LMFDB - Embedded newform 7600.2.a.ch.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.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:	$60.6863055362$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 10x^{4} + 16x^{3} + 15x^{2} - 14x - 3 $ x^6 - 2x^5 - 10x^4 + 16x^3 + 15x^2 - 14*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3800)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.848258$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.848258 q^{3} -1.74484 q^{7} -2.28046 q^{9} +O(q^{10})$	$q-0.848258 q^{3} -1.74484 q^{7} -2.28046 q^{9} -5.92425 q^{11} +6.78582 q^{13} +1.86024 q^{17} +1.00000 q^{19} +1.48007 q^{21} -5.94357 q^{23} +4.47919 q^{27} +3.29977 q^{29} +5.75242 q^{31} +5.02529 q^{33} -4.36379 q^{37} -5.75613 q^{39} +7.12500 q^{41} +6.98455 q^{43} +4.02529 q^{47} -3.95555 q^{49} -1.57796 q^{51} -9.19015 q^{53} -0.848258 q^{57} -2.51553 q^{59} -2.49621 q^{61} +3.97903 q^{63} -6.90485 q^{67} +5.04168 q^{69} -1.27288 q^{71} +12.1217 q^{73} +10.3368 q^{77} +13.8376 q^{79} +3.04187 q^{81} +4.94955 q^{83} -2.79906 q^{87} +15.6067 q^{89} -11.8401 q^{91} -4.87953 q^{93} -15.7764 q^{97} +13.5100 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q - 2 * q^3 - 2 * q^7 + 6 * q^9	$ 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9} - 3 q^{11} - q^{13} - 14 q^{17} + 6 q^{19} + 15 q^{21} - 12 q^{23} - 8 q^{27} + 9 q^{29} - 5 q^{31} + 2 q^{33} - 8 q^{37} - 12 q^{39} + 3 q^{41} - 15 q^{43} - 4 q^{47} + 22 q^{49} - 33 q^{51} + 13 q^{53} - 2 q^{57} + 9 q^{61} - 21 q^{63} + 3 q^{67} - 11 q^{69} - 19 q^{71} + 3 q^{73} - 36 q^{77} + 16 q^{79} + 26 q^{81} - 31 q^{83} + 25 q^{87} + 14 q^{89} - 42 q^{91} + 39 q^{93} - 11 q^{97} + 6 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 2 * q^7 + 6 * q^9 - 3 * q^11 - q^13 - 14 * q^17 + 6 * q^19 + 15 * q^21 - 12 * q^23 - 8 * q^27 + 9 * q^29 - 5 * q^31 + 2 * q^33 - 8 * q^37 - 12 * q^39 + 3 * q^41 - 15 * q^43 - 4 * q^47 + 22 * q^49 - 33 * q^51 + 13 * q^53 - 2 * q^57 + 9 * q^61 - 21 * q^63 + 3 * q^67 - 11 * q^69 - 19 * q^71 + 3 * q^73 - 36 * q^77 + 16 * q^79 + 26 * q^81 - 31 * q^83 + 25 * q^87 + 14 * q^89 - 42 * q^91 + 39 * q^93 - 11 * q^97 + 6 * 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
$3$	−0.848258	−0.489742	−0.244871	−	0.969556i	$-0.578746\pi$
$3$	−0.848258	−0.489742	−0.244871	+	0.969556i	$0.578746\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.74484	−0.659486	−0.329743	−	0.944071i	$-0.606962\pi$
$7$	−1.74484	−0.659486	−0.329743	+	0.944071i	$0.606962\pi$
$8$	0	0
$9$	−2.28046	−0.760153
$10$	0	0
$11$	−5.92425	−1.78623	−0.893115	−	0.449829i	$-0.851485\pi$
$11$	−5.92425	−1.78623	−0.893115	+	0.449829i	$0.851485\pi$
$12$	0	0
$13$	6.78582	1.88205	0.941024	−	0.338339i	$-0.109865\pi$
$13$	6.78582	1.88205	0.941024	+	0.338339i	$0.109865\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.86024	0.451174	0.225587	−	0.974223i	$-0.427570\pi$
$17$	1.86024	0.451174	0.225587	+	0.974223i	$0.427570\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.48007	0.322978
$22$	0	0
$23$	−5.94357	−1.23932	−0.619660	−	0.784871i	$-0.712729\pi$
$23$	−5.94357	−1.23932	−0.619660	+	0.784871i	$0.712729\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.47919	0.862021
$28$	0	0
$29$	3.29977	0.612752	0.306376	−	0.951911i	$-0.400883\pi$
$29$	3.29977	0.612752	0.306376	+	0.951911i	$0.400883\pi$
$30$	0	0
$31$	5.75242	1.03316	0.516582	−	0.856238i	$-0.327204\pi$
$31$	5.75242	1.03316	0.516582	+	0.856238i	$0.327204\pi$
$32$	0	0
$33$	5.02529	0.874791
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.36379	−0.717402	−0.358701	−	0.933453i	$-0.616780\pi$
$37$	−4.36379	−0.717402	−0.358701	+	0.933453i	$0.616780\pi$
$38$	0	0
$39$	−5.75613	−0.921718
$40$	0	0
$41$	7.12500	1.11274	0.556369	−	0.830935i	$-0.312194\pi$
$41$	7.12500	1.11274	0.556369	+	0.830935i	$0.312194\pi$
$42$	0	0
$43$	6.98455	1.06513	0.532567	−	0.846388i	$-0.321227\pi$
$43$	6.98455	1.06513	0.532567	+	0.846388i	$0.321227\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.02529	0.587149	0.293575	−	0.955936i	$-0.405155\pi$
$47$	4.02529	0.587149	0.293575	+	0.955936i	$0.405155\pi$
$48$	0	0
$49$	−3.95555	−0.565078
$50$	0	0
$51$	−1.57796	−0.220959
$52$	0	0
$53$	−9.19015	−1.26236	−0.631182	−	0.775635i	$-0.717430\pi$
$53$	−9.19015	−1.26236	−0.631182	+	0.775635i	$0.717430\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.848258	−0.112354
$58$	0	0
$59$	−2.51553	−0.327494	−0.163747	−	0.986502i	$-0.552358\pi$
$59$	−2.51553	−0.327494	−0.163747	+	0.986502i	$0.552358\pi$
$60$	0	0
$61$	−2.49621	−0.319607	−0.159804	−	0.987149i	$-0.551086\pi$
$61$	−2.49621	−0.319607	−0.159804	+	0.987149i	$0.551086\pi$
$62$	0	0
$63$	3.97903	0.501310
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−6.90485	−0.843561	−0.421781	−	0.906698i	$-0.638595\pi$
$67$	−6.90485	−0.843561	−0.421781	+	0.906698i	$0.638595\pi$
$68$	0	0
$69$	5.04168	0.606947
$70$	0	0
$71$	−1.27288	−0.151063	−0.0755314	−	0.997143i	$-0.524065\pi$
$71$	−1.27288	−0.151063	−0.0755314	+	0.997143i	$0.524065\pi$
$72$	0	0
$73$	12.1217	1.41874	0.709371	−	0.704836i	$-0.248979\pi$
$73$	12.1217	1.41874	0.709371	+	0.704836i	$0.248979\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.3368	1.17799
$78$	0	0
$79$	13.8376	1.55685	0.778424	−	0.627739i	$-0.216020\pi$
$79$	13.8376	1.55685	0.778424	+	0.627739i	$0.216020\pi$
$80$	0	0
$81$	3.04187	0.337985
$82$	0	0
$83$	4.94955	0.543283	0.271642	−	0.962398i	$-0.412433\pi$
$83$	4.94955	0.543283	0.271642	+	0.962398i	$0.412433\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.79906	−0.300091
$88$	0	0
$89$	15.6067	1.65430	0.827151	−	0.561980i	$-0.189960\pi$
$89$	15.6067	1.65430	0.827151	+	0.561980i	$0.189960\pi$
$90$	0	0
$91$	−11.8401	−1.24118
$92$	0	0
$93$	−4.87953	−0.505984
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−15.7764	−1.60185	−0.800924	−	0.598766i	$-0.795658\pi$
$97$	−15.7764	−1.60185	−0.800924	+	0.598766i	$0.795658\pi$
$98$	0	0
$99$	13.5100	1.35781
$100$	0	0
$101$	−6.81483	−0.678101	−0.339050	−	0.940768i	$-0.610106\pi$
$101$	−6.81483	−0.678101	−0.339050	+	0.940768i	$0.610106\pi$
$102$	0	0
$103$	−13.3177	−1.31224	−0.656118	−	0.754659i	$-0.727803\pi$
$103$	−13.3177	−1.31224	−0.656118	+	0.754659i	$0.727803\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.69935	−0.357629	−0.178815	−	0.983883i	$-0.557226\pi$
$107$	−3.69935	−0.357629	−0.178815	+	0.983883i	$0.557226\pi$
$108$	0	0
$109$	12.7753	1.22366	0.611828	−	0.790991i	$-0.290434\pi$
$109$	12.7753	1.22366	0.611828	+	0.790991i	$0.290434\pi$
$110$	0	0
$111$	3.70162	0.351342
$112$	0	0
$113$	−8.81845	−0.829570	−0.414785	−	0.909919i	$-0.636143\pi$
$113$	−8.81845	−0.829570	−0.414785	+	0.909919i	$0.636143\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−15.4748	−1.43064
$118$	0	0
$119$	−3.24581	−0.297543
$120$	0	0
$121$	24.0968	2.19062
$122$	0	0
$123$	−6.04384	−0.544955
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−4.39650	−0.390127	−0.195063	−	0.980791i	$-0.562491\pi$
$127$	−4.39650	−0.390127	−0.195063	+	0.980791i	$0.562491\pi$
$128$	0	0
$129$	−5.92470	−0.521641
$130$	0	0
$131$	−21.0076	−1.83544	−0.917719	−	0.397229i	$-0.869972\pi$
$131$	−21.0076	−1.83544	−0.917719	+	0.397229i	$0.869972\pi$
$132$	0	0
$133$	−1.74484	−0.151296
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.3766	−0.971973	−0.485986	−	0.873966i	$-0.661540\pi$
$137$	−11.3766	−0.971973	−0.485986	+	0.873966i	$0.661540\pi$
$138$	0	0
$139$	−5.18664	−0.439925	−0.219962	−	0.975508i	$-0.570593\pi$
$139$	−5.18664	−0.439925	−0.219962	+	0.975508i	$0.570593\pi$
$140$	0	0
$141$	−3.41449	−0.287552
$142$	0	0
$143$	−40.2009	−3.36177
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.35533	0.276743
$148$	0	0
$149$	0.474871	0.0389030	0.0194515	−	0.999811i	$-0.493808\pi$
$149$	0.474871	0.0389030	0.0194515	+	0.999811i	$0.493808\pi$
$150$	0	0
$151$	−16.6164	−1.35222	−0.676110	−	0.736800i	$-0.736336\pi$
$151$	−16.6164	−1.35222	−0.676110	+	0.736800i	$0.736336\pi$
$152$	0	0
$153$	−4.24220	−0.342962
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.83889	0.545803	0.272901	−	0.962042i	$-0.412017\pi$
$157$	6.83889	0.545803	0.272901	+	0.962042i	$0.412017\pi$
$158$	0	0
$159$	7.79561	0.618232
$160$	0	0
$161$	10.3705	0.817314
$162$	0	0
$163$	−2.42620	−0.190035	−0.0950173	−	0.995476i	$-0.530291\pi$
$163$	−2.42620	−0.190035	−0.0950173	+	0.995476i	$0.530291\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.02699	−0.0794705	−0.0397353	−	0.999210i	$-0.512651\pi$
$167$	−1.02699	−0.0794705	−0.0397353	+	0.999210i	$0.512651\pi$
$168$	0	0
$169$	33.0474	2.54211
$170$	0	0
$171$	−2.28046	−0.174391
$172$	0	0
$173$	−13.2443	−1.00694	−0.503472	−	0.864012i	$-0.667944\pi$
$173$	−13.2443	−1.00694	−0.503472	+	0.864012i	$0.667944\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.13382	0.160387
$178$	0	0
$179$	18.8025	1.40536	0.702681	−	0.711505i	$-0.251986\pi$
$179$	18.8025	1.40536	0.702681	+	0.711505i	$0.251986\pi$
$180$	0	0
$181$	−8.01073	−0.595433	−0.297716	−	0.954654i	$-0.596225\pi$
$181$	−8.01073	−0.595433	−0.297716	+	0.954654i	$0.596225\pi$
$182$	0	0
$183$	2.11743	0.156525
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−11.0205	−0.805901
$188$	0	0
$189$	−7.81545	−0.568490
$190$	0	0
$191$	−23.3198	−1.68736	−0.843682	−	0.536844i	$-0.819616\pi$
$191$	−23.3198	−1.68736	−0.843682	+	0.536844i	$0.819616\pi$
$192$	0	0
$193$	7.55652	0.543930	0.271965	−	0.962307i	$-0.412327\pi$
$193$	7.55652	0.543930	0.271965	+	0.962307i	$0.412327\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−20.5405	−1.46345	−0.731726	−	0.681599i	$-0.761285\pi$
$197$	−20.5405	−1.46345	−0.731726	+	0.681599i	$0.761285\pi$
$198$	0	0
$199$	4.93624	0.349920	0.174960	−	0.984576i	$-0.444020\pi$
$199$	4.93624	0.349920	0.174960	+	0.984576i	$0.444020\pi$
$200$	0	0
$201$	5.85709	0.413127
$202$	0	0
$203$	−5.75756	−0.404102
$204$	0	0
$205$	0	0
$206$	0	0
$207$	13.5541	0.942072
$208$	0	0
$209$	−5.92425	−0.409789
$210$	0	0
$211$	14.6716	1.01003	0.505017	−	0.863109i	$-0.331486\pi$
$211$	14.6716	1.01003	0.505017	+	0.863109i	$0.331486\pi$
$212$	0	0
$213$	1.07973	0.0739818
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−10.0370	−0.681357
$218$	0	0
$219$	−10.2824	−0.694817
$220$	0	0
$221$	12.6233	0.849132
$222$	0	0
$223$	−18.8357	−1.26133	−0.630667	−	0.776054i	$-0.717218\pi$
$223$	−18.8357	−1.26133	−0.630667	+	0.776054i	$0.717218\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.8621	1.65015	0.825076	−	0.565021i	$-0.191132\pi$
$227$	24.8621	1.65015	0.825076	+	0.565021i	$0.191132\pi$
$228$	0	0
$229$	20.2743	1.33976	0.669880	−	0.742469i	$-0.266345\pi$
$229$	20.2743	1.33976	0.669880	+	0.742469i	$0.266345\pi$
$230$	0	0
$231$	−8.76831	−0.576913
$232$	0	0
$233$	−26.9597	−1.76619	−0.883095	−	0.469194i	$-0.844545\pi$
$233$	−26.9597	−1.76619	−0.883095	+	0.469194i	$0.844545\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−11.7378	−0.762453
$238$	0	0
$239$	−12.1013	−0.782767	−0.391384	−	0.920228i	$-0.628003\pi$
$239$	−12.1013	−0.782767	−0.391384	+	0.920228i	$0.628003\pi$
$240$	0	0
$241$	−18.2563	−1.17599	−0.587997	−	0.808863i	$-0.700083\pi$
$241$	−18.2563	−1.17599	−0.587997	+	0.808863i	$0.700083\pi$
$242$	0	0
$243$	−16.0179	−1.02755
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.78582	0.431772
$248$	0	0
$249$	−4.19849	−0.266069
$250$	0	0
$251$	5.65623	0.357018	0.178509	−	0.983938i	$-0.442873\pi$
$251$	5.65623	0.357018	0.178509	+	0.983938i	$0.442873\pi$
$252$	0	0
$253$	35.2112	2.21371
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−8.04957	−0.502119	−0.251059	−	0.967972i	$-0.580779\pi$
$257$	−8.04957	−0.502119	−0.251059	+	0.967972i	$0.580779\pi$
$258$	0	0
$259$	7.61409	0.473116
$260$	0	0
$261$	−7.52500	−0.465786
$262$	0	0
$263$	8.24808	0.508598	0.254299	−	0.967126i	$-0.418155\pi$
$263$	8.24808	0.508598	0.254299	+	0.967126i	$0.418155\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−13.2385	−0.810181
$268$	0	0
$269$	−8.48032	−0.517054	−0.258527	−	0.966004i	$-0.583237\pi$
$269$	−8.48032	−0.517054	−0.258527	+	0.966004i	$0.583237\pi$
$270$	0	0
$271$	−30.2291	−1.83629	−0.918145	−	0.396244i	$-0.870313\pi$
$271$	−30.2291	−1.83629	−0.918145	+	0.396244i	$0.870313\pi$
$272$	0	0
$273$	10.0435	0.607860
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.42364	0.145622	0.0728111	−	0.997346i	$-0.476803\pi$
$277$	2.42364	0.145622	0.0728111	+	0.997346i	$0.476803\pi$
$278$	0	0
$279$	−13.1181	−0.785363
$280$	0	0
$281$	−20.6355	−1.23101	−0.615504	−	0.788133i	$-0.711048\pi$
$281$	−20.6355	−1.23101	−0.615504	+	0.788133i	$0.711048\pi$
$282$	0	0
$283$	30.1197	1.79043	0.895215	−	0.445635i	$-0.147022\pi$
$283$	30.1197	1.79043	0.895215	+	0.445635i	$0.147022\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.4320	−0.733835
$288$	0	0
$289$	−13.5395	−0.796442
$290$	0	0
$291$	13.3824	0.784492
$292$	0	0
$293$	11.8274	0.690963	0.345481	−	0.938426i	$-0.387716\pi$
$293$	11.8274	0.690963	0.345481	+	0.938426i	$0.387716\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−26.5359	−1.53977
$298$	0	0
$299$	−40.3320	−2.33246
$300$	0	0
$301$	−12.1869	−0.702441
$302$	0	0
$303$	5.78073	0.332094
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−7.22041	−0.412091	−0.206045	−	0.978542i	$-0.566059\pi$
$307$	−7.22041	−0.412091	−0.206045	+	0.978542i	$0.566059\pi$
$308$	0	0
$309$	11.2969	0.642657
$310$	0	0
$311$	10.1574	0.575975	0.287987	−	0.957634i	$-0.407014\pi$
$311$	10.1574	0.575975	0.287987	+	0.957634i	$0.407014\pi$
$312$	0	0
$313$	−6.03525	−0.341132	−0.170566	−	0.985346i	$-0.554560\pi$
$313$	−6.03525	−0.341132	−0.170566	+	0.985346i	$0.554560\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−30.5978	−1.71854	−0.859271	−	0.511520i	$-0.829083\pi$
$317$	−30.5978	−1.71854	−0.859271	+	0.511520i	$0.829083\pi$
$318$	0	0
$319$	−19.5487	−1.09452
$320$	0	0
$321$	3.13800	0.175146
$322$	0	0
$323$	1.86024	0.103507
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−10.8368	−0.599276
$328$	0	0
$329$	−7.02348	−0.387217
$330$	0	0
$331$	−19.1478	−1.05246	−0.526228	−	0.850343i	$-0.676394\pi$
$331$	−19.1478	−1.05246	−0.526228	+	0.850343i	$0.676394\pi$
$332$	0	0
$333$	9.95143	0.545335
$334$	0	0
$335$	0	0
$336$	0	0
$337$	10.8090	0.588804	0.294402	−	0.955682i	$-0.404880\pi$
$337$	10.8090	0.588804	0.294402	+	0.955682i	$0.404880\pi$
$338$	0	0
$339$	7.48032	0.406275
$340$	0	0
$341$	−34.0788	−1.84547
$342$	0	0
$343$	19.1156	1.03215
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−19.0847	−1.02452	−0.512261	−	0.858830i	$-0.671192\pi$
$347$	−19.0847	−1.02452	−0.512261	+	0.858830i	$0.671192\pi$
$348$	0	0
$349$	5.18421	0.277504	0.138752	−	0.990327i	$-0.455691\pi$
$349$	5.18421	0.277504	0.138752	+	0.990327i	$0.455691\pi$
$350$	0	0
$351$	30.3950	1.62236
$352$	0	0
$353$	−15.6398	−0.832423	−0.416212	−	0.909268i	$-0.636642\pi$
$353$	−15.6398	−0.832423	−0.416212	+	0.909268i	$0.636642\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.75329	0.145719
$358$	0	0
$359$	17.4768	0.922392	0.461196	−	0.887298i	$-0.347420\pi$
$359$	17.4768	0.922392	0.461196	+	0.887298i	$0.347420\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−20.4403	−1.07284
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2.99560	0.156369	0.0781845	−	0.996939i	$-0.475088\pi$
$367$	2.99560	0.156369	0.0781845	+	0.996939i	$0.475088\pi$
$368$	0	0
$369$	−16.2483	−0.845852
$370$	0	0
$371$	16.0353	0.832511
$372$	0	0
$373$	−3.76504	−0.194946	−0.0974731	−	0.995238i	$-0.531076\pi$
$373$	−3.76504	−0.194946	−0.0974731	+	0.995238i	$0.531076\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	22.3917	1.15323
$378$	0	0
$379$	10.5503	0.541933	0.270967	−	0.962589i	$-0.412657\pi$
$379$	10.5503	0.541933	0.270967	+	0.962589i	$0.412657\pi$
$380$	0	0
$381$	3.72937	0.191061
$382$	0	0
$383$	6.25532	0.319632	0.159816	−	0.987147i	$-0.448910\pi$
$383$	6.25532	0.319632	0.159816	+	0.987147i	$0.448910\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−15.9280	−0.809665
$388$	0	0
$389$	12.8669	0.652379	0.326190	−	0.945304i	$-0.394235\pi$
$389$	12.8669	0.652379	0.326190	+	0.945304i	$0.394235\pi$
$390$	0	0
$391$	−11.0565	−0.559149
$392$	0	0
$393$	17.8198	0.898891
$394$	0	0
$395$	0	0
$396$	0	0
$397$	28.4649	1.42861	0.714306	−	0.699834i	$-0.246743\pi$
$397$	28.4649	1.42861	0.714306	+	0.699834i	$0.246743\pi$
$398$	0	0
$399$	1.48007	0.0740962
$400$	0	0
$401$	17.2837	0.863106	0.431553	−	0.902088i	$-0.357966\pi$
$401$	17.2837	0.863106	0.431553	+	0.902088i	$0.357966\pi$
$402$	0	0
$403$	39.0349	1.94447
$404$	0	0
$405$	0	0
$406$	0	0
$407$	25.8522	1.28144
$408$	0	0
$409$	−28.3677	−1.40269	−0.701346	−	0.712821i	$-0.747417\pi$
$409$	−28.3677	−1.40269	−0.701346	+	0.712821i	$0.747417\pi$
$410$	0	0
$411$	9.65033	0.476016
$412$	0	0
$413$	4.38918	0.215978
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.39961	0.215450
$418$	0	0
$419$	10.0821	0.492542	0.246271	−	0.969201i	$-0.420795\pi$
$419$	10.0821	0.492542	0.246271	+	0.969201i	$0.420795\pi$
$420$	0	0
$421$	20.3802	0.993269	0.496635	−	0.867960i	$-0.334569\pi$
$421$	20.3802	0.993269	0.496635	+	0.867960i	$0.334569\pi$
$422$	0	0
$423$	−9.17952	−0.446323
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.35548	0.210777
$428$	0	0
$429$	34.1008	1.64640
$430$	0	0
$431$	−18.1090	−0.872280	−0.436140	−	0.899879i	$-0.643655\pi$
$431$	−18.1090	−0.872280	−0.436140	+	0.899879i	$0.643655\pi$
$432$	0	0
$433$	−33.8792	−1.62813	−0.814066	−	0.580772i	$-0.802751\pi$
$433$	−33.8792	−1.62813	−0.814066	+	0.580772i	$0.802751\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.94357	−0.284319
$438$	0	0
$439$	−21.7440	−1.03779	−0.518893	−	0.854839i	$-0.673656\pi$
$439$	−21.7440	−1.03779	−0.518893	+	0.854839i	$0.673656\pi$
$440$	0	0
$441$	9.02047	0.429546
$442$	0	0
$443$	−20.6461	−0.980924	−0.490462	−	0.871463i	$-0.663172\pi$
$443$	−20.6461	−0.980924	−0.490462	+	0.871463i	$0.663172\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−0.402813	−0.0190524
$448$	0	0
$449$	16.4288	0.775322	0.387661	−	0.921802i	$-0.373283\pi$
$449$	16.4288	0.775322	0.387661	+	0.921802i	$0.373283\pi$
$450$	0	0
$451$	−42.2103	−1.98761
$452$	0	0
$453$	14.0950	0.662239
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−19.5543	−0.914711	−0.457355	−	0.889284i	$-0.651203\pi$
$457$	−19.5543	−0.914711	−0.457355	+	0.889284i	$0.651203\pi$
$458$	0	0
$459$	8.33237	0.388922
$460$	0	0
$461$	15.4065	0.717554	0.358777	−	0.933423i	$-0.383194\pi$
$461$	15.4065	0.717554	0.358777	+	0.933423i	$0.383194\pi$
$462$	0	0
$463$	−4.44151	−0.206414	−0.103207	−	0.994660i	$-0.532910\pi$
$463$	−4.44151	−0.206414	−0.103207	+	0.994660i	$0.532910\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−12.1240	−0.561030	−0.280515	−	0.959850i	$-0.590505\pi$
$467$	−12.1240	−0.561030	−0.280515	+	0.959850i	$0.590505\pi$
$468$	0	0
$469$	12.0478	0.556317
$470$	0	0
$471$	−5.80114	−0.267303
$472$	0	0
$473$	−41.3783	−1.90257
$474$	0	0
$475$	0	0
$476$	0	0
$477$	20.9577	0.959589
$478$	0	0
$479$	42.5961	1.94626	0.973132	−	0.230249i	$-0.0739539\pi$
$479$	42.5961	1.94626	0.973132	+	0.230249i	$0.0739539\pi$
$480$	0	0
$481$	−29.6119	−1.35019
$482$	0	0
$483$	−8.79690	−0.400273
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−19.3424	−0.876489	−0.438245	−	0.898856i	$-0.644400\pi$
$487$	−19.3424	−0.876489	−0.438245	+	0.898856i	$0.644400\pi$
$488$	0	0
$489$	2.05804	0.0930679
$490$	0	0
$491$	−12.2941	−0.554823	−0.277412	−	0.960751i	$-0.589477\pi$
$491$	−12.2941	−0.554823	−0.277412	+	0.960751i	$0.589477\pi$
$492$	0	0
$493$	6.13837	0.276458
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.22096	0.0996238
$498$	0	0
$499$	−1.75085	−0.0783789	−0.0391894	−	0.999232i	$-0.512478\pi$
$499$	−1.75085	−0.0783789	−0.0391894	+	0.999232i	$0.512478\pi$
$500$	0	0
$501$	0.871149	0.0389201
$502$	0	0
$503$	−2.13252	−0.0950843	−0.0475422	−	0.998869i	$-0.515139\pi$
$503$	−2.13252	−0.0950843	−0.0475422	+	0.998869i	$0.515139\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−28.0327	−1.24498
$508$	0	0
$509$	−35.0078	−1.55169	−0.775846	−	0.630923i	$-0.782676\pi$
$509$	−35.0078	−1.55169	−0.775846	+	0.630923i	$0.782676\pi$
$510$	0	0
$511$	−21.1504	−0.935640
$512$	0	0
$513$	4.47919	0.197761
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−23.8469	−1.04878
$518$	0	0
$519$	11.2346	0.493143
$520$	0	0
$521$	−15.5581	−0.681613	−0.340807	−	0.940133i	$-0.610700\pi$
$521$	−15.5581	−0.681613	−0.340807	+	0.940133i	$0.610700\pi$
$522$	0	0
$523$	−9.91872	−0.433715	−0.216858	−	0.976203i	$-0.569581\pi$
$523$	−9.91872	−0.433715	−0.216858	+	0.976203i	$0.569581\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.7009	0.466137
$528$	0	0
$529$	12.3260	0.535913
$530$	0	0
$531$	5.73656	0.248945
$532$	0	0
$533$	48.3490	2.09423
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−15.9493	−0.688264
$538$	0	0
$539$	23.4337	1.00936
$540$	0	0
$541$	41.7150	1.79347	0.896734	−	0.442569i	$-0.145933\pi$
$541$	41.7150	1.79347	0.896734	+	0.442569i	$0.145933\pi$
$542$	0	0
$543$	6.79516	0.291608
$544$	0	0
$545$	0	0
$546$	0	0
$547$	18.9230	0.809089	0.404545	−	0.914518i	$-0.367430\pi$
$547$	18.9230	0.809089	0.404545	+	0.914518i	$0.367430\pi$
$548$	0	0
$549$	5.69251	0.242951
$550$	0	0
$551$	3.29977	0.140575
$552$	0	0
$553$	−24.1443	−1.02672
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.02768	−0.0859157	−0.0429579	−	0.999077i	$-0.513678\pi$
$557$	−2.02768	−0.0859157	−0.0429579	+	0.999077i	$0.513678\pi$
$558$	0	0
$559$	47.3960	2.00464
$560$	0	0
$561$	9.34825	0.394684
$562$	0	0
$563$	−20.7497	−0.874497	−0.437248	−	0.899341i	$-0.644047\pi$
$563$	−20.7497	−0.874497	−0.437248	+	0.899341i	$0.644047\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−5.30756	−0.222897
$568$	0	0
$569$	18.2161	0.763659	0.381830	−	0.924233i	$-0.375294\pi$
$569$	18.2161	0.763659	0.381830	+	0.924233i	$0.375294\pi$
$570$	0	0
$571$	35.0512	1.46685	0.733424	−	0.679771i	$-0.237921\pi$
$571$	35.0512	1.46685	0.733424	+	0.679771i	$0.237921\pi$
$572$	0	0
$573$	19.7812	0.826372
$574$	0	0
$575$	0	0
$576$	0	0
$577$	21.6181	0.899973	0.449987	−	0.893035i	$-0.351429\pi$
$577$	21.6181	0.899973	0.449987	+	0.893035i	$0.351429\pi$
$578$	0	0
$579$	−6.40987	−0.266385
$580$	0	0
$581$	−8.63615	−0.358288
$582$	0	0
$583$	54.4448	2.25487
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.3305	−0.839130	−0.419565	−	0.907725i	$-0.637817\pi$
$587$	−20.3305	−0.839130	−0.419565	+	0.907725i	$0.637817\pi$
$588$	0	0
$589$	5.75242	0.237024
$590$	0	0
$591$	17.4237	0.716714
$592$	0	0
$593$	−2.11878	−0.0870077	−0.0435039	−	0.999053i	$-0.513852\pi$
$593$	−2.11878	−0.0870077	−0.0435039	+	0.999053i	$0.513852\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−4.18720	−0.171371
$598$	0	0
$599$	4.08633	0.166963	0.0834814	−	0.996509i	$-0.473396\pi$
$599$	4.08633	0.166963	0.0834814	+	0.996509i	$0.473396\pi$
$600$	0	0
$601$	−23.7445	−0.968557	−0.484279	−	0.874914i	$-0.660918\pi$
$601$	−23.7445	−0.968557	−0.484279	+	0.874914i	$0.660918\pi$
$602$	0	0
$603$	15.7462	0.641236
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−23.5940	−0.957653	−0.478826	−	0.877910i	$-0.658938\pi$
$607$	−23.5940	−0.957653	−0.478826	+	0.877910i	$0.658938\pi$
$608$	0	0
$609$	4.88390	0.197905
$610$	0	0
$611$	27.3149	1.10504
$612$	0	0
$613$	−48.4877	−1.95840	−0.979200	−	0.202897i	$-0.934964\pi$
$613$	−48.4877	−1.95840	−0.979200	+	0.202897i	$0.934964\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−31.1353	−1.25346	−0.626730	−	0.779236i	$-0.715607\pi$
$617$	−31.1353	−1.25346	−0.626730	+	0.779236i	$0.715607\pi$
$618$	0	0
$619$	−30.1220	−1.21071	−0.605353	−	0.795957i	$-0.706968\pi$
$619$	−30.1220	−1.21071	−0.605353	+	0.795957i	$0.706968\pi$
$620$	0	0
$621$	−26.6224	−1.06832
$622$	0	0
$623$	−27.2310	−1.09099
$624$	0	0
$625$	0	0
$626$	0	0
$627$	5.02529	0.200691
$628$	0	0
$629$	−8.11769	−0.323673
$630$	0	0
$631$	6.15070	0.244855	0.122428	−	0.992477i	$-0.460932\pi$
$631$	6.15070	0.244855	0.122428	+	0.992477i	$0.460932\pi$
$632$	0	0
$633$	−12.4453	−0.494656
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−26.8417	−1.06351
$638$	0	0
$639$	2.90275	0.114831
$640$	0	0
$641$	−20.9554	−0.827690	−0.413845	−	0.910347i	$-0.635814\pi$
$641$	−20.9554	−0.827690	−0.413845	+	0.910347i	$0.635814\pi$
$642$	0	0
$643$	24.6309	0.971349	0.485675	−	0.874140i	$-0.338574\pi$
$643$	24.6309	0.971349	0.485675	+	0.874140i	$0.338574\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16.1496	−0.634906	−0.317453	−	0.948274i	$-0.602828\pi$
$647$	−16.1496	−0.634906	−0.317453	+	0.948274i	$0.602828\pi$
$648$	0	0
$649$	14.9026	0.584979
$650$	0	0
$651$	8.51398	0.333689
$652$	0	0
$653$	33.3511	1.30513	0.652565	−	0.757733i	$-0.273693\pi$
$653$	33.3511	1.30513	0.652565	+	0.757733i	$0.273693\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−27.6431	−1.07846
$658$	0	0
$659$	11.1727	0.435226	0.217613	−	0.976035i	$-0.430173\pi$
$659$	11.1727	0.435226	0.217613	+	0.976035i	$0.430173\pi$
$660$	0	0
$661$	−33.5484	−1.30488	−0.652440	−	0.757840i	$-0.726255\pi$
$661$	−33.5484	−1.30488	−0.652440	+	0.757840i	$0.726255\pi$
$662$	0	0
$663$	−10.7078	−0.415856
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−19.6124	−0.759396
$668$	0	0
$669$	15.9775	0.617728
$670$	0	0
$671$	14.7882	0.570892
$672$	0	0
$673$	4.13528	0.159403	0.0797017	−	0.996819i	$-0.474603\pi$
$673$	4.13528	0.159403	0.0797017	+	0.996819i	$0.474603\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−17.7150	−0.680842	−0.340421	−	0.940273i	$-0.610570\pi$
$677$	−17.7150	−0.680842	−0.340421	+	0.940273i	$0.610570\pi$
$678$	0	0
$679$	27.5272	1.05640
$680$	0	0
$681$	−21.0894	−0.808149
$682$	0	0
$683$	−13.1004	−0.501273	−0.250637	−	0.968081i	$-0.580640\pi$
$683$	−13.1004	−0.501273	−0.250637	+	0.968081i	$0.580640\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−17.1978	−0.656137
$688$	0	0
$689$	−62.3627	−2.37583
$690$	0	0
$691$	18.6952	0.711200	0.355600	−	0.934638i	$-0.384276\pi$
$691$	18.6952	0.711200	0.355600	+	0.934638i	$0.384276\pi$
$692$	0	0
$693$	−23.5728	−0.895455
$694$	0	0
$695$	0	0
$696$	0	0
$697$	13.2542	0.502039
$698$	0	0
$699$	22.8688	0.864977
$700$	0	0
$701$	−33.3160	−1.25833	−0.629164	−	0.777273i	$-0.716603\pi$
$701$	−33.3160	−1.25833	−0.629164	+	0.777273i	$0.716603\pi$
$702$	0	0
$703$	−4.36379	−0.164583
$704$	0	0
$705$	0	0
$706$	0	0
$707$	11.8908	0.447198
$708$	0	0
$709$	11.1888	0.420206	0.210103	−	0.977679i	$-0.432620\pi$
$709$	11.1888	0.420206	0.210103	+	0.977679i	$0.432620\pi$
$710$	0	0
$711$	−31.5560	−1.18344
$712$	0	0
$713$	−34.1899	−1.28042
$714$	0	0
$715$	0	0
$716$	0	0
$717$	10.2650	0.383354
$718$	0	0
$719$	−18.3734	−0.685212	−0.342606	−	0.939479i	$-0.611310\pi$
$719$	−18.3734	−0.685212	−0.342606	+	0.939479i	$0.611310\pi$
$720$	0	0
$721$	23.2373	0.865401
$722$	0	0
$723$	15.4861	0.575934
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−8.69538	−0.322494	−0.161247	−	0.986914i	$-0.551552\pi$
$727$	−8.69538	−0.322494	−0.161247	+	0.986914i	$0.551552\pi$
$728$	0	0
$729$	4.46167	0.165247
$730$	0	0
$731$	12.9929	0.480562
$732$	0	0
$733$	−41.9016	−1.54767	−0.773836	−	0.633386i	$-0.781665\pi$
$733$	−41.9016	−1.54767	−0.773836	+	0.633386i	$0.781665\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	40.9061	1.50679
$738$	0	0
$739$	−49.4636	−1.81955	−0.909774	−	0.415104i	$-0.863745\pi$
$739$	−49.4636	−1.81955	−0.909774	+	0.415104i	$0.863745\pi$
$740$	0	0
$741$	−5.75613	−0.211457
$742$	0	0
$743$	31.1266	1.14192	0.570962	−	0.820977i	$-0.306570\pi$
$743$	31.1266	1.14192	0.570962	+	0.820977i	$0.306570\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−11.2872	−0.412978
$748$	0	0
$749$	6.45475	0.235852
$750$	0	0
$751$	−6.27888	−0.229119	−0.114560	−	0.993416i	$-0.536546\pi$
$751$	−6.27888	−0.229119	−0.114560	+	0.993416i	$0.536546\pi$
$752$	0	0
$753$	−4.79794	−0.174847
$754$	0	0
$755$	0	0
$756$	0	0
$757$	38.1507	1.38661	0.693304	−	0.720645i	$-0.256154\pi$
$757$	38.1507	1.38661	0.693304	+	0.720645i	$0.256154\pi$
$758$	0	0
$759$	−29.8682	−1.08415
$760$	0	0
$761$	−2.59360	−0.0940180	−0.0470090	−	0.998894i	$-0.514969\pi$
$761$	−2.59360	−0.0940180	−0.0470090	+	0.998894i	$0.514969\pi$
$762$	0	0
$763$	−22.2909	−0.806984
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−17.0699	−0.616359
$768$	0	0
$769$	−52.8561	−1.90604	−0.953020	−	0.302908i	$-0.902043\pi$
$769$	−52.8561	−1.90604	−0.953020	+	0.302908i	$0.902043\pi$
$770$	0	0
$771$	6.82811	0.245908
$772$	0	0
$773$	21.1684	0.761374	0.380687	−	0.924704i	$-0.375688\pi$
$773$	21.1684	0.761374	0.380687	+	0.924704i	$0.375688\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−6.45871	−0.231705
$778$	0	0
$779$	7.12500	0.255280
$780$	0	0
$781$	7.54085	0.269833
$782$	0	0
$783$	14.7803	0.528205
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−48.1097	−1.71492	−0.857462	−	0.514547i	$-0.827960\pi$
$787$	−48.1097	−1.71492	−0.857462	+	0.514547i	$0.827960\pi$
$788$	0	0
$789$	−6.99650	−0.249082
$790$	0	0
$791$	15.3867	0.547090
$792$	0	0
$793$	−16.9389	−0.601517
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.3510	0.508338	0.254169	−	0.967160i	$-0.418198\pi$
$797$	14.3510	0.508338	0.254169	+	0.967160i	$0.418198\pi$
$798$	0	0
$799$	7.48801	0.264907
$800$	0	0
$801$	−35.5903	−1.25752
$802$	0	0
$803$	−71.8122	−2.53420
$804$	0	0
$805$	0	0
$806$	0	0
$807$	7.19350	0.253223
$808$	0	0
$809$	−28.2263	−0.992384	−0.496192	−	0.868213i	$-0.665269\pi$
$809$	−28.2263	−0.992384	−0.496192	+	0.868213i	$0.665269\pi$
$810$	0	0
$811$	17.7490	0.623250	0.311625	−	0.950205i	$-0.399127\pi$
$811$	17.7490	0.623250	0.311625	+	0.950205i	$0.399127\pi$
$812$	0	0
$813$	25.6421	0.899308
$814$	0	0
$815$	0	0
$816$	0	0
$817$	6.98455	0.244359
$818$	0	0
$819$	27.0010	0.943490
$820$	0	0
$821$	−32.2066	−1.12402	−0.562010	−	0.827130i	$-0.689972\pi$
$821$	−32.2066	−1.12402	−0.562010	+	0.827130i	$0.689972\pi$
$822$	0	0
$823$	27.0278	0.942130	0.471065	−	0.882099i	$-0.343870\pi$
$823$	27.0278	0.942130	0.471065	+	0.882099i	$0.343870\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−41.2334	−1.43383	−0.716913	−	0.697163i	$-0.754445\pi$
$827$	−41.2334	−1.43383	−0.716913	+	0.697163i	$0.754445\pi$
$828$	0	0
$829$	−22.0919	−0.767285	−0.383642	−	0.923482i	$-0.625330\pi$
$829$	−22.0919	−0.767285	−0.383642	+	0.923482i	$0.625330\pi$
$830$	0	0
$831$	−2.05587	−0.0713173
$832$	0	0
$833$	−7.35827	−0.254949
$834$	0	0
$835$	0	0
$836$	0	0
$837$	25.7662	0.890609
$838$	0	0
$839$	−22.1096	−0.763310	−0.381655	−	0.924305i	$-0.624646\pi$
$839$	−22.1096	−0.763310	−0.381655	+	0.924305i	$0.624646\pi$
$840$	0	0
$841$	−18.1115	−0.624534
$842$	0	0
$843$	17.5042	0.602877
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−42.0449	−1.44468
$848$	0	0
$849$	−25.5493	−0.876848
$850$	0	0
$851$	25.9365	0.889090
$852$	0	0
$853$	24.0868	0.824717	0.412359	−	0.911022i	$-0.364705\pi$
$853$	24.0868	0.824717	0.412359	+	0.911022i	$0.364705\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−35.7836	−1.22235	−0.611173	−	0.791497i	$-0.709302\pi$
$857$	−35.7836	−1.22235	−0.611173	+	0.791497i	$0.709302\pi$
$858$	0	0
$859$	16.4286	0.560538	0.280269	−	0.959922i	$-0.409576\pi$
$859$	16.4286	0.560538	0.280269	+	0.959922i	$0.409576\pi$
$860$	0	0
$861$	10.5455	0.359390
$862$	0	0
$863$	16.2841	0.554319	0.277159	−	0.960824i	$-0.410607\pi$
$863$	16.2841	0.554319	0.277159	+	0.960824i	$0.410607\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	11.4850	0.390051
$868$	0	0
$869$	−81.9772	−2.78089
$870$	0	0
$871$	−46.8551	−1.58762
$872$	0	0
$873$	35.9774	1.21765
$874$	0	0
$875$	0	0
$876$	0	0
$877$	57.3960	1.93813	0.969063	−	0.246814i	$-0.0793835\pi$
$877$	57.3960	1.93813	0.969063	+	0.246814i	$0.0793835\pi$
$878$	0	0
$879$	−10.0327	−0.338393
$880$	0	0
$881$	51.6832	1.74125	0.870625	−	0.491947i	$-0.163715\pi$
$881$	51.6832	1.74125	0.870625	+	0.491947i	$0.163715\pi$
$882$	0	0
$883$	32.8329	1.10492	0.552458	−	0.833541i	$-0.313690\pi$
$883$	32.8329	1.10492	0.552458	+	0.833541i	$0.313690\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7.04248	0.236463	0.118232	−	0.992986i	$-0.462277\pi$
$887$	7.04248	0.236463	0.118232	+	0.992986i	$0.462277\pi$
$888$	0	0
$889$	7.67118	0.257283
$890$	0	0
$891$	−18.0208	−0.603719
$892$	0	0
$893$	4.02529	0.134701
$894$	0	0
$895$	0	0
$896$	0	0
$897$	34.2119	1.14230
$898$	0	0
$899$	18.9817	0.633074
$900$	0	0
$901$	−17.0959	−0.569546
$902$	0	0
$903$	10.3376	0.344015
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1.94068	−0.0644391	−0.0322195	−	0.999481i	$-0.510258\pi$
$907$	−1.94068	−0.0644391	−0.0322195	+	0.999481i	$0.510258\pi$
$908$	0	0
$909$	15.5409	0.515460
$910$	0	0
$911$	57.4852	1.90457	0.952284	−	0.305212i	$-0.0987274\pi$
$911$	57.4852	1.90457	0.952284	+	0.305212i	$0.0987274\pi$
$912$	0	0
$913$	−29.3224	−0.970429
$914$	0	0
$915$	0	0
$916$	0	0
$917$	36.6547	1.21045
$918$	0	0
$919$	−49.4475	−1.63112	−0.815561	−	0.578672i	$-0.803571\pi$
$919$	−49.4475	−1.63112	−0.815561	+	0.578672i	$0.803571\pi$
$920$	0	0
$921$	6.12477	0.201818
$922$	0	0
$923$	−8.63753	−0.284308
$924$	0	0
$925$	0	0
$926$	0	0
$927$	30.3705	0.997500
$928$	0	0
$929$	43.8092	1.43733	0.718667	−	0.695354i	$-0.244753\pi$
$929$	43.8092	1.43733	0.718667	+	0.695354i	$0.244753\pi$
$930$	0	0
$931$	−3.95555	−0.129638
$932$	0	0
$933$	−8.61612	−0.282079
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−26.4692	−0.864712	−0.432356	−	0.901703i	$-0.642318\pi$
$937$	−26.4692	−0.864712	−0.432356	+	0.901703i	$0.642318\pi$
$938$	0	0
$939$	5.11945	0.167067
$940$	0	0
$941$	−1.38141	−0.0450327	−0.0225164	−	0.999746i	$-0.507168\pi$
$941$	−1.38141	−0.0450327	−0.0225164	+	0.999746i	$0.507168\pi$
$942$	0	0
$943$	−42.3479	−1.37904
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.40723	0.0457289	0.0228645	−	0.999739i	$-0.492721\pi$
$947$	1.40723	0.0457289	0.0228645	+	0.999739i	$0.492721\pi$
$948$	0	0
$949$	82.2559	2.67014
$950$	0	0
$951$	25.9548	0.841642
$952$	0	0
$953$	−22.5899	−0.731760	−0.365880	−	0.930662i	$-0.619232\pi$
$953$	−22.5899	−0.731760	−0.365880	+	0.930662i	$0.619232\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	16.5823	0.536031
$958$	0	0
$959$	19.8504	0.641002
$960$	0	0
$961$	2.09029	0.0674287
$962$	0	0
$963$	8.43621	0.271853
$964$	0	0
$965$	0	0
$966$	0	0
$967$	33.8866	1.08972	0.544860	−	0.838527i	$-0.316583\pi$
$967$	33.8866	1.08972	0.544860	+	0.838527i	$0.316583\pi$
$968$	0	0
$969$	−1.57796	−0.0506915
$970$	0	0
$971$	−19.0982	−0.612890	−0.306445	−	0.951888i	$-0.599140\pi$
$971$	−19.0982	−0.612890	−0.306445	+	0.951888i	$0.599140\pi$
$972$	0	0
$973$	9.04983	0.290124
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−35.4149	−1.13302	−0.566511	−	0.824054i	$-0.691707\pi$
$977$	−35.4149	−1.13302	−0.566511	+	0.824054i	$0.691707\pi$
$978$	0	0
$979$	−92.4578	−2.95496
$980$	0	0
$981$	−29.1336	−0.930166
$982$	0	0
$983$	37.3537	1.19140	0.595700	−	0.803207i	$-0.296875\pi$
$983$	37.3537	1.19140	0.595700	+	0.803207i	$0.296875\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	5.95772	0.189636
$988$	0	0
$989$	−41.5132	−1.32004
$990$	0	0
$991$	−21.4720	−0.682082	−0.341041	−	0.940048i	$-0.610779\pi$
$991$	−21.4720	−0.682082	−0.341041	+	0.940048i	$0.610779\pi$
$992$	0	0
$993$	16.2422	0.515432
$994$	0	0
$995$	0	0
$996$	0	0
$997$	20.3410	0.644204	0.322102	−	0.946705i	$-0.395611\pi$
$997$	20.3410	0.644204	0.322102	+	0.946705i	$0.395611\pi$
$998$	0	0
$999$	−19.5462	−0.618415

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.ch.1.3		6
4.3	odd	2		3800.2.a.bc.1.4	yes	6
5.4	even	2		7600.2.a.cl.1.4		6
20.3	even	4		3800.2.d.q.3649.8		12
20.7	even	4		3800.2.d.q.3649.5		12
20.19	odd	2		3800.2.a.ba.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.ba.1.3	✓	6	20.19	odd	2
3800.2.a.bc.1.4	yes	6	4.3	odd	2
3800.2.d.q.3649.5		12	20.7	even	4
3800.2.d.q.3649.8		12	20.3	even	4
7600.2.a.ch.1.3		6	1.1	even	1	trivial
7600.2.a.cl.1.4		6	5.4	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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q-0.848258 q^{3} -1.74484 q^{7} -2.28046 q^{9} +O(q^{10})\)	\(q-0.848258 q^{3} -1.74484 q^{7} -2.28046 q^{9} -5.92425 q^{11} +6.78582 q^{13} +1.86024 q^{17} +1.00000 q^{19} +1.48007 q^{21} -5.94357 q^{23} +4.47919 q^{27} +3.29977 q^{29} +5.75242 q^{31} +5.02529 q^{33} -4.36379 q^{37} -5.75613 q^{39} +7.12500 q^{41} +6.98455 q^{43} +4.02529 q^{47} -3.95555 q^{49} -1.57796 q^{51} -9.19015 q^{53} -0.848258 q^{57} -2.51553 q^{59} -2.49621 q^{61} +3.97903 q^{63} -6.90485 q^{67} +5.04168 q^{69} -1.27288 q^{71} +12.1217 q^{73} +10.3368 q^{77} +13.8376 q^{79} +3.04187 q^{81} +4.94955 q^{83} -2.79906 q^{87} +15.6067 q^{89} -11.8401 q^{91} -4.87953 q^{93} -15.7764 q^{97} +13.5100 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q - 2 * q^3 - 2 * q^7 + 6 * q^9	\( 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9} - 3 q^{11} - q^{13} - 14 q^{17} + 6 q^{19} + 15 q^{21} - 12 q^{23} - 8 q^{27} + 9 q^{29} - 5 q^{31} + 2 q^{33} - 8 q^{37} - 12 q^{39} + 3 q^{41} - 15 q^{43} - 4 q^{47} + 22 q^{49} - 33 q^{51} + 13 q^{53} - 2 q^{57} + 9 q^{61} - 21 q^{63} + 3 q^{67} - 11 q^{69} - 19 q^{71} + 3 q^{73} - 36 q^{77} + 16 q^{79} + 26 q^{81} - 31 q^{83} + 25 q^{87} + 14 q^{89} - 42 q^{91} + 39 q^{93} - 11 q^{97} + 6 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 2 * q^7 + 6 * q^9 - 3 * q^11 - q^13 - 14 * q^17 + 6 * q^19 + 15 * q^21 - 12 * q^23 - 8 * q^27 + 9 * q^29 - 5 * q^31 + 2 * q^33 - 8 * q^37 - 12 * q^39 + 3 * q^41 - 15 * q^43 - 4 * q^47 + 22 * q^49 - 33 * q^51 + 13 * q^53 - 2 * q^57 + 9 * q^61 - 21 * q^63 + 3 * q^67 - 11 * q^69 - 19 * q^71 + 3 * q^73 - 36 * q^77 + 16 * q^79 + 26 * q^81 - 31 * q^83 + 25 * q^87 + 14 * q^89 - 42 * q^91 + 39 * q^93 - 11 * q^97 + 6 * q^99

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