LMFDB - Embedded newform 784.4.a.bg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,Mod(1,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, 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("784.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	784.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:	$46.2574974445$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{65})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 35x^{2} + 36x + 194 $ x^4 - 2x^3 - 35x^2 + 36*x + 194
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 392)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.11692$ of defining polynomial
Character	$\chi$	$=$	784.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.82220 q^{3} -9.23641 q^{5} +34.1868 q^{9} +O(q^{10})$	$q-7.82220 q^{3} -9.23641 q^{5} +34.1868 q^{9} +71.3113 q^{11} -46.3581 q^{13} +72.2490 q^{15} -83.7908 q^{17} -2.16534 q^{19} -193.245 q^{23} -39.6887 q^{25} -56.2164 q^{27} -135.560 q^{29} +20.5969 q^{31} -557.811 q^{33} -324.805 q^{37} +362.623 q^{39} +431.511 q^{41} +135.693 q^{43} -315.763 q^{45} -592.809 q^{47} +655.428 q^{51} +182.864 q^{53} -658.660 q^{55} +16.9377 q^{57} +208.734 q^{59} +80.2883 q^{61} +428.183 q^{65} +831.121 q^{67} +1511.60 q^{69} +59.1361 q^{71} +367.068 q^{73} +310.453 q^{75} +438.125 q^{79} -483.307 q^{81} -35.0499 q^{83} +773.926 q^{85} +1060.38 q^{87} -824.134 q^{89} -161.113 q^{93} +20.0000 q^{95} -1299.01 q^{97} +2437.90 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 40 q^{9}+O(q^{10}) $ 4 * q + 40 * q^9	$ 4 q + 40 q^{9} + 124 q^{11} + 160 q^{15} - 128 q^{23} - 320 q^{25} - 252 q^{29} - 364 q^{37} + 1128 q^{39} + 1220 q^{43} + 1364 q^{51} - 752 q^{53} + 100 q^{57} + 1100 q^{65} + 2744 q^{67} + 1720 q^{71} + 1688 q^{79} - 1256 q^{81} + 1580 q^{85} + 968 q^{93} + 80 q^{95} + 5140 q^{99}+O(q^{100}) $ 4 * q + 40 * q^9 + 124 * q^11 + 160 * q^15 - 128 * q^23 - 320 * q^25 - 252 * q^29 - 364 * q^37 + 1128 * q^39 + 1220 * q^43 + 1364 * q^51 - 752 * q^53 + 100 * q^57 + 1100 * q^65 + 2744 * q^67 + 1720 * q^71 + 1688 * q^79 - 1256 * q^81 + 1580 * q^85 + 968 * q^93 + 80 * q^95 + 5140 * 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
$3$	−7.82220	−1.50538	−0.752691	−	0.658374i	$-0.771245\pi$
$3$	−7.82220	−1.50538	−0.752691	+	0.658374i	$0.771245\pi$
$4$	0	0
$5$	−9.23641	−0.826130	−0.413065	−	0.910702i	$-0.635542\pi$
$5$	−9.23641	−0.826130	−0.413065	+	0.910702i	$0.635542\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	34.1868	1.26618
$10$	0	0
$11$	71.3113	1.95465	0.977326	−	0.211742i	$-0.0679137\pi$
$11$	71.3113	1.95465	0.977326	+	0.211742i	$0.0679137\pi$
$12$	0	0
$13$	−46.3581	−0.989034	−0.494517	−	0.869168i	$-0.664655\pi$
$13$	−46.3581	−0.989034	−0.494517	+	0.869168i	$0.664655\pi$
$14$	0	0
$15$	72.2490	1.24364
$16$	0	0
$17$	−83.7908	−1.19543	−0.597713	−	0.801710i	$-0.703924\pi$
$17$	−83.7908	−1.19543	−0.597713	+	0.801710i	$0.703924\pi$
$18$	0	0
$19$	−2.16534	−0.0261455	−0.0130727	−	0.999915i	$-0.504161\pi$
$19$	−2.16534	−0.0261455	−0.0130727	+	0.999915i	$0.504161\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−193.245	−1.75193	−0.875965	−	0.482374i	$-0.839775\pi$
$23$	−193.245	−1.75193	−0.875965	+	0.482374i	$0.839775\pi$
$24$	0	0
$25$	−39.6887	−0.317510
$26$	0	0
$27$	−56.2164	−0.400698
$28$	0	0
$29$	−135.560	−0.868032	−0.434016	−	0.900905i	$-0.642904\pi$
$29$	−135.560	−0.868032	−0.434016	+	0.900905i	$0.642904\pi$
$30$	0	0
$31$	20.5969	0.119333	0.0596663	−	0.998218i	$-0.480996\pi$
$31$	20.5969	0.119333	0.0596663	+	0.998218i	$0.480996\pi$
$32$	0	0
$33$	−557.811	−2.94250
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−324.805	−1.44318	−0.721590	−	0.692320i	$-0.756589\pi$
$37$	−324.805	−1.44318	−0.721590	+	0.692320i	$0.756589\pi$
$38$	0	0
$39$	362.623	1.48887
$40$	0	0
$41$	431.511	1.64368	0.821838	−	0.569721i	$-0.192949\pi$
$41$	431.511	1.64368	0.821838	+	0.569721i	$0.192949\pi$
$42$	0	0
$43$	135.693	0.481231	0.240615	−	0.970621i	$-0.422651\pi$
$43$	135.693	0.481231	0.240615	+	0.970621i	$0.422651\pi$
$44$	0	0
$45$	−315.763	−1.04603
$46$	0	0
$47$	−592.809	−1.83979	−0.919894	−	0.392167i	$-0.871725\pi$
$47$	−592.809	−1.83979	−0.919894	+	0.392167i	$0.871725\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	655.428	1.79957
$52$	0	0
$53$	182.864	0.473930	0.236965	−	0.971518i	$-0.423847\pi$
$53$	182.864	0.473930	0.236965	+	0.971518i	$0.423847\pi$
$54$	0	0
$55$	−658.660	−1.61480
$56$	0	0
$57$	16.9377	0.0393589
$58$	0	0
$59$	208.734	0.460591	0.230295	−	0.973121i	$-0.426031\pi$
$59$	208.734	0.460591	0.230295	+	0.973121i	$0.426031\pi$
$60$	0	0
$61$	80.2883	0.168522	0.0842612	−	0.996444i	$-0.473147\pi$
$61$	80.2883	0.168522	0.0842612	+	0.996444i	$0.473147\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	428.183	0.817070
$66$	0	0
$67$	831.121	1.51549	0.757743	−	0.652553i	$-0.226302\pi$
$67$	831.121	1.51549	0.757743	+	0.652553i	$0.226302\pi$
$68$	0	0
$69$	1511.60	2.63733
$70$	0	0
$71$	59.1361	0.0988475	0.0494237	−	0.998778i	$-0.484262\pi$
$71$	59.1361	0.0988475	0.0494237	+	0.998778i	$0.484262\pi$
$72$	0	0
$73$	367.068	0.588522	0.294261	−	0.955725i	$-0.404927\pi$
$73$	367.068	0.588522	0.294261	+	0.955725i	$0.404927\pi$
$74$	0	0
$75$	310.453	0.477974
$76$	0	0
$77$	0	0
$78$	0	0
$79$	438.125	0.623960	0.311980	−	0.950089i	$-0.399008\pi$
$79$	438.125	0.623960	0.311980	+	0.950089i	$0.399008\pi$
$80$	0	0
$81$	−483.307	−0.662973
$82$	0	0
$83$	−35.0499	−0.0463522	−0.0231761	−	0.999731i	$-0.507378\pi$
$83$	−35.0499	−0.0463522	−0.0231761	+	0.999731i	$0.507378\pi$
$84$	0	0
$85$	773.926	0.987577
$86$	0	0
$87$	1060.38	1.30672
$88$	0	0
$89$	−824.134	−0.981551	−0.490776	−	0.871286i	$-0.663287\pi$
$89$	−824.134	−0.981551	−0.490776	+	0.871286i	$0.663287\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−161.113	−0.179641
$94$	0	0
$95$	20.0000	0.0215995
$96$	0	0
$97$	−1299.01	−1.35973	−0.679867	−	0.733335i	$-0.737963\pi$
$97$	−1299.01	−1.35973	−0.679867	+	0.733335i	$0.737963\pi$
$98$	0	0
$99$	2437.90	2.47493
$100$	0	0
$101$	1326.56	1.30690	0.653452	−	0.756968i	$-0.273320\pi$
$101$	1326.56	1.30690	0.653452	+	0.756968i	$0.273320\pi$
$102$	0	0
$103$	−582.760	−0.557486	−0.278743	−	0.960366i	$-0.589918\pi$
$103$	−582.760	−0.557486	−0.278743	+	0.960366i	$0.589918\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−202.109	−0.182604	−0.0913019	−	0.995823i	$-0.529103\pi$
$107$	−202.109	−0.182604	−0.0913019	+	0.995823i	$0.529103\pi$
$108$	0	0
$109$	−992.409	−0.872069	−0.436034	−	0.899930i	$-0.643617\pi$
$109$	−992.409	−0.872069	−0.436034	+	0.899930i	$0.643617\pi$
$110$	0	0
$111$	2540.69	2.17254
$112$	0	0
$113$	−371.405	−0.309193	−0.154597	−	0.987978i	$-0.549408\pi$
$113$	−371.405	−0.309193	−0.154597	+	0.987978i	$0.549408\pi$
$114$	0	0
$115$	1784.89	1.44732
$116$	0	0
$117$	−1584.84	−1.25229
$118$	0	0
$119$	0	0
$120$	0	0
$121$	3754.30	2.82066
$122$	0	0
$123$	−3375.37	−2.47436
$124$	0	0
$125$	1521.13	1.08843
$126$	0	0
$127$	638.872	0.446383	0.223192	−	0.974775i	$-0.428352\pi$
$127$	638.872	0.446383	0.223192	+	0.974775i	$0.428352\pi$
$128$	0	0
$129$	−1061.41	−0.724437
$130$	0	0
$131$	404.132	0.269536	0.134768	−	0.990877i	$-0.456971\pi$
$131$	404.132	0.269536	0.134768	+	0.990877i	$0.456971\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	519.237	0.331028
$136$	0	0
$137$	−1513.32	−0.943732	−0.471866	−	0.881670i	$-0.656419\pi$
$137$	−1513.32	−0.943732	−0.471866	+	0.881670i	$0.656419\pi$
$138$	0	0
$139$	836.293	0.510312	0.255156	−	0.966900i	$-0.417873\pi$
$139$	836.293	0.510312	0.255156	+	0.966900i	$0.417873\pi$
$140$	0	0
$141$	4637.07	2.76958
$142$	0	0
$143$	−3305.86	−1.93322
$144$	0	0
$145$	1252.09	0.717107
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−390.965	−0.214960	−0.107480	−	0.994207i	$-0.534278\pi$
$149$	−390.965	−0.214960	−0.107480	+	0.994207i	$0.534278\pi$
$150$	0	0
$151$	39.8367	0.0214693	0.0107347	−	0.999942i	$-0.496583\pi$
$151$	39.8367	0.0214693	0.0107347	+	0.999942i	$0.496583\pi$
$152$	0	0
$153$	−2864.54	−1.51362
$154$	0	0
$155$	−190.241	−0.0985842
$156$	0	0
$157$	1986.64	1.00988	0.504939	−	0.863155i	$-0.331515\pi$
$157$	1986.64	1.00988	0.504939	+	0.863155i	$0.331515\pi$
$158$	0	0
$159$	−1430.40	−0.713446
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1926.67	0.925816	0.462908	−	0.886406i	$-0.346806\pi$
$163$	1926.67	0.925816	0.462908	+	0.886406i	$0.346806\pi$
$164$	0	0
$165$	5152.17	2.43088
$166$	0	0
$167$	2023.15	0.937459	0.468730	−	0.883342i	$-0.344712\pi$
$167$	2023.15	0.937459	0.468730	+	0.883342i	$0.344712\pi$
$168$	0	0
$169$	−47.9222	−0.0218126
$170$	0	0
$171$	−74.0261	−0.0331048
$172$	0	0
$173$	3685.22	1.61955	0.809776	−	0.586740i	$-0.199589\pi$
$173$	3685.22	1.61955	0.809776	+	0.586740i	$0.199589\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1632.76	−0.693365
$178$	0	0
$179$	2725.56	1.13809	0.569045	−	0.822306i	$-0.307313\pi$
$179$	2725.56	1.13809	0.569045	+	0.822306i	$0.307313\pi$
$180$	0	0
$181$	1149.19	0.471927	0.235963	−	0.971762i	$-0.424175\pi$
$181$	1149.19	0.471927	0.235963	+	0.971762i	$0.424175\pi$
$182$	0	0
$183$	−628.031	−0.253691
$184$	0	0
$185$	3000.04	1.19225
$186$	0	0
$187$	−5975.23	−2.33664
$188$	0	0
$189$	0	0
$190$	0	0
$191$	60.3658	0.0228687	0.0114343	−	0.999935i	$-0.496360\pi$
$191$	60.3658	0.0228687	0.0114343	+	0.999935i	$0.496360\pi$
$192$	0	0
$193$	2475.65	0.923324	0.461662	−	0.887056i	$-0.347253\pi$
$193$	2475.65	0.923324	0.461662	+	0.887056i	$0.347253\pi$
$194$	0	0
$195$	−3349.33	−1.23000
$196$	0	0
$197$	2835.10	1.02534	0.512671	−	0.858585i	$-0.328656\pi$
$197$	2835.10	1.02534	0.512671	+	0.858585i	$0.328656\pi$
$198$	0	0
$199$	239.079	0.0851652	0.0425826	−	0.999093i	$-0.486441\pi$
$199$	239.079	0.0851652	0.0425826	+	0.999093i	$0.486441\pi$
$200$	0	0
$201$	−6501.19	−2.28139
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3985.62	−1.35789
$206$	0	0
$207$	−6606.43	−2.21825
$208$	0	0
$209$	−154.413	−0.0511053
$210$	0	0
$211$	−4541.74	−1.48183	−0.740915	−	0.671599i	$-0.765608\pi$
$211$	−4541.74	−1.48183	−0.740915	+	0.671599i	$0.765608\pi$
$212$	0	0
$213$	−462.575	−0.148803
$214$	0	0
$215$	−1253.31	−0.397559
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−2871.28	−0.885950
$220$	0	0
$221$	3884.39	1.18232
$222$	0	0
$223$	3296.37	0.989872	0.494936	−	0.868929i	$-0.335192\pi$
$223$	3296.37	0.989872	0.494936	+	0.868929i	$0.335192\pi$
$224$	0	0
$225$	−1356.83	−0.402023
$226$	0	0
$227$	2020.02	0.590633	0.295316	−	0.955400i	$-0.404575\pi$
$227$	2020.02	0.590633	0.295316	+	0.955400i	$0.404575\pi$
$228$	0	0
$229$	1445.76	0.417198	0.208599	−	0.978001i	$-0.433110\pi$
$229$	1445.76	0.417198	0.208599	+	0.978001i	$0.433110\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5119.64	1.43948	0.719740	−	0.694244i	$-0.244261\pi$
$233$	5119.64	1.43948	0.719740	+	0.694244i	$0.244261\pi$
$234$	0	0
$235$	5475.42	1.51990
$236$	0	0
$237$	−3427.10	−0.939299
$238$	0	0
$239$	−4316.23	−1.16818	−0.584088	−	0.811691i	$-0.698548\pi$
$239$	−4316.23	−1.16818	−0.584088	+	0.811691i	$0.698548\pi$
$240$	0	0
$241$	−2336.17	−0.624423	−0.312212	−	0.950013i	$-0.601070\pi$
$241$	−2336.17	−0.624423	−0.312212	+	0.950013i	$0.601070\pi$
$242$	0	0
$243$	5298.37	1.39873
$244$	0	0
$245$	0	0
$246$	0	0
$247$	100.381	0.0258587
$248$	0	0
$249$	274.167	0.0697777
$250$	0	0
$251$	1253.17	0.315138	0.157569	−	0.987508i	$-0.449634\pi$
$251$	1253.17	0.315138	0.157569	+	0.987508i	$0.449634\pi$
$252$	0	0
$253$	−13780.6	−3.42441
$254$	0	0
$255$	−6053.80	−1.48668
$256$	0	0
$257$	−1307.75	−0.317414	−0.158707	−	0.987326i	$-0.550732\pi$
$257$	−1307.75	−0.317414	−0.158707	+	0.987326i	$0.550732\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−4634.37	−1.09908
$262$	0	0
$263$	2346.05	0.550051	0.275026	−	0.961437i	$-0.411314\pi$
$263$	2346.05	0.550051	0.275026	+	0.961437i	$0.411314\pi$
$264$	0	0
$265$	−1689.01	−0.391528
$266$	0	0
$267$	6446.54	1.47761
$268$	0	0
$269$	−5808.89	−1.31663	−0.658316	−	0.752741i	$-0.728731\pi$
$269$	−5808.89	−1.31663	−0.658316	+	0.752741i	$0.728731\pi$
$270$	0	0
$271$	−1829.82	−0.410160	−0.205080	−	0.978745i	$-0.565745\pi$
$271$	−1829.82	−0.410160	−0.205080	+	0.978745i	$0.565745\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2830.25	−0.620621
$276$	0	0
$277$	−7151.18	−1.55117	−0.775583	−	0.631246i	$-0.782544\pi$
$277$	−7151.18	−1.55117	−0.775583	+	0.631246i	$0.782544\pi$
$278$	0	0
$279$	704.141	0.151096
$280$	0	0
$281$	1752.44	0.372034	0.186017	−	0.982547i	$-0.440442\pi$
$281$	1752.44	0.372034	0.186017	+	0.982547i	$0.440442\pi$
$282$	0	0
$283$	−4923.10	−1.03409	−0.517046	−	0.855958i	$-0.672968\pi$
$283$	−4923.10	−1.03409	−0.517046	+	0.855958i	$0.672968\pi$
$284$	0	0
$285$	−156.444	−0.0325156
$286$	0	0
$287$	0	0
$288$	0	0
$289$	2107.90	0.429044
$290$	0	0
$291$	10161.1	2.04692
$292$	0	0
$293$	3398.91	0.677702	0.338851	−	0.940840i	$-0.389962\pi$
$293$	3398.91	0.677702	0.338851	+	0.940840i	$0.389962\pi$
$294$	0	0
$295$	−1927.95	−0.380508
$296$	0	0
$297$	−4008.86	−0.783225
$298$	0	0
$299$	8958.49	1.73272
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−10376.6	−1.96739
$304$	0	0
$305$	−741.576	−0.139221
$306$	0	0
$307$	−5931.92	−1.10278	−0.551388	−	0.834249i	$-0.685902\pi$
$307$	−5931.92	−1.10278	−0.551388	+	0.834249i	$0.685902\pi$
$308$	0	0
$309$	4558.47	0.839230
$310$	0	0
$311$	8667.78	1.58040	0.790200	−	0.612849i	$-0.209977\pi$
$311$	8667.78	1.58040	0.790200	+	0.612849i	$0.209977\pi$
$312$	0	0
$313$	6072.41	1.09659	0.548295	−	0.836285i	$-0.315277\pi$
$313$	6072.41	1.09659	0.548295	+	0.836285i	$0.315277\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6386.01	1.13146	0.565732	−	0.824589i	$-0.308594\pi$
$317$	6386.01	1.13146	0.565732	+	0.824589i	$0.308594\pi$
$318$	0	0
$319$	−9666.98	−1.69670
$320$	0	0
$321$	1580.94	0.274889
$322$	0	0
$323$	181.436	0.0312550
$324$	0	0
$325$	1839.90	0.314028
$326$	0	0
$327$	7762.82	1.31280
$328$	0	0
$329$	0	0
$330$	0	0
$331$	4783.37	0.794313	0.397156	−	0.917751i	$-0.369997\pi$
$331$	4783.37	0.794313	0.397156	+	0.917751i	$0.369997\pi$
$332$	0	0
$333$	−11104.1	−1.82732
$334$	0	0
$335$	−7676.57	−1.25199
$336$	0	0
$337$	−4970.91	−0.803509	−0.401754	−	0.915747i	$-0.631599\pi$
$337$	−4970.91	−0.803509	−0.401754	+	0.915747i	$0.631599\pi$
$338$	0	0
$339$	2905.20	0.465454
$340$	0	0
$341$	1468.79	0.233254
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−13961.8	−2.17877
$346$	0	0
$347$	2254.59	0.348797	0.174399	−	0.984675i	$-0.444202\pi$
$347$	2254.59	0.348797	0.174399	+	0.984675i	$0.444202\pi$
$348$	0	0
$349$	−3808.01	−0.584063	−0.292031	−	0.956409i	$-0.594331\pi$
$349$	−3808.01	−0.584063	−0.292031	+	0.956409i	$0.594331\pi$
$350$	0	0
$351$	2606.09	0.396304
$352$	0	0
$353$	4118.26	0.620942	0.310471	−	0.950583i	$-0.399513\pi$
$353$	4118.26	0.620942	0.310471	+	0.950583i	$0.399513\pi$
$354$	0	0
$355$	−546.206	−0.0816608
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9770.58	1.43641	0.718205	−	0.695831i	$-0.244964\pi$
$359$	9770.58	1.43641	0.718205	+	0.695831i	$0.244964\pi$
$360$	0	0
$361$	−6854.31	−0.999316
$362$	0	0
$363$	−29366.9	−4.24617
$364$	0	0
$365$	−3390.39	−0.486195
$366$	0	0
$367$	6189.90	0.880409	0.440204	−	0.897898i	$-0.354906\pi$
$367$	6189.90	0.880409	0.440204	+	0.897898i	$0.354906\pi$
$368$	0	0
$369$	14752.0	2.08118
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−6085.10	−0.844703	−0.422352	−	0.906432i	$-0.638795\pi$
$373$	−6085.10	−0.844703	−0.422352	+	0.906432i	$0.638795\pi$
$374$	0	0
$375$	−11898.6	−1.63851
$376$	0	0
$377$	6284.33	0.858513
$378$	0	0
$379$	8627.76	1.16934	0.584668	−	0.811273i	$-0.301225\pi$
$379$	8627.76	1.16934	0.584668	+	0.811273i	$0.301225\pi$
$380$	0	0
$381$	−4997.38	−0.671978
$382$	0	0
$383$	10179.4	1.35807	0.679036	−	0.734105i	$-0.262398\pi$
$383$	10179.4	1.35807	0.679036	+	0.734105i	$0.262398\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4638.89	0.609323
$388$	0	0
$389$	1359.18	0.177155	0.0885773	−	0.996069i	$-0.471768\pi$
$389$	1359.18	0.177155	0.0885773	+	0.996069i	$0.471768\pi$
$390$	0	0
$391$	16192.2	2.09430
$392$	0	0
$393$	−3161.20	−0.405755
$394$	0	0
$395$	−4046.70	−0.515472
$396$	0	0
$397$	−1643.15	−0.207727	−0.103863	−	0.994592i	$-0.533120\pi$
$397$	−1643.15	−0.207727	−0.103863	+	0.994592i	$0.533120\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12749.3	−1.58771	−0.793855	−	0.608107i	$-0.791929\pi$
$401$	−12749.3	−1.58771	−0.793855	+	0.608107i	$0.791929\pi$
$402$	0	0
$403$	−954.833	−0.118024
$404$	0	0
$405$	4464.03	0.547702
$406$	0	0
$407$	−23162.3	−2.82092
$408$	0	0
$409$	−6255.47	−0.756267	−0.378133	−	0.925751i	$-0.623434\pi$
$409$	−6255.47	−0.756267	−0.378133	+	0.925751i	$0.623434\pi$
$410$	0	0
$411$	11837.5	1.42068
$412$	0	0
$413$	0	0
$414$	0	0
$415$	323.735	0.0382929
$416$	0	0
$417$	−6541.65	−0.768215
$418$	0	0
$419$	3462.47	0.403705	0.201853	−	0.979416i	$-0.435304\pi$
$419$	3462.47	0.403705	0.201853	+	0.979416i	$0.435304\pi$
$420$	0	0
$421$	−4998.81	−0.578687	−0.289343	−	0.957225i	$-0.593437\pi$
$421$	−4998.81	−0.578687	−0.289343	+	0.957225i	$0.593437\pi$
$422$	0	0
$423$	−20266.2	−2.32950
$424$	0	0
$425$	3325.55	0.379559
$426$	0	0
$427$	0	0
$428$	0	0
$429$	25859.1	2.91023
$430$	0	0
$431$	1777.62	0.198666	0.0993328	−	0.995054i	$-0.468329\pi$
$431$	1777.62	0.198666	0.0993328	+	0.995054i	$0.468329\pi$
$432$	0	0
$433$	14968.5	1.66129	0.830645	−	0.556802i	$-0.187972\pi$
$433$	14968.5	1.66129	0.830645	+	0.556802i	$0.187972\pi$
$434$	0	0
$435$	−9794.10	−1.07952
$436$	0	0
$437$	418.442	0.0458050
$438$	0	0
$439$	−7827.08	−0.850948	−0.425474	−	0.904971i	$-0.639893\pi$
$439$	−7827.08	−0.850948	−0.425474	+	0.904971i	$0.639893\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1329.22	−0.142558	−0.0712791	−	0.997456i	$-0.522708\pi$
$443$	−1329.22	−0.142558	−0.0712791	+	0.997456i	$0.522708\pi$
$444$	0	0
$445$	7612.04	0.810889
$446$	0	0
$447$	3058.21	0.323598
$448$	0	0
$449$	2040.55	0.214476	0.107238	−	0.994233i	$-0.465799\pi$
$449$	2040.55	0.214476	0.107238	+	0.994233i	$0.465799\pi$
$450$	0	0
$451$	30771.6	3.21281
$452$	0	0
$453$	−311.611	−0.0323195
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5775.33	−0.591157	−0.295579	−	0.955318i	$-0.595512\pi$
$457$	−5775.33	−0.591157	−0.295579	+	0.955318i	$0.595512\pi$
$458$	0	0
$459$	4710.41	0.479005
$460$	0	0
$461$	−8491.59	−0.857901	−0.428951	−	0.903328i	$-0.641117\pi$
$461$	−8491.59	−0.857901	−0.428951	+	0.903328i	$0.641117\pi$
$462$	0	0
$463$	18801.8	1.88725	0.943623	−	0.331022i	$-0.107394\pi$
$463$	18801.8	1.88725	0.943623	+	0.331022i	$0.107394\pi$
$464$	0	0
$465$	1488.10	0.148407
$466$	0	0
$467$	7219.71	0.715393	0.357696	−	0.933838i	$-0.383562\pi$
$467$	7219.71	0.715393	0.357696	+	0.933838i	$0.383562\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−15539.9	−1.52025
$472$	0	0
$473$	9676.41	0.940638
$474$	0	0
$475$	85.9397	0.00830144
$476$	0	0
$477$	6251.53	0.600079
$478$	0	0
$479$	−19527.0	−1.86265	−0.931325	−	0.364190i	$-0.881346\pi$
$479$	−19527.0	−1.86265	−0.931325	+	0.364190i	$0.881346\pi$
$480$	0	0
$481$	15057.4	1.42735
$482$	0	0
$483$	0	0
$484$	0	0
$485$	11998.2	1.12332
$486$	0	0
$487$	7005.65	0.651861	0.325931	−	0.945394i	$-0.394322\pi$
$487$	7005.65	0.651861	0.325931	+	0.945394i	$0.394322\pi$
$488$	0	0
$489$	−15070.8	−1.39371
$490$	0	0
$491$	10393.5	0.955296	0.477648	−	0.878551i	$-0.341490\pi$
$491$	10393.5	0.955296	0.477648	+	0.878551i	$0.341490\pi$
$492$	0	0
$493$	11358.7	1.03767
$494$	0	0
$495$	−22517.5	−2.04462
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−3873.35	−0.347485	−0.173742	−	0.984791i	$-0.555586\pi$
$499$	−3873.35	−0.347485	−0.173742	+	0.984791i	$0.555586\pi$
$500$	0	0
$501$	−15825.4	−1.41123
$502$	0	0
$503$	20795.6	1.84340	0.921700	−	0.387903i	$-0.126800\pi$
$503$	20795.6	1.84340	0.921700	+	0.387903i	$0.126800\pi$
$504$	0	0
$505$	−12252.6	−1.07967
$506$	0	0
$507$	374.857	0.0328363
$508$	0	0
$509$	−22296.3	−1.94158	−0.970791	−	0.239926i	$-0.922877\pi$
$509$	−22296.3	−1.94158	−0.970791	+	0.239926i	$0.922877\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	121.728	0.0104764
$514$	0	0
$515$	5382.62	0.460556
$516$	0	0
$517$	−42273.9	−3.59614
$518$	0	0
$519$	−28826.5	−2.43804
$520$	0	0
$521$	4035.66	0.339358	0.169679	−	0.985499i	$-0.445727\pi$
$521$	4035.66	0.339358	0.169679	+	0.985499i	$0.445727\pi$
$522$	0	0
$523$	−1196.02	−0.0999967	−0.0499984	−	0.998749i	$-0.515922\pi$
$523$	−1196.02	−0.0999967	−0.0499984	+	0.998749i	$0.515922\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1725.83	−0.142653
$528$	0	0
$529$	25176.7	2.06926
$530$	0	0
$531$	7135.94	0.583189
$532$	0	0
$533$	−20004.1	−1.62565
$534$	0	0
$535$	1866.76	0.150854
$536$	0	0
$537$	−21319.9	−1.71326
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−18676.5	−1.48422	−0.742111	−	0.670277i	$-0.766175\pi$
$541$	−18676.5	−1.48422	−0.742111	+	0.670277i	$0.766175\pi$
$542$	0	0
$543$	−8989.21	−0.710430
$544$	0	0
$545$	9166.29	0.720442
$546$	0	0
$547$	−7882.22	−0.616123	−0.308061	−	0.951366i	$-0.599680\pi$
$547$	−7882.22	−0.616123	−0.308061	+	0.951366i	$0.599680\pi$
$548$	0	0
$549$	2744.80	0.213379
$550$	0	0
$551$	293.535	0.0226951
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−23466.9	−1.79480
$556$	0	0
$557$	19693.2	1.49807	0.749037	−	0.662528i	$-0.230516\pi$
$557$	19693.2	1.49807	0.749037	+	0.662528i	$0.230516\pi$
$558$	0	0
$559$	−6290.46	−0.475953
$560$	0	0
$561$	46739.4	3.51754
$562$	0	0
$563$	−8799.55	−0.658716	−0.329358	−	0.944205i	$-0.606832\pi$
$563$	−8799.55	−0.658716	−0.329358	+	0.944205i	$0.606832\pi$
$564$	0	0
$565$	3430.45	0.255434
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9717.58	0.715961	0.357981	−	0.933729i	$-0.383465\pi$
$569$	9717.58	0.715961	0.357981	+	0.933729i	$0.383465\pi$
$570$	0	0
$571$	4404.59	0.322813	0.161407	−	0.986888i	$-0.448397\pi$
$571$	4404.59	0.322813	0.161407	+	0.986888i	$0.448397\pi$
$572$	0	0
$573$	−472.193	−0.0344261
$574$	0	0
$575$	7669.65	0.556255
$576$	0	0
$577$	−2813.75	−0.203012	−0.101506	−	0.994835i	$-0.532366\pi$
$577$	−2813.75	−0.203012	−0.101506	+	0.994835i	$0.532366\pi$
$578$	0	0
$579$	−19365.1	−1.38996
$580$	0	0
$581$	0	0
$582$	0	0
$583$	13040.3	0.926368
$584$	0	0
$585$	14638.2	1.03456
$586$	0	0
$587$	−18434.9	−1.29624	−0.648119	−	0.761539i	$-0.724444\pi$
$587$	−18434.9	−1.29624	−0.648119	+	0.761539i	$0.724444\pi$
$588$	0	0
$589$	−44.5993	−0.00312001
$590$	0	0
$591$	−22176.7	−1.54353
$592$	0	0
$593$	−13681.5	−0.947439	−0.473720	−	0.880676i	$-0.657089\pi$
$593$	−13681.5	−0.947439	−0.473720	+	0.880676i	$0.657089\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1870.12	−0.128206
$598$	0	0
$599$	−19126.1	−1.30463	−0.652314	−	0.757949i	$-0.726202\pi$
$599$	−19126.1	−1.30463	−0.652314	+	0.757949i	$0.726202\pi$
$600$	0	0
$601$	1663.82	0.112926	0.0564629	−	0.998405i	$-0.482018\pi$
$601$	1663.82	0.112926	0.0564629	+	0.998405i	$0.482018\pi$
$602$	0	0
$603$	28413.3	1.91887
$604$	0	0
$605$	−34676.3	−2.33023
$606$	0	0
$607$	−21403.3	−1.43119	−0.715594	−	0.698516i	$-0.753844\pi$
$607$	−21403.3	−1.43119	−0.715594	+	0.698516i	$0.753844\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	27481.5	1.81961
$612$	0	0
$613$	10314.1	0.679577	0.339789	−	0.940502i	$-0.389644\pi$
$613$	10314.1	0.679577	0.339789	+	0.940502i	$0.389644\pi$
$614$	0	0
$615$	31176.3	2.04414
$616$	0	0
$617$	16999.0	1.10917	0.554583	−	0.832129i	$-0.312878\pi$
$617$	16999.0	1.10917	0.554583	+	0.832129i	$0.312878\pi$
$618$	0	0
$619$	10812.8	0.702108	0.351054	−	0.936355i	$-0.385823\pi$
$619$	10812.8	0.702108	0.351054	+	0.936355i	$0.385823\pi$
$620$	0	0
$621$	10863.5	0.701995
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−9088.72	−0.581678
$626$	0	0
$627$	1207.85	0.0769330
$628$	0	0
$629$	27215.7	1.72522
$630$	0	0
$631$	−4033.27	−0.254456	−0.127228	−	0.991873i	$-0.540608\pi$
$631$	−4033.27	−0.254456	−0.127228	+	0.991873i	$0.540608\pi$
$632$	0	0
$633$	35526.4	2.23072
$634$	0	0
$635$	−5900.88	−0.368771
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2021.67	0.125158
$640$	0	0
$641$	9381.51	0.578077	0.289039	−	0.957317i	$-0.406664\pi$
$641$	9381.51	0.578077	0.289039	+	0.957317i	$0.406664\pi$
$642$	0	0
$643$	22164.5	1.35938	0.679690	−	0.733499i	$-0.262114\pi$
$643$	22164.5	1.35938	0.679690	+	0.733499i	$0.262114\pi$
$644$	0	0
$645$	9803.66	0.598479
$646$	0	0
$647$	−1753.14	−0.106527	−0.0532637	−	0.998580i	$-0.516962\pi$
$647$	−1753.14	−0.106527	−0.0532637	+	0.998580i	$0.516962\pi$
$648$	0	0
$649$	14885.1	0.900294
$650$	0	0
$651$	0	0
$652$	0	0
$653$	19764.5	1.18445	0.592224	−	0.805773i	$-0.298250\pi$
$653$	19764.5	1.18445	0.592224	+	0.805773i	$0.298250\pi$
$654$	0	0
$655$	−3732.73	−0.222672
$656$	0	0
$657$	12548.9	0.745173
$658$	0	0
$659$	−7435.63	−0.439531	−0.219766	−	0.975553i	$-0.570529\pi$
$659$	−7435.63	−0.439531	−0.219766	+	0.975553i	$0.570529\pi$
$660$	0	0
$661$	5536.32	0.325776	0.162888	−	0.986645i	$-0.447919\pi$
$661$	5536.32	0.325776	0.162888	+	0.986645i	$0.447919\pi$
$662$	0	0
$663$	−30384.4	−1.77984
$664$	0	0
$665$	0	0
$666$	0	0
$667$	26196.4	1.52073
$668$	0	0
$669$	−25784.9	−1.49014
$670$	0	0
$671$	5725.46	0.329402
$672$	0	0
$673$	5903.76	0.338147	0.169074	−	0.985603i	$-0.445922\pi$
$673$	5903.76	0.338147	0.169074	+	0.985603i	$0.445922\pi$
$674$	0	0
$675$	2231.15	0.127225
$676$	0	0
$677$	4367.54	0.247944	0.123972	−	0.992286i	$-0.460437\pi$
$677$	4367.54	0.247944	0.123972	+	0.992286i	$0.460437\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−15801.0	−0.889128
$682$	0	0
$683$	−12294.7	−0.688790	−0.344395	−	0.938825i	$-0.611916\pi$
$683$	−12294.7	−0.688790	−0.344395	+	0.938825i	$0.611916\pi$
$684$	0	0
$685$	13977.6	0.779645
$686$	0	0
$687$	−11309.0	−0.628043
$688$	0	0
$689$	−8477.23	−0.468733
$690$	0	0
$691$	20785.6	1.14431	0.572156	−	0.820145i	$-0.306107\pi$
$691$	20785.6	1.14431	0.572156	+	0.820145i	$0.306107\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7724.34	−0.421584
$696$	0	0
$697$	−36156.7	−1.96489
$698$	0	0
$699$	−40046.9	−2.16697
$700$	0	0
$701$	28360.1	1.52803	0.764014	−	0.645200i	$-0.223226\pi$
$701$	28360.1	1.52803	0.764014	+	0.645200i	$0.223226\pi$
$702$	0	0
$703$	703.315	0.0377326
$704$	0	0
$705$	−42829.9	−2.28804
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−12832.7	−0.679749	−0.339875	−	0.940471i	$-0.610385\pi$
$709$	−12832.7	−0.679749	−0.339875	+	0.940471i	$0.610385\pi$
$710$	0	0
$711$	14978.1	0.790044
$712$	0	0
$713$	−3980.25	−0.209062
$714$	0	0
$715$	30534.3	1.59709
$716$	0	0
$717$	33762.4	1.75855
$718$	0	0
$719$	15508.7	0.804420	0.402210	−	0.915547i	$-0.368242\pi$
$719$	15508.7	0.804420	0.402210	+	0.915547i	$0.368242\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	18274.0	0.939996
$724$	0	0
$725$	5380.21	0.275608
$726$	0	0
$727$	−2847.01	−0.145240	−0.0726202	−	0.997360i	$-0.523136\pi$
$727$	−2847.01	−0.145240	−0.0726202	+	0.997360i	$0.523136\pi$
$728$	0	0
$729$	−28395.6	−1.44264
$730$	0	0
$731$	−11369.8	−0.575276
$732$	0	0
$733$	−11423.7	−0.575638	−0.287819	−	0.957685i	$-0.592930\pi$
$733$	−11423.7	−0.575638	−0.287819	+	0.957685i	$0.592930\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	59268.3	2.96225
$738$	0	0
$739$	15908.7	0.791897	0.395948	−	0.918273i	$-0.370416\pi$
$739$	15908.7	0.791897	0.395948	+	0.918273i	$0.370416\pi$
$740$	0	0
$741$	−785.202	−0.0389273
$742$	0	0
$743$	9002.36	0.444501	0.222251	−	0.974990i	$-0.428660\pi$
$743$	9002.36	0.444501	0.222251	+	0.974990i	$0.428660\pi$
$744$	0	0
$745$	3611.11	0.177585
$746$	0	0
$747$	−1198.24	−0.0586900
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−35132.9	−1.70708	−0.853539	−	0.521028i	$-0.825549\pi$
$751$	−35132.9	−1.70708	−0.853539	+	0.521028i	$0.825549\pi$
$752$	0	0
$753$	−9802.56	−0.474403
$754$	0	0
$755$	−367.948	−0.0177364
$756$	0	0
$757$	−7800.49	−0.374523	−0.187261	−	0.982310i	$-0.559961\pi$
$757$	−7800.49	−0.374523	−0.187261	+	0.982310i	$0.559961\pi$
$758$	0	0
$759$	107794.	5.15505
$760$	0	0
$761$	31198.4	1.48612	0.743062	−	0.669223i	$-0.233373\pi$
$761$	31198.4	1.48612	0.743062	+	0.669223i	$0.233373\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	26458.0	1.25045
$766$	0	0
$767$	−9676.52	−0.455540
$768$	0	0
$769$	−16342.0	−0.766328	−0.383164	−	0.923680i	$-0.625166\pi$
$769$	−16342.0	−0.766328	−0.383164	+	0.923680i	$0.625166\pi$
$770$	0	0
$771$	10229.5	0.477829
$772$	0	0
$773$	27643.8	1.28626	0.643131	−	0.765756i	$-0.277635\pi$
$773$	27643.8	1.28626	0.643131	+	0.765756i	$0.277635\pi$
$774$	0	0
$775$	−817.464	−0.0378892
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−934.370	−0.0429747
$780$	0	0
$781$	4217.07	0.193212
$782$	0	0
$783$	7620.71	0.347818
$784$	0	0
$785$	−18349.4	−0.834291
$786$	0	0
$787$	28937.1	1.31067	0.655335	−	0.755338i	$-0.272528\pi$
$787$	28937.1	1.31067	0.655335	+	0.755338i	$0.272528\pi$
$788$	0	0
$789$	−18351.2	−0.828037
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−3722.02	−0.166674
$794$	0	0
$795$	13211.7	0.589399
$796$	0	0
$797$	15972.2	0.709866	0.354933	−	0.934892i	$-0.384504\pi$
$797$	15972.2	0.709866	0.354933	+	0.934892i	$0.384504\pi$
$798$	0	0
$799$	49671.9	2.19933
$800$	0	0
$801$	−28174.5	−1.24282
$802$	0	0
$803$	26176.1	1.15035
$804$	0	0
$805$	0	0
$806$	0	0
$807$	45438.3	1.98204
$808$	0	0
$809$	−32847.1	−1.42749	−0.713747	−	0.700404i	$-0.753003\pi$
$809$	−32847.1	−1.42749	−0.713747	+	0.700404i	$0.753003\pi$
$810$	0	0
$811$	35306.3	1.52870	0.764349	−	0.644803i	$-0.223061\pi$
$811$	35306.3	1.52870	0.764349	+	0.644803i	$0.223061\pi$
$812$	0	0
$813$	14313.2	0.617448
$814$	0	0
$815$	−17795.5	−0.764845
$816$	0	0
$817$	−293.821	−0.0125820
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−23808.6	−1.01209	−0.506044	−	0.862508i	$-0.668893\pi$
$821$	−23808.6	−1.01209	−0.506044	+	0.862508i	$0.668893\pi$
$822$	0	0
$823$	−12326.2	−0.522069	−0.261035	−	0.965329i	$-0.584064\pi$
$823$	−12326.2	−0.522069	−0.261035	+	0.965329i	$0.584064\pi$
$824$	0	0
$825$	22138.8	0.934272
$826$	0	0
$827$	21178.7	0.890515	0.445258	−	0.895402i	$-0.353112\pi$
$827$	21178.7	0.890515	0.445258	+	0.895402i	$0.353112\pi$
$828$	0	0
$829$	−25253.0	−1.05799	−0.528994	−	0.848626i	$-0.677431\pi$
$829$	−25253.0	−1.05799	−0.528994	+	0.848626i	$0.677431\pi$
$830$	0	0
$831$	55938.0	2.33510
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−18686.6	−0.774463
$836$	0	0
$837$	−1157.88	−0.0478163
$838$	0	0
$839$	−6324.59	−0.260249	−0.130125	−	0.991498i	$-0.541538\pi$
$839$	−6324.59	−0.260249	−0.130125	+	0.991498i	$0.541538\pi$
$840$	0	0
$841$	−6012.40	−0.246521
$842$	0	0
$843$	−13707.9	−0.560054
$844$	0	0
$845$	442.629	0.0180200
$846$	0	0
$847$	0	0
$848$	0	0
$849$	38509.5	1.55670
$850$	0	0
$851$	62767.1	2.52835
$852$	0	0
$853$	36635.0	1.47053	0.735264	−	0.677781i	$-0.237058\pi$
$853$	36635.0	1.47053	0.735264	+	0.677781i	$0.237058\pi$
$854$	0	0
$855$	683.735	0.0273488
$856$	0	0
$857$	−25338.0	−1.00995	−0.504977	−	0.863133i	$-0.668499\pi$
$857$	−25338.0	−1.00995	−0.504977	+	0.863133i	$0.668499\pi$
$858$	0	0
$859$	−38112.6	−1.51384	−0.756918	−	0.653510i	$-0.773296\pi$
$859$	−38112.6	−1.51384	−0.756918	+	0.653510i	$0.773296\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	41578.5	1.64003	0.820016	−	0.572341i	$-0.193965\pi$
$863$	41578.5	1.64003	0.820016	+	0.572341i	$0.193965\pi$
$864$	0	0
$865$	−34038.2	−1.33796
$866$	0	0
$867$	−16488.4	−0.645876
$868$	0	0
$869$	31243.2	1.21962
$870$	0	0
$871$	−38529.2	−1.49887
$872$	0	0
$873$	−44408.9	−1.72166
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−5586.92	−0.215116	−0.107558	−	0.994199i	$-0.534303\pi$
$877$	−5586.92	−0.215116	−0.107558	+	0.994199i	$0.534303\pi$
$878$	0	0
$879$	−26587.0	−1.02020
$880$	0	0
$881$	−4331.97	−0.165662	−0.0828308	−	0.996564i	$-0.526396\pi$
$881$	−4331.97	−0.165662	−0.0828308	+	0.996564i	$0.526396\pi$
$882$	0	0
$883$	−29390.0	−1.12010	−0.560052	−	0.828457i	$-0.689219\pi$
$883$	−29390.0	−1.12010	−0.560052	+	0.828457i	$0.689219\pi$
$884$	0	0
$885$	15080.8	0.572810
$886$	0	0
$887$	20903.9	0.791302	0.395651	−	0.918401i	$-0.370519\pi$
$887$	20903.9	0.791302	0.395651	+	0.918401i	$0.370519\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−34465.3	−1.29588
$892$	0	0
$893$	1283.63	0.0481021
$894$	0	0
$895$	−25174.4	−0.940211
$896$	0	0
$897$	−70075.1	−2.60840
$898$	0	0
$899$	−2792.12	−0.103584
$900$	0	0
$901$	−15322.3	−0.566548
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−10614.4	−0.389873
$906$	0	0
$907$	−24727.6	−0.905257	−0.452628	−	0.891699i	$-0.649514\pi$
$907$	−24727.6	−0.905257	−0.452628	+	0.891699i	$0.649514\pi$
$908$	0	0
$909$	45350.7	1.65477
$910$	0	0
$911$	−6937.72	−0.252313	−0.126156	−	0.992010i	$-0.540264\pi$
$911$	−6937.72	−0.252313	−0.126156	+	0.992010i	$0.540264\pi$
$912$	0	0
$913$	−2499.45	−0.0906023
$914$	0	0
$915$	5800.75	0.209581
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−25186.5	−0.904055	−0.452027	−	0.892004i	$-0.649299\pi$
$919$	−25186.5	−0.904055	−0.452027	+	0.892004i	$0.649299\pi$
$920$	0	0
$921$	46400.6	1.66010
$922$	0	0
$923$	−2741.44	−0.0977634
$924$	0	0
$925$	12891.1	0.458224
$926$	0	0
$927$	−19922.7	−0.705876
$928$	0	0
$929$	36116.0	1.27549	0.637743	−	0.770249i	$-0.279868\pi$
$929$	36116.0	1.27549	0.637743	+	0.770249i	$0.279868\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−67801.1	−2.37911
$934$	0	0
$935$	55189.7	1.93037
$936$	0	0
$937$	−45182.4	−1.57529	−0.787644	−	0.616131i	$-0.788699\pi$
$937$	−45182.4	−1.57529	−0.787644	+	0.616131i	$0.788699\pi$
$938$	0	0
$939$	−47499.6	−1.65079
$940$	0	0
$941$	−25425.9	−0.880831	−0.440415	−	0.897794i	$-0.645169\pi$
$941$	−25425.9	−0.880831	−0.440415	+	0.897794i	$0.645169\pi$
$942$	0	0
$943$	−83387.5	−2.87961
$944$	0	0
$945$	0	0
$946$	0	0
$947$	19954.6	0.684729	0.342364	−	0.939567i	$-0.388772\pi$
$947$	19954.6	0.684729	0.342364	+	0.939567i	$0.388772\pi$
$948$	0	0
$949$	−17016.6	−0.582068
$950$	0	0
$951$	−49952.6	−1.70329
$952$	0	0
$953$	27291.3	0.927652	0.463826	−	0.885926i	$-0.346476\pi$
$953$	27291.3	0.927652	0.463826	+	0.885926i	$0.346476\pi$
$954$	0	0
$955$	−557.563	−0.0188925
$956$	0	0
$957$	75617.0	2.55418
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29366.8	−0.985760
$962$	0	0
$963$	−6909.45	−0.231209
$964$	0	0
$965$	−22866.2	−0.762785
$966$	0	0
$967$	−12227.9	−0.406641	−0.203321	−	0.979112i	$-0.565173\pi$
$967$	−12227.9	−0.406641	−0.203321	+	0.979112i	$0.565173\pi$
$968$	0	0
$969$	−1419.23	−0.0470507
$970$	0	0
$971$	−33208.5	−1.09754	−0.548770	−	0.835973i	$-0.684904\pi$
$971$	−33208.5	−1.09754	−0.548770	+	0.835973i	$0.684904\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−14392.0	−0.472732
$976$	0	0
$977$	867.072	0.0283931	0.0141966	−	0.999899i	$-0.495481\pi$
$977$	867.072	0.0283931	0.0141966	+	0.999899i	$0.495481\pi$
$978$	0	0
$979$	−58770.1	−1.91859
$980$	0	0
$981$	−33927.3	−1.10419
$982$	0	0
$983$	−25802.3	−0.837197	−0.418598	−	0.908171i	$-0.637479\pi$
$983$	−25802.3	−0.837197	−0.418598	+	0.908171i	$0.637479\pi$
$984$	0	0
$985$	−26186.1	−0.847065
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−26221.9	−0.843083
$990$	0	0
$991$	−27409.7	−0.878605	−0.439303	−	0.898339i	$-0.644774\pi$
$991$	−27409.7	−0.878605	−0.439303	+	0.898339i	$0.644774\pi$
$992$	0	0
$993$	−37416.4	−1.19574
$994$	0	0
$995$	−2208.23	−0.0703575
$996$	0	0
$997$	−23103.2	−0.733888	−0.366944	−	0.930243i	$-0.619596\pi$
$997$	−23103.2	−0.733888	−0.366944	+	0.930243i	$0.619596\pi$
$998$	0	0
$999$	18259.4	0.578280

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.a.bg.1.1		4
4.3	odd	2		392.4.a.m.1.4	yes	4
7.6	odd	2	inner	784.4.a.bg.1.4		4
28.3	even	6		392.4.i.p.177.4		8
28.11	odd	6		392.4.i.p.177.1		8
28.19	even	6		392.4.i.p.361.4		8
28.23	odd	6		392.4.i.p.361.1		8
28.27	even	2		392.4.a.m.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.4.a.m.1.1	✓	4	28.27	even	2
392.4.a.m.1.4	yes	4	4.3	odd	2
392.4.i.p.177.1		8	28.11	odd	6
392.4.i.p.177.4		8	28.3	even	6
392.4.i.p.361.1		8	28.23	odd	6
392.4.i.p.361.4		8	28.19	even	6
784.4.a.bg.1.1		4	1.1	even	1	trivial
784.4.a.bg.1.4		4	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q-7.82220 q^{3} -9.23641 q^{5} +34.1868 q^{9} +O(q^{10})\)	\(q-7.82220 q^{3} -9.23641 q^{5} +34.1868 q^{9} +71.3113 q^{11} -46.3581 q^{13} +72.2490 q^{15} -83.7908 q^{17} -2.16534 q^{19} -193.245 q^{23} -39.6887 q^{25} -56.2164 q^{27} -135.560 q^{29} +20.5969 q^{31} -557.811 q^{33} -324.805 q^{37} +362.623 q^{39} +431.511 q^{41} +135.693 q^{43} -315.763 q^{45} -592.809 q^{47} +655.428 q^{51} +182.864 q^{53} -658.660 q^{55} +16.9377 q^{57} +208.734 q^{59} +80.2883 q^{61} +428.183 q^{65} +831.121 q^{67} +1511.60 q^{69} +59.1361 q^{71} +367.068 q^{73} +310.453 q^{75} +438.125 q^{79} -483.307 q^{81} -35.0499 q^{83} +773.926 q^{85} +1060.38 q^{87} -824.134 q^{89} -161.113 q^{93} +20.0000 q^{95} -1299.01 q^{97} +2437.90 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 40 q^{9}+O(q^{10}) \) 4 * q + 40 * q^9	\( 4 q + 40 q^{9} + 124 q^{11} + 160 q^{15} - 128 q^{23} - 320 q^{25} - 252 q^{29} - 364 q^{37} + 1128 q^{39} + 1220 q^{43} + 1364 q^{51} - 752 q^{53} + 100 q^{57} + 1100 q^{65} + 2744 q^{67} + 1720 q^{71} + 1688 q^{79} - 1256 q^{81} + 1580 q^{85} + 968 q^{93} + 80 q^{95} + 5140 q^{99}+O(q^{100}) \) 4 * q + 40 * q^9 + 124 * q^11 + 160 * q^15 - 128 * q^23 - 320 * q^25 - 252 * q^29 - 364 * q^37 + 1128 * q^39 + 1220 * q^43 + 1364 * q^51 - 752 * q^53 + 100 * q^57 + 1100 * q^65 + 2744 * q^67 + 1720 * q^71 + 1688 * q^79 - 1256 * q^81 + 1580 * q^85 + 968 * q^93 + 80 * q^95 + 5140 * q^99

Introduction

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