LMFDB - Embedded newform 3465.2.a.bj.1.3

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.7232.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + 4x + 4 $ x^4 - 2x^3 - 5x^2 + 4*x + 4
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			1.3
Root			$-1.63640$ 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)

$f(q)$	$=$	$q+0.222191 q^{2} -1.95063 q^{4} -1.00000 q^{5} +1.00000 q^{7} -0.877796 q^{8} +O(q^{10})$	\(q+0.222191 q^{2} -1.95063 q^{4} -1.00000 q^{5} +1.00000 q^{7} -0.877796 q^{8} -0.222191 q^{10} -1.00000 q^{11} +0.900012 q^{13} +0.222191 q^{14} +3.70622 q^{16} +0.506248 q^{17} -4.95063 q^{19} +1.95063 q^{20} -0.222191 q^{22} +6.60749 q^{23} +1.00000 q^{25} +0.199975 q^{26} -1.95063 q^{28} -3.40626 q^{29} -3.54437 q^{31} +2.57908 q^{32} +0.112484 q^{34} -1.00000 q^{35} +11.4456 q^{37} -1.09999 q^{38} +0.877796 q^{40} -6.80127 q^{41} +9.26309 q^{43} +1.95063 q^{44} +1.46813 q^{46} -12.6581 q^{47} +1.00000 q^{49} +0.222191 q^{50} -1.75559 q^{52} +1.39501 q^{53} +1.00000 q^{55} -0.877796 q^{56} -0.756842 q^{58} -6.93938 q^{59} +5.39501 q^{61} -0.787529 q^{62} -6.83940 q^{64} -0.900012 q^{65} -10.3012 q^{67} -0.987504 q^{68} -0.222191 q^{70} -6.35689 q^{71} +3.55562 q^{73} +2.54312 q^{74} +9.65685 q^{76} -1.00000 q^{77} +1.98875 q^{79} -3.70622 q^{80} -1.51118 q^{82} +4.16310 q^{83} -0.506248 q^{85} +2.05818 q^{86} +0.877796 q^{88} -2.25059 q^{89} +0.900012 q^{91} -12.8888 q^{92} -2.81252 q^{94} +4.95063 q^{95} +0.0506185 q^{97} +0.222191 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 6 q^{4} - 4 q^{5} + 4 q^{7} - 6 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 + 6 * q^4 - 4 * q^5 + 4 * q^7 - 6 * q^8	$ 4 q - 2 q^{2} + 6 q^{4} - 4 q^{5} + 4 q^{7} - 6 q^{8} + 2 q^{10} - 4 q^{11} + 4 q^{13} - 2 q^{14} + 6 q^{16} - 6 q^{17} - 6 q^{19} - 6 q^{20} + 2 q^{22} - 10 q^{23} + 4 q^{25} + 6 q^{28} - 6 q^{29} - 8 q^{31} - 14 q^{32} - 16 q^{34} - 4 q^{35} + 12 q^{37} - 4 q^{38} + 6 q^{40} + 6 q^{43} - 6 q^{44} - 4 q^{46} + 4 q^{49} - 2 q^{50} - 12 q^{52} - 14 q^{53} + 4 q^{55} - 6 q^{56} + 20 q^{58} - 2 q^{59} + 2 q^{61} - 20 q^{62} - 2 q^{64} - 4 q^{65} - 12 q^{67} - 20 q^{68} + 2 q^{70} - 4 q^{71} + 20 q^{73} + 32 q^{74} + 16 q^{76} - 4 q^{77} - 4 q^{79} - 6 q^{80} - 16 q^{82} - 14 q^{83} + 6 q^{85} - 40 q^{86} + 6 q^{88} + 6 q^{89} + 4 q^{91} - 40 q^{92} + 4 q^{94} + 6 q^{95} - 14 q^{97} - 2 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 6 * q^4 - 4 * q^5 + 4 * q^7 - 6 * q^8 + 2 * q^10 - 4 * q^11 + 4 * q^13 - 2 * q^14 + 6 * q^16 - 6 * q^17 - 6 * q^19 - 6 * q^20 + 2 * q^22 - 10 * q^23 + 4 * q^25 + 6 * q^28 - 6 * q^29 - 8 * q^31 - 14 * q^32 - 16 * q^34 - 4 * q^35 + 12 * q^37 - 4 * q^38 + 6 * q^40 + 6 * q^43 - 6 * q^44 - 4 * q^46 + 4 * q^49 - 2 * q^50 - 12 * q^52 - 14 * q^53 + 4 * q^55 - 6 * q^56 + 20 * q^58 - 2 * q^59 + 2 * q^61 - 20 * q^62 - 2 * q^64 - 4 * q^65 - 12 * q^67 - 20 * q^68 + 2 * q^70 - 4 * q^71 + 20 * q^73 + 32 * q^74 + 16 * q^76 - 4 * q^77 - 4 * q^79 - 6 * q^80 - 16 * q^82 - 14 * q^83 + 6 * q^85 - 40 * q^86 + 6 * q^88 + 6 * q^89 + 4 * q^91 - 40 * q^92 + 4 * q^94 + 6 * q^95 - 14 * q^97 - 2 * 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 $

$2$	0.222191	0.157113	0.0785565	−	0.996910i	$-0.474969\pi$
$2$	0.222191	0.157113	0.0785565	+	0.996910i	$0.474969\pi$
$3$	0	0
$3$	0	0
$4$	−1.95063	−0.975315
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−0.877796	−0.310348
$9$	0	0
$10$	−0.222191	−0.0702631
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.900012	0.249619	0.124809	−	0.992181i	$-0.460168\pi$
$13$	0.900012	0.249619	0.124809	+	0.992181i	$0.460168\pi$
$14$	0.222191	0.0593831
$15$	0	0
$16$	3.70622	0.926556
$17$	0.506248	0.122783	0.0613916	−	0.998114i	$-0.480446\pi$
$17$	0.506248	0.122783	0.0613916	+	0.998114i	$0.480446\pi$
$18$	0	0
$19$	−4.95063	−1.13575	−0.567876	−	0.823114i	$-0.692235\pi$
$19$	−4.95063	−1.13575	−0.567876	+	0.823114i	$0.692235\pi$
$20$	1.95063	0.436174
$21$	0	0
$22$	−0.222191	−0.0473714
$23$	6.60749	1.37776	0.688878	−	0.724877i	$-0.258104\pi$
$23$	6.60749	1.37776	0.688878	+	0.724877i	$0.258104\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.199975	0.0392183
$27$	0	0
$28$	−1.95063	−0.368635
$29$	−3.40626	−0.632527	−0.316263	−	0.948671i	$-0.602428\pi$
$29$	−3.40626	−0.632527	−0.316263	+	0.948671i	$0.602428\pi$
$30$	0	0
$31$	−3.54437	−0.636588	−0.318294	−	0.947992i	$-0.603110\pi$
$31$	−3.54437	−0.636588	−0.318294	+	0.947992i	$0.603110\pi$
$32$	2.57908	0.455922
$33$	0	0
$34$	0.112484	0.0192908
$35$	−1.00000	−0.169031
$36$	0	0
$37$	11.4456	1.88165	0.940825	−	0.338892i	$-0.110052\pi$
$37$	11.4456	1.88165	0.940825	+	0.338892i	$0.110052\pi$
$38$	−1.09999	−0.178442
$39$	0	0
$40$	0.877796	0.138792
$41$	−6.80127	−1.06218	−0.531090	−	0.847315i	$-0.678218\pi$
$41$	−6.80127	−1.06218	−0.531090	+	0.847315i	$0.678218\pi$
$42$	0	0
$43$	9.26309	1.41261	0.706304	−	0.707909i	$-0.250361\pi$
$43$	9.26309	1.41261	0.706304	+	0.707909i	$0.250361\pi$
$44$	1.95063	0.294069
$45$	0	0
$46$	1.46813	0.216463
$47$	−12.6581	−1.84637	−0.923187	−	0.384351i	$-0.874425\pi$
$47$	−12.6581	−1.84637	−0.923187	+	0.384351i	$0.874425\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0.222191	0.0314226
$51$	0	0
$52$	−1.75559	−0.243457
$53$	1.39501	0.191620	0.0958099	−	0.995400i	$-0.469456\pi$
$53$	1.39501	0.191620	0.0958099	+	0.995400i	$0.469456\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	−0.877796	−0.117300
$57$	0	0
$58$	−0.756842	−0.0993782
$59$	−6.93938	−0.903431	−0.451715	−	0.892162i	$-0.649188\pi$
$59$	−6.93938	−0.903431	−0.451715	+	0.892162i	$0.649188\pi$
$60$	0	0
$61$	5.39501	0.690761	0.345380	−	0.938463i	$-0.387750\pi$
$61$	5.39501	0.690761	0.345380	+	0.938463i	$0.387750\pi$
$62$	−0.787529	−0.100016
$63$	0	0
$64$	−6.83940	−0.854925
$65$	−0.900012	−0.111633
$66$	0	0
$67$	−10.3012	−1.25849	−0.629247	−	0.777206i	$-0.716636\pi$
$67$	−10.3012	−1.25849	−0.629247	+	0.777206i	$0.716636\pi$
$68$	−0.987504	−0.119752
$69$	0	0
$70$	−0.222191	−0.0265569
$71$	−6.35689	−0.754424	−0.377212	−	0.926127i	$-0.623117\pi$
$71$	−6.35689	−0.754424	−0.377212	+	0.926127i	$0.623117\pi$
$72$	0	0
$73$	3.55562	0.416154	0.208077	−	0.978112i	$-0.433280\pi$
$73$	3.55562	0.416154	0.208077	+	0.978112i	$0.433280\pi$
$74$	2.54312	0.295632
$75$	0	0
$76$	9.65685	1.10772
$77$	−1.00000	−0.113961
$78$	0	0
$79$	1.98875	0.223752	0.111876	−	0.993722i	$-0.464314\pi$
$79$	1.98875	0.223752	0.111876	+	0.993722i	$0.464314\pi$
$80$	−3.70622	−0.414368
$81$	0	0
$82$	−1.51118	−0.166882
$83$	4.16310	0.456960	0.228480	−	0.973549i	$-0.426624\pi$
$83$	4.16310	0.456960	0.228480	+	0.973549i	$0.426624\pi$
$84$	0	0
$85$	−0.506248	−0.0549103
$86$	2.05818	0.221939
$87$	0	0
$88$	0.877796	0.0935734
$89$	−2.25059	−0.238562	−0.119281	−	0.992861i	$-0.538059\pi$
$89$	−2.25059	−0.238562	−0.119281	+	0.992861i	$0.538059\pi$
$90$	0	0
$91$	0.900012	0.0943469
$92$	−12.8888	−1.34375
$93$	0	0
$94$	−2.81252	−0.290089
$95$	4.95063	0.507924
$96$	0	0
$97$	0.0506185	0.00513953	0.00256976	−	0.999997i	$-0.499182\pi$
$97$	0.0506185	0.00513953	0.00256976	+	0.999997i	$0.499182\pi$
$98$	0.222191	0.0224447
$99$	0	0
$100$	−1.95063	−0.195063
$101$	3.25690	0.324074	0.162037	−	0.986785i	$-0.448194\pi$
$101$	3.25690	0.324074	0.162037	+	0.986785i	$0.448194\pi$
$102$	0	0
$103$	−0.0481195	−0.00474136	−0.00237068	−	0.999997i	$-0.500755\pi$
$103$	−0.0481195	−0.00474136	−0.00237068	+	0.999997i	$0.500755\pi$
$104$	−0.790027	−0.0774686
$105$	0	0
$106$	0.309960	0.0301060
$107$	−15.8900	−1.53615	−0.768073	−	0.640362i	$-0.778784\pi$
$107$	−15.8900	−1.53615	−0.768073	+	0.640362i	$0.778784\pi$
$108$	0	0
$109$	−17.4706	−1.67338	−0.836691	−	0.547675i	$-0.815513\pi$
$109$	−17.4706	−1.67338	−0.836691	+	0.547675i	$0.815513\pi$
$110$	0.222191	0.0211851
$111$	0	0
$112$	3.70622	0.350205
$113$	−19.4200	−1.82688	−0.913440	−	0.406973i	$-0.866584\pi$
$113$	−19.4200	−1.82688	−0.913440	+	0.406973i	$0.866584\pi$
$114$	0	0
$115$	−6.60749	−0.616151
$116$	6.64436	0.616913
$117$	0	0
$118$	−1.54187	−0.141941
$119$	0.506248	0.0464077
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.19873	0.108528
$123$	0	0
$124$	6.91376	0.620874
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	10.9394	0.970713	0.485357	−	0.874316i	$-0.338690\pi$
$127$	10.9394	0.970713	0.485357	+	0.874316i	$0.338690\pi$
$128$	−6.67782	−0.590242
$129$	0	0
$130$	−0.199975	−0.0175390
$131$	−11.3469	−0.991383	−0.495691	−	0.868499i	$-0.665085\pi$
$131$	−11.3469	−0.991383	−0.495691	+	0.868499i	$0.665085\pi$
$132$	0	0
$133$	−4.95063	−0.429274
$134$	−2.28884	−0.197726
$135$	0	0
$136$	−0.444383	−0.0381055
$137$	−13.1900	−1.12690	−0.563448	−	0.826152i	$-0.690525\pi$
$137$	−13.1900	−1.12690	−0.563448	+	0.826152i	$0.690525\pi$
$138$	0	0
$139$	−19.5581	−1.65890	−0.829449	−	0.558583i	$-0.811345\pi$
$139$	−19.5581	−1.65890	−0.829449	+	0.558583i	$0.811345\pi$
$140$	1.95063	0.164858
$141$	0	0
$142$	−1.41245	−0.118530
$143$	−0.900012	−0.0752628
$144$	0	0
$145$	3.40626	0.282875
$146$	0.790027	0.0653831
$147$	0	0
$148$	−22.3262	−1.83520
$149$	0.568114	0.0465417	0.0232708	−	0.999729i	$-0.492592\pi$
$149$	0.568114	0.0465417	0.0232708	+	0.999729i	$0.492592\pi$
$150$	0	0
$151$	4.44688	0.361882	0.180941	−	0.983494i	$-0.442086\pi$
$151$	4.44688	0.361882	0.180941	+	0.983494i	$0.442086\pi$
$152$	4.34564	0.352478
$153$	0	0
$154$	−0.222191	−0.0179047
$155$	3.54437	0.284691
$156$	0	0
$157$	8.15186	0.650589	0.325294	−	0.945613i	$-0.394537\pi$
$157$	8.15186	0.650589	0.325294	+	0.945613i	$0.394537\pi$
$158$	0.441884	0.0351544
$159$	0	0
$160$	−2.57908	−0.203894
$161$	6.60749	0.520743
$162$	0	0
$163$	6.63311	0.519545	0.259773	−	0.965670i	$-0.416352\pi$
$163$	6.63311	0.519545	0.259773	+	0.965670i	$0.416352\pi$
$164$	13.2668	1.03596
$165$	0	0
$166$	0.925005	0.0717943
$167$	−3.75559	−0.290616	−0.145308	−	0.989386i	$-0.546417\pi$
$167$	−3.75559	−0.290616	−0.145308	+	0.989386i	$0.546417\pi$
$168$	0	0
$169$	−12.1900	−0.937691
$170$	−0.112484	−0.00862713
$171$	0	0
$172$	−18.0689	−1.37774
$173$	15.7038	1.19394	0.596968	−	0.802265i	$-0.296372\pi$
$173$	15.7038	1.19394	0.596968	+	0.802265i	$0.296372\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−3.70622	−0.279367
$177$	0	0
$178$	−0.500062	−0.0374813
$179$	8.95682	0.669464	0.334732	−	0.942313i	$-0.391354\pi$
$179$	8.95682	0.669464	0.334732	+	0.942313i	$0.391354\pi$
$180$	0	0
$181$	−13.8025	−1.02593	−0.512967	−	0.858408i	$-0.671454\pi$
$181$	−13.8025	−1.02593	−0.512967	+	0.858408i	$0.671454\pi$
$182$	0.199975	0.0148231
$183$	0	0
$184$	−5.80002	−0.427584
$185$	−11.4456	−0.841500
$186$	0	0
$187$	−0.506248	−0.0370205
$188$	24.6913	1.80080
$189$	0	0
$190$	1.09999	0.0798015
$191$	−6.91376	−0.500262	−0.250131	−	0.968212i	$-0.580474\pi$
$191$	−6.91376	−0.500262	−0.250131	+	0.968212i	$0.580474\pi$
$192$	0	0
$193$	2.14317	0.154269	0.0771344	−	0.997021i	$-0.475423\pi$
$193$	2.14317	0.154269	0.0771344	+	0.997021i	$0.475423\pi$
$194$	0.0112470	0.000807487 0
$195$	0	0
$196$	−1.95063	−0.139331
$197$	−18.5901	−1.32449	−0.662243	−	0.749289i	$-0.730395\pi$
$197$	−18.5901	−1.32449	−0.662243	+	0.749289i	$0.730395\pi$
$198$	0	0
$199$	−3.60005	−0.255201	−0.127600	−	0.991826i	$-0.540727\pi$
$199$	−3.60005	−0.255201	−0.127600	+	0.991826i	$0.540727\pi$
$200$	−0.877796	−0.0620696
$201$	0	0
$202$	0.723656	0.0509163
$203$	−3.40626	−0.239073
$204$	0	0
$205$	6.80127	0.475022
$206$	−0.0106917	−0.000744929 0
$207$	0	0
$208$	3.33565	0.231286
$209$	4.95063	0.342442
$210$	0	0
$211$	−4.22191	−0.290649	−0.145324	−	0.989384i	$-0.546423\pi$
$211$	−4.22191	−0.290649	−0.145324	+	0.989384i	$0.546423\pi$
$212$	−2.72116	−0.186890
$213$	0	0
$214$	−3.53062	−0.241349
$215$	−9.26309	−0.631737
$216$	0	0
$217$	−3.54437	−0.240608
$218$	−3.88182	−0.262910
$219$	0	0
$220$	−1.95063	−0.131512
$221$	0.455630	0.0306490
$222$	0	0
$223$	−9.94938	−0.666260	−0.333130	−	0.942881i	$-0.608105\pi$
$223$	−9.94938	−0.666260	−0.333130	+	0.942881i	$0.608105\pi$
$224$	2.57908	0.172322
$225$	0	0
$226$	−4.31496	−0.287027
$227$	−14.8519	−0.985755	−0.492877	−	0.870099i	$-0.664055\pi$
$227$	−14.8519	−0.985755	−0.492877	+	0.870099i	$0.664055\pi$
$228$	0	0
$229$	3.24316	0.214314	0.107157	−	0.994242i	$-0.465825\pi$
$229$	3.24316	0.214314	0.107157	+	0.994242i	$0.465825\pi$
$230$	−1.46813	−0.0968054
$231$	0	0
$232$	2.99000	0.196303
$233$	−11.1694	−0.731733	−0.365866	−	0.930667i	$-0.619227\pi$
$233$	−11.1694	−0.731733	−0.365866	+	0.930667i	$0.619227\pi$
$234$	0	0
$235$	12.6581	0.825724
$236$	13.5362	0.881130
$237$	0	0
$238$	0.112484	0.00729125
$239$	−2.45057	−0.158514	−0.0792571	−	0.996854i	$-0.525255\pi$
$239$	−2.45057	−0.158514	−0.0792571	+	0.996854i	$0.525255\pi$
$240$	0	0
$241$	−26.1012	−1.68133	−0.840664	−	0.541557i	$-0.817835\pi$
$241$	−26.1012	−1.68133	−0.840664	+	0.541557i	$0.817835\pi$
$242$	0.222191	0.0142830
$243$	0	0
$244$	−10.5237	−0.673710
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−4.45563	−0.283505
$248$	3.11123	0.197564
$249$	0	0
$250$	−0.222191	−0.0140526
$251$	9.37758	0.591908	0.295954	−	0.955202i	$-0.404363\pi$
$251$	9.37758	0.591908	0.295954	+	0.955202i	$0.404363\pi$
$252$	0	0
$253$	−6.60749	−0.415409
$254$	2.43064	0.152512
$255$	0	0
$256$	12.1950	0.762190
$257$	13.9718	0.871538	0.435769	−	0.900059i	$-0.356477\pi$
$257$	13.9718	0.871538	0.435769	+	0.900059i	$0.356477\pi$
$258$	0	0
$259$	11.4456	0.711197
$260$	1.75559	0.108877
$261$	0	0
$262$	−2.52118	−0.155759
$263$	18.1232	1.11752	0.558761	−	0.829328i	$-0.311277\pi$
$263$	18.1232	1.11752	0.558761	+	0.829328i	$0.311277\pi$
$264$	0	0
$265$	−1.39501	−0.0856950
$266$	−1.09999	−0.0674446
$267$	0	0
$268$	20.0939	1.22743
$269$	−32.1793	−1.96201	−0.981005	−	0.193984i	$-0.937859\pi$
$269$	−32.1793	−1.96201	−0.981005	+	0.193984i	$0.937859\pi$
$270$	0	0
$271$	−24.0669	−1.46196	−0.730979	−	0.682400i	$-0.760936\pi$
$271$	−24.0669	−1.46196	−0.730979	+	0.682400i	$0.760936\pi$
$272$	1.87627	0.113766
$273$	0	0
$274$	−2.93070	−0.177050
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	16.8443	1.01208	0.506039	−	0.862511i	$-0.331109\pi$
$277$	16.8443	1.01208	0.506039	+	0.862511i	$0.331109\pi$
$278$	−4.34564	−0.260634
$279$	0	0
$280$	0.877796	0.0524584
$281$	0.0444327	0.00265063	0.00132532	−	0.999999i	$-0.499578\pi$
$281$	0.0444327	0.00265063	0.00132532	+	0.999999i	$0.499578\pi$
$282$	0	0
$283$	−0.455630	−0.0270844	−0.0135422	−	0.999908i	$-0.504311\pi$
$283$	−0.455630	−0.0270844	−0.0135422	+	0.999908i	$0.504311\pi$
$284$	12.4000	0.735802
$285$	0	0
$286$	−0.199975	−0.0118248
$287$	−6.80127	−0.401467
$288$	0	0
$289$	−16.7437	−0.984924
$290$	0.756842	0.0444433
$291$	0	0
$292$	−6.93570	−0.405881
$293$	−31.1337	−1.81885	−0.909424	−	0.415870i	$-0.863477\pi$
$293$	−31.1337	−1.81885	−0.909424	+	0.415870i	$0.863477\pi$
$294$	0	0
$295$	6.93938	0.404027
$296$	−10.0469	−0.583966
$297$	0	0
$298$	0.126230	0.00731231
$299$	5.94682	0.343913
$300$	0	0
$301$	9.26309	0.533916
$302$	0.988059	0.0568564
$303$	0	0
$304$	−18.3481	−1.05234
$305$	−5.39501	−0.308918
$306$	0	0
$307$	−27.9038	−1.59255	−0.796276	−	0.604934i	$-0.793200\pi$
$307$	−27.9038	−1.59255	−0.796276	+	0.604934i	$0.793200\pi$
$308$	1.95063	0.111148
$309$	0	0
$310$	0.787529	0.0447286
$311$	20.4274	1.15833	0.579167	−	0.815209i	$-0.303378\pi$
$311$	20.4274	1.15833	0.579167	+	0.815209i	$0.303378\pi$
$312$	0	0
$313$	22.0531	1.24652	0.623258	−	0.782016i	$-0.285809\pi$
$313$	22.0531	1.24652	0.623258	+	0.782016i	$0.285809\pi$
$314$	1.81127	0.102216
$315$	0	0
$316$	−3.87932	−0.218229
$317$	17.9038	1.00558	0.502788	−	0.864410i	$-0.332308\pi$
$317$	17.9038	1.00558	0.502788	+	0.864410i	$0.332308\pi$
$318$	0	0
$319$	3.40626	0.190714
$320$	6.83940	0.382334
$321$	0	0
$322$	1.46813	0.0818155
$323$	−2.50625	−0.139451
$324$	0	0
$325$	0.900012	0.0499237
$326$	1.47382	0.0816273
$327$	0	0
$328$	5.97013	0.329645
$329$	−12.6581	−0.697864
$330$	0	0
$331$	18.5112	1.01747	0.508735	−	0.860923i	$-0.330113\pi$
$331$	18.5112	1.01747	0.508735	+	0.860923i	$0.330113\pi$
$332$	−8.12068	−0.445680
$333$	0	0
$334$	−0.834460	−0.0456596
$335$	10.3012	0.562815
$336$	0	0
$337$	24.8187	1.35196	0.675981	−	0.736919i	$-0.263720\pi$
$337$	24.8187	1.35196	0.675981	+	0.736919i	$0.263720\pi$
$338$	−2.70851	−0.147323
$339$	0	0
$340$	0.987504	0.0535549
$341$	3.54437	0.191938
$342$	0	0
$343$	1.00000	0.0539949
$344$	−8.13110	−0.438400
$345$	0	0
$346$	3.48925	0.187583
$347$	−1.03194	−0.0553972	−0.0276986	−	0.999616i	$-0.508818\pi$
$347$	−1.03194	−0.0553972	−0.0276986	+	0.999616i	$0.508818\pi$
$348$	0	0
$349$	18.6100	0.996170	0.498085	−	0.867128i	$-0.334037\pi$
$349$	18.6100	0.996170	0.498085	+	0.867128i	$0.334037\pi$
$350$	0.222191	0.0118766
$351$	0	0
$352$	−2.57908	−0.137466
$353$	8.89126	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$353$	8.89126	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$354$	0	0
$355$	6.35689	0.337389
$356$	4.39008	0.232674
$357$	0	0
$358$	1.99013	0.105182
$359$	10.1395	0.535141	0.267571	−	0.963538i	$-0.413779\pi$
$359$	10.1395	0.535141	0.267571	+	0.963538i	$0.413779\pi$
$360$	0	0
$361$	5.50875	0.289934
$362$	−3.06680	−0.161188
$363$	0	0
$364$	−1.75559	−0.0920180
$365$	−3.55562	−0.186110
$366$	0	0
$367$	−15.4631	−0.807165	−0.403583	−	0.914943i	$-0.632235\pi$
$367$	−15.4631	−0.807165	−0.403583	+	0.914943i	$0.632235\pi$
$368$	24.4888	1.27657
$369$	0	0
$370$	−2.54312	−0.132211
$371$	1.39501	0.0724255
$372$	0	0
$373$	0.470008	0.0243361	0.0121681	−	0.999926i	$-0.496127\pi$
$373$	0.470008	0.0243361	0.0121681	+	0.999926i	$0.496127\pi$
$374$	−0.112484	−0.00581641
$375$	0	0
$376$	11.1112	0.573018
$377$	−3.06568	−0.157890
$378$	0	0
$379$	−25.2225	−1.29559	−0.647797	−	0.761813i	$-0.724310\pi$
$379$	−25.2225	−1.29559	−0.647797	+	0.761813i	$0.724310\pi$
$380$	−9.65685	−0.495386
$381$	0	0
$382$	−1.53618	−0.0785977
$383$	21.2257	1.08458	0.542290	−	0.840191i	$-0.317557\pi$
$383$	21.2257	1.08458	0.542290	+	0.840191i	$0.317557\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0.476194	0.0242376
$387$	0	0
$388$	−0.0987380	−0.00501266
$389$	−8.93182	−0.452861	−0.226431	−	0.974027i	$-0.572706\pi$
$389$	−8.93182	−0.452861	−0.226431	+	0.974027i	$0.572706\pi$
$390$	0	0
$391$	3.34503	0.169165
$392$	−0.877796	−0.0443354
$393$	0	0
$394$	−4.13055	−0.208094
$395$	−1.98875	−0.100065
$396$	0	0
$397$	−17.6563	−0.886144	−0.443072	−	0.896486i	$-0.646111\pi$
$397$	−17.6563	−0.886144	−0.443072	+	0.896486i	$0.646111\pi$
$398$	−0.799900	−0.0400954
$399$	0	0
$400$	3.70622	0.185311
$401$	25.1494	1.25590	0.627951	−	0.778253i	$-0.283894\pi$
$401$	25.1494	1.25590	0.627951	+	0.778253i	$0.283894\pi$
$402$	0	0
$403$	−3.18998	−0.158904
$404$	−6.35302	−0.316074
$405$	0	0
$406$	−0.756842	−0.0375614
$407$	−11.4456	−0.567339
$408$	0	0
$409$	29.0150	1.43470	0.717350	−	0.696713i	$-0.245355\pi$
$409$	29.0150	1.43470	0.717350	+	0.696713i	$0.245355\pi$
$410$	1.51118	0.0746321
$411$	0	0
$412$	0.0938634	0.00462432
$413$	−6.93938	−0.341465
$414$	0	0
$415$	−4.16310	−0.204359
$416$	2.32121	0.113807
$417$	0	0
$418$	1.09999	0.0538021
$419$	26.1294	1.27650	0.638251	−	0.769828i	$-0.279658\pi$
$419$	26.1294	1.27650	0.638251	+	0.769828i	$0.279658\pi$
$420$	0	0
$421$	37.0963	1.80796	0.903982	−	0.427571i	$-0.140631\pi$
$421$	37.0963	1.80796	0.903982	+	0.427571i	$0.140631\pi$
$422$	−0.938073	−0.0456647
$423$	0	0
$424$	−1.22454	−0.0594688
$425$	0.506248	0.0245566
$426$	0	0
$427$	5.39501	0.261083
$428$	30.9956	1.49823
$429$	0	0
$430$	−2.05818	−0.0992542
$431$	−9.60255	−0.462539	−0.231269	−	0.972890i	$-0.574288\pi$
$431$	−9.60255	−0.462539	−0.231269	+	0.972890i	$0.574288\pi$
$432$	0	0
$433$	−31.1137	−1.49523	−0.747615	−	0.664132i	$-0.768801\pi$
$433$	−31.1137	−1.49523	−0.747615	+	0.664132i	$0.768801\pi$
$434$	−0.787529	−0.0378026
$435$	0	0
$436$	34.0787	1.63208
$437$	−32.7112	−1.56479
$438$	0	0
$439$	−34.0644	−1.62580	−0.812902	−	0.582401i	$-0.802113\pi$
$439$	−34.0644	−1.62580	−0.812902	+	0.582401i	$0.802113\pi$
$440$	−0.877796	−0.0418473
$441$	0	0
$442$	0.101237	0.00481535
$443$	−19.0375	−0.904498	−0.452249	−	0.891892i	$-0.649378\pi$
$443$	−19.0375	−0.904498	−0.452249	+	0.891892i	$0.649378\pi$
$444$	0	0
$445$	2.25059	0.106688
$446$	−2.21067	−0.104678
$447$	0	0
$448$	−6.83940	−0.323131
$449$	−18.6806	−0.881592	−0.440796	−	0.897607i	$-0.645304\pi$
$449$	−18.6806	−0.881592	−0.440796	+	0.897607i	$0.645304\pi$
$450$	0	0
$451$	6.80127	0.320260
$452$	37.8813	1.78178
$453$	0	0
$454$	−3.29996	−0.154875
$455$	−0.900012	−0.0421932
$456$	0	0
$457$	27.2875	1.27646	0.638228	−	0.769847i	$-0.279668\pi$
$457$	27.2875	1.27646	0.638228	+	0.769847i	$0.279668\pi$
$458$	0.720602	0.0336715
$459$	0	0
$460$	12.8888	0.600942
$461$	37.2644	1.73558	0.867788	−	0.496934i	$-0.165541\pi$
$461$	37.2644	1.73558	0.867788	+	0.496934i	$0.165541\pi$
$462$	0	0
$463$	−8.50119	−0.395084	−0.197542	−	0.980294i	$-0.563296\pi$
$463$	−8.50119	−0.395084	−0.197542	+	0.980294i	$0.563296\pi$
$464$	−12.6244	−0.586071
$465$	0	0
$466$	−2.48175	−0.114965
$467$	20.5163	0.949381	0.474691	−	0.880153i	$-0.342560\pi$
$467$	20.5163	0.949381	0.474691	+	0.880153i	$0.342560\pi$
$468$	0	0
$469$	−10.3012	−0.475666
$470$	2.81252	0.129732
$471$	0	0
$472$	6.09136	0.280378
$473$	−9.26309	−0.425917
$474$	0	0
$475$	−4.95063	−0.227151
$476$	−0.987504	−0.0452621
$477$	0	0
$478$	−0.544495	−0.0249046
$479$	−12.2806	−0.561117	−0.280559	−	0.959837i	$-0.590520\pi$
$479$	−12.2806	−0.561117	−0.280559	+	0.959837i	$0.590520\pi$
$480$	0	0
$481$	10.3012	0.469695
$482$	−5.79947	−0.264159
$483$	0	0
$484$	−1.95063	−0.0886650
$485$	−0.0506185	−0.00229847
$486$	0	0
$487$	−20.5901	−0.933024	−0.466512	−	0.884515i	$-0.654490\pi$
$487$	−20.5901	−0.933024	−0.466512	+	0.884515i	$0.654490\pi$
$488$	−4.73572	−0.214376
$489$	0	0
$490$	−0.222191	−0.0100376
$491$	−6.32684	−0.285526	−0.142763	−	0.989757i	$-0.545599\pi$
$491$	−6.32684	−0.285526	−0.142763	+	0.989757i	$0.545599\pi$
$492$	0	0
$493$	−1.72441	−0.0776637
$494$	−0.990002	−0.0445423
$495$	0	0
$496$	−13.1362	−0.589834
$497$	−6.35689	−0.285146
$498$	0	0
$499$	−0.627480	−0.0280899	−0.0140449	−	0.999901i	$-0.504471\pi$
$499$	−0.627480	−0.0280899	−0.0140449	+	0.999901i	$0.504471\pi$
$500$	1.95063	0.0872349
$501$	0	0
$502$	2.08362	0.0929964
$503$	7.07686	0.315542	0.157771	−	0.987476i	$-0.449569\pi$
$503$	7.07686	0.315542	0.157771	+	0.987476i	$0.449569\pi$
$504$	0	0
$505$	−3.25690	−0.144930
$506$	−1.46813	−0.0652662
$507$	0	0
$508$	−21.3387	−0.946752
$509$	−3.06062	−0.135659	−0.0678297	−	0.997697i	$-0.521607\pi$
$509$	−3.06062	−0.135659	−0.0678297	+	0.997697i	$0.521607\pi$
$510$	0	0
$511$	3.55562	0.157291
$512$	16.0653	0.709992
$513$	0	0
$514$	3.10442	0.136930
$515$	0.0481195	0.00212040
$516$	0	0
$517$	12.6581	0.556703
$518$	2.54312	0.111738
$519$	0	0
$520$	0.790027	0.0346450
$521$	36.2018	1.58603	0.793016	−	0.609201i	$-0.208510\pi$
$521$	36.2018	1.58603	0.793016	+	0.609201i	$0.208510\pi$
$522$	0	0
$523$	35.7412	1.56285	0.781426	−	0.623998i	$-0.214493\pi$
$523$	35.7412	1.56285	0.781426	+	0.623998i	$0.214493\pi$
$524$	22.1336	0.966911
$525$	0	0
$526$	4.02681	0.175577
$527$	−1.79433	−0.0781623
$528$	0	0
$529$	20.6589	0.898211
$530$	−0.309960	−0.0134638
$531$	0	0
$532$	9.65685	0.418678
$533$	−6.12123	−0.265140
$534$	0	0
$535$	15.8900	0.686985
$536$	9.04236	0.390571
$537$	0	0
$538$	−7.14997	−0.308257
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	44.1644	1.89878	0.949388	−	0.314105i	$-0.101704\pi$
$541$	44.1644	1.89878	0.949388	+	0.314105i	$0.101704\pi$
$542$	−5.34745	−0.229693
$543$	0	0
$544$	1.30566	0.0559796
$545$	17.4706	0.748359
$546$	0	0
$547$	−11.1644	−0.477353	−0.238677	−	0.971099i	$-0.576714\pi$
$547$	−11.1644	−0.477353	−0.238677	+	0.971099i	$0.576714\pi$
$548$	25.7288	1.09908
$549$	0	0
$550$	−0.222191	−0.00947427
$551$	16.8631	0.718394
$552$	0	0
$553$	1.98875	0.0845704
$554$	3.74267	0.159011
$555$	0	0
$556$	38.1507	1.61795
$557$	15.5137	0.657336	0.328668	−	0.944446i	$-0.393400\pi$
$557$	15.5137	0.657336	0.328668	+	0.944446i	$0.393400\pi$
$558$	0	0
$559$	8.33690	0.352613
$560$	−3.70622	−0.156617
$561$	0	0
$562$	0.00987257	0.000416449 0
$563$	−29.7581	−1.25415	−0.627077	−	0.778957i	$-0.715749\pi$
$563$	−29.7581	−1.25415	−0.627077	+	0.778957i	$0.715749\pi$
$564$	0	0
$565$	19.4200	0.817006
$566$	−0.101237	−0.00425531
$567$	0	0
$568$	5.58005	0.234134
$569$	11.2949	0.473507	0.236753	−	0.971570i	$-0.423917\pi$
$569$	11.2949	0.473507	0.236753	+	0.971570i	$0.423917\pi$
$570$	0	0
$571$	33.3300	1.39482	0.697408	−	0.716675i	$-0.254337\pi$
$571$	33.3300	1.39482	0.697408	+	0.716675i	$0.254337\pi$
$572$	1.75559	0.0734050
$573$	0	0
$574$	−1.51118	−0.0630756
$575$	6.60749	0.275551
$576$	0	0
$577$	−47.6300	−1.98286	−0.991432	−	0.130622i	$-0.958303\pi$
$577$	−47.6300	−1.98286	−0.991432	+	0.130622i	$0.958303\pi$
$578$	−3.72031	−0.154744
$579$	0	0
$580$	−6.64436	−0.275892
$581$	4.16310	0.172715
$582$	0	0
$583$	−1.39501	−0.0577756
$584$	−3.12111	−0.129152
$585$	0	0
$586$	−6.91763	−0.285765
$587$	−20.5844	−0.849607	−0.424804	−	0.905285i	$-0.639657\pi$
$587$	−20.5844	−0.849607	−0.424804	+	0.905285i	$0.639657\pi$
$588$	0	0
$589$	17.5469	0.723006
$590$	1.54187	0.0634778
$591$	0	0
$592$	42.4201	1.74345
$593$	7.47882	0.307118	0.153559	−	0.988139i	$-0.450926\pi$
$593$	7.47882	0.307118	0.153559	+	0.988139i	$0.450926\pi$
$594$	0	0
$595$	−0.506248	−0.0207542
$596$	−1.10818	−0.0453928
$597$	0	0
$598$	1.32133	0.0540333
$599$	28.5512	1.16657	0.583285	−	0.812268i	$-0.301767\pi$
$599$	28.5512	1.16657	0.583285	+	0.812268i	$0.301767\pi$
$600$	0	0
$601$	−23.5950	−0.962460	−0.481230	−	0.876594i	$-0.659810\pi$
$601$	−23.5950	−0.962460	−0.481230	+	0.876594i	$0.659810\pi$
$602$	2.05818	0.0838851
$603$	0	0
$604$	−8.67423	−0.352949
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	20.2401	0.821520	0.410760	−	0.911744i	$-0.365263\pi$
$607$	20.2401	0.821520	0.410760	+	0.911744i	$0.365263\pi$
$608$	−12.7681	−0.517814
$609$	0	0
$610$	−1.19873	−0.0485350
$611$	−11.3925	−0.460889
$612$	0	0
$613$	−31.4644	−1.27083	−0.635417	−	0.772169i	$-0.719172\pi$
$613$	−31.4644	−1.27083	−0.635417	+	0.772169i	$0.719172\pi$
$614$	−6.19998	−0.250211
$615$	0	0
$616$	0.877796	0.0353674
$617$	−35.6813	−1.43647	−0.718237	−	0.695798i	$-0.755051\pi$
$617$	−35.6813	−1.43647	−0.718237	+	0.695798i	$0.755051\pi$
$618$	0	0
$619$	5.41245	0.217545	0.108772	−	0.994067i	$-0.465308\pi$
$619$	5.41245	0.217545	0.108772	+	0.994067i	$0.465308\pi$
$620$	−6.91376	−0.277663
$621$	0	0
$622$	4.53880	0.181989
$623$	−2.25059	−0.0901681
$624$	0	0
$625$	1.00000	0.0400000
$626$	4.90001	0.195844
$627$	0	0
$628$	−15.9013	−0.634529
$629$	5.79433	0.231035
$630$	0	0
$631$	−32.0863	−1.27734	−0.638668	−	0.769483i	$-0.720514\pi$
$631$	−32.0863	−1.27734	−0.638668	+	0.769483i	$0.720514\pi$
$632$	−1.74572	−0.0694410
$633$	0	0
$634$	3.97806	0.157989
$635$	−10.9394	−0.434116
$636$	0	0
$637$	0.900012	0.0356598
$638$	0.756842	0.0299637
$639$	0	0
$640$	6.67782	0.263964
$641$	−33.8075	−1.33532	−0.667658	−	0.744468i	$-0.732703\pi$
$641$	−33.8075	−1.33532	−0.667658	+	0.744468i	$0.732703\pi$
$642$	0	0
$643$	−1.26309	−0.0498114	−0.0249057	−	0.999690i	$-0.507929\pi$
$643$	−1.26309	−0.0498114	−0.0249057	+	0.999690i	$0.507929\pi$
$644$	−12.8888	−0.507889
$645$	0	0
$646$	−0.556867	−0.0219096
$647$	21.8000	0.857047	0.428524	−	0.903531i	$-0.359034\pi$
$647$	21.8000	0.857047	0.428524	+	0.903531i	$0.359034\pi$
$648$	0	0
$649$	6.93938	0.272395
$650$	0.199975	0.00784367
$651$	0	0
$652$	−12.9388	−0.506721
$653$	0.827455	0.0323808	0.0161904	−	0.999869i	$-0.494846\pi$
$653$	0.827455	0.0323808	0.0161904	+	0.999869i	$0.494846\pi$
$654$	0	0
$655$	11.3469	0.443360
$656$	−25.2070	−0.984170
$657$	0	0
$658$	−2.81252	−0.109644
$659$	−19.7769	−0.770399	−0.385199	−	0.922833i	$-0.625867\pi$
$659$	−19.7769	−0.770399	−0.385199	+	0.922833i	$0.625867\pi$
$660$	0	0
$661$	−2.73185	−0.106257	−0.0531283	−	0.998588i	$-0.516919\pi$
$661$	−2.73185	−0.106257	−0.0531283	+	0.998588i	$0.516919\pi$
$662$	4.11304	0.159858
$663$	0	0
$664$	−3.65436	−0.141816
$665$	4.95063	0.191977
$666$	0	0
$667$	−22.5068	−0.871467
$668$	7.32577	0.283443
$669$	0	0
$670$	2.28884	0.0884256
$671$	−5.39501	−0.208272
$672$	0	0
$673$	43.2212	1.66605	0.833027	−	0.553233i	$-0.186606\pi$
$673$	43.2212	1.66605	0.833027	+	0.553233i	$0.186606\pi$
$674$	5.51450	0.212411
$675$	0	0
$676$	23.7781	0.914544
$677$	−2.20504	−0.0847464	−0.0423732	−	0.999102i	$-0.513492\pi$
$677$	−2.20504	−0.0847464	−0.0423732	+	0.999102i	$0.513492\pi$
$678$	0	0
$679$	0.0506185	0.00194256
$680$	0.444383	0.0170413
$681$	0	0
$682$	0.787529	0.0301560
$683$	5.42757	0.207680	0.103840	−	0.994594i	$-0.466887\pi$
$683$	5.42757	0.207680	0.103840	+	0.994594i	$0.466887\pi$
$684$	0	0
$685$	13.1900	0.503963
$686$	0.222191	0.00848331
$687$	0	0
$688$	34.3311	1.30886
$689$	1.25553	0.0478319
$690$	0	0
$691$	7.72378	0.293826	0.146913	−	0.989149i	$-0.453066\pi$
$691$	7.72378	0.293826	0.146913	+	0.989149i	$0.453066\pi$
$692$	−30.6323	−1.16447
$693$	0	0
$694$	−0.229287	−0.00870363
$695$	19.5581	0.741882
$696$	0	0
$697$	−3.44313	−0.130418
$698$	4.13498	0.156511
$699$	0	0
$700$	−1.95063	−0.0737269
$701$	−32.1988	−1.21613	−0.608066	−	0.793887i	$-0.708054\pi$
$701$	−32.1988	−1.21613	−0.608066	+	0.793887i	$0.708054\pi$
$702$	0	0
$703$	−56.6631	−2.13709
$704$	6.83940	0.257769
$705$	0	0
$706$	1.97556	0.0743513
$707$	3.25690	0.122488
$708$	0	0
$709$	−25.0963	−0.942511	−0.471256	−	0.881997i	$-0.656199\pi$
$709$	−25.0963	−0.942511	−0.471256	+	0.881997i	$0.656199\pi$
$710$	1.41245	0.0530082
$711$	0	0
$712$	1.97556	0.0740373
$713$	−23.4194	−0.877062
$714$	0	0
$715$	0.900012	0.0336586
$716$	−17.4714	−0.652939
$717$	0	0
$718$	2.25291	0.0840777
$719$	−9.36433	−0.349230	−0.174615	−	0.984637i	$-0.555868\pi$
$719$	−9.36433	−0.349230	−0.174615	+	0.984637i	$0.555868\pi$
$720$	0	0
$721$	−0.0481195	−0.00179206
$722$	1.22400	0.0455524
$723$	0	0
$724$	26.9236	1.00061
$725$	−3.40626	−0.126505
$726$	0	0
$727$	−13.9145	−0.516061	−0.258030	−	0.966137i	$-0.583073\pi$
$727$	−13.9145	−0.516061	−0.258030	+	0.966137i	$0.583073\pi$
$728$	−0.790027	−0.0292804
$729$	0	0
$730$	−0.790027	−0.0292402
$731$	4.68942	0.173445
$732$	0	0
$733$	−1.16554	−0.0430502	−0.0215251	−	0.999768i	$-0.506852\pi$
$733$	−1.16554	−0.0430502	−0.0215251	+	0.999768i	$0.506852\pi$
$734$	−3.43576	−0.126816
$735$	0	0
$736$	17.0413	0.628149
$737$	10.3012	0.379450
$738$	0	0
$739$	43.5856	1.60332	0.801661	−	0.597779i	$-0.203950\pi$
$739$	43.5856	1.60332	0.801661	+	0.597779i	$0.203950\pi$
$740$	22.3262	0.820728
$741$	0	0
$742$	0.309960	0.0113790
$743$	−0.508255	−0.0186461	−0.00932304	−	0.999957i	$-0.502968\pi$
$743$	−0.508255	−0.0186461	−0.00932304	+	0.999957i	$0.502968\pi$
$744$	0	0
$745$	−0.568114	−0.0208141
$746$	0.104432	0.00382352
$747$	0	0
$748$	0.987504	0.0361067
$749$	−15.8900	−0.580609
$750$	0	0
$751$	−11.1188	−0.405731	−0.202865	−	0.979207i	$-0.565025\pi$
$751$	−11.1188	−0.405731	−0.202865	+	0.979207i	$0.565025\pi$
$752$	−46.9138	−1.71077
$753$	0	0
$754$	−0.681167	−0.0248066
$755$	−4.44688	−0.161839
$756$	0	0
$757$	−3.60255	−0.130937	−0.0654684	−	0.997855i	$-0.520854\pi$
$757$	−3.60255	−0.130937	−0.0654684	+	0.997855i	$0.520854\pi$
$758$	−5.60423	−0.203555
$759$	0	0
$760$	−4.34564	−0.157633
$761$	−41.5980	−1.50793	−0.753963	−	0.656917i	$-0.771860\pi$
$761$	−41.5980	−1.50793	−0.753963	+	0.656917i	$0.771860\pi$
$762$	0	0
$763$	−17.4706	−0.632479
$764$	13.4862	0.487913
$765$	0	0
$766$	4.71616	0.170402
$767$	−6.24553	−0.225513
$768$	0	0
$769$	−27.7113	−0.999297	−0.499648	−	0.866228i	$-0.666537\pi$
$769$	−27.7113	−0.999297	−0.499648	+	0.866228i	$0.666537\pi$
$770$	0.222191	0.00800722
$771$	0	0
$772$	−4.18054	−0.150461
$773$	10.8663	0.390833	0.195416	−	0.980720i	$-0.437394\pi$
$773$	10.8663	0.390833	0.195416	+	0.980720i	$0.437394\pi$
$774$	0	0
$775$	−3.54437	−0.127318
$776$	−0.0444327	−0.00159504
$777$	0	0
$778$	−1.98457	−0.0711504
$779$	33.6706	1.20637
$780$	0	0
$781$	6.35689	0.227467
$782$	0.743236	0.0265781
$783$	0	0
$784$	3.70622	0.132365
$785$	−8.15186	−0.290952
$786$	0	0
$787$	−52.8938	−1.88546	−0.942730	−	0.333558i	$-0.891751\pi$
$787$	−52.8938	−1.88546	−0.942730	+	0.333558i	$0.891751\pi$
$788$	36.2623	1.29179
$789$	0	0
$790$	−0.441884	−0.0157215
$791$	−19.4200	−0.690496
$792$	0	0
$793$	4.85558	0.172427
$794$	−3.92308	−0.139225
$795$	0	0
$796$	7.02237	0.248901
$797$	−7.31621	−0.259153	−0.129577	−	0.991569i	$-0.541362\pi$
$797$	−7.31621	−0.259153	−0.129577	+	0.991569i	$0.541362\pi$
$798$	0	0
$799$	−6.40814	−0.226704
$800$	2.57908	0.0911844
$801$	0	0
$802$	5.58798	0.197319
$803$	−3.55562	−0.125475
$804$	0	0
$805$	−6.60749	−0.232883
$806$	−0.708785	−0.0249659
$807$	0	0
$808$	−2.85890	−0.100576
$809$	44.1282	1.55146	0.775732	−	0.631063i	$-0.217381\pi$
$809$	44.1282	1.55146	0.775732	+	0.631063i	$0.217381\pi$
$810$	0	0
$811$	43.0394	1.51132	0.755659	−	0.654965i	$-0.227317\pi$
$811$	43.0394	1.51132	0.755659	+	0.654965i	$0.227317\pi$
$812$	6.64436	0.233171
$813$	0	0
$814$	−2.54312	−0.0891363
$815$	−6.63311	−0.232348
$816$	0	0
$817$	−45.8581	−1.60437
$818$	6.44688	0.225410
$819$	0	0
$820$	−13.2668	−0.463296
$821$	45.9099	1.60227	0.801134	−	0.598485i	$-0.204230\pi$
$821$	45.9099	1.60227	0.801134	+	0.598485i	$0.204230\pi$
$822$	0	0
$823$	1.96931	0.0686459	0.0343230	−	0.999411i	$-0.489073\pi$
$823$	1.96931	0.0686459	0.0343230	+	0.999411i	$0.489073\pi$
$824$	0.0422391	0.00147147
$825$	0	0
$826$	−1.54187	−0.0536486
$827$	−3.60937	−0.125510	−0.0627550	−	0.998029i	$-0.519989\pi$
$827$	−3.60937	−0.125510	−0.0627550	+	0.998029i	$0.519989\pi$
$828$	0	0
$829$	−5.28622	−0.183598	−0.0917989	−	0.995778i	$-0.529262\pi$
$829$	−5.28622	−0.183598	−0.0917989	+	0.995778i	$0.529262\pi$
$830$	−0.925005	−0.0321074
$831$	0	0
$832$	−6.15554	−0.213405
$833$	0.506248	0.0175405
$834$	0	0
$835$	3.75559	0.129968
$836$	−9.65685	−0.333989
$837$	0	0
$838$	5.80572	0.200555
$839$	−18.3268	−0.632713	−0.316356	−	0.948640i	$-0.602460\pi$
$839$	−18.3268	−0.632713	−0.316356	+	0.948640i	$0.602460\pi$
$840$	0	0
$841$	−17.3974	−0.599910
$842$	8.24248	0.284055
$843$	0	0
$844$	8.23540	0.283474
$845$	12.1900	0.419348
$846$	0	0
$847$	1.00000	0.0343604
$848$	5.17023	0.177547
$849$	0	0
$850$	0.112484	0.00385817
$851$	75.6268	2.59245
$852$	0	0
$853$	38.1150	1.30503	0.652516	−	0.757775i	$-0.273713\pi$
$853$	38.1150	1.30503	0.652516	+	0.757775i	$0.273713\pi$
$854$	1.19873	0.0410195
$855$	0	0
$856$	13.9482	0.476739
$857$	19.0912	0.652144	0.326072	−	0.945345i	$-0.394275\pi$
$857$	19.0912	0.652144	0.326072	+	0.945345i	$0.394275\pi$
$858$	0	0
$859$	16.2357	0.553954	0.276977	−	0.960877i	$-0.410667\pi$
$859$	16.2357	0.553954	0.276977	+	0.960877i	$0.410667\pi$
$860$	18.0689	0.616143
$861$	0	0
$862$	−2.13360	−0.0726708
$863$	−32.0076	−1.08955	−0.544775	−	0.838582i	$-0.683385\pi$
$863$	−32.0076	−1.08955	−0.544775	+	0.838582i	$0.683385\pi$
$864$	0	0
$865$	−15.7038	−0.533945
$866$	−6.91320	−0.234920
$867$	0	0
$868$	6.91376	0.234668
$869$	−1.98875	−0.0674638
$870$	0	0
$871$	−9.27122	−0.314143
$872$	15.3356	0.519331
$873$	0	0
$874$	−7.26815	−0.245849
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−3.88495	−0.131186	−0.0655928	−	0.997846i	$-0.520894\pi$
$877$	−3.88495	−0.131186	−0.0655928	+	0.997846i	$0.520894\pi$
$878$	−7.56881	−0.255435
$879$	0	0
$880$	3.70622	0.124937
$881$	−26.2331	−0.883816	−0.441908	−	0.897060i	$-0.645698\pi$
$881$	−26.2331	−0.883816	−0.441908	+	0.897060i	$0.645698\pi$
$882$	0	0
$883$	−9.26622	−0.311833	−0.155917	−	0.987770i	$-0.549833\pi$
$883$	−9.26622	−0.311833	−0.155917	+	0.987770i	$0.549833\pi$
$884$	−0.888765	−0.0298924
$885$	0	0
$886$	−4.22997	−0.142108
$887$	16.2593	0.545935	0.272968	−	0.962023i	$-0.411995\pi$
$887$	16.2593	0.545935	0.272968	+	0.962023i	$0.411995\pi$
$888$	0	0
$889$	10.9394	0.366895
$890$	0.500062	0.0167621
$891$	0	0
$892$	19.4076	0.649814
$893$	62.6656	2.09702
$894$	0	0
$895$	−8.95682	−0.299393
$896$	−6.67782	−0.223090
$897$	0	0
$898$	−4.15067	−0.138510
$899$	12.0730	0.402659
$900$	0	0
$901$	0.706223	0.0235277
$902$	1.51118	0.0503169
$903$	0	0
$904$	17.0468	0.566968
$905$	13.8025	0.458811
$906$	0	0
$907$	−44.6806	−1.48359	−0.741797	−	0.670624i	$-0.766026\pi$
$907$	−44.6806	−1.48359	−0.741797	+	0.670624i	$0.766026\pi$
$908$	28.9706	0.961422
$909$	0	0
$910$	−0.199975	−0.00662911
$911$	53.3601	1.76790	0.883949	−	0.467583i	$-0.154875\pi$
$911$	53.3601	1.76790	0.883949	+	0.467583i	$0.154875\pi$
$912$	0	0
$913$	−4.16310	−0.137779
$914$	6.06305	0.200548
$915$	0	0
$916$	−6.32620	−0.209024
$917$	−11.3469	−0.374707
$918$	0	0
$919$	42.4769	1.40118	0.700591	−	0.713563i	$-0.252920\pi$
$919$	42.4769	1.40118	0.700591	+	0.713563i	$0.252920\pi$
$920$	5.80002	0.191221
$921$	0	0
$922$	8.27983	0.272682
$923$	−5.72128	−0.188318
$924$	0	0
$925$	11.4456	0.376330
$926$	−1.88889	−0.0620728
$927$	0	0
$928$	−8.78503	−0.288383
$929$	−36.4812	−1.19691	−0.598455	−	0.801157i	$-0.704218\pi$
$929$	−36.4812	−1.19691	−0.598455	+	0.801157i	$0.704218\pi$
$930$	0	0
$931$	−4.95063	−0.162250
$932$	21.7874	0.713670
$933$	0	0
$934$	4.55855	0.149160
$935$	0.506248	0.0165561
$936$	0	0
$937$	−35.4087	−1.15675	−0.578376	−	0.815770i	$-0.696313\pi$
$937$	−35.4087	−1.15675	−0.578376	+	0.815770i	$0.696313\pi$
$938$	−2.28884	−0.0747333
$939$	0	0
$940$	−24.6913	−0.805341
$941$	−16.8161	−0.548191	−0.274095	−	0.961703i	$-0.588378\pi$
$941$	−16.8161	−0.548191	−0.274095	+	0.961703i	$0.588378\pi$
$942$	0	0
$943$	−44.9393	−1.46343
$944$	−25.7189	−0.837079
$945$	0	0
$946$	−2.05818	−0.0669172
$947$	58.8375	1.91196	0.955980	−	0.293431i	$-0.0947972\pi$
$947$	58.8375	1.91196	0.955980	+	0.293431i	$0.0947972\pi$
$948$	0	0
$949$	3.20010	0.103880
$950$	−1.09999	−0.0356883
$951$	0	0
$952$	−0.444383	−0.0144025
$953$	−13.7575	−0.445650	−0.222825	−	0.974858i	$-0.571528\pi$
$953$	−13.7575	−0.445650	−0.222825	+	0.974858i	$0.571528\pi$
$954$	0	0
$955$	6.91376	0.223724
$956$	4.78015	0.154601
$957$	0	0
$958$	−2.72865	−0.0881588
$959$	−13.1900	−0.425927
$960$	0	0
$961$	−18.4374	−0.594756
$962$	2.28884	0.0737952
$963$	0	0
$964$	50.9139	1.63983
$965$	−2.14317	−0.0689911
$966$	0	0
$967$	28.8681	0.928337	0.464168	−	0.885747i	$-0.346353\pi$
$967$	28.8681	0.928337	0.464168	+	0.885747i	$0.346353\pi$
$968$	−0.877796	−0.0282134
$969$	0	0
$970$	−0.0112470	−0.000361119 0
$971$	39.4282	1.26531	0.632656	−	0.774433i	$-0.281965\pi$
$971$	39.4282	1.26531	0.632656	+	0.774433i	$0.281965\pi$
$972$	0	0
$973$	−19.5581	−0.627004
$974$	−4.57493	−0.146590
$975$	0	0
$976$	19.9951	0.640028
$977$	2.81758	0.0901425	0.0450712	−	0.998984i	$-0.485649\pi$
$977$	2.81758	0.0901425	0.0450712	+	0.998984i	$0.485649\pi$
$978$	0	0
$979$	2.25059	0.0719293
$980$	1.95063	0.0623106
$981$	0	0
$982$	−1.40577	−0.0448599
$983$	−49.0000	−1.56286	−0.781429	−	0.623995i	$-0.785509\pi$
$983$	−49.0000	−1.56286	−0.781429	+	0.623995i	$0.785509\pi$
$984$	0	0
$985$	18.5901	0.592328
$986$	−0.383150	−0.0122020
$987$	0	0
$988$	8.69129	0.276507
$989$	61.2057	1.94623
$990$	0	0
$991$	7.64874	0.242970	0.121485	−	0.992593i	$-0.461234\pi$
$991$	7.64874	0.242970	0.121485	+	0.992593i	$0.461234\pi$
$992$	−9.14123	−0.290234
$993$	0	0
$994$	−1.41245	−0.0448001
$995$	3.60005	0.114129
$996$	0	0
$997$	32.7092	1.03591	0.517956	−	0.855408i	$-0.326693\pi$
$997$	32.7092	1.03591	0.517956	+	0.855408i	$0.326693\pi$
$998$	−0.139421	−0.00441329
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3465.2.a.bj.1.3		4
3.2	odd	2		1155.2.a.v.1.2	✓	4
15.14	odd	2		5775.2.a.by.1.3		4
21.20	even	2		8085.2.a.bq.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.v.1.2	✓	4	3.2	odd	2
3465.2.a.bj.1.3		4	1.1	even	1	trivial
5775.2.a.by.1.3		4	15.14	odd	2
8085.2.a.bq.1.2		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)

\(f(q)\)	\(=\)	\(q+0.222191 q^{2} -1.95063 q^{4} -1.00000 q^{5} +1.00000 q^{7} -0.877796 q^{8} +O(q^{10})\)	\(q+0.222191 q^{2} -1.95063 q^{4} -1.00000 q^{5} +1.00000 q^{7} -0.877796 q^{8} -0.222191 q^{10} -1.00000 q^{11} +0.900012 q^{13} +0.222191 q^{14} +3.70622 q^{16} +0.506248 q^{17} -4.95063 q^{19} +1.95063 q^{20} -0.222191 q^{22} +6.60749 q^{23} +1.00000 q^{25} +0.199975 q^{26} -1.95063 q^{28} -3.40626 q^{29} -3.54437 q^{31} +2.57908 q^{32} +0.112484 q^{34} -1.00000 q^{35} +11.4456 q^{37} -1.09999 q^{38} +0.877796 q^{40} -6.80127 q^{41} +9.26309 q^{43} +1.95063 q^{44} +1.46813 q^{46} -12.6581 q^{47} +1.00000 q^{49} +0.222191 q^{50} -1.75559 q^{52} +1.39501 q^{53} +1.00000 q^{55} -0.877796 q^{56} -0.756842 q^{58} -6.93938 q^{59} +5.39501 q^{61} -0.787529 q^{62} -6.83940 q^{64} -0.900012 q^{65} -10.3012 q^{67} -0.987504 q^{68} -0.222191 q^{70} -6.35689 q^{71} +3.55562 q^{73} +2.54312 q^{74} +9.65685 q^{76} -1.00000 q^{77} +1.98875 q^{79} -3.70622 q^{80} -1.51118 q^{82} +4.16310 q^{83} -0.506248 q^{85} +2.05818 q^{86} +0.877796 q^{88} -2.25059 q^{89} +0.900012 q^{91} -12.8888 q^{92} -2.81252 q^{94} +4.95063 q^{95} +0.0506185 q^{97} +0.222191 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 6 q^{4} - 4 q^{5} + 4 q^{7} - 6 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 + 6 * q^4 - 4 * q^5 + 4 * q^7 - 6 * q^8	\( 4 q - 2 q^{2} + 6 q^{4} - 4 q^{5} + 4 q^{7} - 6 q^{8} + 2 q^{10} - 4 q^{11} + 4 q^{13} - 2 q^{14} + 6 q^{16} - 6 q^{17} - 6 q^{19} - 6 q^{20} + 2 q^{22} - 10 q^{23} + 4 q^{25} + 6 q^{28} - 6 q^{29} - 8 q^{31} - 14 q^{32} - 16 q^{34} - 4 q^{35} + 12 q^{37} - 4 q^{38} + 6 q^{40} + 6 q^{43} - 6 q^{44} - 4 q^{46} + 4 q^{49} - 2 q^{50} - 12 q^{52} - 14 q^{53} + 4 q^{55} - 6 q^{56} + 20 q^{58} - 2 q^{59} + 2 q^{61} - 20 q^{62} - 2 q^{64} - 4 q^{65} - 12 q^{67} - 20 q^{68} + 2 q^{70} - 4 q^{71} + 20 q^{73} + 32 q^{74} + 16 q^{76} - 4 q^{77} - 4 q^{79} - 6 q^{80} - 16 q^{82} - 14 q^{83} + 6 q^{85} - 40 q^{86} + 6 q^{88} + 6 q^{89} + 4 q^{91} - 40 q^{92} + 4 q^{94} + 6 q^{95} - 14 q^{97} - 2 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 6 * q^4 - 4 * q^5 + 4 * q^7 - 6 * q^8 + 2 * q^10 - 4 * q^11 + 4 * q^13 - 2 * q^14 + 6 * q^16 - 6 * q^17 - 6 * q^19 - 6 * q^20 + 2 * q^22 - 10 * q^23 + 4 * q^25 + 6 * q^28 - 6 * q^29 - 8 * q^31 - 14 * q^32 - 16 * q^34 - 4 * q^35 + 12 * q^37 - 4 * q^38 + 6 * q^40 + 6 * q^43 - 6 * q^44 - 4 * q^46 + 4 * q^49 - 2 * q^50 - 12 * q^52 - 14 * q^53 + 4 * q^55 - 6 * q^56 + 20 * q^58 - 2 * q^59 + 2 * q^61 - 20 * q^62 - 2 * q^64 - 4 * q^65 - 12 * q^67 - 20 * q^68 + 2 * q^70 - 4 * q^71 + 20 * q^73 + 32 * q^74 + 16 * q^76 - 4 * q^77 - 4 * q^79 - 6 * q^80 - 16 * q^82 - 14 * q^83 + 6 * q^85 - 40 * q^86 + 6 * q^88 + 6 * q^89 + 4 * q^91 - 40 * q^92 + 4 * q^94 + 6 * q^95 - 14 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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