LMFDB - Embedded newform 9200.2.a.dd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9200,2,Mod(1,9200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9200, 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("9200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9200 = 2^{4} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9200.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:	$73.4623698596$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 13x^{6} + 38x^{5} + 41x^{4} - 123x^{3} + 15x^{2} + 32x - 8 $ x^8 - 3x^7 - 13x^6 + 38x^5 + 41x^4 - 123x^3 + 15x^2 + 32*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.89996$ of defining polynomial
Character	$\chi$	$=$	9200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.89996 q^{3} -0.580879 q^{7} +5.40975 q^{9} +O(q^{10})$	$q-2.89996 q^{3} -0.580879 q^{7} +5.40975 q^{9} +0.0809094 q^{11} +3.02066 q^{13} -0.280529 q^{17} -4.72683 q^{19} +1.68452 q^{21} +1.00000 q^{23} -6.98817 q^{27} +1.38057 q^{29} +5.70770 q^{31} -0.234634 q^{33} -2.61536 q^{37} -8.75977 q^{39} +5.31570 q^{41} -7.30240 q^{43} -12.2117 q^{47} -6.66258 q^{49} +0.813523 q^{51} -6.64032 q^{53} +13.7076 q^{57} -3.70211 q^{59} +11.5113 q^{61} -3.14241 q^{63} +5.99193 q^{67} -2.89996 q^{69} +13.8324 q^{71} +1.14465 q^{73} -0.0469985 q^{77} +1.68167 q^{79} +4.03614 q^{81} -9.86877 q^{83} -4.00360 q^{87} +6.69235 q^{89} -1.75463 q^{91} -16.5521 q^{93} -12.8190 q^{97} +0.437699 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 3 q^{3} - 7 q^{7} + 11 q^{9}+O(q^{10}) $ 8 * q - 3 * q^3 - 7 * q^7 + 11 * q^9	$ 8 q - 3 q^{3} - 7 q^{7} + 11 q^{9} - 7 q^{11} - 11 q^{13} + 7 q^{17} - 11 q^{19} + 8 q^{23} - 12 q^{27} + 22 q^{29} - 9 q^{31} + 9 q^{33} - 4 q^{37} + 7 q^{41} - 22 q^{43} - 4 q^{47} + 39 q^{49} + 19 q^{51} - 4 q^{53} - 32 q^{59} + 17 q^{61} - 44 q^{63} + 4 q^{67} - 3 q^{69} - 15 q^{71} - 6 q^{73} - 18 q^{77} + 2 q^{79} + 24 q^{81} - 36 q^{83} + 4 q^{87} + 46 q^{89} + 35 q^{91} - 20 q^{93} - 3 q^{97} - 61 q^{99}+O(q^{100}) $ 8 * q - 3 * q^3 - 7 * q^7 + 11 * q^9 - 7 * q^11 - 11 * q^13 + 7 * q^17 - 11 * q^19 + 8 * q^23 - 12 * q^27 + 22 * q^29 - 9 * q^31 + 9 * q^33 - 4 * q^37 + 7 * q^41 - 22 * q^43 - 4 * q^47 + 39 * q^49 + 19 * q^51 - 4 * q^53 - 32 * q^59 + 17 * q^61 - 44 * q^63 + 4 * q^67 - 3 * q^69 - 15 * q^71 - 6 * q^73 - 18 * q^77 + 2 * q^79 + 24 * q^81 - 36 * q^83 + 4 * q^87 + 46 * q^89 + 35 * q^91 - 20 * q^93 - 3 * q^97 - 61 * q^99

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
$3$	−2.89996	−1.67429	−0.837145	−	0.546980i	$-0.815777\pi$
$3$	−2.89996	−1.67429	−0.837145	+	0.546980i	$0.815777\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.580879	−0.219552	−0.109776	−	0.993956i	$-0.535013\pi$
$7$	−0.580879	−0.219552	−0.109776	+	0.993956i	$0.535013\pi$
$8$	0	0
$9$	5.40975	1.80325
$10$	0	0
$11$	0.0809094	0.0243951	0.0121975	−	0.999926i	$-0.496117\pi$
$11$	0.0809094	0.0243951	0.0121975	+	0.999926i	$0.496117\pi$
$12$	0	0
$13$	3.02066	0.837779	0.418890	−	0.908037i	$-0.362419\pi$
$13$	3.02066	0.837779	0.418890	+	0.908037i	$0.362419\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.280529	−0.0680384	−0.0340192	−	0.999421i	$-0.510831\pi$
$17$	−0.280529	−0.0680384	−0.0340192	+	0.999421i	$0.510831\pi$
$18$	0	0
$19$	−4.72683	−1.08441	−0.542205	−	0.840246i	$-0.682410\pi$
$19$	−4.72683	−1.08441	−0.542205	+	0.840246i	$0.682410\pi$
$20$	0	0
$21$	1.68452	0.367593
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−6.98817	−1.34487
$28$	0	0
$29$	1.38057	0.256366	0.128183	−	0.991751i	$-0.459086\pi$
$29$	1.38057	0.256366	0.128183	+	0.991751i	$0.459086\pi$
$30$	0	0
$31$	5.70770	1.02513	0.512567	−	0.858647i	$-0.328695\pi$
$31$	5.70770	1.02513	0.512567	+	0.858647i	$0.328695\pi$
$32$	0	0
$33$	−0.234634	−0.0408445
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.61536	−0.429962	−0.214981	−	0.976618i	$-0.568969\pi$
$37$	−2.61536	−0.429962	−0.214981	+	0.976618i	$0.568969\pi$
$38$	0	0
$39$	−8.75977	−1.40269
$40$	0	0
$41$	5.31570	0.830172	0.415086	−	0.909782i	$-0.363751\pi$
$41$	5.31570	0.830172	0.415086	+	0.909782i	$0.363751\pi$
$42$	0	0
$43$	−7.30240	−1.11361	−0.556803	−	0.830645i	$-0.687972\pi$
$43$	−7.30240	−1.11361	−0.556803	+	0.830645i	$0.687972\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.2117	−1.78125	−0.890626	−	0.454736i	$-0.849734\pi$
$47$	−12.2117	−1.78125	−0.890626	+	0.454736i	$0.849734\pi$
$48$	0	0
$49$	−6.66258	−0.951797
$50$	0	0
$51$	0.813523	0.113916
$52$	0	0
$53$	−6.64032	−0.912117	−0.456059	−	0.889950i	$-0.650739\pi$
$53$	−6.64032	−0.912117	−0.456059	+	0.889950i	$0.650739\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	13.7076	1.81562
$58$	0	0
$59$	−3.70211	−0.481974	−0.240987	−	0.970528i	$-0.577471\pi$
$59$	−3.70211	−0.481974	−0.240987	+	0.970528i	$0.577471\pi$
$60$	0	0
$61$	11.5113	1.47387	0.736936	−	0.675962i	$-0.236272\pi$
$61$	11.5113	1.47387	0.736936	+	0.675962i	$0.236272\pi$
$62$	0	0
$63$	−3.14241	−0.395906
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.99193	0.732031	0.366015	−	0.930609i	$-0.380722\pi$
$67$	5.99193	0.732031	0.366015	+	0.930609i	$0.380722\pi$
$68$	0	0
$69$	−2.89996	−0.349114
$70$	0	0
$71$	13.8324	1.64160	0.820801	−	0.571214i	$-0.193527\pi$
$71$	13.8324	1.64160	0.820801	+	0.571214i	$0.193527\pi$
$72$	0	0
$73$	1.14465	0.133971	0.0669857	−	0.997754i	$-0.478662\pi$
$73$	1.14465	0.133971	0.0669857	+	0.997754i	$0.478662\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.0469985	−0.00535598
$78$	0	0
$79$	1.68167	0.189203	0.0946015	−	0.995515i	$-0.469842\pi$
$79$	1.68167	0.189203	0.0946015	+	0.995515i	$0.469842\pi$
$80$	0	0
$81$	4.03614	0.448460
$82$	0	0
$83$	−9.86877	−1.08324	−0.541619	−	0.840624i	$-0.682189\pi$
$83$	−9.86877	−1.08324	−0.541619	+	0.840624i	$0.682189\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−4.00360	−0.429231
$88$	0	0
$89$	6.69235	0.709388	0.354694	−	0.934982i	$-0.384585\pi$
$89$	6.69235	0.709388	0.354694	+	0.934982i	$0.384585\pi$
$90$	0	0
$91$	−1.75463	−0.183936
$92$	0	0
$93$	−16.5521	−1.71637
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−12.8190	−1.30158	−0.650788	−	0.759260i	$-0.725561\pi$
$97$	−12.8190	−1.30158	−0.650788	+	0.759260i	$0.725561\pi$
$98$	0	0
$99$	0.437699	0.0439904
$100$	0	0
$101$	−1.01958	−0.101452	−0.0507262	−	0.998713i	$-0.516154\pi$
$101$	−1.01958	−0.101452	−0.0507262	+	0.998713i	$0.516154\pi$
$102$	0	0
$103$	−2.36255	−0.232789	−0.116394	−	0.993203i	$-0.537134\pi$
$103$	−2.36255	−0.232789	−0.116394	+	0.993203i	$0.537134\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.58207	−0.539639	−0.269820	−	0.962911i	$-0.586964\pi$
$107$	−5.58207	−0.539639	−0.269820	+	0.962911i	$0.586964\pi$
$108$	0	0
$109$	6.09973	0.584248	0.292124	−	0.956380i	$-0.405638\pi$
$109$	6.09973	0.584248	0.292124	+	0.956380i	$0.405638\pi$
$110$	0	0
$111$	7.58443	0.719882
$112$	0	0
$113$	17.3938	1.63627	0.818137	−	0.575023i	$-0.195007\pi$
$113$	17.3938	1.63627	0.818137	+	0.575023i	$0.195007\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	16.3410	1.51073
$118$	0	0
$119$	0.162954	0.0149379
$120$	0	0
$121$	−10.9935	−0.999405
$122$	0	0
$123$	−15.4153	−1.38995
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.9365	1.05919	0.529597	−	0.848250i	$-0.322343\pi$
$127$	11.9365	1.05919	0.529597	+	0.848250i	$0.322343\pi$
$128$	0	0
$129$	21.1767	1.86450
$130$	0	0
$131$	9.84214	0.859912	0.429956	−	0.902850i	$-0.358529\pi$
$131$	9.84214	0.859912	0.429956	+	0.902850i	$0.358529\pi$
$132$	0	0
$133$	2.74572	0.238084
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.93256	−0.250545	−0.125273	−	0.992122i	$-0.539981\pi$
$137$	−2.93256	−0.250545	−0.125273	+	0.992122i	$0.539981\pi$
$138$	0	0
$139$	−4.96075	−0.420766	−0.210383	−	0.977619i	$-0.567471\pi$
$139$	−4.96075	−0.420766	−0.210383	+	0.977619i	$0.567471\pi$
$140$	0	0
$141$	35.4133	2.98233
$142$	0	0
$143$	0.244399	0.0204377
$144$	0	0
$145$	0	0
$146$	0	0
$147$	19.3212	1.59359
$148$	0	0
$149$	18.0375	1.47769	0.738846	−	0.673874i	$-0.235371\pi$
$149$	18.0375	1.47769	0.738846	+	0.673874i	$0.235371\pi$
$150$	0	0
$151$	18.1026	1.47317	0.736585	−	0.676345i	$-0.236437\pi$
$151$	18.1026	1.47317	0.736585	+	0.676345i	$0.236437\pi$
$152$	0	0
$153$	−1.51759	−0.122690
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−15.2698	−1.21867	−0.609333	−	0.792915i	$-0.708563\pi$
$157$	−15.2698	−1.21867	−0.609333	+	0.792915i	$0.708563\pi$
$158$	0	0
$159$	19.2566	1.52715
$160$	0	0
$161$	−0.580879	−0.0457797
$162$	0	0
$163$	−16.1488	−1.26487	−0.632435	−	0.774613i	$-0.717945\pi$
$163$	−16.1488	−1.26487	−0.632435	+	0.774613i	$0.717945\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.77915	0.524586	0.262293	−	0.964988i	$-0.415521\pi$
$167$	6.77915	0.524586	0.262293	+	0.964988i	$0.415521\pi$
$168$	0	0
$169$	−3.87563	−0.298126
$170$	0	0
$171$	−25.5710	−1.95546
$172$	0	0
$173$	−21.9428	−1.66828	−0.834140	−	0.551553i	$-0.814035\pi$
$173$	−21.9428	−1.66828	−0.834140	+	0.551553i	$0.814035\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	10.7360	0.806964
$178$	0	0
$179$	−6.76302	−0.505492	−0.252746	−	0.967533i	$-0.581334\pi$
$179$	−6.76302	−0.505492	−0.252746	+	0.967533i	$0.581334\pi$
$180$	0	0
$181$	12.1572	0.903641	0.451820	−	0.892109i	$-0.350775\pi$
$181$	12.1572	0.903641	0.451820	+	0.892109i	$0.350775\pi$
$182$	0	0
$183$	−33.3823	−2.46769
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−0.0226975	−0.00165980
$188$	0	0
$189$	4.05928	0.295269
$190$	0	0
$191$	−8.87427	−0.642119	−0.321060	−	0.947059i	$-0.604039\pi$
$191$	−8.87427	−0.642119	−0.321060	+	0.947059i	$0.604039\pi$
$192$	0	0
$193$	11.7026	0.842374	0.421187	−	0.906974i	$-0.361614\pi$
$193$	11.7026	0.842374	0.421187	+	0.906974i	$0.361614\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.330763	0.0235659	0.0117829	−	0.999931i	$-0.496249\pi$
$197$	0.330763	0.0235659	0.0117829	+	0.999931i	$0.496249\pi$
$198$	0	0
$199$	−9.72600	−0.689458	−0.344729	−	0.938702i	$-0.612029\pi$
$199$	−9.72600	−0.689458	−0.344729	+	0.938702i	$0.612029\pi$
$200$	0	0
$201$	−17.3763	−1.22563
$202$	0	0
$203$	−0.801945	−0.0562855
$204$	0	0
$205$	0	0
$206$	0	0
$207$	5.40975	0.376004
$208$	0	0
$209$	−0.382445	−0.0264543
$210$	0	0
$211$	6.63709	0.456916	0.228458	−	0.973554i	$-0.426632\pi$
$211$	6.63709	0.456916	0.228458	+	0.973554i	$0.426632\pi$
$212$	0	0
$213$	−40.1133	−2.74852
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−3.31548	−0.225070
$218$	0	0
$219$	−3.31944	−0.224307
$220$	0	0
$221$	−0.847383	−0.0570012
$222$	0	0
$223$	27.4323	1.83700	0.918500	−	0.395420i	$-0.129401\pi$
$223$	27.4323	1.83700	0.918500	+	0.395420i	$0.129401\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	20.6195	1.36856	0.684281	−	0.729218i	$-0.260116\pi$
$227$	20.6195	1.36856	0.684281	+	0.729218i	$0.260116\pi$
$228$	0	0
$229$	−22.2949	−1.47329	−0.736644	−	0.676280i	$-0.763591\pi$
$229$	−22.2949	−1.47329	−0.736644	+	0.676280i	$0.763591\pi$
$230$	0	0
$231$	0.136294	0.00896747
$232$	0	0
$233$	25.5331	1.67273	0.836365	−	0.548173i	$-0.184676\pi$
$233$	25.5331	1.67273	0.836365	+	0.548173i	$0.184676\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−4.87678	−0.316781
$238$	0	0
$239$	−21.6017	−1.39730	−0.698649	−	0.715465i	$-0.746215\pi$
$239$	−21.6017	−1.39730	−0.698649	+	0.715465i	$0.746215\pi$
$240$	0	0
$241$	21.6850	1.39685	0.698426	−	0.715682i	$-0.253884\pi$
$241$	21.6850	1.39685	0.698426	+	0.715682i	$0.253884\pi$
$242$	0	0
$243$	9.25988	0.594021
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−14.2781	−0.908496
$248$	0	0
$249$	28.6190	1.81366
$250$	0	0
$251$	−26.5991	−1.67892	−0.839461	−	0.543420i	$-0.817129\pi$
$251$	−26.5991	−1.67892	−0.839461	+	0.543420i	$0.817129\pi$
$252$	0	0
$253$	0.0809094	0.00508673
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.156392	0.00975548	0.00487774	−	0.999988i	$-0.498447\pi$
$257$	0.156392	0.00975548	0.00487774	+	0.999988i	$0.498447\pi$
$258$	0	0
$259$	1.51921	0.0943989
$260$	0	0
$261$	7.46855	0.462292
$262$	0	0
$263$	24.1557	1.48951	0.744753	−	0.667340i	$-0.232567\pi$
$263$	24.1557	1.48951	0.744753	+	0.667340i	$0.232567\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−19.4075	−1.18772
$268$	0	0
$269$	0.270387	0.0164858	0.00824289	−	0.999966i	$-0.497376\pi$
$269$	0.270387	0.0164858	0.00824289	+	0.999966i	$0.497376\pi$
$270$	0	0
$271$	−10.0041	−0.607704	−0.303852	−	0.952719i	$-0.598273\pi$
$271$	−10.0041	−0.607704	−0.303852	+	0.952719i	$0.598273\pi$
$272$	0	0
$273$	5.08837	0.307962
$274$	0	0
$275$	0	0
$276$	0	0
$277$	12.0842	0.726070	0.363035	−	0.931776i	$-0.381741\pi$
$277$	12.0842	0.726070	0.363035	+	0.931776i	$0.381741\pi$
$278$	0	0
$279$	30.8772	1.84857
$280$	0	0
$281$	22.2912	1.32978	0.664890	−	0.746941i	$-0.268478\pi$
$281$	22.2912	1.32978	0.664890	+	0.746941i	$0.268478\pi$
$282$	0	0
$283$	10.5506	0.627165	0.313583	−	0.949561i	$-0.398471\pi$
$283$	10.5506	0.627165	0.313583	+	0.949561i	$0.398471\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.08777	−0.182266
$288$	0	0
$289$	−16.9213	−0.995371
$290$	0	0
$291$	37.1746	2.17922
$292$	0	0
$293$	8.49506	0.496287	0.248143	−	0.968723i	$-0.420180\pi$
$293$	8.49506	0.496287	0.248143	+	0.968723i	$0.420180\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−0.565408	−0.0328083
$298$	0	0
$299$	3.02066	0.174689
$300$	0	0
$301$	4.24181	0.244494
$302$	0	0
$303$	2.95675	0.169861
$304$	0	0
$305$	0	0
$306$	0	0
$307$	16.1607	0.922339	0.461170	−	0.887312i	$-0.347430\pi$
$307$	16.1607	0.922339	0.461170	+	0.887312i	$0.347430\pi$
$308$	0	0
$309$	6.85128	0.389756
$310$	0	0
$311$	−33.3027	−1.88842	−0.944211	−	0.329340i	$-0.893174\pi$
$311$	−33.3027	−1.88842	−0.944211	+	0.329340i	$0.893174\pi$
$312$	0	0
$313$	−8.59232	−0.485666	−0.242833	−	0.970068i	$-0.578077\pi$
$313$	−8.59232	−0.485666	−0.242833	+	0.970068i	$0.578077\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−32.9109	−1.84846	−0.924230	−	0.381837i	$-0.875291\pi$
$317$	−32.9109	−1.84846	−0.924230	+	0.381837i	$0.875291\pi$
$318$	0	0
$319$	0.111701	0.00625407
$320$	0	0
$321$	16.1878	0.903513
$322$	0	0
$323$	1.32602	0.0737815
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−17.6889	−0.978201
$328$	0	0
$329$	7.09349	0.391077
$330$	0	0
$331$	−8.19245	−0.450298	−0.225149	−	0.974324i	$-0.572287\pi$
$331$	−8.19245	−0.450298	−0.225149	+	0.974324i	$0.572287\pi$
$332$	0	0
$333$	−14.1484	−0.775330
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−25.7372	−1.40199	−0.700996	−	0.713165i	$-0.747261\pi$
$337$	−25.7372	−1.40199	−0.700996	+	0.713165i	$0.747261\pi$
$338$	0	0
$339$	−50.4414	−2.73960
$340$	0	0
$341$	0.461806	0.0250082
$342$	0	0
$343$	7.93630	0.428520
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−12.1539	−0.652455	−0.326228	−	0.945291i	$-0.605778\pi$
$347$	−12.1539	−0.652455	−0.326228	+	0.945291i	$0.605778\pi$
$348$	0	0
$349$	4.51456	0.241659	0.120829	−	0.992673i	$-0.461445\pi$
$349$	4.51456	0.241659	0.120829	+	0.992673i	$0.461445\pi$
$350$	0	0
$351$	−21.1089	−1.12671
$352$	0	0
$353$	−29.7767	−1.58485	−0.792426	−	0.609968i	$-0.791182\pi$
$353$	−29.7767	−1.58485	−0.792426	+	0.609968i	$0.791182\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.472558	−0.0250104
$358$	0	0
$359$	4.22436	0.222953	0.111477	−	0.993767i	$-0.464442\pi$
$359$	4.22436	0.222953	0.111477	+	0.993767i	$0.464442\pi$
$360$	0	0
$361$	3.34296	0.175945
$362$	0	0
$363$	31.8805	1.67329
$364$	0	0
$365$	0	0
$366$	0	0
$367$	31.9394	1.66722	0.833611	−	0.552351i	$-0.186269\pi$
$367$	31.9394	1.66722	0.833611	+	0.552351i	$0.186269\pi$
$368$	0	0
$369$	28.7566	1.49701
$370$	0	0
$371$	3.85722	0.200257
$372$	0	0
$373$	−13.3524	−0.691361	−0.345681	−	0.938352i	$-0.612352\pi$
$373$	−13.3524	−0.691361	−0.345681	+	0.938352i	$0.612352\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.17024	0.214778
$378$	0	0
$379$	−20.0481	−1.02980	−0.514901	−	0.857250i	$-0.672171\pi$
$379$	−20.0481	−1.02980	−0.514901	+	0.857250i	$0.672171\pi$
$380$	0	0
$381$	−34.6153	−1.77340
$382$	0	0
$383$	12.7850	0.653281	0.326640	−	0.945149i	$-0.394083\pi$
$383$	12.7850	0.653281	0.326640	+	0.945149i	$0.394083\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−39.5042	−2.00811
$388$	0	0
$389$	5.69492	0.288744	0.144372	−	0.989523i	$-0.453884\pi$
$389$	5.69492	0.288744	0.144372	+	0.989523i	$0.453884\pi$
$390$	0	0
$391$	−0.280529	−0.0141870
$392$	0	0
$393$	−28.5418	−1.43974
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−11.2231	−0.563269	−0.281635	−	0.959522i	$-0.590877\pi$
$397$	−11.2231	−0.563269	−0.281635	+	0.959522i	$0.590877\pi$
$398$	0	0
$399$	−7.96246	−0.398622
$400$	0	0
$401$	−27.1979	−1.35820	−0.679098	−	0.734047i	$-0.737629\pi$
$401$	−27.1979	−1.35820	−0.679098	+	0.734047i	$0.737629\pi$
$402$	0	0
$403$	17.2410	0.858836
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.211607	−0.0104890
$408$	0	0
$409$	−28.2012	−1.39446	−0.697230	−	0.716848i	$-0.745584\pi$
$409$	−28.2012	−1.39446	−0.697230	+	0.716848i	$0.745584\pi$
$410$	0	0
$411$	8.50428	0.419485
$412$	0	0
$413$	2.15048	0.105818
$414$	0	0
$415$	0	0
$416$	0	0
$417$	14.3860	0.704484
$418$	0	0
$419$	−18.5731	−0.907356	−0.453678	−	0.891166i	$-0.649888\pi$
$419$	−18.5731	−0.907356	−0.453678	+	0.891166i	$0.649888\pi$
$420$	0	0
$421$	−16.6531	−0.811622	−0.405811	−	0.913957i	$-0.633011\pi$
$421$	−16.6531	−0.811622	−0.405811	+	0.913957i	$0.633011\pi$
$422$	0	0
$423$	−66.0620	−3.21204
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.68668	−0.323591
$428$	0	0
$429$	−0.708748	−0.0342187
$430$	0	0
$431$	0.446399	0.0215023	0.0107511	−	0.999942i	$-0.496578\pi$
$431$	0.446399	0.0215023	0.0107511	+	0.999942i	$0.496578\pi$
$432$	0	0
$433$	1.87406	0.0900616	0.0450308	−	0.998986i	$-0.485661\pi$
$433$	1.87406	0.0900616	0.0450308	+	0.998986i	$0.485661\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.72683	−0.226115
$438$	0	0
$439$	13.6607	0.651991	0.325995	−	0.945371i	$-0.394301\pi$
$439$	13.6607	0.651991	0.325995	+	0.945371i	$0.394301\pi$
$440$	0	0
$441$	−36.0429	−1.71633
$442$	0	0
$443$	21.6380	1.02805	0.514027	−	0.857774i	$-0.328153\pi$
$443$	21.6380	1.02805	0.514027	+	0.857774i	$0.328153\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−52.3080	−2.47409
$448$	0	0
$449$	−22.3547	−1.05499	−0.527493	−	0.849560i	$-0.676868\pi$
$449$	−22.3547	−1.05499	−0.527493	+	0.849560i	$0.676868\pi$
$450$	0	0
$451$	0.430090	0.0202521
$452$	0	0
$453$	−52.4968	−2.46652
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−13.8614	−0.648408	−0.324204	−	0.945987i	$-0.605096\pi$
$457$	−13.8614	−0.648408	−0.324204	+	0.945987i	$0.605096\pi$
$458$	0	0
$459$	1.96039	0.0915030
$460$	0	0
$461$	−26.4880	−1.23367	−0.616834	−	0.787093i	$-0.711585\pi$
$461$	−26.4880	−1.23367	−0.616834	+	0.787093i	$0.711585\pi$
$462$	0	0
$463$	−31.3980	−1.45919	−0.729594	−	0.683881i	$-0.760291\pi$
$463$	−31.3980	−1.45919	−0.729594	+	0.683881i	$0.760291\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.27210	−0.151415	−0.0757074	−	0.997130i	$-0.524121\pi$
$467$	−3.27210	−0.151415	−0.0757074	+	0.997130i	$0.524121\pi$
$468$	0	0
$469$	−3.48058	−0.160718
$470$	0	0
$471$	44.2819	2.04040
$472$	0	0
$473$	−0.590833	−0.0271665
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−35.9224	−1.64478
$478$	0	0
$479$	30.4865	1.39296	0.696482	−	0.717574i	$-0.254747\pi$
$479$	30.4865	1.39296	0.696482	+	0.717574i	$0.254747\pi$
$480$	0	0
$481$	−7.90010	−0.360214
$482$	0	0
$483$	1.68452	0.0766485
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12.8997	0.584542	0.292271	−	0.956336i	$-0.405589\pi$
$487$	12.8997	0.584542	0.292271	+	0.956336i	$0.405589\pi$
$488$	0	0
$489$	46.8308	2.11776
$490$	0	0
$491$	−1.53828	−0.0694217	−0.0347109	−	0.999397i	$-0.511051\pi$
$491$	−1.53828	−0.0694217	−0.0347109	+	0.999397i	$0.511051\pi$
$492$	0	0
$493$	−0.387291	−0.0174427
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−8.03494	−0.360416
$498$	0	0
$499$	−18.0529	−0.808160	−0.404080	−	0.914724i	$-0.632408\pi$
$499$	−18.0529	−0.808160	−0.404080	+	0.914724i	$0.632408\pi$
$500$	0	0
$501$	−19.6592	−0.878310
$502$	0	0
$503$	−7.35252	−0.327832	−0.163916	−	0.986474i	$-0.552413\pi$
$503$	−7.35252	−0.327832	−0.163916	+	0.986474i	$0.552413\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	11.2392	0.499149
$508$	0	0
$509$	−29.3564	−1.30120	−0.650600	−	0.759421i	$-0.725483\pi$
$509$	−29.3564	−1.30120	−0.650600	+	0.759421i	$0.725483\pi$
$510$	0	0
$511$	−0.664904	−0.0294136
$512$	0	0
$513$	33.0319	1.45839
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−0.988037	−0.0434538
$518$	0	0
$519$	63.6331	2.79318
$520$	0	0
$521$	−38.1800	−1.67270	−0.836349	−	0.548198i	$-0.815314\pi$
$521$	−38.1800	−1.67270	−0.836349	+	0.548198i	$0.815314\pi$
$522$	0	0
$523$	−37.7527	−1.65081	−0.825405	−	0.564540i	$-0.809053\pi$
$523$	−37.7527	−1.65081	−0.825405	+	0.564540i	$0.809053\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.60118	−0.0697484
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−20.0275	−0.869119
$532$	0	0
$533$	16.0569	0.695501
$534$	0	0
$535$	0	0
$536$	0	0
$537$	19.6125	0.846341
$538$	0	0
$539$	−0.539065	−0.0232192
$540$	0	0
$541$	−6.30694	−0.271157	−0.135578	−	0.990767i	$-0.543289\pi$
$541$	−6.30694	−0.271157	−0.135578	+	0.990767i	$0.543289\pi$
$542$	0	0
$543$	−35.2555	−1.51296
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−40.9574	−1.75121	−0.875606	−	0.483027i	$-0.839537\pi$
$547$	−40.9574	−1.75121	−0.875606	+	0.483027i	$0.839537\pi$
$548$	0	0
$549$	62.2733	2.65776
$550$	0	0
$551$	−6.52574	−0.278006
$552$	0	0
$553$	−0.976848	−0.0415398
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−30.6143	−1.29717	−0.648586	−	0.761142i	$-0.724639\pi$
$557$	−30.6143	−1.29717	−0.648586	+	0.761142i	$0.724639\pi$
$558$	0	0
$559$	−22.0581	−0.932956
$560$	0	0
$561$	0.0658217	0.00277899
$562$	0	0
$563$	19.8221	0.835400	0.417700	−	0.908585i	$-0.362836\pi$
$563$	19.8221	0.835400	0.417700	+	0.908585i	$0.362836\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.34451	−0.0984600
$568$	0	0
$569$	34.8830	1.46237	0.731185	−	0.682179i	$-0.238968\pi$
$569$	34.8830	1.46237	0.731185	+	0.682179i	$0.238968\pi$
$570$	0	0
$571$	−10.5730	−0.442466	−0.221233	−	0.975221i	$-0.571008\pi$
$571$	−10.5730	−0.442466	−0.221233	+	0.975221i	$0.571008\pi$
$572$	0	0
$573$	25.7350	1.07509
$574$	0	0
$575$	0	0
$576$	0	0
$577$	12.7839	0.532201	0.266101	−	0.963945i	$-0.414265\pi$
$577$	12.7839	0.532201	0.266101	+	0.963945i	$0.414265\pi$
$578$	0	0
$579$	−33.9371	−1.41038
$580$	0	0
$581$	5.73256	0.237827
$582$	0	0
$583$	−0.537264	−0.0222512
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10.4015	−0.429317	−0.214658	−	0.976689i	$-0.568864\pi$
$587$	−10.4015	−0.429317	−0.214658	+	0.976689i	$0.568864\pi$
$588$	0	0
$589$	−26.9794	−1.11166
$590$	0	0
$591$	−0.959198	−0.0394561
$592$	0	0
$593$	−25.7468	−1.05729	−0.528647	−	0.848842i	$-0.677300\pi$
$593$	−25.7468	−1.05729	−0.528647	+	0.848842i	$0.677300\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	28.2050	1.15435
$598$	0	0
$599$	−21.2360	−0.867680	−0.433840	−	0.900990i	$-0.642842\pi$
$599$	−21.2360	−0.867680	−0.433840	+	0.900990i	$0.642842\pi$
$600$	0	0
$601$	−10.9088	−0.444980	−0.222490	−	0.974935i	$-0.571418\pi$
$601$	−10.9088	−0.444980	−0.222490	+	0.974935i	$0.571418\pi$
$602$	0	0
$603$	32.4148	1.32003
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−6.15713	−0.249910	−0.124955	−	0.992162i	$-0.539879\pi$
$607$	−6.15713	−0.249910	−0.124955	+	0.992162i	$0.539879\pi$
$608$	0	0
$609$	2.32561	0.0942383
$610$	0	0
$611$	−36.8872	−1.49230
$612$	0	0
$613$	−20.3588	−0.822283	−0.411142	−	0.911572i	$-0.634870\pi$
$613$	−20.3588	−0.822283	−0.411142	+	0.911572i	$0.634870\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13.9290	0.560759	0.280380	−	0.959889i	$-0.409540\pi$
$617$	13.9290	0.560759	0.280380	+	0.959889i	$0.409540\pi$
$618$	0	0
$619$	45.4617	1.82726	0.913629	−	0.406548i	$-0.133268\pi$
$619$	45.4617	1.82726	0.913629	+	0.406548i	$0.133268\pi$
$620$	0	0
$621$	−6.98817	−0.280426
$622$	0	0
$623$	−3.88745	−0.155747
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.10907	0.0442922
$628$	0	0
$629$	0.733685	0.0292540
$630$	0	0
$631$	−39.7205	−1.58125	−0.790624	−	0.612303i	$-0.790243\pi$
$631$	−39.7205	−1.58125	−0.790624	+	0.612303i	$0.790243\pi$
$632$	0	0
$633$	−19.2473	−0.765011
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−20.1254	−0.797396
$638$	0	0
$639$	74.8297	2.96022
$640$	0	0
$641$	19.4730	0.769137	0.384568	−	0.923096i	$-0.374350\pi$
$641$	19.4730	0.769137	0.384568	+	0.923096i	$0.374350\pi$
$642$	0	0
$643$	−36.3269	−1.43259	−0.716297	−	0.697795i	$-0.754164\pi$
$643$	−36.3269	−1.43259	−0.716297	+	0.697795i	$0.754164\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.18108	0.164375	0.0821875	−	0.996617i	$-0.473809\pi$
$647$	4.18108	0.164375	0.0821875	+	0.996617i	$0.473809\pi$
$648$	0	0
$649$	−0.299535	−0.0117578
$650$	0	0
$651$	9.61475	0.376832
$652$	0	0
$653$	−30.5378	−1.19504	−0.597519	−	0.801855i	$-0.703847\pi$
$653$	−30.5378	−1.19504	−0.597519	+	0.801855i	$0.703847\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.19228	0.241584
$658$	0	0
$659$	39.7062	1.54673	0.773366	−	0.633960i	$-0.218571\pi$
$659$	39.7062	1.54673	0.773366	+	0.633960i	$0.218571\pi$
$660$	0	0
$661$	4.53963	0.176571	0.0882857	−	0.996095i	$-0.471861\pi$
$661$	4.53963	0.176571	0.0882857	+	0.996095i	$0.471861\pi$
$662$	0	0
$663$	2.45737	0.0954365
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.38057	0.0534560
$668$	0	0
$669$	−79.5524	−3.07567
$670$	0	0
$671$	0.931373	0.0359553
$672$	0	0
$673$	−28.2447	−1.08875	−0.544377	−	0.838841i	$-0.683234\pi$
$673$	−28.2447	−1.08875	−0.544377	+	0.838841i	$0.683234\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.0502	0.386260	0.193130	−	0.981173i	$-0.438136\pi$
$677$	10.0502	0.386260	0.193130	+	0.981173i	$0.438136\pi$
$678$	0	0
$679$	7.44630	0.285763
$680$	0	0
$681$	−59.7956	−2.29137
$682$	0	0
$683$	−13.1814	−0.504372	−0.252186	−	0.967679i	$-0.581149\pi$
$683$	−13.1814	−0.504372	−0.252186	+	0.967679i	$0.581149\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	64.6542	2.46671
$688$	0	0
$689$	−20.0581	−0.764153
$690$	0	0
$691$	−48.3908	−1.84087	−0.920436	−	0.390894i	$-0.872166\pi$
$691$	−48.3908	−1.84087	−0.920436	+	0.390894i	$0.872166\pi$
$692$	0	0
$693$	−0.254250	−0.00965817
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−1.49121	−0.0564836
$698$	0	0
$699$	−74.0449	−2.80064
$700$	0	0
$701$	34.4310	1.30044	0.650220	−	0.759746i	$-0.274677\pi$
$701$	34.4310	1.30044	0.650220	+	0.759746i	$0.274677\pi$
$702$	0	0
$703$	12.3624	0.466256
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.592255	0.0222740
$708$	0	0
$709$	13.7221	0.515346	0.257673	−	0.966232i	$-0.417044\pi$
$709$	13.7221	0.515346	0.257673	+	0.966232i	$0.417044\pi$
$710$	0	0
$711$	9.09743	0.341180
$712$	0	0
$713$	5.70770	0.213755
$714$	0	0
$715$	0	0
$716$	0	0
$717$	62.6440	2.33948
$718$	0	0
$719$	34.7900	1.29745	0.648724	−	0.761024i	$-0.275303\pi$
$719$	34.7900	1.29745	0.648724	+	0.761024i	$0.275303\pi$
$720$	0	0
$721$	1.37235	0.0511091
$722$	0	0
$723$	−62.8855	−2.33874
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−3.35114	−0.124287	−0.0621435	−	0.998067i	$-0.519794\pi$
$727$	−3.35114	−0.124287	−0.0621435	+	0.998067i	$0.519794\pi$
$728$	0	0
$729$	−38.9617	−1.44302
$730$	0	0
$731$	2.04854	0.0757680
$732$	0	0
$733$	−3.20440	−0.118357	−0.0591786	−	0.998247i	$-0.518848\pi$
$733$	−3.20440	−0.118357	−0.0591786	+	0.998247i	$0.518848\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.484803	0.0178580
$738$	0	0
$739$	−38.3376	−1.41027	−0.705136	−	0.709072i	$-0.749114\pi$
$739$	−38.3376	−1.41027	−0.705136	+	0.709072i	$0.749114\pi$
$740$	0	0
$741$	41.4060	1.52109
$742$	0	0
$743$	21.5137	0.789261	0.394631	−	0.918840i	$-0.370873\pi$
$743$	21.5137	0.789261	0.394631	+	0.918840i	$0.370873\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−53.3876	−1.95335
$748$	0	0
$749$	3.24251	0.118479
$750$	0	0
$751$	29.9353	1.09236	0.546178	−	0.837669i	$-0.316082\pi$
$751$	29.9353	1.09236	0.546178	+	0.837669i	$0.316082\pi$
$752$	0	0
$753$	77.1363	2.81100
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−23.2518	−0.845102	−0.422551	−	0.906339i	$-0.638865\pi$
$757$	−23.2518	−0.845102	−0.422551	+	0.906339i	$0.638865\pi$
$758$	0	0
$759$	−0.234634	−0.00851666
$760$	0	0
$761$	40.7791	1.47824	0.739120	−	0.673574i	$-0.235242\pi$
$761$	40.7791	1.47824	0.739120	+	0.673574i	$0.235242\pi$
$762$	0	0
$763$	−3.54320	−0.128272
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−11.1828	−0.403788
$768$	0	0
$769$	−1.08570	−0.0391515	−0.0195757	−	0.999808i	$-0.506232\pi$
$769$	−1.08570	−0.0391515	−0.0195757	+	0.999808i	$0.506232\pi$
$770$	0	0
$771$	−0.453531	−0.0163335
$772$	0	0
$773$	37.7659	1.35835	0.679173	−	0.733978i	$-0.262338\pi$
$773$	37.7659	1.35835	0.679173	+	0.733978i	$0.262338\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−4.40563	−0.158051
$778$	0	0
$779$	−25.1264	−0.900247
$780$	0	0
$781$	1.11917	0.0400470
$782$	0	0
$783$	−9.64767	−0.344780
$784$	0	0
$785$	0	0
$786$	0	0
$787$	38.0489	1.35630	0.678149	−	0.734925i	$-0.262782\pi$
$787$	38.0489	1.35630	0.678149	+	0.734925i	$0.262782\pi$
$788$	0	0
$789$	−70.0506	−2.49387
$790$	0	0
$791$	−10.1037	−0.359246
$792$	0	0
$793$	34.7717	1.23478
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12.8798	0.456227	0.228114	−	0.973634i	$-0.426744\pi$
$797$	12.8798	0.456227	0.228114	+	0.973634i	$0.426744\pi$
$798$	0	0
$799$	3.42573	0.121194
$800$	0	0
$801$	36.2040	1.27920
$802$	0	0
$803$	0.0926131	0.00326825
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−0.784110	−0.0276020
$808$	0	0
$809$	−0.0775700	−0.00272722	−0.00136361	−	0.999999i	$-0.500434\pi$
$809$	−0.0775700	−0.00272722	−0.00136361	+	0.999999i	$0.500434\pi$
$810$	0	0
$811$	0.0163427	0.000573869 0	0.000286934	−	1.00000i	$-0.499909\pi$
$811$	0.0163427	0.000573869 0	0.000286934		1.00000i	$0.499909\pi$
$812$	0	0
$813$	29.0114	1.01747
$814$	0	0
$815$	0	0
$816$	0	0
$817$	34.5173	1.20761
$818$	0	0
$819$	−9.49214	−0.331682
$820$	0	0
$821$	6.85832	0.239357	0.119678	−	0.992813i	$-0.461814\pi$
$821$	6.85832	0.239357	0.119678	+	0.992813i	$0.461814\pi$
$822$	0	0
$823$	−24.4740	−0.853109	−0.426554	−	0.904462i	$-0.640273\pi$
$823$	−24.4740	−0.853109	−0.426554	+	0.904462i	$0.640273\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11.5556	0.401828	0.200914	−	0.979609i	$-0.435609\pi$
$827$	11.5556	0.401828	0.200914	+	0.979609i	$0.435609\pi$
$828$	0	0
$829$	−42.7496	−1.48475	−0.742377	−	0.669983i	$-0.766302\pi$
$829$	−42.7496	−1.48475	−0.742377	+	0.669983i	$0.766302\pi$
$830$	0	0
$831$	−35.0437	−1.21565
$832$	0	0
$833$	1.86905	0.0647587
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−39.8864	−1.37867
$838$	0	0
$839$	−52.4396	−1.81042	−0.905209	−	0.424968i	$-0.860286\pi$
$839$	−52.4396	−1.81042	−0.905209	+	0.424968i	$0.860286\pi$
$840$	0	0
$841$	−27.0940	−0.934277
$842$	0	0
$843$	−64.6434	−2.22644
$844$	0	0
$845$	0	0
$846$	0	0
$847$	6.38586	0.219421
$848$	0	0
$849$	−30.5961	−1.05006
$850$	0	0
$851$	−2.61536	−0.0896534
$852$	0	0
$853$	−23.6690	−0.810410	−0.405205	−	0.914226i	$-0.632800\pi$
$853$	−23.6690	−0.810410	−0.405205	+	0.914226i	$0.632800\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−23.1587	−0.791088	−0.395544	−	0.918447i	$-0.629444\pi$
$857$	−23.1587	−0.791088	−0.395544	+	0.918447i	$0.629444\pi$
$858$	0	0
$859$	11.6044	0.395939	0.197969	−	0.980208i	$-0.436565\pi$
$859$	11.6044	0.395939	0.197969	+	0.980208i	$0.436565\pi$
$860$	0	0
$861$	8.95441	0.305166
$862$	0	0
$863$	−2.71131	−0.0922940	−0.0461470	−	0.998935i	$-0.514694\pi$
$863$	−2.71131	−0.0922940	−0.0461470	+	0.998935i	$0.514694\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	49.0710	1.66654
$868$	0	0
$869$	0.136063	0.00461562
$870$	0	0
$871$	18.0996	0.613280
$872$	0	0
$873$	−69.3477	−2.34706
$874$	0	0
$875$	0	0
$876$	0	0
$877$	48.0862	1.62375	0.811877	−	0.583828i	$-0.198446\pi$
$877$	48.0862	1.62375	0.811877	+	0.583828i	$0.198446\pi$
$878$	0	0
$879$	−24.6353	−0.830928
$880$	0	0
$881$	2.70507	0.0911361	0.0455680	−	0.998961i	$-0.485490\pi$
$881$	2.70507	0.0911361	0.0455680	+	0.998961i	$0.485490\pi$
$882$	0	0
$883$	−13.5827	−0.457094	−0.228547	−	0.973533i	$-0.573398\pi$
$883$	−13.5827	−0.457094	−0.228547	+	0.973533i	$0.573398\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.42335	0.114945	0.0574724	−	0.998347i	$-0.481696\pi$
$887$	3.42335	0.114945	0.0574724	+	0.998347i	$0.481696\pi$
$888$	0	0
$889$	−6.93366	−0.232547
$890$	0	0
$891$	0.326561	0.0109402
$892$	0	0
$893$	57.7225	1.93161
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−8.75977	−0.292480
$898$	0	0
$899$	7.87990	0.262809
$900$	0	0
$901$	1.86280	0.0620590
$902$	0	0
$903$	−12.3011	−0.409354
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−9.90426	−0.328865	−0.164433	−	0.986388i	$-0.552579\pi$
$907$	−9.90426	−0.328865	−0.164433	+	0.986388i	$0.552579\pi$
$908$	0	0
$909$	−5.51570	−0.182944
$910$	0	0
$911$	1.95943	0.0649187	0.0324593	−	0.999473i	$-0.489666\pi$
$911$	1.95943	0.0649187	0.0324593	+	0.999473i	$0.489666\pi$
$912$	0	0
$913$	−0.798476	−0.0264257
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5.71709	−0.188795
$918$	0	0
$919$	55.8522	1.84239	0.921197	−	0.389096i	$-0.127213\pi$
$919$	55.8522	1.84239	0.921197	+	0.389096i	$0.127213\pi$
$920$	0	0
$921$	−46.8653	−1.54426
$922$	0	0
$923$	41.7829	1.37530
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−12.7808	−0.419776
$928$	0	0
$929$	17.0522	0.559464	0.279732	−	0.960078i	$-0.409754\pi$
$929$	17.0522	0.559464	0.279732	+	0.960078i	$0.409754\pi$
$930$	0	0
$931$	31.4929	1.03214
$932$	0	0
$933$	96.5764	3.16177
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−30.9074	−1.00970	−0.504850	−	0.863207i	$-0.668452\pi$
$937$	−30.9074	−1.00970	−0.504850	+	0.863207i	$0.668452\pi$
$938$	0	0
$939$	24.9173	0.813147
$940$	0	0
$941$	−41.4508	−1.35126	−0.675629	−	0.737242i	$-0.736128\pi$
$941$	−41.4508	−1.35126	−0.675629	+	0.737242i	$0.736128\pi$
$942$	0	0
$943$	5.31570	0.173103
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34.4896	1.12076	0.560380	−	0.828236i	$-0.310655\pi$
$947$	34.4896	1.12076	0.560380	+	0.828236i	$0.310655\pi$
$948$	0	0
$949$	3.45760	0.112239
$950$	0	0
$951$	95.4401	3.09486
$952$	0	0
$953$	−28.8622	−0.934939	−0.467470	−	0.884009i	$-0.654834\pi$
$953$	−28.8622	−0.934939	−0.467470	+	0.884009i	$0.654834\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−0.323929	−0.0104711
$958$	0	0
$959$	1.70346	0.0550076
$960$	0	0
$961$	1.57784	0.0508982
$962$	0	0
$963$	−30.1976	−0.973104
$964$	0	0
$965$	0	0
$966$	0	0
$967$	6.47559	0.208241	0.104121	−	0.994565i	$-0.466797\pi$
$967$	6.47559	0.208241	0.104121	+	0.994565i	$0.466797\pi$
$968$	0	0
$969$	−3.84539	−0.123532
$970$	0	0
$971$	−33.9296	−1.08885	−0.544426	−	0.838809i	$-0.683253\pi$
$971$	−33.9296	−1.08885	−0.544426	+	0.838809i	$0.683253\pi$
$972$	0	0
$973$	2.88160	0.0923797
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.90638	−0.0929832	−0.0464916	−	0.998919i	$-0.514804\pi$
$977$	−2.90638	−0.0929832	−0.0464916	+	0.998919i	$0.514804\pi$
$978$	0	0
$979$	0.541474	0.0173056
$980$	0	0
$981$	32.9980	1.05354
$982$	0	0
$983$	−8.66470	−0.276361	−0.138180	−	0.990407i	$-0.544125\pi$
$983$	−8.66470	−0.276361	−0.138180	+	0.990407i	$0.544125\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−20.5708	−0.654776
$988$	0	0
$989$	−7.30240	−0.232203
$990$	0	0
$991$	32.4587	1.03108	0.515542	−	0.856864i	$-0.327590\pi$
$991$	32.4587	1.03108	0.515542	+	0.856864i	$0.327590\pi$
$992$	0	0
$993$	23.7577	0.753929
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−61.7623	−1.95603	−0.978015	−	0.208534i	$-0.933131\pi$
$997$	−61.7623	−1.95603	−0.978015	+	0.208534i	$0.933131\pi$
$998$	0	0
$999$	18.2766	0.578245

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.dd.1.2		8
4.3	odd	2		4600.2.a.bk.1.7		8
5.2	odd	4		1840.2.e.h.369.15		16
5.3	odd	4		1840.2.e.h.369.2		16
5.4	even	2		9200.2.a.de.1.7		8
20.3	even	4		920.2.e.c.369.15	yes	16
20.7	even	4		920.2.e.c.369.2	✓	16
20.19	odd	2		4600.2.a.bj.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.c.369.2	✓	16	20.7	even	4
920.2.e.c.369.15	yes	16	20.3	even	4
1840.2.e.h.369.2		16	5.3	odd	4
1840.2.e.h.369.15		16	5.2	odd	4
4600.2.a.bj.1.2		8	20.19	odd	2
4600.2.a.bk.1.7		8	4.3	odd	2
9200.2.a.dd.1.2		8	1.1	even	1	trivial
9200.2.a.de.1.7		8	5.4	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:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

\(f(q)\)	\(=\)	\(q-2.89996 q^{3} -0.580879 q^{7} +5.40975 q^{9} +O(q^{10})\)	\(q-2.89996 q^{3} -0.580879 q^{7} +5.40975 q^{9} +0.0809094 q^{11} +3.02066 q^{13} -0.280529 q^{17} -4.72683 q^{19} +1.68452 q^{21} +1.00000 q^{23} -6.98817 q^{27} +1.38057 q^{29} +5.70770 q^{31} -0.234634 q^{33} -2.61536 q^{37} -8.75977 q^{39} +5.31570 q^{41} -7.30240 q^{43} -12.2117 q^{47} -6.66258 q^{49} +0.813523 q^{51} -6.64032 q^{53} +13.7076 q^{57} -3.70211 q^{59} +11.5113 q^{61} -3.14241 q^{63} +5.99193 q^{67} -2.89996 q^{69} +13.8324 q^{71} +1.14465 q^{73} -0.0469985 q^{77} +1.68167 q^{79} +4.03614 q^{81} -9.86877 q^{83} -4.00360 q^{87} +6.69235 q^{89} -1.75463 q^{91} -16.5521 q^{93} -12.8190 q^{97} +0.437699 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 3 q^{3} - 7 q^{7} + 11 q^{9}+O(q^{10}) \) 8 * q - 3 * q^3 - 7 * q^7 + 11 * q^9	\( 8 q - 3 q^{3} - 7 q^{7} + 11 q^{9} - 7 q^{11} - 11 q^{13} + 7 q^{17} - 11 q^{19} + 8 q^{23} - 12 q^{27} + 22 q^{29} - 9 q^{31} + 9 q^{33} - 4 q^{37} + 7 q^{41} - 22 q^{43} - 4 q^{47} + 39 q^{49} + 19 q^{51} - 4 q^{53} - 32 q^{59} + 17 q^{61} - 44 q^{63} + 4 q^{67} - 3 q^{69} - 15 q^{71} - 6 q^{73} - 18 q^{77} + 2 q^{79} + 24 q^{81} - 36 q^{83} + 4 q^{87} + 46 q^{89} + 35 q^{91} - 20 q^{93} - 3 q^{97} - 61 q^{99}+O(q^{100}) \) 8 * q - 3 * q^3 - 7 * q^7 + 11 * q^9 - 7 * q^11 - 11 * q^13 + 7 * q^17 - 11 * q^19 + 8 * q^23 - 12 * q^27 + 22 * q^29 - 9 * q^31 + 9 * q^33 - 4 * q^37 + 7 * q^41 - 22 * q^43 - 4 * q^47 + 39 * q^49 + 19 * q^51 - 4 * q^53 - 32 * q^59 + 17 * q^61 - 44 * q^63 + 4 * q^67 - 3 * q^69 - 15 * q^71 - 6 * q^73 - 18 * q^77 + 2 * q^79 + 24 * q^81 - 36 * q^83 + 4 * q^87 + 46 * q^89 + 35 * q^91 - 20 * q^93 - 3 * q^97 - 61 * q^99

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