LMFDB - Embedded newform 784.6.f.b.783.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [784,6,Mod(783,784)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("784.783"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	784.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$125.740914733$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} + 293 x^{10} + 336 x^{9} + 60118 x^{8} + 67616 x^{7} + 5772748 x^{6} + \cdots + 37147165696 $ x^12 - 3x^11 + 293x^10 + 336x^9 + 60118x^8 + 67616x^7 + 5772748x^6 + 13469792x^5 + 396106192x^4 + 376848768x^3 + 4277005888x^2 - 2508651776*x + 37147165696
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{2}\cdot 7^{8} $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			783.5
Root			$5.50778 - 9.53976i$ of defining polynomial
Character	$\chi$	$=$	784.783
Dual form			784.6.f.b.783.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-9.48156 q^{3} -52.4558i q^{5} -153.100 q^{9} +24.4437i q^{11} -58.8178i q^{13} +497.363i q^{15} +385.677i q^{17} +584.926 q^{19} -3247.23i q^{23} +373.386 q^{25} +3755.65 q^{27} -6058.46 q^{29} -9511.10 q^{31} -231.764i q^{33} -5788.62 q^{37} +557.684i q^{39} +8858.89i q^{41} +7390.78i q^{43} +8030.99i q^{45} -2631.31 q^{47} -3656.82i q^{51} +12151.5 q^{53} +1282.21 q^{55} -5546.01 q^{57} -14535.5 q^{59} +20993.4i q^{61} -3085.33 q^{65} -42258.8i q^{67} +30788.8i q^{69} -75175.4i q^{71} -9622.47i q^{73} -3540.28 q^{75} +82968.7i q^{79} +1593.91 q^{81} -81002.0 q^{83} +20231.0 q^{85} +57443.6 q^{87} +30966.3i q^{89} +90180.1 q^{93} -30682.8i q^{95} -131483. i q^{97} -3742.33i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 1088 q^{9} + 4920 q^{25} + 22512 q^{29} + 15876 q^{37} - 45708 q^{53} - 25900 q^{57} + 124440 q^{65} + 186164 q^{81} + 70116 q^{85} + 546364 q^{93}+O(q^{100}) $ 12 * q + 1088 * q^9 + 4920 * q^25 + 22512 * q^29 + 15876 * q^37 - 45708 * q^53 - 25900 * q^57 + 124440 * q^65 + 186164 * q^81 + 70116 * q^85 + 546364 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	−9.48156	−0.608242	−0.304121	−	0.952633i	$-0.598363\pi$
$3$	−9.48156	−0.608242	−0.304121	+	0.952633i	$0.598363\pi$
$4$	0	0
$5$	− 52.4558i	− 0.938358i	−0.883103	−	0.469179i	$-0.844550\pi$
$5$	− 52.4558i	− 0.938358i	0.883103	−	0.469179i	$-0.155450\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−153.100	−0.630041
$10$	0	0
$11$	24.4437i	0.0609095i	0.999536	+	0.0304547i	$0.00969554\pi$
$11$	24.4437i	0.0609095i	−0.999536	+	0.0304547i	$0.990304\pi$
$12$	0	0
$13$	− 58.8178i	− 0.0965273i	−0.998835	−	0.0482636i	$-0.984631\pi$
$13$	− 58.8178i	− 0.0965273i	0.998835	−	0.0482636i	$-0.0153688\pi$
$14$	0	0
$15$	497.363i	0.570749i
$16$	0	0
$17$	385.677i	0.323669i	0.986818	+	0.161835i	$0.0517411\pi$
$17$	385.677i	0.323669i	−0.986818	+	0.161835i	$0.948259\pi$
$18$	0	0
$19$	584.926	0.371721	0.185860	−	0.982576i	$-0.440493\pi$
$19$	584.926	0.371721	0.185860	+	0.982576i	$0.440493\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 3247.23i	− 1.27995i	−0.768395	−	0.639976i	$-0.778944\pi$
$23$	− 3247.23i	− 1.27995i	0.768395	−	0.639976i	$-0.221056\pi$
$24$	0	0
$25$	373.386	0.119484
$26$	0	0
$27$	3755.65	0.991460
$28$	0	0
$29$	−6058.46	−1.33773	−0.668863	−	0.743386i	$-0.733219\pi$
$29$	−6058.46	−1.33773	−0.668863	+	0.743386i	$0.733219\pi$
$30$	0	0
$31$	−9511.10	−1.77757	−0.888785	−	0.458325i	$-0.848449\pi$
$31$	−9511.10	−1.77757	−0.888785	+	0.458325i	$0.848449\pi$
$32$	0	0
$33$	− 231.764i	− 0.0370477i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5788.62	−0.695138	−0.347569	−	0.937654i	$-0.612993\pi$
$37$	−5788.62	−0.695138	−0.347569	+	0.937654i	$0.612993\pi$
$38$	0	0
$39$	557.684i	0.0587120i
$40$	0	0
$41$	8858.89i	0.823037i	0.911401	+	0.411519i	$0.135001\pi$
$41$	8858.89i	0.823037i	−0.911401	+	0.411519i	$0.864999\pi$
$42$	0	0
$43$	7390.78i	0.609563i	0.952422	+	0.304782i	$0.0985835\pi$
$43$	7390.78i	0.609563i	−0.952422	+	0.304782i	$0.901416\pi$
$44$	0	0
$45$	8030.99i	0.591204i
$46$	0	0
$47$	−2631.31	−0.173751	−0.0868755	−	0.996219i	$-0.527688\pi$
$47$	−2631.31	−0.173751	−0.0868755	+	0.996219i	$0.527688\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 3656.82i	− 0.196869i
$52$	0	0
$53$	12151.5	0.594208	0.297104	−	0.954845i	$-0.403979\pi$
$53$	12151.5	0.594208	0.297104	+	0.954845i	$0.403979\pi$
$54$	0	0
$55$	1282.21	0.0571549
$56$	0	0
$57$	−5546.01	−0.226096
$58$	0	0
$59$	−14535.5	−0.543626	−0.271813	−	0.962350i	$-0.587623\pi$
$59$	−14535.5	−0.543626	−0.271813	+	0.962350i	$0.587623\pi$
$60$	0	0
$61$	20993.4i	0.722368i	0.932495	+	0.361184i	$0.117627\pi$
$61$	20993.4i	0.722368i	−0.932495	+	0.361184i	$0.882373\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3085.33	−0.0905772
$66$	0	0
$67$	− 42258.8i	− 1.15008i	−0.818124	−	0.575042i	$-0.804986\pi$
$67$	− 42258.8i	− 1.15008i	0.818124	−	0.575042i	$-0.195014\pi$
$68$	0	0
$69$	30788.8i	0.778521i
$70$	0	0
$71$	− 75175.4i	− 1.76982i	−0.465760	−	0.884911i	$-0.654219\pi$
$71$	− 75175.4i	− 1.76982i	0.465760	−	0.884911i	$-0.345781\pi$
$72$	0	0
$73$	− 9622.47i	− 0.211339i	−0.994401	−	0.105669i	$-0.966301\pi$
$73$	− 9622.47i	− 0.211339i	0.994401	−	0.105669i	$-0.0336985\pi$
$74$	0	0
$75$	−3540.28	−0.0726750
$76$	0	0
$77$	0	0
$78$	0	0
$79$	82968.7i	1.49571i	0.663864	+	0.747854i	$0.268916\pi$
$79$	82968.7i	1.49571i	−0.663864	+	0.747854i	$0.731084\pi$
$80$	0	0
$81$	1593.91	0.0269930
$82$	0	0
$83$	−81002.0	−1.29063	−0.645313	−	0.763918i	$-0.723273\pi$
$83$	−81002.0	−1.29063	−0.645313	+	0.763918i	$0.723273\pi$
$84$	0	0
$85$	20231.0	0.303718
$86$	0	0
$87$	57443.6	0.813662
$88$	0	0
$89$	30966.3i	0.414395i	0.978299	+	0.207197i	$0.0664343\pi$
$89$	30966.3i	0.414395i	−0.978299	+	0.207197i	$0.933566\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	90180.1	1.08119
$94$	0	0
$95$	− 30682.8i	− 0.348807i
$96$	0	0
$97$	− 131483.i	− 1.41886i	−0.704775	−	0.709431i	$-0.748952\pi$
$97$	− 131483.i	− 1.41886i	0.704775	−	0.709431i	$-0.251048\pi$
$98$	0	0
$99$	− 3742.33i	− 0.0383755i
$100$	0	0
$101$	113527.i	1.10738i	0.832722	+	0.553691i	$0.186781\pi$
$101$	113527.i	1.10738i	−0.832722	+	0.553691i	$0.813219\pi$
$102$	0	0
$103$	−105833.	−0.982939	−0.491469	−	0.870895i	$-0.663540\pi$
$103$	−105833.	−0.982939	−0.491469	+	0.870895i	$0.663540\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	33838.6i	0.285728i	0.989742	+	0.142864i	$0.0456312\pi$
$107$	33838.6i	0.285728i	−0.989742	+	0.142864i	$0.954369\pi$
$108$	0	0
$109$	−57043.0	−0.459871	−0.229935	−	0.973206i	$-0.573852\pi$
$109$	−57043.0	−0.459871	−0.229935	+	0.973206i	$0.573852\pi$
$110$	0	0
$111$	54885.2	0.422812
$112$	0	0
$113$	118518.	0.873148	0.436574	−	0.899668i	$-0.356192\pi$
$113$	118518.	0.873148	0.436574	+	0.899668i	$0.356192\pi$
$114$	0	0
$115$	−170336.	−1.20105
$116$	0	0
$117$	9005.00i	0.0608162i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	160454.	0.996290
$122$	0	0
$123$	− 83996.1i	− 0.500606i
$124$	0	0
$125$	− 183511.i	− 1.05048i
$126$	0	0
$127$	− 47301.9i	− 0.260237i	−0.991498	−	0.130119i	$-0.958464\pi$
$127$	− 47301.9i	− 0.260237i	0.991498	−	0.130119i	$-0.0415358\pi$
$128$	0	0
$129$	− 70076.1i	− 0.370762i
$130$	0	0
$131$	237006.	1.20665	0.603325	−	0.797495i	$-0.293842\pi$
$131$	237006.	1.20665	0.603325	+	0.797495i	$0.293842\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 197006.i	− 0.930345i
$136$	0	0
$137$	89347.1	0.406704	0.203352	−	0.979106i	$-0.434816\pi$
$137$	89347.1	0.406704	0.203352	+	0.979106i	$0.434816\pi$
$138$	0	0
$139$	146398.	0.642685	0.321342	−	0.946963i	$-0.395866\pi$
$139$	146398.	0.642685	0.321342	+	0.946963i	$0.395866\pi$
$140$	0	0
$141$	24948.9	0.105683
$142$	0	0
$143$	1437.72	0.00587943
$144$	0	0
$145$	317801.i	1.25527i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	417634.	1.54110	0.770548	−	0.637381i	$-0.219982\pi$
$149$	417634.	1.54110	0.770548	+	0.637381i	$0.219982\pi$
$150$	0	0
$151$	427747.i	1.52667i	0.646004	+	0.763334i	$0.276439\pi$
$151$	427747.i	1.52667i	−0.646004	+	0.763334i	$0.723561\pi$
$152$	0	0
$153$	− 59047.2i	− 0.203925i
$154$	0	0
$155$	498913.i	1.66800i
$156$	0	0
$157$	477093.i	1.54473i	0.635176	+	0.772367i	$0.280927\pi$
$157$	477093.i	1.54473i	−0.635176	+	0.772367i	$0.719073\pi$
$158$	0	0
$159$	−115215.	−0.361423
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 457804.i	− 1.34962i	−0.737993	−	0.674808i	$-0.764226\pi$
$163$	− 457804.i	− 1.34962i	0.737993	−	0.674808i	$-0.235774\pi$
$164$	0	0
$165$	−12157.4	−0.0347640
$166$	0	0
$167$	−1763.27	−0.00489247	−0.00244623	−	0.999997i	$-0.500779\pi$
$167$	−1763.27	−0.00489247	−0.00244623	+	0.999997i	$0.500779\pi$
$168$	0	0
$169$	367833.	0.990682
$170$	0	0
$171$	−89552.1	−0.234199
$172$	0	0
$173$	607348.i	1.54285i	0.636323	+	0.771423i	$0.280455\pi$
$173$	607348.i	1.54285i	−0.636323	+	0.771423i	$0.719545\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	137819.	0.330656
$178$	0	0
$179$	− 267379.i	− 0.623726i	−0.950127	−	0.311863i	$-0.899047\pi$
$179$	− 267379.i	− 0.623726i	0.950127	−	0.311863i	$-0.100953\pi$
$180$	0	0
$181$	170105.i	0.385941i	0.981205	+	0.192971i	$0.0618122\pi$
$181$	170105.i	0.385941i	−0.981205	+	0.192971i	$0.938188\pi$
$182$	0	0
$183$	− 199050.i	− 0.439375i
$184$	0	0
$185$	303647.i	0.652289i
$186$	0	0
$187$	−9427.37	−0.0197145
$188$	0	0
$189$	0	0
$190$	0	0
$191$	728033.i	1.44400i	0.691893	+	0.722000i	$0.256777\pi$
$191$	728033.i	1.44400i	−0.691893	+	0.722000i	$0.743223\pi$
$192$	0	0
$193$	475505.	0.918887	0.459443	−	0.888207i	$-0.348049\pi$
$193$	475505.	0.918887	0.459443	+	0.888207i	$0.348049\pi$
$194$	0	0
$195$	29253.8	0.0550929
$196$	0	0
$197$	−411672.	−0.755763	−0.377881	−	0.925854i	$-0.623347\pi$
$197$	−411672.	−0.755763	−0.377881	+	0.925854i	$0.623347\pi$
$198$	0	0
$199$	727439.	1.30216	0.651079	−	0.759010i	$-0.274317\pi$
$199$	727439.	1.30216	0.651079	+	0.759010i	$0.274317\pi$
$200$	0	0
$201$	400679.i	0.699530i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	464700.	0.772304
$206$	0	0
$207$	497151.i	0.806422i
$208$	0	0
$209$	14297.7i	0.0226413i
$210$	0	0
$211$	− 88502.1i	− 0.136851i	−0.997656	−	0.0684254i	$-0.978202\pi$
$211$	− 88502.1i	− 0.136851i	0.997656	−	0.0684254i	$-0.0217975\pi$
$212$	0	0
$213$	712780.i	1.07648i
$214$	0	0
$215$	387689.	0.571989
$216$	0	0
$217$	0	0
$218$	0	0
$219$	91236.0i	0.128545i
$220$	0	0
$221$	22684.7	0.0312429
$222$	0	0
$223$	−145641.	−0.196120	−0.0980602	−	0.995180i	$-0.531264\pi$
$223$	−145641.	−0.196120	−0.0980602	+	0.995180i	$0.531264\pi$
$224$	0	0
$225$	−57165.4	−0.0752796
$226$	0	0
$227$	970345.	1.24986	0.624930	−	0.780680i	$-0.285127\pi$
$227$	970345.	1.24986	0.624930	+	0.780680i	$0.285127\pi$
$228$	0	0
$229$	1.30878e6i	1.64922i	0.565703	+	0.824609i	$0.308605\pi$
$229$	1.30878e6i	1.64922i	−0.565703	+	0.824609i	$0.691395\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.09583e6	1.32237	0.661183	−	0.750225i	$-0.270055\pi$
$233$	1.09583e6	1.32237	0.661183	+	0.750225i	$0.270055\pi$
$234$	0	0
$235$	138028.i	0.163041i
$236$	0	0
$237$	− 786673.i	− 0.909753i
$238$	0	0
$239$	− 48553.5i	− 0.0549826i	−0.999622	−	0.0274913i	$-0.991248\pi$
$239$	− 48553.5i	− 0.0549826i	0.999622	−	0.0274913i	$-0.00875186\pi$
$240$	0	0
$241$	842646.i	0.934550i	0.884112	+	0.467275i	$0.154764\pi$
$241$	842646.i	0.934550i	−0.884112	+	0.467275i	$0.845236\pi$
$242$	0	0
$243$	−927735.	−1.00788
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 34404.0i	− 0.0358812i
$248$	0	0
$249$	768025.	0.785014
$250$	0	0
$251$	−1.42770e6	−1.43039	−0.715193	−	0.698927i	$-0.753661\pi$
$251$	−1.42770e6	−1.43039	−0.715193	+	0.698927i	$0.753661\pi$
$252$	0	0
$253$	79374.3	0.0779612
$254$	0	0
$255$	−191822.	−0.184734
$256$	0	0
$257$	460344.i	0.434760i	0.976087	+	0.217380i	$0.0697510\pi$
$257$	460344.i	0.434760i	−0.976087	+	0.217380i	$0.930249\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	927550.	0.842822
$262$	0	0
$263$	− 1.11184e6i	− 0.991179i	−0.868557	−	0.495590i	$-0.834952\pi$
$263$	− 1.11184e6i	− 0.991179i	0.868557	−	0.495590i	$-0.165048\pi$
$264$	0	0
$265$	− 637414.i	− 0.557580i
$266$	0	0
$267$	− 293609.i	− 0.252053i
$268$	0	0
$269$	1.28022e6i	1.07871i	0.842079	+	0.539354i	$0.181332\pi$
$269$	1.28022e6i	1.07871i	−0.842079	+	0.539354i	$0.818668\pi$
$270$	0	0
$271$	975526.	0.806892	0.403446	−	0.915003i	$-0.367812\pi$
$271$	975526.	0.806892	0.403446	+	0.915003i	$0.367812\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	9126.93i	0.00727768i
$276$	0	0
$277$	−1.47590e6	−1.15573	−0.577866	−	0.816132i	$-0.696114\pi$
$277$	−1.47590e6	−1.15573	−0.577866	+	0.816132i	$0.696114\pi$
$278$	0	0
$279$	1.45615e6	1.11994
$280$	0	0
$281$	1.59868e6	1.20780	0.603902	−	0.797058i	$-0.293612\pi$
$281$	1.59868e6	1.20780	0.603902	+	0.797058i	$0.293612\pi$
$282$	0	0
$283$	−1.10947e6	−0.823475	−0.411738	−	0.911302i	$-0.635078\pi$
$283$	−1.10947e6	−0.823475	−0.411738	+	0.911302i	$0.635078\pi$
$284$	0	0
$285$	290920.i	0.212159i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.27111e6	0.895238
$290$	0	0
$291$	1.24666e6i	0.863012i
$292$	0	0
$293$	− 1.42205e6i	− 0.967710i	−0.875148	−	0.483855i	$-0.839236\pi$
$293$	− 1.42205e6i	− 0.967710i	0.875148	−	0.483855i	$-0.160764\pi$
$294$	0	0
$295$	762472.i	0.510116i
$296$	0	0
$297$	91801.8i	0.0603893i
$298$	0	0
$299$	−190995.	−0.123550
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 1.07642e6i	− 0.673557i
$304$	0	0
$305$	1.10123e6	0.677840
$306$	0	0
$307$	2.54564e6	1.54153	0.770764	−	0.637121i	$-0.219875\pi$
$307$	2.54564e6	1.54153	0.770764	+	0.637121i	$0.219875\pi$
$308$	0	0
$309$	1.00346e6	0.597865
$310$	0	0
$311$	−2.41831e6	−1.41779	−0.708895	−	0.705314i	$-0.750806\pi$
$311$	−2.41831e6	−1.41779	−0.708895	+	0.705314i	$0.750806\pi$
$312$	0	0
$313$	3.45818e6i	1.99520i	0.0692201	+	0.997601i	$0.477949\pi$
$313$	3.45818e6i	1.99520i	−0.0692201	+	0.997601i	$0.522051\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.12234e6	1.74514	0.872572	−	0.488485i	$-0.162450\pi$
$317$	3.12234e6	1.74514	0.872572	+	0.488485i	$0.162450\pi$
$318$	0	0
$319$	− 148091.i	− 0.0814802i
$320$	0	0
$321$	− 320843.i	− 0.173792i
$322$	0	0
$323$	225592.i	0.120315i
$324$	0	0
$325$	− 21961.7i	− 0.0115334i
$326$	0	0
$327$	540857.	0.279713
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 869474.i	− 0.436201i	−0.975926	−	0.218100i	$-0.930014\pi$
$331$	− 869474.i	− 0.436201i	0.975926	−	0.218100i	$-0.0699860\pi$
$332$	0	0
$333$	886238.	0.437966
$334$	0	0
$335$	−2.21672e6	−1.07919
$336$	0	0
$337$	1.87504e6	0.899365	0.449682	−	0.893188i	$-0.351537\pi$
$337$	1.87504e6	0.899365	0.449682	+	0.893188i	$0.351537\pi$
$338$	0	0
$339$	−1.12373e6	−0.531085
$340$	0	0
$341$	− 232486.i	− 0.108271i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1.61505e6	0.730532
$346$	0	0
$347$	− 3.64180e6i	− 1.62365i	−0.583900	−	0.811826i	$-0.698474\pi$
$347$	− 3.64180e6i	− 1.62365i	0.583900	−	0.811826i	$-0.301526\pi$
$348$	0	0
$349$	− 665541.i	− 0.292490i	−0.989248	−	0.146245i	$-0.953281\pi$
$349$	− 665541.i	− 0.292490i	0.989248	−	0.146245i	$-0.0467188\pi$
$350$	0	0
$351$	− 220899.i	− 0.0957030i
$352$	0	0
$353$	434756.i	0.185699i	0.995680	+	0.0928493i	$0.0295975\pi$
$353$	434756.i	0.185699i	−0.995680	+	0.0928493i	$0.970403\pi$
$354$	0	0
$355$	−3.94339e6	−1.66073
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.11762e6i	0.867186i	0.901109	+	0.433593i	$0.142754\pi$
$359$	2.11762e6i	0.867186i	−0.901109	+	0.433593i	$0.857246\pi$
$360$	0	0
$361$	−2.13396e6	−0.861824
$362$	0	0
$363$	−1.52135e6	−0.605986
$364$	0	0
$365$	−504754.	−0.198312
$366$	0	0
$367$	−245671.	−0.0952114	−0.0476057	−	0.998866i	$-0.515159\pi$
$367$	−245671.	−0.0952114	−0.0476057	+	0.998866i	$0.515159\pi$
$368$	0	0
$369$	− 1.35630e6i	− 0.518547i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−3.48375e6	−1.29651	−0.648254	−	0.761424i	$-0.724501\pi$
$373$	−3.48375e6	−1.29651	−0.648254	+	0.761424i	$0.724501\pi$
$374$	0	0
$375$	1.73997e6i	0.638945i
$376$	0	0
$377$	356345.i	0.129127i
$378$	0	0
$379$	− 3.07711e6i	− 1.10039i	−0.835037	−	0.550194i	$-0.814554\pi$
$379$	− 3.07711e6i	− 1.10039i	0.835037	−	0.550194i	$-0.185446\pi$
$380$	0	0
$381$	448496.i	0.158287i
$382$	0	0
$383$	4.83959e6	1.68582	0.842912	−	0.538052i	$-0.180840\pi$
$383$	4.83959e6	1.68582	0.842912	+	0.538052i	$0.180840\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.13153e6i	− 0.384050i
$388$	0	0
$389$	2.17484e6	0.728707	0.364354	−	0.931261i	$-0.381290\pi$
$389$	2.17484e6	0.728707	0.364354	+	0.931261i	$0.381290\pi$
$390$	0	0
$391$	1.25238e6	0.414281
$392$	0	0
$393$	−2.24719e6	−0.733936
$394$	0	0
$395$	4.35219e6	1.40351
$396$	0	0
$397$	430287.i	0.137020i	0.997650	+	0.0685098i	$0.0218244\pi$
$397$	430287.i	0.137020i	−0.997650	+	0.0685098i	$0.978176\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−89985.2	−0.0279454	−0.0139727	−	0.999902i	$-0.504448\pi$
$401$	−89985.2	−0.0279454	−0.0139727	+	0.999902i	$0.504448\pi$
$402$	0	0
$403$	559422.i	0.171584i
$404$	0	0
$405$	− 83609.7i	− 0.0253291i
$406$	0	0
$407$	− 141495.i	− 0.0423405i
$408$	0	0
$409$	2.38113e6i	0.703843i	0.936030	+	0.351921i	$0.114472\pi$
$409$	2.38113e6i	0.703843i	−0.936030	+	0.351921i	$0.885528\pi$
$410$	0	0
$411$	−847150.	−0.247375
$412$	0	0
$413$	0	0
$414$	0	0
$415$	4.24903e6i	1.21107i
$416$	0	0
$417$	−1.38808e6	−0.390908
$418$	0	0
$419$	−943020.	−0.262413	−0.131207	−	0.991355i	$-0.541885\pi$
$419$	−943020.	−0.262413	−0.131207	+	0.991355i	$0.541885\pi$
$420$	0	0
$421$	−2.41477e6	−0.664005	−0.332002	−	0.943279i	$-0.607724\pi$
$421$	−2.41477e6	−0.664005	−0.332002	+	0.943279i	$0.607724\pi$
$422$	0	0
$423$	402854.	0.109470
$424$	0	0
$425$	144007.i	0.0386732i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−13631.9	−0.00357612
$430$	0	0
$431$	− 4.05254e6i	− 1.05083i	−0.850845	−	0.525417i	$-0.823910\pi$
$431$	− 4.05254e6i	− 1.05083i	0.850845	−	0.525417i	$-0.176090\pi$
$432$	0	0
$433$	4.08792e6i	1.04781i	0.851777	+	0.523905i	$0.175525\pi$
$433$	4.08792e6i	1.04781i	−0.851777	+	0.523905i	$0.824475\pi$
$434$	0	0
$435$	− 3.01325e6i	− 0.763506i
$436$	0	0
$437$	− 1.89939e6i	− 0.475784i
$438$	0	0
$439$	−4.08271e6	−1.01108	−0.505541	−	0.862802i	$-0.668707\pi$
$439$	−4.08271e6	−1.01108	−0.505541	+	0.862802i	$0.668707\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.20981e6i	1.01919i	0.860416	+	0.509593i	$0.170204\pi$
$443$	4.20981e6i	1.01919i	−0.860416	+	0.509593i	$0.829796\pi$
$444$	0	0
$445$	1.62436e6	0.388851
$446$	0	0
$447$	−3.95982e6	−0.937361
$448$	0	0
$449$	−3.96898e6	−0.929102	−0.464551	−	0.885546i	$-0.653784\pi$
$449$	−3.96898e6	−0.929102	−0.464551	+	0.885546i	$0.653784\pi$
$450$	0	0
$451$	−216544.	−0.0501308
$452$	0	0
$453$	− 4.05571e6i	− 0.928584i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.80157e6	−1.52342	−0.761708	−	0.647920i	$-0.775639\pi$
$457$	−6.80157e6	−1.52342	−0.761708	+	0.647920i	$0.775639\pi$
$458$	0	0
$459$	1.44847e6i	0.320905i
$460$	0	0
$461$	1.05680e6i	0.231601i	0.993272	+	0.115801i	$0.0369433\pi$
$461$	1.05680e6i	0.231601i	−0.993272	+	0.115801i	$0.963057\pi$
$462$	0	0
$463$	3.32704e6i	0.721283i	0.932704	+	0.360642i	$0.117442\pi$
$463$	3.32704e6i	0.721283i	−0.932704	+	0.360642i	$0.882558\pi$
$464$	0	0
$465$	− 4.73047e6i	− 1.01455i
$466$	0	0
$467$	592810.	0.125783	0.0628917	−	0.998020i	$-0.479968\pi$
$467$	592810.	0.125783	0.0628917	+	0.998020i	$0.479968\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 4.52358e6i	− 0.939573i
$472$	0	0
$473$	−180658.	−0.0371282
$474$	0	0
$475$	218403.	0.0444145
$476$	0	0
$477$	−1.86039e6	−0.374375
$478$	0	0
$479$	3.56256e6	0.709453	0.354726	−	0.934970i	$-0.384574\pi$
$479$	3.56256e6	0.709453	0.354726	+	0.934970i	$0.384574\pi$
$480$	0	0
$481$	340474.i	0.0670998i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.89704e6	−1.33140
$486$	0	0
$487$	6.19970e6i	1.18454i	0.805741	+	0.592268i	$0.201767\pi$
$487$	6.19970e6i	1.18454i	−0.805741	+	0.592268i	$0.798233\pi$
$488$	0	0
$489$	4.34069e6i	0.820894i
$490$	0	0
$491$	8.52991e6i	1.59676i	0.602152	+	0.798382i	$0.294310\pi$
$491$	8.52991e6i	1.59676i	−0.602152	+	0.798382i	$0.705690\pi$
$492$	0	0
$493$	− 2.33661e6i	− 0.432981i
$494$	0	0
$495$	−196307.	−0.0360100
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 6.69510e6i	− 1.20367i	−0.798622	−	0.601833i	$-0.794437\pi$
$499$	− 6.69510e6i	− 1.20367i	0.798622	−	0.601833i	$-0.205563\pi$
$500$	0	0
$501$	16718.6	0.00297581
$502$	0	0
$503$	8.34500e6	1.47064	0.735320	−	0.677720i	$-0.237032\pi$
$503$	8.34500e6	1.47064	0.735320	+	0.677720i	$0.237032\pi$
$504$	0	0
$505$	5.95518e6	1.03912
$506$	0	0
$507$	−3.48764e6	−0.602575
$508$	0	0
$509$	− 1.07136e7i	− 1.83291i	−0.400142	−	0.916453i	$-0.631039\pi$
$509$	− 1.07136e7i	− 1.83291i	0.400142	−	0.916453i	$-0.368961\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	2.19677e6	0.368546
$514$	0	0
$515$	5.55154e6i	0.922349i
$516$	0	0
$517$	− 64318.9i	− 0.0105831i
$518$	0	0
$519$	− 5.75861e6i	− 0.938425i
$520$	0	0
$521$	− 2.23971e6i	− 0.361491i	−0.983530	−	0.180745i	$-0.942149\pi$
$521$	− 2.23971e6i	− 0.361491i	0.983530	−	0.180745i	$-0.0578510\pi$
$522$	0	0
$523$	1.20123e7	1.92031	0.960156	−	0.279466i	$-0.0901573\pi$
$523$	1.20123e7	1.92031	0.960156	+	0.279466i	$0.0901573\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 3.66821e6i	− 0.575345i
$528$	0	0
$529$	−4.10817e6	−0.638277
$530$	0	0
$531$	2.22539e6	0.342507
$532$	0	0
$533$	521060.	0.0794456
$534$	0	0
$535$	1.77503e6	0.268116
$536$	0	0
$537$	2.53517e6i	0.379377i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−4.72006e6	−0.693353	−0.346677	−	0.937985i	$-0.612690\pi$
$541$	−4.72006e6	−0.693353	−0.346677	+	0.937985i	$0.612690\pi$
$542$	0	0
$543$	− 1.61286e6i	− 0.234746i
$544$	0	0
$545$	2.99224e6i	0.431524i
$546$	0	0
$547$	8.42384e6i	1.20377i	0.798584	+	0.601883i	$0.205583\pi$
$547$	8.42384e6i	1.20377i	−0.798584	+	0.601883i	$0.794417\pi$
$548$	0	0
$549$	− 3.21409e6i	− 0.455122i
$550$	0	0
$551$	−3.54375e6	−0.497260
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 2.87905e6i	− 0.396750i
$556$	0	0
$557$	−1.19369e6	−0.163025	−0.0815125	−	0.996672i	$-0.525975\pi$
$557$	−1.19369e6	−0.163025	−0.0815125	+	0.996672i	$0.525975\pi$
$558$	0	0
$559$	434709.	0.0588395
$560$	0	0
$561$	89386.2	0.0119912
$562$	0	0
$563$	−7.45802e6	−0.991637	−0.495818	−	0.868426i	$-0.665132\pi$
$563$	−7.45802e6	−0.991637	−0.495818	+	0.868426i	$0.665132\pi$
$564$	0	0
$565$	− 6.21695e6i	− 0.819325i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.74821e6	−0.226366	−0.113183	−	0.993574i	$-0.536105\pi$
$569$	−1.74821e6	−0.226366	−0.113183	+	0.993574i	$0.536105\pi$
$570$	0	0
$571$	− 9.44843e6i	− 1.21274i	−0.795181	−	0.606372i	$-0.792624\pi$
$571$	− 9.44843e6i	− 1.21274i	0.795181	−	0.606372i	$-0.207376\pi$
$572$	0	0
$573$	− 6.90289e6i	− 0.878303i
$574$	0	0
$575$	− 1.21247e6i	− 0.152933i
$576$	0	0
$577$	− 1.17492e7i	− 1.46916i	−0.678524	−	0.734578i	$-0.737380\pi$
$577$	− 1.17492e7i	− 1.46916i	0.678524	−	0.734578i	$-0.262620\pi$
$578$	0	0
$579$	−4.50853e6	−0.558906
$580$	0	0
$581$	0	0
$582$	0	0
$583$	297026.i	0.0361929i
$584$	0	0
$585$	472365.	0.0570674
$586$	0	0
$587$	137068.	0.0164188	0.00820941	−	0.999966i	$-0.497387\pi$
$587$	137068.	0.0164188	0.00820941	+	0.999966i	$0.497387\pi$
$588$	0	0
$589$	−5.56328e6	−0.660759
$590$	0	0
$591$	3.90329e6	0.459687
$592$	0	0
$593$	− 1.21101e7i	− 1.41419i	−0.707116	−	0.707097i	$-0.750004\pi$
$593$	− 1.21101e7i	− 1.41419i	0.707116	−	0.707097i	$-0.249996\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.89726e6	−0.792028
$598$	0	0
$599$	1.57170e7i	1.78979i	0.446280	+	0.894893i	$0.352749\pi$
$599$	1.57170e7i	1.78979i	−0.446280	+	0.894893i	$0.647251\pi$
$600$	0	0
$601$	4.02817e6i	0.454906i	0.973789	+	0.227453i	$0.0730398\pi$
$601$	4.02817e6i	0.454906i	−0.973789	+	0.227453i	$0.926960\pi$
$602$	0	0
$603$	6.46982e6i	0.724601i
$604$	0	0
$605$	− 8.41672e6i	− 0.934877i
$606$	0	0
$607$	−441862.	−0.0486760	−0.0243380	−	0.999704i	$-0.507748\pi$
$607$	−441862.	−0.0486760	−0.0243380	+	0.999704i	$0.507748\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	154768.i	0.0167717i
$612$	0	0
$613$	−1.29752e7	−1.39464	−0.697321	−	0.716759i	$-0.745625\pi$
$613$	−1.29752e7	−1.39464	−0.697321	+	0.716759i	$0.745625\pi$
$614$	0	0
$615$	−4.40608e6	−0.469748
$616$	0	0
$617$	2.98260e6	0.315415	0.157707	−	0.987486i	$-0.449590\pi$
$617$	2.98260e6	0.315415	0.157707	+	0.987486i	$0.449590\pi$
$618$	0	0
$619$	8.56365e6	0.898323	0.449161	−	0.893451i	$-0.351723\pi$
$619$	8.56365e6	0.898323	0.449161	+	0.893451i	$0.351723\pi$
$620$	0	0
$621$	− 1.21955e7i	− 1.26902i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−8.45938e6	−0.866240
$626$	0	0
$627$	− 135565.i	− 0.0137714i
$628$	0	0
$629$	− 2.23254e6i	− 0.224995i
$630$	0	0
$631$	− 3.19204e6i	− 0.319150i	−0.987186	−	0.159575i	$-0.948988\pi$
$631$	− 3.19204e6i	− 0.319150i	0.987186	−	0.159575i	$-0.0510124\pi$
$632$	0	0
$633$	839138.i	0.0832385i
$634$	0	0
$635$	−2.48126e6	−0.244196
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.15093e7i	1.11506i
$640$	0	0
$641$	6.80979e6	0.654619	0.327310	−	0.944917i	$-0.393858\pi$
$641$	6.80979e6	0.654619	0.327310	+	0.944917i	$0.393858\pi$
$642$	0	0
$643$	−5.81654e6	−0.554801	−0.277401	−	0.960754i	$-0.589473\pi$
$643$	−5.81654e6	−0.554801	−0.277401	+	0.960754i	$0.589473\pi$
$644$	0	0
$645$	−3.67590e6	−0.347908
$646$	0	0
$647$	−892444.	−0.0838147	−0.0419074	−	0.999122i	$-0.513343\pi$
$647$	−892444.	−0.0838147	−0.0419074	+	0.999122i	$0.513343\pi$
$648$	0	0
$649$	− 355301.i	− 0.0331120i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2.78892e6	−0.255949	−0.127974	−	0.991777i	$-0.540848\pi$
$653$	−2.78892e6	−0.255949	−0.127974	+	0.991777i	$0.540848\pi$
$654$	0	0
$655$	− 1.24324e7i	− 1.13227i
$656$	0	0
$657$	1.47320e6i	0.133152i
$658$	0	0
$659$	− 2.06628e7i	− 1.85342i	−0.375771	−	0.926712i	$-0.622622\pi$
$659$	− 2.06628e7i	− 1.85342i	0.375771	−	0.926712i	$-0.377378\pi$
$660$	0	0
$661$	− 972662.i	− 0.0865881i	−0.999062	−	0.0432941i	$-0.986215\pi$
$661$	− 972662.i	− 0.0865881i	0.999062	−	0.0432941i	$-0.0137852\pi$
$662$	0	0
$663$	−215086.	−0.0190033
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.96732e7i	1.71223i
$668$	0	0
$669$	1.38091e6	0.119289
$670$	0	0
$671$	−513156.	−0.0439991
$672$	0	0
$673$	4.79868e6	0.408399	0.204199	−	0.978929i	$-0.434541\pi$
$673$	4.79868e6	0.408399	0.204199	+	0.978929i	$0.434541\pi$
$674$	0	0
$675$	1.40231e6	0.118463
$676$	0	0
$677$	− 5.14787e6i	− 0.431674i	−0.976429	−	0.215837i	$-0.930752\pi$
$677$	− 5.14787e6i	− 0.431674i	0.976429	−	0.215837i	$-0.0692480\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−9.20039e6	−0.760218
$682$	0	0
$683$	− 9.85451e6i	− 0.808320i	−0.914688	−	0.404160i	$-0.867564\pi$
$683$	− 9.85451e6i	− 0.808320i	0.914688	−	0.404160i	$-0.132436\pi$
$684$	0	0
$685$	− 4.68677e6i	− 0.381634i
$686$	0	0
$687$	− 1.24093e7i	− 1.00312i
$688$	0	0
$689$	− 714721.i	− 0.0573573i
$690$	0	0
$691$	7.61201e6	0.606463	0.303231	−	0.952917i	$-0.401935\pi$
$691$	7.61201e6	0.606463	0.303231	+	0.952917i	$0.401935\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 7.67943e6i	− 0.603069i
$696$	0	0
$697$	−3.41667e6	−0.266392
$698$	0	0
$699$	−1.03901e7	−0.804319
$700$	0	0
$701$	−2.19372e7	−1.68611	−0.843055	−	0.537827i	$-0.819245\pi$
$701$	−2.19372e7	−1.68611	−0.843055	+	0.537827i	$0.819245\pi$
$702$	0	0
$703$	−3.38591e6	−0.258397
$704$	0	0
$705$	− 1.30872e6i	− 0.0991683i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1.13575e7	−0.848528	−0.424264	−	0.905538i	$-0.639467\pi$
$709$	−1.13575e7	−0.848528	−0.424264	+	0.905538i	$0.639467\pi$
$710$	0	0
$711$	− 1.27025e7i	− 0.942357i
$712$	0	0
$713$	3.08847e7i	2.27520i
$714$	0	0
$715$	− 75416.9i	− 0.00551701i
$716$	0	0
$717$	460363.i	0.0334428i
$718$	0	0
$719$	−1.95915e7	−1.41334	−0.706669	−	0.707545i	$-0.749803\pi$
$719$	−1.95915e7	−1.41334	−0.706669	+	0.707545i	$0.749803\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 7.98960e6i	− 0.568433i
$724$	0	0
$725$	−2.26214e6	−0.159836
$726$	0	0
$727$	386859.	0.0271467	0.0135733	−	0.999908i	$-0.495679\pi$
$727$	386859.	0.0271467	0.0135733	+	0.999908i	$0.495679\pi$
$728$	0	0
$729$	8.40905e6	0.586041
$730$	0	0
$731$	−2.85045e6	−0.197297
$732$	0	0
$733$	− 8.30026e6i	− 0.570600i	−0.958438	−	0.285300i	$-0.907907\pi$
$733$	− 8.30026e6i	− 0.570600i	0.958438	−	0.285300i	$-0.0920932\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.03296e6	0.0700511
$738$	0	0
$739$	1.33460e7i	0.898957i	0.893291	+	0.449478i	$0.148390\pi$
$739$	1.33460e7i	0.898957i	−0.893291	+	0.449478i	$0.851610\pi$
$740$	0	0
$741$	326204.i	0.0218245i
$742$	0	0
$743$	2.31448e7i	1.53809i	0.639196	+	0.769044i	$0.279267\pi$
$743$	2.31448e7i	1.53809i	−0.639196	+	0.769044i	$0.720733\pi$
$744$	0	0
$745$	− 2.19073e7i	− 1.44610i
$746$	0	0
$747$	1.24014e7	0.813148
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 944678.i	− 0.0611201i	−0.999533	−	0.0305600i	$-0.990271\pi$
$751$	− 944678.i	− 0.0611201i	0.999533	−	0.0305600i	$-0.00972908\pi$
$752$	0	0
$753$	1.35368e7	0.870022
$754$	0	0
$755$	2.24378e7	1.43256
$756$	0	0
$757$	−2.40741e7	−1.52690	−0.763448	−	0.645869i	$-0.776495\pi$
$757$	−2.40741e7	−1.52690	−0.763448	+	0.645869i	$0.776495\pi$
$758$	0	0
$759$	−752592.	−0.0474193
$760$	0	0
$761$	1.60564e7i	1.00504i	0.864564	+	0.502522i	$0.167594\pi$
$761$	1.60564e7i	1.00504i	−0.864564	+	0.502522i	$0.832406\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−3.09737e6	−0.191355
$766$	0	0
$767$	854946.i	0.0524748i
$768$	0	0
$769$	1.52779e7i	0.931641i	0.884879	+	0.465820i	$0.154241\pi$
$769$	1.52779e7i	0.931641i	−0.884879	+	0.465820i	$0.845759\pi$
$770$	0	0
$771$	− 4.36478e6i	− 0.264439i
$772$	0	0
$773$	1.87480e7i	1.12851i	0.825599	+	0.564257i	$0.190837\pi$
$773$	1.87480e7i	1.12851i	−0.825599	+	0.564257i	$0.809163\pi$
$774$	0	0
$775$	−3.55131e6	−0.212390
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.18179e6i	0.305940i
$780$	0	0
$781$	1.83756e6	0.107799
$782$	0	0
$783$	−2.27534e7	−1.32630
$784$	0	0
$785$	2.50263e7	1.44951
$786$	0	0
$787$	−1.49465e7	−0.860208	−0.430104	−	0.902779i	$-0.641523\pi$
$787$	−1.49465e7	−0.860208	−0.430104	+	0.902779i	$0.641523\pi$
$788$	0	0
$789$	1.05420e7i	0.602877i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1.23479e6	0.0697282
$794$	0	0
$795$	6.04368e6i	0.339144i
$796$	0	0
$797$	− 1.23503e7i	− 0.688702i	−0.938841	−	0.344351i	$-0.888099\pi$
$797$	− 1.23503e7i	− 0.688702i	0.938841	−	0.344351i	$-0.111901\pi$
$798$	0	0
$799$	− 1.01484e6i	− 0.0562379i
$800$	0	0
$801$	− 4.74094e6i	− 0.261086i
$802$	0	0
$803$	235209.	0.0128725
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 1.21385e7i	− 0.656116i
$808$	0	0
$809$	1.04876e7	0.563383	0.281691	−	0.959505i	$-0.409105\pi$
$809$	1.04876e7	0.563383	0.281691	+	0.959505i	$0.409105\pi$
$810$	0	0
$811$	1.19421e7	0.637570	0.318785	−	0.947827i	$-0.396725\pi$
$811$	1.19421e7	0.637570	0.318785	+	0.947827i	$0.396725\pi$
$812$	0	0
$813$	−9.24951e6	−0.490786
$814$	0	0
$815$	−2.40145e7	−1.26642
$816$	0	0
$817$	4.32305e6i	0.226587i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	398785.	0.0206481	0.0103241	−	0.999947i	$-0.496714\pi$
$821$	398785.	0.0206481	0.0103241	+	0.999947i	$0.496714\pi$
$822$	0	0
$823$	− 2.49256e7i	− 1.28276i	−0.767223	−	0.641380i	$-0.778362\pi$
$823$	− 2.49256e7i	− 1.28276i	0.767223	−	0.641380i	$-0.221638\pi$
$824$	0	0
$825$	− 86537.6i	− 0.00442660i
$826$	0	0
$827$	− 4.53496e6i	− 0.230573i	−0.993332	−	0.115287i	$-0.963221\pi$
$827$	− 4.53496e6i	− 0.230573i	0.993332	−	0.115287i	$-0.0367787\pi$
$828$	0	0
$829$	2.70574e7i	1.36742i	0.729756	+	0.683708i	$0.239634\pi$
$829$	2.70574e7i	1.36742i	−0.729756	+	0.683708i	$0.760366\pi$
$830$	0	0
$831$	1.39938e7	0.702965
$832$	0	0
$833$	0	0
$834$	0	0
$835$	92493.8i	0.00459089i
$836$	0	0
$837$	−3.57203e7	−1.76239
$838$	0	0
$839$	3.66855e7	1.79924	0.899620	−	0.436673i	$-0.143843\pi$
$839$	3.66855e7	1.79924	0.899620	+	0.436673i	$0.143843\pi$
$840$	0	0
$841$	1.61938e7	0.789511
$842$	0	0
$843$	−1.51580e7	−0.734638
$844$	0	0
$845$	− 1.92950e7i	− 0.929615i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1.05195e7	0.500873
$850$	0	0
$851$	1.87970e7i	0.889743i
$852$	0	0
$853$	− 2.04043e7i	− 0.960174i	−0.877221	−	0.480087i	$-0.840605\pi$
$853$	− 2.04043e7i	− 0.960174i	0.877221	−	0.480087i	$-0.159395\pi$
$854$	0	0
$855$	4.69753e6i	0.219763i
$856$	0	0
$857$	2.43127e6i	0.113079i	0.998400	+	0.0565395i	$0.0180067\pi$
$857$	2.43127e6i	0.113079i	−0.998400	+	0.0565395i	$0.981993\pi$
$858$	0	0
$859$	−1.66357e7	−0.769234	−0.384617	−	0.923076i	$-0.625666\pi$
$859$	−1.66357e7	−0.769234	−0.384617	+	0.923076i	$0.625666\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 3.03760e7i	− 1.38837i	−0.719799	−	0.694183i	$-0.755766\pi$
$863$	− 3.03760e7i	− 1.38837i	0.719799	−	0.694183i	$-0.244234\pi$
$864$	0	0
$865$	3.18590e7	1.44774
$866$	0	0
$867$	−1.20521e7	−0.544522
$868$	0	0
$869$	−2.02806e6	−0.0911028
$870$	0	0
$871$	−2.48557e6	−0.111015
$872$	0	0
$873$	2.01300e7i	0.893941i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.01722e7	1.32467	0.662336	−	0.749207i	$-0.269565\pi$
$877$	3.01722e7	1.32467	0.662336	+	0.749207i	$0.269565\pi$
$878$	0	0
$879$	1.34832e7i	0.588602i
$880$	0	0
$881$	− 1.00893e7i	− 0.437947i	−0.975731	−	0.218973i	$-0.929729\pi$
$881$	− 1.00893e7i	− 0.437947i	0.975731	−	0.218973i	$-0.0702708\pi$
$882$	0	0
$883$	1.36170e7i	0.587733i	0.955846	+	0.293867i	$0.0949422\pi$
$883$	1.36170e7i	0.587733i	−0.955846	+	0.293867i	$0.905058\pi$
$884$	0	0
$885$	− 7.22943e6i	− 0.310274i
$886$	0	0
$887$	1.01937e7	0.435035	0.217518	−	0.976056i	$-0.430204\pi$
$887$	1.01937e7	0.435035	0.217518	+	0.976056i	$0.430204\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	38961.0i	0.00164413i
$892$	0	0
$893$	−1.53912e6	−0.0645868
$894$	0	0
$895$	−1.40256e7	−0.585279
$896$	0	0
$897$	1.81093e6	0.0751485
$898$	0	0
$899$	5.76226e7	2.37790
$900$	0	0
$901$	4.68654e6i	0.192327i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8.92301e6	0.362151
$906$	0	0
$907$	4.58587e6i	0.185099i	0.995708	+	0.0925495i	$0.0295016\pi$
$907$	4.58587e6i	0.185099i	−0.995708	+	0.0925495i	$0.970498\pi$
$908$	0	0
$909$	− 1.73811e7i	− 0.697696i
$910$	0	0
$911$	2.52740e7i	1.00897i	0.863421	+	0.504484i	$0.168317\pi$
$911$	2.52740e7i	1.00897i	−0.863421	+	0.504484i	$0.831683\pi$
$912$	0	0
$913$	− 1.97999e6i	− 0.0786114i
$914$	0	0
$915$	−1.04414e7	−0.412291
$916$	0	0
$917$	0	0
$918$	0	0
$919$	2.84356e7i	1.11064i	0.831637	+	0.555320i	$0.187404\pi$
$919$	2.84356e7i	1.11064i	−0.831637	+	0.555320i	$0.812596\pi$
$920$	0	0
$921$	−2.41367e7	−0.937623
$922$	0	0
$923$	−4.42165e6	−0.170836
$924$	0	0
$925$	−2.16139e6	−0.0830576
$926$	0	0
$927$	1.62030e7	0.619292
$928$	0	0
$929$	3.73348e7i	1.41930i	0.704554	+	0.709650i	$0.251147\pi$
$929$	3.73348e7i	1.41930i	−0.704554	+	0.709650i	$0.748853\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	2.29294e7	0.862360
$934$	0	0
$935$	494520.i	0.0184993i
$936$	0	0
$937$	− 2.42023e7i	− 0.900548i	−0.892890	−	0.450274i	$-0.851326\pi$
$937$	− 2.42023e7i	− 0.900548i	0.892890	−	0.450274i	$-0.148674\pi$
$938$	0	0
$939$	− 3.27890e7i	− 1.21357i
$940$	0	0
$941$	4.46206e7i	1.64271i	0.570415	+	0.821356i	$0.306782\pi$
$941$	4.46206e7i	1.64271i	−0.570415	+	0.821356i	$0.693218\pi$
$942$	0	0
$943$	2.87669e7	1.05345
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.64895e7i	0.959840i	0.877312	+	0.479920i	$0.159334\pi$
$947$	2.64895e7i	0.959840i	−0.877312	+	0.479920i	$0.840666\pi$
$948$	0	0
$949$	−565972.	−0.0204000
$950$	0	0
$951$	−2.96046e7	−1.06147
$952$	0	0
$953$	3.59018e7	1.28051	0.640257	−	0.768161i	$-0.278828\pi$
$953$	3.59018e7	1.28051	0.640257	+	0.768161i	$0.278828\pi$
$954$	0	0
$955$	3.81896e7	1.35499
$956$	0	0
$957$	1.40413e6i	0.0495597i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	6.18319e7	2.15975
$962$	0	0
$963$	− 5.18069e6i	− 0.180021i
$964$	0	0
$965$	− 2.49430e7i	− 0.862245i
$966$	0	0
$967$	5.76157e6i	0.198141i	0.995080	+	0.0990705i	$0.0315869\pi$
$967$	5.76157e6i	0.198141i	−0.995080	+	0.0990705i	$0.968413\pi$
$968$	0	0
$969$	− 2.13897e6i	− 0.0731804i
$970$	0	0
$971$	−9.22567e6	−0.314015	−0.157007	−	0.987597i	$-0.550185\pi$
$971$	−9.22567e6	−0.314015	−0.157007	+	0.987597i	$0.550185\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	208232.i	0.00701512i
$976$	0	0
$977$	1.19440e7	0.400327	0.200163	−	0.979763i	$-0.435853\pi$
$977$	1.19440e7	0.400327	0.200163	+	0.979763i	$0.435853\pi$
$978$	0	0
$979$	−756931.	−0.0252406
$980$	0	0
$981$	8.73328e6	0.289738
$982$	0	0
$983$	−6.26881e6	−0.206919	−0.103460	−	0.994634i	$-0.532991\pi$
$983$	−6.26881e6	−0.206919	−0.103460	+	0.994634i	$0.532991\pi$
$984$	0	0
$985$	2.15946e7i	0.709176i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.39996e7	0.780212
$990$	0	0
$991$	− 5.21735e6i	− 0.168758i	−0.996434	−	0.0843792i	$-0.973109\pi$
$991$	− 5.21735e6i	− 0.168758i	0.996434	−	0.0843792i	$-0.0268907\pi$
$992$	0	0
$993$	8.24397e6i	0.265316i
$994$	0	0
$995$	− 3.81584e7i	− 1.22189i
$996$	0	0
$997$	3.05016e7i	0.971816i	0.874010	+	0.485908i	$0.161511\pi$
$997$	3.05016e7i	0.971816i	−0.874010	+	0.485908i	$0.838489\pi$
$998$	0	0
$999$	−2.17400e7	−0.689202

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.f.b.783.5		12
4.3	odd	2	inner	784.6.f.b.783.7		12
7.4	even	3		112.6.p.a.47.4	yes	12
7.5	odd	6		112.6.p.a.31.3	✓	12
7.6	odd	2	inner	784.6.f.b.783.8		12
28.11	odd	6		112.6.p.a.47.3	yes	12
28.19	even	6		112.6.p.a.31.4	yes	12
28.27	even	2	inner	784.6.f.b.783.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.6.p.a.31.3	✓	12	7.5	odd	6
112.6.p.a.31.4	yes	12	28.19	even	6
112.6.p.a.47.3	yes	12	28.11	odd	6
112.6.p.a.47.4	yes	12	7.4	even	3
784.6.f.b.783.5		12	1.1	even	1	trivial
784.6.f.b.783.6		12	28.27	even	2	inner
784.6.f.b.783.7		12	4.3	odd	2	inner
784.6.f.b.783.8		12	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	784.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-9.48156 q^{3} -52.4558i q^{5} -153.100 q^{9} +24.4437i q^{11} -58.8178i q^{13} +497.363i q^{15} +385.677i q^{17} +584.926 q^{19} -3247.23i q^{23} +373.386 q^{25} +3755.65 q^{27} -6058.46 q^{29} -9511.10 q^{31} -231.764i q^{33} -5788.62 q^{37} +557.684i q^{39} +8858.89i q^{41} +7390.78i q^{43} +8030.99i q^{45} -2631.31 q^{47} -3656.82i q^{51} +12151.5 q^{53} +1282.21 q^{55} -5546.01 q^{57} -14535.5 q^{59} +20993.4i q^{61} -3085.33 q^{65} -42258.8i q^{67} +30788.8i q^{69} -75175.4i q^{71} -9622.47i q^{73} -3540.28 q^{75} +82968.7i q^{79} +1593.91 q^{81} -81002.0 q^{83} +20231.0 q^{85} +57443.6 q^{87} +30966.3i q^{89} +90180.1 q^{93} -30682.8i q^{95} -131483. i q^{97} -3742.33i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 1088 q^{9} + 4920 q^{25} + 22512 q^{29} + 15876 q^{37} - 45708 q^{53} - 25900 q^{57} + 124440 q^{65} + 186164 q^{81} + 70116 q^{85} + 546364 q^{93}+O(q^{100}) \) 12 * q + 1088 * q^9 + 4920 * q^25 + 22512 * q^29 + 15876 * q^37 - 45708 * q^53 - 25900 * q^57 + 124440 * q^65 + 186164 * q^81 + 70116 * q^85 + 546364 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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