LMFDB - Embedded newform 784.6.a.q.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [784,6,Mod(1,784)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [2,0,-14,0,-42,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$125.740914733$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{193}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 48 $ x^2 - x - 48
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.44622$ 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)

$f(q)$	$=$	$q+6.89244 q^{3} +48.4622 q^{5} -195.494 q^{9} +163.506 q^{11} +120.828 q^{13} +334.023 q^{15} -78.1920 q^{17} -2265.00 q^{19} +2451.95 q^{23} -776.413 q^{25} -3022.30 q^{27} +6985.27 q^{29} +2794.03 q^{31} +1126.95 q^{33} +9459.44 q^{37} +832.803 q^{39} -10088.5 q^{41} +6934.29 q^{43} -9474.08 q^{45} +1165.76 q^{47} -538.934 q^{51} +8562.42 q^{53} +7923.85 q^{55} -15611.4 q^{57} +6220.26 q^{59} +41928.9 q^{61} +5855.62 q^{65} -1812.09 q^{67} +16900.0 q^{69} +56823.3 q^{71} +44299.5 q^{73} -5351.38 q^{75} -34912.4 q^{79} +26674.1 q^{81} -39652.9 q^{83} -3789.36 q^{85} +48145.6 q^{87} +126299. q^{89} +19257.7 q^{93} -109767. q^{95} +145513. q^{97} -31964.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 14 q^{3} - 42 q^{5} - 2 q^{9} + 716 q^{11} + 714 q^{13} + 2224 q^{15} + 1344 q^{17} - 1946 q^{19} + 1792 q^{23} + 4282 q^{25} - 1988 q^{27} - 1200 q^{29} - 6804 q^{31} - 10416 q^{33} + 14640 q^{37}+ \cdots + 74940 q^{99}+O(q^{100}) $ 2 * q - 14 * q^3 - 42 * q^5 - 2 * q^9 + 716 * q^11 + 714 * q^13 + 2224 * q^15 + 1344 * q^17 - 1946 * q^19 + 1792 * q^23 + 4282 * q^25 - 1988 * q^27 - 1200 * q^29 - 6804 * q^31 - 10416 * q^33 + 14640 * q^37 - 11560 * q^39 - 7896 * q^41 - 524 * q^43 - 26978 * q^45 - 18396 * q^47 - 30252 * q^51 + 45132 * q^53 - 42056 * q^55 - 22276 * q^57 + 22582 * q^59 + 52822 * q^61 - 47804 * q^65 - 9848 * q^67 + 30688 * q^69 + 840 * q^71 + 122052 * q^73 - 111034 * q^75 - 31704 * q^79 - 41954 * q^81 + 36974 * q^83 - 132444 * q^85 + 219156 * q^87 + 210588 * q^89 + 219784 * q^93 - 138624 * q^95 + 44240 * q^97 + 74940 * q^99

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
$3$	6.89244	0.442150	0.221075	−	0.975257i	$-0.429043\pi$
$3$	6.89244	0.442150	0.221075	+	0.975257i	$0.429043\pi$
$4$	0	0
$5$	48.4622	0.866919	0.433459	−	0.901173i	$-0.357293\pi$
$5$	48.4622	0.866919	0.433459	+	0.901173i	$0.357293\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−195.494	−0.804503
$10$	0	0
$11$	163.506	0.407429	0.203714	−	0.979030i	$-0.434699\pi$
$11$	163.506	0.407429	0.203714	+	0.979030i	$0.434699\pi$
$12$	0	0
$13$	120.828	0.198295	0.0991473	−	0.995073i	$-0.468389\pi$
$13$	120.828	0.198295	0.0991473	+	0.995073i	$0.468389\pi$
$14$	0	0
$15$	334.023	0.383308
$16$	0	0
$17$	−78.1920	−0.0656206	−0.0328103	−	0.999462i	$-0.510446\pi$
$17$	−78.1920	−0.0656206	−0.0328103	+	0.999462i	$0.510446\pi$
$18$	0	0
$19$	−2265.00	−1.43941	−0.719704	−	0.694281i	$-0.755722\pi$
$19$	−2265.00	−1.43941	−0.719704	+	0.694281i	$0.755722\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2451.95	0.966480	0.483240	−	0.875488i	$-0.339460\pi$
$23$	2451.95	0.966480	0.483240	+	0.875488i	$0.339460\pi$
$24$	0	0
$25$	−776.413	−0.248452
$26$	0	0
$27$	−3022.30	−0.797862
$28$	0	0
$29$	6985.27	1.54237	0.771185	−	0.636611i	$-0.219664\pi$
$29$	6985.27	1.54237	0.771185	+	0.636611i	$0.219664\pi$
$30$	0	0
$31$	2794.03	0.522188	0.261094	−	0.965313i	$-0.415917\pi$
$31$	2794.03	0.522188	0.261094	+	0.965313i	$0.415917\pi$
$32$	0	0
$33$	1126.95	0.180145
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9459.44	1.13595	0.567977	−	0.823044i	$-0.307726\pi$
$37$	9459.44	1.13595	0.567977	+	0.823044i	$0.307726\pi$
$38$	0	0
$39$	832.803	0.0876760
$40$	0	0
$41$	−10088.5	−0.937271	−0.468636	−	0.883392i	$-0.655254\pi$
$41$	−10088.5	−0.937271	−0.468636	+	0.883392i	$0.655254\pi$
$42$	0	0
$43$	6934.29	0.571914	0.285957	−	0.958242i	$-0.407689\pi$
$43$	6934.29	0.571914	0.285957	+	0.958242i	$0.407689\pi$
$44$	0	0
$45$	−9474.08	−0.697439
$46$	0	0
$47$	1165.76	0.0769778	0.0384889	−	0.999259i	$-0.487746\pi$
$47$	1165.76	0.0769778	0.0384889	+	0.999259i	$0.487746\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−538.934	−0.0290142
$52$	0	0
$53$	8562.42	0.418704	0.209352	−	0.977840i	$-0.432865\pi$
$53$	8562.42	0.418704	0.209352	+	0.977840i	$0.432865\pi$
$54$	0	0
$55$	7923.85	0.353207
$56$	0	0
$57$	−15611.4	−0.636435
$58$	0	0
$59$	6220.26	0.232637	0.116318	−	0.993212i	$-0.462891\pi$
$59$	6220.26	0.232637	0.116318	+	0.993212i	$0.462891\pi$
$60$	0	0
$61$	41928.9	1.44274	0.721371	−	0.692549i	$-0.243512\pi$
$61$	41928.9	1.44274	0.721371	+	0.692549i	$0.243512\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5855.62	0.171905
$66$	0	0
$67$	−1812.09	−0.0493166	−0.0246583	−	0.999696i	$-0.507850\pi$
$67$	−1812.09	−0.0493166	−0.0246583	+	0.999696i	$0.507850\pi$
$68$	0	0
$69$	16900.0	0.427329
$70$	0	0
$71$	56823.3	1.33777	0.668884	−	0.743367i	$-0.266772\pi$
$71$	56823.3	1.33777	0.668884	+	0.743367i	$0.266772\pi$
$72$	0	0
$73$	44299.5	0.972953	0.486476	−	0.873694i	$-0.338282\pi$
$73$	44299.5	0.972953	0.486476	+	0.873694i	$0.338282\pi$
$74$	0	0
$75$	−5351.38	−0.109853
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−34912.4	−0.629379	−0.314690	−	0.949195i	$-0.601900\pi$
$79$	−34912.4	−0.629379	−0.314690	+	0.949195i	$0.601900\pi$
$80$	0	0
$81$	26674.1	0.451728
$82$	0	0
$83$	−39652.9	−0.631800	−0.315900	−	0.948793i	$-0.602306\pi$
$83$	−39652.9	−0.631800	−0.315900	+	0.948793i	$0.602306\pi$
$84$	0	0
$85$	−3789.36	−0.0568877
$86$	0	0
$87$	48145.6	0.681960
$88$	0	0
$89$	126299.	1.69015	0.845077	−	0.534645i	$-0.179555\pi$
$89$	126299.	1.69015	0.845077	+	0.534645i	$0.179555\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	19257.7	0.230886
$94$	0	0
$95$	−109767.	−1.24785
$96$	0	0
$97$	145513.	1.57026	0.785130	−	0.619331i	$-0.212596\pi$
$97$	145513.	1.57026	0.785130	+	0.619331i	$0.212596\pi$
$98$	0	0
$99$	−31964.4	−0.327777
$100$	0	0
$101$	−99545.5	−0.970998	−0.485499	−	0.874237i	$-0.661362\pi$
$101$	−99545.5	−0.970998	−0.485499	+	0.874237i	$0.661362\pi$
$102$	0	0
$103$	−129791.	−1.20546	−0.602729	−	0.797946i	$-0.705920\pi$
$103$	−129791.	−1.20546	−0.602729	+	0.797946i	$0.705920\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−218811.	−1.84761	−0.923804	−	0.382865i	$-0.874938\pi$
$107$	−218811.	−1.84761	−0.923804	+	0.382865i	$0.874938\pi$
$108$	0	0
$109$	201627.	1.62549	0.812743	−	0.582623i	$-0.197974\pi$
$109$	201627.	1.62549	0.812743	+	0.582623i	$0.197974\pi$
$110$	0	0
$111$	65198.6	0.502263
$112$	0	0
$113$	31803.9	0.234306	0.117153	−	0.993114i	$-0.462623\pi$
$113$	31803.9	0.234306	0.117153	+	0.993114i	$0.462623\pi$
$114$	0	0
$115$	118827.	0.837859
$116$	0	0
$117$	−23621.3	−0.159529
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−134317.	−0.834002
$122$	0	0
$123$	−69534.1	−0.414415
$124$	0	0
$125$	−189071.	−1.08231
$126$	0	0
$127$	318115.	1.75015	0.875075	−	0.483988i	$-0.160812\pi$
$127$	318115.	1.75015	0.875075	+	0.483988i	$0.160812\pi$
$128$	0	0
$129$	47794.2	0.252872
$130$	0	0
$131$	−305194.	−1.55381	−0.776905	−	0.629618i	$-0.783212\pi$
$131$	−305194.	−1.55381	−0.776905	+	0.629618i	$0.783212\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−146467.	−0.691681
$136$	0	0
$137$	180045.	0.819556	0.409778	−	0.912185i	$-0.365606\pi$
$137$	180045.	0.819556	0.409778	+	0.912185i	$0.365606\pi$
$138$	0	0
$139$	−323204.	−1.41886	−0.709430	−	0.704776i	$-0.751047\pi$
$139$	−323204.	−1.41886	−0.709430	+	0.704776i	$0.751047\pi$
$140$	0	0
$141$	8034.96	0.0340358
$142$	0	0
$143$	19756.2	0.0807909
$144$	0	0
$145$	338522.	1.33711
$146$	0	0
$147$	0	0
$148$	0	0
$149$	294159.	1.08547	0.542733	−	0.839905i	$-0.317390\pi$
$149$	294159.	1.08547	0.542733	+	0.839905i	$0.317390\pi$
$150$	0	0
$151$	334962.	1.19551	0.597754	−	0.801679i	$-0.296060\pi$
$151$	334962.	1.19551	0.597754	+	0.801679i	$0.296060\pi$
$152$	0	0
$153$	15286.1	0.0527919
$154$	0	0
$155$	135405.	0.452694
$156$	0	0
$157$	145150.	0.469967	0.234984	−	0.971999i	$-0.424496\pi$
$157$	145150.	0.469967	0.234984	+	0.971999i	$0.424496\pi$
$158$	0	0
$159$	59016.0	0.185130
$160$	0	0
$161$	0	0
$162$	0	0
$163$	176221.	0.519504	0.259752	−	0.965675i	$-0.416359\pi$
$163$	176221.	0.519504	0.259752	+	0.965675i	$0.416359\pi$
$164$	0	0
$165$	54614.7	0.156171
$166$	0	0
$167$	−45460.7	−0.126138	−0.0630689	−	0.998009i	$-0.520089\pi$
$167$	−45460.7	−0.126138	−0.0630689	+	0.998009i	$0.520089\pi$
$168$	0	0
$169$	−356693.	−0.960679
$170$	0	0
$171$	442794.	1.15801
$172$	0	0
$173$	205842.	0.522900	0.261450	−	0.965217i	$-0.415799\pi$
$173$	205842.	0.522900	0.261450	+	0.965217i	$0.415799\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	42872.8	0.102860
$178$	0	0
$179$	199205.	0.464695	0.232348	−	0.972633i	$-0.425359\pi$
$179$	199205.	0.464695	0.232348	+	0.972633i	$0.425359\pi$
$180$	0	0
$181$	−198411.	−0.450162	−0.225081	−	0.974340i	$-0.572265\pi$
$181$	−198411.	−0.450162	−0.225081	+	0.974340i	$0.572265\pi$
$182$	0	0
$183$	288992.	0.637909
$184$	0	0
$185$	458425.	0.984780
$186$	0	0
$187$	−12784.8	−0.0267357
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−372947.	−0.739714	−0.369857	−	0.929089i	$-0.620593\pi$
$191$	−372947.	−0.739714	−0.369857	+	0.929089i	$0.620593\pi$
$192$	0	0
$193$	175020.	0.338215	0.169108	−	0.985598i	$-0.445911\pi$
$193$	175020.	0.338215	0.169108	+	0.985598i	$0.445911\pi$
$194$	0	0
$195$	40359.5	0.0760080
$196$	0	0
$197$	883770.	1.62246	0.811229	−	0.584728i	$-0.198799\pi$
$197$	883770.	1.62246	0.811229	+	0.584728i	$0.198799\pi$
$198$	0	0
$199$	785843.	1.40671	0.703353	−	0.710841i	$-0.251686\pi$
$199$	785843.	1.40671	0.703353	+	0.710841i	$0.251686\pi$
$200$	0	0
$201$	−12489.7	−0.0218054
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−488909.	−0.812538
$206$	0	0
$207$	−479343.	−0.777536
$208$	0	0
$209$	−370340.	−0.586456
$210$	0	0
$211$	−763861.	−1.18116	−0.590579	−	0.806980i	$-0.701101\pi$
$211$	−763861.	−1.18116	−0.590579	+	0.806980i	$0.701101\pi$
$212$	0	0
$213$	391652.	0.591495
$214$	0	0
$215$	336051.	0.495803
$216$	0	0
$217$	0	0
$218$	0	0
$219$	305332.	0.430191
$220$	0	0
$221$	−9447.82	−0.0130122
$222$	0	0
$223$	−204400.	−0.275245	−0.137623	−	0.990485i	$-0.543946\pi$
$223$	−204400.	−0.275245	−0.137623	+	0.990485i	$0.543946\pi$
$224$	0	0
$225$	151784.	0.199881
$226$	0	0
$227$	1.00806e6	1.29844	0.649218	−	0.760603i	$-0.275096\pi$
$227$	1.00806e6	1.29844	0.649218	+	0.760603i	$0.275096\pi$
$228$	0	0
$229$	244324.	0.307877	0.153938	−	0.988080i	$-0.450804\pi$
$229$	244324.	0.307877	0.153938	+	0.988080i	$0.450804\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20822.9	0.0251276	0.0125638	−	0.999921i	$-0.496001\pi$
$233$	20822.9	0.0251276	0.0125638	+	0.999921i	$0.496001\pi$
$234$	0	0
$235$	56495.5	0.0667335
$236$	0	0
$237$	−240632.	−0.278280
$238$	0	0
$239$	474033.	0.536801	0.268401	−	0.963307i	$-0.413505\pi$
$239$	474033.	0.536801	0.268401	+	0.963307i	$0.413505\pi$
$240$	0	0
$241$	1.70390e6	1.88974	0.944869	−	0.327450i	$-0.106189\pi$
$241$	1.70390e6	1.88974	0.944869	+	0.327450i	$0.106189\pi$
$242$	0	0
$243$	918268.	0.997594
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−273676.	−0.285427
$248$	0	0
$249$	−273305.	−0.279351
$250$	0	0
$251$	−810918.	−0.812442	−0.406221	−	0.913775i	$-0.633154\pi$
$251$	−810918.	−0.812442	−0.406221	+	0.913775i	$0.633154\pi$
$252$	0	0
$253$	400909.	0.393771
$254$	0	0
$255$	−26117.9	−0.0251529
$256$	0	0
$257$	−1.26657e6	−1.19618	−0.598088	−	0.801431i	$-0.704073\pi$
$257$	−1.26657e6	−1.19618	−0.598088	+	0.801431i	$0.704073\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.36558e6	−1.24084
$262$	0	0
$263$	1.96048e6	1.74773	0.873863	−	0.486172i	$-0.161607\pi$
$263$	1.96048e6	1.74773	0.873863	+	0.486172i	$0.161607\pi$
$264$	0	0
$265$	414954.	0.362982
$266$	0	0
$267$	870511.	0.747302
$268$	0	0
$269$	1.49358e6	1.25848	0.629242	−	0.777209i	$-0.283365\pi$
$269$	1.49358e6	1.25848	0.629242	+	0.777209i	$0.283365\pi$
$270$	0	0
$271$	1.23626e6	1.02255	0.511275	−	0.859417i	$-0.329173\pi$
$271$	1.23626e6	1.02255	0.511275	+	0.859417i	$0.329173\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−126948.	−0.101227
$276$	0	0
$277$	−170150.	−0.133239	−0.0666197	−	0.997778i	$-0.521221\pi$
$277$	−170150.	−0.133239	−0.0666197	+	0.997778i	$0.521221\pi$
$278$	0	0
$279$	−546217.	−0.420102
$280$	0	0
$281$	1.29957e6	0.981824	0.490912	−	0.871209i	$-0.336664\pi$
$281$	1.29957e6	0.981824	0.490912	+	0.871209i	$0.336664\pi$
$282$	0	0
$283$	−374032.	−0.277615	−0.138807	−	0.990319i	$-0.544327\pi$
$283$	−374032.	−0.277615	−0.138807	+	0.990319i	$0.544327\pi$
$284$	0	0
$285$	−756561.	−0.551737
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.41374e6	−0.995694
$290$	0	0
$291$	1.00294e6	0.694291
$292$	0	0
$293$	−1.15022e6	−0.782728	−0.391364	−	0.920236i	$-0.627997\pi$
$293$	−1.15022e6	−0.782728	−0.391364	+	0.920236i	$0.627997\pi$
$294$	0	0
$295$	301448.	0.201677
$296$	0	0
$297$	−494163.	−0.325072
$298$	0	0
$299$	296266.	0.191648
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−686112.	−0.429327
$304$	0	0
$305$	2.03197e6	1.25074
$306$	0	0
$307$	2.61214e6	1.58179	0.790897	−	0.611949i	$-0.209614\pi$
$307$	2.61214e6	1.58179	0.790897	+	0.611949i	$0.209614\pi$
$308$	0	0
$309$	−894578.	−0.532994
$310$	0	0
$311$	−1.40646e6	−0.824566	−0.412283	−	0.911056i	$-0.635269\pi$
$311$	−1.40646e6	−0.824566	−0.412283	+	0.911056i	$0.635269\pi$
$312$	0	0
$313$	−2.37930e6	−1.37274	−0.686369	−	0.727253i	$-0.740797\pi$
$313$	−2.37930e6	−1.37274	−0.686369	+	0.727253i	$0.740797\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−563199.	−0.314785	−0.157393	−	0.987536i	$-0.550309\pi$
$317$	−563199.	−0.314785	−0.157393	+	0.987536i	$0.550309\pi$
$318$	0	0
$319$	1.14213e6	0.628405
$320$	0	0
$321$	−1.50814e6	−0.816921
$322$	0	0
$323$	177105.	0.0944547
$324$	0	0
$325$	−93812.8	−0.0492667
$326$	0	0
$327$	1.38970e6	0.718709
$328$	0	0
$329$	0	0
$330$	0	0
$331$	850649.	0.426757	0.213378	−	0.976970i	$-0.431553\pi$
$331$	850649.	0.426757	0.213378	+	0.976970i	$0.431553\pi$
$332$	0	0
$333$	−1.84927e6	−0.913879
$334$	0	0
$335$	−87818.0	−0.0427535
$336$	0	0
$337$	4.01506e6	1.92582	0.962912	−	0.269814i	$-0.0869623\pi$
$337$	4.01506e6	1.92582	0.962912	+	0.269814i	$0.0869623\pi$
$338$	0	0
$339$	219207.	0.103599
$340$	0	0
$341$	456840.	0.212754
$342$	0	0
$343$	0	0
$344$	0	0
$345$	819009.	0.370460
$346$	0	0
$347$	2.39961e6	1.06984	0.534918	−	0.844904i	$-0.320343\pi$
$347$	2.39961e6	1.06984	0.534918	+	0.844904i	$0.320343\pi$
$348$	0	0
$349$	919171.	0.403955	0.201977	−	0.979390i	$-0.435263\pi$
$349$	919171.	0.403955	0.201977	+	0.979390i	$0.435263\pi$
$350$	0	0
$351$	−365179.	−0.158212
$352$	0	0
$353$	−1.07684e6	−0.459953	−0.229976	−	0.973196i	$-0.573865\pi$
$353$	−1.07684e6	−0.459953	−0.229976	+	0.973196i	$0.573865\pi$
$354$	0	0
$355$	2.75378e6	1.15974
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.24219e6	0.508688	0.254344	−	0.967114i	$-0.418140\pi$
$359$	1.24219e6	0.508688	0.254344	+	0.967114i	$0.418140\pi$
$360$	0	0
$361$	2.65411e6	1.07189
$362$	0	0
$363$	−925771.	−0.368754
$364$	0	0
$365$	2.14685e6	0.843471
$366$	0	0
$367$	4.17293e6	1.61725	0.808624	−	0.588326i	$-0.200213\pi$
$367$	4.17293e6	1.61725	0.808624	+	0.588326i	$0.200213\pi$
$368$	0	0
$369$	1.97224e6	0.754037
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−1.72238e6	−0.640999	−0.320500	−	0.947249i	$-0.603851\pi$
$373$	−1.72238e6	−0.640999	−0.320500	+	0.947249i	$0.603851\pi$
$374$	0	0
$375$	−1.30316e6	−0.478542
$376$	0	0
$377$	844020.	0.305844
$378$	0	0
$379$	−1.72998e6	−0.618647	−0.309324	−	0.950957i	$-0.600103\pi$
$379$	−1.72998e6	−0.618647	−0.309324	+	0.950957i	$0.600103\pi$
$380$	0	0
$381$	2.19259e6	0.773830
$382$	0	0
$383$	−3.34881e6	−1.16652	−0.583262	−	0.812284i	$-0.698224\pi$
$383$	−3.34881e6	−1.16652	−0.583262	+	0.812284i	$0.698224\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.35561e6	−0.460106
$388$	0	0
$389$	−5.18691e6	−1.73794	−0.868970	−	0.494864i	$-0.835218\pi$
$389$	−5.18691e6	−1.73794	−0.868970	+	0.494864i	$0.835218\pi$
$390$	0	0
$391$	−191723.	−0.0634209
$392$	0	0
$393$	−2.10353e6	−0.687018
$394$	0	0
$395$	−1.69193e6	−0.545621
$396$	0	0
$397$	1.30271e6	0.414830	0.207415	−	0.978253i	$-0.433495\pi$
$397$	1.30271e6	0.414830	0.207415	+	0.978253i	$0.433495\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.53036e6	−0.475261	−0.237631	−	0.971356i	$-0.576371\pi$
$401$	−1.53036e6	−0.475261	−0.237631	+	0.971356i	$0.576371\pi$
$402$	0	0
$403$	337598.	0.103547
$404$	0	0
$405$	1.29269e6	0.391611
$406$	0	0
$407$	1.54667e6	0.462820
$408$	0	0
$409$	2.93690e6	0.868123	0.434062	−	0.900883i	$-0.357080\pi$
$409$	2.93690e6	0.868123	0.434062	+	0.900883i	$0.357080\pi$
$410$	0	0
$411$	1.24095e6	0.362367
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−1.92167e6	−0.547719
$416$	0	0
$417$	−2.22766e6	−0.627349
$418$	0	0
$419$	−1.16465e6	−0.324085	−0.162043	−	0.986784i	$-0.551808\pi$
$419$	−1.16465e6	−0.324085	−0.162043	+	0.986784i	$0.551808\pi$
$420$	0	0
$421$	−1.58204e6	−0.435022	−0.217511	−	0.976058i	$-0.569794\pi$
$421$	−1.58204e6	−0.435022	−0.217511	+	0.976058i	$0.569794\pi$
$422$	0	0
$423$	−227900.	−0.0619289
$424$	0	0
$425$	60709.3	0.0163036
$426$	0	0
$427$	0	0
$428$	0	0
$429$	136168.	0.0357217
$430$	0	0
$431$	−6.10233e6	−1.58235	−0.791174	−	0.611591i	$-0.790530\pi$
$431$	−6.10233e6	−1.58235	−0.791174	+	0.611591i	$0.790530\pi$
$432$	0	0
$433$	−1.08115e6	−0.277118	−0.138559	−	0.990354i	$-0.544247\pi$
$433$	−1.08115e6	−0.277118	−0.138559	+	0.990354i	$0.544247\pi$
$434$	0	0
$435$	2.33324e6	0.591203
$436$	0	0
$437$	−5.55367e6	−1.39116
$438$	0	0
$439$	−7.04807e6	−1.74546	−0.872728	−	0.488207i	$-0.837651\pi$
$439$	−7.04807e6	−1.74546	−0.872728	+	0.488207i	$0.837651\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.32031e6	0.561742	0.280871	−	0.959746i	$-0.409377\pi$
$443$	2.32031e6	0.561742	0.280871	+	0.959746i	$0.409377\pi$
$444$	0	0
$445$	6.12075e6	1.46523
$446$	0	0
$447$	2.02747e6	0.479939
$448$	0	0
$449$	−5.45039e6	−1.27589	−0.637943	−	0.770084i	$-0.720214\pi$
$449$	−5.45039e6	−1.27589	−0.637943	+	0.770084i	$0.720214\pi$
$450$	0	0
$451$	−1.64952e6	−0.381871
$452$	0	0
$453$	2.30871e6	0.528595
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.62676e6	1.48426	0.742132	−	0.670254i	$-0.233815\pi$
$457$	6.62676e6	1.48426	0.742132	+	0.670254i	$0.233815\pi$
$458$	0	0
$459$	236319.	0.0523561
$460$	0	0
$461$	−606397.	−0.132894	−0.0664469	−	0.997790i	$-0.521166\pi$
$461$	−606397.	−0.132894	−0.0664469	+	0.997790i	$0.521166\pi$
$462$	0	0
$463$	−3.65855e6	−0.793153	−0.396576	−	0.918002i	$-0.629802\pi$
$463$	−3.65855e6	−0.793153	−0.396576	+	0.918002i	$0.629802\pi$
$464$	0	0
$465$	933271.	0.200159
$466$	0	0
$467$	5.62712e6	1.19397	0.596986	−	0.802252i	$-0.296365\pi$
$467$	5.62712e6	1.19397	0.596986	+	0.802252i	$0.296365\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	1.00044e6	0.207796
$472$	0	0
$473$	1.13380e6	0.233014
$474$	0	0
$475$	1.75857e6	0.357624
$476$	0	0
$477$	−1.67390e6	−0.336848
$478$	0	0
$479$	4.25107e6	0.846563	0.423281	−	0.905998i	$-0.360878\pi$
$479$	4.25107e6	0.846563	0.423281	+	0.905998i	$0.360878\pi$
$480$	0	0
$481$	1.14297e6	0.225254
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.05187e6	1.36129
$486$	0	0
$487$	−5.28756e6	−1.01026	−0.505130	−	0.863043i	$-0.668555\pi$
$487$	−5.28756e6	−1.01026	−0.505130	+	0.863043i	$0.668555\pi$
$488$	0	0
$489$	1.21459e6	0.229699
$490$	0	0
$491$	3.33842e6	0.624939	0.312469	−	0.949928i	$-0.398844\pi$
$491$	3.33842e6	0.624939	0.312469	+	0.949928i	$0.398844\pi$
$492$	0	0
$493$	−546192.	−0.101211
$494$	0	0
$495$	−1.54907e6	−0.284156
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−2.52847e6	−0.454576	−0.227288	−	0.973828i	$-0.572986\pi$
$499$	−2.52847e6	−0.454576	−0.227288	+	0.973828i	$0.572986\pi$
$500$	0	0
$501$	−313335.	−0.0557719
$502$	0	0
$503$	−5.94498e6	−1.04768	−0.523842	−	0.851815i	$-0.675502\pi$
$503$	−5.94498e6	−1.04768	−0.523842	+	0.851815i	$0.675502\pi$
$504$	0	0
$505$	−4.82420e6	−0.841776
$506$	0	0
$507$	−2.45849e6	−0.424765
$508$	0	0
$509$	603525.	0.103253	0.0516263	−	0.998666i	$-0.483560\pi$
$509$	603525.	0.103253	0.0516263	+	0.998666i	$0.483560\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	6.84549e6	1.14845
$514$	0	0
$515$	−6.28997e6	−1.04503
$516$	0	0
$517$	190609.	0.0313630
$518$	0	0
$519$	1.41875e6	0.231200
$520$	0	0
$521$	−1.03389e7	−1.66870	−0.834350	−	0.551235i	$-0.814157\pi$
$521$	−1.03389e7	−1.66870	−0.834350	+	0.551235i	$0.814157\pi$
$522$	0	0
$523$	−8.82310e6	−1.41048	−0.705240	−	0.708968i	$-0.749161\pi$
$523$	−8.82310e6	−1.41048	−0.705240	+	0.708968i	$0.749161\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−218471.	−0.0342663
$528$	0	0
$529$	−424266.	−0.0659172
$530$	0	0
$531$	−1.21602e6	−0.187157
$532$	0	0
$533$	−1.21897e6	−0.185856
$534$	0	0
$535$	−1.06041e7	−1.60173
$536$	0	0
$537$	1.37301e6	0.205465
$538$	0	0
$539$	0	0
$540$	0	0
$541$	1.16811e6	0.171590	0.0857949	−	0.996313i	$-0.472657\pi$
$541$	1.16811e6	0.171590	0.0857949	+	0.996313i	$0.472657\pi$
$542$	0	0
$543$	−1.36753e6	−0.199039
$544$	0	0
$545$	9.77130e6	1.40916
$546$	0	0
$547$	−9.17075e6	−1.31050	−0.655249	−	0.755413i	$-0.727436\pi$
$547$	−9.17075e6	−1.31050	−0.655249	+	0.755413i	$0.727436\pi$
$548$	0	0
$549$	−8.19685e6	−1.16069
$550$	0	0
$551$	−1.58216e7	−2.22010
$552$	0	0
$553$	0	0
$554$	0	0
$555$	3.15967e6	0.435421
$556$	0	0
$557$	6.22421e6	0.850054	0.425027	−	0.905181i	$-0.360265\pi$
$557$	6.22421e6	0.850054	0.425027	+	0.905181i	$0.360265\pi$
$558$	0	0
$559$	837859.	0.113407
$560$	0	0
$561$	−88118.8	−0.0118212
$562$	0	0
$563$	−5.64042e6	−0.749964	−0.374982	−	0.927032i	$-0.622351\pi$
$563$	−5.64042e6	−0.749964	−0.374982	+	0.927032i	$0.622351\pi$
$564$	0	0
$565$	1.54129e6	0.203125
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.05286e7	−1.36330	−0.681650	−	0.731678i	$-0.738737\pi$
$569$	−1.05286e7	−1.36330	−0.681650	+	0.731678i	$0.738737\pi$
$570$	0	0
$571$	7.31395e6	0.938776	0.469388	−	0.882992i	$-0.344475\pi$
$571$	7.31395e6	0.938776	0.469388	+	0.882992i	$0.344475\pi$
$572$	0	0
$573$	−2.57052e6	−0.327065
$574$	0	0
$575$	−1.90373e6	−0.240124
$576$	0	0
$577$	−2.66534e6	−0.333282	−0.166641	−	0.986018i	$-0.553292\pi$
$577$	−2.66534e6	−0.333282	−0.166641	+	0.986018i	$0.553292\pi$
$578$	0	0
$579$	1.20631e6	0.149542
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.40000e6	0.170592
$584$	0	0
$585$	−1.14474e6	−0.138298
$586$	0	0
$587$	4.92431e6	0.589861	0.294931	−	0.955519i	$-0.404703\pi$
$587$	4.92431e6	0.589861	0.294931	+	0.955519i	$0.404703\pi$
$588$	0	0
$589$	−6.32847e6	−0.751641
$590$	0	0
$591$	6.09133e6	0.717371
$592$	0	0
$593$	5.08735e6	0.594094	0.297047	−	0.954863i	$-0.403998\pi$
$593$	5.08735e6	0.594094	0.297047	+	0.954863i	$0.403998\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5.41638e6	0.621975
$598$	0	0
$599$	−4.64678e6	−0.529157	−0.264579	−	0.964364i	$-0.585233\pi$
$599$	−4.64678e6	−0.529157	−0.264579	+	0.964364i	$0.585233\pi$
$600$	0	0
$601$	4.35424e6	0.491730	0.245865	−	0.969304i	$-0.420928\pi$
$601$	4.35424e6	0.491730	0.245865	+	0.969304i	$0.420928\pi$
$602$	0	0
$603$	354254.	0.0396754
$604$	0	0
$605$	−6.50929e6	−0.723012
$606$	0	0
$607$	−1.25461e7	−1.38209	−0.691043	−	0.722813i	$-0.742849\pi$
$607$	−1.25461e7	−1.38209	−0.691043	+	0.722813i	$0.742849\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	140857.	0.0152643
$612$	0	0
$613$	−4.07637e6	−0.438149	−0.219075	−	0.975708i	$-0.570304\pi$
$613$	−4.07637e6	−0.438149	−0.219075	+	0.975708i	$0.570304\pi$
$614$	0	0
$615$	−3.36978e6	−0.359264
$616$	0	0
$617$	−8.88533e6	−0.939639	−0.469819	−	0.882763i	$-0.655681\pi$
$617$	−8.88533e6	−0.939639	−0.469819	+	0.882763i	$0.655681\pi$
$618$	0	0
$619$	−1.28530e7	−1.34827	−0.674136	−	0.738607i	$-0.735484\pi$
$619$	−1.28530e7	−1.34827	−0.674136	+	0.738607i	$0.735484\pi$
$620$	0	0
$621$	−7.41053e6	−0.771117
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−6.73652e6	−0.689819
$626$	0	0
$627$	−2.55255e6	−0.259302
$628$	0	0
$629$	−739652.	−0.0745420
$630$	0	0
$631$	288616.	0.0288567	0.0144283	−	0.999896i	$-0.495407\pi$
$631$	288616.	0.0288567	0.0144283	+	0.999896i	$0.495407\pi$
$632$	0	0
$633$	−5.26487e6	−0.522250
$634$	0	0
$635$	1.54166e7	1.51724
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1.11086e7	−1.07624
$640$	0	0
$641$	−1.94591e7	−1.87058	−0.935291	−	0.353880i	$-0.884862\pi$
$641$	−1.94591e7	−1.87058	−0.935291	+	0.353880i	$0.884862\pi$
$642$	0	0
$643$	1.30171e7	1.24161	0.620805	−	0.783965i	$-0.286806\pi$
$643$	1.30171e7	1.24161	0.620805	+	0.783965i	$0.286806\pi$
$644$	0	0
$645$	2.31621e6	0.219219
$646$	0	0
$647$	6.42864e6	0.603751	0.301876	−	0.953347i	$-0.402387\pi$
$647$	6.42864e6	0.603751	0.301876	+	0.953347i	$0.402387\pi$
$648$	0	0
$649$	1.01705e6	0.0947829
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.00202e7	0.919592	0.459796	−	0.888025i	$-0.347923\pi$
$653$	1.00202e7	0.919592	0.459796	+	0.888025i	$0.347923\pi$
$654$	0	0
$655$	−1.47904e7	−1.34703
$656$	0	0
$657$	−8.66030e6	−0.782743
$658$	0	0
$659$	−1.49356e7	−1.33971	−0.669854	−	0.742493i	$-0.733643\pi$
$659$	−1.49356e7	−1.33971	−0.669854	+	0.742493i	$0.733643\pi$
$660$	0	0
$661$	1.35055e6	0.120229	0.0601143	−	0.998191i	$-0.480853\pi$
$661$	1.35055e6	0.120229	0.0601143	+	0.998191i	$0.480853\pi$
$662$	0	0
$663$	−65118.5	−0.00575335
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.71276e7	1.49067
$668$	0	0
$669$	−1.40882e6	−0.121700
$670$	0	0
$671$	6.85561e6	0.587814
$672$	0	0
$673$	6.59401e6	0.561193	0.280596	−	0.959826i	$-0.409468\pi$
$673$	6.59401e6	0.561193	0.280596	+	0.959826i	$0.409468\pi$
$674$	0	0
$675$	2.34655e6	0.198231
$676$	0	0
$677$	−1.42517e7	−1.19508	−0.597538	−	0.801840i	$-0.703854\pi$
$677$	−1.42517e7	−1.19508	−0.597538	+	0.801840i	$0.703854\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	6.94797e6	0.574104
$682$	0	0
$683$	7.65006e6	0.627499	0.313749	−	0.949506i	$-0.398415\pi$
$683$	7.65006e6	0.627499	0.313749	+	0.949506i	$0.398415\pi$
$684$	0	0
$685$	8.72536e6	0.710489
$686$	0	0
$687$	1.68399e6	0.136128
$688$	0	0
$689$	1.03458e6	0.0830266
$690$	0	0
$691$	1.38655e7	1.10469	0.552344	−	0.833616i	$-0.313733\pi$
$691$	1.38655e7	1.10469	0.552344	+	0.833616i	$0.313733\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.56632e7	−1.23004
$696$	0	0
$697$	788837.	0.0615042
$698$	0	0
$699$	143521.	0.0111102
$700$	0	0
$701$	−2.38825e6	−0.183563	−0.0917814	−	0.995779i	$-0.529256\pi$
$701$	−2.38825e6	−0.183563	−0.0917814	+	0.995779i	$0.529256\pi$
$702$	0	0
$703$	−2.14256e7	−1.63510
$704$	0	0
$705$	389392.	0.0295063
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1.55641e7	−1.16281	−0.581404	−	0.813615i	$-0.697496\pi$
$709$	−1.55641e7	−1.16281	−0.581404	+	0.813615i	$0.697496\pi$
$710$	0	0
$711$	6.82518e6	0.506337
$712$	0	0
$713$	6.85083e6	0.504684
$714$	0	0
$715$	957427.	0.0700391
$716$	0	0
$717$	3.26724e6	0.237347
$718$	0	0
$719$	1.82555e7	1.31696	0.658478	−	0.752600i	$-0.271200\pi$
$719$	1.82555e7	1.31696	0.658478	+	0.752600i	$0.271200\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1.17440e7	0.835548
$724$	0	0
$725$	−5.42346e6	−0.383205
$726$	0	0
$727$	−2.11427e6	−0.148362	−0.0741812	−	0.997245i	$-0.523634\pi$
$727$	−2.11427e6	−0.148362	−0.0741812	+	0.997245i	$0.523634\pi$
$728$	0	0
$729$	−152693.	−0.0106414
$730$	0	0
$731$	−542206.	−0.0375293
$732$	0	0
$733$	−2.12932e6	−0.146380	−0.0731898	−	0.997318i	$-0.523318\pi$
$733$	−2.12932e6	−0.146380	−0.0731898	+	0.997318i	$0.523318\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−296288.	−0.0200930
$738$	0	0
$739$	8.08671e6	0.544704	0.272352	−	0.962198i	$-0.412198\pi$
$739$	8.08671e6	0.544704	0.272352	+	0.962198i	$0.412198\pi$
$740$	0	0
$741$	−1.88630e6	−0.126202
$742$	0	0
$743$	1.18944e7	0.790442	0.395221	−	0.918586i	$-0.370668\pi$
$743$	1.18944e7	0.790442	0.395221	+	0.918586i	$0.370668\pi$
$744$	0	0
$745$	1.42556e7	0.941010
$746$	0	0
$747$	7.75191e6	0.508285
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1.85538e7	1.20042	0.600211	−	0.799842i	$-0.295083\pi$
$751$	1.85538e7	1.20042	0.600211	+	0.799842i	$0.295083\pi$
$752$	0	0
$753$	−5.58921e6	−0.359222
$754$	0	0
$755$	1.62330e7	1.03641
$756$	0	0
$757$	9.32534e6	0.591459	0.295730	−	0.955272i	$-0.404437\pi$
$757$	9.32534e6	0.591459	0.295730	+	0.955272i	$0.404437\pi$
$758$	0	0
$759$	2.76324e6	0.174106
$760$	0	0
$761$	−8.00756e6	−0.501232	−0.250616	−	0.968087i	$-0.580633\pi$
$761$	−8.00756e6	−0.501232	−0.250616	+	0.968087i	$0.580633\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	740797.	0.0457663
$766$	0	0
$767$	751584.	0.0461306
$768$	0	0
$769$	−2.02720e7	−1.23618	−0.618089	−	0.786108i	$-0.712093\pi$
$769$	−2.02720e7	−1.23618	−0.618089	+	0.786108i	$0.712093\pi$
$770$	0	0
$771$	−8.72973e6	−0.528889
$772$	0	0
$773$	−2.87881e6	−0.173286	−0.0866431	−	0.996239i	$-0.527614\pi$
$773$	−2.87881e6	−0.173286	−0.0866431	+	0.996239i	$0.527614\pi$
$774$	0	0
$775$	−2.16932e6	−0.129739
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.28503e7	1.34911
$780$	0	0
$781$	9.29094e6	0.545045
$782$	0	0
$783$	−2.11116e7	−1.23060
$784$	0	0
$785$	7.03428e6	0.407423
$786$	0	0
$787$	−2.03508e7	−1.17124	−0.585618	−	0.810587i	$-0.699148\pi$
$787$	−2.03508e7	−1.17124	−0.585618	+	0.810587i	$0.699148\pi$
$788$	0	0
$789$	1.35125e7	0.772758
$790$	0	0
$791$	0	0
$792$	0	0
$793$	5.06620e6	0.286088
$794$	0	0
$795$	2.86005e6	0.160493
$796$	0	0
$797$	2.24765e7	1.25338	0.626689	−	0.779269i	$-0.284410\pi$
$797$	2.24765e7	1.25338	0.626689	+	0.779269i	$0.284410\pi$
$798$	0	0
$799$	−91153.3	−0.00505133
$800$	0	0
$801$	−2.46908e7	−1.35973
$802$	0	0
$803$	7.24322e6	0.396409
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.02944e7	0.556439
$808$	0	0
$809$	−1.15427e6	−0.0620062	−0.0310031	−	0.999519i	$-0.509870\pi$
$809$	−1.15427e6	−0.0620062	−0.0310031	+	0.999519i	$0.509870\pi$
$810$	0	0
$811$	−2.26698e7	−1.21031	−0.605154	−	0.796108i	$-0.706888\pi$
$811$	−2.26698e7	−1.21031	−0.605154	+	0.796108i	$0.706888\pi$
$812$	0	0
$813$	8.52082e6	0.452121
$814$	0	0
$815$	8.54007e6	0.450368
$816$	0	0
$817$	−1.57061e7	−0.823217
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.41657e6	−0.125124	−0.0625622	−	0.998041i	$-0.519927\pi$
$821$	−2.41657e6	−0.125124	−0.0625622	+	0.998041i	$0.519927\pi$
$822$	0	0
$823$	2.10169e6	0.108161	0.0540804	−	0.998537i	$-0.482777\pi$
$823$	2.10169e6	0.108161	0.0540804	+	0.998537i	$0.482777\pi$
$824$	0	0
$825$	−874982.	−0.0447574
$826$	0	0
$827$	−1.53997e7	−0.782977	−0.391489	−	0.920183i	$-0.628040\pi$
$827$	−1.53997e7	−0.782977	−0.391489	+	0.920183i	$0.628040\pi$
$828$	0	0
$829$	2.33839e7	1.18177	0.590883	−	0.806757i	$-0.298779\pi$
$829$	2.33839e7	1.18177	0.590883	+	0.806757i	$0.298779\pi$
$830$	0	0
$831$	−1.17275e6	−0.0589118
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−2.20313e6	−0.109351
$836$	0	0
$837$	−8.44439e6	−0.416634
$838$	0	0
$839$	2.03016e7	0.995695	0.497848	−	0.867265i	$-0.334124\pi$
$839$	2.03016e7	0.995695	0.497848	+	0.867265i	$0.334124\pi$
$840$	0	0
$841$	2.82829e7	1.37890
$842$	0	0
$843$	8.95721e6	0.434114
$844$	0	0
$845$	−1.72862e7	−0.832831
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−2.57800e6	−0.122748
$850$	0	0
$851$	2.31941e7	1.09788
$852$	0	0
$853$	−3.35163e7	−1.57719	−0.788594	−	0.614914i	$-0.789191\pi$
$853$	−3.35163e7	−1.57719	−0.788594	+	0.614914i	$0.789191\pi$
$854$	0	0
$855$	2.14588e7	1.00390
$856$	0	0
$857$	2.30661e7	1.07281	0.536405	−	0.843961i	$-0.319782\pi$
$857$	2.30661e7	1.07281	0.536405	+	0.843961i	$0.319782\pi$
$858$	0	0
$859$	3.85609e7	1.78305	0.891527	−	0.452968i	$-0.149635\pi$
$859$	3.85609e7	1.78305	0.891527	+	0.452968i	$0.149635\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.86892e7	−1.76833	−0.884163	−	0.467179i	$-0.845271\pi$
$863$	−3.86892e7	−1.76833	−0.884163	+	0.467179i	$0.845271\pi$
$864$	0	0
$865$	9.97555e6	0.453311
$866$	0	0
$867$	−9.74414e6	−0.440247
$868$	0	0
$869$	−5.70838e6	−0.256427
$870$	0	0
$871$	−218952.	−0.00977922
$872$	0	0
$873$	−2.84469e7	−1.26328
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.17369e7	−0.954332	−0.477166	−	0.878813i	$-0.658336\pi$
$877$	−2.17369e7	−0.954332	−0.477166	+	0.878813i	$0.658336\pi$
$878$	0	0
$879$	−7.92781e6	−0.346084
$880$	0	0
$881$	−8.42086e6	−0.365525	−0.182762	−	0.983157i	$-0.558504\pi$
$881$	−8.42086e6	−0.365525	−0.182762	+	0.983157i	$0.558504\pi$
$882$	0	0
$883$	−3.22954e7	−1.39392	−0.696962	−	0.717108i	$-0.745465\pi$
$883$	−3.22954e7	−1.39392	−0.696962	+	0.717108i	$0.745465\pi$
$884$	0	0
$885$	2.07771e6	0.0891716
$886$	0	0
$887$	−4.21800e7	−1.80010	−0.900052	−	0.435782i	$-0.856472\pi$
$887$	−4.21800e7	−1.80010	−0.900052	+	0.435782i	$0.856472\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	4.36137e6	0.184047
$892$	0	0
$893$	−2.64045e6	−0.110802
$894$	0	0
$895$	9.65393e6	0.402853
$896$	0	0
$897$	2.04200e6	0.0847371
$898$	0	0
$899$	1.95171e7	0.805407
$900$	0	0
$901$	−669512.	−0.0274756
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9.61542e6	−0.390254
$906$	0	0
$907$	7.64517e6	0.308581	0.154291	−	0.988026i	$-0.450691\pi$
$907$	7.64517e6	0.308581	0.154291	+	0.988026i	$0.450691\pi$
$908$	0	0
$909$	1.94606e7	0.781170
$910$	0	0
$911$	3.93310e7	1.57014	0.785072	−	0.619405i	$-0.212626\pi$
$911$	3.93310e7	1.57014	0.785072	+	0.619405i	$0.212626\pi$
$912$	0	0
$913$	−6.48347e6	−0.257413
$914$	0	0
$915$	1.40052e7	0.553015
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−3.52204e7	−1.37564	−0.687822	−	0.725880i	$-0.741433\pi$
$919$	−3.52204e7	−1.37564	−0.687822	+	0.725880i	$0.741433\pi$
$920$	0	0
$921$	1.80040e7	0.699391
$922$	0	0
$923$	6.86587e6	0.265272
$924$	0	0
$925$	−7.34443e6	−0.282230
$926$	0	0
$927$	2.53734e7	0.969794
$928$	0	0
$929$	1.66300e7	0.632199	0.316099	−	0.948726i	$-0.397627\pi$
$929$	1.66300e7	0.632199	0.316099	+	0.948726i	$0.397627\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−9.69393e6	−0.364582
$934$	0	0
$935$	−619582.	−0.0231777
$936$	0	0
$937$	−2.07121e7	−0.770681	−0.385341	−	0.922774i	$-0.625916\pi$
$937$	−2.07121e7	−0.770681	−0.385341	+	0.922774i	$0.625916\pi$
$938$	0	0
$939$	−1.63992e7	−0.606957
$940$	0	0
$941$	1.09797e7	0.404219	0.202110	−	0.979363i	$-0.435220\pi$
$941$	1.09797e7	0.404219	0.202110	+	0.979363i	$0.435220\pi$
$942$	0	0
$943$	−2.47364e7	−0.905853
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.71723e6	−0.0984581	−0.0492290	−	0.998788i	$-0.515676\pi$
$947$	−2.71723e6	−0.0984581	−0.0492290	+	0.998788i	$0.515676\pi$
$948$	0	0
$949$	5.35264e6	0.192931
$950$	0	0
$951$	−3.88182e6	−0.139182
$952$	0	0
$953$	−9.24083e6	−0.329594	−0.164797	−	0.986328i	$-0.552697\pi$
$953$	−9.24083e6	−0.329594	−0.164797	+	0.986328i	$0.552697\pi$
$954$	0	0
$955$	−1.80739e7	−0.641272
$956$	0	0
$957$	7.87209e6	0.277850
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2.08225e7	−0.727320
$962$	0	0
$963$	4.27763e7	1.48641
$964$	0	0
$965$	8.48183e6	0.293205
$966$	0	0
$967$	1.50444e7	0.517379	0.258690	−	0.965960i	$-0.416709\pi$
$967$	1.50444e7	0.517379	0.258690	+	0.965960i	$0.416709\pi$
$968$	0	0
$969$	1.22068e6	0.0417632
$970$	0	0
$971$	2.27894e7	0.775685	0.387842	−	0.921726i	$-0.373220\pi$
$971$	2.27894e7	0.775685	0.387842	+	0.921726i	$0.373220\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−646600.	−0.0217833
$976$	0	0
$977$	−3.96682e7	−1.32955	−0.664777	−	0.747042i	$-0.731474\pi$
$977$	−3.96682e7	−1.32955	−0.664777	+	0.747042i	$0.731474\pi$
$978$	0	0
$979$	2.06507e7	0.688617
$980$	0	0
$981$	−3.94170e7	−1.30771
$982$	0	0
$983$	2.77093e7	0.914621	0.457310	−	0.889307i	$-0.348813\pi$
$983$	2.77093e7	0.914621	0.457310	+	0.889307i	$0.348813\pi$
$984$	0	0
$985$	4.28294e7	1.40654
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.70025e7	0.552743
$990$	0	0
$991$	−4.78535e7	−1.54785	−0.773926	−	0.633276i	$-0.781710\pi$
$991$	−4.78535e7	−1.54785	−0.773926	+	0.633276i	$0.781710\pi$
$992$	0	0
$993$	5.86305e6	0.188691
$994$	0	0
$995$	3.80837e7	1.21950
$996$	0	0
$997$	5.48289e7	1.74691	0.873457	−	0.486902i	$-0.161873\pi$
$997$	5.48289e7	1.74691	0.873457	+	0.486902i	$0.161873\pi$
$998$	0	0
$999$	−2.85892e7	−0.906335

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.q.1.2		2
4.3	odd	2		392.6.a.e.1.1		2
7.6	odd	2		112.6.a.j.1.1		2
21.20	even	2		1008.6.a.bi.1.2		2
28.3	even	6		392.6.i.k.177.1		4
28.11	odd	6		392.6.i.h.177.2		4
28.19	even	6		392.6.i.k.361.1		4
28.23	odd	6		392.6.i.h.361.2		4
28.27	even	2		56.6.a.d.1.2	✓	2
56.13	odd	2		448.6.a.r.1.2		2
56.27	even	2		448.6.a.x.1.1		2
84.83	odd	2		504.6.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.6.a.d.1.2	✓	2	28.27	even	2
112.6.a.j.1.1		2	7.6	odd	2
392.6.a.e.1.1		2	4.3	odd	2
392.6.i.h.177.2		4	28.11	odd	6
392.6.i.h.361.2		4	28.23	odd	6
392.6.i.k.177.1		4	28.3	even	6
392.6.i.k.361.1		4	28.19	even	6
448.6.a.r.1.2		2	56.13	odd	2
448.6.a.x.1.1		2	56.27	even	2
504.6.a.m.1.2		2	84.83	odd	2
784.6.a.q.1.2		2	1.1	even	1	trivial
1008.6.a.bi.1.2		2	21.20	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+6.89244 q^{3} +48.4622 q^{5} -195.494 q^{9} +163.506 q^{11} +120.828 q^{13} +334.023 q^{15} -78.1920 q^{17} -2265.00 q^{19} +2451.95 q^{23} -776.413 q^{25} -3022.30 q^{27} +6985.27 q^{29} +2794.03 q^{31} +1126.95 q^{33} +9459.44 q^{37} +832.803 q^{39} -10088.5 q^{41} +6934.29 q^{43} -9474.08 q^{45} +1165.76 q^{47} -538.934 q^{51} +8562.42 q^{53} +7923.85 q^{55} -15611.4 q^{57} +6220.26 q^{59} +41928.9 q^{61} +5855.62 q^{65} -1812.09 q^{67} +16900.0 q^{69} +56823.3 q^{71} +44299.5 q^{73} -5351.38 q^{75} -34912.4 q^{79} +26674.1 q^{81} -39652.9 q^{83} -3789.36 q^{85} +48145.6 q^{87} +126299. q^{89} +19257.7 q^{93} -109767. q^{95} +145513. q^{97} -31964.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 14 q^{3} - 42 q^{5} - 2 q^{9} + 716 q^{11} + 714 q^{13} + 2224 q^{15} + 1344 q^{17} - 1946 q^{19} + 1792 q^{23} + 4282 q^{25} - 1988 q^{27} - 1200 q^{29} - 6804 q^{31} - 10416 q^{33} + 14640 q^{37}+ \cdots + 74940 q^{99}+O(q^{100}) \) 2 * q - 14 * q^3 - 42 * q^5 - 2 * q^9 + 716 * q^11 + 714 * q^13 + 2224 * q^15 + 1344 * q^17 - 1946 * q^19 + 1792 * q^23 + 4282 * q^25 - 1988 * q^27 - 1200 * q^29 - 6804 * q^31 - 10416 * q^33 + 14640 * q^37 - 11560 * q^39 - 7896 * q^41 - 524 * q^43 - 26978 * q^45 - 18396 * q^47 - 30252 * q^51 + 45132 * q^53 - 42056 * q^55 - 22276 * q^57 + 22582 * q^59 + 52822 * q^61 - 47804 * q^65 - 9848 * q^67 + 30688 * q^69 + 840 * q^71 + 122052 * q^73 - 111034 * q^75 - 31704 * q^79 - 41954 * q^81 + 36974 * q^83 - 132444 * q^85 + 219156 * q^87 + 210588 * q^89 + 219784 * q^93 - 138624 * q^95 + 44240 * q^97 + 74940 * q^99

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