LMFDB - Embedded newform 3808.2.a.j.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:	6.6.4022000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} + 10x^{3} + 14x^{2} - 8x - 4 $ x^6 - 2x^5 - 7x^4 + 10x^3 + 14x^2 - 8*x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.67090$ 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)

$f(q)$	$=$	$q-1.43012 q^{3} -2.70357 q^{5} -1.00000 q^{7} -0.954755 q^{9} -0.773277 q^{11} +2.59298 q^{13} +3.86643 q^{15} +1.00000 q^{17} +1.10588 q^{19} +1.43012 q^{21} -3.52209 q^{23} +2.30928 q^{25} +5.65578 q^{27} +6.41268 q^{29} +3.65361 q^{31} +1.10588 q^{33} +2.70357 q^{35} +5.57801 q^{37} -3.70828 q^{39} -1.97112 q^{41} +9.97124 q^{43} +2.58124 q^{45} -10.3268 q^{47} +1.00000 q^{49} -1.43012 q^{51} -8.49223 q^{53} +2.09061 q^{55} -1.58154 q^{57} -3.75831 q^{59} +14.4499 q^{61} +0.954755 q^{63} -7.01030 q^{65} -5.38635 q^{67} +5.03701 q^{69} -12.6548 q^{71} -2.56635 q^{73} -3.30254 q^{75} +0.773277 q^{77} +4.96421 q^{79} -5.22418 q^{81} +2.17677 q^{83} -2.70357 q^{85} -9.17091 q^{87} +9.82445 q^{89} -2.59298 q^{91} -5.22511 q^{93} -2.98982 q^{95} +16.3133 q^{97} +0.738290 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 4 q^{9} - 2 q^{11} - 4 q^{13} - 8 q^{15} + 6 q^{17} + 2 q^{19} + 2 q^{21} - 10 q^{23} - 2 q^{27} + 2 q^{29} - 12 q^{31} + 2 q^{33} - 2 q^{35} + 2 q^{37} - 10 q^{39} - 8 q^{43}+ \cdots - 2 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 4 * q^9 - 2 * q^11 - 4 * q^13 - 8 * q^15 + 6 * q^17 + 2 * q^19 + 2 * q^21 - 10 * q^23 - 2 * q^27 + 2 * q^29 - 12 * q^31 + 2 * q^33 - 2 * q^35 + 2 * q^37 - 10 * q^39 - 8 * q^43 + 14 * q^45 - 22 * q^47 + 6 * q^49 - 2 * q^51 - 6 * q^53 - 24 * q^55 - 8 * q^57 - 8 * q^59 - 4 * q^61 - 4 * q^63 - 2 * q^65 + 8 * q^67 + 4 * q^69 - 18 * q^71 + 8 * q^73 - 30 * q^75 + 2 * q^77 - 14 * q^79 - 26 * q^81 + 2 * q^85 - 22 * q^87 + 2 * q^89 + 4 * q^91 - 8 * q^93 - 2 * q^95 + 2 * q^97 - 2 * 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
$3$	−1.43012	−0.825681	−0.412840	−	0.910803i	$-0.635463\pi$
$3$	−1.43012	−0.825681	−0.412840	+	0.910803i	$0.635463\pi$
$4$	0	0
$5$	−2.70357	−1.20907	−0.604536	−	0.796578i	$-0.706641\pi$
$5$	−2.70357	−1.20907	−0.604536	+	0.796578i	$0.706641\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−0.954755	−0.318252
$10$	0	0
$11$	−0.773277	−0.233152	−0.116576	−	0.993182i	$-0.537192\pi$
$11$	−0.773277	−0.233152	−0.116576	+	0.993182i	$0.537192\pi$
$12$	0	0
$13$	2.59298	0.719163	0.359582	−	0.933114i	$-0.382919\pi$
$13$	2.59298	0.719163	0.359582	+	0.933114i	$0.382919\pi$
$14$	0	0
$15$	3.86643	0.998307
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	1.10588	0.253706	0.126853	−	0.991922i	$-0.459512\pi$
$19$	1.10588	0.253706	0.126853	+	0.991922i	$0.459512\pi$
$20$	0	0
$21$	1.43012	0.312078
$22$	0	0
$23$	−3.52209	−0.734406	−0.367203	−	0.930141i	$-0.619685\pi$
$23$	−3.52209	−0.734406	−0.367203	+	0.930141i	$0.619685\pi$
$24$	0	0
$25$	2.30928	0.461855
$26$	0	0
$27$	5.65578	1.08845
$28$	0	0
$29$	6.41268	1.19081	0.595403	−	0.803427i	$-0.296992\pi$
$29$	6.41268	1.19081	0.595403	+	0.803427i	$0.296992\pi$
$30$	0	0
$31$	3.65361	0.656209	0.328104	−	0.944642i	$-0.393590\pi$
$31$	3.65361	0.656209	0.328104	+	0.944642i	$0.393590\pi$
$32$	0	0
$33$	1.10588	0.192509
$34$	0	0
$35$	2.70357	0.456986
$36$	0	0
$37$	5.57801	0.917020	0.458510	−	0.888689i	$-0.348383\pi$
$37$	5.57801	0.917020	0.458510	+	0.888689i	$0.348383\pi$
$38$	0	0
$39$	−3.70828	−0.593799
$40$	0	0
$41$	−1.97112	−0.307838	−0.153919	−	0.988083i	$-0.549189\pi$
$41$	−1.97112	−0.307838	−0.153919	+	0.988083i	$0.549189\pi$
$42$	0	0
$43$	9.97124	1.52060	0.760300	−	0.649572i	$-0.225052\pi$
$43$	9.97124	1.52060	0.760300	+	0.649572i	$0.225052\pi$
$44$	0	0
$45$	2.58124	0.384789
$46$	0	0
$47$	−10.3268	−1.50632	−0.753161	−	0.657836i	$-0.771472\pi$
$47$	−10.3268	−1.50632	−0.753161	+	0.657836i	$0.771472\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.43012	−0.200257
$52$	0	0
$53$	−8.49223	−1.16650	−0.583249	−	0.812293i	$-0.698219\pi$
$53$	−8.49223	−1.16650	−0.583249	+	0.812293i	$0.698219\pi$
$54$	0	0
$55$	2.09061	0.281897
$56$	0	0
$57$	−1.58154	−0.209480
$58$	0	0
$59$	−3.75831	−0.489290	−0.244645	−	0.969613i	$-0.578671\pi$
$59$	−3.75831	−0.489290	−0.244645	+	0.969613i	$0.578671\pi$
$60$	0	0
$61$	14.4499	1.85013	0.925063	−	0.379813i	$-0.124012\pi$
$61$	14.4499	1.85013	0.925063	+	0.379813i	$0.124012\pi$
$62$	0	0
$63$	0.954755	0.120288
$64$	0	0
$65$	−7.01030	−0.869520
$66$	0	0
$67$	−5.38635	−0.658048	−0.329024	−	0.944322i	$-0.606720\pi$
$67$	−5.38635	−0.658048	−0.329024	+	0.944322i	$0.606720\pi$
$68$	0	0
$69$	5.03701	0.606385
$70$	0	0
$71$	−12.6548	−1.50185	−0.750924	−	0.660389i	$-0.770392\pi$
$71$	−12.6548	−1.50185	−0.750924	+	0.660389i	$0.770392\pi$
$72$	0	0
$73$	−2.56635	−0.300369	−0.150185	−	0.988658i	$-0.547987\pi$
$73$	−2.56635	−0.300369	−0.150185	+	0.988658i	$0.547987\pi$
$74$	0	0
$75$	−3.30254	−0.381345
$76$	0	0
$77$	0.773277	0.0881230
$78$	0	0
$79$	4.96421	0.558517	0.279259	−	0.960216i	$-0.409911\pi$
$79$	4.96421	0.558517	0.279259	+	0.960216i	$0.409911\pi$
$80$	0	0
$81$	−5.22418	−0.580464
$82$	0	0
$83$	2.17677	0.238932	0.119466	−	0.992838i	$-0.461882\pi$
$83$	2.17677	0.238932	0.119466	+	0.992838i	$0.461882\pi$
$84$	0	0
$85$	−2.70357	−0.293243
$86$	0	0
$87$	−9.17091	−0.983225
$88$	0	0
$89$	9.82445	1.04139	0.520695	−	0.853743i	$-0.325673\pi$
$89$	9.82445	1.04139	0.520695	+	0.853743i	$0.325673\pi$
$90$	0	0
$91$	−2.59298	−0.271818
$92$	0	0
$93$	−5.22511	−0.541819
$94$	0	0
$95$	−2.98982	−0.306749
$96$	0	0
$97$	16.3133	1.65637	0.828184	−	0.560457i	$-0.189374\pi$
$97$	16.3133	1.65637	0.828184	+	0.560457i	$0.189374\pi$
$98$	0	0
$99$	0.738290	0.0742009
$100$	0	0
$101$	−3.84781	−0.382871	−0.191436	−	0.981505i	$-0.561314\pi$
$101$	−3.84781	−0.382871	−0.191436	+	0.981505i	$0.561314\pi$
$102$	0	0
$103$	−16.8604	−1.66130	−0.830650	−	0.556795i	$-0.812031\pi$
$103$	−16.8604	−1.66130	−0.830650	+	0.556795i	$0.812031\pi$
$104$	0	0
$105$	−3.86643	−0.377325
$106$	0	0
$107$	2.69106	0.260155	0.130077	−	0.991504i	$-0.458477\pi$
$107$	2.69106	0.260155	0.130077	+	0.991504i	$0.458477\pi$
$108$	0	0
$109$	−13.5659	−1.29938	−0.649690	−	0.760199i	$-0.725101\pi$
$109$	−13.5659	−1.29938	−0.649690	+	0.760199i	$0.725101\pi$
$110$	0	0
$111$	−7.97723	−0.757166
$112$	0	0
$113$	17.1133	1.60988	0.804941	−	0.593355i	$-0.202197\pi$
$113$	17.1133	1.60988	0.804941	+	0.593355i	$0.202197\pi$
$114$	0	0
$115$	9.52220	0.887950
$116$	0	0
$117$	−2.47566	−0.228875
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−10.4020	−0.945640
$122$	0	0
$123$	2.81894	0.254176
$124$	0	0
$125$	7.27455	0.650656
$126$	0	0
$127$	−19.9603	−1.77119	−0.885595	−	0.464458i	$-0.846249\pi$
$127$	−19.9603	−1.77119	−0.885595	+	0.464458i	$0.846249\pi$
$128$	0	0
$129$	−14.2601	−1.25553
$130$	0	0
$131$	−1.12571	−0.0983538	−0.0491769	−	0.998790i	$-0.515660\pi$
$131$	−1.12571	−0.0983538	−0.0491769	+	0.998790i	$0.515660\pi$
$132$	0	0
$133$	−1.10588	−0.0958918
$134$	0	0
$135$	−15.2908	−1.31602
$136$	0	0
$137$	−0.123315	−0.0105355	−0.00526774	−	0.999986i	$-0.501677\pi$
$137$	−0.123315	−0.0105355	−0.00526774	+	0.999986i	$0.501677\pi$
$138$	0	0
$139$	−7.04265	−0.597350	−0.298675	−	0.954355i	$-0.596545\pi$
$139$	−7.04265	−0.597350	−0.298675	+	0.954355i	$0.596545\pi$
$140$	0	0
$141$	14.7686	1.24374
$142$	0	0
$143$	−2.00509	−0.167674
$144$	0	0
$145$	−17.3371	−1.43977
$146$	0	0
$147$	−1.43012	−0.117954
$148$	0	0
$149$	5.58666	0.457677	0.228838	−	0.973464i	$-0.426507\pi$
$149$	5.58666	0.457677	0.228838	+	0.973464i	$0.426507\pi$
$150$	0	0
$151$	3.39593	0.276357	0.138179	−	0.990407i	$-0.455875\pi$
$151$	3.39593	0.276357	0.138179	+	0.990407i	$0.455875\pi$
$152$	0	0
$153$	−0.954755	−0.0771874
$154$	0	0
$155$	−9.87779	−0.793403
$156$	0	0
$157$	−8.73362	−0.697019	−0.348509	−	0.937305i	$-0.613312\pi$
$157$	−8.73362	−0.697019	−0.348509	+	0.937305i	$0.613312\pi$
$158$	0	0
$159$	12.1449	0.963155
$160$	0	0
$161$	3.52209	0.277579
$162$	0	0
$163$	−23.8538	−1.86837	−0.934186	−	0.356786i	$-0.883872\pi$
$163$	−23.8538	−1.86837	−0.934186	+	0.356786i	$0.883872\pi$
$164$	0	0
$165$	−2.98982	−0.232757
$166$	0	0
$167$	−1.79550	−0.138940	−0.0694698	−	0.997584i	$-0.522131\pi$
$167$	−1.79550	−0.138940	−0.0694698	+	0.997584i	$0.522131\pi$
$168$	0	0
$169$	−6.27645	−0.482804
$170$	0	0
$171$	−1.05584	−0.0807424
$172$	0	0
$173$	3.73363	0.283862	0.141931	−	0.989877i	$-0.454669\pi$
$173$	3.73363	0.283862	0.141931	+	0.989877i	$0.454669\pi$
$174$	0	0
$175$	−2.30928	−0.174565
$176$	0	0
$177$	5.37484	0.403998
$178$	0	0
$179$	−12.4290	−0.928988	−0.464494	−	0.885576i	$-0.653764\pi$
$179$	−12.4290	−0.928988	−0.464494	+	0.885576i	$0.653764\pi$
$180$	0	0
$181$	−10.4181	−0.774372	−0.387186	−	0.922002i	$-0.626553\pi$
$181$	−10.4181	−0.774372	−0.387186	+	0.922002i	$0.626553\pi$
$182$	0	0
$183$	−20.6652	−1.52761
$184$	0	0
$185$	−15.0805	−1.10874
$186$	0	0
$187$	−0.773277	−0.0565476
$188$	0	0
$189$	−5.65578	−0.411397
$190$	0	0
$191$	−17.2395	−1.24740	−0.623702	−	0.781662i	$-0.714372\pi$
$191$	−17.2395	−1.24740	−0.623702	+	0.781662i	$0.714372\pi$
$192$	0	0
$193$	5.73060	0.412498	0.206249	−	0.978500i	$-0.433874\pi$
$193$	5.73060	0.412498	0.206249	+	0.978500i	$0.433874\pi$
$194$	0	0
$195$	10.0256	0.717946
$196$	0	0
$197$	−4.95548	−0.353063	−0.176532	−	0.984295i	$-0.556488\pi$
$197$	−4.95548	−0.353063	−0.176532	+	0.984295i	$0.556488\pi$
$198$	0	0
$199$	−10.7193	−0.759868	−0.379934	−	0.925014i	$-0.624053\pi$
$199$	−10.7193	−0.759868	−0.379934	+	0.925014i	$0.624053\pi$
$200$	0	0
$201$	7.70314	0.543337
$202$	0	0
$203$	−6.41268	−0.450082
$204$	0	0
$205$	5.32906	0.372198
$206$	0	0
$207$	3.36273	0.233726
$208$	0	0
$209$	−0.855150	−0.0591520
$210$	0	0
$211$	19.7310	1.35834	0.679170	−	0.733981i	$-0.262340\pi$
$211$	19.7310	1.35834	0.679170	+	0.733981i	$0.262340\pi$
$212$	0	0
$213$	18.0979	1.24005
$214$	0	0
$215$	−26.9579	−1.83851
$216$	0	0
$217$	−3.65361	−0.248024
$218$	0	0
$219$	3.67020	0.248009
$220$	0	0
$221$	2.59298	0.174423
$222$	0	0
$223$	11.1872	0.749149	0.374575	−	0.927197i	$-0.377789\pi$
$223$	11.1872	0.749149	0.374575	+	0.927197i	$0.377789\pi$
$224$	0	0
$225$	−2.20479	−0.146986
$226$	0	0
$227$	7.19554	0.477584	0.238792	−	0.971071i	$-0.423249\pi$
$227$	7.19554	0.477584	0.238792	+	0.971071i	$0.423249\pi$
$228$	0	0
$229$	6.99695	0.462371	0.231186	−	0.972910i	$-0.425740\pi$
$229$	6.99695	0.462371	0.231186	+	0.972910i	$0.425740\pi$
$230$	0	0
$231$	−1.10588	−0.0727615
$232$	0	0
$233$	−11.4432	−0.749671	−0.374835	−	0.927091i	$-0.622301\pi$
$233$	−11.4432	−0.749671	−0.374835	+	0.927091i	$0.622301\pi$
$234$	0	0
$235$	27.9193	1.82125
$236$	0	0
$237$	−7.09942	−0.461157
$238$	0	0
$239$	−16.8520	−1.09007	−0.545033	−	0.838415i	$-0.683483\pi$
$239$	−16.8520	−1.09007	−0.545033	+	0.838415i	$0.683483\pi$
$240$	0	0
$241$	−2.35871	−0.151938	−0.0759689	−	0.997110i	$-0.524205\pi$
$241$	−2.35871	−0.151938	−0.0759689	+	0.997110i	$0.524205\pi$
$242$	0	0
$243$	−9.49613	−0.609177
$244$	0	0
$245$	−2.70357	−0.172725
$246$	0	0
$247$	2.86752	0.182456
$248$	0	0
$249$	−3.11304	−0.197281
$250$	0	0
$251$	−4.34850	−0.274475	−0.137237	−	0.990538i	$-0.543822\pi$
$251$	−4.34850	−0.274475	−0.137237	+	0.990538i	$0.543822\pi$
$252$	0	0
$253$	2.72355	0.171228
$254$	0	0
$255$	3.86643	0.242125
$256$	0	0
$257$	−8.33509	−0.519928	−0.259964	−	0.965618i	$-0.583711\pi$
$257$	−8.33509	−0.519928	−0.259964	+	0.965618i	$0.583711\pi$
$258$	0	0
$259$	−5.57801	−0.346601
$260$	0	0
$261$	−6.12254	−0.378976
$262$	0	0
$263$	−1.60262	−0.0988219	−0.0494109	−	0.998779i	$-0.515734\pi$
$263$	−1.60262	−0.0988219	−0.0494109	+	0.998779i	$0.515734\pi$
$264$	0	0
$265$	22.9593	1.41038
$266$	0	0
$267$	−14.0502	−0.859855
$268$	0	0
$269$	27.0032	1.64641	0.823206	−	0.567743i	$-0.192183\pi$
$269$	27.0032	1.64641	0.823206	+	0.567743i	$0.192183\pi$
$270$	0	0
$271$	22.9673	1.39516	0.697582	−	0.716505i	$-0.254260\pi$
$271$	22.9673	1.39516	0.697582	+	0.716505i	$0.254260\pi$
$272$	0	0
$273$	3.70828	0.224435
$274$	0	0
$275$	−1.78571	−0.107682
$276$	0	0
$277$	−22.7050	−1.36421	−0.682107	−	0.731253i	$-0.738936\pi$
$277$	−22.7050	−1.36421	−0.682107	+	0.731253i	$0.738936\pi$
$278$	0	0
$279$	−3.48831	−0.208839
$280$	0	0
$281$	19.8855	1.18627	0.593136	−	0.805102i	$-0.297890\pi$
$281$	19.8855	1.18627	0.593136	+	0.805102i	$0.297890\pi$
$282$	0	0
$283$	23.7051	1.40912	0.704562	−	0.709643i	$-0.251144\pi$
$283$	23.7051	1.40912	0.704562	+	0.709643i	$0.251144\pi$
$284$	0	0
$285$	4.27580	0.253277
$286$	0	0
$287$	1.97112	0.116352
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−23.3300	−1.36763
$292$	0	0
$293$	2.34850	0.137201	0.0686004	−	0.997644i	$-0.478147\pi$
$293$	2.34850	0.137201	0.0686004	+	0.997644i	$0.478147\pi$
$294$	0	0
$295$	10.1608	0.591587
$296$	0	0
$297$	−4.37348	−0.253775
$298$	0	0
$299$	−9.13271	−0.528158
$300$	0	0
$301$	−9.97124	−0.574733
$302$	0	0
$303$	5.50283	0.316129
$304$	0	0
$305$	−39.0664	−2.23694
$306$	0	0
$307$	−0.500806	−0.0285825	−0.0142913	−	0.999898i	$-0.504549\pi$
$307$	−0.500806	−0.0285825	−0.0142913	+	0.999898i	$0.504549\pi$
$308$	0	0
$309$	24.1123	1.37170
$310$	0	0
$311$	−15.0212	−0.851776	−0.425888	−	0.904776i	$-0.640038\pi$
$311$	−15.0212	−0.851776	−0.425888	+	0.904776i	$0.640038\pi$
$312$	0	0
$313$	15.8493	0.895854	0.447927	−	0.894070i	$-0.352162\pi$
$313$	15.8493	0.895854	0.447927	+	0.894070i	$0.352162\pi$
$314$	0	0
$315$	−2.58124	−0.145437
$316$	0	0
$317$	−29.6617	−1.66597	−0.832983	−	0.553298i	$-0.813369\pi$
$317$	−29.6617	−1.66597	−0.832983	+	0.553298i	$0.813369\pi$
$318$	0	0
$319$	−4.95878	−0.277638
$320$	0	0
$321$	−3.84854	−0.214805
$322$	0	0
$323$	1.10588	0.0615327
$324$	0	0
$325$	5.98791	0.332149
$326$	0	0
$327$	19.4009	1.07287
$328$	0	0
$329$	10.3268	0.569336
$330$	0	0
$331$	−2.99226	−0.164469	−0.0822347	−	0.996613i	$-0.526206\pi$
$331$	−2.99226	−0.164469	−0.0822347	+	0.996613i	$0.526206\pi$
$332$	0	0
$333$	−5.32564	−0.291843
$334$	0	0
$335$	14.5624	0.795627
$336$	0	0
$337$	−28.5715	−1.55639	−0.778195	−	0.628022i	$-0.783865\pi$
$337$	−28.5715	−1.55639	−0.778195	+	0.628022i	$0.783865\pi$
$338$	0	0
$339$	−24.4740	−1.32925
$340$	0	0
$341$	−2.82525	−0.152996
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−13.6179	−0.733163
$346$	0	0
$347$	4.47883	0.240436	0.120218	−	0.992748i	$-0.461641\pi$
$347$	4.47883	0.240436	0.120218	+	0.992748i	$0.461641\pi$
$348$	0	0
$349$	−14.2297	−0.761696	−0.380848	−	0.924638i	$-0.624368\pi$
$349$	−14.2297	−0.761696	−0.380848	+	0.924638i	$0.624368\pi$
$350$	0	0
$351$	14.6653	0.782777
$352$	0	0
$353$	−20.6592	−1.09958	−0.549790	−	0.835303i	$-0.685292\pi$
$353$	−20.6592	−1.09958	−0.549790	+	0.835303i	$0.685292\pi$
$354$	0	0
$355$	34.2131	1.81584
$356$	0	0
$357$	1.43012	0.0756900
$358$	0	0
$359$	−15.9587	−0.842268	−0.421134	−	0.906998i	$-0.638368\pi$
$359$	−15.9587	−0.842268	−0.421134	+	0.906998i	$0.638368\pi$
$360$	0	0
$361$	−17.7770	−0.935633
$362$	0	0
$363$	14.8762	0.780797
$364$	0	0
$365$	6.93831	0.363168
$366$	0	0
$367$	33.7047	1.75937	0.879686	−	0.475555i	$-0.157753\pi$
$367$	33.7047	1.75937	0.879686	+	0.475555i	$0.157753\pi$
$368$	0	0
$369$	1.88194	0.0979699
$370$	0	0
$371$	8.49223	0.440895
$372$	0	0
$373$	1.34947	0.0698729	0.0349364	−	0.999390i	$-0.488877\pi$
$373$	1.34947	0.0698729	0.0349364	+	0.999390i	$0.488877\pi$
$374$	0	0
$375$	−10.4035	−0.537234
$376$	0	0
$377$	16.6280	0.856384
$378$	0	0
$379$	−10.6618	−0.547661	−0.273830	−	0.961778i	$-0.588291\pi$
$379$	−10.6618	−0.547661	−0.273830	+	0.961778i	$0.588291\pi$
$380$	0	0
$381$	28.5456	1.46244
$382$	0	0
$383$	−13.6889	−0.699468	−0.349734	−	0.936849i	$-0.613728\pi$
$383$	−13.6889	−0.699468	−0.349734	+	0.936849i	$0.613728\pi$
$384$	0	0
$385$	−2.09061	−0.106547
$386$	0	0
$387$	−9.52009	−0.483933
$388$	0	0
$389$	−22.5941	−1.14556	−0.572782	−	0.819708i	$-0.694136\pi$
$389$	−22.5941	−1.14556	−0.572782	+	0.819708i	$0.694136\pi$
$390$	0	0
$391$	−3.52209	−0.178120
$392$	0	0
$393$	1.60990	0.0812088
$394$	0	0
$395$	−13.4211	−0.675288
$396$	0	0
$397$	−14.1289	−0.709107	−0.354553	−	0.935036i	$-0.615367\pi$
$397$	−14.1289	−0.709107	−0.354553	+	0.935036i	$0.615367\pi$
$398$	0	0
$399$	1.58154	0.0791760
$400$	0	0
$401$	−7.89969	−0.394492	−0.197246	−	0.980354i	$-0.563200\pi$
$401$	−7.89969	−0.394492	−0.197246	+	0.980354i	$0.563200\pi$
$402$	0	0
$403$	9.47375	0.471921
$404$	0	0
$405$	14.1239	0.701823
$406$	0	0
$407$	−4.31335	−0.213805
$408$	0	0
$409$	20.9292	1.03488	0.517441	−	0.855719i	$-0.326885\pi$
$409$	20.9292	1.03488	0.517441	+	0.855719i	$0.326885\pi$
$410$	0	0
$411$	0.176355	0.00869894
$412$	0	0
$413$	3.75831	0.184934
$414$	0	0
$415$	−5.88505	−0.288886
$416$	0	0
$417$	10.0718	0.493220
$418$	0	0
$419$	27.8994	1.36297	0.681487	−	0.731830i	$-0.261333\pi$
$419$	27.8994	1.36297	0.681487	+	0.731830i	$0.261333\pi$
$420$	0	0
$421$	−21.1810	−1.03230	−0.516150	−	0.856498i	$-0.672635\pi$
$421$	−21.1810	−1.03230	−0.516150	+	0.856498i	$0.672635\pi$
$422$	0	0
$423$	9.85959	0.479390
$424$	0	0
$425$	2.30928	0.112016
$426$	0	0
$427$	−14.4499	−0.699282
$428$	0	0
$429$	2.86752	0.138445
$430$	0	0
$431$	9.22545	0.444374	0.222187	−	0.975004i	$-0.428680\pi$
$431$	9.22545	0.444374	0.222187	+	0.975004i	$0.428680\pi$
$432$	0	0
$433$	2.90747	0.139724	0.0698621	−	0.997557i	$-0.477744\pi$
$433$	2.90747	0.139724	0.0698621	+	0.997557i	$0.477744\pi$
$434$	0	0
$435$	24.7942	1.18879
$436$	0	0
$437$	−3.89500	−0.186323
$438$	0	0
$439$	−38.9929	−1.86103	−0.930514	−	0.366256i	$-0.880639\pi$
$439$	−38.9929	−1.86103	−0.930514	+	0.366256i	$0.880639\pi$
$440$	0	0
$441$	−0.954755	−0.0454645
$442$	0	0
$443$	−13.4299	−0.638072	−0.319036	−	0.947743i	$-0.603359\pi$
$443$	−13.4299	−0.638072	−0.319036	+	0.947743i	$0.603359\pi$
$444$	0	0
$445$	−26.5611	−1.25912
$446$	0	0
$447$	−7.98960	−0.377895
$448$	0	0
$449$	6.27564	0.296166	0.148083	−	0.988975i	$-0.452690\pi$
$449$	6.27564	0.296166	0.148083	+	0.988975i	$0.452690\pi$
$450$	0	0
$451$	1.52422	0.0717729
$452$	0	0
$453$	−4.85659	−0.228183
$454$	0	0
$455$	7.01030	0.328648
$456$	0	0
$457$	2.17820	0.101892	0.0509460	−	0.998701i	$-0.483776\pi$
$457$	2.17820	0.101892	0.0509460	+	0.998701i	$0.483776\pi$
$458$	0	0
$459$	5.65578	0.263989
$460$	0	0
$461$	−20.1016	−0.936223	−0.468111	−	0.883669i	$-0.655065\pi$
$461$	−20.1016	−0.936223	−0.468111	+	0.883669i	$0.655065\pi$
$462$	0	0
$463$	3.24000	0.150575	0.0752877	−	0.997162i	$-0.476012\pi$
$463$	3.24000	0.150575	0.0752877	+	0.997162i	$0.476012\pi$
$464$	0	0
$465$	14.1264	0.655098
$466$	0	0
$467$	6.58471	0.304704	0.152352	−	0.988326i	$-0.451315\pi$
$467$	6.58471	0.304704	0.152352	+	0.988326i	$0.451315\pi$
$468$	0	0
$469$	5.38635	0.248719
$470$	0	0
$471$	12.4901	0.575515
$472$	0	0
$473$	−7.71053	−0.354530
$474$	0	0
$475$	2.55378	0.117175
$476$	0	0
$477$	8.10800	0.371240
$478$	0	0
$479$	−27.2824	−1.24657	−0.623283	−	0.781997i	$-0.714201\pi$
$479$	−27.2824	−1.24657	−0.623283	+	0.781997i	$0.714201\pi$
$480$	0	0
$481$	14.4637	0.659487
$482$	0	0
$483$	−5.03701	−0.229192
$484$	0	0
$485$	−44.1042	−2.00267
$486$	0	0
$487$	39.4821	1.78910	0.894552	−	0.446963i	$-0.147495\pi$
$487$	39.4821	1.78910	0.894552	+	0.446963i	$0.147495\pi$
$488$	0	0
$489$	34.1138	1.54268
$490$	0	0
$491$	−12.7766	−0.576601	−0.288300	−	0.957540i	$-0.593090\pi$
$491$	−12.7766	−0.576601	−0.288300	+	0.957540i	$0.593090\pi$
$492$	0	0
$493$	6.41268	0.288813
$494$	0	0
$495$	−1.99602	−0.0897142
$496$	0	0
$497$	12.6548	0.567645
$498$	0	0
$499$	42.0917	1.88428	0.942142	−	0.335213i	$-0.108808\pi$
$499$	42.0917	1.88428	0.942142	+	0.335213i	$0.108808\pi$
$500$	0	0
$501$	2.56778	0.114720
$502$	0	0
$503$	−24.6869	−1.10074	−0.550368	−	0.834922i	$-0.685512\pi$
$503$	−24.6869	−1.10074	−0.550368	+	0.834922i	$0.685512\pi$
$504$	0	0
$505$	10.4028	0.462919
$506$	0	0
$507$	8.97608	0.398642
$508$	0	0
$509$	23.5002	1.04163	0.520815	−	0.853670i	$-0.325628\pi$
$509$	23.5002	1.04163	0.520815	+	0.853670i	$0.325628\pi$
$510$	0	0
$511$	2.56635	0.113529
$512$	0	0
$513$	6.25460	0.276147
$514$	0	0
$515$	45.5831	2.00863
$516$	0	0
$517$	7.98549	0.351202
$518$	0	0
$519$	−5.33954	−0.234380
$520$	0	0
$521$	12.3636	0.541657	0.270829	−	0.962628i	$-0.412702\pi$
$521$	12.3636	0.541657	0.270829	+	0.962628i	$0.412702\pi$
$522$	0	0
$523$	10.7247	0.468960	0.234480	−	0.972121i	$-0.424661\pi$
$523$	10.7247	0.468960	0.234480	+	0.972121i	$0.424661\pi$
$524$	0	0
$525$	3.30254	0.144135
$526$	0	0
$527$	3.65361	0.159154
$528$	0	0
$529$	−10.5949	−0.460647
$530$	0	0
$531$	3.58827	0.155717
$532$	0	0
$533$	−5.11109	−0.221386
$534$	0	0
$535$	−7.27546	−0.314546
$536$	0	0
$537$	17.7750	0.767047
$538$	0	0
$539$	−0.773277	−0.0333074
$540$	0	0
$541$	21.7635	0.935688	0.467844	−	0.883811i	$-0.345031\pi$
$541$	21.7635	0.935688	0.467844	+	0.883811i	$0.345031\pi$
$542$	0	0
$543$	14.8992	0.639384
$544$	0	0
$545$	36.6764	1.57104
$546$	0	0
$547$	−40.0558	−1.71266	−0.856331	−	0.516427i	$-0.827262\pi$
$547$	−40.0558	−1.71266	−0.856331	+	0.516427i	$0.827262\pi$
$548$	0	0
$549$	−13.7962	−0.588806
$550$	0	0
$551$	7.09165	0.302115
$552$	0	0
$553$	−4.96421	−0.211100
$554$	0	0
$555$	21.5670	0.915468
$556$	0	0
$557$	−20.4062	−0.864639	−0.432319	−	0.901721i	$-0.642305\pi$
$557$	−20.4062	−0.864639	−0.432319	+	0.901721i	$0.642305\pi$
$558$	0	0
$559$	25.8552	1.09356
$560$	0	0
$561$	1.10588	0.0466902
$562$	0	0
$563$	−9.04019	−0.380999	−0.190499	−	0.981687i	$-0.561011\pi$
$563$	−9.04019	−0.380999	−0.190499	+	0.981687i	$0.561011\pi$
$564$	0	0
$565$	−46.2669	−1.94646
$566$	0	0
$567$	5.22418	0.219395
$568$	0	0
$569$	39.6612	1.66269	0.831343	−	0.555760i	$-0.187573\pi$
$569$	39.6612	1.66269	0.831343	+	0.555760i	$0.187573\pi$
$570$	0	0
$571$	13.8780	0.580778	0.290389	−	0.956909i	$-0.406215\pi$
$571$	13.8780	0.580778	0.290389	+	0.956909i	$0.406215\pi$
$572$	0	0
$573$	24.6545	1.02996
$574$	0	0
$575$	−8.13347	−0.339189
$576$	0	0
$577$	30.4450	1.26744	0.633720	−	0.773562i	$-0.281527\pi$
$577$	30.4450	1.26744	0.633720	+	0.773562i	$0.281527\pi$
$578$	0	0
$579$	−8.19546	−0.340592
$580$	0	0
$581$	−2.17677	−0.0903077
$582$	0	0
$583$	6.56684	0.271971
$584$	0	0
$585$	6.69312	0.276726
$586$	0	0
$587$	20.3521	0.840020	0.420010	−	0.907520i	$-0.362027\pi$
$587$	20.3521	0.840020	0.420010	+	0.907520i	$0.362027\pi$
$588$	0	0
$589$	4.04045	0.166484
$590$	0	0
$591$	7.08693	0.291517
$592$	0	0
$593$	−0.0332887	−0.00136700	−0.000683502	−	1.00000i	$-0.500218\pi$
$593$	−0.0332887	−0.00136700	−0.000683502		1.00000i	$0.500218\pi$
$594$	0	0
$595$	2.70357	0.110835
$596$	0	0
$597$	15.3298	0.627408
$598$	0	0
$599$	−3.77376	−0.154192	−0.0770958	−	0.997024i	$-0.524565\pi$
$599$	−3.77376	−0.154192	−0.0770958	+	0.997024i	$0.524565\pi$
$600$	0	0
$601$	−8.49210	−0.346400	−0.173200	−	0.984887i	$-0.555411\pi$
$601$	−8.49210	−0.346400	−0.173200	+	0.984887i	$0.555411\pi$
$602$	0	0
$603$	5.14265	0.209425
$604$	0	0
$605$	28.1226	1.14335
$606$	0	0
$607$	−34.1228	−1.38500	−0.692501	−	0.721417i	$-0.743491\pi$
$607$	−34.1228	−1.38500	−0.692501	+	0.721417i	$0.743491\pi$
$608$	0	0
$609$	9.17091	0.371624
$610$	0	0
$611$	−26.7773	−1.08329
$612$	0	0
$613$	−21.0340	−0.849557	−0.424778	−	0.905297i	$-0.639648\pi$
$613$	−21.0340	−0.849557	−0.424778	+	0.905297i	$0.639648\pi$
$614$	0	0
$615$	−7.62121	−0.307317
$616$	0	0
$617$	−34.2110	−1.37728	−0.688641	−	0.725103i	$-0.741792\pi$
$617$	−34.2110	−1.37728	−0.688641	+	0.725103i	$0.741792\pi$
$618$	0	0
$619$	−29.6319	−1.19101	−0.595503	−	0.803353i	$-0.703047\pi$
$619$	−29.6319	−1.19101	−0.595503	+	0.803353i	$0.703047\pi$
$620$	0	0
$621$	−19.9201	−0.799368
$622$	0	0
$623$	−9.82445	−0.393608
$624$	0	0
$625$	−31.2136	−1.24854
$626$	0	0
$627$	1.22297	0.0488406
$628$	0	0
$629$	5.57801	0.222410
$630$	0	0
$631$	10.6285	0.423115	0.211557	−	0.977366i	$-0.432146\pi$
$631$	10.6285	0.423115	0.211557	+	0.977366i	$0.432146\pi$
$632$	0	0
$633$	−28.2177	−1.12155
$634$	0	0
$635$	53.9640	2.14150
$636$	0	0
$637$	2.59298	0.102738
$638$	0	0
$639$	12.0822	0.477966
$640$	0	0
$641$	−9.96712	−0.393678	−0.196839	−	0.980436i	$-0.563068\pi$
$641$	−9.96712	−0.393678	−0.196839	+	0.980436i	$0.563068\pi$
$642$	0	0
$643$	23.7026	0.934739	0.467369	−	0.884062i	$-0.345202\pi$
$643$	23.7026	0.934739	0.467369	+	0.884062i	$0.345202\pi$
$644$	0	0
$645$	38.5531	1.51803
$646$	0	0
$647$	5.42162	0.213146	0.106573	−	0.994305i	$-0.466012\pi$
$647$	5.42162	0.213146	0.106573	+	0.994305i	$0.466012\pi$
$648$	0	0
$649$	2.90621	0.114079
$650$	0	0
$651$	5.22511	0.204788
$652$	0	0
$653$	14.7777	0.578295	0.289147	−	0.957285i	$-0.406628\pi$
$653$	14.7777	0.578295	0.289147	+	0.957285i	$0.406628\pi$
$654$	0	0
$655$	3.04344	0.118917
$656$	0	0
$657$	2.45024	0.0955929
$658$	0	0
$659$	25.0160	0.974484	0.487242	−	0.873267i	$-0.338003\pi$
$659$	25.0160	0.974484	0.487242	+	0.873267i	$0.338003\pi$
$660$	0	0
$661$	32.3276	1.25740	0.628699	−	0.777648i	$-0.283588\pi$
$661$	32.3276	1.25740	0.628699	+	0.777648i	$0.283588\pi$
$662$	0	0
$663$	−3.70828	−0.144017
$664$	0	0
$665$	2.98982	0.115940
$666$	0	0
$667$	−22.5860	−0.874535
$668$	0	0
$669$	−15.9990	−0.618558
$670$	0	0
$671$	−11.1738	−0.431360
$672$	0	0
$673$	4.98644	0.192213	0.0961065	−	0.995371i	$-0.469361\pi$
$673$	4.98644	0.192213	0.0961065	+	0.995371i	$0.469361\pi$
$674$	0	0
$675$	13.0607	0.502708
$676$	0	0
$677$	−36.1402	−1.38898	−0.694490	−	0.719502i	$-0.744370\pi$
$677$	−36.1402	−1.38898	−0.694490	+	0.719502i	$0.744370\pi$
$678$	0	0
$679$	−16.3133	−0.626048
$680$	0	0
$681$	−10.2905	−0.394332
$682$	0	0
$683$	−16.5786	−0.634363	−0.317182	−	0.948365i	$-0.602736\pi$
$683$	−16.5786	−0.634363	−0.317182	+	0.948365i	$0.602736\pi$
$684$	0	0
$685$	0.333389	0.0127382
$686$	0	0
$687$	−10.0065	−0.381771
$688$	0	0
$689$	−22.0202	−0.838903
$690$	0	0
$691$	19.3390	0.735692	0.367846	−	0.929887i	$-0.380095\pi$
$691$	19.3390	0.735692	0.367846	+	0.929887i	$0.380095\pi$
$692$	0	0
$693$	−0.738290	−0.0280453
$694$	0	0
$695$	19.0403	0.722239
$696$	0	0
$697$	−1.97112	−0.0746616
$698$	0	0
$699$	16.3652	0.618989
$700$	0	0
$701$	−5.54756	−0.209528	−0.104764	−	0.994497i	$-0.533409\pi$
$701$	−5.54756	−0.209528	−0.104764	+	0.994497i	$0.533409\pi$
$702$	0	0
$703$	6.16861	0.232653
$704$	0	0
$705$	−39.9279	−1.50377
$706$	0	0
$707$	3.84781	0.144712
$708$	0	0
$709$	−4.66569	−0.175224	−0.0876118	−	0.996155i	$-0.527924\pi$
$709$	−4.66569	−0.175224	−0.0876118	+	0.996155i	$0.527924\pi$
$710$	0	0
$711$	−4.73960	−0.177749
$712$	0	0
$713$	−12.8684	−0.481924
$714$	0	0
$715$	5.42090	0.202730
$716$	0	0
$717$	24.1004	0.900045
$718$	0	0
$719$	−48.6586	−1.81466	−0.907331	−	0.420418i	$-0.861883\pi$
$719$	−48.6586	−1.81466	−0.907331	+	0.420418i	$0.861883\pi$
$720$	0	0
$721$	16.8604	0.627912
$722$	0	0
$723$	3.37324	0.125452
$724$	0	0
$725$	14.8087	0.549980
$726$	0	0
$727$	−24.0680	−0.892631	−0.446316	−	0.894876i	$-0.647264\pi$
$727$	−24.0680	−0.892631	−0.446316	+	0.894876i	$0.647264\pi$
$728$	0	0
$729$	29.2531	1.08345
$730$	0	0
$731$	9.97124	0.368800
$732$	0	0
$733$	−16.8913	−0.623893	−0.311946	−	0.950100i	$-0.600981\pi$
$733$	−16.8913	−0.623893	−0.311946	+	0.950100i	$0.600981\pi$
$734$	0	0
$735$	3.86643	0.142615
$736$	0	0
$737$	4.16514	0.153425
$738$	0	0
$739$	11.6658	0.429134	0.214567	−	0.976709i	$-0.431166\pi$
$739$	11.6658	0.429134	0.214567	+	0.976709i	$0.431166\pi$
$740$	0	0
$741$	−4.10090	−0.150650
$742$	0	0
$743$	−1.95092	−0.0715723	−0.0357862	−	0.999359i	$-0.511394\pi$
$743$	−1.95092	−0.0715723	−0.0357862	+	0.999359i	$0.511394\pi$
$744$	0	0
$745$	−15.1039	−0.553364
$746$	0	0
$747$	−2.07828	−0.0760404
$748$	0	0
$749$	−2.69106	−0.0983292
$750$	0	0
$751$	35.9991	1.31363	0.656813	−	0.754054i	$-0.271904\pi$
$751$	35.9991	1.31363	0.656813	+	0.754054i	$0.271904\pi$
$752$	0	0
$753$	6.21888	0.226629
$754$	0	0
$755$	−9.18113	−0.334136
$756$	0	0
$757$	13.8273	0.502561	0.251280	−	0.967914i	$-0.419148\pi$
$757$	13.8273	0.502561	0.251280	+	0.967914i	$0.419148\pi$
$758$	0	0
$759$	−3.89500	−0.141380
$760$	0	0
$761$	7.47584	0.270999	0.135499	−	0.990777i	$-0.456736\pi$
$761$	7.47584	0.270999	0.135499	+	0.990777i	$0.456736\pi$
$762$	0	0
$763$	13.5659	0.491119
$764$	0	0
$765$	2.58124	0.0933251
$766$	0	0
$767$	−9.74523	−0.351880
$768$	0	0
$769$	20.1327	0.726004	0.363002	−	0.931788i	$-0.381752\pi$
$769$	20.1327	0.726004	0.363002	+	0.931788i	$0.381752\pi$
$770$	0	0
$771$	11.9202	0.429295
$772$	0	0
$773$	31.9802	1.15025	0.575124	−	0.818066i	$-0.304954\pi$
$773$	31.9802	1.15025	0.575124	+	0.818066i	$0.304954\pi$
$774$	0	0
$775$	8.43720	0.303073
$776$	0	0
$777$	7.97723	0.286182
$778$	0	0
$779$	−2.17982	−0.0781003
$780$	0	0
$781$	9.78566	0.350158
$782$	0	0
$783$	36.2687	1.29614
$784$	0	0
$785$	23.6119	0.842746
$786$	0	0
$787$	17.8542	0.636433	0.318216	−	0.948018i	$-0.396916\pi$
$787$	17.8542	0.636433	0.318216	+	0.948018i	$0.396916\pi$
$788$	0	0
$789$	2.29194	0.0815953
$790$	0	0
$791$	−17.1133	−0.608478
$792$	0	0
$793$	37.4684	1.33054
$794$	0	0
$795$	−32.8346	−1.16452
$796$	0	0
$797$	−0.946371	−0.0335222	−0.0167611	−	0.999860i	$-0.505335\pi$
$797$	−0.946371	−0.0335222	−0.0167611	+	0.999860i	$0.505335\pi$
$798$	0	0
$799$	−10.3268	−0.365337
$800$	0	0
$801$	−9.37994	−0.331424
$802$	0	0
$803$	1.98450	0.0700315
$804$	0	0
$805$	−9.52220	−0.335614
$806$	0	0
$807$	−38.6178	−1.35941
$808$	0	0
$809$	−1.06229	−0.0373482	−0.0186741	−	0.999826i	$-0.505944\pi$
$809$	−1.06229	−0.0373482	−0.0186741	+	0.999826i	$0.505944\pi$
$810$	0	0
$811$	55.1600	1.93693	0.968465	−	0.249148i	$-0.0801507\pi$
$811$	55.1600	1.93693	0.968465	+	0.249148i	$0.0801507\pi$
$812$	0	0
$813$	−32.8460	−1.15196
$814$	0	0
$815$	64.4903	2.25900
$816$	0	0
$817$	11.0270	0.385785
$818$	0	0
$819$	2.47566	0.0865066
$820$	0	0
$821$	8.38891	0.292775	0.146388	−	0.989227i	$-0.453235\pi$
$821$	8.38891	0.292775	0.146388	+	0.989227i	$0.453235\pi$
$822$	0	0
$823$	−29.7320	−1.03639	−0.518196	−	0.855262i	$-0.673396\pi$
$823$	−29.7320	−1.03639	−0.518196	+	0.855262i	$0.673396\pi$
$824$	0	0
$825$	2.55378	0.0889112
$826$	0	0
$827$	−7.51576	−0.261349	−0.130674	−	0.991425i	$-0.541714\pi$
$827$	−7.51576	−0.261349	−0.130674	+	0.991425i	$0.541714\pi$
$828$	0	0
$829$	30.9128	1.07364	0.536822	−	0.843695i	$-0.319625\pi$
$829$	30.9128	1.07364	0.536822	+	0.843695i	$0.319625\pi$
$830$	0	0
$831$	32.4709	1.12640
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	4.85424	0.167988
$836$	0	0
$837$	20.6640	0.714253
$838$	0	0
$839$	29.0866	1.00418	0.502091	−	0.864815i	$-0.332564\pi$
$839$	29.0866	1.00418	0.502091	+	0.864815i	$0.332564\pi$
$840$	0	0
$841$	12.1225	0.418018
$842$	0	0
$843$	−28.4387	−0.979482
$844$	0	0
$845$	16.9688	0.583745
$846$	0	0
$847$	10.4020	0.357418
$848$	0	0
$849$	−33.9012	−1.16349
$850$	0	0
$851$	−19.6463	−0.673465
$852$	0	0
$853$	−22.6808	−0.776574	−0.388287	−	0.921538i	$-0.626933\pi$
$853$	−22.6808	−0.776574	−0.388287	+	0.921538i	$0.626933\pi$
$854$	0	0
$855$	2.85454	0.0976233
$856$	0	0
$857$	−1.80427	−0.0616328	−0.0308164	−	0.999525i	$-0.509811\pi$
$857$	−1.80427	−0.0616328	−0.0308164	+	0.999525i	$0.509811\pi$
$858$	0	0
$859$	−20.5554	−0.701340	−0.350670	−	0.936499i	$-0.614046\pi$
$859$	−20.5554	−0.701340	−0.350670	+	0.936499i	$0.614046\pi$
$860$	0	0
$861$	−2.81894	−0.0960694
$862$	0	0
$863$	−48.2711	−1.64317	−0.821583	−	0.570089i	$-0.806909\pi$
$863$	−48.2711	−1.64317	−0.821583	+	0.570089i	$0.806909\pi$
$864$	0	0
$865$	−10.0941	−0.343210
$866$	0	0
$867$	−1.43012	−0.0485694
$868$	0	0
$869$	−3.83871	−0.130219
$870$	0	0
$871$	−13.9667	−0.473244
$872$	0	0
$873$	−15.5752	−0.527142
$874$	0	0
$875$	−7.27455	−0.245925
$876$	0	0
$877$	−36.9852	−1.24890	−0.624451	−	0.781064i	$-0.714677\pi$
$877$	−36.9852	−1.24890	−0.624451	+	0.781064i	$0.714677\pi$
$878$	0	0
$879$	−3.35864	−0.113284
$880$	0	0
$881$	−12.8389	−0.432554	−0.216277	−	0.976332i	$-0.569391\pi$
$881$	−12.8389	−0.432554	−0.216277	+	0.976332i	$0.569391\pi$
$882$	0	0
$883$	19.0185	0.640022	0.320011	−	0.947414i	$-0.396313\pi$
$883$	19.0185	0.640022	0.320011	+	0.947414i	$0.396313\pi$
$884$	0	0
$885$	−14.5312	−0.488462
$886$	0	0
$887$	−52.1581	−1.75130	−0.875649	−	0.482948i	$-0.839566\pi$
$887$	−52.1581	−1.75130	−0.875649	+	0.482948i	$0.839566\pi$
$888$	0	0
$889$	19.9603	0.669447
$890$	0	0
$891$	4.03973	0.135336
$892$	0	0
$893$	−11.4202	−0.382163
$894$	0	0
$895$	33.6027	1.12321
$896$	0	0
$897$	13.0609	0.436090
$898$	0	0
$899$	23.4295	0.781417
$900$	0	0
$901$	−8.49223	−0.282917
$902$	0	0
$903$	14.2601	0.474546
$904$	0	0
$905$	28.1661	0.936272
$906$	0	0
$907$	−46.0632	−1.52950	−0.764752	−	0.644325i	$-0.777139\pi$
$907$	−46.0632	−1.52950	−0.764752	+	0.644325i	$0.777139\pi$
$908$	0	0
$909$	3.67371	0.121849
$910$	0	0
$911$	14.9655	0.495828	0.247914	−	0.968782i	$-0.420255\pi$
$911$	14.9655	0.495828	0.247914	+	0.968782i	$0.420255\pi$
$912$	0	0
$913$	−1.68325	−0.0557073
$914$	0	0
$915$	55.8697	1.84699
$916$	0	0
$917$	1.12571	0.0371742
$918$	0	0
$919$	0.892100	0.0294277	0.0147138	−	0.999892i	$-0.495316\pi$
$919$	0.892100	0.0294277	0.0147138	+	0.999892i	$0.495316\pi$
$920$	0	0
$921$	0.716213	0.0236000
$922$	0	0
$923$	−32.8136	−1.08007
$924$	0	0
$925$	12.8812	0.423530
$926$	0	0
$927$	16.0975	0.528712
$928$	0	0
$929$	−36.0467	−1.18265	−0.591327	−	0.806432i	$-0.701396\pi$
$929$	−36.0467	−1.18265	−0.591327	+	0.806432i	$0.701396\pi$
$930$	0	0
$931$	1.10588	0.0362437
$932$	0	0
$933$	21.4822	0.703295
$934$	0	0
$935$	2.09061	0.0683701
$936$	0	0
$937$	11.9259	0.389602	0.194801	−	0.980843i	$-0.437594\pi$
$937$	11.9259	0.389602	0.194801	+	0.980843i	$0.437594\pi$
$938$	0	0
$939$	−22.6664	−0.739690
$940$	0	0
$941$	19.1916	0.625630	0.312815	−	0.949814i	$-0.398728\pi$
$941$	19.1916	0.625630	0.312815	+	0.949814i	$0.398728\pi$
$942$	0	0
$943$	6.94247	0.226078
$944$	0	0
$945$	15.2908	0.497409
$946$	0	0
$947$	0.910058	0.0295729	0.0147865	−	0.999891i	$-0.495293\pi$
$947$	0.910058	0.0295729	0.0147865	+	0.999891i	$0.495293\pi$
$948$	0	0
$949$	−6.65451	−0.216014
$950$	0	0
$951$	42.4198	1.37556
$952$	0	0
$953$	−16.8492	−0.545798	−0.272899	−	0.962043i	$-0.587982\pi$
$953$	−16.8492	−0.545798	−0.272899	+	0.962043i	$0.587982\pi$
$954$	0	0
$955$	46.6081	1.50820
$956$	0	0
$957$	7.09165	0.229241
$958$	0	0
$959$	0.123315	0.00398204
$960$	0	0
$961$	−17.6511	−0.569390
$962$	0	0
$963$	−2.56930	−0.0827946
$964$	0	0
$965$	−15.4931	−0.498740
$966$	0	0
$967$	20.6236	0.663209	0.331605	−	0.943418i	$-0.392410\pi$
$967$	20.6236	0.663209	0.331605	+	0.943418i	$0.392410\pi$
$968$	0	0
$969$	−1.58154	−0.0508064
$970$	0	0
$971$	−16.8183	−0.539723	−0.269862	−	0.962899i	$-0.586978\pi$
$971$	−16.8183	−0.539723	−0.269862	+	0.962899i	$0.586978\pi$
$972$	0	0
$973$	7.04265	0.225777
$974$	0	0
$975$	−8.56343	−0.274249
$976$	0	0
$977$	−7.51049	−0.240282	−0.120141	−	0.992757i	$-0.538335\pi$
$977$	−7.51049	−0.240282	−0.120141	+	0.992757i	$0.538335\pi$
$978$	0	0
$979$	−7.59702	−0.242802
$980$	0	0
$981$	12.9521	0.413530
$982$	0	0
$983$	30.6340	0.977072	0.488536	−	0.872544i	$-0.337531\pi$
$983$	30.6340	0.977072	0.488536	+	0.872544i	$0.337531\pi$
$984$	0	0
$985$	13.3975	0.426879
$986$	0	0
$987$	−14.7686	−0.470090
$988$	0	0
$989$	−35.1196	−1.11674
$990$	0	0
$991$	21.6616	0.688104	0.344052	−	0.938951i	$-0.388200\pi$
$991$	21.6616	0.688104	0.344052	+	0.938951i	$0.388200\pi$
$992$	0	0
$993$	4.27929	0.135799
$994$	0	0
$995$	28.9802	0.918735
$996$	0	0
$997$	17.9567	0.568693	0.284347	−	0.958722i	$-0.408223\pi$
$997$	17.9567	0.568693	0.284347	+	0.958722i	$0.408223\pi$
$998$	0	0
$999$	31.5480	0.998135

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.j.1.3	✓	6
4.3	odd	2		3808.2.a.n.1.4	yes	6
8.3	odd	2		7616.2.a.bw.1.3		6
8.5	even	2		7616.2.a.ca.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.j.1.3	✓	6	1.1	even	1	trivial
3808.2.a.n.1.4	yes	6	4.3	odd	2
7616.2.a.bw.1.3		6	8.3	odd	2
7616.2.a.ca.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)

\(f(q)\)	\(=\)	\(q-1.43012 q^{3} -2.70357 q^{5} -1.00000 q^{7} -0.954755 q^{9} -0.773277 q^{11} +2.59298 q^{13} +3.86643 q^{15} +1.00000 q^{17} +1.10588 q^{19} +1.43012 q^{21} -3.52209 q^{23} +2.30928 q^{25} +5.65578 q^{27} +6.41268 q^{29} +3.65361 q^{31} +1.10588 q^{33} +2.70357 q^{35} +5.57801 q^{37} -3.70828 q^{39} -1.97112 q^{41} +9.97124 q^{43} +2.58124 q^{45} -10.3268 q^{47} +1.00000 q^{49} -1.43012 q^{51} -8.49223 q^{53} +2.09061 q^{55} -1.58154 q^{57} -3.75831 q^{59} +14.4499 q^{61} +0.954755 q^{63} -7.01030 q^{65} -5.38635 q^{67} +5.03701 q^{69} -12.6548 q^{71} -2.56635 q^{73} -3.30254 q^{75} +0.773277 q^{77} +4.96421 q^{79} -5.22418 q^{81} +2.17677 q^{83} -2.70357 q^{85} -9.17091 q^{87} +9.82445 q^{89} -2.59298 q^{91} -5.22511 q^{93} -2.98982 q^{95} +16.3133 q^{97} +0.738290 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 4 q^{9} - 2 q^{11} - 4 q^{13} - 8 q^{15} + 6 q^{17} + 2 q^{19} + 2 q^{21} - 10 q^{23} - 2 q^{27} + 2 q^{29} - 12 q^{31} + 2 q^{33} - 2 q^{35} + 2 q^{37} - 10 q^{39} - 8 q^{43}+ \cdots - 2 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 + 4 * q^9 - 2 * q^11 - 4 * q^13 - 8 * q^15 + 6 * q^17 + 2 * q^19 + 2 * q^21 - 10 * q^23 - 2 * q^27 + 2 * q^29 - 12 * q^31 + 2 * q^33 - 2 * q^35 + 2 * q^37 - 10 * q^39 - 8 * q^43 + 14 * q^45 - 22 * q^47 + 6 * q^49 - 2 * q^51 - 6 * q^53 - 24 * q^55 - 8 * q^57 - 8 * q^59 - 4 * q^61 - 4 * q^63 - 2 * q^65 + 8 * q^67 + 4 * q^69 - 18 * q^71 + 8 * q^73 - 30 * q^75 + 2 * q^77 - 14 * q^79 - 26 * q^81 + 2 * q^85 - 22 * q^87 + 2 * q^89 + 4 * q^91 - 8 * q^93 - 2 * q^95 + 2 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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