LMFDB - Embedded newform 208.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,4,Mod(1,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("208.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	208.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:	$12.2723972812$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	208.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{3} -18.0000 q^{5} -20.0000 q^{7} -11.0000 q^{9} +O(q^{10})$

$q-4.00000 q^{3} -18.0000 q^{5} -20.0000 q^{7} -11.0000 q^{9} +48.0000 q^{11} +13.0000 q^{13} +72.0000 q^{15} +66.0000 q^{17} +16.0000 q^{19} +80.0000 q^{21} -168.000 q^{23} +199.000 q^{25} +152.000 q^{27} +6.00000 q^{29} -20.0000 q^{31} -192.000 q^{33} +360.000 q^{35} +254.000 q^{37} -52.0000 q^{39} -390.000 q^{41} +124.000 q^{43} +198.000 q^{45} +468.000 q^{47} +57.0000 q^{49} -264.000 q^{51} +558.000 q^{53} -864.000 q^{55} -64.0000 q^{57} +96.0000 q^{59} -826.000 q^{61} +220.000 q^{63} -234.000 q^{65} +160.000 q^{67} +672.000 q^{69} +420.000 q^{71} +362.000 q^{73} -796.000 q^{75} -960.000 q^{77} -776.000 q^{79} -311.000 q^{81} -1188.00 q^{85} -24.0000 q^{87} +1626.00 q^{89} -260.000 q^{91} +80.0000 q^{93} -288.000 q^{95} -1294.00 q^{97} -528.000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−4.00000	−0.769800	−0.384900	−	0.922958i	$-0.625764\pi$
$3$	−4.00000	−0.769800	−0.384900	+	0.922958i	$0.625764\pi$
$4$	0	0
$5$	−18.0000	−1.60997	−0.804984	−	0.593296i	$-0.797826\pi$
$5$	−18.0000	−1.60997	−0.804984	+	0.593296i	$0.797826\pi$
$6$	0	0
$7$	−20.0000	−1.07990	−0.539949	−	0.841698i	$-0.681557\pi$
$7$	−20.0000	−1.07990	−0.539949	+	0.841698i	$0.681557\pi$
$8$	0	0
$9$	−11.0000	−0.407407
$10$	0	0
$11$	48.0000	1.31569	0.657843	−	0.753155i	$-0.271469\pi$
$11$	48.0000	1.31569	0.657843	+	0.753155i	$0.271469\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	72.0000	1.23935
$16$	0	0
$17$	66.0000	0.941609	0.470804	−	0.882238i	$-0.343964\pi$
$17$	66.0000	0.941609	0.470804	+	0.882238i	$0.343964\pi$
$18$	0	0
$19$	16.0000	0.193192	0.0965961	−	0.995324i	$-0.469204\pi$
$19$	16.0000	0.193192	0.0965961	+	0.995324i	$0.469204\pi$
$20$	0	0
$21$	80.0000	0.831306
$22$	0	0
$23$	−168.000	−1.52306	−0.761531	−	0.648129i	$-0.775552\pi$
$23$	−168.000	−1.52306	−0.761531	+	0.648129i	$0.775552\pi$
$24$	0	0
$25$	199.000	1.59200
$26$	0	0
$27$	152.000	1.08342
$28$	0	0
$29$	6.00000	0.0384197	0.0192099	−	0.999815i	$-0.493885\pi$
$29$	6.00000	0.0384197	0.0192099	+	0.999815i	$0.493885\pi$
$30$	0	0
$31$	−20.0000	−0.115874	−0.0579372	−	0.998320i	$-0.518452\pi$
$31$	−20.0000	−0.115874	−0.0579372	+	0.998320i	$0.518452\pi$
$32$	0	0
$33$	−192.000	−1.01282
$34$	0	0
$35$	360.000	1.73860
$36$	0	0
$37$	254.000	1.12858	0.564288	−	0.825578i	$-0.309151\pi$
$37$	254.000	1.12858	0.564288	+	0.825578i	$0.309151\pi$
$38$	0	0
$39$	−52.0000	−0.213504
$40$	0	0
$41$	−390.000	−1.48556	−0.742778	−	0.669538i	$-0.766492\pi$
$41$	−390.000	−1.48556	−0.742778	+	0.669538i	$0.766492\pi$
$42$	0	0
$43$	124.000	0.439763	0.219882	−	0.975527i	$-0.429433\pi$
$43$	124.000	0.439763	0.219882	+	0.975527i	$0.429433\pi$
$44$	0	0
$45$	198.000	0.655913
$46$	0	0
$47$	468.000	1.45244	0.726221	−	0.687461i	$-0.241275\pi$
$47$	468.000	1.45244	0.726221	+	0.687461i	$0.241275\pi$
$48$	0	0
$49$	57.0000	0.166181
$50$	0	0
$51$	−264.000	−0.724851
$52$	0	0
$53$	558.000	1.44617	0.723087	−	0.690757i	$-0.242723\pi$
$53$	558.000	1.44617	0.723087	+	0.690757i	$0.242723\pi$
$54$	0	0
$55$	−864.000	−2.11821
$56$	0	0
$57$	−64.0000	−0.148719
$58$	0	0
$59$	96.0000	0.211833	0.105916	−	0.994375i	$-0.466222\pi$
$59$	96.0000	0.211833	0.105916	+	0.994375i	$0.466222\pi$
$60$	0	0
$61$	−826.000	−1.73375	−0.866873	−	0.498530i	$-0.833873\pi$
$61$	−826.000	−1.73375	−0.866873	+	0.498530i	$0.833873\pi$
$62$	0	0
$63$	220.000	0.439959
$64$	0	0
$65$	−234.000	−0.446525
$66$	0	0
$67$	160.000	0.291748	0.145874	−	0.989303i	$-0.453401\pi$
$67$	160.000	0.291748	0.145874	+	0.989303i	$0.453401\pi$
$68$	0	0
$69$	672.000	1.17245
$70$	0	0
$71$	420.000	0.702040	0.351020	−	0.936368i	$-0.385835\pi$
$71$	420.000	0.702040	0.351020	+	0.936368i	$0.385835\pi$
$72$	0	0
$73$	362.000	0.580396	0.290198	−	0.956967i	$-0.406279\pi$
$73$	362.000	0.580396	0.290198	+	0.956967i	$0.406279\pi$
$74$	0	0
$75$	−796.000	−1.22552
$76$	0	0
$77$	−960.000	−1.42081
$78$	0	0
$79$	−776.000	−1.10515	−0.552575	−	0.833463i	$-0.686355\pi$
$79$	−776.000	−1.10515	−0.552575	+	0.833463i	$0.686355\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−1188.00	−1.51596
$86$	0	0
$87$	−24.0000	−0.0295755
$88$	0	0
$89$	1626.00	1.93658	0.968290	−	0.249828i	$-0.0803741\pi$
$89$	1626.00	1.93658	0.968290	+	0.249828i	$0.0803741\pi$
$90$	0	0
$91$	−260.000	−0.299510
$92$	0	0
$93$	80.0000	0.0892001
$94$	0	0
$95$	−288.000	−0.311033
$96$	0	0
$97$	−1294.00	−1.35449	−0.677246	−	0.735756i	$-0.736827\pi$
$97$	−1294.00	−1.35449	−0.677246	+	0.735756i	$0.736827\pi$
$98$	0	0
$99$	−528.000	−0.536020
$100$	0	0
$101$	222.000	0.218711	0.109356	−	0.994003i	$-0.465121\pi$
$101$	222.000	0.218711	0.109356	+	0.994003i	$0.465121\pi$
$102$	0	0
$103$	−632.000	−0.604590	−0.302295	−	0.953214i	$-0.597753\pi$
$103$	−632.000	−0.604590	−0.302295	+	0.953214i	$0.597753\pi$
$104$	0	0
$105$	−1440.00	−1.33838
$106$	0	0
$107$	948.000	0.856510	0.428255	−	0.903658i	$-0.359128\pi$
$107$	948.000	0.856510	0.428255	+	0.903658i	$0.359128\pi$
$108$	0	0
$109$	758.000	0.666085	0.333042	−	0.942912i	$-0.391925\pi$
$109$	758.000	0.666085	0.333042	+	0.942912i	$0.391925\pi$
$110$	0	0
$111$	−1016.00	−0.868779
$112$	0	0
$113$	642.000	0.534463	0.267231	−	0.963632i	$-0.413891\pi$
$113$	642.000	0.534463	0.267231	+	0.963632i	$0.413891\pi$
$114$	0	0
$115$	3024.00	2.45208
$116$	0	0
$117$	−143.000	−0.112994
$118$	0	0
$119$	−1320.00	−1.01684
$120$	0	0
$121$	973.000	0.731029
$122$	0	0
$123$	1560.00	1.14358
$124$	0	0
$125$	−1332.00	−0.953102
$126$	0	0
$127$	880.000	0.614861	0.307431	−	0.951571i	$-0.400531\pi$
$127$	880.000	0.614861	0.307431	+	0.951571i	$0.400531\pi$
$128$	0	0
$129$	−496.000	−0.338530
$130$	0	0
$131$	−324.000	−0.216092	−0.108046	−	0.994146i	$-0.534459\pi$
$131$	−324.000	−0.216092	−0.108046	+	0.994146i	$0.534459\pi$
$132$	0	0
$133$	−320.000	−0.208628
$134$	0	0
$135$	−2736.00	−1.74428
$136$	0	0
$137$	1722.00	1.07387	0.536936	−	0.843623i	$-0.319582\pi$
$137$	1722.00	1.07387	0.536936	+	0.843623i	$0.319582\pi$
$138$	0	0
$139$	340.000	0.207471	0.103735	−	0.994605i	$-0.466921\pi$
$139$	340.000	0.207471	0.103735	+	0.994605i	$0.466921\pi$
$140$	0	0
$141$	−1872.00	−1.11809
$142$	0	0
$143$	624.000	0.364906
$144$	0	0
$145$	−108.000	−0.0618546
$146$	0	0
$147$	−228.000	−0.127926
$148$	0	0
$149$	750.000	0.412365	0.206183	−	0.978514i	$-0.433896\pi$
$149$	750.000	0.412365	0.206183	+	0.978514i	$0.433896\pi$
$150$	0	0
$151$	−1748.00	−0.942054	−0.471027	−	0.882119i	$-0.656117\pi$
$151$	−1748.00	−0.942054	−0.471027	+	0.882119i	$0.656117\pi$
$152$	0	0
$153$	−726.000	−0.383618
$154$	0	0
$155$	360.000	0.186554
$156$	0	0
$157$	614.000	0.312118	0.156059	−	0.987748i	$-0.450121\pi$
$157$	614.000	0.312118	0.156059	+	0.987748i	$0.450121\pi$
$158$	0	0
$159$	−2232.00	−1.11326
$160$	0	0
$161$	3360.00	1.64475
$162$	0	0
$163$	808.000	0.388267	0.194133	−	0.980975i	$-0.437811\pi$
$163$	808.000	0.388267	0.194133	+	0.980975i	$0.437811\pi$
$164$	0	0
$165$	3456.00	1.63060
$166$	0	0
$167$	2028.00	0.939709	0.469854	−	0.882744i	$-0.344306\pi$
$167$	2028.00	0.939709	0.469854	+	0.882744i	$0.344306\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−176.000	−0.0787079
$172$	0	0
$173$	−1194.00	−0.524729	−0.262365	−	0.964969i	$-0.584502\pi$
$173$	−1194.00	−0.524729	−0.262365	+	0.964969i	$0.584502\pi$
$174$	0	0
$175$	−3980.00	−1.71920
$176$	0	0
$177$	−384.000	−0.163069
$178$	0	0
$179$	2820.00	1.17752	0.588762	−	0.808307i	$-0.299616\pi$
$179$	2820.00	1.17752	0.588762	+	0.808307i	$0.299616\pi$
$180$	0	0
$181$	−754.000	−0.309637	−0.154819	−	0.987943i	$-0.549479\pi$
$181$	−754.000	−0.309637	−0.154819	+	0.987943i	$0.549479\pi$
$182$	0	0
$183$	3304.00	1.33464
$184$	0	0
$185$	−4572.00	−1.81697
$186$	0	0
$187$	3168.00	1.23886
$188$	0	0
$189$	−3040.00	−1.16999
$190$	0	0
$191$	2328.00	0.881928	0.440964	−	0.897525i	$-0.354637\pi$
$191$	2328.00	0.881928	0.440964	+	0.897525i	$0.354637\pi$
$192$	0	0
$193$	2450.00	0.913756	0.456878	−	0.889529i	$-0.348968\pi$
$193$	2450.00	0.913756	0.456878	+	0.889529i	$0.348968\pi$
$194$	0	0
$195$	936.000	0.343735
$196$	0	0
$197$	4542.00	1.64266	0.821330	−	0.570453i	$-0.193232\pi$
$197$	4542.00	1.64266	0.821330	+	0.570453i	$0.193232\pi$
$198$	0	0
$199$	664.000	0.236531	0.118266	−	0.992982i	$-0.462267\pi$
$199$	664.000	0.236531	0.118266	+	0.992982i	$0.462267\pi$
$200$	0	0
$201$	−640.000	−0.224588
$202$	0	0
$203$	−120.000	−0.0414894
$204$	0	0
$205$	7020.00	2.39170
$206$	0	0
$207$	1848.00	0.620507
$208$	0	0
$209$	768.000	0.254180
$210$	0	0
$211$	4156.00	1.35598	0.677988	−	0.735073i	$-0.262852\pi$
$211$	4156.00	1.35598	0.677988	+	0.735073i	$0.262852\pi$
$212$	0	0
$213$	−1680.00	−0.540431
$214$	0	0
$215$	−2232.00	−0.708005
$216$	0	0
$217$	400.000	0.125133
$218$	0	0
$219$	−1448.00	−0.446789
$220$	0	0
$221$	858.000	0.261155
$222$	0	0
$223$	3292.00	0.988559	0.494279	−	0.869303i	$-0.335432\pi$
$223$	3292.00	0.988559	0.494279	+	0.869303i	$0.335432\pi$
$224$	0	0
$225$	−2189.00	−0.648593
$226$	0	0
$227$	−2352.00	−0.687699	−0.343850	−	0.939025i	$-0.611731\pi$
$227$	−2352.00	−0.687699	−0.343850	+	0.939025i	$0.611731\pi$
$228$	0	0
$229$	686.000	0.197957	0.0989785	−	0.995090i	$-0.468442\pi$
$229$	686.000	0.197957	0.0989785	+	0.995090i	$0.468442\pi$
$230$	0	0
$231$	3840.00	1.09374
$232$	0	0
$233$	1818.00	0.511164	0.255582	−	0.966787i	$-0.417733\pi$
$233$	1818.00	0.511164	0.255582	+	0.966787i	$0.417733\pi$
$234$	0	0
$235$	−8424.00	−2.33839
$236$	0	0
$237$	3104.00	0.850745
$238$	0	0
$239$	540.000	0.146149	0.0730747	−	0.997326i	$-0.476719\pi$
$239$	540.000	0.146149	0.0730747	+	0.997326i	$0.476719\pi$
$240$	0	0
$241$	−862.000	−0.230400	−0.115200	−	0.993342i	$-0.536751\pi$
$241$	−862.000	−0.230400	−0.115200	+	0.993342i	$0.536751\pi$
$242$	0	0
$243$	−2860.00	−0.755017
$244$	0	0
$245$	−1026.00	−0.267546
$246$	0	0
$247$	208.000	0.0535819
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4836.00	−1.21612	−0.608059	−	0.793892i	$-0.708052\pi$
$251$	−4836.00	−1.21612	−0.608059	+	0.793892i	$0.708052\pi$
$252$	0	0
$253$	−8064.00	−2.00387
$254$	0	0
$255$	4752.00	1.16699
$256$	0	0
$257$	1410.00	0.342231	0.171116	−	0.985251i	$-0.445263\pi$
$257$	1410.00	0.342231	0.171116	+	0.985251i	$0.445263\pi$
$258$	0	0
$259$	−5080.00	−1.21875
$260$	0	0
$261$	−66.0000	−0.0156525
$262$	0	0
$263$	−8304.00	−1.94695	−0.973473	−	0.228804i	$-0.926519\pi$
$263$	−8304.00	−1.94695	−0.973473	+	0.228804i	$0.926519\pi$
$264$	0	0
$265$	−10044.0	−2.32829
$266$	0	0
$267$	−6504.00	−1.49078
$268$	0	0
$269$	−2634.00	−0.597018	−0.298509	−	0.954407i	$-0.596489\pi$
$269$	−2634.00	−0.597018	−0.298509	+	0.954407i	$0.596489\pi$
$270$	0	0
$271$	−7436.00	−1.66681	−0.833404	−	0.552665i	$-0.813611\pi$
$271$	−7436.00	−1.66681	−0.833404	+	0.552665i	$0.813611\pi$
$272$	0	0
$273$	1040.00	0.230563
$274$	0	0
$275$	9552.00	2.09457
$276$	0	0
$277$	−5074.00	−1.10060	−0.550302	−	0.834966i	$-0.685487\pi$
$277$	−5074.00	−1.10060	−0.550302	+	0.834966i	$0.685487\pi$
$278$	0	0
$279$	220.000	0.0472081
$280$	0	0
$281$	−1638.00	−0.347740	−0.173870	−	0.984769i	$-0.555627\pi$
$281$	−1638.00	−0.347740	−0.173870	+	0.984769i	$0.555627\pi$
$282$	0	0
$283$	4588.00	0.963704	0.481852	−	0.876253i	$-0.339964\pi$
$283$	4588.00	0.963704	0.481852	+	0.876253i	$0.339964\pi$
$284$	0	0
$285$	1152.00	0.239434
$286$	0	0
$287$	7800.00	1.60425
$288$	0	0
$289$	−557.000	−0.113373
$290$	0	0
$291$	5176.00	1.04269
$292$	0	0
$293$	−2850.00	−0.568255	−0.284128	−	0.958786i	$-0.591704\pi$
$293$	−2850.00	−0.568255	−0.284128	+	0.958786i	$0.591704\pi$
$294$	0	0
$295$	−1728.00	−0.341044
$296$	0	0
$297$	7296.00	1.42544
$298$	0	0
$299$	−2184.00	−0.422421
$300$	0	0
$301$	−2480.00	−0.474900
$302$	0	0
$303$	−888.000	−0.168364
$304$	0	0
$305$	14868.0	2.79128
$306$	0	0
$307$	−8120.00	−1.50955	−0.754777	−	0.655982i	$-0.772255\pi$
$307$	−8120.00	−1.50955	−0.754777	+	0.655982i	$0.772255\pi$
$308$	0	0
$309$	2528.00	0.465414
$310$	0	0
$311$	3528.00	0.643262	0.321631	−	0.946865i	$-0.395769\pi$
$311$	3528.00	0.643262	0.321631	+	0.946865i	$0.395769\pi$
$312$	0	0
$313$	−6982.00	−1.26085	−0.630425	−	0.776250i	$-0.717119\pi$
$313$	−6982.00	−1.26085	−0.630425	+	0.776250i	$0.717119\pi$
$314$	0	0
$315$	−3960.00	−0.708320
$316$	0	0
$317$	9270.00	1.64245	0.821223	−	0.570608i	$-0.193292\pi$
$317$	9270.00	1.64245	0.821223	+	0.570608i	$0.193292\pi$
$318$	0	0
$319$	288.000	0.0505483
$320$	0	0
$321$	−3792.00	−0.659342
$322$	0	0
$323$	1056.00	0.181911
$324$	0	0
$325$	2587.00	0.441541
$326$	0	0
$327$	−3032.00	−0.512752
$328$	0	0
$329$	−9360.00	−1.56849
$330$	0	0
$331$	7720.00	1.28196	0.640981	−	0.767557i	$-0.278528\pi$
$331$	7720.00	1.28196	0.640981	+	0.767557i	$0.278528\pi$
$332$	0	0
$333$	−2794.00	−0.459791
$334$	0	0
$335$	−2880.00	−0.469705
$336$	0	0
$337$	−1726.00	−0.278995	−0.139497	−	0.990222i	$-0.544549\pi$
$337$	−1726.00	−0.278995	−0.139497	+	0.990222i	$0.544549\pi$
$338$	0	0
$339$	−2568.00	−0.411430
$340$	0	0
$341$	−960.000	−0.152454
$342$	0	0
$343$	5720.00	0.900440
$344$	0	0
$345$	−12096.0	−1.88761
$346$	0	0
$347$	4020.00	0.621916	0.310958	−	0.950424i	$-0.399350\pi$
$347$	4020.00	0.621916	0.310958	+	0.950424i	$0.399350\pi$
$348$	0	0
$349$	1910.00	0.292951	0.146476	−	0.989214i	$-0.453207\pi$
$349$	1910.00	0.292951	0.146476	+	0.989214i	$0.453207\pi$
$350$	0	0
$351$	1976.00	0.300487
$352$	0	0
$353$	5442.00	0.820534	0.410267	−	0.911965i	$-0.365436\pi$
$353$	5442.00	0.820534	0.410267	+	0.911965i	$0.365436\pi$
$354$	0	0
$355$	−7560.00	−1.13026
$356$	0	0
$357$	5280.00	0.782765
$358$	0	0
$359$	9324.00	1.37076	0.685379	−	0.728187i	$-0.259637\pi$
$359$	9324.00	1.37076	0.685379	+	0.728187i	$0.259637\pi$
$360$	0	0
$361$	−6603.00	−0.962677
$362$	0	0
$363$	−3892.00	−0.562747
$364$	0	0
$365$	−6516.00	−0.934419
$366$	0	0
$367$	−4520.00	−0.642894	−0.321447	−	0.946928i	$-0.604169\pi$
$367$	−4520.00	−0.642894	−0.321447	+	0.946928i	$0.604169\pi$
$368$	0	0
$369$	4290.00	0.605226
$370$	0	0
$371$	−11160.0	−1.56172
$372$	0	0
$373$	−5938.00	−0.824284	−0.412142	−	0.911120i	$-0.635219\pi$
$373$	−5938.00	−0.824284	−0.412142	+	0.911120i	$0.635219\pi$
$374$	0	0
$375$	5328.00	0.733698
$376$	0	0
$377$	78.0000	0.0106557
$378$	0	0
$379$	−2216.00	−0.300338	−0.150169	−	0.988660i	$-0.547982\pi$
$379$	−2216.00	−0.300338	−0.150169	+	0.988660i	$0.547982\pi$
$380$	0	0
$381$	−3520.00	−0.473320
$382$	0	0
$383$	−3828.00	−0.510709	−0.255355	−	0.966847i	$-0.582192\pi$
$383$	−3828.00	−0.510709	−0.255355	+	0.966847i	$0.582192\pi$
$384$	0	0
$385$	17280.0	2.28746
$386$	0	0
$387$	−1364.00	−0.179163
$388$	0	0
$389$	5022.00	0.654564	0.327282	−	0.944927i	$-0.393867\pi$
$389$	5022.00	0.654564	0.327282	+	0.944927i	$0.393867\pi$
$390$	0	0
$391$	−11088.0	−1.43413
$392$	0	0
$393$	1296.00	0.166347
$394$	0	0
$395$	13968.0	1.77926
$396$	0	0
$397$	6086.00	0.769389	0.384695	−	0.923044i	$-0.374307\pi$
$397$	6086.00	0.769389	0.384695	+	0.923044i	$0.374307\pi$
$398$	0	0
$399$	1280.00	0.160602
$400$	0	0
$401$	1122.00	0.139726	0.0698629	−	0.997557i	$-0.477744\pi$
$401$	1122.00	0.139726	0.0698629	+	0.997557i	$0.477744\pi$
$402$	0	0
$403$	−260.000	−0.0321378
$404$	0	0
$405$	5598.00	0.686832
$406$	0	0
$407$	12192.0	1.48485
$408$	0	0
$409$	362.000	0.0437647	0.0218823	−	0.999761i	$-0.493034\pi$
$409$	362.000	0.0437647	0.0218823	+	0.999761i	$0.493034\pi$
$410$	0	0
$411$	−6888.00	−0.826667
$412$	0	0
$413$	−1920.00	−0.228758
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−1360.00	−0.159711
$418$	0	0
$419$	2316.00	0.270033	0.135017	−	0.990843i	$-0.456891\pi$
$419$	2316.00	0.270033	0.135017	+	0.990843i	$0.456891\pi$
$420$	0	0
$421$	5006.00	0.579519	0.289760	−	0.957099i	$-0.406425\pi$
$421$	5006.00	0.579519	0.289760	+	0.957099i	$0.406425\pi$
$422$	0	0
$423$	−5148.00	−0.591736
$424$	0	0
$425$	13134.0	1.49904
$426$	0	0
$427$	16520.0	1.87227
$428$	0	0
$429$	−2496.00	−0.280904
$430$	0	0
$431$	11244.0	1.25662	0.628311	−	0.777962i	$-0.283746\pi$
$431$	11244.0	1.25662	0.628311	+	0.777962i	$0.283746\pi$
$432$	0	0
$433$	13106.0	1.45458	0.727291	−	0.686329i	$-0.240779\pi$
$433$	13106.0	1.45458	0.727291	+	0.686329i	$0.240779\pi$
$434$	0	0
$435$	432.000	0.0476157
$436$	0	0
$437$	−2688.00	−0.294244
$438$	0	0
$439$	13480.0	1.46552	0.732762	−	0.680485i	$-0.238231\pi$
$439$	13480.0	1.46552	0.732762	+	0.680485i	$0.238231\pi$
$440$	0	0
$441$	−627.000	−0.0677033
$442$	0	0
$443$	−14508.0	−1.55597	−0.777986	−	0.628281i	$-0.783759\pi$
$443$	−14508.0	−1.55597	−0.777986	+	0.628281i	$0.783759\pi$
$444$	0	0
$445$	−29268.0	−3.11783
$446$	0	0
$447$	−3000.00	−0.317439
$448$	0	0
$449$	−7566.00	−0.795237	−0.397619	−	0.917551i	$-0.630163\pi$
$449$	−7566.00	−0.795237	−0.397619	+	0.917551i	$0.630163\pi$
$450$	0	0
$451$	−18720.0	−1.95452
$452$	0	0
$453$	6992.00	0.725194
$454$	0	0
$455$	4680.00	0.482202
$456$	0	0
$457$	5402.00	0.552943	0.276471	−	0.961022i	$-0.410835\pi$
$457$	5402.00	0.552943	0.276471	+	0.961022i	$0.410835\pi$
$458$	0	0
$459$	10032.0	1.02016
$460$	0	0
$461$	−4650.00	−0.469788	−0.234894	−	0.972021i	$-0.575474\pi$
$461$	−4650.00	−0.469788	−0.234894	+	0.972021i	$0.575474\pi$
$462$	0	0
$463$	17188.0	1.72526	0.862629	−	0.505838i	$-0.168817\pi$
$463$	17188.0	1.72526	0.862629	+	0.505838i	$0.168817\pi$
$464$	0	0
$465$	−1440.00	−0.143609
$466$	0	0
$467$	14580.0	1.44472	0.722358	−	0.691520i	$-0.243059\pi$
$467$	14580.0	1.44472	0.722358	+	0.691520i	$0.243059\pi$
$468$	0	0
$469$	−3200.00	−0.315058
$470$	0	0
$471$	−2456.00	−0.240269
$472$	0	0
$473$	5952.00	0.578590
$474$	0	0
$475$	3184.00	0.307562
$476$	0	0
$477$	−6138.00	−0.589182
$478$	0	0
$479$	2100.00	0.200316	0.100158	−	0.994972i	$-0.468065\pi$
$479$	2100.00	0.200316	0.100158	+	0.994972i	$0.468065\pi$
$480$	0	0
$481$	3302.00	0.313011
$482$	0	0
$483$	−13440.0	−1.26613
$484$	0	0
$485$	23292.0	2.18069
$486$	0	0
$487$	12004.0	1.11695	0.558473	−	0.829522i	$-0.311387\pi$
$487$	12004.0	1.11695	0.558473	+	0.829522i	$0.311387\pi$
$488$	0	0
$489$	−3232.00	−0.298888
$490$	0	0
$491$	−13236.0	−1.21656	−0.608281	−	0.793721i	$-0.708141\pi$
$491$	−13236.0	−1.21656	−0.608281	+	0.793721i	$0.708141\pi$
$492$	0	0
$493$	396.000	0.0361764
$494$	0	0
$495$	9504.00	0.862976
$496$	0	0
$497$	−8400.00	−0.758132
$498$	0	0
$499$	−18560.0	−1.66505	−0.832525	−	0.553988i	$-0.813105\pi$
$499$	−18560.0	−1.66505	−0.832525	+	0.553988i	$0.813105\pi$
$500$	0	0
$501$	−8112.00	−0.723388
$502$	0	0
$503$	−12432.0	−1.10202	−0.551009	−	0.834499i	$-0.685757\pi$
$503$	−12432.0	−1.10202	−0.551009	+	0.834499i	$0.685757\pi$
$504$	0	0
$505$	−3996.00	−0.352118
$506$	0	0
$507$	−676.000	−0.0592154
$508$	0	0
$509$	−7914.00	−0.689159	−0.344579	−	0.938757i	$-0.611978\pi$
$509$	−7914.00	−0.689159	−0.344579	+	0.938757i	$0.611978\pi$
$510$	0	0
$511$	−7240.00	−0.626769
$512$	0	0
$513$	2432.00	0.209309
$514$	0	0
$515$	11376.0	0.973372
$516$	0	0
$517$	22464.0	1.91096
$518$	0	0
$519$	4776.00	0.403937
$520$	0	0
$521$	−14742.0	−1.23965	−0.619826	−	0.784739i	$-0.712797\pi$
$521$	−14742.0	−1.23965	−0.619826	+	0.784739i	$0.712797\pi$
$522$	0	0
$523$	2500.00	0.209020	0.104510	−	0.994524i	$-0.466673\pi$
$523$	2500.00	0.209020	0.104510	+	0.994524i	$0.466673\pi$
$524$	0	0
$525$	15920.0	1.32344
$526$	0	0
$527$	−1320.00	−0.109108
$528$	0	0
$529$	16057.0	1.31972
$530$	0	0
$531$	−1056.00	−0.0863023
$532$	0	0
$533$	−5070.00	−0.412019
$534$	0	0
$535$	−17064.0	−1.37896
$536$	0	0
$537$	−11280.0	−0.906458
$538$	0	0
$539$	2736.00	0.218642
$540$	0	0
$541$	17894.0	1.42204	0.711020	−	0.703172i	$-0.248234\pi$
$541$	17894.0	1.42204	0.711020	+	0.703172i	$0.248234\pi$
$542$	0	0
$543$	3016.00	0.238359
$544$	0	0
$545$	−13644.0	−1.07238
$546$	0	0
$547$	−17444.0	−1.36353	−0.681766	−	0.731571i	$-0.738788\pi$
$547$	−17444.0	−1.36353	−0.681766	+	0.731571i	$0.738788\pi$
$548$	0	0
$549$	9086.00	0.706341
$550$	0	0
$551$	96.0000	0.00742239
$552$	0	0
$553$	15520.0	1.19345
$554$	0	0
$555$	18288.0	1.39871
$556$	0	0
$557$	−1002.00	−0.0762228	−0.0381114	−	0.999273i	$-0.512134\pi$
$557$	−1002.00	−0.0762228	−0.0381114	+	0.999273i	$0.512134\pi$
$558$	0	0
$559$	1612.00	0.121968
$560$	0	0
$561$	−12672.0	−0.953676
$562$	0	0
$563$	4740.00	0.354826	0.177413	−	0.984136i	$-0.443227\pi$
$563$	4740.00	0.354826	0.177413	+	0.984136i	$0.443227\pi$
$564$	0	0
$565$	−11556.0	−0.860468
$566$	0	0
$567$	6220.00	0.460697
$568$	0	0
$569$	8682.00	0.639663	0.319832	−	0.947474i	$-0.396374\pi$
$569$	8682.00	0.639663	0.319832	+	0.947474i	$0.396374\pi$
$570$	0	0
$571$	−5492.00	−0.402510	−0.201255	−	0.979539i	$-0.564502\pi$
$571$	−5492.00	−0.402510	−0.201255	+	0.979539i	$0.564502\pi$
$572$	0	0
$573$	−9312.00	−0.678908
$574$	0	0
$575$	−33432.0	−2.42471
$576$	0	0
$577$	−17278.0	−1.24661	−0.623304	−	0.781980i	$-0.714210\pi$
$577$	−17278.0	−1.24661	−0.623304	+	0.781980i	$0.714210\pi$
$578$	0	0
$579$	−9800.00	−0.703410
$580$	0	0
$581$	0	0
$582$	0	0
$583$	26784.0	1.90271
$584$	0	0
$585$	2574.00	0.181918
$586$	0	0
$587$	15240.0	1.07159	0.535794	−	0.844349i	$-0.320012\pi$
$587$	15240.0	1.07159	0.535794	+	0.844349i	$0.320012\pi$
$588$	0	0
$589$	−320.000	−0.0223860
$590$	0	0
$591$	−18168.0	−1.26452
$592$	0	0
$593$	−9198.00	−0.636959	−0.318479	−	0.947930i	$-0.603172\pi$
$593$	−9198.00	−0.636959	−0.318479	+	0.947930i	$0.603172\pi$
$594$	0	0
$595$	23760.0	1.63708
$596$	0	0
$597$	−2656.00	−0.182082
$598$	0	0
$599$	−7200.00	−0.491125	−0.245563	−	0.969381i	$-0.578973\pi$
$599$	−7200.00	−0.491125	−0.245563	+	0.969381i	$0.578973\pi$
$600$	0	0
$601$	−14470.0	−0.982103	−0.491051	−	0.871131i	$-0.663387\pi$
$601$	−14470.0	−0.982103	−0.491051	+	0.871131i	$0.663387\pi$
$602$	0	0
$603$	−1760.00	−0.118860
$604$	0	0
$605$	−17514.0	−1.17693
$606$	0	0
$607$	20824.0	1.39245	0.696227	−	0.717821i	$-0.254861\pi$
$607$	20824.0	1.39245	0.696227	+	0.717821i	$0.254861\pi$
$608$	0	0
$609$	480.000	0.0319386
$610$	0	0
$611$	6084.00	0.402835
$612$	0	0
$613$	8606.00	0.567036	0.283518	−	0.958967i	$-0.408498\pi$
$613$	8606.00	0.567036	0.283518	+	0.958967i	$0.408498\pi$
$614$	0	0
$615$	−28080.0	−1.84113
$616$	0	0
$617$	−9654.00	−0.629912	−0.314956	−	0.949106i	$-0.601990\pi$
$617$	−9654.00	−0.629912	−0.314956	+	0.949106i	$0.601990\pi$
$618$	0	0
$619$	−14384.0	−0.933993	−0.466997	−	0.884259i	$-0.654664\pi$
$619$	−14384.0	−0.933993	−0.466997	+	0.884259i	$0.654664\pi$
$620$	0	0
$621$	−25536.0	−1.65012
$622$	0	0
$623$	−32520.0	−2.09131
$624$	0	0
$625$	−899.000	−0.0575360
$626$	0	0
$627$	−3072.00	−0.195668
$628$	0	0
$629$	16764.0	1.06268
$630$	0	0
$631$	−1460.00	−0.0921104	−0.0460552	−	0.998939i	$-0.514665\pi$
$631$	−1460.00	−0.0921104	−0.0460552	+	0.998939i	$0.514665\pi$
$632$	0	0
$633$	−16624.0	−1.04383
$634$	0	0
$635$	−15840.0	−0.989907
$636$	0	0
$637$	741.000	0.0460902
$638$	0	0
$639$	−4620.00	−0.286016
$640$	0	0
$641$	−12462.0	−0.767893	−0.383946	−	0.923355i	$-0.625435\pi$
$641$	−12462.0	−0.767893	−0.383946	+	0.923355i	$0.625435\pi$
$642$	0	0
$643$	9952.00	0.610371	0.305186	−	0.952293i	$-0.401282\pi$
$643$	9952.00	0.610371	0.305186	+	0.952293i	$0.401282\pi$
$644$	0	0
$645$	8928.00	0.545023
$646$	0	0
$647$	−26088.0	−1.58520	−0.792601	−	0.609741i	$-0.791273\pi$
$647$	−26088.0	−1.58520	−0.792601	+	0.609741i	$0.791273\pi$
$648$	0	0
$649$	4608.00	0.278705
$650$	0	0
$651$	−1600.00	−0.0963271
$652$	0	0
$653$	3894.00	0.233360	0.116680	−	0.993170i	$-0.462775\pi$
$653$	3894.00	0.233360	0.116680	+	0.993170i	$0.462775\pi$
$654$	0	0
$655$	5832.00	0.347901
$656$	0	0
$657$	−3982.00	−0.236458
$658$	0	0
$659$	23820.0	1.40804	0.704018	−	0.710182i	$-0.251388\pi$
$659$	23820.0	1.40804	0.704018	+	0.710182i	$0.251388\pi$
$660$	0	0
$661$	7742.00	0.455566	0.227783	−	0.973712i	$-0.426852\pi$
$661$	7742.00	0.455566	0.227783	+	0.973712i	$0.426852\pi$
$662$	0	0
$663$	−3432.00	−0.201037
$664$	0	0
$665$	5760.00	0.335885
$666$	0	0
$667$	−1008.00	−0.0585156
$668$	0	0
$669$	−13168.0	−0.760993
$670$	0	0
$671$	−39648.0	−2.28106
$672$	0	0
$673$	21170.0	1.21255	0.606273	−	0.795257i	$-0.292664\pi$
$673$	21170.0	1.21255	0.606273	+	0.795257i	$0.292664\pi$
$674$	0	0
$675$	30248.0	1.72481
$676$	0	0
$677$	17982.0	1.02083	0.510417	−	0.859927i	$-0.329491\pi$
$677$	17982.0	1.02083	0.510417	+	0.859927i	$0.329491\pi$
$678$	0	0
$679$	25880.0	1.46271
$680$	0	0
$681$	9408.00	0.529391
$682$	0	0
$683$	17520.0	0.981529	0.490764	−	0.871292i	$-0.336718\pi$
$683$	17520.0	0.981529	0.490764	+	0.871292i	$0.336718\pi$
$684$	0	0
$685$	−30996.0	−1.72890
$686$	0	0
$687$	−2744.00	−0.152387
$688$	0	0
$689$	7254.00	0.401096
$690$	0	0
$691$	28096.0	1.54678	0.773388	−	0.633933i	$-0.218560\pi$
$691$	28096.0	1.54678	0.773388	+	0.633933i	$0.218560\pi$
$692$	0	0
$693$	10560.0	0.578847
$694$	0	0
$695$	−6120.00	−0.334021
$696$	0	0
$697$	−25740.0	−1.39881
$698$	0	0
$699$	−7272.00	−0.393494
$700$	0	0
$701$	18342.0	0.988256	0.494128	−	0.869389i	$-0.335487\pi$
$701$	18342.0	0.988256	0.494128	+	0.869389i	$0.335487\pi$
$702$	0	0
$703$	4064.00	0.218032
$704$	0	0
$705$	33696.0	1.80009
$706$	0	0
$707$	−4440.00	−0.236186
$708$	0	0
$709$	−37330.0	−1.97737	−0.988687	−	0.149996i	$-0.952074\pi$
$709$	−37330.0	−1.97737	−0.988687	+	0.149996i	$0.952074\pi$
$710$	0	0
$711$	8536.00	0.450246
$712$	0	0
$713$	3360.00	0.176484
$714$	0	0
$715$	−11232.0	−0.587487
$716$	0	0
$717$	−2160.00	−0.112506
$718$	0	0
$719$	−4800.00	−0.248971	−0.124485	−	0.992221i	$-0.539728\pi$
$719$	−4800.00	−0.248971	−0.124485	+	0.992221i	$0.539728\pi$
$720$	0	0
$721$	12640.0	0.652896
$722$	0	0
$723$	3448.00	0.177362
$724$	0	0
$725$	1194.00	0.0611642
$726$	0	0
$727$	−23960.0	−1.22232	−0.611160	−	0.791507i	$-0.709297\pi$
$727$	−23960.0	−1.22232	−0.611160	+	0.791507i	$0.709297\pi$
$728$	0	0
$729$	19837.0	1.00782
$730$	0	0
$731$	8184.00	0.414085
$732$	0	0
$733$	−21418.0	−1.07925	−0.539626	−	0.841905i	$-0.681434\pi$
$733$	−21418.0	−1.07925	−0.539626	+	0.841905i	$0.681434\pi$
$734$	0	0
$735$	4104.00	0.205957
$736$	0	0
$737$	7680.00	0.383849
$738$	0	0
$739$	−5384.00	−0.268002	−0.134001	−	0.990981i	$-0.542783\pi$
$739$	−5384.00	−0.268002	−0.134001	+	0.990981i	$0.542783\pi$
$740$	0	0
$741$	−832.000	−0.0412473
$742$	0	0
$743$	1524.00	0.0752492	0.0376246	−	0.999292i	$-0.488021\pi$
$743$	1524.00	0.0752492	0.0376246	+	0.999292i	$0.488021\pi$
$744$	0	0
$745$	−13500.0	−0.663895
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−18960.0	−0.924944
$750$	0	0
$751$	19312.0	0.938355	0.469178	−	0.883104i	$-0.344550\pi$
$751$	19312.0	0.938355	0.469178	+	0.883104i	$0.344550\pi$
$752$	0	0
$753$	19344.0	0.936168
$754$	0	0
$755$	31464.0	1.51668
$756$	0	0
$757$	35246.0	1.69226	0.846128	−	0.532980i	$-0.178928\pi$
$757$	35246.0	1.69226	0.846128	+	0.532980i	$0.178928\pi$
$758$	0	0
$759$	32256.0	1.54258
$760$	0	0
$761$	12522.0	0.596481	0.298241	−	0.954491i	$-0.403600\pi$
$761$	12522.0	0.596481	0.298241	+	0.954491i	$0.403600\pi$
$762$	0	0
$763$	−15160.0	−0.719304
$764$	0	0
$765$	13068.0	0.617614
$766$	0	0
$767$	1248.00	0.0587518
$768$	0	0
$769$	24050.0	1.12778	0.563892	−	0.825849i	$-0.309304\pi$
$769$	24050.0	1.12778	0.563892	+	0.825849i	$0.309304\pi$
$770$	0	0
$771$	−5640.00	−0.263450
$772$	0	0
$773$	25806.0	1.20075	0.600373	−	0.799720i	$-0.295019\pi$
$773$	25806.0	1.20075	0.600373	+	0.799720i	$0.295019\pi$
$774$	0	0
$775$	−3980.00	−0.184472
$776$	0	0
$777$	20320.0	0.938193
$778$	0	0
$779$	−6240.00	−0.286998
$780$	0	0
$781$	20160.0	0.923664
$782$	0	0
$783$	912.000	0.0416248
$784$	0	0
$785$	−11052.0	−0.502500
$786$	0	0
$787$	−18632.0	−0.843912	−0.421956	−	0.906616i	$-0.638656\pi$
$787$	−18632.0	−0.843912	−0.421956	+	0.906616i	$0.638656\pi$
$788$	0	0
$789$	33216.0	1.49876
$790$	0	0
$791$	−12840.0	−0.577165
$792$	0	0
$793$	−10738.0	−0.480854
$794$	0	0
$795$	40176.0	1.79232
$796$	0	0
$797$	−16314.0	−0.725058	−0.362529	−	0.931972i	$-0.618087\pi$
$797$	−16314.0	−0.725058	−0.362529	+	0.931972i	$0.618087\pi$
$798$	0	0
$799$	30888.0	1.36763
$800$	0	0
$801$	−17886.0	−0.788977
$802$	0	0
$803$	17376.0	0.763619
$804$	0	0
$805$	−60480.0	−2.64800
$806$	0	0
$807$	10536.0	0.459585
$808$	0	0
$809$	−4278.00	−0.185917	−0.0929583	−	0.995670i	$-0.529632\pi$
$809$	−4278.00	−0.185917	−0.0929583	+	0.995670i	$0.529632\pi$
$810$	0	0
$811$	−18632.0	−0.806730	−0.403365	−	0.915039i	$-0.632159\pi$
$811$	−18632.0	−0.806730	−0.403365	+	0.915039i	$0.632159\pi$
$812$	0	0
$813$	29744.0	1.28311
$814$	0	0
$815$	−14544.0	−0.625097
$816$	0	0
$817$	1984.00	0.0849588
$818$	0	0
$819$	2860.00	0.122023
$820$	0	0
$821$	−46434.0	−1.97388	−0.986941	−	0.161080i	$-0.948502\pi$
$821$	−46434.0	−1.97388	−0.986941	+	0.161080i	$0.948502\pi$
$822$	0	0
$823$	−24968.0	−1.05751	−0.528754	−	0.848775i	$-0.677341\pi$
$823$	−24968.0	−1.05751	−0.528754	+	0.848775i	$0.677341\pi$
$824$	0	0
$825$	−38208.0	−1.61240
$826$	0	0
$827$	14112.0	0.593376	0.296688	−	0.954974i	$-0.404118\pi$
$827$	14112.0	0.593376	0.296688	+	0.954974i	$0.404118\pi$
$828$	0	0
$829$	37190.0	1.55810	0.779048	−	0.626964i	$-0.215703\pi$
$829$	37190.0	1.55810	0.779048	+	0.626964i	$0.215703\pi$
$830$	0	0
$831$	20296.0	0.847245
$832$	0	0
$833$	3762.00	0.156477
$834$	0	0
$835$	−36504.0	−1.51290
$836$	0	0
$837$	−3040.00	−0.125541
$838$	0	0
$839$	−1380.00	−0.0567853	−0.0283927	−	0.999597i	$-0.509039\pi$
$839$	−1380.00	−0.0567853	−0.0283927	+	0.999597i	$0.509039\pi$
$840$	0	0
$841$	−24353.0	−0.998524
$842$	0	0
$843$	6552.00	0.267690
$844$	0	0
$845$	−3042.00	−0.123844
$846$	0	0
$847$	−19460.0	−0.789437
$848$	0	0
$849$	−18352.0	−0.741860
$850$	0	0
$851$	−42672.0	−1.71889
$852$	0	0
$853$	5150.00	0.206721	0.103360	−	0.994644i	$-0.467041\pi$
$853$	5150.00	0.206721	0.103360	+	0.994644i	$0.467041\pi$
$854$	0	0
$855$	3168.00	0.126717
$856$	0	0
$857$	23562.0	0.939163	0.469581	−	0.882889i	$-0.344405\pi$
$857$	23562.0	0.939163	0.469581	+	0.882889i	$0.344405\pi$
$858$	0	0
$859$	34612.0	1.37479	0.687396	−	0.726283i	$-0.258754\pi$
$859$	34612.0	1.37479	0.687396	+	0.726283i	$0.258754\pi$
$860$	0	0
$861$	−31200.0	−1.23495
$862$	0	0
$863$	14940.0	0.589297	0.294649	−	0.955606i	$-0.404797\pi$
$863$	14940.0	0.589297	0.294649	+	0.955606i	$0.404797\pi$
$864$	0	0
$865$	21492.0	0.844798
$866$	0	0
$867$	2228.00	0.0872743
$868$	0	0
$869$	−37248.0	−1.45403
$870$	0	0
$871$	2080.00	0.0809163
$872$	0	0
$873$	14234.0	0.551830
$874$	0	0
$875$	26640.0	1.02925
$876$	0	0
$877$	17030.0	0.655715	0.327858	−	0.944727i	$-0.393673\pi$
$877$	17030.0	0.655715	0.327858	+	0.944727i	$0.393673\pi$
$878$	0	0
$879$	11400.0	0.437443
$880$	0	0
$881$	−27246.0	−1.04193	−0.520965	−	0.853578i	$-0.674428\pi$
$881$	−27246.0	−1.04193	−0.520965	+	0.853578i	$0.674428\pi$
$882$	0	0
$883$	8260.00	0.314803	0.157402	−	0.987535i	$-0.449688\pi$
$883$	8260.00	0.314803	0.157402	+	0.987535i	$0.449688\pi$
$884$	0	0
$885$	6912.00	0.262536
$886$	0	0
$887$	43392.0	1.64257	0.821286	−	0.570517i	$-0.193257\pi$
$887$	43392.0	1.64257	0.821286	+	0.570517i	$0.193257\pi$
$888$	0	0
$889$	−17600.0	−0.663988
$890$	0	0
$891$	−14928.0	−0.561287
$892$	0	0
$893$	7488.00	0.280601
$894$	0	0
$895$	−50760.0	−1.89578
$896$	0	0
$897$	8736.00	0.325180
$898$	0	0
$899$	−120.000	−0.00445186
$900$	0	0
$901$	36828.0	1.36173
$902$	0	0
$903$	9920.00	0.365578
$904$	0	0
$905$	13572.0	0.498507
$906$	0	0
$907$	−1028.00	−0.0376342	−0.0188171	−	0.999823i	$-0.505990\pi$
$907$	−1028.00	−0.0376342	−0.0188171	+	0.999823i	$0.505990\pi$
$908$	0	0
$909$	−2442.00	−0.0891045
$910$	0	0
$911$	21816.0	0.793410	0.396705	−	0.917946i	$-0.370154\pi$
$911$	21816.0	0.793410	0.396705	+	0.917946i	$0.370154\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−59472.0	−2.14873
$916$	0	0
$917$	6480.00	0.233357
$918$	0	0
$919$	−42752.0	−1.53456	−0.767279	−	0.641314i	$-0.778390\pi$
$919$	−42752.0	−1.53456	−0.767279	+	0.641314i	$0.778390\pi$
$920$	0	0
$921$	32480.0	1.16205
$922$	0	0
$923$	5460.00	0.194711
$924$	0	0
$925$	50546.0	1.79669
$926$	0	0
$927$	6952.00	0.246315
$928$	0	0
$929$	24978.0	0.882133	0.441067	−	0.897474i	$-0.354600\pi$
$929$	24978.0	0.882133	0.441067	+	0.897474i	$0.354600\pi$
$930$	0	0
$931$	912.000	0.0321048
$932$	0	0
$933$	−14112.0	−0.495183
$934$	0	0
$935$	−57024.0	−1.99453
$936$	0	0
$937$	33914.0	1.18241	0.591207	−	0.806520i	$-0.298652\pi$
$937$	33914.0	1.18241	0.591207	+	0.806520i	$0.298652\pi$
$938$	0	0
$939$	27928.0	0.970603
$940$	0	0
$941$	−8442.00	−0.292456	−0.146228	−	0.989251i	$-0.546713\pi$
$941$	−8442.00	−0.292456	−0.146228	+	0.989251i	$0.546713\pi$
$942$	0	0
$943$	65520.0	2.26259
$944$	0	0
$945$	54720.0	1.88364
$946$	0	0
$947$	43176.0	1.48155	0.740777	−	0.671751i	$-0.234458\pi$
$947$	43176.0	1.48155	0.740777	+	0.671751i	$0.234458\pi$
$948$	0	0
$949$	4706.00	0.160973
$950$	0	0
$951$	−37080.0	−1.26435
$952$	0	0
$953$	−43926.0	−1.49308	−0.746539	−	0.665342i	$-0.768286\pi$
$953$	−43926.0	−1.49308	−0.746539	+	0.665342i	$0.768286\pi$
$954$	0	0
$955$	−41904.0	−1.41988
$956$	0	0
$957$	−1152.00	−0.0389121
$958$	0	0
$959$	−34440.0	−1.15967
$960$	0	0
$961$	−29391.0	−0.986573
$962$	0	0
$963$	−10428.0	−0.348949
$964$	0	0
$965$	−44100.0	−1.47112
$966$	0	0
$967$	11572.0	0.384830	0.192415	−	0.981314i	$-0.438368\pi$
$967$	11572.0	0.384830	0.192415	+	0.981314i	$0.438368\pi$
$968$	0	0
$969$	−4224.00	−0.140036
$970$	0	0
$971$	−14412.0	−0.476316	−0.238158	−	0.971226i	$-0.576544\pi$
$971$	−14412.0	−0.476316	−0.238158	+	0.971226i	$0.576544\pi$
$972$	0	0
$973$	−6800.00	−0.224047
$974$	0	0
$975$	−10348.0	−0.339899
$976$	0	0
$977$	25602.0	0.838363	0.419181	−	0.907902i	$-0.362317\pi$
$977$	25602.0	0.838363	0.419181	+	0.907902i	$0.362317\pi$
$978$	0	0
$979$	78048.0	2.54793
$980$	0	0
$981$	−8338.00	−0.271368
$982$	0	0
$983$	−32148.0	−1.04309	−0.521547	−	0.853222i	$-0.674645\pi$
$983$	−32148.0	−1.04309	−0.521547	+	0.853222i	$0.674645\pi$
$984$	0	0
$985$	−81756.0	−2.64463
$986$	0	0
$987$	37440.0	1.20742
$988$	0	0
$989$	−20832.0	−0.669787
$990$	0	0
$991$	9736.00	0.312083	0.156041	−	0.987751i	$-0.450127\pi$
$991$	9736.00	0.312083	0.156041	+	0.987751i	$0.450127\pi$
$992$	0	0
$993$	−30880.0	−0.986855
$994$	0	0
$995$	−11952.0	−0.380808
$996$	0	0
$997$	6878.00	0.218484	0.109242	−	0.994015i	$-0.465158\pi$
$997$	6878.00	0.218484	0.109242	+	0.994015i	$0.465158\pi$
$998$	0	0
$999$	38608.0	1.22273

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.4.a.b.1.1		1
3.2	odd	2		1872.4.a.q.1.1		1
4.3	odd	2		26.4.a.c.1.1	✓	1
8.3	odd	2		832.4.a.d.1.1		1
8.5	even	2		832.4.a.o.1.1		1
12.11	even	2		234.4.a.e.1.1		1
20.3	even	4		650.4.b.f.599.1		2
20.7	even	4		650.4.b.f.599.2		2
20.19	odd	2		650.4.a.b.1.1		1
28.27	even	2		1274.4.a.d.1.1		1
52.3	odd	6		338.4.c.a.191.1		2
52.7	even	12		338.4.e.a.23.2		4
52.11	even	12		338.4.e.a.147.1		4
52.15	even	12		338.4.e.a.147.2		4
52.19	even	12		338.4.e.a.23.1		4
52.23	odd	6		338.4.c.e.191.1		2
52.31	even	4		338.4.b.d.337.1		2
52.35	odd	6		338.4.c.a.315.1		2
52.43	odd	6		338.4.c.e.315.1		2
52.47	even	4		338.4.b.d.337.2		2
52.51	odd	2		338.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.4.a.c.1.1	✓	1	4.3	odd	2
208.4.a.b.1.1		1	1.1	even	1	trivial
234.4.a.e.1.1		1	12.11	even	2
338.4.a.c.1.1		1	52.51	odd	2
338.4.b.d.337.1		2	52.31	even	4
338.4.b.d.337.2		2	52.47	even	4
338.4.c.a.191.1		2	52.3	odd	6
338.4.c.a.315.1		2	52.35	odd	6
338.4.c.e.191.1		2	52.23	odd	6
338.4.c.e.315.1		2	52.43	odd	6
338.4.e.a.23.1		4	52.19	even	12
338.4.e.a.23.2		4	52.7	even	12
338.4.e.a.147.1		4	52.11	even	12
338.4.e.a.147.2		4	52.15	even	12
650.4.a.b.1.1		1	20.19	odd	2
650.4.b.f.599.1		2	20.3	even	4
650.4.b.f.599.2		2	20.7	even	4
832.4.a.d.1.1		1	8.3	odd	2
832.4.a.o.1.1		1	8.5	even	2
1274.4.a.d.1.1		1	28.27	even	2
1872.4.a.q.1.1		1	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	208.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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