LMFDB - Embedded newform 232.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [232,4,Mod(1,232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(232, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("232.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 232 = 2^{3} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	232.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:	$13.6884431213$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 40x^{4} - 88x^{3} - 8x^{2} + 48x - 9 $ x^6 - 40x^4 - 88x^3 - 8x^2 + 48x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$7.22553$ of defining polynomial
Character	$\chi$	$=$	232.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.88017 q^{3} +2.64112 q^{5} +3.30580 q^{7} +70.6177 q^{9} +O(q^{10})$	$q-9.88017 q^{3} +2.64112 q^{5} +3.30580 q^{7} +70.6177 q^{9} -28.3440 q^{11} +85.7400 q^{13} -26.0947 q^{15} -111.092 q^{17} +99.8826 q^{19} -32.6619 q^{21} -110.105 q^{23} -118.024 q^{25} -430.950 q^{27} -29.0000 q^{29} -60.0577 q^{31} +280.044 q^{33} +8.73102 q^{35} +119.286 q^{37} -847.126 q^{39} -285.212 q^{41} -249.574 q^{43} +186.510 q^{45} -226.276 q^{47} -332.072 q^{49} +1097.60 q^{51} +387.165 q^{53} -74.8600 q^{55} -986.857 q^{57} +164.811 q^{59} +188.155 q^{61} +233.448 q^{63} +226.450 q^{65} -903.648 q^{67} +1087.86 q^{69} +820.374 q^{71} -394.228 q^{73} +1166.10 q^{75} -93.6997 q^{77} +605.439 q^{79} +2351.18 q^{81} -504.986 q^{83} -293.406 q^{85} +286.525 q^{87} -930.620 q^{89} +283.439 q^{91} +593.380 q^{93} +263.802 q^{95} +12.2577 q^{97} -2001.59 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 5 q^{3} - 5 q^{5} - 38 q^{7} + 47 q^{9}+O(q^{10}) $ 6 * q - 5 * q^3 - 5 * q^5 - 38 * q^7 + 47 * q^9	$ 6 q - 5 q^{3} - 5 q^{5} - 38 q^{7} + 47 q^{9} - 19 q^{11} + 13 q^{13} - 191 q^{15} - 218 q^{17} - 290 q^{19} - 266 q^{21} - 196 q^{23} - 13 q^{25} - 437 q^{27} - 174 q^{29} - 675 q^{31} + 291 q^{33} - 466 q^{35} - 238 q^{37} - 1297 q^{39} - 464 q^{41} - 579 q^{43} - 148 q^{45} - 975 q^{47} + 914 q^{49} - 576 q^{51} + 515 q^{53} - 1605 q^{55} - 340 q^{57} - 108 q^{59} + 1158 q^{61} - 1136 q^{63} + 1239 q^{65} - 80 q^{67} + 2568 q^{69} + 438 q^{71} + 262 q^{73} + 1766 q^{75} + 194 q^{77} - 237 q^{79} + 2554 q^{81} - 1288 q^{83} + 3112 q^{85} + 145 q^{87} - 252 q^{89} + 2450 q^{91} + 2131 q^{93} + 180 q^{95} + 380 q^{97} + 2264 q^{99}+O(q^{100}) $ 6 * q - 5 * q^3 - 5 * q^5 - 38 * q^7 + 47 * q^9 - 19 * q^11 + 13 * q^13 - 191 * q^15 - 218 * q^17 - 290 * q^19 - 266 * q^21 - 196 * q^23 - 13 * q^25 - 437 * q^27 - 174 * q^29 - 675 * q^31 + 291 * q^33 - 466 * q^35 - 238 * q^37 - 1297 * q^39 - 464 * q^41 - 579 * q^43 - 148 * q^45 - 975 * q^47 + 914 * q^49 - 576 * q^51 + 515 * q^53 - 1605 * q^55 - 340 * q^57 - 108 * q^59 + 1158 * q^61 - 1136 * q^63 + 1239 * q^65 - 80 * q^67 + 2568 * q^69 + 438 * q^71 + 262 * q^73 + 1766 * q^75 + 194 * q^77 - 237 * q^79 + 2554 * q^81 - 1288 * q^83 + 3112 * q^85 + 145 * q^87 - 252 * q^89 + 2450 * q^91 + 2131 * q^93 + 180 * q^95 + 380 * q^97 + 2264 * 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
$3$	−9.88017	−1.90144	−0.950719	−	0.310053i	$-0.899653\pi$
$3$	−9.88017	−1.90144	−0.950719	+	0.310053i	$0.899653\pi$
$4$	0	0
$5$	2.64112	0.236229	0.118114	−	0.993000i	$-0.462315\pi$
$5$	2.64112	0.236229	0.118114	+	0.993000i	$0.462315\pi$
$6$	0	0
$7$	3.30580	0.178496	0.0892482	−	0.996009i	$-0.471554\pi$
$7$	3.30580	0.178496	0.0892482	+	0.996009i	$0.471554\pi$
$8$	0	0
$9$	70.6177	2.61547
$10$	0	0
$11$	−28.3440	−0.776913	−0.388457	−	0.921467i	$-0.626992\pi$
$11$	−28.3440	−0.776913	−0.388457	+	0.921467i	$0.626992\pi$
$12$	0	0
$13$	85.7400	1.82923	0.914615	−	0.404325i	$-0.132494\pi$
$13$	85.7400	1.82923	0.914615	+	0.404325i	$0.132494\pi$
$14$	0	0
$15$	−26.0947	−0.449175
$16$	0	0
$17$	−111.092	−1.58492	−0.792462	−	0.609922i	$-0.791201\pi$
$17$	−111.092	−1.58492	−0.792462	+	0.609922i	$0.791201\pi$
$18$	0	0
$19$	99.8826	1.20603	0.603017	−	0.797729i	$-0.293965\pi$
$19$	99.8826	1.20603	0.603017	+	0.797729i	$0.293965\pi$
$20$	0	0
$21$	−32.6619	−0.339400
$22$	0	0
$23$	−110.105	−0.998196	−0.499098	−	0.866546i	$-0.666335\pi$
$23$	−110.105	−0.998196	−0.499098	+	0.866546i	$0.666335\pi$
$24$	0	0
$25$	−118.024	−0.944196
$26$	0	0
$27$	−430.950	−3.07172
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−60.0577	−0.347957	−0.173979	−	0.984749i	$-0.555662\pi$
$31$	−60.0577	−0.347957	−0.173979	+	0.984749i	$0.555662\pi$
$32$	0	0
$33$	280.044	1.47725
$34$	0	0
$35$	8.73102	0.0421660
$36$	0	0
$37$	119.286	0.530014	0.265007	−	0.964247i	$-0.414626\pi$
$37$	119.286	0.530014	0.265007	+	0.964247i	$0.414626\pi$
$38$	0	0
$39$	−847.126	−3.47817
$40$	0	0
$41$	−285.212	−1.08641	−0.543204	−	0.839601i	$-0.682789\pi$
$41$	−285.212	−1.08641	−0.543204	+	0.839601i	$0.682789\pi$
$42$	0	0
$43$	−249.574	−0.885110	−0.442555	−	0.896741i	$-0.645928\pi$
$43$	−249.574	−0.885110	−0.442555	+	0.896741i	$0.645928\pi$
$44$	0	0
$45$	186.510	0.617850
$46$	0	0
$47$	−226.276	−0.702251	−0.351125	−	0.936328i	$-0.614201\pi$
$47$	−226.276	−0.702251	−0.351125	+	0.936328i	$0.614201\pi$
$48$	0	0
$49$	−332.072	−0.968139
$50$	0	0
$51$	1097.60	3.01363
$52$	0	0
$53$	387.165	1.00342	0.501709	−	0.865036i	$-0.332705\pi$
$53$	387.165	1.00342	0.501709	+	0.865036i	$0.332705\pi$
$54$	0	0
$55$	−74.8600	−0.183529
$56$	0	0
$57$	−986.857	−2.29320
$58$	0	0
$59$	164.811	0.363670	0.181835	−	0.983329i	$-0.441796\pi$
$59$	164.811	0.363670	0.181835	+	0.983329i	$0.441796\pi$
$60$	0	0
$61$	188.155	0.394932	0.197466	−	0.980310i	$-0.436729\pi$
$61$	188.155	0.394932	0.197466	+	0.980310i	$0.436729\pi$
$62$	0	0
$63$	233.448	0.466852
$64$	0	0
$65$	226.450	0.432117
$66$	0	0
$67$	−903.648	−1.64773	−0.823867	−	0.566783i	$-0.808188\pi$
$67$	−903.648	−1.64773	−0.823867	+	0.566783i	$0.808188\pi$
$68$	0	0
$69$	1087.86	1.89801
$70$	0	0
$71$	820.374	1.37127	0.685637	−	0.727943i	$-0.259524\pi$
$71$	820.374	1.37127	0.685637	+	0.727943i	$0.259524\pi$
$72$	0	0
$73$	−394.228	−0.632066	−0.316033	−	0.948748i	$-0.602351\pi$
$73$	−394.228	−0.632066	−0.316033	+	0.948748i	$0.602351\pi$
$74$	0	0
$75$	1166.10	1.79533
$76$	0	0
$77$	−93.6997	−0.138676
$78$	0	0
$79$	605.439	0.862243	0.431122	−	0.902294i	$-0.358118\pi$
$79$	605.439	0.862243	0.431122	+	0.902294i	$0.358118\pi$
$80$	0	0
$81$	2351.18	3.22521
$82$	0	0
$83$	−504.986	−0.667824	−0.333912	−	0.942604i	$-0.608369\pi$
$83$	−504.986	−0.667824	−0.333912	+	0.942604i	$0.608369\pi$
$84$	0	0
$85$	−293.406	−0.374405
$86$	0	0
$87$	286.525	0.353088
$88$	0	0
$89$	−930.620	−1.10838	−0.554189	−	0.832391i	$-0.686971\pi$
$89$	−930.620	−1.10838	−0.554189	+	0.832391i	$0.686971\pi$
$90$	0	0
$91$	283.439	0.326511
$92$	0	0
$93$	593.380	0.661620
$94$	0	0
$95$	263.802	0.284900
$96$	0	0
$97$	12.2577	0.0128307	0.00641535	−	0.999979i	$-0.497958\pi$
$97$	12.2577	0.0128307	0.00641535	+	0.999979i	$0.497958\pi$
$98$	0	0
$99$	−2001.59	−2.03199
$100$	0	0
$101$	−1165.44	−1.14818	−0.574089	−	0.818793i	$-0.694644\pi$
$101$	−1165.44	−1.14818	−0.574089	+	0.818793i	$0.694644\pi$
$102$	0	0
$103$	−216.871	−0.207465	−0.103733	−	0.994605i	$-0.533079\pi$
$103$	−216.871	−0.207465	−0.103733	+	0.994605i	$0.533079\pi$
$104$	0	0
$105$	−86.2639	−0.0801761
$106$	0	0
$107$	−1868.17	−1.68787	−0.843937	−	0.536443i	$-0.819768\pi$
$107$	−1868.17	−1.68787	−0.843937	+	0.536443i	$0.819768\pi$
$108$	0	0
$109$	−624.737	−0.548981	−0.274490	−	0.961590i	$-0.588509\pi$
$109$	−624.737	−0.548981	−0.274490	+	0.961590i	$0.588509\pi$
$110$	0	0
$111$	−1178.57	−1.00779
$112$	0	0
$113$	−1181.43	−0.983537	−0.491769	−	0.870726i	$-0.663649\pi$
$113$	−1181.43	−0.983537	−0.491769	+	0.870726i	$0.663649\pi$
$114$	0	0
$115$	−290.801	−0.235803
$116$	0	0
$117$	6054.76	4.78430
$118$	0	0
$119$	−367.247	−0.282903
$120$	0	0
$121$	−527.616	−0.396406
$122$	0	0
$123$	2817.95	2.06574
$124$	0	0
$125$	−641.857	−0.459275
$126$	0	0
$127$	1693.76	1.18344	0.591720	−	0.806144i	$-0.298449\pi$
$127$	1693.76	1.18344	0.591720	+	0.806144i	$0.298449\pi$
$128$	0	0
$129$	2465.84	1.68298
$130$	0	0
$131$	−195.314	−0.130265	−0.0651323	−	0.997877i	$-0.520747\pi$
$131$	−195.314	−0.130265	−0.0651323	+	0.997877i	$0.520747\pi$
$132$	0	0
$133$	330.192	0.215273
$134$	0	0
$135$	−1138.19	−0.725628
$136$	0	0
$137$	−319.479	−0.199233	−0.0996164	−	0.995026i	$-0.531762\pi$
$137$	−319.479	−0.199233	−0.0996164	+	0.995026i	$0.531762\pi$
$138$	0	0
$139$	−1096.70	−0.669213	−0.334607	−	0.942358i	$-0.608603\pi$
$139$	−1096.70	−0.669213	−0.334607	+	0.942358i	$0.608603\pi$
$140$	0	0
$141$	2235.65	1.33529
$142$	0	0
$143$	−2430.22	−1.42115
$144$	0	0
$145$	−76.5925	−0.0438666
$146$	0	0
$147$	3280.92	1.84086
$148$	0	0
$149$	1194.25	0.656625	0.328312	−	0.944569i	$-0.393520\pi$
$149$	1194.25	0.656625	0.328312	+	0.944569i	$0.393520\pi$
$150$	0	0
$151$	2259.35	1.21764	0.608819	−	0.793309i	$-0.291643\pi$
$151$	2259.35	1.21764	0.608819	+	0.793309i	$0.291643\pi$
$152$	0	0
$153$	−7845.04	−4.14532
$154$	0	0
$155$	−158.620	−0.0821976
$156$	0	0
$157$	−2592.33	−1.31777	−0.658887	−	0.752242i	$-0.728972\pi$
$157$	−2592.33	−1.31777	−0.658887	+	0.752242i	$0.728972\pi$
$158$	0	0
$159$	−3825.25	−1.90794
$160$	0	0
$161$	−363.986	−0.178174
$162$	0	0
$163$	498.162	0.239381	0.119690	−	0.992811i	$-0.461810\pi$
$163$	498.162	0.239381	0.119690	+	0.992811i	$0.461810\pi$
$164$	0	0
$165$	739.629	0.348970
$166$	0	0
$167$	551.442	0.255520	0.127760	−	0.991805i	$-0.459221\pi$
$167$	551.442	0.255520	0.127760	+	0.991805i	$0.459221\pi$
$168$	0	0
$169$	5154.35	2.34609
$170$	0	0
$171$	7053.48	3.15434
$172$	0	0
$173$	2539.86	1.11620	0.558099	−	0.829774i	$-0.311531\pi$
$173$	2539.86	1.11620	0.558099	+	0.829774i	$0.311531\pi$
$174$	0	0
$175$	−390.165	−0.168536
$176$	0	0
$177$	−1628.36	−0.691496
$178$	0	0
$179$	739.618	0.308836	0.154418	−	0.988006i	$-0.450650\pi$
$179$	739.618	0.308836	0.154418	+	0.988006i	$0.450650\pi$
$180$	0	0
$181$	−4127.70	−1.69508	−0.847539	−	0.530733i	$-0.821917\pi$
$181$	−4127.70	−1.69508	−0.847539	+	0.530733i	$0.821917\pi$
$182$	0	0
$183$	−1859.01	−0.750938
$184$	0	0
$185$	315.049	0.125205
$186$	0	0
$187$	3148.79	1.23135
$188$	0	0
$189$	−1424.63	−0.548290
$190$	0	0
$191$	−1630.27	−0.617605	−0.308802	−	0.951126i	$-0.599928\pi$
$191$	−1630.27	−0.617605	−0.308802	+	0.951126i	$0.599928\pi$
$192$	0	0
$193$	−210.636	−0.0785592	−0.0392796	−	0.999228i	$-0.512506\pi$
$193$	−210.636	−0.0785592	−0.0392796	+	0.999228i	$0.512506\pi$
$194$	0	0
$195$	−2237.36	−0.821645
$196$	0	0
$197$	−4418.37	−1.59795	−0.798974	−	0.601366i	$-0.794623\pi$
$197$	−4418.37	−1.59795	−0.798974	+	0.601366i	$0.794623\pi$
$198$	0	0
$199$	−1436.52	−0.511721	−0.255860	−	0.966714i	$-0.582359\pi$
$199$	−1436.52	−0.511721	−0.255860	+	0.966714i	$0.582359\pi$
$200$	0	0
$201$	8928.19	3.13307
$202$	0	0
$203$	−95.8682	−0.0331460
$204$	0	0
$205$	−753.280	−0.256641
$206$	0	0
$207$	−7775.37	−2.61075
$208$	0	0
$209$	−2831.07	−0.936983
$210$	0	0
$211$	3571.25	1.16519	0.582595	−	0.812763i	$-0.302037\pi$
$211$	3571.25	1.16519	0.582595	+	0.812763i	$0.302037\pi$
$212$	0	0
$213$	−8105.43	−2.60739
$214$	0	0
$215$	−659.156	−0.209089
$216$	0	0
$217$	−198.539	−0.0621092
$218$	0	0
$219$	3895.03	1.20184
$220$	0	0
$221$	−9525.00	−2.89919
$222$	0	0
$223$	−6382.81	−1.91670	−0.958351	−	0.285594i	$-0.907809\pi$
$223$	−6382.81	−1.91670	−0.958351	+	0.285594i	$0.907809\pi$
$224$	0	0
$225$	−8334.61	−2.46952
$226$	0	0
$227$	4565.57	1.33492	0.667462	−	0.744644i	$-0.267381\pi$
$227$	4565.57	1.33492	0.667462	+	0.744644i	$0.267381\pi$
$228$	0	0
$229$	5084.23	1.46714	0.733570	−	0.679614i	$-0.237853\pi$
$229$	5084.23	1.46714	0.733570	+	0.679614i	$0.237853\pi$
$230$	0	0
$231$	925.769	0.263684
$232$	0	0
$233$	−5743.69	−1.61494	−0.807471	−	0.589907i	$-0.799164\pi$
$233$	−5743.69	−1.61494	−0.807471	+	0.589907i	$0.799164\pi$
$234$	0	0
$235$	−597.623	−0.165892
$236$	0	0
$237$	−5981.84	−1.63950
$238$	0	0
$239$	5554.14	1.50321	0.751605	−	0.659614i	$-0.229280\pi$
$239$	5554.14	1.50321	0.751605	+	0.659614i	$0.229280\pi$
$240$	0	0
$241$	−2222.62	−0.594073	−0.297036	−	0.954866i	$-0.595998\pi$
$241$	−2222.62	−0.594073	−0.297036	+	0.954866i	$0.595998\pi$
$242$	0	0
$243$	−11594.4	−3.06082
$244$	0	0
$245$	−877.041	−0.228702
$246$	0	0
$247$	8563.93	2.20611
$248$	0	0
$249$	4989.34	1.26983
$250$	0	0
$251$	4973.67	1.25074	0.625369	−	0.780329i	$-0.284948\pi$
$251$	4973.67	1.25074	0.625369	+	0.780329i	$0.284948\pi$
$252$	0	0
$253$	3120.82	0.775512
$254$	0	0
$255$	2898.90	0.711908
$256$	0	0
$257$	15.0797	0.00366011	0.00183005	−	0.999998i	$-0.499417\pi$
$257$	15.0797	0.00366011	0.00183005	+	0.999998i	$0.499417\pi$
$258$	0	0
$259$	394.336	0.0946056
$260$	0	0
$261$	−2047.91	−0.485680
$262$	0	0
$263$	3238.30	0.759248	0.379624	−	0.925141i	$-0.376053\pi$
$263$	3238.30	0.759248	0.379624	+	0.925141i	$0.376053\pi$
$264$	0	0
$265$	1022.55	0.237037
$266$	0	0
$267$	9194.68	2.10751
$268$	0	0
$269$	4147.30	0.940020	0.470010	−	0.882661i	$-0.344250\pi$
$269$	4147.30	0.940020	0.470010	+	0.882661i	$0.344250\pi$
$270$	0	0
$271$	1309.45	0.293519	0.146759	−	0.989172i	$-0.453116\pi$
$271$	1309.45	0.293519	0.146759	+	0.989172i	$0.453116\pi$
$272$	0	0
$273$	−2800.43	−0.620841
$274$	0	0
$275$	3345.29	0.733558
$276$	0	0
$277$	6746.76	1.46344	0.731721	−	0.681604i	$-0.238717\pi$
$277$	6746.76	1.46344	0.731721	+	0.681604i	$0.238717\pi$
$278$	0	0
$279$	−4241.13	−0.910072
$280$	0	0
$281$	2896.25	0.614861	0.307430	−	0.951571i	$-0.400531\pi$
$281$	2896.25	0.614861	0.307430	+	0.951571i	$0.400531\pi$
$282$	0	0
$283$	−8577.44	−1.80168	−0.900841	−	0.434150i	$-0.857049\pi$
$283$	−8577.44	−1.80168	−0.900841	+	0.434150i	$0.857049\pi$
$284$	0	0
$285$	−2606.41	−0.541720
$286$	0	0
$287$	−942.856	−0.193920
$288$	0	0
$289$	7428.36	1.51198
$290$	0	0
$291$	−121.108	−0.0243968
$292$	0	0
$293$	−8204.00	−1.63578	−0.817888	−	0.575377i	$-0.804855\pi$
$293$	−8204.00	−1.63578	−0.817888	+	0.575377i	$0.804855\pi$
$294$	0	0
$295$	435.285	0.0859093
$296$	0	0
$297$	12214.9	2.38646
$298$	0	0
$299$	−9440.41	−1.82593
$300$	0	0
$301$	−825.044	−0.157989
$302$	0	0
$303$	11514.8	2.18319
$304$	0	0
$305$	496.941	0.0932943
$306$	0	0
$307$	−2506.45	−0.465963	−0.232981	−	0.972481i	$-0.574848\pi$
$307$	−2506.45	−0.465963	−0.232981	+	0.972481i	$0.574848\pi$
$308$	0	0
$309$	2142.72	0.394482
$310$	0	0
$311$	−6501.50	−1.18542	−0.592711	−	0.805415i	$-0.701942\pi$
$311$	−6501.50	−1.18542	−0.592711	+	0.805415i	$0.701942\pi$
$312$	0	0
$313$	7244.35	1.30823	0.654113	−	0.756397i	$-0.273042\pi$
$313$	7244.35	1.30823	0.654113	+	0.756397i	$0.273042\pi$
$314$	0	0
$315$	616.564	0.110284
$316$	0	0
$317$	8069.20	1.42969	0.714845	−	0.699283i	$-0.246497\pi$
$317$	8069.20	1.42969	0.714845	+	0.699283i	$0.246497\pi$
$318$	0	0
$319$	821.977	0.144269
$320$	0	0
$321$	18457.8	3.20939
$322$	0	0
$323$	−11096.1	−1.91147
$324$	0	0
$325$	−10119.4	−1.72715
$326$	0	0
$327$	6172.50	1.04385
$328$	0	0
$329$	−748.024	−0.125349
$330$	0	0
$331$	6351.97	1.05479	0.527396	−	0.849620i	$-0.323168\pi$
$331$	6351.97	1.05479	0.527396	+	0.849620i	$0.323168\pi$
$332$	0	0
$333$	8423.71	1.38624
$334$	0	0
$335$	−2386.64	−0.389242
$336$	0	0
$337$	−1126.41	−0.182076	−0.0910381	−	0.995847i	$-0.529019\pi$
$337$	−1126.41	−0.182076	−0.0910381	+	0.995847i	$0.529019\pi$
$338$	0	0
$339$	11672.7	1.87014
$340$	0	0
$341$	1702.28	0.270333
$342$	0	0
$343$	−2231.65	−0.351306
$344$	0	0
$345$	2873.16	0.448365
$346$	0	0
$347$	5208.09	0.805720	0.402860	−	0.915262i	$-0.368016\pi$
$347$	5208.09	0.805720	0.402860	+	0.915262i	$0.368016\pi$
$348$	0	0
$349$	9644.73	1.47928	0.739642	−	0.673000i	$-0.234995\pi$
$349$	9644.73	1.47928	0.739642	+	0.673000i	$0.234995\pi$
$350$	0	0
$351$	−36949.6	−5.61888
$352$	0	0
$353$	3082.55	0.464780	0.232390	−	0.972623i	$-0.425345\pi$
$353$	3082.55	0.464780	0.232390	+	0.972623i	$0.425345\pi$
$354$	0	0
$355$	2166.71	0.323935
$356$	0	0
$357$	3628.46	0.537923
$358$	0	0
$359$	5668.89	0.833405	0.416702	−	0.909043i	$-0.363186\pi$
$359$	5668.89	0.833405	0.416702	+	0.909043i	$0.363186\pi$
$360$	0	0
$361$	3117.53	0.454517
$362$	0	0
$363$	5212.94	0.753742
$364$	0	0
$365$	−1041.20	−0.149312
$366$	0	0
$367$	−12831.6	−1.82507	−0.912537	−	0.408994i	$-0.865880\pi$
$367$	−12831.6	−1.82507	−0.912537	+	0.408994i	$0.865880\pi$
$368$	0	0
$369$	−20141.0	−2.84147
$370$	0	0
$371$	1279.89	0.179107
$372$	0	0
$373$	5541.65	0.769265	0.384632	−	0.923070i	$-0.374328\pi$
$373$	5541.65	0.769265	0.384632	+	0.923070i	$0.374328\pi$
$374$	0	0
$375$	6341.65	0.873284
$376$	0	0
$377$	−2486.46	−0.339680
$378$	0	0
$379$	11614.1	1.57408	0.787039	−	0.616903i	$-0.211613\pi$
$379$	11614.1	1.57408	0.787039	+	0.616903i	$0.211613\pi$
$380$	0	0
$381$	−16734.6	−2.25024
$382$	0	0
$383$	2733.89	0.364739	0.182370	−	0.983230i	$-0.441623\pi$
$383$	2733.89	0.364739	0.182370	+	0.983230i	$0.441623\pi$
$384$	0	0
$385$	−247.472	−0.0327593
$386$	0	0
$387$	−17624.4	−2.31498
$388$	0	0
$389$	−8079.05	−1.05302	−0.526509	−	0.850169i	$-0.676499\pi$
$389$	−8079.05	−1.05302	−0.526509	+	0.850169i	$0.676499\pi$
$390$	0	0
$391$	12231.8	1.58206
$392$	0	0
$393$	1929.73	0.247690
$394$	0	0
$395$	1599.04	0.203687
$396$	0	0
$397$	−4371.33	−0.552622	−0.276311	−	0.961068i	$-0.589112\pi$
$397$	−4371.33	−0.552622	−0.276311	+	0.961068i	$0.589112\pi$
$398$	0	0
$399$	−3262.35	−0.409328
$400$	0	0
$401$	−15230.3	−1.89667	−0.948337	−	0.317265i	$-0.897236\pi$
$401$	−15230.3	−1.89667	−0.948337	+	0.317265i	$0.897236\pi$
$402$	0	0
$403$	−5149.35	−0.636494
$404$	0	0
$405$	6209.74	0.761888
$406$	0	0
$407$	−3381.05	−0.411775
$408$	0	0
$409$	−382.195	−0.0462061	−0.0231031	−	0.999733i	$-0.507355\pi$
$409$	−382.195	−0.0462061	−0.0231031	+	0.999733i	$0.507355\pi$
$410$	0	0
$411$	3156.50	0.378829
$412$	0	0
$413$	544.831	0.0649138
$414$	0	0
$415$	−1333.73	−0.157759
$416$	0	0
$417$	10835.6	1.27247
$418$	0	0
$419$	−5109.88	−0.595785	−0.297893	−	0.954599i	$-0.596284\pi$
$419$	−5109.88	−0.595785	−0.297893	+	0.954599i	$0.596284\pi$
$420$	0	0
$421$	9354.78	1.08295	0.541477	−	0.840715i	$-0.317865\pi$
$421$	9354.78	1.08295	0.541477	+	0.840715i	$0.317865\pi$
$422$	0	0
$423$	−15979.1	−1.83672
$424$	0	0
$425$	13111.5	1.49648
$426$	0	0
$427$	622.004	0.0704939
$428$	0	0
$429$	24010.9	2.70224
$430$	0	0
$431$	−3609.19	−0.403361	−0.201680	−	0.979451i	$-0.564640\pi$
$431$	−3609.19	−0.403361	−0.201680	+	0.979451i	$0.564640\pi$
$432$	0	0
$433$	−3781.22	−0.419662	−0.209831	−	0.977738i	$-0.567291\pi$
$433$	−3781.22	−0.419662	−0.209831	+	0.977738i	$0.567291\pi$
$434$	0	0
$435$	756.746	0.0834097
$436$	0	0
$437$	−10997.6	−1.20386
$438$	0	0
$439$	−1990.19	−0.216370	−0.108185	−	0.994131i	$-0.534504\pi$
$439$	−1990.19	−0.216370	−0.108185	+	0.994131i	$0.534504\pi$
$440$	0	0
$441$	−23450.1	−2.53214
$442$	0	0
$443$	−2859.89	−0.306721	−0.153361	−	0.988170i	$-0.549010\pi$
$443$	−2859.89	−0.306721	−0.153361	+	0.988170i	$0.549010\pi$
$444$	0	0
$445$	−2457.88	−0.261831
$446$	0	0
$447$	−11799.4	−1.24853
$448$	0	0
$449$	−1177.30	−0.123742	−0.0618709	−	0.998084i	$-0.519707\pi$
$449$	−1177.30	−0.123742	−0.0618709	+	0.998084i	$0.519707\pi$
$450$	0	0
$451$	8084.07	0.844044
$452$	0	0
$453$	−22322.8	−2.31527
$454$	0	0
$455$	748.597	0.0771314
$456$	0	0
$457$	427.683	0.0437772	0.0218886	−	0.999760i	$-0.493032\pi$
$457$	427.683	0.0437772	0.0218886	+	0.999760i	$0.493032\pi$
$458$	0	0
$459$	47874.9	4.86843
$460$	0	0
$461$	12610.0	1.27399	0.636994	−	0.770869i	$-0.280178\pi$
$461$	12610.0	1.27399	0.636994	+	0.770869i	$0.280178\pi$
$462$	0	0
$463$	−4584.31	−0.460153	−0.230076	−	0.973173i	$-0.573898\pi$
$463$	−4584.31	−0.460153	−0.230076	+	0.973173i	$0.573898\pi$
$464$	0	0
$465$	1567.19	0.156294
$466$	0	0
$467$	−315.535	−0.0312660	−0.0156330	−	0.999878i	$-0.504976\pi$
$467$	−315.535	−0.0312660	−0.0156330	+	0.999878i	$0.504976\pi$
$468$	0	0
$469$	−2987.28	−0.294115
$470$	0	0
$471$	25612.6	2.50566
$472$	0	0
$473$	7073.95	0.687654
$474$	0	0
$475$	−11788.6	−1.13873
$476$	0	0
$477$	27340.7	2.62441
$478$	0	0
$479$	−9507.29	−0.906888	−0.453444	−	0.891285i	$-0.649805\pi$
$479$	−9507.29	−0.906888	−0.453444	+	0.891285i	$0.649805\pi$
$480$	0	0
$481$	10227.6	0.969518
$482$	0	0
$483$	3596.24	0.338788
$484$	0	0
$485$	32.3740	0.00303098
$486$	0	0
$487$	2848.85	0.265079	0.132540	−	0.991178i	$-0.457687\pi$
$487$	2848.85	0.265079	0.132540	+	0.991178i	$0.457687\pi$
$488$	0	0
$489$	−4921.92	−0.455168
$490$	0	0
$491$	16581.9	1.52409	0.762046	−	0.647522i	$-0.224195\pi$
$491$	16581.9	1.52409	0.762046	+	0.647522i	$0.224195\pi$
$492$	0	0
$493$	3221.66	0.294313
$494$	0	0
$495$	−5286.44	−0.480015
$496$	0	0
$497$	2711.99	0.244768
$498$	0	0
$499$	−14103.9	−1.26528	−0.632642	−	0.774444i	$-0.718030\pi$
$499$	−14103.9	−1.26528	−0.632642	+	0.774444i	$0.718030\pi$
$500$	0	0
$501$	−5448.34	−0.485856
$502$	0	0
$503$	−1773.92	−0.157247	−0.0786236	−	0.996904i	$-0.525053\pi$
$503$	−1773.92	−0.157247	−0.0786236	+	0.996904i	$0.525053\pi$
$504$	0	0
$505$	−3078.08	−0.271233
$506$	0	0
$507$	−50925.8	−4.46094
$508$	0	0
$509$	−14432.9	−1.25683	−0.628414	−	0.777879i	$-0.716296\pi$
$509$	−14432.9	−1.25683	−0.628414	+	0.777879i	$0.716296\pi$
$510$	0	0
$511$	−1303.24	−0.112822
$512$	0	0
$513$	−43044.4	−3.70459
$514$	0	0
$515$	−572.782	−0.0490093
$516$	0	0
$517$	6413.58	0.545588
$518$	0	0
$519$	−25094.3	−2.12238
$520$	0	0
$521$	1855.69	0.156044	0.0780222	−	0.996952i	$-0.475140\pi$
$521$	1855.69	0.156044	0.0780222	+	0.996952i	$0.475140\pi$
$522$	0	0
$523$	10294.0	0.860660	0.430330	−	0.902672i	$-0.358397\pi$
$523$	10294.0	0.860660	0.430330	+	0.902672i	$0.358397\pi$
$524$	0	0
$525$	3854.90	0.320460
$526$	0	0
$527$	6671.91	0.551486
$528$	0	0
$529$	−43.8612	−0.00360493
$530$	0	0
$531$	11638.5	0.951167
$532$	0	0
$533$	−24454.1	−1.98729
$534$	0	0
$535$	−4934.05	−0.398725
$536$	0	0
$537$	−7307.55	−0.587233
$538$	0	0
$539$	9412.25	0.752160
$540$	0	0
$541$	−4340.91	−0.344973	−0.172486	−	0.985012i	$-0.555180\pi$
$541$	−4340.91	−0.344973	−0.172486	+	0.985012i	$0.555180\pi$
$542$	0	0
$543$	40782.3	3.22309
$544$	0	0
$545$	−1650.00	−0.129685
$546$	0	0
$547$	−18868.8	−1.47490	−0.737452	−	0.675399i	$-0.763971\pi$
$547$	−18868.8	−1.47490	−0.737452	+	0.675399i	$0.763971\pi$
$548$	0	0
$549$	13287.1	1.03293
$550$	0	0
$551$	−2896.60	−0.223955
$552$	0	0
$553$	2001.46	0.153907
$554$	0	0
$555$	−3112.74	−0.238069
$556$	0	0
$557$	6986.62	0.531477	0.265738	−	0.964045i	$-0.414384\pi$
$557$	6986.62	0.531477	0.265738	+	0.964045i	$0.414384\pi$
$558$	0	0
$559$	−21398.5	−1.61907
$560$	0	0
$561$	−31110.5	−2.34133
$562$	0	0
$563$	−19006.3	−1.42277	−0.711386	−	0.702802i	$-0.751932\pi$
$563$	−19006.3	−1.42277	−0.711386	+	0.702802i	$0.751932\pi$
$564$	0	0
$565$	−3120.30	−0.232340
$566$	0	0
$567$	7772.53	0.575689
$568$	0	0
$569$	−5475.13	−0.403391	−0.201696	−	0.979448i	$-0.564645\pi$
$569$	−5475.13	−0.403391	−0.201696	+	0.979448i	$0.564645\pi$
$570$	0	0
$571$	1264.86	0.0927016	0.0463508	−	0.998925i	$-0.485241\pi$
$571$	1264.86	0.0927016	0.0463508	+	0.998925i	$0.485241\pi$
$572$	0	0
$573$	16107.4	1.17434
$574$	0	0
$575$	12995.1	0.942492
$576$	0	0
$577$	27670.3	1.99642	0.998208	−	0.0598463i	$-0.0190611\pi$
$577$	27670.3	1.99642	0.998208	+	0.0598463i	$0.0190611\pi$
$578$	0	0
$579$	2081.12	0.149375
$580$	0	0
$581$	−1669.38	−0.119204
$582$	0	0
$583$	−10973.8	−0.779569
$584$	0	0
$585$	15991.3	1.13019
$586$	0	0
$587$	6872.78	0.483254	0.241627	−	0.970369i	$-0.422319\pi$
$587$	6872.78	0.483254	0.241627	+	0.970369i	$0.422319\pi$
$588$	0	0
$589$	−5998.72	−0.419648
$590$	0	0
$591$	43654.2	3.03840
$592$	0	0
$593$	−4896.40	−0.339074	−0.169537	−	0.985524i	$-0.554227\pi$
$593$	−4896.40	−0.339074	−0.169537	+	0.985524i	$0.554227\pi$
$594$	0	0
$595$	−969.943	−0.0668299
$596$	0	0
$597$	14193.1	0.973006
$598$	0	0
$599$	−11189.9	−0.763282	−0.381641	−	0.924311i	$-0.624641\pi$
$599$	−11189.9	−0.763282	−0.381641	+	0.924311i	$0.624641\pi$
$600$	0	0
$601$	−6271.18	−0.425635	−0.212818	−	0.977092i	$-0.568264\pi$
$601$	−6271.18	−0.425635	−0.212818	+	0.977092i	$0.568264\pi$
$602$	0	0
$603$	−63813.5	−4.30960
$604$	0	0
$605$	−1393.50	−0.0936426
$606$	0	0
$607$	−22094.6	−1.47742	−0.738710	−	0.674024i	$-0.764564\pi$
$607$	−22094.6	−1.47742	−0.738710	+	0.674024i	$0.764564\pi$
$608$	0	0
$609$	947.194	0.0630250
$610$	0	0
$611$	−19400.9	−1.28458
$612$	0	0
$613$	26196.1	1.72602	0.863009	−	0.505189i	$-0.168577\pi$
$613$	26196.1	1.72602	0.863009	+	0.505189i	$0.168577\pi$
$614$	0	0
$615$	7442.53	0.487987
$616$	0	0
$617$	1836.57	0.119834	0.0599169	−	0.998203i	$-0.480916\pi$
$617$	1836.57	0.119834	0.0599169	+	0.998203i	$0.480916\pi$
$618$	0	0
$619$	−4634.79	−0.300950	−0.150475	−	0.988614i	$-0.548080\pi$
$619$	−4634.79	−0.300950	−0.150475	+	0.988614i	$0.548080\pi$
$620$	0	0
$621$	47449.8	3.06617
$622$	0	0
$623$	−3076.45	−0.197841
$624$	0	0
$625$	13057.8	0.835702
$626$	0	0
$627$	27971.5	1.78162
$628$	0	0
$629$	−13251.7	−0.840031
$630$	0	0
$631$	−16556.8	−1.04456	−0.522278	−	0.852775i	$-0.674918\pi$
$631$	−16556.8	−1.04456	−0.522278	+	0.852775i	$0.674918\pi$
$632$	0	0
$633$	−35284.6	−2.21554
$634$	0	0
$635$	4473.42	0.279563
$636$	0	0
$637$	−28471.8	−1.77095
$638$	0	0
$639$	57932.9	3.58653
$640$	0	0
$641$	−1437.66	−0.0885866	−0.0442933	−	0.999019i	$-0.514104\pi$
$641$	−1437.66	−0.0885866	−0.0442933	+	0.999019i	$0.514104\pi$
$642$	0	0
$643$	−9284.85	−0.569454	−0.284727	−	0.958609i	$-0.591903\pi$
$643$	−9284.85	−0.569454	−0.284727	+	0.958609i	$0.591903\pi$
$644$	0	0
$645$	6512.57	0.397569
$646$	0	0
$647$	14249.5	0.865852	0.432926	−	0.901430i	$-0.357481\pi$
$647$	14249.5	0.865852	0.432926	+	0.901430i	$0.357481\pi$
$648$	0	0
$649$	−4671.40	−0.282540
$650$	0	0
$651$	1961.60	0.118097
$652$	0	0
$653$	−11031.1	−0.661071	−0.330535	−	0.943794i	$-0.607229\pi$
$653$	−11031.1	−0.661071	−0.330535	+	0.943794i	$0.607229\pi$
$654$	0	0
$655$	−515.847	−0.0307723
$656$	0	0
$657$	−27839.4	−1.65315
$658$	0	0
$659$	−23429.9	−1.38497	−0.692487	−	0.721430i	$-0.743485\pi$
$659$	−23429.9	−1.38497	−0.692487	+	0.721430i	$0.743485\pi$
$660$	0	0
$661$	20939.6	1.23216	0.616079	−	0.787684i	$-0.288720\pi$
$661$	20939.6	1.23216	0.616079	+	0.787684i	$0.288720\pi$
$662$	0	0
$663$	94108.6	5.51263
$664$	0	0
$665$	872.077	0.0508537
$666$	0	0
$667$	3193.05	0.185360
$668$	0	0
$669$	63063.2	3.64449
$670$	0	0
$671$	−5333.08	−0.306828
$672$	0	0
$673$	10460.4	0.599138	0.299569	−	0.954075i	$-0.403157\pi$
$673$	10460.4	0.599138	0.299569	+	0.954075i	$0.403157\pi$
$674$	0	0
$675$	50862.6	2.90030
$676$	0	0
$677$	−1898.28	−0.107765	−0.0538824	−	0.998547i	$-0.517160\pi$
$677$	−1898.28	−0.107765	−0.0538824	+	0.998547i	$0.517160\pi$
$678$	0	0
$679$	40.5214	0.00229024
$680$	0	0
$681$	−45108.6	−2.53827
$682$	0	0
$683$	−6001.01	−0.336197	−0.168098	−	0.985770i	$-0.553763\pi$
$683$	−6001.01	−0.336197	−0.168098	+	0.985770i	$0.553763\pi$
$684$	0	0
$685$	−843.781	−0.0470646
$686$	0	0
$687$	−50233.0	−2.78968
$688$	0	0
$689$	33195.5	1.83548
$690$	0	0
$691$	20066.6	1.10473	0.552367	−	0.833601i	$-0.313725\pi$
$691$	20066.6	1.10473	0.552367	+	0.833601i	$0.313725\pi$
$692$	0	0
$693$	−6616.85	−0.362704
$694$	0	0
$695$	−2896.51	−0.158088
$696$	0	0
$697$	31684.7	1.72187
$698$	0	0
$699$	56748.6	3.07071
$700$	0	0
$701$	−7897.60	−0.425518	−0.212759	−	0.977105i	$-0.568245\pi$
$701$	−7897.60	−0.425518	−0.212759	+	0.977105i	$0.568245\pi$
$702$	0	0
$703$	11914.6	0.639215
$704$	0	0
$705$	5904.61	0.315433
$706$	0	0
$707$	−3852.72	−0.204946
$708$	0	0
$709$	−8805.89	−0.466449	−0.233224	−	0.972423i	$-0.574928\pi$
$709$	−8805.89	−0.466449	−0.233224	+	0.972423i	$0.574928\pi$
$710$	0	0
$711$	42754.7	2.25517
$712$	0	0
$713$	6612.66	0.347330
$714$	0	0
$715$	−6418.49	−0.335718
$716$	0	0
$717$	−54875.8	−2.85826
$718$	0	0
$719$	−9315.54	−0.483186	−0.241593	−	0.970378i	$-0.577670\pi$
$719$	−9315.54	−0.483186	−0.241593	+	0.970378i	$0.577670\pi$
$720$	0	0
$721$	−716.932	−0.0370318
$722$	0	0
$723$	21959.9	1.12959
$724$	0	0
$725$	3422.71	0.175333
$726$	0	0
$727$	−12393.7	−0.632264	−0.316132	−	0.948715i	$-0.602384\pi$
$727$	−12393.7	−0.632264	−0.316132	+	0.948715i	$0.602384\pi$
$728$	0	0
$729$	51072.7	2.59476
$730$	0	0
$731$	27725.6	1.40283
$732$	0	0
$733$	−25018.2	−1.26067	−0.630333	−	0.776325i	$-0.717082\pi$
$733$	−25018.2	−1.26067	−0.630333	+	0.776325i	$0.717082\pi$
$734$	0	0
$735$	8665.31	0.434864
$736$	0	0
$737$	25613.0	1.28015
$738$	0	0
$739$	2887.04	0.143710	0.0718548	−	0.997415i	$-0.477108\pi$
$739$	2887.04	0.143710	0.0718548	+	0.997415i	$0.477108\pi$
$740$	0	0
$741$	−84613.1	−4.19479
$742$	0	0
$743$	34004.2	1.67899	0.839497	−	0.543365i	$-0.182850\pi$
$743$	34004.2	1.67899	0.839497	+	0.543365i	$0.182850\pi$
$744$	0	0
$745$	3154.17	0.155114
$746$	0	0
$747$	−35660.9	−1.74667
$748$	0	0
$749$	−6175.79	−0.301279
$750$	0	0
$751$	−37460.2	−1.82016	−0.910081	−	0.414431i	$-0.863981\pi$
$751$	−37460.2	−1.82016	−0.910081	+	0.414431i	$0.863981\pi$
$752$	0	0
$753$	−49140.7	−2.37820
$754$	0	0
$755$	5967.22	0.287641
$756$	0	0
$757$	30483.5	1.46359	0.731797	−	0.681523i	$-0.238682\pi$
$757$	30483.5	1.46359	0.731797	+	0.681523i	$0.238682\pi$
$758$	0	0
$759$	−30834.2	−1.47459
$760$	0	0
$761$	19511.6	0.929430	0.464715	−	0.885460i	$-0.346157\pi$
$761$	19511.6	0.929430	0.464715	+	0.885460i	$0.346157\pi$
$762$	0	0
$763$	−2065.25	−0.0979912
$764$	0	0
$765$	−20719.7	−0.979244
$766$	0	0
$767$	14130.9	0.665236
$768$	0	0
$769$	−17422.4	−0.816995	−0.408497	−	0.912760i	$-0.633947\pi$
$769$	−17422.4	−0.816995	−0.408497	+	0.912760i	$0.633947\pi$
$770$	0	0
$771$	−148.990	−0.00695947
$772$	0	0
$773$	−3685.78	−0.171498	−0.0857491	−	0.996317i	$-0.527328\pi$
$773$	−3685.78	−0.171498	−0.0857491	+	0.996317i	$0.527328\pi$
$774$	0	0
$775$	7088.28	0.328540
$776$	0	0
$777$	−3896.11	−0.179887
$778$	0	0
$779$	−28487.8	−1.31024
$780$	0	0
$781$	−23252.7	−1.06536
$782$	0	0
$783$	12497.5	0.570403
$784$	0	0
$785$	−6846.65	−0.311296
$786$	0	0
$787$	−7764.73	−0.351694	−0.175847	−	0.984418i	$-0.556266\pi$
$787$	−7764.73	−0.351694	−0.175847	+	0.984418i	$0.556266\pi$
$788$	0	0
$789$	−31994.9	−1.44366
$790$	0	0
$791$	−3905.58	−0.175558
$792$	0	0
$793$	16132.4	0.722421
$794$	0	0
$795$	−10103.0	−0.450710
$796$	0	0
$797$	22754.0	1.01128	0.505639	−	0.862745i	$-0.331257\pi$
$797$	22754.0	1.01128	0.505639	+	0.862745i	$0.331257\pi$
$798$	0	0
$799$	25137.4	1.11301
$800$	0	0
$801$	−65718.2	−2.89893
$802$	0	0
$803$	11174.0	0.491061
$804$	0	0
$805$	−961.330	−0.0420900
$806$	0	0
$807$	−40976.0	−1.78739
$808$	0	0
$809$	9866.29	0.428776	0.214388	−	0.976749i	$-0.431224\pi$
$809$	9866.29	0.428776	0.214388	+	0.976749i	$0.431224\pi$
$810$	0	0
$811$	12456.6	0.539346	0.269673	−	0.962952i	$-0.413084\pi$
$811$	12456.6	0.539346	0.269673	+	0.962952i	$0.413084\pi$
$812$	0	0
$813$	−12937.6	−0.558108
$814$	0	0
$815$	1315.71	0.0565487
$816$	0	0
$817$	−24928.1	−1.06747
$818$	0	0
$819$	20015.8	0.853980
$820$	0	0
$821$	22347.4	0.949976	0.474988	−	0.879992i	$-0.342452\pi$
$821$	22347.4	0.949976	0.474988	+	0.879992i	$0.342452\pi$
$822$	0	0
$823$	−37377.5	−1.58311	−0.791554	−	0.611100i	$-0.790727\pi$
$823$	−37377.5	−1.58311	−0.791554	+	0.611100i	$0.790727\pi$
$824$	0	0
$825$	−33052.0	−1.39482
$826$	0	0
$827$	−21218.3	−0.892182	−0.446091	−	0.894988i	$-0.647184\pi$
$827$	−21218.3	−0.892182	−0.446091	+	0.894988i	$0.647184\pi$
$828$	0	0
$829$	12922.1	0.541378	0.270689	−	0.962667i	$-0.412749\pi$
$829$	12922.1	0.541378	0.270689	+	0.962667i	$0.412749\pi$
$830$	0	0
$831$	−66659.1	−2.78265
$832$	0	0
$833$	36890.4	1.53443
$834$	0	0
$835$	1456.42	0.0603613
$836$	0	0
$837$	25881.9	1.06883
$838$	0	0
$839$	25253.6	1.03915	0.519577	−	0.854423i	$-0.326089\pi$
$839$	25253.6	1.03915	0.519577	+	0.854423i	$0.326089\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−28615.4	−1.16912
$844$	0	0
$845$	13613.3	0.554213
$846$	0	0
$847$	−1744.19	−0.0707571
$848$	0	0
$849$	84746.5	3.42579
$850$	0	0
$851$	−13134.0	−0.529058
$852$	0	0
$853$	23958.9	0.961707	0.480854	−	0.876801i	$-0.340327\pi$
$853$	23958.9	0.961707	0.480854	+	0.876801i	$0.340327\pi$
$854$	0	0
$855$	18629.1	0.745147
$856$	0	0
$857$	−1289.87	−0.0514133	−0.0257066	−	0.999670i	$-0.508184\pi$
$857$	−1289.87	−0.0514133	−0.0257066	+	0.999670i	$0.508184\pi$
$858$	0	0
$859$	12961.8	0.514843	0.257422	−	0.966299i	$-0.417127\pi$
$859$	12961.8	0.514843	0.257422	+	0.966299i	$0.417127\pi$
$860$	0	0
$861$	9315.57	0.368727
$862$	0	0
$863$	39133.0	1.54357	0.771785	−	0.635883i	$-0.219364\pi$
$863$	39133.0	1.54357	0.771785	+	0.635883i	$0.219364\pi$
$864$	0	0
$865$	6708.09	0.263678
$866$	0	0
$867$	−73393.5	−2.87494
$868$	0	0
$869$	−17160.6	−0.669888
$870$	0	0
$871$	−77478.8	−3.01409
$872$	0	0
$873$	865.609	0.0335583
$874$	0	0
$875$	−2121.85	−0.0819790
$876$	0	0
$877$	−12957.8	−0.498921	−0.249460	−	0.968385i	$-0.580253\pi$
$877$	−12957.8	−0.498921	−0.249460	+	0.968385i	$0.580253\pi$
$878$	0	0
$879$	81056.8	3.11033
$880$	0	0
$881$	−33767.4	−1.29132	−0.645659	−	0.763626i	$-0.723417\pi$
$881$	−33767.4	−1.29132	−0.645659	+	0.763626i	$0.723417\pi$
$882$	0	0
$883$	−34290.1	−1.30686	−0.653428	−	0.756989i	$-0.726670\pi$
$883$	−34290.1	−1.30686	−0.653428	+	0.756989i	$0.726670\pi$
$884$	0	0
$885$	−4300.68	−0.163351
$886$	0	0
$887$	10538.1	0.398911	0.199456	−	0.979907i	$-0.436083\pi$
$887$	10538.1	0.398911	0.199456	+	0.979907i	$0.436083\pi$
$888$	0	0
$889$	5599.23	0.211240
$890$	0	0
$891$	−66641.9	−2.50571
$892$	0	0
$893$	−22601.1	−0.846938
$894$	0	0
$895$	1953.42	0.0729560
$896$	0	0
$897$	93272.9	3.47190
$898$	0	0
$899$	1741.67	0.0646141
$900$	0	0
$901$	−43010.8	−1.59034
$902$	0	0
$903$	8151.57	0.300407
$904$	0	0
$905$	−10901.7	−0.400427
$906$	0	0
$907$	25835.0	0.945797	0.472899	−	0.881117i	$-0.343208\pi$
$907$	25835.0	0.945797	0.472899	+	0.881117i	$0.343208\pi$
$908$	0	0
$909$	−82300.9	−3.00302
$910$	0	0
$911$	6732.30	0.244842	0.122421	−	0.992478i	$-0.460934\pi$
$911$	6732.30	0.244842	0.122421	+	0.992478i	$0.460934\pi$
$912$	0	0
$913$	14313.3	0.518841
$914$	0	0
$915$	−4909.86	−0.177393
$916$	0	0
$917$	−645.669	−0.0232518
$918$	0	0
$919$	16257.6	0.583557	0.291778	−	0.956486i	$-0.405753\pi$
$919$	16257.6	0.583557	0.291778	+	0.956486i	$0.405753\pi$
$920$	0	0
$921$	24764.1	0.886000
$922$	0	0
$923$	70338.9	2.50838
$924$	0	0
$925$	−14078.7	−0.500437
$926$	0	0
$927$	−15314.9	−0.542619
$928$	0	0
$929$	−27354.8	−0.966073	−0.483037	−	0.875600i	$-0.660466\pi$
$929$	−27354.8	−0.966073	−0.483037	+	0.875600i	$0.660466\pi$
$930$	0	0
$931$	−33168.2	−1.16761
$932$	0	0
$933$	64235.9	2.25401
$934$	0	0
$935$	8316.32	0.290880
$936$	0	0
$937$	38897.4	1.35616	0.678080	−	0.734988i	$-0.262812\pi$
$937$	38897.4	1.35616	0.678080	+	0.734988i	$0.262812\pi$
$938$	0	0
$939$	−71575.4	−2.48751
$940$	0	0
$941$	36998.7	1.28175	0.640874	−	0.767646i	$-0.278572\pi$
$941$	36998.7	1.28175	0.640874	+	0.767646i	$0.278572\pi$
$942$	0	0
$943$	31403.4	1.08445
$944$	0	0
$945$	−3762.63	−0.129522
$946$	0	0
$947$	37594.0	1.29001	0.645006	−	0.764177i	$-0.276855\pi$
$947$	37594.0	1.29001	0.645006	+	0.764177i	$0.276855\pi$
$948$	0	0
$949$	−33801.1	−1.15620
$950$	0	0
$951$	−79725.0	−2.71847
$952$	0	0
$953$	49341.0	1.67714	0.838568	−	0.544797i	$-0.183393\pi$
$953$	49341.0	1.67714	0.838568	+	0.544797i	$0.183393\pi$
$954$	0	0
$955$	−4305.75	−0.145896
$956$	0	0
$957$	−8121.27	−0.274319
$958$	0	0
$959$	−1056.13	−0.0355624
$960$	0	0
$961$	−26184.1	−0.878926
$962$	0	0
$963$	−131926.	−4.41458
$964$	0	0
$965$	−556.315	−0.0185579
$966$	0	0
$967$	44067.6	1.46548	0.732740	−	0.680509i	$-0.238241\pi$
$967$	44067.6	1.46548	0.732740	+	0.680509i	$0.238241\pi$
$968$	0	0
$969$	109632.	3.63454
$970$	0	0
$971$	−51385.3	−1.69828	−0.849141	−	0.528166i	$-0.822880\pi$
$971$	−51385.3	−1.69828	−0.849141	+	0.528166i	$0.822880\pi$
$972$	0	0
$973$	−3625.46	−0.119452
$974$	0	0
$975$	99981.6	3.28407
$976$	0	0
$977$	27297.1	0.893871	0.446935	−	0.894566i	$-0.352515\pi$
$977$	27297.1	0.893871	0.446935	+	0.894566i	$0.352515\pi$
$978$	0	0
$979$	26377.5	0.861113
$980$	0	0
$981$	−44117.4	−1.43584
$982$	0	0
$983$	−3876.96	−0.125794	−0.0628972	−	0.998020i	$-0.520034\pi$
$983$	−3876.96	−0.125794	−0.0628972	+	0.998020i	$0.520034\pi$
$984$	0	0
$985$	−11669.4	−0.377481
$986$	0	0
$987$	7390.60	0.238344
$988$	0	0
$989$	27479.4	0.883514
$990$	0	0
$991$	11942.6	0.382814	0.191407	−	0.981511i	$-0.438695\pi$
$991$	11942.6	0.382814	0.191407	+	0.981511i	$0.438695\pi$
$992$	0	0
$993$	−62758.6	−2.00562
$994$	0	0
$995$	−3794.03	−0.120883
$996$	0	0
$997$	60556.0	1.92360	0.961800	−	0.273755i	$-0.0882656\pi$
$997$	60556.0	1.92360	0.961800	+	0.273755i	$0.0882656\pi$
$998$	0	0
$999$	−51406.3	−1.62805

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	232.4.a.e.1.1	✓	6
3.2	odd	2		2088.4.a.l.1.3		6
4.3	odd	2		464.4.a.n.1.6		6
8.3	odd	2		1856.4.a.bc.1.1		6
8.5	even	2		1856.4.a.bd.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.4.a.e.1.1	✓	6	1.1	even	1	trivial
464.4.a.n.1.6		6	4.3	odd	2
1856.4.a.bc.1.1		6	8.3	odd	2
1856.4.a.bd.1.6		6	8.5	even	2
2088.4.a.l.1.3		6	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 232 = 2^{3} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	232.a (trivial)

\(f(q)\)	\(=\)	\(q-9.88017 q^{3} +2.64112 q^{5} +3.30580 q^{7} +70.6177 q^{9} +O(q^{10})\)	\(q-9.88017 q^{3} +2.64112 q^{5} +3.30580 q^{7} +70.6177 q^{9} -28.3440 q^{11} +85.7400 q^{13} -26.0947 q^{15} -111.092 q^{17} +99.8826 q^{19} -32.6619 q^{21} -110.105 q^{23} -118.024 q^{25} -430.950 q^{27} -29.0000 q^{29} -60.0577 q^{31} +280.044 q^{33} +8.73102 q^{35} +119.286 q^{37} -847.126 q^{39} -285.212 q^{41} -249.574 q^{43} +186.510 q^{45} -226.276 q^{47} -332.072 q^{49} +1097.60 q^{51} +387.165 q^{53} -74.8600 q^{55} -986.857 q^{57} +164.811 q^{59} +188.155 q^{61} +233.448 q^{63} +226.450 q^{65} -903.648 q^{67} +1087.86 q^{69} +820.374 q^{71} -394.228 q^{73} +1166.10 q^{75} -93.6997 q^{77} +605.439 q^{79} +2351.18 q^{81} -504.986 q^{83} -293.406 q^{85} +286.525 q^{87} -930.620 q^{89} +283.439 q^{91} +593.380 q^{93} +263.802 q^{95} +12.2577 q^{97} -2001.59 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 5 q^{3} - 5 q^{5} - 38 q^{7} + 47 q^{9}+O(q^{10}) \) 6 * q - 5 * q^3 - 5 * q^5 - 38 * q^7 + 47 * q^9	\( 6 q - 5 q^{3} - 5 q^{5} - 38 q^{7} + 47 q^{9} - 19 q^{11} + 13 q^{13} - 191 q^{15} - 218 q^{17} - 290 q^{19} - 266 q^{21} - 196 q^{23} - 13 q^{25} - 437 q^{27} - 174 q^{29} - 675 q^{31} + 291 q^{33} - 466 q^{35} - 238 q^{37} - 1297 q^{39} - 464 q^{41} - 579 q^{43} - 148 q^{45} - 975 q^{47} + 914 q^{49} - 576 q^{51} + 515 q^{53} - 1605 q^{55} - 340 q^{57} - 108 q^{59} + 1158 q^{61} - 1136 q^{63} + 1239 q^{65} - 80 q^{67} + 2568 q^{69} + 438 q^{71} + 262 q^{73} + 1766 q^{75} + 194 q^{77} - 237 q^{79} + 2554 q^{81} - 1288 q^{83} + 3112 q^{85} + 145 q^{87} - 252 q^{89} + 2450 q^{91} + 2131 q^{93} + 180 q^{95} + 380 q^{97} + 2264 q^{99}+O(q^{100}) \) 6 * q - 5 * q^3 - 5 * q^5 - 38 * q^7 + 47 * q^9 - 19 * q^11 + 13 * q^13 - 191 * q^15 - 218 * q^17 - 290 * q^19 - 266 * q^21 - 196 * q^23 - 13 * q^25 - 437 * q^27 - 174 * q^29 - 675 * q^31 + 291 * q^33 - 466 * q^35 - 238 * q^37 - 1297 * q^39 - 464 * q^41 - 579 * q^43 - 148 * q^45 - 975 * q^47 + 914 * q^49 - 576 * q^51 + 515 * q^53 - 1605 * q^55 - 340 * q^57 - 108 * q^59 + 1158 * q^61 - 1136 * q^63 + 1239 * q^65 - 80 * q^67 + 2568 * q^69 + 438 * q^71 + 262 * q^73 + 1766 * q^75 + 194 * q^77 - 237 * q^79 + 2554 * q^81 - 1288 * q^83 + 3112 * q^85 + 145 * q^87 - 252 * q^89 + 2450 * q^91 + 2131 * q^93 + 180 * q^95 + 380 * q^97 + 2264 * q^99

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