LMFDB - Embedded newform 232.4.a.e.1.6

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			1.6
Root			$0.488896$ 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)

$f(q)$	$=$	$q+7.88819 q^{3} -14.0062 q^{5} -34.0826 q^{7} +35.2236 q^{9} +O(q^{10})$	$q+7.88819 q^{3} -14.0062 q^{5} -34.0826 q^{7} +35.2236 q^{9} +52.3468 q^{11} -64.2756 q^{13} -110.484 q^{15} -109.902 q^{17} -72.2444 q^{19} -268.850 q^{21} +98.4396 q^{23} +71.1747 q^{25} +64.8692 q^{27} -29.0000 q^{29} +26.6330 q^{31} +412.922 q^{33} +477.368 q^{35} -147.274 q^{37} -507.018 q^{39} +134.137 q^{41} -69.3546 q^{43} -493.350 q^{45} -235.743 q^{47} +818.621 q^{49} -866.929 q^{51} +83.8806 q^{53} -733.182 q^{55} -569.878 q^{57} -164.222 q^{59} +916.499 q^{61} -1200.51 q^{63} +900.259 q^{65} -34.3688 q^{67} +776.511 q^{69} +827.316 q^{71} -360.862 q^{73} +561.440 q^{75} -1784.11 q^{77} +291.765 q^{79} -439.336 q^{81} -1402.06 q^{83} +1539.32 q^{85} -228.758 q^{87} -1039.96 q^{89} +2190.68 q^{91} +210.086 q^{93} +1011.87 q^{95} +472.758 q^{97} +1843.84 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	7.88819	1.51808	0.759042	−	0.651042i	$-0.225668\pi$
$3$	7.88819	1.51808	0.759042	+	0.651042i	$0.225668\pi$
$4$	0	0
$5$	−14.0062	−1.25276	−0.626378	−	0.779519i	$-0.715463\pi$
$5$	−14.0062	−1.25276	−0.626378	+	0.779519i	$0.715463\pi$
$6$	0	0
$7$	−34.0826	−1.84029	−0.920143	−	0.391583i	$-0.871927\pi$
$7$	−34.0826	−1.84029	−0.920143	+	0.391583i	$0.871927\pi$
$8$	0	0
$9$	35.2236	1.30458
$10$	0	0
$11$	52.3468	1.43483	0.717417	−	0.696644i	$-0.245324\pi$
$11$	52.3468	1.43483	0.717417	+	0.696644i	$0.245324\pi$
$12$	0	0
$13$	−64.2756	−1.37130	−0.685648	−	0.727933i	$-0.740481\pi$
$13$	−64.2756	−1.37130	−0.685648	+	0.727933i	$0.740481\pi$
$14$	0	0
$15$	−110.484	−1.90179
$16$	0	0
$17$	−109.902	−1.56795	−0.783976	−	0.620791i	$-0.786812\pi$
$17$	−109.902	−1.56795	−0.783976	+	0.620791i	$0.786812\pi$
$18$	0	0
$19$	−72.2444	−0.872316	−0.436158	−	0.899870i	$-0.643661\pi$
$19$	−72.2444	−0.872316	−0.436158	+	0.899870i	$0.643661\pi$
$20$	0	0
$21$	−268.850	−2.79371
$22$	0	0
$23$	98.4396	0.892438	0.446219	−	0.894924i	$-0.352770\pi$
$23$	98.4396	0.892438	0.446219	+	0.894924i	$0.352770\pi$
$24$	0	0
$25$	71.1747	0.569398
$26$	0	0
$27$	64.8692	0.462373
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	26.6330	0.154304	0.0771520	−	0.997019i	$-0.475417\pi$
$31$	26.6330	0.154304	0.0771520	+	0.997019i	$0.475417\pi$
$32$	0	0
$33$	412.922	2.17820
$34$	0	0
$35$	477.368	2.30543
$36$	0	0
$37$	−147.274	−0.654369	−0.327185	−	0.944960i	$-0.606100\pi$
$37$	−147.274	−0.654369	−0.327185	+	0.944960i	$0.606100\pi$
$38$	0	0
$39$	−507.018	−2.08174
$40$	0	0
$41$	134.137	0.510943	0.255471	−	0.966817i	$-0.417769\pi$
$41$	134.137	0.510943	0.255471	+	0.966817i	$0.417769\pi$
$42$	0	0
$43$	−69.3546	−0.245965	−0.122982	−	0.992409i	$-0.539246\pi$
$43$	−69.3546	−0.245965	−0.122982	+	0.992409i	$0.539246\pi$
$44$	0	0
$45$	−493.350	−1.63432
$46$	0	0
$47$	−235.743	−0.731630	−0.365815	−	0.930688i	$-0.619210\pi$
$47$	−235.743	−0.731630	−0.365815	+	0.930688i	$0.619210\pi$
$48$	0	0
$49$	818.621	2.38665
$50$	0	0
$51$	−866.929	−2.38028
$52$	0	0
$53$	83.8806	0.217394	0.108697	−	0.994075i	$-0.465332\pi$
$53$	83.8806	0.217394	0.108697	+	0.994075i	$0.465332\pi$
$54$	0	0
$55$	−733.182	−1.79750
$56$	0	0
$57$	−569.878	−1.32425
$58$	0	0
$59$	−164.222	−0.362371	−0.181185	−	0.983449i	$-0.557993\pi$
$59$	−164.222	−0.362371	−0.181185	+	0.983449i	$0.557993\pi$
$60$	0	0
$61$	916.499	1.92370	0.961850	−	0.273578i	$-0.0882070\pi$
$61$	916.499	1.92370	0.961850	+	0.273578i	$0.0882070\pi$
$62$	0	0
$63$	−1200.51	−2.40079
$64$	0	0
$65$	900.259	1.71790
$66$	0	0
$67$	−34.3688	−0.0626689	−0.0313345	−	0.999509i	$-0.509976\pi$
$67$	−34.3688	−0.0626689	−0.0313345	+	0.999509i	$0.509976\pi$
$68$	0	0
$69$	776.511	1.35480
$70$	0	0
$71$	827.316	1.38288	0.691439	−	0.722435i	$-0.256977\pi$
$71$	827.316	1.38288	0.691439	+	0.722435i	$0.256977\pi$
$72$	0	0
$73$	−360.862	−0.578571	−0.289286	−	0.957243i	$-0.593418\pi$
$73$	−360.862	−0.578571	−0.289286	+	0.957243i	$0.593418\pi$
$74$	0	0
$75$	561.440	0.864393
$76$	0	0
$77$	−1784.11	−2.64050
$78$	0	0
$79$	291.765	0.415520	0.207760	−	0.978180i	$-0.433383\pi$
$79$	291.765	0.415520	0.207760	+	0.978180i	$0.433383\pi$
$80$	0	0
$81$	−439.336	−0.602656
$82$	0	0
$83$	−1402.06	−1.85417	−0.927084	−	0.374854i	$-0.877693\pi$
$83$	−1402.06	−1.85417	−0.927084	+	0.374854i	$0.877693\pi$
$84$	0	0
$85$	1539.32	1.96426
$86$	0	0
$87$	−228.758	−0.281901
$88$	0	0
$89$	−1039.96	−1.23860	−0.619298	−	0.785156i	$-0.712583\pi$
$89$	−1039.96	−1.23860	−0.619298	+	0.785156i	$0.712583\pi$
$90$	0	0
$91$	2190.68	2.52358
$92$	0	0
$93$	210.086	0.234246
$94$	0	0
$95$	1011.87	1.09280
$96$	0	0
$97$	472.758	0.494858	0.247429	−	0.968906i	$-0.420414\pi$
$97$	472.758	0.494858	0.247429	+	0.968906i	$0.420414\pi$
$98$	0	0
$99$	1843.84	1.87185
$100$	0	0
$101$	−568.150	−0.559733	−0.279867	−	0.960039i	$-0.590290\pi$
$101$	−568.150	−0.559733	−0.279867	+	0.960039i	$0.590290\pi$
$102$	0	0
$103$	−226.793	−0.216957	−0.108478	−	0.994099i	$-0.534598\pi$
$103$	−226.793	−0.216957	−0.108478	+	0.994099i	$0.534598\pi$
$104$	0	0
$105$	3765.57	3.49983
$106$	0	0
$107$	−1795.06	−1.62182	−0.810909	−	0.585172i	$-0.801027\pi$
$107$	−1795.06	−1.62182	−0.810909	+	0.585172i	$0.801027\pi$
$108$	0	0
$109$	574.149	0.504527	0.252264	−	0.967659i	$-0.418825\pi$
$109$	574.149	0.504527	0.252264	+	0.967659i	$0.418825\pi$
$110$	0	0
$111$	−1161.72	−0.993387
$112$	0	0
$113$	2280.06	1.89815	0.949073	−	0.315058i	$-0.102024\pi$
$113$	2280.06	1.89815	0.949073	+	0.315058i	$0.102024\pi$
$114$	0	0
$115$	−1378.77	−1.11801
$116$	0	0
$117$	−2264.02	−1.78896
$118$	0	0
$119$	3745.75	2.88548
$120$	0	0
$121$	1409.19	1.05875
$122$	0	0
$123$	1058.10	0.775654
$124$	0	0
$125$	753.890	0.539440
$126$	0	0
$127$	936.276	0.654181	0.327091	−	0.944993i	$-0.393932\pi$
$127$	936.276	0.654181	0.327091	+	0.944993i	$0.393932\pi$
$128$	0	0
$129$	−547.083	−0.373395
$130$	0	0
$131$	117.142	0.0781279	0.0390639	−	0.999237i	$-0.487562\pi$
$131$	117.142	0.0781279	0.0390639	+	0.999237i	$0.487562\pi$
$132$	0	0
$133$	2462.28	1.60531
$134$	0	0
$135$	−908.573	−0.579241
$136$	0	0
$137$	−2980.30	−1.85857	−0.929287	−	0.369359i	$-0.879577\pi$
$137$	−2980.30	−1.85857	−0.929287	+	0.369359i	$0.879577\pi$
$138$	0	0
$139$	−2075.35	−1.26639	−0.633197	−	0.773991i	$-0.718258\pi$
$139$	−2075.35	−1.26639	−0.633197	+	0.773991i	$0.718258\pi$
$140$	0	0
$141$	−1859.58	−1.11068
$142$	0	0
$143$	−3364.63	−1.96758
$144$	0	0
$145$	406.181	0.232631
$146$	0	0
$147$	6457.44	3.62313
$148$	0	0
$149$	933.824	0.513435	0.256718	−	0.966486i	$-0.417359\pi$
$149$	933.824	0.513435	0.256718	+	0.966486i	$0.417359\pi$
$150$	0	0
$151$	−2409.25	−1.29842	−0.649212	−	0.760607i	$-0.724901\pi$
$151$	−2409.25	−1.29842	−0.649212	+	0.760607i	$0.724901\pi$
$152$	0	0
$153$	−3871.15	−2.04551
$154$	0	0
$155$	−373.028	−0.193305
$156$	0	0
$157$	−2066.46	−1.05046	−0.525228	−	0.850962i	$-0.676020\pi$
$157$	−2066.46	−1.05046	−0.525228	+	0.850962i	$0.676020\pi$
$158$	0	0
$159$	661.666	0.330022
$160$	0	0
$161$	−3355.07	−1.64234
$162$	0	0
$163$	2170.49	1.04298	0.521490	−	0.853257i	$-0.325376\pi$
$163$	2170.49	1.04298	0.521490	+	0.853257i	$0.325376\pi$
$164$	0	0
$165$	−5783.48	−2.72875
$166$	0	0
$167$	−1204.26	−0.558015	−0.279008	−	0.960289i	$-0.590005\pi$
$167$	−1204.26	−0.558015	−0.279008	+	0.960289i	$0.590005\pi$
$168$	0	0
$169$	1934.35	0.880452
$170$	0	0
$171$	−2544.71	−1.13800
$172$	0	0
$173$	2356.19	1.03548	0.517739	−	0.855539i	$-0.326774\pi$
$173$	2356.19	1.03548	0.517739	+	0.855539i	$0.326774\pi$
$174$	0	0
$175$	−2425.82	−1.04785
$176$	0	0
$177$	−1295.41	−0.550109
$178$	0	0
$179$	2111.78	0.881797	0.440899	−	0.897557i	$-0.354660\pi$
$179$	2111.78	0.881797	0.440899	+	0.897557i	$0.354660\pi$
$180$	0	0
$181$	2037.00	0.836513	0.418256	−	0.908329i	$-0.362641\pi$
$181$	2037.00	0.836513	0.418256	+	0.908329i	$0.362641\pi$
$182$	0	0
$183$	7229.52	2.92034
$184$	0	0
$185$	2062.75	0.819765
$186$	0	0
$187$	−5753.03	−2.24975
$188$	0	0
$189$	−2210.91	−0.850899
$190$	0	0
$191$	−3626.31	−1.37377	−0.686887	−	0.726764i	$-0.741023\pi$
$191$	−3626.31	−1.37377	−0.686887	+	0.726764i	$0.741023\pi$
$192$	0	0
$193$	−993.284	−0.370457	−0.185228	−	0.982695i	$-0.559303\pi$
$193$	−993.284	−0.370457	−0.185228	+	0.982695i	$0.559303\pi$
$194$	0	0
$195$	7101.42	2.60791
$196$	0	0
$197$	−2856.69	−1.03315	−0.516576	−	0.856242i	$-0.672793\pi$
$197$	−2856.69	−1.03315	−0.516576	+	0.856242i	$0.672793\pi$
$198$	0	0
$199$	−3264.70	−1.16296	−0.581478	−	0.813562i	$-0.697526\pi$
$199$	−3264.70	−1.16296	−0.581478	+	0.813562i	$0.697526\pi$
$200$	0	0
$201$	−271.108	−0.0951367
$202$	0	0
$203$	988.394	0.341732
$204$	0	0
$205$	−1878.75	−0.640087
$206$	0	0
$207$	3467.40	1.16425
$208$	0	0
$209$	−3781.77	−1.25163
$210$	0	0
$211$	2105.01	0.686799	0.343400	−	0.939189i	$-0.388421\pi$
$211$	2105.01	0.686799	0.343400	+	0.939189i	$0.388421\pi$
$212$	0	0
$213$	6526.03	2.09932
$214$	0	0
$215$	971.398	0.308134
$216$	0	0
$217$	−907.720	−0.283963
$218$	0	0
$219$	−2846.55	−0.878320
$220$	0	0
$221$	7064.03	2.15013
$222$	0	0
$223$	2041.18	0.612950	0.306475	−	0.951879i	$-0.400850\pi$
$223$	2041.18	0.612950	0.306475	+	0.951879i	$0.400850\pi$
$224$	0	0
$225$	2507.03	0.742823
$226$	0	0
$227$	−1127.20	−0.329582	−0.164791	−	0.986329i	$-0.552695\pi$
$227$	−1127.20	−0.329582	−0.164791	+	0.986329i	$0.552695\pi$
$228$	0	0
$229$	−6524.06	−1.88263	−0.941315	−	0.337530i	$-0.890409\pi$
$229$	−6524.06	−1.88263	−0.941315	+	0.337530i	$0.890409\pi$
$230$	0	0
$231$	−14073.4	−4.00850
$232$	0	0
$233$	537.559	0.151145	0.0755723	−	0.997140i	$-0.475922\pi$
$233$	537.559	0.151145	0.0755723	+	0.997140i	$0.475922\pi$
$234$	0	0
$235$	3301.87	0.916554
$236$	0	0
$237$	2301.50	0.630794
$238$	0	0
$239$	−948.923	−0.256823	−0.128411	−	0.991721i	$-0.540988\pi$
$239$	−948.923	−0.256823	−0.128411	+	0.991721i	$0.540988\pi$
$240$	0	0
$241$	−409.231	−0.109381	−0.0546906	−	0.998503i	$-0.517417\pi$
$241$	−409.231	−0.109381	−0.0546906	+	0.998503i	$0.517417\pi$
$242$	0	0
$243$	−5217.04	−1.37725
$244$	0	0
$245$	−11465.8	−2.98989
$246$	0	0
$247$	4643.56	1.19620
$248$	0	0
$249$	−11059.7	−2.81478
$250$	0	0
$251$	−1134.24	−0.285229	−0.142615	−	0.989778i	$-0.545551\pi$
$251$	−1134.24	−0.285229	−0.142615	+	0.989778i	$0.545551\pi$
$252$	0	0
$253$	5153.00	1.28050
$254$	0	0
$255$	12142.4	2.98191
$256$	0	0
$257$	2159.57	0.524166	0.262083	−	0.965045i	$-0.415591\pi$
$257$	2159.57	0.524166	0.262083	+	0.965045i	$0.415591\pi$
$258$	0	0
$259$	5019.47	1.20423
$260$	0	0
$261$	−1021.48	−0.242254
$262$	0	0
$263$	−3759.09	−0.881350	−0.440675	−	0.897667i	$-0.645261\pi$
$263$	−3759.09	−0.881350	−0.440675	+	0.897667i	$0.645261\pi$
$264$	0	0
$265$	−1174.85	−0.272342
$266$	0	0
$267$	−8203.37	−1.88029
$268$	0	0
$269$	5350.57	1.21275	0.606375	−	0.795179i	$-0.292623\pi$
$269$	5350.57	1.21275	0.606375	+	0.795179i	$0.292623\pi$
$270$	0	0
$271$	−5578.98	−1.25055	−0.625274	−	0.780405i	$-0.715013\pi$
$271$	−5578.98	−1.25055	−0.625274	+	0.780405i	$0.715013\pi$
$272$	0	0
$273$	17280.5	3.83100
$274$	0	0
$275$	3725.77	0.816991
$276$	0	0
$277$	−408.588	−0.0886270	−0.0443135	−	0.999018i	$-0.514110\pi$
$277$	−408.588	−0.0886270	−0.0443135	+	0.999018i	$0.514110\pi$
$278$	0	0
$279$	938.109	0.201302
$280$	0	0
$281$	−2837.97	−0.602487	−0.301244	−	0.953547i	$-0.597402\pi$
$281$	−2837.97	−0.602487	−0.301244	+	0.953547i	$0.597402\pi$
$282$	0	0
$283$	5652.04	1.18720	0.593602	−	0.804758i	$-0.297705\pi$
$283$	5652.04	1.18720	0.593602	+	0.804758i	$0.297705\pi$
$284$	0	0
$285$	7981.85	1.65896
$286$	0	0
$287$	−4571.73	−0.940280
$288$	0	0
$289$	7165.48	1.45847
$290$	0	0
$291$	3729.20	0.751236
$292$	0	0
$293$	−1900.37	−0.378911	−0.189455	−	0.981889i	$-0.560672\pi$
$293$	−1900.37	−0.378911	−0.189455	+	0.981889i	$0.560672\pi$
$294$	0	0
$295$	2300.13	0.453962
$296$	0	0
$297$	3395.70	0.663429
$298$	0	0
$299$	−6327.27	−1.22380
$300$	0	0
$301$	2363.78	0.452645
$302$	0	0
$303$	−4481.68	−0.849722
$304$	0	0
$305$	−12836.7	−2.40993
$306$	0	0
$307$	3428.86	0.637445	0.318722	−	0.947848i	$-0.396746\pi$
$307$	3428.86	0.637445	0.318722	+	0.947848i	$0.396746\pi$
$308$	0	0
$309$	−1788.98	−0.329358
$310$	0	0
$311$	−8842.05	−1.61218	−0.806088	−	0.591796i	$-0.798419\pi$
$311$	−8842.05	−1.61218	−0.806088	+	0.591796i	$0.798419\pi$
$312$	0	0
$313$	−5677.71	−1.02531	−0.512657	−	0.858594i	$-0.671339\pi$
$313$	−5677.71	−1.02531	−0.512657	+	0.858594i	$0.671339\pi$
$314$	0	0
$315$	16814.6	3.00761
$316$	0	0
$317$	−1496.83	−0.265207	−0.132603	−	0.991169i	$-0.542334\pi$
$317$	−1496.83	−0.265207	−0.132603	+	0.991169i	$0.542334\pi$
$318$	0	0
$319$	−1518.06	−0.266442
$320$	0	0
$321$	−14159.7	−2.46206
$322$	0	0
$323$	7939.82	1.36775
$324$	0	0
$325$	−4574.80	−0.780812
$326$	0	0
$327$	4529.00	0.765915
$328$	0	0
$329$	8034.72	1.34641
$330$	0	0
$331$	5377.45	0.892965	0.446482	−	0.894792i	$-0.352677\pi$
$331$	5377.45	0.892965	0.446482	+	0.894792i	$0.352677\pi$
$332$	0	0
$333$	−5187.51	−0.853675
$334$	0	0
$335$	481.378	0.0785089
$336$	0	0
$337$	−7525.34	−1.21641	−0.608207	−	0.793779i	$-0.708111\pi$
$337$	−7525.34	−1.21641	−0.608207	+	0.793779i	$0.708111\pi$
$338$	0	0
$339$	17985.6	2.88154
$340$	0	0
$341$	1394.15	0.221401
$342$	0	0
$343$	−16210.4	−2.55183
$344$	0	0
$345$	−10876.0	−1.69723
$346$	0	0
$347$	461.180	0.0713471	0.0356736	−	0.999363i	$-0.488642\pi$
$347$	461.180	0.0713471	0.0356736	+	0.999363i	$0.488642\pi$
$348$	0	0
$349$	1994.75	0.305950	0.152975	−	0.988230i	$-0.451115\pi$
$349$	1994.75	0.305950	0.152975	+	0.988230i	$0.451115\pi$
$350$	0	0
$351$	−4169.51	−0.634051
$352$	0	0
$353$	−1258.03	−0.189683	−0.0948416	−	0.995492i	$-0.530234\pi$
$353$	−1258.03	−0.189683	−0.0948416	+	0.995492i	$0.530234\pi$
$354$	0	0
$355$	−11587.6	−1.73241
$356$	0	0
$357$	29547.2	4.38040
$358$	0	0
$359$	−2861.25	−0.420644	−0.210322	−	0.977632i	$-0.567451\pi$
$359$	−2861.25	−0.420644	−0.210322	+	0.977632i	$0.567451\pi$
$360$	0	0
$361$	−1639.74	−0.239064
$362$	0	0
$363$	11116.0	1.60727
$364$	0	0
$365$	5054.32	0.724809
$366$	0	0
$367$	13458.3	1.91422	0.957109	−	0.289727i	$-0.0935645\pi$
$367$	13458.3	1.91422	0.957109	+	0.289727i	$0.0935645\pi$
$368$	0	0
$369$	4724.78	0.666564
$370$	0	0
$371$	−2858.87	−0.400067
$372$	0	0
$373$	−6997.16	−0.971311	−0.485656	−	0.874150i	$-0.661419\pi$
$373$	−6997.16	−0.971311	−0.485656	+	0.874150i	$0.661419\pi$
$374$	0	0
$375$	5946.83	0.818915
$376$	0	0
$377$	1863.99	0.254643
$378$	0	0
$379$	11477.5	1.55557	0.777785	−	0.628530i	$-0.216343\pi$
$379$	11477.5	1.55557	0.777785	+	0.628530i	$0.216343\pi$
$380$	0	0
$381$	7385.52	0.993102
$382$	0	0
$383$	−8400.58	−1.12076	−0.560378	−	0.828237i	$-0.689344\pi$
$383$	−8400.58	−1.12076	−0.560378	+	0.828237i	$0.689344\pi$
$384$	0	0
$385$	24988.7	3.30791
$386$	0	0
$387$	−2442.92	−0.320880
$388$	0	0
$389$	−2707.10	−0.352842	−0.176421	−	0.984315i	$-0.556452\pi$
$389$	−2707.10	−0.352842	−0.176421	+	0.984315i	$0.556452\pi$
$390$	0	0
$391$	−10818.7	−1.39930
$392$	0	0
$393$	924.039	0.118605
$394$	0	0
$395$	−4086.52	−0.520545
$396$	0	0
$397$	13659.8	1.72687	0.863436	−	0.504458i	$-0.168308\pi$
$397$	13659.8	1.72687	0.863436	+	0.504458i	$0.168308\pi$
$398$	0	0
$399$	19422.9	2.43700
$400$	0	0
$401$	−1701.00	−0.211830	−0.105915	−	0.994375i	$-0.533777\pi$
$401$	−1701.00	−0.211830	−0.105915	+	0.994375i	$0.533777\pi$
$402$	0	0
$403$	−1711.85	−0.211596
$404$	0	0
$405$	6153.44	0.754980
$406$	0	0
$407$	−7709.32	−0.938911
$408$	0	0
$409$	9411.92	1.13787	0.568936	−	0.822382i	$-0.307355\pi$
$409$	9411.92	1.13787	0.568936	+	0.822382i	$0.307355\pi$
$410$	0	0
$411$	−23509.2	−2.82147
$412$	0	0
$413$	5597.10	0.666865
$414$	0	0
$415$	19637.6	2.32282
$416$	0	0
$417$	−16370.7	−1.92249
$418$	0	0
$419$	−8236.14	−0.960291	−0.480145	−	0.877189i	$-0.659416\pi$
$419$	−8236.14	−0.960291	−0.480145	+	0.877189i	$0.659416\pi$
$420$	0	0
$421$	8804.46	1.01925	0.509624	−	0.860397i	$-0.329785\pi$
$421$	8804.46	1.01925	0.509624	+	0.860397i	$0.329785\pi$
$422$	0	0
$423$	−8303.71	−0.954468
$424$	0	0
$425$	−7822.25	−0.892788
$426$	0	0
$427$	−31236.6	−3.54016
$428$	0	0
$429$	−26540.8	−2.98695
$430$	0	0
$431$	4148.48	0.463632	0.231816	−	0.972760i	$-0.425533\pi$
$431$	4148.48	0.463632	0.231816	+	0.972760i	$0.425533\pi$
$432$	0	0
$433$	−2650.09	−0.294123	−0.147062	−	0.989127i	$-0.546982\pi$
$433$	−2650.09	−0.294123	−0.147062	+	0.989127i	$0.546982\pi$
$434$	0	0
$435$	3204.03	0.353153
$436$	0	0
$437$	−7111.71	−0.778488
$438$	0	0
$439$	−13642.1	−1.48315	−0.741576	−	0.670869i	$-0.765921\pi$
$439$	−13642.1	−1.48315	−0.741576	+	0.670869i	$0.765921\pi$
$440$	0	0
$441$	28834.8	3.11357
$442$	0	0
$443$	4229.99	0.453664	0.226832	−	0.973934i	$-0.427163\pi$
$443$	4229.99	0.453664	0.226832	+	0.973934i	$0.427163\pi$
$444$	0	0
$445$	14565.9	1.55166
$446$	0	0
$447$	7366.18	0.779437
$448$	0	0
$449$	269.687	0.0283460	0.0141730	−	0.999900i	$-0.495488\pi$
$449$	269.687	0.0283460	0.0141730	+	0.999900i	$0.495488\pi$
$450$	0	0
$451$	7021.64	0.733118
$452$	0	0
$453$	−19004.6	−1.97112
$454$	0	0
$455$	−30683.1	−3.16142
$456$	0	0
$457$	7387.26	0.756152	0.378076	−	0.925775i	$-0.376586\pi$
$457$	7387.26	0.756152	0.378076	+	0.925775i	$0.376586\pi$
$458$	0	0
$459$	−7129.26	−0.724979
$460$	0	0
$461$	−5551.58	−0.560874	−0.280437	−	0.959872i	$-0.590479\pi$
$461$	−5551.58	−0.560874	−0.280437	+	0.959872i	$0.590479\pi$
$462$	0	0
$463$	−5416.60	−0.543695	−0.271847	−	0.962340i	$-0.587635\pi$
$463$	−5416.60	−0.543695	−0.271847	+	0.962340i	$0.587635\pi$
$464$	0	0
$465$	−2942.52	−0.293454
$466$	0	0
$467$	15494.6	1.53534	0.767671	−	0.640844i	$-0.221415\pi$
$467$	15494.6	1.53534	0.767671	+	0.640844i	$0.221415\pi$
$468$	0	0
$469$	1171.38	0.115329
$470$	0	0
$471$	−16300.6	−1.59468
$472$	0	0
$473$	−3630.50	−0.352918
$474$	0	0
$475$	−5141.98	−0.496695
$476$	0	0
$477$	2954.58	0.283607
$478$	0	0
$479$	−7363.43	−0.702388	−0.351194	−	0.936303i	$-0.614224\pi$
$479$	−7363.43	−0.702388	−0.351194	+	0.936303i	$0.614224\pi$
$480$	0	0
$481$	9466.11	0.897334
$482$	0	0
$483$	−26465.5	−2.49321
$484$	0	0
$485$	−6621.56	−0.619937
$486$	0	0
$487$	−6417.06	−0.597094	−0.298547	−	0.954395i	$-0.596502\pi$
$487$	−6417.06	−0.597094	−0.298547	+	0.954395i	$0.596502\pi$
$488$	0	0
$489$	17121.2	1.58333
$490$	0	0
$491$	1797.95	0.165255	0.0826276	−	0.996580i	$-0.473669\pi$
$491$	1797.95	0.165255	0.0826276	+	0.996580i	$0.473669\pi$
$492$	0	0
$493$	3187.16	0.291161
$494$	0	0
$495$	−25825.3	−2.34497
$496$	0	0
$497$	−28197.0	−2.54489
$498$	0	0
$499$	−21551.3	−1.93340	−0.966701	−	0.255907i	$-0.917626\pi$
$499$	−21551.3	−1.93340	−0.966701	+	0.255907i	$0.917626\pi$
$500$	0	0
$501$	−9499.44	−0.847114
$502$	0	0
$503$	8860.29	0.785409	0.392704	−	0.919665i	$-0.371540\pi$
$503$	8860.29	0.785409	0.392704	+	0.919665i	$0.371540\pi$
$504$	0	0
$505$	7957.65	0.701209
$506$	0	0
$507$	15258.6	1.33660
$508$	0	0
$509$	−19041.4	−1.65814	−0.829071	−	0.559143i	$-0.811130\pi$
$509$	−19041.4	−1.65814	−0.829071	+	0.559143i	$0.811130\pi$
$510$	0	0
$511$	12299.1	1.06474
$512$	0	0
$513$	−4686.44	−0.403336
$514$	0	0
$515$	3176.51	0.271794
$516$	0	0
$517$	−12340.4	−1.04977
$518$	0	0
$519$	18586.1	1.57194
$520$	0	0
$521$	5422.99	0.456018	0.228009	−	0.973659i	$-0.426778\pi$
$521$	5422.99	0.456018	0.228009	+	0.973659i	$0.426778\pi$
$522$	0	0
$523$	−2914.91	−0.243710	−0.121855	−	0.992548i	$-0.538884\pi$
$523$	−2914.91	−0.243710	−0.121855	+	0.992548i	$0.538884\pi$
$524$	0	0
$525$	−19135.3	−1.59073
$526$	0	0
$527$	−2927.02	−0.241941
$528$	0	0
$529$	−2476.64	−0.203554
$530$	0	0
$531$	−5784.48	−0.472740
$532$	0	0
$533$	−8621.73	−0.700654
$534$	0	0
$535$	25142.0	2.03174
$536$	0	0
$537$	16658.1	1.33864
$538$	0	0
$539$	42852.2	3.42445
$540$	0	0
$541$	5943.66	0.472343	0.236172	−	0.971711i	$-0.424107\pi$
$541$	5943.66	0.472343	0.236172	+	0.971711i	$0.424107\pi$
$542$	0	0
$543$	16068.2	1.26990
$544$	0	0
$545$	−8041.67	−0.632050
$546$	0	0
$547$	23439.0	1.83213	0.916067	−	0.401025i	$-0.131346\pi$
$547$	23439.0	1.83213	0.916067	+	0.401025i	$0.131346\pi$
$548$	0	0
$549$	32282.4	2.50961
$550$	0	0
$551$	2095.09	0.161985
$552$	0	0
$553$	−9944.08	−0.764675
$554$	0	0
$555$	16271.4	1.24447
$556$	0	0
$557$	6657.61	0.506449	0.253224	−	0.967408i	$-0.418509\pi$
$557$	6657.61	0.506449	0.253224	+	0.967408i	$0.418509\pi$
$558$	0	0
$559$	4457.81	0.337290
$560$	0	0
$561$	−45381.0	−3.41531
$562$	0	0
$563$	10356.7	0.775284	0.387642	−	0.921810i	$-0.373290\pi$
$563$	10356.7	0.775284	0.387642	+	0.921810i	$0.373290\pi$
$564$	0	0
$565$	−31935.1	−2.37791
$566$	0	0
$567$	14973.7	1.10906
$568$	0	0
$569$	13370.3	0.985082	0.492541	−	0.870289i	$-0.336068\pi$
$569$	13370.3	0.985082	0.492541	+	0.870289i	$0.336068\pi$
$570$	0	0
$571$	−9486.33	−0.695255	−0.347628	−	0.937633i	$-0.613013\pi$
$571$	−9486.33	−0.695255	−0.347628	+	0.937633i	$0.613013\pi$
$572$	0	0
$573$	−28605.1	−2.08550
$574$	0	0
$575$	7006.41	0.508152
$576$	0	0
$577$	21887.7	1.57920	0.789598	−	0.613624i	$-0.210289\pi$
$577$	21887.7	1.57920	0.789598	+	0.613624i	$0.210289\pi$
$578$	0	0
$579$	−7835.22	−0.562384
$580$	0	0
$581$	47785.7	3.41220
$582$	0	0
$583$	4390.89	0.311924
$584$	0	0
$585$	31710.4	2.24113
$586$	0	0
$587$	−867.251	−0.0609800	−0.0304900	−	0.999535i	$-0.509707\pi$
$587$	−867.251	−0.0609800	−0.0304900	+	0.999535i	$0.509707\pi$
$588$	0	0
$589$	−1924.08	−0.134602
$590$	0	0
$591$	−22534.1	−1.56841
$592$	0	0
$593$	16474.5	1.14085	0.570426	−	0.821349i	$-0.306778\pi$
$593$	16474.5	1.14085	0.570426	+	0.821349i	$0.306778\pi$
$594$	0	0
$595$	−52463.8	−3.61480
$596$	0	0
$597$	−25752.6	−1.76547
$598$	0	0
$599$	806.688	0.0550257	0.0275128	−	0.999621i	$-0.491241\pi$
$599$	806.688	0.0550257	0.0275128	+	0.999621i	$0.491241\pi$
$600$	0	0
$601$	22489.3	1.52639	0.763194	−	0.646169i	$-0.223630\pi$
$601$	22489.3	1.52639	0.763194	+	0.646169i	$0.223630\pi$
$602$	0	0
$603$	−1210.59	−0.0817565
$604$	0	0
$605$	−19737.5	−1.32635
$606$	0	0
$607$	12470.7	0.833887	0.416943	−	0.908932i	$-0.363101\pi$
$607$	12470.7	0.833887	0.416943	+	0.908932i	$0.363101\pi$
$608$	0	0
$609$	7796.64	0.518778
$610$	0	0
$611$	15152.5	1.00328
$612$	0	0
$613$	−8451.29	−0.556842	−0.278421	−	0.960459i	$-0.589811\pi$
$613$	−8451.29	−0.556842	−0.278421	+	0.960459i	$0.589811\pi$
$614$	0	0
$615$	−14820.0	−0.971705
$616$	0	0
$617$	540.868	0.0352910	0.0176455	−	0.999844i	$-0.494383\pi$
$617$	540.868	0.0352910	0.0176455	+	0.999844i	$0.494383\pi$
$618$	0	0
$619$	−4740.22	−0.307796	−0.153898	−	0.988087i	$-0.549183\pi$
$619$	−4740.22	−0.307796	−0.153898	+	0.988087i	$0.549183\pi$
$620$	0	0
$621$	6385.70	0.412640
$622$	0	0
$623$	35444.4	2.27937
$624$	0	0
$625$	−19456.0	−1.24518
$626$	0	0
$627$	−29831.3	−1.90008
$628$	0	0
$629$	16185.7	1.02602
$630$	0	0
$631$	−15275.3	−0.963710	−0.481855	−	0.876251i	$-0.660037\pi$
$631$	−15275.3	−0.963710	−0.481855	+	0.876251i	$0.660037\pi$
$632$	0	0
$633$	16604.7	1.04262
$634$	0	0
$635$	−13113.7	−0.819530
$636$	0	0
$637$	−52617.4	−3.27280
$638$	0	0
$639$	29141.0	1.80407
$640$	0	0
$641$	−14059.0	−0.866301	−0.433150	−	0.901322i	$-0.642598\pi$
$641$	−14059.0	−0.866301	−0.433150	+	0.901322i	$0.642598\pi$
$642$	0	0
$643$	22661.3	1.38985	0.694926	−	0.719081i	$-0.255437\pi$
$643$	22661.3	1.38985	0.694926	+	0.719081i	$0.255437\pi$
$644$	0	0
$645$	7662.57	0.467773
$646$	0	0
$647$	23221.3	1.41101	0.705506	−	0.708704i	$-0.250720\pi$
$647$	23221.3	1.41101	0.705506	+	0.708704i	$0.250720\pi$
$648$	0	0
$649$	−8596.50	−0.519941
$650$	0	0
$651$	−7160.27	−0.431080
$652$	0	0
$653$	7048.04	0.422376	0.211188	−	0.977446i	$-0.432267\pi$
$653$	7048.04	0.422376	0.211188	+	0.977446i	$0.432267\pi$
$654$	0	0
$655$	−1640.72	−0.0978752
$656$	0	0
$657$	−12710.9	−0.754791
$658$	0	0
$659$	−7414.70	−0.438294	−0.219147	−	0.975692i	$-0.570327\pi$
$659$	−7414.70	−0.438294	−0.219147	+	0.975692i	$0.570327\pi$
$660$	0	0
$661$	−8930.51	−0.525502	−0.262751	−	0.964864i	$-0.584630\pi$
$661$	−8930.51	−0.525502	−0.262751	+	0.964864i	$0.584630\pi$
$662$	0	0
$663$	55722.4	3.26407
$664$	0	0
$665$	−34487.2	−2.01106
$666$	0	0
$667$	−2854.75	−0.165722
$668$	0	0
$669$	16101.3	0.930509
$670$	0	0
$671$	47975.8	2.76019
$672$	0	0
$673$	27665.3	1.58457	0.792286	−	0.610150i	$-0.208891\pi$
$673$	27665.3	1.58457	0.792286	+	0.610150i	$0.208891\pi$
$674$	0	0
$675$	4617.05	0.263274
$676$	0	0
$677$	7599.94	0.431447	0.215723	−	0.976455i	$-0.430789\pi$
$677$	7599.94	0.431447	0.215723	+	0.976455i	$0.430789\pi$
$678$	0	0
$679$	−16112.8	−0.910681
$680$	0	0
$681$	−8891.60	−0.500333
$682$	0	0
$683$	−20387.4	−1.14217	−0.571086	−	0.820890i	$-0.693478\pi$
$683$	−20387.4	−1.14217	−0.571086	+	0.820890i	$0.693478\pi$
$684$	0	0
$685$	41742.8	2.32834
$686$	0	0
$687$	−51463.1	−2.85799
$688$	0	0
$689$	−5391.48	−0.298112
$690$	0	0
$691$	9323.92	0.513312	0.256656	−	0.966503i	$-0.417379\pi$
$691$	9323.92	0.513312	0.256656	+	0.966503i	$0.417379\pi$
$692$	0	0
$693$	−62842.9	−3.44474
$694$	0	0
$695$	29067.8	1.58648
$696$	0	0
$697$	−14741.9	−0.801134
$698$	0	0
$699$	4240.37	0.229450
$700$	0	0
$701$	−6683.73	−0.360116	−0.180058	−	0.983656i	$-0.557629\pi$
$701$	−6683.73	−0.360116	−0.180058	+	0.983656i	$0.557629\pi$
$702$	0	0
$703$	10639.7	0.570817
$704$	0	0
$705$	26045.8	1.39141
$706$	0	0
$707$	19364.0	1.03007
$708$	0	0
$709$	18275.0	0.968027	0.484014	−	0.875060i	$-0.339179\pi$
$709$	18275.0	0.968027	0.484014	+	0.875060i	$0.339179\pi$
$710$	0	0
$711$	10277.0	0.542078
$712$	0	0
$713$	2621.74	0.137707
$714$	0	0
$715$	47125.7	2.46490
$716$	0	0
$717$	−7485.28	−0.389879
$718$	0	0
$719$	13486.4	0.699524	0.349762	−	0.936839i	$-0.386262\pi$
$719$	13486.4	0.699524	0.349762	+	0.936839i	$0.386262\pi$
$720$	0	0
$721$	7729.67	0.399262
$722$	0	0
$723$	−3228.09	−0.166050
$724$	0	0
$725$	−2064.07	−0.105734
$726$	0	0
$727$	−11101.5	−0.566345	−0.283173	−	0.959069i	$-0.591387\pi$
$727$	−11101.5	−0.566345	−0.283173	+	0.959069i	$0.591387\pi$
$728$	0	0
$729$	−29290.9	−1.48813
$730$	0	0
$731$	7622.23	0.385661
$732$	0	0
$733$	−13965.0	−0.703698	−0.351849	−	0.936057i	$-0.614447\pi$
$733$	−13965.0	−0.703698	−0.351849	+	0.936057i	$0.614447\pi$
$734$	0	0
$735$	−90444.4	−4.53890
$736$	0	0
$737$	−1799.10	−0.0899195
$738$	0	0
$739$	−19734.1	−0.982313	−0.491157	−	0.871071i	$-0.663426\pi$
$739$	−19734.1	−0.982313	−0.491157	+	0.871071i	$0.663426\pi$
$740$	0	0
$741$	36629.3	1.81594
$742$	0	0
$743$	−6019.31	−0.297210	−0.148605	−	0.988897i	$-0.547478\pi$
$743$	−6019.31	−0.297210	−0.148605	+	0.988897i	$0.547478\pi$
$744$	0	0
$745$	−13079.4	−0.643209
$746$	0	0
$747$	−49385.5	−2.41890
$748$	0	0
$749$	61180.1	2.98461
$750$	0	0
$751$	−19954.6	−0.969577	−0.484789	−	0.874631i	$-0.661104\pi$
$751$	−19954.6	−0.969577	−0.484789	+	0.874631i	$0.661104\pi$
$752$	0	0
$753$	−8947.10	−0.433002
$754$	0	0
$755$	33744.6	1.62661
$756$	0	0
$757$	931.258	0.0447122	0.0223561	−	0.999750i	$-0.492883\pi$
$757$	931.258	0.0447122	0.0223561	+	0.999750i	$0.492883\pi$
$758$	0	0
$759$	40647.9	1.94391
$760$	0	0
$761$	17613.1	0.838994	0.419497	−	0.907757i	$-0.362206\pi$
$761$	17613.1	0.838994	0.419497	+	0.907757i	$0.362206\pi$
$762$	0	0
$763$	−19568.5	−0.928474
$764$	0	0
$765$	54220.2	2.56253
$766$	0	0
$767$	10555.5	0.496917
$768$	0	0
$769$	23167.5	1.08640	0.543200	−	0.839603i	$-0.317212\pi$
$769$	23167.5	1.08640	0.543200	+	0.839603i	$0.317212\pi$
$770$	0	0
$771$	17035.1	0.795727
$772$	0	0
$773$	−37068.8	−1.72480	−0.862400	−	0.506227i	$-0.831040\pi$
$773$	−37068.8	−1.72480	−0.862400	+	0.506227i	$0.831040\pi$
$774$	0	0
$775$	1895.59	0.0878603
$776$	0	0
$777$	39594.5	1.82812
$778$	0	0
$779$	−9690.64	−0.445704
$780$	0	0
$781$	43307.4	1.98420
$782$	0	0
$783$	−1881.21	−0.0858606
$784$	0	0
$785$	28943.3	1.31596
$786$	0	0
$787$	−24083.2	−1.09082	−0.545408	−	0.838171i	$-0.683625\pi$
$787$	−24083.2	−1.09082	−0.545408	+	0.838171i	$0.683625\pi$
$788$	0	0
$789$	−29652.4	−1.33796
$790$	0	0
$791$	−77710.4	−3.49313
$792$	0	0
$793$	−58908.5	−2.63796
$794$	0	0
$795$	−9267.46	−0.413437
$796$	0	0
$797$	12694.2	0.564182	0.282091	−	0.959388i	$-0.408972\pi$
$797$	12694.2	0.564182	0.282091	+	0.959388i	$0.408972\pi$
$798$	0	0
$799$	25908.6	1.14716
$800$	0	0
$801$	−36631.0	−1.61584
$802$	0	0
$803$	−18890.0	−0.830154
$804$	0	0
$805$	46992.0	2.05745
$806$	0	0
$807$	42206.3	1.84106
$808$	0	0
$809$	−17334.6	−0.753340	−0.376670	−	0.926348i	$-0.622931\pi$
$809$	−17334.6	−0.753340	−0.376670	+	0.926348i	$0.622931\pi$
$810$	0	0
$811$	33542.9	1.45234	0.726171	−	0.687514i	$-0.241298\pi$
$811$	33542.9	1.45234	0.726171	+	0.687514i	$0.241298\pi$
$812$	0	0
$813$	−44008.1	−1.89844
$814$	0	0
$815$	−30400.4	−1.30660
$816$	0	0
$817$	5010.49	0.214559
$818$	0	0
$819$	77163.5	3.29220
$820$	0	0
$821$	−2118.00	−0.0900350	−0.0450175	−	0.998986i	$-0.514334\pi$
$821$	−2118.00	−0.0900350	−0.0450175	+	0.998986i	$0.514334\pi$
$822$	0	0
$823$	−21234.0	−0.899358	−0.449679	−	0.893190i	$-0.648462\pi$
$823$	−21234.0	−0.899358	−0.449679	+	0.893190i	$0.648462\pi$
$824$	0	0
$825$	29389.6	1.24026
$826$	0	0
$827$	12654.2	0.532080	0.266040	−	0.963962i	$-0.414285\pi$
$827$	12654.2	0.532080	0.266040	+	0.963962i	$0.414285\pi$
$828$	0	0
$829$	40590.0	1.70054	0.850270	−	0.526346i	$-0.176438\pi$
$829$	40590.0	1.70054	0.850270	+	0.526346i	$0.176438\pi$
$830$	0	0
$831$	−3223.02	−0.134543
$832$	0	0
$833$	−89968.2	−3.74215
$834$	0	0
$835$	16867.2	0.699057
$836$	0	0
$837$	1727.66	0.0713461
$838$	0	0
$839$	1994.70	0.0820793	0.0410397	−	0.999158i	$-0.486933\pi$
$839$	1994.70	0.0820793	0.0410397	+	0.999158i	$0.486933\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−22386.4	−0.914626
$844$	0	0
$845$	−27093.0	−1.10299
$846$	0	0
$847$	−48028.9	−1.94840
$848$	0	0
$849$	44584.4	1.80228
$850$	0	0
$851$	−14497.6	−0.583984
$852$	0	0
$853$	3889.79	0.156136	0.0780679	−	0.996948i	$-0.475125\pi$
$853$	3889.79	0.156136	0.0780679	+	0.996948i	$0.475125\pi$
$854$	0	0
$855$	35641.8	1.42564
$856$	0	0
$857$	−26251.7	−1.04637	−0.523186	−	0.852219i	$-0.675257\pi$
$857$	−26251.7	−1.04637	−0.523186	+	0.852219i	$0.675257\pi$
$858$	0	0
$859$	−9648.78	−0.383251	−0.191625	−	0.981468i	$-0.561376\pi$
$859$	−9648.78	−0.383251	−0.191625	+	0.981468i	$0.561376\pi$
$860$	0	0
$861$	−36062.7	−1.42742
$862$	0	0
$863$	22861.6	0.901759	0.450879	−	0.892585i	$-0.351110\pi$
$863$	22861.6	0.901759	0.450879	+	0.892585i	$0.351110\pi$
$864$	0	0
$865$	−33001.3	−1.29720
$866$	0	0
$867$	56522.7	2.21409
$868$	0	0
$869$	15273.0	0.596202
$870$	0	0
$871$	2209.08	0.0859376
$872$	0	0
$873$	16652.2	0.645581
$874$	0	0
$875$	−25694.5	−0.992723
$876$	0	0
$877$	46506.5	1.79067	0.895333	−	0.445397i	$-0.146938\pi$
$877$	46506.5	1.79067	0.895333	+	0.445397i	$0.146938\pi$
$878$	0	0
$879$	−14990.5	−0.575218
$880$	0	0
$881$	10791.0	0.412665	0.206332	−	0.978482i	$-0.433847\pi$
$881$	10791.0	0.412665	0.206332	+	0.978482i	$0.433847\pi$
$882$	0	0
$883$	−9830.93	−0.374674	−0.187337	−	0.982296i	$-0.559986\pi$
$883$	−9830.93	−0.374674	−0.187337	+	0.982296i	$0.559986\pi$
$884$	0	0
$885$	18143.9	0.689152
$886$	0	0
$887$	−8862.90	−0.335498	−0.167749	−	0.985830i	$-0.553650\pi$
$887$	−8862.90	−0.335498	−0.167749	+	0.985830i	$0.553650\pi$
$888$	0	0
$889$	−31910.7	−1.20388
$890$	0	0
$891$	−22997.9	−0.864711
$892$	0	0
$893$	17031.1	0.638213
$894$	0	0
$895$	−29578.1	−1.10468
$896$	0	0
$897$	−49910.7	−1.85783
$898$	0	0
$899$	−772.356	−0.0286535
$900$	0	0
$901$	−9218.66	−0.340864
$902$	0	0
$903$	18646.0	0.687153
$904$	0	0
$905$	−28530.7	−1.04795
$906$	0	0
$907$	11882.9	0.435023	0.217512	−	0.976058i	$-0.430206\pi$
$907$	11882.9	0.435023	0.217512	+	0.976058i	$0.430206\pi$
$908$	0	0
$909$	−20012.3	−0.730215
$910$	0	0
$911$	−14501.2	−0.527382	−0.263691	−	0.964607i	$-0.584940\pi$
$911$	−14501.2	−0.527382	−0.263691	+	0.964607i	$0.584940\pi$
$912$	0	0
$913$	−73393.3	−2.66042
$914$	0	0
$915$	−101258.	−3.65847
$916$	0	0
$917$	−3992.50	−0.143778
$918$	0	0
$919$	−2889.55	−0.103719	−0.0518594	−	0.998654i	$-0.516515\pi$
$919$	−2889.55	−0.103719	−0.0518594	+	0.998654i	$0.516515\pi$
$920$	0	0
$921$	27047.5	0.967694
$922$	0	0
$923$	−53176.2	−1.89633
$924$	0	0
$925$	−10482.2	−0.372596
$926$	0	0
$927$	−7988.45	−0.283037
$928$	0	0
$929$	−40900.7	−1.44446	−0.722232	−	0.691651i	$-0.756884\pi$
$929$	−40900.7	−1.44446	−0.722232	+	0.691651i	$0.756884\pi$
$930$	0	0
$931$	−59140.8	−2.08191
$932$	0	0
$933$	−69747.8	−2.44742
$934$	0	0
$935$	80578.3	2.81839
$936$	0	0
$937$	−40149.3	−1.39981	−0.699903	−	0.714237i	$-0.746774\pi$
$937$	−40149.3	−1.39981	−0.699903	+	0.714237i	$0.746774\pi$
$938$	0	0
$939$	−44786.9	−1.55651
$940$	0	0
$941$	34629.8	1.19968	0.599841	−	0.800119i	$-0.295231\pi$
$941$	34629.8	1.19968	0.599841	+	0.800119i	$0.295231\pi$
$942$	0	0
$943$	13204.4	0.455985
$944$	0	0
$945$	30966.5	1.06597
$946$	0	0
$947$	14988.3	0.514312	0.257156	−	0.966370i	$-0.417215\pi$
$947$	14988.3	0.514312	0.257156	+	0.966370i	$0.417215\pi$
$948$	0	0
$949$	23194.6	0.793392
$950$	0	0
$951$	−11807.3	−0.402606
$952$	0	0
$953$	−7317.35	−0.248722	−0.124361	−	0.992237i	$-0.539688\pi$
$953$	−7317.35	−0.248722	−0.124361	+	0.992237i	$0.539688\pi$
$954$	0	0
$955$	50791.0	1.72100
$956$	0	0
$957$	−11974.7	−0.404481
$958$	0	0
$959$	101576.	3.42031
$960$	0	0
$961$	−29081.7	−0.976190
$962$	0	0
$963$	−63228.3	−2.11579
$964$	0	0
$965$	13912.2	0.464092
$966$	0	0
$967$	51976.9	1.72851	0.864253	−	0.503057i	$-0.167791\pi$
$967$	51976.9	1.72851	0.864253	+	0.503057i	$0.167791\pi$
$968$	0	0
$969$	62630.8	2.07636
$970$	0	0
$971$	5504.72	0.181931	0.0909654	−	0.995854i	$-0.471005\pi$
$971$	5504.72	0.181931	0.0909654	+	0.995854i	$0.471005\pi$
$972$	0	0
$973$	70733.2	2.33053
$974$	0	0
$975$	−36086.9	−1.18534
$976$	0	0
$977$	13265.1	0.434379	0.217190	−	0.976129i	$-0.430311\pi$
$977$	13265.1	0.434379	0.217190	+	0.976129i	$0.430311\pi$
$978$	0	0
$979$	−54438.4	−1.77718
$980$	0	0
$981$	20223.6	0.658195
$982$	0	0
$983$	−30674.1	−0.995271	−0.497635	−	0.867386i	$-0.665798\pi$
$983$	−30674.1	−0.995271	−0.497635	+	0.867386i	$0.665798\pi$
$984$	0	0
$985$	40011.5	1.29429
$986$	0	0
$987$	63379.4	2.04396
$988$	0	0
$989$	−6827.24	−0.219508
$990$	0	0
$991$	39593.5	1.26915	0.634575	−	0.772861i	$-0.281175\pi$
$991$	39593.5	1.26915	0.634575	+	0.772861i	$0.281175\pi$
$992$	0	0
$993$	42418.4	1.35560
$994$	0	0
$995$	45726.2	1.45690
$996$	0	0
$997$	−1791.38	−0.0569043	−0.0284521	−	0.999595i	$-0.509058\pi$
$997$	−1791.38	−0.0569043	−0.0284521	+	0.999595i	$0.509058\pi$
$998$	0	0
$999$	−9553.53	−0.302563

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	232.4.a.e.1.6	✓	6
3.2	odd	2		2088.4.a.l.1.5		6
4.3	odd	2		464.4.a.n.1.1		6
8.3	odd	2		1856.4.a.bc.1.6		6
8.5	even	2		1856.4.a.bd.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.4.a.e.1.6	✓	6	1.1	even	1	trivial
464.4.a.n.1.1		6	4.3	odd	2
1856.4.a.bc.1.6		6	8.3	odd	2
1856.4.a.bd.1.1		6	8.5	even	2
2088.4.a.l.1.5		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)

\(f(q)\)	\(=\)	\(q+7.88819 q^{3} -14.0062 q^{5} -34.0826 q^{7} +35.2236 q^{9} +O(q^{10})\)	\(q+7.88819 q^{3} -14.0062 q^{5} -34.0826 q^{7} +35.2236 q^{9} +52.3468 q^{11} -64.2756 q^{13} -110.484 q^{15} -109.902 q^{17} -72.2444 q^{19} -268.850 q^{21} +98.4396 q^{23} +71.1747 q^{25} +64.8692 q^{27} -29.0000 q^{29} +26.6330 q^{31} +412.922 q^{33} +477.368 q^{35} -147.274 q^{37} -507.018 q^{39} +134.137 q^{41} -69.3546 q^{43} -493.350 q^{45} -235.743 q^{47} +818.621 q^{49} -866.929 q^{51} +83.8806 q^{53} -733.182 q^{55} -569.878 q^{57} -164.222 q^{59} +916.499 q^{61} -1200.51 q^{63} +900.259 q^{65} -34.3688 q^{67} +776.511 q^{69} +827.316 q^{71} -360.862 q^{73} +561.440 q^{75} -1784.11 q^{77} +291.765 q^{79} -439.336 q^{81} -1402.06 q^{83} +1539.32 q^{85} -228.758 q^{87} -1039.96 q^{89} +2190.68 q^{91} +210.086 q^{93} +1011.87 q^{95} +472.758 q^{97} +1843.84 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 5 q^{3} - 5 q^{5} - 38 q^{7} + 47 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more