LMFDB - Embedded newform 392.6.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [392,6,Mod(1,392)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(392, 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("392.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 392 = 2^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	392.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:	$62.8704573667$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.2732674592.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 113x^{2} + 882 $ x^4 - 113*x^2 + 882
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-10.2257$ of defining polynomial
Character	$\chi$	$=$	392.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.21459 q^{3} +57.1619 q^{5} -175.520 q^{9} +O(q^{10})$	$q+8.21459 q^{3} +57.1619 q^{5} -175.520 q^{9} +176.520 q^{11} -252.951 q^{13} +469.561 q^{15} -1946.23 q^{17} -469.593 q^{19} +502.602 q^{23} +142.480 q^{25} -3437.97 q^{27} -3926.33 q^{29} -3945.82 q^{31} +1450.04 q^{33} -2351.04 q^{37} -2077.89 q^{39} +10309.0 q^{41} -4340.36 q^{43} -10033.1 q^{45} -25000.3 q^{47} -15987.5 q^{51} +13821.6 q^{53} +10090.2 q^{55} -3857.51 q^{57} +47285.8 q^{59} +11290.8 q^{61} -14459.2 q^{65} +26445.6 q^{67} +4128.67 q^{69} -63078.1 q^{71} -10793.9 q^{73} +1170.41 q^{75} -61419.5 q^{79} +14409.9 q^{81} +5383.09 q^{83} -111250. q^{85} -32253.2 q^{87} +15237.3 q^{89} -32413.3 q^{93} -26842.8 q^{95} -114148. q^{97} -30983.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 836 q^{9}+O(q^{10}) $ 4 * q + 836 * q^9	$ 4 q + 836 q^{9} - 832 q^{11} - 2736 q^{15} - 5680 q^{23} + 2108 q^{25} + 8904 q^{29} - 6328 q^{37} + 20912 q^{39} - 28128 q^{43} - 30112 q^{51} + 9144 q^{53} - 164624 q^{57} - 83984 q^{65} - 52640 q^{67} - 21600 q^{71} - 282592 q^{79} + 326804 q^{81} - 226592 q^{85} - 255776 q^{93} + 290992 q^{95} - 765312 q^{99}+O(q^{100}) $ 4 * q + 836 * q^9 - 832 * q^11 - 2736 * q^15 - 5680 * q^23 + 2108 * q^25 + 8904 * q^29 - 6328 * q^37 + 20912 * q^39 - 28128 * q^43 - 30112 * q^51 + 9144 * q^53 - 164624 * q^57 - 83984 * q^65 - 52640 * q^67 - 21600 * q^71 - 282592 * q^79 + 326804 * q^81 - 226592 * q^85 - 255776 * q^93 + 290992 * q^95 - 765312 * 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
$3$	8.21459	0.526966	0.263483	−	0.964664i	$-0.415129\pi$
$3$	8.21459	0.526966	0.263483	+	0.964664i	$0.415129\pi$
$4$	0	0
$5$	57.1619	1.02254	0.511271	−	0.859419i	$-0.329175\pi$
$5$	57.1619	1.02254	0.511271	+	0.859419i	$0.329175\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−175.520	−0.722307
$10$	0	0
$11$	176.520	0.439859	0.219929	−	0.975516i	$-0.429417\pi$
$11$	176.520	0.439859	0.219929	+	0.975516i	$0.429417\pi$
$12$	0	0
$13$	−252.951	−0.415124	−0.207562	−	0.978222i	$-0.566553\pi$
$13$	−252.951	−0.415124	−0.207562	+	0.978222i	$0.566553\pi$
$14$	0	0
$15$	469.561	0.538846
$16$	0	0
$17$	−1946.23	−1.63332	−0.816659	−	0.577120i	$-0.804176\pi$
$17$	−1946.23	−1.63332	−0.816659	+	0.577120i	$0.804176\pi$
$18$	0	0
$19$	−469.593	−0.298427	−0.149213	−	0.988805i	$-0.547674\pi$
$19$	−469.593	−0.298427	−0.149213	+	0.988805i	$0.547674\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	502.602	0.198109	0.0990547	−	0.995082i	$-0.468418\pi$
$23$	502.602	0.198109	0.0990547	+	0.995082i	$0.468418\pi$
$24$	0	0
$25$	142.480	0.0455934
$26$	0	0
$27$	−3437.97	−0.907598
$28$	0	0
$29$	−3926.33	−0.866945	−0.433473	−	0.901167i	$-0.642712\pi$
$29$	−3926.33	−0.866945	−0.433473	+	0.901167i	$0.642712\pi$
$30$	0	0
$31$	−3945.82	−0.737451	−0.368726	−	0.929538i	$-0.620206\pi$
$31$	−3945.82	−0.737451	−0.368726	+	0.929538i	$0.620206\pi$
$32$	0	0
$33$	1450.04	0.231791
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2351.04	−0.282329	−0.141165	−	0.989986i	$-0.545085\pi$
$37$	−2351.04	−0.282329	−0.141165	+	0.989986i	$0.545085\pi$
$38$	0	0
$39$	−2077.89	−0.218756
$40$	0	0
$41$	10309.0	0.957758	0.478879	−	0.877881i	$-0.341043\pi$
$41$	10309.0	0.957758	0.478879	+	0.877881i	$0.341043\pi$
$42$	0	0
$43$	−4340.36	−0.357976	−0.178988	−	0.983851i	$-0.557282\pi$
$43$	−4340.36	−0.357976	−0.178988	+	0.983851i	$0.557282\pi$
$44$	0	0
$45$	−10033.1	−0.738589
$46$	0	0
$47$	−25000.3	−1.65082	−0.825410	−	0.564534i	$-0.809056\pi$
$47$	−25000.3	−1.65082	−0.825410	+	0.564534i	$0.809056\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−15987.5	−0.860704
$52$	0	0
$53$	13821.6	0.675879	0.337940	−	0.941168i	$-0.390270\pi$
$53$	13821.6	0.675879	0.337940	+	0.941168i	$0.390270\pi$
$54$	0	0
$55$	10090.2	0.449774
$56$	0	0
$57$	−3857.51	−0.157261
$58$	0	0
$59$	47285.8	1.76848	0.884242	−	0.467029i	$-0.154676\pi$
$59$	47285.8	1.76848	0.884242	+	0.467029i	$0.154676\pi$
$60$	0	0
$61$	11290.8	0.388509	0.194255	−	0.980951i	$-0.437771\pi$
$61$	11290.8	0.388509	0.194255	+	0.980951i	$0.437771\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−14459.2	−0.424482
$66$	0	0
$67$	26445.6	0.719725	0.359862	−	0.933005i	$-0.382824\pi$
$67$	26445.6	0.719725	0.359862	+	0.933005i	$0.382824\pi$
$68$	0	0
$69$	4128.67	0.104397
$70$	0	0
$71$	−63078.1	−1.48502	−0.742510	−	0.669835i	$-0.766365\pi$
$71$	−63078.1	−1.48502	−0.742510	+	0.669835i	$0.766365\pi$
$72$	0	0
$73$	−10793.9	−0.237067	−0.118533	−	0.992950i	$-0.537819\pi$
$73$	−10793.9	−0.237067	−0.118533	+	0.992950i	$0.537819\pi$
$74$	0	0
$75$	1170.41	0.0240262
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−61419.5	−1.10723	−0.553616	−	0.832772i	$-0.686752\pi$
$79$	−61419.5	−1.10723	−0.553616	+	0.832772i	$0.686752\pi$
$80$	0	0
$81$	14409.9	0.244033
$82$	0	0
$83$	5383.09	0.0857702	0.0428851	−	0.999080i	$-0.486345\pi$
$83$	5383.09	0.0857702	0.0428851	+	0.999080i	$0.486345\pi$
$84$	0	0
$85$	−111250.	−1.67014
$86$	0	0
$87$	−32253.2	−0.456851
$88$	0	0
$89$	15237.3	0.203908	0.101954	−	0.994789i	$-0.467491\pi$
$89$	15237.3	0.203908	0.101954	+	0.994789i	$0.467491\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−32413.3	−0.388612
$94$	0	0
$95$	−26842.8	−0.305154
$96$	0	0
$97$	−114148.	−1.23180	−0.615898	−	0.787826i	$-0.711207\pi$
$97$	−114148.	−1.23180	−0.615898	+	0.787826i	$0.711207\pi$
$98$	0	0
$99$	−30983.0	−0.317713
$100$	0	0
$101$	−1671.49	−0.0163042	−0.00815212	−	0.999967i	$-0.502595\pi$
$101$	−1671.49	−0.0163042	−0.00815212	+	0.999967i	$0.502595\pi$
$102$	0	0
$103$	147637.	1.37120	0.685601	−	0.727977i	$-0.259539\pi$
$103$	147637.	1.37120	0.685601	+	0.727977i	$0.259539\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−195451.	−1.65036	−0.825179	−	0.564872i	$-0.808926\pi$
$107$	−195451.	−1.65036	−0.825179	+	0.564872i	$0.808926\pi$
$108$	0	0
$109$	170850.	1.37736	0.688680	−	0.725065i	$-0.258190\pi$
$109$	170850.	1.37736	0.688680	+	0.725065i	$0.258190\pi$
$110$	0	0
$111$	−19312.8	−0.148778
$112$	0	0
$113$	−225942.	−1.66457	−0.832283	−	0.554351i	$-0.812967\pi$
$113$	−225942.	−1.66457	−0.832283	+	0.554351i	$0.812967\pi$
$114$	0	0
$115$	28729.7	0.202575
$116$	0	0
$117$	44398.1	0.299847
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−129892.	−0.806524
$122$	0	0
$123$	84684.0	0.504706
$124$	0	0
$125$	−170486.	−0.975921
$126$	0	0
$127$	−161169.	−0.886688	−0.443344	−	0.896352i	$-0.646208\pi$
$127$	−161169.	−0.886688	−0.443344	+	0.896352i	$0.646208\pi$
$128$	0	0
$129$	−35654.3	−0.188641
$130$	0	0
$131$	−143042.	−0.728257	−0.364129	−	0.931349i	$-0.618633\pi$
$131$	−143042.	−0.728257	−0.364129	+	0.931349i	$0.618633\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−196521.	−0.928057
$136$	0	0
$137$	251450.	1.14459	0.572295	−	0.820048i	$-0.306053\pi$
$137$	251450.	1.14459	0.572295	+	0.820048i	$0.306053\pi$
$138$	0	0
$139$	405952.	1.78212	0.891062	−	0.453881i	$-0.149961\pi$
$139$	405952.	1.78212	0.891062	+	0.453881i	$0.149961\pi$
$140$	0	0
$141$	−205367.	−0.869927
$142$	0	0
$143$	−44651.0	−0.182596
$144$	0	0
$145$	−224436.	−0.886488
$146$	0	0
$147$	0	0
$148$	0	0
$149$	77877.0	0.287371	0.143686	−	0.989623i	$-0.454105\pi$
$149$	77877.0	0.287371	0.143686	+	0.989623i	$0.454105\pi$
$150$	0	0
$151$	−211478.	−0.754786	−0.377393	−	0.926053i	$-0.623179\pi$
$151$	−211478.	−0.754786	−0.377393	+	0.926053i	$0.623179\pi$
$152$	0	0
$153$	341602.	1.17976
$154$	0	0
$155$	−225551.	−0.754075
$156$	0	0
$157$	304925.	0.987288	0.493644	−	0.869664i	$-0.335665\pi$
$157$	304925.	0.987288	0.493644	+	0.869664i	$0.335665\pi$
$158$	0	0
$159$	113539.	0.356166
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−537444.	−1.58440	−0.792199	−	0.610263i	$-0.791064\pi$
$163$	−537444.	−1.58440	−0.792199	+	0.610263i	$0.791064\pi$
$164$	0	0
$165$	82887.2	0.237016
$166$	0	0
$167$	−542515.	−1.50529	−0.752646	−	0.658425i	$-0.771223\pi$
$167$	−542515.	−1.50529	−0.752646	+	0.658425i	$0.771223\pi$
$168$	0	0
$169$	−307309.	−0.827672
$170$	0	0
$171$	82423.2	0.215555
$172$	0	0
$173$	−203550.	−0.517077	−0.258539	−	0.966001i	$-0.583241\pi$
$173$	−203550.	−0.517077	−0.258539	+	0.966001i	$0.583241\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	388434.	0.931932
$178$	0	0
$179$	−188519.	−0.439768	−0.219884	−	0.975526i	$-0.570568\pi$
$179$	−188519.	−0.439768	−0.219884	+	0.975526i	$0.570568\pi$
$180$	0	0
$181$	464405.	1.05366	0.526830	−	0.849971i	$-0.323380\pi$
$181$	464405.	1.05366	0.526830	+	0.849971i	$0.323380\pi$
$182$	0	0
$183$	92749.6	0.204731
$184$	0	0
$185$	−134390.	−0.288694
$186$	0	0
$187$	−343549.	−0.718430
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−638610.	−1.26664	−0.633318	−	0.773891i	$-0.718308\pi$
$191$	−638610.	−1.26664	−0.633318	+	0.773891i	$0.718308\pi$
$192$	0	0
$193$	504542.	0.974998	0.487499	−	0.873123i	$-0.337909\pi$
$193$	504542.	0.974998	0.487499	+	0.873123i	$0.337909\pi$
$194$	0	0
$195$	−118776.	−0.223688
$196$	0	0
$197$	−69194.6	−0.127030	−0.0635151	−	0.997981i	$-0.520231\pi$
$197$	−69194.6	−0.127030	−0.0635151	+	0.997981i	$0.520231\pi$
$198$	0	0
$199$	−846712.	−1.51566	−0.757832	−	0.652449i	$-0.773741\pi$
$199$	−846712.	−1.51566	−0.757832	+	0.652449i	$0.773741\pi$
$200$	0	0
$201$	217240.	0.379271
$202$	0	0
$203$	0	0
$204$	0	0
$205$	589280.	0.979348
$206$	0	0
$207$	−88217.0	−0.143096
$208$	0	0
$209$	−82892.8	−0.131266
$210$	0	0
$211$	625421.	0.967089	0.483544	−	0.875320i	$-0.339349\pi$
$211$	625421.	0.967089	0.483544	+	0.875320i	$0.339349\pi$
$212$	0	0
$213$	−518161.	−0.782556
$214$	0	0
$215$	−248103.	−0.366046
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−88667.3	−0.124926
$220$	0	0
$221$	492300.	0.678030
$222$	0	0
$223$	−584189.	−0.786667	−0.393334	−	0.919396i	$-0.628678\pi$
$223$	−584189.	−0.786667	−0.393334	+	0.919396i	$0.628678\pi$
$224$	0	0
$225$	−25008.1	−0.0329324
$226$	0	0
$227$	984064.	1.26753	0.633766	−	0.773525i	$-0.281508\pi$
$227$	984064.	1.26753	0.633766	+	0.773525i	$0.281508\pi$
$228$	0	0
$229$	1.34694e6	1.69731	0.848653	−	0.528949i	$-0.177414\pi$
$229$	1.34694e6	1.69731	0.848653	+	0.528949i	$0.177414\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.14450e6	1.38111	0.690553	−	0.723282i	$-0.257367\pi$
$233$	1.14450e6	1.38111	0.690553	+	0.723282i	$0.257367\pi$
$234$	0	0
$235$	−1.42906e6	−1.68803
$236$	0	0
$237$	−504536.	−0.583474
$238$	0	0
$239$	1.42563e6	1.61441	0.807204	−	0.590272i	$-0.200979\pi$
$239$	1.42563e6	1.61441	0.807204	+	0.590272i	$0.200979\pi$
$240$	0	0
$241$	94563.3	0.104877	0.0524385	−	0.998624i	$-0.483301\pi$
$241$	94563.3	0.104877	0.0524385	+	0.998624i	$0.483301\pi$
$242$	0	0
$243$	953799.	1.03619
$244$	0	0
$245$	0	0
$246$	0	0
$247$	118784.	0.123884
$248$	0	0
$249$	44219.9	0.0451980
$250$	0	0
$251$	−228497.	−0.228927	−0.114463	−	0.993427i	$-0.536515\pi$
$251$	−228497.	−0.228927	−0.114463	+	0.993427i	$0.536515\pi$
$252$	0	0
$253$	88719.6	0.0871402
$254$	0	0
$255$	−913873.	−0.880106
$256$	0	0
$257$	−1.36203e6	−1.28633	−0.643165	−	0.765727i	$-0.722379\pi$
$257$	−1.36203e6	−1.28633	−0.643165	+	0.765727i	$0.722379\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	689151.	0.626200
$262$	0	0
$263$	−1.22988e6	−1.09641	−0.548206	−	0.836343i	$-0.684689\pi$
$263$	−1.22988e6	−1.09641	−0.548206	+	0.836343i	$0.684689\pi$
$264$	0	0
$265$	790069.	0.691115
$266$	0	0
$267$	125169.	0.107453
$268$	0	0
$269$	1.87483e6	1.57972	0.789860	−	0.613287i	$-0.210153\pi$
$269$	1.87483e6	1.57972	0.789860	+	0.613287i	$0.210153\pi$
$270$	0	0
$271$	18097.8	0.0149693	0.00748465	−	0.999972i	$-0.497618\pi$
$271$	18097.8	0.0149693	0.00748465	+	0.999972i	$0.497618\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	25150.6	0.0200547
$276$	0	0
$277$	141825.	0.111059	0.0555293	−	0.998457i	$-0.482315\pi$
$277$	141825.	0.111059	0.0555293	+	0.998457i	$0.482315\pi$
$278$	0	0
$279$	692573.	0.532666
$280$	0	0
$281$	1.15247e6	0.870693	0.435346	−	0.900263i	$-0.356626\pi$
$281$	1.15247e6	0.870693	0.435346	+	0.900263i	$0.356626\pi$
$282$	0	0
$283$	957483.	0.710665	0.355332	−	0.934740i	$-0.384368\pi$
$283$	957483.	0.710665	0.355332	+	0.934740i	$0.384368\pi$
$284$	0	0
$285$	−220503.	−0.160806
$286$	0	0
$287$	0	0
$288$	0	0
$289$	2.36794e6	1.66773
$290$	0	0
$291$	−937679.	−0.649115
$292$	0	0
$293$	−440238.	−0.299584	−0.149792	−	0.988718i	$-0.547860\pi$
$293$	−440238.	−0.299584	−0.149792	+	0.988718i	$0.547860\pi$
$294$	0	0
$295$	2.70295e6	1.80835
$296$	0	0
$297$	−606873.	−0.399215
$298$	0	0
$299$	−127134.	−0.0822400
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−13730.6	−0.00859178
$304$	0	0
$305$	645405.	0.397267
$306$	0	0
$307$	1.93780e6	1.17345	0.586724	−	0.809787i	$-0.300417\pi$
$307$	1.93780e6	1.17345	0.586724	+	0.809787i	$0.300417\pi$
$308$	0	0
$309$	1.21278e6	0.722577
$310$	0	0
$311$	1.70845e6	1.00162	0.500809	−	0.865558i	$-0.333036\pi$
$311$	1.70845e6	1.00162	0.500809	+	0.865558i	$0.333036\pi$
$312$	0	0
$313$	383059.	0.221006	0.110503	−	0.993876i	$-0.464754\pi$
$313$	383059.	0.221006	0.110503	+	0.993876i	$0.464754\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.21790e6	1.79856	0.899280	−	0.437373i	$-0.144091\pi$
$317$	3.21790e6	1.79856	0.899280	+	0.437373i	$0.144091\pi$
$318$	0	0
$319$	−693077.	−0.381333
$320$	0	0
$321$	−1.60555e6	−0.869683
$322$	0	0
$323$	913934.	0.487426
$324$	0	0
$325$	−36040.3	−0.0189269
$326$	0	0
$327$	1.40346e6	0.725823
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1.66097e6	−0.833279	−0.416640	−	0.909072i	$-0.636792\pi$
$331$	−1.66097e6	−0.833279	−0.416640	+	0.909072i	$0.636792\pi$
$332$	0	0
$333$	412656.	0.203928
$334$	0	0
$335$	1.51168e6	0.735949
$336$	0	0
$337$	1.06508e6	0.510865	0.255433	−	0.966827i	$-0.417782\pi$
$337$	1.06508e6	0.510865	0.255433	+	0.966827i	$0.417782\pi$
$338$	0	0
$339$	−1.85602e6	−0.877170
$340$	0	0
$341$	−696518.	−0.324374
$342$	0	0
$343$	0	0
$344$	0	0
$345$	236003.	0.106750
$346$	0	0
$347$	−178645.	−0.0796464	−0.0398232	−	0.999207i	$-0.512679\pi$
$347$	−178645.	−0.0796464	−0.0398232	+	0.999207i	$0.512679\pi$
$348$	0	0
$349$	−1.87153e6	−0.822494	−0.411247	−	0.911524i	$-0.634907\pi$
$349$	−1.87153e6	−0.822494	−0.411247	+	0.911524i	$0.634907\pi$
$350$	0	0
$351$	869639.	0.376766
$352$	0	0
$353$	−3.41481e6	−1.45858	−0.729290	−	0.684205i	$-0.760149\pi$
$353$	−3.41481e6	−1.45858	−0.729290	+	0.684205i	$0.760149\pi$
$354$	0	0
$355$	−3.60566e6	−1.51850
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.58575e6	0.649380	0.324690	−	0.945820i	$-0.394740\pi$
$359$	1.58575e6	0.649380	0.324690	+	0.945820i	$0.394740\pi$
$360$	0	0
$361$	−2.25558e6	−0.910942
$362$	0	0
$363$	−1.06701e6	−0.425011
$364$	0	0
$365$	−616998.	−0.242411
$366$	0	0
$367$	−220985.	−0.0856443	−0.0428221	−	0.999083i	$-0.513635\pi$
$367$	−220985.	−0.0856443	−0.0428221	+	0.999083i	$0.513635\pi$
$368$	0	0
$369$	−1.80944e6	−0.691795
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−469456.	−0.174712	−0.0873561	−	0.996177i	$-0.527842\pi$
$373$	−469456.	−0.174712	−0.0873561	+	0.996177i	$0.527842\pi$
$374$	0	0
$375$	−1.40048e6	−0.514278
$376$	0	0
$377$	993168.	0.359890
$378$	0	0
$379$	−527577.	−0.188664	−0.0943318	−	0.995541i	$-0.530071\pi$
$379$	−527577.	−0.188664	−0.0943318	+	0.995541i	$0.530071\pi$
$380$	0	0
$381$	−1.32393e6	−0.467255
$382$	0	0
$383$	879752.	0.306453	0.153226	−	0.988191i	$-0.451034\pi$
$383$	879752.	0.306453	0.153226	+	0.988191i	$0.451034\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	761821.	0.258569
$388$	0	0
$389$	−33448.2	−0.0112072	−0.00560362	−	0.999984i	$-0.501784\pi$
$389$	−33448.2	−0.0112072	−0.00560362	+	0.999984i	$0.501784\pi$
$390$	0	0
$391$	−978178.	−0.323576
$392$	0	0
$393$	−1.17503e6	−0.383767
$394$	0	0
$395$	−3.51085e6	−1.13219
$396$	0	0
$397$	4.49449e6	1.43121	0.715607	−	0.698503i	$-0.246150\pi$
$397$	4.49449e6	1.43121	0.715607	+	0.698503i	$0.246150\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.53668e6	−0.477225	−0.238613	−	0.971115i	$-0.576693\pi$
$401$	−1.53668e6	−0.477225	−0.238613	+	0.971115i	$0.576693\pi$
$402$	0	0
$403$	998100.	0.306134
$404$	0	0
$405$	823698.	0.249534
$406$	0	0
$407$	−415007.	−0.124185
$408$	0	0
$409$	2.50479e6	0.740393	0.370197	−	0.928953i	$-0.379290\pi$
$409$	2.50479e6	0.740393	0.370197	+	0.928953i	$0.379290\pi$
$410$	0	0
$411$	2.06556e6	0.603161
$412$	0	0
$413$	0	0
$414$	0	0
$415$	307707.	0.0877036
$416$	0	0
$417$	3.33473e6	0.939120
$418$	0	0
$419$	−3.32003e6	−0.923862	−0.461931	−	0.886916i	$-0.652843\pi$
$419$	−3.32003e6	−0.923862	−0.461931	+	0.886916i	$0.652843\pi$
$420$	0	0
$421$	141680.	0.0389587	0.0194793	−	0.999810i	$-0.493799\pi$
$421$	141680.	0.0389587	0.0194793	+	0.999810i	$0.493799\pi$
$422$	0	0
$423$	4.38806e6	1.19240
$424$	0	0
$425$	−277297.	−0.0744686
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−366790.	−0.0962220
$430$	0	0
$431$	−155475.	−0.0403150	−0.0201575	−	0.999797i	$-0.506417\pi$
$431$	−155475.	−0.0403150	−0.0201575	+	0.999797i	$0.506417\pi$
$432$	0	0
$433$	3.46000e6	0.886864	0.443432	−	0.896308i	$-0.353761\pi$
$433$	3.46000e6	0.886864	0.443432	+	0.896308i	$0.353761\pi$
$434$	0	0
$435$	−1.84365e6	−0.467149
$436$	0	0
$437$	−236018.	−0.0591211
$438$	0	0
$439$	−6.15172e6	−1.52348	−0.761738	−	0.647885i	$-0.775654\pi$
$439$	−6.15172e6	−1.52348	−0.761738	+	0.647885i	$0.775654\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.62933e6	−0.636556	−0.318278	−	0.947997i	$-0.603104\pi$
$443$	−2.62933e6	−0.636556	−0.318278	+	0.947997i	$0.603104\pi$
$444$	0	0
$445$	870995.	0.208505
$446$	0	0
$447$	639728.	0.151435
$448$	0	0
$449$	−3.36346e6	−0.787356	−0.393678	−	0.919248i	$-0.628797\pi$
$449$	−3.36346e6	−0.787356	−0.393678	+	0.919248i	$0.628797\pi$
$450$	0	0
$451$	1.81974e6	0.421278
$452$	0	0
$453$	−1.73721e6	−0.397747
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7.57232e6	−1.69605	−0.848025	−	0.529956i	$-0.822208\pi$
$457$	−7.57232e6	−1.69605	−0.848025	+	0.529956i	$0.822208\pi$
$458$	0	0
$459$	6.69108e6	1.48240
$460$	0	0
$461$	4.61580e6	1.01157	0.505783	−	0.862660i	$-0.331203\pi$
$461$	4.61580e6	1.01157	0.505783	+	0.862660i	$0.331203\pi$
$462$	0	0
$463$	−2.52659e6	−0.547751	−0.273875	−	0.961765i	$-0.588306\pi$
$463$	−2.52659e6	−0.547751	−0.273875	+	0.961765i	$0.588306\pi$
$464$	0	0
$465$	−1.85281e6	−0.397372
$466$	0	0
$467$	4.06866e6	0.863295	0.431647	−	0.902042i	$-0.357932\pi$
$467$	4.06866e6	0.863295	0.431647	+	0.902042i	$0.357932\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	2.50483e6	0.520268
$472$	0	0
$473$	−766162.	−0.157459
$474$	0	0
$475$	−66907.4	−0.0136063
$476$	0	0
$477$	−2.42598e6	−0.488192
$478$	0	0
$479$	5.93903e6	1.18271	0.591353	−	0.806413i	$-0.298594\pi$
$479$	5.93903e6	1.18271	0.591353	+	0.806413i	$0.298594\pi$
$480$	0	0
$481$	594698.	0.117202
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.52491e6	−1.25956
$486$	0	0
$487$	−160923.	−0.0307464	−0.0153732	−	0.999882i	$-0.504894\pi$
$487$	−160923.	−0.0307464	−0.0153732	+	0.999882i	$0.504894\pi$
$488$	0	0
$489$	−4.41488e6	−0.834924
$490$	0	0
$491$	−1.01033e7	−1.89129	−0.945647	−	0.325196i	$-0.894570\pi$
$491$	−1.01033e7	−1.89129	−0.945647	+	0.325196i	$0.894570\pi$
$492$	0	0
$493$	7.64152e6	1.41600
$494$	0	0
$495$	−1.77104e6	−0.324875
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−7.49005e6	−1.34658	−0.673292	−	0.739377i	$-0.735120\pi$
$499$	−7.49005e6	−1.34658	−0.673292	+	0.739377i	$0.735120\pi$
$500$	0	0
$501$	−4.45654e6	−0.793238
$502$	0	0
$503$	3.18116e6	0.560615	0.280308	−	0.959910i	$-0.409563\pi$
$503$	3.18116e6	0.560615	0.280308	+	0.959910i	$0.409563\pi$
$504$	0	0
$505$	−95545.5	−0.0166718
$506$	0	0
$507$	−2.52442e6	−0.436155
$508$	0	0
$509$	6.32304e6	1.08176	0.540881	−	0.841099i	$-0.318091\pi$
$509$	6.32304e6	1.08176	0.540881	+	0.841099i	$0.318091\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1.61445e6	0.270851
$514$	0	0
$515$	8.43919e6	1.40211
$516$	0	0
$517$	−4.41306e6	−0.726128
$518$	0	0
$519$	−1.67208e6	−0.272482
$520$	0	0
$521$	4.16325e6	0.671952	0.335976	−	0.941870i	$-0.390934\pi$
$521$	4.16325e6	0.671952	0.335976	+	0.941870i	$0.390934\pi$
$522$	0	0
$523$	5.58604e6	0.892997	0.446498	−	0.894784i	$-0.352671\pi$
$523$	5.58604e6	0.892997	0.446498	+	0.894784i	$0.352671\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.67946e6	1.20449
$528$	0	0
$529$	−6.18373e6	−0.960753
$530$	0	0
$531$	−8.29964e6	−1.27739
$532$	0	0
$533$	−2.60766e6	−0.397588
$534$	0	0
$535$	−1.11723e7	−1.68756
$536$	0	0
$537$	−1.54861e6	−0.231743
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−1.11301e7	−1.63496	−0.817480	−	0.575957i	$-0.804630\pi$
$541$	−1.11301e7	−1.63496	−0.817480	+	0.575957i	$0.804630\pi$
$542$	0	0
$543$	3.81490e6	0.555244
$544$	0	0
$545$	9.76608e6	1.40841
$546$	0	0
$547$	−1.20844e7	−1.72686	−0.863430	−	0.504468i	$-0.831689\pi$
$547$	−1.20844e7	−1.72686	−0.863430	+	0.504468i	$0.831689\pi$
$548$	0	0
$549$	−1.98177e6	−0.280623
$550$	0	0
$551$	1.84378e6	0.258719
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.10396e6	−0.152132
$556$	0	0
$557$	3.08796e6	0.421730	0.210865	−	0.977515i	$-0.432372\pi$
$557$	3.08796e6	0.421730	0.210865	+	0.977515i	$0.432372\pi$
$558$	0	0
$559$	1.09790e6	0.148605
$560$	0	0
$561$	−2.82211e6	−0.378588
$562$	0	0
$563$	−6.93288e6	−0.921813	−0.460906	−	0.887449i	$-0.652476\pi$
$563$	−6.93288e6	−0.921813	−0.460906	+	0.887449i	$0.652476\pi$
$564$	0	0
$565$	−1.29153e7	−1.70209
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.85938e6	−0.888187	−0.444093	−	0.895981i	$-0.646474\pi$
$569$	−6.85938e6	−0.888187	−0.444093	+	0.895981i	$0.646474\pi$
$570$	0	0
$571$	6.03502e6	0.774619	0.387310	−	0.921950i	$-0.373404\pi$
$571$	6.03502e6	0.774619	0.387310	+	0.921950i	$0.373404\pi$
$572$	0	0
$573$	−5.24592e6	−0.667475
$574$	0	0
$575$	71610.5	0.00903249
$576$	0	0
$577$	−2.63662e6	−0.329691	−0.164846	−	0.986319i	$-0.552713\pi$
$577$	−2.63662e6	−0.329691	−0.164846	+	0.986319i	$0.552713\pi$
$578$	0	0
$579$	4.14460e6	0.513791
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2.43980e6	0.297291
$584$	0	0
$585$	2.53788e6	0.306606
$586$	0	0
$587$	1.90597e6	0.228307	0.114154	−	0.993463i	$-0.463584\pi$
$587$	1.90597e6	0.228307	0.114154	+	0.993463i	$0.463584\pi$
$588$	0	0
$589$	1.85293e6	0.220075
$590$	0	0
$591$	−568406.	−0.0669406
$592$	0	0
$593$	−63087.3	−0.00736724	−0.00368362	−	0.999993i	$-0.501173\pi$
$593$	−63087.3	−0.00736724	−0.00368362	+	0.999993i	$0.501173\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.95539e6	−0.798704
$598$	0	0
$599$	6.34551e6	0.722602	0.361301	−	0.932449i	$-0.382333\pi$
$599$	6.34551e6	0.722602	0.361301	+	0.932449i	$0.382333\pi$
$600$	0	0
$601$	−1.44136e7	−1.62774	−0.813870	−	0.581046i	$-0.802643\pi$
$601$	−1.44136e7	−1.62774	−0.813870	+	0.581046i	$0.802643\pi$
$602$	0	0
$603$	−4.64175e6	−0.519862
$604$	0	0
$605$	−7.42484e6	−0.824705
$606$	0	0
$607$	−53893.8	−0.00593700	−0.00296850	−	0.999996i	$-0.500945\pi$
$607$	−53893.8	−0.00593700	−0.00296850	+	0.999996i	$0.500945\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.32384e6	0.685295
$612$	0	0
$613$	−9.14707e6	−0.983175	−0.491587	−	0.870828i	$-0.663583\pi$
$613$	−9.14707e6	−0.983175	−0.491587	+	0.870828i	$0.663583\pi$
$614$	0	0
$615$	4.84069e6	0.516083
$616$	0	0
$617$	5.57574e6	0.589643	0.294822	−	0.955552i	$-0.404740\pi$
$617$	5.57574e6	0.589643	0.294822	+	0.955552i	$0.404740\pi$
$618$	0	0
$619$	2.70190e6	0.283428	0.141714	−	0.989908i	$-0.454739\pi$
$619$	2.70190e6	0.283428	0.141714	+	0.989908i	$0.454739\pi$
$620$	0	0
$621$	−1.72793e6	−0.179804
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.01906e7	−1.04351
$626$	0	0
$627$	−680930.	−0.0691725
$628$	0	0
$629$	4.57566e6	0.461134
$630$	0	0
$631$	1.42258e7	1.42234	0.711170	−	0.703020i	$-0.248166\pi$
$631$	1.42258e7	1.42234	0.711170	+	0.703020i	$0.248166\pi$
$632$	0	0
$633$	5.13758e6	0.509623
$634$	0	0
$635$	−9.21269e6	−0.906676
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.10715e7	1.07264
$640$	0	0
$641$	1.92872e6	0.185407	0.0927033	−	0.995694i	$-0.470449\pi$
$641$	1.92872e6	0.185407	0.0927033	+	0.995694i	$0.470449\pi$
$642$	0	0
$643$	9.13216e6	0.871056	0.435528	−	0.900175i	$-0.356562\pi$
$643$	9.13216e6	0.871056	0.435528	+	0.900175i	$0.356562\pi$
$644$	0	0
$645$	−2.03806e6	−0.192894
$646$	0	0
$647$	1.08021e7	1.01449	0.507245	−	0.861802i	$-0.330664\pi$
$647$	1.08021e7	1.01449	0.507245	+	0.861802i	$0.330664\pi$
$648$	0	0
$649$	8.34692e6	0.777883
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.05977e6	0.189032	0.0945159	−	0.995523i	$-0.469870\pi$
$653$	2.05977e6	0.189032	0.0945159	+	0.995523i	$0.469870\pi$
$654$	0	0
$655$	−8.17654e6	−0.744674
$656$	0	0
$657$	1.89455e6	0.171235
$658$	0	0
$659$	−1.35721e7	−1.21740	−0.608700	−	0.793401i	$-0.708309\pi$
$659$	−1.35721e7	−1.21740	−0.608700	+	0.793401i	$0.708309\pi$
$660$	0	0
$661$	−3.14262e6	−0.279762	−0.139881	−	0.990168i	$-0.544672\pi$
$661$	−3.14262e6	−0.279762	−0.139881	+	0.990168i	$0.544672\pi$
$662$	0	0
$663$	4.04404e6	0.357299
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.97338e6	−0.171750
$668$	0	0
$669$	−4.79887e6	−0.414547
$670$	0	0
$671$	1.99306e6	0.170889
$672$	0	0
$673$	5.88424e6	0.500787	0.250393	−	0.968144i	$-0.419440\pi$
$673$	5.88424e6	0.500787	0.250393	+	0.968144i	$0.419440\pi$
$674$	0	0
$675$	−489841.	−0.0413805
$676$	0	0
$677$	−8.17770e6	−0.685740	−0.342870	−	0.939383i	$-0.611399\pi$
$677$	−8.17770e6	−0.685740	−0.342870	+	0.939383i	$0.611399\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	8.08369e6	0.667946
$682$	0	0
$683$	3.26901e6	0.268141	0.134071	−	0.990972i	$-0.457195\pi$
$683$	3.26901e6	0.268141	0.134071	+	0.990972i	$0.457195\pi$
$684$	0	0
$685$	1.43733e7	1.17039
$686$	0	0
$687$	1.10646e7	0.894424
$688$	0	0
$689$	−3.49619e6	−0.280574
$690$	0	0
$691$	−1.94889e7	−1.55271	−0.776357	−	0.630293i	$-0.782935\pi$
$691$	−1.94889e7	−1.55271	−0.776357	+	0.630293i	$0.782935\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.32050e7	1.82230
$696$	0	0
$697$	−2.00636e7	−1.56432
$698$	0	0
$699$	9.40162e6	0.727796
$700$	0	0
$701$	1.07266e7	0.824453	0.412227	−	0.911081i	$-0.364751\pi$
$701$	1.07266e7	0.824453	0.412227	+	0.911081i	$0.364751\pi$
$702$	0	0
$703$	1.10403e6	0.0842545
$704$	0	0
$705$	−1.17392e7	−0.889537
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−2.66490e6	−0.199098	−0.0995488	−	0.995033i	$-0.531740\pi$
$709$	−2.66490e6	−0.199098	−0.0995488	+	0.995033i	$0.531740\pi$
$710$	0	0
$711$	1.07804e7	0.799761
$712$	0	0
$713$	−1.98318e6	−0.146096
$714$	0	0
$715$	−2.55234e6	−0.186712
$716$	0	0
$717$	1.17110e7	0.850739
$718$	0	0
$719$	1.01722e7	0.733825	0.366912	−	0.930255i	$-0.380415\pi$
$719$	1.01722e7	0.733825	0.366912	+	0.930255i	$0.380415\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	776799.	0.0552666
$724$	0	0
$725$	−559421.	−0.0395270
$726$	0	0
$727$	−2.80053e7	−1.96519	−0.982594	−	0.185766i	$-0.940523\pi$
$727$	−2.80053e7	−1.96519	−0.982594	+	0.185766i	$0.940523\pi$
$728$	0	0
$729$	4.33346e6	0.302007
$730$	0	0
$731$	8.44731e6	0.584689
$732$	0	0
$733$	2.55563e7	1.75687	0.878433	−	0.477866i	$-0.158590\pi$
$733$	2.55563e7	1.75687	0.878433	+	0.477866i	$0.158590\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.66819e6	0.316577
$738$	0	0
$739$	1.11853e6	0.0753421	0.0376710	−	0.999290i	$-0.488006\pi$
$739$	1.11853e6	0.0753421	0.0376710	+	0.999290i	$0.488006\pi$
$740$	0	0
$741$	975762.	0.0652827
$742$	0	0
$743$	2.08574e7	1.38608	0.693040	−	0.720899i	$-0.256271\pi$
$743$	2.08574e7	1.38608	0.693040	+	0.720899i	$0.256271\pi$
$744$	0	0
$745$	4.45159e6	0.293849
$746$	0	0
$747$	−944842.	−0.0619523
$748$	0	0
$749$	0	0
$750$	0	0
$751$	8.32038e6	0.538324	0.269162	−	0.963095i	$-0.413253\pi$
$751$	8.32038e6	0.538324	0.269162	+	0.963095i	$0.413253\pi$
$752$	0	0
$753$	−1.87701e6	−0.120637
$754$	0	0
$755$	−1.20885e7	−0.771801
$756$	0	0
$757$	2.35804e7	1.49559	0.747793	−	0.663932i	$-0.231114\pi$
$757$	2.35804e7	1.49559	0.747793	+	0.663932i	$0.231114\pi$
$758$	0	0
$759$	728795.	0.0459199
$760$	0	0
$761$	2.93273e7	1.83573	0.917867	−	0.396888i	$-0.129910\pi$
$761$	2.93273e7	1.83573	0.917867	+	0.396888i	$0.129910\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.95266e7	1.20635
$766$	0	0
$767$	−1.19610e7	−0.734140
$768$	0	0
$769$	4.35031e6	0.265280	0.132640	−	0.991164i	$-0.457655\pi$
$769$	4.35031e6	0.265280	0.132640	+	0.991164i	$0.457655\pi$
$770$	0	0
$771$	−1.11885e7	−0.677853
$772$	0	0
$773$	−2.56420e7	−1.54349	−0.771744	−	0.635933i	$-0.780615\pi$
$773$	−2.56420e7	−1.54349	−0.771744	+	0.635933i	$0.780615\pi$
$774$	0	0
$775$	−562199.	−0.0336229
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.84102e6	−0.285820
$780$	0	0
$781$	−1.11346e7	−0.653200
$782$	0	0
$783$	1.34986e7	0.786837
$784$	0	0
$785$	1.74301e7	1.00954
$786$	0	0
$787$	2.26920e7	1.30598	0.652988	−	0.757368i	$-0.273515\pi$
$787$	2.26920e7	1.30598	0.652988	+	0.757368i	$0.273515\pi$
$788$	0	0
$789$	−1.01030e7	−0.577773
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2.85603e6	−0.161280
$794$	0	0
$795$	6.49010e6	0.364194
$796$	0	0
$797$	−3.23986e7	−1.80668	−0.903338	−	0.428929i	$-0.858891\pi$
$797$	−3.23986e7	−1.80668	−0.903338	+	0.428929i	$0.858891\pi$
$798$	0	0
$799$	4.86561e7	2.69632
$800$	0	0
$801$	−2.67447e6	−0.147284
$802$	0	0
$803$	−1.90534e6	−0.104276
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.54009e7	0.832460
$808$	0	0
$809$	8.86367e6	0.476148	0.238074	−	0.971247i	$-0.423484\pi$
$809$	8.86367e6	0.476148	0.238074	+	0.971247i	$0.423484\pi$
$810$	0	0
$811$	3.45729e6	0.184580	0.0922898	−	0.995732i	$-0.470581\pi$
$811$	3.45729e6	0.184580	0.0922898	+	0.995732i	$0.470581\pi$
$812$	0	0
$813$	148666.	0.00788832
$814$	0	0
$815$	−3.07213e7	−1.62011
$816$	0	0
$817$	2.03820e6	0.106830
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.15406e7	1.63310	0.816548	−	0.577278i	$-0.195885\pi$
$821$	3.15406e7	1.63310	0.816548	+	0.577278i	$0.195885\pi$
$822$	0	0
$823$	−3.71909e7	−1.91398	−0.956988	−	0.290126i	$-0.906303\pi$
$823$	−3.71909e7	−1.91398	−0.956988	+	0.290126i	$0.906303\pi$
$824$	0	0
$825$	206602.	0.0105681
$826$	0	0
$827$	−1.55748e7	−0.791879	−0.395939	−	0.918277i	$-0.629581\pi$
$827$	−1.55748e7	−0.791879	−0.395939	+	0.918277i	$0.629581\pi$
$828$	0	0
$829$	−2.23547e7	−1.12975	−0.564876	−	0.825176i	$-0.691076\pi$
$829$	−2.23547e7	−1.12975	−0.564876	+	0.825176i	$0.691076\pi$
$830$	0	0
$831$	1.16503e6	0.0585242
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−3.10112e7	−1.53923
$836$	0	0
$837$	1.35656e7	0.669309
$838$	0	0
$839$	−1.65695e7	−0.812652	−0.406326	−	0.913728i	$-0.633190\pi$
$839$	−1.65695e7	−0.812652	−0.406326	+	0.913728i	$0.633190\pi$
$840$	0	0
$841$	−5.09510e6	−0.248406
$842$	0	0
$843$	9.46710e6	0.458826
$844$	0	0
$845$	−1.75663e7	−0.846330
$846$	0	0
$847$	0	0
$848$	0	0
$849$	7.86533e6	0.374496
$850$	0	0
$851$	−1.18164e6	−0.0559321
$852$	0	0
$853$	2.63671e7	1.24077	0.620384	−	0.784299i	$-0.286977\pi$
$853$	2.63671e7	1.24077	0.620384	+	0.784299i	$0.286977\pi$
$854$	0	0
$855$	4.71146e6	0.220415
$856$	0	0
$857$	1.81803e7	0.845570	0.422785	−	0.906230i	$-0.361053\pi$
$857$	1.81803e7	0.845570	0.422785	+	0.906230i	$0.361053\pi$
$858$	0	0
$859$	3.13487e7	1.44956	0.724782	−	0.688979i	$-0.241941\pi$
$859$	3.13487e7	1.44956	0.724782	+	0.688979i	$0.241941\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.56999e7	1.63170	0.815850	−	0.578264i	$-0.196270\pi$
$863$	3.56999e7	1.63170	0.815850	+	0.578264i	$0.196270\pi$
$864$	0	0
$865$	−1.16353e7	−0.528733
$866$	0	0
$867$	1.94516e7	0.878837
$868$	0	0
$869$	−1.08418e7	−0.487026
$870$	0	0
$871$	−6.68944e6	−0.298775
$872$	0	0
$873$	2.00353e7	0.889735
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.81914e7	−0.798672	−0.399336	−	0.916805i	$-0.630759\pi$
$877$	−1.81914e7	−0.798672	−0.399336	+	0.916805i	$0.630759\pi$
$878$	0	0
$879$	−3.61638e6	−0.157871
$880$	0	0
$881$	−2.04023e7	−0.885603	−0.442802	−	0.896620i	$-0.646015\pi$
$881$	−2.04023e7	−0.885603	−0.442802	+	0.896620i	$0.646015\pi$
$882$	0	0
$883$	4.39975e6	0.189900	0.0949502	−	0.995482i	$-0.469731\pi$
$883$	4.39975e6	0.189900	0.0949502	+	0.995482i	$0.469731\pi$
$884$	0	0
$885$	2.22036e7	0.952940
$886$	0	0
$887$	−2.39608e7	−1.02257	−0.511284	−	0.859412i	$-0.670830\pi$
$887$	−2.39608e7	−1.02257	−0.511284	+	0.859412i	$0.670830\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	2.54365e6	0.107340
$892$	0	0
$893$	1.17399e7	0.492649
$894$	0	0
$895$	−1.07761e7	−0.449681
$896$	0	0
$897$	−1.04435e6	−0.0433377
$898$	0	0
$899$	1.54926e7	0.639330
$900$	0	0
$901$	−2.69000e7	−1.10393
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.65463e7	1.07741
$906$	0	0
$907$	−1.16278e7	−0.469329	−0.234665	−	0.972076i	$-0.575399\pi$
$907$	−1.16278e7	−0.469329	−0.234665	+	0.972076i	$0.575399\pi$
$908$	0	0
$909$	293381.	0.0117767
$910$	0	0
$911$	1.46643e7	0.585417	0.292708	−	0.956202i	$-0.405444\pi$
$911$	1.46643e7	0.585417	0.292708	+	0.956202i	$0.405444\pi$
$912$	0	0
$913$	950225.	0.0377268
$914$	0	0
$915$	5.30174e6	0.209346
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−2.14633e7	−0.838315	−0.419158	−	0.907914i	$-0.637674\pi$
$919$	−2.14633e7	−0.838315	−0.419158	+	0.907914i	$0.637674\pi$
$920$	0	0
$921$	1.59183e7	0.618367
$922$	0	0
$923$	1.59557e7	0.616468
$924$	0	0
$925$	−334975.	−0.0128724
$926$	0	0
$927$	−2.59133e7	−0.990428
$928$	0	0
$929$	−4.12955e7	−1.56987	−0.784934	−	0.619579i	$-0.787303\pi$
$929$	−4.12955e7	−1.56987	−0.784934	+	0.619579i	$0.787303\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1.40342e7	0.527819
$934$	0	0
$935$	−1.96379e7	−0.734625
$936$	0	0
$937$	1.98623e7	0.739061	0.369530	−	0.929219i	$-0.379519\pi$
$937$	1.98623e7	0.739061	0.369530	+	0.929219i	$0.379519\pi$
$938$	0	0
$939$	3.14667e6	0.116463
$940$	0	0
$941$	4.90935e7	1.80738	0.903691	−	0.428186i	$-0.140847\pi$
$941$	4.90935e7	1.80738	0.903691	+	0.428186i	$0.140847\pi$
$942$	0	0
$943$	5.18131e6	0.189741
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.27570e7	−1.54929	−0.774645	−	0.632396i	$-0.782071\pi$
$947$	−4.27570e7	−1.54929	−0.774645	+	0.632396i	$0.782071\pi$
$948$	0	0
$949$	2.73032e6	0.0984121
$950$	0	0
$951$	2.64338e7	0.947781
$952$	0	0
$953$	−2.11234e7	−0.753409	−0.376705	−	0.926333i	$-0.622943\pi$
$953$	−2.11234e7	−0.753409	−0.376705	+	0.926333i	$0.622943\pi$
$954$	0	0
$955$	−3.65041e7	−1.29519
$956$	0	0
$957$	−5.69335e6	−0.200950
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1.30596e7	−0.456166
$962$	0	0
$963$	3.43056e7	1.19206
$964$	0	0
$965$	2.88406e7	0.996977
$966$	0	0
$967$	2.70026e7	0.928623	0.464312	−	0.885672i	$-0.346302\pi$
$967$	2.70026e7	0.928623	0.464312	+	0.885672i	$0.346302\pi$
$968$	0	0
$969$	7.50759e6	0.256857
$970$	0	0
$971$	−2.61997e7	−0.891761	−0.445880	−	0.895093i	$-0.647109\pi$
$971$	−2.61997e7	−0.891761	−0.445880	+	0.895093i	$0.647109\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−296057.	−0.00997386
$976$	0	0
$977$	4.00062e7	1.34088	0.670441	−	0.741963i	$-0.266105\pi$
$977$	4.00062e7	1.34088	0.670441	+	0.741963i	$0.266105\pi$
$978$	0	0
$979$	2.68970e6	0.0896908
$980$	0	0
$981$	−2.99876e7	−0.994877
$982$	0	0
$983$	−3.00329e6	−0.0991319	−0.0495659	−	0.998771i	$-0.515784\pi$
$983$	−3.00329e6	−0.0991319	−0.0495659	+	0.998771i	$0.515784\pi$
$984$	0	0
$985$	−3.95529e6	−0.129894
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.18147e6	−0.0709185
$990$	0	0
$991$	−3.40747e7	−1.10217	−0.551083	−	0.834450i	$-0.685785\pi$
$991$	−3.40747e7	−1.10217	−0.551083	+	0.834450i	$0.685785\pi$
$992$	0	0
$993$	−1.36442e7	−0.439110
$994$	0	0
$995$	−4.83996e7	−1.54983
$996$	0	0
$997$	−5.26699e7	−1.67812	−0.839062	−	0.544035i	$-0.816896\pi$
$997$	−5.26699e7	−1.67812	−0.839062	+	0.544035i	$0.816896\pi$
$998$	0	0
$999$	8.08282e6	0.256241

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.6.a.h.1.3	yes	4
4.3	odd	2		784.6.a.bh.1.2		4
7.2	even	3		392.6.i.m.361.2		8
7.3	odd	6		392.6.i.m.177.3		8
7.4	even	3		392.6.i.m.177.2		8
7.5	odd	6		392.6.i.m.361.3		8
7.6	odd	2	inner	392.6.a.h.1.2	✓	4
28.27	even	2		784.6.a.bh.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.6.a.h.1.2	✓	4	7.6	odd	2	inner
392.6.a.h.1.3	yes	4	1.1	even	1	trivial
392.6.i.m.177.2		8	7.4	even	3
392.6.i.m.177.3		8	7.3	odd	6
392.6.i.m.361.2		8	7.2	even	3
392.6.i.m.361.3		8	7.5	odd	6
784.6.a.bh.1.2		4	4.3	odd	2
784.6.a.bh.1.3		4	28.27	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:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	392.a (trivial)

\(f(q)\)	\(=\)	\(q+8.21459 q^{3} +57.1619 q^{5} -175.520 q^{9} +O(q^{10})\)	\(q+8.21459 q^{3} +57.1619 q^{5} -175.520 q^{9} +176.520 q^{11} -252.951 q^{13} +469.561 q^{15} -1946.23 q^{17} -469.593 q^{19} +502.602 q^{23} +142.480 q^{25} -3437.97 q^{27} -3926.33 q^{29} -3945.82 q^{31} +1450.04 q^{33} -2351.04 q^{37} -2077.89 q^{39} +10309.0 q^{41} -4340.36 q^{43} -10033.1 q^{45} -25000.3 q^{47} -15987.5 q^{51} +13821.6 q^{53} +10090.2 q^{55} -3857.51 q^{57} +47285.8 q^{59} +11290.8 q^{61} -14459.2 q^{65} +26445.6 q^{67} +4128.67 q^{69} -63078.1 q^{71} -10793.9 q^{73} +1170.41 q^{75} -61419.5 q^{79} +14409.9 q^{81} +5383.09 q^{83} -111250. q^{85} -32253.2 q^{87} +15237.3 q^{89} -32413.3 q^{93} -26842.8 q^{95} -114148. q^{97} -30983.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 836 q^{9}+O(q^{10}) \) 4 * q + 836 * q^9	\( 4 q + 836 q^{9} - 832 q^{11} - 2736 q^{15} - 5680 q^{23} + 2108 q^{25} + 8904 q^{29} - 6328 q^{37} + 20912 q^{39} - 28128 q^{43} - 30112 q^{51} + 9144 q^{53} - 164624 q^{57} - 83984 q^{65} - 52640 q^{67} - 21600 q^{71} - 282592 q^{79} + 326804 q^{81} - 226592 q^{85} - 255776 q^{93} + 290992 q^{95} - 765312 q^{99}+O(q^{100}) \) 4 * q + 836 * q^9 - 832 * q^11 - 2736 * q^15 - 5680 * q^23 + 2108 * q^25 + 8904 * q^29 - 6328 * q^37 + 20912 * q^39 - 28128 * q^43 - 30112 * q^51 + 9144 * q^53 - 164624 * q^57 - 83984 * q^65 - 52640 * q^67 - 21600 * q^71 - 282592 * q^79 + 326804 * q^81 - 226592 * q^85 - 255776 * q^93 + 290992 * q^95 - 765312 * q^99

Introduction

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