LMFDB - Embedded newform 784.6.a.bm.1.5

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 = [5,0,13,0,-31,0,0,0,230] 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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 167x^{3} - 387x^{2} + 1720x + 2340 $ x^5 - 167x^3 - 387x^2 + 1720*x + 2340
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{9}\cdot 7 $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.20248$ 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)

$f(q)$	$=$	$q+25.1934 q^{3} -80.6580 q^{5} +391.710 q^{9} +506.725 q^{11} -391.283 q^{13} -2032.05 q^{15} +899.411 q^{17} +336.313 q^{19} -4872.11 q^{23} +3380.71 q^{25} +3746.51 q^{27} +546.591 q^{29} +6099.02 q^{31} +12766.1 q^{33} +14809.7 q^{37} -9857.78 q^{39} +9837.31 q^{41} -18661.0 q^{43} -31594.5 q^{45} +1398.49 q^{47} +22659.3 q^{51} -19760.8 q^{53} -40871.4 q^{55} +8472.89 q^{57} +27170.8 q^{59} -14167.4 q^{61} +31560.1 q^{65} +50446.1 q^{67} -122745. q^{69} +45279.2 q^{71} +7566.83 q^{73} +85171.6 q^{75} -4829.92 q^{79} -797.980 q^{81} -14067.3 q^{83} -72544.7 q^{85} +13770.5 q^{87} +89553.9 q^{89} +153655. q^{93} -27126.4 q^{95} -7209.53 q^{97} +198489. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 13 q^{3} - 31 q^{5} + 230 q^{9} + 351 q^{11} + 54 q^{13} - 607 q^{15} - 111 q^{17} + 1035 q^{19} - 3639 q^{23} + 1540 q^{25} + 3607 q^{27} - 734 q^{29} + 7677 q^{31} + 7439 q^{33} + 13595 q^{37} + 1406 q^{39}+ \cdots + 300154 q^{99}+O(q^{100}) $ 5 * q + 13 * q^3 - 31 * q^5 + 230 * q^9 + 351 * q^11 + 54 * q^13 - 607 * q^15 - 111 * q^17 + 1035 * q^19 - 3639 * q^23 + 1540 * q^25 + 3607 * q^27 - 734 * q^29 + 7677 * q^31 + 7439 * q^33 + 13595 * q^37 + 1406 * q^39 - 5310 * q^41 - 764 * q^43 - 38978 * q^45 + 6675 * q^47 - 20975 * q^51 - 30753 * q^53 + 28267 * q^55 - 14389 * q^57 + 87989 * q^59 - 19899 * q^61 + 119470 * q^65 + 33067 * q^67 - 100399 * q^69 + 108720 * q^71 - 141659 * q^73 + 108788 * q^75 - 118919 * q^79 - 143851 * q^81 + 211004 * q^83 - 143379 * q^85 + 302154 * q^87 + 55861 * q^89 + 410381 * q^93 + 26279 * q^95 - 135470 * q^97 + 300154 * 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
$3$	25.1934	1.61616	0.808080	−	0.589073i	$-0.200507\pi$
$3$	25.1934	1.61616	0.808080	+	0.589073i	$0.200507\pi$
$4$	0	0
$5$	−80.6580	−1.44285	−0.721427	−	0.692491i	$-0.756513\pi$
$5$	−80.6580	−1.44285	−0.721427	+	0.692491i	$0.756513\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	391.710	1.61197
$10$	0	0
$11$	506.725	1.26267	0.631336	−	0.775510i	$-0.282507\pi$
$11$	506.725	1.26267	0.631336	+	0.775510i	$0.282507\pi$
$12$	0	0
$13$	−391.283	−0.642145	−0.321072	−	0.947055i	$-0.604043\pi$
$13$	−391.283	−0.642145	−0.321072	+	0.947055i	$0.604043\pi$
$14$	0	0
$15$	−2032.05	−2.33188
$16$	0	0
$17$	899.411	0.754807	0.377404	−	0.926049i	$-0.376817\pi$
$17$	899.411	0.754807	0.377404	+	0.926049i	$0.376817\pi$
$18$	0	0
$19$	336.313	0.213727	0.106864	−	0.994274i	$-0.465919\pi$
$19$	336.313	0.213727	0.106864	+	0.994274i	$0.465919\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4872.11	−1.92043	−0.960213	−	0.279269i	$-0.909908\pi$
$23$	−4872.11	−1.92043	−0.960213	+	0.279269i	$0.909908\pi$
$24$	0	0
$25$	3380.71	1.08183
$26$	0	0
$27$	3746.51	0.989048
$28$	0	0
$29$	546.591	0.120689	0.0603444	−	0.998178i	$-0.480780\pi$
$29$	546.591	0.120689	0.0603444	+	0.998178i	$0.480780\pi$
$30$	0	0
$31$	6099.02	1.13987	0.569936	−	0.821689i	$-0.306968\pi$
$31$	6099.02	1.13987	0.569936	+	0.821689i	$0.306968\pi$
$32$	0	0
$33$	12766.1	2.04068
$34$	0	0
$35$	0	0
$36$	0	0
$37$	14809.7	1.77845	0.889224	−	0.457471i	$-0.151245\pi$
$37$	14809.7	1.77845	0.889224	+	0.457471i	$0.151245\pi$
$38$	0	0
$39$	−9857.78	−1.03781
$40$	0	0
$41$	9837.31	0.913938	0.456969	−	0.889483i	$-0.348935\pi$
$41$	9837.31	0.913938	0.456969	+	0.889483i	$0.348935\pi$
$42$	0	0
$43$	−18661.0	−1.53909	−0.769546	−	0.638591i	$-0.779518\pi$
$43$	−18661.0	−1.53909	−0.769546	+	0.638591i	$0.779518\pi$
$44$	0	0
$45$	−31594.5	−2.32584
$46$	0	0
$47$	1398.49	0.0923450	0.0461725	−	0.998933i	$-0.485298\pi$
$47$	1398.49	0.0923450	0.0461725	+	0.998933i	$0.485298\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	22659.3	1.21989
$52$	0	0
$53$	−19760.8	−0.966307	−0.483154	−	0.875536i	$-0.660509\pi$
$53$	−19760.8	−0.966307	−0.483154	+	0.875536i	$0.660509\pi$
$54$	0	0
$55$	−40871.4	−1.82185
$56$	0	0
$57$	8472.89	0.345418
$58$	0	0
$59$	27170.8	1.01618	0.508092	−	0.861303i	$-0.330351\pi$
$59$	27170.8	1.01618	0.508092	+	0.861303i	$0.330351\pi$
$60$	0	0
$61$	−14167.4	−0.487491	−0.243746	−	0.969839i	$-0.578376\pi$
$61$	−14167.4	−0.487491	−0.243746	+	0.969839i	$0.578376\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	31560.1	0.926521
$66$	0	0
$67$	50446.1	1.37291	0.686453	−	0.727175i	$-0.259167\pi$
$67$	50446.1	1.37291	0.686453	+	0.727175i	$0.259167\pi$
$68$	0	0
$69$	−122745.	−3.10372
$70$	0	0
$71$	45279.2	1.06599	0.532995	−	0.846118i	$-0.321066\pi$
$71$	45279.2	1.06599	0.532995	+	0.846118i	$0.321066\pi$
$72$	0	0
$73$	7566.83	0.166191	0.0830953	−	0.996542i	$-0.473519\pi$
$73$	7566.83	0.166191	0.0830953	+	0.996542i	$0.473519\pi$
$74$	0	0
$75$	85171.6	1.74840
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4829.92	−0.0870707	−0.0435353	−	0.999052i	$-0.513862\pi$
$79$	−4829.92	−0.0870707	−0.0435353	+	0.999052i	$0.513862\pi$
$80$	0	0
$81$	−797.980	−0.0135139
$82$	0	0
$83$	−14067.3	−0.224138	−0.112069	−	0.993700i	$-0.535748\pi$
$83$	−14067.3	−0.224138	−0.112069	+	0.993700i	$0.535748\pi$
$84$	0	0
$85$	−72544.7	−1.08908
$86$	0	0
$87$	13770.5	0.195052
$88$	0	0
$89$	89553.9	1.19842	0.599210	−	0.800592i	$-0.295481\pi$
$89$	89553.9	1.19842	0.599210	+	0.800592i	$0.295481\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	153655.	1.84222
$94$	0	0
$95$	−27126.4	−0.308377
$96$	0	0
$97$	−7209.53	−0.0777997	−0.0388998	−	0.999243i	$-0.512385\pi$
$97$	−7209.53	−0.0777997	−0.0388998	+	0.999243i	$0.512385\pi$
$98$	0	0
$99$	198489.	2.03539
$100$	0	0
$101$	195218.	1.90422	0.952110	−	0.305756i	$-0.0989091\pi$
$101$	195218.	1.90422	0.952110	+	0.305756i	$0.0989091\pi$
$102$	0	0
$103$	−23633.5	−0.219500	−0.109750	−	0.993959i	$-0.535005\pi$
$103$	−23633.5	−0.219500	−0.109750	+	0.993959i	$0.535005\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	107092.	0.904270	0.452135	−	0.891950i	$-0.350663\pi$
$107$	107092.	0.904270	0.452135	+	0.891950i	$0.350663\pi$
$108$	0	0
$109$	46221.5	0.372630	0.186315	−	0.982490i	$-0.440346\pi$
$109$	46221.5	0.372630	0.186315	+	0.982490i	$0.440346\pi$
$110$	0	0
$111$	373107.	2.87426
$112$	0	0
$113$	55520.3	0.409030	0.204515	−	0.978863i	$-0.434438\pi$
$113$	55520.3	0.409030	0.204515	+	0.978863i	$0.434438\pi$
$114$	0	0
$115$	392974.	2.77089
$116$	0	0
$117$	−153269.	−1.03512
$118$	0	0
$119$	0	0
$120$	0	0
$121$	95719.0	0.594340
$122$	0	0
$123$	247836.	1.47707
$124$	0	0
$125$	−20624.8	−0.118063
$126$	0	0
$127$	230185.	1.26639	0.633195	−	0.773992i	$-0.281743\pi$
$127$	230185.	1.26639	0.633195	+	0.773992i	$0.281743\pi$
$128$	0	0
$129$	−470136.	−2.48742
$130$	0	0
$131$	199001.	1.01316	0.506579	−	0.862194i	$-0.330910\pi$
$131$	199001.	1.01316	0.506579	+	0.862194i	$0.330910\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−302186.	−1.42705
$136$	0	0
$137$	127531.	0.580516	0.290258	−	0.956948i	$-0.406259\pi$
$137$	127531.	0.580516	0.290258	+	0.956948i	$0.406259\pi$
$138$	0	0
$139$	17259.8	0.0757703	0.0378851	−	0.999282i	$-0.487938\pi$
$139$	17259.8	0.0757703	0.0378851	+	0.999282i	$0.487938\pi$
$140$	0	0
$141$	35232.7	0.149244
$142$	0	0
$143$	−198273.	−0.810818
$144$	0	0
$145$	−44086.9	−0.174136
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−834.526	−0.00307946	−0.00153973	−	0.999999i	$-0.500490\pi$
$149$	−834.526	−0.00307946	−0.00153973	+	0.999999i	$0.500490\pi$
$150$	0	0
$151$	−40938.6	−0.146114	−0.0730569	−	0.997328i	$-0.523275\pi$
$151$	−40938.6	−0.146114	−0.0730569	+	0.997328i	$0.523275\pi$
$152$	0	0
$153$	352308.	1.21673
$154$	0	0
$155$	−491935.	−1.64467
$156$	0	0
$157$	−32888.2	−0.106486	−0.0532428	−	0.998582i	$-0.516956\pi$
$157$	−32888.2	−0.106486	−0.0532428	+	0.998582i	$0.516956\pi$
$158$	0	0
$159$	−497843.	−1.56171
$160$	0	0
$161$	0	0
$162$	0	0
$163$	70298.6	0.207242	0.103621	−	0.994617i	$-0.466957\pi$
$163$	70298.6	0.207242	0.103621	+	0.994617i	$0.466957\pi$
$164$	0	0
$165$	−1.02969e6	−2.94440
$166$	0	0
$167$	227937.	0.632447	0.316224	−	0.948685i	$-0.397585\pi$
$167$	227937.	0.632447	0.316224	+	0.948685i	$0.397585\pi$
$168$	0	0
$169$	−218190.	−0.587650
$170$	0	0
$171$	131737.	0.344523
$172$	0	0
$173$	33790.2	0.0858372	0.0429186	−	0.999079i	$-0.486334\pi$
$173$	33790.2	0.0858372	0.0429186	+	0.999079i	$0.486334\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	684526.	1.64232
$178$	0	0
$179$	−615212.	−1.43513	−0.717567	−	0.696490i	$-0.754744\pi$
$179$	−615212.	−1.43513	−0.717567	+	0.696490i	$0.754744\pi$
$180$	0	0
$181$	678787.	1.54006	0.770029	−	0.638008i	$-0.220242\pi$
$181$	678787.	1.54006	0.770029	+	0.638008i	$0.220242\pi$
$182$	0	0
$183$	−356927.	−0.787864
$184$	0	0
$185$	−1.19452e6	−2.56604
$186$	0	0
$187$	455754.	0.953074
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−655175.	−1.29949	−0.649747	−	0.760151i	$-0.725125\pi$
$191$	−655175.	−1.29949	−0.649747	+	0.760151i	$0.725125\pi$
$192$	0	0
$193$	−320460.	−0.619270	−0.309635	−	0.950856i	$-0.600207\pi$
$193$	−320460.	−0.619270	−0.309635	+	0.950856i	$0.600207\pi$
$194$	0	0
$195$	795108.	1.49741
$196$	0	0
$197$	202103.	0.371029	0.185514	−	0.982642i	$-0.440605\pi$
$197$	202103.	0.371029	0.185514	+	0.982642i	$0.440605\pi$
$198$	0	0
$199$	638340.	1.14267	0.571333	−	0.820718i	$-0.306426\pi$
$199$	638340.	1.14267	0.571333	+	0.820718i	$0.306426\pi$
$200$	0	0
$201$	1.27091e6	2.21883
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−793458.	−1.31868
$206$	0	0
$207$	−1.90845e6	−3.09568
$208$	0	0
$209$	170418.	0.269868
$210$	0	0
$211$	517370.	0.800010	0.400005	−	0.916513i	$-0.369008\pi$
$211$	517370.	0.800010	0.400005	+	0.916513i	$0.369008\pi$
$212$	0	0
$213$	1.14074e6	1.72281
$214$	0	0
$215$	1.50516e6	2.22069
$216$	0	0
$217$	0	0
$218$	0	0
$219$	190634.	0.268591
$220$	0	0
$221$	−351925.	−0.484696
$222$	0	0
$223$	755836.	1.01781	0.508904	−	0.860824i	$-0.330051\pi$
$223$	755836.	1.01781	0.508904	+	0.860824i	$0.330051\pi$
$224$	0	0
$225$	1.32426e6	1.74388
$226$	0	0
$227$	−881248.	−1.13510	−0.567550	−	0.823339i	$-0.692108\pi$
$227$	−881248.	−1.13510	−0.567550	+	0.823339i	$0.692108\pi$
$228$	0	0
$229$	279513.	0.352219	0.176109	−	0.984371i	$-0.443649\pi$
$229$	279513.	0.352219	0.176109	+	0.984371i	$0.443649\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.06678e6	−1.28732	−0.643659	−	0.765312i	$-0.722585\pi$
$233$	−1.06678e6	−1.28732	−0.643659	+	0.765312i	$0.722585\pi$
$234$	0	0
$235$	−112799.	−0.133240
$236$	0	0
$237$	−121682.	−0.140720
$238$	0	0
$239$	270341.	0.306138	0.153069	−	0.988215i	$-0.451084\pi$
$239$	270341.	0.306138	0.153069	+	0.988215i	$0.451084\pi$
$240$	0	0
$241$	1.31407e6	1.45739	0.728694	−	0.684839i	$-0.240128\pi$
$241$	1.31407e6	1.45739	0.728694	+	0.684839i	$0.240128\pi$
$242$	0	0
$243$	−930506.	−1.01089
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−131594.	−0.137244
$248$	0	0
$249$	−354404.	−0.362243
$250$	0	0
$251$	−1.31291e6	−1.31538	−0.657690	−	0.753289i	$-0.728466\pi$
$251$	−1.31291e6	−1.31538	−0.657690	+	0.753289i	$0.728466\pi$
$252$	0	0
$253$	−2.46882e6	−2.42487
$254$	0	0
$255$	−1.82765e6	−1.76012
$256$	0	0
$257$	−707440.	−0.668124	−0.334062	−	0.942551i	$-0.608419\pi$
$257$	−707440.	−0.668124	−0.334062	+	0.942551i	$0.608419\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	214105.	0.194547
$262$	0	0
$263$	739442.	0.659196	0.329598	−	0.944121i	$-0.393087\pi$
$263$	739442.	0.659196	0.329598	+	0.944121i	$0.393087\pi$
$264$	0	0
$265$	1.59387e6	1.39424
$266$	0	0
$267$	2.25617e6	1.93684
$268$	0	0
$269$	1.23919e6	1.04414	0.522068	−	0.852904i	$-0.325161\pi$
$269$	1.23919e6	1.04414	0.522068	+	0.852904i	$0.325161\pi$
$270$	0	0
$271$	−514783.	−0.425795	−0.212898	−	0.977075i	$-0.568290\pi$
$271$	−514783.	−0.425795	−0.212898	+	0.977075i	$0.568290\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.71309e6	1.36599
$276$	0	0
$277$	−1.29704e6	−1.01567	−0.507837	−	0.861453i	$-0.669555\pi$
$277$	−1.29704e6	−1.01567	−0.507837	+	0.861453i	$0.669555\pi$
$278$	0	0
$279$	2.38905e6	1.83744
$280$	0	0
$281$	2.61884e6	1.97853	0.989265	−	0.146133i	$-0.0466828\pi$
$281$	2.61884e6	1.97853	0.989265	+	0.146133i	$0.0466828\pi$
$282$	0	0
$283$	41845.7	0.0310588	0.0155294	−	0.999879i	$-0.495057\pi$
$283$	41845.7	0.0310588	0.0155294	+	0.999879i	$0.495057\pi$
$284$	0	0
$285$	−683406.	−0.498387
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−610916.	−0.430266
$290$	0	0
$291$	−181633.	−0.125737
$292$	0	0
$293$	1.94837e6	1.32588	0.662938	−	0.748674i	$-0.269309\pi$
$293$	1.94837e6	1.32588	0.662938	+	0.748674i	$0.269309\pi$
$294$	0	0
$295$	−2.19154e6	−1.46620
$296$	0	0
$297$	1.89845e6	1.24884
$298$	0	0
$299$	1.90638e6	1.23319
$300$	0	0
$301$	0	0
$302$	0	0
$303$	4.91822e6	3.07752
$304$	0	0
$305$	1.14272e6	0.703378
$306$	0	0
$307$	−209113.	−0.126630	−0.0633149	−	0.997994i	$-0.520167\pi$
$307$	−209113.	−0.126630	−0.0633149	+	0.997994i	$0.520167\pi$
$308$	0	0
$309$	−595410.	−0.354748
$310$	0	0
$311$	−93724.0	−0.0549477	−0.0274739	−	0.999623i	$-0.508746\pi$
$311$	−93724.0	−0.0549477	−0.0274739	+	0.999623i	$0.508746\pi$
$312$	0	0
$313$	500114.	0.288542	0.144271	−	0.989538i	$-0.453916\pi$
$313$	500114.	0.288542	0.144271	+	0.989538i	$0.453916\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.47429e6	−0.824015	−0.412007	−	0.911180i	$-0.635172\pi$
$317$	−1.47429e6	−0.824015	−0.412007	+	0.911180i	$0.635172\pi$
$318$	0	0
$319$	276971.	0.152390
$320$	0	0
$321$	2.69802e6	1.46145
$322$	0	0
$323$	302484.	0.161323
$324$	0	0
$325$	−1.32281e6	−0.694689
$326$	0	0
$327$	1.16448e6	0.602229
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2.37637e6	1.19219	0.596093	−	0.802916i	$-0.296719\pi$
$331$	2.37637e6	1.19219	0.596093	+	0.802916i	$0.296719\pi$
$332$	0	0
$333$	5.80109e6	2.86681
$334$	0	0
$335$	−4.06888e6	−1.98090
$336$	0	0
$337$	2.56857e6	1.23202	0.616008	−	0.787740i	$-0.288749\pi$
$337$	2.56857e6	1.23202	0.616008	+	0.787740i	$0.288749\pi$
$338$	0	0
$339$	1.39875e6	0.661059
$340$	0	0
$341$	3.09053e6	1.43928
$342$	0	0
$343$	0	0
$344$	0	0
$345$	9.90038e6	4.47821
$346$	0	0
$347$	612545.	0.273095	0.136548	−	0.990633i	$-0.456399\pi$
$347$	612545.	0.273095	0.136548	+	0.990633i	$0.456399\pi$
$348$	0	0
$349$	−3.83323e6	−1.68462	−0.842310	−	0.538994i	$-0.818805\pi$
$349$	−3.83323e6	−1.68462	−0.842310	+	0.538994i	$0.818805\pi$
$350$	0	0
$351$	−1.46595e6	−0.635112
$352$	0	0
$353$	−2.85850e6	−1.22096	−0.610481	−	0.792031i	$-0.709024\pi$
$353$	−2.85850e6	−1.22096	−0.610481	+	0.792031i	$0.709024\pi$
$354$	0	0
$355$	−3.65213e6	−1.53807
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−387748.	−0.158787	−0.0793933	−	0.996843i	$-0.525298\pi$
$359$	−387748.	−0.158787	−0.0793933	+	0.996843i	$0.525298\pi$
$360$	0	0
$361$	−2.36299e6	−0.954321
$362$	0	0
$363$	2.41149e6	0.960549
$364$	0	0
$365$	−610325.	−0.239789
$366$	0	0
$367$	192310.	0.0745309	0.0372654	−	0.999305i	$-0.488135\pi$
$367$	192310.	0.0745309	0.0372654	+	0.999305i	$0.488135\pi$
$368$	0	0
$369$	3.85337e6	1.47324
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−2.26524e6	−0.843027	−0.421514	−	0.906822i	$-0.638501\pi$
$373$	−2.26524e6	−0.843027	−0.421514	+	0.906822i	$0.638501\pi$
$374$	0	0
$375$	−519609.	−0.190809
$376$	0	0
$377$	−213872.	−0.0774997
$378$	0	0
$379$	−4.46258e6	−1.59583	−0.797917	−	0.602767i	$-0.794065\pi$
$379$	−4.46258e6	−1.59583	−0.797917	+	0.602767i	$0.794065\pi$
$380$	0	0
$381$	5.79915e6	2.04669
$382$	0	0
$383$	−1.80444e6	−0.628557	−0.314279	−	0.949331i	$-0.601763\pi$
$383$	−1.80444e6	−0.628557	−0.314279	+	0.949331i	$0.601763\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7.30971e6	−2.48098
$388$	0	0
$389$	110911.	0.0371622	0.0185811	−	0.999827i	$-0.494085\pi$
$389$	110911.	0.0371622	0.0185811	+	0.999827i	$0.494085\pi$
$390$	0	0
$391$	−4.38203e6	−1.44955
$392$	0	0
$393$	5.01352e6	1.63742
$394$	0	0
$395$	389571.	0.125630
$396$	0	0
$397$	−5.48779e6	−1.74752	−0.873758	−	0.486361i	$-0.838324\pi$
$397$	−5.48779e6	−1.74752	−0.873758	+	0.486361i	$0.838324\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	934438.	0.290195	0.145097	−	0.989417i	$-0.453650\pi$
$401$	934438.	0.290195	0.145097	+	0.989417i	$0.453650\pi$
$402$	0	0
$403$	−2.38645e6	−0.731963
$404$	0	0
$405$	64363.4	0.0194985
$406$	0	0
$407$	7.50443e6	2.24560
$408$	0	0
$409$	1.47693e6	0.436566	0.218283	−	0.975885i	$-0.429954\pi$
$409$	1.47693e6	0.436566	0.218283	+	0.975885i	$0.429954\pi$
$410$	0	0
$411$	3.21294e6	0.938207
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1.13464e6	0.323398
$416$	0	0
$417$	434834.	0.122457
$418$	0	0
$419$	2.66235e6	0.740849	0.370424	−	0.928863i	$-0.379212\pi$
$419$	2.66235e6	0.740849	0.370424	+	0.928863i	$0.379212\pi$
$420$	0	0
$421$	−1.02032e6	−0.280563	−0.140282	−	0.990112i	$-0.544801\pi$
$421$	−1.02032e6	−0.280563	−0.140282	+	0.990112i	$0.544801\pi$
$422$	0	0
$423$	547800.	0.148858
$424$	0	0
$425$	3.04065e6	0.816570
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−4.99518e6	−1.31041
$430$	0	0
$431$	200267.	0.0519298	0.0259649	−	0.999663i	$-0.491734\pi$
$431$	200267.	0.0519298	0.0259649	+	0.999663i	$0.491734\pi$
$432$	0	0
$433$	−5.84752e6	−1.49883	−0.749415	−	0.662101i	$-0.769665\pi$
$433$	−5.84752e6	−1.49883	−0.749415	+	0.662101i	$0.769665\pi$
$434$	0	0
$435$	−1.11070e6	−0.281432
$436$	0	0
$437$	−1.63856e6	−0.410448
$438$	0	0
$439$	6.32555e6	1.56652	0.783262	−	0.621691i	$-0.213554\pi$
$439$	6.32555e6	1.56652	0.783262	+	0.621691i	$0.213554\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.20386e6	−0.533548	−0.266774	−	0.963759i	$-0.585958\pi$
$443$	−2.20386e6	−0.533548	−0.266774	+	0.963759i	$0.585958\pi$
$444$	0	0
$445$	−7.22323e6	−1.72915
$446$	0	0
$447$	−21024.6	−0.00497689
$448$	0	0
$449$	−2.66634e6	−0.624166	−0.312083	−	0.950055i	$-0.601027\pi$
$449$	−2.66634e6	−0.624166	−0.312083	+	0.950055i	$0.601027\pi$
$450$	0	0
$451$	4.98481e6	1.15400
$452$	0	0
$453$	−1.03139e6	−0.236143
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.08812e6	−1.36362	−0.681809	−	0.731530i	$-0.738807\pi$
$457$	−6.08812e6	−1.36362	−0.681809	+	0.731530i	$0.738807\pi$
$458$	0	0
$459$	3.36965e6	0.746541
$460$	0	0
$461$	−381317.	−0.0835668	−0.0417834	−	0.999127i	$-0.513304\pi$
$461$	−381317.	−0.0835668	−0.0417834	+	0.999127i	$0.513304\pi$
$462$	0	0
$463$	99697.8	0.0216139	0.0108069	−	0.999942i	$-0.496560\pi$
$463$	99697.8	0.0216139	0.0108069	+	0.999942i	$0.496560\pi$
$464$	0	0
$465$	−1.23935e7	−2.65805
$466$	0	0
$467$	5.63778e6	1.19623	0.598116	−	0.801409i	$-0.295916\pi$
$467$	5.63778e6	1.19623	0.598116	+	0.801409i	$0.295916\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−828568.	−0.172098
$472$	0	0
$473$	−9.45602e6	−1.94337
$474$	0	0
$475$	1.13698e6	0.231216
$476$	0	0
$477$	−7.74051e6	−1.55766
$478$	0	0
$479$	−1.74079e6	−0.346663	−0.173331	−	0.984864i	$-0.555453\pi$
$479$	−1.74079e6	−0.346663	−0.173331	+	0.984864i	$0.555453\pi$
$480$	0	0
$481$	−5.79478e6	−1.14202
$482$	0	0
$483$	0	0
$484$	0	0
$485$	581506.	0.112254
$486$	0	0
$487$	−4.82471e6	−0.921825	−0.460913	−	0.887446i	$-0.652478\pi$
$487$	−4.82471e6	−0.921825	−0.460913	+	0.887446i	$0.652478\pi$
$488$	0	0
$489$	1.77106e6	0.334936
$490$	0	0
$491$	4.46085e6	0.835053	0.417527	−	0.908665i	$-0.362897\pi$
$491$	4.46085e6	0.835053	0.417527	+	0.908665i	$0.362897\pi$
$492$	0	0
$493$	491610.	0.0910968
$494$	0	0
$495$	−1.60097e7	−2.93678
$496$	0	0
$497$	0	0
$498$	0	0
$499$	740756.	0.133175	0.0665876	−	0.997781i	$-0.478789\pi$
$499$	740756.	0.133175	0.0665876	+	0.997781i	$0.478789\pi$
$500$	0	0
$501$	5.74253e6	1.02214
$502$	0	0
$503$	−3.74899e6	−0.660686	−0.330343	−	0.943861i	$-0.607164\pi$
$503$	−3.74899e6	−0.660686	−0.330343	+	0.943861i	$0.607164\pi$
$504$	0	0
$505$	−1.57459e7	−2.74751
$506$	0	0
$507$	−5.49697e6	−0.949737
$508$	0	0
$509$	2.66456e6	0.455859	0.227929	−	0.973678i	$-0.426804\pi$
$509$	2.66456e6	0.455859	0.227929	+	0.973678i	$0.426804\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1.26000e6	0.211387
$514$	0	0
$515$	1.90623e6	0.316707
$516$	0	0
$517$	708647.	0.116601
$518$	0	0
$519$	851292.	0.138727
$520$	0	0
$521$	−832623.	−0.134386	−0.0671930	−	0.997740i	$-0.521404\pi$
$521$	−832623.	−0.134386	−0.0671930	+	0.997740i	$0.521404\pi$
$522$	0	0
$523$	−1.87301e6	−0.299424	−0.149712	−	0.988730i	$-0.547835\pi$
$523$	−1.87301e6	−0.299424	−0.149712	+	0.988730i	$0.547835\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.48553e6	0.860384
$528$	0	0
$529$	1.73011e7	2.68804
$530$	0	0
$531$	1.06431e7	1.63806
$532$	0	0
$533$	−3.84918e6	−0.586881
$534$	0	0
$535$	−8.63783e6	−1.30473
$536$	0	0
$537$	−1.54993e7	−2.31941
$538$	0	0
$539$	0	0
$540$	0	0
$541$	4.47586e6	0.657481	0.328741	−	0.944420i	$-0.393376\pi$
$541$	4.47586e6	0.657481	0.328741	+	0.944420i	$0.393376\pi$
$542$	0	0
$543$	1.71010e7	2.48898
$544$	0	0
$545$	−3.72813e6	−0.537650
$546$	0	0
$547$	−6.12226e6	−0.874869	−0.437435	−	0.899250i	$-0.644113\pi$
$547$	−6.12226e6	−0.874869	−0.437435	+	0.899250i	$0.644113\pi$
$548$	0	0
$549$	−5.54952e6	−0.785823
$550$	0	0
$551$	183826.	0.0257945
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−3.00940e7	−4.14713
$556$	0	0
$557$	−9.52543e6	−1.30091	−0.650454	−	0.759546i	$-0.725421\pi$
$557$	−9.52543e6	−1.30091	−0.650454	+	0.759546i	$0.725421\pi$
$558$	0	0
$559$	7.30176e6	0.988321
$560$	0	0
$561$	1.14820e7	1.54032
$562$	0	0
$563$	8.09916e6	1.07688	0.538442	−	0.842663i	$-0.319013\pi$
$563$	8.09916e6	1.07688	0.538442	+	0.842663i	$0.319013\pi$
$564$	0	0
$565$	−4.47815e6	−0.590171
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7.39174e6	−0.957119	−0.478560	−	0.878055i	$-0.658841\pi$
$569$	−7.39174e6	−0.957119	−0.478560	+	0.878055i	$0.658841\pi$
$570$	0	0
$571$	−3.46344e6	−0.444546	−0.222273	−	0.974984i	$-0.571348\pi$
$571$	−3.46344e6	−0.444546	−0.222273	+	0.974984i	$0.571348\pi$
$572$	0	0
$573$	−1.65061e7	−2.10019
$574$	0	0
$575$	−1.64712e7	−2.07757
$576$	0	0
$577$	−5.16577e6	−0.645945	−0.322973	−	0.946408i	$-0.604682\pi$
$577$	−5.16577e6	−0.645945	−0.322973	+	0.946408i	$0.604682\pi$
$578$	0	0
$579$	−8.07348e6	−1.00084
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−1.00133e7	−1.22013
$584$	0	0
$585$	1.23624e7	1.49353
$586$	0	0
$587$	4.08305e6	0.489090	0.244545	−	0.969638i	$-0.421361\pi$
$587$	4.08305e6	0.489090	0.244545	+	0.969638i	$0.421361\pi$
$588$	0	0
$589$	2.05118e6	0.243622
$590$	0	0
$591$	5.09168e6	0.599642
$592$	0	0
$593$	7.82184e6	0.913424	0.456712	−	0.889615i	$-0.349027\pi$
$593$	7.82184e6	0.913424	0.456712	+	0.889615i	$0.349027\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.60820e7	1.84673
$598$	0	0
$599$	6.31782e6	0.719449	0.359725	−	0.933059i	$-0.382871\pi$
$599$	6.31782e6	0.719449	0.359725	+	0.933059i	$0.382871\pi$
$600$	0	0
$601$	1.68407e6	0.190184	0.0950919	−	0.995468i	$-0.469685\pi$
$601$	1.68407e6	0.190184	0.0950919	+	0.995468i	$0.469685\pi$
$602$	0	0
$603$	1.97602e7	2.21309
$604$	0	0
$605$	−7.72050e6	−0.857545
$606$	0	0
$607$	5.45513e6	0.600944	0.300472	−	0.953791i	$-0.402856\pi$
$607$	5.45513e6	0.600944	0.300472	+	0.953791i	$0.402856\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−547204.	−0.0592988
$612$	0	0
$613$	−8.13370e6	−0.874253	−0.437127	−	0.899400i	$-0.644004\pi$
$613$	−8.13370e6	−0.874253	−0.437127	+	0.899400i	$0.644004\pi$
$614$	0	0
$615$	−1.99899e7	−2.13120
$616$	0	0
$617$	2.83775e6	0.300097	0.150048	−	0.988679i	$-0.452057\pi$
$617$	2.83775e6	0.300097	0.150048	+	0.988679i	$0.452057\pi$
$618$	0	0
$619$	1.23909e7	1.29980	0.649900	−	0.760020i	$-0.274811\pi$
$619$	1.23909e7	1.29980	0.649900	+	0.760020i	$0.274811\pi$
$620$	0	0
$621$	−1.82534e7	−1.89939
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−8.90116e6	−0.911478
$626$	0	0
$627$	4.29343e6	0.436149
$628$	0	0
$629$	1.33200e7	1.34239
$630$	0	0
$631$	−3.25659e6	−0.325604	−0.162802	−	0.986659i	$-0.552053\pi$
$631$	−3.25659e6	−0.325604	−0.162802	+	0.986659i	$0.552053\pi$
$632$	0	0
$633$	1.30343e7	1.29294
$634$	0	0
$635$	−1.85662e7	−1.82722
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.77363e7	1.71835
$640$	0	0
$641$	2.30150e6	0.221241	0.110621	−	0.993863i	$-0.464716\pi$
$641$	2.30150e6	0.221241	0.110621	+	0.993863i	$0.464716\pi$
$642$	0	0
$643$	−8.68991e6	−0.828872	−0.414436	−	0.910078i	$-0.636021\pi$
$643$	−8.68991e6	−0.828872	−0.414436	+	0.910078i	$0.636021\pi$
$644$	0	0
$645$	3.79202e7	3.58898
$646$	0	0
$647$	−2.28432e6	−0.214534	−0.107267	−	0.994230i	$-0.534210\pi$
$647$	−2.28432e6	−0.214534	−0.107267	+	0.994230i	$0.534210\pi$
$648$	0	0
$649$	1.37681e7	1.28311
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.16767e6	0.198935	0.0994674	−	0.995041i	$-0.468286\pi$
$653$	2.16767e6	0.198935	0.0994674	+	0.995041i	$0.468286\pi$
$654$	0	0
$655$	−1.60510e7	−1.46184
$656$	0	0
$657$	2.96400e6	0.267895
$658$	0	0
$659$	−1.18596e7	−1.06380	−0.531898	−	0.846809i	$-0.678521\pi$
$659$	−1.18596e7	−1.06380	−0.531898	+	0.846809i	$0.678521\pi$
$660$	0	0
$661$	−1.42672e6	−0.127009	−0.0635045	−	0.997982i	$-0.520228\pi$
$661$	−1.42672e6	−0.127009	−0.0635045	+	0.997982i	$0.520228\pi$
$662$	0	0
$663$	−8.86619e6	−0.783346
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.66305e6	−0.231774
$668$	0	0
$669$	1.90421e7	1.64494
$670$	0	0
$671$	−7.17899e6	−0.615541
$672$	0	0
$673$	−2.07459e7	−1.76561	−0.882806	−	0.469738i	$-0.844348\pi$
$673$	−2.07459e7	−1.76561	−0.882806	+	0.469738i	$0.844348\pi$
$674$	0	0
$675$	1.26658e7	1.06998
$676$	0	0
$677$	1.28579e7	1.07820	0.539098	−	0.842243i	$-0.318765\pi$
$677$	1.28579e7	1.07820	0.539098	+	0.842243i	$0.318765\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−2.22017e7	−1.83450
$682$	0	0
$683$	1.92964e7	1.58279	0.791397	−	0.611302i	$-0.209354\pi$
$683$	1.92964e7	1.58279	0.791397	+	0.611302i	$0.209354\pi$
$684$	0	0
$685$	−1.02864e7	−0.837599
$686$	0	0
$687$	7.04189e6	0.569242
$688$	0	0
$689$	7.73208e6	0.620509
$690$	0	0
$691$	1.82999e7	1.45799	0.728993	−	0.684521i	$-0.239989\pi$
$691$	1.82999e7	1.45799	0.728993	+	0.684521i	$0.239989\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.39214e6	−0.109325
$696$	0	0
$697$	8.84779e6	0.689847
$698$	0	0
$699$	−2.68759e7	−2.08051
$700$	0	0
$701$	−1.61515e7	−1.24142	−0.620710	−	0.784040i	$-0.713156\pi$
$701$	−1.61515e7	−1.24142	−0.620710	+	0.784040i	$0.713156\pi$
$702$	0	0
$703$	4.98069e6	0.380103
$704$	0	0
$705$	−2.84179e6	−0.215338
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.49990e6	0.112059	0.0560295	−	0.998429i	$-0.482156\pi$
$709$	1.49990e6	0.112059	0.0560295	+	0.998429i	$0.482156\pi$
$710$	0	0
$711$	−1.89193e6	−0.140356
$712$	0	0
$713$	−2.97151e7	−2.18904
$714$	0	0
$715$	1.59923e7	1.16989
$716$	0	0
$717$	6.81083e6	0.494769
$718$	0	0
$719$	1.75739e7	1.26778	0.633892	−	0.773422i	$-0.281457\pi$
$719$	1.75739e7	1.26778	0.633892	+	0.773422i	$0.281457\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	3.31059e7	2.35537
$724$	0	0
$725$	1.84786e6	0.130564
$726$	0	0
$727$	−2.16304e7	−1.51785	−0.758923	−	0.651180i	$-0.774274\pi$
$727$	−2.16304e7	−1.51785	−0.758923	+	0.651180i	$0.774274\pi$
$728$	0	0
$729$	−2.32487e7	−1.62024
$730$	0	0
$731$	−1.67840e7	−1.16172
$732$	0	0
$733$	1.25536e7	0.862998	0.431499	−	0.902114i	$-0.357985\pi$
$733$	1.25536e7	0.862998	0.431499	+	0.902114i	$0.357985\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.55623e7	1.73353
$738$	0	0
$739$	757732.	0.0510392	0.0255196	−	0.999674i	$-0.491876\pi$
$739$	757732.	0.0510392	0.0255196	+	0.999674i	$0.491876\pi$
$740$	0	0
$741$	−3.31530e6	−0.221808
$742$	0	0
$743$	−2.40878e7	−1.60075	−0.800377	−	0.599497i	$-0.795367\pi$
$743$	−2.40878e7	−1.60075	−0.800377	+	0.599497i	$0.795367\pi$
$744$	0	0
$745$	67311.1	0.00444320
$746$	0	0
$747$	−5.51030e6	−0.361305
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1.37687e7	0.890828	0.445414	−	0.895325i	$-0.353056\pi$
$751$	1.37687e7	0.890828	0.445414	+	0.895325i	$0.353056\pi$
$752$	0	0
$753$	−3.30768e7	−2.12586
$754$	0	0
$755$	3.30203e6	0.210821
$756$	0	0
$757$	2.43930e7	1.54713	0.773564	−	0.633718i	$-0.218472\pi$
$757$	2.43930e7	1.54713	0.773564	+	0.633718i	$0.218472\pi$
$758$	0	0
$759$	−6.21981e7	−3.91897
$760$	0	0
$761$	−2.56002e7	−1.60244	−0.801220	−	0.598370i	$-0.795815\pi$
$761$	−2.56002e7	−1.60244	−0.801220	+	0.598370i	$0.795815\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2.84165e7	−1.75556
$766$	0	0
$767$	−1.06315e7	−0.652537
$768$	0	0
$769$	−2.09807e7	−1.27939	−0.639697	−	0.768627i	$-0.720940\pi$
$769$	−2.09807e7	−1.27939	−0.639697	+	0.768627i	$0.720940\pi$
$770$	0	0
$771$	−1.78229e7	−1.07979
$772$	0	0
$773$	1.12929e7	0.679761	0.339881	−	0.940469i	$-0.389613\pi$
$773$	1.12929e7	0.679761	0.339881	+	0.940469i	$0.389613\pi$
$774$	0	0
$775$	2.06190e7	1.23314
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.30842e6	0.195334
$780$	0	0
$781$	2.29441e7	1.34600
$782$	0	0
$783$	2.04781e6	0.119367
$784$	0	0
$785$	2.65270e6	0.153643
$786$	0	0
$787$	−2.37169e7	−1.36496	−0.682482	−	0.730902i	$-0.739100\pi$
$787$	−2.37169e7	−1.36496	−0.682482	+	0.730902i	$0.739100\pi$
$788$	0	0
$789$	1.86291e7	1.06537
$790$	0	0
$791$	0	0
$792$	0	0
$793$	5.54348e6	0.313040
$794$	0	0
$795$	4.01550e7	2.25332
$796$	0	0
$797$	2.73492e7	1.52510	0.762550	−	0.646929i	$-0.223947\pi$
$797$	2.73492e7	1.52510	0.762550	+	0.646929i	$0.223947\pi$
$798$	0	0
$799$	1.25781e6	0.0697026
$800$	0	0
$801$	3.50791e7	1.93182
$802$	0	0
$803$	3.83430e6	0.209844
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.12195e7	1.68749
$808$	0	0
$809$	−1.09356e7	−0.587452	−0.293726	−	0.955890i	$-0.594895\pi$
$809$	−1.09356e7	−0.587452	−0.293726	+	0.955890i	$0.594895\pi$
$810$	0	0
$811$	3.14053e7	1.67668	0.838342	−	0.545144i	$-0.183525\pi$
$811$	3.14053e7	1.67668	0.838342	+	0.545144i	$0.183525\pi$
$812$	0	0
$813$	−1.29691e7	−0.688153
$814$	0	0
$815$	−5.67014e6	−0.299020
$816$	0	0
$817$	−6.27596e6	−0.328946
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.12501e7	1.61806	0.809028	−	0.587770i	$-0.199994\pi$
$821$	3.12501e7	1.61806	0.809028	+	0.587770i	$0.199994\pi$
$822$	0	0
$823$	−3.66764e7	−1.88750	−0.943749	−	0.330662i	$-0.892728\pi$
$823$	−3.66764e7	−1.88750	−0.943749	+	0.330662i	$0.892728\pi$
$824$	0	0
$825$	4.31586e7	2.20766
$826$	0	0
$827$	1.67620e7	0.852242	0.426121	−	0.904666i	$-0.359880\pi$
$827$	1.67620e7	0.852242	0.426121	+	0.904666i	$0.359880\pi$
$828$	0	0
$829$	1.19048e6	0.0601640	0.0300820	−	0.999547i	$-0.490423\pi$
$829$	1.19048e6	0.0601640	0.0300820	+	0.999547i	$0.490423\pi$
$830$	0	0
$831$	−3.26769e7	−1.64149
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1.83850e7	−0.912529
$836$	0	0
$837$	2.28501e7	1.12739
$838$	0	0
$839$	8.21562e6	0.402935	0.201468	−	0.979495i	$-0.435429\pi$
$839$	8.21562e6	0.402935	0.201468	+	0.979495i	$0.435429\pi$
$840$	0	0
$841$	−2.02124e7	−0.985434
$842$	0	0
$843$	6.59775e7	3.19762
$844$	0	0
$845$	1.75988e7	0.847893
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1.05424e6	0.0501960
$850$	0	0
$851$	−7.21544e7	−3.41538
$852$	0	0
$853$	−2.55133e7	−1.20059	−0.600295	−	0.799779i	$-0.704950\pi$
$853$	−2.55133e7	−1.20059	−0.600295	+	0.799779i	$0.704950\pi$
$854$	0	0
$855$	−1.06257e7	−0.497096
$856$	0	0
$857$	2.87726e7	1.33822	0.669109	−	0.743164i	$-0.266676\pi$
$857$	2.87726e7	1.33822	0.669109	+	0.743164i	$0.266676\pi$
$858$	0	0
$859$	1.03594e7	0.479020	0.239510	−	0.970894i	$-0.423013\pi$
$859$	1.03594e7	0.479020	0.239510	+	0.970894i	$0.423013\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−5.74322e6	−0.262499	−0.131250	−	0.991349i	$-0.541899\pi$
$863$	−5.74322e6	−0.262499	−0.131250	+	0.991349i	$0.541899\pi$
$864$	0	0
$865$	−2.72545e6	−0.123851
$866$	0	0
$867$	−1.53911e7	−0.695379
$868$	0	0
$869$	−2.44744e6	−0.109942
$870$	0	0
$871$	−1.97387e7	−0.881604
$872$	0	0
$873$	−2.82404e6	−0.125411
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−3.61870e7	−1.58874	−0.794372	−	0.607432i	$-0.792200\pi$
$877$	−3.61870e7	−1.58874	−0.794372	+	0.607432i	$0.792200\pi$
$878$	0	0
$879$	4.90862e7	2.14283
$880$	0	0
$881$	2.20285e7	0.956192	0.478096	−	0.878308i	$-0.341327\pi$
$881$	2.20285e7	0.956192	0.478096	+	0.878308i	$0.341327\pi$
$882$	0	0
$883$	3.82050e7	1.64899	0.824497	−	0.565867i	$-0.191458\pi$
$883$	3.82050e7	1.64899	0.824497	+	0.565867i	$0.191458\pi$
$884$	0	0
$885$	−5.52125e7	−2.36962
$886$	0	0
$887$	−3.74392e7	−1.59778	−0.798890	−	0.601477i	$-0.794579\pi$
$887$	−3.74392e7	−1.59778	−0.798890	+	0.601477i	$0.794579\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−404356.	−0.0170636
$892$	0	0
$893$	470329.	0.0197366
$894$	0	0
$895$	4.96218e7	2.07069
$896$	0	0
$897$	4.80282e7	1.99304
$898$	0	0
$899$	3.33367e6	0.137570
$900$	0	0
$901$	−1.77731e7	−0.729376
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.47496e7	−2.22208
$906$	0	0
$907$	2.38414e7	0.962308	0.481154	−	0.876636i	$-0.340218\pi$
$907$	2.38414e7	0.962308	0.481154	+	0.876636i	$0.340218\pi$
$908$	0	0
$909$	7.64689e7	3.06955
$910$	0	0
$911$	−1.78434e7	−0.712332	−0.356166	−	0.934423i	$-0.615916\pi$
$911$	−1.78434e7	−0.712332	−0.356166	+	0.934423i	$0.615916\pi$
$912$	0	0
$913$	−7.12825e6	−0.283013
$914$	0	0
$915$	2.87890e7	1.13677
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1.64290e7	−0.641687	−0.320843	−	0.947132i	$-0.603966\pi$
$919$	−1.64290e7	−0.641687	−0.320843	+	0.947132i	$0.603966\pi$
$920$	0	0
$921$	−5.26829e6	−0.204654
$922$	0	0
$923$	−1.77170e7	−0.684520
$924$	0	0
$925$	5.00672e7	1.92397
$926$	0	0
$927$	−9.25748e6	−0.353829
$928$	0	0
$929$	1.36544e7	0.519080	0.259540	−	0.965732i	$-0.416429\pi$
$929$	1.36544e7	0.519080	0.259540	+	0.965732i	$0.416429\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−2.36123e6	−0.0888043
$934$	0	0
$935$	−3.67602e7	−1.37515
$936$	0	0
$937$	−3.81848e7	−1.42083	−0.710414	−	0.703784i	$-0.751492\pi$
$937$	−3.81848e7	−1.42083	−0.710414	+	0.703784i	$0.751492\pi$
$938$	0	0
$939$	1.25996e7	0.466330
$940$	0	0
$941$	−7.03108e6	−0.258850	−0.129425	−	0.991589i	$-0.541313\pi$
$941$	−7.03108e6	−0.258850	−0.129425	+	0.991589i	$0.541313\pi$
$942$	0	0
$943$	−4.79285e7	−1.75515
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.20199e7	−0.797885	−0.398942	−	0.916976i	$-0.630623\pi$
$947$	−2.20199e7	−0.797885	−0.398942	+	0.916976i	$0.630623\pi$
$948$	0	0
$949$	−2.96077e6	−0.106718
$950$	0	0
$951$	−3.71425e7	−1.33174
$952$	0	0
$953$	578419.	0.0206305	0.0103153	−	0.999947i	$-0.496716\pi$
$953$	578419.	0.0206305	0.0103153	+	0.999947i	$0.496716\pi$
$954$	0	0
$955$	5.28451e7	1.87498
$956$	0	0
$957$	6.97785e6	0.246287
$958$	0	0
$959$	0	0
$960$	0	0
$961$	8.56895e6	0.299309
$962$	0	0
$963$	4.19490e7	1.45766
$964$	0	0
$965$	2.58476e7	0.893516
$966$	0	0
$967$	−2.00036e7	−0.687925	−0.343963	−	0.938983i	$-0.611769\pi$
$967$	−2.00036e7	−0.687925	−0.343963	+	0.938983i	$0.611769\pi$
$968$	0	0
$969$	7.62062e6	0.260724
$970$	0	0
$971$	5.09119e7	1.73289	0.866446	−	0.499272i	$-0.166399\pi$
$971$	5.09119e7	1.73289	0.866446	+	0.499272i	$0.166399\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−3.33262e7	−1.12273
$976$	0	0
$977$	−1.73872e7	−0.582764	−0.291382	−	0.956607i	$-0.594115\pi$
$977$	−1.73872e7	−0.582764	−0.291382	+	0.956607i	$0.594115\pi$
$978$	0	0
$979$	4.53792e7	1.51321
$980$	0	0
$981$	1.81054e7	0.600670
$982$	0	0
$983$	1.10857e7	0.365914	0.182957	−	0.983121i	$-0.441433\pi$
$983$	1.10857e7	0.365914	0.182957	+	0.983121i	$0.441433\pi$
$984$	0	0
$985$	−1.63012e7	−0.535340
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9.09187e7	2.95571
$990$	0	0
$991$	−3.82925e7	−1.23859	−0.619297	−	0.785157i	$-0.712582\pi$
$991$	−3.82925e7	−1.23859	−0.619297	+	0.785157i	$0.712582\pi$
$992$	0	0
$993$	5.98689e7	1.92676
$994$	0	0
$995$	−5.14872e7	−1.64870
$996$	0	0
$997$	−5.27173e7	−1.67964	−0.839818	−	0.542869i	$-0.817338\pi$
$997$	−5.27173e7	−1.67964	−0.839818	+	0.542869i	$0.817338\pi$
$998$	0	0
$999$	5.54846e7	1.75897

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.bm.1.5		5
4.3	odd	2		392.6.a.i.1.1		5
7.3	odd	6		112.6.i.g.65.5		10
7.5	odd	6		112.6.i.g.81.5		10
7.6	odd	2		784.6.a.bj.1.1		5
28.3	even	6		56.6.i.a.9.1	✓	10
28.11	odd	6		392.6.i.p.177.5		10
28.19	even	6		56.6.i.a.25.1	yes	10
28.23	odd	6		392.6.i.p.361.5		10
28.27	even	2		392.6.a.l.1.5		5
84.47	odd	6		504.6.s.d.361.5		10
84.59	odd	6		504.6.s.d.289.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.6.i.a.9.1	✓	10	28.3	even	6
56.6.i.a.25.1	yes	10	28.19	even	6
112.6.i.g.65.5		10	7.3	odd	6
112.6.i.g.81.5		10	7.5	odd	6
392.6.a.i.1.1		5	4.3	odd	2
392.6.a.l.1.5		5	28.27	even	2
392.6.i.p.177.5		10	28.11	odd	6
392.6.i.p.361.5		10	28.23	odd	6
504.6.s.d.289.5		10	84.59	odd	6
504.6.s.d.361.5		10	84.47	odd	6
784.6.a.bj.1.1		5	7.6	odd	2
784.6.a.bm.1.5		5	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+25.1934 q^{3} -80.6580 q^{5} +391.710 q^{9} +506.725 q^{11} -391.283 q^{13} -2032.05 q^{15} +899.411 q^{17} +336.313 q^{19} -4872.11 q^{23} +3380.71 q^{25} +3746.51 q^{27} +546.591 q^{29} +6099.02 q^{31} +12766.1 q^{33} +14809.7 q^{37} -9857.78 q^{39} +9837.31 q^{41} -18661.0 q^{43} -31594.5 q^{45} +1398.49 q^{47} +22659.3 q^{51} -19760.8 q^{53} -40871.4 q^{55} +8472.89 q^{57} +27170.8 q^{59} -14167.4 q^{61} +31560.1 q^{65} +50446.1 q^{67} -122745. q^{69} +45279.2 q^{71} +7566.83 q^{73} +85171.6 q^{75} -4829.92 q^{79} -797.980 q^{81} -14067.3 q^{83} -72544.7 q^{85} +13770.5 q^{87} +89553.9 q^{89} +153655. q^{93} -27126.4 q^{95} -7209.53 q^{97} +198489. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 13 q^{3} - 31 q^{5} + 230 q^{9} + 351 q^{11} + 54 q^{13} - 607 q^{15} - 111 q^{17} + 1035 q^{19} - 3639 q^{23} + 1540 q^{25} + 3607 q^{27} - 734 q^{29} + 7677 q^{31} + 7439 q^{33} + 13595 q^{37} + 1406 q^{39}+ \cdots + 300154 q^{99}+O(q^{100}) \) 5 * q + 13 * q^3 - 31 * q^5 + 230 * q^9 + 351 * q^11 + 54 * q^13 - 607 * q^15 - 111 * q^17 + 1035 * q^19 - 3639 * q^23 + 1540 * q^25 + 3607 * q^27 - 734 * q^29 + 7677 * q^31 + 7439 * q^33 + 13595 * q^37 + 1406 * q^39 - 5310 * q^41 - 764 * q^43 - 38978 * q^45 + 6675 * q^47 - 20975 * q^51 - 30753 * q^53 + 28267 * q^55 - 14389 * q^57 + 87989 * q^59 - 19899 * q^61 + 119470 * q^65 + 33067 * q^67 - 100399 * q^69 + 108720 * q^71 - 141659 * q^73 + 108788 * q^75 - 118919 * q^79 - 143851 * q^81 + 211004 * q^83 - 143379 * q^85 + 302154 * q^87 + 55861 * q^89 + 410381 * q^93 + 26279 * q^95 - 135470 * q^97 + 300154 * q^99

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