LMFDB - Embedded newform 3808.2.a.e.1.3

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 = [5,0,0,0,-2,0,-5] 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:	$0$
Dimension:	$5$
Coefficient field:	5.5.804272.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 9x^{3} - 6x^{2} + 3x + 2 $ x^5 - 9x^3 - 6x^2 + 3*x + 2
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			1.3
Root			$-0.540819$ 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)

$f(q)$	$=$	$q-0.540819 q^{3} +2.15728 q^{5} -1.00000 q^{7} -2.70751 q^{9} +2.06112 q^{11} -5.50921 q^{13} -1.16670 q^{15} -1.00000 q^{17} -2.17414 q^{19} +0.540819 q^{21} -3.57033 q^{23} -0.346161 q^{25} +3.08673 q^{27} +2.95730 q^{29} +10.2904 q^{31} -1.11469 q^{33} -2.15728 q^{35} +2.51088 q^{37} +2.97949 q^{39} +0.794247 q^{41} +5.74068 q^{43} -5.84086 q^{45} +3.00167 q^{47} +1.00000 q^{49} +0.540819 q^{51} +9.28871 q^{53} +4.44641 q^{55} +1.17582 q^{57} +11.7296 q^{59} -0.246361 q^{61} +2.70751 q^{63} -11.8849 q^{65} +12.7180 q^{67} +1.93090 q^{69} +1.63355 q^{71} -7.12369 q^{73} +0.187210 q^{75} -2.06112 q^{77} +5.02051 q^{79} +6.45318 q^{81} -0.0502562 q^{83} -2.15728 q^{85} -1.59936 q^{87} +6.56798 q^{89} +5.50921 q^{91} -5.56524 q^{93} -4.69023 q^{95} +12.9424 q^{97} -5.58052 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{5} - 5 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} + 8 q^{15} - 5 q^{17} - 6 q^{19} + 18 q^{23} + 5 q^{25} + 18 q^{27} - 14 q^{29} + 16 q^{31} - 26 q^{33} + 2 q^{35} + 2 q^{37} + 6 q^{39} - 10 q^{41}+ \cdots - 8 q^{99}+O(q^{100}) $ 5 * q - 2 * q^5 - 5 * q^7 + 3 * q^9 - 4 * q^11 - 6 * q^13 + 8 * q^15 - 5 * q^17 - 6 * q^19 + 18 * q^23 + 5 * q^25 + 18 * q^27 - 14 * q^29 + 16 * q^31 - 26 * q^33 + 2 * q^35 + 2 * q^37 + 6 * q^39 - 10 * q^41 - 20 * q^43 + 6 * q^45 + 26 * q^47 + 5 * q^49 + 4 * q^55 + 12 * q^57 + 8 * q^61 - 3 * q^63 + 2 * q^65 + 12 * q^67 + 32 * q^69 + 10 * q^71 - 14 * q^73 - 4 * q^75 + 4 * q^77 + 34 * q^79 + 33 * q^81 + 20 * q^83 + 2 * q^85 + 14 * q^87 - 4 * q^89 + 6 * q^91 + 16 * q^93 + 42 * q^95 - 12 * q^97 - 8 * 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
$3$	−0.540819	−0.312242	−0.156121	−	0.987738i	$-0.549899\pi$
$3$	−0.540819	−0.312242	−0.156121	+	0.987738i	$0.549899\pi$
$4$	0	0
$5$	2.15728	0.964763	0.482382	−	0.875961i	$-0.339772\pi$
$5$	2.15728	0.964763	0.482382	+	0.875961i	$0.339772\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.70751	−0.902505
$10$	0	0
$11$	2.06112	0.621452	0.310726	−	0.950500i	$-0.399428\pi$
$11$	2.06112	0.621452	0.310726	+	0.950500i	$0.399428\pi$
$12$	0	0
$13$	−5.50921	−1.52798	−0.763990	−	0.645228i	$-0.776762\pi$
$13$	−5.50921	−1.52798	−0.763990	+	0.645228i	$0.776762\pi$
$14$	0	0
$15$	−1.16670	−0.301240
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−2.17414	−0.498783	−0.249391	−	0.968403i	$-0.580231\pi$
$19$	−2.17414	−0.498783	−0.249391	+	0.968403i	$0.580231\pi$
$20$	0	0
$21$	0.540819	0.118016
$22$	0	0
$23$	−3.57033	−0.744466	−0.372233	−	0.928139i	$-0.621408\pi$
$23$	−3.57033	−0.744466	−0.372233	+	0.928139i	$0.621408\pi$
$24$	0	0
$25$	−0.346161	−0.0692322
$26$	0	0
$27$	3.08673	0.594042
$28$	0	0
$29$	2.95730	0.549156	0.274578	−	0.961565i	$-0.411462\pi$
$29$	2.95730	0.549156	0.274578	+	0.961565i	$0.411462\pi$
$30$	0	0
$31$	10.2904	1.84821	0.924104	−	0.382140i	$-0.124813\pi$
$31$	10.2904	1.84821	0.924104	+	0.382140i	$0.124813\pi$
$32$	0	0
$33$	−1.11469	−0.194043
$34$	0	0
$35$	−2.15728	−0.364646
$36$	0	0
$37$	2.51088	0.412787	0.206393	−	0.978469i	$-0.433827\pi$
$37$	2.51088	0.412787	0.206393	+	0.978469i	$0.433827\pi$
$38$	0	0
$39$	2.97949	0.477099
$40$	0	0
$41$	0.794247	0.124041	0.0620203	−	0.998075i	$-0.480246\pi$
$41$	0.794247	0.124041	0.0620203	+	0.998075i	$0.480246\pi$
$42$	0	0
$43$	5.74068	0.875445	0.437722	−	0.899110i	$-0.355785\pi$
$43$	5.74068	0.875445	0.437722	+	0.899110i	$0.355785\pi$
$44$	0	0
$45$	−5.84086	−0.870703
$46$	0	0
$47$	3.00167	0.437839	0.218920	−	0.975743i	$-0.429747\pi$
$47$	3.00167	0.437839	0.218920	+	0.975743i	$0.429747\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.540819	0.0757298
$52$	0	0
$53$	9.28871	1.27590	0.637952	−	0.770077i	$-0.279782\pi$
$53$	9.28871	1.27590	0.637952	+	0.770077i	$0.279782\pi$
$54$	0	0
$55$	4.44641	0.599554
$56$	0	0
$57$	1.17582	0.155741
$58$	0	0
$59$	11.7296	1.52706	0.763531	−	0.645772i	$-0.223464\pi$
$59$	11.7296	1.52706	0.763531	+	0.645772i	$0.223464\pi$
$60$	0	0
$61$	−0.246361	−0.0315433	−0.0157717	−	0.999876i	$-0.505020\pi$
$61$	−0.246361	−0.0315433	−0.0157717	+	0.999876i	$0.505020\pi$
$62$	0	0
$63$	2.70751	0.341115
$64$	0	0
$65$	−11.8849	−1.47414
$66$	0	0
$67$	12.7180	1.55375	0.776873	−	0.629657i	$-0.216805\pi$
$67$	12.7180	1.55375	0.776873	+	0.629657i	$0.216805\pi$
$68$	0	0
$69$	1.93090	0.232454
$70$	0	0
$71$	1.63355	0.193867	0.0969335	−	0.995291i	$-0.469097\pi$
$71$	1.63355	0.193867	0.0969335	+	0.995291i	$0.469097\pi$
$72$	0	0
$73$	−7.12369	−0.833765	−0.416883	−	0.908960i	$-0.636877\pi$
$73$	−7.12369	−0.833765	−0.416883	+	0.908960i	$0.636877\pi$
$74$	0	0
$75$	0.187210	0.0216172
$76$	0	0
$77$	−2.06112	−0.234887
$78$	0	0
$79$	5.02051	0.564852	0.282426	−	0.959289i	$-0.408861\pi$
$79$	5.02051	0.564852	0.282426	+	0.959289i	$0.408861\pi$
$80$	0	0
$81$	6.45318	0.717020
$82$	0	0
$83$	−0.0502562	−0.00551633	−0.00275817	−	0.999996i	$-0.500878\pi$
$83$	−0.0502562	−0.00551633	−0.00275817	+	0.999996i	$0.500878\pi$
$84$	0	0
$85$	−2.15728	−0.233989
$86$	0	0
$87$	−1.59936	−0.171470
$88$	0	0
$89$	6.56798	0.696204	0.348102	−	0.937457i	$-0.386826\pi$
$89$	6.56798	0.696204	0.348102	+	0.937457i	$0.386826\pi$
$90$	0	0
$91$	5.50921	0.577522
$92$	0	0
$93$	−5.56524	−0.577088
$94$	0	0
$95$	−4.69023	−0.481207
$96$	0	0
$97$	12.9424	1.31410	0.657049	−	0.753848i	$-0.271805\pi$
$97$	12.9424	1.31410	0.657049	+	0.753848i	$0.271805\pi$
$98$	0	0
$99$	−5.58052	−0.560864
$100$	0	0
$101$	−17.3438	−1.72578	−0.862888	−	0.505395i	$-0.831347\pi$
$101$	−17.3438	−1.72578	−0.862888	+	0.505395i	$0.831347\pi$
$102$	0	0
$103$	13.8426	1.36395	0.681976	−	0.731375i	$-0.261121\pi$
$103$	13.8426	1.36395	0.681976	+	0.731375i	$0.261121\pi$
$104$	0	0
$105$	1.16670	0.113858
$106$	0	0
$107$	10.5908	1.02386	0.511928	−	0.859028i	$-0.328931\pi$
$107$	10.5908	1.02386	0.511928	+	0.859028i	$0.328931\pi$
$108$	0	0
$109$	−8.60297	−0.824015	−0.412007	−	0.911180i	$-0.635172\pi$
$109$	−8.60297	−0.824015	−0.412007	+	0.911180i	$0.635172\pi$
$110$	0	0
$111$	−1.35793	−0.128889
$112$	0	0
$113$	−1.32777	−0.124906	−0.0624531	−	0.998048i	$-0.519892\pi$
$113$	−1.32777	−0.124906	−0.0624531	+	0.998048i	$0.519892\pi$
$114$	0	0
$115$	−7.70219	−0.718233
$116$	0	0
$117$	14.9163	1.37901
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−6.75177	−0.613797
$122$	0	0
$123$	−0.429544	−0.0387307
$124$	0	0
$125$	−11.5331	−1.03156
$126$	0	0
$127$	17.6334	1.56471	0.782357	−	0.622830i	$-0.214017\pi$
$127$	17.6334	1.56471	0.782357	+	0.622830i	$0.214017\pi$
$128$	0	0
$129$	−3.10467	−0.273351
$130$	0	0
$131$	−10.8112	−0.944581	−0.472290	−	0.881443i	$-0.656573\pi$
$131$	−10.8112	−0.944581	−0.472290	+	0.881443i	$0.656573\pi$
$132$	0	0
$133$	2.17414	0.188522
$134$	0	0
$135$	6.65893	0.573110
$136$	0	0
$137$	−1.34571	−0.114971	−0.0574857	−	0.998346i	$-0.518308\pi$
$137$	−1.34571	−0.114971	−0.0574857	+	0.998346i	$0.518308\pi$
$138$	0	0
$139$	−14.9943	−1.27180	−0.635898	−	0.771773i	$-0.719370\pi$
$139$	−14.9943	−1.27180	−0.635898	+	0.771773i	$0.719370\pi$
$140$	0	0
$141$	−1.62336	−0.136712
$142$	0	0
$143$	−11.3552	−0.949566
$144$	0	0
$145$	6.37970	0.529805
$146$	0	0
$147$	−0.540819	−0.0446060
$148$	0	0
$149$	−4.76119	−0.390052	−0.195026	−	0.980798i	$-0.562479\pi$
$149$	−4.76119	−0.390052	−0.195026	+	0.980798i	$0.562479\pi$
$150$	0	0
$151$	−16.9031	−1.37555	−0.687776	−	0.725923i	$-0.741413\pi$
$151$	−16.9031	−1.37555	−0.687776	+	0.725923i	$0.741413\pi$
$152$	0	0
$153$	2.70751	0.218890
$154$	0	0
$155$	22.1992	1.78308
$156$	0	0
$157$	17.8748	1.42657	0.713283	−	0.700876i	$-0.247208\pi$
$157$	17.8748	1.42657	0.713283	+	0.700876i	$0.247208\pi$
$158$	0	0
$159$	−5.02351	−0.398391
$160$	0	0
$161$	3.57033	0.281382
$162$	0	0
$163$	−6.63872	−0.519984	−0.259992	−	0.965611i	$-0.583720\pi$
$163$	−6.63872	−0.519984	−0.259992	+	0.965611i	$0.583720\pi$
$164$	0	0
$165$	−2.40470	−0.187206
$166$	0	0
$167$	−4.41773	−0.341854	−0.170927	−	0.985284i	$-0.554676\pi$
$167$	−4.41773	−0.341854	−0.170927	+	0.985284i	$0.554676\pi$
$168$	0	0
$169$	17.3514	1.33472
$170$	0	0
$171$	5.88652	0.450154
$172$	0	0
$173$	20.2905	1.54266	0.771329	−	0.636437i	$-0.219592\pi$
$173$	20.2905	1.54266	0.771329	+	0.636437i	$0.219592\pi$
$174$	0	0
$175$	0.346161	0.0261673
$176$	0	0
$177$	−6.34358	−0.476813
$178$	0	0
$179$	−13.9068	−1.03945	−0.519723	−	0.854335i	$-0.673965\pi$
$179$	−13.9068	−1.03945	−0.519723	+	0.854335i	$0.673965\pi$
$180$	0	0
$181$	−10.3073	−0.766132	−0.383066	−	0.923721i	$-0.625132\pi$
$181$	−10.3073	−0.766132	−0.383066	+	0.923721i	$0.625132\pi$
$182$	0	0
$183$	0.133237	0.00984915
$184$	0	0
$185$	5.41667	0.398241
$186$	0	0
$187$	−2.06112	−0.150724
$188$	0	0
$189$	−3.08673	−0.224527
$190$	0	0
$191$	−2.43289	−0.176038	−0.0880190	−	0.996119i	$-0.528054\pi$
$191$	−2.43289	−0.176038	−0.0880190	+	0.996119i	$0.528054\pi$
$192$	0	0
$193$	23.3759	1.68264	0.841318	−	0.540540i	$-0.181780\pi$
$193$	23.3759	1.68264	0.841318	+	0.540540i	$0.181780\pi$
$194$	0	0
$195$	6.42757	0.460288
$196$	0	0
$197$	5.29806	0.377471	0.188736	−	0.982028i	$-0.439561\pi$
$197$	5.29806	0.377471	0.188736	+	0.982028i	$0.439561\pi$
$198$	0	0
$199$	25.1404	1.78216	0.891079	−	0.453848i	$-0.149949\pi$
$199$	25.1404	1.78216	0.891079	+	0.453848i	$0.149949\pi$
$200$	0	0
$201$	−6.87812	−0.485145
$202$	0	0
$203$	−2.95730	−0.207561
$204$	0	0
$205$	1.71341	0.119670
$206$	0	0
$207$	9.66673	0.671884
$208$	0	0
$209$	−4.48118	−0.309970
$210$	0	0
$211$	−17.9219	−1.23379	−0.616897	−	0.787044i	$-0.711610\pi$
$211$	−17.9219	−1.23379	−0.616897	+	0.787044i	$0.711610\pi$
$212$	0	0
$213$	−0.883456	−0.0605334
$214$	0	0
$215$	12.3842	0.844597
$216$	0	0
$217$	−10.2904	−0.698557
$218$	0	0
$219$	3.85263	0.260337
$220$	0	0
$221$	5.50921	0.370590
$222$	0	0
$223$	19.6584	1.31642	0.658212	−	0.752833i	$-0.271313\pi$
$223$	19.6584	1.31642	0.658212	+	0.752833i	$0.271313\pi$
$224$	0	0
$225$	0.937236	0.0624824
$226$	0	0
$227$	−14.1673	−0.940317	−0.470158	−	0.882582i	$-0.655803\pi$
$227$	−14.1673	−0.940317	−0.470158	+	0.882582i	$0.655803\pi$
$228$	0	0
$229$	10.2317	0.676129	0.338065	−	0.941123i	$-0.390228\pi$
$229$	10.2317	0.676129	0.338065	+	0.941123i	$0.390228\pi$
$230$	0	0
$231$	1.11469	0.0733415
$232$	0	0
$233$	−16.1961	−1.06104	−0.530520	−	0.847672i	$-0.678003\pi$
$233$	−16.1961	−1.06104	−0.530520	+	0.847672i	$0.678003\pi$
$234$	0	0
$235$	6.47544	0.422411
$236$	0	0
$237$	−2.71519	−0.176371
$238$	0	0
$239$	15.1591	0.980558	0.490279	−	0.871566i	$-0.336895\pi$
$239$	15.1591	0.980558	0.490279	+	0.871566i	$0.336895\pi$
$240$	0	0
$241$	−9.36071	−0.602976	−0.301488	−	0.953470i	$-0.597483\pi$
$241$	−9.36071	−0.602976	−0.301488	+	0.953470i	$0.597483\pi$
$242$	0	0
$243$	−12.7502	−0.817926
$244$	0	0
$245$	2.15728	0.137823
$246$	0	0
$247$	11.9778	0.762130
$248$	0	0
$249$	0.0271795	0.00172243
$250$	0	0
$251$	−14.7815	−0.933001	−0.466500	−	0.884521i	$-0.654485\pi$
$251$	−14.7815	−0.933001	−0.466500	+	0.884521i	$0.654485\pi$
$252$	0	0
$253$	−7.35890	−0.462650
$254$	0	0
$255$	1.16670	0.0730613
$256$	0	0
$257$	0.885305	0.0552238	0.0276119	−	0.999619i	$-0.491210\pi$
$257$	0.885305	0.0552238	0.0276119	+	0.999619i	$0.491210\pi$
$258$	0	0
$259$	−2.51088	−0.156019
$260$	0	0
$261$	−8.00692	−0.495616
$262$	0	0
$263$	30.6921	1.89255	0.946277	−	0.323356i	$-0.104811\pi$
$263$	30.6921	1.89255	0.946277	+	0.323356i	$0.104811\pi$
$264$	0	0
$265$	20.0383	1.23094
$266$	0	0
$267$	−3.55209	−0.217384
$268$	0	0
$269$	−7.97781	−0.486416	−0.243208	−	0.969974i	$-0.578200\pi$
$269$	−7.97781	−0.486416	−0.243208	+	0.969974i	$0.578200\pi$
$270$	0	0
$271$	17.6999	1.07519	0.537595	−	0.843203i	$-0.319333\pi$
$271$	17.6999	1.07519	0.537595	+	0.843203i	$0.319333\pi$
$272$	0	0
$273$	−2.97949	−0.180327
$274$	0	0
$275$	−0.713481	−0.0430245
$276$	0	0
$277$	−1.73985	−0.104538	−0.0522688	−	0.998633i	$-0.516645\pi$
$277$	−1.73985	−0.104538	−0.0522688	+	0.998633i	$0.516645\pi$
$278$	0	0
$279$	−27.8614	−1.66802
$280$	0	0
$281$	−8.18140	−0.488061	−0.244031	−	0.969768i	$-0.578470\pi$
$281$	−8.18140	−0.488061	−0.244031	+	0.969768i	$0.578470\pi$
$282$	0	0
$283$	−31.5674	−1.87649	−0.938245	−	0.345973i	$-0.887549\pi$
$283$	−31.5674	−1.87649	−0.938245	+	0.345973i	$0.887549\pi$
$284$	0	0
$285$	2.53656	0.150253
$286$	0	0
$287$	−0.794247	−0.0468829
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−6.99947	−0.410316
$292$	0	0
$293$	−28.3285	−1.65497	−0.827485	−	0.561488i	$-0.810229\pi$
$293$	−28.3285	−1.65497	−0.827485	+	0.561488i	$0.810229\pi$
$294$	0	0
$295$	25.3039	1.47325
$296$	0	0
$297$	6.36214	0.369169
$298$	0	0
$299$	19.6697	1.13753
$300$	0	0
$301$	−5.74068	−0.330887
$302$	0	0
$303$	9.37988	0.538860
$304$	0	0
$305$	−0.531469	−0.0304318
$306$	0	0
$307$	9.50837	0.542671	0.271336	−	0.962485i	$-0.412535\pi$
$307$	9.50837	0.542671	0.271336	+	0.962485i	$0.412535\pi$
$308$	0	0
$309$	−7.48634	−0.425883
$310$	0	0
$311$	5.78612	0.328101	0.164050	−	0.986452i	$-0.447544\pi$
$311$	5.78612	0.328101	0.164050	+	0.986452i	$0.447544\pi$
$312$	0	0
$313$	3.22007	0.182009	0.0910047	−	0.995850i	$-0.470992\pi$
$313$	3.22007	0.182009	0.0910047	+	0.995850i	$0.470992\pi$
$314$	0	0
$315$	5.84086	0.329095
$316$	0	0
$317$	15.3336	0.861224	0.430612	−	0.902537i	$-0.358298\pi$
$317$	15.3336	0.861224	0.430612	+	0.902537i	$0.358298\pi$
$318$	0	0
$319$	6.09535	0.341274
$320$	0	0
$321$	−5.72773	−0.319691
$322$	0	0
$323$	2.17414	0.120973
$324$	0	0
$325$	1.90707	0.105785
$326$	0	0
$327$	4.65265	0.257292
$328$	0	0
$329$	−3.00167	−0.165488
$330$	0	0
$331$	21.5074	1.18216	0.591078	−	0.806614i	$-0.298703\pi$
$331$	21.5074	1.18216	0.591078	+	0.806614i	$0.298703\pi$
$332$	0	0
$333$	−6.79826	−0.372542
$334$	0	0
$335$	27.4361	1.49900
$336$	0	0
$337$	10.9215	0.594930	0.297465	−	0.954733i	$-0.403859\pi$
$337$	10.9215	0.594930	0.297465	+	0.954733i	$0.403859\pi$
$338$	0	0
$339$	0.718084	0.0390010
$340$	0	0
$341$	21.2098	1.14857
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	4.16549	0.224263
$346$	0	0
$347$	6.66678	0.357892	0.178946	−	0.983859i	$-0.442731\pi$
$347$	6.66678	0.357892	0.178946	+	0.983859i	$0.442731\pi$
$348$	0	0
$349$	4.61593	0.247085	0.123543	−	0.992339i	$-0.460574\pi$
$349$	4.61593	0.247085	0.123543	+	0.992339i	$0.460574\pi$
$350$	0	0
$351$	−17.0055	−0.907684
$352$	0	0
$353$	−15.2091	−0.809498	−0.404749	−	0.914428i	$-0.632641\pi$
$353$	−15.2091	−0.809498	−0.404749	+	0.914428i	$0.632641\pi$
$354$	0	0
$355$	3.52402	0.187036
$356$	0	0
$357$	−0.540819	−0.0286232
$358$	0	0
$359$	3.42996	0.181027	0.0905133	−	0.995895i	$-0.471149\pi$
$359$	3.42996	0.181027	0.0905133	+	0.995895i	$0.471149\pi$
$360$	0	0
$361$	−14.2731	−0.751216
$362$	0	0
$363$	3.65149	0.191653
$364$	0	0
$365$	−15.3678	−0.804386
$366$	0	0
$367$	20.3618	1.06288	0.531439	−	0.847097i	$-0.321652\pi$
$367$	20.3618	1.06288	0.531439	+	0.847097i	$0.321652\pi$
$368$	0	0
$369$	−2.15044	−0.111947
$370$	0	0
$371$	−9.28871	−0.482246
$372$	0	0
$373$	26.2757	1.36050	0.680252	−	0.732979i	$-0.261871\pi$
$373$	26.2757	1.36050	0.680252	+	0.732979i	$0.261871\pi$
$374$	0	0
$375$	6.23734	0.322095
$376$	0	0
$377$	−16.2924	−0.839099
$378$	0	0
$379$	0.646689	0.0332182	0.0166091	−	0.999862i	$-0.494713\pi$
$379$	0.646689	0.0332182	0.0166091	+	0.999862i	$0.494713\pi$
$380$	0	0
$381$	−9.53649	−0.488569
$382$	0	0
$383$	0.972646	0.0496999	0.0248499	−	0.999691i	$-0.492089\pi$
$383$	0.972646	0.0496999	0.0248499	+	0.999691i	$0.492089\pi$
$384$	0	0
$385$	−4.44641	−0.226610
$386$	0	0
$387$	−15.5430	−0.790093
$388$	0	0
$389$	7.77073	0.393992	0.196996	−	0.980404i	$-0.436881\pi$
$389$	7.77073	0.393992	0.196996	+	0.980404i	$0.436881\pi$
$390$	0	0
$391$	3.57033	0.180559
$392$	0	0
$393$	5.84691	0.294938
$394$	0	0
$395$	10.8306	0.544948
$396$	0	0
$397$	20.8796	1.04792	0.523958	−	0.851744i	$-0.324455\pi$
$397$	20.8796	1.04792	0.523958	+	0.851744i	$0.324455\pi$
$398$	0	0
$399$	−1.17582	−0.0588645
$400$	0	0
$401$	30.7076	1.53347	0.766733	−	0.641966i	$-0.221881\pi$
$401$	30.7076	1.53347	0.766733	+	0.641966i	$0.221881\pi$
$402$	0	0
$403$	−56.6919	−2.82403
$404$	0	0
$405$	13.9213	0.691754
$406$	0	0
$407$	5.17524	0.256527
$408$	0	0
$409$	13.1961	0.652507	0.326253	−	0.945282i	$-0.394214\pi$
$409$	13.1961	0.652507	0.326253	+	0.945282i	$0.394214\pi$
$410$	0	0
$411$	0.727783	0.0358989
$412$	0	0
$413$	−11.7296	−0.577175
$414$	0	0
$415$	−0.108416	−0.00532195
$416$	0	0
$417$	8.10918	0.397108
$418$	0	0
$419$	7.80527	0.381312	0.190656	−	0.981657i	$-0.438938\pi$
$419$	7.80527	0.381312	0.190656	+	0.981657i	$0.438938\pi$
$420$	0	0
$421$	−23.9280	−1.16618	−0.583089	−	0.812408i	$-0.698156\pi$
$421$	−23.9280	−1.16618	−0.583089	+	0.812408i	$0.698156\pi$
$422$	0	0
$423$	−8.12708	−0.395152
$424$	0	0
$425$	0.346161	0.0167913
$426$	0	0
$427$	0.246361	0.0119223
$428$	0	0
$429$	6.14109	0.296494
$430$	0	0
$431$	1.42359	0.0685720	0.0342860	−	0.999412i	$-0.489084\pi$
$431$	1.42359	0.0685720	0.0342860	+	0.999412i	$0.489084\pi$
$432$	0	0
$433$	−19.2384	−0.924539	−0.462270	−	0.886739i	$-0.652965\pi$
$433$	−19.2384	−0.924539	−0.462270	+	0.886739i	$0.652965\pi$
$434$	0	0
$435$	−3.45026	−0.165428
$436$	0	0
$437$	7.76241	0.371327
$438$	0	0
$439$	11.5887	0.553100	0.276550	−	0.961000i	$-0.410809\pi$
$439$	11.5887	0.553100	0.276550	+	0.961000i	$0.410809\pi$
$440$	0	0
$441$	−2.70751	−0.128929
$442$	0	0
$443$	−7.32233	−0.347894	−0.173947	−	0.984755i	$-0.555652\pi$
$443$	−7.32233	−0.347894	−0.173947	+	0.984755i	$0.555652\pi$
$444$	0	0
$445$	14.1689	0.671672
$446$	0	0
$447$	2.57494	0.121791
$448$	0	0
$449$	21.4833	1.01386	0.506929	−	0.861988i	$-0.330780\pi$
$449$	21.4833	1.01386	0.506929	+	0.861988i	$0.330780\pi$
$450$	0	0
$451$	1.63704	0.0770853
$452$	0	0
$453$	9.14150	0.429505
$454$	0	0
$455$	11.8849	0.557172
$456$	0	0
$457$	−16.1809	−0.756913	−0.378456	−	0.925619i	$-0.623545\pi$
$457$	−16.1809	−0.756913	−0.378456	+	0.925619i	$0.623545\pi$
$458$	0	0
$459$	−3.08673	−0.144076
$460$	0	0
$461$	0.312456	0.0145525	0.00727626	−	0.999974i	$-0.497684\pi$
$461$	0.312456	0.0145525	0.00727626	+	0.999974i	$0.497684\pi$
$462$	0	0
$463$	−40.1436	−1.86563	−0.932815	−	0.360355i	$-0.882656\pi$
$463$	−40.1436	−1.86563	−0.932815	+	0.360355i	$0.882656\pi$
$464$	0	0
$465$	−12.0058	−0.556754
$466$	0	0
$467$	28.7018	1.32816	0.664080	−	0.747662i	$-0.268824\pi$
$467$	28.7018	1.32816	0.664080	+	0.747662i	$0.268824\pi$
$468$	0	0
$469$	−12.7180	−0.587261
$470$	0	0
$471$	−9.66704	−0.445434
$472$	0	0
$473$	11.8322	0.544047
$474$	0	0
$475$	0.752604	0.0345318
$476$	0	0
$477$	−25.1493	−1.15151
$478$	0	0
$479$	−34.0619	−1.55633	−0.778164	−	0.628061i	$-0.783849\pi$
$479$	−34.0619	−1.55633	−0.778164	+	0.628061i	$0.783849\pi$
$480$	0	0
$481$	−13.8330	−0.630730
$482$	0	0
$483$	−1.93090	−0.0878592
$484$	0	0
$485$	27.9202	1.26779
$486$	0	0
$487$	31.9012	1.44558	0.722790	−	0.691068i	$-0.242859\pi$
$487$	31.9012	1.44558	0.722790	+	0.691068i	$0.242859\pi$
$488$	0	0
$489$	3.59034	0.162361
$490$	0	0
$491$	−1.99190	−0.0898934	−0.0449467	−	0.998989i	$-0.514312\pi$
$491$	−1.99190	−0.0898934	−0.0449467	+	0.998989i	$0.514312\pi$
$492$	0	0
$493$	−2.95730	−0.133190
$494$	0	0
$495$	−12.0387	−0.541100
$496$	0	0
$497$	−1.63355	−0.0732748
$498$	0	0
$499$	7.89853	0.353587	0.176793	−	0.984248i	$-0.443428\pi$
$499$	7.89853	0.353587	0.176793	+	0.984248i	$0.443428\pi$
$500$	0	0
$501$	2.38919	0.106741
$502$	0	0
$503$	12.2052	0.544204	0.272102	−	0.962268i	$-0.412281\pi$
$503$	12.2052	0.544204	0.272102	+	0.962268i	$0.412281\pi$
$504$	0	0
$505$	−37.4154	−1.66497
$506$	0	0
$507$	−9.38396	−0.416756
$508$	0	0
$509$	20.8644	0.924797	0.462398	−	0.886672i	$-0.346989\pi$
$509$	20.8644	0.924797	0.462398	+	0.886672i	$0.346989\pi$
$510$	0	0
$511$	7.12369	0.315134
$512$	0	0
$513$	−6.71100	−0.296298
$514$	0	0
$515$	29.8623	1.31589
$516$	0	0
$517$	6.18682	0.272096
$518$	0	0
$519$	−10.9735	−0.481683
$520$	0	0
$521$	13.6690	0.598849	0.299425	−	0.954120i	$-0.403205\pi$
$521$	13.6690	0.598849	0.299425	+	0.954120i	$0.403205\pi$
$522$	0	0
$523$	−37.2185	−1.62745	−0.813725	−	0.581251i	$-0.802564\pi$
$523$	−37.2185	−1.62745	−0.813725	+	0.581251i	$0.802564\pi$
$524$	0	0
$525$	−0.187210	−0.00817054
$526$	0	0
$527$	−10.2904	−0.448256
$528$	0	0
$529$	−10.2527	−0.445771
$530$	0	0
$531$	−31.7580	−1.37818
$532$	0	0
$533$	−4.37567	−0.189532
$534$	0	0
$535$	22.8474	0.987778
$536$	0	0
$537$	7.52109	0.324559
$538$	0	0
$539$	2.06112	0.0887789
$540$	0	0
$541$	19.0332	0.818303	0.409151	−	0.912467i	$-0.365825\pi$
$541$	19.0332	0.818303	0.409151	+	0.912467i	$0.365825\pi$
$542$	0	0
$543$	5.57436	0.239219
$544$	0	0
$545$	−18.5590	−0.794979
$546$	0	0
$547$	21.0855	0.901553	0.450776	−	0.892637i	$-0.351147\pi$
$547$	21.0855	0.901553	0.450776	+	0.892637i	$0.351147\pi$
$548$	0	0
$549$	0.667027	0.0284680
$550$	0	0
$551$	−6.42958	−0.273910
$552$	0	0
$553$	−5.02051	−0.213494
$554$	0	0
$555$	−2.92944	−0.124348
$556$	0	0
$557$	−8.00638	−0.339241	−0.169621	−	0.985509i	$-0.554254\pi$
$557$	−8.00638	−0.339241	−0.169621	+	0.985509i	$0.554254\pi$
$558$	0	0
$559$	−31.6266	−1.33766
$560$	0	0
$561$	1.11469	0.0470624
$562$	0	0
$563$	24.4778	1.03162	0.515808	−	0.856704i	$-0.327492\pi$
$563$	24.4778	1.03162	0.515808	+	0.856704i	$0.327492\pi$
$564$	0	0
$565$	−2.86437	−0.120505
$566$	0	0
$567$	−6.45318	−0.271008
$568$	0	0
$569$	2.89788	0.121485	0.0607427	−	0.998153i	$-0.480653\pi$
$569$	2.89788	0.121485	0.0607427	+	0.998153i	$0.480653\pi$
$570$	0	0
$571$	−31.9675	−1.33780	−0.668898	−	0.743354i	$-0.733234\pi$
$571$	−31.9675	−1.33780	−0.668898	+	0.743354i	$0.733234\pi$
$572$	0	0
$573$	1.31576	0.0549665
$574$	0	0
$575$	1.23591	0.0515410
$576$	0	0
$577$	−15.8209	−0.658631	−0.329316	−	0.944220i	$-0.606818\pi$
$577$	−15.8209	−0.658631	−0.329316	+	0.944220i	$0.606818\pi$
$578$	0	0
$579$	−12.6421	−0.525390
$580$	0	0
$581$	0.0502562	0.00208498
$582$	0	0
$583$	19.1452	0.792913
$584$	0	0
$585$	32.1785	1.33042
$586$	0	0
$587$	10.1944	0.420768	0.210384	−	0.977619i	$-0.432529\pi$
$587$	10.1944	0.420768	0.210384	+	0.977619i	$0.432529\pi$
$588$	0	0
$589$	−22.3728	−0.921854
$590$	0	0
$591$	−2.86529	−0.117862
$592$	0	0
$593$	−5.94870	−0.244284	−0.122142	−	0.992513i	$-0.538976\pi$
$593$	−5.94870	−0.244284	−0.122142	+	0.992513i	$0.538976\pi$
$594$	0	0
$595$	2.15728	0.0884397
$596$	0	0
$597$	−13.5964	−0.556465
$598$	0	0
$599$	−19.7001	−0.804924	−0.402462	−	0.915437i	$-0.631846\pi$
$599$	−19.7001	−0.804924	−0.402462	+	0.915437i	$0.631846\pi$
$600$	0	0
$601$	−18.1121	−0.738806	−0.369403	−	0.929269i	$-0.620438\pi$
$601$	−18.1121	−0.738806	−0.369403	+	0.929269i	$0.620438\pi$
$602$	0	0
$603$	−34.4341	−1.40226
$604$	0	0
$605$	−14.5654	−0.592169
$606$	0	0
$607$	−34.2303	−1.38936	−0.694682	−	0.719317i	$-0.744455\pi$
$607$	−34.2303	−1.38936	−0.694682	+	0.719317i	$0.744455\pi$
$608$	0	0
$609$	1.59936	0.0648094
$610$	0	0
$611$	−16.5369	−0.669010
$612$	0	0
$613$	46.9137	1.89482	0.947412	−	0.320015i	$-0.103688\pi$
$613$	46.9137	1.89482	0.947412	+	0.320015i	$0.103688\pi$
$614$	0	0
$615$	−0.926645	−0.0373659
$616$	0	0
$617$	35.8081	1.44158	0.720790	−	0.693153i	$-0.243779\pi$
$617$	35.8081	1.44158	0.720790	+	0.693153i	$0.243779\pi$
$618$	0	0
$619$	1.90730	0.0766607	0.0383304	−	0.999265i	$-0.487796\pi$
$619$	1.90730	0.0766607	0.0383304	+	0.999265i	$0.487796\pi$
$620$	0	0
$621$	−11.0207	−0.442244
$622$	0	0
$623$	−6.56798	−0.263141
$624$	0	0
$625$	−23.1494	−0.925975
$626$	0	0
$627$	2.42351	0.0967855
$628$	0	0
$629$	−2.51088	−0.100116
$630$	0	0
$631$	−13.2851	−0.528873	−0.264437	−	0.964403i	$-0.585186\pi$
$631$	−13.2851	−0.528873	−0.264437	+	0.964403i	$0.585186\pi$
$632$	0	0
$633$	9.69250	0.385242
$634$	0	0
$635$	38.0402	1.50958
$636$	0	0
$637$	−5.50921	−0.218283
$638$	0	0
$639$	−4.42287	−0.174966
$640$	0	0
$641$	30.2084	1.19316	0.596579	−	0.802554i	$-0.296526\pi$
$641$	30.2084	1.19316	0.596579	+	0.802554i	$0.296526\pi$
$642$	0	0
$643$	−20.6626	−0.814854	−0.407427	−	0.913238i	$-0.633574\pi$
$643$	−20.6626	−0.814854	−0.407427	+	0.913238i	$0.633574\pi$
$644$	0	0
$645$	−6.69762	−0.263719
$646$	0	0
$647$	−14.8429	−0.583533	−0.291766	−	0.956490i	$-0.594243\pi$
$647$	−14.8429	−0.583533	−0.291766	+	0.956490i	$0.594243\pi$
$648$	0	0
$649$	24.1761	0.948995
$650$	0	0
$651$	5.56524	0.218119
$652$	0	0
$653$	47.4330	1.85620	0.928099	−	0.372334i	$-0.121442\pi$
$653$	47.4330	1.85620	0.928099	+	0.372334i	$0.121442\pi$
$654$	0	0
$655$	−23.3228	−0.911296
$656$	0	0
$657$	19.2875	0.752477
$658$	0	0
$659$	−5.28888	−0.206025	−0.103013	−	0.994680i	$-0.532848\pi$
$659$	−5.28888	−0.206025	−0.103013	+	0.994680i	$0.532848\pi$
$660$	0	0
$661$	5.29597	0.205989	0.102995	−	0.994682i	$-0.467158\pi$
$661$	5.29597	0.205989	0.102995	+	0.994682i	$0.467158\pi$
$662$	0	0
$663$	−2.97949	−0.115714
$664$	0	0
$665$	4.69023	0.181879
$666$	0	0
$667$	−10.5585	−0.408828
$668$	0	0
$669$	−10.6316	−0.411043
$670$	0	0
$671$	−0.507781	−0.0196027
$672$	0	0
$673$	21.6361	0.834010	0.417005	−	0.908904i	$-0.363080\pi$
$673$	21.6361	0.834010	0.417005	+	0.908904i	$0.363080\pi$
$674$	0	0
$675$	−1.06851	−0.0411268
$676$	0	0
$677$	−25.2318	−0.969738	−0.484869	−	0.874587i	$-0.661133\pi$
$677$	−25.2318	−0.969738	−0.484869	+	0.874587i	$0.661133\pi$
$678$	0	0
$679$	−12.9424	−0.496682
$680$	0	0
$681$	7.66195	0.293606
$682$	0	0
$683$	7.13904	0.273168	0.136584	−	0.990629i	$-0.456388\pi$
$683$	7.13904	0.273168	0.136584	+	0.990629i	$0.456388\pi$
$684$	0	0
$685$	−2.90306	−0.110920
$686$	0	0
$687$	−5.53349	−0.211116
$688$	0	0
$689$	−51.1735	−1.94955
$690$	0	0
$691$	−51.2992	−1.95151	−0.975757	−	0.218856i	$-0.929767\pi$
$691$	−51.2992	−1.95151	−0.975757	+	0.218856i	$0.929767\pi$
$692$	0	0
$693$	5.58052	0.211986
$694$	0	0
$695$	−32.3467	−1.22698
$696$	0	0
$697$	−0.794247	−0.0300843
$698$	0	0
$699$	8.75915	0.331302
$700$	0	0
$701$	−10.7077	−0.404423	−0.202212	−	0.979342i	$-0.564813\pi$
$701$	−10.7077	−0.404423	−0.202212	+	0.979342i	$0.564813\pi$
$702$	0	0
$703$	−5.45902	−0.205891
$704$	0	0
$705$	−3.50204	−0.131895
$706$	0	0
$707$	17.3438	0.652282
$708$	0	0
$709$	−28.4276	−1.06762	−0.533811	−	0.845604i	$-0.679241\pi$
$709$	−28.4276	−1.06762	−0.533811	+	0.845604i	$0.679241\pi$
$710$	0	0
$711$	−13.5931	−0.509782
$712$	0	0
$713$	−36.7401	−1.37593
$714$	0	0
$715$	−24.4962	−0.916106
$716$	0	0
$717$	−8.19831	−0.306171
$718$	0	0
$719$	40.5961	1.51398	0.756990	−	0.653427i	$-0.226669\pi$
$719$	40.5961	1.51398	0.756990	+	0.653427i	$0.226669\pi$
$720$	0	0
$721$	−13.8426	−0.515525
$722$	0	0
$723$	5.06245	0.188274
$724$	0	0
$725$	−1.02370	−0.0380193
$726$	0	0
$727$	6.14972	0.228080	0.114040	−	0.993476i	$-0.463621\pi$
$727$	6.14972	0.228080	0.114040	+	0.993476i	$0.463621\pi$
$728$	0	0
$729$	−12.4640	−0.461629
$730$	0	0
$731$	−5.74068	−0.212327
$732$	0	0
$733$	−12.0501	−0.445080	−0.222540	−	0.974924i	$-0.571435\pi$
$733$	−12.0501	−0.445080	−0.222540	+	0.974924i	$0.571435\pi$
$734$	0	0
$735$	−1.16670	−0.0430342
$736$	0	0
$737$	26.2133	0.965579
$738$	0	0
$739$	27.8334	1.02387	0.511935	−	0.859024i	$-0.328929\pi$
$739$	27.8334	1.02387	0.511935	+	0.859024i	$0.328929\pi$
$740$	0	0
$741$	−6.47783	−0.237969
$742$	0	0
$743$	36.1262	1.32534	0.662671	−	0.748911i	$-0.269423\pi$
$743$	36.1262	1.32534	0.662671	+	0.748911i	$0.269423\pi$
$744$	0	0
$745$	−10.2712	−0.376308
$746$	0	0
$747$	0.136069	0.00497852
$748$	0	0
$749$	−10.5908	−0.386981
$750$	0	0
$751$	−7.52254	−0.274502	−0.137251	−	0.990536i	$-0.543827\pi$
$751$	−7.52254	−0.274502	−0.137251	+	0.990536i	$0.543827\pi$
$752$	0	0
$753$	7.99412	0.291322
$754$	0	0
$755$	−36.4646	−1.32708
$756$	0	0
$757$	−48.6035	−1.76653	−0.883263	−	0.468879i	$-0.844658\pi$
$757$	−48.6035	−1.76653	−0.883263	+	0.468879i	$0.844658\pi$
$758$	0	0
$759$	3.97983	0.144459
$760$	0	0
$761$	27.6245	1.00139	0.500694	−	0.865625i	$-0.333078\pi$
$761$	27.6245	1.00139	0.500694	+	0.865625i	$0.333078\pi$
$762$	0	0
$763$	8.60297	0.311448
$764$	0	0
$765$	5.84086	0.211177
$766$	0	0
$767$	−64.6207	−2.33332
$768$	0	0
$769$	−25.3541	−0.914293	−0.457146	−	0.889391i	$-0.651128\pi$
$769$	−25.3541	−0.914293	−0.457146	+	0.889391i	$0.651128\pi$
$770$	0	0
$771$	−0.478790	−0.0172432
$772$	0	0
$773$	4.20981	0.151416	0.0757082	−	0.997130i	$-0.475878\pi$
$773$	4.20981	0.151416	0.0757082	+	0.997130i	$0.475878\pi$
$774$	0	0
$775$	−3.56213	−0.127956
$776$	0	0
$777$	1.35793	0.0487156
$778$	0	0
$779$	−1.72681	−0.0618693
$780$	0	0
$781$	3.36695	0.120479
$782$	0	0
$783$	9.12838	0.326222
$784$	0	0
$785$	38.5609	1.37630
$786$	0	0
$787$	−25.6320	−0.913683	−0.456842	−	0.889548i	$-0.651019\pi$
$787$	−25.6320	−0.913683	−0.456842	+	0.889548i	$0.651019\pi$
$788$	0	0
$789$	−16.5989	−0.590935
$790$	0	0
$791$	1.32777	0.0472101
$792$	0	0
$793$	1.35726	0.0481976
$794$	0	0
$795$	−10.8371	−0.384352
$796$	0	0
$797$	−42.8427	−1.51757	−0.758783	−	0.651343i	$-0.774206\pi$
$797$	−42.8427	−1.51757	−0.758783	+	0.651343i	$0.774206\pi$
$798$	0	0
$799$	−3.00167	−0.106192
$800$	0	0
$801$	−17.7829	−0.628328
$802$	0	0
$803$	−14.6828	−0.518145
$804$	0	0
$805$	7.70219	0.271467
$806$	0	0
$807$	4.31455	0.151879
$808$	0	0
$809$	−25.3601	−0.891615	−0.445807	−	0.895129i	$-0.647083\pi$
$809$	−25.3601	−0.891615	−0.445807	+	0.895129i	$0.647083\pi$
$810$	0	0
$811$	−24.8332	−0.872011	−0.436005	−	0.899944i	$-0.643607\pi$
$811$	−24.8332	−0.872011	−0.436005	+	0.899944i	$0.643607\pi$
$812$	0	0
$813$	−9.57243	−0.335720
$814$	0	0
$815$	−14.3215	−0.501662
$816$	0	0
$817$	−12.4811	−0.436657
$818$	0	0
$819$	−14.9163	−0.521217
$820$	0	0
$821$	−10.4448	−0.364525	−0.182263	−	0.983250i	$-0.558342\pi$
$821$	−10.4448	−0.364525	−0.182263	+	0.983250i	$0.558342\pi$
$822$	0	0
$823$	−39.3355	−1.37115	−0.685574	−	0.728003i	$-0.740449\pi$
$823$	−39.3355	−1.37115	−0.685574	+	0.728003i	$0.740449\pi$
$824$	0	0
$825$	0.385864	0.0134341
$826$	0	0
$827$	−43.7487	−1.52129	−0.760645	−	0.649168i	$-0.775117\pi$
$827$	−43.7487	−1.52129	−0.760645	+	0.649168i	$0.775117\pi$
$828$	0	0
$829$	16.9080	0.587241	0.293620	−	0.955922i	$-0.405140\pi$
$829$	16.9080	0.587241	0.293620	+	0.955922i	$0.405140\pi$
$830$	0	0
$831$	0.940945	0.0326410
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	−9.53026	−0.329808
$836$	0	0
$837$	31.7637	1.09791
$838$	0	0
$839$	−18.1429	−0.626362	−0.313181	−	0.949693i	$-0.601395\pi$
$839$	−18.1429	−0.626362	−0.313181	+	0.949693i	$0.601395\pi$
$840$	0	0
$841$	−20.2544	−0.698428
$842$	0	0
$843$	4.42466	0.152393
$844$	0	0
$845$	37.4317	1.28769
$846$	0	0
$847$	6.75177	0.231994
$848$	0	0
$849$	17.0723	0.585919
$850$	0	0
$851$	−8.96469	−0.307306
$852$	0	0
$853$	5.51844	0.188948	0.0944738	−	0.995527i	$-0.469883\pi$
$853$	5.51844	0.188948	0.0944738	+	0.995527i	$0.469883\pi$
$854$	0	0
$855$	12.6989	0.434292
$856$	0	0
$857$	−44.9035	−1.53388	−0.766938	−	0.641722i	$-0.778220\pi$
$857$	−44.9035	−1.53388	−0.766938	+	0.641722i	$0.778220\pi$
$858$	0	0
$859$	−12.4901	−0.426157	−0.213079	−	0.977035i	$-0.568349\pi$
$859$	−12.4901	−0.426157	−0.213079	+	0.977035i	$0.568349\pi$
$860$	0	0
$861$	0.429544	0.0146388
$862$	0	0
$863$	28.6033	0.973667	0.486833	−	0.873495i	$-0.338152\pi$
$863$	28.6033	0.973667	0.486833	+	0.873495i	$0.338152\pi$
$864$	0	0
$865$	43.7722	1.48830
$866$	0	0
$867$	−0.540819	−0.0183672
$868$	0	0
$869$	10.3479	0.351028
$870$	0	0
$871$	−70.0659	−2.37409
$872$	0	0
$873$	−35.0416	−1.18598
$874$	0	0
$875$	11.5331	0.389891
$876$	0	0
$877$	44.5608	1.50471	0.752356	−	0.658757i	$-0.228917\pi$
$877$	44.5608	1.50471	0.752356	+	0.658757i	$0.228917\pi$
$878$	0	0
$879$	15.3206	0.516751
$880$	0	0
$881$	−7.02041	−0.236524	−0.118262	−	0.992982i	$-0.537732\pi$
$881$	−7.02041	−0.236524	−0.118262	+	0.992982i	$0.537732\pi$
$882$	0	0
$883$	−32.6634	−1.09921	−0.549605	−	0.835424i	$-0.685222\pi$
$883$	−32.6634	−1.09921	−0.549605	+	0.835424i	$0.685222\pi$
$884$	0	0
$885$	−13.6849	−0.460011
$886$	0	0
$887$	38.5931	1.29583	0.647916	−	0.761712i	$-0.275641\pi$
$887$	38.5931	1.29583	0.647916	+	0.761712i	$0.275641\pi$
$888$	0	0
$889$	−17.6334	−0.591406
$890$	0	0
$891$	13.3008	0.445594
$892$	0	0
$893$	−6.52607	−0.218387
$894$	0	0
$895$	−30.0009	−1.00282
$896$	0	0
$897$	−10.6378	−0.355184
$898$	0	0
$899$	30.4317	1.01495
$900$	0	0
$901$	−9.28871	−0.309452
$902$	0	0
$903$	3.10467	0.103317
$904$	0	0
$905$	−22.2356	−0.739136
$906$	0	0
$907$	25.4634	0.845497	0.422748	−	0.906247i	$-0.361065\pi$
$907$	25.4634	0.845497	0.422748	+	0.906247i	$0.361065\pi$
$908$	0	0
$909$	46.9587	1.55752
$910$	0	0
$911$	8.78190	0.290957	0.145479	−	0.989361i	$-0.453528\pi$
$911$	8.78190	0.290957	0.145479	+	0.989361i	$0.453528\pi$
$912$	0	0
$913$	−0.103584	−0.00342814
$914$	0	0
$915$	0.287429	0.00950210
$916$	0	0
$917$	10.8112	0.357018
$918$	0	0
$919$	14.6730	0.484018	0.242009	−	0.970274i	$-0.422194\pi$
$919$	14.6730	0.484018	0.242009	+	0.970274i	$0.422194\pi$
$920$	0	0
$921$	−5.14231	−0.169445
$922$	0	0
$923$	−8.99958	−0.296225
$924$	0	0
$925$	−0.869170	−0.0285781
$926$	0	0
$927$	−37.4790	−1.23097
$928$	0	0
$929$	−39.7569	−1.30438	−0.652191	−	0.758054i	$-0.726150\pi$
$929$	−39.7569	−1.30438	−0.652191	+	0.758054i	$0.726150\pi$
$930$	0	0
$931$	−2.17414	−0.0712547
$932$	0	0
$933$	−3.12924	−0.102447
$934$	0	0
$935$	−4.44641	−0.145413
$936$	0	0
$937$	−37.9558	−1.23996	−0.619981	−	0.784616i	$-0.712860\pi$
$937$	−37.9558	−1.23996	−0.619981	+	0.784616i	$0.712860\pi$
$938$	0	0
$939$	−1.74148	−0.0568310
$940$	0	0
$941$	3.47508	0.113284	0.0566422	−	0.998395i	$-0.481961\pi$
$941$	3.47508	0.113284	0.0566422	+	0.998395i	$0.481961\pi$
$942$	0	0
$943$	−2.83573	−0.0923440
$944$	0	0
$945$	−6.65893	−0.216615
$946$	0	0
$947$	−8.61706	−0.280017	−0.140008	−	0.990150i	$-0.544713\pi$
$947$	−8.61706	−0.280017	−0.140008	+	0.990150i	$0.544713\pi$
$948$	0	0
$949$	39.2459	1.27398
$950$	0	0
$951$	−8.29273	−0.268910
$952$	0	0
$953$	60.2017	1.95013	0.975063	−	0.221929i	$-0.0712354\pi$
$953$	60.2017	1.95013	0.975063	+	0.221929i	$0.0712354\pi$
$954$	0	0
$955$	−5.24842	−0.169835
$956$	0	0
$957$	−3.29648	−0.106560
$958$	0	0
$959$	1.34571	0.0434551
$960$	0	0
$961$	74.8921	2.41587
$962$	0	0
$963$	−28.6749	−0.924035
$964$	0	0
$965$	50.4283	1.62335
$966$	0	0
$967$	48.5686	1.56186	0.780930	−	0.624618i	$-0.214745\pi$
$967$	48.5686	1.56186	0.780930	+	0.624618i	$0.214745\pi$
$968$	0	0
$969$	−1.17582	−0.0377727
$970$	0	0
$971$	−17.0482	−0.547104	−0.273552	−	0.961857i	$-0.588199\pi$
$971$	−17.0482	−0.547104	−0.273552	+	0.961857i	$0.588199\pi$
$972$	0	0
$973$	14.9943	0.480694
$974$	0	0
$975$	−1.03138	−0.0330307
$976$	0	0
$977$	57.4932	1.83937	0.919686	−	0.392655i	$-0.128443\pi$
$977$	57.4932	1.83937	0.919686	+	0.392655i	$0.128443\pi$
$978$	0	0
$979$	13.5374	0.432658
$980$	0	0
$981$	23.2927	0.743678
$982$	0	0
$983$	14.8492	0.473617	0.236808	−	0.971556i	$-0.423899\pi$
$983$	14.8492	0.473617	0.236808	+	0.971556i	$0.423899\pi$
$984$	0	0
$985$	11.4294	0.364171
$986$	0	0
$987$	1.62336	0.0516722
$988$	0	0
$989$	−20.4961	−0.651739
$990$	0	0
$991$	10.8483	0.344608	0.172304	−	0.985044i	$-0.444879\pi$
$991$	10.8483	0.344608	0.172304	+	0.985044i	$0.444879\pi$
$992$	0	0
$993$	−11.6316	−0.369119
$994$	0	0
$995$	54.2349	1.71936
$996$	0	0
$997$	−20.9973	−0.664990	−0.332495	−	0.943105i	$-0.607890\pi$
$997$	−20.9973	−0.664990	−0.332495	+	0.943105i	$0.607890\pi$
$998$	0	0
$999$	7.75043	0.245213

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.e.1.3	✓	5
4.3	odd	2		3808.2.a.f.1.3	yes	5
8.3	odd	2		7616.2.a.bs.1.3		5
8.5	even	2		7616.2.a.br.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.e.1.3	✓	5	1.1	even	1	trivial
3808.2.a.f.1.3	yes	5	4.3	odd	2
7616.2.a.br.1.3		5	8.5	even	2
7616.2.a.bs.1.3		5	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)

\(f(q)\)	\(=\)	\(q-0.540819 q^{3} +2.15728 q^{5} -1.00000 q^{7} -2.70751 q^{9} +2.06112 q^{11} -5.50921 q^{13} -1.16670 q^{15} -1.00000 q^{17} -2.17414 q^{19} +0.540819 q^{21} -3.57033 q^{23} -0.346161 q^{25} +3.08673 q^{27} +2.95730 q^{29} +10.2904 q^{31} -1.11469 q^{33} -2.15728 q^{35} +2.51088 q^{37} +2.97949 q^{39} +0.794247 q^{41} +5.74068 q^{43} -5.84086 q^{45} +3.00167 q^{47} +1.00000 q^{49} +0.540819 q^{51} +9.28871 q^{53} +4.44641 q^{55} +1.17582 q^{57} +11.7296 q^{59} -0.246361 q^{61} +2.70751 q^{63} -11.8849 q^{65} +12.7180 q^{67} +1.93090 q^{69} +1.63355 q^{71} -7.12369 q^{73} +0.187210 q^{75} -2.06112 q^{77} +5.02051 q^{79} +6.45318 q^{81} -0.0502562 q^{83} -2.15728 q^{85} -1.59936 q^{87} +6.56798 q^{89} +5.50921 q^{91} -5.56524 q^{93} -4.69023 q^{95} +12.9424 q^{97} -5.58052 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{5} - 5 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} + 8 q^{15} - 5 q^{17} - 6 q^{19} + 18 q^{23} + 5 q^{25} + 18 q^{27} - 14 q^{29} + 16 q^{31} - 26 q^{33} + 2 q^{35} + 2 q^{37} + 6 q^{39} - 10 q^{41}+ \cdots - 8 q^{99}+O(q^{100}) \) 5 * q - 2 * q^5 - 5 * q^7 + 3 * q^9 - 4 * q^11 - 6 * q^13 + 8 * q^15 - 5 * q^17 - 6 * q^19 + 18 * q^23 + 5 * q^25 + 18 * q^27 - 14 * q^29 + 16 * q^31 - 26 * q^33 + 2 * q^35 + 2 * q^37 + 6 * q^39 - 10 * q^41 - 20 * q^43 + 6 * q^45 + 26 * q^47 + 5 * q^49 + 4 * q^55 + 12 * q^57 + 8 * q^61 - 3 * q^63 + 2 * q^65 + 12 * q^67 + 32 * q^69 + 10 * q^71 - 14 * q^73 - 4 * q^75 + 4 * q^77 + 34 * q^79 + 33 * q^81 + 20 * q^83 + 2 * q^85 + 14 * q^87 - 4 * q^89 + 6 * q^91 + 16 * q^93 + 42 * q^95 - 12 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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