LMFDB - Embedded newform 3808.2.a.c.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3808,2,Mod(1,3808)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3808, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3808.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3808 = 2^{5} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3808.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$30.4070330897$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.17557$ of defining polynomial
Character	$\chi$	$=$	3808.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.17557 q^{3} -2.17557 q^{5} -1.00000 q^{7} -1.61803 q^{9} +2.52015 q^{11} +4.02967 q^{13} +2.55754 q^{15} +1.00000 q^{17} -5.53077 q^{19} +1.17557 q^{21} +4.07768 q^{23} -0.266893 q^{25} +5.42882 q^{27} -4.42412 q^{29} -2.37425 q^{31} -2.96261 q^{33} +2.17557 q^{35} -6.64584 q^{37} -4.73716 q^{39} +7.85108 q^{41} +9.17848 q^{43} +3.52015 q^{45} +12.6755 q^{47} +1.00000 q^{49} -1.17557 q^{51} -3.95012 q^{53} -5.48276 q^{55} +6.50181 q^{57} -14.9655 q^{59} +13.4252 q^{61} +1.61803 q^{63} -8.76684 q^{65} -2.98751 q^{67} -4.79360 q^{69} +7.86067 q^{71} -2.32553 q^{73} +0.313752 q^{75} -2.52015 q^{77} +1.83099 q^{79} -1.52786 q^{81} +15.0912 q^{83} -2.17557 q^{85} +5.20087 q^{87} +0.541390 q^{89} -4.02967 q^{91} +2.79110 q^{93} +12.0326 q^{95} -12.2218 q^{97} -4.07768 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} - 4 q^{7} - 2 q^{9} - 2 q^{11} - 2 q^{13} + 10 q^{15} + 4 q^{17} - 4 q^{19} + 4 q^{23} - 6 q^{25} + 6 q^{29} + 16 q^{31} + 4 q^{35} - 8 q^{37} - 10 q^{39} - 10 q^{41} - 4 q^{43} + 2 q^{45}+ \cdots - 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^5 - 4 * q^7 - 2 * q^9 - 2 * q^11 - 2 * q^13 + 10 * q^15 + 4 * q^17 - 4 * q^19 + 4 * q^23 - 6 * q^25 + 6 * q^29 + 16 * q^31 + 4 * q^35 - 8 * q^37 - 10 * q^39 - 10 * q^41 - 4 * q^43 + 2 * q^45 + 14 * q^47 + 4 * q^49 - 14 * q^53 + 2 * q^55 - 10 * q^57 - 12 * q^59 - 2 * q^61 + 2 * q^63 - 8 * q^65 - 22 * q^67 - 10 * q^69 + 16 * q^71 - 14 * q^73 - 20 * q^75 + 2 * q^77 + 10 * q^79 - 24 * q^81 + 6 * q^83 - 4 * q^85 - 22 * q^89 + 2 * q^91 - 6 * q^95 - 12 * q^97 - 4 * q^99

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
$3$	−1.17557	−0.678716	−0.339358	−	0.940657i	$-0.610210\pi$
$3$	−1.17557	−0.678716	−0.339358	+	0.940657i	$0.610210\pi$
$4$	0	0
$5$	−2.17557	−0.972945	−0.486472	−	0.873696i	$-0.661717\pi$
$5$	−2.17557	−0.972945	−0.486472	+	0.873696i	$0.661717\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.61803	−0.539345
$10$	0	0
$11$	2.52015	0.759853	0.379926	−	0.925017i	$-0.375949\pi$
$11$	2.52015	0.759853	0.379926	+	0.925017i	$0.375949\pi$
$12$	0	0
$13$	4.02967	1.11763	0.558815	−	0.829292i	$-0.311256\pi$
$13$	4.02967	1.11763	0.558815	+	0.829292i	$0.311256\pi$
$14$	0	0
$15$	2.55754	0.660353
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−5.53077	−1.26885	−0.634423	−	0.772986i	$-0.718762\pi$
$19$	−5.53077	−1.26885	−0.634423	+	0.772986i	$0.718762\pi$
$20$	0	0
$21$	1.17557	0.256531
$22$	0	0
$23$	4.07768	0.850256	0.425128	−	0.905133i	$-0.360229\pi$
$23$	4.07768	0.850256	0.425128	+	0.905133i	$0.360229\pi$
$24$	0	0
$25$	−0.266893	−0.0533786
$26$	0	0
$27$	5.42882	1.04478
$28$	0	0
$29$	−4.42412	−0.821539	−0.410770	−	0.911739i	$-0.634740\pi$
$29$	−4.42412	−0.821539	−0.410770	+	0.911739i	$0.634740\pi$
$30$	0	0
$31$	−2.37425	−0.426428	−0.213214	−	0.977006i	$-0.568393\pi$
$31$	−2.37425	−0.426428	−0.213214	+	0.977006i	$0.568393\pi$
$32$	0	0
$33$	−2.96261	−0.515724
$34$	0	0
$35$	2.17557	0.367739
$36$	0	0
$37$	−6.64584	−1.09257	−0.546285	−	0.837599i	$-0.683958\pi$
$37$	−6.64584	−1.09257	−0.546285	+	0.837599i	$0.683958\pi$
$38$	0	0
$39$	−4.73716	−0.758553
$40$	0	0
$41$	7.85108	1.22613	0.613067	−	0.790031i	$-0.289936\pi$
$41$	7.85108	1.22613	0.613067	+	0.790031i	$0.289936\pi$
$42$	0	0
$43$	9.17848	1.39970	0.699852	−	0.714288i	$-0.253249\pi$
$43$	9.17848	1.39970	0.699852	+	0.714288i	$0.253249\pi$
$44$	0	0
$45$	3.52015	0.524753
$46$	0	0
$47$	12.6755	1.84891	0.924457	−	0.381287i	$-0.124519\pi$
$47$	12.6755	1.84891	0.924457	+	0.381287i	$0.124519\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.17557	−0.164613
$52$	0	0
$53$	−3.95012	−0.542591	−0.271296	−	0.962496i	$-0.587452\pi$
$53$	−3.95012	−0.542591	−0.271296	+	0.962496i	$0.587452\pi$
$54$	0	0
$55$	−5.48276	−0.739295
$56$	0	0
$57$	6.50181	0.861186
$58$	0	0
$59$	−14.9655	−1.94834	−0.974172	−	0.225807i	$-0.927498\pi$
$59$	−14.9655	−1.94834	−0.974172	+	0.225807i	$0.927498\pi$
$60$	0	0
$61$	13.4252	1.71892	0.859458	−	0.511206i	$-0.170801\pi$
$61$	13.4252	1.71892	0.859458	+	0.511206i	$0.170801\pi$
$62$	0	0
$63$	1.61803	0.203853
$64$	0	0
$65$	−8.76684	−1.08739
$66$	0	0
$67$	−2.98751	−0.364983	−0.182491	−	0.983207i	$-0.558416\pi$
$67$	−2.98751	−0.364983	−0.182491	+	0.983207i	$0.558416\pi$
$68$	0	0
$69$	−4.79360	−0.577082
$70$	0	0
$71$	7.86067	0.932889	0.466445	−	0.884550i	$-0.345535\pi$
$71$	7.86067	0.932889	0.466445	+	0.884550i	$0.345535\pi$
$72$	0	0
$73$	−2.32553	−0.272182	−0.136091	−	0.990696i	$-0.543454\pi$
$73$	−2.32553	−0.272182	−0.136091	+	0.990696i	$0.543454\pi$
$74$	0	0
$75$	0.313752	0.0362289
$76$	0	0
$77$	−2.52015	−0.287197
$78$	0	0
$79$	1.83099	0.206003	0.103001	−	0.994681i	$-0.467155\pi$
$79$	1.83099	0.206003	0.103001	+	0.994681i	$0.467155\pi$
$80$	0	0
$81$	−1.52786	−0.169763
$82$	0	0
$83$	15.0912	1.65648	0.828238	−	0.560377i	$-0.189344\pi$
$83$	15.0912	1.65648	0.828238	+	0.560377i	$0.189344\pi$
$84$	0	0
$85$	−2.17557	−0.235974
$86$	0	0
$87$	5.20087	0.557592
$88$	0	0
$89$	0.541390	0.0573872	0.0286936	−	0.999588i	$-0.490865\pi$
$89$	0.541390	0.0573872	0.0286936	+	0.999588i	$0.490865\pi$
$90$	0	0
$91$	−4.02967	−0.422424
$92$	0	0
$93$	2.79110	0.289423
$94$	0	0
$95$	12.0326	1.23452
$96$	0	0
$97$	−12.2218	−1.24093	−0.620467	−	0.784232i	$-0.713057\pi$
$97$	−12.2218	−1.24093	−0.620467	+	0.784232i	$0.713057\pi$
$98$	0	0
$99$	−4.07768	−0.409823
$100$	0	0
$101$	−17.9200	−1.78311	−0.891554	−	0.452915i	$-0.850384\pi$
$101$	−17.9200	−1.78311	−0.891554	+	0.452915i	$0.850384\pi$
$102$	0	0
$103$	8.82798	0.869846	0.434923	−	0.900468i	$-0.356776\pi$
$103$	8.82798	0.869846	0.434923	+	0.900468i	$0.356776\pi$
$104$	0	0
$105$	−2.55754	−0.249590
$106$	0	0
$107$	−12.9490	−1.25182	−0.625912	−	0.779894i	$-0.715273\pi$
$107$	−12.9490	−1.25182	−0.625912	+	0.779894i	$0.715273\pi$
$108$	0	0
$109$	−11.3783	−1.08984	−0.544922	−	0.838487i	$-0.683441\pi$
$109$	−11.3783	−1.08984	−0.544922	+	0.838487i	$0.683441\pi$
$110$	0	0
$111$	7.81266	0.741544
$112$	0	0
$113$	−16.4797	−1.55028	−0.775142	−	0.631787i	$-0.782322\pi$
$113$	−16.4797	−1.55028	−0.775142	+	0.631787i	$0.782322\pi$
$114$	0	0
$115$	−8.87129	−0.827252
$116$	0	0
$117$	−6.52015	−0.602788
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−4.64886	−0.422624
$122$	0	0
$123$	−9.22950	−0.832196
$124$	0	0
$125$	11.4585	1.02488
$126$	0	0
$127$	5.22139	0.463323	0.231662	−	0.972796i	$-0.425584\pi$
$127$	5.22139	0.463323	0.231662	+	0.972796i	$0.425584\pi$
$128$	0	0
$129$	−10.7899	−0.950002
$130$	0	0
$131$	−6.72725	−0.587763	−0.293881	−	0.955842i	$-0.594947\pi$
$131$	−6.72725	−0.587763	−0.293881	+	0.955842i	$0.594947\pi$
$132$	0	0
$133$	5.53077	0.479578
$134$	0	0
$135$	−11.8108	−1.01651
$136$	0	0
$137$	−16.2243	−1.38613	−0.693067	−	0.720873i	$-0.743741\pi$
$137$	−16.2243	−1.38613	−0.693067	+	0.720873i	$0.743741\pi$
$138$	0	0
$139$	−15.4358	−1.30925	−0.654623	−	0.755955i	$-0.727173\pi$
$139$	−15.4358	−1.30925	−0.654623	+	0.755955i	$0.727173\pi$
$140$	0	0
$141$	−14.9010	−1.25489
$142$	0	0
$143$	10.1554	0.849234
$144$	0	0
$145$	9.62500	0.799312
$146$	0	0
$147$	−1.17557	−0.0969594
$148$	0	0
$149$	−16.4544	−1.34800	−0.674000	−	0.738731i	$-0.735425\pi$
$149$	−16.4544	−1.34800	−0.674000	+	0.738731i	$0.735425\pi$
$150$	0	0
$151$	−12.4064	−1.00962	−0.504810	−	0.863230i	$-0.668438\pi$
$151$	−12.4064	−1.00962	−0.504810	+	0.863230i	$0.668438\pi$
$152$	0	0
$153$	−1.61803	−0.130810
$154$	0	0
$155$	5.16535	0.414891
$156$	0	0
$157$	−16.1155	−1.28615	−0.643077	−	0.765801i	$-0.722343\pi$
$157$	−16.1155	−1.28615	−0.643077	+	0.765801i	$0.722343\pi$
$158$	0	0
$159$	4.64365	0.368265
$160$	0	0
$161$	−4.07768	−0.321366
$162$	0	0
$163$	3.92992	0.307815	0.153908	−	0.988085i	$-0.450814\pi$
$163$	3.92992	0.307815	0.153908	+	0.988085i	$0.450814\pi$
$164$	0	0
$165$	6.44537	0.501771
$166$	0	0
$167$	9.61333	0.743902	0.371951	−	0.928252i	$-0.378689\pi$
$167$	9.61333	0.743902	0.371951	+	0.928252i	$0.378689\pi$
$168$	0	0
$169$	3.23826	0.249097
$170$	0	0
$171$	8.94897	0.684345
$172$	0	0
$173$	−13.9259	−1.05876	−0.529382	−	0.848384i	$-0.677576\pi$
$173$	−13.9259	−1.05876	−0.529382	+	0.848384i	$0.677576\pi$
$174$	0	0
$175$	0.266893	0.0201752
$176$	0	0
$177$	17.5930	1.32237
$178$	0	0
$179$	−20.8654	−1.55956	−0.779778	−	0.626056i	$-0.784668\pi$
$179$	−20.8654	−1.55956	−0.779778	+	0.626056i	$0.784668\pi$
$180$	0	0
$181$	13.4046	0.996353	0.498177	−	0.867076i	$-0.334003\pi$
$181$	13.4046	0.996353	0.498177	+	0.867076i	$0.334003\pi$
$182$	0	0
$183$	−15.7822	−1.16666
$184$	0	0
$185$	14.4585	1.06301
$186$	0	0
$187$	2.52015	0.184291
$188$	0	0
$189$	−5.42882	−0.394889
$190$	0	0
$191$	5.66727	0.410069	0.205035	−	0.978755i	$-0.434269\pi$
$191$	5.66727	0.410069	0.205035	+	0.978755i	$0.434269\pi$
$192$	0	0
$193$	8.50181	0.611974	0.305987	−	0.952036i	$-0.401014\pi$
$193$	8.50181	0.611974	0.305987	+	0.952036i	$0.401014\pi$
$194$	0	0
$195$	10.3060	0.738030
$196$	0	0
$197$	−25.3214	−1.80407	−0.902036	−	0.431661i	$-0.857928\pi$
$197$	−25.3214	−1.80407	−0.902036	+	0.431661i	$0.857928\pi$
$198$	0	0
$199$	9.41756	0.667593	0.333797	−	0.942645i	$-0.391670\pi$
$199$	9.41756	0.667593	0.333797	+	0.942645i	$0.391670\pi$
$200$	0	0
$201$	3.51203	0.247720
$202$	0	0
$203$	4.42412	0.310513
$204$	0	0
$205$	−17.0806	−1.19296
$206$	0	0
$207$	−6.59783	−0.458581
$208$	0	0
$209$	−13.9383	−0.964136
$210$	0	0
$211$	−6.23826	−0.429460	−0.214730	−	0.976673i	$-0.568887\pi$
$211$	−6.23826	−0.429460	−0.214730	+	0.976673i	$0.568887\pi$
$212$	0	0
$213$	−9.24077	−0.633167
$214$	0	0
$215$	−19.9684	−1.36183
$216$	0	0
$217$	2.37425	0.161175
$218$	0	0
$219$	2.73382	0.184734
$220$	0	0
$221$	4.02967	0.271065
$222$	0	0
$223$	3.24128	0.217052	0.108526	−	0.994094i	$-0.465387\pi$
$223$	3.24128	0.217052	0.108526	+	0.994094i	$0.465387\pi$
$224$	0	0
$225$	0.431842	0.0287895
$226$	0	0
$227$	−19.3529	−1.28450	−0.642248	−	0.766497i	$-0.721998\pi$
$227$	−19.3529	−1.28450	−0.642248	+	0.766497i	$0.721998\pi$
$228$	0	0
$229$	−24.1263	−1.59431	−0.797155	−	0.603774i	$-0.793663\pi$
$229$	−24.1263	−1.59431	−0.797155	+	0.603774i	$0.793663\pi$
$230$	0	0
$231$	2.96261	0.194925
$232$	0	0
$233$	4.00290	0.262239	0.131119	−	0.991367i	$-0.458143\pi$
$233$	4.00290	0.262239	0.131119	+	0.991367i	$0.458143\pi$
$234$	0	0
$235$	−27.5765	−1.79889
$236$	0	0
$237$	−2.15246	−0.139817
$238$	0	0
$239$	23.0575	1.49146	0.745732	−	0.666246i	$-0.232100\pi$
$239$	23.0575	1.49146	0.745732	+	0.666246i	$0.232100\pi$
$240$	0	0
$241$	27.0009	1.73928	0.869641	−	0.493684i	$-0.164350\pi$
$241$	27.0009	1.73928	0.869641	+	0.493684i	$0.164350\pi$
$242$	0	0
$243$	−14.4904	−0.929557
$244$	0	0
$245$	−2.17557	−0.138992
$246$	0	0
$247$	−22.2872	−1.41810
$248$	0	0
$249$	−17.7408	−1.12428
$250$	0	0
$251$	−2.77997	−0.175470	−0.0877349	−	0.996144i	$-0.527963\pi$
$251$	−2.77997	−0.175470	−0.0877349	+	0.996144i	$0.527963\pi$
$252$	0	0
$253$	10.2764	0.646069
$254$	0	0
$255$	2.55754	0.160159
$256$	0	0
$257$	−12.7206	−0.793491	−0.396745	−	0.917929i	$-0.629860\pi$
$257$	−12.7206	−0.793491	−0.396745	+	0.917929i	$0.629860\pi$
$258$	0	0
$259$	6.64584	0.412953
$260$	0	0
$261$	7.15838	0.443093
$262$	0	0
$263$	−4.92303	−0.303567	−0.151783	−	0.988414i	$-0.548502\pi$
$263$	−4.92303	−0.303567	−0.151783	+	0.988414i	$0.548502\pi$
$264$	0	0
$265$	8.59377	0.527911
$266$	0	0
$267$	−0.636442	−0.0389496
$268$	0	0
$269$	20.9845	1.27944	0.639722	−	0.768606i	$-0.279049\pi$
$269$	20.9845	1.27944	0.639722	+	0.768606i	$0.279049\pi$
$270$	0	0
$271$	−0.0814132	−0.00494550	−0.00247275	−	0.999997i	$-0.500787\pi$
$271$	−0.0814132	−0.00494550	−0.00247275	+	0.999997i	$0.500787\pi$
$272$	0	0
$273$	4.73716	0.286706
$274$	0	0
$275$	−0.672610	−0.0405599
$276$	0	0
$277$	33.0028	1.98295	0.991473	−	0.130314i	$-0.0415985\pi$
$277$	33.0028	1.98295	0.991473	+	0.130314i	$0.0415985\pi$
$278$	0	0
$279$	3.84162	0.229992
$280$	0	0
$281$	−30.8965	−1.84313	−0.921565	−	0.388225i	$-0.873088\pi$
$281$	−30.8965	−1.84313	−0.921565	+	0.388225i	$0.873088\pi$
$282$	0	0
$283$	10.6294	0.631850	0.315925	−	0.948784i	$-0.397685\pi$
$283$	10.6294	0.631850	0.315925	+	0.948784i	$0.397685\pi$
$284$	0	0
$285$	−14.1451	−0.837886
$286$	0	0
$287$	−7.85108	−0.463435
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	14.3676	0.842242
$292$	0	0
$293$	−5.78819	−0.338150	−0.169075	−	0.985603i	$-0.554078\pi$
$293$	−5.78819	−0.338150	−0.169075	+	0.985603i	$0.554078\pi$
$294$	0	0
$295$	32.5585	1.89563
$296$	0	0
$297$	13.6814	0.793877
$298$	0	0
$299$	16.4317	0.950271
$300$	0	0
$301$	−9.17848	−0.529039
$302$	0	0
$303$	21.0662	1.21022
$304$	0	0
$305$	−29.2074	−1.67241
$306$	0	0
$307$	20.3889	1.16366	0.581829	−	0.813311i	$-0.302337\pi$
$307$	20.3889	1.16366	0.581829	+	0.813311i	$0.302337\pi$
$308$	0	0
$309$	−10.3779	−0.590379
$310$	0	0
$311$	−14.6981	−0.833453	−0.416727	−	0.909032i	$-0.636823\pi$
$311$	−14.6981	−0.833453	−0.416727	+	0.909032i	$0.636823\pi$
$312$	0	0
$313$	−24.0685	−1.36043	−0.680216	−	0.733012i	$-0.738114\pi$
$313$	−24.0685	−1.36043	−0.680216	+	0.733012i	$0.738114\pi$
$314$	0	0
$315$	−3.52015	−0.198338
$316$	0	0
$317$	23.4037	1.31448	0.657242	−	0.753679i	$-0.271723\pi$
$317$	23.4037	1.31448	0.657242	+	0.753679i	$0.271723\pi$
$318$	0	0
$319$	−11.1494	−0.624249
$320$	0	0
$321$	15.2224	0.849633
$322$	0	0
$323$	−5.53077	−0.307740
$324$	0	0
$325$	−1.07549	−0.0596575
$326$	0	0
$327$	13.3760	0.739695
$328$	0	0
$329$	−12.6755	−0.698824
$330$	0	0
$331$	10.7359	0.590098	0.295049	−	0.955482i	$-0.404664\pi$
$331$	10.7359	0.590098	0.295049	+	0.955482i	$0.404664\pi$
$332$	0	0
$333$	10.7532	0.589272
$334$	0	0
$335$	6.49955	0.355108
$336$	0	0
$337$	24.4398	1.33132	0.665662	−	0.746254i	$-0.268149\pi$
$337$	24.4398	1.33132	0.665662	+	0.746254i	$0.268149\pi$
$338$	0	0
$339$	19.3731	1.05220
$340$	0	0
$341$	−5.98346	−0.324022
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	10.4288	0.561469
$346$	0	0
$347$	−9.29772	−0.499128	−0.249564	−	0.968358i	$-0.580287\pi$
$347$	−9.29772	−0.499128	−0.249564	+	0.968358i	$0.580287\pi$
$348$	0	0
$349$	25.6240	1.37162	0.685810	−	0.727781i	$-0.259448\pi$
$349$	25.6240	1.37162	0.685810	+	0.727781i	$0.259448\pi$
$350$	0	0
$351$	21.8764	1.16768
$352$	0	0
$353$	−0.389753	−0.0207444	−0.0103722	−	0.999946i	$-0.503302\pi$
$353$	−0.389753	−0.0207444	−0.0103722	+	0.999946i	$0.503302\pi$
$354$	0	0
$355$	−17.1014	−0.907650
$356$	0	0
$357$	1.17557	0.0622178
$358$	0	0
$359$	−25.9258	−1.36831	−0.684154	−	0.729337i	$-0.739829\pi$
$359$	−25.9258	−1.36831	−0.684154	+	0.729337i	$0.739829\pi$
$360$	0	0
$361$	11.5894	0.609968
$362$	0	0
$363$	5.46506	0.286841
$364$	0	0
$365$	5.05934	0.264818
$366$	0	0
$367$	0.430802	0.0224877	0.0112438	−	0.999937i	$-0.496421\pi$
$367$	0.430802	0.0224877	0.0112438	+	0.999937i	$0.496421\pi$
$368$	0	0
$369$	−12.7033	−0.661309
$370$	0	0
$371$	3.95012	0.205080
$372$	0	0
$373$	−14.8730	−0.770097	−0.385048	−	0.922896i	$-0.625815\pi$
$373$	−14.8730	−0.770097	−0.385048	+	0.922896i	$0.625815\pi$
$374$	0	0
$375$	−13.4703	−0.695602
$376$	0	0
$377$	−17.8278	−0.918177
$378$	0	0
$379$	−10.4504	−0.536800	−0.268400	−	0.963308i	$-0.586495\pi$
$379$	−10.4504	−0.536800	−0.268400	+	0.963308i	$0.586495\pi$
$380$	0	0
$381$	−6.13811	−0.314465
$382$	0	0
$383$	14.9425	0.763525	0.381762	−	0.924260i	$-0.375317\pi$
$383$	14.9425	0.763525	0.381762	+	0.924260i	$0.375317\pi$
$384$	0	0
$385$	5.48276	0.279427
$386$	0	0
$387$	−14.8511	−0.754923
$388$	0	0
$389$	36.4058	1.84585	0.922924	−	0.384983i	$-0.125793\pi$
$389$	36.4058	1.84585	0.922924	+	0.384983i	$0.125793\pi$
$390$	0	0
$391$	4.07768	0.206217
$392$	0	0
$393$	7.90836	0.398924
$394$	0	0
$395$	−3.98346	−0.200429
$396$	0	0
$397$	22.7225	1.14041	0.570204	−	0.821503i	$-0.306864\pi$
$397$	22.7225	1.14041	0.570204	+	0.821503i	$0.306864\pi$
$398$	0	0
$399$	−6.50181	−0.325498
$400$	0	0
$401$	−12.1845	−0.608466	−0.304233	−	0.952598i	$-0.598400\pi$
$401$	−12.1845	−0.608466	−0.304233	+	0.952598i	$0.598400\pi$
$402$	0	0
$403$	−9.56745	−0.476588
$404$	0	0
$405$	3.32398	0.165170
$406$	0	0
$407$	−16.7485	−0.830192
$408$	0	0
$409$	−21.0211	−1.03943	−0.519714	−	0.854340i	$-0.673961\pi$
$409$	−21.0211	−1.03943	−0.519714	+	0.854340i	$0.673961\pi$
$410$	0	0
$411$	19.0728	0.940792
$412$	0	0
$413$	14.9655	0.736405
$414$	0	0
$415$	−32.8320	−1.61166
$416$	0	0
$417$	18.1459	0.888606
$418$	0	0
$419$	−37.3931	−1.82677	−0.913386	−	0.407095i	$-0.866542\pi$
$419$	−37.3931	−1.82677	−0.913386	+	0.407095i	$0.866542\pi$
$420$	0	0
$421$	−9.79464	−0.477362	−0.238681	−	0.971098i	$-0.576715\pi$
$421$	−9.79464	−0.477362	−0.238681	+	0.971098i	$0.576715\pi$
$422$	0	0
$423$	−20.5094	−0.997202
$424$	0	0
$425$	−0.266893	−0.0129462
$426$	0	0
$427$	−13.4252	−0.649689
$428$	0	0
$429$	−11.9383	−0.576389
$430$	0	0
$431$	−34.6036	−1.66680	−0.833399	−	0.552671i	$-0.813609\pi$
$431$	−34.6036	−1.66680	−0.833399	+	0.552671i	$0.813609\pi$
$432$	0	0
$433$	−15.9460	−0.766314	−0.383157	−	0.923683i	$-0.625163\pi$
$433$	−15.9460	−0.766314	−0.383157	+	0.923683i	$0.625163\pi$
$434$	0	0
$435$	−11.3149	−0.542506
$436$	0	0
$437$	−22.5527	−1.07884
$438$	0	0
$439$	18.0832	0.863062	0.431531	−	0.902098i	$-0.357973\pi$
$439$	18.0832	0.863062	0.431531	+	0.902098i	$0.357973\pi$
$440$	0	0
$441$	−1.61803	−0.0770492
$442$	0	0
$443$	−11.2442	−0.534227	−0.267114	−	0.963665i	$-0.586070\pi$
$443$	−11.2442	−0.534227	−0.267114	+	0.963665i	$0.586070\pi$
$444$	0	0
$445$	−1.17783	−0.0558346
$446$	0	0
$447$	19.3434	0.914909
$448$	0	0
$449$	−8.94086	−0.421945	−0.210973	−	0.977492i	$-0.567663\pi$
$449$	−8.94086	−0.421945	−0.210973	+	0.977492i	$0.567663\pi$
$450$	0	0
$451$	19.7859	0.931681
$452$	0	0
$453$	14.5846	0.685246
$454$	0	0
$455$	8.76684	0.410996
$456$	0	0
$457$	−18.6795	−0.873788	−0.436894	−	0.899513i	$-0.643922\pi$
$457$	−18.6795	−0.873788	−0.436894	+	0.899513i	$0.643922\pi$
$458$	0	0
$459$	5.42882	0.253396
$460$	0	0
$461$	−18.8427	−0.877593	−0.438797	−	0.898586i	$-0.644595\pi$
$461$	−18.8427	−0.877593	−0.438797	+	0.898586i	$0.644595\pi$
$462$	0	0
$463$	−21.7443	−1.01054	−0.505272	−	0.862960i	$-0.668608\pi$
$463$	−21.7443	−1.01054	−0.505272	+	0.862960i	$0.668608\pi$
$464$	0	0
$465$	−6.07223	−0.281593
$466$	0	0
$467$	−2.80821	−0.129949	−0.0649743	−	0.997887i	$-0.520697\pi$
$467$	−2.80821	−0.129949	−0.0649743	+	0.997887i	$0.520697\pi$
$468$	0	0
$469$	2.98751	0.137951
$470$	0	0
$471$	18.9449	0.872934
$472$	0	0
$473$	23.1311	1.06357
$474$	0	0
$475$	1.47612	0.0677292
$476$	0	0
$477$	6.39144	0.292644
$478$	0	0
$479$	32.6476	1.49171	0.745853	−	0.666110i	$-0.232042\pi$
$479$	32.6476	1.49171	0.745853	+	0.666110i	$0.232042\pi$
$480$	0	0
$481$	−26.7806	−1.22109
$482$	0	0
$483$	4.79360	0.218117
$484$	0	0
$485$	26.5894	1.20736
$486$	0	0
$487$	0.454307	0.0205866	0.0102933	−	0.999947i	$-0.496723\pi$
$487$	0.454307	0.0205866	0.0102933	+	0.999947i	$0.496723\pi$
$488$	0	0
$489$	−4.61990	−0.208919
$490$	0	0
$491$	−14.4214	−0.650830	−0.325415	−	0.945571i	$-0.605504\pi$
$491$	−14.4214	−0.650830	−0.325415	+	0.945571i	$0.605504\pi$
$492$	0	0
$493$	−4.42412	−0.199253
$494$	0	0
$495$	8.87129	0.398735
$496$	0	0
$497$	−7.86067	−0.352599
$498$	0	0
$499$	27.9216	1.24994	0.624971	−	0.780648i	$-0.285111\pi$
$499$	27.9216	1.24994	0.624971	+	0.780648i	$0.285111\pi$
$500$	0	0
$501$	−11.3012	−0.504898
$502$	0	0
$503$	−7.62718	−0.340079	−0.170039	−	0.985437i	$-0.554390\pi$
$503$	−7.62718	−0.340079	−0.170039	+	0.985437i	$0.554390\pi$
$504$	0	0
$505$	38.9862	1.73487
$506$	0	0
$507$	−3.80680	−0.169066
$508$	0	0
$509$	15.7129	0.696462	0.348231	−	0.937409i	$-0.386782\pi$
$509$	15.7129	0.696462	0.348231	+	0.937409i	$0.386782\pi$
$510$	0	0
$511$	2.32553	0.102875
$512$	0	0
$513$	−30.0256	−1.32566
$514$	0	0
$515$	−19.2059	−0.846312
$516$	0	0
$517$	31.9442	1.40490
$518$	0	0
$519$	16.3708	0.718600
$520$	0	0
$521$	30.5865	1.34002	0.670008	−	0.742353i	$-0.266290\pi$
$521$	30.5865	1.34002	0.670008	+	0.742353i	$0.266290\pi$
$522$	0	0
$523$	−41.3364	−1.80752	−0.903758	−	0.428044i	$-0.859203\pi$
$523$	−41.3364	−1.80752	−0.903758	+	0.428044i	$0.859203\pi$
$524$	0	0
$525$	−0.313752	−0.0136932
$526$	0	0
$527$	−2.37425	−0.103424
$528$	0	0
$529$	−6.37250	−0.277065
$530$	0	0
$531$	24.2147	1.05083
$532$	0	0
$533$	31.6373	1.37036
$534$	0	0
$535$	28.1714	1.21796
$536$	0	0
$537$	24.5288	1.05850
$538$	0	0
$539$	2.52015	0.108550
$540$	0	0
$541$	−20.0312	−0.861210	−0.430605	−	0.902540i	$-0.641700\pi$
$541$	−20.0312	−0.861210	−0.430605	+	0.902540i	$0.641700\pi$
$542$	0	0
$543$	−15.7580	−0.676241
$544$	0	0
$545$	24.7543	1.06036
$546$	0	0
$547$	27.7371	1.18595	0.592975	−	0.805221i	$-0.297953\pi$
$547$	27.7371	1.18595	0.592975	+	0.805221i	$0.297953\pi$
$548$	0	0
$549$	−21.7224	−0.927088
$550$	0	0
$551$	24.4688	1.04241
$552$	0	0
$553$	−1.83099	−0.0778618
$554$	0	0
$555$	−16.9970	−0.721482
$556$	0	0
$557$	−13.9444	−0.590843	−0.295421	−	0.955367i	$-0.595460\pi$
$557$	−13.9444	−0.590843	−0.295421	+	0.955367i	$0.595460\pi$
$558$	0	0
$559$	36.9862	1.56435
$560$	0	0
$561$	−2.96261	−0.125082
$562$	0	0
$563$	7.89034	0.332538	0.166269	−	0.986080i	$-0.446828\pi$
$563$	7.89034	0.332538	0.166269	+	0.986080i	$0.446828\pi$
$564$	0	0
$565$	35.8528	1.50834
$566$	0	0
$567$	1.52786	0.0641643
$568$	0	0
$569$	0.961968	0.0403278	0.0201639	−	0.999797i	$-0.493581\pi$
$569$	0.961968	0.0403278	0.0201639	+	0.999797i	$0.493581\pi$
$570$	0	0
$571$	−15.5389	−0.650282	−0.325141	−	0.945666i	$-0.605412\pi$
$571$	−15.5389	−0.650282	−0.325141	+	0.945666i	$0.605412\pi$
$572$	0	0
$573$	−6.66227	−0.278320
$574$	0	0
$575$	−1.08831	−0.0453855
$576$	0	0
$577$	32.5245	1.35401	0.677006	−	0.735978i	$-0.263277\pi$
$577$	32.5245	1.35401	0.677006	+	0.735978i	$0.263277\pi$
$578$	0	0
$579$	−9.99448	−0.415356
$580$	0	0
$581$	−15.0912	−0.626089
$582$	0	0
$583$	−9.95489	−0.412290
$584$	0	0
$585$	14.1850	0.586479
$586$	0	0
$587$	−39.9568	−1.64919	−0.824596	−	0.565722i	$-0.808598\pi$
$587$	−39.9568	−1.64919	−0.824596	+	0.565722i	$0.808598\pi$
$588$	0	0
$589$	13.1314	0.541071
$590$	0	0
$591$	29.7670	1.22445
$592$	0	0
$593$	20.7726	0.853030	0.426515	−	0.904480i	$-0.359741\pi$
$593$	20.7726	0.853030	0.426515	+	0.904480i	$0.359741\pi$
$594$	0	0
$595$	2.17557	0.0891897
$596$	0	0
$597$	−11.0710	−0.453106
$598$	0	0
$599$	−3.41839	−0.139671	−0.0698357	−	0.997559i	$-0.522248\pi$
$599$	−3.41839	−0.139671	−0.0698357	+	0.997559i	$0.522248\pi$
$600$	0	0
$601$	29.0130	1.18347	0.591733	−	0.806134i	$-0.298444\pi$
$601$	29.0130	1.18347	0.591733	+	0.806134i	$0.298444\pi$
$602$	0	0
$603$	4.83390	0.196852
$604$	0	0
$605$	10.1139	0.411189
$606$	0	0
$607$	−30.1536	−1.22390	−0.611949	−	0.790897i	$-0.709614\pi$
$607$	−30.1536	−1.22390	−0.611949	+	0.790897i	$0.709614\pi$
$608$	0	0
$609$	−5.20087	−0.210750
$610$	0	0
$611$	51.0782	2.06640
$612$	0	0
$613$	3.95385	0.159695	0.0798473	−	0.996807i	$-0.474557\pi$
$613$	3.95385	0.159695	0.0798473	+	0.996807i	$0.474557\pi$
$614$	0	0
$615$	20.0794	0.809681
$616$	0	0
$617$	−15.0610	−0.606334	−0.303167	−	0.952937i	$-0.598044\pi$
$617$	−15.0610	−0.606334	−0.303167	+	0.952937i	$0.598044\pi$
$618$	0	0
$619$	31.4922	1.26578	0.632890	−	0.774242i	$-0.281868\pi$
$619$	31.4922	1.26578	0.632890	+	0.774242i	$0.281868\pi$
$620$	0	0
$621$	22.1370	0.888328
$622$	0	0
$623$	−0.541390	−0.0216903
$624$	0	0
$625$	−23.5943	−0.943772
$626$	0	0
$627$	16.3855	0.654374
$628$	0	0
$629$	−6.64584	−0.264987
$630$	0	0
$631$	−26.8663	−1.06953	−0.534765	−	0.845001i	$-0.679600\pi$
$631$	−26.8663	−1.06953	−0.534765	+	0.845001i	$0.679600\pi$
$632$	0	0
$633$	7.33351	0.291481
$634$	0	0
$635$	−11.3595	−0.450788
$636$	0	0
$637$	4.02967	0.159661
$638$	0	0
$639$	−12.7188	−0.503149
$640$	0	0
$641$	24.0224	0.948826	0.474413	−	0.880302i	$-0.342660\pi$
$641$	24.0224	0.948826	0.474413	+	0.880302i	$0.342660\pi$
$642$	0	0
$643$	−30.8396	−1.21620	−0.608098	−	0.793862i	$-0.708067\pi$
$643$	−30.8396	−1.21620	−0.608098	+	0.793862i	$0.708067\pi$
$644$	0	0
$645$	23.4743	0.924299
$646$	0	0
$647$	−5.14996	−0.202466	−0.101233	−	0.994863i	$-0.532279\pi$
$647$	−5.14996	−0.202466	−0.101233	+	0.994863i	$0.532279\pi$
$648$	0	0
$649$	−37.7153	−1.48045
$650$	0	0
$651$	−2.79110	−0.109392
$652$	0	0
$653$	−29.1906	−1.14232	−0.571159	−	0.820839i	$-0.693506\pi$
$653$	−29.1906	−1.14232	−0.571159	+	0.820839i	$0.693506\pi$
$654$	0	0
$655$	14.6356	0.571861
$656$	0	0
$657$	3.76278	0.146800
$658$	0	0
$659$	9.65041	0.375927	0.187963	−	0.982176i	$-0.439811\pi$
$659$	9.65041	0.375927	0.187963	+	0.982176i	$0.439811\pi$
$660$	0	0
$661$	26.8629	1.04484	0.522422	−	0.852687i	$-0.325029\pi$
$661$	26.8629	1.04484	0.522422	+	0.852687i	$0.325029\pi$
$662$	0	0
$663$	−4.73716	−0.183976
$664$	0	0
$665$	−12.0326	−0.466603
$666$	0	0
$667$	−18.0402	−0.698519
$668$	0	0
$669$	−3.81035	−0.147317
$670$	0	0
$671$	33.8334	1.30612
$672$	0	0
$673$	−34.6136	−1.33426	−0.667129	−	0.744942i	$-0.732477\pi$
$673$	−34.6136	−1.33426	−0.667129	+	0.744942i	$0.732477\pi$
$674$	0	0
$675$	−1.44892	−0.0557688
$676$	0	0
$677$	−26.8266	−1.03103	−0.515516	−	0.856880i	$-0.672400\pi$
$677$	−26.8266	−1.03103	−0.515516	+	0.856880i	$0.672400\pi$
$678$	0	0
$679$	12.2218	0.469029
$680$	0	0
$681$	22.7507	0.871808
$682$	0	0
$683$	25.0616	0.958956	0.479478	−	0.877554i	$-0.340826\pi$
$683$	25.0616	0.958956	0.479478	+	0.877554i	$0.340826\pi$
$684$	0	0
$685$	35.2971	1.34863
$686$	0	0
$687$	28.3622	1.08208
$688$	0	0
$689$	−15.9177	−0.606416
$690$	0	0
$691$	−14.8332	−0.564281	−0.282140	−	0.959373i	$-0.591044\pi$
$691$	−14.8332	−0.564281	−0.282140	+	0.959373i	$0.591044\pi$
$692$	0	0
$693$	4.07768	0.154898
$694$	0	0
$695$	33.5816	1.27382
$696$	0	0
$697$	7.85108	0.297381
$698$	0	0
$699$	−4.70570	−0.177986
$700$	0	0
$701$	−30.8526	−1.16529	−0.582644	−	0.812728i	$-0.697982\pi$
$701$	−30.8526	−1.16529	−0.582644	+	0.812728i	$0.697982\pi$
$702$	0	0
$703$	36.7566	1.38630
$704$	0	0
$705$	32.4181	1.22094
$706$	0	0
$707$	17.9200	0.673951
$708$	0	0
$709$	26.7378	1.00416	0.502079	−	0.864822i	$-0.332569\pi$
$709$	26.7378	1.00416	0.502079	+	0.864822i	$0.332569\pi$
$710$	0	0
$711$	−2.96261	−0.111107
$712$	0	0
$713$	−9.68144	−0.362573
$714$	0	0
$715$	−22.0937	−0.826258
$716$	0	0
$717$	−27.1057	−1.01228
$718$	0	0
$719$	34.0545	1.27002	0.635009	−	0.772505i	$-0.280997\pi$
$719$	34.0545	1.27002	0.635009	+	0.772505i	$0.280997\pi$
$720$	0	0
$721$	−8.82798	−0.328771
$722$	0	0
$723$	−31.7415	−1.18048
$724$	0	0
$725$	1.18077	0.0438526
$726$	0	0
$727$	−3.66296	−0.135852	−0.0679258	−	0.997690i	$-0.521638\pi$
$727$	−3.66296	−0.135852	−0.0679258	+	0.997690i	$0.521638\pi$
$728$	0	0
$729$	21.6180	0.800668
$730$	0	0
$731$	9.17848	0.339478
$732$	0	0
$733$	10.2365	0.378092	0.189046	−	0.981968i	$-0.439460\pi$
$733$	10.2365	0.378092	0.189046	+	0.981968i	$0.439460\pi$
$734$	0	0
$735$	2.55754	0.0943362
$736$	0	0
$737$	−7.52897	−0.277333
$738$	0	0
$739$	−40.2175	−1.47942	−0.739712	−	0.672924i	$-0.765038\pi$
$739$	−40.2175	−1.47942	−0.739712	+	0.672924i	$0.765038\pi$
$740$	0	0
$741$	26.2002	0.962487
$742$	0	0
$743$	6.00709	0.220379	0.110189	−	0.993911i	$-0.464854\pi$
$743$	6.00709	0.220379	0.110189	+	0.993911i	$0.464854\pi$
$744$	0	0
$745$	35.7978	1.31153
$746$	0	0
$747$	−24.4181	−0.893411
$748$	0	0
$749$	12.9490	0.473145
$750$	0	0
$751$	13.5938	0.496046	0.248023	−	0.968754i	$-0.420219\pi$
$751$	13.5938	0.496046	0.248023	+	0.968754i	$0.420219\pi$
$752$	0	0
$753$	3.26805	0.119094
$754$	0	0
$755$	26.9911	0.982305
$756$	0	0
$757$	−24.7793	−0.900620	−0.450310	−	0.892872i	$-0.648686\pi$
$757$	−24.7793	−0.900620	−0.450310	+	0.892872i	$0.648686\pi$
$758$	0	0
$759$	−12.0806	−0.438498
$760$	0	0
$761$	16.9499	0.614435	0.307217	−	0.951639i	$-0.400602\pi$
$761$	16.9499	0.614435	0.307217	+	0.951639i	$0.400602\pi$
$762$	0	0
$763$	11.3783	0.411922
$764$	0	0
$765$	3.52015	0.127271
$766$	0	0
$767$	−60.3061	−2.17753
$768$	0	0
$769$	−15.1183	−0.545180	−0.272590	−	0.962130i	$-0.587880\pi$
$769$	−15.1183	−0.545180	−0.272590	+	0.962130i	$0.587880\pi$
$770$	0	0
$771$	14.9540	0.538555
$772$	0	0
$773$	6.71800	0.241630	0.120815	−	0.992675i	$-0.461449\pi$
$773$	6.71800	0.241630	0.120815	+	0.992675i	$0.461449\pi$
$774$	0	0
$775$	0.633670	0.0227621
$776$	0	0
$777$	−7.81266	−0.280277
$778$	0	0
$779$	−43.4225	−1.55577
$780$	0	0
$781$	19.8100	0.708859
$782$	0	0
$783$	−24.0178	−0.858326
$784$	0	0
$785$	35.0603	1.25136
$786$	0	0
$787$	45.8812	1.63549	0.817745	−	0.575581i	$-0.195224\pi$
$787$	45.8812	1.63549	0.817745	+	0.575581i	$0.195224\pi$
$788$	0	0
$789$	5.78737	0.206036
$790$	0	0
$791$	16.4797	0.585952
$792$	0	0
$793$	54.0990	1.92111
$794$	0	0
$795$	−10.1026	−0.358302
$796$	0	0
$797$	−13.0889	−0.463633	−0.231816	−	0.972760i	$-0.574467\pi$
$797$	−13.0889	−0.463633	−0.231816	+	0.972760i	$0.574467\pi$
$798$	0	0
$799$	12.6755	0.448427
$800$	0	0
$801$	−0.875988	−0.0309515
$802$	0	0
$803$	−5.86067	−0.206818
$804$	0	0
$805$	8.87129	0.312672
$806$	0	0
$807$	−24.6687	−0.868380
$808$	0	0
$809$	−42.4362	−1.49198	−0.745989	−	0.665959i	$-0.768023\pi$
$809$	−42.4362	−1.49198	−0.745989	+	0.665959i	$0.768023\pi$
$810$	0	0
$811$	27.2657	0.957428	0.478714	−	0.877971i	$-0.341103\pi$
$811$	27.2657	0.957428	0.478714	+	0.877971i	$0.341103\pi$
$812$	0	0
$813$	0.0957070	0.00335659
$814$	0	0
$815$	−8.54982	−0.299487
$816$	0	0
$817$	−50.7640	−1.77601
$818$	0	0
$819$	6.52015	0.227832
$820$	0	0
$821$	−17.4287	−0.608266	−0.304133	−	0.952630i	$-0.598367\pi$
$821$	−17.4287	−0.608266	−0.304133	+	0.952630i	$0.598367\pi$
$822$	0	0
$823$	35.4081	1.23425	0.617124	−	0.786866i	$-0.288298\pi$
$823$	35.4081	1.23425	0.617124	+	0.786866i	$0.288298\pi$
$824$	0	0
$825$	0.790700	0.0275286
$826$	0	0
$827$	11.3274	0.393892	0.196946	−	0.980414i	$-0.436898\pi$
$827$	11.3274	0.393892	0.196946	+	0.980414i	$0.436898\pi$
$828$	0	0
$829$	34.4437	1.19628	0.598139	−	0.801392i	$-0.295907\pi$
$829$	34.4437	1.19628	0.598139	+	0.801392i	$0.295907\pi$
$830$	0	0
$831$	−38.7971	−1.34586
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	−20.9145	−0.723776
$836$	0	0
$837$	−12.8894	−0.445522
$838$	0	0
$839$	−5.61657	−0.193905	−0.0969527	−	0.995289i	$-0.530910\pi$
$839$	−5.61657	−0.193905	−0.0969527	+	0.995289i	$0.530910\pi$
$840$	0	0
$841$	−9.42712	−0.325073
$842$	0	0
$843$	36.3210	1.25096
$844$	0	0
$845$	−7.04506	−0.242358
$846$	0	0
$847$	4.64886	0.159737
$848$	0	0
$849$	−12.4956	−0.428847
$850$	0	0
$851$	−27.0996	−0.928964
$852$	0	0
$853$	−13.7807	−0.471841	−0.235921	−	0.971772i	$-0.575811\pi$
$853$	−13.7807	−0.471841	−0.235921	+	0.971772i	$0.575811\pi$
$854$	0	0
$855$	−19.4691	−0.665830
$856$	0	0
$857$	−15.2314	−0.520296	−0.260148	−	0.965569i	$-0.583771\pi$
$857$	−15.2314	−0.520296	−0.260148	+	0.965569i	$0.583771\pi$
$858$	0	0
$859$	−26.3088	−0.897645	−0.448822	−	0.893621i	$-0.648156\pi$
$859$	−26.3088	−0.897645	−0.448822	+	0.893621i	$0.648156\pi$
$860$	0	0
$861$	9.22950	0.314541
$862$	0	0
$863$	−18.2580	−0.621508	−0.310754	−	0.950490i	$-0.600582\pi$
$863$	−18.2580	−0.621508	−0.310754	+	0.950490i	$0.600582\pi$
$864$	0	0
$865$	30.2967	1.03012
$866$	0	0
$867$	−1.17557	−0.0399245
$868$	0	0
$869$	4.61437	0.156532
$870$	0	0
$871$	−12.0387	−0.407916
$872$	0	0
$873$	19.7753	0.669291
$874$	0	0
$875$	−11.4585	−0.387368
$876$	0	0
$877$	48.2333	1.62872	0.814362	−	0.580357i	$-0.197087\pi$
$877$	48.2333	1.62872	0.814362	+	0.580357i	$0.197087\pi$
$878$	0	0
$879$	6.80443	0.229508
$880$	0	0
$881$	23.6432	0.796560	0.398280	−	0.917264i	$-0.369607\pi$
$881$	23.6432	0.796560	0.398280	+	0.917264i	$0.369607\pi$
$882$	0	0
$883$	16.3388	0.549843	0.274922	−	0.961467i	$-0.411348\pi$
$883$	16.3388	0.549843	0.274922	+	0.961467i	$0.411348\pi$
$884$	0	0
$885$	−38.2749	−1.28660
$886$	0	0
$887$	−51.2333	−1.72025	−0.860123	−	0.510086i	$-0.829614\pi$
$887$	−51.2333	−1.72025	−0.860123	+	0.510086i	$0.829614\pi$
$888$	0	0
$889$	−5.22139	−0.175120
$890$	0	0
$891$	−3.85044	−0.128995
$892$	0	0
$893$	−70.1053	−2.34599
$894$	0	0
$895$	45.3942	1.51736
$896$	0	0
$897$	−19.3167	−0.644964
$898$	0	0
$899$	10.5040	0.350327
$900$	0	0
$901$	−3.95012	−0.131598
$902$	0	0
$903$	10.7899	0.359067
$904$	0	0
$905$	−29.1626	−0.969397
$906$	0	0
$907$	−24.1841	−0.803019	−0.401509	−	0.915855i	$-0.631514\pi$
$907$	−24.1841	−0.803019	−0.401509	+	0.915855i	$0.631514\pi$
$908$	0	0
$909$	28.9952	0.961710
$910$	0	0
$911$	26.2366	0.869256	0.434628	−	0.900610i	$-0.356880\pi$
$911$	26.2366	0.869256	0.434628	+	0.900610i	$0.356880\pi$
$912$	0	0
$913$	38.0321	1.25868
$914$	0	0
$915$	34.3353	1.13509
$916$	0	0
$917$	6.72725	0.222154
$918$	0	0
$919$	50.3065	1.65946	0.829728	−	0.558167i	$-0.188495\pi$
$919$	50.3065	1.65946	0.829728	+	0.558167i	$0.188495\pi$
$920$	0	0
$921$	−23.9686	−0.789793
$922$	0	0
$923$	31.6759	1.04263
$924$	0	0
$925$	1.77373	0.0583198
$926$	0	0
$927$	−14.2840	−0.469147
$928$	0	0
$929$	−27.6357	−0.906700	−0.453350	−	0.891333i	$-0.649771\pi$
$929$	−27.6357	−0.906700	−0.453350	+	0.891333i	$0.649771\pi$
$930$	0	0
$931$	−5.53077	−0.181264
$932$	0	0
$933$	17.2787	0.565678
$934$	0	0
$935$	−5.48276	−0.179305
$936$	0	0
$937$	−43.3450	−1.41602	−0.708010	−	0.706203i	$-0.750407\pi$
$937$	−43.3450	−1.41602	−0.708010	+	0.706203i	$0.750407\pi$
$938$	0	0
$939$	28.2942	0.923347
$940$	0	0
$941$	37.6971	1.22889	0.614446	−	0.788959i	$-0.289380\pi$
$941$	37.6971	1.22889	0.614446	+	0.788959i	$0.289380\pi$
$942$	0	0
$943$	32.0142	1.04253
$944$	0	0
$945$	11.8108	0.384205
$946$	0	0
$947$	39.4813	1.28297	0.641485	−	0.767136i	$-0.278319\pi$
$947$	39.4813	1.28297	0.641485	+	0.767136i	$0.278319\pi$
$948$	0	0
$949$	−9.37111	−0.304199
$950$	0	0
$951$	−27.5127	−0.892162
$952$	0	0
$953$	−2.13405	−0.0691288	−0.0345644	−	0.999402i	$-0.511004\pi$
$953$	−2.13405	−0.0691288	−0.0345644	+	0.999402i	$0.511004\pi$
$954$	0	0
$955$	−12.3295	−0.398975
$956$	0	0
$957$	13.1070	0.423688
$958$	0	0
$959$	16.2243	0.523910
$960$	0	0
$961$	−25.3629	−0.818159
$962$	0	0
$963$	20.9519	0.675165
$964$	0	0
$965$	−18.4963	−0.595416
$966$	0	0
$967$	−2.45495	−0.0789459	−0.0394729	−	0.999221i	$-0.512568\pi$
$967$	−2.45495	−0.0789459	−0.0394729	+	0.999221i	$0.512568\pi$
$968$	0	0
$969$	6.50181	0.208868
$970$	0	0
$971$	−37.5190	−1.20404	−0.602021	−	0.798480i	$-0.705638\pi$
$971$	−37.5190	−1.20404	−0.602021	+	0.798480i	$0.705638\pi$
$972$	0	0
$973$	15.4358	0.494849
$974$	0	0
$975$	1.26432	0.0404905
$976$	0	0
$977$	17.0841	0.546570	0.273285	−	0.961933i	$-0.411890\pi$
$977$	17.0841	0.546570	0.273285	+	0.961933i	$0.411890\pi$
$978$	0	0
$979$	1.36438	0.0436059
$980$	0	0
$981$	18.4105	0.587802
$982$	0	0
$983$	−20.0052	−0.638068	−0.319034	−	0.947743i	$-0.603358\pi$
$983$	−20.0052	−0.638068	−0.319034	+	0.947743i	$0.603358\pi$
$984$	0	0
$985$	55.0884	1.75526
$986$	0	0
$987$	14.9010	0.474303
$988$	0	0
$989$	37.4269	1.19011
$990$	0	0
$991$	12.8254	0.407411	0.203705	−	0.979032i	$-0.434702\pi$
$991$	12.8254	0.407411	0.203705	+	0.979032i	$0.434702\pi$
$992$	0	0
$993$	−12.6208	−0.400509
$994$	0	0
$995$	−20.4886	−0.649531
$996$	0	0
$997$	−35.0182	−1.10904	−0.554518	−	0.832171i	$-0.687098\pi$
$997$	−35.0182	−1.10904	−0.554518	+	0.832171i	$0.687098\pi$
$998$	0	0
$999$	−36.0791	−1.14149

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.c.1.2	✓	4
4.3	odd	2		3808.2.a.d.1.3	yes	4
8.3	odd	2		7616.2.a.bm.1.2		4
8.5	even	2		7616.2.a.bl.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.c.1.2	✓	4	1.1	even	1	trivial
3808.2.a.d.1.3	yes	4	4.3	odd	2
7616.2.a.bl.1.3		4	8.5	even	2
7616.2.a.bm.1.2		4	8.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3808 = 2^{5} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3808.a (trivial)

\(f(q)\)	\(=\)	\(q-1.17557 q^{3} -2.17557 q^{5} -1.00000 q^{7} -1.61803 q^{9} +2.52015 q^{11} +4.02967 q^{13} +2.55754 q^{15} +1.00000 q^{17} -5.53077 q^{19} +1.17557 q^{21} +4.07768 q^{23} -0.266893 q^{25} +5.42882 q^{27} -4.42412 q^{29} -2.37425 q^{31} -2.96261 q^{33} +2.17557 q^{35} -6.64584 q^{37} -4.73716 q^{39} +7.85108 q^{41} +9.17848 q^{43} +3.52015 q^{45} +12.6755 q^{47} +1.00000 q^{49} -1.17557 q^{51} -3.95012 q^{53} -5.48276 q^{55} +6.50181 q^{57} -14.9655 q^{59} +13.4252 q^{61} +1.61803 q^{63} -8.76684 q^{65} -2.98751 q^{67} -4.79360 q^{69} +7.86067 q^{71} -2.32553 q^{73} +0.313752 q^{75} -2.52015 q^{77} +1.83099 q^{79} -1.52786 q^{81} +15.0912 q^{83} -2.17557 q^{85} +5.20087 q^{87} +0.541390 q^{89} -4.02967 q^{91} +2.79110 q^{93} +12.0326 q^{95} -12.2218 q^{97} -4.07768 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} - 4 q^{7} - 2 q^{9} - 2 q^{11} - 2 q^{13} + 10 q^{15} + 4 q^{17} - 4 q^{19} + 4 q^{23} - 6 q^{25} + 6 q^{29} + 16 q^{31} + 4 q^{35} - 8 q^{37} - 10 q^{39} - 10 q^{41} - 4 q^{43} + 2 q^{45}+ \cdots - 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^5 - 4 * q^7 - 2 * q^9 - 2 * q^11 - 2 * q^13 + 10 * q^15 + 4 * q^17 - 4 * q^19 + 4 * q^23 - 6 * q^25 + 6 * q^29 + 16 * q^31 + 4 * q^35 - 8 * q^37 - 10 * q^39 - 10 * q^41 - 4 * q^43 + 2 * q^45 + 14 * q^47 + 4 * q^49 - 14 * q^53 + 2 * q^55 - 10 * q^57 - 12 * q^59 - 2 * q^61 + 2 * q^63 - 8 * q^65 - 22 * q^67 - 10 * q^69 + 16 * q^71 - 14 * q^73 - 20 * q^75 + 2 * q^77 + 10 * q^79 - 24 * q^81 + 6 * q^83 - 4 * q^85 - 22 * q^89 + 2 * q^91 - 6 * q^95 - 12 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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