LMFDB - Embedded newform 3465.2.a.bh.1.2

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:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 385)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.17009$ 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)

$f(q)$	$=$	$q+1.53919 q^{2} +0.369102 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.51026 q^{8} +O(q^{10})$	\(q+1.53919 q^{2} +0.369102 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.51026 q^{8} +1.53919 q^{10} +1.00000 q^{11} -0.0917087 q^{13} -1.53919 q^{14} -4.60197 q^{16} +5.51026 q^{17} +0.921622 q^{19} +0.369102 q^{20} +1.53919 q^{22} +5.70928 q^{23} +1.00000 q^{25} -0.141157 q^{26} -0.369102 q^{28} -1.41855 q^{29} +0.879362 q^{31} -2.06278 q^{32} +8.48133 q^{34} -1.00000 q^{35} -8.78765 q^{37} +1.41855 q^{38} -2.51026 q^{40} +1.61757 q^{41} +3.86603 q^{43} +0.369102 q^{44} +8.78765 q^{46} +5.90829 q^{47} +1.00000 q^{49} +1.53919 q^{50} -0.0338499 q^{52} +10.0494 q^{53} +1.00000 q^{55} +2.51026 q^{56} -2.18342 q^{58} +2.14116 q^{59} -3.03612 q^{61} +1.35350 q^{62} +6.02893 q^{64} -0.0917087 q^{65} -1.52586 q^{67} +2.03385 q^{68} -1.53919 q^{70} -4.09890 q^{71} +14.1906 q^{73} -13.5259 q^{74} +0.340173 q^{76} -1.00000 q^{77} +14.5464 q^{79} -4.60197 q^{80} +2.48974 q^{82} +8.52359 q^{83} +5.51026 q^{85} +5.95055 q^{86} -2.51026 q^{88} +2.83710 q^{89} +0.0917087 q^{91} +2.10731 q^{92} +9.09398 q^{94} +0.921622 q^{95} +14.2557 q^{97} +1.53919 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 3 q^{7} + 9 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 - 3 * q^7 + 9 * q^8	$ 3 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 3 q^{7} + 9 q^{8} + 3 q^{10} + 3 q^{11} + 2 q^{13} - 3 q^{14} + 5 q^{16} + 6 q^{19} + 5 q^{20} + 3 q^{22} + 10 q^{23} + 3 q^{25} + 20 q^{26} - 5 q^{28} + 10 q^{29} - 10 q^{31} + 11 q^{32} - 6 q^{34} - 3 q^{35} - 16 q^{37} - 10 q^{38} + 9 q^{40} - 2 q^{43} + 5 q^{44} + 16 q^{46} + 20 q^{47} + 3 q^{49} + 3 q^{50} + 32 q^{52} + 12 q^{53} + 3 q^{55} - 9 q^{56} - 2 q^{58} - 14 q^{59} + 10 q^{61} - 6 q^{62} + 33 q^{64} + 2 q^{65} - 2 q^{67} - 26 q^{68} - 3 q^{70} + 24 q^{71} + 4 q^{73} - 38 q^{74} - 10 q^{76} - 3 q^{77} + 8 q^{79} + 5 q^{80} + 24 q^{82} + 10 q^{83} + 36 q^{86} + 9 q^{88} - 20 q^{89} - 2 q^{91} + 18 q^{92} + 38 q^{94} + 6 q^{95} + 3 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 - 3 * q^7 + 9 * q^8 + 3 * q^10 + 3 * q^11 + 2 * q^13 - 3 * q^14 + 5 * q^16 + 6 * q^19 + 5 * q^20 + 3 * q^22 + 10 * q^23 + 3 * q^25 + 20 * q^26 - 5 * q^28 + 10 * q^29 - 10 * q^31 + 11 * q^32 - 6 * q^34 - 3 * q^35 - 16 * q^37 - 10 * q^38 + 9 * q^40 - 2 * q^43 + 5 * q^44 + 16 * q^46 + 20 * q^47 + 3 * q^49 + 3 * q^50 + 32 * q^52 + 12 * q^53 + 3 * q^55 - 9 * q^56 - 2 * q^58 - 14 * q^59 + 10 * q^61 - 6 * q^62 + 33 * q^64 + 2 * q^65 - 2 * q^67 - 26 * q^68 - 3 * q^70 + 24 * q^71 + 4 * q^73 - 38 * q^74 - 10 * q^76 - 3 * q^77 + 8 * q^79 + 5 * q^80 + 24 * q^82 + 10 * q^83 + 36 * q^86 + 9 * q^88 - 20 * q^89 - 2 * q^91 + 18 * q^92 + 38 * q^94 + 6 * q^95 + 3 * q^98

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 $

$2$	1.53919	1.08837	0.544185	−	0.838965i	$-0.316839\pi$
$2$	1.53919	1.08837	0.544185	+	0.838965i	$0.316839\pi$
$3$	0	0
$3$	0	0
$4$	0.369102	0.184551
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.51026	−0.887511
$9$	0	0
$10$	1.53919	0.486734
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−0.0917087	−0.0254354	−0.0127177	−	0.999919i	$-0.504048\pi$
$13$	−0.0917087	−0.0254354	−0.0127177	+	0.999919i	$0.504048\pi$
$14$	−1.53919	−0.411366
$15$	0	0
$16$	−4.60197	−1.15049
$17$	5.51026	1.33643	0.668217	−	0.743966i	$-0.267058\pi$
$17$	5.51026	1.33643	0.668217	+	0.743966i	$0.267058\pi$
$18$	0	0
$19$	0.921622	0.211435	0.105717	−	0.994396i	$-0.466286\pi$
$19$	0.921622	0.211435	0.105717	+	0.994396i	$0.466286\pi$
$20$	0.369102	0.0825338
$21$	0	0
$22$	1.53919	0.328156
$23$	5.70928	1.19047	0.595233	−	0.803553i	$-0.297060\pi$
$23$	5.70928	1.19047	0.595233	+	0.803553i	$0.297060\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.141157	−0.0276832
$27$	0	0
$28$	−0.369102	−0.0697538
$29$	−1.41855	−0.263418	−0.131709	−	0.991288i	$-0.542046\pi$
$29$	−1.41855	−0.263418	−0.131709	+	0.991288i	$0.542046\pi$
$30$	0	0
$31$	0.879362	0.157938	0.0789690	−	0.996877i	$-0.474837\pi$
$31$	0.879362	0.157938	0.0789690	+	0.996877i	$0.474837\pi$
$32$	−2.06278	−0.364651
$33$	0	0
$34$	8.48133	1.45454
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−8.78765	−1.44468	−0.722341	−	0.691537i	$-0.756934\pi$
$37$	−8.78765	−1.44468	−0.722341	+	0.691537i	$0.756934\pi$
$38$	1.41855	0.230119
$39$	0	0
$40$	−2.51026	−0.396907
$41$	1.61757	0.252621	0.126311	−	0.991991i	$-0.459686\pi$
$41$	1.61757	0.252621	0.126311	+	0.991991i	$0.459686\pi$
$42$	0	0
$43$	3.86603	0.589564	0.294782	−	0.955565i	$-0.404753\pi$
$43$	3.86603	0.589564	0.294782	+	0.955565i	$0.404753\pi$
$44$	0.369102	0.0556443
$45$	0	0
$46$	8.78765	1.29567
$47$	5.90829	0.861813	0.430906	−	0.902397i	$-0.358194\pi$
$47$	5.90829	0.861813	0.430906	+	0.902397i	$0.358194\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.53919	0.217674
$51$	0	0
$52$	−0.0338499	−0.00469414
$53$	10.0494	1.38040	0.690199	−	0.723620i	$-0.257523\pi$
$53$	10.0494	1.38040	0.690199	+	0.723620i	$0.257523\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	2.51026	0.335448
$57$	0	0
$58$	−2.18342	−0.286697
$59$	2.14116	0.278755	0.139377	−	0.990239i	$-0.455490\pi$
$59$	2.14116	0.278755	0.139377	+	0.990239i	$0.455490\pi$
$60$	0	0
$61$	−3.03612	−0.388735	−0.194367	−	0.980929i	$-0.562265\pi$
$61$	−3.03612	−0.388735	−0.194367	+	0.980929i	$0.562265\pi$
$62$	1.35350	0.171895
$63$	0	0
$64$	6.02893	0.753616
$65$	−0.0917087	−0.0113751
$66$	0	0
$67$	−1.52586	−0.186413	−0.0932066	−	0.995647i	$-0.529712\pi$
$67$	−1.52586	−0.186413	−0.0932066	+	0.995647i	$0.529712\pi$
$68$	2.03385	0.246641
$69$	0	0
$70$	−1.53919	−0.183968
$71$	−4.09890	−0.486450	−0.243225	−	0.969970i	$-0.578205\pi$
$71$	−4.09890	−0.486450	−0.243225	+	0.969970i	$0.578205\pi$
$72$	0	0
$73$	14.1906	1.66088	0.830442	−	0.557105i	$-0.188088\pi$
$73$	14.1906	1.66088	0.830442	+	0.557105i	$0.188088\pi$
$74$	−13.5259	−1.57235
$75$	0	0
$76$	0.340173	0.0390205
$77$	−1.00000	−0.113961
$78$	0	0
$79$	14.5464	1.63660	0.818298	−	0.574795i	$-0.194918\pi$
$79$	14.5464	1.63660	0.818298	+	0.574795i	$0.194918\pi$
$80$	−4.60197	−0.514516
$81$	0	0
$82$	2.48974	0.274946
$83$	8.52359	0.935586	0.467793	−	0.883838i	$-0.345049\pi$
$83$	8.52359	0.935586	0.467793	+	0.883838i	$0.345049\pi$
$84$	0	0
$85$	5.51026	0.597672
$86$	5.95055	0.641664
$87$	0	0
$88$	−2.51026	−0.267595
$89$	2.83710	0.300732	0.150366	−	0.988630i	$-0.451955\pi$
$89$	2.83710	0.300732	0.150366	+	0.988630i	$0.451955\pi$
$90$	0	0
$91$	0.0917087	0.00961369
$92$	2.10731	0.219702
$93$	0	0
$94$	9.09398	0.937972
$95$	0.921622	0.0945564
$96$	0	0
$97$	14.2557	1.44744	0.723721	−	0.690093i	$-0.242430\pi$
$97$	14.2557	1.44744	0.723721	+	0.690093i	$0.242430\pi$
$98$	1.53919	0.155482
$99$	0	0
$100$	0.369102	0.0369102
$101$	−9.03612	−0.899127	−0.449564	−	0.893248i	$-0.648421\pi$
$101$	−9.03612	−0.899127	−0.449564	+	0.893248i	$0.648421\pi$
$102$	0	0
$103$	3.32684	0.327803	0.163902	−	0.986477i	$-0.447592\pi$
$103$	3.32684	0.327803	0.163902	+	0.986477i	$0.447592\pi$
$104$	0.230213	0.0225742
$105$	0	0
$106$	15.4680	1.50238
$107$	−8.09890	−0.782950	−0.391475	−	0.920189i	$-0.628035\pi$
$107$	−8.09890	−0.782950	−0.391475	+	0.920189i	$0.628035\pi$
$108$	0	0
$109$	15.1773	1.45372	0.726860	−	0.686786i	$-0.240979\pi$
$109$	15.1773	1.45372	0.726860	+	0.686786i	$0.240979\pi$
$110$	1.53919	0.146756
$111$	0	0
$112$	4.60197	0.434845
$113$	7.07838	0.665878	0.332939	−	0.942948i	$-0.391960\pi$
$113$	7.07838	0.665878	0.332939	+	0.942948i	$0.391960\pi$
$114$	0	0
$115$	5.70928	0.532393
$116$	−0.523590	−0.0486142
$117$	0	0
$118$	3.29565	0.303389
$119$	−5.51026	−0.505125
$120$	0	0
$121$	1.00000	0.0909091
$122$	−4.67316	−0.423088
$123$	0	0
$124$	0.324575	0.0291477
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.65983	−0.857171	−0.428586	−	0.903501i	$-0.640988\pi$
$127$	−9.65983	−0.857171	−0.428586	+	0.903501i	$0.640988\pi$
$128$	13.4052	1.18487
$129$	0	0
$130$	−0.141157	−0.0123803
$131$	−4.68035	−0.408924	−0.204462	−	0.978875i	$-0.565544\pi$
$131$	−4.68035	−0.408924	−0.204462	+	0.978875i	$0.565544\pi$
$132$	0	0
$133$	−0.921622	−0.0799148
$134$	−2.34858	−0.202887
$135$	0	0
$136$	−13.8322	−1.18610
$137$	−8.88655	−0.759229	−0.379615	−	0.925145i	$-0.623943\pi$
$137$	−8.88655	−0.759229	−0.379615	+	0.925145i	$0.623943\pi$
$138$	0	0
$139$	−15.0205	−1.27402	−0.637012	−	0.770854i	$-0.719830\pi$
$139$	−15.0205	−1.27402	−0.637012	+	0.770854i	$0.719830\pi$
$140$	−0.369102	−0.0311948
$141$	0	0
$142$	−6.30898	−0.529438
$143$	−0.0917087	−0.00766907
$144$	0	0
$145$	−1.41855	−0.117804
$146$	21.8420	1.80766
$147$	0	0
$148$	−3.24354	−0.266618
$149$	13.7009	1.12242	0.561209	−	0.827674i	$-0.310336\pi$
$149$	13.7009	1.12242	0.561209	+	0.827674i	$0.310336\pi$
$150$	0	0
$151$	1.05559	0.0859028	0.0429514	−	0.999077i	$-0.486324\pi$
$151$	1.05559	0.0859028	0.0429514	+	0.999077i	$0.486324\pi$
$152$	−2.31351	−0.187651
$153$	0	0
$154$	−1.53919	−0.124031
$155$	0.879362	0.0706320
$156$	0	0
$157$	−17.7587	−1.41730	−0.708650	−	0.705560i	$-0.750696\pi$
$157$	−17.7587	−1.41730	−0.708650	+	0.705560i	$0.750696\pi$
$158$	22.3896	1.78122
$159$	0	0
$160$	−2.06278	−0.163077
$161$	−5.70928	−0.449954
$162$	0	0
$163$	−11.4680	−0.898243	−0.449122	−	0.893471i	$-0.648263\pi$
$163$	−11.4680	−0.898243	−0.449122	+	0.893471i	$0.648263\pi$
$164$	0.597048	0.0466216
$165$	0	0
$166$	13.1194	1.01826
$167$	5.60197	0.433493	0.216747	−	0.976228i	$-0.430455\pi$
$167$	5.60197	0.433493	0.216747	+	0.976228i	$0.430455\pi$
$168$	0	0
$169$	−12.9916	−0.999353
$170$	8.48133	0.650488
$171$	0	0
$172$	1.42696	0.108805
$173$	−21.6092	−1.64291	−0.821457	−	0.570271i	$-0.806838\pi$
$173$	−21.6092	−1.64291	−0.821457	+	0.570271i	$0.806838\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−4.60197	−0.346886
$177$	0	0
$178$	4.36683	0.327308
$179$	2.05786	0.153812	0.0769058	−	0.997038i	$-0.475496\pi$
$179$	2.05786	0.153812	0.0769058	+	0.997038i	$0.475496\pi$
$180$	0	0
$181$	20.2823	1.50757	0.753786	−	0.657120i	$-0.228225\pi$
$181$	20.2823	1.50757	0.753786	+	0.657120i	$0.228225\pi$
$182$	0.141157	0.0104633
$183$	0	0
$184$	−14.3318	−1.05655
$185$	−8.78765	−0.646081
$186$	0	0
$187$	5.51026	0.402950
$188$	2.18076	0.159049
$189$	0	0
$190$	1.41855	0.102912
$191$	20.2823	1.46758	0.733788	−	0.679378i	$-0.237750\pi$
$191$	20.2823	1.46758	0.733788	+	0.679378i	$0.237750\pi$
$192$	0	0
$193$	−24.3051	−1.74952	−0.874760	−	0.484557i	$-0.838981\pi$
$193$	−24.3051	−1.74952	−0.874760	+	0.484557i	$0.838981\pi$
$194$	21.9421	1.57535
$195$	0	0
$196$	0.369102	0.0263645
$197$	14.1483	1.00803	0.504014	−	0.863696i	$-0.331856\pi$
$197$	14.1483	1.00803	0.504014	+	0.863696i	$0.331856\pi$
$198$	0	0
$199$	10.4813	0.743002	0.371501	−	0.928433i	$-0.378843\pi$
$199$	10.4813	0.743002	0.371501	+	0.928433i	$0.378843\pi$
$200$	−2.51026	−0.177502
$201$	0	0
$202$	−13.9083	−0.978584
$203$	1.41855	0.0995627
$204$	0	0
$205$	1.61757	0.112976
$206$	5.12064	0.356772
$207$	0	0
$208$	0.422041	0.0292633
$209$	0.921622	0.0637499
$210$	0	0
$211$	−2.65368	−0.182687	−0.0913436	−	0.995819i	$-0.529116\pi$
$211$	−2.65368	−0.182687	−0.0913436	+	0.995819i	$0.529116\pi$
$212$	3.70928	0.254754
$213$	0	0
$214$	−12.4657	−0.852140
$215$	3.86603	0.263661
$216$	0	0
$217$	−0.879362	−0.0596950
$218$	23.3607	1.58219
$219$	0	0
$220$	0.369102	0.0248849
$221$	−0.505339	−0.0339928
$222$	0	0
$223$	−8.67316	−0.580798	−0.290399	−	0.956906i	$-0.593788\pi$
$223$	−8.67316	−0.580798	−0.290399	+	0.956906i	$0.593788\pi$
$224$	2.06278	0.137825
$225$	0	0
$226$	10.8950	0.724722
$227$	−9.67420	−0.642099	−0.321050	−	0.947062i	$-0.604036\pi$
$227$	−9.67420	−0.642099	−0.321050	+	0.947062i	$0.604036\pi$
$228$	0	0
$229$	−13.5486	−0.895320	−0.447660	−	0.894204i	$-0.647742\pi$
$229$	−13.5486	−0.895320	−0.447660	+	0.894204i	$0.647742\pi$
$230$	8.78765	0.579441
$231$	0	0
$232$	3.56093	0.233787
$233$	−8.38962	−0.549622	−0.274811	−	0.961498i	$-0.588615\pi$
$233$	−8.38962	−0.549622	−0.274811	+	0.961498i	$0.588615\pi$
$234$	0	0
$235$	5.90829	0.385414
$236$	0.790306	0.0514446
$237$	0	0
$238$	−8.48133	−0.549763
$239$	29.4908	1.90760	0.953800	−	0.300442i	$-0.0971341\pi$
$239$	29.4908	1.90760	0.953800	+	0.300442i	$0.0971341\pi$
$240$	0	0
$241$	3.64423	0.234745	0.117373	−	0.993088i	$-0.462553\pi$
$241$	3.64423	0.234745	0.117373	+	0.993088i	$0.462553\pi$
$242$	1.53919	0.0989428
$243$	0	0
$244$	−1.12064	−0.0717415
$245$	1.00000	0.0638877
$246$	0	0
$247$	−0.0845208	−0.00537793
$248$	−2.20743	−0.140172
$249$	0	0
$250$	1.53919	0.0973469
$251$	−23.1350	−1.46027	−0.730135	−	0.683303i	$-0.760543\pi$
$251$	−23.1350	−1.46027	−0.730135	+	0.683303i	$0.760543\pi$
$252$	0	0
$253$	5.70928	0.358939
$254$	−14.8683	−0.932920
$255$	0	0
$256$	8.57531	0.535957
$257$	−20.8104	−1.29812	−0.649060	−	0.760737i	$-0.724838\pi$
$257$	−20.8104	−1.29812	−0.649060	+	0.760737i	$0.724838\pi$
$258$	0	0
$259$	8.78765	0.546038
$260$	−0.0338499	−0.00209928
$261$	0	0
$262$	−7.20394	−0.445061
$263$	23.7009	1.46146	0.730729	−	0.682668i	$-0.239180\pi$
$263$	23.7009	1.46146	0.730729	+	0.682668i	$0.239180\pi$
$264$	0	0
$265$	10.0494	0.617333
$266$	−1.41855	−0.0869769
$267$	0	0
$268$	−0.563198	−0.0344028
$269$	3.50307	0.213586	0.106793	−	0.994281i	$-0.465942\pi$
$269$	3.50307	0.213586	0.106793	+	0.994281i	$0.465942\pi$
$270$	0	0
$271$	−8.49693	−0.516152	−0.258076	−	0.966125i	$-0.583088\pi$
$271$	−8.49693	−0.516152	−0.258076	+	0.966125i	$0.583088\pi$
$272$	−25.3580	−1.53756
$273$	0	0
$274$	−13.6781	−0.826323
$275$	1.00000	0.0603023
$276$	0	0
$277$	−25.9649	−1.56008	−0.780041	−	0.625729i	$-0.784802\pi$
$277$	−25.9649	−1.56008	−0.780041	+	0.625729i	$0.784802\pi$
$278$	−23.1194	−1.38661
$279$	0	0
$280$	2.51026	0.150017
$281$	−11.6742	−0.696425	−0.348212	−	0.937416i	$-0.613211\pi$
$281$	−11.6742	−0.696425	−0.348212	+	0.937416i	$0.613211\pi$
$282$	0	0
$283$	−14.2557	−0.847411	−0.423705	−	0.905800i	$-0.639271\pi$
$283$	−14.2557	−0.847411	−0.423705	+	0.905800i	$0.639271\pi$
$284$	−1.51291	−0.0897748
$285$	0	0
$286$	−0.141157	−0.00834679
$287$	−1.61757	−0.0954819
$288$	0	0
$289$	13.3630	0.786056
$290$	−2.18342	−0.128215
$291$	0	0
$292$	5.23779	0.306518
$293$	−25.1122	−1.46707	−0.733536	−	0.679651i	$-0.762131\pi$
$293$	−25.1122	−1.46707	−0.733536	+	0.679651i	$0.762131\pi$
$294$	0	0
$295$	2.14116	0.124663
$296$	22.0593	1.28217
$297$	0	0
$298$	21.0882	1.22161
$299$	−0.523590	−0.0302800
$300$	0	0
$301$	−3.86603	−0.222834
$302$	1.62475	0.0934941
$303$	0	0
$304$	−4.24128	−0.243254
$305$	−3.03612	−0.173848
$306$	0	0
$307$	8.02666	0.458106	0.229053	−	0.973414i	$-0.426437\pi$
$307$	8.02666	0.458106	0.229053	+	0.973414i	$0.426437\pi$
$308$	−0.369102	−0.0210316
$309$	0	0
$310$	1.35350	0.0768739
$311$	26.3968	1.49683	0.748413	−	0.663233i	$-0.230816\pi$
$311$	26.3968	1.49683	0.748413	+	0.663233i	$0.230816\pi$
$312$	0	0
$313$	25.7321	1.45446	0.727231	−	0.686393i	$-0.240807\pi$
$313$	25.7321	1.45446	0.727231	+	0.686393i	$0.240807\pi$
$314$	−27.3340	−1.54255
$315$	0	0
$316$	5.36910	0.302036
$317$	−6.31351	−0.354602	−0.177301	−	0.984157i	$-0.556737\pi$
$317$	−6.31351	−0.354602	−0.177301	+	0.984157i	$0.556737\pi$
$318$	0	0
$319$	−1.41855	−0.0794236
$320$	6.02893	0.337027
$321$	0	0
$322$	−8.78765	−0.489717
$323$	5.07838	0.282568
$324$	0	0
$325$	−0.0917087	−0.00508709
$326$	−17.6514	−0.977622
$327$	0	0
$328$	−4.06051	−0.224204
$329$	−5.90829	−0.325735
$330$	0	0
$331$	3.50307	0.192546	0.0962731	−	0.995355i	$-0.469308\pi$
$331$	3.50307	0.192546	0.0962731	+	0.995355i	$0.469308\pi$
$332$	3.14608	0.172663
$333$	0	0
$334$	8.62249	0.471802
$335$	−1.52586	−0.0833665
$336$	0	0
$337$	7.57918	0.412864	0.206432	−	0.978461i	$-0.433815\pi$
$337$	7.57918	0.412864	0.206432	+	0.978461i	$0.433815\pi$
$338$	−19.9965	−1.08767
$339$	0	0
$340$	2.03385	0.110301
$341$	0.879362	0.0476201
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−9.70474	−0.523245
$345$	0	0
$346$	−33.2606	−1.78810
$347$	35.4824	1.90479	0.952397	−	0.304861i	$-0.0986100\pi$
$347$	35.4824	1.90479	0.952397	+	0.304861i	$0.0986100\pi$
$348$	0	0
$349$	−13.6586	−0.731128	−0.365564	−	0.930786i	$-0.619124\pi$
$349$	−13.6586	−0.731128	−0.365564	+	0.930786i	$0.619124\pi$
$350$	−1.53919	−0.0822731
$351$	0	0
$352$	−2.06278	−0.109947
$353$	−26.4657	−1.40863	−0.704314	−	0.709888i	$-0.748745\pi$
$353$	−26.4657	−1.40863	−0.704314	+	0.709888i	$0.748745\pi$
$354$	0	0
$355$	−4.09890	−0.217547
$356$	1.04718	0.0555005
$357$	0	0
$358$	3.16743	0.167404
$359$	15.3958	0.812557	0.406279	−	0.913749i	$-0.366826\pi$
$359$	15.3958	0.812557	0.406279	+	0.913749i	$0.366826\pi$
$360$	0	0
$361$	−18.1506	−0.955295
$362$	31.2183	1.64080
$363$	0	0
$364$	0.0338499	0.00177422
$365$	14.1906	0.742770
$366$	0	0
$367$	−34.6875	−1.81067	−0.905337	−	0.424693i	$-0.860382\pi$
$367$	−34.6875	−1.81067	−0.905337	+	0.424693i	$0.860382\pi$
$368$	−26.2739	−1.36962
$369$	0	0
$370$	−13.5259	−0.703176
$371$	−10.0494	−0.521741
$372$	0	0
$373$	36.3584	1.88257	0.941284	−	0.337616i	$-0.109621\pi$
$373$	36.3584	1.88257	0.941284	+	0.337616i	$0.109621\pi$
$374$	8.48133	0.438559
$375$	0	0
$376$	−14.8313	−0.764868
$377$	0.130094	0.00670016
$378$	0	0
$379$	−33.1461	−1.70260	−0.851300	−	0.524680i	$-0.824185\pi$
$379$	−33.1461	−1.70260	−0.851300	+	0.524680i	$0.824185\pi$
$380$	0.340173	0.0174505
$381$	0	0
$382$	31.2183	1.59727
$383$	34.2628	1.75075	0.875375	−	0.483445i	$-0.160615\pi$
$383$	34.2628	1.75075	0.875375	+	0.483445i	$0.160615\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	−37.4101	−1.90413
$387$	0	0
$388$	5.26180	0.267127
$389$	−23.2762	−1.18015	−0.590074	−	0.807349i	$-0.700902\pi$
$389$	−23.2762	−1.18015	−0.590074	+	0.807349i	$0.700902\pi$
$390$	0	0
$391$	31.4596	1.59098
$392$	−2.51026	−0.126787
$393$	0	0
$394$	21.7770	1.09711
$395$	14.5464	0.731908
$396$	0	0
$397$	−29.8576	−1.49851	−0.749256	−	0.662281i	$-0.769588\pi$
$397$	−29.8576	−1.49851	−0.749256	+	0.662281i	$0.769588\pi$
$398$	16.1327	0.808662
$399$	0	0
$400$	−4.60197	−0.230098
$401$	5.51745	0.275528	0.137764	−	0.990465i	$-0.456008\pi$
$401$	5.51745	0.275528	0.137764	+	0.990465i	$0.456008\pi$
$402$	0	0
$403$	−0.0806452	−0.00401722
$404$	−3.33525	−0.165935
$405$	0	0
$406$	2.18342	0.108361
$407$	−8.78765	−0.435588
$408$	0	0
$409$	−3.43415	−0.169808	−0.0849039	−	0.996389i	$-0.527058\pi$
$409$	−3.43415	−0.169808	−0.0849039	+	0.996389i	$0.527058\pi$
$410$	2.48974	0.122960
$411$	0	0
$412$	1.22795	0.0604965
$413$	−2.14116	−0.105359
$414$	0	0
$415$	8.52359	0.418407
$416$	0.189175	0.00927506
$417$	0	0
$418$	1.41855	0.0693836
$419$	−18.2134	−0.889782	−0.444891	−	0.895585i	$-0.646758\pi$
$419$	−18.2134	−0.889782	−0.444891	+	0.895585i	$0.646758\pi$
$420$	0	0
$421$	10.6576	0.519418	0.259709	−	0.965687i	$-0.416373\pi$
$421$	10.6576	0.519418	0.259709	+	0.965687i	$0.416373\pi$
$422$	−4.08452	−0.198831
$423$	0	0
$424$	−25.2267	−1.22512
$425$	5.51026	0.267287
$426$	0	0
$427$	3.03612	0.146928
$428$	−2.98932	−0.144494
$429$	0	0
$430$	5.95055	0.286961
$431$	7.61038	0.366579	0.183290	−	0.983059i	$-0.441325\pi$
$431$	7.61038	0.366579	0.183290	+	0.983059i	$0.441325\pi$
$432$	0	0
$433$	33.5318	1.61144	0.805718	−	0.592299i	$-0.201780\pi$
$433$	33.5318	1.61144	0.805718	+	0.592299i	$0.201780\pi$
$434$	−1.35350	−0.0649703
$435$	0	0
$436$	5.60197	0.268286
$437$	5.26180	0.251706
$438$	0	0
$439$	−13.7587	−0.656668	−0.328334	−	0.944562i	$-0.606487\pi$
$439$	−13.7587	−0.656668	−0.328334	+	0.944562i	$0.606487\pi$
$440$	−2.51026	−0.119672
$441$	0	0
$442$	−0.777812	−0.0369967
$443$	11.1689	0.530649	0.265324	−	0.964159i	$-0.414521\pi$
$443$	11.1689	0.530649	0.265324	+	0.964159i	$0.414521\pi$
$444$	0	0
$445$	2.83710	0.134492
$446$	−13.3496	−0.632123
$447$	0	0
$448$	−6.02893	−0.284840
$449$	19.0700	0.899967	0.449984	−	0.893037i	$-0.351430\pi$
$449$	19.0700	0.899967	0.449984	+	0.893037i	$0.351430\pi$
$450$	0	0
$451$	1.61757	0.0761682
$452$	2.61265	0.122889
$453$	0	0
$454$	−14.8904	−0.698842
$455$	0.0917087	0.00429937
$456$	0	0
$457$	−0.787653	−0.0368449	−0.0184224	−	0.999830i	$-0.505864\pi$
$457$	−0.787653	−0.0368449	−0.0184224	+	0.999830i	$0.505864\pi$
$458$	−20.8539	−0.974440
$459$	0	0
$460$	2.10731	0.0982537
$461$	−25.5864	−1.19168	−0.595838	−	0.803105i	$-0.703180\pi$
$461$	−25.5864	−1.19168	−0.595838	+	0.803105i	$0.703180\pi$
$462$	0	0
$463$	−14.2595	−0.662696	−0.331348	−	0.943509i	$-0.607503\pi$
$463$	−14.2595	−0.662696	−0.331348	+	0.943509i	$0.607503\pi$
$464$	6.52813	0.303061
$465$	0	0
$466$	−12.9132	−0.598193
$467$	−18.3474	−0.849015	−0.424507	−	0.905425i	$-0.639553\pi$
$467$	−18.3474	−0.849015	−0.424507	+	0.905425i	$0.639553\pi$
$468$	0	0
$469$	1.52586	0.0704576
$470$	9.09398	0.419474
$471$	0	0
$472$	−5.37486	−0.247398
$473$	3.86603	0.177760
$474$	0	0
$475$	0.921622	0.0422869
$476$	−2.03385	−0.0932214
$477$	0	0
$478$	45.3919	2.07618
$479$	−12.1711	−0.556113	−0.278057	−	0.960565i	$-0.589690\pi$
$479$	−12.1711	−0.556113	−0.278057	+	0.960565i	$0.589690\pi$
$480$	0	0
$481$	0.805905	0.0367461
$482$	5.60916	0.255490
$483$	0	0
$484$	0.369102	0.0167774
$485$	14.2557	0.647316
$486$	0	0
$487$	−4.23287	−0.191809	−0.0959047	−	0.995391i	$-0.530574\pi$
$487$	−4.23287	−0.191809	−0.0959047	+	0.995391i	$0.530574\pi$
$488$	7.62144	0.345006
$489$	0	0
$490$	1.53919	0.0695335
$491$	−22.0183	−0.993670	−0.496835	−	0.867845i	$-0.665505\pi$
$491$	−22.0183	−0.993670	−0.496835	+	0.867845i	$0.665505\pi$
$492$	0	0
$493$	−7.81658	−0.352041
$494$	−0.130094	−0.00585318
$495$	0	0
$496$	−4.04680	−0.181706
$497$	4.09890	0.183861
$498$	0	0
$499$	32.9939	1.47701	0.738504	−	0.674249i	$-0.235533\pi$
$499$	32.9939	1.47701	0.738504	+	0.674249i	$0.235533\pi$
$500$	0.369102	0.0165068
$501$	0	0
$502$	−35.6092	−1.58931
$503$	29.0349	1.29460	0.647301	−	0.762235i	$-0.275898\pi$
$503$	29.0349	1.29460	0.647301	+	0.762235i	$0.275898\pi$
$504$	0	0
$505$	−9.03612	−0.402102
$506$	8.78765	0.390659
$507$	0	0
$508$	−3.56547	−0.158192
$509$	21.5031	0.953107	0.476553	−	0.879146i	$-0.341886\pi$
$509$	21.5031	0.953107	0.476553	+	0.879146i	$0.341886\pi$
$510$	0	0
$511$	−14.1906	−0.627755
$512$	−13.6114	−0.601546
$513$	0	0
$514$	−32.0312	−1.41284
$515$	3.32684	0.146598
$516$	0	0
$517$	5.90829	0.259846
$518$	13.5259	0.594292
$519$	0	0
$520$	0.230213	0.0100955
$521$	−24.5958	−1.07756	−0.538781	−	0.842446i	$-0.681115\pi$
$521$	−24.5958	−1.07756	−0.538781	+	0.842446i	$0.681115\pi$
$522$	0	0
$523$	38.0677	1.66458	0.832292	−	0.554337i	$-0.187028\pi$
$523$	38.0677	1.66458	0.832292	+	0.554337i	$0.187028\pi$
$524$	−1.72753	−0.0754674
$525$	0	0
$526$	36.4801	1.59061
$527$	4.84551	0.211074
$528$	0	0
$529$	9.59583	0.417210
$530$	15.4680	0.671887
$531$	0	0
$532$	−0.340173	−0.0147484
$533$	−0.148345	−0.00642554
$534$	0	0
$535$	−8.09890	−0.350146
$536$	3.83030	0.165444
$537$	0	0
$538$	5.39189	0.232461
$539$	1.00000	0.0430730
$540$	0	0
$541$	2.13009	0.0915799	0.0457899	−	0.998951i	$-0.485420\pi$
$541$	2.13009	0.0915799	0.0457899	+	0.998951i	$0.485420\pi$
$542$	−13.0784	−0.561764
$543$	0	0
$544$	−11.3664	−0.487332
$545$	15.1773	0.650123
$546$	0	0
$547$	−9.13170	−0.390443	−0.195222	−	0.980759i	$-0.562543\pi$
$547$	−9.13170	−0.390443	−0.195222	+	0.980759i	$0.562543\pi$
$548$	−3.28005	−0.140117
$549$	0	0
$550$	1.53919	0.0656312
$551$	−1.30737	−0.0556957
$552$	0	0
$553$	−14.5464	−0.618575
$554$	−39.9649	−1.69795
$555$	0	0
$556$	−5.54411	−0.235123
$557$	11.7093	0.496138	0.248069	−	0.968742i	$-0.420204\pi$
$557$	11.7093	0.496138	0.248069	+	0.968742i	$0.420204\pi$
$558$	0	0
$559$	−0.354549	−0.0149958
$560$	4.60197	0.194469
$561$	0	0
$562$	−17.9688	−0.757968
$563$	−12.5958	−0.530851	−0.265425	−	0.964131i	$-0.585512\pi$
$563$	−12.5958	−0.530851	−0.265425	+	0.964131i	$0.585512\pi$
$564$	0	0
$565$	7.07838	0.297790
$566$	−21.9421	−0.922297
$567$	0	0
$568$	10.2893	0.431729
$569$	−7.54411	−0.316266	−0.158133	−	0.987418i	$-0.550547\pi$
$569$	−7.54411	−0.316266	−0.158133	+	0.987418i	$0.550547\pi$
$570$	0	0
$571$	36.8104	1.54047	0.770234	−	0.637761i	$-0.220139\pi$
$571$	36.8104	1.54047	0.770234	+	0.637761i	$0.220139\pi$
$572$	−0.0338499	−0.00141534
$573$	0	0
$574$	−2.48974	−0.103920
$575$	5.70928	0.238093
$576$	0	0
$577$	−39.5174	−1.64513	−0.822566	−	0.568669i	$-0.807459\pi$
$577$	−39.5174	−1.64513	−0.822566	+	0.568669i	$0.807459\pi$
$578$	20.5681	0.855521
$579$	0	0
$580$	−0.523590	−0.0217409
$581$	−8.52359	−0.353618
$582$	0	0
$583$	10.0494	0.416206
$584$	−35.6221	−1.47405
$585$	0	0
$586$	−38.6525	−1.59672
$587$	−1.56812	−0.0647232	−0.0323616	−	0.999476i	$-0.510303\pi$
$587$	−1.56812	−0.0647232	−0.0323616	+	0.999476i	$0.510303\pi$
$588$	0	0
$589$	0.810439	0.0333936
$590$	3.29565	0.135680
$591$	0	0
$592$	40.4405	1.66209
$593$	−4.95547	−0.203497	−0.101748	−	0.994810i	$-0.532444\pi$
$593$	−4.95547	−0.203497	−0.101748	+	0.994810i	$0.532444\pi$
$594$	0	0
$595$	−5.51026	−0.225899
$596$	5.05702	0.207144
$597$	0	0
$598$	−0.805905	−0.0329559
$599$	12.3668	0.505295	0.252648	−	0.967558i	$-0.418699\pi$
$599$	12.3668	0.505295	0.252648	+	0.967558i	$0.418699\pi$
$600$	0	0
$601$	24.9516	1.01780	0.508898	−	0.860827i	$-0.330053\pi$
$601$	24.9516	1.01780	0.508898	+	0.860827i	$0.330053\pi$
$602$	−5.95055	−0.242526
$603$	0	0
$604$	0.389621	0.0158535
$605$	1.00000	0.0406558
$606$	0	0
$607$	−19.2762	−0.782396	−0.391198	−	0.920307i	$-0.627939\pi$
$607$	−19.2762	−0.782396	−0.391198	+	0.920307i	$0.627939\pi$
$608$	−1.90110	−0.0770999
$609$	0	0
$610$	−4.67316	−0.189211
$611$	−0.541842	−0.0219206
$612$	0	0
$613$	−42.6986	−1.72458	−0.862290	−	0.506415i	$-0.830971\pi$
$613$	−42.6986	−1.72458	−0.862290	+	0.506415i	$0.830971\pi$
$614$	12.3545	0.498589
$615$	0	0
$616$	2.51026	0.101141
$617$	−31.7770	−1.27929	−0.639646	−	0.768669i	$-0.720919\pi$
$617$	−31.7770	−1.27929	−0.639646	+	0.768669i	$0.720919\pi$
$618$	0	0
$619$	−19.8420	−0.797518	−0.398759	−	0.917056i	$-0.630559\pi$
$619$	−19.8420	−0.797518	−0.398759	+	0.917056i	$0.630559\pi$
$620$	0.324575	0.0130352
$621$	0	0
$622$	40.6297	1.62910
$623$	−2.83710	−0.113666
$624$	0	0
$625$	1.00000	0.0400000
$626$	39.6065	1.58299
$627$	0	0
$628$	−6.55479	−0.261564
$629$	−48.4222	−1.93072
$630$	0	0
$631$	−3.63317	−0.144634	−0.0723170	−	0.997382i	$-0.523039\pi$
$631$	−3.63317	−0.144634	−0.0723170	+	0.997382i	$0.523039\pi$
$632$	−36.5152	−1.45250
$633$	0	0
$634$	−9.71769	−0.385939
$635$	−9.65983	−0.383339
$636$	0	0
$637$	−0.0917087	−0.00363363
$638$	−2.18342	−0.0864423
$639$	0	0
$640$	13.4052	0.529888
$641$	−37.5402	−1.48275	−0.741375	−	0.671091i	$-0.765826\pi$
$641$	−37.5402	−1.48275	−0.741375	+	0.671091i	$0.765826\pi$
$642$	0	0
$643$	−34.8710	−1.37518	−0.687588	−	0.726101i	$-0.741330\pi$
$643$	−34.8710	−1.37518	−0.687588	+	0.726101i	$0.741330\pi$
$644$	−2.10731	−0.0830395
$645$	0	0
$646$	7.81658	0.307539
$647$	−30.6342	−1.20436	−0.602178	−	0.798362i	$-0.705700\pi$
$647$	−30.6342	−1.20436	−0.602178	+	0.798362i	$0.705700\pi$
$648$	0	0
$649$	2.14116	0.0840478
$650$	−0.141157	−0.00553664
$651$	0	0
$652$	−4.23287	−0.165772
$653$	−5.40417	−0.211482	−0.105741	−	0.994394i	$-0.533721\pi$
$653$	−5.40417	−0.211482	−0.105741	+	0.994394i	$0.533721\pi$
$654$	0	0
$655$	−4.68035	−0.182876
$656$	−7.44399	−0.290639
$657$	0	0
$658$	−9.09398	−0.354520
$659$	19.4101	0.756112	0.378056	−	0.925783i	$-0.376593\pi$
$659$	19.4101	0.756112	0.378056	+	0.925783i	$0.376593\pi$
$660$	0	0
$661$	−26.3090	−1.02330	−0.511650	−	0.859194i	$-0.670966\pi$
$661$	−26.3090	−1.02330	−0.511650	+	0.859194i	$0.670966\pi$
$662$	5.39189	0.209562
$663$	0	0
$664$	−21.3964	−0.830342
$665$	−0.921622	−0.0357390
$666$	0	0
$667$	−8.09890	−0.313591
$668$	2.06770	0.0800017
$669$	0	0
$670$	−2.34858	−0.0907337
$671$	−3.03612	−0.117208
$672$	0	0
$673$	−15.7938	−0.608806	−0.304403	−	0.952543i	$-0.598457\pi$
$673$	−15.7938	−0.608806	−0.304403	+	0.952543i	$0.598457\pi$
$674$	11.6658	0.449350
$675$	0	0
$676$	−4.79523	−0.184432
$677$	15.7081	0.603710	0.301855	−	0.953354i	$-0.402394\pi$
$677$	15.7081	0.603710	0.301855	+	0.953354i	$0.402394\pi$
$678$	0	0
$679$	−14.2557	−0.547082
$680$	−13.8322	−0.530440
$681$	0	0
$682$	1.35350	0.0518283
$683$	44.0326	1.68486	0.842431	−	0.538805i	$-0.181124\pi$
$683$	44.0326	1.68486	0.842431	+	0.538805i	$0.181124\pi$
$684$	0	0
$685$	−8.88655	−0.339538
$686$	−1.53919	−0.0587665
$687$	0	0
$688$	−17.7914	−0.678289
$689$	−0.921622	−0.0351110
$690$	0	0
$691$	4.63809	0.176441	0.0882205	−	0.996101i	$-0.471882\pi$
$691$	4.63809	0.176441	0.0882205	+	0.996101i	$0.471882\pi$
$692$	−7.97599	−0.303202
$693$	0	0
$694$	54.6141	2.07312
$695$	−15.0205	−0.569761
$696$	0	0
$697$	8.91321	0.337612
$698$	−21.0232	−0.795739
$699$	0	0
$700$	−0.369102	−0.0139508
$701$	14.6491	0.553291	0.276645	−	0.960972i	$-0.410777\pi$
$701$	14.6491	0.553291	0.276645	+	0.960972i	$0.410777\pi$
$702$	0	0
$703$	−8.09890	−0.305456
$704$	6.02893	0.227224
$705$	0	0
$706$	−40.7358	−1.53311
$707$	9.03612	0.339838
$708$	0	0
$709$	−25.5174	−0.958328	−0.479164	−	0.877725i	$-0.659060\pi$
$709$	−25.5174	−0.958328	−0.479164	+	0.877725i	$0.659060\pi$
$710$	−6.30898	−0.236772
$711$	0	0
$712$	−7.12186	−0.266903
$713$	5.02052	0.188020
$714$	0	0
$715$	−0.0917087	−0.00342971
$716$	0.759561	0.0283861
$717$	0	0
$718$	23.6970	0.884364
$719$	−39.2918	−1.46534	−0.732668	−	0.680586i	$-0.761725\pi$
$719$	−39.2918	−1.46534	−0.732668	+	0.680586i	$0.761725\pi$
$720$	0	0
$721$	−3.32684	−0.123898
$722$	−27.9372	−1.03972
$723$	0	0
$724$	7.48625	0.278224
$725$	−1.41855	−0.0526837
$726$	0	0
$727$	−37.7081	−1.39851	−0.699257	−	0.714870i	$-0.746486\pi$
$727$	−37.7081	−1.39851	−0.699257	+	0.714870i	$0.746486\pi$
$728$	−0.230213	−0.00853225
$729$	0	0
$730$	21.8420	0.808410
$731$	21.3028	0.787914
$732$	0	0
$733$	−4.34736	−0.160573	−0.0802867	−	0.996772i	$-0.525584\pi$
$733$	−4.34736	−0.160573	−0.0802867	+	0.996772i	$0.525584\pi$
$734$	−53.3907	−1.97069
$735$	0	0
$736$	−11.7770	−0.434105
$737$	−1.52586	−0.0562057
$738$	0	0
$739$	38.1568	1.40362	0.701809	−	0.712365i	$-0.252376\pi$
$739$	38.1568	1.40362	0.701809	+	0.712365i	$0.252376\pi$
$740$	−3.24354	−0.119235
$741$	0	0
$742$	−15.4680	−0.567848
$743$	29.2618	1.07351	0.536756	−	0.843738i	$-0.319650\pi$
$743$	29.2618	1.07351	0.536756	+	0.843738i	$0.319650\pi$
$744$	0	0
$745$	13.7009	0.501961
$746$	55.9625	2.04893
$747$	0	0
$748$	2.03385	0.0743649
$749$	8.09890	0.295927
$750$	0	0
$751$	41.6886	1.52124	0.760619	−	0.649199i	$-0.224896\pi$
$751$	41.6886	1.52124	0.760619	+	0.649199i	$0.224896\pi$
$752$	−27.1898	−0.991509
$753$	0	0
$754$	0.200238	0.00729225
$755$	1.05559	0.0384169
$756$	0	0
$757$	39.7419	1.44444	0.722222	−	0.691661i	$-0.243121\pi$
$757$	39.7419	1.44444	0.722222	+	0.691661i	$0.243121\pi$
$758$	−51.0181	−1.85306
$759$	0	0
$760$	−2.31351	−0.0839199
$761$	36.4112	1.31990	0.659952	−	0.751308i	$-0.270577\pi$
$761$	36.4112	1.31990	0.659952	+	0.751308i	$0.270577\pi$
$762$	0	0
$763$	−15.1773	−0.549454
$764$	7.48625	0.270843
$765$	0	0
$766$	52.7370	1.90546
$767$	−0.196363	−0.00709025
$768$	0	0
$769$	−10.5347	−0.379889	−0.189945	−	0.981795i	$-0.560831\pi$
$769$	−10.5347	−0.379889	−0.189945	+	0.981795i	$0.560831\pi$
$770$	−1.53919	−0.0554685
$771$	0	0
$772$	−8.97107	−0.322876
$773$	1.52198	0.0547419	0.0273709	−	0.999625i	$-0.491286\pi$
$773$	1.52198	0.0547419	0.0273709	+	0.999625i	$0.491286\pi$
$774$	0	0
$775$	0.879362	0.0315876
$776$	−35.7854	−1.28462
$777$	0	0
$778$	−35.8264	−1.28444
$779$	1.49079	0.0534129
$780$	0	0
$781$	−4.09890	−0.146670
$782$	48.4222	1.73158
$783$	0	0
$784$	−4.60197	−0.164356
$785$	−17.7587	−0.633836
$786$	0	0
$787$	−15.6020	−0.556150	−0.278075	−	0.960559i	$-0.589696\pi$
$787$	−15.6020	−0.556150	−0.278075	+	0.960559i	$0.589696\pi$
$788$	5.22219	0.186033
$789$	0	0
$790$	22.3896	0.796587
$791$	−7.07838	−0.251678
$792$	0	0
$793$	0.278438	0.00988764
$794$	−45.9565	−1.63094
$795$	0	0
$796$	3.86868	0.137122
$797$	−12.6491	−0.448056	−0.224028	−	0.974583i	$-0.571921\pi$
$797$	−12.6491	−0.448056	−0.224028	+	0.974583i	$0.571921\pi$
$798$	0	0
$799$	32.5562	1.15176
$800$	−2.06278	−0.0729303
$801$	0	0
$802$	8.49239	0.299877
$803$	14.1906	0.500776
$804$	0	0
$805$	−5.70928	−0.201226
$806$	−0.124128	−0.00437223
$807$	0	0
$808$	22.6830	0.797985
$809$	49.9299	1.75544	0.877720	−	0.479174i	$-0.159064\pi$
$809$	49.9299	1.75544	0.877720	+	0.479174i	$0.159064\pi$
$810$	0	0
$811$	7.95896	0.279477	0.139738	−	0.990188i	$-0.455374\pi$
$811$	7.95896	0.279477	0.139738	+	0.990188i	$0.455374\pi$
$812$	0.523590	0.0183744
$813$	0	0
$814$	−13.5259	−0.474081
$815$	−11.4680	−0.401706
$816$	0	0
$817$	3.56302	0.124654
$818$	−5.28580	−0.184814
$819$	0	0
$820$	0.597048	0.0208498
$821$	−4.92162	−0.171766	−0.0858829	−	0.996305i	$-0.527371\pi$
$821$	−4.92162	−0.171766	−0.0858829	+	0.996305i	$0.527371\pi$
$822$	0	0
$823$	−4.04945	−0.141155	−0.0705774	−	0.997506i	$-0.522484\pi$
$823$	−4.04945	−0.141155	−0.0705774	+	0.997506i	$0.522484\pi$
$824$	−8.35124	−0.290929
$825$	0	0
$826$	−3.29565	−0.114670
$827$	−33.3256	−1.15885	−0.579423	−	0.815027i	$-0.696722\pi$
$827$	−33.3256	−1.15885	−0.579423	+	0.815027i	$0.696722\pi$
$828$	0	0
$829$	−0.156755	−0.00544434	−0.00272217	−	0.999996i	$-0.500866\pi$
$829$	−0.156755	−0.00544434	−0.00272217	+	0.999996i	$0.500866\pi$
$830$	13.1194	0.455382
$831$	0	0
$832$	−0.552906	−0.0191686
$833$	5.51026	0.190919
$834$	0	0
$835$	5.60197	0.193864
$836$	0.340173	0.0117651
$837$	0	0
$838$	−28.0338	−0.968413
$839$	49.6775	1.71506	0.857529	−	0.514435i	$-0.171998\pi$
$839$	49.6775	1.71506	0.857529	+	0.514435i	$0.171998\pi$
$840$	0	0
$841$	−26.9877	−0.930611
$842$	16.4040	0.565319
$843$	0	0
$844$	−0.979481	−0.0337151
$845$	−12.9916	−0.446924
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	−46.2472	−1.58814
$849$	0	0
$850$	8.48133	0.290907
$851$	−50.1711	−1.71984
$852$	0	0
$853$	39.4257	1.34991	0.674956	−	0.737858i	$-0.264163\pi$
$853$	39.4257	1.34991	0.674956	+	0.737858i	$0.264163\pi$
$854$	4.67316	0.159912
$855$	0	0
$856$	20.3303	0.694876
$857$	−33.2423	−1.13554	−0.567768	−	0.823189i	$-0.692193\pi$
$857$	−33.2423	−1.13554	−0.567768	+	0.823189i	$0.692193\pi$
$858$	0	0
$859$	7.71646	0.263282	0.131641	−	0.991297i	$-0.457975\pi$
$859$	7.71646	0.263282	0.131641	+	0.991297i	$0.457975\pi$
$860$	1.42696	0.0486590
$861$	0	0
$862$	11.7138	0.398974
$863$	24.6453	0.838935	0.419467	−	0.907770i	$-0.362217\pi$
$863$	24.6453	0.838935	0.419467	+	0.907770i	$0.362217\pi$
$864$	0	0
$865$	−21.6092	−0.734733
$866$	51.6118	1.75384
$867$	0	0
$868$	−0.324575	−0.0110168
$869$	14.5464	0.493452
$870$	0	0
$871$	0.139935	0.00474150
$872$	−38.0989	−1.29019
$873$	0	0
$874$	8.09890	0.273949
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	6.17954	0.208668	0.104334	−	0.994542i	$-0.466729\pi$
$877$	6.17954	0.208668	0.104334	+	0.994542i	$0.466729\pi$
$878$	−21.1773	−0.714698
$879$	0	0
$880$	−4.60197	−0.155132
$881$	49.3295	1.66195	0.830976	−	0.556308i	$-0.187782\pi$
$881$	49.3295	1.66195	0.830976	+	0.556308i	$0.187782\pi$
$882$	0	0
$883$	−22.1529	−0.745504	−0.372752	−	0.927931i	$-0.621586\pi$
$883$	−22.1529	−0.745504	−0.372752	+	0.927931i	$0.621586\pi$
$884$	−0.186522	−0.00627341
$885$	0	0
$886$	17.1910	0.577543
$887$	−38.7358	−1.30062	−0.650310	−	0.759669i	$-0.725361\pi$
$887$	−38.7358	−1.30062	−0.650310	+	0.759669i	$0.725361\pi$
$888$	0	0
$889$	9.65983	0.323980
$890$	4.36683	0.146377
$891$	0	0
$892$	−3.20128	−0.107187
$893$	5.44521	0.182217
$894$	0	0
$895$	2.05786	0.0687866
$896$	−13.4052	−0.447837
$897$	0	0
$898$	29.3523	0.979498
$899$	−1.24742	−0.0416038
$900$	0	0
$901$	55.3751	1.84481
$902$	2.48974	0.0828993
$903$	0	0
$904$	−17.7686	−0.590974
$905$	20.2823	0.674207
$906$	0	0
$907$	18.4352	0.612131	0.306065	−	0.952011i	$-0.400987\pi$
$907$	18.4352	0.612131	0.306065	+	0.952011i	$0.400987\pi$
$908$	−3.57077	−0.118500
$909$	0	0
$910$	0.141157	0.00467931
$911$	11.9011	0.394301	0.197151	−	0.980373i	$-0.436831\pi$
$911$	11.9011	0.394301	0.197151	+	0.980373i	$0.436831\pi$
$912$	0	0
$913$	8.52359	0.282090
$914$	−1.21235	−0.0401009
$915$	0	0
$916$	−5.00084	−0.165232
$917$	4.68035	0.154559
$918$	0	0
$919$	56.3812	1.85984	0.929922	−	0.367756i	$-0.119874\pi$
$919$	56.3812	1.85984	0.929922	+	0.367756i	$0.119874\pi$
$920$	−14.3318	−0.472504
$921$	0	0
$922$	−39.3823	−1.29699
$923$	0.375905	0.0123731
$924$	0	0
$925$	−8.78765	−0.288936
$926$	−21.9481	−0.721260
$927$	0	0
$928$	2.92616	0.0960558
$929$	−19.2351	−0.631084	−0.315542	−	0.948912i	$-0.602186\pi$
$929$	−19.2351	−0.631084	−0.315542	+	0.948912i	$0.602186\pi$
$930$	0	0
$931$	0.921622	0.0302049
$932$	−3.09663	−0.101433
$933$	0	0
$934$	−28.2401	−0.924043
$935$	5.51026	0.180205
$936$	0	0
$937$	17.6358	0.576137	0.288069	−	0.957610i	$-0.406987\pi$
$937$	17.6358	0.576137	0.288069	+	0.957610i	$0.406987\pi$
$938$	2.34858	0.0766840
$939$	0	0
$940$	2.18076	0.0711287
$941$	20.1990	0.658469	0.329235	−	0.944248i	$-0.393209\pi$
$941$	20.1990	0.658469	0.329235	+	0.944248i	$0.393209\pi$
$942$	0	0
$943$	9.23513	0.300737
$944$	−9.85354	−0.320705
$945$	0	0
$946$	5.95055	0.193469
$947$	−17.6925	−0.574928	−0.287464	−	0.957792i	$-0.592812\pi$
$947$	−17.6925	−0.574928	−0.287464	+	0.957792i	$0.592812\pi$
$948$	0	0
$949$	−1.30140	−0.0422453
$950$	1.41855	0.0460239
$951$	0	0
$952$	13.8322	0.448304
$953$	−39.7093	−1.28631	−0.643155	−	0.765736i	$-0.722375\pi$
$953$	−39.7093	−1.28631	−0.643155	+	0.765736i	$0.722375\pi$
$954$	0	0
$955$	20.2823	0.656320
$956$	10.8851	0.352050
$957$	0	0
$958$	−18.7337	−0.605257
$959$	8.88655	0.286962
$960$	0	0
$961$	−30.2267	−0.975056
$962$	1.24044	0.0399934
$963$	0	0
$964$	1.34509	0.0433225
$965$	−24.3051	−0.782409
$966$	0	0
$967$	3.01664	0.0970087	0.0485044	−	0.998823i	$-0.484555\pi$
$967$	3.01664	0.0970087	0.0485044	+	0.998823i	$0.484555\pi$
$968$	−2.51026	−0.0806828
$969$	0	0
$970$	21.9421	0.704520
$971$	18.2134	0.584496	0.292248	−	0.956343i	$-0.405597\pi$
$971$	18.2134	0.584496	0.292248	+	0.956343i	$0.405597\pi$
$972$	0	0
$973$	15.0205	0.481536
$974$	−6.51518	−0.208760
$975$	0	0
$976$	13.9721	0.447237
$977$	30.0845	0.962489	0.481245	−	0.876586i	$-0.340185\pi$
$977$	30.0845	0.962489	0.481245	+	0.876586i	$0.340185\pi$
$978$	0	0
$979$	2.83710	0.0906742
$980$	0.369102	0.0117905
$981$	0	0
$982$	−33.8902	−1.08148
$983$	−5.92267	−0.188904	−0.0944519	−	0.995529i	$-0.530110\pi$
$983$	−5.92267	−0.188904	−0.0944519	+	0.995529i	$0.530110\pi$
$984$	0	0
$985$	14.1483	0.450804
$986$	−12.0312	−0.383151
$987$	0	0
$988$	−0.0311968	−0.000992504 0
$989$	22.0722	0.701856
$990$	0	0
$991$	9.24742	0.293754	0.146877	−	0.989155i	$-0.453078\pi$
$991$	9.24742	0.293754	0.146877	+	0.989155i	$0.453078\pi$
$992$	−1.81393	−0.0575923
$993$	0	0
$994$	6.30898	0.200109
$995$	10.4813	0.332281
$996$	0	0
$997$	32.1496	1.01819	0.509094	−	0.860711i	$-0.329981\pi$
$997$	32.1496	1.01819	0.509094	+	0.860711i	$0.329981\pi$
$998$	50.7838	1.60753
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3465.2.a.bh.1.2		3
3.2	odd	2		385.2.a.f.1.2	✓	3
12.11	even	2		6160.2.a.bn.1.1		3
15.2	even	4		1925.2.b.n.1849.2		6
15.8	even	4		1925.2.b.n.1849.5		6
15.14	odd	2		1925.2.a.v.1.2		3
21.20	even	2		2695.2.a.g.1.2		3
33.32	even	2		4235.2.a.q.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
385.2.a.f.1.2	✓	3	3.2	odd	2
1925.2.a.v.1.2		3	15.14	odd	2
1925.2.b.n.1849.2		6	15.2	even	4
1925.2.b.n.1849.5		6	15.8	even	4
2695.2.a.g.1.2		3	21.20	even	2
3465.2.a.bh.1.2		3	1.1	even	1	trivial
4235.2.a.q.1.2		3	33.32	even	2
6160.2.a.bn.1.1		3	12.11	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)

\(f(q)\)	\(=\)	\(q+1.53919 q^{2} +0.369102 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.51026 q^{8} +O(q^{10})\)	\(q+1.53919 q^{2} +0.369102 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.51026 q^{8} +1.53919 q^{10} +1.00000 q^{11} -0.0917087 q^{13} -1.53919 q^{14} -4.60197 q^{16} +5.51026 q^{17} +0.921622 q^{19} +0.369102 q^{20} +1.53919 q^{22} +5.70928 q^{23} +1.00000 q^{25} -0.141157 q^{26} -0.369102 q^{28} -1.41855 q^{29} +0.879362 q^{31} -2.06278 q^{32} +8.48133 q^{34} -1.00000 q^{35} -8.78765 q^{37} +1.41855 q^{38} -2.51026 q^{40} +1.61757 q^{41} +3.86603 q^{43} +0.369102 q^{44} +8.78765 q^{46} +5.90829 q^{47} +1.00000 q^{49} +1.53919 q^{50} -0.0338499 q^{52} +10.0494 q^{53} +1.00000 q^{55} +2.51026 q^{56} -2.18342 q^{58} +2.14116 q^{59} -3.03612 q^{61} +1.35350 q^{62} +6.02893 q^{64} -0.0917087 q^{65} -1.52586 q^{67} +2.03385 q^{68} -1.53919 q^{70} -4.09890 q^{71} +14.1906 q^{73} -13.5259 q^{74} +0.340173 q^{76} -1.00000 q^{77} +14.5464 q^{79} -4.60197 q^{80} +2.48974 q^{82} +8.52359 q^{83} +5.51026 q^{85} +5.95055 q^{86} -2.51026 q^{88} +2.83710 q^{89} +0.0917087 q^{91} +2.10731 q^{92} +9.09398 q^{94} +0.921622 q^{95} +14.2557 q^{97} +1.53919 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 3 q^{7} + 9 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 - 3 * q^7 + 9 * q^8	\( 3 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 3 q^{7} + 9 q^{8} + 3 q^{10} + 3 q^{11} + 2 q^{13} - 3 q^{14} + 5 q^{16} + 6 q^{19} + 5 q^{20} + 3 q^{22} + 10 q^{23} + 3 q^{25} + 20 q^{26} - 5 q^{28} + 10 q^{29} - 10 q^{31} + 11 q^{32} - 6 q^{34} - 3 q^{35} - 16 q^{37} - 10 q^{38} + 9 q^{40} - 2 q^{43} + 5 q^{44} + 16 q^{46} + 20 q^{47} + 3 q^{49} + 3 q^{50} + 32 q^{52} + 12 q^{53} + 3 q^{55} - 9 q^{56} - 2 q^{58} - 14 q^{59} + 10 q^{61} - 6 q^{62} + 33 q^{64} + 2 q^{65} - 2 q^{67} - 26 q^{68} - 3 q^{70} + 24 q^{71} + 4 q^{73} - 38 q^{74} - 10 q^{76} - 3 q^{77} + 8 q^{79} + 5 q^{80} + 24 q^{82} + 10 q^{83} + 36 q^{86} + 9 q^{88} - 20 q^{89} - 2 q^{91} + 18 q^{92} + 38 q^{94} + 6 q^{95} + 3 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 - 3 * q^7 + 9 * q^8 + 3 * q^10 + 3 * q^11 + 2 * q^13 - 3 * q^14 + 5 * q^16 + 6 * q^19 + 5 * q^20 + 3 * q^22 + 10 * q^23 + 3 * q^25 + 20 * q^26 - 5 * q^28 + 10 * q^29 - 10 * q^31 + 11 * q^32 - 6 * q^34 - 3 * q^35 - 16 * q^37 - 10 * q^38 + 9 * q^40 - 2 * q^43 + 5 * q^44 + 16 * q^46 + 20 * q^47 + 3 * q^49 + 3 * q^50 + 32 * q^52 + 12 * q^53 + 3 * q^55 - 9 * q^56 - 2 * q^58 - 14 * q^59 + 10 * q^61 - 6 * q^62 + 33 * q^64 + 2 * q^65 - 2 * q^67 - 26 * q^68 - 3 * q^70 + 24 * q^71 + 4 * q^73 - 38 * q^74 - 10 * q^76 - 3 * q^77 + 8 * q^79 + 5 * q^80 + 24 * q^82 + 10 * q^83 + 36 * q^86 + 9 * q^88 - 20 * q^89 - 2 * q^91 + 18 * q^92 + 38 * q^94 + 6 * q^95 + 3 * q^98

Introduction

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