LMFDB - Embedded newform 3465.2.a.bl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3465,2,Mod(1,3465)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3465, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3465.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3465 = 3^{2} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3465.a (trivial)

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:	yes
Analytic conductor:	$27.6681643004$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13448.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 2 $ x^4 - 7*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1155)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.58874$ of defining polynomial
Character	$\chi$	$=$	3465.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.58874 q^{2} +4.70156 q^{4} +1.00000 q^{5} -1.00000 q^{7} -6.99364 q^{8} +O(q^{10})$	\(q-2.58874 q^{2} +4.70156 q^{4} +1.00000 q^{5} -1.00000 q^{7} -6.99364 q^{8} -2.58874 q^{10} +1.00000 q^{11} +6.40490 q^{13} +2.58874 q^{14} +8.70156 q^{16} +3.70156 q^{17} +7.24672 q^{19} +4.70156 q^{20} -2.58874 q^{22} +7.33388 q^{23} +1.00000 q^{25} -16.5806 q^{26} -4.70156 q^{28} +8.10646 q^{29} +6.77258 q^{31} -8.53879 q^{32} -9.58237 q^{34} -1.00000 q^{35} -3.95005 q^{37} -18.7598 q^{38} -6.99364 q^{40} +2.31773 q^{41} -7.65161 q^{43} +4.70156 q^{44} -18.9855 q^{46} -1.45307 q^{47} +1.00000 q^{49} -2.58874 q^{50} +30.1130 q^{52} -9.10823 q^{53} +1.00000 q^{55} +6.99364 q^{56} -20.9855 q^{58} +2.92898 q^{59} -8.42419 q^{61} -17.5324 q^{62} +4.70156 q^{64} +6.40490 q^{65} -8.72263 q^{67} +17.4031 q^{68} +2.58874 q^{70} +4.40490 q^{71} +3.40667 q^{73} +10.2256 q^{74} +34.0709 q^{76} -1.00000 q^{77} +4.40490 q^{79} +8.70156 q^{80} -6.00000 q^{82} +8.96620 q^{83} +3.70156 q^{85} +19.8080 q^{86} -6.99364 q^{88} -0.206355 q^{89} -6.40490 q^{91} +34.4807 q^{92} +3.76162 q^{94} +7.24672 q^{95} -13.9644 q^{97} -2.58874 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{4} + 4 q^{5} - 4 q^{7}+O(q^{10}) $ 4 * q + 6 * q^4 + 4 * q^5 - 4 * q^7	$ 4 q + 6 q^{4} + 4 q^{5} - 4 q^{7} + 4 q^{11} + 8 q^{13} + 22 q^{16} + 2 q^{17} + 10 q^{19} + 6 q^{20} + 2 q^{23} + 4 q^{25} - 20 q^{26} - 6 q^{28} + 2 q^{29} + 24 q^{31} - 4 q^{35} + 8 q^{37} - 16 q^{38} + 6 q^{43} + 6 q^{44} - 12 q^{46} - 4 q^{47} + 4 q^{49} + 12 q^{52} - 14 q^{53} + 4 q^{55} - 20 q^{58} + 2 q^{59} + 6 q^{61} - 8 q^{62} + 6 q^{64} + 8 q^{65} - 8 q^{67} + 44 q^{68} + 4 q^{73} + 36 q^{74} + 56 q^{76} - 4 q^{77} + 22 q^{80} - 24 q^{82} - 6 q^{83} + 2 q^{85} + 36 q^{86} - 18 q^{89} - 8 q^{91} + 44 q^{92} - 36 q^{94} + 10 q^{95} - 6 q^{97}+O(q^{100}) $ 4 * q + 6 * q^4 + 4 * q^5 - 4 * q^7 + 4 * q^11 + 8 * q^13 + 22 * q^16 + 2 * q^17 + 10 * q^19 + 6 * q^20 + 2 * q^23 + 4 * q^25 - 20 * q^26 - 6 * q^28 + 2 * q^29 + 24 * q^31 - 4 * q^35 + 8 * q^37 - 16 * q^38 + 6 * q^43 + 6 * q^44 - 12 * q^46 - 4 * q^47 + 4 * q^49 + 12 * q^52 - 14 * q^53 + 4 * q^55 - 20 * q^58 + 2 * q^59 + 6 * q^61 - 8 * q^62 + 6 * q^64 + 8 * q^65 - 8 * q^67 + 44 * q^68 + 4 * q^73 + 36 * q^74 + 56 * q^76 - 4 * q^77 + 22 * q^80 - 24 * q^82 - 6 * q^83 + 2 * q^85 + 36 * q^86 - 18 * q^89 - 8 * q^91 + 44 * q^92 - 36 * q^94 + 10 * q^95 - 6 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	−2.58874	−1.83051	−0.915257	−	0.402871i	$-0.868012\pi$
$2$	−2.58874	−1.83051	−0.915257	+	0.402871i	$0.868012\pi$
$3$	0	0
$3$	0	0
$4$	4.70156	2.35078
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−6.99364	−2.47262
$9$	0	0
$10$	−2.58874	−0.818631
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	6.40490	1.77640	0.888200	−	0.459458i	$-0.151956\pi$
$13$	6.40490	1.77640	0.888200	+	0.459458i	$0.151956\pi$
$14$	2.58874	0.691869
$15$	0	0
$16$	8.70156	2.17539
$17$	3.70156	0.897761	0.448880	−	0.893592i	$-0.351823\pi$
$17$	3.70156	0.897761	0.448880	+	0.893592i	$0.351823\pi$
$18$	0	0
$19$	7.24672	1.66251	0.831255	−	0.555891i	$-0.187623\pi$
$19$	7.24672	1.66251	0.831255	+	0.555891i	$0.187623\pi$
$20$	4.70156	1.05130
$21$	0	0
$22$	−2.58874	−0.551921
$23$	7.33388	1.52922	0.764610	−	0.644493i	$-0.222932\pi$
$23$	7.33388	1.52922	0.764610	+	0.644493i	$0.222932\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−16.5806	−3.25172
$27$	0	0
$28$	−4.70156	−0.888512
$29$	8.10646	1.50533	0.752666	−	0.658403i	$-0.228768\pi$
$29$	8.10646	1.50533	0.752666	+	0.658403i	$0.228768\pi$
$30$	0	0
$31$	6.77258	1.21639	0.608195	−	0.793787i	$-0.291894\pi$
$31$	6.77258	1.21639	0.608195	+	0.793787i	$0.291894\pi$
$32$	−8.53879	−1.50946
$33$	0	0
$34$	−9.58237	−1.64336
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−3.95005	−0.649385	−0.324692	−	0.945820i	$-0.605261\pi$
$37$	−3.95005	−0.649385	−0.324692	+	0.945820i	$0.605261\pi$
$38$	−18.7598	−3.04325
$39$	0	0
$40$	−6.99364	−1.10579
$41$	2.31773	0.361969	0.180984	−	0.983486i	$-0.442072\pi$
$41$	2.31773	0.361969	0.180984	+	0.983486i	$0.442072\pi$
$42$	0	0
$43$	−7.65161	−1.16686	−0.583430	−	0.812163i	$-0.698290\pi$
$43$	−7.65161	−1.16686	−0.583430	+	0.812163i	$0.698290\pi$
$44$	4.70156	0.708787
$45$	0	0
$46$	−18.9855	−2.79926
$47$	−1.45307	−0.211952	−0.105976	−	0.994369i	$-0.533797\pi$
$47$	−1.45307	−0.211952	−0.105976	+	0.994369i	$0.533797\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.58874	−0.366103
$51$	0	0
$52$	30.1130	4.17593
$53$	−9.10823	−1.25111	−0.625556	−	0.780179i	$-0.715128\pi$
$53$	−9.10823	−1.25111	−0.625556	+	0.780179i	$0.715128\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	6.99364	0.934564
$57$	0	0
$58$	−20.9855	−2.75553
$59$	2.92898	0.381321	0.190661	−	0.981656i	$-0.438937\pi$
$59$	2.92898	0.381321	0.190661	+	0.981656i	$0.438937\pi$
$60$	0	0
$61$	−8.42419	−1.07861	−0.539304	−	0.842111i	$-0.681312\pi$
$61$	−8.42419	−1.07861	−0.539304	+	0.842111i	$0.681312\pi$
$62$	−17.5324	−2.22662
$63$	0	0
$64$	4.70156	0.587695
$65$	6.40490	0.794430
$66$	0	0
$67$	−8.72263	−1.06564	−0.532819	−	0.846229i	$-0.678868\pi$
$67$	−8.72263	−1.06564	−0.532819	+	0.846229i	$0.678868\pi$
$68$	17.4031	2.11044
$69$	0	0
$70$	2.58874	0.309413
$71$	4.40490	0.522765	0.261383	−	0.965235i	$-0.415822\pi$
$71$	4.40490	0.522765	0.261383	+	0.965235i	$0.415822\pi$
$72$	0	0
$73$	3.40667	0.398721	0.199360	−	0.979926i	$-0.436114\pi$
$73$	3.40667	0.398721	0.199360	+	0.979926i	$0.436114\pi$
$74$	10.2256	1.18871
$75$	0	0
$76$	34.0709	3.90820
$77$	−1.00000	−0.113961
$78$	0	0
$79$	4.40490	0.495590	0.247795	−	0.968813i	$-0.420294\pi$
$79$	4.40490	0.495590	0.247795	+	0.968813i	$0.420294\pi$
$80$	8.70156	0.972864
$81$	0	0
$82$	−6.00000	−0.662589
$83$	8.96620	0.984169	0.492084	−	0.870548i	$-0.336235\pi$
$83$	8.96620	0.984169	0.492084	+	0.870548i	$0.336235\pi$
$84$	0	0
$85$	3.70156	0.401491
$86$	19.8080	2.13595
$87$	0	0
$88$	−6.99364	−0.745524
$89$	−0.206355	−0.0218736	−0.0109368	−	0.999940i	$-0.503481\pi$
$89$	−0.206355	−0.0218736	−0.0109368	+	0.999940i	$0.503481\pi$
$90$	0	0
$91$	−6.40490	−0.671416
$92$	34.4807	3.59486
$93$	0	0
$94$	3.76162	0.387982
$95$	7.24672	0.743497
$96$	0	0
$97$	−13.9644	−1.41787	−0.708936	−	0.705272i	$-0.750825\pi$
$97$	−13.9644	−1.41787	−0.708936	+	0.705272i	$0.750825\pi$
$98$	−2.58874	−0.261502
$99$	0	0
$100$	4.70156	0.470156
$101$	−3.77080	−0.375209	−0.187605	−	0.982245i	$-0.560072\pi$
$101$	−3.77080	−0.375209	−0.187605	+	0.982245i	$0.560072\pi$
$102$	0	0
$103$	−8.56131	−0.843570	−0.421785	−	0.906696i	$-0.638596\pi$
$103$	−8.56131	−0.843570	−0.421785	+	0.906696i	$0.638596\pi$
$104$	−44.7935	−4.39237
$105$	0	0
$106$	23.5788	2.29018
$107$	−14.1630	−1.36919	−0.684593	−	0.728925i	$-0.740020\pi$
$107$	−14.1630	−1.36919	−0.684593	+	0.728925i	$0.740020\pi$
$108$	0	0
$109$	7.14026	0.683913	0.341956	−	0.939716i	$-0.388911\pi$
$109$	7.14026	0.683913	0.341956	+	0.939716i	$0.388911\pi$
$110$	−2.58874	−0.246826
$111$	0	0
$112$	−8.70156	−0.822220
$113$	1.10823	0.104254	0.0521269	−	0.998640i	$-0.483400\pi$
$113$	1.10823	0.104254	0.0521269	+	0.998640i	$0.483400\pi$
$114$	0	0
$115$	7.33388	0.683888
$116$	38.1130	3.53871
$117$	0	0
$118$	−7.58237	−0.698014
$119$	−3.70156	−0.339322
$120$	0	0
$121$	1.00000	0.0909091
$122$	21.8080	1.97441
$123$	0	0
$124$	31.8417	2.85947
$125$	1.00000	0.0894427
$126$	0	0
$127$	11.6516	1.03391	0.516957	−	0.856011i	$-0.327065\pi$
$127$	11.6516	1.03391	0.516957	+	0.856011i	$0.327065\pi$
$128$	4.90647	0.433675
$129$	0	0
$130$	−16.5806	−1.45421
$131$	13.8116	1.20672	0.603361	−	0.797468i	$-0.293828\pi$
$131$	13.8116	1.20672	0.603361	+	0.797468i	$0.293828\pi$
$132$	0	0
$133$	−7.24672	−0.628370
$134$	22.5806	1.95067
$135$	0	0
$136$	−25.8874	−2.21982
$137$	−9.40312	−0.803363	−0.401682	−	0.915779i	$-0.631574\pi$
$137$	−9.40312	−0.803363	−0.401682	+	0.915779i	$0.631574\pi$
$138$	0	0
$139$	16.3128	1.38363	0.691817	−	0.722072i	$-0.256810\pi$
$139$	16.3128	1.38363	0.691817	+	0.722072i	$0.256810\pi$
$140$	−4.70156	−0.397355
$141$	0	0
$142$	−11.4031	−0.956929
$143$	6.40490	0.535604
$144$	0	0
$145$	8.10646	0.673205
$146$	−8.81898	−0.729864
$147$	0	0
$148$	−18.5714	−1.52656
$149$	−9.90010	−0.811048	−0.405524	−	0.914084i	$-0.632911\pi$
$149$	−9.90010	−0.811048	−0.405524	+	0.914084i	$0.632911\pi$
$150$	0	0
$151$	−17.7581	−1.44513	−0.722566	−	0.691302i	$-0.757037\pi$
$151$	−17.7581	−1.44513	−0.722566	+	0.691302i	$0.757037\pi$
$152$	−50.6809	−4.11076
$153$	0	0
$154$	2.58874	0.208606
$155$	6.77258	0.543987
$156$	0	0
$157$	17.0547	1.36112	0.680558	−	0.732694i	$-0.261737\pi$
$157$	17.0547	1.36112	0.680558	+	0.732694i	$0.261737\pi$
$158$	−11.4031	−0.907184
$159$	0	0
$160$	−8.53879	−0.675051
$161$	−7.33388	−0.577991
$162$	0	0
$163$	−19.9338	−1.56133	−0.780667	−	0.624947i	$-0.785120\pi$
$163$	−19.9338	−1.56133	−0.780667	+	0.624947i	$0.785120\pi$
$164$	10.8970	0.850910
$165$	0	0
$166$	−23.2111	−1.80153
$167$	2.95183	0.228419	0.114210	−	0.993457i	$-0.463566\pi$
$167$	2.95183	0.228419	0.114210	+	0.993457i	$0.463566\pi$
$168$	0	0
$169$	28.0227	2.15559
$170$	−9.58237	−0.734934
$171$	0	0
$172$	−35.9745	−2.74303
$173$	−20.2094	−1.53649	−0.768245	−	0.640156i	$-0.778870\pi$
$173$	−20.2094	−1.53649	−0.768245	+	0.640156i	$0.778870\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	8.70156	0.655905
$177$	0	0
$178$	0.534199	0.0400399
$179$	−5.95005	−0.444728	−0.222364	−	0.974964i	$-0.571377\pi$
$179$	−5.95005	−0.444728	−0.222364	+	0.974964i	$0.571377\pi$
$180$	0	0
$181$	−20.0709	−1.49186	−0.745929	−	0.666026i	$-0.767994\pi$
$181$	−20.0709	−1.49186	−0.745929	+	0.666026i	$0.767994\pi$
$182$	16.5806	1.22904
$183$	0	0
$184$	−51.2905	−3.78119
$185$	−3.95005	−0.290414
$186$	0	0
$187$	3.70156	0.270685
$188$	−6.83171	−0.498253
$189$	0	0
$190$	−18.7598	−1.36098
$191$	−19.7159	−1.42660	−0.713298	−	0.700861i	$-0.752799\pi$
$191$	−19.7159	−1.42660	−0.713298	+	0.700861i	$0.752799\pi$
$192$	0	0
$193$	−5.49029	−0.395200	−0.197600	−	0.980283i	$-0.563315\pi$
$193$	−5.49029	−0.395200	−0.197600	+	0.980283i	$0.563315\pi$
$194$	36.1502	2.59544
$195$	0	0
$196$	4.70156	0.335826
$197$	−20.4421	−1.45644	−0.728220	−	0.685343i	$-0.759652\pi$
$197$	−20.4421	−1.45644	−0.728220	+	0.685343i	$0.759652\pi$
$198$	0	0
$199$	22.9391	1.62611	0.813055	−	0.582187i	$-0.197803\pi$
$199$	22.9391	1.62611	0.813055	+	0.582187i	$0.197803\pi$
$200$	−6.99364	−0.494525
$201$	0	0
$202$	9.76162	0.686825
$203$	−8.10646	−0.568962
$204$	0	0
$205$	2.31773	0.161877
$206$	22.1630	1.54417
$207$	0	0
$208$	55.7326	3.86436
$209$	7.24672	0.501266
$210$	0	0
$211$	−6.59333	−0.453903	−0.226952	−	0.973906i	$-0.572876\pi$
$211$	−6.59333	−0.453903	−0.226952	+	0.973906i	$0.572876\pi$
$212$	−42.8229	−2.94109
$213$	0	0
$214$	36.6642	2.50631
$215$	−7.65161	−0.521836
$216$	0	0
$217$	−6.77258	−0.459753
$218$	−18.4843	−1.25191
$219$	0	0
$220$	4.70156	0.316979
$221$	23.7081	1.59478
$222$	0	0
$223$	11.0162	0.737696	0.368848	−	0.929490i	$-0.379752\pi$
$223$	11.0162	0.737696	0.368848	+	0.929490i	$0.379752\pi$
$224$	8.53879	0.570522
$225$	0	0
$226$	−2.86893	−0.190838
$227$	21.0048	1.39414	0.697068	−	0.717005i	$-0.254487\pi$
$227$	21.0048	1.39414	0.697068	+	0.717005i	$0.254487\pi$
$228$	0	0
$229$	−19.5661	−1.29296	−0.646482	−	0.762929i	$-0.723760\pi$
$229$	−19.5661	−1.29296	−0.646482	+	0.762929i	$0.723760\pi$
$230$	−18.9855	−1.25187
$231$	0	0
$232$	−56.6936	−3.72212
$233$	−16.3886	−1.07365	−0.536827	−	0.843692i	$-0.680377\pi$
$233$	−16.3886	−1.07365	−0.536827	+	0.843692i	$0.680377\pi$
$234$	0	0
$235$	−1.45307	−0.0947880
$236$	13.7708	0.896403
$237$	0	0
$238$	9.58237	0.621133
$239$	−18.0066	−1.16475	−0.582374	−	0.812921i	$-0.697876\pi$
$239$	−18.0066	−1.16475	−0.582374	+	0.812921i	$0.697876\pi$
$240$	0	0
$241$	27.0740	1.74399	0.871996	−	0.489513i	$-0.162826\pi$
$241$	27.0740	1.74399	0.871996	+	0.489513i	$0.162826\pi$
$242$	−2.58874	−0.166410
$243$	0	0
$244$	−39.6069	−2.53557
$245$	1.00000	0.0638877
$246$	0	0
$247$	46.4145	2.95328
$248$	−47.3649	−3.00768
$249$	0	0
$250$	−2.58874	−0.163726
$251$	−8.58415	−0.541827	−0.270913	−	0.962604i	$-0.587326\pi$
$251$	−8.58415	−0.541827	−0.270913	+	0.962604i	$0.587326\pi$
$252$	0	0
$253$	7.33388	0.461077
$254$	−30.1630	−1.89259
$255$	0	0
$256$	−22.1047	−1.38154
$257$	−0.0885359	−0.00552272	−0.00276136	−	0.999996i	$-0.500879\pi$
$257$	−0.0885359	−0.00552272	−0.00276136	+	0.999996i	$0.500879\pi$
$258$	0	0
$259$	3.95005	0.245444
$260$	30.1130	1.86753
$261$	0	0
$262$	−35.7545	−2.20892
$263$	17.5452	1.08188	0.540940	−	0.841061i	$-0.318068\pi$
$263$	17.5452	1.08188	0.540940	+	0.841061i	$0.318068\pi$
$264$	0	0
$265$	−9.10823	−0.559514
$266$	18.7598	1.15024
$267$	0	0
$268$	−41.0100	−2.50508
$269$	−12.2520	−0.747020	−0.373510	−	0.927626i	$-0.621846\pi$
$269$	−12.2520	−0.747020	−0.373510	+	0.927626i	$0.621846\pi$
$270$	0	0
$271$	−1.91448	−0.116296	−0.0581482	−	0.998308i	$-0.518520\pi$
$271$	−1.91448	−0.116296	−0.0581482	+	0.998308i	$0.518520\pi$
$272$	32.2094	1.95298
$273$	0	0
$274$	24.3422	1.47057
$275$	1.00000	0.0603023
$276$	0	0
$277$	5.62877	0.338200	0.169100	−	0.985599i	$-0.445914\pi$
$277$	5.62877	0.338200	0.169100	+	0.985599i	$0.445914\pi$
$278$	−42.2296	−2.53276
$279$	0	0
$280$	6.99364	0.417950
$281$	29.3032	1.74808	0.874042	−	0.485850i	$-0.161490\pi$
$281$	29.3032	1.74808	0.874042	+	0.485850i	$0.161490\pi$
$282$	0	0
$283$	6.89833	0.410063	0.205032	−	0.978755i	$-0.434270\pi$
$283$	6.89833	0.410063	0.205032	+	0.978755i	$0.434270\pi$
$284$	20.7099	1.22891
$285$	0	0
$286$	−16.5806	−0.980431
$287$	−2.31773	−0.136811
$288$	0	0
$289$	−3.29844	−0.194026
$290$	−20.9855	−1.23231
$291$	0	0
$292$	16.0167	0.937305
$293$	28.7243	1.67809	0.839045	−	0.544062i	$-0.183114\pi$
$293$	28.7243	1.67809	0.839045	+	0.544062i	$0.183114\pi$
$294$	0	0
$295$	2.92898	0.170532
$296$	27.6252	1.60568
$297$	0	0
$298$	25.6288	1.48463
$299$	46.9728	2.71651
$300$	0	0
$301$	7.65161	0.441032
$302$	45.9710	2.64533
$303$	0	0
$304$	63.0578	3.61661
$305$	−8.42419	−0.482368
$306$	0	0
$307$	15.7616	0.899563	0.449782	−	0.893139i	$-0.351502\pi$
$307$	15.7616	0.899563	0.449782	+	0.893139i	$0.351502\pi$
$308$	−4.70156	−0.267896
$309$	0	0
$310$	−17.5324	−0.995775
$311$	−18.7549	−1.06349	−0.531747	−	0.846903i	$-0.678464\pi$
$311$	−18.7549	−1.06349	−0.531747	+	0.846903i	$0.678464\pi$
$312$	0	0
$313$	8.45786	0.478067	0.239033	−	0.971011i	$-0.423169\pi$
$313$	8.45786	0.478067	0.239033	+	0.971011i	$0.423169\pi$
$314$	−44.1502	−2.49154
$315$	0	0
$316$	20.7099	1.16502
$317$	12.8519	0.721836	0.360918	−	0.932597i	$-0.382463\pi$
$317$	12.8519	0.721836	0.360918	+	0.932597i	$0.382463\pi$
$318$	0	0
$319$	8.10646	0.453875
$320$	4.70156	0.262825
$321$	0	0
$322$	18.9855	1.05802
$323$	26.8242	1.49254
$324$	0	0
$325$	6.40490	0.355280
$326$	51.6033	2.85804
$327$	0	0
$328$	−16.2094	−0.895013
$329$	1.45307	0.0801104
$330$	0	0
$331$	−6.64984	−0.365508	−0.182754	−	0.983159i	$-0.558501\pi$
$331$	−6.64984	−0.365508	−0.182754	+	0.983159i	$0.558501\pi$
$332$	42.1552	2.31356
$333$	0	0
$334$	−7.64150	−0.418124
$335$	−8.72263	−0.476568
$336$	0	0
$337$	−18.0228	−0.981767	−0.490883	−	0.871225i	$-0.663326\pi$
$337$	−18.0228	−0.981767	−0.490883	+	0.871225i	$0.663326\pi$
$338$	−72.5435	−3.94584
$339$	0	0
$340$	17.4031	0.943817
$341$	6.77258	0.366756
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	53.5126	2.88521
$345$	0	0
$346$	52.3168	2.81257
$347$	2.00355	0.107556	0.0537780	−	0.998553i	$-0.482874\pi$
$347$	2.00355	0.107556	0.0537780	+	0.998553i	$0.482874\pi$
$348$	0	0
$349$	1.15955	0.0620693	0.0310347	−	0.999518i	$-0.490120\pi$
$349$	1.15955	0.0620693	0.0310347	+	0.999518i	$0.490120\pi$
$350$	2.58874	0.138374
$351$	0	0
$352$	−8.53879	−0.455119
$353$	−29.6160	−1.57630	−0.788151	−	0.615481i	$-0.788962\pi$
$353$	−29.6160	−1.57630	−0.788151	+	0.615481i	$0.788962\pi$
$354$	0	0
$355$	4.40490	0.233788
$356$	−0.970191	−0.0514200
$357$	0	0
$358$	15.4031	0.814080
$359$	22.4650	1.18566	0.592828	−	0.805329i	$-0.298012\pi$
$359$	22.4650	1.18566	0.592828	+	0.805329i	$0.298012\pi$
$360$	0	0
$361$	33.5149	1.76394
$362$	51.9583	2.73087
$363$	0	0
$364$	−30.1130	−1.57835
$365$	3.40667	0.178313
$366$	0	0
$367$	5.12233	0.267384	0.133692	−	0.991023i	$-0.457317\pi$
$367$	5.12233	0.267384	0.133692	+	0.991023i	$0.457317\pi$
$368$	63.8162	3.32665
$369$	0	0
$370$	10.2256	0.531606
$371$	9.10823	0.472876
$372$	0	0
$373$	−26.2358	−1.35844	−0.679218	−	0.733936i	$-0.737681\pi$
$373$	−26.2358	−1.35844	−0.679218	+	0.733936i	$0.737681\pi$
$374$	−9.58237	−0.495493
$375$	0	0
$376$	10.1623	0.524078
$377$	51.9210	2.67407
$378$	0	0
$379$	−9.60167	−0.493205	−0.246602	−	0.969117i	$-0.579314\pi$
$379$	−9.60167	−0.493205	−0.246602	+	0.969117i	$0.579314\pi$
$380$	34.0709	1.74780
$381$	0	0
$382$	51.0394	2.61140
$383$	−10.1244	−0.517332	−0.258666	−	0.965967i	$-0.583283\pi$
$383$	−10.1244	−0.517332	−0.258666	+	0.965967i	$0.583283\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	14.2129	0.723419
$387$	0	0
$388$	−65.6546	−3.33311
$389$	−16.9342	−0.858597	−0.429298	−	0.903163i	$-0.641239\pi$
$389$	−16.9342	−0.858597	−0.429298	+	0.903163i	$0.641239\pi$
$390$	0	0
$391$	27.1468	1.37287
$392$	−6.99364	−0.353232
$393$	0	0
$394$	52.9193	2.66603
$395$	4.40490	0.221634
$396$	0	0
$397$	13.7616	0.690676	0.345338	−	0.938478i	$-0.387764\pi$
$397$	13.7616	0.690676	0.345338	+	0.938478i	$0.387764\pi$
$398$	−59.3833	−2.97662
$399$	0	0
$400$	8.70156	0.435078
$401$	−35.6118	−1.77837	−0.889184	−	0.457550i	$-0.848727\pi$
$401$	−35.6118	−1.77837	−0.889184	+	0.457550i	$0.848727\pi$
$402$	0	0
$403$	43.3777	2.16080
$404$	−17.7287	−0.882034
$405$	0	0
$406$	20.9855	1.04149
$407$	−3.95005	−0.195797
$408$	0	0
$409$	9.17748	0.453797	0.226898	−	0.973918i	$-0.427141\pi$
$409$	9.17748	0.453797	0.226898	+	0.973918i	$0.427141\pi$
$410$	−6.00000	−0.296319
$411$	0	0
$412$	−40.2515	−1.98305
$413$	−2.92898	−0.144126
$414$	0	0
$415$	8.96620	0.440134
$416$	−54.6901	−2.68140
$417$	0	0
$418$	−18.7598	−0.917574
$419$	−37.8554	−1.84935	−0.924677	−	0.380751i	$-0.875665\pi$
$419$	−37.8554	−1.84935	−0.924677	+	0.380751i	$0.875665\pi$
$420$	0	0
$421$	3.10469	0.151313	0.0756566	−	0.997134i	$-0.475895\pi$
$421$	3.10469	0.151313	0.0756566	+	0.997134i	$0.475895\pi$
$422$	17.0684	0.830877
$423$	0	0
$424$	63.6997	3.09353
$425$	3.70156	0.179552
$426$	0	0
$427$	8.42419	0.407675
$428$	−66.5881	−3.21866
$429$	0	0
$430$	19.8080	0.955228
$431$	10.4548	0.503592	0.251796	−	0.967780i	$-0.418979\pi$
$431$	10.4548	0.503592	0.251796	+	0.967780i	$0.418979\pi$
$432$	0	0
$433$	−7.82212	−0.375907	−0.187954	−	0.982178i	$-0.560185\pi$
$433$	−7.82212	−0.375907	−0.187954	+	0.982178i	$0.560185\pi$
$434$	17.5324	0.841583
$435$	0	0
$436$	33.5704	1.60773
$437$	53.1466	2.54235
$438$	0	0
$439$	−29.8146	−1.42297	−0.711486	−	0.702700i	$-0.751978\pi$
$439$	−29.8146	−1.42297	−0.711486	+	0.702700i	$0.751978\pi$
$440$	−6.99364	−0.333408
$441$	0	0
$442$	−61.3741	−2.91927
$443$	11.2774	0.535804	0.267902	−	0.963446i	$-0.413670\pi$
$443$	11.2774	0.535804	0.267902	+	0.963446i	$0.413670\pi$
$444$	0	0
$445$	−0.206355	−0.00978216
$446$	−28.5179	−1.35036
$447$	0	0
$448$	−4.70156	−0.222128
$449$	−11.2787	−0.532277	−0.266138	−	0.963935i	$-0.585748\pi$
$449$	−11.2787	−0.532277	−0.266138	+	0.963935i	$0.585748\pi$
$450$	0	0
$451$	2.31773	0.109138
$452$	5.21043	0.245078
$453$	0	0
$454$	−54.3759	−2.55199
$455$	−6.40490	−0.300266
$456$	0	0
$457$	−0.403250	−0.0188632	−0.00943162	−	0.999956i	$-0.503002\pi$
$457$	−0.403250	−0.0188632	−0.00943162	+	0.999956i	$0.503002\pi$
$458$	50.6515	2.36679
$459$	0	0
$460$	34.4807	1.60767
$461$	−34.9355	−1.62711	−0.813555	−	0.581487i	$-0.802471\pi$
$461$	−34.9355	−1.62711	−0.813555	+	0.581487i	$0.802471\pi$
$462$	0	0
$463$	4.36413	0.202818	0.101409	−	0.994845i	$-0.467665\pi$
$463$	4.36413	0.202818	0.101409	+	0.994845i	$0.467665\pi$
$464$	70.5389	3.27468
$465$	0	0
$466$	42.4258	1.96534
$467$	30.2551	1.40004	0.700018	−	0.714125i	$-0.253175\pi$
$467$	30.2551	1.40004	0.700018	+	0.714125i	$0.253175\pi$
$468$	0	0
$469$	8.72263	0.402774
$470$	3.76162	0.173511
$471$	0	0
$472$	−20.4843	−0.942864
$473$	−7.65161	−0.351822
$474$	0	0
$475$	7.24672	0.332502
$476$	−17.4031	−0.797671
$477$	0	0
$478$	46.6143	2.13209
$479$	−11.8151	−0.539846	−0.269923	−	0.962882i	$-0.586998\pi$
$479$	−11.8151	−0.539846	−0.269923	+	0.962882i	$0.586998\pi$
$480$	0	0
$481$	−25.2997	−1.15357
$482$	−70.0876	−3.19240
$483$	0	0
$484$	4.70156	0.213707
$485$	−13.9644	−0.634092
$486$	0	0
$487$	36.3387	1.64666	0.823331	−	0.567561i	$-0.192113\pi$
$487$	36.3387	1.64666	0.823331	+	0.567561i	$0.192113\pi$
$488$	58.9157	2.66699
$489$	0	0
$490$	−2.58874	−0.116947
$491$	41.4554	1.87085	0.935427	−	0.353519i	$-0.115015\pi$
$491$	41.4554	1.87085	0.935427	+	0.353519i	$0.115015\pi$
$492$	0	0
$493$	30.0066	1.35143
$494$	−120.155	−5.40602
$495$	0	0
$496$	58.9320	2.64613
$497$	−4.40490	−0.197587
$498$	0	0
$499$	2.41146	0.107952	0.0539759	−	0.998542i	$-0.482811\pi$
$499$	2.41146	0.107952	0.0539759	+	0.998542i	$0.482811\pi$
$500$	4.70156	0.210260
$501$	0	0
$502$	22.2221	0.991821
$503$	24.3335	1.08498	0.542488	−	0.840063i	$-0.317482\pi$
$503$	24.3335	1.08498	0.542488	+	0.840063i	$0.317482\pi$
$504$	0	0
$505$	−3.77080	−0.167799
$506$	−18.9855	−0.844008
$507$	0	0
$508$	54.7808	2.43050
$509$	−21.3388	−0.945826	−0.472913	−	0.881109i	$-0.656797\pi$
$509$	−21.3388	−0.945826	−0.472913	+	0.881109i	$0.656797\pi$
$510$	0	0
$511$	−3.40667	−0.150702
$512$	47.4103	2.09526
$513$	0	0
$514$	0.229196	0.0101094
$515$	−8.56131	−0.377256
$516$	0	0
$517$	−1.45307	−0.0639060
$518$	−10.2256	−0.449289
$519$	0	0
$520$	−44.7935	−1.96433
$521$	−2.03557	−0.0891800	−0.0445900	−	0.999005i	$-0.514198\pi$
$521$	−2.03557	−0.0891800	−0.0445900	+	0.999005i	$0.514198\pi$
$522$	0	0
$523$	−11.1647	−0.488200	−0.244100	−	0.969750i	$-0.578493\pi$
$523$	−11.1647	−0.488200	−0.244100	+	0.969750i	$0.578493\pi$
$524$	64.9359	2.83674
$525$	0	0
$526$	−45.4198	−1.98040
$527$	25.0691	1.09203
$528$	0	0
$529$	30.7858	1.33851
$530$	23.5788	1.02420
$531$	0	0
$532$	−34.0709	−1.47716
$533$	14.8448	0.643001
$534$	0	0
$535$	−14.1630	−0.612319
$536$	61.0029	2.63492
$537$	0	0
$538$	31.7173	1.36743
$539$	1.00000	0.0430730
$540$	0	0
$541$	−37.8212	−1.62606	−0.813029	−	0.582223i	$-0.802183\pi$
$541$	−37.8212	−1.62606	−0.813029	+	0.582223i	$0.802183\pi$
$542$	4.95609	0.212882
$543$	0	0
$544$	−31.6069	−1.35513
$545$	7.14026	0.305855
$546$	0	0
$547$	−32.2871	−1.38050	−0.690248	−	0.723573i	$-0.742499\pi$
$547$	−32.2871	−1.38050	−0.690248	+	0.723573i	$0.742499\pi$
$548$	−44.2094	−1.88853
$549$	0	0
$550$	−2.58874	−0.110384
$551$	58.7452	2.50263
$552$	0	0
$553$	−4.40490	−0.187315
$554$	−14.5714	−0.619080
$555$	0	0
$556$	76.6957	3.25262
$557$	−17.9416	−0.760209	−0.380105	−	0.924943i	$-0.624112\pi$
$557$	−17.9416	−0.760209	−0.380105	+	0.924943i	$0.624112\pi$
$558$	0	0
$559$	−49.0078	−2.07281
$560$	−8.70156	−0.367708
$561$	0	0
$562$	−75.8584	−3.19989
$563$	1.40667	0.0592841	0.0296421	−	0.999561i	$-0.490563\pi$
$563$	1.40667	0.0592841	0.0296421	+	0.999561i	$0.490563\pi$
$564$	0	0
$565$	1.10823	0.0466237
$566$	−17.8580	−0.750626
$567$	0	0
$568$	−30.8062	−1.29260
$569$	−9.36755	−0.392708	−0.196354	−	0.980533i	$-0.562910\pi$
$569$	−9.36755	−0.392708	−0.196354	+	0.980533i	$0.562910\pi$
$570$	0	0
$571$	−24.8562	−1.04020	−0.520100	−	0.854106i	$-0.674105\pi$
$571$	−24.8562	−1.04020	−0.520100	+	0.854106i	$0.674105\pi$
$572$	30.1130	1.25909
$573$	0	0
$574$	6.00000	0.250435
$575$	7.33388	0.305844
$576$	0	0
$577$	6.17433	0.257041	0.128520	−	0.991707i	$-0.458977\pi$
$577$	6.17433	0.257041	0.128520	+	0.991707i	$0.458977\pi$
$578$	8.53879	0.355167
$579$	0	0
$580$	38.1130	1.58256
$581$	−8.96620	−0.371981
$582$	0	0
$583$	−9.10823	−0.377224
$584$	−23.8250	−0.985886
$585$	0	0
$586$	−74.3596	−3.07177
$587$	−2.04995	−0.0846104	−0.0423052	−	0.999105i	$-0.513470\pi$
$587$	−2.04995	−0.0846104	−0.0423052	+	0.999105i	$0.513470\pi$
$588$	0	0
$589$	49.0790	2.02226
$590$	−7.58237	−0.312161
$591$	0	0
$592$	−34.3716	−1.41267
$593$	7.85797	0.322688	0.161344	−	0.986898i	$-0.448417\pi$
$593$	7.85797	0.322688	0.161344	+	0.986898i	$0.448417\pi$
$594$	0	0
$595$	−3.70156	−0.151749
$596$	−46.5460	−1.90660
$597$	0	0
$598$	−121.600	−4.97260
$599$	11.2190	0.458394	0.229197	−	0.973380i	$-0.426390\pi$
$599$	11.2190	0.458394	0.229197	+	0.973380i	$0.426390\pi$
$600$	0	0
$601$	19.5047	0.795612	0.397806	−	0.917470i	$-0.369772\pi$
$601$	19.5047	0.795612	0.397806	+	0.917470i	$0.369772\pi$
$602$	−19.8080	−0.807315
$603$	0	0
$604$	−83.4907	−3.39719
$605$	1.00000	0.0406558
$606$	0	0
$607$	−16.9483	−0.687909	−0.343955	−	0.938986i	$-0.611767\pi$
$607$	−16.9483	−0.687909	−0.343955	+	0.938986i	$0.611767\pi$
$608$	−61.8782	−2.50949
$609$	0	0
$610$	21.8080	0.882981
$611$	−9.30678	−0.376512
$612$	0	0
$613$	−12.3964	−0.500687	−0.250344	−	0.968157i	$-0.580544\pi$
$613$	−12.3964	−0.500687	−0.250344	+	0.968157i	$0.580544\pi$
$614$	−40.8027	−1.64666
$615$	0	0
$616$	6.99364	0.281782
$617$	21.0192	0.846200	0.423100	−	0.906083i	$-0.360942\pi$
$617$	21.0192	0.846200	0.423100	+	0.906083i	$0.360942\pi$
$618$	0	0
$619$	11.5289	0.463385	0.231692	−	0.972789i	$-0.425574\pi$
$619$	11.5289	0.463385	0.231692	+	0.972789i	$0.425574\pi$
$620$	31.8417	1.27879
$621$	0	0
$622$	48.5516	1.94674
$623$	0.206355	0.00826744
$624$	0	0
$625$	1.00000	0.0400000
$626$	−21.8952	−0.875108
$627$	0	0
$628$	80.1839	3.19969
$629$	−14.6214	−0.582992
$630$	0	0
$631$	33.9988	1.35347	0.676734	−	0.736227i	$-0.263395\pi$
$631$	33.9988	1.35347	0.676734	+	0.736227i	$0.263395\pi$
$632$	−30.8062	−1.22541
$633$	0	0
$634$	−33.2703	−1.32133
$635$	11.6516	0.462380
$636$	0	0
$637$	6.40490	0.253771
$638$	−20.9855	−0.830824
$639$	0	0
$640$	4.90647	0.193945
$641$	20.5293	0.810858	0.405429	−	0.914127i	$-0.367122\pi$
$641$	20.5293	0.810858	0.405429	+	0.914127i	$0.367122\pi$
$642$	0	0
$643$	−16.9584	−0.668774	−0.334387	−	0.942436i	$-0.608529\pi$
$643$	−16.9584	−0.668774	−0.334387	+	0.942436i	$0.608529\pi$
$644$	−34.4807	−1.35873
$645$	0	0
$646$	−69.4407	−2.73211
$647$	2.25506	0.0886554	0.0443277	−	0.999017i	$-0.485885\pi$
$647$	2.25506	0.0886554	0.0443277	+	0.999017i	$0.485885\pi$
$648$	0	0
$649$	2.92898	0.114973
$650$	−16.5806	−0.650345
$651$	0	0
$652$	−93.7199	−3.67035
$653$	−1.98917	−0.0778422	−0.0389211	−	0.999242i	$-0.512392\pi$
$653$	−1.98917	−0.0778422	−0.0389211	+	0.999242i	$0.512392\pi$
$654$	0	0
$655$	13.8116	0.539663
$656$	20.1679	0.787424
$657$	0	0
$658$	−3.76162	−0.146643
$659$	−3.41324	−0.132961	−0.0664804	−	0.997788i	$-0.521177\pi$
$659$	−3.41324	−0.132961	−0.0664804	+	0.997788i	$0.521177\pi$
$660$	0	0
$661$	15.4988	0.602832	0.301416	−	0.953493i	$-0.402541\pi$
$661$	15.4988	0.602832	0.301416	+	0.953493i	$0.402541\pi$
$662$	17.2147	0.669068
$663$	0	0
$664$	−62.7064	−2.43348
$665$	−7.24672	−0.281016
$666$	0	0
$667$	59.4518	2.30198
$668$	13.8782	0.536963
$669$	0	0
$670$	22.5806	0.872365
$671$	−8.42419	−0.325212
$672$	0	0
$673$	−40.6449	−1.56675	−0.783373	−	0.621551i	$-0.786503\pi$
$673$	−40.6449	−1.56675	−0.783373	+	0.621551i	$0.786503\pi$
$674$	46.6564	1.79714
$675$	0	0
$676$	131.751	5.06733
$677$	−41.1756	−1.58251	−0.791253	−	0.611489i	$-0.790571\pi$
$677$	−41.1756	−1.58251	−0.791253	+	0.611489i	$0.790571\pi$
$678$	0	0
$679$	13.9644	0.535906
$680$	−25.8874	−0.992736
$681$	0	0
$682$	−17.5324	−0.671351
$683$	−4.86111	−0.186005	−0.0930027	−	0.995666i	$-0.529647\pi$
$683$	−4.86111	−0.186005	−0.0930027	+	0.995666i	$0.529647\pi$
$684$	0	0
$685$	−9.40312	−0.359275
$686$	2.58874	0.0988385
$687$	0	0
$688$	−66.5810	−2.53838
$689$	−58.3373	−2.22247
$690$	0	0
$691$	41.8838	1.59334	0.796668	−	0.604417i	$-0.206594\pi$
$691$	41.8838	1.59334	0.796668	+	0.604417i	$0.206594\pi$
$692$	−95.0156	−3.61195
$693$	0	0
$694$	−5.18666	−0.196883
$695$	16.3128	0.618780
$696$	0	0
$697$	8.57923	0.324961
$698$	−3.00177	−0.113619
$699$	0	0
$700$	−4.70156	−0.177702
$701$	−4.10646	−0.155099	−0.0775494	−	0.996989i	$-0.524710\pi$
$701$	−4.10646	−0.155099	−0.0775494	+	0.996989i	$0.524710\pi$
$702$	0	0
$703$	−28.6249	−1.07961
$704$	4.70156	0.177197
$705$	0	0
$706$	76.6682	2.88544
$707$	3.77080	0.141816
$708$	0	0
$709$	9.52094	0.357567	0.178783	−	0.983888i	$-0.442784\pi$
$709$	9.52094	0.357567	0.178783	+	0.983888i	$0.442784\pi$
$710$	−11.4031	−0.427952
$711$	0	0
$712$	1.44317	0.0540851
$713$	49.6693	1.86013
$714$	0	0
$715$	6.40490	0.239530
$716$	−27.9745	−1.04546
$717$	0	0
$718$	−58.1559	−2.17036
$719$	−17.4969	−0.652523	−0.326261	−	0.945280i	$-0.605789\pi$
$719$	−17.4969	−0.652523	−0.326261	+	0.945280i	$0.605789\pi$
$720$	0	0
$721$	8.56131	0.318840
$722$	−86.7613	−3.22892
$723$	0	0
$724$	−94.3645	−3.50703
$725$	8.10646	0.301066
$726$	0	0
$727$	−46.8908	−1.73908	−0.869542	−	0.493860i	$-0.835586\pi$
$727$	−46.8908	−1.73908	−0.869542	+	0.493860i	$0.835586\pi$
$728$	44.7935	1.66016
$729$	0	0
$730$	−8.81898	−0.326405
$731$	−28.3229	−1.04756
$732$	0	0
$733$	−15.8038	−0.583725	−0.291863	−	0.956460i	$-0.594275\pi$
$733$	−15.8038	−0.583725	−0.291863	+	0.956460i	$0.594275\pi$
$734$	−13.2604	−0.489449
$735$	0	0
$736$	−62.6225	−2.30830
$737$	−8.72263	−0.321302
$738$	0	0
$739$	−5.69073	−0.209337	−0.104668	−	0.994507i	$-0.533378\pi$
$739$	−5.69073	−0.209337	−0.104668	+	0.994507i	$0.533378\pi$
$740$	−18.5714	−0.682699
$741$	0	0
$742$	−23.5788	−0.865606
$743$	−22.2909	−0.817774	−0.408887	−	0.912585i	$-0.634083\pi$
$743$	−22.2909	−0.817774	−0.408887	+	0.912585i	$0.634083\pi$
$744$	0	0
$745$	−9.90010	−0.362712
$746$	67.9175	2.48664
$747$	0	0
$748$	17.4031	0.636321
$749$	14.1630	0.517504
$750$	0	0
$751$	38.6786	1.41140	0.705701	−	0.708510i	$-0.250632\pi$
$751$	38.6786	1.41140	0.705701	+	0.708510i	$0.250632\pi$
$752$	−12.6440	−0.461079
$753$	0	0
$754$	−134.410	−4.89492
$755$	−17.7581	−0.646283
$756$	0	0
$757$	43.7967	1.59182	0.795908	−	0.605417i	$-0.206994\pi$
$757$	43.7967	1.59182	0.795908	+	0.605417i	$0.206994\pi$
$758$	24.8562	0.902818
$759$	0	0
$760$	−50.6809	−1.83839
$761$	−40.1966	−1.45713	−0.728564	−	0.684978i	$-0.759812\pi$
$761$	−40.1966	−1.45713	−0.728564	+	0.684978i	$0.759812\pi$
$762$	0	0
$763$	−7.14026	−0.258495
$764$	−92.6957	−3.35361
$765$	0	0
$766$	26.2094	0.946983
$767$	18.7598	0.677379
$768$	0	0
$769$	1.32679	0.0478452	0.0239226	−	0.999714i	$-0.492384\pi$
$769$	1.32679	0.0478452	0.0239226	+	0.999714i	$0.492384\pi$
$770$	2.58874	0.0932916
$771$	0	0
$772$	−25.8129	−0.929028
$773$	9.62314	0.346120	0.173060	−	0.984911i	$-0.444635\pi$
$773$	9.62314	0.346120	0.173060	+	0.984911i	$0.444635\pi$
$774$	0	0
$775$	6.77258	0.243278
$776$	97.6621	3.50587
$777$	0	0
$778$	43.8381	1.57167
$779$	16.7959	0.601777
$780$	0	0
$781$	4.40490	0.157620
$782$	−70.2760	−2.51306
$783$	0	0
$784$	8.70156	0.310770
$785$	17.0547	0.608710
$786$	0	0
$787$	44.1489	1.57374	0.786869	−	0.617120i	$-0.211701\pi$
$787$	44.1489	1.57374	0.786869	+	0.617120i	$0.211701\pi$
$788$	−96.1099	−3.42377
$789$	0	0
$790$	−11.4031	−0.405705
$791$	−1.10823	−0.0394042
$792$	0	0
$793$	−53.9561	−1.91604
$794$	−35.6252	−1.26429
$795$	0	0
$796$	107.850	3.82263
$797$	18.5257	0.656215	0.328108	−	0.944640i	$-0.393589\pi$
$797$	18.5257	0.656215	0.328108	+	0.944640i	$0.393589\pi$
$798$	0	0
$799$	−5.37864	−0.190282
$800$	−8.53879	−0.301892
$801$	0	0
$802$	92.1895	3.25533
$803$	3.40667	0.120219
$804$	0	0
$805$	−7.33388	−0.258485
$806$	−112.293	−3.95537
$807$	0	0
$808$	26.3716	0.927751
$809$	−38.1516	−1.34134	−0.670670	−	0.741756i	$-0.733993\pi$
$809$	−38.1516	−1.34134	−0.670670	+	0.741756i	$0.733993\pi$
$810$	0	0
$811$	−37.3355	−1.31103	−0.655514	−	0.755183i	$-0.727548\pi$
$811$	−37.3355	−1.31103	−0.655514	+	0.755183i	$0.727548\pi$
$812$	−38.1130	−1.33750
$813$	0	0
$814$	10.2256	0.358409
$815$	−19.9338	−0.698250
$816$	0	0
$817$	−55.4491	−1.93992
$818$	−23.7581	−0.830682
$819$	0	0
$820$	10.8970	0.380538
$821$	19.9039	0.694652	0.347326	−	0.937744i	$-0.387090\pi$
$821$	19.9039	0.694652	0.347326	+	0.937744i	$0.387090\pi$
$822$	0	0
$823$	−50.8399	−1.77217	−0.886084	−	0.463525i	$-0.846585\pi$
$823$	−50.8399	−1.77217	−0.886084	+	0.463525i	$0.846585\pi$
$824$	59.8746	2.08583
$825$	0	0
$826$	7.58237	0.263824
$827$	43.7003	1.51961	0.759804	−	0.650152i	$-0.225295\pi$
$827$	43.7003	1.51961	0.759804	+	0.650152i	$0.225295\pi$
$828$	0	0
$829$	51.5197	1.78935	0.894677	−	0.446715i	$-0.147406\pi$
$829$	51.5197	1.78935	0.894677	+	0.446715i	$0.147406\pi$
$830$	−23.2111	−0.805671
$831$	0	0
$832$	30.1130	1.04398
$833$	3.70156	0.128252
$834$	0	0
$835$	2.95183	0.102152
$836$	34.0709	1.17837
$837$	0	0
$838$	97.9976	3.38527
$839$	18.7834	0.648475	0.324238	−	0.945976i	$-0.394892\pi$
$839$	18.7834	0.648475	0.324238	+	0.945976i	$0.394892\pi$
$840$	0	0
$841$	36.7147	1.26602
$842$	−8.03722	−0.276981
$843$	0	0
$844$	−30.9989	−1.06703
$845$	28.0227	0.964011
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	−79.2559	−2.72166
$849$	0	0
$850$	−9.58237	−0.328673
$851$	−28.9692	−0.993052
$852$	0	0
$853$	−10.4506	−0.357821	−0.178911	−	0.983865i	$-0.557257\pi$
$853$	−10.4506	−0.357821	−0.178911	+	0.983865i	$0.557257\pi$
$854$	−21.8080	−0.746255
$855$	0	0
$856$	99.0507	3.38548
$857$	−33.2646	−1.13630	−0.568149	−	0.822926i	$-0.692340\pi$
$857$	−33.2646	−1.13630	−0.568149	+	0.822926i	$0.692340\pi$
$858$	0	0
$859$	27.5569	0.940230	0.470115	−	0.882605i	$-0.344212\pi$
$859$	27.5569	0.940230	0.470115	+	0.882605i	$0.344212\pi$
$860$	−35.9745	−1.22672
$861$	0	0
$862$	−27.0649	−0.921832
$863$	10.2042	0.347354	0.173677	−	0.984803i	$-0.444435\pi$
$863$	10.2042	0.347354	0.173677	+	0.984803i	$0.444435\pi$
$864$	0	0
$865$	−20.2094	−0.687139
$866$	20.2494	0.688103
$867$	0	0
$868$	−31.8417	−1.08078
$869$	4.40490	0.149426
$870$	0	0
$871$	−55.8676	−1.89300
$872$	−49.9364	−1.69106
$873$	0	0
$874$	−137.583	−4.65380
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	57.5319	1.94271	0.971357	−	0.237625i	$-0.0763688\pi$
$877$	57.5319	1.94271	0.971357	+	0.237625i	$0.0763688\pi$
$878$	77.1821	2.60477
$879$	0	0
$880$	8.70156	0.293330
$881$	1.05474	0.0355351	0.0177675	−	0.999842i	$-0.494344\pi$
$881$	1.05474	0.0355351	0.0177675	+	0.999842i	$0.494344\pi$
$882$	0	0
$883$	14.4063	0.484810	0.242405	−	0.970175i	$-0.422064\pi$
$883$	14.4063	0.484810	0.242405	+	0.970175i	$0.422064\pi$
$884$	111.465	3.74898
$885$	0	0
$886$	−29.1942	−0.980797
$887$	16.3792	0.549958	0.274979	−	0.961450i	$-0.411329\pi$
$887$	16.3792	0.549958	0.274979	+	0.961450i	$0.411329\pi$
$888$	0	0
$889$	−11.6516	−0.390783
$890$	0.534199	0.0179064
$891$	0	0
$892$	51.7931	1.73416
$893$	−10.5300	−0.352373
$894$	0	0
$895$	−5.95005	−0.198888
$896$	−4.90647	−0.163914
$897$	0	0
$898$	29.1977	0.974340
$899$	54.9016	1.83107
$900$	0	0
$901$	−33.7147	−1.12320
$902$	−6.00000	−0.199778
$903$	0	0
$904$	−7.75058	−0.257780
$905$	−20.0709	−0.667179
$906$	0	0
$907$	−26.0372	−0.864552	−0.432276	−	0.901741i	$-0.642289\pi$
$907$	−26.0372	−0.864552	−0.432276	+	0.901741i	$0.642289\pi$
$908$	98.7553	3.27731
$909$	0	0
$910$	16.5806	0.549642
$911$	−25.5381	−0.846114	−0.423057	−	0.906103i	$-0.639043\pi$
$911$	−25.5381	−0.846114	−0.423057	+	0.906103i	$0.639043\pi$
$912$	0	0
$913$	8.96620	0.296738
$914$	1.04391	0.0345294
$915$	0	0
$916$	−91.9912	−3.03948
$917$	−13.8116	−0.456098
$918$	0	0
$919$	−28.3971	−0.936733	−0.468367	−	0.883534i	$-0.655157\pi$
$919$	−28.3971	−0.936733	−0.468367	+	0.883534i	$0.655157\pi$
$920$	−51.2905	−1.69100
$921$	0	0
$922$	90.4390	2.97845
$923$	28.2129	0.928640
$924$	0	0
$925$	−3.95005	−0.129877
$926$	−11.2976	−0.371262
$927$	0	0
$928$	−69.2194	−2.27224
$929$	10.7485	0.352646	0.176323	−	0.984332i	$-0.443580\pi$
$929$	10.7485	0.352646	0.176323	+	0.984332i	$0.443580\pi$
$930$	0	0
$931$	7.24672	0.237502
$932$	−77.0521	−2.52393
$933$	0	0
$934$	−78.3224	−2.56279
$935$	3.70156	0.121054
$936$	0	0
$937$	−34.9763	−1.14263	−0.571313	−	0.820732i	$-0.693566\pi$
$937$	−34.9763	−1.14263	−0.571313	+	0.820732i	$0.693566\pi$
$938$	−22.5806	−0.737283
$939$	0	0
$940$	−6.83171	−0.222826
$941$	−25.8979	−0.844248	−0.422124	−	0.906538i	$-0.638715\pi$
$941$	−25.8979	−0.844248	−0.422124	+	0.906538i	$0.638715\pi$
$942$	0	0
$943$	16.9980	0.553530
$944$	25.4867	0.829523
$945$	0	0
$946$	19.8080	0.644014
$947$	15.9623	0.518704	0.259352	−	0.965783i	$-0.416491\pi$
$947$	15.9623	0.518704	0.259352	+	0.965783i	$0.416491\pi$
$948$	0	0
$949$	21.8194	0.708287
$950$	−18.7598	−0.608650
$951$	0	0
$952$	25.8874	0.839015
$953$	30.8611	0.999690	0.499845	−	0.866115i	$-0.333390\pi$
$953$	30.8611	0.999690	0.499845	+	0.866115i	$0.333390\pi$
$954$	0	0
$955$	−19.7159	−0.637993
$956$	−84.6590	−2.73807
$957$	0	0
$958$	30.5862	0.988196
$959$	9.40312	0.303643
$960$	0	0
$961$	14.8678	0.479607
$962$	65.4942	2.11162
$963$	0	0
$964$	127.290	4.09974
$965$	−5.49029	−0.176739
$966$	0	0
$967$	−38.9934	−1.25394	−0.626972	−	0.779042i	$-0.715706\pi$
$967$	−38.9934	−1.25394	−0.626972	+	0.779042i	$0.715706\pi$
$968$	−6.99364	−0.224784
$969$	0	0
$970$	36.1502	1.16071
$971$	1.98071	0.0635639	0.0317819	−	0.999495i	$-0.489882\pi$
$971$	1.98071	0.0635639	0.0317819	+	0.999495i	$0.489882\pi$
$972$	0	0
$973$	−16.3128	−0.522965
$974$	−94.0713	−3.01424
$975$	0	0
$976$	−73.3036	−2.34639
$977$	−9.66926	−0.309347	−0.154674	−	0.987966i	$-0.549433\pi$
$977$	−9.66926	−0.309347	−0.154674	+	0.987966i	$0.549433\pi$
$978$	0	0
$979$	−0.206355	−0.00659513
$980$	4.70156	0.150186
$981$	0	0
$982$	−107.317	−3.42463
$983$	31.1289	0.992858	0.496429	−	0.868077i	$-0.334644\pi$
$983$	31.1289	0.992858	0.496429	+	0.868077i	$0.334644\pi$
$984$	0	0
$985$	−20.4421	−0.651340
$986$	−77.6791	−2.47381
$987$	0	0
$988$	218.221	6.94252
$989$	−56.1160	−1.78439
$990$	0	0
$991$	30.4473	0.967191	0.483595	−	0.875292i	$-0.339331\pi$
$991$	30.4473	0.967191	0.483595	+	0.875292i	$0.339331\pi$
$992$	−57.8296	−1.83609
$993$	0	0
$994$	11.4031	0.361685
$995$	22.9391	0.727218
$996$	0	0
$997$	−27.1612	−0.860204	−0.430102	−	0.902780i	$-0.641522\pi$
$997$	−27.1612	−0.860204	−0.430102	+	0.902780i	$0.641522\pi$
$998$	−6.24264	−0.197607
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3465.2.a.bl.1.1		4
3.2	odd	2		1155.2.a.u.1.4	✓	4
15.14	odd	2		5775.2.a.bz.1.1		4
21.20	even	2		8085.2.a.bn.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.u.1.4	✓	4	3.2	odd	2
3465.2.a.bl.1.1		4	1.1	even	1	trivial
5775.2.a.bz.1.1		4	15.14	odd	2
8085.2.a.bn.1.4		4	21.20	even	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:	\( N \)	\(=\)	\( 3465 = 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3465.a (trivial)

\(f(q)\)	\(=\)	\(q-2.58874 q^{2} +4.70156 q^{4} +1.00000 q^{5} -1.00000 q^{7} -6.99364 q^{8} +O(q^{10})\)	\(q-2.58874 q^{2} +4.70156 q^{4} +1.00000 q^{5} -1.00000 q^{7} -6.99364 q^{8} -2.58874 q^{10} +1.00000 q^{11} +6.40490 q^{13} +2.58874 q^{14} +8.70156 q^{16} +3.70156 q^{17} +7.24672 q^{19} +4.70156 q^{20} -2.58874 q^{22} +7.33388 q^{23} +1.00000 q^{25} -16.5806 q^{26} -4.70156 q^{28} +8.10646 q^{29} +6.77258 q^{31} -8.53879 q^{32} -9.58237 q^{34} -1.00000 q^{35} -3.95005 q^{37} -18.7598 q^{38} -6.99364 q^{40} +2.31773 q^{41} -7.65161 q^{43} +4.70156 q^{44} -18.9855 q^{46} -1.45307 q^{47} +1.00000 q^{49} -2.58874 q^{50} +30.1130 q^{52} -9.10823 q^{53} +1.00000 q^{55} +6.99364 q^{56} -20.9855 q^{58} +2.92898 q^{59} -8.42419 q^{61} -17.5324 q^{62} +4.70156 q^{64} +6.40490 q^{65} -8.72263 q^{67} +17.4031 q^{68} +2.58874 q^{70} +4.40490 q^{71} +3.40667 q^{73} +10.2256 q^{74} +34.0709 q^{76} -1.00000 q^{77} +4.40490 q^{79} +8.70156 q^{80} -6.00000 q^{82} +8.96620 q^{83} +3.70156 q^{85} +19.8080 q^{86} -6.99364 q^{88} -0.206355 q^{89} -6.40490 q^{91} +34.4807 q^{92} +3.76162 q^{94} +7.24672 q^{95} -13.9644 q^{97} -2.58874 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{4} + 4 q^{5} - 4 q^{7}+O(q^{10}) \) 4 * q + 6 * q^4 + 4 * q^5 - 4 * q^7	\( 4 q + 6 q^{4} + 4 q^{5} - 4 q^{7} + 4 q^{11} + 8 q^{13} + 22 q^{16} + 2 q^{17} + 10 q^{19} + 6 q^{20} + 2 q^{23} + 4 q^{25} - 20 q^{26} - 6 q^{28} + 2 q^{29} + 24 q^{31} - 4 q^{35} + 8 q^{37} - 16 q^{38} + 6 q^{43} + 6 q^{44} - 12 q^{46} - 4 q^{47} + 4 q^{49} + 12 q^{52} - 14 q^{53} + 4 q^{55} - 20 q^{58} + 2 q^{59} + 6 q^{61} - 8 q^{62} + 6 q^{64} + 8 q^{65} - 8 q^{67} + 44 q^{68} + 4 q^{73} + 36 q^{74} + 56 q^{76} - 4 q^{77} + 22 q^{80} - 24 q^{82} - 6 q^{83} + 2 q^{85} + 36 q^{86} - 18 q^{89} - 8 q^{91} + 44 q^{92} - 36 q^{94} + 10 q^{95} - 6 q^{97}+O(q^{100}) \) 4 * q + 6 * q^4 + 4 * q^5 - 4 * q^7 + 4 * q^11 + 8 * q^13 + 22 * q^16 + 2 * q^17 + 10 * q^19 + 6 * q^20 + 2 * q^23 + 4 * q^25 - 20 * q^26 - 6 * q^28 + 2 * q^29 + 24 * q^31 - 4 * q^35 + 8 * q^37 - 16 * q^38 + 6 * q^43 + 6 * q^44 - 12 * q^46 - 4 * q^47 + 4 * q^49 + 12 * q^52 - 14 * q^53 + 4 * q^55 - 20 * q^58 + 2 * q^59 + 6 * q^61 - 8 * q^62 + 6 * q^64 + 8 * q^65 - 8 * q^67 + 44 * q^68 + 4 * q^73 + 36 * q^74 + 56 * q^76 - 4 * q^77 + 22 * q^80 - 24 * q^82 - 6 * q^83 + 2 * q^85 + 36 * q^86 - 18 * q^89 - 8 * q^91 + 44 * q^92 - 36 * q^94 + 10 * q^95 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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