LMFDB - Embedded newform 3808.2.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3808,2,Mod(1,3808)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$30.4070330897$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.109859312.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 7x^{4} + 15x^{3} + 13x^{2} - 9x - 2 $ x^6 - 3x^5 - 7x^4 + 15x^3 + 13x^2 - 9*x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.10086$ 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)

$f(q)$	$=$	$q-1.07243 q^{3} +3.18060 q^{5} -1.00000 q^{7} -1.84989 q^{9} +0.565788 q^{11} +6.01588 q^{13} -3.41098 q^{15} -1.00000 q^{17} -5.60956 q^{19} +1.07243 q^{21} -7.60956 q^{23} +5.11621 q^{25} +5.20118 q^{27} -8.65334 q^{29} -7.61740 q^{31} -0.606769 q^{33} -3.18060 q^{35} -0.737022 q^{37} -6.45162 q^{39} -6.04166 q^{41} +9.59679 q^{43} -5.88376 q^{45} +2.91237 q^{47} +1.00000 q^{49} +1.07243 q^{51} +7.85269 q^{53} +1.79954 q^{55} +6.01588 q^{57} -9.84464 q^{59} -11.6846 q^{61} +1.84989 q^{63} +19.1341 q^{65} +8.79143 q^{67} +8.16074 q^{69} -15.2894 q^{71} -5.75194 q^{73} -5.48679 q^{75} -0.565788 q^{77} +5.13920 q^{79} -0.0282405 q^{81} -3.50982 q^{83} -3.18060 q^{85} +9.28012 q^{87} -7.43833 q^{89} -6.01588 q^{91} +8.16914 q^{93} -17.8418 q^{95} +6.51390 q^{97} -1.04664 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} + 4 q^{5} - 6 q^{7} + 8 q^{9} - 8 q^{11} + 8 q^{13} - 6 q^{17} - 6 q^{19} + 4 q^{21} - 18 q^{23} + 12 q^{25} - 22 q^{27} - 8 q^{29} + 4 q^{31} - 6 q^{33} - 4 q^{35} - 8 q^{37} - 14 q^{39}+ \cdots - 28 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 + 4 * q^5 - 6 * q^7 + 8 * q^9 - 8 * q^11 + 8 * q^13 - 6 * q^17 - 6 * q^19 + 4 * q^21 - 18 * q^23 + 12 * q^25 - 22 * q^27 - 8 * q^29 + 4 * q^31 - 6 * q^33 - 4 * q^35 - 8 * q^37 - 14 * q^39 + 16 * q^41 - 12 * q^43 + 4 * q^45 - 18 * q^47 + 6 * q^49 + 4 * q^51 - 2 * q^53 + 8 * q^55 + 8 * q^57 - 16 * q^59 + 6 * q^61 - 8 * q^63 - 22 * q^65 - 12 * q^67 + 16 * q^69 + 2 * q^71 + 8 * q^73 - 24 * q^75 + 8 * q^77 + 18 * q^79 + 18 * q^81 - 28 * q^83 - 4 * q^85 + 2 * q^87 - 2 * q^89 - 8 * q^91 - 32 * q^93 - 26 * q^95 - 18 * q^97 - 28 * 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
$3$	−1.07243	−0.619169	−0.309584	−	0.950872i	$-0.600190\pi$
$3$	−1.07243	−0.619169	−0.309584	+	0.950872i	$0.600190\pi$
$4$	0	0
$5$	3.18060	1.42241	0.711204	−	0.702986i	$-0.248150\pi$
$5$	3.18060	1.42241	0.711204	+	0.702986i	$0.248150\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.84989	−0.616630
$10$	0	0
$11$	0.565788	0.170591	0.0852957	−	0.996356i	$-0.472817\pi$
$11$	0.565788	0.170591	0.0852957	+	0.996356i	$0.472817\pi$
$12$	0	0
$13$	6.01588	1.66850	0.834252	−	0.551383i	$-0.185900\pi$
$13$	6.01588	1.66850	0.834252	+	0.551383i	$0.185900\pi$
$14$	0	0
$15$	−3.41098	−0.880710
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−5.60956	−1.28692	−0.643461	−	0.765479i	$-0.722502\pi$
$19$	−5.60956	−1.28692	−0.643461	+	0.765479i	$0.722502\pi$
$20$	0	0
$21$	1.07243	0.234024
$22$	0	0
$23$	−7.60956	−1.58670	−0.793352	−	0.608763i	$-0.791666\pi$
$23$	−7.60956	−1.58670	−0.793352	+	0.608763i	$0.791666\pi$
$24$	0	0
$25$	5.11621	1.02324
$26$	0	0
$27$	5.20118	1.00097
$28$	0	0
$29$	−8.65334	−1.60689	−0.803443	−	0.595382i	$-0.797001\pi$
$29$	−8.65334	−1.60689	−0.803443	+	0.595382i	$0.797001\pi$
$30$	0	0
$31$	−7.61740	−1.36813	−0.684063	−	0.729423i	$-0.739789\pi$
$31$	−7.61740	−1.36813	−0.684063	+	0.729423i	$0.739789\pi$
$32$	0	0
$33$	−0.606769	−0.105625
$34$	0	0
$35$	−3.18060	−0.537619
$36$	0	0
$37$	−0.737022	−0.121166	−0.0605828	−	0.998163i	$-0.519296\pi$
$37$	−0.737022	−0.121166	−0.0605828	+	0.998163i	$0.519296\pi$
$38$	0	0
$39$	−6.45162	−1.03309
$40$	0	0
$41$	−6.04166	−0.943549	−0.471775	−	0.881719i	$-0.656386\pi$
$41$	−6.04166	−0.943549	−0.471775	+	0.881719i	$0.656386\pi$
$42$	0	0
$43$	9.59679	1.46350	0.731748	−	0.681575i	$-0.238705\pi$
$43$	9.59679	1.46350	0.731748	+	0.681575i	$0.238705\pi$
$44$	0	0
$45$	−5.88376	−0.877099
$46$	0	0
$47$	2.91237	0.424813	0.212407	−	0.977181i	$-0.431870\pi$
$47$	2.91237	0.424813	0.212407	+	0.977181i	$0.431870\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.07243	0.150171
$52$	0	0
$53$	7.85269	1.07865	0.539325	−	0.842098i	$-0.318680\pi$
$53$	7.85269	1.07865	0.539325	+	0.842098i	$0.318680\pi$
$54$	0	0
$55$	1.79954	0.242650
$56$	0	0
$57$	6.01588	0.796822
$58$	0	0
$59$	−9.84464	−1.28166	−0.640832	−	0.767681i	$-0.721410\pi$
$59$	−9.84464	−1.28166	−0.640832	+	0.767681i	$0.721410\pi$
$60$	0	0
$61$	−11.6846	−1.49606	−0.748029	−	0.663666i	$-0.769000\pi$
$61$	−11.6846	−1.49606	−0.748029	+	0.663666i	$0.769000\pi$
$62$	0	0
$63$	1.84989	0.233064
$64$	0	0
$65$	19.1341	2.37329
$66$	0	0
$67$	8.79143	1.07404	0.537022	−	0.843568i	$-0.319549\pi$
$67$	8.79143	1.07404	0.537022	+	0.843568i	$0.319549\pi$
$68$	0	0
$69$	8.16074	0.982438
$70$	0	0
$71$	−15.2894	−1.81452	−0.907262	−	0.420566i	$-0.861831\pi$
$71$	−15.2894	−1.81452	−0.907262	+	0.420566i	$0.861831\pi$
$72$	0	0
$73$	−5.75194	−0.673213	−0.336607	−	0.941645i	$-0.609279\pi$
$73$	−5.75194	−0.673213	−0.336607	+	0.941645i	$0.609279\pi$
$74$	0	0
$75$	−5.48679	−0.633560
$76$	0	0
$77$	−0.565788	−0.0644775
$78$	0	0
$79$	5.13920	0.578205	0.289103	−	0.957298i	$-0.406643\pi$
$79$	5.13920	0.578205	0.289103	+	0.957298i	$0.406643\pi$
$80$	0	0
$81$	−0.0282405	−0.00313783
$82$	0	0
$83$	−3.50982	−0.385252	−0.192626	−	0.981272i	$-0.561700\pi$
$83$	−3.50982	−0.385252	−0.192626	+	0.981272i	$0.561700\pi$
$84$	0	0
$85$	−3.18060	−0.344984
$86$	0	0
$87$	9.28012	0.994933
$88$	0	0
$89$	−7.43833	−0.788461	−0.394231	−	0.919011i	$-0.628989\pi$
$89$	−7.43833	−0.788461	−0.394231	+	0.919011i	$0.628989\pi$
$90$	0	0
$91$	−6.01588	−0.630635
$92$	0	0
$93$	8.16914	0.847101
$94$	0	0
$95$	−17.8418	−1.83053
$96$	0	0
$97$	6.51390	0.661386	0.330693	−	0.943738i	$-0.392718\pi$
$97$	6.51390	0.661386	0.330693	+	0.943738i	$0.392718\pi$
$98$	0	0
$99$	−1.04664	−0.105192
$100$	0	0
$101$	4.43574	0.441373	0.220686	−	0.975345i	$-0.429170\pi$
$101$	4.43574	0.441373	0.220686	+	0.975345i	$0.429170\pi$
$102$	0	0
$103$	−2.38757	−0.235254	−0.117627	−	0.993058i	$-0.537529\pi$
$103$	−2.38757	−0.235254	−0.117627	+	0.993058i	$0.537529\pi$
$104$	0	0
$105$	3.41098	0.332877
$106$	0	0
$107$	−16.9307	−1.63675	−0.818375	−	0.574685i	$-0.805125\pi$
$107$	−16.9307	−1.63675	−0.818375	+	0.574685i	$0.805125\pi$
$108$	0	0
$109$	2.76993	0.265311	0.132655	−	0.991162i	$-0.457650\pi$
$109$	2.76993	0.265311	0.132655	+	0.991162i	$0.457650\pi$
$110$	0	0
$111$	0.790406	0.0750220
$112$	0	0
$113$	17.6149	1.65708	0.828538	−	0.559934i	$-0.189173\pi$
$113$	17.6149	1.65708	0.828538	+	0.559934i	$0.189173\pi$
$114$	0	0
$115$	−24.2030	−2.25694
$116$	0	0
$117$	−11.1287	−1.02885
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−10.6799	−0.970899
$122$	0	0
$123$	6.47927	0.584217
$124$	0	0
$125$	0.369615	0.0330594
$126$	0	0
$127$	7.65985	0.679702	0.339851	−	0.940479i	$-0.389623\pi$
$127$	7.65985	0.679702	0.339851	+	0.940479i	$0.389623\pi$
$128$	0	0
$129$	−10.2919	−0.906151
$130$	0	0
$131$	17.6689	1.54374	0.771871	−	0.635779i	$-0.219321\pi$
$131$	17.6689	1.54374	0.771871	+	0.635779i	$0.219321\pi$
$132$	0	0
$133$	5.60956	0.486411
$134$	0	0
$135$	16.5429	1.42378
$136$	0	0
$137$	0.948655	0.0810490	0.0405245	−	0.999179i	$-0.487097\pi$
$137$	0.948655	0.0810490	0.0405245	+	0.999179i	$0.487097\pi$
$138$	0	0
$139$	−7.82961	−0.664099	−0.332050	−	0.943262i	$-0.607740\pi$
$139$	−7.82961	−0.664099	−0.332050	+	0.943262i	$0.607740\pi$
$140$	0	0
$141$	−3.12332	−0.263031
$142$	0	0
$143$	3.40371	0.284632
$144$	0	0
$145$	−27.5228	−2.28564
$146$	0	0
$147$	−1.07243	−0.0884527
$148$	0	0
$149$	−4.30332	−0.352542	−0.176271	−	0.984342i	$-0.556403\pi$
$149$	−4.30332	−0.352542	−0.176271	+	0.984342i	$0.556403\pi$
$150$	0	0
$151$	17.2449	1.40337	0.701686	−	0.712487i	$-0.252431\pi$
$151$	17.2449	1.40337	0.701686	+	0.712487i	$0.252431\pi$
$152$	0	0
$153$	1.84989	0.149555
$154$	0	0
$155$	−24.2279	−1.94603
$156$	0	0
$157$	2.72069	0.217135	0.108567	−	0.994089i	$-0.465374\pi$
$157$	2.72069	0.217135	0.108567	+	0.994089i	$0.465374\pi$
$158$	0	0
$159$	−8.42147	−0.667866
$160$	0	0
$161$	7.60956	0.599718
$162$	0	0
$163$	−10.4756	−0.820510	−0.410255	−	0.911971i	$-0.634560\pi$
$163$	−10.4756	−0.820510	−0.410255	+	0.911971i	$0.634560\pi$
$164$	0	0
$165$	−1.92989	−0.150242
$166$	0	0
$167$	−0.251538	−0.0194646	−0.00973230	−	0.999953i	$-0.503098\pi$
$167$	−0.251538	−0.0194646	−0.00973230	+	0.999953i	$0.503098\pi$
$168$	0	0
$169$	23.1908	1.78391
$170$	0	0
$171$	10.3771	0.793555
$172$	0	0
$173$	7.41094	0.563443	0.281722	−	0.959496i	$-0.409094\pi$
$173$	7.41094	0.563443	0.281722	+	0.959496i	$0.409094\pi$
$174$	0	0
$175$	−5.11621	−0.386749
$176$	0	0
$177$	10.5577	0.793566
$178$	0	0
$179$	−18.2380	−1.36317	−0.681585	−	0.731739i	$-0.738709\pi$
$179$	−18.2380	−1.36317	−0.681585	+	0.731739i	$0.738709\pi$
$180$	0	0
$181$	−4.71719	−0.350626	−0.175313	−	0.984513i	$-0.556094\pi$
$181$	−4.71719	−0.350626	−0.175313	+	0.984513i	$0.556094\pi$
$182$	0	0
$183$	12.5309	0.926313
$184$	0	0
$185$	−2.34417	−0.172347
$186$	0	0
$187$	−0.565788	−0.0413745
$188$	0	0
$189$	−5.20118	−0.378330
$190$	0	0
$191$	4.31459	0.312193	0.156096	−	0.987742i	$-0.450109\pi$
$191$	4.31459	0.312193	0.156096	+	0.987742i	$0.450109\pi$
$192$	0	0
$193$	−7.64419	−0.550241	−0.275120	−	0.961410i	$-0.588718\pi$
$193$	−7.64419	−0.550241	−0.275120	+	0.961410i	$0.588718\pi$
$194$	0	0
$195$	−20.5200	−1.46947
$196$	0	0
$197$	−20.9446	−1.49224	−0.746119	−	0.665812i	$-0.768085\pi$
$197$	−20.9446	−1.49224	−0.746119	+	0.665812i	$0.768085\pi$
$198$	0	0
$199$	0.0936490	0.00663860	0.00331930	−	0.999994i	$-0.498943\pi$
$199$	0.0936490	0.00663860	0.00331930	+	0.999994i	$0.498943\pi$
$200$	0	0
$201$	−9.42821	−0.665015
$202$	0	0
$203$	8.65334	0.607346
$204$	0	0
$205$	−19.2161	−1.34211
$206$	0	0
$207$	14.0769	0.978409
$208$	0	0
$209$	−3.17382	−0.219538
$210$	0	0
$211$	−26.4236	−1.81908	−0.909539	−	0.415618i	$-0.863565\pi$
$211$	−26.4236	−1.81908	−0.909539	+	0.415618i	$0.863565\pi$
$212$	0	0
$213$	16.3969	1.12350
$214$	0	0
$215$	30.5235	2.08169
$216$	0	0
$217$	7.61740	0.517103
$218$	0	0
$219$	6.16856	0.416833
$220$	0	0
$221$	−6.01588	−0.404672
$222$	0	0
$223$	−6.69432	−0.448285	−0.224143	−	0.974556i	$-0.571958\pi$
$223$	−6.69432	−0.448285	−0.224143	+	0.974556i	$0.571958\pi$
$224$	0	0
$225$	−9.46442	−0.630961
$226$	0	0
$227$	10.1542	0.673955	0.336978	−	0.941513i	$-0.390595\pi$
$227$	10.1542	0.673955	0.336978	+	0.941513i	$0.390595\pi$
$228$	0	0
$229$	−24.2389	−1.60175	−0.800875	−	0.598831i	$-0.795632\pi$
$229$	−24.2389	−1.60175	−0.800875	+	0.598831i	$0.795632\pi$
$230$	0	0
$231$	0.606769	0.0399225
$232$	0	0
$233$	−19.4343	−1.27318	−0.636592	−	0.771201i	$-0.719656\pi$
$233$	−19.4343	−1.27318	−0.636592	+	0.771201i	$0.719656\pi$
$234$	0	0
$235$	9.26309	0.604258
$236$	0	0
$237$	−5.51144	−0.358007
$238$	0	0
$239$	−7.98354	−0.516412	−0.258206	−	0.966090i	$-0.583131\pi$
$239$	−7.98354	−0.516412	−0.258206	+	0.966090i	$0.583131\pi$
$240$	0	0
$241$	22.8402	1.47127	0.735634	−	0.677380i	$-0.236885\pi$
$241$	22.8402	1.47127	0.735634	+	0.677380i	$0.236885\pi$
$242$	0	0
$243$	−15.5732	−0.999024
$244$	0	0
$245$	3.18060	0.203201
$246$	0	0
$247$	−33.7465	−2.14724
$248$	0	0
$249$	3.76404	0.238536
$250$	0	0
$251$	21.7023	1.36984	0.684918	−	0.728620i	$-0.259838\pi$
$251$	21.7023	1.36984	0.684918	+	0.728620i	$0.259838\pi$
$252$	0	0
$253$	−4.30540	−0.270678
$254$	0	0
$255$	3.41098	0.213604
$256$	0	0
$257$	−23.3082	−1.45392	−0.726962	−	0.686678i	$-0.759068\pi$
$257$	−23.3082	−1.45392	−0.726962	+	0.686678i	$0.759068\pi$
$258$	0	0
$259$	0.737022	0.0457963
$260$	0	0
$261$	16.0077	0.990853
$262$	0	0
$263$	−8.21633	−0.506641	−0.253320	−	0.967382i	$-0.581523\pi$
$263$	−8.21633	−0.506641	−0.253320	+	0.967382i	$0.581523\pi$
$264$	0	0
$265$	24.9762	1.53428
$266$	0	0
$267$	7.97710	0.488191
$268$	0	0
$269$	8.59407	0.523990	0.261995	−	0.965069i	$-0.415620\pi$
$269$	8.59407	0.523990	0.261995	+	0.965069i	$0.415620\pi$
$270$	0	0
$271$	0.00265971	0.000161566 0	8.07829e−5	−	1.00000i	$-0.499974\pi$
$271$	0.00265971	0.000161566 0	8.07829e−5		1.00000i	$0.499974\pi$
$272$	0	0
$273$	6.45162	0.390470
$274$	0	0
$275$	2.89469	0.174556
$276$	0	0
$277$	−16.3080	−0.979854	−0.489927	−	0.871763i	$-0.662977\pi$
$277$	−16.3080	−0.979854	−0.489927	+	0.871763i	$0.662977\pi$
$278$	0	0
$279$	14.0913	0.843627
$280$	0	0
$281$	16.0529	0.957638	0.478819	−	0.877914i	$-0.341065\pi$
$281$	16.0529	0.957638	0.478819	+	0.877914i	$0.341065\pi$
$282$	0	0
$283$	20.8537	1.23962	0.619811	−	0.784751i	$-0.287209\pi$
$283$	20.8537	1.23962	0.619811	+	0.784751i	$0.287209\pi$
$284$	0	0
$285$	19.1341	1.13341
$286$	0	0
$287$	6.04166	0.356628
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−6.98571	−0.409510
$292$	0	0
$293$	−5.48705	−0.320557	−0.160279	−	0.987072i	$-0.551239\pi$
$293$	−5.48705	−0.320557	−0.160279	+	0.987072i	$0.551239\pi$
$294$	0	0
$295$	−31.3119	−1.82305
$296$	0	0
$297$	2.94276	0.170756
$298$	0	0
$299$	−45.7782	−2.64742
$300$	0	0
$301$	−9.59679	−0.553150
$302$	0	0
$303$	−4.75703	−0.273284
$304$	0	0
$305$	−37.1640	−2.12800
$306$	0	0
$307$	−9.38674	−0.535730	−0.267865	−	0.963457i	$-0.586318\pi$
$307$	−9.38674	−0.535730	−0.267865	+	0.963457i	$0.586318\pi$
$308$	0	0
$309$	2.56050	0.145662
$310$	0	0
$311$	21.2513	1.20505	0.602525	−	0.798100i	$-0.294161\pi$
$311$	21.2513	1.20505	0.602525	+	0.798100i	$0.294161\pi$
$312$	0	0
$313$	−14.3710	−0.812295	−0.406147	−	0.913808i	$-0.633128\pi$
$313$	−14.3710	−0.812295	−0.406147	+	0.913808i	$0.633128\pi$
$314$	0	0
$315$	5.88376	0.331512
$316$	0	0
$317$	24.3225	1.36609	0.683045	−	0.730377i	$-0.260655\pi$
$317$	24.3225	1.36609	0.683045	+	0.730377i	$0.260655\pi$
$318$	0	0
$319$	−4.89595	−0.274121
$320$	0	0
$321$	18.1570	1.01342
$322$	0	0
$323$	5.60956	0.312125
$324$	0	0
$325$	30.7785	1.70728
$326$	0	0
$327$	−2.97056	−0.164272
$328$	0	0
$329$	−2.91237	−0.160564
$330$	0	0
$331$	−17.3559	−0.953969	−0.476984	−	0.878912i	$-0.658270\pi$
$331$	−17.3559	−0.953969	−0.476984	+	0.878912i	$0.658270\pi$
$332$	0	0
$333$	1.36341	0.0747143
$334$	0	0
$335$	27.9620	1.52773
$336$	0	0
$337$	−19.1221	−1.04165	−0.520824	−	0.853664i	$-0.674375\pi$
$337$	−19.1221	−1.04165	−0.520824	+	0.853664i	$0.674375\pi$
$338$	0	0
$339$	−18.8908	−1.02601
$340$	0	0
$341$	−4.30983	−0.233390
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	25.9560	1.39743
$346$	0	0
$347$	16.5618	0.889086	0.444543	−	0.895758i	$-0.353366\pi$
$347$	16.5618	0.889086	0.444543	+	0.895758i	$0.353366\pi$
$348$	0	0
$349$	15.5311	0.831359	0.415680	−	0.909511i	$-0.363544\pi$
$349$	15.5311	0.831359	0.415680	+	0.909511i	$0.363544\pi$
$350$	0	0
$351$	31.2896	1.67012
$352$	0	0
$353$	14.4021	0.766546	0.383273	−	0.923635i	$-0.374797\pi$
$353$	14.4021	0.766546	0.383273	+	0.923635i	$0.374797\pi$
$354$	0	0
$355$	−48.6296	−2.58099
$356$	0	0
$357$	−1.07243	−0.0567591
$358$	0	0
$359$	−1.13178	−0.0597333	−0.0298667	−	0.999554i	$-0.509508\pi$
$359$	−1.13178	−0.0597333	−0.0298667	+	0.999554i	$0.509508\pi$
$360$	0	0
$361$	12.4672	0.656169
$362$	0	0
$363$	11.4535	0.601150
$364$	0	0
$365$	−18.2946	−0.957583
$366$	0	0
$367$	15.9100	0.830496	0.415248	−	0.909708i	$-0.363695\pi$
$367$	15.9100	0.830496	0.415248	+	0.909708i	$0.363695\pi$
$368$	0	0
$369$	11.1764	0.581821
$370$	0	0
$371$	−7.85269	−0.407691
$372$	0	0
$373$	13.0538	0.675900	0.337950	−	0.941164i	$-0.390266\pi$
$373$	13.0538	0.675900	0.337950	+	0.941164i	$0.390266\pi$
$374$	0	0
$375$	−0.396387	−0.0204693
$376$	0	0
$377$	−52.0574	−2.68109
$378$	0	0
$379$	−30.4198	−1.56256	−0.781279	−	0.624182i	$-0.785432\pi$
$379$	−30.4198	−1.56256	−0.781279	+	0.624182i	$0.785432\pi$
$380$	0	0
$381$	−8.21467	−0.420850
$382$	0	0
$383$	−35.4221	−1.80998	−0.904992	−	0.425428i	$-0.860124\pi$
$383$	−35.4221	−1.80998	−0.904992	+	0.425428i	$0.860124\pi$
$384$	0	0
$385$	−1.79954	−0.0917132
$386$	0	0
$387$	−17.7530	−0.902435
$388$	0	0
$389$	−12.7473	−0.646312	−0.323156	−	0.946346i	$-0.604744\pi$
$389$	−12.7473	−0.646312	−0.323156	+	0.946346i	$0.604744\pi$
$390$	0	0
$391$	7.60956	0.384832
$392$	0	0
$393$	−18.9487	−0.955837
$394$	0	0
$395$	16.3457	0.822443
$396$	0	0
$397$	−33.9721	−1.70501	−0.852505	−	0.522719i	$-0.824918\pi$
$397$	−33.9721	−1.70501	−0.852505	+	0.522719i	$0.824918\pi$
$398$	0	0
$399$	−6.01588	−0.301171
$400$	0	0
$401$	16.2997	0.813967	0.406984	−	0.913435i	$-0.366581\pi$
$401$	16.2997	0.813967	0.406984	+	0.913435i	$0.366581\pi$
$402$	0	0
$403$	−45.8253	−2.28272
$404$	0	0
$405$	−0.0898217	−0.00446328
$406$	0	0
$407$	−0.416998	−0.0206698
$408$	0	0
$409$	−18.2156	−0.900703	−0.450351	−	0.892851i	$-0.648701\pi$
$409$	−18.2156	−0.900703	−0.450351	+	0.892851i	$0.648701\pi$
$410$	0	0
$411$	−1.01737	−0.0501830
$412$	0	0
$413$	9.84464	0.484423
$414$	0	0
$415$	−11.1633	−0.547986
$416$	0	0
$417$	8.39673	0.411190
$418$	0	0
$419$	−21.7259	−1.06138	−0.530690	−	0.847566i	$-0.678067\pi$
$419$	−21.7259	−1.06138	−0.530690	+	0.847566i	$0.678067\pi$
$420$	0	0
$421$	10.7609	0.524453	0.262226	−	0.965006i	$-0.415543\pi$
$421$	10.7609	0.524453	0.262226	+	0.965006i	$0.415543\pi$
$422$	0	0
$423$	−5.38757	−0.261953
$424$	0	0
$425$	−5.11621	−0.248173
$426$	0	0
$427$	11.6846	0.565457
$428$	0	0
$429$	−3.65025	−0.176236
$430$	0	0
$431$	−6.71437	−0.323420	−0.161710	−	0.986838i	$-0.551701\pi$
$431$	−6.71437	−0.323420	−0.161710	+	0.986838i	$0.551701\pi$
$432$	0	0
$433$	−32.1887	−1.54689	−0.773446	−	0.633862i	$-0.781469\pi$
$433$	−32.1887	−1.54689	−0.773446	+	0.633862i	$0.781469\pi$
$434$	0	0
$435$	29.5163	1.41520
$436$	0	0
$437$	42.6863	2.04196
$438$	0	0
$439$	25.8839	1.23537	0.617685	−	0.786426i	$-0.288071\pi$
$439$	25.8839	1.23537	0.617685	+	0.786426i	$0.288071\pi$
$440$	0	0
$441$	−1.84989	−0.0880900
$442$	0	0
$443$	−3.01136	−0.143074	−0.0715371	−	0.997438i	$-0.522790\pi$
$443$	−3.01136	−0.143074	−0.0715371	+	0.997438i	$0.522790\pi$
$444$	0	0
$445$	−23.6583	−1.12151
$446$	0	0
$447$	4.61502	0.218283
$448$	0	0
$449$	−1.06964	−0.0504795	−0.0252397	−	0.999681i	$-0.508035\pi$
$449$	−1.06964	−0.0504795	−0.0252397	+	0.999681i	$0.508035\pi$
$450$	0	0
$451$	−3.41830	−0.160961
$452$	0	0
$453$	−18.4940	−0.868924
$454$	0	0
$455$	−19.1341	−0.897020
$456$	0	0
$457$	−37.6134	−1.75948	−0.879741	−	0.475453i	$-0.842284\pi$
$457$	−37.6134	−1.75948	−0.879741	+	0.475453i	$0.842284\pi$
$458$	0	0
$459$	−5.20118	−0.242770
$460$	0	0
$461$	6.64704	0.309583	0.154792	−	0.987947i	$-0.450529\pi$
$461$	6.64704	0.309583	0.154792	+	0.987947i	$0.450529\pi$
$462$	0	0
$463$	35.8158	1.66450	0.832250	−	0.554401i	$-0.187052\pi$
$463$	35.8158	1.66450	0.832250	+	0.554401i	$0.187052\pi$
$464$	0	0
$465$	25.9828	1.20492
$466$	0	0
$467$	−25.8823	−1.19769	−0.598846	−	0.800865i	$-0.704374\pi$
$467$	−25.8823	−1.19769	−0.598846	+	0.800865i	$0.704374\pi$
$468$	0	0
$469$	−8.79143	−0.405951
$470$	0	0
$471$	−2.91776	−0.134443
$472$	0	0
$473$	5.42974	0.249660
$474$	0	0
$475$	−28.6997	−1.31683
$476$	0	0
$477$	−14.5266	−0.665127
$478$	0	0
$479$	−28.2837	−1.29231	−0.646157	−	0.763204i	$-0.723625\pi$
$479$	−28.2837	−1.29231	−0.646157	+	0.763204i	$0.723625\pi$
$480$	0	0
$481$	−4.43383	−0.202165
$482$	0	0
$483$	−8.16074	−0.371327
$484$	0	0
$485$	20.7181	0.940760
$486$	0	0
$487$	35.2167	1.59582	0.797910	−	0.602777i	$-0.205939\pi$
$487$	35.2167	1.59582	0.797910	+	0.602777i	$0.205939\pi$
$488$	0	0
$489$	11.2343	0.508034
$490$	0	0
$491$	−2.01739	−0.0910433	−0.0455217	−	0.998963i	$-0.514495\pi$
$491$	−2.01739	−0.0910433	−0.0455217	+	0.998963i	$0.514495\pi$
$492$	0	0
$493$	8.65334	0.389727
$494$	0	0
$495$	−3.32896	−0.149625
$496$	0	0
$497$	15.2894	0.685826
$498$	0	0
$499$	10.8472	0.485587	0.242794	−	0.970078i	$-0.421936\pi$
$499$	10.8472	0.485587	0.242794	+	0.970078i	$0.421936\pi$
$500$	0	0
$501$	0.269757	0.0120519
$502$	0	0
$503$	−11.9677	−0.533613	−0.266806	−	0.963750i	$-0.585968\pi$
$503$	−11.9677	−0.533613	−0.266806	+	0.963750i	$0.585968\pi$
$504$	0	0
$505$	14.1083	0.627812
$506$	0	0
$507$	−24.8705	−1.10454
$508$	0	0
$509$	32.4608	1.43880	0.719400	−	0.694596i	$-0.244417\pi$
$509$	32.4608	1.43880	0.719400	+	0.694596i	$0.244417\pi$
$510$	0	0
$511$	5.75194	0.254451
$512$	0	0
$513$	−29.1763	−1.28817
$514$	0	0
$515$	−7.59390	−0.334627
$516$	0	0
$517$	1.64779	0.0724695
$518$	0	0
$519$	−7.94773	−0.348867
$520$	0	0
$521$	−16.8689	−0.739038	−0.369519	−	0.929223i	$-0.620478\pi$
$521$	−16.8689	−0.739038	−0.369519	+	0.929223i	$0.620478\pi$
$522$	0	0
$523$	36.4122	1.59220	0.796098	−	0.605168i	$-0.206894\pi$
$523$	36.4122	1.59220	0.796098	+	0.605168i	$0.206894\pi$
$524$	0	0
$525$	5.48679	0.239463
$526$	0	0
$527$	7.61740	0.331819
$528$	0	0
$529$	34.9055	1.51763
$530$	0	0
$531$	18.2115	0.790312
$532$	0	0
$533$	−36.3459	−1.57432
$534$	0	0
$535$	−53.8497	−2.32812
$536$	0	0
$537$	19.5590	0.844032
$538$	0	0
$539$	0.565788	0.0243702
$540$	0	0
$541$	21.5010	0.924400	0.462200	−	0.886776i	$-0.347060\pi$
$541$	21.5010	0.924400	0.462200	+	0.886776i	$0.347060\pi$
$542$	0	0
$543$	5.05886	0.217097
$544$	0	0
$545$	8.81002	0.377380
$546$	0	0
$547$	−27.2708	−1.16602	−0.583008	−	0.812466i	$-0.698124\pi$
$547$	−27.2708	−1.16602	−0.583008	+	0.812466i	$0.698124\pi$
$548$	0	0
$549$	21.6152	0.922514
$550$	0	0
$551$	48.5415	2.06794
$552$	0	0
$553$	−5.13920	−0.218541
$554$	0	0
$555$	2.51396	0.106712
$556$	0	0
$557$	−10.3029	−0.436546	−0.218273	−	0.975888i	$-0.570042\pi$
$557$	−10.3029	−0.436546	−0.218273	+	0.975888i	$0.570042\pi$
$558$	0	0
$559$	57.7331	2.44185
$560$	0	0
$561$	0.606769	0.0256178
$562$	0	0
$563$	20.2550	0.853648	0.426824	−	0.904335i	$-0.359632\pi$
$563$	20.2550	0.853648	0.426824	+	0.904335i	$0.359632\pi$
$564$	0	0
$565$	56.0261	2.35704
$566$	0	0
$567$	0.0282405	0.00118599
$568$	0	0
$569$	−16.2918	−0.682988	−0.341494	−	0.939884i	$-0.610933\pi$
$569$	−16.2918	−0.682988	−0.341494	+	0.939884i	$0.610933\pi$
$570$	0	0
$571$	36.5486	1.52951	0.764756	−	0.644320i	$-0.222860\pi$
$571$	36.5486	1.52951	0.764756	+	0.644320i	$0.222860\pi$
$572$	0	0
$573$	−4.62710	−0.193300
$574$	0	0
$575$	−38.9321	−1.62358
$576$	0	0
$577$	−16.0398	−0.667745	−0.333872	−	0.942618i	$-0.608355\pi$
$577$	−16.0398	−0.667745	−0.333872	+	0.942618i	$0.608355\pi$
$578$	0	0
$579$	8.19787	0.340692
$580$	0	0
$581$	3.50982	0.145612
$582$	0	0
$583$	4.44295	0.184008
$584$	0	0
$585$	−35.3960	−1.46344
$586$	0	0
$587$	23.8841	0.985802	0.492901	−	0.870085i	$-0.335936\pi$
$587$	23.8841	0.985802	0.492901	+	0.870085i	$0.335936\pi$
$588$	0	0
$589$	42.7303	1.76067
$590$	0	0
$591$	22.4616	0.923948
$592$	0	0
$593$	14.9443	0.613690	0.306845	−	0.951759i	$-0.400727\pi$
$593$	14.9443	0.613690	0.306845	+	0.951759i	$0.400727\pi$
$594$	0	0
$595$	3.18060	0.130392
$596$	0	0
$597$	−0.100432	−0.00411042
$598$	0	0
$599$	11.1611	0.456029	0.228014	−	0.973658i	$-0.426777\pi$
$599$	11.1611	0.456029	0.228014	+	0.973658i	$0.426777\pi$
$600$	0	0
$601$	23.9318	0.976198	0.488099	−	0.872788i	$-0.337691\pi$
$601$	23.9318	0.976198	0.488099	+	0.872788i	$0.337691\pi$
$602$	0	0
$603$	−16.2632	−0.662288
$604$	0	0
$605$	−33.9684	−1.38101
$606$	0	0
$607$	10.9074	0.442717	0.221359	−	0.975192i	$-0.428951\pi$
$607$	10.9074	0.442717	0.221359	+	0.975192i	$0.428951\pi$
$608$	0	0
$609$	−9.28012	−0.376049
$610$	0	0
$611$	17.5205	0.708803
$612$	0	0
$613$	−15.6985	−0.634056	−0.317028	−	0.948416i	$-0.602685\pi$
$613$	−15.6985	−0.634056	−0.317028	+	0.948416i	$0.602685\pi$
$614$	0	0
$615$	20.6080	0.830994
$616$	0	0
$617$	32.8872	1.32399	0.661995	−	0.749508i	$-0.269710\pi$
$617$	32.8872	1.32399	0.661995	+	0.749508i	$0.269710\pi$
$618$	0	0
$619$	19.8331	0.797159	0.398580	−	0.917134i	$-0.369503\pi$
$619$	19.8331	0.797159	0.398580	+	0.917134i	$0.369503\pi$
$620$	0	0
$621$	−39.5787	−1.58824
$622$	0	0
$623$	7.43833	0.298010
$624$	0	0
$625$	−24.4054	−0.976218
$626$	0	0
$627$	3.40371	0.135931
$628$	0	0
$629$	0.737022	0.0293870
$630$	0	0
$631$	29.8673	1.18900	0.594499	−	0.804096i	$-0.297350\pi$
$631$	29.8673	1.18900	0.594499	+	0.804096i	$0.297350\pi$
$632$	0	0
$633$	28.3376	1.12632
$634$	0	0
$635$	24.3629	0.966813
$636$	0	0
$637$	6.01588	0.238358
$638$	0	0
$639$	28.2838	1.11889
$640$	0	0
$641$	−41.4059	−1.63543	−0.817717	−	0.575620i	$-0.804761\pi$
$641$	−41.4059	−1.63543	−0.817717	+	0.575620i	$0.804761\pi$
$642$	0	0
$643$	43.7843	1.72668	0.863341	−	0.504620i	$-0.168367\pi$
$643$	43.7843	1.72668	0.863341	+	0.504620i	$0.168367\pi$
$644$	0	0
$645$	−32.7344	−1.28892
$646$	0	0
$647$	0.708087	0.0278378	0.0139189	−	0.999903i	$-0.495569\pi$
$647$	0.708087	0.0278378	0.0139189	+	0.999903i	$0.495569\pi$
$648$	0	0
$649$	−5.56998	−0.218641
$650$	0	0
$651$	−8.16914	−0.320174
$652$	0	0
$653$	−20.1332	−0.787872	−0.393936	−	0.919138i	$-0.628887\pi$
$653$	−20.1332	−0.787872	−0.393936	+	0.919138i	$0.628887\pi$
$654$	0	0
$655$	56.1978	2.19583
$656$	0	0
$657$	10.6404	0.415123
$658$	0	0
$659$	4.34634	0.169310	0.0846548	−	0.996410i	$-0.473021\pi$
$659$	4.34634	0.169310	0.0846548	+	0.996410i	$0.473021\pi$
$660$	0	0
$661$	9.79150	0.380845	0.190423	−	0.981702i	$-0.439014\pi$
$661$	9.79150	0.380845	0.190423	+	0.981702i	$0.439014\pi$
$662$	0	0
$663$	6.45162	0.250560
$664$	0	0
$665$	17.8418	0.691874
$666$	0	0
$667$	65.8482	2.54965
$668$	0	0
$669$	7.17921	0.277564
$670$	0	0
$671$	−6.61099	−0.255215
$672$	0	0
$673$	15.4496	0.595538	0.297769	−	0.954638i	$-0.403758\pi$
$673$	15.4496	0.595538	0.297769	+	0.954638i	$0.403758\pi$
$674$	0	0
$675$	26.6103	1.02423
$676$	0	0
$677$	−6.37751	−0.245108	−0.122554	−	0.992462i	$-0.539108\pi$
$677$	−6.37751	−0.245108	−0.122554	+	0.992462i	$0.539108\pi$
$678$	0	0
$679$	−6.51390	−0.249980
$680$	0	0
$681$	−10.8896	−0.417292
$682$	0	0
$683$	33.4023	1.27810	0.639051	−	0.769164i	$-0.279327\pi$
$683$	33.4023	1.27810	0.639051	+	0.769164i	$0.279327\pi$
$684$	0	0
$685$	3.01729	0.115285
$686$	0	0
$687$	25.9946	0.991754
$688$	0	0
$689$	47.2408	1.79973
$690$	0	0
$691$	−6.17478	−0.234900	−0.117450	−	0.993079i	$-0.537472\pi$
$691$	−6.17478	−0.234900	−0.117450	+	0.993079i	$0.537472\pi$
$692$	0	0
$693$	1.04664	0.0397587
$694$	0	0
$695$	−24.9029	−0.944619
$696$	0	0
$697$	6.04166	0.228844
$698$	0	0
$699$	20.8420	0.788316
$700$	0	0
$701$	−33.2071	−1.25422	−0.627108	−	0.778932i	$-0.715762\pi$
$701$	−33.2071	−1.25422	−0.627108	+	0.778932i	$0.715762\pi$
$702$	0	0
$703$	4.13437	0.155931
$704$	0	0
$705$	−9.93404	−0.374138
$706$	0	0
$707$	−4.43574	−0.166823
$708$	0	0
$709$	−17.3087	−0.650042	−0.325021	−	0.945707i	$-0.605371\pi$
$709$	−17.3087	−0.650042	−0.325021	+	0.945707i	$0.605371\pi$
$710$	0	0
$711$	−9.50695	−0.356539
$712$	0	0
$713$	57.9651	2.17081
$714$	0	0
$715$	10.8258	0.404863
$716$	0	0
$717$	8.56180	0.319746
$718$	0	0
$719$	22.6523	0.844789	0.422394	−	0.906412i	$-0.361190\pi$
$719$	22.6523	0.844789	0.422394	+	0.906412i	$0.361190\pi$
$720$	0	0
$721$	2.38757	0.0889177
$722$	0	0
$723$	−24.4946	−0.910963
$724$	0	0
$725$	−44.2723	−1.64423
$726$	0	0
$727$	20.1743	0.748223	0.374112	−	0.927384i	$-0.377948\pi$
$727$	20.1743	0.748223	0.374112	+	0.927384i	$0.377948\pi$
$728$	0	0
$729$	16.7860	0.621703
$730$	0	0
$731$	−9.59679	−0.354950
$732$	0	0
$733$	32.0124	1.18240	0.591202	−	0.806523i	$-0.298653\pi$
$733$	32.0124	1.18240	0.591202	+	0.806523i	$0.298653\pi$
$734$	0	0
$735$	−3.41098	−0.125816
$736$	0	0
$737$	4.97408	0.183223
$738$	0	0
$739$	−33.2350	−1.22257	−0.611285	−	0.791410i	$-0.709347\pi$
$739$	−33.2350	−1.22257	−0.611285	+	0.791410i	$0.709347\pi$
$740$	0	0
$741$	36.1908	1.32950
$742$	0	0
$743$	39.5480	1.45088	0.725438	−	0.688288i	$-0.241637\pi$
$743$	39.5480	1.45088	0.725438	+	0.688288i	$0.241637\pi$
$744$	0	0
$745$	−13.6871	−0.501458
$746$	0	0
$747$	6.49277	0.237558
$748$	0	0
$749$	16.9307	0.618633
$750$	0	0
$751$	−13.9905	−0.510521	−0.255261	−	0.966872i	$-0.582161\pi$
$751$	−13.9905	−0.510521	−0.255261	+	0.966872i	$0.582161\pi$
$752$	0	0
$753$	−23.2742	−0.848160
$754$	0	0
$755$	54.8492	1.99617
$756$	0	0
$757$	43.9379	1.59695	0.798474	−	0.602029i	$-0.205641\pi$
$757$	43.9379	1.59695	0.798474	+	0.602029i	$0.205641\pi$
$758$	0	0
$759$	4.61725	0.167595
$760$	0	0
$761$	−7.93480	−0.287636	−0.143818	−	0.989604i	$-0.545938\pi$
$761$	−7.93480	−0.287636	−0.143818	+	0.989604i	$0.545938\pi$
$762$	0	0
$763$	−2.76993	−0.100278
$764$	0	0
$765$	5.88376	0.212728
$766$	0	0
$767$	−59.2242	−2.13846
$768$	0	0
$769$	3.46389	0.124911	0.0624555	−	0.998048i	$-0.480107\pi$
$769$	3.46389	0.124911	0.0624555	+	0.998048i	$0.480107\pi$
$770$	0	0
$771$	24.9964	0.900225
$772$	0	0
$773$	36.9208	1.32795	0.663974	−	0.747755i	$-0.268868\pi$
$773$	36.9208	1.32795	0.663974	+	0.747755i	$0.268868\pi$
$774$	0	0
$775$	−38.9722	−1.39992
$776$	0	0
$777$	−0.790406	−0.0283556
$778$	0	0
$779$	33.8911	1.21428
$780$	0	0
$781$	−8.65058	−0.309542
$782$	0	0
$783$	−45.0076	−1.60844
$784$	0	0
$785$	8.65343	0.308854
$786$	0	0
$787$	−27.6999	−0.987396	−0.493698	−	0.869633i	$-0.664355\pi$
$787$	−27.6999	−0.987396	−0.493698	+	0.869633i	$0.664355\pi$
$788$	0	0
$789$	8.81146	0.313696
$790$	0	0
$791$	−17.6149	−0.626315
$792$	0	0
$793$	−70.2930	−2.49618
$794$	0	0
$795$	−26.7853	−0.949978
$796$	0	0
$797$	−0.871164	−0.0308582	−0.0154291	−	0.999881i	$-0.504911\pi$
$797$	−0.871164	−0.0308582	−0.0154291	+	0.999881i	$0.504911\pi$
$798$	0	0
$799$	−2.91237	−0.103032
$800$	0	0
$801$	13.7601	0.486189
$802$	0	0
$803$	−3.25437	−0.114844
$804$	0	0
$805$	24.2030	0.853043
$806$	0	0
$807$	−9.21655	−0.324438
$808$	0	0
$809$	−44.9196	−1.57929	−0.789645	−	0.613564i	$-0.789735\pi$
$809$	−44.9196	−1.57929	−0.789645	+	0.613564i	$0.789735\pi$
$810$	0	0
$811$	−37.1509	−1.30454	−0.652272	−	0.757985i	$-0.726184\pi$
$811$	−37.1509	−1.30454	−0.652272	+	0.757985i	$0.726184\pi$
$812$	0	0
$813$	−0.00285236	−0.000100037 0
$814$	0	0
$815$	−33.3186	−1.16710
$816$	0	0
$817$	−53.8338	−1.88341
$818$	0	0
$819$	11.1287	0.388869
$820$	0	0
$821$	40.4231	1.41078	0.705388	−	0.708821i	$-0.250773\pi$
$821$	40.4231	1.41078	0.705388	+	0.708821i	$0.250773\pi$
$822$	0	0
$823$	17.5691	0.612422	0.306211	−	0.951964i	$-0.400939\pi$
$823$	17.5691	0.612422	0.306211	+	0.951964i	$0.400939\pi$
$824$	0	0
$825$	−3.10436	−0.108080
$826$	0	0
$827$	36.8149	1.28018	0.640089	−	0.768300i	$-0.278897\pi$
$827$	36.8149	1.28018	0.640089	+	0.768300i	$0.278897\pi$
$828$	0	0
$829$	−37.8449	−1.31441	−0.657204	−	0.753713i	$-0.728261\pi$
$829$	−37.8449	−1.31441	−0.657204	+	0.753713i	$0.728261\pi$
$830$	0	0
$831$	17.4892	0.606695
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	−0.800041	−0.0276866
$836$	0	0
$837$	−39.6194	−1.36945
$838$	0	0
$839$	10.3934	0.358819	0.179409	−	0.983774i	$-0.442581\pi$
$839$	10.3934	0.358819	0.179409	+	0.983774i	$0.442581\pi$
$840$	0	0
$841$	45.8803	1.58208
$842$	0	0
$843$	−17.2157	−0.592940
$844$	0	0
$845$	73.7606	2.53744
$846$	0	0
$847$	10.6799	0.366965
$848$	0	0
$849$	−22.3641	−0.767536
$850$	0	0
$851$	5.60841	0.192254
$852$	0	0
$853$	34.9766	1.19758	0.598788	−	0.800908i	$-0.295649\pi$
$853$	34.9766	1.19758	0.598788	+	0.800908i	$0.295649\pi$
$854$	0	0
$855$	33.0053	1.12876
$856$	0	0
$857$	−27.3992	−0.935938	−0.467969	−	0.883745i	$-0.655014\pi$
$857$	−27.3992	−0.935938	−0.467969	+	0.883745i	$0.655014\pi$
$858$	0	0
$859$	19.7022	0.672229	0.336114	−	0.941821i	$-0.390887\pi$
$859$	19.7022	0.672229	0.336114	+	0.941821i	$0.390887\pi$
$860$	0	0
$861$	−6.47927	−0.220813
$862$	0	0
$863$	40.5207	1.37934	0.689670	−	0.724123i	$-0.257755\pi$
$863$	40.5207	1.37934	0.689670	+	0.724123i	$0.257755\pi$
$864$	0	0
$865$	23.5712	0.801446
$866$	0	0
$867$	−1.07243	−0.0364217
$868$	0	0
$869$	2.90770	0.0986368
$870$	0	0
$871$	52.8882	1.79205
$872$	0	0
$873$	−12.0500	−0.407830
$874$	0	0
$875$	−0.369615	−0.0124953
$876$	0	0
$877$	39.1528	1.32210	0.661048	−	0.750344i	$-0.270112\pi$
$877$	39.1528	1.32210	0.661048	+	0.750344i	$0.270112\pi$
$878$	0	0
$879$	5.88449	0.198479
$880$	0	0
$881$	10.6853	0.359998	0.179999	−	0.983667i	$-0.442391\pi$
$881$	10.6853	0.359998	0.179999	+	0.983667i	$0.442391\pi$
$882$	0	0
$883$	−23.0187	−0.774640	−0.387320	−	0.921945i	$-0.626599\pi$
$883$	−23.0187	−0.774640	−0.387320	+	0.921945i	$0.626599\pi$
$884$	0	0
$885$	33.5798	1.12877
$886$	0	0
$887$	−27.5360	−0.924567	−0.462284	−	0.886732i	$-0.652970\pi$
$887$	−27.5360	−0.924567	−0.462284	+	0.886732i	$0.652970\pi$
$888$	0	0
$889$	−7.65985	−0.256903
$890$	0	0
$891$	−0.0159781	−0.000535288 0
$892$	0	0
$893$	−16.3372	−0.546702
$894$	0	0
$895$	−58.0077	−1.93898
$896$	0	0
$897$	49.0940	1.63920
$898$	0	0
$899$	65.9160	2.19842
$900$	0	0
$901$	−7.85269	−0.261611
$902$	0	0
$903$	10.2919	0.342493
$904$	0	0
$905$	−15.0035	−0.498732
$906$	0	0
$907$	20.3518	0.675769	0.337885	−	0.941188i	$-0.390289\pi$
$907$	20.3518	0.675769	0.337885	+	0.941188i	$0.390289\pi$
$908$	0	0
$909$	−8.20563	−0.272164
$910$	0	0
$911$	−57.8477	−1.91658	−0.958290	−	0.285798i	$-0.907741\pi$
$911$	−57.8477	−1.91658	−0.958290	+	0.285798i	$0.907741\pi$
$912$	0	0
$913$	−1.98581	−0.0657207
$914$	0	0
$915$	39.8558	1.31759
$916$	0	0
$917$	−17.6689	−0.583480
$918$	0	0
$919$	21.3389	0.703905	0.351952	−	0.936018i	$-0.385518\pi$
$919$	21.3389	0.703905	0.351952	+	0.936018i	$0.385518\pi$
$920$	0	0
$921$	10.0666	0.331707
$922$	0	0
$923$	−91.9794	−3.02754
$924$	0	0
$925$	−3.77076	−0.123982
$926$	0	0
$927$	4.41674	0.145065
$928$	0	0
$929$	8.02916	0.263428	0.131714	−	0.991288i	$-0.457952\pi$
$929$	8.02916	0.263428	0.131714	+	0.991288i	$0.457952\pi$
$930$	0	0
$931$	−5.60956	−0.183846
$932$	0	0
$933$	−22.7906	−0.746130
$934$	0	0
$935$	−1.79954	−0.0588514
$936$	0	0
$937$	59.4541	1.94228	0.971141	−	0.238507i	$-0.0766580\pi$
$937$	59.4541	1.94228	0.971141	+	0.238507i	$0.0766580\pi$
$938$	0	0
$939$	15.4119	0.502948
$940$	0	0
$941$	8.40071	0.273855	0.136928	−	0.990581i	$-0.456277\pi$
$941$	8.40071	0.273855	0.136928	+	0.990581i	$0.456277\pi$
$942$	0	0
$943$	45.9744	1.49713
$944$	0	0
$945$	−16.5429	−0.538139
$946$	0	0
$947$	−30.3153	−0.985115	−0.492557	−	0.870280i	$-0.663938\pi$
$947$	−30.3153	−0.985115	−0.492557	+	0.870280i	$0.663938\pi$
$948$	0	0
$949$	−34.6029	−1.12326
$950$	0	0
$951$	−26.0843	−0.845840
$952$	0	0
$953$	12.9838	0.420588	0.210294	−	0.977638i	$-0.432558\pi$
$953$	12.9838	0.420588	0.210294	+	0.977638i	$0.432558\pi$
$954$	0	0
$955$	13.7230	0.444065
$956$	0	0
$957$	5.25058	0.169727
$958$	0	0
$959$	−0.948655	−0.0306337
$960$	0	0
$961$	27.0248	0.871767
$962$	0	0
$963$	31.3199	1.00927
$964$	0	0
$965$	−24.3131	−0.782666
$966$	0	0
$967$	10.8013	0.347348	0.173674	−	0.984803i	$-0.444436\pi$
$967$	10.8013	0.347348	0.173674	+	0.984803i	$0.444436\pi$
$968$	0	0
$969$	−6.01588	−0.193258
$970$	0	0
$971$	34.2280	1.09843	0.549215	−	0.835681i	$-0.314927\pi$
$971$	34.2280	1.09843	0.549215	+	0.835681i	$0.314927\pi$
$972$	0	0
$973$	7.82961	0.251006
$974$	0	0
$975$	−33.0078	−1.05710
$976$	0	0
$977$	27.9190	0.893209	0.446604	−	0.894732i	$-0.352633\pi$
$977$	27.9190	0.893209	0.446604	+	0.894732i	$0.352633\pi$
$978$	0	0
$979$	−4.20852	−0.134505
$980$	0	0
$981$	−5.12406	−0.163599
$982$	0	0
$983$	31.5704	1.00694	0.503470	−	0.864013i	$-0.332056\pi$
$983$	31.5704	1.00694	0.503470	+	0.864013i	$0.332056\pi$
$984$	0	0
$985$	−66.6163	−2.12257
$986$	0	0
$987$	3.12332	0.0994165
$988$	0	0
$989$	−73.0274	−2.32214
$990$	0	0
$991$	−13.5078	−0.429088	−0.214544	−	0.976714i	$-0.568827\pi$
$991$	−13.5078	−0.429088	−0.214544	+	0.976714i	$0.568827\pi$
$992$	0	0
$993$	18.6131	0.590668
$994$	0	0
$995$	0.297860	0.00944279
$996$	0	0
$997$	−13.5349	−0.428654	−0.214327	−	0.976762i	$-0.568756\pi$
$997$	−13.5349	−0.428654	−0.214327	+	0.976762i	$0.568756\pi$
$998$	0	0
$999$	−3.83338	−0.121283

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.h.1.3	✓	6
4.3	odd	2		3808.2.a.p.1.4	yes	6
8.3	odd	2		7616.2.a.bu.1.3		6
8.5	even	2		7616.2.a.cc.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.h.1.3	✓	6	1.1	even	1	trivial
3808.2.a.p.1.4	yes	6	4.3	odd	2
7616.2.a.bu.1.3		6	8.3	odd	2
7616.2.a.cc.1.4		6	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.07243 q^{3} +3.18060 q^{5} -1.00000 q^{7} -1.84989 q^{9} +0.565788 q^{11} +6.01588 q^{13} -3.41098 q^{15} -1.00000 q^{17} -5.60956 q^{19} +1.07243 q^{21} -7.60956 q^{23} +5.11621 q^{25} +5.20118 q^{27} -8.65334 q^{29} -7.61740 q^{31} -0.606769 q^{33} -3.18060 q^{35} -0.737022 q^{37} -6.45162 q^{39} -6.04166 q^{41} +9.59679 q^{43} -5.88376 q^{45} +2.91237 q^{47} +1.00000 q^{49} +1.07243 q^{51} +7.85269 q^{53} +1.79954 q^{55} +6.01588 q^{57} -9.84464 q^{59} -11.6846 q^{61} +1.84989 q^{63} +19.1341 q^{65} +8.79143 q^{67} +8.16074 q^{69} -15.2894 q^{71} -5.75194 q^{73} -5.48679 q^{75} -0.565788 q^{77} +5.13920 q^{79} -0.0282405 q^{81} -3.50982 q^{83} -3.18060 q^{85} +9.28012 q^{87} -7.43833 q^{89} -6.01588 q^{91} +8.16914 q^{93} -17.8418 q^{95} +6.51390 q^{97} -1.04664 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} + 4 q^{5} - 6 q^{7} + 8 q^{9} - 8 q^{11} + 8 q^{13} - 6 q^{17} - 6 q^{19} + 4 q^{21} - 18 q^{23} + 12 q^{25} - 22 q^{27} - 8 q^{29} + 4 q^{31} - 6 q^{33} - 4 q^{35} - 8 q^{37} - 14 q^{39}+ \cdots - 28 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 + 4 * q^5 - 6 * q^7 + 8 * q^9 - 8 * q^11 + 8 * q^13 - 6 * q^17 - 6 * q^19 + 4 * q^21 - 18 * q^23 + 12 * q^25 - 22 * q^27 - 8 * q^29 + 4 * q^31 - 6 * q^33 - 4 * q^35 - 8 * q^37 - 14 * q^39 + 16 * q^41 - 12 * q^43 + 4 * q^45 - 18 * q^47 + 6 * q^49 + 4 * q^51 - 2 * q^53 + 8 * q^55 + 8 * q^57 - 16 * q^59 + 6 * q^61 - 8 * q^63 - 22 * q^65 - 12 * q^67 + 16 * q^69 + 2 * q^71 + 8 * q^73 - 24 * q^75 + 8 * q^77 + 18 * q^79 + 18 * q^81 - 28 * q^83 - 4 * q^85 + 2 * q^87 - 2 * q^89 - 8 * q^91 - 32 * q^93 - 26 * q^95 - 18 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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