LMFDB - Embedded newform 784.6.a.be.1.2

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:	$\Q(\sqrt{86}, \sqrt{134})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 110x^{2} + 144 $ x^4 - 110*x^2 + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 392)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.15111$ 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)

$f(q)$	$=$	$q-2.30222 q^{3} -81.0956 q^{5} -237.700 q^{9} +O(q^{10})$	$q-2.30222 q^{3} -81.0956 q^{5} -237.700 q^{9} +732.099 q^{11} -919.678 q^{13} +186.700 q^{15} -1728.85 q^{17} -2443.34 q^{19} -2424.30 q^{23} +3451.50 q^{25} +1106.68 q^{27} +1010.00 q^{29} +6128.04 q^{31} -1685.45 q^{33} -1238.60 q^{37} +2117.30 q^{39} -15388.6 q^{41} -12980.9 q^{43} +19276.4 q^{45} -6557.64 q^{47} +3980.19 q^{51} -31549.8 q^{53} -59370.0 q^{55} +5625.11 q^{57} -21881.4 q^{59} -27003.0 q^{61} +74581.9 q^{65} -8264.07 q^{67} +5581.26 q^{69} -17131.8 q^{71} -9282.54 q^{73} -7946.10 q^{75} -39144.8 q^{79} +55213.2 q^{81} +108829. q^{83} +140202. q^{85} -2325.24 q^{87} +17842.6 q^{89} -14108.1 q^{93} +198144. q^{95} +26643.9 q^{97} -174020. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 92 q^{9}+O(q^{10}) $ 4 * q - 92 * q^9	$ 4 q - 92 q^{9} + 352 q^{11} - 112 q^{15} - 1968 q^{23} + 924 q^{25} + 4040 q^{29} + 10504 q^{37} + 9328 q^{39} - 28736 q^{43} - 23584 q^{51} - 28296 q^{53} + 62864 q^{57} + 146320 q^{65} + 72576 q^{67} - 176736 q^{71} - 135968 q^{79} - 27340 q^{81} + 298016 q^{85} + 395296 q^{93} + 367472 q^{95} - 561248 q^{99}+O(q^{100}) $ 4 * q - 92 * q^9 + 352 * q^11 - 112 * q^15 - 1968 * q^23 + 924 * q^25 + 4040 * q^29 + 10504 * q^37 + 9328 * q^39 - 28736 * q^43 - 23584 * q^51 - 28296 * q^53 + 62864 * q^57 + 146320 * q^65 + 72576 * q^67 - 176736 * q^71 - 135968 * q^79 - 27340 * q^81 + 298016 * q^85 + 395296 * q^93 + 367472 * q^95 - 561248 * 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
$3$	−2.30222	−0.147687	−0.0738437	−	0.997270i	$-0.523527\pi$
$3$	−2.30222	−0.147687	−0.0738437	+	0.997270i	$0.523527\pi$
$4$	0	0
$5$	−81.0956	−1.45068	−0.725341	−	0.688390i	$-0.758318\pi$
$5$	−81.0956	−1.45068	−0.725341	+	0.688390i	$0.758318\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−237.700	−0.978188
$10$	0	0
$11$	732.099	1.82427	0.912133	−	0.409894i	$-0.134434\pi$
$11$	732.099	1.82427	0.912133	+	0.409894i	$0.134434\pi$
$12$	0	0
$13$	−919.678	−1.50931	−0.754653	−	0.656124i	$-0.772195\pi$
$13$	−919.678	−1.50931	−0.754653	+	0.656124i	$0.772195\pi$
$14$	0	0
$15$	186.700	0.214247
$16$	0	0
$17$	−1728.85	−1.45089	−0.725446	−	0.688279i	$-0.758366\pi$
$17$	−1728.85	−1.45089	−0.725446	+	0.688279i	$0.758366\pi$
$18$	0	0
$19$	−2443.34	−1.55275	−0.776373	−	0.630274i	$-0.782943\pi$
$19$	−2443.34	−1.55275	−0.776373	+	0.630274i	$0.782943\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2424.30	−0.955579	−0.477789	−	0.878474i	$-0.658562\pi$
$23$	−2424.30	−0.955579	−0.477789	+	0.878474i	$0.658562\pi$
$24$	0	0
$25$	3451.50	1.10448
$26$	0	0
$27$	1106.68	0.292153
$28$	0	0
$29$	1010.00	0.223011	0.111506	−	0.993764i	$-0.464433\pi$
$29$	1010.00	0.223011	0.111506	+	0.993764i	$0.464433\pi$
$30$	0	0
$31$	6128.04	1.14530	0.572648	−	0.819802i	$-0.305916\pi$
$31$	6128.04	1.14530	0.572648	+	0.819802i	$0.305916\pi$
$32$	0	0
$33$	−1685.45	−0.269421
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1238.60	−0.148739	−0.0743696	−	0.997231i	$-0.523694\pi$
$37$	−1238.60	−0.148739	−0.0743696	+	0.997231i	$0.523694\pi$
$38$	0	0
$39$	2117.30	0.222906
$40$	0	0
$41$	−15388.6	−1.42968	−0.714841	−	0.699287i	$-0.753501\pi$
$41$	−15388.6	−1.42968	−0.714841	+	0.699287i	$0.753501\pi$
$42$	0	0
$43$	−12980.9	−1.07062	−0.535308	−	0.844657i	$-0.679804\pi$
$43$	−12980.9	−1.07062	−0.535308	+	0.844657i	$0.679804\pi$
$44$	0	0
$45$	19276.4	1.41904
$46$	0	0
$47$	−6557.64	−0.433015	−0.216507	−	0.976281i	$-0.569467\pi$
$47$	−6557.64	−0.433015	−0.216507	+	0.976281i	$0.569467\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	3980.19	0.214278
$52$	0	0
$53$	−31549.8	−1.54279	−0.771395	−	0.636357i	$-0.780441\pi$
$53$	−31549.8	−1.54279	−0.771395	+	0.636357i	$0.780441\pi$
$54$	0	0
$55$	−59370.0	−2.64643
$56$	0	0
$57$	5625.11	0.229321
$58$	0	0
$59$	−21881.4	−0.818362	−0.409181	−	0.912453i	$-0.634186\pi$
$59$	−21881.4	−0.818362	−0.409181	+	0.912453i	$0.634186\pi$
$60$	0	0
$61$	−27003.0	−0.929153	−0.464577	−	0.885533i	$-0.653793\pi$
$61$	−27003.0	−0.929153	−0.464577	+	0.885533i	$0.653793\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	74581.9	2.18952
$66$	0	0
$67$	−8264.07	−0.224909	−0.112455	−	0.993657i	$-0.535871\pi$
$67$	−8264.07	−0.224909	−0.112455	+	0.993657i	$0.535871\pi$
$68$	0	0
$69$	5581.26	0.141127
$70$	0	0
$71$	−17131.8	−0.403327	−0.201664	−	0.979455i	$-0.564635\pi$
$71$	−17131.8	−0.403327	−0.201664	+	0.979455i	$0.564635\pi$
$72$	0	0
$73$	−9282.54	−0.203873	−0.101937	−	0.994791i	$-0.532504\pi$
$73$	−9282.54	−0.203873	−0.101937	+	0.994791i	$0.532504\pi$
$74$	0	0
$75$	−7946.10	−0.163118
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−39144.8	−0.705678	−0.352839	−	0.935684i	$-0.614784\pi$
$79$	−39144.8	−0.705678	−0.352839	+	0.935684i	$0.614784\pi$
$80$	0	0
$81$	55213.2	0.935041
$82$	0	0
$83$	108829.	1.73399	0.866997	−	0.498313i	$-0.166047\pi$
$83$	108829.	1.73399	0.866997	+	0.498313i	$0.166047\pi$
$84$	0	0
$85$	140202.	2.10478
$86$	0	0
$87$	−2325.24	−0.0329359
$88$	0	0
$89$	17842.6	0.238772	0.119386	−	0.992848i	$-0.461907\pi$
$89$	17842.6	0.238772	0.119386	+	0.992848i	$0.461907\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−14108.1	−0.169146
$94$	0	0
$95$	198144.	2.25254
$96$	0	0
$97$	26643.9	0.287520	0.143760	−	0.989613i	$-0.454081\pi$
$97$	26643.9	0.287520	0.143760	+	0.989613i	$0.454081\pi$
$98$	0	0
$99$	−174020.	−1.78448
$100$	0	0
$101$	−125773.	−1.22683	−0.613416	−	0.789760i	$-0.710205\pi$
$101$	−125773.	−1.22683	−0.613416	+	0.789760i	$0.710205\pi$
$102$	0	0
$103$	42069.8	0.390731	0.195365	−	0.980731i	$-0.437411\pi$
$103$	42069.8	0.390731	0.195365	+	0.980731i	$0.437411\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−88464.9	−0.746984	−0.373492	−	0.927633i	$-0.621840\pi$
$107$	−88464.9	−0.746984	−0.373492	+	0.927633i	$0.621840\pi$
$108$	0	0
$109$	−56483.8	−0.455363	−0.227681	−	0.973736i	$-0.573114\pi$
$109$	−56483.8	−0.455363	−0.227681	+	0.973736i	$0.573114\pi$
$110$	0	0
$111$	2851.52	0.0219669
$112$	0	0
$113$	7649.01	0.0563520	0.0281760	−	0.999603i	$-0.491030\pi$
$113$	7649.01	0.0563520	0.0281760	+	0.999603i	$0.491030\pi$
$114$	0	0
$115$	196600.	1.38624
$116$	0	0
$117$	218607.	1.47639
$118$	0	0
$119$	0	0
$120$	0	0
$121$	374918.	2.32795
$122$	0	0
$123$	35427.9	0.211146
$124$	0	0
$125$	−26477.5	−0.151566
$126$	0	0
$127$	−239070.	−1.31527	−0.657636	−	0.753336i	$-0.728444\pi$
$127$	−239070.	−1.31527	−0.657636	+	0.753336i	$0.728444\pi$
$128$	0	0
$129$	29884.9	0.158116
$130$	0	0
$131$	195792.	0.996818	0.498409	−	0.866942i	$-0.333918\pi$
$131$	195792.	0.996818	0.498409	+	0.866942i	$0.333918\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−89746.6	−0.423822
$136$	0	0
$137$	−35505.9	−0.161622	−0.0808109	−	0.996729i	$-0.525751\pi$
$137$	−35505.9	−0.161622	−0.0808109	+	0.996729i	$0.525751\pi$
$138$	0	0
$139$	116321.	0.510646	0.255323	−	0.966856i	$-0.417818\pi$
$139$	116321.	0.510646	0.255323	+	0.966856i	$0.417818\pi$
$140$	0	0
$141$	15097.1	0.0639508
$142$	0	0
$143$	−673296.	−2.75338
$144$	0	0
$145$	−81906.6	−0.323518
$146$	0	0
$147$	0	0
$148$	0	0
$149$	441169.	1.62794	0.813971	−	0.580905i	$-0.197301\pi$
$149$	441169.	1.62794	0.813971	+	0.580905i	$0.197301\pi$
$150$	0	0
$151$	139896.	0.499302	0.249651	−	0.968336i	$-0.419684\pi$
$151$	139896.	0.499302	0.249651	+	0.968336i	$0.419684\pi$
$152$	0	0
$153$	410947.	1.41925
$154$	0	0
$155$	−496957.	−1.66146
$156$	0	0
$157$	224372.	0.726474	0.363237	−	0.931697i	$-0.381671\pi$
$157$	224372.	0.726474	0.363237	+	0.931697i	$0.381671\pi$
$158$	0	0
$159$	72634.5	0.227851
$160$	0	0
$161$	0	0
$162$	0	0
$163$	506920.	1.49441	0.747206	−	0.664592i	$-0.231395\pi$
$163$	506920.	1.49441	0.747206	+	0.664592i	$0.231395\pi$
$164$	0	0
$165$	136683.	0.390844
$166$	0	0
$167$	−200404.	−0.556053	−0.278026	−	0.960573i	$-0.589680\pi$
$167$	−200404.	−0.556053	−0.278026	+	0.960573i	$0.589680\pi$
$168$	0	0
$169$	474515.	1.27801
$170$	0	0
$171$	580782.	1.51888
$172$	0	0
$173$	−447074.	−1.13570	−0.567851	−	0.823131i	$-0.692225\pi$
$173$	−447074.	−1.13570	−0.567851	+	0.823131i	$0.692225\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	50375.8	0.120862
$178$	0	0
$179$	503571.	1.17470	0.587352	−	0.809332i	$-0.300170\pi$
$179$	503571.	1.17470	0.587352	+	0.809332i	$0.300170\pi$
$180$	0	0
$181$	−33323.4	−0.0756054	−0.0378027	−	0.999285i	$-0.512036\pi$
$181$	−33323.4	−0.0756054	−0.0378027	+	0.999285i	$0.512036\pi$
$182$	0	0
$183$	62166.8	0.137224
$184$	0	0
$185$	100445.	0.215773
$186$	0	0
$187$	−1.26569e6	−2.64681
$188$	0	0
$189$	0	0
$190$	0	0
$191$	241698.	0.479390	0.239695	−	0.970848i	$-0.422953\pi$
$191$	241698.	0.479390	0.239695	+	0.970848i	$0.422953\pi$
$192$	0	0
$193$	306901.	0.593068	0.296534	−	0.955022i	$-0.404169\pi$
$193$	306901.	0.593068	0.296534	+	0.955022i	$0.404169\pi$
$194$	0	0
$195$	−171704.	−0.323365
$196$	0	0
$197$	−335701.	−0.616293	−0.308147	−	0.951339i	$-0.599709\pi$
$197$	−335701.	−0.616293	−0.308147	+	0.951339i	$0.599709\pi$
$198$	0	0
$199$	−89682.8	−0.160538	−0.0802688	−	0.996773i	$-0.525578\pi$
$199$	−89682.8	−0.160538	−0.0802688	+	0.996773i	$0.525578\pi$
$200$	0	0
$201$	19025.7	0.0332162
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1.24795e6	2.07401
$206$	0	0
$207$	576255.	0.934736
$208$	0	0
$209$	−1.78877e6	−2.83262
$210$	0	0
$211$	883193.	1.36568	0.682841	−	0.730567i	$-0.260744\pi$
$211$	883193.	1.36568	0.682841	+	0.730567i	$0.260744\pi$
$212$	0	0
$213$	39441.2	0.0595664
$214$	0	0
$215$	1.05269e6	1.55312
$216$	0	0
$217$	0	0
$218$	0	0
$219$	21370.4	0.0301095
$220$	0	0
$221$	1.58999e6	2.18984
$222$	0	0
$223$	−550430.	−0.741208	−0.370604	−	0.928791i	$-0.620849\pi$
$223$	−550430.	−0.741208	−0.370604	+	0.928791i	$0.620849\pi$
$224$	0	0
$225$	−820420.	−1.08039
$226$	0	0
$227$	1.25332e6	1.61435	0.807173	−	0.590315i	$-0.200997\pi$
$227$	1.25332e6	1.61435	0.807173	+	0.590315i	$0.200997\pi$
$228$	0	0
$229$	357868.	0.450956	0.225478	−	0.974248i	$-0.427606\pi$
$229$	357868.	0.450956	0.225478	+	0.974248i	$0.427606\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−771152.	−0.930572	−0.465286	−	0.885160i	$-0.654049\pi$
$233$	−771152.	−0.930572	−0.465286	+	0.885160i	$0.654049\pi$
$234$	0	0
$235$	531795.	0.628167
$236$	0	0
$237$	90119.9	0.104220
$238$	0	0
$239$	−1.49577e6	−1.69384	−0.846918	−	0.531723i	$-0.821545\pi$
$239$	−1.49577e6	−1.69384	−0.846918	+	0.531723i	$0.821545\pi$
$240$	0	0
$241$	−891400.	−0.988622	−0.494311	−	0.869285i	$-0.664580\pi$
$241$	−891400.	−0.988622	−0.494311	+	0.869285i	$0.664580\pi$
$242$	0	0
$243$	−396035.	−0.430247
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.24709e6	2.34357
$248$	0	0
$249$	−250547.	−0.256089
$250$	0	0
$251$	1.36229e6	1.36485	0.682425	−	0.730956i	$-0.260926\pi$
$251$	1.36229e6	1.36485	0.682425	+	0.730956i	$0.260926\pi$
$252$	0	0
$253$	−1.77483e6	−1.74323
$254$	0	0
$255$	−322776.	−0.310850
$256$	0	0
$257$	−144356.	−0.136333	−0.0681667	−	0.997674i	$-0.521715\pi$
$257$	−144356.	−0.136333	−0.0681667	+	0.997674i	$0.521715\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−240077.	−0.218147
$262$	0	0
$263$	−827054.	−0.737300	−0.368650	−	0.929568i	$-0.620180\pi$
$263$	−827054.	−0.737300	−0.368650	+	0.929568i	$0.620180\pi$
$264$	0	0
$265$	2.55855e6	2.23810
$266$	0	0
$267$	−41077.7	−0.0352637
$268$	0	0
$269$	1.13644e6	0.957558	0.478779	−	0.877935i	$-0.341079\pi$
$269$	1.13644e6	0.957558	0.478779	+	0.877935i	$0.341079\pi$
$270$	0	0
$271$	−580417.	−0.480084	−0.240042	−	0.970763i	$-0.577161\pi$
$271$	−580417.	−0.480084	−0.240042	+	0.970763i	$0.577161\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.52684e6	2.01486
$276$	0	0
$277$	789917.	0.618560	0.309280	−	0.950971i	$-0.399912\pi$
$277$	789917.	0.618560	0.309280	+	0.950971i	$0.399912\pi$
$278$	0	0
$279$	−1.45663e6	−1.12031
$280$	0	0
$281$	−1.93626e6	−1.46284	−0.731421	−	0.681927i	$-0.761142\pi$
$281$	−1.93626e6	−1.46284	−0.731421	+	0.681927i	$0.761142\pi$
$282$	0	0
$283$	−2.05996e6	−1.52895	−0.764474	−	0.644654i	$-0.777001\pi$
$283$	−2.05996e6	−1.52895	−0.764474	+	0.644654i	$0.777001\pi$
$284$	0	0
$285$	−456172.	−0.332672
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.56907e6	1.10509
$290$	0	0
$291$	−61340.1	−0.0424631
$292$	0	0
$293$	5616.09	0.00382177	0.00191089	−	0.999998i	$-0.499392\pi$
$293$	5616.09	0.00382177	0.00191089	+	0.999998i	$0.499392\pi$
$294$	0	0
$295$	1.77449e6	1.18718
$296$	0	0
$297$	810197.	0.532966
$298$	0	0
$299$	2.22957e6	1.44226
$300$	0	0
$301$	0	0
$302$	0	0
$303$	289558.	0.181188
$304$	0	0
$305$	2.18982e6	1.34791
$306$	0	0
$307$	1.53244e6	0.927978	0.463989	−	0.885841i	$-0.346418\pi$
$307$	1.53244e6	0.927978	0.463989	+	0.885841i	$0.346418\pi$
$308$	0	0
$309$	−96853.9	−0.0577060
$310$	0	0
$311$	−755438.	−0.442892	−0.221446	−	0.975173i	$-0.571078\pi$
$311$	−755438.	−0.442892	−0.221446	+	0.975173i	$0.571078\pi$
$312$	0	0
$313$	−1.11217e6	−0.641670	−0.320835	−	0.947135i	$-0.603964\pi$
$313$	−1.11217e6	−0.641670	−0.320835	+	0.947135i	$0.603964\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−221462.	−0.123780	−0.0618901	−	0.998083i	$-0.519713\pi$
$317$	−221462.	−0.123780	−0.0618901	+	0.998083i	$0.519713\pi$
$318$	0	0
$319$	739420.	0.406832
$320$	0	0
$321$	203665.	0.110320
$322$	0	0
$323$	4.22417e6	2.25287
$324$	0	0
$325$	−3.17427e6	−1.66700
$326$	0	0
$327$	130038.	0.0672513
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.22590e6	0.615013	0.307507	−	0.951546i	$-0.400505\pi$
$331$	1.22590e6	0.615013	0.307507	+	0.951546i	$0.400505\pi$
$332$	0	0
$333$	294414.	0.145495
$334$	0	0
$335$	670180.	0.326272
$336$	0	0
$337$	−914368.	−0.438578	−0.219289	−	0.975660i	$-0.570374\pi$
$337$	−914368.	−0.438578	−0.219289	+	0.975660i	$0.570374\pi$
$338$	0	0
$339$	−17609.7	−0.00832247
$340$	0	0
$341$	4.48634e6	2.08932
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−452616.	−0.204730
$346$	0	0
$347$	2.18915e6	0.976005	0.488003	−	0.872842i	$-0.337726\pi$
$347$	2.18915e6	0.976005	0.488003	+	0.872842i	$0.337726\pi$
$348$	0	0
$349$	165745.	0.0728411	0.0364205	−	0.999337i	$-0.488404\pi$
$349$	165745.	0.0728411	0.0364205	+	0.999337i	$0.488404\pi$
$350$	0	0
$351$	−1.01779e6	−0.440949
$352$	0	0
$353$	582347.	0.248740	0.124370	−	0.992236i	$-0.460309\pi$
$353$	582347.	0.248740	0.124370	+	0.992236i	$0.460309\pi$
$354$	0	0
$355$	1.38932e6	0.585100
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.92164e6	0.786928	0.393464	−	0.919340i	$-0.371277\pi$
$359$	1.92164e6	0.786928	0.393464	+	0.919340i	$0.371277\pi$
$360$	0	0
$361$	3.49383e6	1.41102
$362$	0	0
$363$	−863144.	−0.343809
$364$	0	0
$365$	752774.	0.295755
$366$	0	0
$367$	1.89745e6	0.735370	0.367685	−	0.929950i	$-0.380150\pi$
$367$	1.89745e6	0.735370	0.367685	+	0.929950i	$0.380150\pi$
$368$	0	0
$369$	3.65787e6	1.39850
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−3.67351e6	−1.36713	−0.683564	−	0.729890i	$-0.739571\pi$
$373$	−3.67351e6	−1.36713	−0.683564	+	0.729890i	$0.739571\pi$
$374$	0	0
$375$	60956.9	0.0223844
$376$	0	0
$377$	−928875.	−0.336592
$378$	0	0
$379$	−2.94056e6	−1.05156	−0.525778	−	0.850622i	$-0.676226\pi$
$379$	−2.94056e6	−1.05156	−0.525778	+	0.850622i	$0.676226\pi$
$380$	0	0
$381$	550391.	0.194249
$382$	0	0
$383$	−3.24999e6	−1.13210	−0.566050	−	0.824371i	$-0.691529\pi$
$383$	−3.24999e6	−1.13210	−0.566050	+	0.824371i	$0.691529\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.08556e6	1.04726
$388$	0	0
$389$	−3.86718e6	−1.29575	−0.647873	−	0.761748i	$-0.724341\pi$
$389$	−3.86718e6	−1.29575	−0.647873	+	0.761748i	$0.724341\pi$
$390$	0	0
$391$	4.19125e6	1.38644
$392$	0	0
$393$	−450755.	−0.147217
$394$	0	0
$395$	3.17447e6	1.02371
$396$	0	0
$397$	−5.85455e6	−1.86431	−0.932153	−	0.362064i	$-0.882072\pi$
$397$	−5.85455e6	−1.86431	−0.932153	+	0.362064i	$0.882072\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.54533e6	1.41158	0.705789	−	0.708422i	$-0.250593\pi$
$401$	4.54533e6	1.41158	0.705789	+	0.708422i	$0.250593\pi$
$402$	0	0
$403$	−5.63583e6	−1.72860
$404$	0	0
$405$	−4.47755e6	−1.35645
$406$	0	0
$407$	−906776.	−0.271340
$408$	0	0
$409$	−2.31306e6	−0.683721	−0.341861	−	0.939751i	$-0.611057\pi$
$409$	−2.31306e6	−0.683721	−0.341861	+	0.939751i	$0.611057\pi$
$410$	0	0
$411$	81742.5	0.0238695
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−8.82552e6	−2.51547
$416$	0	0
$417$	−267796.	−0.0754160
$418$	0	0
$419$	−6.55278e6	−1.82344	−0.911718	−	0.410816i	$-0.865244\pi$
$419$	−6.55278e6	−1.82344	−0.911718	+	0.410816i	$0.865244\pi$
$420$	0	0
$421$	−2.62048e6	−0.720570	−0.360285	−	0.932842i	$-0.617321\pi$
$421$	−2.62048e6	−0.720570	−0.360285	+	0.932842i	$0.617321\pi$
$422$	0	0
$423$	1.55875e6	0.423570
$424$	0	0
$425$	−5.96712e6	−1.60248
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1.55007e6	0.406639
$430$	0	0
$431$	3.73182e6	0.967671	0.483836	−	0.875159i	$-0.339243\pi$
$431$	3.73182e6	0.967671	0.483836	+	0.875159i	$0.339243\pi$
$432$	0	0
$433$	−2.49129e6	−0.638564	−0.319282	−	0.947660i	$-0.603442\pi$
$433$	−2.49129e6	−0.638564	−0.319282	+	0.947660i	$0.603442\pi$
$434$	0	0
$435$	188567.	0.0477796
$436$	0	0
$437$	5.92339e6	1.48377
$438$	0	0
$439$	−5.13887e6	−1.27264	−0.636322	−	0.771424i	$-0.719545\pi$
$439$	−5.13887e6	−1.27264	−0.636322	+	0.771424i	$0.719545\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.04437e6	−0.737034	−0.368517	−	0.929621i	$-0.620134\pi$
$443$	−3.04437e6	−0.737034	−0.368517	+	0.929621i	$0.620134\pi$
$444$	0	0
$445$	−1.44696e6	−0.346383
$446$	0	0
$447$	−1.01567e6	−0.240427
$448$	0	0
$449$	2.03852e6	0.477199	0.238599	−	0.971118i	$-0.423312\pi$
$449$	2.03852e6	0.477199	0.238599	+	0.971118i	$0.423312\pi$
$450$	0	0
$451$	−1.12660e7	−2.60812
$452$	0	0
$453$	−322071.	−0.0737405
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.39673e6	−0.312841	−0.156420	−	0.987691i	$-0.549995\pi$
$457$	−1.39673e6	−0.312841	−0.156420	+	0.987691i	$0.549995\pi$
$458$	0	0
$459$	−1.91328e6	−0.423883
$460$	0	0
$461$	8.37595e6	1.83562	0.917808	−	0.397024i	$-0.129957\pi$
$461$	8.37595e6	1.83562	0.917808	+	0.397024i	$0.129957\pi$
$462$	0	0
$463$	−6.39026e6	−1.38537	−0.692685	−	0.721240i	$-0.743572\pi$
$463$	−6.39026e6	−1.38537	−0.692685	+	0.721240i	$0.743572\pi$
$464$	0	0
$465$	1.14410e6	0.245377
$466$	0	0
$467$	−2.86526e6	−0.607955	−0.303978	−	0.952679i	$-0.598315\pi$
$467$	−2.86526e6	−0.607955	−0.303978	+	0.952679i	$0.598315\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−516554.	−0.107291
$472$	0	0
$473$	−9.50330e6	−1.95309
$474$	0	0
$475$	−8.43319e6	−1.71498
$476$	0	0
$477$	7.49938e6	1.50914
$478$	0	0
$479$	5.39613e6	1.07459	0.537296	−	0.843394i	$-0.319446\pi$
$479$	5.39613e6	1.07459	0.537296	+	0.843394i	$0.319446\pi$
$480$	0	0
$481$	1.13911e6	0.224493
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.16070e6	−0.417101
$486$	0	0
$487$	4.10377e6	0.784081	0.392040	−	0.919948i	$-0.371769\pi$
$487$	4.10377e6	0.784081	0.392040	+	0.919948i	$0.371769\pi$
$488$	0	0
$489$	−1.16704e6	−0.220706
$490$	0	0
$491$	320967.	0.0600837	0.0300418	−	0.999549i	$-0.490436\pi$
$491$	320967.	0.0600837	0.0300418	+	0.999549i	$0.490436\pi$
$492$	0	0
$493$	−1.74614e6	−0.323565
$494$	0	0
$495$	1.41122e7	2.58871
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−4.87878e6	−0.877123	−0.438561	−	0.898701i	$-0.644512\pi$
$499$	−4.87878e6	−0.877123	−0.438561	+	0.898701i	$0.644512\pi$
$500$	0	0
$501$	461374.	0.0821219
$502$	0	0
$503$	5.03631e6	0.887548	0.443774	−	0.896139i	$-0.353639\pi$
$503$	5.03631e6	0.887548	0.443774	+	0.896139i	$0.353639\pi$
$504$	0	0
$505$	1.01997e7	1.77974
$506$	0	0
$507$	−1.09244e6	−0.188746
$508$	0	0
$509$	4.54564e6	0.777680	0.388840	−	0.921305i	$-0.372876\pi$
$509$	4.54564e6	0.777680	0.388840	+	0.921305i	$0.372876\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2.70399e6	−0.453640
$514$	0	0
$515$	−3.41168e6	−0.566826
$516$	0	0
$517$	−4.80084e6	−0.789934
$518$	0	0
$519$	1.02926e6	0.167729
$520$	0	0
$521$	−2.89492e6	−0.467242	−0.233621	−	0.972328i	$-0.575057\pi$
$521$	−2.89492e6	−0.467242	−0.233621	+	0.972328i	$0.575057\pi$
$522$	0	0
$523$	−2.14411e6	−0.342762	−0.171381	−	0.985205i	$-0.554823\pi$
$523$	−2.14411e6	−0.342762	−0.171381	+	0.985205i	$0.554823\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.05945e7	−1.66170
$528$	0	0
$529$	−559122.	−0.0868695
$530$	0	0
$531$	5.20121e6	0.800512
$532$	0	0
$533$	1.41526e7	2.15783
$534$	0	0
$535$	7.17411e6	1.08364
$536$	0	0
$537$	−1.15933e6	−0.173489
$538$	0	0
$539$	0	0
$540$	0	0
$541$	1.14490e7	1.68180	0.840901	−	0.541189i	$-0.182026\pi$
$541$	1.14490e7	1.68180	0.840901	+	0.541189i	$0.182026\pi$
$542$	0	0
$543$	76717.7	0.0111660
$544$	0	0
$545$	4.58058e6	0.660586
$546$	0	0
$547$	6.26051e6	0.894626	0.447313	−	0.894377i	$-0.352381\pi$
$547$	6.26051e6	0.894626	0.447313	+	0.894377i	$0.352381\pi$
$548$	0	0
$549$	6.41861e6	0.908887
$550$	0	0
$551$	−2.46778e6	−0.346280
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−231246.	−0.0318670
$556$	0	0
$557$	3.29773e6	0.450378	0.225189	−	0.974315i	$-0.427700\pi$
$557$	3.29773e6	0.450378	0.225189	+	0.974315i	$0.427700\pi$
$558$	0	0
$559$	1.19382e7	1.61589
$560$	0	0
$561$	2.91389e6	0.390901
$562$	0	0
$563$	3.69396e6	0.491158	0.245579	−	0.969377i	$-0.421022\pi$
$563$	3.69396e6	0.491158	0.245579	+	0.969377i	$0.421022\pi$
$564$	0	0
$565$	−620301.	−0.0817488
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.08934e7	−1.41053	−0.705267	−	0.708942i	$-0.749173\pi$
$569$	−1.08934e7	−1.41053	−0.705267	+	0.708942i	$0.749173\pi$
$570$	0	0
$571$	−6.07769e6	−0.780096	−0.390048	−	0.920795i	$-0.627542\pi$
$571$	−6.07769e6	−0.780096	−0.390048	+	0.920795i	$0.627542\pi$
$572$	0	0
$573$	−556441.	−0.0707998
$574$	0	0
$575$	−8.36746e6	−1.05542
$576$	0	0
$577$	2.98523e6	0.373283	0.186641	−	0.982428i	$-0.440240\pi$
$577$	2.98523e6	0.373283	0.186641	+	0.982428i	$0.440240\pi$
$578$	0	0
$579$	−706553.	−0.0875887
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−2.30976e7	−2.81446
$584$	0	0
$585$	−1.77281e7	−2.14177
$586$	0	0
$587$	201262.	0.0241083	0.0120541	−	0.999927i	$-0.496163\pi$
$587$	201262.	0.0241083	0.0120541	+	0.999927i	$0.496163\pi$
$588$	0	0
$589$	−1.49729e7	−1.77835
$590$	0	0
$591$	772858.	0.0910188
$592$	0	0
$593$	1.62091e7	1.89288	0.946441	−	0.322877i	$-0.104650\pi$
$593$	1.62091e7	1.89288	0.946441	+	0.322877i	$0.104650\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	206469.	0.0237094
$598$	0	0
$599$	−5.05095e6	−0.575183	−0.287591	−	0.957753i	$-0.592855\pi$
$599$	−5.05095e6	−0.575183	−0.287591	+	0.957753i	$0.592855\pi$
$600$	0	0
$601$	6.01246e6	0.678994	0.339497	−	0.940607i	$-0.389743\pi$
$601$	6.01246e6	0.678994	0.339497	+	0.940607i	$0.389743\pi$
$602$	0	0
$603$	1.96437e6	0.220004
$604$	0	0
$605$	−3.04042e7	−3.37711
$606$	0	0
$607$	−9.55760e6	−1.05288	−0.526438	−	0.850214i	$-0.676473\pi$
$607$	−9.55760e6	−1.05288	−0.526438	+	0.850214i	$0.676473\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.03092e6	0.653552
$612$	0	0
$613$	−1.30424e7	−1.40187	−0.700934	−	0.713226i	$-0.747233\pi$
$613$	−1.30424e7	−1.40187	−0.700934	+	0.713226i	$0.747233\pi$
$614$	0	0
$615$	−2.87305e6	−0.306306
$616$	0	0
$617$	3.67329e6	0.388456	0.194228	−	0.980956i	$-0.437780\pi$
$617$	3.67329e6	0.388456	0.194228	+	0.980956i	$0.437780\pi$
$618$	0	0
$619$	−1.11671e7	−1.17142	−0.585712	−	0.810519i	$-0.699185\pi$
$619$	−1.11671e7	−1.17142	−0.585712	+	0.810519i	$0.699185\pi$
$620$	0	0
$621$	−2.68291e6	−0.279176
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−8.63872e6	−0.884605
$626$	0	0
$627$	4.11814e6	0.418343
$628$	0	0
$629$	2.14135e6	0.215804
$630$	0	0
$631$	−1.52709e7	−1.52684	−0.763418	−	0.645904i	$-0.776480\pi$
$631$	−1.52709e7	−1.52684	−0.763418	+	0.645904i	$0.776480\pi$
$632$	0	0
$633$	−2.03330e6	−0.201694
$634$	0	0
$635$	1.93875e7	1.90804
$636$	0	0
$637$	0	0
$638$	0	0
$639$	4.07223e6	0.394530
$640$	0	0
$641$	−3.78973e6	−0.364303	−0.182152	−	0.983270i	$-0.558306\pi$
$641$	−3.78973e6	−0.364303	−0.182152	+	0.983270i	$0.558306\pi$
$642$	0	0
$643$	1.15155e7	1.09838	0.549192	−	0.835697i	$-0.314936\pi$
$643$	1.15155e7	1.09838	0.549192	+	0.835697i	$0.314936\pi$
$644$	0	0
$645$	−2.42353e6	−0.229377
$646$	0	0
$647$	−1.75894e7	−1.65193	−0.825963	−	0.563724i	$-0.809368\pi$
$647$	−1.75894e7	−1.65193	−0.825963	+	0.563724i	$0.809368\pi$
$648$	0	0
$649$	−1.60194e7	−1.49291
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.79853e7	−1.65057	−0.825286	−	0.564715i	$-0.808986\pi$
$653$	−1.79853e7	−1.65057	−0.825286	+	0.564715i	$0.808986\pi$
$654$	0	0
$655$	−1.58778e7	−1.44607
$656$	0	0
$657$	2.20646e6	0.199426
$658$	0	0
$659$	1.27198e7	1.14095	0.570476	−	0.821314i	$-0.306759\pi$
$659$	1.27198e7	1.14095	0.570476	+	0.821314i	$0.306759\pi$
$660$	0	0
$661$	2.00171e7	1.78196	0.890979	−	0.454045i	$-0.150020\pi$
$661$	2.00171e7	1.78196	0.890979	+	0.454045i	$0.150020\pi$
$662$	0	0
$663$	−3.66049e6	−0.323412
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.44854e6	−0.213105
$668$	0	0
$669$	1.26721e6	0.109467
$670$	0	0
$671$	−1.97689e7	−1.69502
$672$	0	0
$673$	869505.	0.0740005	0.0370002	−	0.999315i	$-0.488220\pi$
$673$	869505.	0.0740005	0.0370002	+	0.999315i	$0.488220\pi$
$674$	0	0
$675$	3.81969e6	0.322677
$676$	0	0
$677$	1.00374e7	0.841689	0.420845	−	0.907133i	$-0.361734\pi$
$677$	1.00374e7	0.841689	0.420845	+	0.907133i	$0.361734\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−2.88541e6	−0.238418
$682$	0	0
$683$	−1.04666e7	−0.858525	−0.429263	−	0.903180i	$-0.641227\pi$
$683$	−1.04666e7	−0.858525	−0.429263	+	0.903180i	$0.641227\pi$
$684$	0	0
$685$	2.87938e6	0.234462
$686$	0	0
$687$	−823890.	−0.0666005
$688$	0	0
$689$	2.90156e7	2.32854
$690$	0	0
$691$	7.68589e6	0.612349	0.306175	−	0.951975i	$-0.400951\pi$
$691$	7.68589e6	0.612349	0.306175	+	0.951975i	$0.400951\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.43310e6	−0.740785
$696$	0	0
$697$	2.66046e7	2.07431
$698$	0	0
$699$	1.77536e6	0.137434
$700$	0	0
$701$	7.25870e6	0.557910	0.278955	−	0.960304i	$-0.410012\pi$
$701$	7.25870e6	0.557910	0.278955	+	0.960304i	$0.410012\pi$
$702$	0	0
$703$	3.02632e6	0.230954
$704$	0	0
$705$	−1.22431e6	−0.0927723
$706$	0	0
$707$	0	0
$708$	0	0
$709$	2.86619e6	0.214136	0.107068	−	0.994252i	$-0.465854\pi$
$709$	2.86619e6	0.214136	0.107068	+	0.994252i	$0.465854\pi$
$710$	0	0
$711$	9.30471e6	0.690286
$712$	0	0
$713$	−1.48562e7	−1.09442
$714$	0	0
$715$	5.46013e7	3.99428
$716$	0	0
$717$	3.44360e6	0.250158
$718$	0	0
$719$	−1.16044e7	−0.837147	−0.418574	−	0.908183i	$-0.637470\pi$
$719$	−1.16044e7	−0.837147	−0.418574	+	0.908183i	$0.637470\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	2.05220e6	0.146007
$724$	0	0
$725$	3.48601e6	0.246311
$726$	0	0
$727$	−2.19891e7	−1.54302	−0.771509	−	0.636218i	$-0.780498\pi$
$727$	−2.19891e7	−1.54302	−0.771509	+	0.636218i	$0.780498\pi$
$728$	0	0
$729$	−1.25051e7	−0.871499
$730$	0	0
$731$	2.24420e7	1.55335
$732$	0	0
$733$	−1.49448e7	−1.02738	−0.513689	−	0.857976i	$-0.671722\pi$
$733$	−1.49448e7	−1.02738	−0.513689	+	0.857976i	$0.671722\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.05012e6	−0.410294
$738$	0	0
$739$	−1.33592e7	−0.899850	−0.449925	−	0.893066i	$-0.648549\pi$
$739$	−1.33592e7	−0.899850	−0.449925	+	0.893066i	$0.648549\pi$
$740$	0	0
$741$	−5.17329e6	−0.346116
$742$	0	0
$743$	−7.12788e6	−0.473683	−0.236842	−	0.971548i	$-0.576112\pi$
$743$	−7.12788e6	−0.473683	−0.236842	+	0.971548i	$0.576112\pi$
$744$	0	0
$745$	−3.57768e7	−2.36163
$746$	0	0
$747$	−2.58685e7	−1.69617
$748$	0	0
$749$	0	0
$750$	0	0
$751$	7.67411e6	0.496510	0.248255	−	0.968695i	$-0.420143\pi$
$751$	7.67411e6	0.496510	0.248255	+	0.968695i	$0.420143\pi$
$752$	0	0
$753$	−3.13629e6	−0.201571
$754$	0	0
$755$	−1.13450e7	−0.724328
$756$	0	0
$757$	−4.39278e6	−0.278612	−0.139306	−	0.990249i	$-0.544487\pi$
$757$	−4.39278e6	−0.278612	−0.139306	+	0.990249i	$0.544487\pi$
$758$	0	0
$759$	4.08604e6	0.257453
$760$	0	0
$761$	−6.78783e6	−0.424883	−0.212441	−	0.977174i	$-0.568141\pi$
$761$	−6.78783e6	−0.424883	−0.212441	+	0.977174i	$0.568141\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−3.33260e7	−2.05887
$766$	0	0
$767$	2.01239e7	1.23516
$768$	0	0
$769$	−8.12821e6	−0.495655	−0.247827	−	0.968804i	$-0.579717\pi$
$769$	−8.12821e6	−0.495655	−0.247827	+	0.968804i	$0.579717\pi$
$770$	0	0
$771$	332339.	0.0201347
$772$	0	0
$773$	1.55236e7	0.934422	0.467211	−	0.884146i	$-0.345259\pi$
$773$	1.55236e7	0.934422	0.467211	+	0.884146i	$0.345259\pi$
$774$	0	0
$775$	2.11509e7	1.26495
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.75996e7	2.21993
$780$	0	0
$781$	−1.25422e7	−0.735777
$782$	0	0
$783$	1.11774e6	0.0651535
$784$	0	0
$785$	−1.81956e7	−1.05388
$786$	0	0
$787$	1.87463e7	1.07889	0.539446	−	0.842020i	$-0.318634\pi$
$787$	1.87463e7	1.07889	0.539446	+	0.842020i	$0.318634\pi$
$788$	0	0
$789$	1.90406e6	0.108890
$790$	0	0
$791$	0	0
$792$	0	0
$793$	2.48341e7	1.40238
$794$	0	0
$795$	−5.89034e6	−0.330539
$796$	0	0
$797$	−1.85929e7	−1.03682	−0.518408	−	0.855133i	$-0.673475\pi$
$797$	−1.85929e7	−1.03682	−0.518408	+	0.855133i	$0.673475\pi$
$798$	0	0
$799$	1.13372e7	0.628258
$800$	0	0
$801$	−4.24119e6	−0.233564
$802$	0	0
$803$	−6.79575e6	−0.371919
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.61633e6	−0.141419
$808$	0	0
$809$	−4.49558e6	−0.241498	−0.120749	−	0.992683i	$-0.538530\pi$
$809$	−4.49558e6	−0.241498	−0.120749	+	0.992683i	$0.538530\pi$
$810$	0	0
$811$	2.75509e7	1.47090	0.735452	−	0.677577i	$-0.236970\pi$
$811$	2.75509e7	1.47090	0.735452	+	0.677577i	$0.236970\pi$
$812$	0	0
$813$	1.33625e6	0.0709023
$814$	0	0
$815$	−4.11090e7	−2.16792
$816$	0	0
$817$	3.17168e7	1.66239
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.25426e7	−1.16720	−0.583602	−	0.812040i	$-0.698357\pi$
$821$	−2.25426e7	−1.16720	−0.583602	+	0.812040i	$0.698357\pi$
$822$	0	0
$823$	−2.70610e7	−1.39266	−0.696329	−	0.717723i	$-0.745184\pi$
$823$	−2.70610e7	−1.39266	−0.696329	+	0.717723i	$0.745184\pi$
$824$	0	0
$825$	−5.81733e6	−0.297570
$826$	0	0
$827$	2.90437e7	1.47668	0.738342	−	0.674426i	$-0.235609\pi$
$827$	2.90437e7	1.47668	0.738342	+	0.674426i	$0.235609\pi$
$828$	0	0
$829$	−2.64956e7	−1.33902	−0.669511	−	0.742802i	$-0.733497\pi$
$829$	−2.64956e7	−1.33902	−0.669511	+	0.742802i	$0.733497\pi$
$830$	0	0
$831$	−1.81856e6	−0.0913535
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.62519e7	0.806656
$836$	0	0
$837$	6.78176e6	0.334602
$838$	0	0
$839$	2.65076e7	1.30007	0.650034	−	0.759905i	$-0.274755\pi$
$839$	2.65076e7	1.30007	0.650034	+	0.759905i	$0.274755\pi$
$840$	0	0
$841$	−1.94910e7	−0.950266
$842$	0	0
$843$	4.45769e6	0.216043
$844$	0	0
$845$	−3.84811e7	−1.85398
$846$	0	0
$847$	0	0
$848$	0	0
$849$	4.74248e6	0.225806
$850$	0	0
$851$	3.00273e6	0.142132
$852$	0	0
$853$	2.78106e7	1.30869	0.654347	−	0.756194i	$-0.272943\pi$
$853$	2.78106e7	1.30869	0.654347	+	0.756194i	$0.272943\pi$
$854$	0	0
$855$	−4.70989e7	−2.20341
$856$	0	0
$857$	−1.92803e7	−0.896732	−0.448366	−	0.893850i	$-0.647994\pi$
$857$	−1.92803e7	−0.896732	−0.448366	+	0.893850i	$0.647994\pi$
$858$	0	0
$859$	−2.43578e7	−1.12630	−0.563150	−	0.826354i	$-0.690411\pi$
$859$	−2.43578e7	−1.12630	−0.563150	+	0.826354i	$0.690411\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−6.03724e6	−0.275938	−0.137969	−	0.990437i	$-0.544057\pi$
$863$	−6.03724e6	−0.275938	−0.137969	+	0.990437i	$0.544057\pi$
$864$	0	0
$865$	3.62557e7	1.64754
$866$	0	0
$867$	−3.61233e6	−0.163207
$868$	0	0
$869$	−2.86579e7	−1.28734
$870$	0	0
$871$	7.60029e6	0.339457
$872$	0	0
$873$	−6.33325e6	−0.281249
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.04735e7	1.33790	0.668949	−	0.743309i	$-0.266745\pi$
$877$	3.04735e7	1.33790	0.668949	+	0.743309i	$0.266745\pi$
$878$	0	0
$879$	−12929.5	−0.000564428 0
$880$	0	0
$881$	398779.	0.0173098	0.00865491	−	0.999963i	$-0.497245\pi$
$881$	398779.	0.0173098	0.00865491	+	0.999963i	$0.497245\pi$
$882$	0	0
$883$	−1.86675e6	−0.0805721	−0.0402860	−	0.999188i	$-0.512827\pi$
$883$	−1.86675e6	−0.0805721	−0.0402860	+	0.999188i	$0.512827\pi$
$884$	0	0
$885$	−4.08526e6	−0.175332
$886$	0	0
$887$	4.00861e7	1.71074	0.855371	−	0.518015i	$-0.173329\pi$
$887$	4.00861e7	1.71074	0.855371	+	0.518015i	$0.173329\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	4.04216e7	1.70576
$892$	0	0
$893$	1.60226e7	0.672362
$894$	0	0
$895$	−4.08374e7	−1.70412
$896$	0	0
$897$	−5.13297e6	−0.213004
$898$	0	0
$899$	6.18932e6	0.255413
$900$	0	0
$901$	5.45448e7	2.23842
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.70238e6	0.109679
$906$	0	0
$907$	−3.24212e7	−1.30861	−0.654307	−	0.756229i	$-0.727040\pi$
$907$	−3.24212e7	−1.30861	−0.654307	+	0.756229i	$0.727040\pi$
$908$	0	0
$909$	2.98963e7	1.20007
$910$	0	0
$911$	−1.27656e7	−0.509620	−0.254810	−	0.966991i	$-0.582013\pi$
$911$	−1.27656e7	−0.509620	−0.254810	+	0.966991i	$0.582013\pi$
$912$	0	0
$913$	7.96733e7	3.16327
$914$	0	0
$915$	−5.04145e6	−0.199069
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−4.41131e7	−1.72297	−0.861487	−	0.507780i	$-0.830466\pi$
$919$	−4.41131e7	−1.72297	−0.861487	+	0.507780i	$0.830466\pi$
$920$	0	0
$921$	−3.52801e6	−0.137051
$922$	0	0
$923$	1.57558e7	0.608745
$924$	0	0
$925$	−4.27501e6	−0.164279
$926$	0	0
$927$	−9.99998e6	−0.382208
$928$	0	0
$929$	−1.66127e7	−0.631541	−0.315770	−	0.948836i	$-0.602263\pi$
$929$	−1.66127e7	−0.631541	−0.315770	+	0.948836i	$0.602263\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1.73918e6	0.0654096
$934$	0	0
$935$	1.02642e8	3.83969
$936$	0	0
$937$	4.38174e7	1.63041	0.815207	−	0.579170i	$-0.196623\pi$
$937$	4.38174e7	1.63041	0.815207	+	0.579170i	$0.196623\pi$
$938$	0	0
$939$	2.56047e6	0.0947666
$940$	0	0
$941$	1.59498e7	0.587195	0.293597	−	0.955929i	$-0.405147\pi$
$941$	1.59498e7	0.587195	0.293597	+	0.955929i	$0.405147\pi$
$942$	0	0
$943$	3.73065e7	1.36617
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.61841e7	0.948774	0.474387	−	0.880316i	$-0.342670\pi$
$947$	2.61841e7	0.948774	0.474387	+	0.880316i	$0.342670\pi$
$948$	0	0
$949$	8.53695e6	0.307707
$950$	0	0
$951$	509854.	0.0182808
$952$	0	0
$953$	9.43052e6	0.336359	0.168180	−	0.985756i	$-0.446211\pi$
$953$	9.43052e6	0.336359	0.168180	+	0.985756i	$0.446211\pi$
$954$	0	0
$955$	−1.96006e7	−0.695442
$956$	0	0
$957$	−1.70231e6	−0.0600839
$958$	0	0
$959$	0	0
$960$	0	0
$961$	8.92374e6	0.311701
$962$	0	0
$963$	2.10281e7	0.730691
$964$	0	0
$965$	−2.48883e7	−0.860354
$966$	0	0
$967$	−1.95159e7	−0.671155	−0.335578	−	0.942013i	$-0.608931\pi$
$967$	−1.95159e7	−0.671155	−0.335578	+	0.942013i	$0.608931\pi$
$968$	0	0
$969$	−9.72497e6	−0.332720
$970$	0	0
$971$	−3.11647e6	−0.106075	−0.0530377	−	0.998593i	$-0.516890\pi$
$971$	−3.11647e6	−0.106075	−0.0530377	+	0.998593i	$0.516890\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	7.30786e6	0.246195
$976$	0	0
$977$	−9.42502e6	−0.315897	−0.157949	−	0.987447i	$-0.550488\pi$
$977$	−9.42502e6	−0.315897	−0.157949	+	0.987447i	$0.550488\pi$
$978$	0	0
$979$	1.30626e7	0.435585
$980$	0	0
$981$	1.34262e7	0.445430
$982$	0	0
$983$	−1.87703e7	−0.619566	−0.309783	−	0.950807i	$-0.600256\pi$
$983$	−1.87703e7	−0.619566	−0.309783	+	0.950807i	$0.600256\pi$
$984$	0	0
$985$	2.72239e7	0.894046
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.14696e7	1.02306
$990$	0	0
$991$	2.53382e7	0.819581	0.409791	−	0.912180i	$-0.365602\pi$
$991$	2.53382e7	0.819581	0.409791	+	0.912180i	$0.365602\pi$
$992$	0	0
$993$	−2.82229e6	−0.0908297
$994$	0	0
$995$	7.27288e6	0.232889
$996$	0	0
$997$	−5.56262e6	−0.177232	−0.0886158	−	0.996066i	$-0.528244\pi$
$997$	−5.56262e6	−0.177232	−0.0886158	+	0.996066i	$0.528244\pi$
$998$	0	0
$999$	−1.37072e6	−0.0434547

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.be.1.2		4
4.3	odd	2		392.6.a.g.1.3	yes	4
7.6	odd	2	inner	784.6.a.be.1.3		4
28.3	even	6		392.6.i.n.177.3		8
28.11	odd	6		392.6.i.n.177.2		8
28.19	even	6		392.6.i.n.361.3		8
28.23	odd	6		392.6.i.n.361.2		8
28.27	even	2		392.6.a.g.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.6.a.g.1.2	✓	4	28.27	even	2
392.6.a.g.1.3	yes	4	4.3	odd	2
392.6.i.n.177.2		8	28.11	odd	6
392.6.i.n.177.3		8	28.3	even	6
392.6.i.n.361.2		8	28.23	odd	6
392.6.i.n.361.3		8	28.19	even	6
784.6.a.be.1.2		4	1.1	even	1	trivial
784.6.a.be.1.3		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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30222 q^{3} -81.0956 q^{5} -237.700 q^{9} +O(q^{10})\)	\(q-2.30222 q^{3} -81.0956 q^{5} -237.700 q^{9} +732.099 q^{11} -919.678 q^{13} +186.700 q^{15} -1728.85 q^{17} -2443.34 q^{19} -2424.30 q^{23} +3451.50 q^{25} +1106.68 q^{27} +1010.00 q^{29} +6128.04 q^{31} -1685.45 q^{33} -1238.60 q^{37} +2117.30 q^{39} -15388.6 q^{41} -12980.9 q^{43} +19276.4 q^{45} -6557.64 q^{47} +3980.19 q^{51} -31549.8 q^{53} -59370.0 q^{55} +5625.11 q^{57} -21881.4 q^{59} -27003.0 q^{61} +74581.9 q^{65} -8264.07 q^{67} +5581.26 q^{69} -17131.8 q^{71} -9282.54 q^{73} -7946.10 q^{75} -39144.8 q^{79} +55213.2 q^{81} +108829. q^{83} +140202. q^{85} -2325.24 q^{87} +17842.6 q^{89} -14108.1 q^{93} +198144. q^{95} +26643.9 q^{97} -174020. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 92 q^{9}+O(q^{10}) \) 4 * q - 92 * q^9	\( 4 q - 92 q^{9} + 352 q^{11} - 112 q^{15} - 1968 q^{23} + 924 q^{25} + 4040 q^{29} + 10504 q^{37} + 9328 q^{39} - 28736 q^{43} - 23584 q^{51} - 28296 q^{53} + 62864 q^{57} + 146320 q^{65} + 72576 q^{67} - 176736 q^{71} - 135968 q^{79} - 27340 q^{81} + 298016 q^{85} + 395296 q^{93} + 367472 q^{95} - 561248 q^{99}+O(q^{100}) \) 4 * q - 92 * q^9 + 352 * q^11 - 112 * q^15 - 1968 * q^23 + 924 * q^25 + 4040 * q^29 + 10504 * q^37 + 9328 * q^39 - 28736 * q^43 - 23584 * q^51 - 28296 * q^53 + 62864 * q^57 + 146320 * q^65 + 72576 * q^67 - 176736 * q^71 - 135968 * q^79 - 27340 * q^81 + 298016 * q^85 + 395296 * q^93 + 367472 * q^95 - 561248 * q^99

Introduction

L-functions

Modular forms

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