LMFDB - Embedded newform 784.6.a.bg.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,6,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, 6, names="a")

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

chi := DirichletCharacter("784.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
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:	$125.740914733$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{793})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 399x^{2} + 400x + 38414 $ x^4 - 2x^3 - 399x^2 + 400*x + 38414
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 98)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-14.9943$ 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)

$f(q)$	$=$	$q+24.8621 q^{3} -84.4857 q^{5} +375.122 q^{9} +O(q^{10})$	$q+24.8621 q^{3} -84.4857 q^{5} +375.122 q^{9} +434.365 q^{11} +848.786 q^{13} -2100.49 q^{15} -301.445 q^{17} -2909.92 q^{19} +1709.46 q^{23} +4012.83 q^{25} +3284.82 q^{27} +3841.63 q^{29} +969.851 q^{31} +10799.2 q^{33} +2507.10 q^{37} +21102.6 q^{39} -13284.0 q^{41} -7781.29 q^{43} -31692.4 q^{45} +5489.13 q^{47} -7494.54 q^{51} +22859.9 q^{53} -36697.6 q^{55} -72346.6 q^{57} +11109.9 q^{59} +22205.2 q^{61} -71710.3 q^{65} +7218.58 q^{67} +42500.7 q^{69} -46153.3 q^{71} +67681.2 q^{73} +99767.1 q^{75} +69949.5 q^{79} -9487.24 q^{81} +3915.73 q^{83} +25467.8 q^{85} +95510.9 q^{87} +52951.2 q^{89} +24112.5 q^{93} +245847. q^{95} -64.5382 q^{97} +162940. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 712 q^{9}+O(q^{10}) $ 4 * q + 712 * q^9	$ 4 q + 712 q^{9} - 628 q^{11} - 5248 q^{15} - 2624 q^{23} + 4224 q^{25} + 17732 q^{29} + 2932 q^{37} + 41832 q^{39} - 9836 q^{43} + 51236 q^{51} + 1552 q^{53} - 104092 q^{57} - 142548 q^{65} - 13704 q^{67} + 56664 q^{71} + 218296 q^{79} - 127048 q^{81} - 103924 q^{85} - 88056 q^{93} + 396752 q^{95} + 354500 q^{99}+O(q^{100}) $ 4 * q + 712 * q^9 - 628 * q^11 - 5248 * q^15 - 2624 * q^23 + 4224 * q^25 + 17732 * q^29 + 2932 * q^37 + 41832 * q^39 - 9836 * q^43 + 51236 * q^51 + 1552 * q^53 - 104092 * q^57 - 142548 * q^65 - 13704 * q^67 + 56664 * q^71 + 218296 * q^79 - 127048 * q^81 - 103924 * q^85 - 88056 * q^93 + 396752 * q^95 + 354500 * 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
$3$	24.8621	1.59490	0.797451	−	0.603384i	$-0.206181\pi$
$3$	24.8621	1.59490	0.797451	+	0.603384i	$0.206181\pi$
$4$	0	0
$5$	−84.4857	−1.51133	−0.755663	−	0.654961i	$-0.772685\pi$
$5$	−84.4857	−1.51133	−0.755663	+	0.654961i	$0.772685\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	375.122	1.54371
$10$	0	0
$11$	434.365	1.08236	0.541182	−	0.840905i	$-0.317977\pi$
$11$	434.365	1.08236	0.541182	+	0.840905i	$0.317977\pi$
$12$	0	0
$13$	848.786	1.39296	0.696482	−	0.717574i	$-0.254747\pi$
$13$	848.786	1.39296	0.696482	+	0.717574i	$0.254747\pi$
$14$	0	0
$15$	−2100.49	−2.41042
$16$	0	0
$17$	−301.445	−0.252980	−0.126490	−	0.991968i	$-0.540371\pi$
$17$	−301.445	−0.252980	−0.126490	+	0.991968i	$0.540371\pi$
$18$	0	0
$19$	−2909.92	−1.84926	−0.924628	−	0.380870i	$-0.875624\pi$
$19$	−2909.92	−1.84926	−0.924628	+	0.380870i	$0.875624\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1709.46	0.673814	0.336907	−	0.941538i	$-0.390619\pi$
$23$	1709.46	0.673814	0.336907	+	0.941538i	$0.390619\pi$
$24$	0	0
$25$	4012.83	1.28410
$26$	0	0
$27$	3284.82	0.867166
$28$	0	0
$29$	3841.63	0.848245	0.424122	−	0.905605i	$-0.360583\pi$
$29$	3841.63	0.848245	0.424122	+	0.905605i	$0.360583\pi$
$30$	0	0
$31$	969.851	0.181260	0.0906298	−	0.995885i	$-0.471112\pi$
$31$	969.851	0.181260	0.0906298	+	0.995885i	$0.471112\pi$
$32$	0	0
$33$	10799.2	1.72626
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2507.10	0.301069	0.150535	−	0.988605i	$-0.451900\pi$
$37$	2507.10	0.301069	0.150535	+	0.988605i	$0.451900\pi$
$38$	0	0
$39$	21102.6	2.22164
$40$	0	0
$41$	−13284.0	−1.23415	−0.617076	−	0.786904i	$-0.711683\pi$
$41$	−13284.0	−1.23415	−0.617076	+	0.786904i	$0.711683\pi$
$42$	0	0
$43$	−7781.29	−0.641771	−0.320886	−	0.947118i	$-0.603981\pi$
$43$	−7781.29	−0.641771	−0.320886	+	0.947118i	$0.603981\pi$
$44$	0	0
$45$	−31692.4	−2.33305
$46$	0	0
$47$	5489.13	0.362459	0.181230	−	0.983441i	$-0.441992\pi$
$47$	5489.13	0.362459	0.181230	+	0.983441i	$0.441992\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−7494.54	−0.403478
$52$	0	0
$53$	22859.9	1.11785	0.558926	−	0.829218i	$-0.311214\pi$
$53$	22859.9	1.11785	0.558926	+	0.829218i	$0.311214\pi$
$54$	0	0
$55$	−36697.6	−1.63580
$56$	0	0
$57$	−72346.6	−2.94938
$58$	0	0
$59$	11109.9	0.415508	0.207754	−	0.978181i	$-0.433385\pi$
$59$	11109.9	0.415508	0.207754	+	0.978181i	$0.433385\pi$
$60$	0	0
$61$	22205.2	0.764064	0.382032	−	0.924149i	$-0.375224\pi$
$61$	22205.2	0.764064	0.382032	+	0.924149i	$0.375224\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−71710.3	−2.10522
$66$	0	0
$67$	7218.58	0.196456	0.0982278	−	0.995164i	$-0.468683\pi$
$67$	7218.58	0.196456	0.0982278	+	0.995164i	$0.468683\pi$
$68$	0	0
$69$	42500.7	1.07467
$70$	0	0
$71$	−46153.3	−1.08657	−0.543284	−	0.839549i	$-0.682819\pi$
$71$	−46153.3	−1.08657	−0.543284	+	0.839549i	$0.682819\pi$
$72$	0	0
$73$	67681.2	1.48649	0.743244	−	0.669021i	$-0.233286\pi$
$73$	67681.2	1.48649	0.743244	+	0.669021i	$0.233286\pi$
$74$	0	0
$75$	99767.1	2.04802
$76$	0	0
$77$	0	0
$78$	0	0
$79$	69949.5	1.26101	0.630503	−	0.776187i	$-0.282849\pi$
$79$	69949.5	1.26101	0.630503	+	0.776187i	$0.282849\pi$
$80$	0	0
$81$	−9487.24	−0.160667
$82$	0	0
$83$	3915.73	0.0623903	0.0311951	−	0.999513i	$-0.490069\pi$
$83$	3915.73	0.0623903	0.0311951	+	0.999513i	$0.490069\pi$
$84$	0	0
$85$	25467.8	0.382335
$86$	0	0
$87$	95510.9	1.35287
$88$	0	0
$89$	52951.2	0.708599	0.354300	−	0.935132i	$-0.384719\pi$
$89$	52951.2	0.708599	0.354300	+	0.935132i	$0.384719\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	24112.5	0.289091
$94$	0	0
$95$	245847.	2.79483
$96$	0	0
$97$	−64.5382	−0.000696447 0	−0.000348223	−	1.00000i	$-0.500111\pi$
$97$	−64.5382	−0.000696447 0	−0.000348223		1.00000i	$0.500111\pi$
$98$	0	0
$99$	162940.	1.67086
$100$	0	0
$101$	72493.6	0.707125	0.353563	−	0.935411i	$-0.384970\pi$
$101$	72493.6	0.707125	0.353563	+	0.935411i	$0.384970\pi$
$102$	0	0
$103$	78987.2	0.733607	0.366804	−	0.930298i	$-0.380452\pi$
$103$	78987.2	0.733607	0.366804	+	0.930298i	$0.380452\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−126386.	−1.06718	−0.533592	−	0.845742i	$-0.679158\pi$
$107$	−126386.	−1.06718	−0.533592	+	0.845742i	$0.679158\pi$
$108$	0	0
$109$	179718.	1.44886	0.724428	−	0.689351i	$-0.242104\pi$
$109$	179718.	1.44886	0.724428	+	0.689351i	$0.242104\pi$
$110$	0	0
$111$	62331.6	0.480176
$112$	0	0
$113$	258027.	1.90094	0.950471	−	0.310812i	$-0.100601\pi$
$113$	258027.	1.90094	0.950471	+	0.310812i	$0.100601\pi$
$114$	0	0
$115$	−144425.	−1.01835
$116$	0	0
$117$	318398.	2.15033
$118$	0	0
$119$	0	0
$120$	0	0
$121$	27622.3	0.171513
$122$	0	0
$123$	−330267.	−1.96835
$124$	0	0
$125$	−75008.6	−0.429374
$126$	0	0
$127$	71222.2	0.391838	0.195919	−	0.980620i	$-0.437231\pi$
$127$	71222.2	0.391838	0.195919	+	0.980620i	$0.437231\pi$
$128$	0	0
$129$	−193459.	−1.02356
$130$	0	0
$131$	−136703.	−0.695983	−0.347991	−	0.937498i	$-0.613136\pi$
$131$	−136703.	−0.695983	−0.347991	+	0.937498i	$0.613136\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−277520.	−1.31057
$136$	0	0
$137$	144193.	0.656361	0.328181	−	0.944615i	$-0.393565\pi$
$137$	144193.	0.656361	0.328181	+	0.944615i	$0.393565\pi$
$138$	0	0
$139$	217792.	0.956102	0.478051	−	0.878332i	$-0.341343\pi$
$139$	217792.	0.956102	0.478051	+	0.878332i	$0.341343\pi$
$140$	0	0
$141$	136471.	0.578087
$142$	0	0
$143$	368683.	1.50770
$144$	0	0
$145$	−324563.	−1.28197
$146$	0	0
$147$	0	0
$148$	0	0
$149$	285078.	1.05196	0.525978	−	0.850498i	$-0.323699\pi$
$149$	285078.	1.05196	0.525978	+	0.850498i	$0.323699\pi$
$150$	0	0
$151$	−101076.	−0.360750	−0.180375	−	0.983598i	$-0.557731\pi$
$151$	−101076.	−0.360750	−0.180375	+	0.983598i	$0.557731\pi$
$152$	0	0
$153$	−113079.	−0.390528
$154$	0	0
$155$	−81938.5	−0.273942
$156$	0	0
$157$	271885.	0.880310	0.440155	−	0.897922i	$-0.354923\pi$
$157$	271885.	0.880310	0.440155	+	0.897922i	$0.354923\pi$
$158$	0	0
$159$	568344.	1.78286
$160$	0	0
$161$	0	0
$162$	0	0
$163$	381176.	1.12372	0.561858	−	0.827234i	$-0.310087\pi$
$163$	381176.	1.12372	0.561858	+	0.827234i	$0.310087\pi$
$164$	0	0
$165$	−912379.	−2.60895
$166$	0	0
$167$	−216934.	−0.601917	−0.300958	−	0.953637i	$-0.597306\pi$
$167$	−216934.	−0.601917	−0.300958	+	0.953637i	$0.597306\pi$
$168$	0	0
$169$	349146.	0.940350
$170$	0	0
$171$	−1.09157e6	−2.85472
$172$	0	0
$173$	278202.	0.706716	0.353358	−	0.935488i	$-0.385040\pi$
$173$	278202.	0.706716	0.353358	+	0.935488i	$0.385040\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	276215.	0.662695
$178$	0	0
$179$	−360162.	−0.840166	−0.420083	−	0.907486i	$-0.637999\pi$
$179$	−360162.	−0.840166	−0.420083	+	0.907486i	$0.637999\pi$
$180$	0	0
$181$	−309871.	−0.703047	−0.351524	−	0.936179i	$-0.614336\pi$
$181$	−309871.	−0.703047	−0.351524	+	0.936179i	$0.614336\pi$
$182$	0	0
$183$	552066.	1.21861
$184$	0	0
$185$	−211814.	−0.455014
$186$	0	0
$187$	−130937.	−0.273816
$188$	0	0
$189$	0	0
$190$	0	0
$191$	960954.	1.90598	0.952992	−	0.302995i	$-0.0979867\pi$
$191$	960954.	1.90598	0.952992	+	0.302995i	$0.0979867\pi$
$192$	0	0
$193$	513006.	0.991356	0.495678	−	0.868506i	$-0.334920\pi$
$193$	513006.	0.991356	0.495678	+	0.868506i	$0.334920\pi$
$194$	0	0
$195$	−1.78287e6	−3.35762
$196$	0	0
$197$	−768131.	−1.41017	−0.705083	−	0.709125i	$-0.749090\pi$
$197$	−768131.	−1.41017	−0.705083	+	0.709125i	$0.749090\pi$
$198$	0	0
$199$	−1.04213e6	−1.86547	−0.932735	−	0.360563i	$-0.882585\pi$
$199$	−1.04213e6	−1.86547	−0.932735	+	0.360563i	$0.882585\pi$
$200$	0	0
$201$	179469.	0.313327
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1.12231e6	1.86520
$206$	0	0
$207$	641256.	1.04017
$208$	0	0
$209$	−1.26397e6	−2.00157
$210$	0	0
$211$	813317.	1.25763	0.628817	−	0.777554i	$-0.283540\pi$
$211$	813317.	1.25763	0.628817	+	0.777554i	$0.283540\pi$
$212$	0	0
$213$	−1.14747e6	−1.73297
$214$	0	0
$215$	657407.	0.969925
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.68269e6	2.37080
$220$	0	0
$221$	−255863.	−0.352392
$222$	0	0
$223$	238869.	0.321661	0.160830	−	0.986982i	$-0.448583\pi$
$223$	238869.	0.321661	0.160830	+	0.986982i	$0.448583\pi$
$224$	0	0
$225$	1.50530e6	1.98229
$226$	0	0
$227$	585236.	0.753818	0.376909	−	0.926250i	$-0.376987\pi$
$227$	585236.	0.753818	0.376909	+	0.926250i	$0.376987\pi$
$228$	0	0
$229$	−1.48546e6	−1.87186	−0.935930	−	0.352187i	$-0.885438\pi$
$229$	−1.48546e6	−1.87186	−0.935930	+	0.352187i	$0.885438\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.12071e6	−1.35239	−0.676196	−	0.736722i	$-0.736373\pi$
$233$	−1.12071e6	−1.35239	−0.676196	+	0.736722i	$0.736373\pi$
$234$	0	0
$235$	−463753.	−0.547794
$236$	0	0
$237$	1.73909e6	2.01118
$238$	0	0
$239$	−13757.0	−0.0155786	−0.00778931	−	0.999970i	$-0.502479\pi$
$239$	−13757.0	−0.0155786	−0.00778931	+	0.999970i	$0.502479\pi$
$240$	0	0
$241$	−1.49876e6	−1.66223	−0.831114	−	0.556102i	$-0.812296\pi$
$241$	−1.49876e6	−1.66223	−0.831114	+	0.556102i	$0.812296\pi$
$242$	0	0
$243$	−1.03408e6	−1.12341
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.46990e6	−2.57595
$248$	0	0
$249$	97353.0	0.0995064
$250$	0	0
$251$	966842.	0.968660	0.484330	−	0.874886i	$-0.339064\pi$
$251$	966842.	0.968660	0.484330	+	0.874886i	$0.339064\pi$
$252$	0	0
$253$	742531.	0.729312
$254$	0	0
$255$	633182.	0.609786
$256$	0	0
$257$	1.71485e6	1.61955	0.809774	−	0.586743i	$-0.199590\pi$
$257$	1.71485e6	1.61955	0.809774	+	0.586743i	$0.199590\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1.44108e6	1.30944
$262$	0	0
$263$	477988.	0.426116	0.213058	−	0.977040i	$-0.431658\pi$
$263$	477988.	0.426116	0.213058	+	0.977040i	$0.431658\pi$
$264$	0	0
$265$	−1.93133e6	−1.68944
$266$	0	0
$267$	1.31648e6	1.13015
$268$	0	0
$269$	−720098.	−0.606751	−0.303376	−	0.952871i	$-0.598114\pi$
$269$	−720098.	−0.606751	−0.303376	+	0.952871i	$0.598114\pi$
$270$	0	0
$271$	381619.	0.315651	0.157826	−	0.987467i	$-0.449552\pi$
$271$	381619.	0.315651	0.157826	+	0.987467i	$0.449552\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.74303e6	1.38987
$276$	0	0
$277$	722576.	0.565828	0.282914	−	0.959145i	$-0.408699\pi$
$277$	722576.	0.565828	0.282914	+	0.959145i	$0.408699\pi$
$278$	0	0
$279$	363812.	0.279812
$280$	0	0
$281$	−51828.2	−0.0391561	−0.0195781	−	0.999808i	$-0.506232\pi$
$281$	−51828.2	−0.0391561	−0.0195781	+	0.999808i	$0.506232\pi$
$282$	0	0
$283$	−62161.2	−0.0461374	−0.0230687	−	0.999734i	$-0.507344\pi$
$283$	−62161.2	−0.0461374	−0.0230687	+	0.999734i	$0.507344\pi$
$284$	0	0
$285$	6.11225e6	4.45748
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.32899e6	−0.936001
$290$	0	0
$291$	−1604.55	−0.00111076
$292$	0	0
$293$	2.00409e6	1.36379	0.681896	−	0.731449i	$-0.261156\pi$
$293$	2.00409e6	1.36379	0.681896	+	0.731449i	$0.261156\pi$
$294$	0	0
$295$	−938626.	−0.627968
$296$	0	0
$297$	1.42681e6	0.938589
$298$	0	0
$299$	1.45097e6	0.938598
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1.80234e6	1.12780
$304$	0	0
$305$	−1.87602e6	−1.15475
$306$	0	0
$307$	−2.11066e6	−1.27812	−0.639061	−	0.769156i	$-0.720677\pi$
$307$	−2.11066e6	−1.27812	−0.639061	+	0.769156i	$0.720677\pi$
$308$	0	0
$309$	1.96378e6	1.17003
$310$	0	0
$311$	−1.04343e6	−0.611733	−0.305867	−	0.952074i	$-0.598946\pi$
$311$	−1.04343e6	−0.611733	−0.305867	+	0.952074i	$0.598946\pi$
$312$	0	0
$313$	−919919.	−0.530748	−0.265374	−	0.964145i	$-0.585496\pi$
$313$	−919919.	−0.530748	−0.265374	+	0.964145i	$0.585496\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.95297e6	1.09156	0.545779	−	0.837929i	$-0.316234\pi$
$317$	1.95297e6	1.09156	0.545779	+	0.837929i	$0.316234\pi$
$318$	0	0
$319$	1.66867e6	0.918110
$320$	0	0
$321$	−3.14221e6	−1.70205
$322$	0	0
$323$	877181.	0.467825
$324$	0	0
$325$	3.40603e6	1.78871
$326$	0	0
$327$	4.46816e6	2.31078
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−3.85038e6	−1.93167	−0.965836	−	0.259155i	$-0.916556\pi$
$331$	−3.85038e6	−1.93167	−0.965836	+	0.259155i	$0.916556\pi$
$332$	0	0
$333$	940466.	0.464764
$334$	0	0
$335$	−609866.	−0.296908
$336$	0	0
$337$	3.34682e6	1.60531	0.802653	−	0.596447i	$-0.203421\pi$
$337$	3.34682e6	1.60531	0.802653	+	0.596447i	$0.203421\pi$
$338$	0	0
$339$	6.41508e6	3.03182
$340$	0	0
$341$	421270.	0.196189
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−3.59070e6	−1.62417
$346$	0	0
$347$	1.20659e6	0.537942	0.268971	−	0.963148i	$-0.413316\pi$
$347$	1.20659e6	0.537942	0.268971	+	0.963148i	$0.413316\pi$
$348$	0	0
$349$	398849.	0.175285	0.0876426	−	0.996152i	$-0.472067\pi$
$349$	398849.	0.175285	0.0876426	+	0.996152i	$0.472067\pi$
$350$	0	0
$351$	2.78811e6	1.20793
$352$	0	0
$353$	841946.	0.359623	0.179812	−	0.983701i	$-0.442451\pi$
$353$	841946.	0.359623	0.179812	+	0.983701i	$0.442451\pi$
$354$	0	0
$355$	3.89929e6	1.64216
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.06994e6	0.438151	0.219075	−	0.975708i	$-0.429696\pi$
$359$	1.06994e6	0.438151	0.219075	+	0.975708i	$0.429696\pi$
$360$	0	0
$361$	5.99154e6	2.41975
$362$	0	0
$363$	686747.	0.273546
$364$	0	0
$365$	−5.71809e6	−2.24657
$366$	0	0
$367$	−3.33520e6	−1.29258	−0.646290	−	0.763092i	$-0.723680\pi$
$367$	−3.33520e6	−1.29258	−0.646290	+	0.763092i	$0.723680\pi$
$368$	0	0
$369$	−4.98311e6	−1.90517
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−719121.	−0.267627	−0.133813	−	0.991007i	$-0.542722\pi$
$373$	−719121.	−0.267627	−0.133813	+	0.991007i	$0.542722\pi$
$374$	0	0
$375$	−1.86487e6	−0.684810
$376$	0	0
$377$	3.26073e6	1.18157
$378$	0	0
$379$	−1.33953e6	−0.479022	−0.239511	−	0.970894i	$-0.576987\pi$
$379$	−1.33953e6	−0.479022	−0.239511	+	0.970894i	$0.576987\pi$
$380$	0	0
$381$	1.77073e6	0.624942
$382$	0	0
$383$	−4.24875e6	−1.48001	−0.740005	−	0.672601i	$-0.765177\pi$
$383$	−4.24875e6	−1.48001	−0.740005	+	0.672601i	$0.765177\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.91893e6	−0.990710
$388$	0	0
$389$	−1.80973e6	−0.606374	−0.303187	−	0.952931i	$-0.598051\pi$
$389$	−1.80973e6	−0.606374	−0.303187	+	0.952931i	$0.598051\pi$
$390$	0	0
$391$	−515309.	−0.170461
$392$	0	0
$393$	−3.39871e6	−1.11002
$394$	0	0
$395$	−5.90973e6	−1.90579
$396$	0	0
$397$	3.94167e6	1.25517	0.627587	−	0.778546i	$-0.284043\pi$
$397$	3.94167e6	1.25517	0.627587	+	0.778546i	$0.284043\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	164541.	0.0510992	0.0255496	−	0.999674i	$-0.491866\pi$
$401$	164541.	0.0510992	0.0255496	+	0.999674i	$0.491866\pi$
$402$	0	0
$403$	823197.	0.252488
$404$	0	0
$405$	801536.	0.242820
$406$	0	0
$407$	1.08900e6	0.325867
$408$	0	0
$409$	−2.39073e6	−0.706680	−0.353340	−	0.935495i	$-0.614954\pi$
$409$	−2.39073e6	−0.706680	−0.353340	+	0.935495i	$0.614954\pi$
$410$	0	0
$411$	3.58494e6	1.04683
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−330823.	−0.0942920
$416$	0	0
$417$	5.41475e6	1.52489
$418$	0	0
$419$	3.45750e6	0.962114	0.481057	−	0.876689i	$-0.340253\pi$
$419$	3.45750e6	0.962114	0.481057	+	0.876689i	$0.340253\pi$
$420$	0	0
$421$	−3.69055e6	−1.01481	−0.507406	−	0.861707i	$-0.669396\pi$
$421$	−3.69055e6	−1.01481	−0.507406	+	0.861707i	$0.669396\pi$
$422$	0	0
$423$	2.05909e6	0.559532
$424$	0	0
$425$	−1.20965e6	−0.324853
$426$	0	0
$427$	0	0
$428$	0	0
$429$	9.16623e6	2.40463
$430$	0	0
$431$	5.26181e6	1.36440	0.682200	−	0.731166i	$-0.261023\pi$
$431$	5.26181e6	1.36440	0.682200	+	0.731166i	$0.261023\pi$
$432$	0	0
$433$	−5.01353e6	−1.28506	−0.642530	−	0.766260i	$-0.722115\pi$
$433$	−5.01353e6	−1.28506	−0.642530	+	0.766260i	$0.722115\pi$
$434$	0	0
$435$	−8.06930e6	−2.04462
$436$	0	0
$437$	−4.97440e6	−1.24605
$438$	0	0
$439$	−5.94160e6	−1.47144	−0.735720	−	0.677286i	$-0.763156\pi$
$439$	−5.94160e6	−1.47144	−0.735720	+	0.677286i	$0.763156\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.26333e6	0.305850	0.152925	−	0.988238i	$-0.451131\pi$
$443$	1.26333e6	0.305850	0.152925	+	0.988238i	$0.451131\pi$
$444$	0	0
$445$	−4.47362e6	−1.07092
$446$	0	0
$447$	7.08762e6	1.67777
$448$	0	0
$449$	6.37580e6	1.49252	0.746258	−	0.665657i	$-0.231849\pi$
$449$	6.37580e6	1.49252	0.746258	+	0.665657i	$0.231849\pi$
$450$	0	0
$451$	−5.77010e6	−1.33580
$452$	0	0
$453$	−2.51296e6	−0.575361
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.04066e6	−0.233087	−0.116544	−	0.993186i	$-0.537181\pi$
$457$	−1.04066e6	−0.233087	−0.116544	+	0.993186i	$0.537181\pi$
$458$	0	0
$459$	−990193.	−0.219375
$460$	0	0
$461$	−4.79707e6	−1.05129	−0.525647	−	0.850703i	$-0.676177\pi$
$461$	−4.79707e6	−1.05129	−0.525647	+	0.850703i	$0.676177\pi$
$462$	0	0
$463$	−1.28428e6	−0.278424	−0.139212	−	0.990263i	$-0.544457\pi$
$463$	−1.28428e6	−0.278424	−0.139212	+	0.990263i	$0.544457\pi$
$464$	0	0
$465$	−2.03716e6	−0.436911
$466$	0	0
$467$	1.08659e6	0.230554	0.115277	−	0.993333i	$-0.463224\pi$
$467$	1.08659e6	0.230554	0.115277	+	0.993333i	$0.463224\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	6.75961e6	1.40401
$472$	0	0
$473$	−3.37992e6	−0.694631
$474$	0	0
$475$	−1.16770e7	−2.37464
$476$	0	0
$477$	8.57524e6	1.72564
$478$	0	0
$479$	−1.29336e6	−0.257562	−0.128781	−	0.991673i	$-0.541106\pi$
$479$	−1.29336e6	−0.257562	−0.128781	+	0.991673i	$0.541106\pi$
$480$	0	0
$481$	2.12799e6	0.419379
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5452.56	0.00105256
$486$	0	0
$487$	−9.61626e6	−1.83732	−0.918658	−	0.395054i	$-0.870726\pi$
$487$	−9.61626e6	−1.83732	−0.918658	+	0.395054i	$0.870726\pi$
$488$	0	0
$489$	9.47682e6	1.79222
$490$	0	0
$491$	−7.37274e6	−1.38015	−0.690074	−	0.723739i	$-0.742422\pi$
$491$	−7.37274e6	−1.38015	−0.690074	+	0.723739i	$0.742422\pi$
$492$	0	0
$493$	−1.15804e6	−0.214589
$494$	0	0
$495$	−1.37661e7	−2.52521
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−5.42014e6	−0.974449	−0.487224	−	0.873277i	$-0.661991\pi$
$499$	−5.42014e6	−0.974449	−0.487224	+	0.873277i	$0.661991\pi$
$500$	0	0
$501$	−5.39342e6	−0.959998
$502$	0	0
$503$	−410131.	−0.0722774	−0.0361387	−	0.999347i	$-0.511506\pi$
$503$	−410131.	−0.0722774	−0.0361387	+	0.999347i	$0.511506\pi$
$504$	0	0
$505$	−6.12467e6	−1.06870
$506$	0	0
$507$	8.68047e6	1.49977
$508$	0	0
$509$	−5.12382e6	−0.876597	−0.438298	−	0.898830i	$-0.644419\pi$
$509$	−5.12382e6	−0.876597	−0.438298	+	0.898830i	$0.644419\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−9.55856e6	−1.60361
$514$	0	0
$515$	−6.67328e6	−1.10872
$516$	0	0
$517$	2.38429e6	0.392313
$518$	0	0
$519$	6.91667e6	1.12714
$520$	0	0
$521$	6.67402e6	1.07719	0.538596	−	0.842564i	$-0.318955\pi$
$521$	6.67402e6	1.07719	0.538596	+	0.842564i	$0.318955\pi$
$522$	0	0
$523$	−9.40416e6	−1.50337	−0.751685	−	0.659522i	$-0.770759\pi$
$523$	−9.40416e6	−1.50337	−0.751685	+	0.659522i	$0.770759\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−292357.	−0.0458550
$528$	0	0
$529$	−3.51408e6	−0.545975
$530$	0	0
$531$	4.16756e6	0.641425
$532$	0	0
$533$	−1.12753e7	−1.71913
$534$	0	0
$535$	1.06778e7	1.61286
$536$	0	0
$537$	−8.95436e6	−1.33998
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−4.60948e6	−0.677109	−0.338554	−	0.940947i	$-0.609938\pi$
$541$	−4.60948e6	−0.677109	−0.338554	+	0.940947i	$0.609938\pi$
$542$	0	0
$543$	−7.70403e6	−1.12129
$544$	0	0
$545$	−1.51836e7	−2.18969
$546$	0	0
$547$	31688.1	0.00452822	0.00226411	−	0.999997i	$-0.499279\pi$
$547$	31688.1	0.00452822	0.00226411	+	0.999997i	$0.499279\pi$
$548$	0	0
$549$	8.32965e6	1.17949
$550$	0	0
$551$	−1.11789e7	−1.56862
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−5.26612e6	−0.725702
$556$	0	0
$557$	−1.08620e6	−0.148345	−0.0741725	−	0.997245i	$-0.523632\pi$
$557$	−1.08620e6	−0.148345	−0.0741725	+	0.997245i	$0.523632\pi$
$558$	0	0
$559$	−6.60465e6	−0.893965
$560$	0	0
$561$	−3.25537e6	−0.436710
$562$	0	0
$563$	7.56924e6	1.00642	0.503212	−	0.864163i	$-0.332151\pi$
$563$	7.56924e6	1.00642	0.503212	+	0.864163i	$0.332151\pi$
$564$	0	0
$565$	−2.17996e7	−2.87294
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.45816e7	1.88809	0.944046	−	0.329814i	$-0.106986\pi$
$569$	1.45816e7	1.88809	0.944046	+	0.329814i	$0.106986\pi$
$570$	0	0
$571$	1.15216e7	1.47884	0.739419	−	0.673245i	$-0.235100\pi$
$571$	1.15216e7	1.47884	0.739419	+	0.673245i	$0.235100\pi$
$572$	0	0
$573$	2.38913e7	3.03986
$574$	0	0
$575$	6.85977e6	0.865247
$576$	0	0
$577$	−1.53853e7	−1.92383	−0.961916	−	0.273347i	$-0.911869\pi$
$577$	−1.53853e7	−1.92383	−0.961916	+	0.273347i	$0.911869\pi$
$578$	0	0
$579$	1.27544e7	1.58112
$580$	0	0
$581$	0	0
$582$	0	0
$583$	9.92954e6	1.20992
$584$	0	0
$585$	−2.69001e7	−3.24986
$586$	0	0
$587$	5.34856e6	0.640680	0.320340	−	0.947303i	$-0.396203\pi$
$587$	5.34856e6	0.640680	0.320340	+	0.947303i	$0.396203\pi$
$588$	0	0
$589$	−2.82219e6	−0.335195
$590$	0	0
$591$	−1.90973e7	−2.24907
$592$	0	0
$593$	5.19309e6	0.606441	0.303221	−	0.952920i	$-0.401938\pi$
$593$	5.19309e6	0.606441	0.303221	+	0.952920i	$0.401938\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.59094e7	−2.97524
$598$	0	0
$599$	1.12929e7	1.28599	0.642994	−	0.765871i	$-0.277692\pi$
$599$	1.12929e7	1.28599	0.642994	+	0.765871i	$0.277692\pi$
$600$	0	0
$601$	3.49678e6	0.394895	0.197447	−	0.980313i	$-0.436735\pi$
$601$	3.49678e6	0.394895	0.197447	+	0.980313i	$0.436735\pi$
$602$	0	0
$603$	2.70785e6	0.303271
$604$	0	0
$605$	−2.33369e6	−0.259211
$606$	0	0
$607$	8.43932e6	0.929685	0.464842	−	0.885393i	$-0.346111\pi$
$607$	8.43932e6	0.929685	0.464842	+	0.885393i	$0.346111\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.65910e6	0.504893
$612$	0	0
$613$	1.58823e7	1.70711	0.853557	−	0.520999i	$-0.174441\pi$
$613$	1.58823e7	1.70711	0.853557	+	0.520999i	$0.174441\pi$
$614$	0	0
$615$	2.79028e7	2.97482
$616$	0	0
$617$	−1.68397e6	−0.178082	−0.0890412	−	0.996028i	$-0.528380\pi$
$617$	−1.68397e6	−0.178082	−0.0890412	+	0.996028i	$0.528380\pi$
$618$	0	0
$619$	1.45879e7	1.53026	0.765131	−	0.643875i	$-0.222674\pi$
$619$	1.45879e7	1.53026	0.765131	+	0.643875i	$0.222674\pi$
$620$	0	0
$621$	5.61527e6	0.584308
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−6.20293e6	−0.635180
$626$	0	0
$627$	−3.14249e7	−3.19231
$628$	0	0
$629$	−755752.	−0.0761645
$630$	0	0
$631$	7.42899e6	0.742773	0.371387	−	0.928478i	$-0.378882\pi$
$631$	7.42899e6	0.742773	0.371387	+	0.928478i	$0.378882\pi$
$632$	0	0
$633$	2.02207e7	2.00580
$634$	0	0
$635$	−6.01725e6	−0.592194
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1.73131e7	−1.67735
$640$	0	0
$641$	1.05422e7	1.01342	0.506708	−	0.862118i	$-0.330862\pi$
$641$	1.05422e7	1.01342	0.506708	+	0.862118i	$0.330862\pi$
$642$	0	0
$643$	−8.53482e6	−0.814080	−0.407040	−	0.913410i	$-0.633439\pi$
$643$	−8.53482e6	−0.814080	−0.407040	+	0.913410i	$0.633439\pi$
$644$	0	0
$645$	1.63445e7	1.54694
$646$	0	0
$647$	−9.49608e6	−0.891834	−0.445917	−	0.895074i	$-0.647122\pi$
$647$	−9.49608e6	−0.891834	−0.445917	+	0.895074i	$0.647122\pi$
$648$	0	0
$649$	4.82575e6	0.449731
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.63052e6	0.608506	0.304253	−	0.952591i	$-0.401593\pi$
$653$	6.63052e6	0.608506	0.304253	+	0.952591i	$0.401593\pi$
$654$	0	0
$655$	1.15494e7	1.05186
$656$	0	0
$657$	2.53887e7	2.29471
$658$	0	0
$659$	−7.54914e6	−0.677149	−0.338574	−	0.940940i	$-0.609945\pi$
$659$	−7.54914e6	−0.677149	−0.338574	+	0.940940i	$0.609945\pi$
$660$	0	0
$661$	−1.86067e6	−0.165641	−0.0828203	−	0.996564i	$-0.526393\pi$
$661$	−1.86067e6	−0.165641	−0.0828203	+	0.996564i	$0.526393\pi$
$662$	0	0
$663$	−6.36127e6	−0.562030
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.56713e6	0.571559
$668$	0	0
$669$	5.93878e6	0.513017
$670$	0	0
$671$	9.64516e6	0.826996
$672$	0	0
$673$	−1.14956e6	−0.0978350	−0.0489175	−	0.998803i	$-0.515577\pi$
$673$	−1.14956e6	−0.0978350	−0.0489175	+	0.998803i	$0.515577\pi$
$674$	0	0
$675$	1.31814e7	1.11353
$676$	0	0
$677$	2.01579e6	0.169034	0.0845169	−	0.996422i	$-0.473065\pi$
$677$	2.01579e6	0.169034	0.0845169	+	0.996422i	$0.473065\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1.45502e7	1.20227
$682$	0	0
$683$	7.46429e6	0.612261	0.306131	−	0.951989i	$-0.400966\pi$
$683$	7.46429e6	0.612261	0.306131	+	0.951989i	$0.400966\pi$
$684$	0	0
$685$	−1.21822e7	−0.991975
$686$	0	0
$687$	−3.69317e7	−2.98543
$688$	0	0
$689$	1.94032e7	1.55713
$690$	0	0
$691$	−1.88570e7	−1.50237	−0.751187	−	0.660090i	$-0.770518\pi$
$691$	−1.88570e7	−1.50237	−0.751187	+	0.660090i	$0.770518\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.84003e7	−1.44498
$696$	0	0
$697$	4.00439e6	0.312216
$698$	0	0
$699$	−2.78631e7	−2.15693
$700$	0	0
$701$	−1.66789e7	−1.28195	−0.640976	−	0.767561i	$-0.721470\pi$
$701$	−1.66789e7	−1.28195	−0.640976	+	0.767561i	$0.721470\pi$
$702$	0	0
$703$	−7.29545e6	−0.556755
$704$	0	0
$705$	−1.15299e7	−0.873677
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1.59824e7	−1.19406	−0.597031	−	0.802218i	$-0.703653\pi$
$709$	−1.59824e7	−1.19406	−0.597031	+	0.802218i	$0.703653\pi$
$710$	0	0
$711$	2.62396e7	1.94663
$712$	0	0
$713$	1.65792e6	0.122135
$714$	0	0
$715$	−3.11485e7	−2.27862
$716$	0	0
$717$	−342027.	−0.0248463
$718$	0	0
$719$	1.17813e7	0.849909	0.424954	−	0.905215i	$-0.360290\pi$
$719$	1.17813e7	0.849909	0.424954	+	0.905215i	$0.360290\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−3.72623e7	−2.65109
$724$	0	0
$725$	1.54158e7	1.08923
$726$	0	0
$727$	1.26528e7	0.887870	0.443935	−	0.896059i	$-0.353582\pi$
$727$	1.26528e7	0.887870	0.443935	+	0.896059i	$0.353582\pi$
$728$	0	0
$729$	−2.34040e7	−1.63107
$730$	0	0
$731$	2.34563e6	0.162355
$732$	0	0
$733$	−5.34339e6	−0.367330	−0.183665	−	0.982989i	$-0.558796\pi$
$733$	−5.34339e6	−0.367330	−0.183665	+	0.982989i	$0.558796\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.13550e6	0.212637
$738$	0	0
$739$	4.22491e6	0.284581	0.142290	−	0.989825i	$-0.454553\pi$
$739$	4.22491e6	0.284581	0.142290	+	0.989825i	$0.454553\pi$
$740$	0	0
$741$	−6.14068e7	−4.10839
$742$	0	0
$743$	−1.70227e7	−1.13124	−0.565622	−	0.824665i	$-0.691364\pi$
$743$	−1.70227e7	−1.13124	−0.565622	+	0.824665i	$0.691364\pi$
$744$	0	0
$745$	−2.40850e7	−1.58985
$746$	0	0
$747$	1.46887e6	0.0963126
$748$	0	0
$749$	0	0
$750$	0	0
$751$	2.39433e6	0.154912	0.0774560	−	0.996996i	$-0.475320\pi$
$751$	2.39433e6	0.154912	0.0774560	+	0.996996i	$0.475320\pi$
$752$	0	0
$753$	2.40377e7	1.54492
$754$	0	0
$755$	8.53950e6	0.545211
$756$	0	0
$757$	3.82315e6	0.242483	0.121242	−	0.992623i	$-0.461312\pi$
$757$	3.82315e6	0.242483	0.121242	+	0.992623i	$0.461312\pi$
$758$	0	0
$759$	1.84608e7	1.16318
$760$	0	0
$761$	1.83640e7	1.14949	0.574745	−	0.818333i	$-0.305101\pi$
$761$	1.83640e7	1.14949	0.574745	+	0.818333i	$0.305101\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	9.55352e6	0.590215
$766$	0	0
$767$	9.42992e6	0.578788
$768$	0	0
$769$	1.24704e7	0.760441	0.380220	−	0.924896i	$-0.375848\pi$
$769$	1.24704e7	0.760441	0.380220	+	0.924896i	$0.375848\pi$
$770$	0	0
$771$	4.26347e7	2.58302
$772$	0	0
$773$	1.65412e7	0.995679	0.497840	−	0.867269i	$-0.334127\pi$
$773$	1.65412e7	0.995679	0.497840	+	0.867269i	$0.334127\pi$
$774$	0	0
$775$	3.89185e6	0.232756
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.86553e7	2.28226
$780$	0	0
$781$	−2.00474e7	−1.17606
$782$	0	0
$783$	1.26191e7	0.735569
$784$	0	0
$785$	−2.29704e7	−1.33043
$786$	0	0
$787$	−7.66364e6	−0.441061	−0.220530	−	0.975380i	$-0.570779\pi$
$787$	−7.66364e6	−0.441061	−0.220530	+	0.975380i	$0.570779\pi$
$788$	0	0
$789$	1.18838e7	0.679613
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1.88475e7	1.06431
$794$	0	0
$795$	−4.80169e7	−2.69449
$796$	0	0
$797$	−8.13707e6	−0.453756	−0.226878	−	0.973923i	$-0.572852\pi$
$797$	−8.13707e6	−0.453756	−0.226878	+	0.973923i	$0.572852\pi$
$798$	0	0
$799$	−1.65467e6	−0.0916949
$800$	0	0
$801$	1.98631e7	1.09387
$802$	0	0
$803$	2.93984e7	1.60892
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.79031e7	−0.967709
$808$	0	0
$809$	1.39535e7	0.749569	0.374784	−	0.927112i	$-0.377717\pi$
$809$	1.39535e7	0.749569	0.374784	+	0.927112i	$0.377717\pi$
$810$	0	0
$811$	−1.92110e7	−1.02564	−0.512822	−	0.858495i	$-0.671400\pi$
$811$	−1.92110e7	−1.02564	−0.512822	+	0.858495i	$0.671400\pi$
$812$	0	0
$813$	9.48784e6	0.503432
$814$	0	0
$815$	−3.22039e7	−1.69830
$816$	0	0
$817$	2.26429e7	1.18680
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3.37694e7	−1.74850	−0.874249	−	0.485477i	$-0.838646\pi$
$821$	−3.37694e7	−1.74850	−0.874249	+	0.485477i	$0.838646\pi$
$822$	0	0
$823$	1.27140e7	0.654311	0.327156	−	0.944971i	$-0.393910\pi$
$823$	1.27140e7	0.654311	0.327156	+	0.944971i	$0.393910\pi$
$824$	0	0
$825$	4.33354e7	2.21670
$826$	0	0
$827$	−2.85821e7	−1.45322	−0.726608	−	0.687053i	$-0.758904\pi$
$827$	−2.85821e7	−1.45322	−0.726608	+	0.687053i	$0.758904\pi$
$828$	0	0
$829$	−8.04571e6	−0.406610	−0.203305	−	0.979115i	$-0.565168\pi$
$829$	−8.04571e6	−0.406610	−0.203305	+	0.979115i	$0.565168\pi$
$830$	0	0
$831$	1.79647e7	0.902440
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.83278e7	0.909692
$836$	0	0
$837$	3.18579e6	0.157182
$838$	0	0
$839$	−4.26854e6	−0.209351	−0.104675	−	0.994506i	$-0.533380\pi$
$839$	−4.26854e6	−0.209351	−0.104675	+	0.994506i	$0.533380\pi$
$840$	0	0
$841$	−5.75299e6	−0.280481
$842$	0	0
$843$	−1.28855e6	−0.0624502
$844$	0	0
$845$	−2.94978e7	−1.42118
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.54546e6	−0.0735847
$850$	0	0
$851$	4.28578e6	0.202865
$852$	0	0
$853$	6.37238e6	0.299867	0.149934	−	0.988696i	$-0.452094\pi$
$853$	6.37238e6	0.299867	0.149934	+	0.988696i	$0.452094\pi$
$854$	0	0
$855$	9.22224e7	4.31441
$856$	0	0
$857$	−2.06675e7	−0.961251	−0.480625	−	0.876926i	$-0.659590\pi$
$857$	−2.06675e7	−0.961251	−0.480625	+	0.876926i	$0.659590\pi$
$858$	0	0
$859$	2.10547e7	0.973569	0.486784	−	0.873522i	$-0.338170\pi$
$859$	2.10547e7	0.973569	0.486784	+	0.873522i	$0.338170\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.07059e7	0.946384	0.473192	−	0.880959i	$-0.343102\pi$
$863$	2.07059e7	0.946384	0.473192	+	0.880959i	$0.343102\pi$
$864$	0	0
$865$	−2.35041e7	−1.06808
$866$	0	0
$867$	−3.30414e7	−1.49283
$868$	0	0
$869$	3.03836e7	1.36487
$870$	0	0
$871$	6.12703e6	0.273656
$872$	0	0
$873$	−24209.7	−0.00107511
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.47616e6	0.372135	0.186067	−	0.982537i	$-0.440426\pi$
$877$	8.47616e6	0.372135	0.186067	+	0.982537i	$0.440426\pi$
$878$	0	0
$879$	4.98258e7	2.17511
$880$	0	0
$881$	652358.	0.0283169	0.0141585	−	0.999900i	$-0.495493\pi$
$881$	652358.	0.0283169	0.0141585	+	0.999900i	$0.495493\pi$
$882$	0	0
$883$	−1.20036e7	−0.518097	−0.259049	−	0.965864i	$-0.583409\pi$
$883$	−1.20036e7	−0.518097	−0.259049	+	0.965864i	$0.583409\pi$
$884$	0	0
$885$	−2.33362e7	−1.00155
$886$	0	0
$887$	−3.03300e6	−0.129439	−0.0647193	−	0.997904i	$-0.520615\pi$
$887$	−3.03300e6	−0.129439	−0.0647193	+	0.997904i	$0.520615\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4.12093e6	−0.173900
$892$	0	0
$893$	−1.59729e7	−0.670280
$894$	0	0
$895$	3.04285e7	1.26976
$896$	0	0
$897$	3.60740e7	1.49697
$898$	0	0
$899$	3.72581e6	0.153752
$900$	0	0
$901$	−6.89100e6	−0.282794
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.61797e7	1.06253
$906$	0	0
$907$	−3.95629e7	−1.59687	−0.798436	−	0.602079i	$-0.794339\pi$
$907$	−3.95629e7	−1.59687	−0.798436	+	0.602079i	$0.794339\pi$
$908$	0	0
$909$	2.71939e7	1.09160
$910$	0	0
$911$	8.51152e6	0.339790	0.169895	−	0.985462i	$-0.445657\pi$
$911$	8.51152e6	0.339790	0.169895	+	0.985462i	$0.445657\pi$
$912$	0	0
$913$	1.70086e6	0.0675290
$914$	0	0
$915$	−4.66417e7	−1.84171
$916$	0	0
$917$	0	0
$918$	0	0
$919$	5.51885e6	0.215556	0.107778	−	0.994175i	$-0.465626\pi$
$919$	5.51885e6	0.215556	0.107778	+	0.994175i	$0.465626\pi$
$920$	0	0
$921$	−5.24754e7	−2.03848
$922$	0	0
$923$	−3.91743e7	−1.51355
$924$	0	0
$925$	1.00605e7	0.386605
$926$	0	0
$927$	2.96298e7	1.13248
$928$	0	0
$929$	−4.13526e7	−1.57204	−0.786019	−	0.618202i	$-0.787861\pi$
$929$	−4.13526e7	−1.57204	−0.786019	+	0.618202i	$0.787861\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−2.59418e7	−0.975655
$934$	0	0
$935$	1.10623e7	0.413826
$936$	0	0
$937$	1.56526e7	0.582421	0.291210	−	0.956659i	$-0.405942\pi$
$937$	1.56526e7	0.582421	0.291210	+	0.956659i	$0.405942\pi$
$938$	0	0
$939$	−2.28711e7	−0.846492
$940$	0	0
$941$	−3.34176e6	−0.123027	−0.0615137	−	0.998106i	$-0.519593\pi$
$941$	−3.34176e6	−0.123027	−0.0615137	+	0.998106i	$0.519593\pi$
$942$	0	0
$943$	−2.27085e7	−0.831588
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.50563e7	−0.545562	−0.272781	−	0.962076i	$-0.587943\pi$
$947$	−1.50563e7	−0.545562	−0.272781	+	0.962076i	$0.587943\pi$
$948$	0	0
$949$	5.74469e7	2.07062
$950$	0	0
$951$	4.85548e7	1.74093
$952$	0	0
$953$	1.17556e7	0.419289	0.209644	−	0.977778i	$-0.432769\pi$
$953$	1.17556e7	0.419289	0.209644	+	0.977778i	$0.432769\pi$
$954$	0	0
$955$	−8.11868e7	−2.88056
$956$	0	0
$957$	4.14866e7	1.46429
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2.76885e7	−0.967145
$962$	0	0
$963$	−4.74101e7	−1.64742
$964$	0	0
$965$	−4.33417e7	−1.49826
$966$	0	0
$967$	3.35850e7	1.15499	0.577496	−	0.816394i	$-0.304030\pi$
$967$	3.35850e7	1.15499	0.577496	+	0.816394i	$0.304030\pi$
$968$	0	0
$969$	2.18085e7	0.746134
$970$	0	0
$971$	−2.91560e7	−0.992384	−0.496192	−	0.868213i	$-0.665269\pi$
$971$	−2.91560e7	−0.992384	−0.496192	+	0.868213i	$0.665269\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	8.46810e7	2.85282
$976$	0	0
$977$	−3.52569e7	−1.18170	−0.590850	−	0.806782i	$-0.701207\pi$
$977$	−3.52569e7	−1.18170	−0.590850	+	0.806782i	$0.701207\pi$
$978$	0	0
$979$	2.30002e7	0.766963
$980$	0	0
$981$	6.74161e7	2.23661
$982$	0	0
$983$	3.61304e7	1.19258	0.596291	−	0.802768i	$-0.296640\pi$
$983$	3.61304e7	1.19258	0.596291	+	0.802768i	$0.296640\pi$
$984$	0	0
$985$	6.48961e7	2.13122
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.33018e7	−0.432434
$990$	0	0
$991$	2.32916e6	0.0753383	0.0376692	−	0.999290i	$-0.488007\pi$
$991$	2.32916e6	0.0753383	0.0376692	+	0.999290i	$0.488007\pi$
$992$	0	0
$993$	−9.57283e7	−3.08083
$994$	0	0
$995$	8.80449e7	2.81933
$996$	0	0
$997$	−3.06550e7	−0.976704	−0.488352	−	0.872647i	$-0.662402\pi$
$997$	−3.06550e7	−0.976704	−0.488352	+	0.872647i	$0.662402\pi$
$998$	0	0
$999$	8.23536e6	0.261077

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.bg.1.4		4
4.3	odd	2		98.6.a.i.1.1	✓	4
7.6	odd	2	inner	784.6.a.bg.1.1		4
12.11	even	2		882.6.a.bv.1.4		4
28.3	even	6		98.6.c.i.79.1		8
28.11	odd	6		98.6.c.i.79.4		8
28.19	even	6		98.6.c.i.67.1		8
28.23	odd	6		98.6.c.i.67.4		8
28.27	even	2		98.6.a.i.1.4	yes	4
84.83	odd	2		882.6.a.bv.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.6.a.i.1.1	✓	4	4.3	odd	2
98.6.a.i.1.4	yes	4	28.27	even	2
98.6.c.i.67.1		8	28.19	even	6
98.6.c.i.67.4		8	28.23	odd	6
98.6.c.i.79.1		8	28.3	even	6
98.6.c.i.79.4		8	28.11	odd	6
784.6.a.bg.1.1		4	7.6	odd	2	inner
784.6.a.bg.1.4		4	1.1	even	1	trivial
882.6.a.bv.1.1		4	84.83	odd	2
882.6.a.bv.1.4		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q+24.8621 q^{3} -84.4857 q^{5} +375.122 q^{9} +O(q^{10})\)	\(q+24.8621 q^{3} -84.4857 q^{5} +375.122 q^{9} +434.365 q^{11} +848.786 q^{13} -2100.49 q^{15} -301.445 q^{17} -2909.92 q^{19} +1709.46 q^{23} +4012.83 q^{25} +3284.82 q^{27} +3841.63 q^{29} +969.851 q^{31} +10799.2 q^{33} +2507.10 q^{37} +21102.6 q^{39} -13284.0 q^{41} -7781.29 q^{43} -31692.4 q^{45} +5489.13 q^{47} -7494.54 q^{51} +22859.9 q^{53} -36697.6 q^{55} -72346.6 q^{57} +11109.9 q^{59} +22205.2 q^{61} -71710.3 q^{65} +7218.58 q^{67} +42500.7 q^{69} -46153.3 q^{71} +67681.2 q^{73} +99767.1 q^{75} +69949.5 q^{79} -9487.24 q^{81} +3915.73 q^{83} +25467.8 q^{85} +95510.9 q^{87} +52951.2 q^{89} +24112.5 q^{93} +245847. q^{95} -64.5382 q^{97} +162940. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 712 q^{9}+O(q^{10}) \) 4 * q + 712 * q^9	\( 4 q + 712 q^{9} - 628 q^{11} - 5248 q^{15} - 2624 q^{23} + 4224 q^{25} + 17732 q^{29} + 2932 q^{37} + 41832 q^{39} - 9836 q^{43} + 51236 q^{51} + 1552 q^{53} - 104092 q^{57} - 142548 q^{65} - 13704 q^{67} + 56664 q^{71} + 218296 q^{79} - 127048 q^{81} - 103924 q^{85} - 88056 q^{93} + 396752 q^{95} + 354500 q^{99}+O(q^{100}) \) 4 * q + 712 * q^9 - 628 * q^11 - 5248 * q^15 - 2624 * q^23 + 4224 * q^25 + 17732 * q^29 + 2932 * q^37 + 41832 * q^39 - 9836 * q^43 + 51236 * q^51 + 1552 * q^53 - 104092 * q^57 - 142548 * q^65 - 13704 * q^67 + 56664 * q^71 + 218296 * q^79 - 127048 * q^81 - 103924 * q^85 - 88056 * q^93 + 396752 * q^95 + 354500 * q^99

Introduction

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