LMFDB - Embedded newform 4600.2.a.bh.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(1,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, 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("4600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.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:	$36.7311849298$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 7x^{5} + 24x^{4} + x^{3} - 35x^{2} + 17x - 2 $ x^7 - 3x^6 - 7x^5 + 24x^4 + x^3 - 35x^2 + 17*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.78665$ of defining polynomial
Character	$\chi$	$=$	4600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.78665 q^{3} -1.75578 q^{7} +0.192116 q^{9} +O(q^{10})$	$q-1.78665 q^{3} -1.75578 q^{7} +0.192116 q^{9} -4.77574 q^{11} -1.72967 q^{13} +7.81453 q^{17} -2.43970 q^{19} +3.13696 q^{21} +1.00000 q^{23} +5.01670 q^{27} +7.86235 q^{29} +6.14217 q^{31} +8.53258 q^{33} +6.83324 q^{37} +3.09031 q^{39} -2.50416 q^{41} +3.26755 q^{43} -8.46487 q^{47} -3.91724 q^{49} -13.9618 q^{51} +2.76559 q^{53} +4.35889 q^{57} +1.91706 q^{59} -3.50364 q^{61} -0.337313 q^{63} +12.7649 q^{67} -1.78665 q^{69} -13.3973 q^{71} -0.0111737 q^{73} +8.38515 q^{77} -16.6960 q^{79} -9.53944 q^{81} +2.64145 q^{83} -14.0473 q^{87} -13.1729 q^{89} +3.03692 q^{91} -10.9739 q^{93} -15.8568 q^{97} -0.917494 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 3 q^{3} - 4 q^{7} + 2 q^{9}+O(q^{10}) $ 7 * q - 3 * q^3 - 4 * q^7 + 2 * q^9	$ 7 q - 3 q^{3} - 4 q^{7} + 2 q^{9} - 7 q^{11} + 7 q^{13} - 7 q^{19} - 6 q^{21} + 7 q^{23} - 11 q^{29} - 10 q^{31} + 19 q^{33} + 19 q^{37} - 24 q^{39} - 16 q^{41} - 6 q^{43} - 6 q^{47} - 17 q^{49} - 7 q^{51} + 15 q^{53} + 8 q^{57} - 11 q^{59} + 5 q^{61} - 13 q^{63} - 9 q^{67} - 3 q^{69} - 14 q^{71} + 10 q^{73} + 6 q^{77} - 32 q^{79} - 5 q^{81} - q^{83} - 10 q^{87} - 24 q^{89} - 7 q^{91} + 26 q^{93} - 7 q^{97} - 61 q^{99}+O(q^{100}) $ 7 * q - 3 * q^3 - 4 * q^7 + 2 * q^9 - 7 * q^11 + 7 * q^13 - 7 * q^19 - 6 * q^21 + 7 * q^23 - 11 * q^29 - 10 * q^31 + 19 * q^33 + 19 * q^37 - 24 * q^39 - 16 * q^41 - 6 * q^43 - 6 * q^47 - 17 * q^49 - 7 * q^51 + 15 * q^53 + 8 * q^57 - 11 * q^59 + 5 * q^61 - 13 * q^63 - 9 * q^67 - 3 * q^69 - 14 * q^71 + 10 * q^73 + 6 * q^77 - 32 * q^79 - 5 * q^81 - q^83 - 10 * q^87 - 24 * q^89 - 7 * q^91 + 26 * q^93 - 7 * q^97 - 61 * q^99

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	0
$2$	0	0
$3$	−1.78665	−1.03152	−0.515761	−	0.856732i	$-0.672491\pi$
$3$	−1.78665	−1.03152	−0.515761	+	0.856732i	$0.672491\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.75578	−0.663622	−0.331811	−	0.943346i	$-0.607660\pi$
$7$	−1.75578	−0.663622	−0.331811	+	0.943346i	$0.607660\pi$
$8$	0	0
$9$	0.192116	0.0640385
$10$	0	0
$11$	−4.77574	−1.43994	−0.719970	−	0.694005i	$-0.755845\pi$
$11$	−4.77574	−1.43994	−0.719970	+	0.694005i	$0.755845\pi$
$12$	0	0
$13$	−1.72967	−0.479724	−0.239862	−	0.970807i	$-0.577102\pi$
$13$	−1.72967	−0.479724	−0.239862	+	0.970807i	$0.577102\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.81453	1.89530	0.947651	−	0.319309i	$-0.103451\pi$
$17$	7.81453	1.89530	0.947651	+	0.319309i	$0.103451\pi$
$18$	0	0
$19$	−2.43970	−0.559706	−0.279853	−	0.960043i	$-0.590286\pi$
$19$	−2.43970	−0.559706	−0.279853	+	0.960043i	$0.590286\pi$
$20$	0	0
$21$	3.13696	0.684541
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.01670	0.965465
$28$	0	0
$29$	7.86235	1.46000	0.730001	−	0.683446i	$-0.239520\pi$
$29$	7.86235	1.46000	0.730001	+	0.683446i	$0.239520\pi$
$30$	0	0
$31$	6.14217	1.10317	0.551583	−	0.834120i	$-0.314024\pi$
$31$	6.14217	1.10317	0.551583	+	0.834120i	$0.314024\pi$
$32$	0	0
$33$	8.53258	1.48533
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.83324	1.12338	0.561689	−	0.827349i	$-0.310152\pi$
$37$	6.83324	1.12338	0.561689	+	0.827349i	$0.310152\pi$
$38$	0	0
$39$	3.09031	0.494846
$40$	0	0
$41$	−2.50416	−0.391084	−0.195542	−	0.980695i	$-0.562647\pi$
$41$	−2.50416	−0.391084	−0.195542	+	0.980695i	$0.562647\pi$
$42$	0	0
$43$	3.26755	0.498297	0.249148	−	0.968465i	$-0.419849\pi$
$43$	3.26755	0.498297	0.249148	+	0.968465i	$0.419849\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.46487	−1.23473	−0.617364	−	0.786678i	$-0.711799\pi$
$47$	−8.46487	−1.23473	−0.617364	+	0.786678i	$0.711799\pi$
$48$	0	0
$49$	−3.91724	−0.559605
$50$	0	0
$51$	−13.9618	−1.95505
$52$	0	0
$53$	2.76559	0.379883	0.189942	−	0.981795i	$-0.439170\pi$
$53$	2.76559	0.379883	0.189942	+	0.981795i	$0.439170\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.35889	0.577349
$58$	0	0
$59$	1.91706	0.249580	0.124790	−	0.992183i	$-0.460174\pi$
$59$	1.91706	0.249580	0.124790	+	0.992183i	$0.460174\pi$
$60$	0	0
$61$	−3.50364	−0.448595	−0.224297	−	0.974521i	$-0.572009\pi$
$61$	−3.50364	−0.448595	−0.224297	+	0.974521i	$0.572009\pi$
$62$	0	0
$63$	−0.337313	−0.0424974
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.7649	1.55948	0.779741	−	0.626102i	$-0.215351\pi$
$67$	12.7649	1.55948	0.779741	+	0.626102i	$0.215351\pi$
$68$	0	0
$69$	−1.78665	−0.215087
$70$	0	0
$71$	−13.3973	−1.58997	−0.794984	−	0.606630i	$-0.792521\pi$
$71$	−13.3973	−1.58997	−0.794984	+	0.606630i	$0.792521\pi$
$72$	0	0
$73$	−0.0111737	−0.00130778	−0.000653890	−	1.00000i	$-0.500208\pi$
$73$	−0.0111737	−0.00130778	−0.000653890		1.00000i	$0.500208\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.38515	0.955577
$78$	0	0
$79$	−16.6960	−1.87844	−0.939222	−	0.343310i	$-0.888452\pi$
$79$	−16.6960	−1.87844	−0.939222	+	0.343310i	$0.888452\pi$
$80$	0	0
$81$	−9.53944	−1.05994
$82$	0	0
$83$	2.64145	0.289937	0.144969	−	0.989436i	$-0.453692\pi$
$83$	2.64145	0.289937	0.144969	+	0.989436i	$0.453692\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−14.0473	−1.50602
$88$	0	0
$89$	−13.1729	−1.39632	−0.698160	−	0.715942i	$-0.745998\pi$
$89$	−13.1729	−1.39632	−0.698160	+	0.715942i	$0.745998\pi$
$90$	0	0
$91$	3.03692	0.318356
$92$	0	0
$93$	−10.9739	−1.13794
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−15.8568	−1.61001	−0.805005	−	0.593268i	$-0.797838\pi$
$97$	−15.8568	−1.61001	−0.805005	+	0.593268i	$0.797838\pi$
$98$	0	0
$99$	−0.917494	−0.0922117
$100$	0	0
$101$	15.6987	1.56208	0.781040	−	0.624481i	$-0.214689\pi$
$101$	15.6987	1.56208	0.781040	+	0.624481i	$0.214689\pi$
$102$	0	0
$103$	−8.37467	−0.825180	−0.412590	−	0.910917i	$-0.635376\pi$
$103$	−8.37467	−0.825180	−0.412590	+	0.910917i	$0.635376\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.52262	0.630565	0.315283	−	0.948998i	$-0.397901\pi$
$107$	6.52262	0.630565	0.315283	+	0.948998i	$0.397901\pi$
$108$	0	0
$109$	−8.11474	−0.777251	−0.388626	−	0.921396i	$-0.627050\pi$
$109$	−8.11474	−0.777251	−0.388626	+	0.921396i	$0.627050\pi$
$110$	0	0
$111$	−12.2086	−1.15879
$112$	0	0
$113$	12.9224	1.21563	0.607817	−	0.794077i	$-0.292045\pi$
$113$	12.9224	1.21563	0.607817	+	0.794077i	$0.292045\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.332296	−0.0307208
$118$	0	0
$119$	−13.7206	−1.25776
$120$	0	0
$121$	11.8077	1.07343
$122$	0	0
$123$	4.47406	0.403412
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−7.05406	−0.625947	−0.312973	−	0.949762i	$-0.601325\pi$
$127$	−7.05406	−0.625947	−0.312973	+	0.949762i	$0.601325\pi$
$128$	0	0
$129$	−5.83797	−0.514004
$130$	0	0
$131$	−11.1585	−0.974923	−0.487461	−	0.873145i	$-0.662077\pi$
$131$	−11.1585	−0.974923	−0.487461	+	0.873145i	$0.662077\pi$
$132$	0	0
$133$	4.28358	0.371434
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.71462	−0.231926	−0.115963	−	0.993254i	$-0.536995\pi$
$137$	−2.71462	−0.231926	−0.115963	+	0.993254i	$0.536995\pi$
$138$	0	0
$139$	13.0913	1.11039	0.555197	−	0.831719i	$-0.312643\pi$
$139$	13.0913	1.11039	0.555197	+	0.831719i	$0.312643\pi$
$140$	0	0
$141$	15.1238	1.27365
$142$	0	0
$143$	8.26046	0.690774
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.99873	0.577245
$148$	0	0
$149$	−12.0898	−0.990435	−0.495217	−	0.868769i	$-0.664912\pi$
$149$	−12.0898	−0.990435	−0.495217	+	0.868769i	$0.664912\pi$
$150$	0	0
$151$	14.9430	1.21604	0.608021	−	0.793921i	$-0.291964\pi$
$151$	14.9430	1.21604	0.608021	+	0.793921i	$0.291964\pi$
$152$	0	0
$153$	1.50129	0.121372
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−4.78395	−0.381800	−0.190900	−	0.981609i	$-0.561141\pi$
$157$	−4.78395	−0.381800	−0.190900	+	0.981609i	$0.561141\pi$
$158$	0	0
$159$	−4.94114	−0.391858
$160$	0	0
$161$	−1.75578	−0.138375
$162$	0	0
$163$	−5.49063	−0.430059	−0.215030	−	0.976608i	$-0.568985\pi$
$163$	−5.49063	−0.430059	−0.215030	+	0.976608i	$0.568985\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.26932	0.562517	0.281258	−	0.959632i	$-0.409248\pi$
$167$	7.26932	0.562517	0.281258	+	0.959632i	$0.409248\pi$
$168$	0	0
$169$	−10.0082	−0.769865
$170$	0	0
$171$	−0.468705	−0.0358427
$172$	0	0
$173$	7.50892	0.570893	0.285446	−	0.958395i	$-0.407858\pi$
$173$	7.50892	0.570893	0.285446	+	0.958395i	$0.407858\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3.42511	−0.257447
$178$	0	0
$179$	−5.73414	−0.428590	−0.214295	−	0.976769i	$-0.568745\pi$
$179$	−5.73414	−0.428590	−0.214295	+	0.976769i	$0.568745\pi$
$180$	0	0
$181$	25.2262	1.87505	0.937523	−	0.347923i	$-0.113113\pi$
$181$	25.2262	1.87505	0.937523	+	0.347923i	$0.113113\pi$
$182$	0	0
$183$	6.25977	0.462736
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−37.3202	−2.72912
$188$	0	0
$189$	−8.80823	−0.640704
$190$	0	0
$191$	3.63151	0.262767	0.131383	−	0.991332i	$-0.458058\pi$
$191$	3.63151	0.262767	0.131383	+	0.991332i	$0.458058\pi$
$192$	0	0
$193$	−21.7050	−1.56236	−0.781179	−	0.624307i	$-0.785381\pi$
$193$	−21.7050	−1.56236	−0.781179	+	0.624307i	$0.785381\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−7.26526	−0.517628	−0.258814	−	0.965927i	$-0.583332\pi$
$197$	−7.26526	−0.517628	−0.258814	+	0.965927i	$0.583332\pi$
$198$	0	0
$199$	6.52705	0.462690	0.231345	−	0.972872i	$-0.425687\pi$
$199$	6.52705	0.462690	0.231345	+	0.972872i	$0.425687\pi$
$200$	0	0
$201$	−22.8064	−1.60864
$202$	0	0
$203$	−13.8046	−0.968890
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.192116	0.0133530
$208$	0	0
$209$	11.6514	0.805944
$210$	0	0
$211$	−22.7856	−1.56863	−0.784313	−	0.620365i	$-0.786984\pi$
$211$	−22.7856	−1.56863	−0.784313	+	0.620365i	$0.786984\pi$
$212$	0	0
$213$	23.9363	1.64009
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−10.7843	−0.732086
$218$	0	0
$219$	0.0199634	0.00134900
$220$	0	0
$221$	−13.5165	−0.909221
$222$	0	0
$223$	−6.86315	−0.459591	−0.229795	−	0.973239i	$-0.573806\pi$
$223$	−6.86315	−0.459591	−0.229795	+	0.973239i	$0.573806\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.5001	−1.16152	−0.580761	−	0.814074i	$-0.697245\pi$
$227$	−17.5001	−1.16152	−0.580761	+	0.814074i	$0.697245\pi$
$228$	0	0
$229$	−23.8643	−1.57700	−0.788498	−	0.615038i	$-0.789141\pi$
$229$	−23.8643	−1.57700	−0.788498	+	0.615038i	$0.789141\pi$
$230$	0	0
$231$	−14.9813	−0.985699
$232$	0	0
$233$	21.8957	1.43444	0.717218	−	0.696849i	$-0.245415\pi$
$233$	21.8957	1.43444	0.717218	+	0.696849i	$0.245415\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	29.8299	1.93766
$238$	0	0
$239$	−19.3443	−1.25128	−0.625639	−	0.780113i	$-0.715162\pi$
$239$	−19.3443	−1.25128	−0.625639	+	0.780113i	$0.715162\pi$
$240$	0	0
$241$	11.4761	0.739240	0.369620	−	0.929183i	$-0.379488\pi$
$241$	11.4761	0.739240	0.369620	+	0.929183i	$0.379488\pi$
$242$	0	0
$243$	1.99352	0.127884
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.21988	0.268504
$248$	0	0
$249$	−4.71935	−0.299077
$250$	0	0
$251$	17.0040	1.07328	0.536641	−	0.843811i	$-0.319693\pi$
$251$	17.0040	1.07328	0.536641	+	0.843811i	$0.319693\pi$
$252$	0	0
$253$	−4.77574	−0.300248
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.6462	0.913607	0.456803	−	0.889568i	$-0.348994\pi$
$257$	14.6462	0.913607	0.456803	+	0.889568i	$0.348994\pi$
$258$	0	0
$259$	−11.9977	−0.745499
$260$	0	0
$261$	1.51048	0.0934964
$262$	0	0
$263$	−10.2641	−0.632912	−0.316456	−	0.948607i	$-0.602493\pi$
$263$	−10.2641	−0.632912	−0.316456	+	0.948607i	$0.602493\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	23.5353	1.44034
$268$	0	0
$269$	−2.61485	−0.159430	−0.0797150	−	0.996818i	$-0.525401\pi$
$269$	−2.61485	−0.159430	−0.0797150	+	0.996818i	$0.525401\pi$
$270$	0	0
$271$	9.87492	0.599859	0.299929	−	0.953961i	$-0.403037\pi$
$271$	9.87492	0.599859	0.299929	+	0.953961i	$0.403037\pi$
$272$	0	0
$273$	−5.42591	−0.328391
$274$	0	0
$275$	0	0
$276$	0	0
$277$	4.21949	0.253525	0.126762	−	0.991933i	$-0.459541\pi$
$277$	4.21949	0.253525	0.126762	+	0.991933i	$0.459541\pi$
$278$	0	0
$279$	1.18001	0.0706451
$280$	0	0
$281$	−13.4058	−0.799724	−0.399862	−	0.916575i	$-0.630942\pi$
$281$	−13.4058	−0.799724	−0.399862	+	0.916575i	$0.630942\pi$
$282$	0	0
$283$	5.66084	0.336502	0.168251	−	0.985744i	$-0.446188\pi$
$283$	5.66084	0.336502	0.168251	+	0.985744i	$0.446188\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.39676	0.259532
$288$	0	0
$289$	44.0668	2.59217
$290$	0	0
$291$	28.3305	1.66076
$292$	0	0
$293$	−5.70404	−0.333233	−0.166617	−	0.986022i	$-0.553284\pi$
$293$	−5.70404	−0.333233	−0.166617	+	0.986022i	$0.553284\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−23.9585	−1.39021
$298$	0	0
$299$	−1.72967	−0.100029
$300$	0	0
$301$	−5.73710	−0.330681
$302$	0	0
$303$	−28.0481	−1.61132
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−14.8943	−0.850065	−0.425033	−	0.905178i	$-0.639737\pi$
$307$	−14.8943	−0.850065	−0.425033	+	0.905178i	$0.639737\pi$
$308$	0	0
$309$	14.9626	0.851192
$310$	0	0
$311$	−10.0355	−0.569059	−0.284530	−	0.958667i	$-0.591837\pi$
$311$	−10.0355	−0.569059	−0.284530	+	0.958667i	$0.591837\pi$
$312$	0	0
$313$	23.5043	1.32854	0.664272	−	0.747491i	$-0.268742\pi$
$313$	23.5043	1.32854	0.664272	+	0.747491i	$0.268742\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.94093	0.446007	0.223004	−	0.974818i	$-0.428414\pi$
$317$	7.94093	0.446007	0.223004	+	0.974818i	$0.428414\pi$
$318$	0	0
$319$	−37.5486	−2.10232
$320$	0	0
$321$	−11.6536	−0.650442
$322$	0	0
$323$	−19.0651	−1.06081
$324$	0	0
$325$	0	0
$326$	0	0
$327$	14.4982	0.801752
$328$	0	0
$329$	14.8625	0.819393
$330$	0	0
$331$	6.31442	0.347072	0.173536	−	0.984828i	$-0.444481\pi$
$331$	6.31442	0.347072	0.173536	+	0.984828i	$0.444481\pi$
$332$	0	0
$333$	1.31277	0.0719394
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−23.4516	−1.27749	−0.638746	−	0.769418i	$-0.720547\pi$
$337$	−23.4516	−1.27749	−0.638746	+	0.769418i	$0.720547\pi$
$338$	0	0
$339$	−23.0877	−1.25395
$340$	0	0
$341$	−29.3334	−1.58849
$342$	0	0
$343$	19.1683	1.03499
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.4757	0.830780	0.415390	−	0.909643i	$-0.363645\pi$
$347$	15.4757	0.830780	0.415390	+	0.909643i	$0.363645\pi$
$348$	0	0
$349$	−25.5514	−1.36773	−0.683867	−	0.729607i	$-0.739703\pi$
$349$	−25.5514	−1.36773	−0.683867	+	0.729607i	$0.739703\pi$
$350$	0	0
$351$	−8.67724	−0.463157
$352$	0	0
$353$	−23.5917	−1.25566	−0.627829	−	0.778352i	$-0.716056\pi$
$353$	−23.5917	−1.25566	−0.627829	+	0.778352i	$0.716056\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	24.5139	1.29741
$358$	0	0
$359$	3.14840	0.166166	0.0830832	−	0.996543i	$-0.473523\pi$
$359$	3.14840	0.166166	0.0830832	+	0.996543i	$0.473523\pi$
$360$	0	0
$361$	−13.0479	−0.686729
$362$	0	0
$363$	−21.0963	−1.10727
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−32.4297	−1.69282	−0.846408	−	0.532534i	$-0.821240\pi$
$367$	−32.4297	−1.69282	−0.846408	+	0.532534i	$0.821240\pi$
$368$	0	0
$369$	−0.481088	−0.0250445
$370$	0	0
$371$	−4.85577	−0.252099
$372$	0	0
$373$	7.99302	0.413863	0.206931	−	0.978355i	$-0.433652\pi$
$373$	7.99302	0.413863	0.206931	+	0.978355i	$0.433652\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−13.5993	−0.700398
$378$	0	0
$379$	8.96061	0.460275	0.230138	−	0.973158i	$-0.426082\pi$
$379$	8.96061	0.460275	0.230138	+	0.973158i	$0.426082\pi$
$380$	0	0
$381$	12.6031	0.645678
$382$	0	0
$383$	−36.7195	−1.87628	−0.938141	−	0.346255i	$-0.887453\pi$
$383$	−36.7195	−1.87628	−0.938141	+	0.346255i	$0.887453\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.627747	0.0319102
$388$	0	0
$389$	0.0627294	0.00318051	0.00159025	−	0.999999i	$-0.499494\pi$
$389$	0.0627294	0.00318051	0.00159025	+	0.999999i	$0.499494\pi$
$390$	0	0
$391$	7.81453	0.395198
$392$	0	0
$393$	19.9363	1.00565
$394$	0	0
$395$	0	0
$396$	0	0
$397$	20.9714	1.05252	0.526262	−	0.850323i	$-0.323593\pi$
$397$	20.9714	1.05252	0.526262	+	0.850323i	$0.323593\pi$
$398$	0	0
$399$	−7.65326	−0.383142
$400$	0	0
$401$	−13.8567	−0.691972	−0.345986	−	0.938240i	$-0.612456\pi$
$401$	−13.8567	−0.691972	−0.345986	+	0.938240i	$0.612456\pi$
$402$	0	0
$403$	−10.6239	−0.529215
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−32.6338	−1.61760
$408$	0	0
$409$	12.5868	0.622378	0.311189	−	0.950348i	$-0.399273\pi$
$409$	12.5868	0.622378	0.311189	+	0.950348i	$0.399273\pi$
$410$	0	0
$411$	4.85007	0.239236
$412$	0	0
$413$	−3.36593	−0.165627
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−23.3896	−1.14540
$418$	0	0
$419$	−32.4927	−1.58737	−0.793685	−	0.608328i	$-0.791840\pi$
$419$	−32.4927	−1.58737	−0.793685	+	0.608328i	$0.791840\pi$
$420$	0	0
$421$	18.8356	0.917992	0.458996	−	0.888438i	$-0.348209\pi$
$421$	18.8356	0.917992	0.458996	+	0.888438i	$0.348209\pi$
$422$	0	0
$423$	−1.62623	−0.0790702
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.15162	0.297698
$428$	0	0
$429$	−14.7585	−0.712549
$430$	0	0
$431$	1.66739	0.0803155	0.0401577	−	0.999193i	$-0.487214\pi$
$431$	1.66739	0.0803155	0.0401577	+	0.999193i	$0.487214\pi$
$432$	0	0
$433$	5.83595	0.280458	0.140229	−	0.990119i	$-0.455216\pi$
$433$	5.83595	0.280458	0.140229	+	0.990119i	$0.455216\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.43970	−0.116707
$438$	0	0
$439$	30.5742	1.45923	0.729614	−	0.683860i	$-0.239700\pi$
$439$	30.5742	1.45923	0.729614	+	0.683860i	$0.239700\pi$
$440$	0	0
$441$	−0.752562	−0.0358363
$442$	0	0
$443$	−34.9173	−1.65897	−0.829485	−	0.558530i	$-0.811366\pi$
$443$	−34.9173	−1.65897	−0.829485	+	0.558530i	$0.811366\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	21.6002	1.02166
$448$	0	0
$449$	−33.1265	−1.56334	−0.781668	−	0.623694i	$-0.785631\pi$
$449$	−33.1265	−1.56334	−0.781668	+	0.623694i	$0.785631\pi$
$450$	0	0
$451$	11.9592	0.563138
$452$	0	0
$453$	−26.6978	−1.25437
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.27252	−0.199860	−0.0999299	−	0.994994i	$-0.531862\pi$
$457$	−4.27252	−0.199860	−0.0999299	+	0.994994i	$0.531862\pi$
$458$	0	0
$459$	39.2032	1.82985
$460$	0	0
$461$	−38.4702	−1.79173	−0.895867	−	0.444322i	$-0.853444\pi$
$461$	−38.4702	−1.79173	−0.895867	+	0.444322i	$0.853444\pi$
$462$	0	0
$463$	8.94727	0.415815	0.207908	−	0.978148i	$-0.433335\pi$
$463$	8.94727	0.415815	0.207908	+	0.978148i	$0.433335\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	16.3171	0.755064	0.377532	−	0.925996i	$-0.376773\pi$
$467$	16.3171	0.755064	0.377532	+	0.925996i	$0.376773\pi$
$468$	0	0
$469$	−22.4124	−1.03491
$470$	0	0
$471$	8.54723	0.393836
$472$	0	0
$473$	−15.6050	−0.717518
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.531313	0.0243272
$478$	0	0
$479$	11.9574	0.546349	0.273174	−	0.961965i	$-0.411926\pi$
$479$	11.9574	0.546349	0.273174	+	0.961965i	$0.411926\pi$
$480$	0	0
$481$	−11.8192	−0.538911
$482$	0	0
$483$	3.13696	0.142737
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−1.19495	−0.0541484	−0.0270742	−	0.999633i	$-0.508619\pi$
$487$	−1.19495	−0.0541484	−0.0270742	+	0.999633i	$0.508619\pi$
$488$	0	0
$489$	9.80982	0.443616
$490$	0	0
$491$	−13.6912	−0.617874	−0.308937	−	0.951083i	$-0.599973\pi$
$491$	−13.6912	−0.617874	−0.308937	+	0.951083i	$0.599973\pi$
$492$	0	0
$493$	61.4406	2.76714
$494$	0	0
$495$	0	0
$496$	0	0
$497$	23.5227	1.05514
$498$	0	0
$499$	11.9774	0.536184	0.268092	−	0.963393i	$-0.413607\pi$
$499$	11.9774	0.536184	0.268092	+	0.963393i	$0.413607\pi$
$500$	0	0
$501$	−12.9877	−0.580249
$502$	0	0
$503$	−32.3666	−1.44316	−0.721578	−	0.692333i	$-0.756583\pi$
$503$	−32.3666	−1.44316	−0.721578	+	0.692333i	$0.756583\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	17.8812	0.794133
$508$	0	0
$509$	15.6507	0.693706	0.346853	−	0.937919i	$-0.387250\pi$
$509$	15.6507	0.693706	0.346853	+	0.937919i	$0.387250\pi$
$510$	0	0
$511$	0.0196185	0.000867872 0
$512$	0	0
$513$	−12.2393	−0.540377
$514$	0	0
$515$	0	0
$516$	0	0
$517$	40.4261	1.77794
$518$	0	0
$519$	−13.4158	−0.588889
$520$	0	0
$521$	21.7184	0.951499	0.475749	−	0.879581i	$-0.342177\pi$
$521$	21.7184	0.951499	0.475749	+	0.879581i	$0.342177\pi$
$522$	0	0
$523$	−8.89014	−0.388739	−0.194369	−	0.980928i	$-0.562266\pi$
$523$	−8.89014	−0.388739	−0.194369	+	0.980928i	$0.562266\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	47.9981	2.09083
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0.368297	0.0159827
$532$	0	0
$533$	4.33137	0.187613
$534$	0	0
$535$	0	0
$536$	0	0
$537$	10.2449	0.442100
$538$	0	0
$539$	18.7077	0.805798
$540$	0	0
$541$	−29.0112	−1.24729	−0.623645	−	0.781708i	$-0.714349\pi$
$541$	−29.0112	−1.24729	−0.623645	+	0.781708i	$0.714349\pi$
$542$	0	0
$543$	−45.0703	−1.93415
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−24.9089	−1.06503	−0.532514	−	0.846421i	$-0.678753\pi$
$547$	−24.9089	−1.06503	−0.532514	+	0.846421i	$0.678753\pi$
$548$	0	0
$549$	−0.673103	−0.0287273
$550$	0	0
$551$	−19.1818	−0.817172
$552$	0	0
$553$	29.3145	1.24658
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−15.3536	−0.650552	−0.325276	−	0.945619i	$-0.605457\pi$
$557$	−15.3536	−0.650552	−0.325276	+	0.945619i	$0.605457\pi$
$558$	0	0
$559$	−5.65178	−0.239045
$560$	0	0
$561$	66.6781	2.81515
$562$	0	0
$563$	16.8264	0.709146	0.354573	−	0.935028i	$-0.384626\pi$
$563$	16.8264	0.709146	0.354573	+	0.935028i	$0.384626\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	16.7492	0.703398
$568$	0	0
$569$	1.75646	0.0736344	0.0368172	−	0.999322i	$-0.488278\pi$
$569$	1.75646	0.0736344	0.0368172	+	0.999322i	$0.488278\pi$
$570$	0	0
$571$	13.4273	0.561913	0.280956	−	0.959721i	$-0.409348\pi$
$571$	13.4273	0.561913	0.280956	+	0.959721i	$0.409348\pi$
$572$	0	0
$573$	−6.48823	−0.271050
$574$	0	0
$575$	0	0
$576$	0	0
$577$	20.4806	0.852620	0.426310	−	0.904577i	$-0.359813\pi$
$577$	20.4806	0.852620	0.426310	+	0.904577i	$0.359813\pi$
$578$	0	0
$579$	38.7792	1.61161
$580$	0	0
$581$	−4.63781	−0.192409
$582$	0	0
$583$	−13.2078	−0.547010
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−45.3569	−1.87208	−0.936040	−	0.351893i	$-0.885538\pi$
$587$	−45.3569	−1.87208	−0.936040	+	0.351893i	$0.885538\pi$
$588$	0	0
$589$	−14.9851	−0.617449
$590$	0	0
$591$	12.9805	0.533945
$592$	0	0
$593$	28.8520	1.18481	0.592404	−	0.805641i	$-0.298179\pi$
$593$	28.8520	1.18481	0.592404	+	0.805641i	$0.298179\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−11.6615	−0.477275
$598$	0	0
$599$	8.78767	0.359054	0.179527	−	0.983753i	$-0.442543\pi$
$599$	8.78767	0.359054	0.179527	+	0.983753i	$0.442543\pi$
$600$	0	0
$601$	−44.7810	−1.82665	−0.913327	−	0.407227i	$-0.866496\pi$
$601$	−44.7810	−1.82665	−0.913327	+	0.407227i	$0.866496\pi$
$602$	0	0
$603$	2.45234	0.0998669
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−6.96387	−0.282655	−0.141327	−	0.989963i	$-0.545137\pi$
$607$	−6.96387	−0.282655	−0.141327	+	0.989963i	$0.545137\pi$
$608$	0	0
$609$	24.6639	0.999432
$610$	0	0
$611$	14.6414	0.592329
$612$	0	0
$613$	−30.6214	−1.23679	−0.618393	−	0.785869i	$-0.712216\pi$
$613$	−30.6214	−1.23679	−0.618393	+	0.785869i	$0.712216\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−10.6041	−0.426905	−0.213452	−	0.976953i	$-0.568471\pi$
$617$	−10.6041	−0.426905	−0.213452	+	0.976953i	$0.568471\pi$
$618$	0	0
$619$	21.0095	0.844445	0.422223	−	0.906492i	$-0.361250\pi$
$619$	21.0095	0.844445	0.422223	+	0.906492i	$0.361250\pi$
$620$	0	0
$621$	5.01670	0.201313
$622$	0	0
$623$	23.1286	0.926629
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−20.8170	−0.831349
$628$	0	0
$629$	53.3985	2.12914
$630$	0	0
$631$	5.45489	0.217156	0.108578	−	0.994088i	$-0.465370\pi$
$631$	5.45489	0.217156	0.108578	+	0.994088i	$0.465370\pi$
$632$	0	0
$633$	40.7099	1.61807
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.77552	0.268456
$638$	0	0
$639$	−2.57383	−0.101819
$640$	0	0
$641$	−28.7734	−1.13648	−0.568241	−	0.822862i	$-0.692376\pi$
$641$	−28.7734	−1.13648	−0.568241	+	0.822862i	$0.692376\pi$
$642$	0	0
$643$	11.5319	0.454775	0.227388	−	0.973804i	$-0.426982\pi$
$643$	11.5319	0.454775	0.227388	+	0.973804i	$0.426982\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.18564	0.125240	0.0626202	−	0.998037i	$-0.480054\pi$
$647$	3.18564	0.125240	0.0626202	+	0.998037i	$0.480054\pi$
$648$	0	0
$649$	−9.15538	−0.359380
$650$	0	0
$651$	19.2678	0.755163
$652$	0	0
$653$	28.6814	1.12239	0.561194	−	0.827684i	$-0.310342\pi$
$653$	28.6814	1.12239	0.561194	+	0.827684i	$0.310342\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−0.00214664	−8.37483e−5 0
$658$	0	0
$659$	24.0012	0.934954	0.467477	−	0.884005i	$-0.345163\pi$
$659$	24.0012	0.934954	0.467477	+	0.884005i	$0.345163\pi$
$660$	0	0
$661$	31.9043	1.24093	0.620467	−	0.784232i	$-0.286943\pi$
$661$	31.9043	1.24093	0.620467	+	0.784232i	$0.286943\pi$
$662$	0	0
$663$	24.1493	0.937882
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.86235	0.304431
$668$	0	0
$669$	12.2620	0.474078
$670$	0	0
$671$	16.7325	0.645950
$672$	0	0
$673$	1.87848	0.0724101	0.0362051	−	0.999344i	$-0.488473\pi$
$673$	1.87848	0.0724101	0.0362051	+	0.999344i	$0.488473\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−34.6119	−1.33024	−0.665121	−	0.746736i	$-0.731620\pi$
$677$	−34.6119	−1.33024	−0.665121	+	0.746736i	$0.731620\pi$
$678$	0	0
$679$	27.8410	1.06844
$680$	0	0
$681$	31.2665	1.19814
$682$	0	0
$683$	16.0493	0.614110	0.307055	−	0.951692i	$-0.400656\pi$
$683$	16.0493	0.614110	0.307055	+	0.951692i	$0.400656\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	42.6371	1.62671
$688$	0	0
$689$	−4.78356	−0.182239
$690$	0	0
$691$	−19.5985	−0.745564	−0.372782	−	0.927919i	$-0.621596\pi$
$691$	−19.5985	−0.745564	−0.372782	+	0.927919i	$0.621596\pi$
$692$	0	0
$693$	1.61092	0.0611937
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−19.5688	−0.741223
$698$	0	0
$699$	−39.1199	−1.47965
$700$	0	0
$701$	−51.5536	−1.94715	−0.973575	−	0.228366i	$-0.926662\pi$
$701$	−51.5536	−1.94715	−0.973575	+	0.228366i	$0.926662\pi$
$702$	0	0
$703$	−16.6711	−0.628761
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−27.5635	−1.03663
$708$	0	0
$709$	8.64772	0.324772	0.162386	−	0.986727i	$-0.448081\pi$
$709$	8.64772	0.324772	0.162386	+	0.986727i	$0.448081\pi$
$710$	0	0
$711$	−3.20756	−0.120293
$712$	0	0
$713$	6.14217	0.230026
$714$	0	0
$715$	0	0
$716$	0	0
$717$	34.5615	1.29072
$718$	0	0
$719$	−7.50754	−0.279984	−0.139992	−	0.990153i	$-0.544708\pi$
$719$	−7.50754	−0.279984	−0.139992	+	0.990153i	$0.544708\pi$
$720$	0	0
$721$	14.7041	0.547608
$722$	0	0
$723$	−20.5037	−0.762543
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−2.79852	−0.103791	−0.0518956	−	0.998653i	$-0.516526\pi$
$727$	−2.79852	−0.103791	−0.0518956	+	0.998653i	$0.516526\pi$
$728$	0	0
$729$	25.0566	0.928022
$730$	0	0
$731$	25.5344	0.944423
$732$	0	0
$733$	1.38900	0.0513038	0.0256519	−	0.999671i	$-0.491834\pi$
$733$	1.38900	0.0513038	0.0256519	+	0.999671i	$0.491834\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−60.9619	−2.24556
$738$	0	0
$739$	6.31935	0.232461	0.116230	−	0.993222i	$-0.462919\pi$
$739$	6.31935	0.232461	0.116230	+	0.993222i	$0.462919\pi$
$740$	0	0
$741$	−7.53944	−0.276968
$742$	0	0
$743$	39.7611	1.45869	0.729347	−	0.684144i	$-0.239824\pi$
$743$	39.7611	1.45869	0.729347	+	0.684144i	$0.239824\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.507464	0.0185671
$748$	0	0
$749$	−11.4523	−0.418457
$750$	0	0
$751$	7.27267	0.265384	0.132692	−	0.991157i	$-0.457638\pi$
$751$	7.27267	0.265384	0.132692	+	0.991157i	$0.457638\pi$
$752$	0	0
$753$	−30.3801	−1.10711
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−15.8682	−0.576739	−0.288369	−	0.957519i	$-0.593113\pi$
$757$	−15.8682	−0.576739	−0.288369	+	0.957519i	$0.593113\pi$
$758$	0	0
$759$	8.53258	0.309713
$760$	0	0
$761$	−4.80457	−0.174166	−0.0870828	−	0.996201i	$-0.527754\pi$
$761$	−4.80457	−0.174166	−0.0870828	+	0.996201i	$0.527754\pi$
$762$	0	0
$763$	14.2477	0.515801
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.31588	−0.119729
$768$	0	0
$769$	30.1358	1.08673	0.543363	−	0.839498i	$-0.317151\pi$
$769$	30.1358	1.08673	0.543363	+	0.839498i	$0.317151\pi$
$770$	0	0
$771$	−26.1677	−0.942406
$772$	0	0
$773$	−3.35253	−0.120582	−0.0602910	−	0.998181i	$-0.519203\pi$
$773$	−3.35253	−0.120582	−0.0602910	+	0.998181i	$0.519203\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	21.4356	0.768999
$778$	0	0
$779$	6.10941	0.218892
$780$	0	0
$781$	63.9821	2.28946
$782$	0	0
$783$	39.4431	1.40958
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−2.36807	−0.0844126	−0.0422063	−	0.999109i	$-0.513439\pi$
$787$	−2.36807	−0.0844126	−0.0422063	+	0.999109i	$0.513439\pi$
$788$	0	0
$789$	18.3384	0.652863
$790$	0	0
$791$	−22.6888	−0.806722
$792$	0	0
$793$	6.06013	0.215202
$794$	0	0
$795$	0	0
$796$	0	0
$797$	33.0367	1.17022	0.585110	−	0.810954i	$-0.301051\pi$
$797$	33.0367	1.17022	0.585110	+	0.810954i	$0.301051\pi$
$798$	0	0
$799$	−66.1490	−2.34018
$800$	0	0
$801$	−2.53071	−0.0894183
$802$	0	0
$803$	0.0533626	0.00188313
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.67181	0.164456
$808$	0	0
$809$	−16.7157	−0.587692	−0.293846	−	0.955853i	$-0.594935\pi$
$809$	−16.7157	−0.587692	−0.293846	+	0.955853i	$0.594935\pi$
$810$	0	0
$811$	−5.85752	−0.205685	−0.102843	−	0.994698i	$-0.532794\pi$
$811$	−5.85752	−0.205685	−0.102843	+	0.994698i	$0.532794\pi$
$812$	0	0
$813$	−17.6430	−0.618768
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−7.97185	−0.278900
$818$	0	0
$819$	0.583439	0.0203870
$820$	0	0
$821$	26.3966	0.921249	0.460625	−	0.887595i	$-0.347625\pi$
$821$	26.3966	0.921249	0.460625	+	0.887595i	$0.347625\pi$
$822$	0	0
$823$	25.5803	0.891672	0.445836	−	0.895115i	$-0.352906\pi$
$823$	25.5803	0.891672	0.445836	+	0.895115i	$0.352906\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−9.74132	−0.338739	−0.169369	−	0.985553i	$-0.554173\pi$
$827$	−9.74132	−0.338739	−0.169369	+	0.985553i	$0.554173\pi$
$828$	0	0
$829$	−54.7679	−1.90217	−0.951085	−	0.308931i	$-0.900029\pi$
$829$	−54.7679	−1.90217	−0.951085	+	0.308931i	$0.900029\pi$
$830$	0	0
$831$	−7.53875	−0.261516
$832$	0	0
$833$	−30.6114	−1.06062
$834$	0	0
$835$	0	0
$836$	0	0
$837$	30.8134	1.06507
$838$	0	0
$839$	−51.2171	−1.76821	−0.884106	−	0.467286i	$-0.845232\pi$
$839$	−51.2171	−1.76821	−0.884106	+	0.467286i	$0.845232\pi$
$840$	0	0
$841$	32.8166	1.13161
$842$	0	0
$843$	23.9515	0.824933
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−20.7318	−0.712352
$848$	0	0
$849$	−10.1139	−0.347109
$850$	0	0
$851$	6.83324	0.234240
$852$	0	0
$853$	23.2520	0.796135	0.398067	−	0.917356i	$-0.369681\pi$
$853$	23.2520	0.796135	0.398067	+	0.917356i	$0.369681\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−48.8722	−1.66944	−0.834721	−	0.550674i	$-0.814371\pi$
$857$	−48.8722	−1.66944	−0.834721	+	0.550674i	$0.814371\pi$
$858$	0	0
$859$	−12.7171	−0.433903	−0.216952	−	0.976182i	$-0.569611\pi$
$859$	−12.7171	−0.433903	−0.216952	+	0.976182i	$0.569611\pi$
$860$	0	0
$861$	−7.85546	−0.267713
$862$	0	0
$863$	13.5511	0.461285	0.230642	−	0.973039i	$-0.425917\pi$
$863$	13.5511	0.461285	0.230642	+	0.973039i	$0.425917\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−78.7320	−2.67388
$868$	0	0
$869$	79.7357	2.70485
$870$	0	0
$871$	−22.0791	−0.748121
$872$	0	0
$873$	−3.04633	−0.103103
$874$	0	0
$875$	0	0
$876$	0	0
$877$	4.88341	0.164901	0.0824505	−	0.996595i	$-0.473725\pi$
$877$	4.88341	0.164901	0.0824505	+	0.996595i	$0.473725\pi$
$878$	0	0
$879$	10.1911	0.343738
$880$	0	0
$881$	−22.5717	−0.760461	−0.380230	−	0.924892i	$-0.624155\pi$
$881$	−22.5717	−0.760461	−0.380230	+	0.924892i	$0.624155\pi$
$882$	0	0
$883$	−0.202906	−0.00682834	−0.00341417	−	0.999994i	$-0.501087\pi$
$883$	−0.202906	−0.00682834	−0.00341417	+	0.999994i	$0.501087\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−19.1859	−0.644201	−0.322100	−	0.946705i	$-0.604389\pi$
$887$	−19.1859	−0.644201	−0.322100	+	0.946705i	$0.604389\pi$
$888$	0	0
$889$	12.3854	0.415392
$890$	0	0
$891$	45.5579	1.52625
$892$	0	0
$893$	20.6518	0.691085
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.09031	0.103183
$898$	0	0
$899$	48.2919	1.61062
$900$	0	0
$901$	21.6118	0.719993
$902$	0	0
$903$	10.2502	0.341105
$904$	0	0
$905$	0	0
$906$	0	0
$907$	12.9713	0.430704	0.215352	−	0.976536i	$-0.430910\pi$
$907$	12.9713	0.430704	0.215352	+	0.976536i	$0.430910\pi$
$908$	0	0
$909$	3.01597	0.100033
$910$	0	0
$911$	31.3377	1.03826	0.519132	−	0.854694i	$-0.326255\pi$
$911$	31.3377	1.03826	0.519132	+	0.854694i	$0.326255\pi$
$912$	0	0
$913$	−12.6149	−0.417492
$914$	0	0
$915$	0	0
$916$	0	0
$917$	19.5919	0.646980
$918$	0	0
$919$	38.4577	1.26860	0.634302	−	0.773085i	$-0.281288\pi$
$919$	38.4577	1.26860	0.634302	+	0.773085i	$0.281288\pi$
$920$	0	0
$921$	26.6110	0.876861
$922$	0	0
$923$	23.1729	0.762746
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.60890	−0.0528433
$928$	0	0
$929$	−54.3396	−1.78282	−0.891412	−	0.453193i	$-0.850285\pi$
$929$	−54.3396	−1.78282	−0.891412	+	0.453193i	$0.850285\pi$
$930$	0	0
$931$	9.55689	0.313215
$932$	0	0
$933$	17.9299	0.586998
$934$	0	0
$935$	0	0
$936$	0	0
$937$	43.1182	1.40861	0.704304	−	0.709898i	$-0.251259\pi$
$937$	43.1182	1.40861	0.704304	+	0.709898i	$0.251259\pi$
$938$	0	0
$939$	−41.9940	−1.37042
$940$	0	0
$941$	9.22344	0.300676	0.150338	−	0.988635i	$-0.451964\pi$
$941$	9.22344	0.300676	0.150338	+	0.988635i	$0.451964\pi$
$942$	0	0
$943$	−2.50416	−0.0815467
$944$	0	0
$945$	0	0
$946$	0	0
$947$	45.8127	1.48871	0.744357	−	0.667782i	$-0.232756\pi$
$947$	45.8127	1.48871	0.744357	+	0.667782i	$0.232756\pi$
$948$	0	0
$949$	0.0193268	0.000627374 0
$950$	0	0
$951$	−14.1877	−0.460066
$952$	0	0
$953$	−21.2363	−0.687913	−0.343956	−	0.938986i	$-0.611767\pi$
$953$	−21.2363	−0.687913	−0.343956	+	0.938986i	$0.611767\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	67.0861	2.16859
$958$	0	0
$959$	4.76627	0.153911
$960$	0	0
$961$	6.72623	0.216975
$962$	0	0
$963$	1.25310	0.0403805
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−5.88651	−0.189297	−0.0946486	−	0.995511i	$-0.530173\pi$
$967$	−5.88651	−0.189297	−0.0946486	+	0.995511i	$0.530173\pi$
$968$	0	0
$969$	34.0627	1.09425
$970$	0	0
$971$	27.1841	0.872378	0.436189	−	0.899855i	$-0.356328\pi$
$971$	27.1841	0.872378	0.436189	+	0.899855i	$0.356328\pi$
$972$	0	0
$973$	−22.9855	−0.736882
$974$	0	0
$975$	0	0
$976$	0	0
$977$	48.5280	1.55255	0.776274	−	0.630395i	$-0.217107\pi$
$977$	48.5280	1.55255	0.776274	+	0.630395i	$0.217107\pi$
$978$	0	0
$979$	62.9102	2.01062
$980$	0	0
$981$	−1.55897	−0.0497740
$982$	0	0
$983$	−50.7935	−1.62006	−0.810030	−	0.586388i	$-0.800549\pi$
$983$	−50.7935	−1.62006	−0.810030	+	0.586388i	$0.800549\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−26.5540	−0.845223
$988$	0	0
$989$	3.26755	0.103902
$990$	0	0
$991$	−9.42740	−0.299471	−0.149736	−	0.988726i	$-0.547842\pi$
$991$	−9.42740	−0.299471	−0.149736	+	0.988726i	$0.547842\pi$
$992$	0	0
$993$	−11.2816	−0.358012
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−6.78817	−0.214984	−0.107492	−	0.994206i	$-0.534282\pi$
$997$	−6.78817	−0.214984	−0.107492	+	0.994206i	$0.534282\pi$
$998$	0	0
$999$	34.2803	1.08458

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.a.bh.1.2		7
4.3	odd	2		9200.2.a.dc.1.6		7
5.2	odd	4		920.2.e.b.369.12	yes	14
5.3	odd	4		920.2.e.b.369.3	✓	14
5.4	even	2		4600.2.a.bi.1.6		7
20.3	even	4		1840.2.e.g.369.12		14
20.7	even	4		1840.2.e.g.369.3		14
20.19	odd	2		9200.2.a.cz.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.b.369.3	✓	14	5.3	odd	4
920.2.e.b.369.12	yes	14	5.2	odd	4
1840.2.e.g.369.3		14	20.7	even	4
1840.2.e.g.369.12		14	20.3	even	4
4600.2.a.bh.1.2		7	1.1	even	1	trivial
4600.2.a.bi.1.6		7	5.4	even	2
9200.2.a.cz.1.2		7	20.19	odd	2
9200.2.a.dc.1.6		7	4.3	odd	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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.a (trivial)

\(f(q)\)	\(=\)	\(q-1.78665 q^{3} -1.75578 q^{7} +0.192116 q^{9} +O(q^{10})\)	\(q-1.78665 q^{3} -1.75578 q^{7} +0.192116 q^{9} -4.77574 q^{11} -1.72967 q^{13} +7.81453 q^{17} -2.43970 q^{19} +3.13696 q^{21} +1.00000 q^{23} +5.01670 q^{27} +7.86235 q^{29} +6.14217 q^{31} +8.53258 q^{33} +6.83324 q^{37} +3.09031 q^{39} -2.50416 q^{41} +3.26755 q^{43} -8.46487 q^{47} -3.91724 q^{49} -13.9618 q^{51} +2.76559 q^{53} +4.35889 q^{57} +1.91706 q^{59} -3.50364 q^{61} -0.337313 q^{63} +12.7649 q^{67} -1.78665 q^{69} -13.3973 q^{71} -0.0111737 q^{73} +8.38515 q^{77} -16.6960 q^{79} -9.53944 q^{81} +2.64145 q^{83} -14.0473 q^{87} -13.1729 q^{89} +3.03692 q^{91} -10.9739 q^{93} -15.8568 q^{97} -0.917494 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 3 q^{3} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) 7 * q - 3 * q^3 - 4 * q^7 + 2 * q^9	\( 7 q - 3 q^{3} - 4 q^{7} + 2 q^{9} - 7 q^{11} + 7 q^{13} - 7 q^{19} - 6 q^{21} + 7 q^{23} - 11 q^{29} - 10 q^{31} + 19 q^{33} + 19 q^{37} - 24 q^{39} - 16 q^{41} - 6 q^{43} - 6 q^{47} - 17 q^{49} - 7 q^{51} + 15 q^{53} + 8 q^{57} - 11 q^{59} + 5 q^{61} - 13 q^{63} - 9 q^{67} - 3 q^{69} - 14 q^{71} + 10 q^{73} + 6 q^{77} - 32 q^{79} - 5 q^{81} - q^{83} - 10 q^{87} - 24 q^{89} - 7 q^{91} + 26 q^{93} - 7 q^{97} - 61 q^{99}+O(q^{100}) \) 7 * q - 3 * q^3 - 4 * q^7 + 2 * q^9 - 7 * q^11 + 7 * q^13 - 7 * q^19 - 6 * q^21 + 7 * q^23 - 11 * q^29 - 10 * q^31 + 19 * q^33 + 19 * q^37 - 24 * q^39 - 16 * q^41 - 6 * q^43 - 6 * q^47 - 17 * q^49 - 7 * q^51 + 15 * q^53 + 8 * q^57 - 11 * q^59 + 5 * q^61 - 13 * q^63 - 9 * q^67 - 3 * q^69 - 14 * q^71 + 10 * q^73 + 6 * q^77 - 32 * q^79 - 5 * q^81 - q^83 - 10 * q^87 - 24 * q^89 - 7 * q^91 + 26 * q^93 - 7 * q^97 - 61 * q^99

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