LMFDB - Embedded newform 7605.2.a.cf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7605,2,Mod(1,7605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7605, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7605.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7605 = 3^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7605.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:	$60.7262307372$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4752.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 3x^{2} + 4x + 1 $ x^4 - 2x^3 - 3x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.21969$ of defining polynomial
Character	$\chi$	$=$	7605.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.21969 q^{2} -0.512364 q^{4} +1.00000 q^{5} +3.60020 q^{7} +3.06430 q^{8} +O(q^{10})$	$q-1.21969 q^{2} -0.512364 q^{4} +1.00000 q^{5} +3.60020 q^{7} +3.06430 q^{8} -1.21969 q^{10} -5.37182 q^{11} -4.39111 q^{14} -2.71276 q^{16} +1.13186 q^{17} +2.26795 q^{19} -0.512364 q^{20} +6.55193 q^{22} +3.89287 q^{23} +1.00000 q^{25} -1.84461 q^{28} +0.0247279 q^{29} +5.46410 q^{31} -2.81988 q^{32} -1.38051 q^{34} +3.60020 q^{35} -8.70406 q^{37} -2.76619 q^{38} +3.06430 q^{40} -3.73205 q^{41} -1.13186 q^{43} +2.75232 q^{44} -4.74809 q^{46} -2.58535 q^{47} +5.96141 q^{49} -1.21969 q^{50} +4.43937 q^{53} -5.37182 q^{55} +11.0321 q^{56} -0.0301603 q^{58} +0.171425 q^{59} +3.36023 q^{61} -6.66449 q^{62} +8.86488 q^{64} +6.39980 q^{67} -0.579922 q^{68} -4.39111 q^{70} +10.7973 q^{71} -4.70308 q^{73} +10.6162 q^{74} -1.16202 q^{76} -19.3396 q^{77} -11.9826 q^{79} -2.71276 q^{80} +4.55193 q^{82} +12.1286 q^{83} +1.13186 q^{85} +1.38051 q^{86} -16.4608 q^{88} +16.1540 q^{89} -1.99457 q^{92} +3.15332 q^{94} +2.26795 q^{95} +12.1682 q^{97} -7.27105 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 2 q^{4} + 4 q^{5} + 10 q^{7} - 6 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 + 10 * q^7 - 6 * q^8	$ 4 q - 2 q^{2} + 2 q^{4} + 4 q^{5} + 10 q^{7} - 6 q^{8} - 2 q^{10} - 2 q^{14} + 2 q^{16} + 2 q^{17} + 16 q^{19} + 2 q^{20} + 12 q^{22} + 10 q^{23} + 4 q^{25} + 8 q^{28} - 8 q^{29} + 8 q^{31} - 4 q^{32} - 4 q^{34} + 10 q^{35} - 2 q^{37} - 8 q^{38} - 6 q^{40} - 8 q^{41} - 2 q^{43} - 12 q^{44} + 16 q^{46} - 8 q^{47} + 12 q^{49} - 2 q^{50} + 12 q^{53} - 12 q^{56} + 22 q^{58} - 12 q^{59} + 28 q^{61} - 4 q^{62} + 4 q^{64} + 30 q^{67} - 14 q^{68} - 2 q^{70} - 4 q^{71} - 8 q^{73} + 10 q^{74} + 20 q^{76} - 18 q^{77} - 8 q^{79} + 2 q^{80} + 4 q^{82} + 12 q^{83} + 2 q^{85} + 4 q^{86} - 18 q^{88} + 12 q^{89} - 22 q^{92} - 32 q^{94} + 16 q^{95} + 2 q^{97} + 24 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 + 10 * q^7 - 6 * q^8 - 2 * q^10 - 2 * q^14 + 2 * q^16 + 2 * q^17 + 16 * q^19 + 2 * q^20 + 12 * q^22 + 10 * q^23 + 4 * q^25 + 8 * q^28 - 8 * q^29 + 8 * q^31 - 4 * q^32 - 4 * q^34 + 10 * q^35 - 2 * q^37 - 8 * q^38 - 6 * q^40 - 8 * q^41 - 2 * q^43 - 12 * q^44 + 16 * q^46 - 8 * q^47 + 12 * q^49 - 2 * q^50 + 12 * q^53 - 12 * q^56 + 22 * q^58 - 12 * q^59 + 28 * q^61 - 4 * q^62 + 4 * q^64 + 30 * q^67 - 14 * q^68 - 2 * q^70 - 4 * q^71 - 8 * q^73 + 10 * q^74 + 20 * q^76 - 18 * q^77 - 8 * q^79 + 2 * q^80 + 4 * q^82 + 12 * q^83 + 2 * q^85 + 4 * q^86 - 18 * q^88 + 12 * q^89 - 22 * q^92 - 32 * q^94 + 16 * q^95 + 2 * q^97 + 24 * 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 $

$2$	−1.21969	−0.862449	−0.431224	−	0.902245i	$-0.641918\pi$
$2$	−1.21969	−0.862449	−0.431224	+	0.902245i	$0.641918\pi$
$3$	0	0
$3$	0	0
$4$	−0.512364	−0.256182
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.60020	1.36075	0.680373	−	0.732866i	$-0.261818\pi$
$7$	3.60020	1.36075	0.680373	+	0.732866i	$0.261818\pi$
$8$	3.06430	1.08339
$9$	0	0
$10$	−1.21969	−0.385699
$11$	−5.37182	−1.61966	−0.809832	−	0.586662i	$-0.800442\pi$
$11$	−5.37182	−1.61966	−0.809832	+	0.586662i	$0.800442\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−4.39111	−1.17357
$15$	0	0
$16$	−2.71276	−0.678189
$17$	1.13186	0.274515	0.137258	−	0.990535i	$-0.456171\pi$
$17$	1.13186	0.274515	0.137258	+	0.990535i	$0.456171\pi$
$18$	0	0
$19$	2.26795	0.520303	0.260152	−	0.965568i	$-0.416227\pi$
$19$	2.26795	0.520303	0.260152	+	0.965568i	$0.416227\pi$
$20$	−0.512364	−0.114568
$21$	0	0
$22$	6.55193	1.39688
$23$	3.89287	0.811720	0.405860	−	0.913935i	$-0.366972\pi$
$23$	3.89287	0.811720	0.405860	+	0.913935i	$0.366972\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	−1.84461	−0.348599
$29$	0.0247279	0.00459185	0.00229593	−	0.999997i	$-0.499269\pi$
$29$	0.0247279	0.00459185	0.00229593	+	0.999997i	$0.499269\pi$
$30$	0	0
$31$	5.46410	0.981382	0.490691	−	0.871334i	$-0.336744\pi$
$31$	5.46410	0.981382	0.490691	+	0.871334i	$0.336744\pi$
$32$	−2.81988	−0.498490
$33$	0	0
$34$	−1.38051	−0.236755
$35$	3.60020	0.608544
$36$	0	0
$37$	−8.70406	−1.43094	−0.715470	−	0.698644i	$-0.753787\pi$
$37$	−8.70406	−1.43094	−0.715470	+	0.698644i	$0.753787\pi$
$38$	−2.76619	−0.448735
$39$	0	0
$40$	3.06430	0.484508
$41$	−3.73205	−0.582848	−0.291424	−	0.956594i	$-0.594129\pi$
$41$	−3.73205	−0.582848	−0.291424	+	0.956594i	$0.594129\pi$
$42$	0	0
$43$	−1.13186	−0.172606	−0.0863031	−	0.996269i	$-0.527505\pi$
$43$	−1.13186	−0.172606	−0.0863031	+	0.996269i	$0.527505\pi$
$44$	2.75232	0.414929
$45$	0	0
$46$	−4.74809	−0.700067
$47$	−2.58535	−0.377113	−0.188556	−	0.982062i	$-0.560381\pi$
$47$	−2.58535	−0.377113	−0.188556	+	0.982062i	$0.560381\pi$
$48$	0	0
$49$	5.96141	0.851630
$50$	−1.21969	−0.172490
$51$	0	0
$52$	0	0
$53$	4.43937	0.609795	0.304897	−	0.952385i	$-0.401378\pi$
$53$	4.43937	0.609795	0.304897	+	0.952385i	$0.401378\pi$
$54$	0	0
$55$	−5.37182	−0.724336
$56$	11.0321	1.47422
$57$	0	0
$58$	−0.0301603	−0.00396024
$59$	0.171425	0.0223176	0.0111588	−	0.999938i	$-0.496448\pi$
$59$	0.171425	0.0223176	0.0111588	+	0.999938i	$0.496448\pi$
$60$	0	0
$61$	3.36023	0.430234	0.215117	−	0.976588i	$-0.430987\pi$
$61$	3.36023	0.430234	0.215117	+	0.976588i	$0.430987\pi$
$62$	−6.66449	−0.846391
$63$	0	0
$64$	8.86488	1.10811
$65$	0	0
$66$	0	0
$67$	6.39980	0.781861	0.390930	−	0.920420i	$-0.372153\pi$
$67$	6.39980	0.781861	0.390930	+	0.920420i	$0.372153\pi$
$68$	−0.579922	−0.0703258
$69$	0	0
$70$	−4.39111	−0.524838
$71$	10.7973	1.28141	0.640703	−	0.767788i	$-0.278643\pi$
$71$	10.7973	1.28141	0.640703	+	0.767788i	$0.278643\pi$
$72$	0	0
$73$	−4.70308	−0.550454	−0.275227	−	0.961379i	$-0.588753\pi$
$73$	−4.70308	−0.550454	−0.275227	+	0.961379i	$0.588753\pi$
$74$	10.6162	1.23411
$75$	0	0
$76$	−1.16202	−0.133292
$77$	−19.3396	−2.20395
$78$	0	0
$79$	−11.9826	−1.34815	−0.674075	−	0.738663i	$-0.735457\pi$
$79$	−11.9826	−1.34815	−0.674075	+	0.738663i	$0.735457\pi$
$80$	−2.71276	−0.303295
$81$	0	0
$82$	4.55193	0.502677
$83$	12.1286	1.33129	0.665643	−	0.746270i	$-0.268157\pi$
$83$	12.1286	1.33129	0.665643	+	0.746270i	$0.268157\pi$
$84$	0	0
$85$	1.13186	0.122767
$86$	1.38051	0.148864
$87$	0	0
$88$	−16.4608	−1.75473
$89$	16.1540	1.71232	0.856162	−	0.516707i	$-0.172842\pi$
$89$	16.1540	1.71232	0.856162	+	0.516707i	$0.172842\pi$
$90$	0	0
$91$	0	0
$92$	−1.99457	−0.207948
$93$	0	0
$94$	3.15332	0.325240
$95$	2.26795	0.232687
$96$	0	0
$97$	12.1682	1.23549	0.617745	−	0.786379i	$-0.288046\pi$
$97$	12.1682	1.23549	0.617745	+	0.786379i	$0.288046\pi$
$98$	−7.27105	−0.734487
$99$	0	0
$100$	−0.512364	−0.0512364
$101$	4.05441	0.403429	0.201714	−	0.979444i	$-0.435349\pi$
$101$	4.05441	0.403429	0.201714	+	0.979444i	$0.435349\pi$
$102$	0	0
$103$	−17.9035	−1.76408	−0.882041	−	0.471173i	$-0.843831\pi$
$103$	−17.9035	−1.76408	−0.882041	+	0.471173i	$0.843831\pi$
$104$	0	0
$105$	0	0
$106$	−5.41465	−0.525917
$107$	9.13186	0.882810	0.441405	−	0.897308i	$-0.354480\pi$
$107$	9.13186	0.882810	0.441405	+	0.897308i	$0.354480\pi$
$108$	0	0
$109$	7.37605	0.706498	0.353249	−	0.935529i	$-0.385077\pi$
$109$	7.37605	0.706498	0.353249	+	0.935529i	$0.385077\pi$
$110$	6.55193	0.624702
$111$	0	0
$112$	−9.76645	−0.922843
$113$	−7.07588	−0.665643	−0.332821	−	0.942990i	$-0.608001\pi$
$113$	−7.07588	−0.665643	−0.332821	+	0.942990i	$0.608001\pi$
$114$	0	0
$115$	3.89287	0.363012
$116$	−0.0126697	−0.00117635
$117$	0	0
$118$	−0.209084	−0.0192478
$119$	4.07490	0.373545
$120$	0	0
$121$	17.8564	1.62331
$122$	−4.09843	−0.371055
$123$	0	0
$124$	−2.79961	−0.251412
$125$	1.00000	0.0894427
$126$	0	0
$127$	−11.4361	−1.01479	−0.507395	−	0.861713i	$-0.669392\pi$
$127$	−11.4361	−1.01479	−0.507395	+	0.861713i	$0.669392\pi$
$128$	−5.17262	−0.457199
$129$	0	0
$130$	0	0
$131$	10.5680	0.923328	0.461664	−	0.887055i	$-0.347253\pi$
$131$	10.5680	0.923328	0.461664	+	0.887055i	$0.347253\pi$
$132$	0	0
$133$	8.16506	0.708001
$134$	−7.80576	−0.674315
$135$	0	0
$136$	3.46834	0.297408
$137$	−3.78672	−0.323522	−0.161761	−	0.986830i	$-0.551717\pi$
$137$	−3.78672	−0.323522	−0.161761	+	0.986830i	$0.551717\pi$
$138$	0	0
$139$	2.01386	0.170814	0.0854068	−	0.996346i	$-0.472781\pi$
$139$	2.01386	0.170814	0.0854068	+	0.996346i	$0.472781\pi$
$140$	−1.84461	−0.155898
$141$	0	0
$142$	−13.1694	−1.10515
$143$	0	0
$144$	0	0
$145$	0.0247279	0.00205354
$146$	5.73629	0.474739
$147$	0	0
$148$	4.45965	0.366581
$149$	5.51780	0.452035	0.226018	−	0.974123i	$-0.427429\pi$
$149$	5.51780	0.452035	0.226018	+	0.974123i	$0.427429\pi$
$150$	0	0
$151$	−4.88961	−0.397911	−0.198956	−	0.980009i	$-0.563755\pi$
$151$	−4.88961	−0.397911	−0.198956	+	0.980009i	$0.563755\pi$
$152$	6.94967	0.563693
$153$	0	0
$154$	23.5882	1.90079
$155$	5.46410	0.438887
$156$	0	0
$157$	10.0405	0.801323	0.400661	−	0.916226i	$-0.368780\pi$
$157$	10.0405	0.801323	0.400661	+	0.916226i	$0.368780\pi$
$158$	14.6150	1.16271
$159$	0	0
$160$	−2.81988	−0.222931
$161$	14.0151	1.10454
$162$	0	0
$163$	6.78124	0.531148	0.265574	−	0.964090i	$-0.414439\pi$
$163$	6.78124	0.531148	0.265574	+	0.964090i	$0.414439\pi$
$164$	1.91217	0.149315
$165$	0	0
$166$	−14.7931	−1.14817
$167$	−10.4898	−0.811726	−0.405863	−	0.913934i	$-0.633029\pi$
$167$	−10.4898	−0.811726	−0.405863	+	0.913934i	$0.633029\pi$
$168$	0	0
$169$	0	0
$170$	−1.38051	−0.105880
$171$	0	0
$172$	0.579922	0.0442186
$173$	4.45845	0.338970	0.169485	−	0.985533i	$-0.445790\pi$
$173$	4.45845	0.338970	0.169485	+	0.985533i	$0.445790\pi$
$174$	0	0
$175$	3.60020	0.272149
$176$	14.5724	1.09844
$177$	0	0
$178$	−19.7029	−1.47679
$179$	−18.6313	−1.39257	−0.696284	−	0.717766i	$-0.745165\pi$
$179$	−18.6313	−1.39257	−0.696284	+	0.717766i	$0.745165\pi$
$180$	0	0
$181$	18.0900	1.34462	0.672310	−	0.740270i	$-0.265302\pi$
$181$	18.0900	1.34462	0.672310	+	0.740270i	$0.265302\pi$
$182$	0	0
$183$	0	0
$184$	11.9289	0.879412
$185$	−8.70406	−0.639935
$186$	0	0
$187$	−6.08012	−0.444622
$188$	1.32464	0.0966095
$189$	0	0
$190$	−2.76619	−0.200680
$191$	−27.3363	−1.97799	−0.988994	−	0.147958i	$-0.952730\pi$
$191$	−27.3363	−1.97799	−0.988994	+	0.147958i	$0.952730\pi$
$192$	0	0
$193$	21.7674	1.56685	0.783425	−	0.621486i	$-0.213471\pi$
$193$	21.7674	1.56685	0.783425	+	0.621486i	$0.213471\pi$
$194$	−14.8413	−1.06555
$195$	0	0
$196$	−3.05441	−0.218172
$197$	1.69672	0.120886	0.0604432	−	0.998172i	$-0.480749\pi$
$197$	1.69672	0.120886	0.0604432	+	0.998172i	$0.480749\pi$
$198$	0	0
$199$	25.3255	1.79527	0.897637	−	0.440735i	$-0.145282\pi$
$199$	25.3255	1.79527	0.897637	+	0.440735i	$0.145282\pi$
$200$	3.06430	0.216679
$201$	0	0
$202$	−4.94511	−0.347937
$203$	0.0890252	0.00624834
$204$	0	0
$205$	−3.73205	−0.260658
$206$	21.8366	1.52143
$207$	0	0
$208$	0	0
$209$	−12.1830	−0.842716
$210$	0	0
$211$	−0.335507	−0.0230973	−0.0115486	−	0.999933i	$-0.503676\pi$
$211$	−0.335507	−0.0230973	−0.0115486	+	0.999933i	$0.503676\pi$
$212$	−2.27458	−0.156218
$213$	0	0
$214$	−11.1380	−0.761378
$215$	−1.13186	−0.0771919
$216$	0	0
$217$	19.6718	1.33541
$218$	−8.99648	−0.609318
$219$	0	0
$220$	2.75232	0.185562
$221$	0	0
$222$	0	0
$223$	−12.2968	−0.823452	−0.411726	−	0.911308i	$-0.635074\pi$
$223$	−12.2968	−0.823452	−0.411726	+	0.911308i	$0.635074\pi$
$224$	−10.1521	−0.678318
$225$	0	0
$226$	8.63036	0.574083
$227$	7.63227	0.506571	0.253286	−	0.967392i	$-0.418489\pi$
$227$	7.63227	0.506571	0.253286	+	0.967392i	$0.418489\pi$
$228$	0	0
$229$	−14.4008	−0.951631	−0.475815	−	0.879545i	$-0.657847\pi$
$229$	−14.4008	−0.951631	−0.475815	+	0.879545i	$0.657847\pi$
$230$	−4.74809	−0.313080
$231$	0	0
$232$	0.0757736	0.00497478
$233$	−9.49617	−0.622115	−0.311057	−	0.950391i	$-0.600683\pi$
$233$	−9.49617	−0.622115	−0.311057	+	0.950391i	$0.600683\pi$
$234$	0	0
$235$	−2.58535	−0.168650
$236$	−0.0878318	−0.00571736
$237$	0	0
$238$	−4.97010	−0.322164
$239$	19.9143	1.28815	0.644076	−	0.764962i	$-0.277242\pi$
$239$	19.9143	1.28815	0.644076	+	0.764962i	$0.277242\pi$
$240$	0	0
$241$	23.2664	1.49872	0.749360	−	0.662163i	$-0.230361\pi$
$241$	23.2664	1.49872	0.749360	+	0.662163i	$0.230361\pi$
$242$	−21.7792	−1.40002
$243$	0	0
$244$	−1.72166	−0.110218
$245$	5.96141	0.380860
$246$	0	0
$247$	0	0
$248$	16.7436	1.06322
$249$	0	0
$250$	−1.21969	−0.0771398
$251$	−11.8402	−0.747344	−0.373672	−	0.927561i	$-0.621901\pi$
$251$	−11.8402	−0.747344	−0.373672	+	0.927561i	$0.621901\pi$
$252$	0	0
$253$	−20.9118	−1.31471
$254$	13.9485	0.875205
$255$	0	0
$256$	−11.4208	−0.713800
$257$	5.55002	0.346201	0.173100	−	0.984904i	$-0.444621\pi$
$257$	5.55002	0.346201	0.173100	+	0.984904i	$0.444621\pi$
$258$	0	0
$259$	−31.3363	−1.94714
$260$	0	0
$261$	0	0
$262$	−12.8896	−0.796323
$263$	−6.85967	−0.422985	−0.211493	−	0.977380i	$-0.567832\pi$
$263$	−6.85967	−0.422985	−0.211493	+	0.977380i	$0.567832\pi$
$264$	0	0
$265$	4.43937	0.272709
$266$	−9.95882	−0.610614
$267$	0	0
$268$	−3.27903	−0.200299
$269$	1.42199	0.0867001	0.0433501	−	0.999060i	$-0.486197\pi$
$269$	1.42199	0.0867001	0.0433501	+	0.999060i	$0.486197\pi$
$270$	0	0
$271$	−9.96947	−0.605602	−0.302801	−	0.953054i	$-0.597922\pi$
$271$	−9.96947	−0.605602	−0.302801	+	0.953054i	$0.597922\pi$
$272$	−3.07045	−0.186173
$273$	0	0
$274$	4.61862	0.279021
$275$	−5.37182	−0.323933
$276$	0	0
$277$	−17.5237	−1.05290	−0.526449	−	0.850206i	$-0.676477\pi$
$277$	−17.5237	−1.05290	−0.526449	+	0.850206i	$0.676477\pi$
$278$	−2.45628	−0.147318
$279$	0	0
$280$	11.0321	0.659292
$281$	−10.7352	−0.640406	−0.320203	−	0.947349i	$-0.603751\pi$
$281$	−10.7352	−0.640406	−0.320203	+	0.947349i	$0.603751\pi$
$282$	0	0
$283$	1.31838	0.0783698	0.0391849	−	0.999232i	$-0.487524\pi$
$283$	1.31838	0.0783698	0.0391849	+	0.999232i	$0.487524\pi$
$284$	−5.53216	−0.328273
$285$	0	0
$286$	0	0
$287$	−13.4361	−0.793109
$288$	0	0
$289$	−15.7189	−0.924641
$290$	−0.0301603	−0.00177107
$291$	0	0
$292$	2.40969	0.141016
$293$	−18.7427	−1.09496	−0.547479	−	0.836820i	$-0.684412\pi$
$293$	−18.7427	−1.09496	−0.547479	+	0.836820i	$0.684412\pi$
$294$	0	0
$295$	0.171425	0.00998072
$296$	−26.6718	−1.55027
$297$	0	0
$298$	−6.72998	−0.389857
$299$	0	0
$300$	0	0
$301$	−4.07490	−0.234873
$302$	5.96380	0.343178
$303$	0	0
$304$	−6.15239	−0.352864
$305$	3.36023	0.192406
$306$	0	0
$307$	14.3043	0.816387	0.408194	−	0.912895i	$-0.366159\pi$
$307$	14.3043	0.816387	0.408194	+	0.912895i	$0.366159\pi$
$308$	9.90891	0.564612
$309$	0	0
$310$	−6.66449	−0.378518
$311$	−2.76102	−0.156563	−0.0782815	−	0.996931i	$-0.524943\pi$
$311$	−2.76102	−0.156563	−0.0782815	+	0.996931i	$0.524943\pi$
$312$	0	0
$313$	−16.3858	−0.926179	−0.463090	−	0.886311i	$-0.653259\pi$
$313$	−16.3858	−0.926179	−0.463090	+	0.886311i	$0.653259\pi$
$314$	−12.2463	−0.691100
$315$	0	0
$316$	6.13946	0.345372
$317$	−1.78575	−0.100297	−0.0501487	−	0.998742i	$-0.515970\pi$
$317$	−1.78575	−0.100297	−0.0501487	+	0.998742i	$0.515970\pi$
$318$	0	0
$319$	−0.132834	−0.00743725
$320$	8.86488	0.495562
$321$	0	0
$322$	−17.0940	−0.952613
$323$	2.56699	0.142831
$324$	0	0
$325$	0	0
$326$	−8.27099	−0.458088
$327$	0	0
$328$	−11.4361	−0.631454
$329$	−9.30778	−0.513155
$330$	0	0
$331$	7.22440	0.397089	0.198545	−	0.980092i	$-0.436379\pi$
$331$	7.22440	0.397089	0.198545	+	0.980092i	$0.436379\pi$
$332$	−6.21425	−0.341052
$333$	0	0
$334$	12.7943	0.700072
$335$	6.39980	0.349659
$336$	0	0
$337$	−4.36219	−0.237624	−0.118812	−	0.992917i	$-0.537909\pi$
$337$	−4.36219	−0.237624	−0.118812	+	0.992917i	$0.537909\pi$
$338$	0	0
$339$	0	0
$340$	−0.579922	−0.0314507
$341$	−29.3521	−1.58951
$342$	0	0
$343$	−3.73913	−0.201894
$344$	−3.46834	−0.187000
$345$	0	0
$346$	−5.43792	−0.292344
$347$	26.7072	1.43372	0.716858	−	0.697219i	$-0.245580\pi$
$347$	26.7072	1.43372	0.716858	+	0.697219i	$0.245580\pi$
$348$	0	0
$349$	23.5711	1.26173	0.630865	−	0.775892i	$-0.282700\pi$
$349$	23.5711	1.26173	0.630865	+	0.775892i	$0.282700\pi$
$350$	−4.39111	−0.234715
$351$	0	0
$352$	15.1479	0.807385
$353$	−5.73727	−0.305364	−0.152682	−	0.988275i	$-0.548791\pi$
$353$	−5.73727	−0.305364	−0.152682	+	0.988275i	$0.548791\pi$
$354$	0	0
$355$	10.7973	0.573063
$356$	−8.27675	−0.438667
$357$	0	0
$358$	22.7243	1.20102
$359$	24.7583	1.30669	0.653347	−	0.757059i	$-0.273364\pi$
$359$	24.7583	1.30669	0.653347	+	0.757059i	$0.273364\pi$
$360$	0	0
$361$	−13.8564	−0.729285
$362$	−22.0641	−1.15967
$363$	0	0
$364$	0	0
$365$	−4.70308	−0.246171
$366$	0	0
$367$	26.0535	1.35998	0.679992	−	0.733220i	$-0.261983\pi$
$367$	26.0535	1.35998	0.679992	+	0.733220i	$0.261983\pi$
$368$	−10.5604	−0.550499
$369$	0	0
$370$	10.6162	0.551912
$371$	15.9826	0.829776
$372$	0	0
$373$	13.2045	0.683702	0.341851	−	0.939754i	$-0.388946\pi$
$373$	13.2045	0.683702	0.341851	+	0.939754i	$0.388946\pi$
$374$	7.41584	0.383464
$375$	0	0
$376$	−7.92229	−0.408561
$377$	0	0
$378$	0	0
$379$	25.9977	1.33541	0.667707	−	0.744425i	$-0.267276\pi$
$379$	25.9977	1.33541	0.667707	+	0.744425i	$0.267276\pi$
$380$	−1.16202	−0.0596101
$381$	0	0
$382$	33.3418	1.70591
$383$	9.60020	0.490547	0.245274	−	0.969454i	$-0.421122\pi$
$383$	9.60020	0.490547	0.245274	+	0.969454i	$0.421122\pi$
$384$	0	0
$385$	−19.3396	−0.985637
$386$	−26.5494	−1.35133
$387$	0	0
$388$	−6.23453	−0.316510
$389$	−5.63129	−0.285518	−0.142759	−	0.989758i	$-0.545597\pi$
$389$	−5.63129	−0.285518	−0.142759	+	0.989758i	$0.545597\pi$
$390$	0	0
$391$	4.40617	0.222829
$392$	18.2675	0.922650
$393$	0	0
$394$	−2.06947	−0.104258
$395$	−11.9826	−0.602911
$396$	0	0
$397$	−16.7658	−0.841452	−0.420726	−	0.907188i	$-0.638225\pi$
$397$	−16.7658	−0.841452	−0.420726	+	0.907188i	$0.638225\pi$
$398$	−30.8891	−1.54833
$399$	0	0
$400$	−2.71276	−0.135638
$401$	13.8780	0.693036	0.346518	−	0.938043i	$-0.387364\pi$
$401$	13.8780	0.693036	0.346518	+	0.938043i	$0.387364\pi$
$402$	0	0
$403$	0	0
$404$	−2.07733	−0.103351
$405$	0	0
$406$	−0.108583	−0.00538888
$407$	46.7566	2.31764
$408$	0	0
$409$	29.4251	1.45498	0.727489	−	0.686120i	$-0.240687\pi$
$409$	29.4251	1.45498	0.727489	+	0.686120i	$0.240687\pi$
$410$	4.55193	0.224804
$411$	0	0
$412$	9.17310	0.451926
$413$	0.617162	0.0303686
$414$	0	0
$415$	12.1286	0.595369
$416$	0	0
$417$	0	0
$418$	14.8595	0.726800
$419$	−6.96793	−0.340406	−0.170203	−	0.985409i	$-0.554442\pi$
$419$	−6.96793	−0.340406	−0.170203	+	0.985409i	$0.554442\pi$
$420$	0	0
$421$	7.12125	0.347069	0.173534	−	0.984828i	$-0.444481\pi$
$421$	7.12125	0.347069	0.173534	+	0.984828i	$0.444481\pi$
$422$	0.409213	0.0199202
$423$	0	0
$424$	13.6036	0.660647
$425$	1.13186	0.0549030
$426$	0	0
$427$	12.0975	0.585439
$428$	−4.67883	−0.226160
$429$	0	0
$430$	1.38051	0.0665740
$431$	−30.2144	−1.45537	−0.727687	−	0.685909i	$-0.759405\pi$
$431$	−30.2144	−1.45537	−0.727687	+	0.685909i	$0.759405\pi$
$432$	0	0
$433$	1.20013	0.0576745	0.0288373	−	0.999584i	$-0.490820\pi$
$433$	1.20013	0.0576745	0.0288373	+	0.999584i	$0.490820\pi$
$434$	−23.9935	−1.15172
$435$	0	0
$436$	−3.77922	−0.180992
$437$	8.82884	0.422341
$438$	0	0
$439$	−16.5541	−0.790084	−0.395042	−	0.918663i	$-0.629270\pi$
$439$	−16.5541	−0.790084	−0.395042	+	0.918663i	$0.629270\pi$
$440$	−16.4608	−0.784740
$441$	0	0
$442$	0	0
$443$	−4.55949	−0.216628	−0.108314	−	0.994117i	$-0.534545\pi$
$443$	−4.55949	−0.216628	−0.108314	+	0.994117i	$0.534545\pi$
$444$	0	0
$445$	16.1540	0.765775
$446$	14.9982	0.710185
$447$	0	0
$448$	31.9153	1.50786
$449$	−13.8522	−0.653724	−0.326862	−	0.945072i	$-0.605991\pi$
$449$	−13.8522	−0.653724	−0.326862	+	0.945072i	$0.605991\pi$
$450$	0	0
$451$	20.0479	0.944018
$452$	3.62542	0.170526
$453$	0	0
$454$	−9.30897	−0.436892
$455$	0	0
$456$	0	0
$457$	40.1146	1.87648	0.938240	−	0.345984i	$-0.112455\pi$
$457$	40.1146	1.87648	0.938240	+	0.345984i	$0.112455\pi$
$458$	17.5644	0.820733
$459$	0	0
$460$	−1.99457	−0.0929972
$461$	7.53900	0.351126	0.175563	−	0.984468i	$-0.443825\pi$
$461$	7.53900	0.351126	0.175563	+	0.984468i	$0.443825\pi$
$462$	0	0
$463$	23.3031	1.08299	0.541494	−	0.840705i	$-0.317859\pi$
$463$	23.3031	1.08299	0.541494	+	0.840705i	$0.317859\pi$
$464$	−0.0670807	−0.00311414
$465$	0	0
$466$	11.5824	0.536542
$467$	22.6297	1.04718	0.523589	−	0.851971i	$-0.324593\pi$
$467$	22.6297	1.04718	0.523589	+	0.851971i	$0.324593\pi$
$468$	0	0
$469$	23.0405	1.06391
$470$	3.15332	0.145452
$471$	0	0
$472$	0.525296	0.0241787
$473$	6.08012	0.279564
$474$	0	0
$475$	2.26795	0.104061
$476$	−2.08783	−0.0956956
$477$	0	0
$478$	−24.2893	−1.11096
$479$	−20.6448	−0.943286	−0.471643	−	0.881790i	$-0.656339\pi$
$479$	−20.6448	−0.943286	−0.471643	+	0.881790i	$0.656339\pi$
$480$	0	0
$481$	0	0
$482$	−28.3777	−1.29257
$483$	0	0
$484$	−9.14898	−0.415863
$485$	12.1682	0.552528
$486$	0	0
$487$	3.03605	0.137576	0.0687882	−	0.997631i	$-0.478087\pi$
$487$	3.03605	0.137576	0.0687882	+	0.997631i	$0.478087\pi$
$488$	10.2968	0.466112
$489$	0	0
$490$	−7.27105	−0.328473
$491$	−10.6680	−0.481441	−0.240720	−	0.970595i	$-0.577384\pi$
$491$	−10.6680	−0.481441	−0.240720	+	0.970595i	$0.577384\pi$
$492$	0	0
$493$	0.0279884	0.00126053
$494$	0	0
$495$	0	0
$496$	−14.8228	−0.665562
$497$	38.8725	1.74367
$498$	0	0
$499$	33.9143	1.51821	0.759107	−	0.650966i	$-0.225636\pi$
$499$	33.9143	1.51821	0.759107	+	0.650966i	$0.225636\pi$
$500$	−0.512364	−0.0229136
$501$	0	0
$502$	14.4413	0.644546
$503$	12.6276	0.563037	0.281518	−	0.959556i	$-0.409162\pi$
$503$	12.6276	0.563037	0.281518	+	0.959556i	$0.409162\pi$
$504$	0	0
$505$	4.05441	0.180419
$506$	25.5058	1.13387
$507$	0	0
$508$	5.85945	0.259971
$509$	24.1526	1.07055	0.535273	−	0.844679i	$-0.320209\pi$
$509$	24.1526	1.07055	0.535273	+	0.844679i	$0.320209\pi$
$510$	0	0
$511$	−16.9320	−0.749029
$512$	24.2750	1.07281
$513$	0	0
$514$	−6.76929	−0.298581
$515$	−17.9035	−0.788921
$516$	0	0
$517$	13.8880	0.610796
$518$	38.2205	1.67931
$519$	0	0
$520$	0	0
$521$	24.7521	1.08441	0.542205	−	0.840246i	$-0.317590\pi$
$521$	24.7521	1.08441	0.542205	+	0.840246i	$0.317590\pi$
$522$	0	0
$523$	37.0326	1.61932	0.809662	−	0.586897i	$-0.199650\pi$
$523$	37.0326	1.61932	0.809662	+	0.586897i	$0.199650\pi$
$524$	−5.41465	−0.236540
$525$	0	0
$526$	8.36665	0.364803
$527$	6.18457	0.269404
$528$	0	0
$529$	−7.84554	−0.341111
$530$	−5.41465	−0.235197
$531$	0	0
$532$	−4.18348	−0.181377
$533$	0	0
$534$	0	0
$535$	9.13186	0.394805
$536$	19.6109	0.847062
$537$	0	0
$538$	−1.73438	−0.0747744
$539$	−32.0236	−1.37935
$540$	0	0
$541$	−8.38144	−0.360346	−0.180173	−	0.983635i	$-0.557666\pi$
$541$	−8.38144	−0.360346	−0.180173	+	0.983635i	$0.557666\pi$
$542$	12.1596	0.522301
$543$	0	0
$544$	−3.19170	−0.136843
$545$	7.37605	0.315955
$546$	0	0
$547$	−22.7842	−0.974181	−0.487091	−	0.873351i	$-0.661942\pi$
$547$	−22.7842	−0.974181	−0.487091	+	0.873351i	$0.661942\pi$
$548$	1.94018	0.0828804
$549$	0	0
$550$	6.55193	0.279375
$551$	0.0560816	0.00238915
$552$	0	0
$553$	−43.1398	−1.83449
$554$	21.3735	0.908071
$555$	0	0
$556$	−1.03183	−0.0437594
$557$	28.1527	1.19287	0.596435	−	0.802662i	$-0.296583\pi$
$557$	28.1527	1.19287	0.596435	+	0.802662i	$0.296583\pi$
$558$	0	0
$559$	0	0
$560$	−9.76645	−0.412708
$561$	0	0
$562$	13.0935	0.552317
$563$	18.1303	0.764101	0.382050	−	0.924142i	$-0.375218\pi$
$563$	18.1303	0.764101	0.382050	+	0.924142i	$0.375218\pi$
$564$	0	0
$565$	−7.07588	−0.297684
$566$	−1.60801	−0.0675899
$567$	0	0
$568$	33.0862	1.38827
$569$	−40.5985	−1.70198	−0.850988	−	0.525185i	$-0.823996\pi$
$569$	−40.5985	−1.70198	−0.850988	+	0.525185i	$0.823996\pi$
$570$	0	0
$571$	24.7159	1.03433	0.517164	−	0.855886i	$-0.326988\pi$
$571$	24.7159	1.03433	0.517164	+	0.855886i	$0.326988\pi$
$572$	0	0
$573$	0	0
$574$	16.3879	0.684016
$575$	3.89287	0.162344
$576$	0	0
$577$	23.0691	0.960379	0.480189	−	0.877165i	$-0.340568\pi$
$577$	23.0691	0.960379	0.480189	+	0.877165i	$0.340568\pi$
$578$	19.1721	0.797456
$579$	0	0
$580$	−0.0126697	−0.000526079 0
$581$	43.6653	1.81154
$582$	0	0
$583$	−23.8475	−0.987662
$584$	−14.4116	−0.596358
$585$	0	0
$586$	22.8602	0.944345
$587$	20.3523	0.840030	0.420015	−	0.907517i	$-0.362025\pi$
$587$	20.3523	0.840030	0.420015	+	0.907517i	$0.362025\pi$
$588$	0	0
$589$	12.3923	0.510616
$590$	−0.209084	−0.00860786
$591$	0	0
$592$	23.6120	0.970447
$593$	−10.3834	−0.426395	−0.213198	−	0.977009i	$-0.568388\pi$
$593$	−10.3834	−0.426395	−0.213198	+	0.977009i	$0.568388\pi$
$594$	0	0
$595$	4.07490	0.167055
$596$	−2.82712	−0.115803
$597$	0	0
$598$	0	0
$599$	31.5965	1.29100	0.645499	−	0.763761i	$-0.276649\pi$
$599$	31.5965	1.29100	0.645499	+	0.763761i	$0.276649\pi$
$600$	0	0
$601$	−43.8845	−1.79009	−0.895044	−	0.445979i	$-0.852856\pi$
$601$	−43.8845	−1.79009	−0.895044	+	0.445979i	$0.852856\pi$
$602$	4.97010	0.202566
$603$	0	0
$604$	2.50526	0.101938
$605$	17.8564	0.725966
$606$	0	0
$607$	2.17540	0.0882968	0.0441484	−	0.999025i	$-0.485943\pi$
$607$	2.17540	0.0882968	0.0441484	+	0.999025i	$0.485943\pi$
$608$	−6.39535	−0.259366
$609$	0	0
$610$	−4.09843	−0.165941
$611$	0	0
$612$	0	0
$613$	−14.7620	−0.596231	−0.298116	−	0.954530i	$-0.596358\pi$
$613$	−14.7620	−0.596231	−0.298116	+	0.954530i	$0.596358\pi$
$614$	−17.4467	−0.704092
$615$	0	0
$616$	−59.2622	−2.38774
$617$	20.2972	0.817134	0.408567	−	0.912728i	$-0.366029\pi$
$617$	20.2972	0.817134	0.408567	+	0.912728i	$0.366029\pi$
$618$	0	0
$619$	9.94207	0.399605	0.199803	−	0.979836i	$-0.435970\pi$
$619$	9.94207	0.399605	0.199803	+	0.979836i	$0.435970\pi$
$620$	−2.79961	−0.112435
$621$	0	0
$622$	3.36758	0.135028
$623$	58.1577	2.33004
$624$	0	0
$625$	1.00000	0.0400000
$626$	19.9855	0.798782
$627$	0	0
$628$	−5.14441	−0.205284
$629$	−9.85174	−0.392815
$630$	0	0
$631$	0.973420	0.0387512	0.0193756	−	0.999812i	$-0.493832\pi$
$631$	0.973420	0.0387512	0.0193756	+	0.999812i	$0.493832\pi$
$632$	−36.7183	−1.46058
$633$	0	0
$634$	2.17805	0.0865014
$635$	−11.4361	−0.453828
$636$	0	0
$637$	0	0
$638$	0.162015	0.00641425
$639$	0	0
$640$	−5.17262	−0.204466
$641$	12.6209	0.498497	0.249249	−	0.968440i	$-0.419816\pi$
$641$	12.6209	0.498497	0.249249	+	0.968440i	$0.419816\pi$
$642$	0	0
$643$	9.96043	0.392801	0.196401	−	0.980524i	$-0.437075\pi$
$643$	9.96043	0.392801	0.196401	+	0.980524i	$0.437075\pi$
$644$	−7.18083	−0.282964
$645$	0	0
$646$	−3.13092	−0.123185
$647$	36.2763	1.42617	0.713084	−	0.701079i	$-0.247298\pi$
$647$	36.2763	1.42617	0.713084	+	0.701079i	$0.247298\pi$
$648$	0	0
$649$	−0.920861	−0.0361470
$650$	0	0
$651$	0	0
$652$	−3.47447	−0.136071
$653$	−13.7554	−0.538289	−0.269145	−	0.963100i	$-0.586741\pi$
$653$	−13.7554	−0.538289	−0.269145	+	0.963100i	$0.586741\pi$
$654$	0	0
$655$	10.5680	0.412925
$656$	10.1241	0.395281
$657$	0	0
$658$	11.3526	0.442570
$659$	2.58183	0.100574	0.0502869	−	0.998735i	$-0.483986\pi$
$659$	2.58183	0.100574	0.0502869	+	0.998735i	$0.483986\pi$
$660$	0	0
$661$	24.8765	0.967582	0.483791	−	0.875183i	$-0.339259\pi$
$661$	24.8765	0.967582	0.483791	+	0.875183i	$0.339259\pi$
$662$	−8.81151	−0.342469
$663$	0	0
$664$	37.1656	1.44231
$665$	8.16506	0.316627
$666$	0	0
$667$	0.0962625	0.00372730
$668$	5.37460	0.207949
$669$	0	0
$670$	−7.80576	−0.301563
$671$	−18.0506	−0.696834
$672$	0	0
$673$	43.3222	1.66995	0.834974	−	0.550289i	$-0.185483\pi$
$673$	43.3222	1.66995	0.834974	+	0.550289i	$0.185483\pi$
$674$	5.32051	0.204938
$675$	0	0
$676$	0	0
$677$	41.3625	1.58969	0.794845	−	0.606813i	$-0.207552\pi$
$677$	41.3625	1.58969	0.794845	+	0.606813i	$0.207552\pi$
$678$	0	0
$679$	43.8078	1.68119
$680$	3.46834	0.133005
$681$	0	0
$682$	35.8004	1.37087
$683$	−2.62688	−0.100515	−0.0502574	−	0.998736i	$-0.516004\pi$
$683$	−2.62688	−0.100515	−0.0502574	+	0.998736i	$0.516004\pi$
$684$	0	0
$685$	−3.78672	−0.144683
$686$	4.56057	0.174123
$687$	0	0
$688$	3.07045	0.117060
$689$	0	0
$690$	0	0
$691$	−15.2753	−0.581099	−0.290550	−	0.956860i	$-0.593838\pi$
$691$	−15.2753	−0.581099	−0.290550	+	0.956860i	$0.593838\pi$
$692$	−2.28435	−0.0868380
$693$	0	0
$694$	−32.5744	−1.23651
$695$	2.01386	0.0763902
$696$	0	0
$697$	−4.22414	−0.160001
$698$	−28.7493	−1.08818
$699$	0	0
$700$	−1.84461	−0.0697197
$701$	48.1947	1.82029	0.910144	−	0.414292i	$-0.135971\pi$
$701$	48.1947	1.82029	0.910144	+	0.414292i	$0.135971\pi$
$702$	0	0
$703$	−19.7404	−0.744522
$704$	−47.6205	−1.79477
$705$	0	0
$706$	6.99767	0.263361
$707$	14.5967	0.548964
$708$	0	0
$709$	−38.8699	−1.45979	−0.729896	−	0.683559i	$-0.760431\pi$
$709$	−38.8699	−1.45979	−0.729896	+	0.683559i	$0.760431\pi$
$710$	−13.1694	−0.494237
$711$	0	0
$712$	49.5008	1.85512
$713$	21.2711	0.796607
$714$	0	0
$715$	0	0
$716$	9.54600	0.356751
$717$	0	0
$718$	−30.1974	−1.12696
$719$	−6.61660	−0.246758	−0.123379	−	0.992360i	$-0.539373\pi$
$719$	−6.61660	−0.246758	−0.123379	+	0.992360i	$0.539373\pi$
$720$	0	0
$721$	−64.4560	−2.40047
$722$	16.9005	0.628971
$723$	0	0
$724$	−9.26867	−0.344467
$725$	0.0247279	0.000918370 0
$726$	0	0
$727$	−18.3735	−0.681435	−0.340717	−	0.940166i	$-0.610670\pi$
$727$	−18.3735	−0.681435	−0.340717	+	0.940166i	$0.610670\pi$
$728$	0	0
$729$	0	0
$730$	5.73629	0.212310
$731$	−1.28110	−0.0473830
$732$	0	0
$733$	0.791131	0.0292211	0.0146105	−	0.999893i	$-0.495349\pi$
$733$	0.791131	0.0292211	0.0146105	+	0.999893i	$0.495349\pi$
$734$	−31.7772	−1.17292
$735$	0	0
$736$	−10.9774	−0.404634
$737$	−34.3786	−1.26635
$738$	0	0
$739$	31.1853	1.14717	0.573585	−	0.819146i	$-0.305552\pi$
$739$	31.1853	1.14717	0.573585	+	0.819146i	$0.305552\pi$
$740$	4.45965	0.163940
$741$	0	0
$742$	−19.4938	−0.715639
$743$	−5.56304	−0.204088	−0.102044	−	0.994780i	$-0.532538\pi$
$743$	−5.56304	−0.204088	−0.102044	+	0.994780i	$0.532538\pi$
$744$	0	0
$745$	5.51780	0.202156
$746$	−16.1053	−0.589658
$747$	0	0
$748$	3.11523	0.113904
$749$	32.8765	1.20128
$750$	0	0
$751$	−35.2097	−1.28482	−0.642410	−	0.766361i	$-0.722065\pi$
$751$	−35.2097	−1.28482	−0.642410	+	0.766361i	$0.722065\pi$
$752$	7.01343	0.255754
$753$	0	0
$754$	0	0
$755$	−4.88961	−0.177951
$756$	0	0
$757$	50.0446	1.81890	0.909451	−	0.415810i	$-0.136502\pi$
$757$	50.0446	1.81890	0.909451	+	0.415810i	$0.136502\pi$
$758$	−31.7091	−1.15173
$759$	0	0
$760$	6.94967	0.252091
$761$	−44.8209	−1.62476	−0.812379	−	0.583130i	$-0.801828\pi$
$761$	−44.8209	−1.62476	−0.812379	+	0.583130i	$0.801828\pi$
$762$	0	0
$763$	26.5552	0.961364
$764$	14.0061	0.506725
$765$	0	0
$766$	−11.7092	−0.423072
$767$	0	0
$768$	0	0
$769$	−39.3633	−1.41948	−0.709739	−	0.704465i	$-0.751187\pi$
$769$	−39.3633	−1.41948	−0.709739	+	0.704465i	$0.751187\pi$
$770$	23.5882	0.850061
$771$	0	0
$772$	−11.1528	−0.401399
$773$	−48.7805	−1.75451	−0.877256	−	0.480022i	$-0.840629\pi$
$773$	−48.7805	−1.75451	−0.877256	+	0.480022i	$0.840629\pi$
$774$	0	0
$775$	5.46410	0.196276
$776$	37.2869	1.33852
$777$	0	0
$778$	6.86841	0.246244
$779$	−8.46410	−0.303258
$780$	0	0
$781$	−58.0013	−2.07545
$782$	−5.37415	−0.192179
$783$	0	0
$784$	−16.1718	−0.577566
$785$	10.0405	0.358363
$786$	0	0
$787$	39.8608	1.42088	0.710442	−	0.703756i	$-0.248495\pi$
$787$	39.8608	1.42088	0.710442	+	0.703756i	$0.248495\pi$
$788$	−0.869338	−0.0309689
$789$	0	0
$790$	14.6150	0.519980
$791$	−25.4745	−0.905771
$792$	0	0
$793$	0	0
$794$	20.4490	0.725709
$795$	0	0
$796$	−12.9759	−0.459917
$797$	−26.2118	−0.928470	−0.464235	−	0.885712i	$-0.653671\pi$
$797$	−26.2118	−0.928470	−0.464235	+	0.885712i	$0.653671\pi$
$798$	0	0
$799$	−2.92625	−0.103523
$800$	−2.81988	−0.0996979
$801$	0	0
$802$	−16.9269	−0.597708
$803$	25.2641	0.891551
$804$	0	0
$805$	14.0151	0.493968
$806$	0	0
$807$	0	0
$808$	12.4239	0.437072
$809$	−22.2136	−0.780990	−0.390495	−	0.920605i	$-0.627696\pi$
$809$	−22.2136	−0.780990	−0.390495	+	0.920605i	$0.627696\pi$
$810$	0	0
$811$	19.0950	0.670515	0.335257	−	0.942127i	$-0.391177\pi$
$811$	19.0950	0.670515	0.335257	+	0.942127i	$0.391177\pi$
$812$	−0.0456133	−0.00160071
$813$	0	0
$814$	−57.0284	−1.99885
$815$	6.78124	0.237537
$816$	0	0
$817$	−2.56699	−0.0898076
$818$	−35.8894	−1.25484
$819$	0	0
$820$	1.91217	0.0667758
$821$	34.1584	1.19214	0.596068	−	0.802934i	$-0.296729\pi$
$821$	34.1584	1.19214	0.596068	+	0.802934i	$0.296729\pi$
$822$	0	0
$823$	4.07940	0.142199	0.0710995	−	0.997469i	$-0.477349\pi$
$823$	4.07940	0.142199	0.0710995	+	0.997469i	$0.477349\pi$
$824$	−54.8616	−1.91119
$825$	0	0
$826$	−0.752744	−0.0261913
$827$	54.8780	1.90830	0.954148	−	0.299337i	$-0.0967654\pi$
$827$	54.8780	1.90830	0.954148	+	0.299337i	$0.0967654\pi$
$828$	0	0
$829$	8.14950	0.283044	0.141522	−	0.989935i	$-0.454800\pi$
$829$	8.14950	0.283044	0.141522	+	0.989935i	$0.454800\pi$
$830$	−14.7931	−0.513476
$831$	0	0
$832$	0	0
$833$	6.74745	0.233785
$834$	0	0
$835$	−10.4898	−0.363015
$836$	6.24213	0.215889
$837$	0	0
$838$	8.49869	0.293582
$839$	21.8865	0.755606	0.377803	−	0.925886i	$-0.376680\pi$
$839$	21.8865	0.755606	0.377803	+	0.925886i	$0.376680\pi$
$840$	0	0
$841$	−28.9994	−0.999979
$842$	−8.68570	−0.299329
$843$	0	0
$844$	0.171902	0.00591710
$845$	0	0
$846$	0	0
$847$	64.2866	2.20891
$848$	−12.0429	−0.413556
$849$	0	0
$850$	−1.38051	−0.0473511
$851$	−33.8838	−1.16152
$852$	0	0
$853$	−19.2240	−0.658217	−0.329108	−	0.944292i	$-0.606748\pi$
$853$	−19.2240	−0.658217	−0.329108	+	0.944292i	$0.606748\pi$
$854$	−14.7552	−0.504911
$855$	0	0
$856$	27.9827	0.956430
$857$	27.8197	0.950302	0.475151	−	0.879904i	$-0.342393\pi$
$857$	27.8197	0.950302	0.475151	+	0.879904i	$0.342393\pi$
$858$	0	0
$859$	45.7355	1.56048	0.780238	−	0.625482i	$-0.215098\pi$
$859$	45.7355	1.56048	0.780238	+	0.625482i	$0.215098\pi$
$860$	0.579922	0.0197752
$861$	0	0
$862$	36.8521	1.25519
$863$	−54.8186	−1.86605	−0.933024	−	0.359814i	$-0.882840\pi$
$863$	−54.8186	−1.86605	−0.933024	+	0.359814i	$0.882840\pi$
$864$	0	0
$865$	4.45845	0.151592
$866$	−1.46378	−0.0497413
$867$	0	0
$868$	−10.0791	−0.342108
$869$	64.3684	2.18355
$870$	0	0
$871$	0	0
$872$	22.6024	0.765415
$873$	0	0
$874$	−10.7684	−0.364247
$875$	3.60020	0.121709
$876$	0	0
$877$	−27.3794	−0.924537	−0.462269	−	0.886740i	$-0.652964\pi$
$877$	−27.3794	−0.924537	−0.462269	+	0.886740i	$0.652964\pi$
$878$	20.1908	0.681407
$879$	0	0
$880$	14.5724	0.491236
$881$	34.4426	1.16040	0.580200	−	0.814474i	$-0.302974\pi$
$881$	34.4426	1.16040	0.580200	+	0.814474i	$0.302974\pi$
$882$	0	0
$883$	−17.3592	−0.584183	−0.292092	−	0.956390i	$-0.594351\pi$
$883$	−17.3592	−0.584183	−0.292092	+	0.956390i	$0.594351\pi$
$884$	0	0
$885$	0	0
$886$	5.56115	0.186831
$887$	31.1427	1.04567	0.522835	−	0.852434i	$-0.324874\pi$
$887$	31.1427	1.04567	0.522835	+	0.852434i	$0.324874\pi$
$888$	0	0
$889$	−41.1722	−1.38087
$890$	−19.7029	−0.660442
$891$	0	0
$892$	6.30042	0.210954
$893$	−5.86345	−0.196213
$894$	0	0
$895$	−18.6313	−0.622775
$896$	−18.6224	−0.622132
$897$	0	0
$898$	16.8953	0.563804
$899$	0.135116	0.00450636
$900$	0	0
$901$	5.02473	0.167398
$902$	−24.4521	−0.814167
$903$	0	0
$904$	−21.6826	−0.721152
$905$	18.0900	0.601332
$906$	0	0
$907$	17.6057	0.584587	0.292294	−	0.956329i	$-0.405582\pi$
$907$	17.6057	0.584587	0.292294	+	0.956329i	$0.405582\pi$
$908$	−3.91050	−0.129774
$909$	0	0
$910$	0	0
$911$	−50.0232	−1.65734	−0.828671	−	0.559737i	$-0.810902\pi$
$911$	−50.0232	−1.65734	−0.828671	+	0.559737i	$0.810902\pi$
$912$	0	0
$913$	−65.1526	−2.15624
$914$	−48.9272	−1.61837
$915$	0	0
$916$	7.37844	0.243791
$917$	38.0468	1.25641
$918$	0	0
$919$	−7.61556	−0.251214	−0.125607	−	0.992080i	$-0.540088\pi$
$919$	−7.61556	−0.251214	−0.125607	+	0.992080i	$0.540088\pi$
$920$	11.9289	0.393285
$921$	0	0
$922$	−9.19522	−0.302828
$923$	0	0
$924$	0	0
$925$	−8.70406	−0.286188
$926$	−28.4225	−0.934022
$927$	0	0
$928$	−0.0697297	−0.00228899
$929$	14.1239	0.463391	0.231695	−	0.972788i	$-0.425573\pi$
$929$	14.1239	0.463391	0.231695	+	0.972788i	$0.425573\pi$
$930$	0	0
$931$	13.5202	0.443106
$932$	4.86550	0.159375
$933$	0	0
$934$	−27.6012	−0.903138
$935$	−6.08012	−0.198841
$936$	0	0
$937$	−23.9317	−0.781815	−0.390908	−	0.920430i	$-0.627839\pi$
$937$	−23.9317	−0.781815	−0.390908	+	0.920430i	$0.627839\pi$
$938$	−28.1023	−0.917571
$939$	0	0
$940$	1.32464	0.0432051
$941$	25.3591	0.826683	0.413342	−	0.910576i	$-0.364362\pi$
$941$	25.3591	0.826683	0.413342	+	0.910576i	$0.364362\pi$
$942$	0	0
$943$	−14.5284	−0.473110
$944$	−0.465033	−0.0151355
$945$	0	0
$946$	−7.41584	−0.241110
$947$	−41.4223	−1.34604	−0.673021	−	0.739623i	$-0.735004\pi$
$947$	−41.4223	−1.34604	−0.673021	+	0.739623i	$0.735004\pi$
$948$	0	0
$949$	0	0
$950$	−2.76619	−0.0897470
$951$	0	0
$952$	12.4867	0.404696
$953$	−24.3026	−0.787237	−0.393619	−	0.919274i	$-0.628777\pi$
$953$	−24.3026	−0.787237	−0.393619	+	0.919274i	$0.628777\pi$
$954$	0	0
$955$	−27.3363	−0.884583
$956$	−10.2034	−0.330001
$957$	0	0
$958$	25.1802	0.813536
$959$	−13.6329	−0.440231
$960$	0	0
$961$	−1.14359	−0.0368901
$962$	0	0
$963$	0	0
$964$	−11.9209	−0.383945
$965$	21.7674	0.700717
$966$	0	0
$967$	23.6784	0.761445	0.380722	−	0.924689i	$-0.375675\pi$
$967$	23.6784	0.761445	0.380722	+	0.924689i	$0.375675\pi$
$968$	54.7173	1.75868
$969$	0	0
$970$	−14.8413	−0.476527
$971$	16.9722	0.544663	0.272332	−	0.962203i	$-0.412205\pi$
$971$	16.9722	0.544663	0.272332	+	0.962203i	$0.412205\pi$
$972$	0	0
$973$	7.25030	0.232434
$974$	−3.70303	−0.118653
$975$	0	0
$976$	−9.11550	−0.291780
$977$	−24.9994	−0.799802	−0.399901	−	0.916558i	$-0.630956\pi$
$977$	−24.9994	−0.799802	−0.399901	+	0.916558i	$0.630956\pi$
$978$	0	0
$979$	−86.7765	−2.77339
$980$	−3.05441	−0.0975696
$981$	0	0
$982$	13.0116	0.415218
$983$	27.3418	0.872068	0.436034	−	0.899930i	$-0.356383\pi$
$983$	27.3418	0.872068	0.436034	+	0.899930i	$0.356383\pi$
$984$	0	0
$985$	1.69672	0.0540620
$986$	−0.0341370	−0.00108715
$987$	0	0
$988$	0	0
$989$	−4.40617	−0.140108
$990$	0	0
$991$	16.0760	0.510672	0.255336	−	0.966852i	$-0.417814\pi$
$991$	16.0760	0.510672	0.255336	+	0.966852i	$0.417814\pi$
$992$	−15.4081	−0.489208
$993$	0	0
$994$	−47.4123	−1.50383
$995$	25.3255	0.802871
$996$	0	0
$997$	−34.5612	−1.09456	−0.547282	−	0.836948i	$-0.684337\pi$
$997$	−34.5612	−1.09456	−0.547282	+	0.836948i	$0.684337\pi$
$998$	−41.3649	−1.30938
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.cf.1.2		4
3.2	odd	2		845.2.a.m.1.3		4
13.2	odd	12		585.2.bu.c.316.2		8
13.7	odd	12		585.2.bu.c.361.2		8
13.12	even	2		7605.2.a.cj.1.3		4
15.14	odd	2		4225.2.a.bi.1.2		4
39.2	even	12		65.2.m.a.56.3	yes	8
39.5	even	4		845.2.c.g.506.3		8
39.8	even	4		845.2.c.g.506.6		8
39.11	even	12		845.2.m.g.316.2		8
39.17	odd	6		845.2.e.n.146.3		8
39.20	even	12		65.2.m.a.36.3	✓	8
39.23	odd	6		845.2.e.n.191.3		8
39.29	odd	6		845.2.e.m.191.2		8
39.32	even	12		845.2.m.g.361.2		8
39.35	odd	6		845.2.e.m.146.2		8
39.38	odd	2		845.2.a.l.1.2		4
156.59	odd	12		1040.2.da.b.881.4		8
156.119	odd	12		1040.2.da.b.641.4		8
195.2	odd	12		325.2.m.b.199.2		8
195.59	even	12		325.2.n.d.101.2		8
195.98	odd	12		325.2.m.b.49.2		8
195.119	even	12		325.2.n.d.251.2		8
195.137	odd	12		325.2.m.c.49.3		8
195.158	odd	12		325.2.m.c.199.3		8
195.194	odd	2		4225.2.a.bl.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.m.a.36.3	✓	8	39.20	even	12
65.2.m.a.56.3	yes	8	39.2	even	12
325.2.m.b.49.2		8	195.98	odd	12
325.2.m.b.199.2		8	195.2	odd	12
325.2.m.c.49.3		8	195.137	odd	12
325.2.m.c.199.3		8	195.158	odd	12
325.2.n.d.101.2		8	195.59	even	12
325.2.n.d.251.2		8	195.119	even	12
585.2.bu.c.316.2		8	13.2	odd	12
585.2.bu.c.361.2		8	13.7	odd	12
845.2.a.l.1.2		4	39.38	odd	2
845.2.a.m.1.3		4	3.2	odd	2
845.2.c.g.506.3		8	39.5	even	4
845.2.c.g.506.6		8	39.8	even	4
845.2.e.m.146.2		8	39.35	odd	6
845.2.e.m.191.2		8	39.29	odd	6
845.2.e.n.146.3		8	39.17	odd	6
845.2.e.n.191.3		8	39.23	odd	6
845.2.m.g.316.2		8	39.11	even	12
845.2.m.g.361.2		8	39.32	even	12
1040.2.da.b.641.4		8	156.119	odd	12
1040.2.da.b.881.4		8	156.59	odd	12
4225.2.a.bi.1.2		4	15.14	odd	2
4225.2.a.bl.1.3		4	195.194	odd	2
7605.2.a.cf.1.2		4	1.1	even	1	trivial
7605.2.a.cj.1.3		4	13.12	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:	\( N \)	\(=\)	\( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7605.a (trivial)

\(f(q)\)	\(=\)	\(q-1.21969 q^{2} -0.512364 q^{4} +1.00000 q^{5} +3.60020 q^{7} +3.06430 q^{8} +O(q^{10})\)	\(q-1.21969 q^{2} -0.512364 q^{4} +1.00000 q^{5} +3.60020 q^{7} +3.06430 q^{8} -1.21969 q^{10} -5.37182 q^{11} -4.39111 q^{14} -2.71276 q^{16} +1.13186 q^{17} +2.26795 q^{19} -0.512364 q^{20} +6.55193 q^{22} +3.89287 q^{23} +1.00000 q^{25} -1.84461 q^{28} +0.0247279 q^{29} +5.46410 q^{31} -2.81988 q^{32} -1.38051 q^{34} +3.60020 q^{35} -8.70406 q^{37} -2.76619 q^{38} +3.06430 q^{40} -3.73205 q^{41} -1.13186 q^{43} +2.75232 q^{44} -4.74809 q^{46} -2.58535 q^{47} +5.96141 q^{49} -1.21969 q^{50} +4.43937 q^{53} -5.37182 q^{55} +11.0321 q^{56} -0.0301603 q^{58} +0.171425 q^{59} +3.36023 q^{61} -6.66449 q^{62} +8.86488 q^{64} +6.39980 q^{67} -0.579922 q^{68} -4.39111 q^{70} +10.7973 q^{71} -4.70308 q^{73} +10.6162 q^{74} -1.16202 q^{76} -19.3396 q^{77} -11.9826 q^{79} -2.71276 q^{80} +4.55193 q^{82} +12.1286 q^{83} +1.13186 q^{85} +1.38051 q^{86} -16.4608 q^{88} +16.1540 q^{89} -1.99457 q^{92} +3.15332 q^{94} +2.26795 q^{95} +12.1682 q^{97} -7.27105 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 2 q^{4} + 4 q^{5} + 10 q^{7} - 6 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 + 10 * q^7 - 6 * q^8	\( 4 q - 2 q^{2} + 2 q^{4} + 4 q^{5} + 10 q^{7} - 6 q^{8} - 2 q^{10} - 2 q^{14} + 2 q^{16} + 2 q^{17} + 16 q^{19} + 2 q^{20} + 12 q^{22} + 10 q^{23} + 4 q^{25} + 8 q^{28} - 8 q^{29} + 8 q^{31} - 4 q^{32} - 4 q^{34} + 10 q^{35} - 2 q^{37} - 8 q^{38} - 6 q^{40} - 8 q^{41} - 2 q^{43} - 12 q^{44} + 16 q^{46} - 8 q^{47} + 12 q^{49} - 2 q^{50} + 12 q^{53} - 12 q^{56} + 22 q^{58} - 12 q^{59} + 28 q^{61} - 4 q^{62} + 4 q^{64} + 30 q^{67} - 14 q^{68} - 2 q^{70} - 4 q^{71} - 8 q^{73} + 10 q^{74} + 20 q^{76} - 18 q^{77} - 8 q^{79} + 2 q^{80} + 4 q^{82} + 12 q^{83} + 2 q^{85} + 4 q^{86} - 18 q^{88} + 12 q^{89} - 22 q^{92} - 32 q^{94} + 16 q^{95} + 2 q^{97} + 24 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 + 10 * q^7 - 6 * q^8 - 2 * q^10 - 2 * q^14 + 2 * q^16 + 2 * q^17 + 16 * q^19 + 2 * q^20 + 12 * q^22 + 10 * q^23 + 4 * q^25 + 8 * q^28 - 8 * q^29 + 8 * q^31 - 4 * q^32 - 4 * q^34 + 10 * q^35 - 2 * q^37 - 8 * q^38 - 6 * q^40 - 8 * q^41 - 2 * q^43 - 12 * q^44 + 16 * q^46 - 8 * q^47 + 12 * q^49 - 2 * q^50 + 12 * q^53 - 12 * q^56 + 22 * q^58 - 12 * q^59 + 28 * q^61 - 4 * q^62 + 4 * q^64 + 30 * q^67 - 14 * q^68 - 2 * q^70 - 4 * q^71 - 8 * q^73 + 10 * q^74 + 20 * q^76 - 18 * q^77 - 8 * q^79 + 2 * q^80 + 4 * q^82 + 12 * q^83 + 2 * q^85 + 4 * q^86 - 18 * q^88 + 12 * q^89 - 22 * q^92 - 32 * q^94 + 16 * q^95 + 2 * q^97 + 24 * q^98

Introduction

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