LMFDB - Embedded newform 4368.2.a.bn.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.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:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1373.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 8x - 5 $ x^3 - 8*x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1092)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.661120$ of defining polynomial
Character	$\chi$	$=$	4368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.90180 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.90180 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.32224 q^{11} +1.00000 q^{13} +3.90180 q^{15} +4.00000 q^{17} -7.22404 q^{19} -1.00000 q^{21} +2.57956 q^{23} +10.2240 q^{25} -1.00000 q^{27} +9.22404 q^{29} -4.57956 q^{31} -1.32224 q^{33} -3.90180 q^{35} -5.80360 q^{37} -1.00000 q^{39} -11.1258 q^{41} -8.57956 q^{43} -3.90180 q^{45} -11.7054 q^{47} +1.00000 q^{49} -4.00000 q^{51} -1.42044 q^{53} -5.15912 q^{55} +7.22404 q^{57} -3.32224 q^{59} +2.00000 q^{61} +1.00000 q^{63} -3.90180 q^{65} -7.80360 q^{67} -2.57956 q^{69} +14.4814 q^{71} +14.3832 q^{73} -10.2240 q^{75} +1.32224 q^{77} +12.3832 q^{79} +1.00000 q^{81} +11.9018 q^{83} -15.6072 q^{85} -9.22404 q^{87} +1.25732 q^{89} +1.00000 q^{91} +4.57956 q^{93} +28.1868 q^{95} -2.57956 q^{97} +1.32224 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + q^5 + 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{13} - q^{15} + 12 q^{17} - 5 q^{19} - 3 q^{21} - q^{23} + 14 q^{25} - 3 q^{27} + 11 q^{29} - 5 q^{31} + q^{35} + 8 q^{37} - 3 q^{39} - 4 q^{41} - 17 q^{43} + q^{45} + 3 q^{47} + 3 q^{49} - 12 q^{51} - 13 q^{53} + 2 q^{55} + 5 q^{57} - 6 q^{59} + 6 q^{61} + 3 q^{63} + q^{65} + 2 q^{67} + q^{69} + 22 q^{71} + 9 q^{73} - 14 q^{75} + 3 q^{79} + 3 q^{81} + 23 q^{83} + 4 q^{85} - 11 q^{87} - q^{89} + 3 q^{91} + 5 q^{93} + 25 q^{95} + q^{97}+O(q^{100}) $ 3 * q - 3 * q^3 + q^5 + 3 * q^7 + 3 * q^9 + 3 * q^13 - q^15 + 12 * q^17 - 5 * q^19 - 3 * q^21 - q^23 + 14 * q^25 - 3 * q^27 + 11 * q^29 - 5 * q^31 + q^35 + 8 * q^37 - 3 * q^39 - 4 * q^41 - 17 * q^43 + q^45 + 3 * q^47 + 3 * q^49 - 12 * q^51 - 13 * q^53 + 2 * q^55 + 5 * q^57 - 6 * q^59 + 6 * q^61 + 3 * q^63 + q^65 + 2 * q^67 + q^69 + 22 * q^71 + 9 * q^73 - 14 * q^75 + 3 * q^79 + 3 * q^81 + 23 * q^83 + 4 * q^85 - 11 * q^87 - q^89 + 3 * q^91 + 5 * q^93 + 25 * q^95 + q^97

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−3.90180	−1.74494	−0.872469	−	0.488670i	$-0.837482\pi$
$5$	−3.90180	−1.74494	−0.872469	+	0.488670i	$0.837482\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.32224	0.398671	0.199335	−	0.979931i	$-0.436122\pi$
$11$	1.32224	0.398671	0.199335	+	0.979931i	$0.436122\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	3.90180	1.00744
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	−7.22404	−1.65731	−0.828654	−	0.559761i	$-0.810893\pi$
$19$	−7.22404	−1.65731	−0.828654	+	0.559761i	$0.810893\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	2.57956	0.537875	0.268938	−	0.963158i	$-0.413327\pi$
$23$	2.57956	0.537875	0.268938	+	0.963158i	$0.413327\pi$
$24$	0	0
$25$	10.2240	2.04481
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	9.22404	1.71286	0.856431	−	0.516262i	$-0.172677\pi$
$29$	9.22404	1.71286	0.856431	+	0.516262i	$0.172677\pi$
$30$	0	0
$31$	−4.57956	−0.822513	−0.411257	−	0.911520i	$-0.634910\pi$
$31$	−4.57956	−0.822513	−0.411257	+	0.911520i	$0.634910\pi$
$32$	0	0
$33$	−1.32224	−0.230173
$34$	0	0
$35$	−3.90180	−0.659525
$36$	0	0
$37$	−5.80360	−0.954106	−0.477053	−	0.878875i	$-0.658295\pi$
$37$	−5.80360	−0.954106	−0.477053	+	0.878875i	$0.658295\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−11.1258	−1.73756	−0.868782	−	0.495195i	$-0.835097\pi$
$41$	−11.1258	−1.73756	−0.868782	+	0.495195i	$0.835097\pi$
$42$	0	0
$43$	−8.57956	−1.30837	−0.654185	−	0.756334i	$-0.726988\pi$
$43$	−8.57956	−1.30837	−0.654185	+	0.756334i	$0.726988\pi$
$44$	0	0
$45$	−3.90180	−0.581646
$46$	0	0
$47$	−11.7054	−1.70741	−0.853704	−	0.520759i	$-0.825649\pi$
$47$	−11.7054	−1.70741	−0.853704	+	0.520759i	$0.825649\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−1.42044	−0.195113	−0.0975563	−	0.995230i	$-0.531103\pi$
$53$	−1.42044	−0.195113	−0.0975563	+	0.995230i	$0.531103\pi$
$54$	0	0
$55$	−5.15912	−0.695655
$56$	0	0
$57$	7.22404	0.956848
$58$	0	0
$59$	−3.32224	−0.432519	−0.216259	−	0.976336i	$-0.569386\pi$
$59$	−3.32224	−0.432519	−0.216259	+	0.976336i	$0.569386\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−3.90180	−0.483959
$66$	0	0
$67$	−7.80360	−0.953361	−0.476681	−	0.879077i	$-0.658160\pi$
$67$	−7.80360	−0.953361	−0.476681	+	0.879077i	$0.658160\pi$
$68$	0	0
$69$	−2.57956	−0.310542
$70$	0	0
$71$	14.4814	1.71862	0.859311	−	0.511454i	$-0.170893\pi$
$71$	14.4814	1.71862	0.859311	+	0.511454i	$0.170893\pi$
$72$	0	0
$73$	14.3832	1.68342	0.841711	−	0.539929i	$-0.181549\pi$
$73$	14.3832	1.68342	0.841711	+	0.539929i	$0.181549\pi$
$74$	0	0
$75$	−10.2240	−1.18057
$76$	0	0
$77$	1.32224	0.150683
$78$	0	0
$79$	12.3832	1.39321	0.696607	−	0.717453i	$-0.254692\pi$
$79$	12.3832	1.39321	0.696607	+	0.717453i	$0.254692\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.9018	1.30639	0.653196	−	0.757189i	$-0.273428\pi$
$83$	11.9018	1.30639	0.653196	+	0.757189i	$0.273428\pi$
$84$	0	0
$85$	−15.6072	−1.69284
$86$	0	0
$87$	−9.22404	−0.988921
$88$	0	0
$89$	1.25732	0.133275	0.0666377	−	0.997777i	$-0.478773\pi$
$89$	1.25732	0.133275	0.0666377	+	0.997777i	$0.478773\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	4.57956	0.474878
$94$	0	0
$95$	28.1868	2.89190
$96$	0	0
$97$	−2.57956	−0.261915	−0.130957	−	0.991388i	$-0.541805\pi$
$97$	−2.57956	−0.261915	−0.130957	+	0.991388i	$0.541805\pi$
$98$	0	0
$99$	1.32224	0.132890
$100$	0	0
$101$	15.8036	1.57252	0.786258	−	0.617898i	$-0.212015\pi$
$101$	15.8036	1.57252	0.786258	+	0.617898i	$0.212015\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	3.90180	0.380777
$106$	0	0
$107$	−3.15912	−0.305403	−0.152702	−	0.988272i	$-0.548797\pi$
$107$	−3.15912	−0.305403	−0.152702	+	0.988272i	$0.548797\pi$
$108$	0	0
$109$	−3.35552	−0.321400	−0.160700	−	0.987003i	$-0.551375\pi$
$109$	−3.35552	−0.321400	−0.160700	+	0.987003i	$0.551375\pi$
$110$	0	0
$111$	5.80360	0.550853
$112$	0	0
$113$	6.38316	0.600477	0.300239	−	0.953864i	$-0.402934\pi$
$113$	6.38316	0.600477	0.300239	+	0.953864i	$0.402934\pi$
$114$	0	0
$115$	−10.0649	−0.938559
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	−9.25168	−0.841062
$122$	0	0
$123$	11.1258	1.00318
$124$	0	0
$125$	−20.3832	−1.82313
$126$	0	0
$127$	22.2517	1.97452	0.987259	−	0.159124i	$-0.0508669\pi$
$127$	22.2517	1.97452	0.987259	+	0.159124i	$0.0508669\pi$
$128$	0	0
$129$	8.57956	0.755388
$130$	0	0
$131$	5.35552	0.467914	0.233957	−	0.972247i	$-0.424833\pi$
$131$	5.35552	0.467914	0.233957	+	0.972247i	$0.424833\pi$
$132$	0	0
$133$	−7.22404	−0.626404
$134$	0	0
$135$	3.90180	0.335813
$136$	0	0
$137$	17.1258	1.46316	0.731580	−	0.681756i	$-0.238783\pi$
$137$	17.1258	1.46316	0.731580	+	0.681756i	$0.238783\pi$
$138$	0	0
$139$	−14.4481	−1.22547	−0.612735	−	0.790288i	$-0.709931\pi$
$139$	−14.4481	−1.22547	−0.612735	+	0.790288i	$0.709931\pi$
$140$	0	0
$141$	11.7054	0.985772
$142$	0	0
$143$	1.32224	0.110571
$144$	0	0
$145$	−35.9904	−2.98884
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−9.32224	−0.763708	−0.381854	−	0.924223i	$-0.624714\pi$
$149$	−9.32224	−0.763708	−0.381854	+	0.924223i	$0.624714\pi$
$150$	0	0
$151$	−15.6072	−1.27010	−0.635048	−	0.772473i	$-0.719020\pi$
$151$	−15.6072	−1.27010	−0.635048	+	0.772473i	$0.719020\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	17.8685	1.43523
$156$	0	0
$157$	7.15912	0.571360	0.285680	−	0.958325i	$-0.407781\pi$
$157$	7.15912	0.571360	0.285680	+	0.958325i	$0.407781\pi$
$158$	0	0
$159$	1.42044	0.112648
$160$	0	0
$161$	2.57956	0.203298
$162$	0	0
$163$	−6.64448	−0.520436	−0.260218	−	0.965550i	$-0.583794\pi$
$163$	−6.64448	−0.520436	−0.260218	+	0.965550i	$0.583794\pi$
$164$	0	0
$165$	5.15912	0.401637
$166$	0	0
$167$	−2.54628	−0.197037	−0.0985186	−	0.995135i	$-0.531410\pi$
$167$	−2.54628	−0.197037	−0.0985186	+	0.995135i	$0.531410\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−7.22404	−0.552436
$172$	0	0
$173$	22.2517	1.69176	0.845882	−	0.533370i	$-0.179075\pi$
$173$	22.2517	1.69176	0.845882	+	0.533370i	$0.179075\pi$
$174$	0	0
$175$	10.2240	0.772865
$176$	0	0
$177$	3.32224	0.249715
$178$	0	0
$179$	−2.38316	−0.178126	−0.0890628	−	0.996026i	$-0.528387\pi$
$179$	−2.38316	−0.178126	−0.0890628	+	0.996026i	$0.528387\pi$
$180$	0	0
$181$	17.6072	1.30873	0.654366	−	0.756178i	$-0.272935\pi$
$181$	17.6072	1.30873	0.654366	+	0.756178i	$0.272935\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	22.6445	1.66486
$186$	0	0
$187$	5.28896	0.386767
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	2.19640	0.158926	0.0794630	−	0.996838i	$-0.474679\pi$
$191$	2.19640	0.158926	0.0794630	+	0.996838i	$0.474679\pi$
$192$	0	0
$193$	−10.9627	−0.789114	−0.394557	−	0.918872i	$-0.629102\pi$
$193$	−10.9627	−0.789114	−0.394557	+	0.918872i	$0.629102\pi$
$194$	0	0
$195$	3.90180	0.279414
$196$	0	0
$197$	−2.67776	−0.190782	−0.0953912	−	0.995440i	$-0.530410\pi$
$197$	−2.67776	−0.190782	−0.0953912	+	0.995440i	$0.530410\pi$
$198$	0	0
$199$	15.6072	1.10636	0.553182	−	0.833060i	$-0.313413\pi$
$199$	15.6072	1.10636	0.553182	+	0.833060i	$0.313413\pi$
$200$	0	0
$201$	7.80360	0.550423
$202$	0	0
$203$	9.22404	0.647401
$204$	0	0
$205$	43.4108	3.03194
$206$	0	0
$207$	2.57956	0.179292
$208$	0	0
$209$	−9.55192	−0.660720
$210$	0	0
$211$	−16.5130	−1.13680	−0.568401	−	0.822752i	$-0.692438\pi$
$211$	−16.5130	−1.13680	−0.568401	+	0.822752i	$0.692438\pi$
$212$	0	0
$213$	−14.4814	−0.992246
$214$	0	0
$215$	33.4757	2.28303
$216$	0	0
$217$	−4.57956	−0.310881
$218$	0	0
$219$	−14.3832	−0.971924
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	3.61684	0.242202	0.121101	−	0.992640i	$-0.461358\pi$
$223$	3.61684	0.242202	0.121101	+	0.992640i	$0.461358\pi$
$224$	0	0
$225$	10.2240	0.681603
$226$	0	0
$227$	25.5739	1.69740	0.848700	−	0.528874i	$-0.177386\pi$
$227$	25.5739	1.69740	0.848700	+	0.528874i	$0.177386\pi$
$228$	0	0
$229$	12.2517	0.809614	0.404807	−	0.914402i	$-0.367339\pi$
$229$	12.2517	0.809614	0.404807	+	0.914402i	$0.367339\pi$
$230$	0	0
$231$	−1.32224	−0.0869970
$232$	0	0
$233$	1.22404	0.0801895	0.0400948	−	0.999196i	$-0.487234\pi$
$233$	1.22404	0.0801895	0.0400948	+	0.999196i	$0.487234\pi$
$234$	0	0
$235$	45.6721	2.97932
$236$	0	0
$237$	−12.3832	−0.804373
$238$	0	0
$239$	18.4814	1.19546	0.597730	−	0.801698i	$-0.296070\pi$
$239$	18.4814	1.19546	0.597730	+	0.801698i	$0.296070\pi$
$240$	0	0
$241$	14.7760	0.951803	0.475902	−	0.879499i	$-0.342122\pi$
$241$	14.7760	0.951803	0.475902	+	0.879499i	$0.342122\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−3.90180	−0.249277
$246$	0	0
$247$	−7.22404	−0.459655
$248$	0	0
$249$	−11.9018	−0.754246
$250$	0	0
$251$	0.196401	0.0123967	0.00619835	−	0.999981i	$-0.498027\pi$
$251$	0.196401	0.0123967	0.00619835	+	0.999981i	$0.498027\pi$
$252$	0	0
$253$	3.41080	0.214435
$254$	0	0
$255$	15.6072	0.977361
$256$	0	0
$257$	8.19640	0.511277	0.255639	−	0.966772i	$-0.417714\pi$
$257$	8.19640	0.511277	0.255639	+	0.966772i	$0.417714\pi$
$258$	0	0
$259$	−5.80360	−0.360618
$260$	0	0
$261$	9.22404	0.570954
$262$	0	0
$263$	−3.86852	−0.238543	−0.119272	−	0.992862i	$-0.538056\pi$
$263$	−3.86852	−0.238543	−0.119272	+	0.992862i	$0.538056\pi$
$264$	0	0
$265$	5.54228	0.340459
$266$	0	0
$267$	−1.25732	−0.0769466
$268$	0	0
$269$	5.35552	0.326532	0.163266	−	0.986582i	$-0.447797\pi$
$269$	5.35552	0.326532	0.163266	+	0.986582i	$0.447797\pi$
$270$	0	0
$271$	6.64448	0.403623	0.201812	−	0.979424i	$-0.435317\pi$
$271$	6.64448	0.403623	0.201812	+	0.979424i	$0.435317\pi$
$272$	0	0
$273$	−1.00000	−0.0605228
$274$	0	0
$275$	13.5186	0.815205
$276$	0	0
$277$	−7.86852	−0.472774	−0.236387	−	0.971659i	$-0.575963\pi$
$277$	−7.86852	−0.472774	−0.236387	+	0.971659i	$0.575963\pi$
$278$	0	0
$279$	−4.57956	−0.274171
$280$	0	0
$281$	2.28496	0.136309	0.0681546	−	0.997675i	$-0.478289\pi$
$281$	2.28496	0.136309	0.0681546	+	0.997675i	$0.478289\pi$
$282$	0	0
$283$	11.6072	0.689976	0.344988	−	0.938607i	$-0.387883\pi$
$283$	11.6072	0.689976	0.344988	+	0.938607i	$0.387883\pi$
$284$	0	0
$285$	−28.1868	−1.66964
$286$	0	0
$287$	−11.1258	−0.656738
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	2.57956	0.151216
$292$	0	0
$293$	−14.1535	−0.826855	−0.413428	−	0.910537i	$-0.635669\pi$
$293$	−14.1535	−0.826855	−0.413428	+	0.910537i	$0.635669\pi$
$294$	0	0
$295$	12.9627	0.754719
$296$	0	0
$297$	−1.32224	−0.0767242
$298$	0	0
$299$	2.57956	0.149180
$300$	0	0
$301$	−8.57956	−0.494518
$302$	0	0
$303$	−15.8036	−0.907893
$304$	0	0
$305$	−7.80360	−0.446833
$306$	0	0
$307$	−15.0276	−0.857673	−0.428836	−	0.903382i	$-0.641076\pi$
$307$	−15.0276	−0.857673	−0.428836	+	0.903382i	$0.641076\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	3.80360	0.215682	0.107841	−	0.994168i	$-0.465606\pi$
$311$	3.80360	0.215682	0.107841	+	0.994168i	$0.465606\pi$
$312$	0	0
$313$	−33.2144	−1.87739	−0.938694	−	0.344750i	$-0.887964\pi$
$313$	−33.2144	−1.87739	−0.938694	+	0.344750i	$0.887964\pi$
$314$	0	0
$315$	−3.90180	−0.219842
$316$	0	0
$317$	27.7703	1.55974	0.779868	−	0.625944i	$-0.215286\pi$
$317$	27.7703	1.55974	0.779868	+	0.625944i	$0.215286\pi$
$318$	0	0
$319$	12.1964	0.682867
$320$	0	0
$321$	3.15912	0.176325
$322$	0	0
$323$	−28.8962	−1.60783
$324$	0	0
$325$	10.2240	0.567128
$326$	0	0
$327$	3.35552	0.185561
$328$	0	0
$329$	−11.7054	−0.645340
$330$	0	0
$331$	10.4481	0.574278	0.287139	−	0.957889i	$-0.407296\pi$
$331$	10.4481	0.574278	0.287139	+	0.957889i	$0.407296\pi$
$332$	0	0
$333$	−5.80360	−0.318035
$334$	0	0
$335$	30.4481	1.66356
$336$	0	0
$337$	−18.5130	−1.00847	−0.504234	−	0.863567i	$-0.668225\pi$
$337$	−18.5130	−1.00847	−0.504234	+	0.863567i	$0.668225\pi$
$338$	0	0
$339$	−6.38316	−0.346686
$340$	0	0
$341$	−6.05528	−0.327912
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	10.0649	0.541877
$346$	0	0
$347$	25.4108	1.36412	0.682062	−	0.731295i	$-0.261084\pi$
$347$	25.4108	1.36412	0.682062	+	0.731295i	$0.261084\pi$
$348$	0	0
$349$	−22.5130	−1.20509	−0.602547	−	0.798084i	$-0.705847\pi$
$349$	−22.5130	−1.20509	−0.602547	+	0.798084i	$0.705847\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−1.83688	−0.0977671	−0.0488836	−	0.998804i	$-0.515566\pi$
$353$	−1.83688	−0.0977671	−0.0488836	+	0.998804i	$0.515566\pi$
$354$	0	0
$355$	−56.5034	−2.99889
$356$	0	0
$357$	−4.00000	−0.211702
$358$	0	0
$359$	30.4814	1.60874	0.804372	−	0.594126i	$-0.202502\pi$
$359$	30.4814	1.60874	0.804372	+	0.594126i	$0.202502\pi$
$360$	0	0
$361$	33.1868	1.74667
$362$	0	0
$363$	9.25168	0.485587
$364$	0	0
$365$	−56.1202	−2.93747
$366$	0	0
$367$	−10.4481	−0.545385	−0.272693	−	0.962101i	$-0.587914\pi$
$367$	−10.4481	−0.545385	−0.272693	+	0.962101i	$0.587914\pi$
$368$	0	0
$369$	−11.1258	−0.579188
$370$	0	0
$371$	−1.42044	−0.0737456
$372$	0	0
$373$	−4.25168	−0.220144	−0.110072	−	0.993924i	$-0.535108\pi$
$373$	−4.25168	−0.220144	−0.110072	+	0.993924i	$0.535108\pi$
$374$	0	0
$375$	20.3832	1.05258
$376$	0	0
$377$	9.22404	0.475062
$378$	0	0
$379$	29.1591	1.49780	0.748902	−	0.662681i	$-0.230582\pi$
$379$	29.1591	1.49780	0.748902	+	0.662681i	$0.230582\pi$
$380$	0	0
$381$	−22.2517	−1.13999
$382$	0	0
$383$	−10.9294	−0.558468	−0.279234	−	0.960223i	$-0.590081\pi$
$383$	−10.9294	−0.558468	−0.279234	+	0.960223i	$0.590081\pi$
$384$	0	0
$385$	−5.15912	−0.262933
$386$	0	0
$387$	−8.57956	−0.436123
$388$	0	0
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	0	0
$391$	10.3182	0.521816
$392$	0	0
$393$	−5.35552	−0.270150
$394$	0	0
$395$	−48.3166	−2.43107
$396$	0	0
$397$	36.6348	1.83865	0.919325	−	0.393499i	$-0.128736\pi$
$397$	36.6348	1.83865	0.919325	+	0.393499i	$0.128736\pi$
$398$	0	0
$399$	7.22404	0.361654
$400$	0	0
$401$	−5.12584	−0.255972	−0.127986	−	0.991776i	$-0.540851\pi$
$401$	−5.12584	−0.255972	−0.127986	+	0.991776i	$0.540851\pi$
$402$	0	0
$403$	−4.57956	−0.228124
$404$	0	0
$405$	−3.90180	−0.193882
$406$	0	0
$407$	−7.67375	−0.380374
$408$	0	0
$409$	18.3166	0.905698	0.452849	−	0.891587i	$-0.350408\pi$
$409$	18.3166	0.905698	0.452849	+	0.891587i	$0.350408\pi$
$410$	0	0
$411$	−17.1258	−0.844755
$412$	0	0
$413$	−3.32224	−0.163477
$414$	0	0
$415$	−46.4384	−2.27957
$416$	0	0
$417$	14.4481	0.707526
$418$	0	0
$419$	−10.5146	−0.513674	−0.256837	−	0.966455i	$-0.582680\pi$
$419$	−10.5146	−0.513674	−0.256837	+	0.966455i	$0.582680\pi$
$420$	0	0
$421$	21.7370	1.05940	0.529699	−	0.848185i	$-0.322305\pi$
$421$	21.7370	1.05940	0.529699	+	0.848185i	$0.322305\pi$
$422$	0	0
$423$	−11.7054	−0.569136
$424$	0	0
$425$	40.8962	1.98376
$426$	0	0
$427$	2.00000	0.0967868
$428$	0	0
$429$	−1.32224	−0.0638384
$430$	0	0
$431$	−0.929439	−0.0447695	−0.0223848	−	0.999749i	$-0.507126\pi$
$431$	−0.929439	−0.0447695	−0.0223848	+	0.999749i	$0.507126\pi$
$432$	0	0
$433$	4.44808	0.213761	0.106881	−	0.994272i	$-0.465914\pi$
$433$	4.44808	0.213761	0.106881	+	0.994272i	$0.465914\pi$
$434$	0	0
$435$	35.9904	1.72561
$436$	0	0
$437$	−18.6348	−0.891425
$438$	0	0
$439$	33.6625	1.60662	0.803311	−	0.595560i	$-0.203070\pi$
$439$	33.6625	1.60662	0.803311	+	0.595560i	$0.203070\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−16.6348	−0.790345	−0.395173	−	0.918607i	$-0.629315\pi$
$443$	−16.6348	−0.790345	−0.395173	+	0.918607i	$0.629315\pi$
$444$	0	0
$445$	−4.90580	−0.232557
$446$	0	0
$447$	9.32224	0.440927
$448$	0	0
$449$	27.3775	1.29203	0.646013	−	0.763327i	$-0.276435\pi$
$449$	27.3775	1.29203	0.646013	+	0.763327i	$0.276435\pi$
$450$	0	0
$451$	−14.7110	−0.692716
$452$	0	0
$453$	15.6072	0.733290
$454$	0	0
$455$	−3.90180	−0.182919
$456$	0	0
$457$	41.4108	1.93712	0.968558	−	0.248789i	$-0.0800326\pi$
$457$	41.4108	1.93712	0.968558	+	0.248789i	$0.0800326\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	29.4441	1.37135	0.685674	−	0.727909i	$-0.259508\pi$
$461$	29.4441	1.37135	0.685674	+	0.727909i	$0.259508\pi$
$462$	0	0
$463$	9.48536	0.440822	0.220411	−	0.975407i	$-0.429260\pi$
$463$	9.48536	0.440822	0.220411	+	0.975407i	$0.429260\pi$
$464$	0	0
$465$	−17.8685	−0.828633
$466$	0	0
$467$	7.41080	0.342931	0.171465	−	0.985190i	$-0.445150\pi$
$467$	7.41080	0.342931	0.171465	+	0.985190i	$0.445150\pi$
$468$	0	0
$469$	−7.80360	−0.360337
$470$	0	0
$471$	−7.15912	−0.329875
$472$	0	0
$473$	−11.3442	−0.521609
$474$	0	0
$475$	−73.8589	−3.38888
$476$	0	0
$477$	−1.42044	−0.0650375
$478$	0	0
$479$	−6.61284	−0.302148	−0.151074	−	0.988522i	$-0.548273\pi$
$479$	−6.61284	−0.302148	−0.151074	+	0.988522i	$0.548273\pi$
$480$	0	0
$481$	−5.80360	−0.264621
$482$	0	0
$483$	−2.57956	−0.117374
$484$	0	0
$485$	10.0649	0.457025
$486$	0	0
$487$	17.5519	0.795353	0.397677	−	0.917526i	$-0.369817\pi$
$487$	17.5519	0.795353	0.397677	+	0.917526i	$0.369817\pi$
$488$	0	0
$489$	6.64448	0.300474
$490$	0	0
$491$	−2.12984	−0.0961185	−0.0480593	−	0.998844i	$-0.515304\pi$
$491$	−2.12984	−0.0961185	−0.0480593	+	0.998844i	$0.515304\pi$
$492$	0	0
$493$	36.8962	1.66172
$494$	0	0
$495$	−5.15912	−0.231885
$496$	0	0
$497$	14.4814	0.649578
$498$	0	0
$499$	16.8962	0.756376	0.378188	−	0.925729i	$-0.376547\pi$
$499$	16.8962	0.756376	0.378188	+	0.925729i	$0.376547\pi$
$500$	0	0
$501$	2.54628	0.113759
$502$	0	0
$503$	−23.8036	−1.06135	−0.530675	−	0.847575i	$-0.678062\pi$
$503$	−23.8036	−1.06135	−0.530675	+	0.847575i	$0.678062\pi$
$504$	0	0
$505$	−61.6625	−2.74394
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−9.19076	−0.407373	−0.203687	−	0.979036i	$-0.565292\pi$
$509$	−9.19076	−0.407373	−0.203687	+	0.979036i	$0.565292\pi$
$510$	0	0
$511$	14.3832	0.636273
$512$	0	0
$513$	7.22404	0.318949
$514$	0	0
$515$	15.6072	0.687735
$516$	0	0
$517$	−15.4774	−0.680693
$518$	0	0
$519$	−22.2517	−0.976740
$520$	0	0
$521$	12.9627	0.567907	0.283953	−	0.958838i	$-0.408354\pi$
$521$	12.9627	0.567907	0.283953	+	0.958838i	$0.408354\pi$
$522$	0	0
$523$	16.7663	0.733140	0.366570	−	0.930391i	$-0.380532\pi$
$523$	16.7663	0.733140	0.366570	+	0.930391i	$0.380532\pi$
$524$	0	0
$525$	−10.2240	−0.446214
$526$	0	0
$527$	−18.3182	−0.797955
$528$	0	0
$529$	−16.3459	−0.710690
$530$	0	0
$531$	−3.32224	−0.144173
$532$	0	0
$533$	−11.1258	−0.481914
$534$	0	0
$535$	12.3262	0.532910
$536$	0	0
$537$	2.38316	0.102841
$538$	0	0
$539$	1.32224	0.0569529
$540$	0	0
$541$	−30.7663	−1.32275	−0.661374	−	0.750057i	$-0.730026\pi$
$541$	−30.7663	−1.32275	−0.661374	+	0.750057i	$0.730026\pi$
$542$	0	0
$543$	−17.6072	−0.755597
$544$	0	0
$545$	13.0926	0.560824
$546$	0	0
$547$	−21.7387	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$547$	−21.7387	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	−66.6348	−2.83874
$552$	0	0
$553$	12.3832	0.526585
$554$	0	0
$555$	−22.6445	−0.961205
$556$	0	0
$557$	42.4148	1.79717	0.898586	−	0.438797i	$-0.144595\pi$
$557$	42.4148	1.79717	0.898586	+	0.438797i	$0.144595\pi$
$558$	0	0
$559$	−8.57956	−0.362877
$560$	0	0
$561$	−5.28896	−0.223300
$562$	0	0
$563$	−32.0000	−1.34864	−0.674320	−	0.738440i	$-0.735563\pi$
$563$	−32.0000	−1.34864	−0.674320	+	0.738440i	$0.735563\pi$
$564$	0	0
$565$	−24.9058	−1.04780
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	11.9351	0.500344	0.250172	−	0.968201i	$-0.419513\pi$
$569$	11.9351	0.500344	0.250172	+	0.968201i	$0.419513\pi$
$570$	0	0
$571$	21.2793	0.890512	0.445256	−	0.895403i	$-0.353113\pi$
$571$	21.2793	0.890512	0.445256	+	0.895403i	$0.353113\pi$
$572$	0	0
$573$	−2.19640	−0.0917560
$574$	0	0
$575$	26.3735	1.09985
$576$	0	0
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	0	0
$579$	10.9627	0.455595
$580$	0	0
$581$	11.9018	0.493770
$582$	0	0
$583$	−1.87816	−0.0777856
$584$	0	0
$585$	−3.90180	−0.161320
$586$	0	0
$587$	26.2200	1.08222	0.541108	−	0.840953i	$-0.318005\pi$
$587$	26.2200	1.08222	0.541108	+	0.840953i	$0.318005\pi$
$588$	0	0
$589$	33.0829	1.36316
$590$	0	0
$591$	2.67776	0.110148
$592$	0	0
$593$	−19.5090	−0.801139	−0.400569	−	0.916266i	$-0.631188\pi$
$593$	−19.5090	−0.801139	−0.400569	+	0.916266i	$0.631188\pi$
$594$	0	0
$595$	−15.6072	−0.639833
$596$	0	0
$597$	−15.6072	−0.638760
$598$	0	0
$599$	14.9724	0.611754	0.305877	−	0.952071i	$-0.401050\pi$
$599$	14.9724	0.611754	0.305877	+	0.952071i	$0.401050\pi$
$600$	0	0
$601$	−25.2144	−1.02852	−0.514258	−	0.857635i	$-0.671933\pi$
$601$	−25.2144	−1.02852	−0.514258	+	0.857635i	$0.671933\pi$
$602$	0	0
$603$	−7.80360	−0.317787
$604$	0	0
$605$	36.0982	1.46760
$606$	0	0
$607$	10.0553	0.408131	0.204066	−	0.978957i	$-0.434584\pi$
$607$	10.0553	0.408131	0.204066	+	0.978957i	$0.434584\pi$
$608$	0	0
$609$	−9.22404	−0.373777
$610$	0	0
$611$	−11.7054	−0.473550
$612$	0	0
$613$	44.0553	1.77938	0.889688	−	0.456569i	$-0.150922\pi$
$613$	44.0553	1.77938	0.889688	+	0.456569i	$0.150922\pi$
$614$	0	0
$615$	−43.4108	−1.75049
$616$	0	0
$617$	28.8629	1.16198	0.580988	−	0.813912i	$-0.302666\pi$
$617$	28.8629	1.16198	0.580988	+	0.813912i	$0.302666\pi$
$618$	0	0
$619$	25.8589	1.03936	0.519678	−	0.854362i	$-0.326052\pi$
$619$	25.8589	1.03936	0.519678	+	0.854362i	$0.326052\pi$
$620$	0	0
$621$	−2.57956	−0.103514
$622$	0	0
$623$	1.25732	0.0503734
$624$	0	0
$625$	28.4108	1.13643
$626$	0	0
$627$	9.55192	0.381467
$628$	0	0
$629$	−23.2144	−0.925619
$630$	0	0
$631$	−12.1964	−0.485531	−0.242766	−	0.970085i	$-0.578055\pi$
$631$	−12.1964	−0.485531	−0.242766	+	0.970085i	$0.578055\pi$
$632$	0	0
$633$	16.5130	0.656333
$634$	0	0
$635$	−86.8216	−3.44541
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	14.4814	0.572874
$640$	0	0
$641$	−35.4757	−1.40121	−0.700603	−	0.713551i	$-0.747086\pi$
$641$	−35.4757	−1.40121	−0.700603	+	0.713551i	$0.747086\pi$
$642$	0	0
$643$	11.6072	0.457743	0.228872	−	0.973457i	$-0.426496\pi$
$643$	11.6072	0.457743	0.228872	+	0.973457i	$0.426496\pi$
$644$	0	0
$645$	−33.4757	−1.31811
$646$	0	0
$647$	−42.0553	−1.65336	−0.826682	−	0.562670i	$-0.809774\pi$
$647$	−42.0553	−1.65336	−0.826682	+	0.562670i	$0.809774\pi$
$648$	0	0
$649$	−4.39280	−0.172433
$650$	0	0
$651$	4.57956	0.179487
$652$	0	0
$653$	−8.25168	−0.322913	−0.161457	−	0.986880i	$-0.551619\pi$
$653$	−8.25168	−0.322913	−0.161457	+	0.986880i	$0.551619\pi$
$654$	0	0
$655$	−20.8962	−0.816481
$656$	0	0
$657$	14.3832	0.561140
$658$	0	0
$659$	−8.32788	−0.324408	−0.162204	−	0.986757i	$-0.551860\pi$
$659$	−8.32788	−0.324408	−0.162204	+	0.986757i	$0.551860\pi$
$660$	0	0
$661$	−43.5423	−1.69360	−0.846799	−	0.531913i	$-0.821473\pi$
$661$	−43.5423	−1.69360	−0.846799	+	0.531913i	$0.821473\pi$
$662$	0	0
$663$	−4.00000	−0.155347
$664$	0	0
$665$	28.1868	1.09304
$666$	0	0
$667$	23.7940	0.921306
$668$	0	0
$669$	−3.61684	−0.139835
$670$	0	0
$671$	2.64448	0.102089
$672$	0	0
$673$	12.1315	0.467634	0.233817	−	0.972281i	$-0.424878\pi$
$673$	12.1315	0.467634	0.233817	+	0.972281i	$0.424878\pi$
$674$	0	0
$675$	−10.2240	−0.393523
$676$	0	0
$677$	−24.7663	−0.951847	−0.475923	−	0.879487i	$-0.657886\pi$
$677$	−24.7663	−0.951847	−0.475923	+	0.879487i	$0.657886\pi$
$678$	0	0
$679$	−2.57956	−0.0989944
$680$	0	0
$681$	−25.5739	−0.979995
$682$	0	0
$683$	−35.4441	−1.35623	−0.678115	−	0.734956i	$-0.737203\pi$
$683$	−35.4441	−1.35623	−0.678115	+	0.734956i	$0.737203\pi$
$684$	0	0
$685$	−66.8216	−2.55312
$686$	0	0
$687$	−12.2517	−0.467431
$688$	0	0
$689$	−1.42044	−0.0541145
$690$	0	0
$691$	−3.48700	−0.132652	−0.0663258	−	0.997798i	$-0.521128\pi$
$691$	−3.48700	−0.132652	−0.0663258	+	0.997798i	$0.521128\pi$
$692$	0	0
$693$	1.32224	0.0502278
$694$	0	0
$695$	56.3735	2.13837
$696$	0	0
$697$	−44.5034	−1.68568
$698$	0	0
$699$	−1.22404	−0.0462975
$700$	0	0
$701$	−30.1868	−1.14014	−0.570069	−	0.821597i	$-0.693084\pi$
$701$	−30.1868	−1.14014	−0.570069	+	0.821597i	$0.693084\pi$
$702$	0	0
$703$	41.9254	1.58125
$704$	0	0
$705$	−45.6721	−1.72011
$706$	0	0
$707$	15.8036	0.594356
$708$	0	0
$709$	17.6738	0.663752	0.331876	−	0.943323i	$-0.392318\pi$
$709$	17.6738	0.663752	0.331876	+	0.943323i	$0.392318\pi$
$710$	0	0
$711$	12.3832	0.464405
$712$	0	0
$713$	−11.8132	−0.442409
$714$	0	0
$715$	−5.15912	−0.192940
$716$	0	0
$717$	−18.4814	−0.690199
$718$	0	0
$719$	−39.0180	−1.45513	−0.727563	−	0.686041i	$-0.759347\pi$
$719$	−39.0180	−1.45513	−0.727563	+	0.686041i	$0.759347\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	0	0
$723$	−14.7760	−0.549524
$724$	0	0
$725$	94.3070	3.50247
$726$	0	0
$727$	16.7663	0.621828	0.310914	−	0.950438i	$-0.399365\pi$
$727$	16.7663	0.621828	0.310914	+	0.950438i	$0.399365\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−34.3182	−1.26931
$732$	0	0
$733$	−21.6168	−0.798436	−0.399218	−	0.916856i	$-0.630718\pi$
$733$	−21.6168	−0.798436	−0.399218	+	0.916856i	$0.630718\pi$
$734$	0	0
$735$	3.90180	0.143920
$736$	0	0
$737$	−10.3182	−0.380077
$738$	0	0
$739$	26.3815	0.970460	0.485230	−	0.874387i	$-0.338736\pi$
$739$	26.3815	0.970460	0.485230	+	0.874387i	$0.338736\pi$
$740$	0	0
$741$	7.22404	0.265382
$742$	0	0
$743$	9.45209	0.346763	0.173382	−	0.984855i	$-0.444531\pi$
$743$	9.45209	0.346763	0.173382	+	0.984855i	$0.444531\pi$
$744$	0	0
$745$	36.3735	1.33262
$746$	0	0
$747$	11.9018	0.435464
$748$	0	0
$749$	−3.15912	−0.115432
$750$	0	0
$751$	−9.86852	−0.360107	−0.180054	−	0.983657i	$-0.557627\pi$
$751$	−9.86852	−0.360107	−0.180054	+	0.983657i	$0.557627\pi$
$752$	0	0
$753$	−0.196401	−0.00715724
$754$	0	0
$755$	60.8962	2.21624
$756$	0	0
$757$	−38.5130	−1.39978	−0.699889	−	0.714251i	$-0.746767\pi$
$757$	−38.5130	−1.39978	−0.699889	+	0.714251i	$0.746767\pi$
$758$	0	0
$759$	−3.41080	−0.123804
$760$	0	0
$761$	22.3499	0.810183	0.405091	−	0.914276i	$-0.367240\pi$
$761$	22.3499	0.810183	0.405091	+	0.914276i	$0.367240\pi$
$762$	0	0
$763$	−3.35552	−0.121478
$764$	0	0
$765$	−15.6072	−0.564279
$766$	0	0
$767$	−3.32224	−0.119959
$768$	0	0
$769$	−22.3832	−0.807157	−0.403579	−	0.914945i	$-0.632234\pi$
$769$	−22.3832	−0.807157	−0.403579	+	0.914945i	$0.632234\pi$
$770$	0	0
$771$	−8.19640	−0.295186
$772$	0	0
$773$	33.7703	1.21463	0.607317	−	0.794460i	$-0.292246\pi$
$773$	33.7703	1.21463	0.607317	+	0.794460i	$0.292246\pi$
$774$	0	0
$775$	−46.8216	−1.68188
$776$	0	0
$777$	5.80360	0.208203
$778$	0	0
$779$	80.3735	2.87968
$780$	0	0
$781$	19.1478	0.685164
$782$	0	0
$783$	−9.22404	−0.329640
$784$	0	0
$785$	−27.9334	−0.996987
$786$	0	0
$787$	45.2793	1.61403	0.807017	−	0.590529i	$-0.201081\pi$
$787$	45.2793	1.61403	0.807017	+	0.590529i	$0.201081\pi$
$788$	0	0
$789$	3.86852	0.137723
$790$	0	0
$791$	6.38316	0.226959
$792$	0	0
$793$	2.00000	0.0710221
$794$	0	0
$795$	−5.54228	−0.196564
$796$	0	0
$797$	48.3735	1.71348	0.856739	−	0.515750i	$-0.172487\pi$
$797$	48.3735	1.71348	0.856739	+	0.515750i	$0.172487\pi$
$798$	0	0
$799$	−46.8216	−1.65643
$800$	0	0
$801$	1.25732	0.0444252
$802$	0	0
$803$	19.0180	0.671131
$804$	0	0
$805$	−10.0649	−0.354742
$806$	0	0
$807$	−5.35552	−0.188523
$808$	0	0
$809$	−3.93508	−0.138350	−0.0691750	−	0.997605i	$-0.522037\pi$
$809$	−3.93508	−0.138350	−0.0691750	+	0.997605i	$0.522037\pi$
$810$	0	0
$811$	−33.2890	−1.16893	−0.584467	−	0.811418i	$-0.698696\pi$
$811$	−33.2890	−1.16893	−0.584467	+	0.811418i	$0.698696\pi$
$812$	0	0
$813$	−6.64448	−0.233032
$814$	0	0
$815$	25.9254	0.908128
$816$	0	0
$817$	61.9791	2.16837
$818$	0	0
$819$	1.00000	0.0349428
$820$	0	0
$821$	27.5739	0.962336	0.481168	−	0.876628i	$-0.340213\pi$
$821$	27.5739	0.962336	0.481168	+	0.876628i	$0.340213\pi$
$822$	0	0
$823$	−9.68176	−0.337485	−0.168743	−	0.985660i	$-0.553971\pi$
$823$	−9.68176	−0.337485	−0.168743	+	0.985660i	$0.553971\pi$
$824$	0	0
$825$	−13.5186	−0.470659
$826$	0	0
$827$	13.1924	0.458745	0.229372	−	0.973339i	$-0.426333\pi$
$827$	13.1924	0.458745	0.229372	+	0.973339i	$0.426333\pi$
$828$	0	0
$829$	2.89616	0.100588	0.0502939	−	0.998734i	$-0.483984\pi$
$829$	2.89616	0.100588	0.0502939	+	0.998734i	$0.483984\pi$
$830$	0	0
$831$	7.86852	0.272956
$832$	0	0
$833$	4.00000	0.138592
$834$	0	0
$835$	9.93508	0.343818
$836$	0	0
$837$	4.57956	0.158293
$838$	0	0
$839$	2.99600	0.103433	0.0517166	−	0.998662i	$-0.483531\pi$
$839$	2.99600	0.103433	0.0517166	+	0.998662i	$0.483531\pi$
$840$	0	0
$841$	56.0829	1.93389
$842$	0	0
$843$	−2.28496	−0.0786981
$844$	0	0
$845$	−3.90180	−0.134226
$846$	0	0
$847$	−9.25168	−0.317891
$848$	0	0
$849$	−11.6072	−0.398358
$850$	0	0
$851$	−14.9707	−0.513190
$852$	0	0
$853$	46.1868	1.58140	0.790702	−	0.612201i	$-0.209716\pi$
$853$	46.1868	1.58140	0.790702	+	0.612201i	$0.209716\pi$
$854$	0	0
$855$	28.1868	0.963967
$856$	0	0
$857$	20.8329	0.711637	0.355819	−	0.934555i	$-0.384202\pi$
$857$	20.8329	0.711637	0.355819	+	0.934555i	$0.384202\pi$
$858$	0	0
$859$	15.8702	0.541483	0.270741	−	0.962652i	$-0.412731\pi$
$859$	15.8702	0.541483	0.270741	+	0.962652i	$0.412731\pi$
$860$	0	0
$861$	11.1258	0.379168
$862$	0	0
$863$	14.4148	0.490686	0.245343	−	0.969436i	$-0.421099\pi$
$863$	14.4148	0.490686	0.245343	+	0.969436i	$0.421099\pi$
$864$	0	0
$865$	−86.8216	−2.95202
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	16.3735	0.555433
$870$	0	0
$871$	−7.80360	−0.264415
$872$	0	0
$873$	−2.57956	−0.0873048
$874$	0	0
$875$	−20.3832	−0.689077
$876$	0	0
$877$	−24.8409	−0.838817	−0.419408	−	0.907798i	$-0.637762\pi$
$877$	−24.8409	−0.838817	−0.419408	+	0.907798i	$0.637762\pi$
$878$	0	0
$879$	14.1535	0.477385
$880$	0	0
$881$	−25.0926	−0.845390	−0.422695	−	0.906272i	$-0.638916\pi$
$881$	−25.0926	−0.845390	−0.422695	+	0.906272i	$0.638916\pi$
$882$	0	0
$883$	3.67375	0.123632	0.0618158	−	0.998088i	$-0.480311\pi$
$883$	3.67375	0.123632	0.0618158	+	0.998088i	$0.480311\pi$
$884$	0	0
$885$	−12.9627	−0.435737
$886$	0	0
$887$	−14.3182	−0.480759	−0.240380	−	0.970679i	$-0.577272\pi$
$887$	−14.3182	−0.480759	−0.240380	+	0.970679i	$0.577272\pi$
$888$	0	0
$889$	22.2517	0.746297
$890$	0	0
$891$	1.32224	0.0442967
$892$	0	0
$893$	84.5603	2.82970
$894$	0	0
$895$	9.29861	0.310818
$896$	0	0
$897$	−2.57956	−0.0861290
$898$	0	0
$899$	−42.2420	−1.40885
$900$	0	0
$901$	−5.68176	−0.189287
$902$	0	0
$903$	8.57956	0.285510
$904$	0	0
$905$	−68.6998	−2.28366
$906$	0	0
$907$	7.09420	0.235559	0.117779	−	0.993040i	$-0.462422\pi$
$907$	7.09420	0.235559	0.117779	+	0.993040i	$0.462422\pi$
$908$	0	0
$909$	15.8036	0.524172
$910$	0	0
$911$	27.6721	0.916818	0.458409	−	0.888741i	$-0.348419\pi$
$911$	27.6721	0.916818	0.458409	+	0.888741i	$0.348419\pi$
$912$	0	0
$913$	15.7370	0.520820
$914$	0	0
$915$	7.80360	0.257979
$916$	0	0
$917$	5.35552	0.176855
$918$	0	0
$919$	−31.6738	−1.04482	−0.522410	−	0.852694i	$-0.674967\pi$
$919$	−31.6738	−1.04482	−0.522410	+	0.852694i	$0.674967\pi$
$920$	0	0
$921$	15.0276	0.495178
$922$	0	0
$923$	14.4814	0.476660
$924$	0	0
$925$	−59.3362	−1.95096
$926$	0	0
$927$	−4.00000	−0.131377
$928$	0	0
$929$	−27.8352	−0.913245	−0.456622	−	0.889661i	$-0.650941\pi$
$929$	−27.8352	−0.913245	−0.456622	+	0.889661i	$0.650941\pi$
$930$	0	0
$931$	−7.22404	−0.236758
$932$	0	0
$933$	−3.80360	−0.124524
$934$	0	0
$935$	−20.6365	−0.674885
$936$	0	0
$937$	−29.7370	−0.971467	−0.485733	−	0.874107i	$-0.661447\pi$
$937$	−29.7370	−0.971467	−0.485733	+	0.874107i	$0.661447\pi$
$938$	0	0
$939$	33.2144	1.08391
$940$	0	0
$941$	−3.70540	−0.120793	−0.0603963	−	0.998174i	$-0.519236\pi$
$941$	−3.70540	−0.120793	−0.0603963	+	0.998174i	$0.519236\pi$
$942$	0	0
$943$	−28.6998	−0.934593
$944$	0	0
$945$	3.90180	0.126926
$946$	0	0
$947$	46.7883	1.52042	0.760208	−	0.649680i	$-0.225097\pi$
$947$	46.7883	1.52042	0.760208	+	0.649680i	$0.225097\pi$
$948$	0	0
$949$	14.3832	0.466897
$950$	0	0
$951$	−27.7703	−0.900514
$952$	0	0
$953$	−59.1495	−1.91604	−0.958020	−	0.286702i	$-0.907441\pi$
$953$	−59.1495	−1.91604	−0.958020	+	0.286702i	$0.907441\pi$
$954$	0	0
$955$	−8.56992	−0.277316
$956$	0	0
$957$	−12.1964	−0.394254
$958$	0	0
$959$	17.1258	0.553022
$960$	0	0
$961$	−10.0276	−0.323472
$962$	0	0
$963$	−3.15912	−0.101801
$964$	0	0
$965$	42.7743	1.37695
$966$	0	0
$967$	32.8329	1.05583	0.527917	−	0.849296i	$-0.322973\pi$
$967$	32.8329	1.05583	0.527917	+	0.849296i	$0.322973\pi$
$968$	0	0
$969$	28.8962	0.928278
$970$	0	0
$971$	−5.94472	−0.190775	−0.0953876	−	0.995440i	$-0.530409\pi$
$971$	−5.94472	−0.190775	−0.0953876	+	0.995440i	$0.530409\pi$
$972$	0	0
$973$	−14.4481	−0.463184
$974$	0	0
$975$	−10.2240	−0.327431
$976$	0	0
$977$	20.0333	0.640921	0.320461	−	0.947262i	$-0.396162\pi$
$977$	20.0333	0.640921	0.320461	+	0.947262i	$0.396162\pi$
$978$	0	0
$979$	1.66248	0.0531330
$980$	0	0
$981$	−3.35552	−0.107133
$982$	0	0
$983$	−20.4052	−0.650823	−0.325412	−	0.945572i	$-0.605503\pi$
$983$	−20.4052	−0.650823	−0.325412	+	0.945572i	$0.605503\pi$
$984$	0	0
$985$	10.4481	0.332904
$986$	0	0
$987$	11.7054	0.372587
$988$	0	0
$989$	−22.1315	−0.703740
$990$	0	0
$991$	16.3928	0.520735	0.260367	−	0.965510i	$-0.416156\pi$
$991$	16.3928	0.520735	0.260367	+	0.965510i	$0.416156\pi$
$992$	0	0
$993$	−10.4481	−0.331560
$994$	0	0
$995$	−60.8962	−1.93054
$996$	0	0
$997$	25.4774	0.806876	0.403438	−	0.915007i	$-0.367815\pi$
$997$	25.4774	0.806876	0.403438	+	0.915007i	$0.367815\pi$
$998$	0	0
$999$	5.80360	0.183618

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.bn.1.1		3
4.3	odd	2		1092.2.a.h.1.1	✓	3
12.11	even	2		3276.2.a.r.1.3		3
28.27	even	2		7644.2.a.t.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1092.2.a.h.1.1	✓	3	4.3	odd	2
3276.2.a.r.1.3		3	12.11	even	2
4368.2.a.bn.1.1		3	1.1	even	1	trivial
7644.2.a.t.1.3		3	28.27	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.90180 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.90180 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.32224 q^{11} +1.00000 q^{13} +3.90180 q^{15} +4.00000 q^{17} -7.22404 q^{19} -1.00000 q^{21} +2.57956 q^{23} +10.2240 q^{25} -1.00000 q^{27} +9.22404 q^{29} -4.57956 q^{31} -1.32224 q^{33} -3.90180 q^{35} -5.80360 q^{37} -1.00000 q^{39} -11.1258 q^{41} -8.57956 q^{43} -3.90180 q^{45} -11.7054 q^{47} +1.00000 q^{49} -4.00000 q^{51} -1.42044 q^{53} -5.15912 q^{55} +7.22404 q^{57} -3.32224 q^{59} +2.00000 q^{61} +1.00000 q^{63} -3.90180 q^{65} -7.80360 q^{67} -2.57956 q^{69} +14.4814 q^{71} +14.3832 q^{73} -10.2240 q^{75} +1.32224 q^{77} +12.3832 q^{79} +1.00000 q^{81} +11.9018 q^{83} -15.6072 q^{85} -9.22404 q^{87} +1.25732 q^{89} +1.00000 q^{91} +4.57956 q^{93} +28.1868 q^{95} -2.57956 q^{97} +1.32224 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + q^5 + 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{13} - q^{15} + 12 q^{17} - 5 q^{19} - 3 q^{21} - q^{23} + 14 q^{25} - 3 q^{27} + 11 q^{29} - 5 q^{31} + q^{35} + 8 q^{37} - 3 q^{39} - 4 q^{41} - 17 q^{43} + q^{45} + 3 q^{47} + 3 q^{49} - 12 q^{51} - 13 q^{53} + 2 q^{55} + 5 q^{57} - 6 q^{59} + 6 q^{61} + 3 q^{63} + q^{65} + 2 q^{67} + q^{69} + 22 q^{71} + 9 q^{73} - 14 q^{75} + 3 q^{79} + 3 q^{81} + 23 q^{83} + 4 q^{85} - 11 q^{87} - q^{89} + 3 q^{91} + 5 q^{93} + 25 q^{95} + q^{97}+O(q^{100}) \) 3 * q - 3 * q^3 + q^5 + 3 * q^7 + 3 * q^9 + 3 * q^13 - q^15 + 12 * q^17 - 5 * q^19 - 3 * q^21 - q^23 + 14 * q^25 - 3 * q^27 + 11 * q^29 - 5 * q^31 + q^35 + 8 * q^37 - 3 * q^39 - 4 * q^41 - 17 * q^43 + q^45 + 3 * q^47 + 3 * q^49 - 12 * q^51 - 13 * q^53 + 2 * q^55 + 5 * q^57 - 6 * q^59 + 6 * q^61 + 3 * q^63 + q^65 + 2 * q^67 + q^69 + 22 * q^71 + 9 * q^73 - 14 * q^75 + 3 * q^79 + 3 * q^81 + 23 * q^83 + 4 * q^85 - 11 * q^87 - q^89 + 3 * q^91 + 5 * q^93 + 25 * q^95 + q^97

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