LMFDB - Embedded newform 2151.2.b.a.2150.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,2,Mod(2150,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2151.2150");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.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:	$17.1758214748$
Analytic rank:	$0$
Dimension:	$80$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2150.10
Character	$\chi$	$=$	2151.2150
Dual form			2151.2.b.a.2150.71

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23937i q^{2} -3.01478 q^{4} +3.12574i q^{5} +3.32975i q^{7} +2.27247i q^{8} +O(q^{10})$	\(q-2.23937i q^{2} -3.01478 q^{4} +3.12574i q^{5} +3.32975i q^{7} +2.27247i q^{8} +6.99969 q^{10} +5.13357i q^{11} +4.49241i q^{13} +7.45655 q^{14} -0.940657 q^{16} -4.62052i q^{17} -7.16135i q^{19} -9.42342i q^{20} +11.4960 q^{22} -5.78220 q^{23} -4.77024 q^{25} +10.0602 q^{26} -10.0385i q^{28} -7.08816i q^{29} +0.814919 q^{31} +6.65142i q^{32} -10.3470 q^{34} -10.4079 q^{35} +3.00053i q^{37} -16.0369 q^{38} -7.10315 q^{40} -2.79238 q^{41} +12.8215i q^{43} -15.4766i q^{44} +12.9485i q^{46} +8.09301 q^{47} -4.08724 q^{49} +10.6823i q^{50} -13.5436i q^{52} -11.6959 q^{53} -16.0462 q^{55} -7.56676 q^{56} -15.8730 q^{58} -5.80523 q^{59} -6.39543 q^{61} -1.82491i q^{62} +13.0137 q^{64} -14.0421 q^{65} +4.20704 q^{67} +13.9298i q^{68} +23.3072i q^{70} -11.5462i q^{71} +11.2343i q^{73} +6.71930 q^{74} +21.5899i q^{76} -17.0935 q^{77} -0.0404472i q^{79} -2.94025i q^{80} +6.25316i q^{82} -10.8950i q^{83} +14.4425 q^{85} +28.7121 q^{86} -11.6659 q^{88} -1.00491 q^{89} -14.9586 q^{91} +17.4321 q^{92} -18.1232i q^{94} +22.3845 q^{95} +9.55401i q^{97} +9.15284i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 80 q^{4}+O(q^{10}) $ 80 * q - 80 * q^4	$ 80 q - 80 q^{4} + 16 q^{10} + 56 q^{16} + 40 q^{22} - 64 q^{25} - 8 q^{31} + 32 q^{34} - 24 q^{40} - 104 q^{49} - 24 q^{55} + 56 q^{58} + 40 q^{61} - 80 q^{64} - 8 q^{67} - 8 q^{85} - 120 q^{88} + 32 q^{91}+O(q^{100}) $ 80 * q - 80 * q^4 + 16 * q^10 + 56 * q^16 + 40 * q^22 - 64 * q^25 - 8 * q^31 + 32 * q^34 - 24 * q^40 - 104 * q^49 - 24 * q^55 + 56 * q^58 + 40 * q^61 - 80 * q^64 - 8 * q^67 - 8 * q^85 - 120 * q^88 + 32 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2151\mathbb{Z}\right)^\times$.

$n$	$479$	$1441$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	2.23937i		−	1.58347i	−0.610862	−	0.791737i	$-0.709177\pi$
$2$		−	2.23937i		−	1.58347i	0.610862	−	0.791737i	$-0.290823\pi$
$3$		0			0
$3$		0			0
$4$	−3.01478			−1.50739
$5$			3.12574i			1.39787i	0.715184	+	0.698936i	$0.246343\pi$
$5$			3.12574i			1.39787i	−0.715184	+	0.698936i	$0.753657\pi$
$6$		0			0
$7$			3.32975i			1.25853i	0.777192	+	0.629264i	$0.216643\pi$
$7$			3.32975i			1.25853i	−0.777192	+	0.629264i	$0.783357\pi$
$8$			2.27247i			0.803440i
$9$		0			0
$10$	6.99969			2.21350
$11$			5.13357i			1.54783i	0.633290	+	0.773915i	$0.281704\pi$
$11$			5.13357i			1.54783i	−0.633290	+	0.773915i	$0.718296\pi$
$12$		0			0
$13$			4.49241i			1.24597i	0.782233	+	0.622985i	$0.214080\pi$
$13$			4.49241i			1.24597i	−0.782233	+	0.622985i	$0.785920\pi$
$14$	7.45655			1.99285
$15$		0			0
$16$	−0.940657			−0.235164
$17$		−	4.62052i		−	1.12064i	−0.828276	−	0.560320i	$-0.810678\pi$
$17$		−	4.62052i		−	1.12064i	0.828276	−	0.560320i	$-0.189322\pi$
$18$		0			0
$19$		−	7.16135i		−	1.64293i	−0.570262	−	0.821463i	$-0.693158\pi$
$19$		−	7.16135i		−	1.64293i	0.570262	−	0.821463i	$-0.306842\pi$
$20$		−	9.42342i		−	2.10714i
$21$		0			0
$22$	11.4960			2.45095
$23$	−5.78220			−1.20567			−0.602836	−	0.797865i	$-0.705963\pi$
$23$	−5.78220			−1.20567			−0.602836	+	0.797865i	$0.705963\pi$
$24$		0			0
$25$	−4.77024			−0.954048
$26$	10.0602			1.97296
$27$		0			0
$28$		−	10.0385i		−	1.89709i
$29$		−	7.08816i		−	1.31624i	−0.752914	−	0.658119i	$-0.771352\pi$
$29$		−	7.08816i		−	1.31624i	0.752914	−	0.658119i	$-0.228648\pi$
$30$		0			0
$31$	0.814919			0.146364			0.0731819	−	0.997319i	$-0.476685\pi$
$31$	0.814919			0.146364			0.0731819	+	0.997319i	$0.476685\pi$
$32$			6.65142i			1.17582i
$33$		0			0
$34$	−10.3470			−1.77450
$35$	−10.4079			−1.75926
$36$		0			0
$37$			3.00053i			0.493284i	0.969107	+	0.246642i	$0.0793272\pi$
$37$			3.00053i			0.493284i	−0.969107	+	0.246642i	$0.920673\pi$
$38$	−16.0369			−2.60153
$39$		0			0
$40$	−7.10315			−1.12311
$41$	−2.79238			−0.436096			−0.218048	−	0.975938i	$-0.569969\pi$
$41$	−2.79238			−0.436096			−0.218048	+	0.975938i	$0.569969\pi$
$42$		0			0
$43$			12.8215i			1.95526i	0.210328	+	0.977631i	$0.432547\pi$
$43$			12.8215i			1.95526i	−0.210328	+	0.977631i	$0.567453\pi$
$44$		−	15.4766i		−	2.33318i
$45$		0			0
$46$			12.9485i			1.90915i
$47$	8.09301			1.18049			0.590243	−	0.807225i	$-0.299032\pi$
$47$	8.09301			1.18049			0.590243	+	0.807225i	$0.299032\pi$
$48$		0			0
$49$	−4.08724			−0.583891
$50$			10.6823i			1.51071i
$51$		0			0
$52$		−	13.5436i		−	1.87816i
$53$	−11.6959			−1.60655			−0.803277	−	0.595606i	$-0.796912\pi$
$53$	−11.6959			−1.60655			−0.803277	+	0.595606i	$0.796912\pi$
$54$		0			0
$55$	−16.0462			−2.16367
$56$	−7.56676			−1.01115
$57$		0			0
$58$	−15.8730			−2.08423
$59$	−5.80523			−0.755777			−0.377889	−	0.925851i	$-0.623350\pi$
$59$	−5.80523			−0.755777			−0.377889	+	0.925851i	$0.623350\pi$
$60$		0			0
$61$	−6.39543			−0.818851			−0.409426	−	0.912343i	$-0.634271\pi$
$61$	−6.39543			−0.818851			−0.409426	+	0.912343i	$0.634271\pi$
$62$		−	1.82491i		−	0.231763i
$63$		0			0
$64$	13.0137			1.62671
$65$	−14.0421			−1.74171
$66$		0			0
$67$	4.20704			0.513972			0.256986	−	0.966415i	$-0.417271\pi$
$67$	4.20704			0.513972			0.256986	+	0.966415i	$0.417271\pi$
$68$			13.9298i			1.68924i
$69$		0			0
$70$			23.3072i			2.78574i
$71$		−	11.5462i		−	1.37028i	−0.728409	−	0.685142i	$-0.759740\pi$
$71$		−	11.5462i		−	1.37028i	0.728409	−	0.685142i	$-0.240260\pi$
$72$		0			0
$73$			11.2343i			1.31488i	0.753509	+	0.657438i	$0.228360\pi$
$73$			11.2343i			1.31488i	−0.753509	+	0.657438i	$0.771640\pi$
$74$	6.71930			0.781103
$75$		0			0
$76$			21.5899i			2.47653i
$77$	−17.0935			−1.94799
$78$		0			0
$79$		−	0.0404472i		−	0.00455066i	−0.999997	−	0.00227533i	$-0.999276\pi$
$79$		−	0.0404472i		−	0.00455066i	0.999997	−	0.00227533i	$-0.000724261\pi$
$80$		−	2.94025i		−	0.328730i
$81$		0			0
$82$			6.25316i			0.690546i
$83$		−	10.8950i		−	1.19588i	−0.801541	−	0.597940i	$-0.795986\pi$
$83$		−	10.8950i		−	1.19588i	0.801541	−	0.597940i	$-0.204014\pi$
$84$		0			0
$85$	14.4425			1.56651
$86$	28.7121			3.09611
$87$		0			0
$88$	−11.6659			−1.24359
$89$	−1.00491			−0.106520			−0.0532599	−	0.998581i	$-0.516961\pi$
$89$	−1.00491			−0.106520			−0.0532599	+	0.998581i	$0.516961\pi$
$90$		0			0
$91$	−14.9586			−1.56809
$92$	17.4321			1.81742
$93$		0			0
$94$		−	18.1232i		−	1.86927i
$95$	22.3845			2.29660
$96$		0			0
$97$			9.55401i			0.970063i	0.874497	+	0.485031i	$0.161192\pi$
$97$			9.55401i			0.970063i	−0.874497	+	0.485031i	$0.838808\pi$
$98$			9.15284i			0.924577i
$99$		0			0
$100$	14.3812			1.43812
$101$		−	7.20538i		−	0.716962i	−0.933537	−	0.358481i	$-0.883295\pi$
$101$		−	7.20538i		−	0.716962i	0.933537	−	0.358481i	$-0.116705\pi$
$102$		0			0
$103$		−	13.7617i		−	1.35598i	−0.735070	−	0.677991i	$-0.762851\pi$
$103$		−	13.7617i		−	1.35598i	0.735070	−	0.677991i	$-0.237149\pi$
$104$	−10.2089			−1.00106
$105$		0			0
$106$			26.1914i			2.54394i
$107$	−0.648291			−0.0626727			−0.0313363	−	0.999509i	$-0.509976\pi$
$107$	−0.648291			−0.0626727			−0.0313363	+	0.999509i	$0.509976\pi$
$108$		0			0
$109$	−16.1215			−1.54416			−0.772080	−	0.635526i	$-0.780783\pi$
$109$	−16.1215			−1.54416			−0.772080	+	0.635526i	$0.780783\pi$
$110$			35.9334i			3.42611i
$111$		0			0
$112$		−	3.13215i		−	0.295961i
$113$			5.54129i			0.521281i	0.965436	+	0.260641i	$0.0839337\pi$
$113$			5.54129i			0.521281i	−0.965436	+	0.260641i	$0.916066\pi$
$114$		0			0
$115$		−	18.0736i		−	1.68538i
$116$			21.3693i			1.98409i
$117$		0			0
$118$			13.0001i			1.19675i
$119$	15.3852			1.41036
$120$		0			0
$121$	−15.3535			−1.39578
$122$			14.3217i			1.29663i
$123$		0			0
$124$	−2.45680			−0.220627
$125$			0.718172i			0.0642352i
$126$		0			0
$127$	−15.4280			−1.36902			−0.684508	−	0.729005i	$-0.739983\pi$
$127$	−15.4280			−1.36902			−0.684508	+	0.729005i	$0.739983\pi$
$128$		−	15.8396i		−	1.40004i
$129$		0			0
$130$			31.4455i			2.75795i
$131$	−8.00203			−0.699140			−0.349570	−	0.936910i	$-0.613672\pi$
$131$	−8.00203			−0.699140			−0.349570	+	0.936910i	$0.613672\pi$
$132$		0			0
$133$	23.8455			2.06767
$134$		−	9.42113i		−	0.813862i
$135$		0			0
$136$	10.5000			0.900367
$137$	11.3664			0.971096			0.485548	−	0.874210i	$-0.338620\pi$
$137$	11.3664			0.971096			0.485548	+	0.874210i	$0.338620\pi$
$138$		0			0
$139$			16.3554i			1.38725i	0.720337	+	0.693624i	$0.243987\pi$
$139$			16.3554i			1.38725i	−0.720337	+	0.693624i	$0.756013\pi$
$140$	31.3776			2.65189
$141$		0			0
$142$	−25.8563			−2.16981
$143$	−23.0621			−1.92855
$144$		0			0
$145$	22.1557			1.83993
$146$	25.1578			2.08207
$147$		0			0
$148$		−	9.04594i		−	0.743572i
$149$	9.53653			0.781263			0.390632	−	0.920547i	$-0.372257\pi$
$149$	9.53653			0.781263			0.390632	+	0.920547i	$0.372257\pi$
$150$		0			0
$151$			6.35110i			0.516845i	0.966032	+	0.258423i	$0.0832027\pi$
$151$			6.35110i			0.516845i	−0.966032	+	0.258423i	$0.916797\pi$
$152$	16.2740			1.31999
$153$		0			0
$154$			38.2787i			3.08459i
$155$			2.54722i			0.204598i
$156$		0			0
$157$	−14.3467			−1.14499			−0.572495	−	0.819908i	$-0.694025\pi$
$157$	−14.3467			−1.14499			−0.572495	+	0.819908i	$0.694025\pi$
$158$	−0.0905763			−0.00720586
$159$		0			0
$160$	−20.7906			−1.64364
$161$		−	19.2533i		−	1.51737i
$162$		0			0
$163$	18.7252			1.46667			0.733335	−	0.679867i	$-0.237963\pi$
$163$	18.7252			1.46667			0.733335	+	0.679867i	$0.237963\pi$
$164$	8.41840			0.657367
$165$		0			0
$166$	−24.3979			−1.89365
$167$	9.21494			0.713073			0.356537	−	0.934281i	$-0.383957\pi$
$167$	9.21494			0.713073			0.356537	+	0.934281i	$0.383957\pi$
$168$		0			0
$169$	−7.18176			−0.552443
$170$		−	32.3422i		−	2.48053i
$171$		0			0
$172$		−	38.6540i		−	2.94734i
$173$	−10.8971			−0.828489			−0.414244	−	0.910166i	$-0.635954\pi$
$173$	−10.8971			−0.828489			−0.414244	+	0.910166i	$0.635954\pi$
$174$		0			0
$175$		−	15.8837i		−	1.20070i
$176$		−	4.82893i		−	0.363994i
$177$		0			0
$178$			2.25036i			0.168671i
$179$	−19.8784			−1.48578			−0.742890	−	0.669413i	$-0.766546\pi$
$179$	−19.8784			−1.48578			−0.742890	+	0.669413i	$0.766546\pi$
$180$		0			0
$181$		−	4.12684i		−	0.306746i	−0.988168	−	0.153373i	$-0.950986\pi$
$181$		−	4.12684i		−	0.306746i	0.988168	−	0.153373i	$-0.0490135\pi$
$182$			33.4979i			2.48303i
$183$		0			0
$184$		−	13.1399i		−	0.968685i
$185$	−9.37887			−0.689548
$186$		0			0
$187$	23.7197			1.73456
$188$	−24.3986			−1.77945
$189$		0			0
$190$		−	50.1272i		−	3.63661i
$191$	−7.24029			−0.523889			−0.261944	−	0.965083i	$-0.584364\pi$
$191$	−7.24029			−0.523889			−0.261944	+	0.965083i	$0.584364\pi$
$192$		0			0
$193$	20.9582			1.50860			0.754301	−	0.656529i	$-0.227976\pi$
$193$	20.9582			1.50860			0.754301	+	0.656529i	$0.227976\pi$
$194$	21.3950			1.53607
$195$		0			0
$196$	12.3221			0.880152
$197$			13.2349i			0.942948i	0.881880	+	0.471474i	$0.156278\pi$
$197$			13.2349i			0.942948i	−0.881880	+	0.471474i	$0.843722\pi$
$198$		0			0
$199$		−	17.4304i		−	1.23561i	−0.786332	−	0.617804i	$-0.788023\pi$
$199$		−	17.4304i		−	1.23561i	0.786332	−	0.617804i	$-0.211977\pi$
$200$		−	10.8402i		−	0.766520i
$201$		0			0
$202$	−16.1355			−1.13529
$203$	23.6018			1.65652
$204$		0			0
$205$		−	8.72823i		−	0.609606i
$206$	−30.8176			−2.14716
$207$		0			0
$208$		−	4.22582i		−	0.293008i
$209$	36.7633			2.54297
$210$		0			0
$211$	16.9397			1.16618			0.583090	−	0.812408i	$-0.301844\pi$
$211$	16.9397			1.16618			0.583090	+	0.812408i	$0.301844\pi$
$212$	35.2605			2.42170
$213$		0			0
$214$			1.45176i			0.0992405i
$215$	−40.0767			−2.73321
$216$		0			0
$217$			2.71348i			0.184203i
$218$			36.1020i			2.44514i
$219$		0			0
$220$	48.3758			3.26149
$221$	20.7573			1.39628
$222$		0			0
$223$			8.23139i			0.551215i	0.961270	+	0.275607i	$0.0888790\pi$
$223$			8.23139i			0.551215i	−0.961270	+	0.275607i	$0.911121\pi$
$224$	−22.1476			−1.47980
$225$		0			0
$226$	12.4090			0.825435
$227$	−6.43312			−0.426981			−0.213491	−	0.976945i	$-0.568483\pi$
$227$	−6.43312			−0.426981			−0.213491	+	0.976945i	$0.568483\pi$
$228$		0			0
$229$			27.5067i			1.81769i	0.417133	+	0.908845i	$0.363035\pi$
$229$			27.5067i			1.81769i	−0.417133	+	0.908845i	$0.636965\pi$
$230$	−40.4736			−2.66875
$231$		0			0
$232$	16.1076			1.05752
$233$	9.60307			0.629118			0.314559	−	0.949238i	$-0.398143\pi$
$233$	9.60307			0.629118			0.314559	+	0.949238i	$0.398143\pi$
$234$		0			0
$235$			25.2966i			1.65017i
$236$	17.5015			1.13925
$237$		0			0
$238$		−	34.4531i		−	2.23326i
$239$	−12.8787	+	8.55215i	−0.833054	+	0.553192i
$239$	−12.8787	+	8.55215i	−0.833054	+	0.553192i
$240$		0			0
$241$	11.9114			0.767280			0.383640	−	0.923483i	$-0.374670\pi$
$241$	11.9114			0.767280			0.383640	+	0.923483i	$0.374670\pi$
$242$			34.3823i			2.21018i
$243$		0			0
$244$	19.2808			1.23433
$245$		−	12.7756i		−	0.816205i
$246$		0			0
$247$	32.1717			2.04704
$248$			1.85188i			0.117594i
$249$		0			0
$250$	1.60825			0.101715
$251$		−	1.49442i		−	0.0943269i	−0.998887	−	0.0471635i	$-0.984982\pi$
$251$		−	1.49442i		−	0.0943269i	0.998887	−	0.0471635i	$-0.0150182\pi$
$252$		0			0
$253$		−	29.6833i		−	1.86617i
$254$			34.5491i			2.16780i
$255$		0			0
$256$	−9.44341			−0.590213
$257$			13.2790i			0.828322i	0.910204	+	0.414161i	$0.135925\pi$
$257$			13.2790i			0.828322i	−0.910204	+	0.414161i	$0.864075\pi$
$258$		0			0
$259$	−9.99102			−0.620812
$260$	42.3339			2.62543
$261$		0			0
$262$			17.9195i			1.10707i
$263$			3.45306i			0.212925i	0.994317	+	0.106462i	$0.0339523\pi$
$263$			3.45306i			0.212925i	−0.994317	+	0.106462i	$0.966048\pi$
$264$		0			0
$265$		−	36.5583i		−	2.24576i
$266$		−	53.3989i		−	3.27410i
$267$		0			0
$268$	−12.6833			−0.774757
$269$			9.36503i			0.570996i	0.958379	+	0.285498i	$0.0921590\pi$
$269$			9.36503i			0.570996i	−0.958379	+	0.285498i	$0.907841\pi$
$270$		0			0
$271$	6.50011			0.394853			0.197427	−	0.980318i	$-0.436742\pi$
$271$	6.50011			0.394853			0.197427	+	0.980318i	$0.436742\pi$
$272$			4.34632i			0.263535i
$273$		0			0
$274$		−	25.4535i		−	1.53770i
$275$		−	24.4884i		−	1.47670i
$276$		0			0
$277$		−	15.7854i		−	0.948452i	−0.880403	−	0.474226i	$-0.842728\pi$
$277$		−	15.7854i		−	0.948452i	0.880403	−	0.474226i	$-0.157272\pi$
$278$	36.6259			2.19667
$279$		0			0
$280$		−	23.6517i		−	1.41346i
$281$	−4.36923			−0.260646			−0.130323	−	0.991472i	$-0.541602\pi$
$281$	−4.36923			−0.260646			−0.130323	+	0.991472i	$0.541602\pi$
$282$		0			0
$283$	−25.7554			−1.53100			−0.765500	−	0.643436i	$-0.777508\pi$
$283$	−25.7554			−1.53100			−0.765500	+	0.643436i	$0.777508\pi$
$284$			34.8093i			2.06555i
$285$		0			0
$286$			51.6446i			3.05381i
$287$		−	9.29791i		−	0.548838i
$288$		0			0
$289$	−4.34918			−0.255834
$290$		−	49.6149i		−	2.91349i
$291$		0			0
$292$		−	33.8690i		−	1.98203i
$293$		−	11.3001i		−	0.660156i	−0.943954	−	0.330078i	$-0.892925\pi$
$293$		−	11.3001i		−	0.660156i	0.943954	−	0.330078i	$-0.107075\pi$
$294$		0			0
$295$		−	18.1456i		−	1.05648i
$296$	−6.81862			−0.396324
$297$		0			0
$298$		−	21.3558i		−	1.23711i
$299$		−	25.9760i		−	1.50223i
$300$		0			0
$301$	−42.6924			−2.46075
$302$	14.2225			0.818411
$303$		0			0
$304$			6.73637i			0.386358i
$305$		−	19.9905i		−	1.14465i
$306$		0			0
$307$	7.93893			0.453098			0.226549	−	0.974000i	$-0.427256\pi$
$307$	7.93893			0.453098			0.226549	+	0.974000i	$0.427256\pi$
$308$	51.5332			2.93638
$309$		0			0
$310$	5.70418			0.323975
$311$			19.1961i			1.08851i	0.838919	+	0.544257i	$0.183188\pi$
$311$			19.1961i			1.08851i	−0.838919	+	0.544257i	$0.816812\pi$
$312$		0			0
$313$			8.53465i			0.482407i	0.970475	+	0.241203i	$0.0775421\pi$
$313$			8.53465i			0.482407i	−0.970475	+	0.241203i	$0.922458\pi$
$314$			32.1276i			1.81306i
$315$		0			0
$316$			0.121939i			0.00685963i
$317$	6.74808			0.379010			0.189505	−	0.981880i	$-0.439312\pi$
$317$	6.74808			0.379010			0.189505	+	0.981880i	$0.439312\pi$
$318$		0			0
$319$	36.3876			2.03731
$320$			40.6774i			2.27393i
$321$		0			0
$322$	−43.1152			−2.40272
$323$	−33.0891			−1.84113
$324$		0			0
$325$		−	21.4299i		−	1.18872i
$326$		−	41.9326i		−	2.32243i
$327$		0			0
$328$		−	6.34559i		−	0.350377i
$329$			26.9477i			1.48567i
$330$		0			0
$331$			5.62110i			0.308964i	0.987996	+	0.154482i	$0.0493708\pi$
$331$			5.62110i			0.308964i	−0.987996	+	0.154482i	$0.950629\pi$
$332$			32.8460i			1.80266i
$333$		0			0
$334$		−	20.6357i		−	1.12913i
$335$			13.1501i			0.718468i
$336$		0			0
$337$	23.4365			1.27667			0.638334	−	0.769760i	$-0.279624\pi$
$337$	23.4365			1.27667			0.638334	+	0.769760i	$0.279624\pi$
$338$			16.0826i			0.874779i
$339$		0			0
$340$	−43.5411			−2.36135
$341$			4.18344i			0.226546i
$342$		0			0
$343$			9.69877i			0.523684i
$344$	−29.1365			−1.57093
$345$		0			0
$346$			24.4026i			1.31189i
$347$		−	11.8063i		−	0.633794i	−0.948460	−	0.316897i	$-0.897359\pi$
$347$		−	11.8063i		−	0.633794i	0.948460	−	0.316897i	$-0.102641\pi$
$348$		0			0
$349$	−7.39350			−0.395765			−0.197882	−	0.980226i	$-0.563406\pi$
$349$	−7.39350			−0.395765			−0.197882	+	0.980226i	$0.563406\pi$
$350$	−35.5695			−1.90127
$351$		0			0
$352$	−34.1455			−1.81996
$353$	−9.88919			−0.526348			−0.263174	−	0.964748i	$-0.584769\pi$
$353$	−9.88919			−0.526348			−0.263174	+	0.964748i	$0.584769\pi$
$354$		0			0
$355$	36.0905			1.91548
$356$	3.02957			0.160567
$357$		0			0
$358$			44.5151i			2.35270i
$359$		−	20.7315i		−	1.09416i	−0.837079	−	0.547082i	$-0.815738\pi$
$359$		−	20.7315i		−	1.09416i	0.837079	−	0.547082i	$-0.184262\pi$
$360$		0			0
$361$	−32.2849			−1.69921
$362$	−9.24153			−0.485724
$363$		0			0
$364$	45.0969			2.36372
$365$	−35.1155			−1.83803
$366$		0			0
$367$	−3.53521			−0.184536			−0.0922682	−	0.995734i	$-0.529412\pi$
$367$	−3.53521			−0.184536			−0.0922682	+	0.995734i	$0.529412\pi$
$368$	5.43907			0.283531
$369$		0			0
$370$			21.0028i			1.09188i
$371$		−	38.9444i		−	2.02189i
$372$		0			0
$373$	−28.0750			−1.45367			−0.726833	−	0.686814i	$-0.759009\pi$
$373$	−28.0750			−1.45367			−0.726833	+	0.686814i	$0.759009\pi$
$374$		−	53.1173i		−	2.74663i
$375$		0			0
$376$			18.3911i			0.948450i
$377$	31.8429			1.63999
$378$		0			0
$379$		−	21.2464i		−	1.09136i	−0.837995	−	0.545678i	$-0.816272\pi$
$379$		−	21.2464i		−	1.09136i	0.837995	−	0.545678i	$-0.183728\pi$
$380$	−67.4844			−3.46187
$381$		0			0
$382$			16.2137i			0.829564i
$383$			4.31160i			0.220313i	0.993914	+	0.110156i	$0.0351352\pi$
$383$			4.31160i			0.220313i	−0.993914	+	0.110156i	$0.964865\pi$
$384$		0			0
$385$		−	53.4298i		−	2.72304i
$386$		−	46.9331i		−	2.38883i
$387$		0			0
$388$		−	28.8032i		−	1.46226i
$389$			31.7291i			1.60873i	0.594137	+	0.804364i	$0.297494\pi$
$389$			31.7291i			1.60873i	−0.594137	+	0.804364i	$0.702506\pi$
$390$		0			0
$391$			26.7168i			1.35112i
$392$		−	9.28813i		−	0.469121i
$393$		0			0
$394$	29.6379			1.49313
$395$	0.126427			0.00636125
$396$		0			0
$397$			12.6744i			0.636109i	0.948072	+	0.318055i	$0.103030\pi$
$397$			12.6744i			0.636109i	−0.948072	+	0.318055i	$0.896970\pi$
$398$	−39.0331			−1.95655
$399$		0			0
$400$	4.48716			0.224358
$401$			23.3579i			1.16644i	0.812314	+	0.583220i	$0.198207\pi$
$401$			23.3579i			1.16644i	−0.812314	+	0.583220i	$0.801793\pi$
$402$		0			0
$403$			3.66095i			0.182365i
$404$			21.7226i			1.08074i
$405$		0			0
$406$		−	52.8532i		−	2.62306i
$407$	−15.4034			−0.763520
$408$		0			0
$409$	14.8212			0.732859			0.366430	−	0.930446i	$-0.380580\pi$
$409$	14.8212			0.732859			0.366430	+	0.930446i	$0.380580\pi$
$410$	−19.5458			−0.965296
$411$		0			0
$412$			41.4885i			2.04399i
$413$		−	19.3300i		−	0.951166i
$414$		0			0
$415$	34.0549			1.67169
$416$	−29.8809			−1.46503
$417$		0			0
$418$		−	82.3266i		−	4.02673i
$419$		−	26.3416i		−	1.28687i	−0.765500	−	0.643436i	$-0.777508\pi$
$419$		−	26.3416i		−	1.28687i	0.765500	−	0.643436i	$-0.222492\pi$
$420$		0			0
$421$	−2.52877			−0.123245			−0.0616223	−	0.998100i	$-0.519627\pi$
$421$	−2.52877			−0.123245			−0.0616223	+	0.998100i	$0.519627\pi$
$422$		−	37.9343i		−	1.84661i
$423$		0			0
$424$		−	26.5786i		−	1.29077i
$425$			22.0410i			1.06914i
$426$		0			0
$427$		−	21.2952i		−	1.03055i
$428$	1.95446			0.0944722
$429$		0			0
$430$			89.7465i			4.32796i
$431$			32.8758i			1.58357i	0.610798	+	0.791786i	$0.290849\pi$
$431$			32.8758i			1.58357i	−0.610798	+	0.791786i	$0.709151\pi$
$432$		0			0
$433$			17.8850i			0.859499i	0.902948	+	0.429750i	$0.141398\pi$
$433$			17.8850i			0.859499i	−0.902948	+	0.429750i	$0.858602\pi$
$434$	6.07648			0.291680
$435$		0			0
$436$	48.6028			2.32765
$437$			41.4083i			1.98083i
$438$		0			0
$439$	−30.8250			−1.47120			−0.735598	−	0.677418i	$-0.763099\pi$
$439$	−30.8250			−1.47120			−0.735598	+	0.677418i	$0.763099\pi$
$440$		−	36.4645i		−	1.73838i
$441$		0			0
$442$		−	46.4832i		−	2.21098i
$443$		−	17.6409i		−	0.838145i	−0.907953	−	0.419073i	$-0.862355\pi$
$443$		−	17.6409i		−	0.838145i	0.907953	−	0.419073i	$-0.137645\pi$
$444$		0			0
$445$		−	3.14107i		−	0.148901i
$446$	18.4331			0.872834
$447$		0			0
$448$			43.3323i			2.04726i
$449$	41.5985			1.96315			0.981576	−	0.191071i	$-0.0611962\pi$
$449$	41.5985			1.96315			0.981576	+	0.191071i	$0.0611962\pi$
$450$		0			0
$451$		−	14.3349i		−	0.675002i
$452$		−	16.7058i		−	0.785774i
$453$		0			0
$454$			14.4061i			0.676113i
$455$		−	46.7567i		−	2.19199i
$456$		0			0
$457$	22.3461			1.04531			0.522654	−	0.852545i	$-0.324942\pi$
$457$	22.3461			1.04531			0.522654	+	0.852545i	$0.324942\pi$
$458$	61.5976			2.87827
$459$		0			0
$460$			54.4881i			2.54052i
$461$	−18.9690			−0.883474			−0.441737	−	0.897144i	$-0.645638\pi$
$461$	−18.9690			−0.883474			−0.441737	+	0.897144i	$0.645638\pi$
$462$		0			0
$463$		−	0.0819870i		−	0.00381026i	−0.999998	−	0.00190513i	$-0.999394\pi$
$463$		−	0.0819870i		−	0.00381026i	0.999998	−	0.00190513i	$-0.000606422\pi$
$464$			6.66753i			0.309532i
$465$		0			0
$466$		−	21.5048i		−	0.996193i
$467$	9.09152			0.420705			0.210353	−	0.977626i	$-0.432539\pi$
$467$	9.09152			0.420705			0.210353	+	0.977626i	$0.432539\pi$
$468$		0			0
$469$			14.0084i			0.646848i
$470$	56.6485			2.61300
$471$		0			0
$472$		−	13.1922i		−	0.607221i
$473$	−65.8201			−3.02641
$474$		0			0
$475$			34.1613i			1.56743i
$476$	−46.3829			−2.12596
$477$		0			0
$478$	19.1514	+	28.8402i	0.875966	+	1.31912i
$479$			43.2259i			1.97504i	0.157485	+	0.987521i	$0.449662\pi$
$479$			43.2259i			1.97504i	−0.157485	+	0.987521i	$0.550338\pi$
$480$		0			0
$481$	−13.4796			−0.614618
$482$		−	26.6740i		−	1.21497i
$483$		0			0
$484$	46.2875			2.10398
$485$	−29.8633			−1.35602
$486$		0			0
$487$	−7.32683			−0.332010			−0.166005	−	0.986125i	$-0.553087\pi$
$487$	−7.32683			−0.332010			−0.166005	+	0.986125i	$0.553087\pi$
$488$		−	14.5334i		−	0.657898i
$489$		0			0
$490$	−28.6094			−1.29244
$491$	−40.2721			−1.81745			−0.908727	−	0.417390i	$-0.862945\pi$
$491$	−40.2721			−1.81745			−0.908727	+	0.417390i	$0.862945\pi$
$492$		0			0
$493$	−32.7510			−1.47503
$494$		−	72.0444i		−	3.24143i
$495$		0			0
$496$	−0.766559			−0.0344195
$497$	38.4460			1.72454
$498$		0			0
$499$			42.3491i			1.89581i	0.318556	+	0.947904i	$0.396802\pi$
$499$			42.3491i			1.89581i	−0.318556	+	0.947904i	$0.603198\pi$
$500$		−	2.16513i		−	0.0968276i
$501$		0			0
$502$	−3.34656			−0.149364
$503$			20.6471i			0.920608i	0.887761	+	0.460304i	$0.152260\pi$
$503$			20.6471i			0.920608i	−0.887761	+	0.460304i	$0.847740\pi$
$504$		0			0
$505$	22.5221			1.00222
$506$	−66.4720			−2.95504
$507$		0			0
$508$	46.5121			2.06364
$509$			15.8045i			0.700521i	0.936652	+	0.350261i	$0.113907\pi$
$509$			15.8045i			0.700521i	−0.936652	+	0.350261i	$0.886093\pi$
$510$		0			0
$511$	−37.4074			−1.65481
$512$		−	10.5319i		−	0.465450i
$513$		0			0
$514$	29.7366			1.31163
$515$	43.0155			1.89549
$516$		0			0
$517$			41.5460i			1.82719i
$518$			22.3736i			0.983039i
$519$		0			0
$520$		−	31.9103i		−	1.39936i
$521$	45.3218			1.98559			0.992793	−	0.119842i	$-0.0382388\pi$
$521$	45.3218			1.98559			0.992793	+	0.119842i	$0.0382388\pi$
$522$		0			0
$523$	26.7173			1.16827			0.584134	−	0.811657i	$-0.301434\pi$
$523$	26.7173			1.16827			0.584134	+	0.811657i	$0.301434\pi$
$524$	24.1244			1.05388
$525$		0			0
$526$	7.73267			0.337161
$527$		−	3.76535i		−	0.164021i
$528$		0			0
$529$	10.4338			0.453645
$530$	−81.8675			−3.55610
$531$		0			0
$532$	−71.8890			−3.11678
$533$		−	12.5445i		−	0.543363i
$534$		0			0
$535$		−	2.02639i		−	0.0876084i
$536$			9.56038i			0.412946i
$537$		0			0
$538$	20.9718			0.904158
$539$		−	20.9821i		−	0.903764i
$540$		0			0
$541$		−	10.2178i		−	0.439296i	−0.975579	−	0.219648i	$-0.929509\pi$
$541$		−	10.2178i		−	0.439296i	0.975579	−	0.219648i	$-0.0704909\pi$
$542$		−	14.5561i		−	0.625240i
$543$		0			0
$544$	30.7330			1.31767
$545$		−	50.3916i		−	2.15854i
$546$		0			0
$547$			36.0904i			1.54311i	0.636160	+	0.771557i	$0.280522\pi$
$547$			36.0904i			1.54311i	−0.636160	+	0.771557i	$0.719478\pi$
$548$	−34.2672			−1.46382
$549$		0			0
$550$	−54.8385			−2.33832
$551$	−50.7608			−2.16248
$552$		0			0
$553$	0.134679			0.00572714
$554$	−35.3493			−1.50185
$555$		0			0
$556$		−	49.3080i		−	2.09113i
$557$	6.60090			0.279689			0.139845	−	0.990173i	$-0.455340\pi$
$557$	6.60090			0.279689			0.139845	+	0.990173i	$0.455340\pi$
$558$		0			0
$559$	−57.5995			−2.43620
$560$	9.79029			0.413715
$561$		0			0
$562$			9.78433i			0.412727i
$563$		−	42.3974i		−	1.78684i	−0.449225	−	0.893419i	$-0.648300\pi$
$563$		−	42.3974i		−	1.78684i	0.449225	−	0.893419i	$-0.351700\pi$
$564$		0			0
$565$	−17.3206			−0.728685
$566$			57.6759i			2.42430i
$567$		0			0
$568$	26.2384			1.10094
$569$			40.1001i			1.68108i	0.541748	+	0.840541i	$0.317763\pi$
$569$			40.1001i			1.68108i	−0.541748	+	0.840541i	$0.682237\pi$
$570$		0			0
$571$	−4.11867			−0.172361			−0.0861806	−	0.996280i	$-0.527466\pi$
$571$	−4.11867			−0.172361			−0.0861806	+	0.996280i	$0.527466\pi$
$572$	69.5272			2.90708
$573$		0			0
$574$	−20.8215			−0.869072
$575$	27.5825			1.15027
$576$		0			0
$577$	−10.8120			−0.450111			−0.225056	−	0.974346i	$-0.572256\pi$
$577$	−10.8120			−0.450111			−0.225056	+	0.974346i	$0.572256\pi$
$578$			9.73942i			0.405106i
$579$		0			0
$580$	−66.7947			−2.77350
$581$	36.2776			1.50505
$582$		0			0
$583$		−	60.0416i		−	2.48667i
$584$	−25.5296			−1.05642
$585$		0			0
$586$	−25.3050			−1.04534
$587$			18.7281i			0.772992i	0.922291	+	0.386496i	$0.126315\pi$
$587$			18.7281i			0.772992i	−0.922291	+	0.386496i	$0.873685\pi$
$588$		0			0
$589$		−	5.83592i		−	0.240465i
$590$	−40.6348			−1.67291
$591$		0			0
$592$		−	2.82247i		−	0.116003i
$593$	−6.48770			−0.266418			−0.133209	−	0.991088i	$-0.542528\pi$
$593$	−6.48770			−0.266418			−0.133209	+	0.991088i	$0.542528\pi$
$594$		0			0
$595$			48.0900i			1.97150i
$596$	−28.7506			−1.17767
$597$		0			0
$598$	−58.1699			−2.37875
$599$			20.9820i			0.857303i	0.903470	+	0.428651i	$0.141011\pi$
$599$			20.9820i			0.857303i	−0.903470	+	0.428651i	$0.858989\pi$
$600$		0			0
$601$			32.6564i			1.33208i	0.745915	+	0.666041i	$0.232013\pi$
$601$			32.6564i			1.33208i	−0.745915	+	0.666041i	$0.767987\pi$
$602$			95.6042i			3.89653i
$603$		0			0
$604$		−	19.1472i		−	0.779087i
$605$		−	47.9911i		−	1.95112i
$606$		0			0
$607$		−	40.6155i		−	1.64853i	−0.566203	−	0.824266i	$-0.691588\pi$
$607$		−	40.6155i		−	1.64853i	0.566203	−	0.824266i	$-0.308412\pi$
$608$	47.6331			1.93178
$609$		0			0
$610$	−44.7660			−1.81252
$611$			36.3571i			1.47085i
$612$		0			0
$613$	30.6112			1.23638			0.618188	−	0.786031i	$-0.287867\pi$
$613$	30.6112			1.23638			0.618188	+	0.786031i	$0.287867\pi$
$614$		−	17.7782i		−	0.717470i
$615$		0			0
$616$		−	38.8445i		−	1.56509i
$617$	0.414979			0.0167064			0.00835320	−	0.999965i	$-0.497341\pi$
$617$	0.414979			0.0167064			0.00835320	+	0.999965i	$0.497341\pi$
$618$		0			0
$619$		−	8.53208i		−	0.342933i	−0.985190	−	0.171467i	$-0.945149\pi$
$619$		−	8.53208i		−	0.342933i	0.985190	−	0.171467i	$-0.0548506\pi$
$620$		−	7.67932i		−	0.308409i
$621$		0			0
$622$	42.9873			1.72363
$623$		−	3.34608i		−	0.134058i
$624$		0			0
$625$	−26.0960			−1.04384
$626$	19.1122			0.763879
$627$		0			0
$628$	43.2521			1.72595
$629$	13.8640			0.552794
$630$		0			0
$631$	28.9696			1.15326			0.576632	−	0.817004i	$-0.304367\pi$
$631$	28.9696			1.15326			0.576632	+	0.817004i	$0.304367\pi$
$632$	0.0919151			0.00365619
$633$		0			0
$634$		−	15.1115i		−	0.600153i
$635$		−	48.2240i		−	1.91371i
$636$		0			0
$637$		−	18.3616i		−	0.727511i
$638$		−	81.4853i		−	3.22603i
$639$		0			0
$640$	49.5105			1.95707
$641$			24.2869i			0.959276i	0.877467	+	0.479638i	$0.159232\pi$
$641$			24.2869i			0.959276i	−0.877467	+	0.479638i	$0.840768\pi$
$642$		0			0
$643$	14.6456			0.577565			0.288782	−	0.957395i	$-0.406750\pi$
$643$	14.6456			0.577565			0.288782	+	0.957395i	$0.406750\pi$
$644$			58.0444i			2.28727i
$645$		0			0
$646$			74.0988i			2.91538i
$647$		−	34.4332i		−	1.35371i	−0.736118	−	0.676853i	$-0.763343\pi$
$647$		−	34.4332i		−	1.35371i	0.736118	−	0.676853i	$-0.236657\pi$
$648$		0			0
$649$		−	29.8016i		−	1.16981i
$650$	−47.9894			−1.88230
$651$		0			0
$652$	−56.4523			−2.21084
$653$	−10.6586			−0.417101			−0.208551	−	0.978012i	$-0.566875\pi$
$653$	−10.6586			−0.417101			−0.208551	+	0.978012i	$0.566875\pi$
$654$		0			0
$655$		−	25.0122i		−	0.977309i
$656$	2.62667			0.102554
$657$		0			0
$658$	60.3459			2.35253
$659$	−6.55953			−0.255523			−0.127762	−	0.991805i	$-0.540779\pi$
$659$	−6.55953			−0.255523			−0.127762	+	0.991805i	$0.540779\pi$
$660$		0			0
$661$	−9.11330			−0.354466			−0.177233	−	0.984169i	$-0.556715\pi$
$661$	−9.11330			−0.354466			−0.177233	+	0.984169i	$0.556715\pi$
$662$	12.5877			0.489236
$663$		0			0
$664$	24.7585			0.960818
$665$			74.5348i			2.89034i
$666$		0			0
$667$			40.9852i			1.58695i
$668$	−27.7810			−1.07488
$669$		0			0
$670$	29.4480			1.13768
$671$		−	32.8314i		−	1.26744i
$672$		0			0
$673$		−	37.8088i		−	1.45742i	−0.684822	−	0.728711i	$-0.740120\pi$
$673$		−	37.8088i		−	1.45742i	0.684822	−	0.728711i	$-0.259880\pi$
$674$		−	52.4830i		−	2.02157i
$675$		0			0
$676$	21.6514			0.832747
$677$	23.1563			0.889969			0.444984	−	0.895538i	$-0.353209\pi$
$677$	23.1563			0.889969			0.444984	+	0.895538i	$0.353209\pi$
$678$		0			0
$679$	−31.8125			−1.22085
$680$			32.8202i			1.25860i
$681$		0			0
$682$	9.36828			0.358730
$683$	−8.21022			−0.314155			−0.157078	−	0.987586i	$-0.550207\pi$
$683$	−8.21022			−0.314155			−0.157078	+	0.987586i	$0.550207\pi$
$684$		0			0
$685$			35.5283i			1.35747i
$686$	21.7191			0.829241
$687$		0			0
$688$		−	12.0606i		−	0.459808i
$689$		−	52.5427i		−	2.00172i
$690$		0			0
$691$	−13.4824			−0.512896			−0.256448	−	0.966558i	$-0.582552\pi$
$691$	−13.4824			−0.512896			−0.256448	+	0.966558i	$0.582552\pi$
$692$	32.8523			1.24886
$693$		0			0
$694$	−26.4386			−1.00360
$695$	−51.1228			−1.93920
$696$		0			0
$697$			12.9022i			0.488706i
$698$			16.5568i			0.626684i
$699$		0			0
$700$			47.8859i			1.80992i
$701$	−24.9560			−0.942576			−0.471288	−	0.881979i	$-0.656211\pi$
$701$	−24.9560			−0.942576			−0.471288	+	0.881979i	$0.656211\pi$
$702$		0			0
$703$	21.4878			0.810429
$704$			66.8067i			2.51787i
$705$		0			0
$706$			22.1456i			0.833459i
$707$	23.9921			0.902317
$708$		0			0
$709$		−	30.7177i		−	1.15363i	−0.816876	−	0.576813i	$-0.804296\pi$
$709$		−	30.7177i		−	1.15363i	0.816876	−	0.576813i	$-0.195704\pi$
$710$		−	80.8199i		−	3.03312i
$711$		0			0
$712$		−	2.28362i		−	0.0855822i
$713$	−4.71202			−0.176467
$714$		0			0
$715$		−	72.0861i		−	2.69587i
$716$	59.9290			2.23965
$717$		0			0
$718$	−46.4254			−1.73258
$719$		−	25.2171i		−	0.940439i	−0.882549	−	0.470220i	$-0.844175\pi$
$719$		−	25.2171i		−	0.940439i	0.882549	−	0.470220i	$-0.155825\pi$
$720$		0			0
$721$	45.8231			1.70654
$722$			72.2979i			2.69065i
$723$		0			0
$724$			12.4415i			0.462385i
$725$			33.8122i			1.25575i
$726$		0			0
$727$	7.84152			0.290826			0.145413	−	0.989371i	$-0.453549\pi$
$727$	7.84152			0.290826			0.145413	+	0.989371i	$0.453549\pi$
$728$		−	33.9930i		−	1.25986i
$729$		0			0
$730$			78.6366i			2.91047i
$731$	59.2420			2.19114
$732$		0			0
$733$	−16.9156			−0.624791			−0.312395	−	0.949952i	$-0.601131\pi$
$733$	−16.9156			−0.624791			−0.312395	+	0.949952i	$0.601131\pi$
$734$			7.91665i			0.292209i
$735$		0			0
$736$		−	38.4599i		−	1.41765i
$737$			21.5972i			0.795541i
$738$		0			0
$739$	−19.3608			−0.712199			−0.356100	−	0.934448i	$-0.615894\pi$
$739$	−19.3608			−0.712199			−0.356100	+	0.934448i	$0.615894\pi$
$740$	28.2752			1.03942
$741$		0			0
$742$	−87.2109			−3.20161
$743$	24.8071			0.910084			0.455042	−	0.890470i	$-0.349624\pi$
$743$	24.8071			0.910084			0.455042	+	0.890470i	$0.349624\pi$
$744$		0			0
$745$			29.8087i			1.09211i
$746$			62.8703i			2.30184i
$747$		0			0
$748$	−71.5098			−2.61466
$749$		−	2.15865i		−	0.0788753i
$750$		0			0
$751$	43.1677			1.57521			0.787605	−	0.616180i	$-0.211321\pi$
$751$	43.1677			1.57521			0.787605	+	0.616180i	$0.211321\pi$
$752$	−7.61275			−0.277608
$753$		0			0
$754$		−	71.3081i		−	2.59689i
$755$	−19.8519			−0.722484
$756$		0			0
$757$	−12.9969			−0.472380			−0.236190	−	0.971707i	$-0.575899\pi$
$757$	−12.9969			−0.472380			−0.236190	+	0.971707i	$0.575899\pi$
$758$	−47.5786			−1.72813
$759$		0			0
$760$			50.8681i			1.84518i
$761$			7.85186i			0.284630i	0.989821	+	0.142315i	$0.0454546\pi$
$761$			7.85186i			0.284630i	−0.989821	+	0.142315i	$0.954545\pi$
$762$		0			0
$763$		−	53.6806i		−	1.94337i
$764$	21.8279			0.789705
$765$		0			0
$766$	9.65528			0.348859
$767$		−	26.0795i		−	0.941676i
$768$		0			0
$769$		−	16.8206i		−	0.606566i	−0.952901	−	0.303283i	$-0.901917\pi$
$769$		−	16.8206i		−	0.606566i	0.952901	−	0.303283i	$-0.0980827\pi$
$770$	−119.649			−4.31186
$771$		0			0
$772$	−63.1843			−2.27405
$773$	−8.39126			−0.301813			−0.150906	−	0.988548i	$-0.548219\pi$
$773$	−8.39126			−0.301813			−0.150906	+	0.988548i	$0.548219\pi$
$774$		0			0
$775$	−3.88736			−0.139638
$776$	−21.7112			−0.779387
$777$		0			0
$778$	71.0532			2.54738
$779$			19.9972i			0.716473i
$780$		0			0
$781$	59.2733			2.12097
$782$	59.8287			2.13947
$783$		0			0
$784$	3.84469			0.137310
$785$		−	44.8440i		−	1.60055i
$786$		0			0
$787$		−	27.0294i		−	0.963495i	−0.876310	−	0.481747i	$-0.840002\pi$
$787$		−	27.0294i		−	0.963495i	0.876310	−	0.481747i	$-0.159998\pi$
$788$		−	39.9003i		−	1.42139i
$789$		0			0
$790$		−	0.283118i		−	0.0100729i
$791$	−18.4511			−0.656047
$792$		0			0
$793$		−	28.7309i		−	1.02026i
$794$	28.3826			1.00726
$795$		0			0
$796$			52.5488i			1.86254i
$797$			37.3780i			1.32400i	0.749506	+	0.661998i	$0.230291\pi$
$797$			37.3780i			1.32400i	−0.749506	+	0.661998i	$0.769709\pi$
$798$		0			0
$799$		−	37.3939i		−	1.32290i
$800$		−	31.7289i		−	1.12179i
$801$		0			0
$802$	52.3071			1.84703
$803$	−57.6721			−2.03520
$804$		0			0
$805$	60.1807			2.12109
$806$	8.19823			0.288770
$807$		0			0
$808$	16.3740			0.576036
$809$	−0.295416			−0.0103863			−0.00519314	−	0.999987i	$-0.501653\pi$
$809$	−0.295416			−0.0103863			−0.00519314	+	0.999987i	$0.501653\pi$
$810$		0			0
$811$		−	18.4753i		−	0.648755i	−0.945928	−	0.324378i	$-0.894845\pi$
$811$		−	18.4753i		−	0.648755i	0.945928	−	0.324378i	$-0.105155\pi$
$812$	−71.1543			−2.49703
$813$		0			0
$814$			34.4940i			1.20901i
$815$			58.5300i			2.05022i
$816$		0			0
$817$	91.8193			3.21235
$818$		−	33.1901i		−	1.16046i
$819$		0			0
$820$			26.3137i			0.918915i
$821$	4.05916			0.141666			0.0708329	−	0.997488i	$-0.477434\pi$
$821$	4.05916			0.141666			0.0708329	+	0.997488i	$0.477434\pi$
$822$		0			0
$823$		−	38.9639i		−	1.35820i	−0.734047	−	0.679098i	$-0.762371\pi$
$823$		−	38.9639i		−	1.35820i	0.734047	−	0.679098i	$-0.237629\pi$
$824$	31.2731			1.08945
$825$		0			0
$826$	−43.2870			−1.50615
$827$			7.08062i			0.246217i	0.992393	+	0.123109i	$0.0392863\pi$
$827$			7.08062i			0.246217i	−0.992393	+	0.123109i	$0.960714\pi$
$828$		0			0
$829$			27.2913i			0.947867i	0.880561	+	0.473934i	$0.157166\pi$
$829$			27.2913i			0.947867i	−0.880561	+	0.473934i	$0.842834\pi$
$830$		−	76.2615i		−	2.64708i
$831$		0			0
$832$			58.4628i			2.02683i
$833$			18.8852i			0.654332i
$834$		0			0
$835$			28.8035i			0.996786i
$836$	−110.833			−3.83325
$837$		0			0
$838$	−58.9887			−2.03773
$839$			36.7413i			1.26845i	0.773148	+	0.634225i	$0.218681\pi$
$839$			36.7413i			1.26845i	−0.773148	+	0.634225i	$0.781319\pi$
$840$		0			0
$841$	−21.2420			−0.732484
$842$			5.66285i			0.195155i
$843$		0			0
$844$	−51.0696			−1.75789
$845$		−	22.4483i		−	0.772245i
$846$		0			0
$847$		−	51.1234i		−	1.75662i
$848$	11.0018			0.377804
$849$		0			0
$850$	49.3579			1.69296
$851$		−	17.3497i		−	0.594739i
$852$		0			0
$853$	19.0944			0.653781			0.326891	−	0.945062i	$-0.393999\pi$
$853$	19.0944			0.653781			0.326891	+	0.945062i	$0.393999\pi$
$854$	−47.6878			−1.63184
$855$		0			0
$856$		−	1.47322i		−	0.0503537i
$857$	5.38273			0.183871			0.0919353	−	0.995765i	$-0.470695\pi$
$857$	5.38273			0.183871			0.0919353	+	0.995765i	$0.470695\pi$
$858$		0			0
$859$	9.20166			0.313957			0.156978	−	0.987602i	$-0.449825\pi$
$859$	9.20166			0.313957			0.156978	+	0.987602i	$0.449825\pi$
$860$	120.822			4.12001
$861$		0			0
$862$	73.6211			2.50755
$863$	−54.2628			−1.84713			−0.923564	−	0.383445i	$-0.874738\pi$
$863$	−54.2628			−1.84713			−0.923564	+	0.383445i	$0.874738\pi$
$864$		0			0
$865$		−	34.0614i		−	1.15812i
$866$	40.0512			1.36099
$867$		0			0
$868$		−	8.18054i		−	0.277666i
$869$	0.207638			0.00704365
$870$		0			0
$871$			18.8998i			0.640394i
$872$		−	36.6356i		−	1.24064i
$873$		0			0
$874$	92.7286			3.13659
$875$	−2.39133			−0.0808418
$876$		0			0
$877$	−6.30079			−0.212763			−0.106381	−	0.994325i	$-0.533926\pi$
$877$	−6.30079			−0.212763			−0.106381	+	0.994325i	$0.533926\pi$
$878$			69.0286i			2.32960i
$879$		0			0
$880$	15.0940			0.508818
$881$	20.8072			0.701013			0.350506	−	0.936560i	$-0.386009\pi$
$881$	20.8072			0.701013			0.350506	+	0.936560i	$0.386009\pi$
$882$		0			0
$883$	19.6236			0.660388			0.330194	−	0.943913i	$-0.392886\pi$
$883$	19.6236			0.660388			0.330194	+	0.943913i	$0.392886\pi$
$884$	−62.5786			−2.10475
$885$		0			0
$886$	−39.5046			−1.32718
$887$		−	13.2504i		−	0.444906i	−0.974943	−	0.222453i	$-0.928594\pi$
$887$		−	13.2504i		−	0.444906i	0.974943	−	0.222453i	$-0.0714064\pi$
$888$		0			0
$889$		−	51.3715i		−	1.72295i
$890$	−7.03402			−0.235781
$891$		0			0
$892$		−	24.8158i		−	0.830896i
$893$		−	57.9568i		−	1.93945i
$894$		0			0
$895$		−	62.1347i		−	2.07693i
$896$	52.7420			1.76199
$897$		0			0
$898$		−	93.1544i		−	3.10860i
$899$		−	5.77628i		−	0.192650i
$900$		0			0
$901$			54.0410i			1.80037i
$902$	−32.1010			−1.06885
$903$		0			0
$904$	−12.5924			−0.418818
$905$	12.8994			0.428791
$906$		0			0
$907$			29.9021i			0.992882i	0.868070	+	0.496441i	$0.165360\pi$
$907$			29.9021i			0.992882i	−0.868070	+	0.496441i	$0.834640\pi$
$908$	19.3944			0.643627
$909$		0			0
$910$	−104.706			−3.47096
$911$	18.8551			0.624697			0.312348	−	0.949968i	$-0.398884\pi$
$911$	18.8551			0.624697			0.312348	+	0.949968i	$0.398884\pi$
$912$		0			0
$913$	55.9302			1.85102
$914$		−	50.0412i		−	1.65522i
$915$		0			0
$916$		−	82.9265i		−	2.73997i
$917$		−	26.6448i		−	0.879887i
$918$		0			0
$919$	−6.74535			−0.222509			−0.111254	−	0.993792i	$-0.535487\pi$
$919$	−6.74535			−0.222509			−0.111254	+	0.993792i	$0.535487\pi$
$920$	41.0718			1.35410
$921$		0			0
$922$			42.4786i			1.39896i
$923$	51.8704			1.70733
$924$		0			0
$925$		−	14.3132i		−	0.470617i
$926$	−0.183599			−0.00603345
$927$		0			0
$928$	47.1464			1.54766
$929$	19.6837			0.645803			0.322901	−	0.946433i	$-0.395342\pi$
$929$	19.6837			0.645803			0.322901	+	0.946433i	$0.395342\pi$
$930$		0			0
$931$			29.2701i			0.959290i
$932$	−28.9512			−0.948327
$933$		0			0
$934$		−	20.3593i		−	0.666176i
$935$			74.1417i			2.42469i
$936$		0			0
$937$	6.62002			0.216266			0.108133	−	0.994136i	$-0.465513\pi$
$937$	6.62002			0.216266			0.108133	+	0.994136i	$0.465513\pi$
$938$	31.3700			1.02427
$939$		0			0
$940$		−	76.2638i		−	2.48745i
$941$	7.64326			0.249163			0.124582	−	0.992209i	$-0.460241\pi$
$941$	7.64326			0.249163			0.124582	+	0.992209i	$0.460241\pi$
$942$		0			0
$943$	16.1461			0.525788
$944$	5.46074			0.177732
$945$		0			0
$946$			147.396i			4.79224i
$947$	−27.2178			−0.884460			−0.442230	−	0.896902i	$-0.645813\pi$
$947$	−27.2178			−0.884460			−0.442230	+	0.896902i	$0.645813\pi$
$948$		0			0
$949$	−50.4691			−1.63830
$950$	76.4999			2.48198
$951$		0			0
$952$			34.9623i			1.13314i
$953$	39.0826			1.26601			0.633006	−	0.774147i	$-0.281821\pi$
$953$	39.0826			1.26601			0.633006	+	0.774147i	$0.281821\pi$
$954$		0			0
$955$		−	22.6312i		−	0.732330i
$956$	38.8264	−	25.7829i	1.25574	−	0.833877i
$957$		0			0
$958$	96.7989			3.12743
$959$			37.8472i			1.22215i
$960$		0			0
$961$	−30.3359			−0.978578
$962$			30.1859i			0.973231i
$963$		0			0
$964$	−35.9102			−1.15659
$965$			65.5097i			2.10883i
$966$		0			0
$967$	−11.9183			−0.383267			−0.191633	−	0.981467i	$-0.561378\pi$
$967$	−11.9183			−0.383267			−0.191633	+	0.981467i	$0.561378\pi$
$968$		−	34.8905i		−	1.12142i
$969$		0			0
$970$			66.8751i			2.14723i
$971$			18.5413i			0.595019i	0.954719	+	0.297510i	$0.0961560\pi$
$971$			18.5413i			0.595019i	−0.954719	+	0.297510i	$0.903844\pi$
$972$		0			0
$973$	−54.4595			−1.74589
$974$			16.4075i			0.525730i
$975$		0			0
$976$	6.01591			0.192565
$977$	10.3670			0.331669			0.165834	−	0.986154i	$-0.446968\pi$
$977$	10.3670			0.331669			0.165834	+	0.986154i	$0.446968\pi$
$978$		0			0
$979$		−	5.15875i		−	0.164874i
$980$			38.5157i			1.23034i
$981$		0			0
$982$			90.1842i			2.87789i
$983$		−	31.3206i		−	0.998971i	−0.866322	−	0.499486i	$-0.833522\pi$
$983$		−	31.3206i		−	0.998971i	0.866322	−	0.499486i	$-0.166478\pi$
$984$		0			0
$985$	−41.3689			−1.31812
$986$			73.3416i			2.33567i
$987$		0			0
$988$	−96.9907			−3.08568
$989$		−	74.1365i		−	2.35740i
$990$		0			0
$991$		−	14.9562i		−	0.475101i	−0.971375	−	0.237550i	$-0.923656\pi$
$991$		−	14.9562i		−	0.475101i	0.971375	−	0.237550i	$-0.0763445\pi$
$992$			5.42037i			0.172097i
$993$		0			0
$994$		−	86.0949i		−	2.73076i
$995$	54.4828			1.72722
$996$		0			0
$997$		−	34.6345i		−	1.09688i	−0.836188	−	0.548442i	$-0.815221\pi$
$997$		−	34.6345i		−	1.09688i	0.836188	−	0.548442i	$-0.184779\pi$
$998$	94.8354			3.00196
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.b.a.2150.10	yes	80
3.2	odd	2	inner	2151.2.b.a.2150.72	yes	80
239.238	odd	2	inner	2151.2.b.a.2150.9	✓	80
717.716	even	2	inner	2151.2.b.a.2150.71	yes	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.b.a.2150.9	✓	80	239.238	odd	2	inner
2151.2.b.a.2150.10	yes	80	1.1	even	1	trivial
2151.2.b.a.2150.71	yes	80	717.716	even	2	inner
2151.2.b.a.2150.72	yes	80	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23937i q^{2} -3.01478 q^{4} +3.12574i q^{5} +3.32975i q^{7} +2.27247i q^{8} +O(q^{10})\)	\(q-2.23937i q^{2} -3.01478 q^{4} +3.12574i q^{5} +3.32975i q^{7} +2.27247i q^{8} +6.99969 q^{10} +5.13357i q^{11} +4.49241i q^{13} +7.45655 q^{14} -0.940657 q^{16} -4.62052i q^{17} -7.16135i q^{19} -9.42342i q^{20} +11.4960 q^{22} -5.78220 q^{23} -4.77024 q^{25} +10.0602 q^{26} -10.0385i q^{28} -7.08816i q^{29} +0.814919 q^{31} +6.65142i q^{32} -10.3470 q^{34} -10.4079 q^{35} +3.00053i q^{37} -16.0369 q^{38} -7.10315 q^{40} -2.79238 q^{41} +12.8215i q^{43} -15.4766i q^{44} +12.9485i q^{46} +8.09301 q^{47} -4.08724 q^{49} +10.6823i q^{50} -13.5436i q^{52} -11.6959 q^{53} -16.0462 q^{55} -7.56676 q^{56} -15.8730 q^{58} -5.80523 q^{59} -6.39543 q^{61} -1.82491i q^{62} +13.0137 q^{64} -14.0421 q^{65} +4.20704 q^{67} +13.9298i q^{68} +23.3072i q^{70} -11.5462i q^{71} +11.2343i q^{73} +6.71930 q^{74} +21.5899i q^{76} -17.0935 q^{77} -0.0404472i q^{79} -2.94025i q^{80} +6.25316i q^{82} -10.8950i q^{83} +14.4425 q^{85} +28.7121 q^{86} -11.6659 q^{88} -1.00491 q^{89} -14.9586 q^{91} +17.4321 q^{92} -18.1232i q^{94} +22.3845 q^{95} +9.55401i q^{97} +9.15284i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 80 q^{4}+O(q^{10}) \) 80 * q - 80 * q^4	\( 80 q - 80 q^{4} + 16 q^{10} + 56 q^{16} + 40 q^{22} - 64 q^{25} - 8 q^{31} + 32 q^{34} - 24 q^{40} - 104 q^{49} - 24 q^{55} + 56 q^{58} + 40 q^{61} - 80 q^{64} - 8 q^{67} - 8 q^{85} - 120 q^{88} + 32 q^{91}+O(q^{100}) \) 80 * q - 80 * q^4 + 16 * q^10 + 56 * q^16 + 40 * q^22 - 64 * q^25 - 8 * q^31 + 32 * q^34 - 24 * q^40 - 104 * q^49 - 24 * q^55 + 56 * q^58 + 40 * q^61 - 80 * q^64 - 8 * q^67 - 8 * q^85 - 120 * q^88 + 32 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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