LMFDB - Embedded newform 3808.2.a.e.1.5

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			1.5
Root			$3.24142$ 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)

$f(q)$	$=$	$q+3.24142 q^{3} +1.62440 q^{5} -1.00000 q^{7} +7.50679 q^{9} -4.09420 q^{11} +1.35392 q^{13} +5.26537 q^{15} -1.00000 q^{17} +2.21410 q^{19} -3.24142 q^{21} +9.44813 q^{23} -2.36131 q^{25} +14.6084 q^{27} -4.61365 q^{29} +8.18757 q^{31} -13.2710 q^{33} -1.62440 q^{35} +6.03699 q^{37} +4.38863 q^{39} +2.10159 q^{41} -11.2636 q^{43} +12.1941 q^{45} +13.3909 q^{47} +1.00000 q^{49} -3.24142 q^{51} -3.20334 q^{53} -6.65064 q^{55} +7.17682 q^{57} -9.76477 q^{59} +5.80716 q^{61} -7.50679 q^{63} +2.19932 q^{65} +11.3165 q^{67} +30.6253 q^{69} -5.22311 q^{71} -11.4529 q^{73} -7.65400 q^{75} +4.09420 q^{77} +3.61137 q^{79} +24.8315 q^{81} +16.8458 q^{83} -1.62440 q^{85} -14.9548 q^{87} +10.5918 q^{89} -1.35392 q^{91} +26.5393 q^{93} +3.59659 q^{95} -9.74339 q^{97} -30.7343 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.24142	1.87143	0.935717	−	0.352753i	$-0.114754\pi$
$3$	3.24142	1.87143	0.935717	+	0.352753i	$0.114754\pi$
$4$	0	0
$5$	1.62440	0.726455	0.363228	−	0.931700i	$-0.381675\pi$
$5$	1.62440	0.726455	0.363228	+	0.931700i	$0.381675\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	7.50679	2.50226
$10$	0	0
$11$	−4.09420	−1.23445	−0.617224	−	0.786787i	$-0.711743\pi$
$11$	−4.09420	−1.23445	−0.617224	+	0.786787i	$0.711743\pi$
$12$	0	0
$13$	1.35392	0.375511	0.187755	−	0.982216i	$-0.439879\pi$
$13$	1.35392	0.375511	0.187755	+	0.982216i	$0.439879\pi$
$14$	0	0
$15$	5.26537	1.35951
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	2.21410	0.507949	0.253974	−	0.967211i	$-0.418262\pi$
$19$	2.21410	0.507949	0.253974	+	0.967211i	$0.418262\pi$
$20$	0	0
$21$	−3.24142	−0.707335
$22$	0	0
$23$	9.44813	1.97007	0.985035	−	0.172353i	$-0.0551371\pi$
$23$	9.44813	1.97007	0.985035	+	0.172353i	$0.0551371\pi$
$24$	0	0
$25$	−2.36131	−0.472263
$26$	0	0
$27$	14.6084	2.81138
$28$	0	0
$29$	−4.61365	−0.856733	−0.428366	−	0.903605i	$-0.640911\pi$
$29$	−4.61365	−0.856733	−0.428366	+	0.903605i	$0.640911\pi$
$30$	0	0
$31$	8.18757	1.47053	0.735266	−	0.677779i	$-0.237057\pi$
$31$	8.18757	1.47053	0.735266	+	0.677779i	$0.237057\pi$
$32$	0	0
$33$	−13.2710	−2.31019
$34$	0	0
$35$	−1.62440	−0.274574
$36$	0	0
$37$	6.03699	0.992475	0.496237	−	0.868187i	$-0.334715\pi$
$37$	6.03699	0.992475	0.496237	+	0.868187i	$0.334715\pi$
$38$	0	0
$39$	4.38863	0.702744
$40$	0	0
$41$	2.10159	0.328213	0.164107	−	0.986443i	$-0.447526\pi$
$41$	2.10159	0.328213	0.164107	+	0.986443i	$0.447526\pi$
$42$	0	0
$43$	−11.2636	−1.71769	−0.858844	−	0.512238i	$-0.828817\pi$
$43$	−11.2636	−1.71769	−0.858844	+	0.512238i	$0.828817\pi$
$44$	0	0
$45$	12.1941	1.81778
$46$	0	0
$47$	13.3909	1.95327	0.976633	−	0.214915i	$-0.0689475\pi$
$47$	13.3909	1.95327	0.976633	+	0.214915i	$0.0689475\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.24142	−0.453889
$52$	0	0
$53$	−3.20334	−0.440013	−0.220006	−	0.975498i	$-0.570608\pi$
$53$	−3.20334	−0.440013	−0.220006	+	0.975498i	$0.570608\pi$
$54$	0	0
$55$	−6.65064	−0.896771
$56$	0	0
$57$	7.17682	0.950592
$58$	0	0
$59$	−9.76477	−1.27126	−0.635632	−	0.771992i	$-0.719260\pi$
$59$	−9.76477	−1.27126	−0.635632	+	0.771992i	$0.719260\pi$
$60$	0	0
$61$	5.80716	0.743531	0.371765	−	0.928327i	$-0.378753\pi$
$61$	5.80716	0.743531	0.371765	+	0.928327i	$0.378753\pi$
$62$	0	0
$63$	−7.50679	−0.945766
$64$	0	0
$65$	2.19932	0.272792
$66$	0	0
$67$	11.3165	1.38253	0.691264	−	0.722602i	$-0.257054\pi$
$67$	11.3165	1.38253	0.691264	+	0.722602i	$0.257054\pi$
$68$	0	0
$69$	30.6253	3.68686
$70$	0	0
$71$	−5.22311	−0.619869	−0.309935	−	0.950758i	$-0.600307\pi$
$71$	−5.22311	−0.619869	−0.309935	+	0.950758i	$0.600307\pi$
$72$	0	0
$73$	−11.4529	−1.34047	−0.670233	−	0.742151i	$-0.733806\pi$
$73$	−11.4529	−1.34047	−0.670233	+	0.742151i	$0.733806\pi$
$74$	0	0
$75$	−7.65400	−0.883808
$76$	0	0
$77$	4.09420	0.466578
$78$	0	0
$79$	3.61137	0.406311	0.203155	−	0.979147i	$-0.434880\pi$
$79$	3.61137	0.406311	0.203155	+	0.979147i	$0.434880\pi$
$80$	0	0
$81$	24.8315	2.75905
$82$	0	0
$83$	16.8458	1.84906	0.924532	−	0.381105i	$-0.124456\pi$
$83$	16.8458	1.84906	0.924532	+	0.381105i	$0.124456\pi$
$84$	0	0
$85$	−1.62440	−0.176191
$86$	0	0
$87$	−14.9548	−1.60332
$88$	0	0
$89$	10.5918	1.12273	0.561365	−	0.827568i	$-0.310276\pi$
$89$	10.5918	1.12273	0.561365	+	0.827568i	$0.310276\pi$
$90$	0	0
$91$	−1.35392	−0.141930
$92$	0	0
$93$	26.5393	2.75200
$94$	0	0
$95$	3.59659	0.369002
$96$	0	0
$97$	−9.74339	−0.989291	−0.494646	−	0.869095i	$-0.664702\pi$
$97$	−9.74339	−0.989291	−0.494646	+	0.869095i	$0.664702\pi$
$98$	0	0
$99$	−30.7343	−3.08891
$100$	0	0
$101$	−13.2925	−1.32266	−0.661328	−	0.750097i	$-0.730007\pi$
$101$	−13.2925	−1.32266	−0.661328	+	0.750097i	$0.730007\pi$
$102$	0	0
$103$	−5.88466	−0.579833	−0.289917	−	0.957052i	$-0.593628\pi$
$103$	−5.88466	−0.579833	−0.289917	+	0.957052i	$0.593628\pi$
$104$	0	0
$105$	−5.26537	−0.513847
$106$	0	0
$107$	−3.83676	−0.370913	−0.185457	−	0.982652i	$-0.559376\pi$
$107$	−3.83676	−0.370913	−0.185457	+	0.982652i	$0.559376\pi$
$108$	0	0
$109$	−13.3774	−1.28132	−0.640662	−	0.767823i	$-0.721340\pi$
$109$	−13.3774	−1.28132	−0.640662	+	0.767823i	$0.721340\pi$
$110$	0	0
$111$	19.5684	1.85735
$112$	0	0
$113$	6.03956	0.568154	0.284077	−	0.958801i	$-0.408313\pi$
$113$	6.03956	0.568154	0.284077	+	0.958801i	$0.408313\pi$
$114$	0	0
$115$	15.3476	1.43117
$116$	0	0
$117$	10.1636	0.939627
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	5.76249	0.523863
$122$	0	0
$123$	6.81213	0.614229
$124$	0	0
$125$	−11.9577	−1.06953
$126$	0	0
$127$	−9.48337	−0.841513	−0.420757	−	0.907174i	$-0.638235\pi$
$127$	−9.48337	−0.841513	−0.420757	+	0.907174i	$0.638235\pi$
$128$	0	0
$129$	−36.5101	−3.21454
$130$	0	0
$131$	18.2476	1.59430	0.797150	−	0.603781i	$-0.206340\pi$
$131$	18.2476	1.59430	0.797150	+	0.603781i	$0.206340\pi$
$132$	0	0
$133$	−2.21410	−0.191987
$134$	0	0
$135$	23.7299	2.04234
$136$	0	0
$137$	−15.8622	−1.35520	−0.677599	−	0.735432i	$-0.736979\pi$
$137$	−15.8622	−1.35520	−0.677599	+	0.735432i	$0.736979\pi$
$138$	0	0
$139$	−0.230917	−0.0195861	−0.00979306	−	0.999952i	$-0.503117\pi$
$139$	−0.230917	−0.0195861	−0.00979306	+	0.999952i	$0.503117\pi$
$140$	0	0
$141$	43.4055	3.65541
$142$	0	0
$143$	−5.54324	−0.463549
$144$	0	0
$145$	−7.49442	−0.622378
$146$	0	0
$147$	3.24142	0.267348
$148$	0	0
$149$	13.6523	1.11844	0.559218	−	0.829020i	$-0.311101\pi$
$149$	13.6523	1.11844	0.559218	+	0.829020i	$0.311101\pi$
$150$	0	0
$151$	4.65673	0.378960	0.189480	−	0.981885i	$-0.439320\pi$
$151$	4.65673	0.378960	0.189480	+	0.981885i	$0.439320\pi$
$152$	0	0
$153$	−7.50679	−0.606888
$154$	0	0
$155$	13.2999	1.06828
$156$	0	0
$157$	14.0126	1.11833	0.559163	−	0.829058i	$-0.311123\pi$
$157$	14.0126	1.11833	0.559163	+	0.829058i	$0.311123\pi$
$158$	0	0
$159$	−10.3834	−0.823454
$160$	0	0
$161$	−9.44813	−0.744617
$162$	0	0
$163$	−6.78657	−0.531566	−0.265783	−	0.964033i	$-0.585630\pi$
$163$	−6.78657	−0.531566	−0.265783	+	0.964033i	$0.585630\pi$
$164$	0	0
$165$	−21.5575	−1.67825
$166$	0	0
$167$	−9.09038	−0.703435	−0.351717	−	0.936106i	$-0.614402\pi$
$167$	−9.09038	−0.703435	−0.351717	+	0.936106i	$0.614402\pi$
$168$	0	0
$169$	−11.1669	−0.858992
$170$	0	0
$171$	16.6208	1.27102
$172$	0	0
$173$	−8.32312	−0.632795	−0.316398	−	0.948627i	$-0.602473\pi$
$173$	−8.32312	−0.632795	−0.316398	+	0.948627i	$0.602473\pi$
$174$	0	0
$175$	2.36131	0.178499
$176$	0	0
$177$	−31.6517	−2.37909
$178$	0	0
$179$	−15.0534	−1.12514	−0.562572	−	0.826748i	$-0.690188\pi$
$179$	−15.0534	−1.12514	−0.562572	+	0.826748i	$0.690188\pi$
$180$	0	0
$181$	−4.34907	−0.323264	−0.161632	−	0.986851i	$-0.551676\pi$
$181$	−4.34907	−0.323264	−0.161632	+	0.986851i	$0.551676\pi$
$182$	0	0
$183$	18.8234	1.39147
$184$	0	0
$185$	9.80651	0.720989
$186$	0	0
$187$	4.09420	0.299398
$188$	0	0
$189$	−14.6084	−1.06260
$190$	0	0
$191$	−15.9697	−1.15553	−0.577763	−	0.816204i	$-0.696074\pi$
$191$	−15.9697	−1.15553	−0.577763	+	0.816204i	$0.696074\pi$
$192$	0	0
$193$	−13.2046	−0.950484	−0.475242	−	0.879855i	$-0.657640\pi$
$193$	−13.2046	−0.950484	−0.475242	+	0.879855i	$0.657640\pi$
$194$	0	0
$195$	7.12891	0.510512
$196$	0	0
$197$	−1.01159	−0.0720727	−0.0360363	−	0.999350i	$-0.511473\pi$
$197$	−1.01159	−0.0720727	−0.0360363	+	0.999350i	$0.511473\pi$
$198$	0	0
$199$	5.35418	0.379548	0.189774	−	0.981828i	$-0.439225\pi$
$199$	5.35418	0.379548	0.189774	+	0.981828i	$0.439225\pi$
$200$	0	0
$201$	36.6814	2.58731
$202$	0	0
$203$	4.61365	0.323815
$204$	0	0
$205$	3.41383	0.238432
$206$	0	0
$207$	70.9251	4.92963
$208$	0	0
$209$	−9.06496	−0.627037
$210$	0	0
$211$	−7.67244	−0.528193	−0.264096	−	0.964496i	$-0.585074\pi$
$211$	−7.67244	−0.528193	−0.264096	+	0.964496i	$0.585074\pi$
$212$	0	0
$213$	−16.9303	−1.16004
$214$	0	0
$215$	−18.2967	−1.24782
$216$	0	0
$217$	−8.18757	−0.555809
$218$	0	0
$219$	−37.1238	−2.50859
$220$	0	0
$221$	−1.35392	−0.0910748
$222$	0	0
$223$	14.5413	0.973760	0.486880	−	0.873469i	$-0.338135\pi$
$223$	14.5413	0.973760	0.486880	+	0.873469i	$0.338135\pi$
$224$	0	0
$225$	−17.7259	−1.18173
$226$	0	0
$227$	−20.5194	−1.36192	−0.680959	−	0.732321i	$-0.738437\pi$
$227$	−20.5194	−1.36192	−0.680959	+	0.732321i	$0.738437\pi$
$228$	0	0
$229$	−6.16014	−0.407073	−0.203537	−	0.979067i	$-0.565244\pi$
$229$	−6.16014	−0.407073	−0.203537	+	0.979067i	$0.565244\pi$
$230$	0	0
$231$	13.2710	0.873169
$232$	0	0
$233$	19.7323	1.29271	0.646354	−	0.763038i	$-0.276293\pi$
$233$	19.7323	1.29271	0.646354	+	0.763038i	$0.276293\pi$
$234$	0	0
$235$	21.7522	1.41896
$236$	0	0
$237$	11.7059	0.760383
$238$	0	0
$239$	−1.49538	−0.0967278	−0.0483639	−	0.998830i	$-0.515401\pi$
$239$	−1.49538	−0.0967278	−0.0483639	+	0.998830i	$0.515401\pi$
$240$	0	0
$241$	−1.45403	−0.0936623	−0.0468311	−	0.998903i	$-0.514912\pi$
$241$	−1.45403	−0.0936623	−0.0468311	+	0.998903i	$0.514912\pi$
$242$	0	0
$243$	36.6641	2.35200
$244$	0	0
$245$	1.62440	0.103779
$246$	0	0
$247$	2.99772	0.190740
$248$	0	0
$249$	54.6042	3.46040
$250$	0	0
$251$	13.2196	0.834416	0.417208	−	0.908811i	$-0.363009\pi$
$251$	13.2196	0.834416	0.417208	+	0.908811i	$0.363009\pi$
$252$	0	0
$253$	−38.6825	−2.43195
$254$	0	0
$255$	−5.26537	−0.329730
$256$	0	0
$257$	−11.2710	−0.703067	−0.351533	−	0.936175i	$-0.614340\pi$
$257$	−11.2710	−0.703067	−0.351533	+	0.936175i	$0.614340\pi$
$258$	0	0
$259$	−6.03699	−0.375120
$260$	0	0
$261$	−34.6337	−2.14377
$262$	0	0
$263$	−5.79789	−0.357513	−0.178757	−	0.983893i	$-0.557207\pi$
$263$	−5.79789	−0.357513	−0.178757	+	0.983893i	$0.557207\pi$
$264$	0	0
$265$	−5.20352	−0.319650
$266$	0	0
$267$	34.3325	2.10111
$268$	0	0
$269$	1.00228	0.0611101	0.0305550	−	0.999533i	$-0.490273\pi$
$269$	1.00228	0.0611101	0.0305550	+	0.999533i	$0.490273\pi$
$270$	0	0
$271$	−2.73679	−0.166248	−0.0831242	−	0.996539i	$-0.526490\pi$
$271$	−2.73679	−0.166248	−0.0831242	+	0.996539i	$0.526490\pi$
$272$	0	0
$273$	−4.38863	−0.265612
$274$	0	0
$275$	9.66769	0.582984
$276$	0	0
$277$	−27.6507	−1.66137	−0.830685	−	0.556742i	$-0.812051\pi$
$277$	−27.6507	−1.66137	−0.830685	+	0.556742i	$0.812051\pi$
$278$	0	0
$279$	61.4624	3.67966
$280$	0	0
$281$	2.34156	0.139686	0.0698429	−	0.997558i	$-0.477750\pi$
$281$	2.34156	0.139686	0.0698429	+	0.997558i	$0.477750\pi$
$282$	0	0
$283$	−13.4033	−0.796741	−0.398370	−	0.917225i	$-0.630424\pi$
$283$	−13.4033	−0.796741	−0.398370	+	0.917225i	$0.630424\pi$
$284$	0	0
$285$	11.6580	0.690563
$286$	0	0
$287$	−2.10159	−0.124053
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−31.5824	−1.85139
$292$	0	0
$293$	−18.2617	−1.06686	−0.533430	−	0.845845i	$-0.679097\pi$
$293$	−18.2617	−1.06686	−0.533430	+	0.845845i	$0.679097\pi$
$294$	0	0
$295$	−15.8619	−0.923517
$296$	0	0
$297$	−59.8096	−3.47051
$298$	0	0
$299$	12.7920	0.739783
$300$	0	0
$301$	11.2636	0.649225
$302$	0	0
$303$	−43.0866	−2.47526
$304$	0	0
$305$	9.43317	0.540142
$306$	0	0
$307$	−1.21053	−0.0690885	−0.0345443	−	0.999403i	$-0.510998\pi$
$307$	−1.21053	−0.0690885	−0.0345443	+	0.999403i	$0.510998\pi$
$308$	0	0
$309$	−19.0747	−1.08512
$310$	0	0
$311$	32.4812	1.84184	0.920919	−	0.389754i	$-0.127440\pi$
$311$	32.4812	1.84184	0.920919	+	0.389754i	$0.127440\pi$
$312$	0	0
$313$	6.92111	0.391205	0.195602	−	0.980683i	$-0.437334\pi$
$313$	6.92111	0.391205	0.195602	+	0.980683i	$0.437334\pi$
$314$	0	0
$315$	−12.1941	−0.687057
$316$	0	0
$317$	−26.8899	−1.51029	−0.755143	−	0.655560i	$-0.772433\pi$
$317$	−26.8899	−1.51029	−0.755143	+	0.655560i	$0.772433\pi$
$318$	0	0
$319$	18.8892	1.05759
$320$	0	0
$321$	−12.4365	−0.694140
$322$	0	0
$323$	−2.21410	−0.123196
$324$	0	0
$325$	−3.19704	−0.177340
$326$	0	0
$327$	−43.3618	−2.39791
$328$	0	0
$329$	−13.3909	−0.738265
$330$	0	0
$331$	−30.4343	−1.67282	−0.836409	−	0.548106i	$-0.815349\pi$
$331$	−30.4343	−1.67282	−0.836409	+	0.548106i	$0.815349\pi$
$332$	0	0
$333$	45.3184	2.48343
$334$	0	0
$335$	18.3825	1.00435
$336$	0	0
$337$	−1.25586	−0.0684110	−0.0342055	−	0.999415i	$-0.510890\pi$
$337$	−1.25586	−0.0684110	−0.0342055	+	0.999415i	$0.510890\pi$
$338$	0	0
$339$	19.5767	1.06326
$340$	0	0
$341$	−33.5216	−1.81529
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	49.7479	2.67834
$346$	0	0
$347$	−19.0615	−1.02327	−0.511637	−	0.859202i	$-0.670961\pi$
$347$	−19.0615	−1.02327	−0.511637	+	0.859202i	$0.670961\pi$
$348$	0	0
$349$	32.4482	1.73691	0.868456	−	0.495765i	$-0.165112\pi$
$349$	32.4482	1.73691	0.868456	+	0.495765i	$0.165112\pi$
$350$	0	0
$351$	19.7786	1.05571
$352$	0	0
$353$	12.0907	0.643524	0.321762	−	0.946821i	$-0.395725\pi$
$353$	12.0907	0.643524	0.321762	+	0.946821i	$0.395725\pi$
$354$	0	0
$355$	−8.48444	−0.450307
$356$	0	0
$357$	3.24142	0.171554
$358$	0	0
$359$	−1.88383	−0.0994248	−0.0497124	−	0.998764i	$-0.515830\pi$
$359$	−1.88383	−0.0994248	−0.0497124	+	0.998764i	$0.515830\pi$
$360$	0	0
$361$	−14.0978	−0.741988
$362$	0	0
$363$	18.6786	0.980374
$364$	0	0
$365$	−18.6042	−0.973788
$366$	0	0
$367$	23.7651	1.24053	0.620264	−	0.784393i	$-0.287025\pi$
$367$	23.7651	1.24053	0.620264	+	0.784393i	$0.287025\pi$
$368$	0	0
$369$	15.7762	0.821276
$370$	0	0
$371$	3.20334	0.166309
$372$	0	0
$373$	2.80740	0.145362	0.0726808	−	0.997355i	$-0.476845\pi$
$373$	2.80740	0.145362	0.0726808	+	0.997355i	$0.476845\pi$
$374$	0	0
$375$	−38.7600	−2.00156
$376$	0	0
$377$	−6.24653	−0.321713
$378$	0	0
$379$	−21.7445	−1.11694	−0.558471	−	0.829524i	$-0.688612\pi$
$379$	−21.7445	−1.11694	−0.558471	+	0.829524i	$0.688612\pi$
$380$	0	0
$381$	−30.7396	−1.57484
$382$	0	0
$383$	−15.0120	−0.767076	−0.383538	−	0.923525i	$-0.625294\pi$
$383$	−15.0120	−0.767076	−0.383538	+	0.923525i	$0.625294\pi$
$384$	0	0
$385$	6.65064	0.338948
$386$	0	0
$387$	−84.5537	−4.29810
$388$	0	0
$389$	3.71823	0.188522	0.0942608	−	0.995548i	$-0.469951\pi$
$389$	3.71823	0.188522	0.0942608	+	0.995548i	$0.469951\pi$
$390$	0	0
$391$	−9.44813	−0.477812
$392$	0	0
$393$	59.1481	2.98363
$394$	0	0
$395$	5.86632	0.295166
$396$	0	0
$397$	−20.4693	−1.02732	−0.513662	−	0.857993i	$-0.671711\pi$
$397$	−20.4693	−1.02732	−0.513662	+	0.857993i	$0.671711\pi$
$398$	0	0
$399$	−7.17682	−0.359290
$400$	0	0
$401$	−29.1262	−1.45449	−0.727247	−	0.686376i	$-0.759200\pi$
$401$	−29.1262	−1.45449	−0.727247	+	0.686376i	$0.759200\pi$
$402$	0	0
$403$	11.0854	0.552201
$404$	0	0
$405$	40.3364	2.00433
$406$	0	0
$407$	−24.7166	−1.22516
$408$	0	0
$409$	0.929890	0.0459801	0.0229901	−	0.999736i	$-0.492681\pi$
$409$	0.929890	0.0459801	0.0229901	+	0.999736i	$0.492681\pi$
$410$	0	0
$411$	−51.4159	−2.53616
$412$	0	0
$413$	9.76477	0.480493
$414$	0	0
$415$	27.3643	1.34326
$416$	0	0
$417$	−0.748498	−0.0366541
$418$	0	0
$419$	17.0047	0.830734	0.415367	−	0.909654i	$-0.363653\pi$
$419$	17.0047	0.830734	0.415367	+	0.909654i	$0.363653\pi$
$420$	0	0
$421$	27.4283	1.33677	0.668387	−	0.743813i	$-0.266985\pi$
$421$	27.4283	1.33677	0.668387	+	0.743813i	$0.266985\pi$
$422$	0	0
$423$	100.523	4.88758
$424$	0	0
$425$	2.36131	0.114541
$426$	0	0
$427$	−5.80716	−0.281028
$428$	0	0
$429$	−17.9679	−0.867501
$430$	0	0
$431$	32.4402	1.56259	0.781294	−	0.624163i	$-0.214560\pi$
$431$	32.4402	1.56259	0.781294	+	0.624163i	$0.214560\pi$
$432$	0	0
$433$	−12.6152	−0.606249	−0.303125	−	0.952951i	$-0.598030\pi$
$433$	−12.6152	−0.606249	−0.303125	+	0.952951i	$0.598030\pi$
$434$	0	0
$435$	−24.2926	−1.16474
$436$	0	0
$437$	20.9191	1.00070
$438$	0	0
$439$	7.95275	0.379564	0.189782	−	0.981826i	$-0.439222\pi$
$439$	7.95275	0.379564	0.189782	+	0.981826i	$0.439222\pi$
$440$	0	0
$441$	7.50679	0.357466
$442$	0	0
$443$	33.1406	1.57456	0.787279	−	0.616597i	$-0.211489\pi$
$443$	33.1406	1.57456	0.787279	+	0.616597i	$0.211489\pi$
$444$	0	0
$445$	17.2054	0.815613
$446$	0	0
$447$	44.2527	2.09308
$448$	0	0
$449$	−31.4955	−1.48636	−0.743182	−	0.669089i	$-0.766684\pi$
$449$	−31.4955	−1.48636	−0.743182	+	0.669089i	$0.766684\pi$
$450$	0	0
$451$	−8.60434	−0.405162
$452$	0	0
$453$	15.0944	0.709198
$454$	0	0
$455$	−2.19932	−0.103106
$456$	0	0
$457$	−18.1593	−0.849456	−0.424728	−	0.905321i	$-0.639630\pi$
$457$	−18.1593	−0.849456	−0.424728	+	0.905321i	$0.639630\pi$
$458$	0	0
$459$	−14.6084	−0.681861
$460$	0	0
$461$	−13.0704	−0.608750	−0.304375	−	0.952552i	$-0.598448\pi$
$461$	−13.0704	−0.608750	−0.304375	+	0.952552i	$0.598448\pi$
$462$	0	0
$463$	−24.2777	−1.12828	−0.564141	−	0.825679i	$-0.690792\pi$
$463$	−24.2777	−1.12828	−0.564141	+	0.825679i	$0.690792\pi$
$464$	0	0
$465$	43.1106	1.99921
$466$	0	0
$467$	3.72413	0.172332	0.0861661	−	0.996281i	$-0.472538\pi$
$467$	3.72413	0.172332	0.0861661	+	0.996281i	$0.472538\pi$
$468$	0	0
$469$	−11.3165	−0.522547
$470$	0	0
$471$	45.4206	2.09287
$472$	0	0
$473$	46.1156	2.12040
$474$	0	0
$475$	−5.22818	−0.239885
$476$	0	0
$477$	−24.0468	−1.10103
$478$	0	0
$479$	−2.09832	−0.0958748	−0.0479374	−	0.998850i	$-0.515265\pi$
$479$	−2.09832	−0.0958748	−0.0479374	+	0.998850i	$0.515265\pi$
$480$	0	0
$481$	8.17362	0.372685
$482$	0	0
$483$	−30.6253	−1.39350
$484$	0	0
$485$	−15.8272	−0.718676
$486$	0	0
$487$	6.20278	0.281075	0.140537	−	0.990075i	$-0.455117\pi$
$487$	6.20278	0.281075	0.140537	+	0.990075i	$0.455117\pi$
$488$	0	0
$489$	−21.9981	−0.994789
$490$	0	0
$491$	−4.27086	−0.192741	−0.0963706	−	0.995346i	$-0.530723\pi$
$491$	−4.27086	−0.192741	−0.0963706	+	0.995346i	$0.530723\pi$
$492$	0	0
$493$	4.61365	0.207788
$494$	0	0
$495$	−49.9249	−2.24396
$496$	0	0
$497$	5.22311	0.234289
$498$	0	0
$499$	−10.5594	−0.472703	−0.236352	−	0.971668i	$-0.575952\pi$
$499$	−10.5594	−0.472703	−0.236352	+	0.971668i	$0.575952\pi$
$500$	0	0
$501$	−29.4657	−1.31643
$502$	0	0
$503$	35.4808	1.58201	0.791005	−	0.611809i	$-0.209558\pi$
$503$	35.4808	1.58201	0.791005	+	0.611809i	$0.209558\pi$
$504$	0	0
$505$	−21.5924	−0.960851
$506$	0	0
$507$	−36.1966	−1.60755
$508$	0	0
$509$	8.18931	0.362985	0.181492	−	0.983392i	$-0.441907\pi$
$509$	8.18931	0.362985	0.181492	+	0.983392i	$0.441907\pi$
$510$	0	0
$511$	11.4529	0.506648
$512$	0	0
$513$	32.3444	1.42804
$514$	0	0
$515$	−9.55907	−0.421223
$516$	0	0
$517$	−54.8251	−2.41121
$518$	0	0
$519$	−26.9787	−1.18423
$520$	0	0
$521$	−13.6179	−0.596612	−0.298306	−	0.954470i	$-0.596422\pi$
$521$	−13.6179	−0.596612	−0.298306	+	0.954470i	$0.596422\pi$
$522$	0	0
$523$	28.3223	1.23845	0.619223	−	0.785215i	$-0.287448\pi$
$523$	28.3223	1.23845	0.619223	+	0.785215i	$0.287448\pi$
$524$	0	0
$525$	7.65400	0.334048
$526$	0	0
$527$	−8.18757	−0.356656
$528$	0	0
$529$	66.2671	2.88118
$530$	0	0
$531$	−73.3020	−3.18104
$532$	0	0
$533$	2.84539	0.123248
$534$	0	0
$535$	−6.23244	−0.269452
$536$	0	0
$537$	−48.7943	−2.10563
$538$	0	0
$539$	−4.09420	−0.176350
$540$	0	0
$541$	0.161634	0.00694919	0.00347460	−	0.999994i	$-0.498894\pi$
$541$	0.161634	0.00694919	0.00347460	+	0.999994i	$0.498894\pi$
$542$	0	0
$543$	−14.0972	−0.604967
$544$	0	0
$545$	−21.7303	−0.930825
$546$	0	0
$547$	14.7419	0.630320	0.315160	−	0.949039i	$-0.397942\pi$
$547$	14.7419	0.630320	0.315160	+	0.949039i	$0.397942\pi$
$548$	0	0
$549$	43.5931	1.86051
$550$	0	0
$551$	−10.2151	−0.435176
$552$	0	0
$553$	−3.61137	−0.153571
$554$	0	0
$555$	31.7870	1.34928
$556$	0	0
$557$	−37.9015	−1.60594	−0.802968	−	0.596022i	$-0.796747\pi$
$557$	−37.9015	−1.60594	−0.802968	+	0.596022i	$0.796747\pi$
$558$	0	0
$559$	−15.2501	−0.645010
$560$	0	0
$561$	13.2710	0.560303
$562$	0	0
$563$	25.3900	1.07006	0.535031	−	0.844833i	$-0.320300\pi$
$563$	25.3900	1.07006	0.535031	+	0.844833i	$0.320300\pi$
$564$	0	0
$565$	9.81069	0.412739
$566$	0	0
$567$	−24.8315	−1.04282
$568$	0	0
$569$	−11.2373	−0.471090	−0.235545	−	0.971863i	$-0.575688\pi$
$569$	−11.2373	−0.471090	−0.235545	+	0.971863i	$0.575688\pi$
$570$	0	0
$571$	46.4722	1.94480	0.972400	−	0.233321i	$-0.0749593\pi$
$571$	46.4722	1.94480	0.972400	+	0.233321i	$0.0749593\pi$
$572$	0	0
$573$	−51.7644	−2.16249
$574$	0	0
$575$	−22.3100	−0.930391
$576$	0	0
$577$	25.3878	1.05691	0.528454	−	0.848962i	$-0.322772\pi$
$577$	25.3878	1.05691	0.528454	+	0.848962i	$0.322772\pi$
$578$	0	0
$579$	−42.8015	−1.77877
$580$	0	0
$581$	−16.8458	−0.698880
$582$	0	0
$583$	13.1151	0.543173
$584$	0	0
$585$	16.5098	0.682597
$586$	0	0
$587$	33.7564	1.39328	0.696638	−	0.717422i	$-0.254678\pi$
$587$	33.7564	1.39328	0.696638	+	0.717422i	$0.254678\pi$
$588$	0	0
$589$	18.1281	0.746955
$590$	0	0
$591$	−3.27898	−0.134879
$592$	0	0
$593$	38.6382	1.58668	0.793341	−	0.608778i	$-0.208340\pi$
$593$	38.6382	1.58668	0.793341	+	0.608778i	$0.208340\pi$
$594$	0	0
$595$	1.62440	0.0665940
$596$	0	0
$597$	17.3551	0.710298
$598$	0	0
$599$	6.98723	0.285490	0.142745	−	0.989759i	$-0.454407\pi$
$599$	6.98723	0.285490	0.142745	+	0.989759i	$0.454407\pi$
$600$	0	0
$601$	32.3708	1.32043	0.660217	−	0.751075i	$-0.270464\pi$
$601$	32.3708	1.32043	0.660217	+	0.751075i	$0.270464\pi$
$602$	0	0
$603$	84.9504	3.45945
$604$	0	0
$605$	9.36060	0.380563
$606$	0	0
$607$	−6.22386	−0.252619	−0.126309	−	0.991991i	$-0.540313\pi$
$607$	−6.22386	−0.252619	−0.126309	+	0.991991i	$0.540313\pi$
$608$	0	0
$609$	14.9548	0.605997
$610$	0	0
$611$	18.1303	0.733473
$612$	0	0
$613$	−6.77332	−0.273572	−0.136786	−	0.990601i	$-0.543677\pi$
$613$	−6.77332	−0.273572	−0.136786	+	0.990601i	$0.543677\pi$
$614$	0	0
$615$	11.0657	0.446210
$616$	0	0
$617$	29.1688	1.17429	0.587146	−	0.809481i	$-0.300252\pi$
$617$	29.1688	1.17429	0.587146	+	0.809481i	$0.300252\pi$
$618$	0	0
$619$	−18.1270	−0.728587	−0.364293	−	0.931284i	$-0.618689\pi$
$619$	−18.1270	−0.728587	−0.364293	+	0.931284i	$0.618689\pi$
$620$	0	0
$621$	138.022	5.53862
$622$	0	0
$623$	−10.5918	−0.424352
$624$	0	0
$625$	−7.61763	−0.304705
$626$	0	0
$627$	−29.3833	−1.17346
$628$	0	0
$629$	−6.03699	−0.240711
$630$	0	0
$631$	5.05518	0.201243	0.100622	−	0.994925i	$-0.467917\pi$
$631$	5.05518	0.201243	0.100622	+	0.994925i	$0.467917\pi$
$632$	0	0
$633$	−24.8696	−0.988477
$634$	0	0
$635$	−15.4048	−0.611322
$636$	0	0
$637$	1.35392	0.0536444
$638$	0	0
$639$	−39.2088	−1.55108
$640$	0	0
$641$	44.7693	1.76828	0.884141	−	0.467221i	$-0.154745\pi$
$641$	44.7693	1.76828	0.884141	+	0.467221i	$0.154745\pi$
$642$	0	0
$643$	−17.0710	−0.673216	−0.336608	−	0.941645i	$-0.609280\pi$
$643$	−17.0710	−0.673216	−0.336608	+	0.941645i	$0.609280\pi$
$644$	0	0
$645$	−59.3072	−2.33522
$646$	0	0
$647$	34.2438	1.34626	0.673132	−	0.739523i	$-0.264949\pi$
$647$	34.2438	1.34626	0.673132	+	0.739523i	$0.264949\pi$
$648$	0	0
$649$	39.9789	1.56931
$650$	0	0
$651$	−26.5393	−1.04016
$652$	0	0
$653$	11.3503	0.444171	0.222085	−	0.975027i	$-0.428714\pi$
$653$	11.3503	0.444171	0.222085	+	0.975027i	$0.428714\pi$
$654$	0	0
$655$	29.6415	1.15819
$656$	0	0
$657$	−85.9748	−3.35420
$658$	0	0
$659$	−24.0841	−0.938184	−0.469092	−	0.883149i	$-0.655419\pi$
$659$	−24.0841	−0.938184	−0.469092	+	0.883149i	$0.655419\pi$
$660$	0	0
$661$	−13.3308	−0.518508	−0.259254	−	0.965809i	$-0.583477\pi$
$661$	−13.3308	−0.518508	−0.259254	+	0.965809i	$0.583477\pi$
$662$	0	0
$663$	−4.38863	−0.170440
$664$	0	0
$665$	−3.59659	−0.139470
$666$	0	0
$667$	−43.5903	−1.68782
$668$	0	0
$669$	47.1345	1.82233
$670$	0	0
$671$	−23.7757	−0.917850
$672$	0	0
$673$	−40.8553	−1.57485	−0.787427	−	0.616407i	$-0.788588\pi$
$673$	−40.8553	−1.57485	−0.787427	+	0.616407i	$0.788588\pi$
$674$	0	0
$675$	−34.4950	−1.32771
$676$	0	0
$677$	12.6455	0.486005	0.243003	−	0.970026i	$-0.421868\pi$
$677$	12.6455	0.486005	0.243003	+	0.970026i	$0.421868\pi$
$678$	0	0
$679$	9.74339	0.373917
$680$	0	0
$681$	−66.5118	−2.54874
$682$	0	0
$683$	−5.62494	−0.215232	−0.107616	−	0.994193i	$-0.534322\pi$
$683$	−5.62494	−0.215232	−0.107616	+	0.994193i	$0.534322\pi$
$684$	0	0
$685$	−25.7666	−0.984490
$686$	0	0
$687$	−19.9676	−0.761810
$688$	0	0
$689$	−4.33708	−0.165230
$690$	0	0
$691$	2.55299	0.0971204	0.0485602	−	0.998820i	$-0.484537\pi$
$691$	2.55299	0.0971204	0.0485602	+	0.998820i	$0.484537\pi$
$692$	0	0
$693$	30.7343	1.16750
$694$	0	0
$695$	−0.375102	−0.0142284
$696$	0	0
$697$	−2.10159	−0.0796034
$698$	0	0
$699$	63.9607	2.41922
$700$	0	0
$701$	−6.32431	−0.238866	−0.119433	−	0.992842i	$-0.538108\pi$
$701$	−6.32431	−0.238866	−0.119433	+	0.992842i	$0.538108\pi$
$702$	0	0
$703$	13.3665	0.504126
$704$	0	0
$705$	70.5081	2.65549
$706$	0	0
$707$	13.2925	0.499917
$708$	0	0
$709$	17.0885	0.641773	0.320887	−	0.947118i	$-0.396019\pi$
$709$	17.0885	0.641773	0.320887	+	0.947118i	$0.396019\pi$
$710$	0	0
$711$	27.1098	1.01670
$712$	0	0
$713$	77.3572	2.89705
$714$	0	0
$715$	−9.00446	−0.336748
$716$	0	0
$717$	−4.84714	−0.181020
$718$	0	0
$719$	−18.7941	−0.700903	−0.350451	−	0.936581i	$-0.613972\pi$
$719$	−18.7941	−0.700903	−0.350451	+	0.936581i	$0.613972\pi$
$720$	0	0
$721$	5.88466	0.219156
$722$	0	0
$723$	−4.71312	−0.175283
$724$	0	0
$725$	10.8943	0.404603
$726$	0	0
$727$	4.62746	0.171623	0.0858115	−	0.996311i	$-0.472652\pi$
$727$	4.62746	0.171623	0.0858115	+	0.996311i	$0.472652\pi$
$728$	0	0
$729$	44.3492	1.64256
$730$	0	0
$731$	11.2636	0.416600
$732$	0	0
$733$	36.1332	1.33461	0.667306	−	0.744784i	$-0.267447\pi$
$733$	36.1332	1.33461	0.667306	+	0.744784i	$0.267447\pi$
$734$	0	0
$735$	5.26537	0.194216
$736$	0	0
$737$	−46.3320	−1.70666
$738$	0	0
$739$	−15.3540	−0.564808	−0.282404	−	0.959296i	$-0.591132\pi$
$739$	−15.3540	−0.564808	−0.282404	+	0.959296i	$0.591132\pi$
$740$	0	0
$741$	9.71686	0.356958
$742$	0	0
$743$	7.92951	0.290905	0.145453	−	0.989365i	$-0.453536\pi$
$743$	7.92951	0.290905	0.145453	+	0.989365i	$0.453536\pi$
$744$	0	0
$745$	22.1768	0.812494
$746$	0	0
$747$	126.458	4.62684
$748$	0	0
$749$	3.83676	0.140192
$750$	0	0
$751$	−1.98294	−0.0723584	−0.0361792	−	0.999345i	$-0.511519\pi$
$751$	−1.98294	−0.0723584	−0.0361792	+	0.999345i	$0.511519\pi$
$752$	0	0
$753$	42.8503	1.56155
$754$	0	0
$755$	7.56442	0.275297
$756$	0	0
$757$	−42.4999	−1.54469	−0.772343	−	0.635205i	$-0.780915\pi$
$757$	−42.4999	−1.54469	−0.772343	+	0.635205i	$0.780915\pi$
$758$	0	0
$759$	−125.386	−4.55123
$760$	0	0
$761$	−1.06912	−0.0387555	−0.0193778	−	0.999812i	$-0.506169\pi$
$761$	−1.06912	−0.0387555	−0.0193778	+	0.999812i	$0.506169\pi$
$762$	0	0
$763$	13.3774	0.484295
$764$	0	0
$765$	−12.1941	−0.440877
$766$	0	0
$767$	−13.2208	−0.477374
$768$	0	0
$769$	41.9408	1.51242	0.756211	−	0.654328i	$-0.227048\pi$
$769$	41.9408	1.51242	0.756211	+	0.654328i	$0.227048\pi$
$770$	0	0
$771$	−36.5341	−1.31574
$772$	0	0
$773$	−26.8593	−0.966063	−0.483032	−	0.875603i	$-0.660464\pi$
$773$	−26.8593	−0.966063	−0.483032	+	0.875603i	$0.660464\pi$
$774$	0	0
$775$	−19.3334	−0.694477
$776$	0	0
$777$	−19.5684	−0.702012
$778$	0	0
$779$	4.65313	0.166716
$780$	0	0
$781$	21.3845	0.765197
$782$	0	0
$783$	−67.3979	−2.40860
$784$	0	0
$785$	22.7621	0.812414
$786$	0	0
$787$	−42.3511	−1.50966	−0.754828	−	0.655923i	$-0.772279\pi$
$787$	−42.3511	−1.50966	−0.754828	+	0.655923i	$0.772279\pi$
$788$	0	0
$789$	−18.7934	−0.669062
$790$	0	0
$791$	−6.03956	−0.214742
$792$	0	0
$793$	7.86245	0.279204
$794$	0	0
$795$	−16.8668	−0.598203
$796$	0	0
$797$	37.5313	1.32943	0.664713	−	0.747099i	$-0.268554\pi$
$797$	37.5313	1.32943	0.664713	+	0.747099i	$0.268554\pi$
$798$	0	0
$799$	−13.3909	−0.473736
$800$	0	0
$801$	79.5105	2.80937
$802$	0	0
$803$	46.8907	1.65474
$804$	0	0
$805$	−15.3476	−0.540931
$806$	0	0
$807$	3.24881	0.114363
$808$	0	0
$809$	−9.69463	−0.340845	−0.170423	−	0.985371i	$-0.554513\pi$
$809$	−9.69463	−0.340845	−0.170423	+	0.985371i	$0.554513\pi$
$810$	0	0
$811$	−11.0051	−0.386442	−0.193221	−	0.981155i	$-0.561893\pi$
$811$	−11.0051	−0.386442	−0.193221	+	0.981155i	$0.561893\pi$
$812$	0	0
$813$	−8.87109	−0.311123
$814$	0	0
$815$	−11.0241	−0.386159
$816$	0	0
$817$	−24.9388	−0.872497
$818$	0	0
$819$	−10.1636	−0.355146
$820$	0	0
$821$	25.4707	0.888934	0.444467	−	0.895795i	$-0.353393\pi$
$821$	25.4707	0.888934	0.444467	+	0.895795i	$0.353393\pi$
$822$	0	0
$823$	−41.4662	−1.44542	−0.722710	−	0.691151i	$-0.757104\pi$
$823$	−41.4662	−1.44542	−0.722710	+	0.691151i	$0.757104\pi$
$824$	0	0
$825$	31.3370	1.09102
$826$	0	0
$827$	16.2258	0.564225	0.282113	−	0.959381i	$-0.408965\pi$
$827$	16.2258	0.564225	0.282113	+	0.959381i	$0.408965\pi$
$828$	0	0
$829$	−5.82578	−0.202338	−0.101169	−	0.994869i	$-0.532258\pi$
$829$	−5.82578	−0.202338	−0.101169	+	0.994869i	$0.532258\pi$
$830$	0	0
$831$	−89.6275	−3.10914
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	−14.7664	−0.511014
$836$	0	0
$837$	119.607	4.13423
$838$	0	0
$839$	17.0624	0.589059	0.294529	−	0.955642i	$-0.404837\pi$
$839$	17.0624	0.589059	0.294529	+	0.955642i	$0.404837\pi$
$840$	0	0
$841$	−7.71427	−0.266009
$842$	0	0
$843$	7.58997	0.261413
$844$	0	0
$845$	−18.1395	−0.624019
$846$	0	0
$847$	−5.76249	−0.198001
$848$	0	0
$849$	−43.4455	−1.49105
$850$	0	0
$851$	57.0382	1.95525
$852$	0	0
$853$	−15.4224	−0.528054	−0.264027	−	0.964515i	$-0.585051\pi$
$853$	−15.4224	−0.528054	−0.264027	+	0.964515i	$0.585051\pi$
$854$	0	0
$855$	26.9988	0.923340
$856$	0	0
$857$	−30.2969	−1.03492	−0.517461	−	0.855707i	$-0.673123\pi$
$857$	−30.2969	−1.03492	−0.517461	+	0.855707i	$0.673123\pi$
$858$	0	0
$859$	−46.7848	−1.59628	−0.798138	−	0.602475i	$-0.794181\pi$
$859$	−46.7848	−1.59628	−0.798138	+	0.602475i	$0.794181\pi$
$860$	0	0
$861$	−6.81213	−0.232157
$862$	0	0
$863$	−54.8342	−1.86658	−0.933289	−	0.359126i	$-0.883075\pi$
$863$	−54.8342	−1.86658	−0.933289	+	0.359126i	$0.883075\pi$
$864$	0	0
$865$	−13.5201	−0.459697
$866$	0	0
$867$	3.24142	0.110084
$868$	0	0
$869$	−14.7857	−0.501569
$870$	0	0
$871$	15.3217	0.519155
$872$	0	0
$873$	−73.1415	−2.47547
$874$	0	0
$875$	11.9577	0.404245
$876$	0	0
$877$	−18.5624	−0.626807	−0.313403	−	0.949620i	$-0.601469\pi$
$877$	−18.5624	−0.626807	−0.313403	+	0.949620i	$0.601469\pi$
$878$	0	0
$879$	−59.1937	−1.99656
$880$	0	0
$881$	−40.8016	−1.37464	−0.687321	−	0.726354i	$-0.741213\pi$
$881$	−40.8016	−1.37464	−0.687321	+	0.726354i	$0.741213\pi$
$882$	0	0
$883$	24.8705	0.836959	0.418480	−	0.908226i	$-0.362563\pi$
$883$	24.8705	0.836959	0.418480	+	0.908226i	$0.362563\pi$
$884$	0	0
$885$	−51.4151	−1.72830
$886$	0	0
$887$	11.5951	0.389327	0.194663	−	0.980870i	$-0.437639\pi$
$887$	11.5951	0.389327	0.194663	+	0.980870i	$0.437639\pi$
$888$	0	0
$889$	9.48337	0.318062
$890$	0	0
$891$	−101.665	−3.40591
$892$	0	0
$893$	29.6488	0.992159
$894$	0	0
$895$	−24.4528	−0.817366
$896$	0	0
$897$	41.4644	1.38445
$898$	0	0
$899$	−37.7746	−1.25985
$900$	0	0
$901$	3.20334	0.106719
$902$	0	0
$903$	36.5101	1.21498
$904$	0	0
$905$	−7.06465	−0.234837
$906$	0	0
$907$	7.10890	0.236047	0.118024	−	0.993011i	$-0.462344\pi$
$907$	7.10890	0.236047	0.118024	+	0.993011i	$0.462344\pi$
$908$	0	0
$909$	−99.7842	−3.30963
$910$	0	0
$911$	−2.86216	−0.0948276	−0.0474138	−	0.998875i	$-0.515098\pi$
$911$	−2.86216	−0.0948276	−0.0474138	+	0.998875i	$0.515098\pi$
$912$	0	0
$913$	−68.9700	−2.28257
$914$	0	0
$915$	30.5768	1.01084
$916$	0	0
$917$	−18.2476	−0.602589
$918$	0	0
$919$	34.3235	1.13223	0.566114	−	0.824327i	$-0.308446\pi$
$919$	34.3235	1.13223	0.566114	+	0.824327i	$0.308446\pi$
$920$	0	0
$921$	−3.92383	−0.129295
$922$	0	0
$923$	−7.07170	−0.232768
$924$	0	0
$925$	−14.2552	−0.468709
$926$	0	0
$927$	−44.1749	−1.45089
$928$	0	0
$929$	−8.79083	−0.288418	−0.144209	−	0.989547i	$-0.546064\pi$
$929$	−8.79083	−0.288418	−0.144209	+	0.989547i	$0.546064\pi$
$930$	0	0
$931$	2.21410	0.0725641
$932$	0	0
$933$	105.285	3.44688
$934$	0	0
$935$	6.65064	0.217499
$936$	0	0
$937$	33.0259	1.07891	0.539455	−	0.842014i	$-0.318630\pi$
$937$	33.0259	1.07891	0.539455	+	0.842014i	$0.318630\pi$
$938$	0	0
$939$	22.4342	0.732113
$940$	0	0
$941$	−20.7139	−0.675255	−0.337628	−	0.941280i	$-0.609624\pi$
$941$	−20.7139	−0.675255	−0.337628	+	0.941280i	$0.609624\pi$
$942$	0	0
$943$	19.8561	0.646603
$944$	0	0
$945$	−23.7299	−0.771934
$946$	0	0
$947$	3.48351	0.113199	0.0565994	−	0.998397i	$-0.481974\pi$
$947$	3.48351	0.113199	0.0565994	+	0.998397i	$0.481974\pi$
$948$	0	0
$949$	−15.5064	−0.503359
$950$	0	0
$951$	−87.1614	−2.82640
$952$	0	0
$953$	32.4814	1.05218	0.526088	−	0.850430i	$-0.323658\pi$
$953$	32.4814	1.05218	0.526088	+	0.850430i	$0.323658\pi$
$954$	0	0
$955$	−25.9412	−0.839438
$956$	0	0
$957$	61.2278	1.97921
$958$	0	0
$959$	15.8622	0.512216
$960$	0	0
$961$	36.0363	1.16246
$962$	0	0
$963$	−28.8017	−0.928123
$964$	0	0
$965$	−21.4495	−0.690484
$966$	0	0
$967$	−4.50312	−0.144811	−0.0724053	−	0.997375i	$-0.523068\pi$
$967$	−4.50312	−0.144811	−0.0724053	+	0.997375i	$0.523068\pi$
$968$	0	0
$969$	−7.17682	−0.230553
$970$	0	0
$971$	−50.6585	−1.62571	−0.812854	−	0.582467i	$-0.802088\pi$
$971$	−50.6585	−1.62571	−0.812854	+	0.582467i	$0.802088\pi$
$972$	0	0
$973$	0.230917	0.00740286
$974$	0	0
$975$	−10.3629	−0.331880
$976$	0	0
$977$	−25.9677	−0.830779	−0.415390	−	0.909644i	$-0.636355\pi$
$977$	−25.9677	−0.830779	−0.415390	+	0.909644i	$0.636355\pi$
$978$	0	0
$979$	−43.3650	−1.38595
$980$	0	0
$981$	−100.421	−3.20621
$982$	0	0
$983$	40.0810	1.27838	0.639192	−	0.769047i	$-0.279269\pi$
$983$	40.0810	1.27838	0.639192	+	0.769047i	$0.279269\pi$
$984$	0	0
$985$	−1.64323	−0.0523576
$986$	0	0
$987$	−43.4055	−1.38161
$988$	0	0
$989$	−106.420	−3.38396
$990$	0	0
$991$	−5.14278	−0.163366	−0.0816829	−	0.996658i	$-0.526029\pi$
$991$	−5.14278	−0.163366	−0.0816829	+	0.996658i	$0.526029\pi$
$992$	0	0
$993$	−98.6502	−3.13057
$994$	0	0
$995$	8.69735	0.275725
$996$	0	0
$997$	−13.0029	−0.411806	−0.205903	−	0.978572i	$-0.566013\pi$
$997$	−13.0029	−0.411806	−0.205903	+	0.978572i	$0.566013\pi$
$998$	0	0
$999$	88.1906	2.79023

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.e.1.5	✓	5
4.3	odd	2		3808.2.a.f.1.1	yes	5
8.3	odd	2		7616.2.a.bs.1.5		5
8.5	even	2		7616.2.a.br.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.e.1.5	✓	5	1.1	even	1	trivial
3808.2.a.f.1.1	yes	5	4.3	odd	2
7616.2.a.br.1.1		5	8.5	even	2
7616.2.a.bs.1.5		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)

\(f(q)\)	\(=\)	\(q+3.24142 q^{3} +1.62440 q^{5} -1.00000 q^{7} +7.50679 q^{9} -4.09420 q^{11} +1.35392 q^{13} +5.26537 q^{15} -1.00000 q^{17} +2.21410 q^{19} -3.24142 q^{21} +9.44813 q^{23} -2.36131 q^{25} +14.6084 q^{27} -4.61365 q^{29} +8.18757 q^{31} -13.2710 q^{33} -1.62440 q^{35} +6.03699 q^{37} +4.38863 q^{39} +2.10159 q^{41} -11.2636 q^{43} +12.1941 q^{45} +13.3909 q^{47} +1.00000 q^{49} -3.24142 q^{51} -3.20334 q^{53} -6.65064 q^{55} +7.17682 q^{57} -9.76477 q^{59} +5.80716 q^{61} -7.50679 q^{63} +2.19932 q^{65} +11.3165 q^{67} +30.6253 q^{69} -5.22311 q^{71} -11.4529 q^{73} -7.65400 q^{75} +4.09420 q^{77} +3.61137 q^{79} +24.8315 q^{81} +16.8458 q^{83} -1.62440 q^{85} -14.9548 q^{87} +10.5918 q^{89} -1.35392 q^{91} +26.5393 q^{93} +3.59659 q^{95} -9.74339 q^{97} -30.7343 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{5} - 5 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} + 8 q^{15} - 5 q^{17} - 6 q^{19} + 18 q^{23} + 5 q^{25} + 18 q^{27} - 14 q^{29} + 16 q^{31} - 26 q^{33} + 2 q^{35} + 2 q^{37} + 6 q^{39} - 10 q^{41}+ \cdots - 8 q^{99}+O(q^{100}) \) 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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