LMFDB - Embedded newform 3100.3.d.f.1301.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3100,3,Mod(1301,3100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3100, 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("3100.1301");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3100 = 2^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3100.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:	$84.4688819517$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1301.4
Character	$\chi$	$=$	3100.1301
Dual form			3100.3.d.f.1301.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.71258i q^{3} +1.89499 q^{7} -4.78325 q^{9} +O(q^{10})$	$q-3.71258i q^{3} +1.89499 q^{7} -4.78325 q^{9} +3.40869i q^{11} -11.5742i q^{13} +23.9546i q^{17} +5.61015 q^{19} -7.03530i q^{21} -5.50461i q^{23} -15.6550i q^{27} -50.5543i q^{29} +(-30.5754 - 5.11298i) q^{31} +12.6550 q^{33} +58.2789i q^{37} -42.9701 q^{39} +35.1169 q^{41} +17.2118i q^{43} -69.6158 q^{47} -45.4090 q^{49} +88.9334 q^{51} -14.5516i q^{53} -20.8281i q^{57} +45.0393 q^{59} -89.1838i q^{61} -9.06420 q^{63} +129.892 q^{67} -20.4363 q^{69} -115.347 q^{71} -97.3909i q^{73} +6.45943i q^{77} -50.8005i q^{79} -101.170 q^{81} -59.9529i q^{83} -187.687 q^{87} -137.342i q^{89} -21.9329i q^{91} +(-18.9823 + 113.514i) q^{93} -139.105 q^{97} -16.3046i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 4 q^{7} - 72 q^{9}+O(q^{10}) $ 22 * q - 4 * q^7 - 72 * q^9	$ 22 q - 4 q^{7} - 72 q^{9} - 28 q^{19} - 18 q^{31} + 34 q^{33} - 62 q^{39} + 64 q^{41} + 96 q^{47} + 150 q^{49} - 130 q^{51} + 40 q^{59} + 4 q^{63} - 110 q^{67} + 100 q^{69} + 132 q^{71} + 234 q^{81} + 62 q^{87} - 16 q^{93} - 186 q^{97}+O(q^{100}) $ 22 * q - 4 * q^7 - 72 * q^9 - 28 * q^19 - 18 * q^31 + 34 * q^33 - 62 * q^39 + 64 * q^41 + 96 * q^47 + 150 * q^49 - 130 * q^51 + 40 * q^59 + 4 * q^63 - 110 * q^67 + 100 * q^69 + 132 * q^71 + 234 * q^81 + 62 * q^87 - 16 * q^93 - 186 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3100\mathbb{Z}\right)^\times$.

$n$	$1551$	$1801$	$2977$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		−	3.71258i		−	1.23753i	−0.785578	−	0.618763i	$-0.787634\pi$
$3$		−	3.71258i		−	1.23753i	0.785578	−	0.618763i	$-0.212366\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.89499			0.270713			0.135356	−	0.990797i	$-0.456782\pi$
$7$	1.89499			0.270713			0.135356	+	0.990797i	$0.456782\pi$
$8$		0			0
$9$	−4.78325			−0.531472
$10$		0			0
$11$			3.40869i			0.309881i	0.987924	+	0.154941i	$0.0495186\pi$
$11$			3.40869i			0.309881i	−0.987924	+	0.154941i	$0.950481\pi$
$12$		0			0
$13$		−	11.5742i		−	0.890322i	−0.895451	−	0.445161i	$-0.853146\pi$
$13$		−	11.5742i		−	0.890322i	0.895451	−	0.445161i	$-0.146854\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			23.9546i			1.40910i	0.709657	+	0.704548i	$0.248850\pi$
$17$			23.9546i			1.40910i	−0.709657	+	0.704548i	$0.751150\pi$
$18$		0			0
$19$	5.61015			0.295271			0.147636	−	0.989042i	$-0.452834\pi$
$19$	5.61015			0.295271			0.147636	+	0.989042i	$0.452834\pi$
$20$		0			0
$21$		−	7.03530i		−	0.335014i
$22$		0			0
$23$		−	5.50461i		−	0.239331i	−0.992814	−	0.119665i	$-0.961818\pi$
$23$		−	5.50461i		−	0.239331i	0.992814	−	0.119665i	$-0.0381822\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	15.6550i		−	0.579816i
$28$		0			0
$29$		−	50.5543i		−	1.74325i	−0.490171	−	0.871626i	$-0.663066\pi$
$29$		−	50.5543i		−	1.74325i	0.490171	−	0.871626i	$-0.336934\pi$
$30$		0			0
$31$	−30.5754	−	5.11298i	−0.986304	−	0.164935i
$31$	−30.5754	−	5.11298i	−0.986304	−	0.164935i
$32$		0			0
$33$	12.6550			0.383486
$34$		0			0
$35$		0			0
$36$		0			0
$37$			58.2789i			1.57510i	0.616248	+	0.787552i	$0.288652\pi$
$37$			58.2789i			1.57510i	−0.616248	+	0.787552i	$0.711348\pi$
$38$		0			0
$39$	−42.9701			−1.10180
$40$		0			0
$41$	35.1169			0.856509			0.428254	−	0.903658i	$-0.359129\pi$
$41$	35.1169			0.856509			0.428254	+	0.903658i	$0.359129\pi$
$42$		0			0
$43$			17.2118i			0.400275i	0.979768	+	0.200138i	$0.0641389\pi$
$43$			17.2118i			0.400275i	−0.979768	+	0.200138i	$0.935861\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−69.6158			−1.48119			−0.740594	−	0.671953i	$-0.765456\pi$
$47$	−69.6158			−1.48119			−0.740594	+	0.671953i	$0.765456\pi$
$48$		0			0
$49$	−45.4090			−0.926715
$50$		0			0
$51$	88.9334			1.74379
$52$		0			0
$53$		−	14.5516i		−	0.274559i	−0.990532	−	0.137280i	$-0.956164\pi$
$53$		−	14.5516i		−	0.274559i	0.990532	−	0.137280i	$-0.0438359\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	20.8281i		−	0.365406i
$58$		0			0
$59$	45.0393			0.763378			0.381689	−	0.924291i	$-0.375342\pi$
$59$	45.0393			0.763378			0.381689	+	0.924291i	$0.375342\pi$
$60$		0			0
$61$		−	89.1838i		−	1.46203i	−0.682362	−	0.731014i	$-0.739047\pi$
$61$		−	89.1838i		−	1.46203i	0.682362	−	0.731014i	$-0.260953\pi$
$62$		0			0
$63$	−9.06420			−0.143876
$64$		0			0
$65$		0			0
$66$		0			0
$67$	129.892			1.93869			0.969345	−	0.245705i	$-0.0790193\pi$
$67$	129.892			1.93869			0.969345	+	0.245705i	$0.0790193\pi$
$68$		0			0
$69$	−20.4363			−0.296178
$70$		0			0
$71$	−115.347			−1.62460			−0.812300	−	0.583240i	$-0.801785\pi$
$71$	−115.347			−1.62460			−0.812300	+	0.583240i	$0.801785\pi$
$72$		0			0
$73$		−	97.3909i		−	1.33412i	−0.745003	−	0.667061i	$-0.767552\pi$
$73$		−	97.3909i		−	1.33412i	0.745003	−	0.667061i	$-0.232448\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			6.45943i			0.0838888i
$78$		0			0
$79$		−	50.8005i		−	0.643044i	−0.946902	−	0.321522i	$-0.895806\pi$
$79$		−	50.8005i		−	0.643044i	0.946902	−	0.321522i	$-0.104194\pi$
$80$		0			0
$81$	−101.170			−1.24901
$82$		0			0
$83$		−	59.9529i		−	0.722324i	−0.932503	−	0.361162i	$-0.882380\pi$
$83$		−	59.9529i		−	0.722324i	0.932503	−	0.361162i	$-0.117620\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−187.687			−2.15732
$88$		0			0
$89$		−	137.342i		−	1.54317i	−0.636127	−	0.771585i	$-0.719464\pi$
$89$		−	137.342i		−	1.54317i	0.636127	−	0.771585i	$-0.280536\pi$
$90$		0			0
$91$		−	21.9329i		−	0.241021i
$92$		0			0
$93$	−18.9823	+	113.514i	−0.204111	+	1.22058i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−139.105			−1.43408			−0.717038	−	0.697034i	$-0.754503\pi$
$97$	−139.105			−1.43408			−0.717038	+	0.697034i	$0.754503\pi$
$98$		0			0
$99$		−	16.3046i		−	0.164693i
$100$		0			0
$101$	122.873			1.21656			0.608282	−	0.793721i	$-0.291859\pi$
$101$	122.873			1.21656			0.608282	+	0.793721i	$0.291859\pi$
$102$		0			0
$103$	−78.4122			−0.761284			−0.380642	−	0.924723i	$-0.624297\pi$
$103$	−78.4122			−0.761284			−0.380642	+	0.924723i	$0.624297\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	39.2421			0.366749			0.183374	−	0.983043i	$-0.441298\pi$
$107$	39.2421			0.366749			0.183374	+	0.983043i	$0.441298\pi$
$108$		0			0
$109$	−123.349			−1.13164			−0.565821	−	0.824528i	$-0.691441\pi$
$109$	−123.349			−1.13164			−0.565821	+	0.824528i	$0.691441\pi$
$110$		0			0
$111$	216.365			1.94923
$112$		0			0
$113$	−191.900			−1.69823			−0.849117	−	0.528205i	$-0.822865\pi$
$113$	−191.900			−1.69823			−0.849117	+	0.528205i	$0.822865\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			55.3622i			0.473181i
$118$		0			0
$119$			45.3937i			0.381460i
$120$		0			0
$121$	109.381			0.903974
$122$		0			0
$123$		−	130.374i		−	1.05995i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			202.585i			1.59516i	0.603215	+	0.797578i	$0.293886\pi$
$127$			202.585i			1.59516i	−0.603215	+	0.797578i	$0.706114\pi$
$128$		0			0
$129$	63.9003			0.495351
$130$		0			0
$131$	−101.047			−0.771352			−0.385676	−	0.922634i	$-0.626032\pi$
$131$	−101.047			−0.771352			−0.385676	+	0.922634i	$0.626032\pi$
$132$		0			0
$133$	10.6312			0.0799336
$134$		0			0
$135$		0			0
$136$		0			0
$137$			56.5585i			0.412836i	0.978464	+	0.206418i	$0.0661806\pi$
$137$			56.5585i			0.412836i	−0.978464	+	0.206418i	$0.933819\pi$
$138$		0			0
$139$		−	193.443i		−	1.39168i	−0.718199	−	0.695838i	$-0.755033\pi$
$139$		−	193.443i		−	1.39168i	0.718199	−	0.695838i	$-0.244967\pi$
$140$		0			0
$141$			258.454i			1.83301i
$142$		0			0
$143$	39.4528			0.275894
$144$		0			0
$145$		0			0
$146$		0			0
$147$			168.585i			1.14683i
$148$		0			0
$149$	−218.954			−1.46949			−0.734744	−	0.678344i	$-0.762698\pi$
$149$	−218.954			−1.46949			−0.734744	+	0.678344i	$0.762698\pi$
$150$		0			0
$151$		−	100.926i		−	0.668381i	−0.942506	−	0.334191i	$-0.891537\pi$
$151$		−	100.926i		−	0.668381i	0.942506	−	0.334191i	$-0.108463\pi$
$152$		0			0
$153$		−	114.581i		−	0.748895i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−240.496			−1.53182			−0.765911	−	0.642947i	$-0.777712\pi$
$157$	−240.496			−1.53182			−0.765911	+	0.642947i	$0.777712\pi$
$158$		0			0
$159$	−54.0241			−0.339774
$160$		0			0
$161$		−	10.4312i		−	0.0647899i
$162$		0			0
$163$	105.969			0.650116			0.325058	−	0.945694i	$-0.394616\pi$
$163$	105.969			0.650116			0.325058	+	0.945694i	$0.394616\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	14.5818i		−	0.0873161i	−0.999047	−	0.0436581i	$-0.986099\pi$
$167$		−	14.5818i		−	0.0873161i	0.999047	−	0.0436581i	$-0.0139012\pi$
$168$		0			0
$169$	35.0383			0.207327
$170$		0			0
$171$	−26.8347			−0.156928
$172$		0			0
$173$	21.2450			0.122803			0.0614017	−	0.998113i	$-0.480443\pi$
$173$	21.2450			0.122803			0.0614017	+	0.998113i	$0.480443\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	167.212i		−	0.944701i
$178$		0			0
$179$			176.383i			0.985380i	0.870205	+	0.492690i	$0.163986\pi$
$179$			176.383i			0.985380i	−0.870205	+	0.492690i	$0.836014\pi$
$180$		0			0
$181$		−	188.312i		−	1.04040i	−0.854046	−	0.520198i	$-0.825858\pi$
$181$		−	188.312i		−	1.04040i	0.854046	−	0.520198i	$-0.174142\pi$
$182$		0			0
$183$	−331.102			−1.80930
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−81.6539			−0.436652
$188$		0			0
$189$		−	29.6661i		−	0.156964i
$190$		0			0
$191$	129.437			0.677678			0.338839	−	0.940844i	$-0.389966\pi$
$191$	129.437			0.677678			0.338839	+	0.940844i	$0.389966\pi$
$192$		0			0
$193$	−69.1622			−0.358353			−0.179177	−	0.983817i	$-0.557343\pi$
$193$	−69.1622			−0.358353			−0.179177	+	0.983817i	$0.557343\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	100.047i		−	0.507854i	−0.967223	−	0.253927i	$-0.918278\pi$
$197$		−	100.047i		−	0.507854i	0.967223	−	0.253927i	$-0.0817224\pi$
$198$		0			0
$199$		−	90.7335i		−	0.455947i	−0.973667	−	0.227974i	$-0.926790\pi$
$199$		−	90.7335i		−	0.455947i	0.973667	−	0.227974i	$-0.0732100\pi$
$200$		0			0
$201$		−	482.235i		−	2.39918i
$202$		0			0
$203$		−	95.7999i		−	0.471921i
$204$		0			0
$205$		0			0
$206$		0			0
$207$			26.3299i			0.127198i
$208$		0			0
$209$			19.1233i			0.0914989i
$210$		0			0
$211$	173.015			0.819976			0.409988	−	0.912091i	$-0.365533\pi$
$211$	173.015			0.819976			0.409988	+	0.912091i	$0.365533\pi$
$212$		0			0
$213$			428.233i			2.01049i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−57.9401	−	9.68903i	−0.267005	−	0.0446499i
$218$		0			0
$219$	−361.571			−1.65101
$220$		0			0
$221$	277.255			1.25455
$222$		0			0
$223$		−	121.642i		−	0.545481i	−0.962088	−	0.272741i	$-0.912070\pi$
$223$		−	121.642i		−	0.545481i	0.962088	−	0.272741i	$-0.0879301\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−317.136			−1.39708			−0.698538	−	0.715573i	$-0.746166\pi$
$227$	−317.136			−1.39708			−0.698538	+	0.715573i	$0.746166\pi$
$228$		0			0
$229$			15.2026i			0.0663868i	0.999449	+	0.0331934i	$0.0105677\pi$
$229$			15.2026i			0.0663868i	−0.999449	+	0.0331934i	$0.989432\pi$
$230$		0			0
$231$	23.9812			0.103815
$232$		0			0
$233$	358.323			1.53787			0.768934	−	0.639329i	$-0.220788\pi$
$233$	358.323			1.53787			0.768934	+	0.639329i	$0.220788\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−188.601			−0.795784
$238$		0			0
$239$			197.899i			0.828030i	0.910270	+	0.414015i	$0.135874\pi$
$239$			197.899i			0.828030i	−0.910270	+	0.414015i	$0.864126\pi$
$240$		0			0
$241$		−	272.538i		−	1.13086i	−0.824795	−	0.565431i	$-0.808710\pi$
$241$		−	272.538i		−	1.13086i	0.824795	−	0.565431i	$-0.191290\pi$
$242$		0			0
$243$			234.706i			0.965867i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	64.9329i		−	0.262886i
$248$		0			0
$249$	−222.580			−0.893895
$250$		0			0
$251$			365.989i			1.45813i	0.684447	+	0.729063i	$0.260044\pi$
$251$			365.989i			1.45813i	−0.684447	+	0.729063i	$0.739956\pi$
$252$		0			0
$253$	18.7635			0.0741641
$254$		0			0
$255$		0			0
$256$		0			0
$257$	37.3786			0.145442			0.0727210	−	0.997352i	$-0.476832\pi$
$257$	37.3786			0.145442			0.0727210	+	0.997352i	$0.476832\pi$
$258$		0			0
$259$			110.438i			0.426401i
$260$		0			0
$261$			241.814i			0.926490i
$262$		0			0
$263$			353.167i			1.34284i	0.741076	+	0.671421i	$0.234316\pi$
$263$			353.167i			1.34284i	−0.741076	+	0.671421i	$0.765684\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−509.893			−1.90971
$268$		0			0
$269$			445.514i			1.65619i	0.560590	+	0.828093i	$0.310574\pi$
$269$			445.514i			1.65619i	−0.560590	+	0.828093i	$0.689426\pi$
$270$		0			0
$271$			322.564i			1.19027i	0.803625	+	0.595136i	$0.202902\pi$
$271$			322.564i			1.19027i	−0.803625	+	0.595136i	$0.797098\pi$
$272$		0			0
$273$	−81.4278			−0.298270
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	265.856i		−	0.959770i	−0.877332	−	0.479885i	$-0.840678\pi$
$277$		−	265.856i		−	0.959770i	0.877332	−	0.479885i	$-0.159322\pi$
$278$		0			0
$279$	146.250	+	24.4566i	0.524193	+	0.0876582i
$280$		0			0
$281$	233.766			0.831906			0.415953	−	0.909386i	$-0.363448\pi$
$281$	233.766			0.831906			0.415953	+	0.909386i	$0.363448\pi$
$282$		0			0
$283$	−484.480			−1.71194			−0.855971	−	0.517024i	$-0.827040\pi$
$283$	−484.480			−1.71194			−0.855971	+	0.517024i	$0.827040\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	66.5461			0.231868
$288$		0			0
$289$	−284.824			−0.985549
$290$		0			0
$291$			516.440i			1.77471i
$292$		0			0
$293$	228.425			0.779608			0.389804	−	0.920898i	$-0.372543\pi$
$293$	228.425			0.779608			0.389804	+	0.920898i	$0.372543\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	53.3632			0.179674
$298$		0			0
$299$	−63.7113			−0.213081
$300$		0			0
$301$			32.6162i			0.108360i
$302$		0			0
$303$		−	456.176i		−	1.50553i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−436.892			−1.42310			−0.711550	−	0.702636i	$-0.752006\pi$
$307$	−436.892			−1.42310			−0.711550	+	0.702636i	$0.752006\pi$
$308$		0			0
$309$			291.112i			0.942109i
$310$		0			0
$311$	249.738			0.803017			0.401509	−	0.915855i	$-0.368486\pi$
$311$	249.738			0.803017			0.401509	+	0.915855i	$0.368486\pi$
$312$		0			0
$313$			404.244i			1.29151i	0.763543	+	0.645757i	$0.223458\pi$
$313$			404.244i			1.29151i	−0.763543	+	0.645757i	$0.776542\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−166.480			−0.525172			−0.262586	−	0.964909i	$-0.584575\pi$
$317$	−166.480			−0.525172			−0.262586	+	0.964909i	$0.584575\pi$
$318$		0			0
$319$	172.324			0.540201
$320$		0			0
$321$		−	145.689i		−	0.453861i
$322$		0			0
$323$			134.389i			0.416065i
$324$		0			0
$325$		0			0
$326$		0			0
$327$			457.943i			1.40044i
$328$		0			0
$329$	−131.921			−0.400976
$330$		0			0
$331$			80.8625i			0.244298i	0.992512	+	0.122149i	$0.0389785\pi$
$331$			80.8625i			0.244298i	−0.992512	+	0.122149i	$0.961021\pi$
$332$		0			0
$333$		−	278.762i		−	0.837124i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	427.937i		−	1.26984i	−0.772576	−	0.634922i	$-0.781032\pi$
$337$		−	427.937i		−	1.26984i	0.772576	−	0.634922i	$-0.218968\pi$
$338$		0			0
$339$			712.446i			2.10161i
$340$		0			0
$341$	17.4286	−	104.222i	0.0511102	−	0.305637i
$342$		0			0
$343$	−178.904			−0.521586
$344$		0			0
$345$		0			0
$346$		0			0
$347$			98.0564i			0.282583i	0.989968	+	0.141292i	$0.0451255\pi$
$347$			98.0564i			0.282583i	−0.989968	+	0.141292i	$0.954874\pi$
$348$		0			0
$349$	−147.109			−0.421516			−0.210758	−	0.977538i	$-0.567593\pi$
$349$	−147.109			−0.421516			−0.210758	+	0.977538i	$0.567593\pi$
$350$		0			0
$351$	−181.194			−0.516223
$352$		0			0
$353$			579.359i			1.64124i	0.571471	+	0.820622i	$0.306373\pi$
$353$			579.359i			1.64124i	−0.571471	+	0.820622i	$0.693627\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	168.528			0.472067
$358$		0			0
$359$	195.353			0.544158			0.272079	−	0.962275i	$-0.412289\pi$
$359$	195.353			0.544158			0.272079	+	0.962275i	$0.412289\pi$
$360$		0			0
$361$	−329.526			−0.912815
$362$		0			0
$363$		−	406.085i		−	1.11869i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	554.974i		−	1.51219i	−0.654461	−	0.756096i	$-0.727104\pi$
$367$		−	554.974i		−	1.51219i	0.654461	−	0.756096i	$-0.272896\pi$
$368$		0			0
$369$	−167.973			−0.455210
$370$		0			0
$371$		−	27.5752i		−	0.0743267i
$372$		0			0
$373$	−296.717			−0.795488			−0.397744	−	0.917496i	$-0.630207\pi$
$373$	−296.717			−0.795488			−0.397744	+	0.917496i	$0.630207\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−585.125			−1.55206
$378$		0			0
$379$	120.224			0.317214			0.158607	−	0.987342i	$-0.449300\pi$
$379$	120.224			0.317214			0.158607	+	0.987342i	$0.449300\pi$
$380$		0			0
$381$	752.113			1.97405
$382$		0			0
$383$		−	279.975i		−	0.731006i	−0.930810	−	0.365503i	$-0.880897\pi$
$383$		−	279.975i		−	0.731006i	0.930810	−	0.365503i	$-0.119103\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	82.3285i		−	0.212735i
$388$		0			0
$389$		−	12.7032i		−	0.0326559i	−0.999867	−	0.0163280i	$-0.994802\pi$
$389$		−	12.7032i		−	0.0326559i	0.999867	−	0.0163280i	$-0.00519758\pi$
$390$		0			0
$391$	131.861			0.337240
$392$		0			0
$393$			375.145i			0.954568i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	161.546			0.406918			0.203459	−	0.979083i	$-0.434782\pi$
$397$	161.546			0.406918			0.203459	+	0.979083i	$0.434782\pi$
$398$		0			0
$399$		−	39.4691i		−	0.0989200i
$400$		0			0
$401$		−	425.417i		−	1.06089i	−0.847719	−	0.530446i	$-0.822025\pi$
$401$		−	425.417i		−	1.06089i	0.847719	−	0.530446i	$-0.177975\pi$
$402$		0			0
$403$	−59.1785	+	353.886i	−0.146845	+	0.878128i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−198.655			−0.488095
$408$		0			0
$409$			410.270i			1.00311i	0.865127	+	0.501553i	$0.167238\pi$
$409$			410.270i			1.00311i	−0.865127	+	0.501553i	$0.832762\pi$
$410$		0			0
$411$	209.978			0.510895
$412$		0			0
$413$	85.3490			0.206656
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−718.173			−1.72224
$418$		0			0
$419$	10.6185			0.0253426			0.0126713	−	0.999920i	$-0.495966\pi$
$419$	10.6185			0.0253426			0.0126713	+	0.999920i	$0.495966\pi$
$420$		0			0
$421$	−617.483			−1.46670			−0.733352	−	0.679849i	$-0.762045\pi$
$421$	−617.483			−1.46670			−0.733352	+	0.679849i	$0.762045\pi$
$422$		0			0
$423$	332.990			0.787210
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	169.002i		−	0.395790i
$428$		0			0
$429$		−	146.472i		−	0.341426i
$430$		0			0
$431$	−11.9121			−0.0276382			−0.0138191	−	0.999905i	$-0.504399\pi$
$431$	−11.9121			−0.0276382			−0.0138191	+	0.999905i	$0.504399\pi$
$432$		0			0
$433$		−	90.1959i		−	0.208305i	−0.994561	−	0.104152i	$-0.966787\pi$
$433$		−	90.1959i		−	0.208305i	0.994561	−	0.104152i	$-0.0332129\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	30.8817i		−	0.0706674i
$438$		0			0
$439$	−640.634			−1.45930			−0.729651	−	0.683819i	$-0.760318\pi$
$439$	−640.634			−1.45930			−0.729651	+	0.683819i	$0.760318\pi$
$440$		0			0
$441$	217.203			0.492523
$442$		0			0
$443$	−462.897			−1.04491			−0.522457	−	0.852666i	$-0.674984\pi$
$443$	−462.897			−1.04491			−0.522457	+	0.852666i	$0.674984\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			812.883i			1.81853i
$448$		0			0
$449$		−	830.540i		−	1.84976i	−0.380265	−	0.924878i	$-0.624167\pi$
$449$		−	830.540i		−	1.84976i	0.380265	−	0.924878i	$-0.375833\pi$
$450$		0			0
$451$			119.703i			0.265416i
$452$		0			0
$453$	−374.694			−0.827139
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	307.310i		−	0.672450i	−0.941782	−	0.336225i	$-0.890850\pi$
$457$		−	307.310i		−	0.672450i	0.941782	−	0.336225i	$-0.109150\pi$
$458$		0			0
$459$	375.010			0.817016
$460$		0			0
$461$		−	512.468i		−	1.11165i	−0.831301	−	0.555823i	$-0.812403\pi$
$461$		−	512.468i		−	1.11165i	0.831301	−	0.555823i	$-0.187597\pi$
$462$		0			0
$463$			413.732i			0.893589i	0.894637	+	0.446795i	$0.147435\pi$
$463$			413.732i			0.893589i	−0.894637	+	0.446795i	$0.852565\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	242.115			0.518447			0.259224	−	0.965817i	$-0.416533\pi$
$467$	242.115			0.518447			0.259224	+	0.965817i	$0.416533\pi$
$468$		0			0
$469$	246.144			0.524828
$470$		0			0
$471$			892.861i			1.89567i
$472$		0			0
$473$	−58.6699			−0.124038
$474$		0			0
$475$		0			0
$476$		0			0
$477$			69.6041i			0.145921i
$478$		0			0
$479$	−799.525			−1.66915			−0.834577	−	0.550891i	$-0.814288\pi$
$479$	−799.525			−1.66915			−0.834577	+	0.550891i	$0.814288\pi$
$480$		0			0
$481$	674.530			1.40235
$482$		0			0
$483$	−38.7265			−0.0801792
$484$		0			0
$485$		0			0
$486$		0			0
$487$			921.729i			1.89267i	0.323191	+	0.946334i	$0.395244\pi$
$487$			921.729i			1.89267i	−0.323191	+	0.946334i	$0.604756\pi$
$488$		0			0
$489$		−	393.418i		−	0.804536i
$490$		0			0
$491$		−	416.555i		−	0.848381i	−0.905573	−	0.424191i	$-0.860559\pi$
$491$		−	416.555i		−	0.848381i	0.905573	−	0.424191i	$-0.139441\pi$
$492$		0			0
$493$	1211.01			2.45641
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−218.580			−0.439800
$498$		0			0
$499$			601.620i			1.20565i	0.797873	+	0.602826i	$0.205959\pi$
$499$			601.620i			1.20565i	−0.797873	+	0.602826i	$0.794041\pi$
$500$		0			0
$501$	−54.1361			−0.108056
$502$		0			0
$503$	−566.179			−1.12560			−0.562802	−	0.826592i	$-0.690277\pi$
$503$	−566.179			−1.12560			−0.562802	+	0.826592i	$0.690277\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	130.082i		−	0.256573i
$508$		0			0
$509$			72.8890i			0.143200i	0.997433	+	0.0716002i	$0.0228106\pi$
$509$			72.8890i			0.143200i	−0.997433	+	0.0716002i	$0.977189\pi$
$510$		0			0
$511$		−	184.555i		−	0.361164i
$512$		0			0
$513$		−	87.8271i		−	0.171203i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	237.299i		−	0.458992i
$518$		0			0
$519$		−	78.8737i		−	0.151972i
$520$		0			0
$521$	−487.256			−0.935232			−0.467616	−	0.883932i	$-0.654887\pi$
$521$	−487.256			−0.935232			−0.467616	+	0.883932i	$0.654887\pi$
$522$		0			0
$523$			275.461i			0.526694i	0.964701	+	0.263347i	$0.0848265\pi$
$523$			275.461i			0.526694i	−0.964701	+	0.263347i	$0.915174\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	122.479	−	732.423i	0.232409	−	1.38980i
$528$		0			0
$529$	498.699			0.942721
$530$		0			0
$531$	−215.434			−0.405714
$532$		0			0
$533$		−	406.449i		−	0.762568i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	654.836			1.21943
$538$		0			0
$539$		−	154.785i		−	0.287171i
$540$		0			0
$541$	477.793			0.883166			0.441583	−	0.897220i	$-0.354417\pi$
$541$	477.793			0.883166			0.441583	+	0.897220i	$0.354417\pi$
$542$		0			0
$543$	−699.122			−1.28752
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−28.1488			−0.0514603			−0.0257301	−	0.999669i	$-0.508191\pi$
$547$	−28.1488			−0.0514603			−0.0257301	+	0.999669i	$0.508191\pi$
$548$		0			0
$549$			426.588i			0.777027i
$550$		0			0
$551$		−	283.617i		−	0.514732i
$552$		0			0
$553$		−	96.2663i		−	0.174080i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			212.771i			0.381995i	0.981590	+	0.190998i	$0.0611723\pi$
$557$			212.771i			0.381995i	−0.981590	+	0.190998i	$0.938828\pi$
$558$		0			0
$559$	199.213			0.356374
$560$		0			0
$561$			303.147i			0.540368i
$562$		0			0
$563$	218.412			0.387943			0.193972	−	0.981007i	$-0.437863\pi$
$563$	218.412			0.387943			0.193972	+	0.981007i	$0.437863\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−191.716			−0.338123
$568$		0			0
$569$		−	779.749i		−	1.37038i	−0.728362	−	0.685192i	$-0.759718\pi$
$569$		−	779.749i		−	1.37038i	0.728362	−	0.685192i	$-0.240282\pi$
$570$		0			0
$571$		−	239.080i		−	0.418704i	−0.977840	−	0.209352i	$-0.932864\pi$
$571$		−	239.080i		−	0.418704i	0.977840	−	0.209352i	$-0.0671355\pi$
$572$		0			0
$573$		−	480.544i		−	0.838645i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	813.231			1.40941			0.704706	−	0.709499i	$-0.251079\pi$
$577$	813.231			1.40941			0.704706	+	0.709499i	$0.251079\pi$
$578$		0			0
$579$			256.770i			0.443472i
$580$		0			0
$581$		−	113.610i		−	0.195542i
$582$		0			0
$583$	49.6021			0.0850807
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	329.899i		−	0.562008i	−0.959707	−	0.281004i	$-0.909333\pi$
$587$		−	329.899i		−	0.562008i	0.959707	−	0.281004i	$-0.0906674\pi$
$588$		0			0
$589$	−171.533	−	28.6846i	−0.291227	−	0.0487004i
$590$		0			0
$591$	−371.434			−0.628483
$592$		0			0
$593$	540.585			0.911610			0.455805	−	0.890080i	$-0.349351\pi$
$593$	540.585			0.911610			0.455805	+	0.890080i	$0.349351\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−336.855			−0.564247
$598$		0			0
$599$	1191.88			1.98978			0.994888	−	0.100987i	$-0.0322000\pi$
$599$	1191.88			1.98978			0.994888	+	0.100987i	$0.0322000\pi$
$600$		0			0
$601$			241.037i			0.401061i	0.979687	+	0.200530i	$0.0642665\pi$
$601$			241.037i			0.401061i	−0.979687	+	0.200530i	$0.935733\pi$
$602$		0			0
$603$	−621.307			−1.03036
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−1000.58			−1.64840			−0.824200	−	0.566298i	$-0.808375\pi$
$607$	−1000.58			−1.64840			−0.824200	+	0.566298i	$0.808375\pi$
$608$		0			0
$609$	−355.665			−0.584014
$610$		0			0
$611$			805.746i			1.31873i
$612$		0			0
$613$		−	70.5466i		−	0.115084i	−0.998343	−	0.0575421i	$-0.981674\pi$
$613$		−	70.5466i		−	0.115084i	0.998343	−	0.0575421i	$-0.0183263\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1196.37			1.93901			0.969505	−	0.245071i	$-0.0788114\pi$
$617$	1196.37			1.93901			0.969505	+	0.245071i	$0.0788114\pi$
$618$		0			0
$619$			126.041i			0.203620i	0.994804	+	0.101810i	$0.0324633\pi$
$619$			126.041i			0.203620i	−0.994804	+	0.101810i	$0.967537\pi$
$620$		0			0
$621$	−86.1748			−0.138768
$622$		0			0
$623$		−	260.262i		−	0.417755i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	70.9967			0.113232
$628$		0			0
$629$	−1396.05			−2.21947
$630$		0			0
$631$		−	790.291i		−	1.25244i	−0.779645	−	0.626221i	$-0.784601\pi$
$631$		−	790.291i		−	1.25244i	0.779645	−	0.626221i	$-0.215399\pi$
$632$		0			0
$633$		−	642.331i		−	1.01474i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			525.572i			0.825074i
$638$		0			0
$639$	551.731			0.863429
$640$		0			0
$641$		−	857.148i		−	1.33720i	−0.743620	−	0.668602i	$-0.766893\pi$
$641$		−	857.148i		−	1.33720i	0.743620	−	0.668602i	$-0.233107\pi$
$642$		0			0
$643$			570.044i			0.886539i	0.896388	+	0.443269i	$0.146181\pi$
$643$			570.044i			0.886539i	−0.896388	+	0.443269i	$0.853819\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			955.524i			1.47685i	0.674334	+	0.738427i	$0.264431\pi$
$647$			955.524i			1.47685i	−0.674334	+	0.738427i	$0.735569\pi$
$648$		0			0
$649$			153.525i			0.236557i
$650$		0			0
$651$	−35.9713	+	215.107i	−0.0552554	+	0.330426i
$652$		0			0
$653$	1059.17			1.62201			0.811003	−	0.585042i	$-0.198922\pi$
$653$	1059.17			1.62201			0.811003	+	0.585042i	$0.198922\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			465.845i			0.709048i
$658$		0			0
$659$	319.186			0.484349			0.242174	−	0.970233i	$-0.422139\pi$
$659$	319.186			0.484349			0.242174	+	0.970233i	$0.422139\pi$
$660$		0			0
$661$	428.054			0.647585			0.323793	−	0.946128i	$-0.395042\pi$
$661$	428.054			0.647585			0.323793	+	0.946128i	$0.395042\pi$
$662$		0			0
$663$		−	1029.33i		−	1.55254i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−278.282			−0.417214
$668$		0			0
$669$	−451.607			−0.675047
$670$		0			0
$671$	304.000			0.453055
$672$		0			0
$673$		−	198.634i		−	0.295147i	−0.989051	−	0.147574i	$-0.952854\pi$
$673$		−	198.634i		−	0.295147i	0.989051	−	0.147574i	$-0.0471463\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	485.301i		−	0.716841i	−0.933560	−	0.358420i	$-0.883315\pi$
$677$		−	485.301i		−	0.716841i	0.933560	−	0.358420i	$-0.116685\pi$
$678$		0			0
$679$	−263.603			−0.388223
$680$		0			0
$681$			1177.39i			1.72892i
$682$		0			0
$683$	1067.43			1.56286			0.781430	−	0.623993i	$-0.214491\pi$
$683$	1067.43			1.56286			0.781430	+	0.623993i	$0.214491\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	56.4408			0.0821555
$688$		0			0
$689$	−168.423			−0.244446
$690$		0			0
$691$	−678.114			−0.981351			−0.490676	−	0.871342i	$-0.663250\pi$
$691$	−678.114			−0.981351			−0.490676	+	0.871342i	$0.663250\pi$
$692$		0			0
$693$		−	30.8971i		−	0.0445845i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			841.211i			1.20690i
$698$		0			0
$699$		−	1330.30i		−	1.90315i
$700$		0			0
$701$	628.214			0.896169			0.448084	−	0.893991i	$-0.352106\pi$
$701$	628.214			0.896169			0.448084	+	0.893991i	$0.352106\pi$
$702$		0			0
$703$			326.953i			0.465083i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	232.843			0.329339
$708$		0			0
$709$		−	721.643i		−	1.01783i	−0.860816	−	0.508916i	$-0.830046\pi$
$709$		−	721.643i		−	1.01783i	0.860816	−	0.508916i	$-0.169954\pi$
$710$		0			0
$711$			242.991i			0.341760i
$712$		0			0
$713$	−28.1449	+	168.306i	−0.0394739	+	0.236053i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	734.716			1.02471
$718$		0			0
$719$		−	1288.89i		−	1.79261i	−0.443436	−	0.896306i	$-0.646241\pi$
$719$		−	1288.89i		−	1.79261i	0.443436	−	0.896306i	$-0.353759\pi$
$720$		0			0
$721$	−148.590			−0.206089
$722$		0			0
$723$	−1011.82			−1.39947
$724$		0			0
$725$		0			0
$726$		0			0
$727$	855.982			1.17742			0.588708	−	0.808345i	$-0.299637\pi$
$727$	855.982			1.17742			0.588708	+	0.808345i	$0.299637\pi$
$728$		0			0
$729$	−39.1648			−0.0537240
$730$		0			0
$731$	−412.303			−0.564026
$732$		0			0
$733$	−597.482			−0.815118			−0.407559	−	0.913179i	$-0.633620\pi$
$733$	−597.482			−0.815118			−0.407559	+	0.913179i	$0.633620\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			442.763i			0.600763i
$738$		0			0
$739$			48.7312i			0.0659421i	0.999456	+	0.0329711i	$0.0104969\pi$
$739$			48.7312i			0.0659421i	−0.999456	+	0.0329711i	$0.989503\pi$
$740$		0			0
$741$	−241.069			−0.325329
$742$		0			0
$743$		−	1007.44i		−	1.35590i	−0.735107	−	0.677952i	$-0.762868\pi$
$743$		−	1007.44i		−	1.35590i	0.735107	−	0.677952i	$-0.237132\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			286.770i			0.383895i
$748$		0			0
$749$	74.3633			0.0992835
$750$		0			0
$751$	−839.666			−1.11806			−0.559032	−	0.829146i	$-0.688827\pi$
$751$	−839.666			−1.11806			−0.559032	+	0.829146i	$0.688827\pi$
$752$		0			0
$753$	1358.76			1.80447
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	444.202i		−	0.586793i	−0.955991	−	0.293396i	$-0.905214\pi$
$757$		−	444.202i		−	0.586793i	0.955991	−	0.293396i	$-0.0947855\pi$
$758$		0			0
$759$		−	69.6610i		−	0.0917800i
$760$		0			0
$761$		−	1219.01i		−	1.60186i	−0.598758	−	0.800930i	$-0.704339\pi$
$761$		−	1219.01i		−	1.60186i	0.598758	−	0.800930i	$-0.295661\pi$
$762$		0			0
$763$	−233.745			−0.306350
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	521.293i		−	0.679652i
$768$		0			0
$769$	1094.38			1.42313			0.711563	−	0.702622i	$-0.247988\pi$
$769$	1094.38			1.42313			0.711563	+	0.702622i	$0.247988\pi$
$770$		0			0
$771$		−	138.771i		−	0.179988i
$772$		0			0
$773$			1135.47i			1.46891i	0.678659	+	0.734454i	$0.262562\pi$
$773$			1135.47i			1.46891i	−0.678659	+	0.734454i	$0.737438\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	410.009			0.527682
$778$		0			0
$779$	197.011			0.252902
$780$		0			0
$781$		−	393.181i		−	0.503433i
$782$		0			0
$783$	−791.430			−1.01077
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	659.487i		−	0.837975i	−0.907992	−	0.418988i	$-0.862385\pi$
$787$		−	659.487i		−	0.837975i	0.907992	−	0.418988i	$-0.137615\pi$
$788$		0			0
$789$	1311.16			1.66180
$790$		0			0
$791$	−363.649			−0.459733
$792$		0			0
$793$	−1032.23			−1.30168
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	613.414i		−	0.769654i	−0.922989	−	0.384827i	$-0.874261\pi$
$797$		−	613.414i		−	0.769654i	0.922989	−	0.384827i	$-0.125739\pi$
$798$		0			0
$799$		−	1667.62i		−	2.08713i
$800$		0			0
$801$			656.941i			0.820151i
$802$		0			0
$803$	331.976			0.413419
$804$		0			0
$805$		0			0
$806$		0			0
$807$	1654.01			2.04957
$808$		0			0
$809$		−	1054.11i		−	1.30298i	−0.758658	−	0.651489i	$-0.774145\pi$
$809$		−	1054.11i		−	1.30298i	0.758658	−	0.651489i	$-0.225855\pi$
$810$		0			0
$811$	−638.218			−0.786951			−0.393476	−	0.919335i	$-0.628727\pi$
$811$	−638.218			−0.786951			−0.393476	+	0.919335i	$0.628727\pi$
$812$		0			0
$813$	1197.54			1.47299
$814$		0			0
$815$		0			0
$816$		0			0
$817$			96.5610i			0.118190i
$818$		0			0
$819$			104.911i			0.128096i
$820$		0			0
$821$		−	1418.85i		−	1.72820i	−0.503320	−	0.864100i	$-0.667888\pi$
$821$		−	1418.85i		−	1.72820i	0.503320	−	0.864100i	$-0.332112\pi$
$822$		0			0
$823$		−	524.149i		−	0.636876i	−0.947944	−	0.318438i	$-0.896842\pi$
$823$		−	524.149i		−	0.636876i	0.947944	−	0.318438i	$-0.103158\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	207.485i		−	0.250888i	−0.992101	−	0.125444i	$-0.959964\pi$
$827$		−	207.485i		−	0.250888i	0.992101	−	0.125444i	$-0.0400356\pi$
$828$		0			0
$829$		−	432.539i		−	0.521760i	−0.965371	−	0.260880i	$-0.915987\pi$
$829$		−	432.539i		−	0.521760i	0.965371	−	0.260880i	$-0.0840127\pi$
$830$		0			0
$831$	−987.012			−1.18774
$832$		0			0
$833$		−	1087.76i		−	1.30583i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−80.0438	+	478.659i	−0.0956318	+	0.571875i
$838$		0			0
$839$	−605.791			−0.722040			−0.361020	−	0.932558i	$-0.617571\pi$
$839$	−605.791			−0.722040			−0.361020	+	0.932558i	$0.617571\pi$
$840$		0			0
$841$	−1714.74			−2.03893
$842$		0			0
$843$		−	867.873i		−	1.02951i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	207.275			0.244717
$848$		0			0
$849$			1798.67i			2.11857i
$850$		0			0
$851$	320.802			0.376971
$852$		0			0
$853$	294.886			0.345704			0.172852	−	0.984948i	$-0.444702\pi$
$853$	294.886			0.345704			0.172852	+	0.984948i	$0.444702\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	190.854			0.222700			0.111350	−	0.993781i	$-0.464483\pi$
$857$	190.854			0.222700			0.111350	+	0.993781i	$0.464483\pi$
$858$		0			0
$859$		−	659.154i		−	0.767351i	−0.923468	−	0.383675i	$-0.874658\pi$
$859$		−	659.154i		−	0.767351i	0.923468	−	0.383675i	$-0.125342\pi$
$860$		0			0
$861$		−	247.058i		−	0.286943i
$862$		0			0
$863$			1027.02i			1.19006i	0.803704	+	0.595029i	$0.202859\pi$
$863$			1027.02i			1.19006i	−0.803704	+	0.595029i	$0.797141\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			1057.43i			1.21964i
$868$		0			0
$869$	173.163			0.199267
$870$		0			0
$871$		−	1503.40i		−	1.72606i
$872$		0			0
$873$	665.376			0.762172
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1133.59			−1.29258			−0.646288	−	0.763093i	$-0.723680\pi$
$877$	−1133.59			−1.29258			−0.646288	+	0.763093i	$0.723680\pi$
$878$		0			0
$879$		−	848.047i		−	0.964786i
$880$		0			0
$881$			1122.92i			1.27460i	0.770617	+	0.637299i	$0.219948\pi$
$881$			1122.92i			1.27460i	−0.770617	+	0.637299i	$0.780052\pi$
$882$		0			0
$883$		−	764.741i		−	0.866071i	−0.901377	−	0.433036i	$-0.857442\pi$
$883$		−	764.741i		−	0.866071i	0.901377	−	0.433036i	$-0.142558\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−105.527			−0.118971			−0.0594856	−	0.998229i	$-0.518946\pi$
$887$	−105.527			−0.118971			−0.0594856	+	0.998229i	$0.518946\pi$
$888$		0			0
$889$			383.896i			0.431829i
$890$		0			0
$891$		−	344.857i		−	0.387045i
$892$		0			0
$893$	−390.555			−0.437352
$894$		0			0
$895$		0			0
$896$		0			0
$897$			236.533i			0.263694i
$898$		0			0
$899$	−258.483	+	1545.72i	−0.287523	+	1.71938i
$900$		0			0
$901$	348.579			0.386880
$902$		0			0
$903$	121.090			0.134098
$904$		0			0
$905$		0			0
$906$		0			0
$907$	968.281			1.06756			0.533782	−	0.845622i	$-0.320770\pi$
$907$	968.281			1.06756			0.533782	+	0.845622i	$0.320770\pi$
$908$		0			0
$909$	−587.732			−0.646570
$910$		0			0
$911$		−	321.781i		−	0.353218i	−0.984281	−	0.176609i	$-0.943487\pi$
$911$		−	321.781i		−	0.353218i	0.984281	−	0.176609i	$-0.0565128\pi$
$912$		0			0
$913$	204.361			0.223835
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−191.483			−0.208815
$918$		0			0
$919$	1599.83			1.74084			0.870421	−	0.492308i	$-0.163847\pi$
$919$	1599.83			1.74084			0.870421	+	0.492308i	$0.163847\pi$
$920$		0			0
$921$			1621.99i			1.76112i
$922$		0			0
$923$			1335.04i			1.44642i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	375.065			0.404601
$928$		0			0
$929$			269.851i			0.290474i	0.989397	+	0.145237i	$0.0463945\pi$
$929$			269.851i			0.290474i	−0.989397	+	0.145237i	$0.953606\pi$
$930$		0			0
$931$	−254.751			−0.273632
$932$		0			0
$933$		−	927.174i		−	0.993755i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	260.190			0.277684			0.138842	−	0.990315i	$-0.455662\pi$
$937$	260.190			0.277684			0.138842	+	0.990315i	$0.455662\pi$
$938$		0			0
$939$	1500.79			1.59828
$940$		0			0
$941$			501.960i			0.533433i	0.963775	+	0.266716i	$0.0859387\pi$
$941$			501.960i			0.533433i	−0.963775	+	0.266716i	$0.914061\pi$
$942$		0			0
$943$		−	193.304i		−	0.204989i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1611.04i		−	1.70120i	−0.525814	−	0.850599i	$-0.676239\pi$
$947$		−	1611.04i		−	1.70120i	0.525814	−	0.850599i	$-0.323761\pi$
$948$		0			0
$949$	−1127.22			−1.18780
$950$		0			0
$951$			618.069i			0.649915i
$952$		0			0
$953$		−	37.8714i		−	0.0397391i	−0.999803	−	0.0198696i	$-0.993675\pi$
$953$		−	37.8714i		−	0.0397391i	0.999803	−	0.0198696i	$-0.00632510\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	639.767i		−	0.668513i
$958$		0			0
$959$			107.178i			0.111760i
$960$		0			0
$961$	908.715	+	312.663i	0.945593	+	0.325352i
$962$		0			0
$963$	−187.705			−0.194917
$964$		0			0
$965$		0			0
$966$		0			0
$967$			1211.42i			1.25277i	0.779516	+	0.626383i	$0.215465\pi$
$967$			1211.42i			1.25277i	−0.779516	+	0.626383i	$0.784535\pi$
$968$		0			0
$969$	498.930			0.514892
$970$		0			0
$971$	−97.7075			−0.100626			−0.0503128	−	0.998734i	$-0.516022\pi$
$971$	−97.7075			−0.100626			−0.0503128	+	0.998734i	$0.516022\pi$
$972$		0			0
$973$		−	366.572i		−	0.376744i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1032.42			1.05672			0.528360	−	0.849020i	$-0.322807\pi$
$977$	1032.42			1.05672			0.528360	+	0.849020i	$0.322807\pi$
$978$		0			0
$979$	468.157			0.478199
$980$		0			0
$981$	590.009			0.601436
$982$		0			0
$983$			195.733i			0.199118i	0.995032	+	0.0995588i	$0.0317431\pi$
$983$			195.733i			0.199118i	−0.995032	+	0.0995588i	$0.968257\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$			489.768i			0.496219i
$988$		0			0
$989$	94.7444			0.0957982
$990$		0			0
$991$			958.065i			0.966766i	0.875409	+	0.483383i	$0.160592\pi$
$991$			958.065i			0.966766i	−0.875409	+	0.483383i	$0.839408\pi$
$992$		0			0
$993$	300.208			0.302325
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1764.19			1.76950			0.884750	−	0.466067i	$-0.154329\pi$
$997$	1764.19			1.76950			0.884750	+	0.466067i	$0.154329\pi$
$998$		0			0
$999$	912.357			0.913271

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3100.3.d.f.1301.4	✓	22
5.2	odd	4		3100.3.f.d.1549.7		44
5.3	odd	4		3100.3.f.d.1549.38		44
5.4	even	2		3100.3.d.g.1301.19	yes	22
31.30	odd	2	inner	3100.3.d.f.1301.19	yes	22
155.92	even	4		3100.3.f.d.1549.37		44
155.123	even	4		3100.3.f.d.1549.8		44
155.154	odd	2		3100.3.d.g.1301.4	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3100.3.d.f.1301.4	✓	22	1.1	even	1	trivial
3100.3.d.f.1301.19	yes	22	31.30	odd	2	inner
3100.3.d.g.1301.4	yes	22	155.154	odd	2
3100.3.d.g.1301.19	yes	22	5.4	even	2
3100.3.f.d.1549.7		44	5.2	odd	4
3100.3.f.d.1549.8		44	155.123	even	4
3100.3.f.d.1549.37		44	155.92	even	4
3100.3.f.d.1549.38		44	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3100 = 2^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3100.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.71258i q^{3} +1.89499 q^{7} -4.78325 q^{9} +O(q^{10})\)	\(q-3.71258i q^{3} +1.89499 q^{7} -4.78325 q^{9} +3.40869i q^{11} -11.5742i q^{13} +23.9546i q^{17} +5.61015 q^{19} -7.03530i q^{21} -5.50461i q^{23} -15.6550i q^{27} -50.5543i q^{29} +(-30.5754 - 5.11298i) q^{31} +12.6550 q^{33} +58.2789i q^{37} -42.9701 q^{39} +35.1169 q^{41} +17.2118i q^{43} -69.6158 q^{47} -45.4090 q^{49} +88.9334 q^{51} -14.5516i q^{53} -20.8281i q^{57} +45.0393 q^{59} -89.1838i q^{61} -9.06420 q^{63} +129.892 q^{67} -20.4363 q^{69} -115.347 q^{71} -97.3909i q^{73} +6.45943i q^{77} -50.8005i q^{79} -101.170 q^{81} -59.9529i q^{83} -187.687 q^{87} -137.342i q^{89} -21.9329i q^{91} +(-18.9823 + 113.514i) q^{93} -139.105 q^{97} -16.3046i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 4 q^{7} - 72 q^{9}+O(q^{10}) \) 22 * q - 4 * q^7 - 72 * q^9	\( 22 q - 4 q^{7} - 72 q^{9} - 28 q^{19} - 18 q^{31} + 34 q^{33} - 62 q^{39} + 64 q^{41} + 96 q^{47} + 150 q^{49} - 130 q^{51} + 40 q^{59} + 4 q^{63} - 110 q^{67} + 100 q^{69} + 132 q^{71} + 234 q^{81} + 62 q^{87} - 16 q^{93} - 186 q^{97}+O(q^{100}) \) 22 * q - 4 * q^7 - 72 * q^9 - 28 * q^19 - 18 * q^31 + 34 * q^33 - 62 * q^39 + 64 * q^41 + 96 * q^47 + 150 * q^49 - 130 * q^51 + 40 * q^59 + 4 * q^63 - 110 * q^67 + 100 * q^69 + 132 * q^71 + 234 * q^81 + 62 * q^87 - 16 * q^93 - 186 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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