LMFDB - Embedded newform 2496.4.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,4,Mod(1,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2496.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2496.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:	$147.268767374$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{43}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 43 $ x^2 - 43
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 312)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$6.55744$ of defining polynomial
Character	$\chi$	$=$	2496.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +7.11488 q^{5} +8.88512 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +7.11488 q^{5} +8.88512 q^{7} +9.00000 q^{9} -26.0000 q^{11} +13.0000 q^{13} -21.3446 q^{15} +16.2298 q^{17} -35.5744 q^{19} -26.6554 q^{21} +153.379 q^{23} -74.3785 q^{25} -27.0000 q^{27} -223.379 q^{29} +126.723 q^{31} +78.0000 q^{33} +63.2166 q^{35} -217.608 q^{37} -39.0000 q^{39} +105.804 q^{41} +183.608 q^{43} +64.0339 q^{45} +96.6893 q^{47} -264.055 q^{49} -48.6893 q^{51} -386.460 q^{53} -184.987 q^{55} +106.723 q^{57} -34.5273 q^{59} -274.919 q^{61} +79.9661 q^{63} +92.4934 q^{65} -93.9397 q^{67} -460.136 q^{69} -741.812 q^{71} +640.689 q^{73} +223.136 q^{75} -231.013 q^{77} -182.945 q^{79} +81.0000 q^{81} +288.825 q^{83} +115.473 q^{85} +670.136 q^{87} +963.115 q^{89} +115.507 q^{91} -380.169 q^{93} -253.107 q^{95} -481.932 q^{97} -234.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 12 q^{5} + 44 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 - 12 * q^5 + 44 * q^7 + 18 * q^9	$ 2 q - 6 q^{3} - 12 q^{5} + 44 q^{7} + 18 q^{9} - 52 q^{11} + 26 q^{13} + 36 q^{15} - 20 q^{17} + 60 q^{19} - 132 q^{21} - 8 q^{23} + 166 q^{25} - 54 q^{27} - 132 q^{29} - 140 q^{31} + 156 q^{33} - 608 q^{35} - 68 q^{37} - 78 q^{39} + 28 q^{41} - 108 q^{45} + 36 q^{47} + 626 q^{49} + 60 q^{51} - 668 q^{53} + 312 q^{55} - 180 q^{57} + 508 q^{59} - 340 q^{61} + 396 q^{63} - 156 q^{65} + 940 q^{67} + 24 q^{69} + 300 q^{71} + 1124 q^{73} - 498 q^{75} - 1144 q^{77} - 1520 q^{79} + 162 q^{81} - 524 q^{83} + 808 q^{85} + 396 q^{87} + 1900 q^{89} + 572 q^{91} + 420 q^{93} - 2080 q^{95} - 1436 q^{97} - 468 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 12 * q^5 + 44 * q^7 + 18 * q^9 - 52 * q^11 + 26 * q^13 + 36 * q^15 - 20 * q^17 + 60 * q^19 - 132 * q^21 - 8 * q^23 + 166 * q^25 - 54 * q^27 - 132 * q^29 - 140 * q^31 + 156 * q^33 - 608 * q^35 - 68 * q^37 - 78 * q^39 + 28 * q^41 - 108 * q^45 + 36 * q^47 + 626 * q^49 + 60 * q^51 - 668 * q^53 + 312 * q^55 - 180 * q^57 + 508 * q^59 - 340 * q^61 + 396 * q^63 - 156 * q^65 + 940 * q^67 + 24 * q^69 + 300 * q^71 + 1124 * q^73 - 498 * q^75 - 1144 * q^77 - 1520 * q^79 + 162 * q^81 - 524 * q^83 + 808 * q^85 + 396 * q^87 + 1900 * q^89 + 572 * q^91 + 420 * q^93 - 2080 * q^95 - 1436 * q^97 - 468 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	7.11488	0.636374	0.318187	−	0.948028i	$-0.396926\pi$
$5$	7.11488	0.636374	0.318187	+	0.948028i	$0.396926\pi$
$6$	0	0
$7$	8.88512	0.479752	0.239876	−	0.970804i	$-0.422893\pi$
$7$	8.88512	0.479752	0.239876	+	0.970804i	$0.422893\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−26.0000	−0.712663	−0.356332	−	0.934360i	$-0.615973\pi$
$11$	−26.0000	−0.712663	−0.356332	+	0.934360i	$0.615973\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−21.3446	−0.367411
$16$	0	0
$17$	16.2298	0.231547	0.115773	−	0.993276i	$-0.463065\pi$
$17$	16.2298	0.231547	0.115773	+	0.993276i	$0.463065\pi$
$18$	0	0
$19$	−35.5744	−0.429543	−0.214772	−	0.976664i	$-0.568901\pi$
$19$	−35.5744	−0.429543	−0.214772	+	0.976664i	$0.568901\pi$
$20$	0	0
$21$	−26.6554	−0.276985
$22$	0	0
$23$	153.379	1.39051	0.695253	−	0.718765i	$-0.255292\pi$
$23$	153.379	1.39051	0.695253	+	0.718765i	$0.255292\pi$
$24$	0	0
$25$	−74.3785	−0.595028
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−223.379	−1.43036	−0.715178	−	0.698942i	$-0.753655\pi$
$29$	−223.379	−1.43036	−0.715178	+	0.698942i	$0.753655\pi$
$30$	0	0
$31$	126.723	0.734198	0.367099	−	0.930182i	$-0.380351\pi$
$31$	126.723	0.734198	0.367099	+	0.930182i	$0.380351\pi$
$32$	0	0
$33$	78.0000	0.411456
$34$	0	0
$35$	63.2166	0.305301
$36$	0	0
$37$	−217.608	−0.966881	−0.483440	−	0.875377i	$-0.660613\pi$
$37$	−217.608	−0.966881	−0.483440	+	0.875377i	$0.660613\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	105.804	0.403020	0.201510	−	0.979486i	$-0.435415\pi$
$41$	105.804	0.403020	0.201510	+	0.979486i	$0.435415\pi$
$42$	0	0
$43$	183.608	0.651163	0.325581	−	0.945514i	$-0.394440\pi$
$43$	183.608	0.651163	0.325581	+	0.945514i	$0.394440\pi$
$44$	0	0
$45$	64.0339	0.212125
$46$	0	0
$47$	96.6893	0.300076	0.150038	−	0.988680i	$-0.452060\pi$
$47$	96.6893	0.300076	0.150038	+	0.988680i	$0.452060\pi$
$48$	0	0
$49$	−264.055	−0.769838
$50$	0	0
$51$	−48.6893	−0.133684
$52$	0	0
$53$	−386.460	−1.00159	−0.500795	−	0.865566i	$-0.666959\pi$
$53$	−386.460	−1.00159	−0.500795	+	0.865566i	$0.666959\pi$
$54$	0	0
$55$	−184.987	−0.453520
$56$	0	0
$57$	106.723	0.247997
$58$	0	0
$59$	−34.5273	−0.0761876	−0.0380938	−	0.999274i	$-0.512129\pi$
$59$	−34.5273	−0.0761876	−0.0380938	+	0.999274i	$0.512129\pi$
$60$	0	0
$61$	−274.919	−0.577045	−0.288523	−	0.957473i	$-0.593164\pi$
$61$	−274.919	−0.577045	−0.288523	+	0.957473i	$0.593164\pi$
$62$	0	0
$63$	79.9661	0.159917
$64$	0	0
$65$	92.4934	0.176498
$66$	0	0
$67$	−93.9397	−0.171292	−0.0856460	−	0.996326i	$-0.527295\pi$
$67$	−93.9397	−0.171292	−0.0856460	+	0.996326i	$0.527295\pi$
$68$	0	0
$69$	−460.136	−0.802809
$70$	0	0
$71$	−741.812	−1.23996	−0.619978	−	0.784619i	$-0.712858\pi$
$71$	−741.812	−1.23996	−0.619978	+	0.784619i	$0.712858\pi$
$72$	0	0
$73$	640.689	1.02722	0.513610	−	0.858024i	$-0.328308\pi$
$73$	640.689	1.02722	0.513610	+	0.858024i	$0.328308\pi$
$74$	0	0
$75$	223.136	0.343540
$76$	0	0
$77$	−231.013	−0.341901
$78$	0	0
$79$	−182.945	−0.260544	−0.130272	−	0.991478i	$-0.541585\pi$
$79$	−182.945	−0.260544	−0.130272	+	0.991478i	$0.541585\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	288.825	0.381960	0.190980	−	0.981594i	$-0.438834\pi$
$83$	288.825	0.381960	0.190980	+	0.981594i	$0.438834\pi$
$84$	0	0
$85$	115.473	0.147350
$86$	0	0
$87$	670.136	0.825817
$88$	0	0
$89$	963.115	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$89$	963.115	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$90$	0	0
$91$	115.507	0.133059
$92$	0	0
$93$	−380.169	−0.423890
$94$	0	0
$95$	−253.107	−0.273350
$96$	0	0
$97$	−481.932	−0.504462	−0.252231	−	0.967667i	$-0.581164\pi$
$97$	−481.932	−0.504462	−0.252231	+	0.967667i	$0.581164\pi$
$98$	0	0
$99$	−234.000	−0.237554
$100$	0	0
$101$	−1138.85	−1.12198	−0.560990	−	0.827823i	$-0.689579\pi$
$101$	−1138.85	−1.12198	−0.560990	+	0.827823i	$0.689579\pi$
$102$	0	0
$103$	−1090.96	−1.04365	−0.521823	−	0.853054i	$-0.674748\pi$
$103$	−1090.96	−1.04365	−0.521823	+	0.853054i	$0.674748\pi$
$104$	0	0
$105$	−189.650	−0.176266
$106$	0	0
$107$	−124.595	−0.112571	−0.0562853	−	0.998415i	$-0.517926\pi$
$107$	−124.595	−0.112571	−0.0562853	+	0.998415i	$0.517926\pi$
$108$	0	0
$109$	1400.89	1.23102	0.615510	−	0.788129i	$-0.288950\pi$
$109$	1400.89	1.23102	0.615510	+	0.788129i	$0.288950\pi$
$110$	0	0
$111$	652.825	0.558229
$112$	0	0
$113$	−541.812	−0.451056	−0.225528	−	0.974237i	$-0.572411\pi$
$113$	−541.812	−0.451056	−0.225528	+	0.974237i	$0.572411\pi$
$114$	0	0
$115$	1091.27	0.884882
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	144.203	0.111085
$120$	0	0
$121$	−655.000	−0.492111
$122$	0	0
$123$	−317.412	−0.232684
$124$	0	0
$125$	−1418.55	−1.01503
$126$	0	0
$127$	−1552.07	−1.08444	−0.542219	−	0.840237i	$-0.682416\pi$
$127$	−1552.07	−1.08444	−0.542219	+	0.840237i	$0.682416\pi$
$128$	0	0
$129$	−550.825	−0.375949
$130$	0	0
$131$	2281.97	1.52196	0.760981	−	0.648774i	$-0.224718\pi$
$131$	2281.97	1.52196	0.760981	+	0.648774i	$0.224718\pi$
$132$	0	0
$133$	−316.083	−0.206074
$134$	0	0
$135$	−192.102	−0.122470
$136$	0	0
$137$	−1443.06	−0.899921	−0.449960	−	0.893049i	$-0.648562\pi$
$137$	−1443.06	−0.899921	−0.449960	+	0.893049i	$0.648562\pi$
$138$	0	0
$139$	3060.41	1.86748	0.933742	−	0.357946i	$-0.116523\pi$
$139$	3060.41	1.86748	0.933742	+	0.357946i	$0.116523\pi$
$140$	0	0
$141$	−290.068	−0.173249
$142$	0	0
$143$	−338.000	−0.197657
$144$	0	0
$145$	−1589.31	−0.910242
$146$	0	0
$147$	792.164	0.444466
$148$	0	0
$149$	−302.520	−0.166331	−0.0831657	−	0.996536i	$-0.526503\pi$
$149$	−302.520	−0.166331	−0.0831657	+	0.996536i	$0.526503\pi$
$150$	0	0
$151$	383.574	0.206721	0.103360	−	0.994644i	$-0.467040\pi$
$151$	383.574	0.206721	0.103360	+	0.994644i	$0.467040\pi$
$152$	0	0
$153$	146.068	0.0771822
$154$	0	0
$155$	901.620	0.467225
$156$	0	0
$157$	2583.38	1.31322	0.656612	−	0.754229i	$-0.271989\pi$
$157$	2583.38	1.31322	0.656612	+	0.754229i	$0.271989\pi$
$158$	0	0
$159$	1159.38	0.578269
$160$	0	0
$161$	1362.79	0.667097
$162$	0	0
$163$	−1303.45	−0.626346	−0.313173	−	0.949696i	$-0.601392\pi$
$163$	−1303.45	−0.626346	−0.313173	+	0.949696i	$0.601392\pi$
$164$	0	0
$165$	554.960	0.261840
$166$	0	0
$167$	1040.04	0.481921	0.240961	−	0.970535i	$-0.422538\pi$
$167$	1040.04	0.481921	0.240961	+	0.970535i	$0.422538\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−320.169	−0.143181
$172$	0	0
$173$	−2989.10	−1.31362	−0.656811	−	0.754055i	$-0.728095\pi$
$173$	−2989.10	−1.31362	−0.656811	+	0.754055i	$0.728095\pi$
$174$	0	0
$175$	−660.862	−0.285466
$176$	0	0
$177$	103.582	0.0439870
$178$	0	0
$179$	1786.06	0.745788	0.372894	−	0.927874i	$-0.378365\pi$
$179$	1786.06	0.745788	0.372894	+	0.927874i	$0.378365\pi$
$180$	0	0
$181$	1320.84	0.542416	0.271208	−	0.962521i	$-0.412577\pi$
$181$	1320.84	0.542416	0.271208	+	0.962521i	$0.412577\pi$
$182$	0	0
$183$	824.757	0.333157
$184$	0	0
$185$	−1548.26	−0.615298
$186$	0	0
$187$	−421.974	−0.165015
$188$	0	0
$189$	−239.898	−0.0923282
$190$	0	0
$191$	−436.595	−0.165397	−0.0826987	−	0.996575i	$-0.526354\pi$
$191$	−436.595	−0.165397	−0.0826987	+	0.996575i	$0.526354\pi$
$192$	0	0
$193$	−2402.19	−0.895924	−0.447962	−	0.894053i	$-0.647850\pi$
$193$	−2402.19	−0.895924	−0.447962	+	0.894053i	$0.647850\pi$
$194$	0	0
$195$	−277.480	−0.101901
$196$	0	0
$197$	−1129.94	−0.408654	−0.204327	−	0.978903i	$-0.565501\pi$
$197$	−1129.94	−0.408654	−0.204327	+	0.978903i	$0.565501\pi$
$198$	0	0
$199$	1859.61	0.662433	0.331216	−	0.943555i	$-0.392541\pi$
$199$	1859.61	0.662433	0.331216	+	0.943555i	$0.392541\pi$
$200$	0	0
$201$	281.819	0.0988955
$202$	0	0
$203$	−1984.75	−0.686216
$204$	0	0
$205$	752.783	0.256472
$206$	0	0
$207$	1380.41	0.463502
$208$	0	0
$209$	924.934	0.306120
$210$	0	0
$211$	−859.729	−0.280503	−0.140252	−	0.990116i	$-0.544791\pi$
$211$	−859.729	−0.280503	−0.140252	+	0.990116i	$0.544791\pi$
$212$	0	0
$213$	2225.43	0.715889
$214$	0	0
$215$	1306.35	0.414383
$216$	0	0
$217$	1125.95	0.352233
$218$	0	0
$219$	−1922.07	−0.593065
$220$	0	0
$221$	210.987	0.0642195
$222$	0	0
$223$	265.345	0.0796807	0.0398403	−	0.999206i	$-0.487315\pi$
$223$	265.345	0.0796807	0.0398403	+	0.999206i	$0.487315\pi$
$224$	0	0
$225$	−669.407	−0.198343
$226$	0	0
$227$	−1519.43	−0.444265	−0.222133	−	0.975016i	$-0.571302\pi$
$227$	−1519.43	−0.444265	−0.222133	+	0.975016i	$0.571302\pi$
$228$	0	0
$229$	−2069.00	−0.597045	−0.298522	−	0.954403i	$-0.596494\pi$
$229$	−2069.00	−0.597045	−0.298522	+	0.954403i	$0.596494\pi$
$230$	0	0
$231$	693.040	0.197397
$232$	0	0
$233$	−4624.94	−1.30039	−0.650193	−	0.759769i	$-0.725312\pi$
$233$	−4624.94	−1.30039	−0.650193	+	0.759769i	$0.725312\pi$
$234$	0	0
$235$	687.932	0.190961
$236$	0	0
$237$	548.836	0.150425
$238$	0	0
$239$	2098.35	0.567913	0.283957	−	0.958837i	$-0.408353\pi$
$239$	2098.35	0.567913	0.283957	+	0.958837i	$0.408353\pi$
$240$	0	0
$241$	−1345.96	−0.359755	−0.179878	−	0.983689i	$-0.557570\pi$
$241$	−1345.96	−0.359755	−0.179878	+	0.983689i	$0.557570\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−1878.72	−0.489905
$246$	0	0
$247$	−462.467	−0.119134
$248$	0	0
$249$	−866.475	−0.220524
$250$	0	0
$251$	−4117.24	−1.03537	−0.517685	−	0.855571i	$-0.673206\pi$
$251$	−4117.24	−1.03537	−0.517685	+	0.855571i	$0.673206\pi$
$252$	0	0
$253$	−3987.84	−0.990962
$254$	0	0
$255$	−346.418	−0.0850727
$256$	0	0
$257$	3393.64	0.823694	0.411847	−	0.911253i	$-0.364884\pi$
$257$	3393.64	0.823694	0.411847	+	0.911253i	$0.364884\pi$
$258$	0	0
$259$	−1933.48	−0.463862
$260$	0	0
$261$	−2010.41	−0.476786
$262$	0	0
$263$	−6335.19	−1.48534	−0.742671	−	0.669657i	$-0.766441\pi$
$263$	−6335.19	−1.48534	−0.742671	+	0.669657i	$0.766441\pi$
$264$	0	0
$265$	−2749.61	−0.637386
$266$	0	0
$267$	−2889.34	−0.662266
$268$	0	0
$269$	−619.702	−0.140461	−0.0702303	−	0.997531i	$-0.522373\pi$
$269$	−619.702	−0.140461	−0.0702303	+	0.997531i	$0.522373\pi$
$270$	0	0
$271$	5165.75	1.15792	0.578961	−	0.815355i	$-0.303458\pi$
$271$	5165.75	1.15792	0.578961	+	0.815355i	$0.303458\pi$
$272$	0	0
$273$	−346.520	−0.0768217
$274$	0	0
$275$	1933.84	0.424055
$276$	0	0
$277$	−3534.38	−0.766643	−0.383322	−	0.923615i	$-0.625220\pi$
$277$	−3534.38	−0.766643	−0.383322	+	0.923615i	$0.625220\pi$
$278$	0	0
$279$	1140.51	0.244733
$280$	0	0
$281$	4115.81	0.873767	0.436883	−	0.899518i	$-0.356082\pi$
$281$	4115.81	0.873767	0.436883	+	0.899518i	$0.356082\pi$
$282$	0	0
$283$	−3944.56	−0.828550	−0.414275	−	0.910152i	$-0.635965\pi$
$283$	−3944.56	−0.828550	−0.414275	+	0.910152i	$0.635965\pi$
$284$	0	0
$285$	759.322	0.157819
$286$	0	0
$287$	940.083	0.193350
$288$	0	0
$289$	−4649.60	−0.946386
$290$	0	0
$291$	1445.80	0.291251
$292$	0	0
$293$	−772.441	−0.154015	−0.0770076	−	0.997031i	$-0.524537\pi$
$293$	−772.441	−0.154015	−0.0770076	+	0.997031i	$0.524537\pi$
$294$	0	0
$295$	−245.657	−0.0484838
$296$	0	0
$297$	702.000	0.137152
$298$	0	0
$299$	1993.92	0.385657
$300$	0	0
$301$	1631.38	0.312396
$302$	0	0
$303$	3416.55	0.647775
$304$	0	0
$305$	−1956.01	−0.367217
$306$	0	0
$307$	−4885.74	−0.908286	−0.454143	−	0.890929i	$-0.650054\pi$
$307$	−4885.74	−0.908286	−0.454143	+	0.890929i	$0.650054\pi$
$308$	0	0
$309$	3272.88	0.602549
$310$	0	0
$311$	177.544	0.0323717	0.0161859	−	0.999869i	$-0.494848\pi$
$311$	177.544	0.0323717	0.0161859	+	0.999869i	$0.494848\pi$
$312$	0	0
$313$	−6677.03	−1.20578	−0.602888	−	0.797826i	$-0.705983\pi$
$313$	−6677.03	−1.20578	−0.602888	+	0.797826i	$0.705983\pi$
$314$	0	0
$315$	568.949	0.101767
$316$	0	0
$317$	−1806.53	−0.320079	−0.160040	−	0.987111i	$-0.551162\pi$
$317$	−1806.53	−0.320079	−0.160040	+	0.987111i	$0.551162\pi$
$318$	0	0
$319$	5807.84	1.01936
$320$	0	0
$321$	373.785	0.0649927
$322$	0	0
$323$	−577.364	−0.0994593
$324$	0	0
$325$	−966.921	−0.165031
$326$	0	0
$327$	−4202.68	−0.710730
$328$	0	0
$329$	859.096	0.143962
$330$	0	0
$331$	−11447.2	−1.90088	−0.950442	−	0.310901i	$-0.899369\pi$
$331$	−11447.2	−1.90088	−0.950442	+	0.310901i	$0.899369\pi$
$332$	0	0
$333$	−1958.47	−0.322294
$334$	0	0
$335$	−668.370	−0.109006
$336$	0	0
$337$	−821.752	−0.132830	−0.0664149	−	0.997792i	$-0.521156\pi$
$337$	−821.752	−0.132830	−0.0664149	+	0.997792i	$0.521156\pi$
$338$	0	0
$339$	1625.43	0.260417
$340$	0	0
$341$	−3294.80	−0.523236
$342$	0	0
$343$	−5393.75	−0.849083
$344$	0	0
$345$	−3273.81	−0.510887
$346$	0	0
$347$	4819.55	0.745611	0.372805	−	0.927910i	$-0.378396\pi$
$347$	4819.55	0.745611	0.372805	+	0.927910i	$0.378396\pi$
$348$	0	0
$349$	−7922.72	−1.21517	−0.607584	−	0.794255i	$-0.707861\pi$
$349$	−7922.72	−1.21517	−0.607584	+	0.794255i	$0.707861\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	−2860.73	−0.431336	−0.215668	−	0.976467i	$-0.569193\pi$
$353$	−2860.73	−0.431336	−0.215668	+	0.976467i	$0.569193\pi$
$354$	0	0
$355$	−5277.90	−0.789075
$356$	0	0
$357$	−432.610	−0.0641349
$358$	0	0
$359$	−11785.8	−1.73267	−0.866334	−	0.499464i	$-0.833530\pi$
$359$	−11785.8	−1.73267	−0.866334	+	0.499464i	$0.833530\pi$
$360$	0	0
$361$	−5593.46	−0.815493
$362$	0	0
$363$	1965.00	0.284121
$364$	0	0
$365$	4558.43	0.653696
$366$	0	0
$367$	2017.19	0.286911	0.143456	−	0.989657i	$-0.454179\pi$
$367$	2017.19	0.286911	0.143456	+	0.989657i	$0.454179\pi$
$368$	0	0
$369$	952.237	0.134340
$370$	0	0
$371$	−3433.74	−0.480515
$372$	0	0
$373$	−5148.21	−0.714650	−0.357325	−	0.933980i	$-0.616311\pi$
$373$	−5148.21	−0.714650	−0.357325	+	0.933980i	$0.616311\pi$
$374$	0	0
$375$	4255.66	0.586030
$376$	0	0
$377$	−2903.92	−0.396710
$378$	0	0
$379$	3419.10	0.463397	0.231698	−	0.972788i	$-0.425572\pi$
$379$	3419.10	0.463397	0.231698	+	0.972788i	$0.425572\pi$
$380$	0	0
$381$	4656.20	0.626101
$382$	0	0
$383$	−5427.04	−0.724044	−0.362022	−	0.932170i	$-0.617913\pi$
$383$	−5427.04	−0.724044	−0.362022	+	0.932170i	$0.617913\pi$
$384$	0	0
$385$	−1643.63	−0.217577
$386$	0	0
$387$	1652.47	0.217054
$388$	0	0
$389$	−6843.88	−0.892027	−0.446014	−	0.895026i	$-0.647157\pi$
$389$	−6843.88	−0.892027	−0.446014	+	0.895026i	$0.647157\pi$
$390$	0	0
$391$	2489.30	0.321967
$392$	0	0
$393$	−6845.92	−0.878705
$394$	0	0
$395$	−1301.63	−0.165803
$396$	0	0
$397$	8382.62	1.05973	0.529863	−	0.848083i	$-0.322243\pi$
$397$	8382.62	1.05973	0.529863	+	0.848083i	$0.322243\pi$
$398$	0	0
$399$	948.248	0.118977
$400$	0	0
$401$	3525.30	0.439015	0.219508	−	0.975611i	$-0.429555\pi$
$401$	3525.30	0.439015	0.219508	+	0.975611i	$0.429555\pi$
$402$	0	0
$403$	1647.40	0.203630
$404$	0	0
$405$	576.305	0.0707082
$406$	0	0
$407$	5657.82	0.689060
$408$	0	0
$409$	940.772	0.113736	0.0568682	−	0.998382i	$-0.481889\pi$
$409$	940.772	0.113736	0.0568682	+	0.998382i	$0.481889\pi$
$410$	0	0
$411$	4329.19	0.519569
$412$	0	0
$413$	−306.779	−0.0365511
$414$	0	0
$415$	2054.95	0.243069
$416$	0	0
$417$	−9181.22	−1.07819
$418$	0	0
$419$	−532.791	−0.0621206	−0.0310603	−	0.999518i	$-0.509888\pi$
$419$	−532.791	−0.0621206	−0.0310603	+	0.999518i	$0.509888\pi$
$420$	0	0
$421$	−14425.3	−1.66994	−0.834970	−	0.550295i	$-0.814515\pi$
$421$	−14425.3	−1.66994	−0.834970	+	0.550295i	$0.814515\pi$
$422$	0	0
$423$	870.203	0.100025
$424$	0	0
$425$	−1207.15	−0.137777
$426$	0	0
$427$	−2442.69	−0.276838
$428$	0	0
$429$	1014.00	0.114117
$430$	0	0
$431$	−15224.9	−1.70152	−0.850762	−	0.525551i	$-0.823859\pi$
$431$	−15224.9	−1.70152	−0.850762	+	0.525551i	$0.823859\pi$
$432$	0	0
$433$	10115.5	1.12268	0.561338	−	0.827587i	$-0.310287\pi$
$433$	10115.5	1.12268	0.561338	+	0.827587i	$0.310287\pi$
$434$	0	0
$435$	4767.93	0.525528
$436$	0	0
$437$	−5456.35	−0.597283
$438$	0	0
$439$	−15925.0	−1.73134	−0.865669	−	0.500617i	$-0.833106\pi$
$439$	−15925.0	−1.73134	−0.865669	+	0.500617i	$0.833106\pi$
$440$	0	0
$441$	−2376.49	−0.256613
$442$	0	0
$443$	−2387.09	−0.256014	−0.128007	−	0.991773i	$-0.540858\pi$
$443$	−2387.09	−0.256014	−0.128007	+	0.991773i	$0.540858\pi$
$444$	0	0
$445$	6852.44	0.729971
$446$	0	0
$447$	907.559	0.0960315
$448$	0	0
$449$	−13985.3	−1.46995	−0.734973	−	0.678096i	$-0.762805\pi$
$449$	−13985.3	−1.46995	−0.734973	+	0.678096i	$0.762805\pi$
$450$	0	0
$451$	−2750.91	−0.287218
$452$	0	0
$453$	−1150.72	−0.119350
$454$	0	0
$455$	821.815	0.0846754
$456$	0	0
$457$	−6520.34	−0.667414	−0.333707	−	0.942677i	$-0.608300\pi$
$457$	−6520.34	−0.667414	−0.333707	+	0.942677i	$0.608300\pi$
$458$	0	0
$459$	−438.203	−0.0445612
$460$	0	0
$461$	−6543.86	−0.661124	−0.330562	−	0.943784i	$-0.607238\pi$
$461$	−6543.86	−0.661124	−0.330562	+	0.943784i	$0.607238\pi$
$462$	0	0
$463$	13172.7	1.32221	0.661107	−	0.750292i	$-0.270087\pi$
$463$	13172.7	1.32221	0.661107	+	0.750292i	$0.270087\pi$
$464$	0	0
$465$	−2704.86	−0.269752
$466$	0	0
$467$	3166.27	0.313742	0.156871	−	0.987619i	$-0.449859\pi$
$467$	3166.27	0.313742	0.156871	+	0.987619i	$0.449859\pi$
$468$	0	0
$469$	−834.666	−0.0821776
$470$	0	0
$471$	−7750.14	−0.758190
$472$	0	0
$473$	−4773.82	−0.464060
$474$	0	0
$475$	2645.97	0.255590
$476$	0	0
$477$	−3478.14	−0.333863
$478$	0	0
$479$	−6470.71	−0.617232	−0.308616	−	0.951187i	$-0.599866\pi$
$479$	−6470.71	−0.617232	−0.308616	+	0.951187i	$0.599866\pi$
$480$	0	0
$481$	−2828.91	−0.268164
$482$	0	0
$483$	−4088.36	−0.385149
$484$	0	0
$485$	−3428.89	−0.321026
$486$	0	0
$487$	7620.36	0.709058	0.354529	−	0.935045i	$-0.384641\pi$
$487$	7620.36	0.709058	0.354529	+	0.935045i	$0.384641\pi$
$488$	0	0
$489$	3910.36	0.361621
$490$	0	0
$491$	14472.0	1.33017	0.665084	−	0.746769i	$-0.268396\pi$
$491$	14472.0	1.33017	0.665084	+	0.746769i	$0.268396\pi$
$492$	0	0
$493$	−3625.38	−0.331194
$494$	0	0
$495$	−1664.88	−0.151173
$496$	0	0
$497$	−6591.09	−0.594871
$498$	0	0
$499$	12338.1	1.10687	0.553436	−	0.832892i	$-0.313316\pi$
$499$	12338.1	1.10687	0.553436	+	0.832892i	$0.313316\pi$
$500$	0	0
$501$	−3120.12	−0.278237
$502$	0	0
$503$	−12491.1	−1.10725	−0.553627	−	0.832765i	$-0.686757\pi$
$503$	−12491.1	−1.10725	−0.553627	+	0.832765i	$0.686757\pi$
$504$	0	0
$505$	−8102.79	−0.713999
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	21986.9	1.91464	0.957318	−	0.289036i	$-0.0933346\pi$
$509$	21986.9	1.91464	0.957318	+	0.289036i	$0.0933346\pi$
$510$	0	0
$511$	5692.60	0.492810
$512$	0	0
$513$	960.508	0.0826657
$514$	0	0
$515$	−7762.05	−0.664149
$516$	0	0
$517$	−2513.92	−0.213853
$518$	0	0
$519$	8967.29	0.758421
$520$	0	0
$521$	−6385.80	−0.536981	−0.268491	−	0.963282i	$-0.586525\pi$
$521$	−6385.80	−0.536981	−0.268491	+	0.963282i	$0.586525\pi$
$522$	0	0
$523$	11858.5	0.991467	0.495733	−	0.868475i	$-0.334899\pi$
$523$	11858.5	0.991467	0.495733	+	0.868475i	$0.334899\pi$
$524$	0	0
$525$	1982.59	0.164814
$526$	0	0
$527$	2056.69	0.170001
$528$	0	0
$529$	11358.0	0.933506
$530$	0	0
$531$	−310.746	−0.0253959
$532$	0	0
$533$	1375.45	0.111778
$534$	0	0
$535$	−886.479	−0.0716370
$536$	0	0
$537$	−5358.17	−0.430581
$538$	0	0
$539$	6865.42	0.548636
$540$	0	0
$541$	−19726.4	−1.56766	−0.783829	−	0.620976i	$-0.786736\pi$
$541$	−19726.4	−1.56766	−0.783829	+	0.620976i	$0.786736\pi$
$542$	0	0
$543$	−3962.52	−0.313164
$544$	0	0
$545$	9967.18	0.783389
$546$	0	0
$547$	9070.36	0.708996	0.354498	−	0.935057i	$-0.384652\pi$
$547$	9070.36	0.708996	0.354498	+	0.935057i	$0.384652\pi$
$548$	0	0
$549$	−2474.27	−0.192348
$550$	0	0
$551$	7946.55	0.614400
$552$	0	0
$553$	−1625.49	−0.124996
$554$	0	0
$555$	4644.77	0.355242
$556$	0	0
$557$	−21557.9	−1.63992	−0.819961	−	0.572419i	$-0.806005\pi$
$557$	−21557.9	−1.63992	−0.819961	+	0.572419i	$0.806005\pi$
$558$	0	0
$559$	2386.91	0.180600
$560$	0	0
$561$	1265.92	0.0952713
$562$	0	0
$563$	−22899.8	−1.71423	−0.857113	−	0.515128i	$-0.827745\pi$
$563$	−22899.8	−1.71423	−0.857113	+	0.515128i	$0.827745\pi$
$564$	0	0
$565$	−3854.92	−0.287040
$566$	0	0
$567$	719.695	0.0533057
$568$	0	0
$569$	−14009.0	−1.03214	−0.516071	−	0.856546i	$-0.672606\pi$
$569$	−14009.0	−1.03214	−0.516071	+	0.856546i	$0.672606\pi$
$570$	0	0
$571$	22800.2	1.67103	0.835517	−	0.549465i	$-0.185168\pi$
$571$	22800.2	1.67103	0.835517	+	0.549465i	$0.185168\pi$
$572$	0	0
$573$	1309.79	0.0954923
$574$	0	0
$575$	−11408.1	−0.827390
$576$	0	0
$577$	16722.7	1.20655	0.603273	−	0.797535i	$-0.293863\pi$
$577$	16722.7	1.20655	0.603273	+	0.797535i	$0.293863\pi$
$578$	0	0
$579$	7206.57	0.517262
$580$	0	0
$581$	2566.24	0.183246
$582$	0	0
$583$	10047.9	0.713797
$584$	0	0
$585$	832.441	0.0588328
$586$	0	0
$587$	−4728.47	−0.332479	−0.166239	−	0.986085i	$-0.553162\pi$
$587$	−4728.47	−0.332479	−0.166239	+	0.986085i	$0.553162\pi$
$588$	0	0
$589$	−4508.10	−0.315370
$590$	0	0
$591$	3389.82	0.235937
$592$	0	0
$593$	15903.8	1.10133	0.550667	−	0.834725i	$-0.314374\pi$
$593$	15903.8	1.10133	0.550667	+	0.834725i	$0.314374\pi$
$594$	0	0
$595$	1025.99	0.0706915
$596$	0	0
$597$	−5578.82	−0.382456
$598$	0	0
$599$	−7515.22	−0.512627	−0.256313	−	0.966594i	$-0.582508\pi$
$599$	−7515.22	−0.512627	−0.256313	+	0.966594i	$0.582508\pi$
$600$	0	0
$601$	26704.5	1.81247	0.906237	−	0.422769i	$-0.138942\pi$
$601$	26704.5	1.81247	0.906237	+	0.422769i	$0.138942\pi$
$602$	0	0
$603$	−845.457	−0.0570973
$604$	0	0
$605$	−4660.24	−0.313167
$606$	0	0
$607$	−1113.76	−0.0744748	−0.0372374	−	0.999306i	$-0.511856\pi$
$607$	−1113.76	−0.0744748	−0.0372374	+	0.999306i	$0.511856\pi$
$608$	0	0
$609$	5954.24	0.396187
$610$	0	0
$611$	1256.96	0.0832261
$612$	0	0
$613$	12195.8	0.803559	0.401780	−	0.915736i	$-0.368392\pi$
$613$	12195.8	0.803559	0.401780	+	0.915736i	$0.368392\pi$
$614$	0	0
$615$	−2258.35	−0.148074
$616$	0	0
$617$	−447.009	−0.0291668	−0.0145834	−	0.999894i	$-0.504642\pi$
$617$	−447.009	−0.0291668	−0.0145834	+	0.999894i	$0.504642\pi$
$618$	0	0
$619$	22476.3	1.45945	0.729725	−	0.683740i	$-0.239648\pi$
$619$	22476.3	1.45945	0.729725	+	0.683740i	$0.239648\pi$
$620$	0	0
$621$	−4141.22	−0.267603
$622$	0	0
$623$	8557.39	0.550313
$624$	0	0
$625$	−795.520	−0.0509132
$626$	0	0
$627$	−2774.80	−0.176738
$628$	0	0
$629$	−3531.73	−0.223878
$630$	0	0
$631$	−7165.04	−0.452037	−0.226019	−	0.974123i	$-0.572571\pi$
$631$	−7165.04	−0.452037	−0.226019	+	0.974123i	$0.572571\pi$
$632$	0	0
$633$	2579.19	0.161949
$634$	0	0
$635$	−11042.8	−0.690109
$636$	0	0
$637$	−3432.71	−0.213515
$638$	0	0
$639$	−6676.30	−0.413319
$640$	0	0
$641$	12463.4	0.767980	0.383990	−	0.923337i	$-0.374550\pi$
$641$	12463.4	0.767980	0.383990	+	0.923337i	$0.374550\pi$
$642$	0	0
$643$	−11009.8	−0.675249	−0.337625	−	0.941281i	$-0.609623\pi$
$643$	−11009.8	−0.675249	−0.337625	+	0.941281i	$0.609623\pi$
$644$	0	0
$645$	−3919.05	−0.239244
$646$	0	0
$647$	−30825.9	−1.87309	−0.936547	−	0.350541i	$-0.885998\pi$
$647$	−30825.9	−1.87309	−0.936547	+	0.350541i	$0.885998\pi$
$648$	0	0
$649$	897.710	0.0542961
$650$	0	0
$651$	−3377.85	−0.203362
$652$	0	0
$653$	−15675.9	−0.939426	−0.469713	−	0.882819i	$-0.655643\pi$
$653$	−15675.9	−0.939426	−0.469713	+	0.882819i	$0.655643\pi$
$654$	0	0
$655$	16236.0	0.968537
$656$	0	0
$657$	5766.20	0.342406
$658$	0	0
$659$	−22568.8	−1.33408	−0.667038	−	0.745024i	$-0.732438\pi$
$659$	−22568.8	−1.33408	−0.667038	+	0.745024i	$0.732438\pi$
$660$	0	0
$661$	−31243.5	−1.83847	−0.919237	−	0.393705i	$-0.871193\pi$
$661$	−31243.5	−1.83847	−0.919237	+	0.393705i	$0.871193\pi$
$662$	0	0
$663$	−632.960	−0.0370771
$664$	0	0
$665$	−2248.89	−0.131140
$666$	0	0
$667$	−34261.5	−1.98892
$668$	0	0
$669$	−796.034	−0.0460037
$670$	0	0
$671$	7147.89	0.411239
$672$	0	0
$673$	−19818.1	−1.13511	−0.567556	−	0.823335i	$-0.692111\pi$
$673$	−19818.1	−1.13511	−0.567556	+	0.823335i	$0.692111\pi$
$674$	0	0
$675$	2008.22	0.114513
$676$	0	0
$677$	−3421.40	−0.194232	−0.0971159	−	0.995273i	$-0.530962\pi$
$677$	−3421.40	−0.194232	−0.0971159	+	0.995273i	$0.530962\pi$
$678$	0	0
$679$	−4282.03	−0.242016
$680$	0	0
$681$	4558.29	0.256497
$682$	0	0
$683$	−11002.1	−0.616373	−0.308186	−	0.951326i	$-0.599722\pi$
$683$	−11002.1	−0.616373	−0.308186	+	0.951326i	$0.599722\pi$
$684$	0	0
$685$	−10267.2	−0.572686
$686$	0	0
$687$	6206.99	0.344704
$688$	0	0
$689$	−5023.97	−0.277791
$690$	0	0
$691$	9817.44	0.540482	0.270241	−	0.962793i	$-0.412897\pi$
$691$	9817.44	0.540482	0.270241	+	0.962793i	$0.412897\pi$
$692$	0	0
$693$	−2079.12	−0.113967
$694$	0	0
$695$	21774.4	1.18842
$696$	0	0
$697$	1717.18	0.0933180
$698$	0	0
$699$	13874.8	0.750778
$700$	0	0
$701$	5776.14	0.311215	0.155608	−	0.987819i	$-0.450266\pi$
$701$	5776.14	0.311215	0.155608	+	0.987819i	$0.450266\pi$
$702$	0	0
$703$	7741.28	0.415317
$704$	0	0
$705$	−2063.80	−0.110251
$706$	0	0
$707$	−10118.8	−0.538271
$708$	0	0
$709$	35654.9	1.88864	0.944322	−	0.329022i	$-0.106719\pi$
$709$	35654.9	1.88864	0.944322	+	0.329022i	$0.106719\pi$
$710$	0	0
$711$	−1646.51	−0.0868480
$712$	0	0
$713$	19436.6	1.02091
$714$	0	0
$715$	−2404.83	−0.125784
$716$	0	0
$717$	−6295.06	−0.327885
$718$	0	0
$719$	14353.1	0.744478	0.372239	−	0.928137i	$-0.378590\pi$
$719$	14353.1	0.744478	0.372239	+	0.928137i	$0.378590\pi$
$720$	0	0
$721$	−9693.32	−0.500691
$722$	0	0
$723$	4037.89	0.207705
$724$	0	0
$725$	16614.6	0.851103
$726$	0	0
$727$	−28167.8	−1.43698	−0.718491	−	0.695536i	$-0.755167\pi$
$727$	−28167.8	−1.43698	−0.718491	+	0.695536i	$0.755167\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	2979.92	0.150775
$732$	0	0
$733$	23945.2	1.20660	0.603299	−	0.797515i	$-0.293853\pi$
$733$	23945.2	1.20660	0.603299	+	0.797515i	$0.293853\pi$
$734$	0	0
$735$	5636.15	0.282847
$736$	0	0
$737$	2442.43	0.122073
$738$	0	0
$739$	4021.49	0.200180	0.100090	−	0.994978i	$-0.468087\pi$
$739$	4021.49	0.200180	0.100090	+	0.994978i	$0.468087\pi$
$740$	0	0
$741$	1387.40	0.0687820
$742$	0	0
$743$	13584.9	0.670770	0.335385	−	0.942081i	$-0.391134\pi$
$743$	13584.9	0.670770	0.335385	+	0.942081i	$0.391134\pi$
$744$	0	0
$745$	−2152.39	−0.105849
$746$	0	0
$747$	2599.42	0.127320
$748$	0	0
$749$	−1107.04	−0.0540060
$750$	0	0
$751$	3537.89	0.171903	0.0859517	−	0.996299i	$-0.472607\pi$
$751$	3537.89	0.171903	0.0859517	+	0.996299i	$0.472607\pi$
$752$	0	0
$753$	12351.7	0.597772
$754$	0	0
$755$	2729.08	0.131552
$756$	0	0
$757$	4421.46	0.212286	0.106143	−	0.994351i	$-0.466150\pi$
$757$	4421.46	0.212286	0.106143	+	0.994351i	$0.466150\pi$
$758$	0	0
$759$	11963.5	0.572132
$760$	0	0
$761$	1093.63	0.0520947	0.0260473	−	0.999661i	$-0.491708\pi$
$761$	1093.63	0.0520947	0.0260473	+	0.999661i	$0.491708\pi$
$762$	0	0
$763$	12447.1	0.590584
$764$	0	0
$765$	1039.25	0.0491168
$766$	0	0
$767$	−448.855	−0.0211307
$768$	0	0
$769$	−15814.2	−0.741580	−0.370790	−	0.928717i	$-0.620913\pi$
$769$	−15814.2	−0.741580	−0.370790	+	0.928717i	$0.620913\pi$
$770$	0	0
$771$	−10180.9	−0.475560
$772$	0	0
$773$	37380.8	1.73932	0.869659	−	0.493653i	$-0.164339\pi$
$773$	37380.8	1.73932	0.869659	+	0.493653i	$0.164339\pi$
$774$	0	0
$775$	−9425.48	−0.436869
$776$	0	0
$777$	5800.43	0.267811
$778$	0	0
$779$	−3763.92	−0.173115
$780$	0	0
$781$	19287.1	0.883671
$782$	0	0
$783$	6031.22	0.275272
$784$	0	0
$785$	18380.4	0.835701
$786$	0	0
$787$	−1979.23	−0.0896469	−0.0448234	−	0.998995i	$-0.514273\pi$
$787$	−1979.23	−0.0896469	−0.0448234	+	0.998995i	$0.514273\pi$
$788$	0	0
$789$	19005.6	0.857562
$790$	0	0
$791$	−4814.06	−0.216395
$792$	0	0
$793$	−3573.95	−0.160044
$794$	0	0
$795$	8248.84	0.367995
$796$	0	0
$797$	23575.5	1.04779	0.523894	−	0.851783i	$-0.324479\pi$
$797$	23575.5	1.04779	0.523894	+	0.851783i	$0.324479\pi$
$798$	0	0
$799$	1569.24	0.0694816
$800$	0	0
$801$	8668.03	0.382359
$802$	0	0
$803$	−16657.9	−0.732061
$804$	0	0
$805$	9696.06	0.424523
$806$	0	0
$807$	1859.11	0.0810950
$808$	0	0
$809$	−39145.7	−1.70122	−0.850611	−	0.525796i	$-0.823768\pi$
$809$	−39145.7	−1.70122	−0.850611	+	0.525796i	$0.823768\pi$
$810$	0	0
$811$	39342.9	1.70347	0.851735	−	0.523973i	$-0.175551\pi$
$811$	39342.9	1.70347	0.851735	+	0.523973i	$0.175551\pi$
$812$	0	0
$813$	−15497.3	−0.668527
$814$	0	0
$815$	−9273.91	−0.398590
$816$	0	0
$817$	−6531.75	−0.279703
$818$	0	0
$819$	1039.56	0.0443530
$820$	0	0
$821$	−216.569	−0.00920622	−0.00460311	−	0.999989i	$-0.501465\pi$
$821$	−216.569	−0.00920622	−0.00460311	+	0.999989i	$0.501465\pi$
$822$	0	0
$823$	45061.7	1.90857	0.954284	−	0.298901i	$-0.0966201\pi$
$823$	45061.7	1.90857	0.954284	+	0.298901i	$0.0966201\pi$
$824$	0	0
$825$	−5801.52	−0.244828
$826$	0	0
$827$	−1228.67	−0.0516625	−0.0258313	−	0.999666i	$-0.508223\pi$
$827$	−1228.67	−0.0516625	−0.0258313	+	0.999666i	$0.508223\pi$
$828$	0	0
$829$	29560.3	1.23845	0.619224	−	0.785215i	$-0.287447\pi$
$829$	29560.3	1.23845	0.619224	+	0.785215i	$0.287447\pi$
$830$	0	0
$831$	10603.1	0.442622
$832$	0	0
$833$	−4285.54	−0.178254
$834$	0	0
$835$	7399.77	0.306682
$836$	0	0
$837$	−3421.53	−0.141297
$838$	0	0
$839$	23680.0	0.974402	0.487201	−	0.873290i	$-0.338018\pi$
$839$	23680.0	0.974402	0.487201	+	0.873290i	$0.338018\pi$
$840$	0	0
$841$	25509.0	1.04592
$842$	0	0
$843$	−12347.4	−0.504470
$844$	0	0
$845$	1202.41	0.0489518
$846$	0	0
$847$	−5819.76	−0.236091
$848$	0	0
$849$	11833.7	0.478363
$850$	0	0
$851$	−33376.4	−1.34445
$852$	0	0
$853$	−15779.1	−0.633374	−0.316687	−	0.948530i	$-0.602570\pi$
$853$	−15779.1	−0.633374	−0.316687	+	0.948530i	$0.602570\pi$
$854$	0	0
$855$	−2277.97	−0.0911167
$856$	0	0
$857$	−6759.13	−0.269414	−0.134707	−	0.990886i	$-0.543009\pi$
$857$	−6759.13	−0.269414	−0.134707	+	0.990886i	$0.543009\pi$
$858$	0	0
$859$	20653.1	0.820344	0.410172	−	0.912008i	$-0.365469\pi$
$859$	20653.1	0.820344	0.410172	+	0.912008i	$0.365469\pi$
$860$	0	0
$861$	−2820.25	−0.111630
$862$	0	0
$863$	6087.19	0.240105	0.120052	−	0.992768i	$-0.461694\pi$
$863$	6087.19	0.240105	0.120052	+	0.992768i	$0.461694\pi$
$864$	0	0
$865$	−21267.1	−0.835955
$866$	0	0
$867$	13948.8	0.546396
$868$	0	0
$869$	4756.58	0.185680
$870$	0	0
$871$	−1221.22	−0.0475078
$872$	0	0
$873$	−4337.39	−0.168154
$874$	0	0
$875$	−12604.0	−0.486964
$876$	0	0
$877$	6841.87	0.263436	0.131718	−	0.991287i	$-0.457951\pi$
$877$	6841.87	0.263436	0.131718	+	0.991287i	$0.457951\pi$
$878$	0	0
$879$	2317.32	0.0889207
$880$	0	0
$881$	9416.61	0.360106	0.180053	−	0.983657i	$-0.442373\pi$
$881$	9416.61	0.360106	0.180053	+	0.983657i	$0.442373\pi$
$882$	0	0
$883$	21561.3	0.821740	0.410870	−	0.911694i	$-0.365225\pi$
$883$	21561.3	0.821740	0.410870	+	0.911694i	$0.365225\pi$
$884$	0	0
$885$	736.972	0.0279922
$886$	0	0
$887$	−25672.9	−0.971828	−0.485914	−	0.874007i	$-0.661513\pi$
$887$	−25672.9	−0.971828	−0.485914	+	0.874007i	$0.661513\pi$
$888$	0	0
$889$	−13790.3	−0.520261
$890$	0	0
$891$	−2106.00	−0.0791848
$892$	0	0
$893$	−3439.66	−0.128896
$894$	0	0
$895$	12707.6	0.474600
$896$	0	0
$897$	−5981.76	−0.222659
$898$	0	0
$899$	−28307.2	−1.05017
$900$	0	0
$901$	−6272.14	−0.231915
$902$	0	0
$903$	−4894.15	−0.180362
$904$	0	0
$905$	9397.61	0.345179
$906$	0	0
$907$	17612.3	0.644770	0.322385	−	0.946609i	$-0.395515\pi$
$907$	17612.3	0.644770	0.322385	+	0.946609i	$0.395515\pi$
$908$	0	0
$909$	−10249.7	−0.373993
$910$	0	0
$911$	−8830.95	−0.321166	−0.160583	−	0.987022i	$-0.551337\pi$
$911$	−8830.95	−0.321166	−0.160583	+	0.987022i	$0.551337\pi$
$912$	0	0
$913$	−7509.45	−0.272209
$914$	0	0
$915$	5868.04	0.212013
$916$	0	0
$917$	20275.6	0.730163
$918$	0	0
$919$	−30983.7	−1.11214	−0.556071	−	0.831135i	$-0.687692\pi$
$919$	−30983.7	−1.11214	−0.556071	+	0.831135i	$0.687692\pi$
$920$	0	0
$921$	14657.2	0.524399
$922$	0	0
$923$	−9643.55	−0.343902
$924$	0	0
$925$	16185.4	0.575321
$926$	0	0
$927$	−9818.64	−0.347882
$928$	0	0
$929$	−15131.2	−0.534378	−0.267189	−	0.963644i	$-0.586095\pi$
$929$	−15131.2	−0.534378	−0.267189	+	0.963644i	$0.586095\pi$
$930$	0	0
$931$	9393.58	0.330679
$932$	0	0
$933$	−532.632	−0.0186898
$934$	0	0
$935$	−3002.29	−0.105011
$936$	0	0
$937$	40264.3	1.40382	0.701909	−	0.712267i	$-0.252331\pi$
$937$	40264.3	1.40382	0.701909	+	0.712267i	$0.252331\pi$
$938$	0	0
$939$	20031.1	0.696155
$940$	0	0
$941$	36254.5	1.25597	0.627983	−	0.778227i	$-0.283881\pi$
$941$	36254.5	1.25597	0.627983	+	0.778227i	$0.283881\pi$
$942$	0	0
$943$	16228.1	0.560402
$944$	0	0
$945$	−1706.85	−0.0587553
$946$	0	0
$947$	−47707.4	−1.63705	−0.818524	−	0.574473i	$-0.805207\pi$
$947$	−47707.4	−1.63705	−0.818524	+	0.574473i	$0.805207\pi$
$948$	0	0
$949$	8328.96	0.284899
$950$	0	0
$951$	5419.60	0.184798
$952$	0	0
$953$	−28934.4	−0.983502	−0.491751	−	0.870736i	$-0.663643\pi$
$953$	−28934.4	−0.983502	−0.491751	+	0.870736i	$0.663643\pi$
$954$	0	0
$955$	−3106.32	−0.105255
$956$	0	0
$957$	−17423.5	−0.588529
$958$	0	0
$959$	−12821.8	−0.431738
$960$	0	0
$961$	−13732.2	−0.460953
$962$	0	0
$963$	−1121.36	−0.0375236
$964$	0	0
$965$	−17091.3	−0.570143
$966$	0	0
$967$	9108.15	0.302894	0.151447	−	0.988465i	$-0.451607\pi$
$967$	9108.15	0.302894	0.151447	+	0.988465i	$0.451607\pi$
$968$	0	0
$969$	1732.09	0.0574229
$970$	0	0
$971$	34912.3	1.15385	0.576925	−	0.816797i	$-0.304252\pi$
$971$	34912.3	1.15385	0.576925	+	0.816797i	$0.304252\pi$
$972$	0	0
$973$	27192.1	0.895928
$974$	0	0
$975$	2900.76	0.0952808
$976$	0	0
$977$	−41510.7	−1.35931	−0.679655	−	0.733532i	$-0.737870\pi$
$977$	−41510.7	−1.35931	−0.679655	+	0.733532i	$0.737870\pi$
$978$	0	0
$979$	−25041.0	−0.817480
$980$	0	0
$981$	12608.0	0.410340
$982$	0	0
$983$	−18673.8	−0.605901	−0.302950	−	0.953006i	$-0.597972\pi$
$983$	−18673.8	−0.605901	−0.302950	+	0.953006i	$0.597972\pi$
$984$	0	0
$985$	−8039.38	−0.260057
$986$	0	0
$987$	−2577.29	−0.0831165
$988$	0	0
$989$	28161.6	0.905446
$990$	0	0
$991$	−5954.58	−0.190871	−0.0954356	−	0.995436i	$-0.530424\pi$
$991$	−5954.58	−0.190871	−0.0954356	+	0.995436i	$0.530424\pi$
$992$	0	0
$993$	34341.5	1.09748
$994$	0	0
$995$	13230.9	0.421555
$996$	0	0
$997$	8918.13	0.283290	0.141645	−	0.989918i	$-0.454761\pi$
$997$	8918.13	0.283290	0.141645	+	0.989918i	$0.454761\pi$
$998$	0	0
$999$	5875.42	0.186076

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.4.a.u.1.2		2
4.3	odd	2		2496.4.a.bd.1.2		2
8.3	odd	2		624.4.a.l.1.1		2
8.5	even	2		312.4.a.f.1.1	✓	2
24.5	odd	2		936.4.a.c.1.2		2
24.11	even	2		1872.4.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.f.1.1	✓	2	8.5	even	2
624.4.a.l.1.1		2	8.3	odd	2
936.4.a.c.1.2		2	24.5	odd	2
1872.4.a.v.1.2		2	24.11	even	2
2496.4.a.u.1.2		2	1.1	even	1	trivial
2496.4.a.bd.1.2		2	4.3	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:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2496.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +7.11488 q^{5} +8.88512 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +7.11488 q^{5} +8.88512 q^{7} +9.00000 q^{9} -26.0000 q^{11} +13.0000 q^{13} -21.3446 q^{15} +16.2298 q^{17} -35.5744 q^{19} -26.6554 q^{21} +153.379 q^{23} -74.3785 q^{25} -27.0000 q^{27} -223.379 q^{29} +126.723 q^{31} +78.0000 q^{33} +63.2166 q^{35} -217.608 q^{37} -39.0000 q^{39} +105.804 q^{41} +183.608 q^{43} +64.0339 q^{45} +96.6893 q^{47} -264.055 q^{49} -48.6893 q^{51} -386.460 q^{53} -184.987 q^{55} +106.723 q^{57} -34.5273 q^{59} -274.919 q^{61} +79.9661 q^{63} +92.4934 q^{65} -93.9397 q^{67} -460.136 q^{69} -741.812 q^{71} +640.689 q^{73} +223.136 q^{75} -231.013 q^{77} -182.945 q^{79} +81.0000 q^{81} +288.825 q^{83} +115.473 q^{85} +670.136 q^{87} +963.115 q^{89} +115.507 q^{91} -380.169 q^{93} -253.107 q^{95} -481.932 q^{97} -234.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 12 q^{5} + 44 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 - 12 * q^5 + 44 * q^7 + 18 * q^9	\( 2 q - 6 q^{3} - 12 q^{5} + 44 q^{7} + 18 q^{9} - 52 q^{11} + 26 q^{13} + 36 q^{15} - 20 q^{17} + 60 q^{19} - 132 q^{21} - 8 q^{23} + 166 q^{25} - 54 q^{27} - 132 q^{29} - 140 q^{31} + 156 q^{33} - 608 q^{35} - 68 q^{37} - 78 q^{39} + 28 q^{41} - 108 q^{45} + 36 q^{47} + 626 q^{49} + 60 q^{51} - 668 q^{53} + 312 q^{55} - 180 q^{57} + 508 q^{59} - 340 q^{61} + 396 q^{63} - 156 q^{65} + 940 q^{67} + 24 q^{69} + 300 q^{71} + 1124 q^{73} - 498 q^{75} - 1144 q^{77} - 1520 q^{79} + 162 q^{81} - 524 q^{83} + 808 q^{85} + 396 q^{87} + 1900 q^{89} + 572 q^{91} + 420 q^{93} - 2080 q^{95} - 1436 q^{97} - 468 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 - 12 * q^5 + 44 * q^7 + 18 * q^9 - 52 * q^11 + 26 * q^13 + 36 * q^15 - 20 * q^17 + 60 * q^19 - 132 * q^21 - 8 * q^23 + 166 * q^25 - 54 * q^27 - 132 * q^29 - 140 * q^31 + 156 * q^33 - 608 * q^35 - 68 * q^37 - 78 * q^39 + 28 * q^41 - 108 * q^45 + 36 * q^47 + 626 * q^49 + 60 * q^51 - 668 * q^53 + 312 * q^55 - 180 * q^57 + 508 * q^59 - 340 * q^61 + 396 * q^63 - 156 * q^65 + 940 * q^67 + 24 * q^69 + 300 * q^71 + 1124 * q^73 - 498 * q^75 - 1144 * q^77 - 1520 * q^79 + 162 * q^81 - 524 * q^83 + 808 * q^85 + 396 * q^87 + 1900 * q^89 + 572 * q^91 + 420 * q^93 - 2080 * q^95 - 1436 * q^97 - 468 * q^99

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