LMFDB - Embedded newform 3150.2.b.f.251.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,2,Mod(251,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3150.251");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3150.b (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:	$25.1528766367$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.7442857984.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 26x^{6} + 205x^{4} + 540x^{2} + 324 $ x^8 + 26x^6 + 205x^4 + 540*x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			251.3
Root			$2.73923i$ of defining polynomial
Character	$\chi$	$=$	3150.251
Dual form			3150.2.b.f.251.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +(1.80230 + 1.93693i) q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +(1.80230 + 1.93693i) q^{7} +1.00000i q^{8} -3.87386i q^{11} -1.60461i q^{13} +(1.93693 - 1.80230i) q^{14} +1.00000 q^{16} -8.11650 q^{17} -2.63803i q^{19} -3.87386 q^{22} +5.47847i q^{23} -1.60461 q^{26} +(-1.80230 - 1.93693i) q^{28} -5.47847i q^{29} +3.73074i q^{31} -1.00000i q^{32} +8.11650i q^{34} -4.51190 q^{37} -2.63803 q^{38} -1.60461 q^{41} +10.1165 q^{43} +3.87386i q^{44} +5.47847 q^{46} -11.1097 q^{47} +(-0.503406 + 6.98188i) q^{49} +1.60461i q^{52} -2.26926i q^{53} +(-1.93693 + 1.80230i) q^{56} -5.47847 q^{58} +4.61142 q^{59} -11.8472i q^{61} +3.73074 q^{62} -1.00000 q^{64} -6.90729 q^{67} +8.11650 q^{68} -2.63803i q^{71} -13.7477i q^{73} +4.51190i q^{74} +2.63803i q^{76} +(7.50341 - 6.98188i) q^{77} -8.01698 q^{79} +1.60461i q^{82} -3.20921 q^{83} -10.1165i q^{86} +3.87386 q^{88} -17.8376 q^{89} +(3.10801 - 2.89199i) q^{91} -5.47847i q^{92} +11.1097i q^{94} +8.68768i q^{97} +(6.98188 + 0.503406i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} + 4 q^{7}+O(q^{10}) $ 8 * q - 8 * q^4 + 4 * q^7	$ 8 q - 8 q^{4} + 4 q^{7} + 8 q^{16} + 8 q^{26} - 4 q^{28} + 8 q^{37} - 8 q^{38} + 8 q^{41} + 16 q^{43} - 8 q^{46} - 40 q^{47} + 4 q^{49} + 8 q^{58} + 40 q^{62} - 8 q^{64} - 32 q^{67} + 52 q^{77} + 8 q^{79} + 16 q^{83} + 8 q^{89} - 4 q^{91} - 4 q^{98}+O(q^{100}) $ 8 * q - 8 * q^4 + 4 * q^7 + 8 * q^16 + 8 * q^26 - 4 * q^28 + 8 * q^37 - 8 * q^38 + 8 * q^41 + 16 * q^43 - 8 * q^46 - 40 * q^47 + 4 * q^49 + 8 * q^58 + 40 * q^62 - 8 * q^64 - 32 * q^67 + 52 * q^77 + 8 * q^79 + 16 * q^83 + 8 * q^89 - 4 * q^91 - 4 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.80230	+	1.93693i	0.681207	+	0.732091i
$7$	1.80230	+	1.93693i	0.681207	+	0.732091i
$8$			1.00000i			0.353553i
$9$		0			0
$10$		0			0
$11$		−	3.87386i		−	1.16801i	−0.811749	−	0.584007i	$-0.801484\pi$
$11$		−	3.87386i		−	1.16801i	0.811749	−	0.584007i	$-0.198516\pi$
$12$		0			0
$13$		−	1.60461i		−	0.445038i	−0.974928	−	0.222519i	$-0.928572\pi$
$13$		−	1.60461i		−	0.445038i	0.974928	−	0.222519i	$-0.0714279\pi$
$14$	1.93693	−	1.80230i	0.517667	−	0.481686i
$15$		0			0
$16$	1.00000			0.250000
$17$	−8.11650			−1.96854			−0.984271	−	0.176667i	$-0.943468\pi$
$17$	−8.11650			−1.96854			−0.984271	+	0.176667i	$0.943468\pi$
$18$		0			0
$19$		−	2.63803i		−	0.605207i	−0.953117	−	0.302603i	$-0.902144\pi$
$19$		−	2.63803i		−	0.605207i	0.953117	−	0.302603i	$-0.0978557\pi$
$20$		0			0
$21$		0			0
$22$	−3.87386			−0.825910
$23$			5.47847i			1.14234i	0.820832	+	0.571170i	$0.193510\pi$
$23$			5.47847i			1.14234i	−0.820832	+	0.571170i	$0.806490\pi$
$24$		0			0
$25$		0			0
$26$	−1.60461			−0.314689
$27$		0			0
$28$	−1.80230	−	1.93693i	−0.340603	−	0.366046i
$29$		−	5.47847i		−	1.01733i	−0.860966	−	0.508663i	$-0.830140\pi$
$29$		−	5.47847i		−	1.01733i	0.860966	−	0.508663i	$-0.169860\pi$
$30$		0			0
$31$			3.73074i			0.670061i	0.942207	+	0.335031i	$0.108747\pi$
$31$			3.73074i			0.670061i	−0.942207	+	0.335031i	$0.891253\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$			8.11650i			1.39197i
$35$		0			0
$36$		0			0
$37$	−4.51190			−0.741751			−0.370876	−	0.928683i	$-0.620942\pi$
$37$	−4.51190			−0.741751			−0.370876	+	0.928683i	$0.620942\pi$
$38$	−2.63803			−0.427946
$39$		0			0
$40$		0			0
$41$	−1.60461			−0.250597			−0.125299	−	0.992119i	$-0.539989\pi$
$41$	−1.60461			−0.250597			−0.125299	+	0.992119i	$0.539989\pi$
$42$		0			0
$43$	10.1165			1.54275			0.771376	−	0.636379i	$-0.219569\pi$
$43$	10.1165			1.54275			0.771376	+	0.636379i	$0.219569\pi$
$44$			3.87386i			0.584007i
$45$		0			0
$46$	5.47847			0.807756
$47$	−11.1097			−1.62052			−0.810258	−	0.586074i	$-0.800673\pi$
$47$	−11.1097			−1.62052			−0.810258	+	0.586074i	$0.800673\pi$
$48$		0			0
$49$	−0.503406	+	6.98188i	−0.0719152	+	0.997411i
$50$		0			0
$51$		0			0
$52$			1.60461i			0.222519i
$53$		−	2.26926i		−	0.311706i	−0.987780	−	0.155853i	$-0.950187\pi$
$53$		−	2.26926i		−	0.311706i	0.987780	−	0.155853i	$-0.0498127\pi$
$54$		0			0
$55$		0			0
$56$	−1.93693	+	1.80230i	−0.258833	+	0.240843i
$57$		0			0
$58$	−5.47847			−0.719358
$59$	4.61142			0.600356			0.300178	−	0.953883i	$-0.402954\pi$
$59$	4.61142			0.600356			0.300178	+	0.953883i	$0.402954\pi$
$60$		0			0
$61$		−	11.8472i		−	1.51688i	−0.651740	−	0.758442i	$-0.725961\pi$
$61$		−	11.8472i		−	1.51688i	0.651740	−	0.758442i	$-0.274039\pi$
$62$	3.73074			0.473805
$63$		0			0
$64$	−1.00000			−0.125000
$65$		0			0
$66$		0			0
$67$	−6.90729			−0.843860			−0.421930	−	0.906628i	$-0.638647\pi$
$67$	−6.90729			−0.843860			−0.421930	+	0.906628i	$0.638647\pi$
$68$	8.11650			0.984271
$69$		0			0
$70$		0			0
$71$		−	2.63803i		−	0.313077i	−0.987672	−	0.156539i	$-0.949966\pi$
$71$		−	2.63803i		−	0.313077i	0.987672	−	0.156539i	$-0.0500335\pi$
$72$		0			0
$73$		−	13.7477i		−	1.60905i	−0.593919	−	0.804525i	$-0.702420\pi$
$73$		−	13.7477i		−	1.60905i	0.593919	−	0.804525i	$-0.297580\pi$
$74$			4.51190i			0.524497i
$75$		0			0
$76$			2.63803i			0.302603i
$77$	7.50341	−	6.98188i	0.855092	−	0.795659i
$78$		0			0
$79$	−8.01698			−0.901981			−0.450990	−	0.892529i	$-0.648929\pi$
$79$	−8.01698			−0.901981			−0.450990	+	0.892529i	$0.648929\pi$
$80$		0			0
$81$		0			0
$82$			1.60461i			0.177199i
$83$	−3.20921			−0.352257			−0.176128	−	0.984367i	$-0.556357\pi$
$83$	−3.20921			−0.352257			−0.176128	+	0.984367i	$0.556357\pi$
$84$		0			0
$85$		0			0
$86$		−	10.1165i		−	1.09089i
$87$		0			0
$88$	3.87386			0.412955
$89$	−17.8376			−1.89078			−0.945392	−	0.325937i	$-0.894320\pi$
$89$	−17.8376			−1.89078			−0.945392	+	0.325937i	$0.894320\pi$
$90$		0			0
$91$	3.10801	−	2.89199i	0.325808	−	0.303163i
$92$		−	5.47847i		−	0.571170i
$93$		0			0
$94$			11.1097i			1.14588i
$95$		0			0
$96$		0			0
$97$			8.68768i			0.882100i	0.897482	+	0.441050i	$0.145394\pi$
$97$			8.68768i			0.882100i	−0.897482	+	0.441050i	$0.854606\pi$
$98$	6.98188	+	0.503406i	0.705276	+	0.0508517i
$99$		0			0
$100$		0			0
$101$	6.21603			0.618518			0.309259	−	0.950978i	$-0.399919\pi$
$101$	6.21603			0.618518			0.309259	+	0.950978i	$0.399919\pi$
$102$		0			0
$103$			12.8807i			1.26917i	0.772853	+	0.634585i	$0.218829\pi$
$103$			12.8807i			1.26917i	−0.772853	+	0.634585i	$0.781171\pi$
$104$	1.60461			0.157345
$105$		0			0
$106$	−2.26926			−0.220410
$107$		−	19.9638i		−	1.92997i	−0.262308	−	0.964984i	$-0.584484\pi$
$107$		−	19.9638i		−	1.92997i	0.262308	−	0.964984i	$-0.415516\pi$
$108$		0			0
$109$	−15.0238			−1.43902			−0.719509	−	0.694483i	$-0.755633\pi$
$109$	−15.0238			−1.43902			−0.719509	+	0.694483i	$0.755633\pi$
$110$		0			0
$111$		0			0
$112$	1.80230	+	1.93693i	0.170302	+	0.183023i
$113$		−	10.2330i		−	0.962640i	−0.876545	−	0.481320i	$-0.840157\pi$
$113$		−	10.2330i		−	0.962640i	0.876545	−	0.481320i	$-0.159843\pi$
$114$		0			0
$115$		0			0
$116$			5.47847i			0.508663i
$117$		0			0
$118$		−	4.61142i		−	0.424515i
$119$	−14.6284	−	15.7211i	−1.34098	−	1.44115i
$120$		0			0
$121$	−4.00681			−0.364256
$122$	−11.8472			−1.07260
$123$		0			0
$124$		−	3.73074i		−	0.335031i
$125$		0			0
$126$		0			0
$127$	2.81382			0.249686			0.124843	−	0.992177i	$-0.460157\pi$
$127$	2.81382			0.249686			0.124843	+	0.992177i	$0.460157\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$		0			0
$131$	−8.62840			−0.753867			−0.376933	−	0.926240i	$-0.623021\pi$
$131$	−8.62840			−0.753867			−0.376933	+	0.926240i	$0.623021\pi$
$132$		0			0
$133$	5.10969	−	4.75454i	0.443066	−	0.412271i
$134$			6.90729i			0.596699i
$135$		0			0
$136$		−	8.11650i		−	0.695984i
$137$		−	0.723932i		−	0.0618496i	−0.999522	−	0.0309248i	$-0.990155\pi$
$137$		−	0.723932i		−	0.0618496i	0.999522	−	0.0309248i	$-0.00984525\pi$
$138$		0			0
$139$			2.84043i			0.240923i	0.992718	+	0.120461i	$0.0384374\pi$
$139$			2.84043i			0.240923i	−0.992718	+	0.120461i	$0.961563\pi$
$140$		0			0
$141$		0			0
$142$	−2.63803			−0.221379
$143$	−6.21603			−0.519810
$144$		0			0
$145$		0			0
$146$	−13.7477			−1.13777
$147$		0			0
$148$	4.51190			0.370876
$149$		−	3.00681i		−	0.246328i	−0.992386	−	0.123164i	$-0.960696\pi$
$149$		−	3.00681i		−	0.246328i	0.992386	−	0.123164i	$-0.0393041\pi$
$150$		0			0
$151$	−15.2262			−1.23909			−0.619545	−	0.784961i	$-0.712683\pi$
$151$	−15.2262			−1.23909			−0.619545	+	0.784961i	$0.712683\pi$
$152$	2.63803			0.213973
$153$		0			0
$154$	−6.98188	−	7.50341i	−0.562616	−	0.604642i
$155$		0			0
$156$		0			0
$157$		−	2.64767i		−	0.211307i	−0.994403	−	0.105653i	$-0.966307\pi$
$157$		−	2.64767i		−	0.211307i	0.994403	−	0.105653i	$-0.0336934\pi$
$158$			8.01698i			0.637797i
$159$		0			0
$160$		0			0
$161$	−10.6114	+	9.87386i	−0.836297	+	0.778169i
$162$		0			0
$163$	9.32572			0.730446			0.365223	−	0.930920i	$-0.380993\pi$
$163$	9.32572			0.730446			0.365223	+	0.930920i	$0.380993\pi$
$164$	1.60461			0.125299
$165$		0			0
$166$			3.20921i			0.249083i
$167$	−17.3257			−1.34070			−0.670352	−	0.742043i	$-0.733857\pi$
$167$	−17.3257			−1.34070			−0.670352	+	0.742043i	$0.733857\pi$
$168$		0			0
$169$	10.4252			0.801941
$170$		0			0
$171$		0			0
$172$	−10.1165			−0.771376
$173$	−2.99319			−0.227568			−0.113784	−	0.993506i	$-0.536297\pi$
$173$	−2.99319			−0.227568			−0.113784	+	0.993506i	$0.536297\pi$
$174$		0			0
$175$		0			0
$176$		−	3.87386i		−	0.292003i
$177$		0			0
$178$			17.8376i			1.33699i
$179$		−	7.38858i		−	0.552248i	−0.961122	−	0.276124i	$-0.910950\pi$
$179$		−	7.38858i		−	0.552248i	0.961122	−	0.276124i	$-0.0890501\pi$
$180$		0			0
$181$			11.1097i			0.825777i	0.910782	+	0.412888i	$0.135480\pi$
$181$			11.1097i			0.825777i	−0.910782	+	0.412888i	$0.864520\pi$
$182$	−2.89199	−	3.10801i	−0.214368	−	0.230381i
$183$		0			0
$184$	−5.47847			−0.403878
$185$		0			0
$186$		0			0
$187$			31.4422i			2.29928i
$188$	11.1097			0.810258
$189$		0			0
$190$		0			0
$191$		−	9.36197i		−	0.677408i	−0.940893	−	0.338704i	$-0.890011\pi$
$191$		−	9.36197i		−	0.677408i	0.940893	−	0.338704i	$-0.109989\pi$
$192$		0			0
$193$	−17.4422			−1.25552			−0.627759	−	0.778408i	$-0.716028\pi$
$193$	−17.4422			−1.25552			−0.627759	+	0.778408i	$0.716028\pi$
$194$	8.68768			0.623739
$195$		0			0
$196$	0.503406	−	6.98188i	0.0359576	−	0.498705i
$197$			5.27607i			0.375904i	0.982178	+	0.187952i	$0.0601850\pi$
$197$			5.27607i			0.375904i	−0.982178	+	0.187952i	$0.939815\pi$
$198$		0			0
$199$			12.2160i			0.865971i	0.901401	+	0.432986i	$0.142540\pi$
$199$			12.2160i			0.865971i	−0.901401	+	0.432986i	$0.857460\pi$
$200$		0			0
$201$		0			0
$202$		−	6.21603i		−	0.437358i
$203$	10.6114	−	9.87386i	0.744776	−	0.693009i
$204$		0			0
$205$		0			0
$206$	12.8807			0.897439
$207$		0			0
$208$		−	1.60461i		−	0.111259i
$209$	−10.2194			−0.706889
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$			2.26926i			0.155853i
$213$		0			0
$214$	−19.9638			−1.36469
$215$		0			0
$216$		0			0
$217$	−7.22619	+	6.72393i	−0.490546	+	0.456450i
$218$			15.0238i			1.01754i
$219$		0			0
$220$		0			0
$221$			13.0238i			0.876075i
$222$		0			0
$223$		−	3.65784i		−	0.244947i	−0.992472	−	0.122473i	$-0.960917\pi$
$223$		−	3.65784i		−	0.244947i	0.992472	−	0.122473i	$-0.0390826\pi$
$224$	1.93693	−	1.80230i	0.129417	−	0.120421i
$225$		0			0
$226$	−10.2330			−0.680689
$227$	−4.23301			−0.280955			−0.140477	−	0.990084i	$-0.544864\pi$
$227$	−4.23301			−0.280955			−0.140477	+	0.990084i	$0.544864\pi$
$228$		0			0
$229$			7.59497i			0.501890i	0.968001	+	0.250945i	$0.0807413\pi$
$229$			7.59497i			0.501890i	−0.968001	+	0.250945i	$0.919259\pi$
$230$		0			0
$231$		0			0
$232$	5.47847			0.359679
$233$			9.94677i			0.651634i	0.945433	+	0.325817i	$0.105639\pi$
$233$			9.94677i			0.651634i	−0.945433	+	0.325817i	$0.894361\pi$
$234$		0			0
$235$		0			0
$236$	−4.61142			−0.300178
$237$		0			0
$238$	−15.7211	+	14.6284i	−1.01905	+	0.948218i
$239$			9.36197i			0.605575i	0.953058	+	0.302788i	$0.0979173\pi$
$239$			9.36197i			0.605575i	−0.953058	+	0.302788i	$0.902083\pi$
$240$		0			0
$241$			23.0408i			1.48419i	0.670296	+	0.742093i	$0.266167\pi$
$241$			23.0408i			1.48419i	−0.670296	+	0.742093i	$0.733833\pi$
$242$			4.00681i			0.257568i
$243$		0			0
$244$			11.8472i			0.758442i
$245$		0			0
$246$		0			0
$247$	−4.23301			−0.269340
$248$	−3.73074			−0.236902
$249$		0			0
$250$		0			0
$251$	23.8376			1.50462			0.752308	−	0.658811i	$-0.228940\pi$
$251$	23.8376			1.50462			0.752308	+	0.658811i	$0.228940\pi$
$252$		0			0
$253$	21.2228			1.33427
$254$		−	2.81382i		−	0.176555i
$255$		0			0
$256$	1.00000			0.0625000
$257$	29.7441			1.85539			0.927694	−	0.373341i	$-0.121788\pi$
$257$	29.7441			1.85539			0.927694	+	0.373341i	$0.121788\pi$
$258$		0			0
$259$	−8.13181	−	8.73923i	−0.505286	−	0.543030i
$260$		0			0
$261$		0			0
$262$			8.62840i			0.533064i
$263$		−	22.7545i		−	1.40310i	−0.712618	−	0.701552i	$-0.752491\pi$
$263$		−	22.7545i		−	1.40310i	0.712618	−	0.701552i	$-0.247509\pi$
$264$		0			0
$265$		0			0
$266$	−4.75454	−	5.10969i	−0.291519	−	0.313295i
$267$		0			0
$268$	6.90729			0.421930
$269$	−0.202401			−0.0123406			−0.00617030	−	0.999981i	$-0.501964\pi$
$269$	−0.202401			−0.0123406			−0.00617030	+	0.999981i	$0.501964\pi$
$270$		0			0
$271$			11.4785i			0.697267i	0.937259	+	0.348634i	$0.113354\pi$
$271$			11.4785i			0.697267i	−0.937259	+	0.348634i	$0.886646\pi$
$272$	−8.11650			−0.492135
$273$		0			0
$274$	−0.723932			−0.0437343
$275$		0			0
$276$		0			0
$277$	−23.7211			−1.42526			−0.712632	−	0.701538i	$-0.752497\pi$
$277$	−23.7211			−1.42526			−0.712632	+	0.701538i	$0.752497\pi$
$278$	2.84043			0.170358
$279$		0			0
$280$		0			0
$281$			5.57194i			0.332394i	0.986093	+	0.166197i	$0.0531488\pi$
$281$			5.57194i			0.332394i	−0.986093	+	0.166197i	$0.946851\pi$
$282$		0			0
$283$		−	32.1798i		−	1.91289i	−0.291914	−	0.956445i	$-0.594292\pi$
$283$		−	32.1798i		−	1.91289i	0.291914	−	0.956445i	$-0.405708\pi$
$284$			2.63803i			0.156539i
$285$		0			0
$286$			6.21603i			0.367561i
$287$	−2.89199	−	3.10801i	−0.170709	−	0.183460i
$288$		0			0
$289$	48.8776			2.87515
$290$		0			0
$291$		0			0
$292$			13.7477i			0.804525i
$293$	−15.8146			−0.923898			−0.461949	−	0.886907i	$-0.652850\pi$
$293$	−15.8146			−0.923898			−0.461949	+	0.886907i	$0.652850\pi$
$294$		0			0
$295$		0			0
$296$		−	4.51190i		−	0.262249i
$297$		0			0
$298$	−3.00681			−0.174180
$299$	8.79079			0.508384
$300$		0			0
$301$	18.2330	+	19.5950i	1.05093	+	1.12944i
$302$			15.2262i			0.876169i
$303$		0			0
$304$		−	2.63803i		−	0.151302i
$305$		0			0
$306$		0			0
$307$		−	6.72393i		−	0.383755i	−0.981419	−	0.191878i	$-0.938542\pi$
$307$		−	6.72393i		−	0.383755i	0.981419	−	0.191878i	$-0.0614576\pi$
$308$	−7.50341	+	6.98188i	−0.427546	+	0.397829i
$309$		0			0
$310$		0			0
$311$	1.02379			0.0580540			0.0290270	−	0.999579i	$-0.490759\pi$
$311$	1.02379			0.0580540			0.0290270	+	0.999579i	$0.490759\pi$
$312$		0			0
$313$		−	23.2568i		−	1.31455i	−0.753650	−	0.657276i	$-0.771709\pi$
$313$		−	23.2568i		−	1.31455i	0.753650	−	0.657276i	$-0.228291\pi$
$314$	−2.64767			−0.149417
$315$		0			0
$316$	8.01698			0.450990
$317$		−	4.46830i		−	0.250965i	−0.992096	−	0.125482i	$-0.959952\pi$
$317$		−	4.46830i		−	0.250965i	0.992096	−	0.125482i	$-0.0400478\pi$
$318$		0			0
$319$	−21.2228			−1.18825
$320$		0			0
$321$		0			0
$322$	9.87386	+	10.6114i	0.550249	+	0.591351i
$323$			21.4116i			1.19137i
$324$		0			0
$325$		0			0
$326$		−	9.32572i		−	0.516504i
$327$		0			0
$328$		−	1.60461i		−	0.0885996i
$329$	−20.0230	−	21.5187i	−1.10391	−	1.18636i
$330$		0			0
$331$	−14.2160			−0.781383			−0.390692	−	0.920522i	$-0.627764\pi$
$331$	−14.2160			−0.781383			−0.390692	+	0.920522i	$0.627764\pi$
$332$	3.20921			0.176128
$333$		0			0
$334$			17.3257i			0.948021i
$335$		0			0
$336$		0			0
$337$	−23.4286			−1.27624			−0.638118	−	0.769938i	$-0.720287\pi$
$337$	−23.4286			−1.27624			−0.638118	+	0.769938i	$0.720287\pi$
$338$		−	10.4252i		−	0.567058i
$339$		0			0
$340$		0			0
$341$	14.4524			0.782641
$342$		0			0
$343$	−14.4307	+	11.6084i	−0.779185	+	0.626794i
$344$			10.1165i			0.545445i
$345$		0			0
$346$			2.99319i			0.160915i
$347$			20.1798i			1.08331i	0.840602	+	0.541654i	$0.182202\pi$
$347$			20.1798i			1.08331i	−0.840602	+	0.541654i	$0.817798\pi$
$348$		0			0
$349$		−	27.0372i		−	1.44727i	−0.690184	−	0.723634i	$-0.742470\pi$
$349$		−	27.0372i		−	1.44727i	0.690184	−	0.723634i	$-0.257530\pi$
$350$		0			0
$351$		0			0
$352$	−3.87386			−0.206478
$353$	−12.3495			−0.657298			−0.328649	−	0.944452i	$-0.606593\pi$
$353$	−12.3495			−0.657298			−0.328649	+	0.944452i	$0.606593\pi$
$354$		0			0
$355$		0			0
$356$	17.8376			0.945392
$357$		0			0
$358$	−7.38858			−0.390499
$359$		−	25.5757i		−	1.34983i	−0.737894	−	0.674917i	$-0.764179\pi$
$359$		−	25.5757i		−	1.34983i	0.737894	−	0.674917i	$-0.235821\pi$
$360$		0			0
$361$	12.0408			0.633725
$362$	11.1097			0.583912
$363$		0			0
$364$	−3.10801	+	2.89199i	−0.162904	+	0.151581i
$365$		0			0
$366$		0			0
$367$		−	20.8444i		−	1.08807i	−0.839062	−	0.544035i	$-0.816896\pi$
$367$		−	20.8444i		−	1.08807i	0.839062	−	0.544035i	$-0.183104\pi$
$368$			5.47847i			0.285585i
$369$		0			0
$370$		0			0
$371$	4.39539	−	4.08989i	0.228197	−	0.212336i
$372$		0			0
$373$	−7.48810			−0.387719			−0.193860	−	0.981029i	$-0.562101\pi$
$373$	−7.48810			−0.387719			−0.193860	+	0.981029i	$0.562101\pi$
$374$	31.4422			1.62584
$375$		0			0
$376$		−	11.1097i		−	0.572939i
$377$	−8.79079			−0.452749
$378$		0			0
$379$	32.6651			1.67789			0.838946	−	0.544215i	$-0.183173\pi$
$379$	32.6651			1.67789			0.838946	+	0.544215i	$0.183173\pi$
$380$		0			0
$381$		0			0
$382$	−9.36197			−0.479000
$383$	0.890309			0.0454927			0.0227463	−	0.999741i	$-0.492759\pi$
$383$	0.890309			0.0454927			0.0227463	+	0.999741i	$0.492759\pi$
$384$		0			0
$385$		0			0
$386$			17.4422i			0.887786i
$387$		0			0
$388$		−	8.68768i		−	0.441050i
$389$		−	13.5317i		−	0.686084i	−0.939320	−	0.343042i	$-0.888543\pi$
$389$		−	13.5317i		−	0.686084i	0.939320	−	0.343042i	$-0.111457\pi$
$390$		0			0
$391$		−	44.4660i		−	2.24874i
$392$	−6.98188	−	0.503406i	−0.352638	−	0.0254258i
$393$		0			0
$394$	5.27607			0.265804
$395$		0			0
$396$		0			0
$397$		−	25.2991i		−	1.26973i	−0.772625	−	0.634863i	$-0.781057\pi$
$397$		−	25.2991i		−	1.26973i	0.772625	−	0.634863i	$-0.218943\pi$
$398$	12.2160			0.612334
$399$		0			0
$400$		0			0
$401$			32.7619i			1.63605i	0.575182	+	0.818025i	$0.304931\pi$
$401$			32.7619i			1.63605i	−0.575182	+	0.818025i	$0.695069\pi$
$402$		0			0
$403$	5.98638			0.298203
$404$	−6.21603			−0.309259
$405$		0			0
$406$	−9.87386	−	10.6114i	−0.490032	−	0.526636i
$407$			17.4785i			0.866376i
$408$		0			0
$409$			12.0000i			0.593362i	0.954977	+	0.296681i	$0.0958798\pi$
$409$			12.0000i			0.593362i	−0.954977	+	0.296681i	$0.904120\pi$
$410$		0			0
$411$		0			0
$412$		−	12.8807i		−	0.634585i
$413$	8.31117	+	8.93200i	0.408966	+	0.439515i
$414$		0			0
$415$		0			0
$416$	−1.60461			−0.0786723
$417$		0			0
$418$			10.2194i			0.499846i
$419$	4.17937			0.204176			0.102088	−	0.994775i	$-0.467448\pi$
$419$	4.17937			0.204176			0.102088	+	0.994775i	$0.467448\pi$
$420$		0			0
$421$	−32.6514			−1.59133			−0.795667	−	0.605735i	$-0.792879\pi$
$421$	−32.6514			−1.59133			−0.795667	+	0.605735i	$0.792879\pi$
$422$		−	4.00000i		−	0.194717i
$423$		0			0
$424$	2.26926			0.110205
$425$		0			0
$426$		0			0
$427$	22.9473	−	21.3523i	1.11050	−	1.03331i
$428$			19.9638i			0.964984i
$429$		0			0
$430$		0			0
$431$			13.3087i			0.641059i	0.947239	+	0.320530i	$0.103861\pi$
$431$			13.3087i			0.641059i	−0.947239	+	0.320530i	$0.896139\pi$
$432$		0			0
$433$			6.95358i			0.334168i	0.985943	+	0.167084i	$0.0534351\pi$
$433$			6.95358i			0.334168i	−0.985943	+	0.167084i	$0.946565\pi$
$434$	6.72393	+	7.22619i	0.322759	+	0.346868i
$435$		0			0
$436$	15.0238			0.719509
$437$	14.4524			0.691352
$438$		0			0
$439$		−	26.9739i		−	1.28739i	−0.765280	−	0.643697i	$-0.777400\pi$
$439$		−	26.9739i		−	1.28739i	0.765280	−	0.643697i	$-0.222600\pi$
$440$		0			0
$441$		0			0
$442$	13.0238			0.619479
$443$			2.80441i			0.133242i	0.997778	+	0.0666208i	$0.0212218\pi$
$443$			2.80441i			0.133242i	−0.997778	+	0.0666208i	$0.978778\pi$
$444$		0			0
$445$		0			0
$446$	−3.65784			−0.173204
$447$		0			0
$448$	−1.80230	−	1.93693i	−0.0851508	−	0.0915114i
$449$			30.1226i			1.42157i	0.703409	+	0.710786i	$0.251660\pi$
$449$			30.1226i			1.42157i	−0.703409	+	0.710786i	$0.748340\pi$
$450$		0			0
$451$			6.21603i			0.292701i
$452$			10.2330i			0.481320i
$453$		0			0
$454$			4.23301i			0.198665i
$455$		0			0
$456$		0			0
$457$	22.0340			1.03071			0.515353	−	0.856978i	$-0.327661\pi$
$457$	22.0340			1.03071			0.515353	+	0.856978i	$0.327661\pi$
$458$	7.59497			0.354890
$459$		0			0
$460$		0			0
$461$	15.4116			0.717790			0.358895	−	0.933378i	$-0.383154\pi$
$461$	15.4116			0.717790			0.358895	+	0.933378i	$0.383154\pi$
$462$		0			0
$463$	25.0605			1.16466			0.582329	−	0.812953i	$-0.302142\pi$
$463$	25.0605			1.16466			0.582329	+	0.812953i	$0.302142\pi$
$464$		−	5.47847i		−	0.254332i
$465$		0			0
$466$	9.94677			0.460775
$467$	−9.62764			−0.445514			−0.222757	−	0.974874i	$-0.571506\pi$
$467$	−9.62764			−0.445514			−0.222757	+	0.974874i	$0.571506\pi$
$468$		0			0
$469$	−12.4490	−	13.3789i	−0.574843	−	0.617782i
$470$		0			0
$471$		0			0
$472$			4.61142i			0.212258i
$473$		−	39.1899i		−	1.80196i
$474$		0			0
$475$		0			0
$476$	14.6284	+	15.7211i	0.670492	+	0.720576i
$477$		0			0
$478$	9.36197			0.428206
$479$	−37.0238			−1.69166			−0.845830	−	0.533452i	$-0.820894\pi$
$479$	−37.0238			−1.69166			−0.845830	+	0.533452i	$0.820894\pi$
$480$		0			0
$481$			7.23982i			0.330107i
$482$	23.0408			1.04948
$483$		0			0
$484$	4.00681			0.182128
$485$		0			0
$486$		0			0
$487$	11.6386			0.527394			0.263697	−	0.964606i	$-0.415058\pi$
$487$	11.6386			0.527394			0.263697	+	0.964606i	$0.415058\pi$
$488$	11.8472			0.536300
$489$		0			0
$490$		0			0
$491$		−	32.1069i		−	1.44896i	−0.689294	−	0.724481i	$-0.742079\pi$
$491$		−	32.1069i		−	1.44896i	0.689294	−	0.724481i	$-0.257921\pi$
$492$		0			0
$493$			44.4660i			2.00265i
$494$			4.23301i			0.190452i
$495$		0			0
$496$			3.73074i			0.167515i
$497$	5.10969	−	4.75454i	0.229201	−	0.213270i
$498$		0			0
$499$	13.6122			0.609365			0.304682	−	0.952454i	$-0.401450\pi$
$499$	13.6122			0.609365			0.304682	+	0.952454i	$0.401450\pi$
$500$		0			0
$501$		0			0
$502$		−	23.8376i		−	1.06392i
$503$	−3.07573			−0.137140			−0.0685700	−	0.997646i	$-0.521844\pi$
$503$	−3.07573			−0.137140			−0.0685700	+	0.997646i	$0.521844\pi$
$504$		0			0
$505$		0			0
$506$		−	21.2228i		−	0.943470i
$507$		0			0
$508$	−2.81382			−0.124843
$509$	−14.7908			−0.655590			−0.327795	−	0.944749i	$-0.606306\pi$
$509$	−14.7908			−0.655590			−0.327795	+	0.944749i	$0.606306\pi$
$510$		0			0
$511$	26.6284	−	24.7776i	1.17797	−	1.09610i
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$		−	29.7441i		−	1.31196i
$515$		0			0
$516$		0			0
$517$			43.0374i			1.89278i
$518$	−8.73923	+	8.13181i	−0.383980	+	0.357291i
$519$		0			0
$520$		0			0
$521$	27.0332			1.18435			0.592173	−	0.805811i	$-0.298270\pi$
$521$	27.0332			1.18435			0.592173	+	0.805811i	$0.298270\pi$
$522$		0			0
$523$		−	3.51472i		−	0.153688i	−0.997043	−	0.0768440i	$-0.975516\pi$
$523$		−	3.51472i		−	0.153688i	0.997043	−	0.0768440i	$-0.0244843\pi$
$524$	8.62840			0.376933
$525$		0			0
$526$	−22.7545			−0.992145
$527$		−	30.2806i		−	1.31904i
$528$		0			0
$529$	−7.01362			−0.304940
$530$		0			0
$531$		0			0
$532$	−5.10969	+	4.75454i	−0.221533	+	0.206135i
$533$			2.57476i			0.111525i
$534$		0			0
$535$		0			0
$536$		−	6.90729i		−	0.298350i
$537$		0			0
$538$			0.202401i			0.00872612i
$539$	27.0468	+	1.95013i	1.16499	+	0.0839979i
$540$		0			0
$541$	−39.6616			−1.70519			−0.852593	−	0.522576i	$-0.824971\pi$
$541$	−39.6616			−1.70519			−0.852593	+	0.522576i	$0.824971\pi$
$542$	11.4785			0.493042
$543$		0			0
$544$			8.11650i			0.347992i
$545$		0			0
$546$		0			0
$547$	34.3495			1.46868			0.734339	−	0.678782i	$-0.237492\pi$
$547$	34.3495			1.46868			0.734339	+	0.678782i	$0.237492\pi$
$548$			0.723932i			0.0309248i
$549$		0			0
$550$		0			0
$551$	−14.4524			−0.615692
$552$		0			0
$553$	−14.4490	−	15.5283i	−0.614435	−	0.660332i
$554$			23.7211i			1.00781i
$555$		0			0
$556$		−	2.84043i		−	0.120461i
$557$			38.4660i			1.62986i	0.579561	+	0.814929i	$0.303224\pi$
$557$			38.4660i			1.62986i	−0.579561	+	0.814929i	$0.696776\pi$
$558$		0			0
$559$		−	16.2330i		−	0.686583i
$560$		0			0
$561$		0			0
$562$	5.57194			0.235038
$563$	−10.2194			−0.430696			−0.215348	−	0.976537i	$-0.569089\pi$
$563$	−10.2194			−0.430696			−0.215348	+	0.976537i	$0.569089\pi$
$564$		0			0
$565$		0			0
$566$	−32.1798			−1.35262
$567$		0			0
$568$	2.63803			0.110689
$569$			2.48129i			0.104021i	0.998647	+	0.0520106i	$0.0165629\pi$
$569$			2.48129i			0.104021i	−0.998647	+	0.0520106i	$0.983437\pi$
$570$		0			0
$571$	20.6345			0.863525			0.431762	−	0.901987i	$-0.357892\pi$
$571$	20.6345			0.863525			0.431762	+	0.901987i	$0.357892\pi$
$572$	6.21603			0.259905
$573$		0			0
$574$	−3.10801	+	2.89199i	−0.129726	+	0.120709i
$575$		0			0
$576$		0			0
$577$			2.50455i			0.104266i	0.998640	+	0.0521329i	$0.0166019\pi$
$577$			2.50455i			0.104266i	−0.998640	+	0.0521329i	$0.983398\pi$
$578$		−	48.8776i		−	2.03304i
$579$		0			0
$580$		0			0
$581$	−5.78397	−	6.21603i	−0.239960	−	0.257884i
$582$		0			0
$583$	−8.79079			−0.364077
$584$	13.7477			0.568885
$585$		0			0
$586$			15.8146i			0.653294i
$587$	31.4422			1.29776			0.648880	−	0.760891i	$-0.275238\pi$
$587$	31.4422			1.29776			0.648880	+	0.760891i	$0.275238\pi$
$588$		0			0
$589$	9.84183			0.405526
$590$		0			0
$591$		0			0
$592$	−4.51190			−0.185438
$593$	27.1267			1.11396			0.556979	−	0.830526i	$-0.311960\pi$
$593$	27.1267			1.11396			0.556979	+	0.830526i	$0.311960\pi$
$594$		0			0
$595$		0			0
$596$			3.00681i			0.123164i
$597$		0			0
$598$		−	8.79079i		−	0.359482i
$599$			33.7747i			1.38000i	0.723810	+	0.689999i	$0.242389\pi$
$599$			33.7747i			1.38000i	−0.723810	+	0.689999i	$0.757611\pi$
$600$		0			0
$601$			24.4886i			0.998912i	0.866339	+	0.499456i	$0.166467\pi$
$601$			24.4886i			0.998912i	−0.866339	+	0.499456i	$0.833533\pi$
$602$	19.5950	−	18.2330i	0.798631	−	0.743122i
$603$		0			0
$604$	15.2262			0.619545
$605$		0			0
$606$		0			0
$607$		−	27.3331i		−	1.10941i	−0.832045	−	0.554707i	$-0.812830\pi$
$607$		−	27.3331i		−	1.10941i	0.832045	−	0.554707i	$-0.187170\pi$
$608$	−2.63803			−0.106986
$609$		0			0
$610$		0			0
$611$			17.8267i			0.721190i
$612$		0			0
$613$	−11.3163			−0.457061			−0.228531	−	0.973537i	$-0.573392\pi$
$613$	−11.3163			−0.457061			−0.228531	+	0.973537i	$0.573392\pi$
$614$	−6.72393			−0.271356
$615$		0			0
$616$	6.98188	+	7.50341i	0.281308	+	0.302321i
$617$			10.6843i			0.430135i	0.976599	+	0.215067i	$0.0689971\pi$
$617$			10.6843i			0.430135i	−0.976599	+	0.215067i	$0.931003\pi$
$618$		0			0
$619$		−	9.56437i		−	0.384424i	−0.981353	−	0.192212i	$-0.938434\pi$
$619$		−	9.56437i		−	0.384424i	0.981353	−	0.192212i	$-0.0615662\pi$
$620$		0			0
$621$		0			0
$622$		−	1.02379i		−	0.0410504i
$623$	−32.1488	−	34.5502i	−1.28801	−	1.38423i
$624$		0			0
$625$		0			0
$626$	−23.2568			−0.929529
$627$		0			0
$628$			2.64767i			0.105653i
$629$	36.6208			1.46017
$630$		0			0
$631$	17.1956			0.684546			0.342273	−	0.939601i	$-0.388803\pi$
$631$	17.1956			0.684546			0.342273	+	0.939601i	$0.388803\pi$
$632$		−	8.01698i		−	0.318898i
$633$		0			0
$634$	−4.46830			−0.177459
$635$		0			0
$636$		0			0
$637$	11.2032	+	0.807769i	0.443885	+	0.0320050i
$638$			21.2228i			0.840220i
$639$		0			0
$640$		0			0
$641$			16.5289i			0.652851i	0.945223	+	0.326426i	$0.105844\pi$
$641$			16.5289i			0.652851i	−0.945223	+	0.326426i	$0.894156\pi$
$642$		0			0
$643$			37.4286i			1.47604i	0.674779	+	0.738020i	$0.264239\pi$
$643$			37.4286i			1.47604i	−0.674779	+	0.738020i	$0.735761\pi$
$644$	10.6114	−	9.87386i	0.418148	−	0.389085i
$645$		0			0
$646$	21.4116			0.842429
$647$	−5.52812			−0.217333			−0.108666	−	0.994078i	$-0.534658\pi$
$647$	−5.52812			−0.217333			−0.108666	+	0.994078i	$0.534658\pi$
$648$		0			0
$649$		−	17.8640i		−	0.701223i
$650$		0			0
$651$		0			0
$652$	−9.32572			−0.365223
$653$			18.5159i			0.724583i	0.932065	+	0.362291i	$0.118005\pi$
$653$			18.5159i			0.724583i	−0.932065	+	0.362291i	$0.881995\pi$
$654$		0			0
$655$		0			0
$656$	−1.60461			−0.0626494
$657$		0			0
$658$	−21.5187	+	20.0230i	−0.838887	+	0.780579i
$659$			43.3693i			1.68943i	0.535217	+	0.844714i	$0.320230\pi$
$659$			43.3693i			1.68943i	−0.535217	+	0.844714i	$0.679770\pi$
$660$		0			0
$661$		−	36.1142i		−	1.40468i	−0.711842	−	0.702340i	$-0.752139\pi$
$661$		−	36.1142i		−	1.40468i	0.711842	−	0.702340i	$-0.247861\pi$
$662$			14.2160i			0.552522i
$663$		0			0
$664$		−	3.20921i		−	0.124542i
$665$		0			0
$666$		0			0
$667$	30.0136			1.16213
$668$	17.3257			0.670352
$669$		0			0
$670$		0			0
$671$	−45.8946			−1.77174
$672$		0			0
$673$	33.8743			1.30576			0.652879	−	0.757463i	$-0.273561\pi$
$673$	33.8743			1.30576			0.652879	+	0.757463i	$0.273561\pi$
$674$			23.4286i			0.902436i
$675$		0			0
$676$	−10.4252			−0.400971
$677$	−4.38446			−0.168509			−0.0842543	−	0.996444i	$-0.526851\pi$
$677$	−4.38446			−0.168509			−0.0842543	+	0.996444i	$0.526851\pi$
$678$		0			0
$679$	−16.8274	+	15.6578i	−0.645778	+	0.600893i
$680$		0			0
$681$		0			0
$682$		−	14.4524i		−	0.553411i
$683$			1.95013i			0.0746195i	0.999304	+	0.0373098i	$0.0118788\pi$
$683$			1.95013i			0.0746195i	−0.999304	+	0.0373098i	$0.988121\pi$
$684$		0			0
$685$		0			0
$686$	11.6084	+	14.4307i	0.443211	+	0.550967i
$687$		0			0
$688$	10.1165			0.385688
$689$	−3.64126			−0.138721
$690$		0			0
$691$		−	12.1471i		−	0.462098i	−0.972942	−	0.231049i	$-0.925784\pi$
$691$		−	12.1471i		−	0.462098i	0.972942	−	0.231049i	$-0.0742157\pi$
$692$	2.99319			0.113784
$693$		0			0
$694$	20.1798			0.766014
$695$		0			0
$696$		0			0
$697$	13.0238			0.493311
$698$	−27.0372			−1.02337
$699$		0			0
$700$		0			0
$701$		−	11.0600i		−	0.417732i	−0.977944	−	0.208866i	$-0.933023\pi$
$701$		−	11.0600i		−	0.417732i	0.977944	−	0.208866i	$-0.0669773\pi$
$702$		0			0
$703$			11.9025i			0.448913i
$704$			3.87386i			0.146002i
$705$		0			0
$706$			12.3495i			0.464780i
$707$	11.2032	+	12.0400i	0.421338	+	0.452811i
$708$		0			0
$709$	14.2330			0.534532			0.267266	−	0.963623i	$-0.413880\pi$
$709$	14.2330			0.534532			0.267266	+	0.963623i	$0.413880\pi$
$710$		0			0
$711$		0			0
$712$		−	17.8376i		−	0.668493i
$713$	−20.4388			−0.765438
$714$		0			0
$715$		0			0
$716$			7.38858i			0.276124i
$717$		0			0
$718$	−25.5757			−0.954477
$719$	20.4660			0.763254			0.381627	−	0.924317i	$-0.375364\pi$
$719$	20.4660			0.763254			0.381627	+	0.924317i	$0.375364\pi$
$720$		0			0
$721$	−24.9490	+	23.2149i	−0.929149	+	0.864567i
$722$		−	12.0408i		−	0.448111i
$723$		0			0
$724$		−	11.1097i		−	0.412888i
$725$		0			0
$726$		0			0
$727$		−	25.5717i		−	0.948402i	−0.880417	−	0.474201i	$-0.842737\pi$
$727$		−	25.5717i		−	0.948402i	0.880417	−	0.474201i	$-0.157263\pi$
$728$	2.89199	+	3.10801i	0.107184	+	0.115191i
$729$		0			0
$730$		0			0
$731$	−82.1106			−3.03697
$732$		0			0
$733$			22.6624i			0.837053i	0.908204	+	0.418527i	$0.137453\pi$
$733$			22.6624i			0.837053i	−0.908204	+	0.418527i	$0.862547\pi$
$734$	−20.8444			−0.769382
$735$		0			0
$736$	5.47847			0.201939
$737$			26.7579i			0.985640i
$738$		0			0
$739$	−44.8675			−1.65048			−0.825238	−	0.564785i	$-0.808959\pi$
$739$	−44.8675			−1.65048			−0.825238	+	0.564785i	$0.808959\pi$
$740$		0			0
$741$		0			0
$742$	−4.08989	−	4.39539i	−0.150145	−	0.161360i
$743$		−	5.37536i		−	0.197203i	−0.995127	−	0.0986015i	$-0.968563\pi$
$743$		−	5.37536i		−	0.197203i	0.995127	−	0.0986015i	$-0.0314369\pi$
$744$		0			0
$745$		0			0
$746$			7.48810i			0.274159i
$747$		0			0
$748$		−	31.4422i		−	1.14964i
$749$	38.6684	−	35.9807i	1.41291	−	1.31471i
$750$		0			0
$751$	13.6276			0.497280			0.248640	−	0.968596i	$-0.420016\pi$
$751$	13.6276			0.497280			0.248640	+	0.968596i	$0.420016\pi$
$752$	−11.1097			−0.405129
$753$		0			0
$754$			8.79079i			0.320142i
$755$		0			0
$756$		0			0
$757$	4.54586			0.165222			0.0826110	−	0.996582i	$-0.473674\pi$
$757$	4.54586			0.165222			0.0826110	+	0.996582i	$0.473674\pi$
$758$		−	32.6651i		−	1.18645i
$759$		0			0
$760$		0			0
$761$	15.6522			0.567392			0.283696	−	0.958914i	$-0.408439\pi$
$761$	15.6522			0.567392			0.283696	+	0.958914i	$0.408439\pi$
$762$		0			0
$763$	−27.0774	−	29.1001i	−0.980269	−	1.05349i
$764$			9.36197i			0.338704i
$765$		0			0
$766$		−	0.890309i		−	0.0321682i
$767$		−	7.39951i		−	0.267181i
$768$		0			0
$769$		−	17.1922i		−	0.619968i	−0.950742	−	0.309984i	$-0.899676\pi$
$769$		−	17.1922i		−	0.619968i	0.950742	−	0.309984i	$-0.100324\pi$
$770$		0			0
$771$		0			0
$772$	17.4422			0.627759
$773$	42.6990			1.53578			0.767889	−	0.640584i	$-0.221307\pi$
$773$	42.6990			1.53578			0.767889	+	0.640584i	$0.221307\pi$
$774$		0			0
$775$		0			0
$776$	−8.68768			−0.311870
$777$		0			0
$778$	−13.5317			−0.485135
$779$			4.23301i			0.151663i
$780$		0			0
$781$	−10.2194			−0.365678
$782$	−44.4660			−1.59010
$783$		0			0
$784$	−0.503406	+	6.98188i	−0.0179788	+	0.249353i
$785$		0			0
$786$		0			0
$787$		−	30.3992i		−	1.08361i	−0.840503	−	0.541806i	$-0.817741\pi$
$787$		−	30.3992i		−	1.08361i	0.840503	−	0.541806i	$-0.182259\pi$
$788$		−	5.27607i		−	0.187952i
$789$		0			0
$790$		0			0
$791$	19.8206	−	18.4430i	0.704741	−	0.655757i
$792$		0			0
$793$	−19.0102			−0.675071
$794$	−25.2991			−0.897831
$795$		0			0
$796$		−	12.2160i		−	0.432986i
$797$	2.99319			0.106024			0.0530121	−	0.998594i	$-0.483118\pi$
$797$	2.99319			0.106024			0.0530121	+	0.998594i	$0.483118\pi$
$798$		0			0
$799$	90.1718			3.19005
$800$		0			0
$801$		0			0
$802$	32.7619			1.15686
$803$	−53.2568			−1.87939
$804$		0			0
$805$		0			0
$806$		−	5.98638i		−	0.210861i
$807$		0			0
$808$			6.21603i			0.218679i
$809$		−	25.0334i		−	0.880128i	−0.897966	−	0.440064i	$-0.854956\pi$
$809$		−	25.0334i		−	0.880128i	0.897966	−	0.440064i	$-0.145044\pi$
$810$		0			0
$811$		−	19.4782i		−	0.683974i	−0.939705	−	0.341987i	$-0.888900\pi$
$811$		−	19.4782i		−	0.683974i	0.939705	−	0.341987i	$-0.111100\pi$
$812$	−10.6114	+	9.87386i	−0.372388	+	0.346505i
$813$		0			0
$814$	17.4785			0.612620
$815$		0			0
$816$		0			0
$817$		−	26.6877i		−	0.933684i
$818$	12.0000			0.419570
$819$		0			0
$820$		0			0
$821$		−	45.3787i		−	1.58373i	−0.610697	−	0.791864i	$-0.709111\pi$
$821$		−	45.3787i		−	1.58373i	0.610697	−	0.791864i	$-0.290889\pi$
$822$		0			0
$823$	−25.8512			−0.901117			−0.450559	−	0.892747i	$-0.648775\pi$
$823$	−25.8512			−0.901117			−0.450559	+	0.892747i	$0.648775\pi$
$824$	−12.8807			−0.448720
$825$		0			0
$826$	8.93200	−	8.31117i	0.310784	−	0.289183i
$827$		−	19.3429i		−	0.672619i	−0.941751	−	0.336310i	$-0.890821\pi$
$827$		−	19.3429i		−	0.672619i	0.941751	−	0.336310i	$-0.109179\pi$
$828$		0			0
$829$		−	43.4764i		−	1.51000i	−0.655726	−	0.754999i	$-0.727637\pi$
$829$		−	43.4764i		−	1.51000i	0.655726	−	0.754999i	$-0.272363\pi$
$830$		0			0
$831$		0			0
$832$			1.60461i			0.0556297i
$833$	4.08590	−	56.6684i	0.141568	−	1.96344i
$834$		0			0
$835$		0			0
$836$	10.2194			0.353445
$837$		0			0
$838$		−	4.17937i		−	0.144374i
$839$	24.2670			0.837789			0.418894	−	0.908035i	$-0.362418\pi$
$839$	24.2670			0.837789			0.418894	+	0.908035i	$0.362418\pi$
$840$		0			0
$841$	−1.01362			−0.0349526
$842$			32.6514i			1.12524i
$843$		0			0
$844$	−4.00000			−0.137686
$845$		0			0
$846$		0			0
$847$	−7.22149	−	7.76092i	−0.248133	−	0.266668i
$848$		−	2.26926i		−	0.0779266i
$849$		0			0
$850$		0			0
$851$		−	24.7183i		−	0.847332i
$852$		0			0
$853$			16.2439i			0.556182i	0.960555	+	0.278091i	$0.0897017\pi$
$853$			16.2439i			0.556182i	−0.960555	+	0.278091i	$0.910298\pi$
$854$	−21.3523	−	22.9473i	−0.730662	−	0.785241i
$855$		0			0
$856$	19.9638			0.682347
$857$	40.5825			1.38627			0.693136	−	0.720807i	$-0.256228\pi$
$857$	40.5825			1.38627			0.693136	+	0.720807i	$0.256228\pi$
$858$		0			0
$859$		−	5.05310i		−	0.172410i	−0.996277	−	0.0862048i	$-0.972526\pi$
$859$		−	5.05310i		−	0.172410i	0.996277	−	0.0862048i	$-0.0274739\pi$
$860$		0			0
$861$		0			0
$862$	13.3087			0.453297
$863$			48.3289i			1.64514i	0.568666	+	0.822568i	$0.307460\pi$
$863$			48.3289i			1.64514i	−0.568666	+	0.822568i	$0.692540\pi$
$864$		0			0
$865$		0			0
$866$	6.95358			0.236292
$867$		0			0
$868$	7.22619	−	6.72393i	0.245273	−	0.228225i
$869$			31.0567i			1.05353i
$870$		0			0
$871$			11.0835i			0.375549i
$872$		−	15.0238i		−	0.508770i
$873$		0			0
$874$		−	14.4524i		−	0.488859i
$875$		0			0
$876$		0			0
$877$	55.3963			1.87060			0.935301	−	0.353854i	$-0.115129\pi$
$877$	55.3963			1.87060			0.935301	+	0.353854i	$0.115129\pi$
$878$	−26.9739			−0.910326
$879$		0			0
$880$		0			0
$881$	49.8716			1.68022			0.840108	−	0.542419i	$-0.182491\pi$
$881$	49.8716			1.68022			0.840108	+	0.542419i	$0.182491\pi$
$882$		0			0
$883$	43.3393			1.45848			0.729242	−	0.684255i	$-0.239873\pi$
$883$	43.3393			1.45848			0.729242	+	0.684255i	$0.239873\pi$
$884$		−	13.0238i		−	0.438038i
$885$		0			0
$886$	2.80441			0.0942160
$887$	−33.1539			−1.11320			−0.556600	−	0.830781i	$-0.687894\pi$
$887$	−33.1539			−1.11320			−0.556600	+	0.830781i	$0.687894\pi$
$888$		0			0
$889$	5.07136	+	5.45018i	0.170088	+	0.182793i
$890$		0			0
$891$		0			0
$892$			3.65784i			0.122473i
$893$			29.3077i			0.980746i
$894$		0			0
$895$		0			0
$896$	−1.93693	+	1.80230i	−0.0647083	+	0.0602107i
$897$		0			0
$898$	30.1226			1.00520
$899$	20.4388			0.681671
$900$		0			0
$901$			18.4184i			0.613607i
$902$	6.21603			0.206971
$903$		0			0
$904$	10.2330			0.340345
$905$		0			0
$906$		0			0
$907$	−50.5553			−1.67866			−0.839330	−	0.543622i	$-0.817052\pi$
$907$	−50.5553			−1.67866			−0.839330	+	0.543622i	$0.817052\pi$
$908$	4.23301			0.140477
$909$		0			0
$910$		0			0
$911$		−	21.6755i		−	0.718140i	−0.933311	−	0.359070i	$-0.883094\pi$
$911$		−	21.6755i		−	0.718140i	0.933311	−	0.359070i	$-0.116906\pi$
$912$		0			0
$913$			12.4321i			0.411441i
$914$		−	22.0340i		−	0.728819i
$915$		0			0
$916$		−	7.59497i		−	0.250945i
$917$	−15.5510	−	16.7126i	−0.513539	−	0.551899i
$918$		0			0
$919$	−25.6616			−0.846498			−0.423249	−	0.906013i	$-0.639110\pi$
$919$	−25.6616			−0.846498			−0.423249	+	0.906013i	$0.639110\pi$
$920$		0			0
$921$		0			0
$922$		−	15.4116i		−	0.507554i
$923$	−4.23301			−0.139331
$924$		0			0
$925$		0			0
$926$		−	25.0605i		−	0.823538i
$927$		0			0
$928$	−5.47847			−0.179840
$929$	31.6182			1.03736			0.518680	−	0.854968i	$-0.326424\pi$
$929$	31.6182			1.03736			0.518680	+	0.854968i	$0.326424\pi$
$930$		0			0
$931$	18.4184	+	1.32800i	0.603640	+	0.0435235i
$932$		−	9.94677i		−	0.325817i
$933$		0			0
$934$			9.62764i			0.315026i
$935$		0			0
$936$		0			0
$937$		−	38.7013i		−	1.26432i	−0.774839	−	0.632158i	$-0.782169\pi$
$937$		−	38.7013i		−	1.26432i	0.774839	−	0.632158i	$-0.217831\pi$
$938$	−13.3789	+	12.4490i	−0.436838	+	0.406475i
$939$		0			0
$940$		0			0
$941$	32.9932			1.07555			0.537774	−	0.843089i	$-0.319266\pi$
$941$	32.9932			1.07555			0.537774	+	0.843089i	$0.319266\pi$
$942$		0			0
$943$		−	8.79079i		−	0.286267i
$944$	4.61142			0.150089
$945$		0			0
$946$	−39.1899			−1.27418
$947$		−	31.8452i		−	1.03483i	−0.855735	−	0.517415i	$-0.826894\pi$
$947$		−	31.8452i		−	1.03483i	0.855735	−	0.517415i	$-0.173106\pi$
$948$		0			0
$949$	−22.0597			−0.716088
$950$		0			0
$951$		0			0
$952$	15.7211	−	14.6284i	0.509524	−	0.474109i
$953$			45.9003i			1.48686i	0.668817	+	0.743428i	$0.266801\pi$
$953$			45.9003i			1.48686i	−0.668817	+	0.743428i	$0.733199\pi$
$954$		0			0
$955$		0			0
$956$		−	9.36197i		−	0.302788i
$957$		0			0
$958$			37.0238i			1.19618i
$959$	1.40221	−	1.30474i	0.0452796	−	0.0421324i
$960$		0			0
$961$	17.0816			0.551018
$962$	7.23982			0.233421
$963$		0			0
$964$		−	23.0408i		−	0.742093i
$965$		0			0
$966$		0			0
$967$	41.2662			1.32703			0.663516	−	0.748162i	$-0.269064\pi$
$967$	41.2662			1.32703			0.663516	+	0.748162i	$0.269064\pi$
$968$		−	4.00681i		−	0.128784i
$969$		0			0
$970$		0			0
$971$	11.0008			0.353031			0.176516	−	0.984298i	$-0.443517\pi$
$971$	11.0008			0.353031			0.176516	+	0.984298i	$0.443517\pi$
$972$		0			0
$973$	−5.50173	+	5.11933i	−0.176377	+	0.164118i
$974$		−	11.6386i		−	0.372924i
$975$		0			0
$976$		−	11.8472i		−	0.379221i
$977$		−	2.48528i		−	0.0795112i	−0.999209	−	0.0397556i	$-0.987342\pi$
$977$		−	2.48528i		−	0.0795112i	0.999209	−	0.0397556i	$-0.0126579\pi$
$978$		0			0
$979$			69.1005i			2.20846i
$980$		0			0
$981$		0			0
$982$	−32.1069			−1.02457
$983$	31.7408			1.01237			0.506187	−	0.862424i	$-0.331055\pi$
$983$	31.7408			1.01237			0.506187	+	0.862424i	$0.331055\pi$
$984$		0			0
$985$		0			0
$986$	44.4660			1.41609
$987$		0			0
$988$	4.23301			0.134670
$989$			55.4230i			1.76235i
$990$		0			0
$991$	57.3368			1.82136			0.910682	−	0.413108i	$-0.135557\pi$
$991$	57.3368			1.82136			0.910682	+	0.413108i	$0.135557\pi$
$992$	3.73074			0.118451
$993$		0			0
$994$	−4.75454	−	5.10969i	−0.150805	−	0.162070i
$995$		0			0
$996$		0			0
$997$			52.6760i			1.66827i	0.551564	+	0.834133i	$0.314031\pi$
$997$			52.6760i			1.66827i	−0.551564	+	0.834133i	$0.685969\pi$
$998$		−	13.6122i		−	0.430886i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.2.b.f.251.3		8
3.2	odd	2		3150.2.b.e.251.7		8
5.2	odd	4		3150.2.d.f.3149.2		8
5.3	odd	4		3150.2.d.a.3149.7		8
5.4	even	2		630.2.b.b.251.6	yes	8
7.6	odd	2		3150.2.b.e.251.3		8
15.2	even	4		3150.2.d.c.3149.2		8
15.8	even	4		3150.2.d.d.3149.7		8
15.14	odd	2		630.2.b.a.251.2	✓	8
20.19	odd	2		5040.2.f.i.881.6		8
21.20	even	2	inner	3150.2.b.f.251.7		8
35.13	even	4		3150.2.d.c.3149.1		8
35.27	even	4		3150.2.d.d.3149.8		8
35.34	odd	2		630.2.b.a.251.6	yes	8
60.59	even	2		5040.2.f.f.881.6		8
105.62	odd	4		3150.2.d.a.3149.8		8
105.83	odd	4		3150.2.d.f.3149.1		8
105.104	even	2		630.2.b.b.251.2	yes	8
140.139	even	2		5040.2.f.f.881.5		8
420.419	odd	2		5040.2.f.i.881.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.b.a.251.2	✓	8	15.14	odd	2
630.2.b.a.251.6	yes	8	35.34	odd	2
630.2.b.b.251.2	yes	8	105.104	even	2
630.2.b.b.251.6	yes	8	5.4	even	2
3150.2.b.e.251.3		8	7.6	odd	2
3150.2.b.e.251.7		8	3.2	odd	2
3150.2.b.f.251.3		8	1.1	even	1	trivial
3150.2.b.f.251.7		8	21.20	even	2	inner
3150.2.d.a.3149.7		8	5.3	odd	4
3150.2.d.a.3149.8		8	105.62	odd	4
3150.2.d.c.3149.1		8	35.13	even	4
3150.2.d.c.3149.2		8	15.2	even	4
3150.2.d.d.3149.7		8	15.8	even	4
3150.2.d.d.3149.8		8	35.27	even	4
3150.2.d.f.3149.1		8	105.83	odd	4
3150.2.d.f.3149.2		8	5.2	odd	4
5040.2.f.f.881.5		8	140.139	even	2
5040.2.f.f.881.6		8	60.59	even	2
5040.2.f.i.881.5		8	420.419	odd	2
5040.2.f.i.881.6		8	20.19	odd	2

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:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3150.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(1.80230 + 1.93693i) q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(1.80230 + 1.93693i) q^{7} +1.00000i q^{8} -3.87386i q^{11} -1.60461i q^{13} +(1.93693 - 1.80230i) q^{14} +1.00000 q^{16} -8.11650 q^{17} -2.63803i q^{19} -3.87386 q^{22} +5.47847i q^{23} -1.60461 q^{26} +(-1.80230 - 1.93693i) q^{28} -5.47847i q^{29} +3.73074i q^{31} -1.00000i q^{32} +8.11650i q^{34} -4.51190 q^{37} -2.63803 q^{38} -1.60461 q^{41} +10.1165 q^{43} +3.87386i q^{44} +5.47847 q^{46} -11.1097 q^{47} +(-0.503406 + 6.98188i) q^{49} +1.60461i q^{52} -2.26926i q^{53} +(-1.93693 + 1.80230i) q^{56} -5.47847 q^{58} +4.61142 q^{59} -11.8472i q^{61} +3.73074 q^{62} -1.00000 q^{64} -6.90729 q^{67} +8.11650 q^{68} -2.63803i q^{71} -13.7477i q^{73} +4.51190i q^{74} +2.63803i q^{76} +(7.50341 - 6.98188i) q^{77} -8.01698 q^{79} +1.60461i q^{82} -3.20921 q^{83} -10.1165i q^{86} +3.87386 q^{88} -17.8376 q^{89} +(3.10801 - 2.89199i) q^{91} -5.47847i q^{92} +11.1097i q^{94} +8.68768i q^{97} +(6.98188 + 0.503406i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} + 4 q^{7}+O(q^{10}) \) 8 * q - 8 * q^4 + 4 * q^7	\( 8 q - 8 q^{4} + 4 q^{7} + 8 q^{16} + 8 q^{26} - 4 q^{28} + 8 q^{37} - 8 q^{38} + 8 q^{41} + 16 q^{43} - 8 q^{46} - 40 q^{47} + 4 q^{49} + 8 q^{58} + 40 q^{62} - 8 q^{64} - 32 q^{67} + 52 q^{77} + 8 q^{79} + 16 q^{83} + 8 q^{89} - 4 q^{91} - 4 q^{98}+O(q^{100}) \) 8 * q - 8 * q^4 + 4 * q^7 + 8 * q^16 + 8 * q^26 - 4 * q^28 + 8 * q^37 - 8 * q^38 + 8 * q^41 + 16 * q^43 - 8 * q^46 - 40 * q^47 + 4 * q^49 + 8 * q^58 + 40 * q^62 - 8 * q^64 - 32 * q^67 + 52 * q^77 + 8 * q^79 + 16 * q^83 + 8 * q^89 - 4 * q^91 - 4 * q^98

Introduction

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