\pi$
$462$	0	0
$463$	− 568.468i	− 1.22779i	−0.789386	−	0.613897i	$-0.789601\pi$
$463$	− 568.468i	− 1.22779i	0.789386	−	0.613897i	$-0.210399\pi$
$464$	0	0
$465$	36.5461	0.0785939
$466$	0	0
$467$	165.113i	0.353561i	0.984250	+	0.176780i	$0.0565682\pi$
$467$	165.113i	0.353561i	−0.984250	+	0.176780i	$0.943432\pi$
$468$	0	0
$469$	18.5574	0.0395679
$470$	0	0
$471$	− 577.814i	− 1.22678i
$472$	0	0
$473$	−366.143	−0.774088
$474$	0	0
$475$	− 589.255i	− 1.24054i
$476$	0	0
$477$	260.619	0.546371
$478$	0	0
$479$	− 400.122i	− 0.835329i	−0.908601	−	0.417664i	$-0.862849\pi$
$479$	− 400.122i	− 0.835329i	0.908601	−	0.417664i	$-0.137151\pi$
$480$	0	0
$481$	−303.456	−0.630887
$482$	0	0
$483$	− 39.3396i	− 0.0814484i
$484$	0	0
$485$	−275.234	−0.567493
$486$	0	0
$487$	22.9865i	0.0472002i	0.999721	+	0.0236001i	$0.00751284\pi$
$487$	22.9865i	0.0472002i	−0.999721	+	0.0236001i	$0.992487\pi$
$488$	0	0
$489$	246.111	0.503295
$490$	0	0
$491$	381.074i	0.776117i	0.921635	+	0.388059i	$0.126854\pi$
$491$	381.074i	0.776117i	−0.921635	+	0.388059i	$0.873146\pi$
$492$	0	0
$493$	154.907	0.314213
$494$	0	0
$495$	90.5229i	0.182875i
$496$	0	0
$497$	−85.9324	−0.172902
$498$	0	0
$499$	− 915.621i	− 1.83491i	−0.397837	−	0.917456i	$-0.630239\pi$
$499$	− 915.621i	− 1.83491i	0.397837	−	0.917456i	$-0.369761\pi$
$500$	0	0
$501$	50.9524	0.101701
$502$	0	0
$503$	456.416i	0.907387i	0.891158	+	0.453694i	$0.149894\pi$
$503$	456.416i	0.907387i	−0.891158	+	0.453694i	$0.850106\pi$
$504$	0	0
$505$	−59.5492	−0.117919
$506$	0	0
$507$	− 99.0430i	− 0.195351i
$508$	0	0
$509$	229.681	0.451240	0.225620	−	0.974215i	$-0.427559\pi$
$509$	229.681	0.451240	0.225620	+	0.974215i	$0.427559\pi$
$510$	0	0
$511$	93.6404i	0.183249i
$512$	0	0
$513$	−796.915	−1.55344
$514$	0	0
$515$	111.764i	0.217017i
$516$	0	0
$517$	1043.32	2.01802
$518$	0	0
$519$	− 478.951i	− 0.922835i
$520$	0	0
$521$	−831.060	−1.59512	−0.797562	−	0.603237i	$-0.793877\pi$
$521$	−831.060	−1.59512	−0.797562	+	0.603237i	$0.793877\pi$
$522$	0	0
$523$	− 679.074i	− 1.29842i	−0.760609	−	0.649211i	$-0.775099\pi$
$523$	− 679.074i	− 1.29842i	0.760609	−	0.649211i	$-0.224901\pi$
$524$	0	0
$525$	35.1836	0.0670164
$526$	0	0
$527$	245.587i	0.466010i
$528$	0	0
$529$	−58.6766	−0.110920
$530$	0	0
$531$	− 256.413i	− 0.482887i
$532$	0	0
$533$	182.088	0.341628
$534$	0	0
$535$	60.0741i	0.112288i
$536$	0	0
$537$	158.231	0.294658
$538$	0	0
$539$	− 693.011i	− 1.28573i
$540$	0	0
$541$	−33.1593	−0.0612925	−0.0306463	−	0.999530i	$-0.509757\pi$
$541$	−33.1593	−0.0612925	−0.0306463	+	0.999530i	$0.509757\pi$
$542$	0	0
$543$	76.7565i	0.141356i
$544$	0	0
$545$	−11.4890	−0.0210808
$546$	0	0
$547$	299.078i	0.546761i	0.961906	+	0.273380i	$0.0881418\pi$
$547$	299.078i	0.546761i	−0.961906	+	0.273380i	$0.911858\pi$
$548$	0	0
$549$	−313.326	−0.570722
$550$	0	0
$551$	146.360i	0.265626i
$552$	0	0
$553$	56.0266	0.101314
$554$	0	0
$555$	115.335i	0.207812i
$556$	0	0
$557$	389.777	0.699778	0.349889	−	0.936791i	$-0.386219\pi$
$557$	389.777	0.699778	0.349889	+	0.936791i	$0.386219\pi$
$558$	0	0
$559$	− 288.736i	− 0.516522i
$560$	0	0
$561$	965.317	1.72071
$562$	0	0
$563$	− 781.149i	− 1.38748i	−0.720227	−	0.693738i	$-0.755963\pi$
$563$	− 781.149i	− 1.38748i	0.720227	−	0.693738i	$-0.244037\pi$
$564$	0	0
$565$	217.021	0.384108
$566$	0	0
$567$	− 25.9574i	− 0.0457803i
$568$	0	0
$569$	974.448	1.71256	0.856282	−	0.516509i	$-0.172769\pi$
$569$	974.448	1.71256	0.856282	+	0.516509i	$0.172769\pi$
$570$	0	0
$571$	− 248.880i	− 0.435866i	−0.975964	−	0.217933i	$-0.930069\pi$
$571$	− 248.880i	− 0.435866i	0.975964	−	0.217933i	$-0.0699315\pi$
$572$	0	0
$573$	−589.088	−1.02808
$574$	0	0
$575$	− 525.592i	− 0.914073i
$576$	0	0
$577$	−1123.31	−1.94682	−0.973409	−	0.229075i	$-0.926430\pi$
$577$	−1123.31	−1.94682	−0.973409	+	0.229075i	$0.926430\pi$
$578$	0	0
$579$	− 123.358i	− 0.213053i
$580$	0	0
$581$	−56.9788	−0.0980702
$582$	0	0
$583$	− 1069.88i	− 1.83512i
$584$	0	0
$585$	−71.3852	−0.122026
$586$	0	0
$587$	430.729i	0.733781i	0.930264	+	0.366890i	$0.119578\pi$
$587$	430.729i	0.733781i	−0.930264	+	0.366890i	$0.880422\pi$
$588$	0	0
$589$	−232.037	−0.393950
$590$	0	0
$591$	− 401.580i	− 0.679493i
$592$	0	0
$593$	−29.7378	−0.0501481	−0.0250741	−	0.999686i	$-0.507982\pi$
$593$	−29.7378	−0.0501481	−0.0250741	+	0.999686i	$0.507982\pi$
$594$	0	0
$595$	− 36.1934i	− 0.0608292i
$596$	0	0
$597$	−136.320	−0.228342
$598$	0	0
$599$	39.8356i	0.0665035i	0.999447	+	0.0332518i	$0.0105863\pi$
$599$	39.8356i	0.0665035i	−0.999447	+	0.0332518i	$0.989414\pi$
$600$	0	0
$601$	51.8730	0.0863111	0.0431555	−	0.999068i	$-0.486259\pi$
$601$	51.8730	0.0863111	0.0431555	+	0.999068i	$0.486259\pi$
$602$	0	0
$603$	− 93.4814i	− 0.155027i
$604$	0	0
$605$	151.171	0.249869
$606$	0	0
$607$	223.429i	0.368087i	0.982918	+	0.184044i	$0.0589188\pi$
$607$	223.429i	0.368087i	−0.982918	+	0.184044i	$0.941081\pi$
$608$	0	0
$609$	−8.73896	−0.0143497
$610$	0	0
$611$	822.746i	1.34656i
$612$	0	0
$613$	−315.690	−0.514992	−0.257496	−	0.966279i	$-0.582897\pi$
$613$	−315.690	−0.514992	−0.257496	+	0.966279i	$0.582897\pi$
$614$	0	0
$615$	− 69.2066i	− 0.112531i
$616$	0	0
$617$	−696.107	−1.12821	−0.564106	−	0.825702i	$-0.690779\pi$
$617$	−696.107	−1.12821	−0.564106	+	0.825702i	$0.690779\pi$
$618$	0	0
$619$	922.143i	1.48973i	0.667215	+	0.744865i	$0.267486\pi$
$619$	922.143i	1.48973i	−0.667215	+	0.744865i	$0.732514\pi$
$620$	0	0
$621$	−710.817	−1.14463
$622$	0	0
$623$	38.3259i	0.0615183i
$624$	0	0
$625$	387.092	0.619347
$626$	0	0
$627$	912.056i	1.45463i
$628$	0	0
$629$	−775.044	−1.23218
$630$	0	0
$631$	− 935.862i	− 1.48314i	−0.670875	−	0.741570i	$-0.734081\pi$
$631$	− 935.862i	− 1.48314i	0.670875	−	0.741570i	$-0.265919\pi$
$632$	0	0
$633$	−527.437	−0.833234
$634$	0	0
$635$	243.052i	0.382758i
$636$	0	0
$637$	546.499	0.857927
$638$	0	0
$639$	432.879i	0.677432i
$640$	0	0
$641$	−359.262	−0.560472	−0.280236	−	0.959931i	$-0.590413\pi$
$641$	−359.262	−0.560472	−0.280236	+	0.959931i	$0.590413\pi$
$642$	0	0
$643$	− 487.138i	− 0.757602i	−0.925478	−	0.378801i	$-0.876336\pi$
$643$	− 487.138i	− 0.757602i	0.925478	−	0.378801i	$-0.123664\pi$
$644$	0	0
$645$	−109.741	−0.170140
$646$	0	0
$647$	− 511.959i	− 0.791282i	−0.918405	−	0.395641i	$-0.870522\pi$
$647$	− 511.959i	− 0.791282i	0.918405	−	0.395641i	$-0.129478\pi$
$648$	0	0
$649$	−1052.61	−1.62190
$650$	0	0
$651$	− 13.8546i	− 0.0212820i
$652$	0	0
$653$	1077.84	1.65060	0.825299	−	0.564696i	$-0.191007\pi$
$653$	1077.84	1.65060	0.825299	+	0.564696i	$0.191007\pi$
$654$	0	0
$655$	220.787i	0.337080i
$656$	0	0
$657$	471.707	0.717972
$658$	0	0
$659$	894.373i	1.35717i	0.734523	+	0.678584i	$0.237406\pi$
$659$	894.373i	1.35717i	−0.734523	+	0.678584i	$0.762594\pi$
$660$	0	0
$661$	−933.964	−1.41296	−0.706478	−	0.707735i	$-0.749717\pi$
$661$	−933.964	−1.41296	−0.706478	+	0.707735i	$0.749717\pi$
$662$	0	0
$663$	761.237i	1.14817i
$664$	0	0
$665$	34.1964	0.0514232
$666$	0	0
$667$	130.547i	0.195723i
$668$	0	0
$669$	485.372	0.725518
$670$	0	0
$671$	1286.25i	1.91691i
$672$	0	0
$673$	−629.790	−0.935795	−0.467898	−	0.883783i	$-0.654988\pi$
$673$	−629.790	−0.935795	−0.467898	+	0.883783i	$0.654988\pi$
$674$	0	0
$675$	− 635.723i	− 0.941812i
$676$	0	0
$677$	685.354	1.01234	0.506170	−	0.862434i	$-0.331061\pi$
$677$	685.354	1.01234	0.506170	+	0.862434i	$0.331061\pi$
$678$	0	0
$679$	104.341i	0.153669i
$680$	0	0
$681$	209.043	0.306965
$682$	0	0
$683$	− 258.863i	− 0.379009i	−0.981880	−	0.189504i	$-0.939312\pi$
$683$	− 258.863i	− 0.379009i	0.981880	−	0.189504i	$-0.0606881\pi$
$684$	0	0
$685$	58.8013	0.0858413
$686$	0	0
$687$	− 735.035i	− 1.06992i
$688$	0	0
$689$	843.691	1.22452
$690$	0	0
$691$	− 214.788i	− 0.310837i	−0.987849	−	0.155419i	$-0.950327\pi$
$691$	− 214.788i	− 0.310837i	0.987849	−	0.155419i	$-0.0496726\pi$
$692$	0	0
$693$	34.3171	0.0495197
$694$	0	0
$695$	− 142.214i	− 0.204624i
$696$	0	0
$697$	465.063	0.667235
$698$	0	0
$699$	37.6183i	0.0538173i
$700$	0	0
$701$	376.058	0.536460	0.268230	−	0.963355i	$-0.413561\pi$
$701$	376.058	0.536460	0.268230	+	0.963355i	$0.413561\pi$
$702$	0	0
$703$	− 732.281i	− 1.04165i
$704$	0	0
$705$	312.703	0.443551
$706$	0	0
$707$	22.5750i	0.0319307i
$708$	0	0
$709$	−39.7841	−0.0561130	−0.0280565	−	0.999606i	$-0.508932\pi$
$709$	−39.7841	−0.0561130	−0.0280565	+	0.999606i	$0.508932\pi$
$710$	0	0
$711$	− 282.230i	− 0.396948i
$712$	0	0
$713$	−206.968	−0.290277
$714$	0	0
$715$	293.046i	0.409855i
$716$	0	0
$717$	186.781	0.260504
$718$	0	0
$719$	− 503.430i	− 0.700180i	−0.936716	−	0.350090i	$-0.886151\pi$
$719$	− 503.430i	− 0.700180i	0.936716	−	0.350090i	$-0.113849\pi$
$720$	0	0
$721$	42.3695	0.0587650
$722$	0	0
$723$	− 730.583i	− 1.01049i
$724$	0	0
$725$	−116.756	−0.161043
$726$	0	0
$727$	249.531i	0.343234i	0.985164	+	0.171617i	$0.0548992\pi$
$727$	249.531i	0.343234i	−0.985164	+	0.171617i	$0.945101\pi$
$728$	0	0
$729$	−750.823	−1.02994
$730$	0	0
$731$	− 737.447i	− 1.00882i
$732$	0	0
$733$	141.209	0.192645	0.0963223	−	0.995350i	$-0.469292\pi$
$733$	141.209	0.192645	0.0963223	+	0.995350i	$0.469292\pi$
$734$	0	0
$735$	− 207.709i	− 0.282598i
$736$	0	0
$737$	−383.755	−0.520698
$738$	0	0
$739$	1418.57i	1.91959i	0.280709	+	0.959793i	$0.409430\pi$
$739$	1418.57i	1.91959i	−0.280709	+	0.959793i	$0.590570\pi$
$740$	0	0
$741$	−719.235	−0.970628
$742$	0	0
$743$	886.796i	1.19353i	0.802415	+	0.596767i	$0.203548\pi$
$743$	886.796i	1.19353i	−0.802415	+	0.596767i	$0.796452\pi$
$744$	0	0
$745$	332.645	0.446503
$746$	0	0
$747$	287.027i	0.384239i
$748$	0	0
$749$	22.7740	0.0304059
$750$	0	0
$751$	1471.29i	1.95911i	0.201182	+	0.979554i	$0.435522\pi$
$751$	1471.29i	1.95911i	−0.201182	+	0.979554i	$0.564478\pi$
$752$	0	0
$753$	−224.493	−0.298132
$754$	0	0
$755$	356.002i	0.471526i
$756$	0	0
$757$	−211.707	−0.279666	−0.139833	−	0.990175i	$-0.544657\pi$
$757$	−211.707	−0.279666	−0.139833	+	0.990175i	$0.544657\pi$
$758$	0	0
$759$	813.518i	1.07183i
$760$	0	0
$761$	−508.153	−0.667743	−0.333872	−	0.942619i	$-0.608355\pi$
$761$	−508.153	−0.667743	−0.333872	+	0.942619i	$0.608355\pi$
$762$	0	0
$763$	4.35548i	0.00570836i
$764$	0	0
$765$	−182.322	−0.238329
$766$	0	0
$767$	− 830.076i	− 1.08224i
$768$	0	0
$769$	−278.591	−0.362277	−0.181139	−	0.983458i	$-0.557978\pi$
$769$	−278.591	−0.362277	−0.181139	+	0.983458i	$0.557978\pi$
$770$	0	0
$771$	1059.62i	1.37435i
$772$	0	0
$773$	552.597	0.714874	0.357437	−	0.933937i	$-0.383651\pi$
$773$	552.597	0.714874	0.357437	+	0.933937i	$0.383651\pi$
$774$	0	0
$775$	− 185.103i	− 0.238842i
$776$	0	0
$777$	43.7235	0.0562722
$778$	0	0
$779$	439.403i	0.564060i
$780$	0	0
$781$	1777.03	2.27533
$782$	0	0
$783$	157.902i	0.201663i
$784$	0	0
$785$	448.007	0.570710
$786$	0	0
$787$	− 1140.18i	− 1.44877i	−0.689395	−	0.724386i	$-0.742123\pi$
$787$	− 1140.18i	− 1.44877i	0.689395	−	0.724386i	$-0.257877\pi$
$788$	0	0
$789$	363.788	0.461075
$790$	0	0
$791$	− 82.2723i	− 0.104011i
$792$	0	0
$793$	−1014.32	−1.27909
$794$	0	0
$795$	− 320.664i	− 0.403351i
$796$	0	0
$797$	778.570	0.976875	0.488438	−	0.872599i	$-0.337567\pi$
$797$	778.570	0.976875	0.488438	+	0.872599i	$0.337567\pi$
$798$	0	0
$799$	2101.34i	2.62996i
$800$	0	0
$801$	193.064	0.241029
$802$	0	0
$803$	− 1936.43i	− 2.41149i
$804$	0	0
$805$	30.5019	0.0378905
$806$	0	0
$807$	809.957i	1.00366i
$808$	0	0
$809$	−199.801	−0.246973	−0.123486	−	0.992346i	$-0.539408\pi$
$809$	−199.801	−0.246973	−0.123486	+	0.992346i	$0.539408\pi$
$810$	0	0
$811$	1021.54i	1.25961i	0.776754	+	0.629804i	$0.216865\pi$
$811$	1021.54i	1.25961i	−0.776754	+	0.629804i	$0.783135\pi$
$812$	0	0
$813$	754.457	0.927991
$814$	0	0
$815$	190.822i	0.234137i
$816$	0	0
$817$	696.759	0.852826
$818$	0	0
$819$	27.0620i	0.0330428i
$820$	0	0
$821$	148.801	0.181244	0.0906219	−	0.995885i	$-0.471115\pi$
$821$	148.801	0.181244	0.0906219	+	0.995885i	$0.471115\pi$
$822$	0	0
$823$	− 791.060i	− 0.961190i	−0.876943	−	0.480595i	$-0.840421\pi$
$823$	− 791.060i	− 0.961190i	0.876943	−	0.480595i	$-0.159579\pi$
$824$	0	0
$825$	−727.575	−0.881909
$826$	0	0
$827$	925.059i	1.11857i	0.828975	+	0.559286i	$0.188925\pi$
$827$	925.059i	1.11857i	−0.828975	+	0.559286i	$0.811075\pi$
$828$	0	0
$829$	877.725	1.05878	0.529388	−	0.848380i	$-0.322422\pi$
$829$	877.725	1.05878	0.529388	+	0.848380i	$0.322422\pi$
$830$	0	0
$831$	734.657i	0.884064i
$832$	0	0
$833$	1395.79	1.67562
$834$	0	0
$835$	39.5058i	0.0473123i
$836$	0	0
$837$	−250.335	−0.299086
$838$	0	0
$839$	− 787.049i	− 0.938079i	−0.883177	−	0.469040i	$-0.844600\pi$
$839$	− 787.049i	− 0.938079i	0.883177	−	0.469040i	$-0.155400\pi$
$840$	0	0
$841$	29.0000	0.0344828
$842$	0	0
$843$	− 68.8040i	− 0.0816181i
$844$	0	0
$845$	76.7928	0.0908790
$846$	0	0
$847$	− 57.3087i	− 0.0676608i
$848$	0	0
$849$	645.526	0.760337
$850$	0	0
$851$	− 653.166i	− 0.767528i
$852$	0	0
$853$	−327.020	−0.383377	−0.191688	−	0.981456i	$-0.561396\pi$
$853$	−327.020	−0.383377	−0.191688	+	0.981456i	$0.561396\pi$
$854$	0	0
$855$	− 172.262i	− 0.201476i
$856$	0	0
$857$	212.896	0.248420	0.124210	−	0.992256i	$-0.460360\pi$
$857$	212.896	0.248420	0.124210	+	0.992256i	$0.460360\pi$
$858$	0	0
$859$	1527.79i	1.77857i	0.457358	+	0.889283i	$0.348796\pi$
$859$	1527.79i	1.77857i	−0.457358	+	0.889283i	$0.651204\pi$
$860$	0	0
$861$	−26.2361	−0.0304717
$862$	0	0
$863$	− 557.729i	− 0.646268i	−0.946353	−	0.323134i	$-0.895264\pi$
$863$	− 557.729i	− 0.646268i	0.946353	−	0.323134i	$-0.104736\pi$
$864$	0	0
$865$	371.354	0.429311
$866$	0	0
$867$	1265.19i	1.45927i
$868$	0	0
$869$	−1158.60	−1.33325
$870$	0	0
$871$	− 302.624i	− 0.347444i
$872$	0	0
$873$	525.610	0.602074
$874$	0	0
$875$	58.7351i	0.0671258i
$876$	0	0
$877$	135.773	0.154816	0.0774079	−	0.997000i	$-0.475336\pi$
$877$	135.773	0.154816	0.0774079	+	0.997000i	$0.475336\pi$
$878$	0	0
$879$	− 464.554i	− 0.528502i
$880$	0	0
$881$	560.838	0.636592	0.318296	−	0.947991i	$-0.396889\pi$
$881$	560.838	0.636592	0.318296	+	0.947991i	$0.396889\pi$
$882$	0	0
$883$	− 1230.79i	− 1.39388i	−0.717131	−	0.696938i	$-0.754545\pi$
$883$	− 1230.79i	− 1.39388i	0.717131	−	0.696938i	$-0.245455\pi$
$884$	0	0
$885$	−315.489	−0.356485
$886$	0	0
$887$	− 772.049i	− 0.870404i	−0.900333	−	0.435202i	$-0.856677\pi$
$887$	− 772.049i	− 0.870404i	0.900333	−	0.435202i	$-0.143323\pi$
$888$	0	0
$889$	92.1406	0.103645
$890$	0	0
$891$	536.783i	0.602450i
$892$	0	0
$893$	−1985.40	−2.22329
$894$	0	0
$895$	122.684i	0.137077i
$896$	0	0
$897$	−641.530	−0.715195
$898$	0	0
$899$	45.9761i	0.0511414i
$900$	0	0
$901$	2154.83	2.39160
$902$	0	0
$903$	41.6025i	0.0460714i
$904$	0	0
$905$	−59.5130	−0.0657602
$906$	0	0
$907$	1495.84i	1.64922i	0.565705	+	0.824608i	$0.308604\pi$
$907$	1495.84i	1.64922i	−0.565705	+	0.824608i	$0.691396\pi$
$908$	0	0
$909$	113.720	0.125105
$910$	0	0
$911$	− 199.158i	− 0.218615i	−0.994008	−	0.109307i	$-0.965137\pi$
$911$	− 199.158i	− 0.218615i	0.994008	−	0.109307i	$-0.0348633\pi$
$912$	0	0
$913$	1178.29	1.29056
$914$	0	0
$915$	385.515i	0.421328i
$916$	0	0
$917$	83.7002	0.0912761
$918$	0	0
$919$	− 791.506i	− 0.861268i	−0.902527	−	0.430634i	$-0.858290\pi$
$919$	− 791.506i	− 0.861268i	0.902527	−	0.430634i	$-0.141710\pi$
$920$	0	0
$921$	645.618	0.700996
$922$	0	0
$923$	1401.34i	1.51825i
$924$	0	0
$925$	584.163	0.631528
$926$	0	0
$927$	− 213.434i	− 0.230241i
$928$	0	0
$929$	12.2055	0.0131383	0.00656913	−	0.999978i	$-0.497909\pi$
$929$	12.2055	0.0131383	0.00656913	+	0.999978i	$0.497909\pi$
$930$	0	0
$931$	1318.78i	1.41652i
$932$	0	0
$933$	−1171.54	−1.25567
$934$	0	0
$935$	748.457i	0.800488i
$936$	0	0
$937$	89.2686	0.0952706	0.0476353	−	0.998865i	$-0.484831\pi$
$937$	89.2686	0.0952706	0.0476353	+	0.998865i	$0.484831\pi$
$938$	0	0
$939$	− 879.926i	− 0.937089i
$940$	0	0
$941$	−481.158	−0.511326	−0.255663	−	0.966766i	$-0.582294\pi$
$941$	−481.158	−0.511326	−0.255663	+	0.966766i	$0.582294\pi$
$942$	0	0
$943$	391.930i	0.415620i
$944$	0	0
$945$	36.8931	0.0390403
$946$	0	0
$947$	− 796.225i	− 0.840786i	−0.907342	−	0.420393i	$-0.861892\pi$
$947$	− 796.225i	− 0.840786i	0.907342	−	0.420393i	$-0.138108\pi$
$948$	0	0
$949$	1527.04	1.60910
$950$	0	0
$951$	− 565.195i	− 0.594316i
$952$	0	0
$953$	−95.6047	−0.100320	−0.0501598	−	0.998741i	$-0.515973\pi$
$953$	−95.6047	−0.100320	−0.0501598	+	0.998741i	$0.515973\pi$
$954$	0	0
$955$	− 456.748i	− 0.478270i
$956$	0	0
$957$	180.716	0.188836
$958$	0	0
$959$	− 22.2915i	− 0.0232445i
$960$	0	0
$961$	888.110	0.924152
$962$	0	0
$963$	− 114.723i	− 0.119130i
$964$	0	0
$965$	95.6451	0.0991141
$966$	0	0
$967$	640.927i	0.662800i	0.943490	+	0.331400i	$0.107521\pi$
$967$	640.927i	0.662800i	−0.943490	+	0.331400i	$0.892479\pi$
$968$	0	0
$969$	−1836.97	−1.89573
$970$	0	0
$971$	1512.11i	1.55727i	0.627476	+	0.778636i	$0.284088\pi$
$971$	1512.11i	1.55727i	−0.627476	+	0.778636i	$0.715912\pi$
$972$	0	0
$973$	−53.9131	−0.0554092
$974$	0	0
$975$	− 573.756i	− 0.588468i
$976$	0	0
$977$	−1540.22	−1.57648	−0.788240	−	0.615369i	$-0.789007\pi$
$977$	−1540.22	−1.57648	−0.788240	+	0.615369i	$0.789007\pi$
$978$	0	0
$979$	− 792.556i	− 0.809557i
$980$	0	0
$981$	21.9405	0.0223654
$982$	0	0
$983$	− 755.412i	− 0.768476i	−0.923234	−	0.384238i	$-0.874464\pi$
$983$	− 755.412i	− 0.768476i	0.923234	−	0.384238i	$-0.125536\pi$
$984$	0	0
$985$	311.364	0.316106
$986$	0	0
$987$	− 118.545i	− 0.120107i
$988$	0	0
$989$	621.481	0.628394
$990$	0	0
$991$	1510.68i	1.52440i	0.647344	+	0.762198i	$0.275880\pi$
$991$	1510.68i	1.52440i	−0.647344	+	0.762198i	$0.724120\pi$
$992$	0	0
$993$	354.477	0.356976
$994$	0	0
$995$	− 105.695i	− 0.106227i
$996$	0	0
$997$	−1362.77	−1.36687	−0.683436	−	0.730011i	$-0.739515\pi$
$997$	−1362.77	−1.36687	−0.683436	+	0.730011i	$0.739515\pi$
$998$	0	0
$999$	− 790.029i	− 0.790819i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	464.3.d.b.175.15	yes	20
4.3	odd	2	inner	464.3.d.b.175.6	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
464.3.d.b.175.6	✓	20	4.3	odd	2	inner
464.3.d.b.175.15	yes	20	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:	\( N \)	\(=\)	\( 464 = 2^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	464.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.34966i q^{3} -1.82181 q^{5} +0.690645i q^{7} +3.47908 q^{9} +O(q^{10})\)	\(q+2.34966i q^{3} -1.82181 q^{5} +0.690645i q^{7} +3.47908 q^{9} -14.2821i q^{11} +11.2627 q^{13} -4.28063i q^{15} +28.7655 q^{17} +27.1784i q^{19} -1.62278 q^{21} +24.2420i q^{23} -21.6810 q^{25} +29.3216i q^{27} +5.38516 q^{29} +8.53755i q^{31} +33.5582 q^{33} -1.25822i q^{35} -26.9435 q^{37} +26.4635i q^{39} +16.1674 q^{41} -25.6365i q^{43} -6.33821 q^{45} +73.0506i q^{47} +48.5230 q^{49} +67.5893i q^{51} +74.9103 q^{53} +26.0192i q^{55} -63.8601 q^{57} -73.7015i q^{59} -90.0602 q^{61} +2.40281i q^{63} -20.5184 q^{65} -26.8696i q^{67} -56.9607 q^{69} +124.423i q^{71} +135.584 q^{73} -50.9431i q^{75} +9.86386 q^{77} -81.1222i q^{79} -37.5843 q^{81} +82.5008i q^{83} -52.4052 q^{85} +12.6533i q^{87} +55.4929 q^{89} +7.77851i q^{91} -20.0604 q^{93} -49.5137i q^{95} +151.078 q^{97} -49.6885i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 8 q^{5} - 68 q^{9}+O(q^{10}) \) 20 * q - 8 * q^5 - 68 * q^9	\( 20 q - 8 q^{5} - 68 q^{9} + 16 q^{13} + 40 q^{17} - 48 q^{21} + 188 q^{25} - 120 q^{33} - 80 q^{37} - 72 q^{41} + 72 q^{45} - 28 q^{49} + 96 q^{53} + 104 q^{57} - 96 q^{61} - 80 q^{65} + 352 q^{69} - 312 q^{73} - 192 q^{77} + 60 q^{81} + 336 q^{85} - 152 q^{89} - 472 q^{93} + 56 q^{97}+O(q^{100}) \) 20 * q - 8 * q^5 - 68 * q^9 + 16 * q^13 + 40 * q^17 - 48 * q^21 + 188 * q^25 - 120 * q^33 - 80 * q^37 - 72 * q^41 + 72 * q^45 - 28 * q^49 + 96 * q^53 + 104 * q^57 - 96 * q^61 - 80 * q^65 + 352 * q^69 - 312 * q^73 - 192 * q^77 + 60 * q^81 + 336 * q^85 - 152 * q^89 - 472 * q^93 + 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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