LMFDB - Embedded newform 7605.2.a.bp.1.1

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:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2535)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.80194$ 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)

$f(q)$	$=$	$q-1.80194 q^{2} +1.24698 q^{4} +1.00000 q^{5} -4.00000 q^{7} +1.35690 q^{8} +O(q^{10})$	$q-1.80194 q^{2} +1.24698 q^{4} +1.00000 q^{5} -4.00000 q^{7} +1.35690 q^{8} -1.80194 q^{10} -4.09783 q^{11} +7.20775 q^{14} -4.93900 q^{16} -6.69202 q^{17} +3.82908 q^{19} +1.24698 q^{20} +7.38404 q^{22} -6.45473 q^{23} +1.00000 q^{25} -4.98792 q^{28} -0.219833 q^{29} +8.51573 q^{31} +6.18598 q^{32} +12.0586 q^{34} -4.00000 q^{35} +2.93362 q^{37} -6.89977 q^{38} +1.35690 q^{40} +7.48188 q^{41} +6.31767 q^{43} -5.10992 q^{44} +11.6310 q^{46} +12.1468 q^{47} +9.00000 q^{49} -1.80194 q^{50} -0.225209 q^{53} -4.09783 q^{55} -5.42758 q^{56} +0.396125 q^{58} -1.87800 q^{59} +10.7899 q^{61} -15.3448 q^{62} -1.26875 q^{64} +8.05429 q^{67} -8.34481 q^{68} +7.20775 q^{70} +4.98792 q^{71} -4.49396 q^{73} -5.28621 q^{74} +4.77479 q^{76} +16.3913 q^{77} -13.5646 q^{79} -4.93900 q^{80} -13.4819 q^{82} -14.3937 q^{83} -6.69202 q^{85} -11.3840 q^{86} -5.56033 q^{88} +4.81163 q^{89} -8.04892 q^{92} -21.8877 q^{94} +3.82908 q^{95} +3.50604 q^{97} -16.2174 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} - q^{4} + 3 q^{5} - 12 q^{7}+O(q^{10}) $ 3 * q - q^2 - q^4 + 3 * q^5 - 12 * q^7	$ 3 q - q^{2} - q^{4} + 3 q^{5} - 12 q^{7} - q^{10} + 6 q^{11} + 4 q^{14} - 5 q^{16} - 15 q^{17} + q^{19} - q^{20} + 12 q^{22} + 3 q^{23} + 3 q^{25} + 4 q^{28} - 2 q^{29} + 13 q^{31} + 4 q^{32} + 5 q^{34} - 12 q^{35} + 2 q^{37} + 2 q^{38} - 6 q^{41} + 2 q^{43} - 16 q^{44} + 20 q^{46} + 9 q^{47} + 27 q^{49} - q^{50} + q^{53} + 6 q^{55} + 10 q^{58} + 14 q^{59} + 9 q^{61} - 23 q^{62} + 4 q^{64} + 12 q^{67} - 2 q^{68} + 4 q^{70} - 4 q^{71} - 4 q^{73} - 24 q^{74} + 16 q^{76} - 24 q^{77} - 19 q^{79} - 5 q^{80} - 12 q^{82} - 11 q^{83} - 15 q^{85} - 24 q^{86} - 14 q^{88} - 12 q^{89} - 15 q^{92} - 24 q^{94} + q^{95} + 20 q^{97} - 9 q^{98}+O(q^{100}) $ 3 * q - q^2 - q^4 + 3 * q^5 - 12 * q^7 - q^10 + 6 * q^11 + 4 * q^14 - 5 * q^16 - 15 * q^17 + q^19 - q^20 + 12 * q^22 + 3 * q^23 + 3 * q^25 + 4 * q^28 - 2 * q^29 + 13 * q^31 + 4 * q^32 + 5 * q^34 - 12 * q^35 + 2 * q^37 + 2 * q^38 - 6 * q^41 + 2 * q^43 - 16 * q^44 + 20 * q^46 + 9 * q^47 + 27 * q^49 - q^50 + q^53 + 6 * q^55 + 10 * q^58 + 14 * q^59 + 9 * q^61 - 23 * q^62 + 4 * q^64 + 12 * q^67 - 2 * q^68 + 4 * q^70 - 4 * q^71 - 4 * q^73 - 24 * q^74 + 16 * q^76 - 24 * q^77 - 19 * q^79 - 5 * q^80 - 12 * q^82 - 11 * q^83 - 15 * q^85 - 24 * q^86 - 14 * q^88 - 12 * q^89 - 15 * q^92 - 24 * q^94 + q^95 + 20 * q^97 - 9 * 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 $

$2$	−1.80194	−1.27416	−0.637081	−	0.770797i	$-0.719858\pi$
$2$	−1.80194	−1.27416	−0.637081	+	0.770797i	$0.719858\pi$
$3$	0	0
$3$	0	0
$4$	1.24698	0.623490
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	1.35690	0.479735
$9$	0	0
$10$	−1.80194	−0.569823
$11$	−4.09783	−1.23554	−0.617772	−	0.786357i	$-0.711964\pi$
$11$	−4.09783	−1.23554	−0.617772	+	0.786357i	$0.711964\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	7.20775	1.92635
$15$	0	0
$16$	−4.93900	−1.23475
$17$	−6.69202	−1.62305	−0.811527	−	0.584315i	$-0.801363\pi$
$17$	−6.69202	−1.62305	−0.811527	+	0.584315i	$0.801363\pi$
$18$	0	0
$19$	3.82908	0.878452	0.439226	−	0.898377i	$-0.355253\pi$
$19$	3.82908	0.878452	0.439226	+	0.898377i	$0.355253\pi$
$20$	1.24698	0.278833
$21$	0	0
$22$	7.38404	1.57428
$23$	−6.45473	−1.34590	−0.672952	−	0.739686i	$-0.734974\pi$
$23$	−6.45473	−1.34590	−0.672952	+	0.739686i	$0.734974\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	−4.98792	−0.942628
$29$	−0.219833	−0.0408219	−0.0204109	−	0.999792i	$-0.506497\pi$
$29$	−0.219833	−0.0408219	−0.0204109	+	0.999792i	$0.506497\pi$
$30$	0	0
$31$	8.51573	1.52947	0.764735	−	0.644345i	$-0.222870\pi$
$31$	8.51573	1.52947	0.764735	+	0.644345i	$0.222870\pi$
$32$	6.18598	1.09354
$33$	0	0
$34$	12.0586	2.06803
$35$	−4.00000	−0.676123
$36$	0	0
$37$	2.93362	0.482285	0.241142	−	0.970490i	$-0.422478\pi$
$37$	2.93362	0.482285	0.241142	+	0.970490i	$0.422478\pi$
$38$	−6.89977	−1.11929
$39$	0	0
$40$	1.35690	0.214544
$41$	7.48188	1.16847	0.584236	−	0.811583i	$-0.301394\pi$
$41$	7.48188	1.16847	0.584236	+	0.811583i	$0.301394\pi$
$42$	0	0
$43$	6.31767	0.963435	0.481718	−	0.876327i	$-0.340013\pi$
$43$	6.31767	0.963435	0.481718	+	0.876327i	$0.340013\pi$
$44$	−5.10992	−0.770349
$45$	0	0
$46$	11.6310	1.71490
$47$	12.1468	1.77179	0.885893	−	0.463890i	$-0.153547\pi$
$47$	12.1468	1.77179	0.885893	+	0.463890i	$0.153547\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	−1.80194	−0.254832
$51$	0	0
$52$	0	0
$53$	−0.225209	−0.0309349	−0.0154674	−	0.999880i	$-0.504924\pi$
$53$	−0.225209	−0.0309349	−0.0154674	+	0.999880i	$0.504924\pi$
$54$	0	0
$55$	−4.09783	−0.552552
$56$	−5.42758	−0.725291
$57$	0	0
$58$	0.396125	0.0520137
$59$	−1.87800	−0.244495	−0.122248	−	0.992500i	$-0.539010\pi$
$59$	−1.87800	−0.244495	−0.122248	+	0.992500i	$0.539010\pi$
$60$	0	0
$61$	10.7899	1.38150	0.690750	−	0.723094i	$-0.257281\pi$
$61$	10.7899	1.38150	0.690750	+	0.723094i	$0.257281\pi$
$62$	−15.3448	−1.94879
$63$	0	0
$64$	−1.26875	−0.158594
$65$	0	0
$66$	0	0
$67$	8.05429	0.983989	0.491994	−	0.870598i	$-0.336268\pi$
$67$	8.05429	0.983989	0.491994	+	0.870598i	$0.336268\pi$
$68$	−8.34481	−1.01196
$69$	0	0
$70$	7.20775	0.861491
$71$	4.98792	0.591957	0.295979	−	0.955195i	$-0.404354\pi$
$71$	4.98792	0.591957	0.295979	+	0.955195i	$0.404354\pi$
$72$	0	0
$73$	−4.49396	−0.525978	−0.262989	−	0.964799i	$-0.584708\pi$
$73$	−4.49396	−0.525978	−0.262989	+	0.964799i	$0.584708\pi$
$74$	−5.28621	−0.614509
$75$	0	0
$76$	4.77479	0.547706
$77$	16.3913	1.86797
$78$	0	0
$79$	−13.5646	−1.52614	−0.763071	−	0.646315i	$-0.776309\pi$
$79$	−13.5646	−1.52614	−0.763071	+	0.646315i	$0.776309\pi$
$80$	−4.93900	−0.552197
$81$	0	0
$82$	−13.4819	−1.48882
$83$	−14.3937	−1.57992	−0.789959	−	0.613160i	$-0.789898\pi$
$83$	−14.3937	−1.57992	−0.789959	+	0.613160i	$0.789898\pi$
$84$	0	0
$85$	−6.69202	−0.725852
$86$	−11.3840	−1.22757
$87$	0	0
$88$	−5.56033	−0.592734
$89$	4.81163	0.510031	0.255016	−	0.966937i	$-0.417919\pi$
$89$	4.81163	0.510031	0.255016	+	0.966937i	$0.417919\pi$
$90$	0	0
$91$	0	0
$92$	−8.04892	−0.839158
$93$	0	0
$94$	−21.8877	−2.25754
$95$	3.82908	0.392856
$96$	0	0
$97$	3.50604	0.355985	0.177992	−	0.984032i	$-0.443040\pi$
$97$	3.50604	0.355985	0.177992	+	0.984032i	$0.443040\pi$
$98$	−16.2174	−1.63821
$99$	0	0
$100$	1.24698	0.124698
$101$	15.9215	1.58425	0.792126	−	0.610357i	$-0.208974\pi$
$101$	15.9215	1.58425	0.792126	+	0.610357i	$0.208974\pi$
$102$	0	0
$103$	12.2741	1.20941	0.604703	−	0.796451i	$-0.293292\pi$
$103$	12.2741	1.20941	0.604703	+	0.796451i	$0.293292\pi$
$104$	0	0
$105$	0	0
$106$	0.405813	0.0394161
$107$	−4.84117	−0.468013	−0.234007	−	0.972235i	$-0.575184\pi$
$107$	−4.84117	−0.468013	−0.234007	+	0.972235i	$0.575184\pi$
$108$	0	0
$109$	−9.52111	−0.911957	−0.455978	−	0.889991i	$-0.650711\pi$
$109$	−9.52111	−0.911957	−0.455978	+	0.889991i	$0.650711\pi$
$110$	7.38404	0.704041
$111$	0	0
$112$	19.7560	1.86677
$113$	−7.64310	−0.719003	−0.359501	−	0.933145i	$-0.617053\pi$
$113$	−7.64310	−0.719003	−0.359501	+	0.933145i	$0.617053\pi$
$114$	0	0
$115$	−6.45473	−0.601907
$116$	−0.274127	−0.0254520
$117$	0	0
$118$	3.38404	0.311526
$119$	26.7681	2.45383
$120$	0	0
$121$	5.79225	0.526568
$122$	−19.4426	−1.76025
$123$	0	0
$124$	10.6189	0.953609
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.10992	−0.808374	−0.404187	−	0.914676i	$-0.632446\pi$
$127$	−9.10992	−0.808374	−0.404187	+	0.914676i	$0.632446\pi$
$128$	−10.0858	−0.891463
$129$	0	0
$130$	0	0
$131$	0.0978347	0.00854786	0.00427393	−	0.999991i	$-0.498640\pi$
$131$	0.0978347	0.00854786	0.00427393	+	0.999991i	$0.498640\pi$
$132$	0	0
$133$	−15.3163	−1.32810
$134$	−14.5133	−1.25376
$135$	0	0
$136$	−9.08038	−0.778636
$137$	9.50365	0.811951	0.405976	−	0.913884i	$-0.366932\pi$
$137$	9.50365	0.811951	0.405976	+	0.913884i	$0.366932\pi$
$138$	0	0
$139$	−8.41119	−0.713428	−0.356714	−	0.934214i	$-0.616103\pi$
$139$	−8.41119	−0.713428	−0.356714	+	0.934214i	$0.616103\pi$
$140$	−4.98792	−0.421556
$141$	0	0
$142$	−8.98792	−0.754249
$143$	0	0
$144$	0	0
$145$	−0.219833	−0.0182561
$146$	8.09783	0.670182
$147$	0	0
$148$	3.65817	0.300700
$149$	−6.76809	−0.554463	−0.277232	−	0.960803i	$-0.589417\pi$
$149$	−6.76809	−0.554463	−0.277232	+	0.960803i	$0.589417\pi$
$150$	0	0
$151$	6.35690	0.517317	0.258658	−	0.965969i	$-0.416720\pi$
$151$	6.35690	0.517317	0.258658	+	0.965969i	$0.416720\pi$
$152$	5.19567	0.421424
$153$	0	0
$154$	−29.5362	−2.38009
$155$	8.51573	0.684000
$156$	0	0
$157$	12.5918	1.00493	0.502467	−	0.864596i	$-0.332426\pi$
$157$	12.5918	1.00493	0.502467	+	0.864596i	$0.332426\pi$
$158$	24.4426	1.94455
$159$	0	0
$160$	6.18598	0.489045
$161$	25.8189	2.03482
$162$	0	0
$163$	−14.5918	−1.14292	−0.571459	−	0.820631i	$-0.693622\pi$
$163$	−14.5918	−1.14292	−0.571459	+	0.820631i	$0.693622\pi$
$164$	9.32975	0.728531
$165$	0	0
$166$	25.9366	2.01307
$167$	−18.1981	−1.40821	−0.704104	−	0.710097i	$-0.748651\pi$
$167$	−18.1981	−1.40821	−0.704104	+	0.710097i	$0.748651\pi$
$168$	0	0
$169$	0	0
$170$	12.0586	0.924853
$171$	0	0
$172$	7.87800	0.600692
$173$	−18.8509	−1.43320	−0.716602	−	0.697482i	$-0.754304\pi$
$173$	−18.8509	−1.43320	−0.716602	+	0.697482i	$0.754304\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	20.2392	1.52559
$177$	0	0
$178$	−8.67025	−0.649863
$179$	−12.5918	−0.941155	−0.470577	−	0.882359i	$-0.655954\pi$
$179$	−12.5918	−0.941155	−0.470577	+	0.882359i	$0.655954\pi$
$180$	0	0
$181$	−9.33513	−0.693874	−0.346937	−	0.937888i	$-0.612778\pi$
$181$	−9.33513	−0.693874	−0.346937	+	0.937888i	$0.612778\pi$
$182$	0	0
$183$	0	0
$184$	−8.75840	−0.645678
$185$	2.93362	0.215684
$186$	0	0
$187$	27.4228	2.00535
$188$	15.1468	1.10469
$189$	0	0
$190$	−6.89977	−0.500562
$191$	1.10992	0.0803107	0.0401554	−	0.999193i	$-0.487215\pi$
$191$	1.10992	0.0803107	0.0401554	+	0.999193i	$0.487215\pi$
$192$	0	0
$193$	4.61596	0.332264	0.166132	−	0.986104i	$-0.446872\pi$
$193$	4.61596	0.332264	0.166132	+	0.986104i	$0.446872\pi$
$194$	−6.31767	−0.453582
$195$	0	0
$196$	11.2228	0.801630
$197$	0.762709	0.0543408	0.0271704	−	0.999631i	$-0.491350\pi$
$197$	0.762709	0.0543408	0.0271704	+	0.999631i	$0.491350\pi$
$198$	0	0
$199$	−17.8562	−1.26579	−0.632897	−	0.774236i	$-0.718134\pi$
$199$	−17.8562	−1.26579	−0.632897	+	0.774236i	$0.718134\pi$
$200$	1.35690	0.0959470
$201$	0	0
$202$	−28.6896	−2.01860
$203$	0.879330	0.0617169
$204$	0	0
$205$	7.48188	0.522557
$206$	−22.1172	−1.54098
$207$	0	0
$208$	0	0
$209$	−15.6910	−1.08537
$210$	0	0
$211$	−12.3773	−0.852091	−0.426046	−	0.904702i	$-0.640094\pi$
$211$	−12.3773	−0.852091	−0.426046	+	0.904702i	$0.640094\pi$
$212$	−0.280831	−0.0192876
$213$	0	0
$214$	8.72348	0.596325
$215$	6.31767	0.430861
$216$	0	0
$217$	−34.0629	−2.31234
$218$	17.1564	1.16198
$219$	0	0
$220$	−5.10992	−0.344510
$221$	0	0
$222$	0	0
$223$	−24.9879	−1.67331	−0.836657	−	0.547727i	$-0.815493\pi$
$223$	−24.9879	−1.67331	−0.836657	+	0.547727i	$0.815493\pi$
$224$	−24.7439	−1.65327
$225$	0	0
$226$	13.7724	0.916126
$227$	3.50066	0.232347	0.116174	−	0.993229i	$-0.462937\pi$
$227$	3.50066	0.232347	0.116174	+	0.993229i	$0.462937\pi$
$228$	0	0
$229$	−13.4034	−0.885723	−0.442861	−	0.896590i	$-0.646037\pi$
$229$	−13.4034	−0.885723	−0.442861	+	0.896590i	$0.646037\pi$
$230$	11.6310	0.766927
$231$	0	0
$232$	−0.298290	−0.0195837
$233$	−16.6649	−1.09175	−0.545876	−	0.837866i	$-0.683803\pi$
$233$	−16.6649	−1.09175	−0.545876	+	0.837866i	$0.683803\pi$
$234$	0	0
$235$	12.1468	0.792367
$236$	−2.34183	−0.152440
$237$	0	0
$238$	−48.2344	−3.12657
$239$	25.0858	1.62266	0.811331	−	0.584587i	$-0.198743\pi$
$239$	25.0858	1.62266	0.811331	+	0.584587i	$0.198743\pi$
$240$	0	0
$241$	10.4004	0.669951	0.334976	−	0.942227i	$-0.391272\pi$
$241$	10.4004	0.669951	0.334976	+	0.942227i	$0.391272\pi$
$242$	−10.4373	−0.670933
$243$	0	0
$244$	13.4547	0.861351
$245$	9.00000	0.574989
$246$	0	0
$247$	0	0
$248$	11.5550	0.733741
$249$	0	0
$250$	−1.80194	−0.113965
$251$	17.4819	1.10345	0.551723	−	0.834027i	$-0.313971\pi$
$251$	17.4819	1.10345	0.551723	+	0.834027i	$0.313971\pi$
$252$	0	0
$253$	26.4504	1.66292
$254$	16.4155	1.03000
$255$	0	0
$256$	20.7114	1.29446
$257$	17.4088	1.08593	0.542965	−	0.839755i	$-0.317301\pi$
$257$	17.4088	1.08593	0.542965	+	0.839755i	$0.317301\pi$
$258$	0	0
$259$	−11.7345	−0.729146
$260$	0	0
$261$	0	0
$262$	−0.176292	−0.0108914
$263$	−8.85756	−0.546181	−0.273090	−	0.961988i	$-0.588046\pi$
$263$	−8.85756	−0.546181	−0.273090	+	0.961988i	$0.588046\pi$
$264$	0	0
$265$	−0.225209	−0.0138345
$266$	27.5991	1.69221
$267$	0	0
$268$	10.0435	0.613507
$269$	−18.9095	−1.15293	−0.576465	−	0.817122i	$-0.695568\pi$
$269$	−18.9095	−1.15293	−0.576465	+	0.817122i	$0.695568\pi$
$270$	0	0
$271$	−22.4741	−1.36520	−0.682602	−	0.730790i	$-0.739152\pi$
$271$	−22.4741	−1.36520	−0.682602	+	0.730790i	$0.739152\pi$
$272$	33.0519	2.00407
$273$	0	0
$274$	−17.1250	−1.03456
$275$	−4.09783	−0.247109
$276$	0	0
$277$	−10.4397	−0.627259	−0.313629	−	0.949545i	$-0.601545\pi$
$277$	−10.4397	−0.627259	−0.313629	+	0.949545i	$0.601545\pi$
$278$	15.1564	0.909023
$279$	0	0
$280$	−5.42758	−0.324360
$281$	−3.06638	−0.182925	−0.0914623	−	0.995809i	$-0.529154\pi$
$281$	−3.06638	−0.182925	−0.0914623	+	0.995809i	$0.529154\pi$
$282$	0	0
$283$	27.3793	1.62753	0.813764	−	0.581196i	$-0.197415\pi$
$283$	27.3793	1.62753	0.813764	+	0.581196i	$0.197415\pi$
$284$	6.21983	0.369079
$285$	0	0
$286$	0	0
$287$	−29.9275	−1.76657
$288$	0	0
$289$	27.7832	1.63430
$290$	0.396125	0.0232612
$291$	0	0
$292$	−5.60388	−0.327942
$293$	−5.10454	−0.298210	−0.149105	−	0.988821i	$-0.547639\pi$
$293$	−5.10454	−0.298210	−0.149105	+	0.988821i	$0.547639\pi$
$294$	0	0
$295$	−1.87800	−0.109342
$296$	3.98062	0.231369
$297$	0	0
$298$	12.1957	0.706476
$299$	0	0
$300$	0	0
$301$	−25.2707	−1.45658
$302$	−11.4547	−0.659146
$303$	0	0
$304$	−18.9119	−1.08467
$305$	10.7899	0.617825
$306$	0	0
$307$	−27.3599	−1.56151	−0.780755	−	0.624837i	$-0.785166\pi$
$307$	−27.3599	−1.56151	−0.780755	+	0.624837i	$0.785166\pi$
$308$	20.4397	1.16466
$309$	0	0
$310$	−15.3448	−0.871527
$311$	−12.0435	−0.682927	−0.341463	−	0.939895i	$-0.610922\pi$
$311$	−12.0435	−0.682927	−0.341463	+	0.939895i	$0.610922\pi$
$312$	0	0
$313$	17.5362	0.991203	0.495602	−	0.868550i	$-0.334948\pi$
$313$	17.5362	0.991203	0.495602	+	0.868550i	$0.334948\pi$
$314$	−22.6896	−1.28045
$315$	0	0
$316$	−16.9148	−0.951534
$317$	16.7482	0.940675	0.470337	−	0.882487i	$-0.344132\pi$
$317$	16.7482	0.940675	0.470337	+	0.882487i	$0.344132\pi$
$318$	0	0
$319$	0.900837	0.0504372
$320$	−1.26875	−0.0709253
$321$	0	0
$322$	−46.5241	−2.59269
$323$	−25.6243	−1.42578
$324$	0	0
$325$	0	0
$326$	26.2935	1.45626
$327$	0	0
$328$	10.1521	0.560558
$329$	−48.5870	−2.67869
$330$	0	0
$331$	31.1269	1.71089	0.855445	−	0.517894i	$-0.173284\pi$
$331$	31.1269	1.71089	0.855445	+	0.517894i	$0.173284\pi$
$332$	−17.9487	−0.985062
$333$	0	0
$334$	32.7918	1.79429
$335$	8.05429	0.440053
$336$	0	0
$337$	−6.91425	−0.376643	−0.188322	−	0.982107i	$-0.560305\pi$
$337$	−6.91425	−0.376643	−0.188322	+	0.982107i	$0.560305\pi$
$338$	0	0
$339$	0	0
$340$	−8.34481	−0.452561
$341$	−34.8961	−1.88973
$342$	0	0
$343$	−8.00000	−0.431959
$344$	8.57242	0.462194
$345$	0	0
$346$	33.9681	1.82613
$347$	0.217440	0.0116728	0.00583639	−	0.999983i	$-0.498142\pi$
$347$	0.217440	0.0116728	0.00583639	+	0.999983i	$0.498142\pi$
$348$	0	0
$349$	5.68771	0.304456	0.152228	−	0.988345i	$-0.451355\pi$
$349$	5.68771	0.304456	0.152228	+	0.988345i	$0.451355\pi$
$350$	7.20775	0.385270
$351$	0	0
$352$	−25.3491	−1.35111
$353$	−12.2784	−0.653515	−0.326758	−	0.945108i	$-0.605956\pi$
$353$	−12.2784	−0.653515	−0.326758	+	0.945108i	$0.605956\pi$
$354$	0	0
$355$	4.98792	0.264731
$356$	6.00000	0.317999
$357$	0	0
$358$	22.6896	1.19918
$359$	−4.93362	−0.260387	−0.130193	−	0.991489i	$-0.541560\pi$
$359$	−4.93362	−0.260387	−0.130193	+	0.991489i	$0.541560\pi$
$360$	0	0
$361$	−4.33811	−0.228322
$362$	16.8213	0.884109
$363$	0	0
$364$	0	0
$365$	−4.49396	−0.235225
$366$	0	0
$367$	−30.3913	−1.58641	−0.793207	−	0.608951i	$-0.791590\pi$
$367$	−30.3913	−1.58641	−0.793207	+	0.608951i	$0.791590\pi$
$368$	31.8799	1.66186
$369$	0	0
$370$	−5.28621	−0.274817
$371$	0.900837	0.0467691
$372$	0	0
$373$	15.7754	0.816818	0.408409	−	0.912799i	$-0.366084\pi$
$373$	15.7754	0.816818	0.408409	+	0.912799i	$0.366084\pi$
$374$	−49.4142	−2.55515
$375$	0	0
$376$	16.4819	0.849988
$377$	0	0
$378$	0	0
$379$	8.95944	0.460216	0.230108	−	0.973165i	$-0.426092\pi$
$379$	8.95944	0.460216	0.230108	+	0.973165i	$0.426092\pi$
$380$	4.77479	0.244942
$381$	0	0
$382$	−2.00000	−0.102329
$383$	−13.1927	−0.674115	−0.337057	−	0.941484i	$-0.609432\pi$
$383$	−13.1927	−0.674115	−0.337057	+	0.941484i	$0.609432\pi$
$384$	0	0
$385$	16.3913	0.835380
$386$	−8.31767	−0.423358
$387$	0	0
$388$	4.37196	0.221953
$389$	−9.42758	−0.477997	−0.238999	−	0.971020i	$-0.576819\pi$
$389$	−9.42758	−0.477997	−0.238999	+	0.971020i	$0.576819\pi$
$390$	0	0
$391$	43.1952	2.18447
$392$	12.2121	0.616802
$393$	0	0
$394$	−1.37435	−0.0692390
$395$	−13.5646	−0.682511
$396$	0	0
$397$	9.95646	0.499700	0.249850	−	0.968285i	$-0.419619\pi$
$397$	9.95646	0.499700	0.249850	+	0.968285i	$0.419619\pi$
$398$	32.1758	1.61283
$399$	0	0
$400$	−4.93900	−0.246950
$401$	4.81163	0.240281	0.120141	−	0.992757i	$-0.461665\pi$
$401$	4.81163	0.240281	0.120141	+	0.992757i	$0.461665\pi$
$402$	0	0
$403$	0	0
$404$	19.8538	0.987765
$405$	0	0
$406$	−1.58450	−0.0786373
$407$	−12.0215	−0.595884
$408$	0	0
$409$	19.9758	0.987742	0.493871	−	0.869535i	$-0.335582\pi$
$409$	19.9758	0.987742	0.493871	+	0.869535i	$0.335582\pi$
$410$	−13.4819	−0.665822
$411$	0	0
$412$	15.3056	0.754052
$413$	7.51201	0.369642
$414$	0	0
$415$	−14.3937	−0.706560
$416$	0	0
$417$	0	0
$418$	28.2741	1.38293
$419$	23.8103	1.16321	0.581605	−	0.813472i	$-0.302425\pi$
$419$	23.8103	1.16321	0.581605	+	0.813472i	$0.302425\pi$
$420$	0	0
$421$	14.5090	0.707127	0.353563	−	0.935411i	$-0.384970\pi$
$421$	14.5090	0.707127	0.353563	+	0.935411i	$0.384970\pi$
$422$	22.3032	1.08570
$423$	0	0
$424$	−0.305586	−0.0148405
$425$	−6.69202	−0.324611
$426$	0	0
$427$	−43.1594	−2.08863
$428$	−6.03684	−0.291801
$429$	0	0
$430$	−11.3840	−0.548987
$431$	3.93230	0.189412	0.0947060	−	0.995505i	$-0.469809\pi$
$431$	3.93230	0.189412	0.0947060	+	0.995505i	$0.469809\pi$
$432$	0	0
$433$	−31.1051	−1.49482	−0.747409	−	0.664365i	$-0.768702\pi$
$433$	−31.1051	−1.49482	−0.747409	+	0.664365i	$0.768702\pi$
$434$	61.3793	2.94630
$435$	0	0
$436$	−11.8726	−0.568596
$437$	−24.7157	−1.18231
$438$	0	0
$439$	14.5972	0.696685	0.348342	−	0.937367i	$-0.386745\pi$
$439$	14.5972	0.696685	0.348342	+	0.937367i	$0.386745\pi$
$440$	−5.56033	−0.265079
$441$	0	0
$442$	0	0
$443$	−15.0513	−0.715109	−0.357555	−	0.933892i	$-0.616389\pi$
$443$	−15.0513	−0.715109	−0.357555	+	0.933892i	$0.616389\pi$
$444$	0	0
$445$	4.81163	0.228093
$446$	45.0267	2.13207
$447$	0	0
$448$	5.07500	0.239771
$449$	21.4034	1.01009	0.505045	−	0.863093i	$-0.331476\pi$
$449$	21.4034	1.01009	0.505045	+	0.863093i	$0.331476\pi$
$450$	0	0
$451$	−30.6595	−1.44370
$452$	−9.53079	−0.448291
$453$	0	0
$454$	−6.30798	−0.296048
$455$	0	0
$456$	0	0
$457$	40.6353	1.90084	0.950420	−	0.310968i	$-0.100653\pi$
$457$	40.6353	1.90084	0.950420	+	0.310968i	$0.100653\pi$
$458$	24.1521	1.12855
$459$	0	0
$460$	−8.04892	−0.375283
$461$	−29.3008	−1.36467	−0.682337	−	0.731038i	$-0.739036\pi$
$461$	−29.3008	−1.36467	−0.682337	+	0.731038i	$0.739036\pi$
$462$	0	0
$463$	−28.1909	−1.31014	−0.655071	−	0.755568i	$-0.727361\pi$
$463$	−28.1909	−1.31014	−0.655071	+	0.755568i	$0.727361\pi$
$464$	1.08575	0.0504048
$465$	0	0
$466$	30.0291	1.39107
$467$	27.9028	1.29119	0.645593	−	0.763682i	$-0.276610\pi$
$467$	27.9028	1.29119	0.645593	+	0.763682i	$0.276610\pi$
$468$	0	0
$469$	−32.2172	−1.48765
$470$	−21.8877	−1.00960
$471$	0	0
$472$	−2.54825	−0.117293
$473$	−25.8888	−1.19037
$474$	0	0
$475$	3.82908	0.175690
$476$	33.3793	1.52994
$477$	0	0
$478$	−45.2030	−2.06754
$479$	33.1400	1.51421	0.757104	−	0.653295i	$-0.226614\pi$
$479$	33.1400	1.51421	0.757104	+	0.653295i	$0.226614\pi$
$480$	0	0
$481$	0	0
$482$	−18.7409	−0.853626
$483$	0	0
$484$	7.22282	0.328310
$485$	3.50604	0.159201
$486$	0	0
$487$	23.9866	1.08694	0.543468	−	0.839430i	$-0.317111\pi$
$487$	23.9866	1.08694	0.543468	+	0.839430i	$0.317111\pi$
$488$	14.6407	0.662754
$489$	0	0
$490$	−16.2174	−0.732629
$491$	23.8974	1.07847	0.539237	−	0.842154i	$-0.318713\pi$
$491$	23.8974	1.07847	0.539237	+	0.842154i	$0.318713\pi$
$492$	0	0
$493$	1.47112	0.0662561
$494$	0	0
$495$	0	0
$496$	−42.0592	−1.88851
$497$	−19.9517	−0.894955
$498$	0	0
$499$	−17.6122	−0.788432	−0.394216	−	0.919018i	$-0.628984\pi$
$499$	−17.6122	−0.788432	−0.394216	+	0.919018i	$0.628984\pi$
$500$	1.24698	0.0557666
$501$	0	0
$502$	−31.5013	−1.40597
$503$	−28.8907	−1.28817	−0.644086	−	0.764953i	$-0.722762\pi$
$503$	−28.8907	−1.28817	−0.644086	+	0.764953i	$0.722762\pi$
$504$	0	0
$505$	15.9215	0.708499
$506$	−47.6620	−2.11883
$507$	0	0
$508$	−11.3599	−0.504013
$509$	−23.2573	−1.03086	−0.515430	−	0.856932i	$-0.672368\pi$
$509$	−23.2573	−1.03086	−0.515430	+	0.856932i	$0.672368\pi$
$510$	0	0
$511$	17.9758	0.795204
$512$	−17.1491	−0.757892
$513$	0	0
$514$	−31.3696	−1.38365
$515$	12.2741	0.540863
$516$	0	0
$517$	−49.7754	−2.18912
$518$	21.1448	0.929051
$519$	0	0
$520$	0	0
$521$	10.0435	0.440015	0.220008	−	0.975498i	$-0.429392\pi$
$521$	10.0435	0.440015	0.220008	+	0.975498i	$0.429392\pi$
$522$	0	0
$523$	26.5327	1.16019	0.580097	−	0.814547i	$-0.303015\pi$
$523$	26.5327	1.16019	0.580097	+	0.814547i	$0.303015\pi$
$524$	0.121998	0.00532950
$525$	0	0
$526$	15.9608	0.695923
$527$	−56.9874	−2.48241
$528$	0	0
$529$	18.6635	0.811459
$530$	0.405813	0.0176274
$531$	0	0
$532$	−19.0992	−0.828054
$533$	0	0
$534$	0	0
$535$	−4.84117	−0.209302
$536$	10.9288	0.472054
$537$	0	0
$538$	34.0737	1.46902
$539$	−36.8805	−1.58856
$540$	0	0
$541$	−7.01746	−0.301704	−0.150852	−	0.988556i	$-0.548202\pi$
$541$	−7.01746	−0.301704	−0.150852	+	0.988556i	$0.548202\pi$
$542$	40.4969	1.73949
$543$	0	0
$544$	−41.3967	−1.77487
$545$	−9.52111	−0.407839
$546$	0	0
$547$	−39.4336	−1.68606	−0.843028	−	0.537869i	$-0.819230\pi$
$547$	−39.4336	−1.68606	−0.843028	+	0.537869i	$0.819230\pi$
$548$	11.8509	0.506243
$549$	0	0
$550$	7.38404	0.314857
$551$	−0.841757	−0.0358601
$552$	0	0
$553$	54.2586	2.30731
$554$	18.8116	0.799229
$555$	0	0
$556$	−10.4886	−0.444815
$557$	3.32437	0.140858	0.0704291	−	0.997517i	$-0.477563\pi$
$557$	3.32437	0.140858	0.0704291	+	0.997517i	$0.477563\pi$
$558$	0	0
$559$	0	0
$560$	19.7560	0.834844
$561$	0	0
$562$	5.52542	0.233076
$563$	−30.8944	−1.30204	−0.651022	−	0.759059i	$-0.725659\pi$
$563$	−30.8944	−1.30204	−0.651022	+	0.759059i	$0.725659\pi$
$564$	0	0
$565$	−7.64310	−0.321548
$566$	−49.3357	−2.07373
$567$	0	0
$568$	6.76809	0.283983
$569$	38.9724	1.63381	0.816904	−	0.576774i	$-0.195689\pi$
$569$	38.9724	1.63381	0.816904	+	0.576774i	$0.195689\pi$
$570$	0	0
$571$	−28.7724	−1.20409	−0.602044	−	0.798463i	$-0.705647\pi$
$571$	−28.7724	−1.20409	−0.602044	+	0.798463i	$0.705647\pi$
$572$	0	0
$573$	0	0
$574$	53.9275	2.25089
$575$	−6.45473	−0.269181
$576$	0	0
$577$	−39.2271	−1.63305	−0.816523	−	0.577312i	$-0.804101\pi$
$577$	−39.2271	−1.63305	−0.816523	+	0.577312i	$0.804101\pi$
$578$	−50.0635	−2.08237
$579$	0	0
$580$	−0.274127	−0.0113825
$581$	57.5749	2.38861
$582$	0	0
$583$	0.922871	0.0382214
$584$	−6.09783	−0.252330
$585$	0	0
$586$	9.19806	0.379968
$587$	−16.5483	−0.683020	−0.341510	−	0.939878i	$-0.610938\pi$
$587$	−16.5483	−0.683020	−0.341510	+	0.939878i	$0.610938\pi$
$588$	0	0
$589$	32.6075	1.34357
$590$	3.38404	0.139319
$591$	0	0
$592$	−14.4892	−0.595501
$593$	0.787937	0.0323567	0.0161784	−	0.999869i	$-0.494850\pi$
$593$	0.787937	0.0323567	0.0161784	+	0.999869i	$0.494850\pi$
$594$	0	0
$595$	26.7681	1.09738
$596$	−8.43967	−0.345702
$597$	0	0
$598$	0	0
$599$	41.5314	1.69693	0.848463	−	0.529254i	$-0.177528\pi$
$599$	41.5314	1.69693	0.848463	+	0.529254i	$0.177528\pi$
$600$	0	0
$601$	−11.3797	−0.464189	−0.232094	−	0.972693i	$-0.574558\pi$
$601$	−11.3797	−0.464189	−0.232094	+	0.972693i	$0.574558\pi$
$602$	45.5362	1.85592
$603$	0	0
$604$	7.92692	0.322542
$605$	5.79225	0.235488
$606$	0	0
$607$	2.67025	0.108382	0.0541911	−	0.998531i	$-0.482742\pi$
$607$	2.67025	0.108382	0.0541911	+	0.998531i	$0.482742\pi$
$608$	23.6866	0.960620
$609$	0	0
$610$	−19.4426	−0.787210
$611$	0	0
$612$	0	0
$613$	−36.6413	−1.47993	−0.739964	−	0.672646i	$-0.765158\pi$
$613$	−36.6413	−1.47993	−0.739964	+	0.672646i	$0.765158\pi$
$614$	49.3008	1.98962
$615$	0	0
$616$	22.2413	0.896129
$617$	38.0017	1.52989	0.764945	−	0.644096i	$-0.222766\pi$
$617$	38.0017	1.52989	0.764945	+	0.644096i	$0.222766\pi$
$618$	0	0
$619$	−15.8060	−0.635296	−0.317648	−	0.948209i	$-0.602893\pi$
$619$	−15.8060	−0.635296	−0.317648	+	0.948209i	$0.602893\pi$
$620$	10.6189	0.426467
$621$	0	0
$622$	21.7017	0.870159
$623$	−19.2465	−0.771095
$624$	0	0
$625$	1.00000	0.0400000
$626$	−31.5991	−1.26295
$627$	0	0
$628$	15.7017	0.626566
$629$	−19.6319	−0.782774
$630$	0	0
$631$	−38.5864	−1.53610	−0.768051	−	0.640389i	$-0.778773\pi$
$631$	−38.5864	−1.53610	−0.768051	+	0.640389i	$0.778773\pi$
$632$	−18.4058	−0.732144
$633$	0	0
$634$	−30.1793	−1.19857
$635$	−9.10992	−0.361516
$636$	0	0
$637$	0	0
$638$	−1.62325	−0.0642652
$639$	0	0
$640$	−10.0858	−0.398674
$641$	−19.2379	−0.759851	−0.379925	−	0.925017i	$-0.624050\pi$
$641$	−19.2379	−0.759851	−0.379925	+	0.925017i	$0.624050\pi$
$642$	0	0
$643$	15.7802	0.622309	0.311155	−	0.950359i	$-0.399284\pi$
$643$	15.7802	0.622309	0.311155	+	0.950359i	$0.399284\pi$
$644$	32.1957	1.26869
$645$	0	0
$646$	46.1734	1.81667
$647$	−9.60148	−0.377473	−0.188737	−	0.982028i	$-0.560439\pi$
$647$	−9.60148	−0.377473	−0.188737	+	0.982028i	$0.560439\pi$
$648$	0	0
$649$	7.69574	0.302084
$650$	0	0
$651$	0	0
$652$	−18.1957	−0.712597
$653$	23.7235	0.928372	0.464186	−	0.885738i	$-0.346347\pi$
$653$	23.7235	0.928372	0.464186	+	0.885738i	$0.346347\pi$
$654$	0	0
$655$	0.0978347	0.00382272
$656$	−36.9530	−1.44277
$657$	0	0
$658$	87.5508	3.41308
$659$	−1.92154	−0.0748527	−0.0374263	−	0.999299i	$-0.511916\pi$
$659$	−1.92154	−0.0748527	−0.0374263	+	0.999299i	$0.511916\pi$
$660$	0	0
$661$	17.1618	0.667517	0.333759	−	0.942659i	$-0.391683\pi$
$661$	17.1618	0.667517	0.333759	+	0.942659i	$0.391683\pi$
$662$	−56.0887	−2.17995
$663$	0	0
$664$	−19.5308	−0.757942
$665$	−15.3163	−0.593942
$666$	0	0
$667$	1.41896	0.0549423
$668$	−22.6926	−0.878004
$669$	0	0
$670$	−14.5133	−0.560699
$671$	−44.2150	−1.70690
$672$	0	0
$673$	−5.32975	−0.205447	−0.102723	−	0.994710i	$-0.532756\pi$
$673$	−5.32975	−0.205447	−0.102723	+	0.994710i	$0.532756\pi$
$674$	12.4590	0.479904
$675$	0	0
$676$	0	0
$677$	24.2717	0.932839	0.466419	−	0.884564i	$-0.345544\pi$
$677$	24.2717	0.932839	0.466419	+	0.884564i	$0.345544\pi$
$678$	0	0
$679$	−14.0242	−0.538198
$680$	−9.08038	−0.348217
$681$	0	0
$682$	62.8805	2.40782
$683$	4.08682	0.156378	0.0781889	−	0.996939i	$-0.475086\pi$
$683$	4.08682	0.156378	0.0781889	+	0.996939i	$0.475086\pi$
$684$	0	0
$685$	9.50365	0.363116
$686$	14.4155	0.550386
$687$	0	0
$688$	−31.2030	−1.18960
$689$	0	0
$690$	0	0
$691$	−23.4905	−0.893621	−0.446810	−	0.894629i	$-0.647440\pi$
$691$	−23.4905	−0.893621	−0.446810	+	0.894629i	$0.647440\pi$
$692$	−23.5066	−0.893588
$693$	0	0
$694$	−0.391813	−0.0148730
$695$	−8.41119	−0.319055
$696$	0	0
$697$	−50.0689	−1.89649
$698$	−10.2489	−0.387927
$699$	0	0
$700$	−4.98792	−0.188526
$701$	28.9288	1.09263	0.546314	−	0.837581i	$-0.316031\pi$
$701$	28.9288	1.09263	0.546314	+	0.837581i	$0.316031\pi$
$702$	0	0
$703$	11.2331	0.423664
$704$	5.19913	0.195949
$705$	0	0
$706$	22.1250	0.832685
$707$	−63.6862	−2.39516
$708$	0	0
$709$	48.2103	1.81057	0.905287	−	0.424800i	$-0.139656\pi$
$709$	48.2103	1.81057	0.905287	+	0.424800i	$0.139656\pi$
$710$	−8.98792	−0.337311
$711$	0	0
$712$	6.52888	0.244680
$713$	−54.9667	−2.05852
$714$	0	0
$715$	0	0
$716$	−15.7017	−0.586800
$717$	0	0
$718$	8.89008	0.331775
$719$	18.4155	0.686782	0.343391	−	0.939192i	$-0.388424\pi$
$719$	18.4155	0.686782	0.343391	+	0.939192i	$0.388424\pi$
$720$	0	0
$721$	−49.0965	−1.82845
$722$	7.81700	0.290919
$723$	0	0
$724$	−11.6407	−0.432624
$725$	−0.219833	−0.00816438
$726$	0	0
$727$	−18.4650	−0.684829	−0.342415	−	0.939549i	$-0.611245\pi$
$727$	−18.4650	−0.684829	−0.342415	+	0.939549i	$0.611245\pi$
$728$	0	0
$729$	0	0
$730$	8.09783	0.299714
$731$	−42.2780	−1.56371
$732$	0	0
$733$	−11.1293	−0.411070	−0.205535	−	0.978650i	$-0.565893\pi$
$733$	−11.1293	−0.411070	−0.205535	+	0.978650i	$0.565893\pi$
$734$	54.7633	2.02135
$735$	0	0
$736$	−39.9288	−1.47180
$737$	−33.0052	−1.21576
$738$	0	0
$739$	5.22090	0.192054	0.0960269	−	0.995379i	$-0.469387\pi$
$739$	5.22090	0.192054	0.0960269	+	0.995379i	$0.469387\pi$
$740$	3.65817	0.134477
$741$	0	0
$742$	−1.62325	−0.0595915
$743$	−41.0097	−1.50450	−0.752250	−	0.658878i	$-0.771031\pi$
$743$	−41.0097	−1.50450	−0.752250	+	0.658878i	$0.771031\pi$
$744$	0	0
$745$	−6.76809	−0.247963
$746$	−28.4263	−1.04076
$747$	0	0
$748$	34.1957	1.25032
$749$	19.3647	0.707569
$750$	0	0
$751$	31.1879	1.13806	0.569031	−	0.822316i	$-0.307318\pi$
$751$	31.1879	1.13806	0.569031	+	0.822316i	$0.307318\pi$
$752$	−59.9928	−2.18771
$753$	0	0
$754$	0	0
$755$	6.35690	0.231351
$756$	0	0
$757$	−35.5206	−1.29102	−0.645510	−	0.763752i	$-0.723355\pi$
$757$	−35.5206	−1.29102	−0.645510	+	0.763752i	$0.723355\pi$
$758$	−16.1444	−0.586390
$759$	0	0
$760$	5.19567	0.188467
$761$	−2.58104	−0.0935626	−0.0467813	−	0.998905i	$-0.514896\pi$
$761$	−2.58104	−0.0935626	−0.0467813	+	0.998905i	$0.514896\pi$
$762$	0	0
$763$	38.0844	1.37875
$764$	1.38404	0.0500729
$765$	0	0
$766$	23.7724	0.858932
$767$	0	0
$768$	0	0
$769$	3.38165	0.121945	0.0609727	−	0.998139i	$-0.480580\pi$
$769$	3.38165	0.121945	0.0609727	+	0.998139i	$0.480580\pi$
$770$	−29.5362	−1.06441
$771$	0	0
$772$	5.75600	0.207163
$773$	3.21877	0.115771	0.0578855	−	0.998323i	$-0.481564\pi$
$773$	3.21877	0.115771	0.0578855	+	0.998323i	$0.481564\pi$
$774$	0	0
$775$	8.51573	0.305894
$776$	4.75733	0.170778
$777$	0	0
$778$	16.9879	0.609046
$779$	28.6487	1.02645
$780$	0	0
$781$	−20.4397	−0.731389
$782$	−77.8351	−2.78338
$783$	0	0
$784$	−44.4510	−1.58754
$785$	12.5918	0.449420
$786$	0	0
$787$	42.9530	1.53111	0.765555	−	0.643371i	$-0.222465\pi$
$787$	42.9530	1.53111	0.765555	+	0.643371i	$0.222465\pi$
$788$	0.951083	0.0338809
$789$	0	0
$790$	24.4426	0.869630
$791$	30.5724	1.08703
$792$	0	0
$793$	0	0
$794$	−17.9409	−0.636699
$795$	0	0
$796$	−22.2664	−0.789210
$797$	20.8821	0.739680	0.369840	−	0.929095i	$-0.379412\pi$
$797$	20.8821	0.739680	0.369840	+	0.929095i	$0.379412\pi$
$798$	0	0
$799$	−81.2863	−2.87570
$800$	6.18598	0.218707
$801$	0	0
$802$	−8.67025	−0.306157
$803$	18.4155	0.649869
$804$	0	0
$805$	25.8189	0.909997
$806$	0	0
$807$	0	0
$808$	21.6039	0.760022
$809$	4.49396	0.157999	0.0789996	−	0.996875i	$-0.474827\pi$
$809$	4.49396	0.157999	0.0789996	+	0.996875i	$0.474827\pi$
$810$	0	0
$811$	12.4397	0.436816	0.218408	−	0.975858i	$-0.429914\pi$
$811$	12.4397	0.436816	0.218408	+	0.975858i	$0.429914\pi$
$812$	1.09651	0.0384798
$813$	0	0
$814$	21.6620	0.759253
$815$	−14.5918	−0.511128
$816$	0	0
$817$	24.1909	0.846332
$818$	−35.9952	−1.25854
$819$	0	0
$820$	9.32975	0.325809
$821$	−35.1159	−1.22555	−0.612776	−	0.790256i	$-0.709947\pi$
$821$	−35.1159	−1.22555	−0.612776	+	0.790256i	$0.709947\pi$
$822$	0	0
$823$	−51.4577	−1.79370	−0.896852	−	0.442332i	$-0.854151\pi$
$823$	−51.4577	−1.79370	−0.896852	+	0.442332i	$0.854151\pi$
$824$	16.6547	0.580194
$825$	0	0
$826$	−13.5362	−0.470984
$827$	−14.4638	−0.502957	−0.251478	−	0.967863i	$-0.580917\pi$
$827$	−14.4638	−0.502957	−0.251478	+	0.967863i	$0.580917\pi$
$828$	0	0
$829$	0.289192	0.0100441	0.00502203	−	0.999987i	$-0.498401\pi$
$829$	0.289192	0.0100441	0.00502203	+	0.999987i	$0.498401\pi$
$830$	25.9366	0.900273
$831$	0	0
$832$	0	0
$833$	−60.2282	−2.08678
$834$	0	0
$835$	−18.1981	−0.629770
$836$	−19.5663	−0.676715
$837$	0	0
$838$	−42.9047	−1.48212
$839$	−31.8888	−1.10092	−0.550461	−	0.834861i	$-0.685548\pi$
$839$	−31.8888	−1.10092	−0.550461	+	0.834861i	$0.685548\pi$
$840$	0	0
$841$	−28.9517	−0.998334
$842$	−26.1444	−0.900994
$843$	0	0
$844$	−15.4343	−0.531270
$845$	0	0
$846$	0	0
$847$	−23.1690	−0.796096
$848$	1.11231	0.0381969
$849$	0	0
$850$	12.0586	0.413607
$851$	−18.9358	−0.649109
$852$	0	0
$853$	−44.0146	−1.50703	−0.753515	−	0.657430i	$-0.771643\pi$
$853$	−44.0146	−1.50703	−0.753515	+	0.657430i	$0.771643\pi$
$854$	77.7706	2.66126
$855$	0	0
$856$	−6.56896	−0.224522
$857$	−30.1957	−1.03146	−0.515732	−	0.856750i	$-0.672480\pi$
$857$	−30.1957	−1.03146	−0.515732	+	0.856750i	$0.672480\pi$
$858$	0	0
$859$	−40.0498	−1.36648	−0.683240	−	0.730194i	$-0.739430\pi$
$859$	−40.0498	−1.36648	−0.683240	+	0.730194i	$0.739430\pi$
$860$	7.87800	0.268638
$861$	0	0
$862$	−7.08575	−0.241342
$863$	26.5730	0.904556	0.452278	−	0.891877i	$-0.350612\pi$
$863$	26.5730	0.904556	0.452278	+	0.891877i	$0.350612\pi$
$864$	0	0
$865$	−18.8509	−0.640948
$866$	56.0495	1.90464
$867$	0	0
$868$	−42.4758	−1.44172
$869$	55.5857	1.88562
$870$	0	0
$871$	0	0
$872$	−12.9191	−0.437498
$873$	0	0
$874$	44.5362	1.50646
$875$	−4.00000	−0.135225
$876$	0	0
$877$	−8.53750	−0.288291	−0.144145	−	0.989557i	$-0.546043\pi$
$877$	−8.53750	−0.288291	−0.144145	+	0.989557i	$0.546043\pi$
$878$	−26.3032	−0.887690
$879$	0	0
$880$	20.2392	0.682264
$881$	−10.0785	−0.339552	−0.169776	−	0.985483i	$-0.554304\pi$
$881$	−10.0785	−0.339552	−0.169776	+	0.985483i	$0.554304\pi$
$882$	0	0
$883$	−49.0965	−1.65223	−0.826115	−	0.563502i	$-0.809454\pi$
$883$	−49.0965	−1.65223	−0.826115	+	0.563502i	$0.809454\pi$
$884$	0	0
$885$	0	0
$886$	27.1215	0.911165
$887$	33.0484	1.10966	0.554829	−	0.831964i	$-0.312784\pi$
$887$	33.0484	1.10966	0.554829	+	0.831964i	$0.312784\pi$
$888$	0	0
$889$	36.4397	1.22215
$890$	−8.67025	−0.290627
$891$	0	0
$892$	−31.1594	−1.04329
$893$	46.5109	1.55643
$894$	0	0
$895$	−12.5918	−0.420897
$896$	40.3430	1.34777
$897$	0	0
$898$	−38.5676	−1.28702
$899$	−1.87203	−0.0624358
$900$	0	0
$901$	1.50711	0.0502090
$902$	55.2465	1.83951
$903$	0	0
$904$	−10.3709	−0.344931
$905$	−9.33513	−0.310310
$906$	0	0
$907$	−22.6305	−0.751435	−0.375718	−	0.926734i	$-0.622604\pi$
$907$	−22.6305	−0.751435	−0.375718	+	0.926734i	$0.622604\pi$
$908$	4.36526	0.144866
$909$	0	0
$910$	0	0
$911$	−13.8586	−0.459157	−0.229578	−	0.973290i	$-0.573735\pi$
$911$	−13.8586	−0.459157	−0.229578	+	0.973290i	$0.573735\pi$
$912$	0	0
$913$	58.9831	1.95206
$914$	−73.2223	−2.42198
$915$	0	0
$916$	−16.7138	−0.552239
$917$	−0.391339	−0.0129231
$918$	0	0
$919$	13.8086	0.455505	0.227753	−	0.973719i	$-0.426862\pi$
$919$	13.8086	0.455505	0.227753	+	0.973719i	$0.426862\pi$
$920$	−8.75840	−0.288756
$921$	0	0
$922$	52.7982	1.73882
$923$	0	0
$924$	0	0
$925$	2.93362	0.0964570
$926$	50.7982	1.66933
$927$	0	0
$928$	−1.35988	−0.0446402
$929$	−42.2210	−1.38523	−0.692613	−	0.721309i	$-0.743541\pi$
$929$	−42.2210	−1.38523	−0.692613	+	0.721309i	$0.743541\pi$
$930$	0	0
$931$	34.4618	1.12944
$932$	−20.7808	−0.680696
$933$	0	0
$934$	−50.2790	−1.64518
$935$	27.4228	0.896821
$936$	0	0
$937$	22.7681	0.743801	0.371900	−	0.928273i	$-0.378706\pi$
$937$	22.7681	0.743801	0.371900	+	0.928273i	$0.378706\pi$
$938$	58.0533	1.89551
$939$	0	0
$940$	15.1468	0.494033
$941$	56.1581	1.83070	0.915351	−	0.402657i	$-0.131913\pi$
$941$	56.1581	1.83070	0.915351	+	0.402657i	$0.131913\pi$
$942$	0	0
$943$	−48.2935	−1.57265
$944$	9.27545	0.301890
$945$	0	0
$946$	46.6499	1.51672
$947$	32.5478	1.05766	0.528830	−	0.848728i	$-0.322631\pi$
$947$	32.5478	1.05766	0.528830	+	0.848728i	$0.322631\pi$
$948$	0	0
$949$	0	0
$950$	−6.89977	−0.223858
$951$	0	0
$952$	36.3215	1.17719
$953$	−0.485335	−0.0157216	−0.00786078	−	0.999969i	$-0.502502\pi$
$953$	−0.485335	−0.0157216	−0.00786078	+	0.999969i	$0.502502\pi$
$954$	0	0
$955$	1.10992	0.0359160
$956$	31.2814	1.01171
$957$	0	0
$958$	−59.7163	−1.92935
$959$	−38.0146	−1.22756
$960$	0	0
$961$	41.5176	1.33928
$962$	0	0
$963$	0	0
$964$	12.9691	0.417708
$965$	4.61596	0.148593
$966$	0	0
$967$	−46.7875	−1.50458	−0.752292	−	0.658830i	$-0.771052\pi$
$967$	−46.7875	−1.50458	−0.752292	+	0.658830i	$0.771052\pi$
$968$	7.85948	0.252613
$969$	0	0
$970$	−6.31767	−0.202848
$971$	−16.1909	−0.519590	−0.259795	−	0.965664i	$-0.583655\pi$
$971$	−16.1909	−0.519590	−0.259795	+	0.965664i	$0.583655\pi$
$972$	0	0
$973$	33.6448	1.07860
$974$	−43.2223	−1.38493
$975$	0	0
$976$	−53.2911	−1.70581
$977$	29.3217	0.938085	0.469042	−	0.883176i	$-0.344599\pi$
$977$	29.3217	0.938085	0.469042	+	0.883176i	$0.344599\pi$
$978$	0	0
$979$	−19.7172	−0.630166
$980$	11.2228	0.358500
$981$	0	0
$982$	−43.0616	−1.37415
$983$	−40.4607	−1.29050	−0.645248	−	0.763973i	$-0.723246\pi$
$983$	−40.4607	−1.29050	−0.645248	+	0.763973i	$0.723246\pi$
$984$	0	0
$985$	0.762709	0.0243019
$986$	−2.65087	−0.0844210
$987$	0	0
$988$	0	0
$989$	−40.7788	−1.29669
$990$	0	0
$991$	−8.31527	−0.264143	−0.132072	−	0.991240i	$-0.542163\pi$
$991$	−8.31527	−0.264143	−0.132072	+	0.991240i	$0.542163\pi$
$992$	52.6781	1.67253
$993$	0	0
$994$	35.9517	1.14032
$995$	−17.8562	−0.566081
$996$	0	0
$997$	12.4310	0.393695	0.196848	−	0.980434i	$-0.436930\pi$
$997$	12.4310	0.393695	0.196848	+	0.980434i	$0.436930\pi$
$998$	31.7362	1.00459
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.bp.1.1		3
3.2	odd	2		2535.2.a.be.1.3	yes	3
13.12	even	2		7605.2.a.ca.1.3		3
39.38	odd	2		2535.2.a.x.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2535.2.a.x.1.1	✓	3	39.38	odd	2
2535.2.a.be.1.3	yes	3	3.2	odd	2
7605.2.a.bp.1.1		3	1.1	even	1	trivial
7605.2.a.ca.1.3		3	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)

\(f(q)\)	\(=\)	\(q-1.80194 q^{2} +1.24698 q^{4} +1.00000 q^{5} -4.00000 q^{7} +1.35690 q^{8} +O(q^{10})\)	\(q-1.80194 q^{2} +1.24698 q^{4} +1.00000 q^{5} -4.00000 q^{7} +1.35690 q^{8} -1.80194 q^{10} -4.09783 q^{11} +7.20775 q^{14} -4.93900 q^{16} -6.69202 q^{17} +3.82908 q^{19} +1.24698 q^{20} +7.38404 q^{22} -6.45473 q^{23} +1.00000 q^{25} -4.98792 q^{28} -0.219833 q^{29} +8.51573 q^{31} +6.18598 q^{32} +12.0586 q^{34} -4.00000 q^{35} +2.93362 q^{37} -6.89977 q^{38} +1.35690 q^{40} +7.48188 q^{41} +6.31767 q^{43} -5.10992 q^{44} +11.6310 q^{46} +12.1468 q^{47} +9.00000 q^{49} -1.80194 q^{50} -0.225209 q^{53} -4.09783 q^{55} -5.42758 q^{56} +0.396125 q^{58} -1.87800 q^{59} +10.7899 q^{61} -15.3448 q^{62} -1.26875 q^{64} +8.05429 q^{67} -8.34481 q^{68} +7.20775 q^{70} +4.98792 q^{71} -4.49396 q^{73} -5.28621 q^{74} +4.77479 q^{76} +16.3913 q^{77} -13.5646 q^{79} -4.93900 q^{80} -13.4819 q^{82} -14.3937 q^{83} -6.69202 q^{85} -11.3840 q^{86} -5.56033 q^{88} +4.81163 q^{89} -8.04892 q^{92} -21.8877 q^{94} +3.82908 q^{95} +3.50604 q^{97} -16.2174 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} - q^{4} + 3 q^{5} - 12 q^{7}+O(q^{10}) \) 3 * q - q^2 - q^4 + 3 * q^5 - 12 * q^7	\( 3 q - q^{2} - q^{4} + 3 q^{5} - 12 q^{7} - q^{10} + 6 q^{11} + 4 q^{14} - 5 q^{16} - 15 q^{17} + q^{19} - q^{20} + 12 q^{22} + 3 q^{23} + 3 q^{25} + 4 q^{28} - 2 q^{29} + 13 q^{31} + 4 q^{32} + 5 q^{34} - 12 q^{35} + 2 q^{37} + 2 q^{38} - 6 q^{41} + 2 q^{43} - 16 q^{44} + 20 q^{46} + 9 q^{47} + 27 q^{49} - q^{50} + q^{53} + 6 q^{55} + 10 q^{58} + 14 q^{59} + 9 q^{61} - 23 q^{62} + 4 q^{64} + 12 q^{67} - 2 q^{68} + 4 q^{70} - 4 q^{71} - 4 q^{73} - 24 q^{74} + 16 q^{76} - 24 q^{77} - 19 q^{79} - 5 q^{80} - 12 q^{82} - 11 q^{83} - 15 q^{85} - 24 q^{86} - 14 q^{88} - 12 q^{89} - 15 q^{92} - 24 q^{94} + q^{95} + 20 q^{97} - 9 q^{98}+O(q^{100}) \) 3 * q - q^2 - q^4 + 3 * q^5 - 12 * q^7 - q^10 + 6 * q^11 + 4 * q^14 - 5 * q^16 - 15 * q^17 + q^19 - q^20 + 12 * q^22 + 3 * q^23 + 3 * q^25 + 4 * q^28 - 2 * q^29 + 13 * q^31 + 4 * q^32 + 5 * q^34 - 12 * q^35 + 2 * q^37 + 2 * q^38 - 6 * q^41 + 2 * q^43 - 16 * q^44 + 20 * q^46 + 9 * q^47 + 27 * q^49 - q^50 + q^53 + 6 * q^55 + 10 * q^58 + 14 * q^59 + 9 * q^61 - 23 * q^62 + 4 * q^64 + 12 * q^67 - 2 * q^68 + 4 * q^70 - 4 * q^71 - 4 * q^73 - 24 * q^74 + 16 * q^76 - 24 * q^77 - 19 * q^79 - 5 * q^80 - 12 * q^82 - 11 * q^83 - 15 * q^85 - 24 * q^86 - 14 * q^88 - 12 * q^89 - 15 * q^92 - 24 * q^94 + q^95 + 20 * q^97 - 9 * q^98

Introduction

L-functions

Modular forms

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Database

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