LMFDB - Embedded newform 4600.2.a.bj.1.1

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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 13x^{6} + 38x^{5} + 41x^{4} - 123x^{3} + 15x^{2} + 32x - 8 $ x^8 - 3x^7 - 13x^6 + 38x^5 + 41x^4 - 123x^3 + 15x^2 + 32*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
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			1.1
Root			$3.20935$ 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)

$f(q)$	$=$	$q-3.20935 q^{3} -4.54713 q^{7} +7.29996 q^{9} +O(q^{10})$	$q-3.20935 q^{3} -4.54713 q^{7} +7.29996 q^{9} +4.59577 q^{11} +5.17990 q^{13} +3.33965 q^{17} -4.43893 q^{19} +14.5934 q^{21} +1.00000 q^{23} -13.8001 q^{27} +4.13029 q^{29} +7.82535 q^{31} -14.7494 q^{33} -1.03174 q^{37} -16.6241 q^{39} -5.75780 q^{41} +1.67719 q^{43} +6.20177 q^{47} +13.6764 q^{49} -10.7181 q^{51} -7.35094 q^{53} +14.2461 q^{57} +1.83951 q^{59} +0.524294 q^{61} -33.1939 q^{63} -2.55477 q^{67} -3.20935 q^{69} -6.58764 q^{71} -2.41579 q^{73} -20.8975 q^{77} -9.85496 q^{79} +22.3895 q^{81} -10.7875 q^{83} -13.2556 q^{87} +5.13994 q^{89} -23.5537 q^{91} -25.1143 q^{93} -3.36751 q^{97} +33.5489 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 3 q^{3} - 7 q^{7} + 11 q^{9}+O(q^{10}) $ 8 * q - 3 * q^3 - 7 * q^7 + 11 * q^9	$ 8 q - 3 q^{3} - 7 q^{7} + 11 q^{9} + 7 q^{11} + 11 q^{13} - 7 q^{17} + 11 q^{19} + 8 q^{23} - 12 q^{27} + 22 q^{29} + 9 q^{31} - 9 q^{33} + 4 q^{37} + 7 q^{41} - 22 q^{43} - 4 q^{47} + 39 q^{49} - 19 q^{51} + 4 q^{53} + 32 q^{59} + 17 q^{61} - 44 q^{63} + 4 q^{67} - 3 q^{69} + 15 q^{71} + 6 q^{73} + 18 q^{77} - 2 q^{79} + 24 q^{81} - 36 q^{83} + 4 q^{87} + 46 q^{89} - 35 q^{91} + 20 q^{93} + 3 q^{97} + 61 q^{99}+O(q^{100}) $ 8 * q - 3 * q^3 - 7 * q^7 + 11 * q^9 + 7 * q^11 + 11 * q^13 - 7 * q^17 + 11 * q^19 + 8 * q^23 - 12 * q^27 + 22 * q^29 + 9 * q^31 - 9 * q^33 + 4 * q^37 + 7 * q^41 - 22 * q^43 - 4 * q^47 + 39 * q^49 - 19 * q^51 + 4 * q^53 + 32 * q^59 + 17 * q^61 - 44 * q^63 + 4 * q^67 - 3 * q^69 + 15 * q^71 + 6 * q^73 + 18 * q^77 - 2 * q^79 + 24 * q^81 - 36 * q^83 + 4 * q^87 + 46 * q^89 - 35 * q^91 + 20 * q^93 + 3 * 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
$3$	−3.20935	−1.85292	−0.926461	−	0.376391i	$-0.877165\pi$
$3$	−3.20935	−1.85292	−0.926461	+	0.376391i	$0.877165\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.54713	−1.71865	−0.859327	−	0.511427i	$-0.829117\pi$
$7$	−4.54713	−1.71865	−0.859327	+	0.511427i	$0.829117\pi$
$8$	0	0
$9$	7.29996	2.43332
$10$	0	0
$11$	4.59577	1.38568	0.692838	−	0.721093i	$-0.256360\pi$
$11$	4.59577	1.38568	0.692838	+	0.721093i	$0.256360\pi$
$12$	0	0
$13$	5.17990	1.43665	0.718323	−	0.695710i	$-0.244910\pi$
$13$	5.17990	1.43665	0.718323	+	0.695710i	$0.244910\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.33965	0.809984	0.404992	−	0.914320i	$-0.367274\pi$
$17$	3.33965	0.809984	0.404992	+	0.914320i	$0.367274\pi$
$18$	0	0
$19$	−4.43893	−1.01836	−0.509181	−	0.860660i	$-0.670051\pi$
$19$	−4.43893	−1.01836	−0.509181	+	0.860660i	$0.670051\pi$
$20$	0	0
$21$	14.5934	3.18453
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−13.8001	−2.65583
$28$	0	0
$29$	4.13029	0.766976	0.383488	−	0.923546i	$-0.374723\pi$
$29$	4.13029	0.766976	0.383488	+	0.923546i	$0.374723\pi$
$30$	0	0
$31$	7.82535	1.40547	0.702737	−	0.711450i	$-0.251961\pi$
$31$	7.82535	1.40547	0.702737	+	0.711450i	$0.251961\pi$
$32$	0	0
$33$	−14.7494	−2.56755
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.03174	−0.169618	−0.0848089	−	0.996397i	$-0.527028\pi$
$37$	−1.03174	−0.169618	−0.0848089	+	0.996397i	$0.527028\pi$
$38$	0	0
$39$	−16.6241	−2.66199
$40$	0	0
$41$	−5.75780	−0.899218	−0.449609	−	0.893225i	$-0.648437\pi$
$41$	−5.75780	−0.899218	−0.449609	+	0.893225i	$0.648437\pi$
$42$	0	0
$43$	1.67719	0.255770	0.127885	−	0.991789i	$-0.459181\pi$
$43$	1.67719	0.255770	0.127885	+	0.991789i	$0.459181\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.20177	0.904621	0.452310	−	0.891861i	$-0.350600\pi$
$47$	6.20177	0.904621	0.452310	+	0.891861i	$0.350600\pi$
$48$	0	0
$49$	13.6764	1.95377
$50$	0	0
$51$	−10.7181	−1.50084
$52$	0	0
$53$	−7.35094	−1.00973	−0.504865	−	0.863198i	$-0.668458\pi$
$53$	−7.35094	−1.00973	−0.504865	+	0.863198i	$0.668458\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	14.2461	1.88694
$58$	0	0
$59$	1.83951	0.239484	0.119742	−	0.992805i	$-0.461793\pi$
$59$	1.83951	0.239484	0.119742	+	0.992805i	$0.461793\pi$
$60$	0	0
$61$	0.524294	0.0671289	0.0335645	−	0.999437i	$-0.489314\pi$
$61$	0.524294	0.0671289	0.0335645	+	0.999437i	$0.489314\pi$
$62$	0	0
$63$	−33.1939	−4.18203
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.55477	−0.312115	−0.156057	−	0.987748i	$-0.549878\pi$
$67$	−2.55477	−0.312115	−0.156057	+	0.987748i	$0.549878\pi$
$68$	0	0
$69$	−3.20935	−0.386361
$70$	0	0
$71$	−6.58764	−0.781809	−0.390904	−	0.920431i	$-0.627838\pi$
$71$	−6.58764	−0.781809	−0.390904	+	0.920431i	$0.627838\pi$
$72$	0	0
$73$	−2.41579	−0.282747	−0.141374	−	0.989956i	$-0.545152\pi$
$73$	−2.41579	−0.282747	−0.141374	+	0.989956i	$0.545152\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−20.8975	−2.38150
$78$	0	0
$79$	−9.85496	−1.10877	−0.554385	−	0.832261i	$-0.687046\pi$
$79$	−9.85496	−1.10877	−0.554385	+	0.832261i	$0.687046\pi$
$80$	0	0
$81$	22.3895	2.48772
$82$	0	0
$83$	−10.7875	−1.18408	−0.592040	−	0.805908i	$-0.701677\pi$
$83$	−10.7875	−1.18408	−0.592040	+	0.805908i	$0.701677\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−13.2556	−1.42115
$88$	0	0
$89$	5.13994	0.544832	0.272416	−	0.962180i	$-0.412177\pi$
$89$	5.13994	0.544832	0.272416	+	0.962180i	$0.412177\pi$
$90$	0	0
$91$	−23.5537	−2.46909
$92$	0	0
$93$	−25.1143	−2.60423
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.36751	−0.341919	−0.170959	−	0.985278i	$-0.554687\pi$
$97$	−3.36751	−0.341919	−0.170959	+	0.985278i	$0.554687\pi$
$98$	0	0
$99$	33.5489	3.37179
$100$	0	0
$101$	−4.18121	−0.416046	−0.208023	−	0.978124i	$-0.566703\pi$
$101$	−4.18121	−0.416046	−0.208023	+	0.978124i	$0.566703\pi$
$102$	0	0
$103$	9.92121	0.977566	0.488783	−	0.872405i	$-0.337441\pi$
$103$	9.92121	0.977566	0.488783	+	0.872405i	$0.337441\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.6829	1.12943	0.564713	−	0.825287i	$-0.308987\pi$
$107$	11.6829	1.12943	0.564713	+	0.825287i	$0.308987\pi$
$108$	0	0
$109$	−5.58937	−0.535365	−0.267682	−	0.963507i	$-0.586258\pi$
$109$	−5.58937	−0.535365	−0.267682	+	0.963507i	$0.586258\pi$
$110$	0	0
$111$	3.31123	0.314288
$112$	0	0
$113$	−4.02597	−0.378731	−0.189366	−	0.981907i	$-0.560643\pi$
$113$	−4.02597	−0.378731	−0.189366	+	0.981907i	$0.560643\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	37.8130	3.49582
$118$	0	0
$119$	−15.1858	−1.39208
$120$	0	0
$121$	10.1211	0.920098
$122$	0	0
$123$	18.4788	1.66618
$124$	0	0
$125$	0	0
$126$	0	0
$127$	15.8137	1.40324	0.701618	−	0.712553i	$-0.252461\pi$
$127$	15.8137	1.40324	0.701618	+	0.712553i	$0.252461\pi$
$128$	0	0
$129$	−5.38271	−0.473921
$130$	0	0
$131$	15.5004	1.35428	0.677139	−	0.735855i	$-0.263220\pi$
$131$	15.5004	1.35428	0.677139	+	0.735855i	$0.263220\pi$
$132$	0	0
$133$	20.1844	1.75021
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.94249	−0.422265	−0.211133	−	0.977457i	$-0.567715\pi$
$137$	−4.94249	−0.422265	−0.211133	+	0.977457i	$0.567715\pi$
$138$	0	0
$139$	23.0723	1.95697	0.978483	−	0.206326i	$-0.0661506\pi$
$139$	23.0723	1.95697	0.978483	+	0.206326i	$0.0661506\pi$
$140$	0	0
$141$	−19.9037	−1.67619
$142$	0	0
$143$	23.8056	1.99072
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−43.8924	−3.62018
$148$	0	0
$149$	−14.7115	−1.20521	−0.602606	−	0.798039i	$-0.705871\pi$
$149$	−14.7115	−1.20521	−0.602606	+	0.798039i	$0.705871\pi$
$150$	0	0
$151$	−11.5461	−0.939606	−0.469803	−	0.882771i	$-0.655675\pi$
$151$	−11.5461	−0.939606	−0.469803	+	0.882771i	$0.655675\pi$
$152$	0	0
$153$	24.3793	1.97095
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.02677	0.560797	0.280399	−	0.959884i	$-0.409533\pi$
$157$	7.02677	0.560797	0.280399	+	0.959884i	$0.409533\pi$
$158$	0	0
$159$	23.5918	1.87095
$160$	0	0
$161$	−4.54713	−0.358364
$162$	0	0
$163$	13.4203	1.05116	0.525580	−	0.850744i	$-0.323848\pi$
$163$	13.4203	1.05116	0.525580	+	0.850744i	$0.323848\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.77046	0.678679	0.339339	−	0.940664i	$-0.389797\pi$
$167$	8.77046	0.678679	0.339339	+	0.940664i	$0.389797\pi$
$168$	0	0
$169$	13.8313	1.06395
$170$	0	0
$171$	−32.4040	−2.47800
$172$	0	0
$173$	20.5416	1.56175	0.780873	−	0.624690i	$-0.214775\pi$
$173$	20.5416	1.56175	0.780873	+	0.624690i	$0.214775\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−5.90365	−0.443745
$178$	0	0
$179$	−15.4880	−1.15763	−0.578814	−	0.815460i	$-0.696484\pi$
$179$	−15.4880	−1.15763	−0.578814	+	0.815460i	$0.696484\pi$
$180$	0	0
$181$	−3.31887	−0.246690	−0.123345	−	0.992364i	$-0.539362\pi$
$181$	−3.31887	−0.246690	−0.123345	+	0.992364i	$0.539362\pi$
$182$	0	0
$183$	−1.68264	−0.124385
$184$	0	0
$185$	0	0
$186$	0	0
$187$	15.3482	1.12237
$188$	0	0
$189$	62.7508	4.56445
$190$	0	0
$191$	−5.46525	−0.395451	−0.197726	−	0.980257i	$-0.563356\pi$
$191$	−5.46525	−0.395451	−0.197726	+	0.980257i	$0.563356\pi$
$192$	0	0
$193$	−1.88244	−0.135501	−0.0677506	−	0.997702i	$-0.521582\pi$
$193$	−1.88244	−0.135501	−0.0677506	+	0.997702i	$0.521582\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.27284	0.518169	0.259084	−	0.965855i	$-0.416579\pi$
$197$	7.27284	0.518169	0.259084	+	0.965855i	$0.416579\pi$
$198$	0	0
$199$	−7.84296	−0.555972	−0.277986	−	0.960585i	$-0.589667\pi$
$199$	−7.84296	−0.555972	−0.277986	+	0.960585i	$0.589667\pi$
$200$	0	0
$201$	8.19916	0.578324
$202$	0	0
$203$	−18.7810	−1.31817
$204$	0	0
$205$	0	0
$206$	0	0
$207$	7.29996	0.507382
$208$	0	0
$209$	−20.4003	−1.41112
$210$	0	0
$211$	6.03882	0.415730	0.207865	−	0.978158i	$-0.433349\pi$
$211$	6.03882	0.415730	0.207865	+	0.978158i	$0.433349\pi$
$212$	0	0
$213$	21.1421	1.44863
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−35.5829	−2.41552
$218$	0	0
$219$	7.75314	0.523908
$220$	0	0
$221$	17.2990	1.16366
$222$	0	0
$223$	−13.7931	−0.923654	−0.461827	−	0.886970i	$-0.652806\pi$
$223$	−13.7931	−0.923654	−0.461827	+	0.886970i	$0.652806\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.2668	1.21241	0.606204	−	0.795309i	$-0.292692\pi$
$227$	18.2668	1.21241	0.606204	+	0.795309i	$0.292692\pi$
$228$	0	0
$229$	15.5985	1.03078	0.515390	−	0.856956i	$-0.327647\pi$
$229$	15.5985	1.03078	0.515390	+	0.856956i	$0.327647\pi$
$230$	0	0
$231$	67.0676	4.41273
$232$	0	0
$233$	−26.9422	−1.76504	−0.882520	−	0.470275i	$-0.844155\pi$
$233$	−26.9422	−1.76504	−0.882520	+	0.470275i	$0.844155\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	31.6281	2.05446
$238$	0	0
$239$	−4.91116	−0.317677	−0.158838	−	0.987305i	$-0.550775\pi$
$239$	−4.91116	−0.317677	−0.158838	+	0.987305i	$0.550775\pi$
$240$	0	0
$241$	20.3100	1.30828	0.654140	−	0.756373i	$-0.273031\pi$
$241$	20.3100	1.30828	0.654140	+	0.756373i	$0.273031\pi$
$242$	0	0
$243$	−30.4556	−1.95373
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−22.9932	−1.46302
$248$	0	0
$249$	34.6209	2.19401
$250$	0	0
$251$	−15.1382	−0.955514	−0.477757	−	0.878492i	$-0.658550\pi$
$251$	−15.1382	−0.955514	−0.477757	+	0.878492i	$0.658550\pi$
$252$	0	0
$253$	4.59577	0.288933
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−21.7394	−1.35607	−0.678033	−	0.735031i	$-0.737168\pi$
$257$	−21.7394	−1.35607	−0.678033	+	0.735031i	$0.737168\pi$
$258$	0	0
$259$	4.69147	0.291514
$260$	0	0
$261$	30.1510	1.86630
$262$	0	0
$263$	−20.5960	−1.27000	−0.635002	−	0.772511i	$-0.719001\pi$
$263$	−20.5960	−1.27000	−0.635002	+	0.772511i	$0.719001\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−16.4959	−1.00953
$268$	0	0
$269$	15.8810	0.968284	0.484142	−	0.874989i	$-0.339132\pi$
$269$	15.8810	0.968284	0.484142	+	0.874989i	$0.339132\pi$
$270$	0	0
$271$	−3.55030	−0.215666	−0.107833	−	0.994169i	$-0.534391\pi$
$271$	−3.55030	−0.215666	−0.107833	+	0.994169i	$0.534391\pi$
$272$	0	0
$273$	75.5921	4.57504
$274$	0	0
$275$	0	0
$276$	0	0
$277$	23.3134	1.40077	0.700383	−	0.713767i	$-0.253013\pi$
$277$	23.3134	1.40077	0.700383	+	0.713767i	$0.253013\pi$
$278$	0	0
$279$	57.1247	3.41997
$280$	0	0
$281$	−8.93301	−0.532899	−0.266449	−	0.963849i	$-0.585850\pi$
$281$	−8.93301	−0.532899	−0.266449	+	0.963849i	$0.585850\pi$
$282$	0	0
$283$	14.6498	0.870843	0.435421	−	0.900227i	$-0.356599\pi$
$283$	14.6498	0.870843	0.435421	+	0.900227i	$0.356599\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	26.1815	1.54544
$288$	0	0
$289$	−5.84675	−0.343927
$290$	0	0
$291$	10.8075	0.633549
$292$	0	0
$293$	27.4766	1.60520	0.802601	−	0.596516i	$-0.203449\pi$
$293$	27.4766	1.60520	0.802601	+	0.596516i	$0.203449\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−63.4220	−3.68012
$298$	0	0
$299$	5.17990	0.299561
$300$	0	0
$301$	−7.62641	−0.439579
$302$	0	0
$303$	13.4190	0.770900
$304$	0	0
$305$	0	0
$306$	0	0
$307$	19.1369	1.09220	0.546101	−	0.837719i	$-0.316112\pi$
$307$	19.1369	1.09220	0.546101	+	0.837719i	$0.316112\pi$
$308$	0	0
$309$	−31.8407	−1.81135
$310$	0	0
$311$	15.3439	0.870074	0.435037	−	0.900413i	$-0.356735\pi$
$311$	15.3439	0.870074	0.435037	+	0.900413i	$0.356735\pi$
$312$	0	0
$313$	8.41254	0.475505	0.237753	−	0.971326i	$-0.423589\pi$
$313$	8.41254	0.475505	0.237753	+	0.971326i	$0.423589\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−16.5135	−0.927493	−0.463747	−	0.885968i	$-0.653495\pi$
$317$	−16.5135	−0.927493	−0.463747	+	0.885968i	$0.653495\pi$
$318$	0	0
$319$	18.9819	1.06278
$320$	0	0
$321$	−37.4945	−2.09274
$322$	0	0
$323$	−14.8245	−0.824856
$324$	0	0
$325$	0	0
$326$	0	0
$327$	17.9383	0.991989
$328$	0	0
$329$	−28.2002	−1.55473
$330$	0	0
$331$	−2.78794	−0.153239	−0.0766196	−	0.997060i	$-0.524413\pi$
$331$	−2.78794	−0.153239	−0.0766196	+	0.997060i	$0.524413\pi$
$332$	0	0
$333$	−7.53169	−0.412734
$334$	0	0
$335$	0	0
$336$	0	0
$337$	8.70062	0.473953	0.236977	−	0.971515i	$-0.423844\pi$
$337$	8.70062	0.473953	0.236977	+	0.971515i	$0.423844\pi$
$338$	0	0
$339$	12.9208	0.701759
$340$	0	0
$341$	35.9635	1.94753
$342$	0	0
$343$	−30.3584	−1.63920
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.77786	−0.0954405	−0.0477203	−	0.998861i	$-0.515196\pi$
$347$	−1.77786	−0.0954405	−0.0477203	+	0.998861i	$0.515196\pi$
$348$	0	0
$349$	23.9187	1.28034	0.640170	−	0.768233i	$-0.278864\pi$
$349$	23.9187	1.28034	0.640170	+	0.768233i	$0.278864\pi$
$350$	0	0
$351$	−71.4831	−3.81548
$352$	0	0
$353$	−8.55855	−0.455526	−0.227763	−	0.973717i	$-0.573141\pi$
$353$	−8.55855	−0.455526	−0.227763	+	0.973717i	$0.573141\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	48.7367	2.57942
$358$	0	0
$359$	28.6262	1.51083	0.755417	−	0.655245i	$-0.227435\pi$
$359$	28.6262	1.51083	0.755417	+	0.655245i	$0.227435\pi$
$360$	0	0
$361$	0.704135	0.0370597
$362$	0	0
$363$	−32.4821	−1.70487
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−25.3094	−1.32114	−0.660570	−	0.750764i	$-0.729685\pi$
$367$	−25.3094	−1.32114	−0.660570	+	0.750764i	$0.729685\pi$
$368$	0	0
$369$	−42.0317	−2.18808
$370$	0	0
$371$	33.4257	1.73537
$372$	0	0
$373$	24.0094	1.24316	0.621579	−	0.783351i	$-0.286491\pi$
$373$	24.0094	1.24316	0.621579	+	0.783351i	$0.286491\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	21.3945	1.10187
$378$	0	0
$379$	−25.6358	−1.31682	−0.658411	−	0.752659i	$-0.728771\pi$
$379$	−25.6358	−1.31682	−0.658411	+	0.752659i	$0.728771\pi$
$380$	0	0
$381$	−50.7517	−2.60009
$382$	0	0
$383$	17.7477	0.906865	0.453433	−	0.891291i	$-0.350199\pi$
$383$	17.7477	0.906865	0.453433	+	0.891291i	$0.350199\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	12.2434	0.622369
$388$	0	0
$389$	20.4043	1.03454	0.517269	−	0.855823i	$-0.326949\pi$
$389$	20.4043	1.03454	0.517269	+	0.855823i	$0.326949\pi$
$390$	0	0
$391$	3.33965	0.168893
$392$	0	0
$393$	−49.7463	−2.50937
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.2458	1.11648	0.558242	−	0.829678i	$-0.311476\pi$
$397$	22.2458	1.11648	0.558242	+	0.829678i	$0.311476\pi$
$398$	0	0
$399$	−64.7789	−3.24300
$400$	0	0
$401$	−8.44952	−0.421949	−0.210975	−	0.977492i	$-0.567664\pi$
$401$	−8.44952	−0.421949	−0.210975	+	0.977492i	$0.567664\pi$
$402$	0	0
$403$	40.5345	2.01917
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.74166	−0.235035
$408$	0	0
$409$	3.18182	0.157331	0.0786654	−	0.996901i	$-0.474934\pi$
$409$	3.18182	0.157331	0.0786654	+	0.996901i	$0.474934\pi$
$410$	0	0
$411$	15.8622	0.782425
$412$	0	0
$413$	−8.36450	−0.411590
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−74.0472	−3.62611
$418$	0	0
$419$	33.7228	1.64747	0.823733	−	0.566977i	$-0.191887\pi$
$419$	33.7228	1.64747	0.823733	+	0.566977i	$0.191887\pi$
$420$	0	0
$421$	−23.6641	−1.15332	−0.576659	−	0.816985i	$-0.695644\pi$
$421$	−23.6641	−1.15332	−0.576659	+	0.816985i	$0.695644\pi$
$422$	0	0
$423$	45.2726	2.20123
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.38403	−0.115371
$428$	0	0
$429$	−76.4006	−3.68866
$430$	0	0
$431$	36.4849	1.75742	0.878708	−	0.477359i	$-0.158406\pi$
$431$	36.4849	1.75742	0.878708	+	0.477359i	$0.158406\pi$
$432$	0	0
$433$	−28.1736	−1.35393	−0.676967	−	0.736013i	$-0.736706\pi$
$433$	−28.1736	−1.35393	−0.676967	+	0.736013i	$0.736706\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.43893	−0.212343
$438$	0	0
$439$	−26.6445	−1.27167	−0.635837	−	0.771824i	$-0.719345\pi$
$439$	−26.6445	−1.27167	−0.635837	+	0.771824i	$0.719345\pi$
$440$	0	0
$441$	99.8370	4.75414
$442$	0	0
$443$	5.98936	0.284563	0.142281	−	0.989826i	$-0.454556\pi$
$443$	5.98936	0.284563	0.142281	+	0.989826i	$0.454556\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	47.2144	2.23316
$448$	0	0
$449$	35.0989	1.65642	0.828209	−	0.560420i	$-0.189360\pi$
$449$	35.0989	1.65642	0.828209	+	0.560420i	$0.189360\pi$
$450$	0	0
$451$	−26.4615	−1.24602
$452$	0	0
$453$	37.0555	1.74102
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.90460	0.276205	0.138103	−	0.990418i	$-0.455900\pi$
$457$	5.90460	0.276205	0.138103	+	0.990418i	$0.455900\pi$
$458$	0	0
$459$	−46.0875	−2.15118
$460$	0	0
$461$	−6.25369	−0.291263	−0.145632	−	0.989339i	$-0.546521\pi$
$461$	−6.25369	−0.291263	−0.145632	+	0.989339i	$0.546521\pi$
$462$	0	0
$463$	31.7798	1.47693	0.738466	−	0.674291i	$-0.235550\pi$
$463$	31.7798	1.47693	0.738466	+	0.674291i	$0.235550\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9.63128	−0.445682	−0.222841	−	0.974855i	$-0.571533\pi$
$467$	−9.63128	−0.445682	−0.222841	+	0.974855i	$0.571533\pi$
$468$	0	0
$469$	11.6169	0.536417
$470$	0	0
$471$	−22.5514	−1.03911
$472$	0	0
$473$	7.70799	0.354414
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−53.6616	−2.45699
$478$	0	0
$479$	0.656246	0.0299846	0.0149923	−	0.999888i	$-0.495228\pi$
$479$	0.656246	0.0299846	0.0149923	+	0.999888i	$0.495228\pi$
$480$	0	0
$481$	−5.34433	−0.243680
$482$	0	0
$483$	14.5934	0.664020
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−33.5157	−1.51874	−0.759371	−	0.650658i	$-0.774493\pi$
$487$	−33.5157	−1.51874	−0.759371	+	0.650658i	$0.774493\pi$
$488$	0	0
$489$	−43.0705	−1.94772
$490$	0	0
$491$	14.7081	0.663768	0.331884	−	0.943320i	$-0.392316\pi$
$491$	14.7081	0.663768	0.331884	+	0.943320i	$0.392316\pi$
$492$	0	0
$493$	13.7937	0.621238
$494$	0	0
$495$	0	0
$496$	0	0
$497$	29.9548	1.34366
$498$	0	0
$499$	11.8889	0.532221	0.266110	−	0.963943i	$-0.414261\pi$
$499$	11.8889	0.532221	0.266110	+	0.963943i	$0.414261\pi$
$500$	0	0
$501$	−28.1475	−1.25754
$502$	0	0
$503$	−27.2549	−1.21524	−0.607618	−	0.794230i	$-0.707875\pi$
$503$	−27.2549	−1.21524	−0.607618	+	0.794230i	$0.707875\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−44.3897	−1.97142
$508$	0	0
$509$	−9.46671	−0.419605	−0.209802	−	0.977744i	$-0.567282\pi$
$509$	−9.46671	−0.419605	−0.209802	+	0.977744i	$0.567282\pi$
$510$	0	0
$511$	10.9849	0.485944
$512$	0	0
$513$	61.2577	2.70459
$514$	0	0
$515$	0	0
$516$	0	0
$517$	28.5019	1.25351
$518$	0	0
$519$	−65.9251	−2.89379
$520$	0	0
$521$	4.85164	0.212554	0.106277	−	0.994337i	$-0.466107\pi$
$521$	4.85164	0.212554	0.106277	+	0.994337i	$0.466107\pi$
$522$	0	0
$523$	−39.1914	−1.71372	−0.856860	−	0.515549i	$-0.827588\pi$
$523$	−39.1914	−1.71372	−0.856860	+	0.515549i	$0.827588\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	26.1339	1.13841
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	13.4284	0.582741
$532$	0	0
$533$	−29.8248	−1.29186
$534$	0	0
$535$	0	0
$536$	0	0
$537$	49.7065	2.14499
$538$	0	0
$539$	62.8535	2.70729
$540$	0	0
$541$	−35.9793	−1.54687	−0.773436	−	0.633875i	$-0.781464\pi$
$541$	−35.9793	−1.54687	−0.773436	+	0.633875i	$0.781464\pi$
$542$	0	0
$543$	10.6514	0.457097
$544$	0	0
$545$	0	0
$546$	0	0
$547$	11.0470	0.472335	0.236167	−	0.971712i	$-0.424109\pi$
$547$	11.0470	0.472335	0.236167	+	0.971712i	$0.424109\pi$
$548$	0	0
$549$	3.82732	0.163346
$550$	0	0
$551$	−18.3341	−0.781059
$552$	0	0
$553$	44.8118	1.90559
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.20298	0.0933433	0.0466716	−	0.998910i	$-0.485139\pi$
$557$	2.20298	0.0933433	0.0466716	+	0.998910i	$0.485139\pi$
$558$	0	0
$559$	8.68769	0.367450
$560$	0	0
$561$	−49.2580	−2.07967
$562$	0	0
$563$	−44.9181	−1.89307	−0.946535	−	0.322601i	$-0.895443\pi$
$563$	−44.9181	−1.89307	−0.946535	+	0.322601i	$0.895443\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−101.808	−4.27554
$568$	0	0
$569$	−37.4663	−1.57067	−0.785336	−	0.619070i	$-0.787510\pi$
$569$	−37.4663	−1.57067	−0.785336	+	0.619070i	$0.787510\pi$
$570$	0	0
$571$	−23.8787	−0.999291	−0.499645	−	0.866230i	$-0.666536\pi$
$571$	−23.8787	−0.999291	−0.499645	+	0.866230i	$0.666536\pi$
$572$	0	0
$573$	17.5399	0.732741
$574$	0	0
$575$	0	0
$576$	0	0
$577$	32.6048	1.35735	0.678677	−	0.734437i	$-0.262553\pi$
$577$	32.6048	1.35735	0.678677	+	0.734437i	$0.262553\pi$
$578$	0	0
$579$	6.04143	0.251073
$580$	0	0
$581$	49.0521	2.03502
$582$	0	0
$583$	−33.7832	−1.39916
$584$	0	0
$585$	0	0
$586$	0	0
$587$	40.1563	1.65743	0.828714	−	0.559673i	$-0.189073\pi$
$587$	40.1563	1.65743	0.828714	+	0.559673i	$0.189073\pi$
$588$	0	0
$589$	−34.7362	−1.43128
$590$	0	0
$591$	−23.3411	−0.960126
$592$	0	0
$593$	−4.08558	−0.167775	−0.0838873	−	0.996475i	$-0.526734\pi$
$593$	−4.08558	−0.167775	−0.0838873	+	0.996475i	$0.526734\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	25.1708	1.03017
$598$	0	0
$599$	−2.18194	−0.0891516	−0.0445758	−	0.999006i	$-0.514194\pi$
$599$	−2.18194	−0.0891516	−0.0445758	+	0.999006i	$0.514194\pi$
$600$	0	0
$601$	−17.9240	−0.731136	−0.365568	−	0.930785i	$-0.619125\pi$
$601$	−17.9240	−0.731136	−0.365568	+	0.930785i	$0.619125\pi$
$602$	0	0
$603$	−18.6497	−0.759475
$604$	0	0
$605$	0	0
$606$	0	0
$607$	17.4158	0.706886	0.353443	−	0.935456i	$-0.385011\pi$
$607$	17.4158	0.706886	0.353443	+	0.935456i	$0.385011\pi$
$608$	0	0
$609$	60.2748	2.44246
$610$	0	0
$611$	32.1245	1.29962
$612$	0	0
$613$	5.74163	0.231902	0.115951	−	0.993255i	$-0.463008\pi$
$613$	5.74163	0.231902	0.115951	+	0.993255i	$0.463008\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.74232	0.231177	0.115589	−	0.993297i	$-0.463125\pi$
$617$	5.74232	0.231177	0.115589	+	0.993297i	$0.463125\pi$
$618$	0	0
$619$	9.23964	0.371373	0.185686	−	0.982609i	$-0.440549\pi$
$619$	9.23964	0.371373	0.185686	+	0.982609i	$0.440549\pi$
$620$	0	0
$621$	−13.8001	−0.553779
$622$	0	0
$623$	−23.3720	−0.936378
$624$	0	0
$625$	0	0
$626$	0	0
$627$	65.4718	2.61469
$628$	0	0
$629$	−3.44566	−0.137388
$630$	0	0
$631$	40.0251	1.59338	0.796688	−	0.604391i	$-0.206584\pi$
$631$	40.0251	1.59338	0.796688	+	0.604391i	$0.206584\pi$
$632$	0	0
$633$	−19.3807	−0.770314
$634$	0	0
$635$	0	0
$636$	0	0
$637$	70.8423	2.80687
$638$	0	0
$639$	−48.0895	−1.90239
$640$	0	0
$641$	32.5523	1.28574	0.642869	−	0.765977i	$-0.277744\pi$
$641$	32.5523	1.28574	0.642869	+	0.765977i	$0.277744\pi$
$642$	0	0
$643$	−20.0972	−0.792555	−0.396277	−	0.918131i	$-0.629698\pi$
$643$	−20.0972	−0.792555	−0.396277	+	0.918131i	$0.629698\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−0.167428	−0.00658228	−0.00329114	−	0.999995i	$-0.501048\pi$
$647$	−0.167428	−0.00658228	−0.00329114	+	0.999995i	$0.501048\pi$
$648$	0	0
$649$	8.45397	0.331847
$650$	0	0
$651$	114.198	4.47577
$652$	0	0
$653$	37.2262	1.45677	0.728387	−	0.685166i	$-0.240270\pi$
$653$	37.2262	1.45677	0.728387	+	0.685166i	$0.240270\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−17.6352	−0.688014
$658$	0	0
$659$	5.23004	0.203734	0.101867	−	0.994798i	$-0.467518\pi$
$659$	5.23004	0.203734	0.101867	+	0.994798i	$0.467518\pi$
$660$	0	0
$661$	31.8801	1.23999	0.619996	−	0.784605i	$-0.287134\pi$
$661$	31.8801	1.23999	0.619996	+	0.784605i	$0.287134\pi$
$662$	0	0
$663$	−55.5187	−2.15617
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.13029	0.159926
$668$	0	0
$669$	44.2670	1.71146
$670$	0	0
$671$	2.40953	0.0930189
$672$	0	0
$673$	22.5996	0.871151	0.435576	−	0.900152i	$-0.356545\pi$
$673$	22.5996	0.871151	0.435576	+	0.900152i	$0.356545\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	11.6369	0.447242	0.223621	−	0.974676i	$-0.428212\pi$
$677$	11.6369	0.447242	0.223621	+	0.974676i	$0.428212\pi$
$678$	0	0
$679$	15.3125	0.587640
$680$	0	0
$681$	−58.6245	−2.24650
$682$	0	0
$683$	24.6575	0.943492	0.471746	−	0.881735i	$-0.343624\pi$
$683$	24.6575	0.943492	0.471746	+	0.881735i	$0.343624\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−50.0612	−1.90995
$688$	0	0
$689$	−38.0771	−1.45062
$690$	0	0
$691$	−49.7207	−1.89146	−0.945732	−	0.324949i	$-0.894653\pi$
$691$	−49.7207	−1.89146	−0.945732	+	0.324949i	$0.894653\pi$
$692$	0	0
$693$	−152.551	−5.79494
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−19.2290	−0.728352
$698$	0	0
$699$	86.4670	3.27048
$700$	0	0
$701$	−36.2499	−1.36914	−0.684570	−	0.728947i	$-0.740010\pi$
$701$	−36.2499	−1.36914	−0.684570	+	0.728947i	$0.740010\pi$
$702$	0	0
$703$	4.57984	0.172732
$704$	0	0
$705$	0	0
$706$	0	0
$707$	19.0125	0.715038
$708$	0	0
$709$	42.1097	1.58146	0.790732	−	0.612162i	$-0.209700\pi$
$709$	42.1097	1.58146	0.790732	+	0.612162i	$0.209700\pi$
$710$	0	0
$711$	−71.9408	−2.69799
$712$	0	0
$713$	7.82535	0.293062
$714$	0	0
$715$	0	0
$716$	0	0
$717$	15.7617	0.588630
$718$	0	0
$719$	27.9125	1.04096	0.520481	−	0.853873i	$-0.325753\pi$
$719$	27.9125	1.04096	0.520481	+	0.853873i	$0.325753\pi$
$720$	0	0
$721$	−45.1130	−1.68010
$722$	0	0
$723$	−65.1819	−2.42414
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−3.14225	−0.116539	−0.0582697	−	0.998301i	$-0.518558\pi$
$727$	−3.14225	−0.116539	−0.0582697	+	0.998301i	$0.518558\pi$
$728$	0	0
$729$	30.5744	1.13239
$730$	0	0
$731$	5.60123	0.207169
$732$	0	0
$733$	−43.2304	−1.59675	−0.798375	−	0.602161i	$-0.794307\pi$
$733$	−43.2304	−1.59675	−0.798375	+	0.602161i	$0.794307\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−11.7411	−0.432490
$738$	0	0
$739$	21.2903	0.783177	0.391589	−	0.920140i	$-0.371926\pi$
$739$	21.2903	0.783177	0.391589	+	0.920140i	$0.371926\pi$
$740$	0	0
$741$	73.7934	2.71087
$742$	0	0
$743$	23.6168	0.866417	0.433208	−	0.901294i	$-0.357381\pi$
$743$	23.6168	0.866417	0.433208	+	0.901294i	$0.357381\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−78.7482	−2.88125
$748$	0	0
$749$	−53.1235	−1.94109
$750$	0	0
$751$	−24.1779	−0.882263	−0.441131	−	0.897443i	$-0.645423\pi$
$751$	−24.1779	−0.882263	−0.441131	+	0.897443i	$0.645423\pi$
$752$	0	0
$753$	48.5838	1.77049
$754$	0	0
$755$	0	0
$756$	0	0
$757$	17.9076	0.650864	0.325432	−	0.945565i	$-0.394490\pi$
$757$	17.9076	0.650864	0.325432	+	0.945565i	$0.394490\pi$
$758$	0	0
$759$	−14.7494	−0.535371
$760$	0	0
$761$	13.7932	0.500002	0.250001	−	0.968246i	$-0.419569\pi$
$761$	13.7932	0.500002	0.250001	+	0.968246i	$0.419569\pi$
$762$	0	0
$763$	25.4156	0.920106
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.52849	0.344054
$768$	0	0
$769$	8.37214	0.301907	0.150954	−	0.988541i	$-0.451766\pi$
$769$	8.37214	0.301907	0.150954	+	0.988541i	$0.451766\pi$
$770$	0	0
$771$	69.7695	2.51269
$772$	0	0
$773$	−3.64496	−0.131100	−0.0655501	−	0.997849i	$-0.520880\pi$
$773$	−3.64496	−0.131100	−0.0655501	+	0.997849i	$0.520880\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−15.0566	−0.540153
$778$	0	0
$779$	25.5585	0.915729
$780$	0	0
$781$	−30.2753	−1.08333
$782$	0	0
$783$	−56.9984	−2.03696
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−3.54385	−0.126325	−0.0631624	−	0.998003i	$-0.520119\pi$
$787$	−3.54385	−0.126325	−0.0631624	+	0.998003i	$0.520119\pi$
$788$	0	0
$789$	66.0999	2.35322
$790$	0	0
$791$	18.3066	0.650908
$792$	0	0
$793$	2.71579	0.0964404
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9.76387	−0.345854	−0.172927	−	0.984935i	$-0.555323\pi$
$797$	−9.76387	−0.345854	−0.172927	+	0.984935i	$0.555323\pi$
$798$	0	0
$799$	20.7117	0.732728
$800$	0	0
$801$	37.5213	1.32575
$802$	0	0
$803$	−11.1024	−0.391796
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−50.9679	−1.79415
$808$	0	0
$809$	6.55235	0.230368	0.115184	−	0.993344i	$-0.463254\pi$
$809$	6.55235	0.230368	0.115184	+	0.993344i	$0.463254\pi$
$810$	0	0
$811$	48.5149	1.70359	0.851795	−	0.523875i	$-0.175514\pi$
$811$	48.5149	1.70359	0.851795	+	0.523875i	$0.175514\pi$
$812$	0	0
$813$	11.3942	0.399611
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−7.44495	−0.260466
$818$	0	0
$819$	−171.941	−6.00810
$820$	0	0
$821$	−32.5956	−1.13759	−0.568797	−	0.822478i	$-0.692591\pi$
$821$	−32.5956	−1.13759	−0.568797	+	0.822478i	$0.692591\pi$
$822$	0	0
$823$	−7.63895	−0.266277	−0.133139	−	0.991097i	$-0.542506\pi$
$823$	−7.63895	−0.266277	−0.133139	+	0.991097i	$0.542506\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.35631	0.0819370	0.0409685	−	0.999160i	$-0.486956\pi$
$827$	2.35631	0.0819370	0.0409685	+	0.999160i	$0.486956\pi$
$828$	0	0
$829$	−36.1206	−1.25452	−0.627261	−	0.778809i	$-0.715824\pi$
$829$	−36.1206	−1.25452	−0.627261	+	0.778809i	$0.715824\pi$
$830$	0	0
$831$	−74.8209	−2.59551
$832$	0	0
$833$	45.6743	1.58252
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−107.991	−3.73270
$838$	0	0
$839$	−0.821644	−0.0283663	−0.0141832	−	0.999899i	$-0.504515\pi$
$839$	−0.821644	−0.0283663	−0.0141832	+	0.999899i	$0.504515\pi$
$840$	0	0
$841$	−11.9407	−0.411748
$842$	0	0
$843$	28.6692	0.987420
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−46.0218	−1.58133
$848$	0	0
$849$	−47.0166	−1.61360
$850$	0	0
$851$	−1.03174	−0.0353677
$852$	0	0
$853$	4.87966	0.167076	0.0835382	−	0.996505i	$-0.473378\pi$
$853$	4.87966	0.167076	0.0835382	+	0.996505i	$0.473378\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−10.5428	−0.360136	−0.180068	−	0.983654i	$-0.557632\pi$
$857$	−10.5428	−0.360136	−0.180068	+	0.983654i	$0.557632\pi$
$858$	0	0
$859$	−46.8583	−1.59878	−0.799392	−	0.600810i	$-0.794845\pi$
$859$	−46.8583	−1.59878	−0.799392	+	0.600810i	$0.794845\pi$
$860$	0	0
$861$	−84.0257	−2.86359
$862$	0	0
$863$	31.1007	1.05868	0.529340	−	0.848410i	$-0.322440\pi$
$863$	31.1007	1.05868	0.529340	+	0.848410i	$0.322440\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	18.7643	0.637269
$868$	0	0
$869$	−45.2911	−1.53639
$870$	0	0
$871$	−13.2334	−0.448398
$872$	0	0
$873$	−24.5827	−0.831998
$874$	0	0
$875$	0	0
$876$	0	0
$877$	37.8579	1.27837	0.639185	−	0.769053i	$-0.279272\pi$
$877$	37.8579	1.27837	0.639185	+	0.769053i	$0.279272\pi$
$878$	0	0
$879$	−88.1823	−2.97431
$880$	0	0
$881$	−15.8882	−0.535286	−0.267643	−	0.963518i	$-0.586245\pi$
$881$	−15.8882	−0.535286	−0.267643	+	0.963518i	$0.586245\pi$
$882$	0	0
$883$	31.7598	1.06880	0.534401	−	0.845231i	$-0.320537\pi$
$883$	31.7598	1.06880	0.534401	+	0.845231i	$0.320537\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−6.46849	−0.217191	−0.108595	−	0.994086i	$-0.534635\pi$
$887$	−6.46849	−0.217191	−0.108595	+	0.994086i	$0.534635\pi$
$888$	0	0
$889$	−71.9068	−2.41168
$890$	0	0
$891$	102.897	3.44718
$892$	0	0
$893$	−27.5292	−0.921231
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−16.6241	−0.555064
$898$	0	0
$899$	32.3210	1.07796
$900$	0	0
$901$	−24.5496	−0.817864
$902$	0	0
$903$	24.4759	0.814506
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−35.2611	−1.17082	−0.585412	−	0.810736i	$-0.699067\pi$
$907$	−35.2611	−1.17082	−0.585412	+	0.810736i	$0.699067\pi$
$908$	0	0
$909$	−30.5226	−1.01237
$910$	0	0
$911$	16.2888	0.539673	0.269837	−	0.962906i	$-0.413030\pi$
$911$	16.2888	0.539673	0.269837	+	0.962906i	$0.413030\pi$
$912$	0	0
$913$	−49.5768	−1.64075
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−70.4824	−2.32753
$918$	0	0
$919$	−42.9436	−1.41658	−0.708289	−	0.705923i	$-0.750533\pi$
$919$	−42.9436	−1.41658	−0.708289	+	0.705923i	$0.750533\pi$
$920$	0	0
$921$	−61.4172	−2.02376
$922$	0	0
$923$	−34.1233	−1.12318
$924$	0	0
$925$	0	0
$926$	0	0
$927$	72.4244	2.37873
$928$	0	0
$929$	−3.19236	−0.104738	−0.0523690	−	0.998628i	$-0.516677\pi$
$929$	−3.19236	−0.104738	−0.0523690	+	0.998628i	$0.516677\pi$
$930$	0	0
$931$	−60.7086	−1.98964
$932$	0	0
$933$	−49.2441	−1.61218
$934$	0	0
$935$	0	0
$936$	0	0
$937$	5.68073	0.185581	0.0927907	−	0.995686i	$-0.470421\pi$
$937$	5.68073	0.185581	0.0927907	+	0.995686i	$0.470421\pi$
$938$	0	0
$939$	−26.9988	−0.881074
$940$	0	0
$941$	54.0630	1.76240	0.881202	−	0.472739i	$-0.156735\pi$
$941$	54.0630	1.76240	0.881202	+	0.472739i	$0.156735\pi$
$942$	0	0
$943$	−5.75780	−0.187500
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−19.3780	−0.629700	−0.314850	−	0.949141i	$-0.601954\pi$
$947$	−19.3780	−0.629700	−0.314850	+	0.949141i	$0.601954\pi$
$948$	0	0
$949$	−12.5136	−0.406207
$950$	0	0
$951$	52.9978	1.71857
$952$	0	0
$953$	10.7090	0.346898	0.173449	−	0.984843i	$-0.444509\pi$
$953$	10.7090	0.346898	0.173449	+	0.984843i	$0.444509\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−60.9195	−1.96925
$958$	0	0
$959$	22.4741	0.725728
$960$	0	0
$961$	30.2360	0.975356
$962$	0	0
$963$	85.2845	2.74825
$964$	0	0
$965$	0	0
$966$	0	0
$967$	32.6034	1.04845	0.524227	−	0.851579i	$-0.324355\pi$
$967$	32.6034	1.04845	0.524227	+	0.851579i	$0.324355\pi$
$968$	0	0
$969$	47.5770	1.52839
$970$	0	0
$971$	−42.8194	−1.37414	−0.687070	−	0.726591i	$-0.741104\pi$
$971$	−42.8194	−1.37414	−0.687070	+	0.726591i	$0.741104\pi$
$972$	0	0
$973$	−104.913	−3.36335
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−7.76025	−0.248272	−0.124136	−	0.992265i	$-0.539616\pi$
$977$	−7.76025	−0.248272	−0.124136	+	0.992265i	$0.539616\pi$
$978$	0	0
$979$	23.6220	0.754961
$980$	0	0
$981$	−40.8022	−1.30271
$982$	0	0
$983$	−1.15071	−0.0367021	−0.0183510	−	0.999832i	$-0.505842\pi$
$983$	−1.15071	−0.0367021	−0.0183510	+	0.999832i	$0.505842\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	90.5046	2.88079
$988$	0	0
$989$	1.67719	0.0533316
$990$	0	0
$991$	31.0124	0.985142	0.492571	−	0.870272i	$-0.336057\pi$
$991$	31.0124	0.985142	0.492571	+	0.870272i	$0.336057\pi$
$992$	0	0
$993$	8.94750	0.283940
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−24.0635	−0.762097	−0.381049	−	0.924555i	$-0.624437\pi$
$997$	−24.0635	−0.762097	−0.381049	+	0.924555i	$0.624437\pi$
$998$	0	0
$999$	14.2382	0.450476

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.a.bj.1.1		8
4.3	odd	2		9200.2.a.de.1.8		8
5.2	odd	4		920.2.e.c.369.16	yes	16
5.3	odd	4		920.2.e.c.369.1	✓	16
5.4	even	2		4600.2.a.bk.1.8		8
20.3	even	4		1840.2.e.h.369.16		16
20.7	even	4		1840.2.e.h.369.1		16
20.19	odd	2		9200.2.a.dd.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.c.369.1	✓	16	5.3	odd	4
920.2.e.c.369.16	yes	16	5.2	odd	4
1840.2.e.h.369.1		16	20.7	even	4
1840.2.e.h.369.16		16	20.3	even	4
4600.2.a.bj.1.1		8	1.1	even	1	trivial
4600.2.a.bk.1.8		8	5.4	even	2
9200.2.a.dd.1.1		8	20.19	odd	2
9200.2.a.de.1.8		8	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)

\(f(q)\)	\(=\)	\(q-3.20935 q^{3} -4.54713 q^{7} +7.29996 q^{9} +O(q^{10})\)	\(q-3.20935 q^{3} -4.54713 q^{7} +7.29996 q^{9} +4.59577 q^{11} +5.17990 q^{13} +3.33965 q^{17} -4.43893 q^{19} +14.5934 q^{21} +1.00000 q^{23} -13.8001 q^{27} +4.13029 q^{29} +7.82535 q^{31} -14.7494 q^{33} -1.03174 q^{37} -16.6241 q^{39} -5.75780 q^{41} +1.67719 q^{43} +6.20177 q^{47} +13.6764 q^{49} -10.7181 q^{51} -7.35094 q^{53} +14.2461 q^{57} +1.83951 q^{59} +0.524294 q^{61} -33.1939 q^{63} -2.55477 q^{67} -3.20935 q^{69} -6.58764 q^{71} -2.41579 q^{73} -20.8975 q^{77} -9.85496 q^{79} +22.3895 q^{81} -10.7875 q^{83} -13.2556 q^{87} +5.13994 q^{89} -23.5537 q^{91} -25.1143 q^{93} -3.36751 q^{97} +33.5489 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 3 q^{3} - 7 q^{7} + 11 q^{9}+O(q^{10}) \) 8 * q - 3 * q^3 - 7 * q^7 + 11 * q^9	\( 8 q - 3 q^{3} - 7 q^{7} + 11 q^{9} + 7 q^{11} + 11 q^{13} - 7 q^{17} + 11 q^{19} + 8 q^{23} - 12 q^{27} + 22 q^{29} + 9 q^{31} - 9 q^{33} + 4 q^{37} + 7 q^{41} - 22 q^{43} - 4 q^{47} + 39 q^{49} - 19 q^{51} + 4 q^{53} + 32 q^{59} + 17 q^{61} - 44 q^{63} + 4 q^{67} - 3 q^{69} + 15 q^{71} + 6 q^{73} + 18 q^{77} - 2 q^{79} + 24 q^{81} - 36 q^{83} + 4 q^{87} + 46 q^{89} - 35 q^{91} + 20 q^{93} + 3 q^{97} + 61 q^{99}+O(q^{100}) \) 8 * q - 3 * q^3 - 7 * q^7 + 11 * q^9 + 7 * q^11 + 11 * q^13 - 7 * q^17 + 11 * q^19 + 8 * q^23 - 12 * q^27 + 22 * q^29 + 9 * q^31 - 9 * q^33 + 4 * q^37 + 7 * q^41 - 22 * q^43 - 4 * q^47 + 39 * q^49 - 19 * q^51 + 4 * q^53 + 32 * q^59 + 17 * q^61 - 44 * q^63 + 4 * q^67 - 3 * q^69 + 15 * q^71 + 6 * q^73 + 18 * q^77 - 2 * q^79 + 24 * q^81 - 36 * q^83 + 4 * q^87 + 46 * q^89 - 35 * q^91 + 20 * q^93 + 3 * q^97 + 61 * q^99

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