LMFDB - Embedded newform 676.6.d.a.337.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [676,6,Mod(337,676)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(676, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("676.337");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 676 = 2^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	676.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$108.419462194$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	676.337
Dual form			676.6.d.a.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-12.0000 q^{3} +54.0000i q^{5} +88.0000i q^{7} -99.0000 q^{9} +O(q^{10})$	$q-12.0000 q^{3} +54.0000i q^{5} +88.0000i q^{7} -99.0000 q^{9} -540.000i q^{11} -648.000i q^{15} -594.000 q^{17} +836.000i q^{19} -1056.00i q^{21} +4104.00 q^{23} +209.000 q^{25} +4104.00 q^{27} -594.000 q^{29} +4256.00i q^{31} +6480.00i q^{33} -4752.00 q^{35} +298.000i q^{37} +17226.0i q^{41} +12100.0 q^{43} -5346.00i q^{45} +1296.00i q^{47} +9063.00 q^{49} +7128.00 q^{51} +19494.0 q^{53} +29160.0 q^{55} -10032.0i q^{57} +7668.00i q^{59} -34738.0 q^{61} -8712.00i q^{63} +21812.0i q^{67} -49248.0 q^{69} -46872.0i q^{71} -67562.0i q^{73} -2508.00 q^{75} +47520.0 q^{77} -76912.0 q^{79} -25191.0 q^{81} +67716.0i q^{83} -32076.0i q^{85} +7128.00 q^{87} -29754.0i q^{89} -51072.0i q^{93} -45144.0 q^{95} -122398. i q^{97} +53460.0i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 24 q^{3} - 198 q^{9}+O(q^{10}) $ 2 * q - 24 * q^3 - 198 * q^9	$ 2 q - 24 q^{3} - 198 q^{9} - 1188 q^{17} + 8208 q^{23} + 418 q^{25} + 8208 q^{27} - 1188 q^{29} - 9504 q^{35} + 24200 q^{43} + 18126 q^{49} + 14256 q^{51} + 38988 q^{53} + 58320 q^{55} - 69476 q^{61} - 98496 q^{69} - 5016 q^{75} + 95040 q^{77} - 153824 q^{79} - 50382 q^{81} + 14256 q^{87} - 90288 q^{95}+O(q^{100}) $ 2 * q - 24 * q^3 - 198 * q^9 - 1188 * q^17 + 8208 * q^23 + 418 * q^25 + 8208 * q^27 - 1188 * q^29 - 9504 * q^35 + 24200 * q^43 + 18126 * q^49 + 14256 * q^51 + 38988 * q^53 + 58320 * q^55 - 69476 * q^61 - 98496 * q^69 - 5016 * q^75 + 95040 * q^77 - 153824 * q^79 - 50382 * q^81 + 14256 * q^87 - 90288 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/676\mathbb{Z}\right)^\times$.

$n$	$339$	$509$
$\chi(n)$	$1$	$-1$

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
$3$	−12.0000	−0.769800	−0.384900	−	0.922958i	$-0.625764\pi$
$3$	−12.0000	−0.769800	−0.384900	+	0.922958i	$0.625764\pi$
$4$	0	0
$5$	54.0000i	0.965981i	0.875625	+	0.482991i	$0.160450\pi$
$5$	54.0000i	0.965981i	−0.875625	+	0.482991i	$0.839550\pi$
$6$	0	0
$7$	88.0000i	0.678793i	0.940643	+	0.339397i	$0.110223\pi$
$7$	88.0000i	0.678793i	−0.940643	+	0.339397i	$0.889777\pi$
$8$	0	0
$9$	−99.0000	−0.407407
$10$	0	0
$11$	− 540.000i	− 1.34559i	−0.739830	−	0.672794i	$-0.765094\pi$
$11$	− 540.000i	− 1.34559i	0.739830	−	0.672794i	$-0.234906\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 648.000i	− 0.743613i
$16$	0	0
$17$	−594.000	−0.498499	−0.249249	−	0.968439i	$-0.580184\pi$
$17$	−594.000	−0.498499	−0.249249	+	0.968439i	$0.580184\pi$
$18$	0	0
$19$	836.000i	0.531279i	0.964072	+	0.265639i	$0.0855830\pi$
$19$	836.000i	0.531279i	−0.964072	+	0.265639i	$0.914417\pi$
$20$	0	0
$21$	− 1056.00i	− 0.522535i
$22$	0	0
$23$	4104.00	1.61766	0.808831	−	0.588041i	$-0.200101\pi$
$23$	4104.00	1.61766	0.808831	+	0.588041i	$0.200101\pi$
$24$	0	0
$25$	209.000	0.0668800
$26$	0	0
$27$	4104.00	1.08342
$28$	0	0
$29$	−594.000	−0.131157	−0.0655785	−	0.997847i	$-0.520889\pi$
$29$	−594.000	−0.131157	−0.0655785	+	0.997847i	$0.520889\pi$
$30$	0	0
$31$	4256.00i	0.795422i	0.917511	+	0.397711i	$0.130195\pi$
$31$	4256.00i	0.795422i	−0.917511	+	0.397711i	$0.869805\pi$
$32$	0	0
$33$	6480.00i	1.03583i
$34$	0	0
$35$	−4752.00	−0.655702
$36$	0	0
$37$	298.000i	0.0357859i	0.999840	+	0.0178930i	$0.00569581\pi$
$37$	298.000i	0.0357859i	−0.999840	+	0.0178930i	$0.994304\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	17226.0i	1.60039i	0.599742	+	0.800193i	$0.295270\pi$
$41$	17226.0i	1.60039i	−0.599742	+	0.800193i	$0.704730\pi$
$42$	0	0
$43$	12100.0	0.997963	0.498981	−	0.866613i	$-0.333708\pi$
$43$	12100.0	0.997963	0.498981	+	0.866613i	$0.333708\pi$
$44$	0	0
$45$	− 5346.00i	− 0.393548i
$46$	0	0
$47$	1296.00i	0.0855777i	0.999084	+	0.0427888i	$0.0136243\pi$
$47$	1296.00i	0.0855777i	−0.999084	+	0.0427888i	$0.986376\pi$
$48$	0	0
$49$	9063.00	0.539240
$50$	0	0
$51$	7128.00	0.383745
$52$	0	0
$53$	19494.0	0.953260	0.476630	−	0.879104i	$-0.341858\pi$
$53$	19494.0	0.953260	0.476630	+	0.879104i	$0.341858\pi$
$54$	0	0
$55$	29160.0	1.29981
$56$	0	0
$57$	− 10032.0i	− 0.408978i
$58$	0	0
$59$	7668.00i	0.286782i	0.989666	+	0.143391i	$0.0458007\pi$
$59$	7668.00i	0.286782i	−0.989666	+	0.143391i	$0.954199\pi$
$60$	0	0
$61$	−34738.0	−1.19531	−0.597655	−	0.801754i	$-0.703901\pi$
$61$	−34738.0	−1.19531	−0.597655	+	0.801754i	$0.703901\pi$
$62$	0	0
$63$	− 8712.00i	− 0.276545i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	21812.0i	0.593620i	0.954937	+	0.296810i	$0.0959228\pi$
$67$	21812.0i	0.593620i	−0.954937	+	0.296810i	$0.904077\pi$
$68$	0	0
$69$	−49248.0	−1.24528
$70$	0	0
$71$	− 46872.0i	− 1.10349i	−0.834014	−	0.551744i	$-0.813963\pi$
$71$	− 46872.0i	− 1.10349i	0.834014	−	0.551744i	$-0.186037\pi$
$72$	0	0
$73$	− 67562.0i	− 1.48387i	−0.670473	−	0.741934i	$-0.733909\pi$
$73$	− 67562.0i	− 1.48387i	0.670473	−	0.741934i	$-0.266091\pi$
$74$	0	0
$75$	−2508.00	−0.0514842
$76$	0	0
$77$	47520.0	0.913376
$78$	0	0
$79$	−76912.0	−1.38652	−0.693260	−	0.720687i	$-0.743826\pi$
$79$	−76912.0	−1.38652	−0.693260	+	0.720687i	$0.743826\pi$
$80$	0	0
$81$	−25191.0	−0.426612
$82$	0	0
$83$	67716.0i	1.07894i	0.842006	+	0.539468i	$0.181375\pi$
$83$	67716.0i	1.07894i	−0.842006	+	0.539468i	$0.818625\pi$
$84$	0	0
$85$	− 32076.0i	− 0.481541i
$86$	0	0
$87$	7128.00	0.100965
$88$	0	0
$89$	− 29754.0i	− 0.398172i	−0.979982	−	0.199086i	$-0.936203\pi$
$89$	− 29754.0i	− 0.398172i	0.979982	−	0.199086i	$-0.0637973\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 51072.0i	− 0.612316i
$94$	0	0
$95$	−45144.0	−0.513205
$96$	0	0
$97$	− 122398.i	− 1.32082i	−0.750903	−	0.660412i	$-0.770382\pi$
$97$	− 122398.i	− 1.32082i	0.750903	−	0.660412i	$-0.229618\pi$
$98$	0	0
$99$	53460.0i	0.548202i
$100$	0	0
$101$	−11286.0	−0.110087	−0.0550436	−	0.998484i	$-0.517530\pi$
$101$	−11286.0	−0.110087	−0.0550436	+	0.998484i	$0.517530\pi$
$102$	0	0
$103$	27256.0	0.253145	0.126572	−	0.991957i	$-0.459602\pi$
$103$	27256.0	0.253145	0.126572	+	0.991957i	$0.459602\pi$
$104$	0	0
$105$	57024.0	0.504759
$106$	0	0
$107$	122364.	1.03322	0.516612	−	0.856220i	$-0.327193\pi$
$107$	122364.	1.03322	0.516612	+	0.856220i	$0.327193\pi$
$108$	0	0
$109$	99902.0i	0.805393i	0.915334	+	0.402697i	$0.131927\pi$
$109$	99902.0i	0.805393i	−0.915334	+	0.402697i	$0.868073\pi$
$110$	0	0
$111$	− 3576.00i	− 0.0275480i
$112$	0	0
$113$	−29646.0	−0.218409	−0.109204	−	0.994019i	$-0.534830\pi$
$113$	−29646.0	−0.218409	−0.109204	+	0.994019i	$0.534830\pi$
$114$	0	0
$115$	221616.i	1.56263i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 52272.0i	− 0.338378i
$120$	0	0
$121$	−130549.	−0.810607
$122$	0	0
$123$	− 206712.i	− 1.23198i
$124$	0	0
$125$	180036.i	1.03059i
$126$	0	0
$127$	−336512.	−1.85136	−0.925681	−	0.378305i	$-0.876507\pi$
$127$	−336512.	−1.85136	−0.925681	+	0.378305i	$0.876507\pi$
$128$	0	0
$129$	−145200.	−0.768232
$130$	0	0
$131$	100980.	0.514111	0.257056	−	0.966397i	$-0.417248\pi$
$131$	100980.	0.514111	0.257056	+	0.966397i	$0.417248\pi$
$132$	0	0
$133$	−73568.0	−0.360628
$134$	0	0
$135$	221616.i	1.04657i
$136$	0	0
$137$	317142.i	1.44362i	0.692092	+	0.721809i	$0.256689\pi$
$137$	317142.i	1.44362i	−0.692092	+	0.721809i	$0.743311\pi$
$138$	0	0
$139$	−148324.	−0.651140	−0.325570	−	0.945518i	$-0.605556\pi$
$139$	−148324.	−0.651140	−0.325570	+	0.945518i	$0.605556\pi$
$140$	0	0
$141$	− 15552.0i	− 0.0658777i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 32076.0i	− 0.126695i
$146$	0	0
$147$	−108756.	−0.415107
$148$	0	0
$149$	196614.i	0.725519i	0.931883	+	0.362759i	$0.118165\pi$
$149$	196614.i	0.725519i	−0.931883	+	0.362759i	$0.881835\pi$
$150$	0	0
$151$	− 74360.0i	− 0.265398i	−0.991156	−	0.132699i	$-0.957636\pi$
$151$	− 74360.0i	− 0.265398i	0.991156	−	0.132699i	$-0.0423643\pi$
$152$	0	0
$153$	58806.0	0.203092
$154$	0	0
$155$	−229824.	−0.768362
$156$	0	0
$157$	120878.	0.391380	0.195690	−	0.980666i	$-0.437305\pi$
$157$	120878.	0.391380	0.195690	+	0.980666i	$0.437305\pi$
$158$	0	0
$159$	−233928.	−0.733820
$160$	0	0
$161$	361152.i	1.09806i
$162$	0	0
$163$	111340.i	0.328233i	0.986441	+	0.164116i	$0.0524773\pi$
$163$	111340.i	0.328233i	−0.986441	+	0.164116i	$0.947523\pi$
$164$	0	0
$165$	−349920.	−1.00060
$166$	0	0
$167$	491832.i	1.36466i	0.731043	+	0.682332i	$0.239034\pi$
$167$	491832.i	1.36466i	−0.731043	+	0.682332i	$0.760966\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 82764.0i	− 0.216447i
$172$	0	0
$173$	−707454.	−1.79714	−0.898572	−	0.438826i	$-0.855395\pi$
$173$	−707454.	−1.79714	−0.898572	+	0.438826i	$0.855395\pi$
$174$	0	0
$175$	18392.0i	0.0453977i
$176$	0	0
$177$	− 92016.0i	− 0.220765i
$178$	0	0
$179$	−493668.	−1.15160	−0.575801	−	0.817590i	$-0.695310\pi$
$179$	−493668.	−1.15160	−0.575801	+	0.817590i	$0.695310\pi$
$180$	0	0
$181$	559450.	1.26930	0.634651	−	0.772799i	$-0.281144\pi$
$181$	559450.	1.26930	0.634651	+	0.772799i	$0.281144\pi$
$182$	0	0
$183$	416856.	0.920149
$184$	0	0
$185$	−16092.0	−0.0345685
$186$	0	0
$187$	320760.i	0.670774i
$188$	0	0
$189$	361152.i	0.735420i
$190$	0	0
$191$	−724032.	−1.43607	−0.718033	−	0.696009i	$-0.754957\pi$
$191$	−724032.	−1.43607	−0.718033	+	0.696009i	$0.754957\pi$
$192$	0	0
$193$	− 7106.00i	− 0.0137319i	−0.999976	−	0.00686597i	$-0.997814\pi$
$193$	− 7106.00i	− 0.0137319i	0.999976	−	0.00686597i	$-0.00218552\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 530442.i	− 0.973806i	−0.873456	−	0.486903i	$-0.838127\pi$
$197$	− 530442.i	− 0.973806i	0.873456	−	0.486903i	$-0.161873\pi$
$198$	0	0
$199$	−56168.0	−0.100544	−0.0502720	−	0.998736i	$-0.516009\pi$
$199$	−56168.0	−0.100544	−0.0502720	+	0.998736i	$0.516009\pi$
$200$	0	0
$201$	− 261744.i	− 0.456969i
$202$	0	0
$203$	− 52272.0i	− 0.0890285i
$204$	0	0
$205$	−930204.	−1.54594
$206$	0	0
$207$	−406296.	−0.659047
$208$	0	0
$209$	451440.	0.714882
$210$	0	0
$211$	−339196.	−0.524499	−0.262249	−	0.965000i	$-0.584464\pi$
$211$	−339196.	−0.524499	−0.262249	+	0.965000i	$0.584464\pi$
$212$	0	0
$213$	562464.i	0.849465i
$214$	0	0
$215$	653400.i	0.964013i
$216$	0	0
$217$	−374528.	−0.539927
$218$	0	0
$219$	810744.i	1.14228i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	779360.i	1.04948i	0.851261	+	0.524742i	$0.175838\pi$
$223$	779360.i	1.04948i	−0.851261	+	0.524742i	$0.824162\pi$
$224$	0	0
$225$	−20691.0	−0.0272474
$226$	0	0
$227$	− 744876.i	− 0.959443i	−0.877421	−	0.479722i	$-0.840738\pi$
$227$	− 744876.i	− 0.959443i	0.877421	−	0.479722i	$-0.159262\pi$
$228$	0	0
$229$	272746.i	0.343692i	0.985124	+	0.171846i	$0.0549732\pi$
$229$	272746.i	0.343692i	−0.985124	+	0.171846i	$0.945027\pi$
$230$	0	0
$231$	−570240.	−0.703117
$232$	0	0
$233$	153846.	0.185651	0.0928253	−	0.995682i	$-0.470410\pi$
$233$	153846.	0.185651	0.0928253	+	0.995682i	$0.470410\pi$
$234$	0	0
$235$	−69984.0	−0.0826664
$236$	0	0
$237$	922944.	1.06734
$238$	0	0
$239$	1.15474e6i	1.30764i	0.756650	+	0.653820i	$0.226834\pi$
$239$	1.15474e6i	1.30764i	−0.756650	+	0.653820i	$0.773166\pi$
$240$	0	0
$241$	− 657074.i	− 0.728738i	−0.931255	−	0.364369i	$-0.881285\pi$
$241$	− 657074.i	− 0.728738i	0.931255	−	0.364369i	$-0.118715\pi$
$242$	0	0
$243$	−694980.	−0.755017
$244$	0	0
$245$	489402.i	0.520895i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 812592.i	− 0.830566i
$250$	0	0
$251$	−1.34190e6	−1.34442	−0.672211	−	0.740359i	$-0.734655\pi$
$251$	−1.34190e6	−1.34442	−0.672211	+	0.740359i	$0.734655\pi$
$252$	0	0
$253$	− 2.21616e6i	− 2.17671i
$254$	0	0
$255$	384912.i	0.370690i
$256$	0	0
$257$	−132354.	−0.124998	−0.0624992	−	0.998045i	$-0.519907\pi$
$257$	−132354.	−0.124998	−0.0624992	+	0.998045i	$0.519907\pi$
$258$	0	0
$259$	−26224.0	−0.0242912
$260$	0	0
$261$	58806.0	0.0534343
$262$	0	0
$263$	943272.	0.840906	0.420453	−	0.907314i	$-0.361871\pi$
$263$	943272.	0.840906	0.420453	+	0.907314i	$0.361871\pi$
$264$	0	0
$265$	1.05268e6i	0.920831i
$266$	0	0
$267$	357048.i	0.306513i
$268$	0	0
$269$	967518.	0.815227	0.407613	−	0.913155i	$-0.366361\pi$
$269$	967518.	0.815227	0.407613	+	0.913155i	$0.366361\pi$
$270$	0	0
$271$	518320.i	0.428721i	0.976755	+	0.214360i	$0.0687667\pi$
$271$	518320.i	0.428721i	−0.976755	+	0.214360i	$0.931233\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 112860.i	− 0.0899929i
$276$	0	0
$277$	−2.22273e6	−1.74055	−0.870275	−	0.492566i	$-0.836059\pi$
$277$	−2.22273e6	−1.74055	−0.870275	+	0.492566i	$0.836059\pi$
$278$	0	0
$279$	− 421344.i	− 0.324061i
$280$	0	0
$281$	196614.i	0.148542i	0.997238	+	0.0742709i	$0.0236629\pi$
$281$	196614.i	0.148542i	−0.997238	+	0.0742709i	$0.976337\pi$
$282$	0	0
$283$	1.55228e6	1.15213	0.576067	−	0.817403i	$-0.304587\pi$
$283$	1.55228e6	1.15213	0.576067	+	0.817403i	$0.304587\pi$
$284$	0	0
$285$	541728.	0.395066
$286$	0	0
$287$	−1.51589e6	−1.08633
$288$	0	0
$289$	−1.06702e6	−0.751499
$290$	0	0
$291$	1.46878e6i	1.01677i
$292$	0	0
$293$	1.07217e6i	0.729616i	0.931083	+	0.364808i	$0.118865\pi$
$293$	1.07217e6i	0.729616i	−0.931083	+	0.364808i	$0.881135\pi$
$294$	0	0
$295$	−414072.	−0.277026
$296$	0	0
$297$	− 2.21616e6i	− 1.45784i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1.06480e6i	0.677410i
$302$	0	0
$303$	135432.	0.0847451
$304$	0	0
$305$	− 1.87585e6i	− 1.15465i
$306$	0	0
$307$	− 1.58589e6i	− 0.960346i	−0.877174	−	0.480173i	$-0.840574\pi$
$307$	− 1.58589e6i	− 0.960346i	0.877174	−	0.480173i	$-0.159426\pi$
$308$	0	0
$309$	−327072.	−0.194871
$310$	0	0
$311$	730728.	0.428405	0.214203	−	0.976789i	$-0.431285\pi$
$311$	730728.	0.428405	0.214203	+	0.976789i	$0.431285\pi$
$312$	0	0
$313$	584858.	0.337435	0.168717	−	0.985664i	$-0.446038\pi$
$313$	584858.	0.337435	0.168717	+	0.985664i	$0.446038\pi$
$314$	0	0
$315$	470448.	0.267138
$316$	0	0
$317$	− 2.48287e6i	− 1.38773i	−0.720105	−	0.693865i	$-0.755906\pi$
$317$	− 2.48287e6i	− 1.38773i	0.720105	−	0.693865i	$-0.244094\pi$
$318$	0	0
$319$	320760.i	0.176483i
$320$	0	0
$321$	−1.46837e6	−0.795376
$322$	0	0
$323$	− 496584.i	− 0.264842i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 1.19882e6i	− 0.619992i
$328$	0	0
$329$	−114048.	−0.0580895
$330$	0	0
$331$	377948.i	0.189610i	0.995496	+	0.0948052i	$0.0302228\pi$
$331$	377948.i	0.189610i	−0.995496	+	0.0948052i	$0.969777\pi$
$332$	0	0
$333$	− 29502.0i	− 0.0145794i
$334$	0	0
$335$	−1.17785e6	−0.573426
$336$	0	0
$337$	−639122.	−0.306555	−0.153278	−	0.988183i	$-0.548983\pi$
$337$	−639122.	−0.306555	−0.153278	+	0.988183i	$0.548983\pi$
$338$	0	0
$339$	355752.	0.168131
$340$	0	0
$341$	2.29824e6	1.07031
$342$	0	0
$343$	2.27656e6i	1.04483i
$344$	0	0
$345$	− 2.65939e6i	− 1.20291i
$346$	0	0
$347$	−2.90466e6	−1.29501	−0.647503	−	0.762063i	$-0.724187\pi$
$347$	−2.90466e6	−1.29501	−0.647503	+	0.762063i	$0.724187\pi$
$348$	0	0
$349$	3.99157e6i	1.75420i	0.480304	+	0.877102i	$0.340526\pi$
$349$	3.99157e6i	1.75420i	−0.480304	+	0.877102i	$0.659474\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.42922e6i	0.610466i	0.952278	+	0.305233i	$0.0987344\pi$
$353$	1.42922e6i	0.610466i	−0.952278	+	0.305233i	$0.901266\pi$
$354$	0	0
$355$	2.53109e6	1.06595
$356$	0	0
$357$	627264.i	0.260483i
$358$	0	0
$359$	− 1.16186e6i	− 0.475794i	−0.971290	−	0.237897i	$-0.923542\pi$
$359$	− 1.16186e6i	− 0.475794i	0.971290	−	0.237897i	$-0.0764581\pi$
$360$	0	0
$361$	1.77720e6	0.717743
$362$	0	0
$363$	1.56659e6	0.624005
$364$	0	0
$365$	3.64835e6	1.43339
$366$	0	0
$367$	−1.08923e6	−0.422139	−0.211069	−	0.977471i	$-0.567695\pi$
$367$	−1.08923e6	−0.422139	−0.211069	+	0.977471i	$0.567695\pi$
$368$	0	0
$369$	− 1.70537e6i	− 0.652009i
$370$	0	0
$371$	1.71547e6i	0.647066i
$372$	0	0
$373$	3.50577e6	1.30470	0.652350	−	0.757918i	$-0.273783\pi$
$373$	3.50577e6	1.30470	0.652350	+	0.757918i	$0.273783\pi$
$374$	0	0
$375$	− 2.16043e6i	− 0.793346i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	4.04385e6i	1.44610i	0.690798	+	0.723048i	$0.257260\pi$
$379$	4.04385e6i	1.44610i	−0.690798	+	0.723048i	$0.742740\pi$
$380$	0	0
$381$	4.03814e6	1.42518
$382$	0	0
$383$	5.18746e6i	1.80700i	0.428591	+	0.903499i	$0.359010\pi$
$383$	5.18746e6i	1.80700i	−0.428591	+	0.903499i	$0.640990\pi$
$384$	0	0
$385$	2.56608e6i	0.882304i
$386$	0	0
$387$	−1.19790e6	−0.406577
$388$	0	0
$389$	950346.	0.318425	0.159213	−	0.987244i	$-0.449104\pi$
$389$	950346.	0.318425	0.159213	+	0.987244i	$0.449104\pi$
$390$	0	0
$391$	−2.43778e6	−0.806403
$392$	0	0
$393$	−1.21176e6	−0.395763
$394$	0	0
$395$	− 4.15325e6i	− 1.33935i
$396$	0	0
$397$	520738.i	0.165822i	0.996557	+	0.0829112i	$0.0264218\pi$
$397$	520738.i	0.165822i	−0.996557	+	0.0829112i	$0.973578\pi$
$398$	0	0
$399$	882816.	0.277612
$400$	0	0
$401$	− 764370.i	− 0.237379i	−0.992931	−	0.118690i	$-0.962131\pi$
$401$	− 764370.i	− 0.237379i	0.992931	−	0.118690i	$-0.0378693\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 1.36031e6i	− 0.412099i
$406$	0	0
$407$	160920.	0.0481531
$408$	0	0
$409$	2.64051e6i	0.780511i	0.920707	+	0.390255i	$0.127613\pi$
$409$	2.64051e6i	0.780511i	−0.920707	+	0.390255i	$0.872387\pi$
$410$	0	0
$411$	− 3.80570e6i	− 1.11130i
$412$	0	0
$413$	−674784.	−0.194666
$414$	0	0
$415$	−3.65666e6	−1.04223
$416$	0	0
$417$	1.77989e6	0.501248
$418$	0	0
$419$	−4.98020e6	−1.38584	−0.692918	−	0.721016i	$-0.743675\pi$
$419$	−4.98020e6	−1.38584	−0.692918	+	0.721016i	$0.743675\pi$
$420$	0	0
$421$	− 237994.i	− 0.0654426i	−0.999465	−	0.0327213i	$-0.989583\pi$
$421$	− 237994.i	− 0.0654426i	0.999465	−	0.0327213i	$-0.0104174\pi$
$422$	0	0
$423$	− 128304.i	− 0.0348650i
$424$	0	0
$425$	−124146.	−0.0333396
$426$	0	0
$427$	− 3.05694e6i	− 0.811368i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 3.88238e6i	− 1.00671i	−0.864079	−	0.503356i	$-0.832098\pi$
$431$	− 3.88238e6i	− 1.00671i	0.864079	−	0.503356i	$-0.167902\pi$
$432$	0	0
$433$	66958.0	0.0171626	0.00858129	−	0.999963i	$-0.497268\pi$
$433$	66958.0	0.0171626	0.00858129	+	0.999963i	$0.497268\pi$
$434$	0	0
$435$	384912.i	0.0975300i
$436$	0	0
$437$	3.43094e6i	0.859429i
$438$	0	0
$439$	6.50135e6	1.61006	0.805031	−	0.593233i	$-0.202149\pi$
$439$	6.50135e6	1.61006	0.805031	+	0.593233i	$0.202149\pi$
$440$	0	0
$441$	−897237.	−0.219690
$442$	0	0
$443$	−4.60760e6	−1.11549	−0.557745	−	0.830012i	$-0.688333\pi$
$443$	−4.60760e6	−1.11549	−0.557745	+	0.830012i	$0.688333\pi$
$444$	0	0
$445$	1.60672e6	0.384626
$446$	0	0
$447$	− 2.35937e6i	− 0.558505i
$448$	0	0
$449$	− 3.77671e6i	− 0.884092i	−0.896992	−	0.442046i	$-0.854253\pi$
$449$	− 3.77671e6i	− 0.884092i	0.896992	−	0.442046i	$-0.145747\pi$
$450$	0	0
$451$	9.30204e6	2.15346
$452$	0	0
$453$	892320.i	0.204303i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 3.18069e6i	− 0.712412i	−0.934407	−	0.356206i	$-0.884070\pi$
$457$	− 3.18069e6i	− 0.712412i	0.934407	−	0.356206i	$-0.115930\pi$
$458$	0	0
$459$	−2.43778e6	−0.540085
$460$	0	0
$461$	6.68547e6i	1.46514i	0.680691	+	0.732571i	$0.261680\pi$
$461$	6.68547e6i	1.46514i	−0.680691	+	0.732571i	$0.738320\pi$
$462$	0	0
$463$	4.35122e6i	0.943318i	0.881781	+	0.471659i	$0.156345\pi$
$463$	4.35122e6i	0.943318i	−0.881781	+	0.471659i	$0.843655\pi$
$464$	0	0
$465$	2.75789e6	0.591486
$466$	0	0
$467$	−7.07994e6	−1.50223	−0.751117	−	0.660170i	$-0.770484\pi$
$467$	−7.07994e6	−1.50223	−0.751117	+	0.660170i	$0.770484\pi$
$468$	0	0
$469$	−1.91946e6	−0.402945
$470$	0	0
$471$	−1.45054e6	−0.301284
$472$	0	0
$473$	− 6.53400e6i	− 1.34285i
$474$	0	0
$475$	174724.i	0.0355319i
$476$	0	0
$477$	−1.92991e6	−0.388365
$478$	0	0
$479$	− 3.22186e6i	− 0.641604i	−0.947146	−	0.320802i	$-0.896048\pi$
$479$	− 3.22186e6i	− 0.641604i	0.947146	−	0.320802i	$-0.103952\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 4.33382e6i	− 0.845286i
$484$	0	0
$485$	6.60949e6	1.27589
$486$	0	0
$487$	2.29710e6i	0.438891i	0.975625	+	0.219446i	$0.0704248\pi$
$487$	2.29710e6i	0.438891i	−0.975625	+	0.219446i	$0.929575\pi$
$488$	0	0
$489$	− 1.33608e6i	− 0.252674i
$490$	0	0
$491$	−2.82150e6	−0.528173	−0.264087	−	0.964499i	$-0.585070\pi$
$491$	−2.82150e6	−0.528173	−0.264087	+	0.964499i	$0.585070\pi$
$492$	0	0
$493$	352836.	0.0653816
$494$	0	0
$495$	−2.88684e6	−0.529553
$496$	0	0
$497$	4.12474e6	0.749040
$498$	0	0
$499$	− 4.13628e6i	− 0.743634i	−0.928306	−	0.371817i	$-0.878735\pi$
$499$	− 4.13628e6i	− 0.743634i	0.928306	−	0.371817i	$-0.121265\pi$
$500$	0	0
$501$	− 5.90198e6i	− 1.05052i
$502$	0	0
$503$	8.33263e6	1.46846	0.734230	−	0.678901i	$-0.237543\pi$
$503$	8.33263e6	1.46846	0.734230	+	0.678901i	$0.237543\pi$
$504$	0	0
$505$	− 609444.i	− 0.106342i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4.34101e6i	0.742670i	0.928499	+	0.371335i	$0.121100\pi$
$509$	4.34101e6i	0.742670i	−0.928499	+	0.371335i	$0.878900\pi$
$510$	0	0
$511$	5.94546e6	1.00724
$512$	0	0
$513$	3.43094e6i	0.575599i
$514$	0	0
$515$	1.47182e6i	0.244533i
$516$	0	0
$517$	699840.	0.115152
$518$	0	0
$519$	8.48945e6	1.38344
$520$	0	0
$521$	−6.74185e6	−1.08814	−0.544070	−	0.839040i	$-0.683117\pi$
$521$	−6.74185e6	−1.08814	−0.544070	+	0.839040i	$0.683117\pi$
$522$	0	0
$523$	−7.72196e6	−1.23445	−0.617224	−	0.786787i	$-0.711743\pi$
$523$	−7.72196e6	−1.23445	−0.617224	+	0.786787i	$0.711743\pi$
$524$	0	0
$525$	− 220704.i	− 0.0349472i
$526$	0	0
$527$	− 2.52806e6i	− 0.396517i
$528$	0	0
$529$	1.04065e7	1.61683
$530$	0	0
$531$	− 759132.i	− 0.116837i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	6.60766e6i	0.998075i
$536$	0	0
$537$	5.92402e6	0.886504
$538$	0	0
$539$	− 4.89402e6i	− 0.725594i
$540$	0	0
$541$	682066.i	0.100192i	0.998744	+	0.0500960i	$0.0159527\pi$
$541$	682066.i	0.100192i	−0.998744	+	0.0500960i	$0.984047\pi$
$542$	0	0
$543$	−6.71340e6	−0.977109
$544$	0	0
$545$	−5.39471e6	−0.777995
$546$	0	0
$547$	2.15772e6	0.308337	0.154169	−	0.988045i	$-0.450730\pi$
$547$	2.15772e6	0.308337	0.154169	+	0.988045i	$0.450730\pi$
$548$	0	0
$549$	3.43906e6	0.486978
$550$	0	0
$551$	− 496584.i	− 0.0696809i
$552$	0	0
$553$	− 6.76826e6i	− 0.941161i
$554$	0	0
$555$	193104.	0.0266109
$556$	0	0
$557$	2.67597e6i	0.365463i	0.983163	+	0.182731i	$0.0584939\pi$
$557$	2.67597e6i	0.365463i	−0.983163	+	0.182731i	$0.941506\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	− 3.84912e6i	− 0.516362i
$562$	0	0
$563$	3.55331e6	0.472457	0.236228	−	0.971698i	$-0.424089\pi$
$563$	3.55331e6	0.472457	0.236228	+	0.971698i	$0.424089\pi$
$564$	0	0
$565$	− 1.60088e6i	− 0.210979i
$566$	0	0
$567$	− 2.21681e6i	− 0.289581i
$568$	0	0
$569$	1.29225e7	1.67327	0.836633	−	0.547764i	$-0.184521\pi$
$569$	1.29225e7	1.67327	0.836633	+	0.547764i	$0.184521\pi$
$570$	0	0
$571$	6.08357e6	0.780851	0.390426	−	0.920634i	$-0.372328\pi$
$571$	6.08357e6	0.780851	0.390426	+	0.920634i	$0.372328\pi$
$572$	0	0
$573$	8.68838e6	1.10548
$574$	0	0
$575$	857736.	0.108189
$576$	0	0
$577$	− 1.58241e7i	− 1.97869i	−0.145579	−	0.989347i	$-0.546505\pi$
$577$	− 1.58241e7i	− 1.97869i	0.145579	−	0.989347i	$-0.453495\pi$
$578$	0	0
$579$	85272.0i	0.0105709i
$580$	0	0
$581$	−5.95901e6	−0.732375
$582$	0	0
$583$	− 1.05268e7i	− 1.28269i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.60220e6i	0.551278i	0.961261	+	0.275639i	$0.0888894\pi$
$587$	4.60220e6i	0.551278i	−0.961261	+	0.275639i	$0.911111\pi$
$588$	0	0
$589$	−3.55802e6	−0.422590
$590$	0	0
$591$	6.36530e6i	0.749636i
$592$	0	0
$593$	− 8.61122e6i	− 1.00561i	−0.864401	−	0.502803i	$-0.832302\pi$
$593$	− 8.61122e6i	− 1.00561i	0.864401	−	0.502803i	$-0.167698\pi$
$594$	0	0
$595$	2.82269e6	0.326867
$596$	0	0
$597$	674016.	0.0773988
$598$	0	0
$599$	−7.98228e6	−0.908992	−0.454496	−	0.890749i	$-0.650181\pi$
$599$	−7.98228e6	−0.908992	−0.454496	+	0.890749i	$0.650181\pi$
$600$	0	0
$601$	1.01740e7	1.14896	0.574481	−	0.818518i	$-0.305204\pi$
$601$	1.01740e7	1.14896	0.574481	+	0.818518i	$0.305204\pi$
$602$	0	0
$603$	− 2.15939e6i	− 0.241845i
$604$	0	0
$605$	− 7.04965e6i	− 0.783031i
$606$	0	0
$607$	−9.95843e6	−1.09703	−0.548516	−	0.836140i	$-0.684807\pi$
$607$	−9.95843e6	−1.09703	−0.548516	+	0.836140i	$0.684807\pi$
$608$	0	0
$609$	627264.i	0.0685342i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	4.19586e6i	0.450993i	0.974244	+	0.225497i	$0.0724005\pi$
$613$	4.19586e6i	0.450993i	−0.974244	+	0.225497i	$0.927600\pi$
$614$	0	0
$615$	1.11624e7	1.19007
$616$	0	0
$617$	9.12551e6i	0.965038i	0.875885	+	0.482519i	$0.160278\pi$
$617$	9.12551e6i	0.965038i	−0.875885	+	0.482519i	$0.839722\pi$
$618$	0	0
$619$	− 6.45734e6i	− 0.677372i	−0.940900	−	0.338686i	$-0.890018\pi$
$619$	− 6.45734e6i	− 0.677372i	0.940900	−	0.338686i	$-0.109982\pi$
$620$	0	0
$621$	1.68428e7	1.75261
$622$	0	0
$623$	2.61835e6	0.270276
$624$	0	0
$625$	−9.06882e6	−0.928647
$626$	0	0
$627$	−5.41728e6	−0.550316
$628$	0	0
$629$	− 177012.i	− 0.0178392i
$630$	0	0
$631$	1.40514e7i	1.40490i	0.711733	+	0.702450i	$0.247910\pi$
$631$	1.40514e7i	1.40490i	−0.711733	+	0.702450i	$0.752090\pi$
$632$	0	0
$633$	4.07035e6	0.403759
$634$	0	0
$635$	− 1.81716e7i	− 1.78838i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	4.64033e6i	0.449569i
$640$	0	0
$641$	−8.47168e6	−0.814375	−0.407188	−	0.913345i	$-0.633490\pi$
$641$	−8.47168e6	−0.814375	−0.407188	+	0.913345i	$0.633490\pi$
$642$	0	0
$643$	488564.i	0.0466009i	0.999729	+	0.0233004i	$0.00741743\pi$
$643$	488564.i	0.0466009i	−0.999729	+	0.0233004i	$0.992583\pi$
$644$	0	0
$645$	− 7.84080e6i	− 0.742098i
$646$	0	0
$647$	−2.48119e6	−0.233023	−0.116512	−	0.993189i	$-0.537171\pi$
$647$	−2.48119e6	−0.233023	−0.116512	+	0.993189i	$0.537171\pi$
$648$	0	0
$649$	4.14072e6	0.385891
$650$	0	0
$651$	4.49434e6	0.415636
$652$	0	0
$653$	−5.29130e6	−0.485601	−0.242800	−	0.970076i	$-0.578066\pi$
$653$	−5.29130e6	−0.485601	−0.242800	+	0.970076i	$0.578066\pi$
$654$	0	0
$655$	5.45292e6i	0.496622i
$656$	0	0
$657$	6.68864e6i	0.604539i
$658$	0	0
$659$	4.72468e6	0.423798	0.211899	−	0.977292i	$-0.432035\pi$
$659$	4.72468e6	0.423798	0.211899	+	0.977292i	$0.432035\pi$
$660$	0	0
$661$	6.17420e6i	0.549639i	0.961496	+	0.274819i	$0.0886180\pi$
$661$	6.17420e6i	0.549639i	−0.961496	+	0.274819i	$0.911382\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 3.97267e6i	− 0.348360i
$666$	0	0
$667$	−2.43778e6	−0.212168
$668$	0	0
$669$	− 9.35232e6i	− 0.807893i
$670$	0	0
$671$	1.87585e7i	1.60839i
$672$	0	0
$673$	9.40925e6	0.800787	0.400394	−	0.916343i	$-0.368873\pi$
$673$	9.40925e6	0.800787	0.400394	+	0.916343i	$0.368873\pi$
$674$	0	0
$675$	857736.	0.0724593
$676$	0	0
$677$	1.50086e7	1.25854	0.629272	−	0.777185i	$-0.283353\pi$
$677$	1.50086e7	1.25854	0.629272	+	0.777185i	$0.283353\pi$
$678$	0	0
$679$	1.07710e7	0.896567
$680$	0	0
$681$	8.93851e6i	0.738580i
$682$	0	0
$683$	1.29707e7i	1.06393i	0.846768	+	0.531963i	$0.178545\pi$
$683$	1.29707e7i	1.06393i	−0.846768	+	0.531963i	$0.821455\pi$
$684$	0	0
$685$	−1.71257e7	−1.39451
$686$	0	0
$687$	− 3.27295e6i	− 0.264574i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	2.26556e7i	1.80501i	0.430677	+	0.902506i	$0.358275\pi$
$691$	2.26556e7i	1.80501i	−0.430677	+	0.902506i	$0.641725\pi$
$692$	0	0
$693$	−4.70448e6	−0.372116
$694$	0	0
$695$	− 8.00950e6i	− 0.628989i
$696$	0	0
$697$	− 1.02322e7i	− 0.797791i
$698$	0	0
$699$	−1.84615e6	−0.142914
$700$	0	0
$701$	−1.90169e7	−1.46166	−0.730828	−	0.682562i	$-0.760866\pi$
$701$	−1.90169e7	−1.46166	−0.730828	+	0.682562i	$0.760866\pi$
$702$	0	0
$703$	−249128.	−0.0190123
$704$	0	0
$705$	839808.	0.0636366
$706$	0	0
$707$	− 993168.i	− 0.0747264i
$708$	0	0
$709$	− 1.51311e7i	− 1.13046i	−0.824933	−	0.565231i	$-0.808787\pi$
$709$	− 1.51311e7i	− 1.13046i	0.824933	−	0.565231i	$-0.191213\pi$
$710$	0	0
$711$	7.61429e6	0.564879
$712$	0	0
$713$	1.74666e7i	1.28672i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 1.38568e7i	− 1.00662i
$718$	0	0
$719$	1.50323e7	1.08443	0.542217	−	0.840238i	$-0.317585\pi$
$719$	1.50323e7	1.08443	0.542217	+	0.840238i	$0.317585\pi$
$720$	0	0
$721$	2.39853e6i	0.171833i
$722$	0	0
$723$	7.88489e6i	0.560983i
$724$	0	0
$725$	−124146.	−0.00877178
$726$	0	0
$727$	7.41230e6	0.520136	0.260068	−	0.965590i	$-0.416255\pi$
$727$	7.41230e6	0.520136	0.260068	+	0.965590i	$0.416255\pi$
$728$	0	0
$729$	1.44612e7	1.00782
$730$	0	0
$731$	−7.18740e6	−0.497483
$732$	0	0
$733$	− 2.77928e6i	− 0.191061i	−0.995426	−	0.0955306i	$-0.969545\pi$
$733$	− 2.77928e6i	− 0.191061i	0.995426	−	0.0955306i	$-0.0304548\pi$
$734$	0	0
$735$	− 5.87282e6i	− 0.400985i
$736$	0	0
$737$	1.17785e7	0.798768
$738$	0	0
$739$	1.21046e7i	0.815342i	0.913129	+	0.407671i	$0.133659\pi$
$739$	1.21046e7i	0.815342i	−0.913129	+	0.407671i	$0.866341\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	4.46926e6i	0.297005i	0.988912	+	0.148502i	$0.0474452\pi$
$743$	4.46926e6i	0.297005i	−0.988912	+	0.148502i	$0.952555\pi$
$744$	0	0
$745$	−1.06172e7	−0.700838
$746$	0	0
$747$	− 6.70388e6i	− 0.439567i
$748$	0	0
$749$	1.07680e7i	0.701345i
$750$	0	0
$751$	−2.88463e7	−1.86634	−0.933168	−	0.359442i	$-0.882967\pi$
$751$	−2.88463e7	−1.86634	−0.933168	+	0.359442i	$0.882967\pi$
$752$	0	0
$753$	1.61028e7	1.03494
$754$	0	0
$755$	4.01544e6	0.256369
$756$	0	0
$757$	9.60868e6	0.609430	0.304715	−	0.952444i	$-0.401439\pi$
$757$	9.60868e6	0.609430	0.304715	+	0.952444i	$0.401439\pi$
$758$	0	0
$759$	2.65939e7i	1.67563i
$760$	0	0
$761$	− 4.54588e6i	− 0.284549i	−0.989827	−	0.142274i	$-0.954558\pi$
$761$	− 4.54588e6i	− 0.284549i	0.989827	−	0.142274i	$-0.0454415\pi$
$762$	0	0
$763$	−8.79138e6	−0.546696
$764$	0	0
$765$	3.17552e6i	0.196183i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 2.15923e7i	− 1.31669i	−0.752716	−	0.658345i	$-0.771257\pi$
$769$	− 2.15923e7i	− 1.31669i	0.752716	−	0.658345i	$-0.228743\pi$
$770$	0	0
$771$	1.58825e6	0.0962238
$772$	0	0
$773$	− 1.48400e7i	− 0.893276i	−0.894715	−	0.446638i	$-0.852621\pi$
$773$	− 1.48400e7i	− 0.893276i	0.894715	−	0.446638i	$-0.147379\pi$
$774$	0	0
$775$	889504.i	0.0531978i
$776$	0	0
$777$	314688.	0.0186994
$778$	0	0
$779$	−1.44009e7	−0.850251
$780$	0	0
$781$	−2.53109e7	−1.48484
$782$	0	0
$783$	−2.43778e6	−0.142098
$784$	0	0
$785$	6.52741e6i	0.378065i
$786$	0	0
$787$	2.48785e7i	1.43182i	0.698194	+	0.715909i	$0.253987\pi$
$787$	2.48785e7i	1.43182i	−0.698194	+	0.715909i	$0.746013\pi$
$788$	0	0
$789$	−1.13193e7	−0.647330
$790$	0	0
$791$	− 2.60885e6i	− 0.148254i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 1.26321e7i	− 0.708856i
$796$	0	0
$797$	−3.16080e7	−1.76259	−0.881294	−	0.472568i	$-0.843327\pi$
$797$	−3.16080e7	−1.76259	−0.881294	+	0.472568i	$0.843327\pi$
$798$	0	0
$799$	− 769824.i	− 0.0426604i
$800$	0	0
$801$	2.94565e6i	0.162218i
$802$	0	0
$803$	−3.64835e7	−1.99668
$804$	0	0
$805$	−1.95022e7	−1.06070
$806$	0	0
$807$	−1.16102e7	−0.627562
$808$	0	0
$809$	−3.10009e6	−0.166534	−0.0832669	−	0.996527i	$-0.526535\pi$
$809$	−3.10009e6	−0.166534	−0.0832669	+	0.996527i	$0.526535\pi$
$810$	0	0
$811$	1.87180e6i	0.0999328i	0.998751	+	0.0499664i	$0.0159114\pi$
$811$	1.87180e6i	0.0999328i	−0.998751	+	0.0499664i	$0.984089\pi$
$812$	0	0
$813$	− 6.21984e6i	− 0.330030i
$814$	0	0
$815$	−6.01236e6	−0.317067
$816$	0	0
$817$	1.01156e7i	0.530196i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 2.00184e7i	− 1.03650i	−0.855228	−	0.518252i	$-0.826583\pi$
$821$	− 2.00184e7i	− 1.03650i	0.855228	−	0.518252i	$-0.173417\pi$
$822$	0	0
$823$	−1.53118e7	−0.787999	−0.394000	−	0.919111i	$-0.628909\pi$
$823$	−1.53118e7	−0.787999	−0.394000	+	0.919111i	$0.628909\pi$
$824$	0	0
$825$	1.35432e6i	0.0692766i
$826$	0	0
$827$	− 9.59310e6i	− 0.487748i	−0.969807	−	0.243874i	$-0.921582\pi$
$827$	− 9.59310e6i	− 0.487748i	0.969807	−	0.243874i	$-0.0784183\pi$
$828$	0	0
$829$	−2.52209e7	−1.27460	−0.637302	−	0.770615i	$-0.719949\pi$
$829$	−2.52209e7	−1.27460	−0.637302	+	0.770615i	$0.719949\pi$
$830$	0	0
$831$	2.66727e7	1.33988
$832$	0	0
$833$	−5.38342e6	−0.268810
$834$	0	0
$835$	−2.65589e7	−1.31824
$836$	0	0
$837$	1.74666e7i	0.861778i
$838$	0	0
$839$	1.77623e7i	0.871154i	0.900151	+	0.435577i	$0.143456\pi$
$839$	1.77623e7i	0.871154i	−0.900151	+	0.435577i	$0.856544\pi$
$840$	0	0
$841$	−2.01583e7	−0.982798
$842$	0	0
$843$	− 2.35937e6i	− 0.114348i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 1.14883e7i	− 0.550234i
$848$	0	0
$849$	−1.86273e7	−0.886913
$850$	0	0
$851$	1.22299e6i	0.0578895i
$852$	0	0
$853$	486970.i	0.0229155i	0.999934	+	0.0114578i	$0.00364720\pi$
$853$	486970.i	0.0229155i	−0.999934	+	0.0114578i	$0.996353\pi$
$854$	0	0
$855$	4.46926e6	0.209084
$856$	0	0
$857$	1.92634e6	0.0895945	0.0447972	−	0.998996i	$-0.485736\pi$
$857$	1.92634e6	0.0895945	0.0447972	+	0.998996i	$0.485736\pi$
$858$	0	0
$859$	2.23538e7	1.03364	0.516820	−	0.856094i	$-0.327116\pi$
$859$	2.23538e7	1.03364	0.516820	+	0.856094i	$0.327116\pi$
$860$	0	0
$861$	1.81907e7	0.836258
$862$	0	0
$863$	1.85838e7i	0.849390i	0.905337	+	0.424695i	$0.139619\pi$
$863$	1.85838e7i	0.849390i	−0.905337	+	0.424695i	$0.860381\pi$
$864$	0	0
$865$	− 3.82025e7i	− 1.73601i
$866$	0	0
$867$	1.28043e7	0.578504
$868$	0	0
$869$	4.15325e7i	1.86569i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	1.21174e7i	0.538114i
$874$	0	0
$875$	−1.58432e7	−0.699555
$876$	0	0
$877$	− 2.91048e7i	− 1.27781i	−0.769286	−	0.638905i	$-0.779388\pi$
$877$	− 2.91048e7i	− 1.27781i	0.769286	−	0.638905i	$-0.220612\pi$
$878$	0	0
$879$	− 1.28660e7i	− 0.561659i
$880$	0	0
$881$	3.14696e6	0.136600	0.0683001	−	0.997665i	$-0.478242\pi$
$881$	3.14696e6	0.136600	0.0683001	+	0.997665i	$0.478242\pi$
$882$	0	0
$883$	−1.59995e7	−0.690566	−0.345283	−	0.938499i	$-0.612217\pi$
$883$	−1.59995e7	−0.690566	−0.345283	+	0.938499i	$0.612217\pi$
$884$	0	0
$885$	4.96886e6	0.213255
$886$	0	0
$887$	−3.45874e7	−1.47608	−0.738039	−	0.674758i	$-0.764248\pi$
$887$	−3.45874e7	−1.47608	−0.738039	+	0.674758i	$0.764248\pi$
$888$	0	0
$889$	− 2.96131e7i	− 1.25669i
$890$	0	0
$891$	1.36031e7i	0.574044i
$892$	0	0
$893$	−1.08346e6	−0.0454656
$894$	0	0
$895$	− 2.66581e7i	− 1.11243i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 2.52806e6i	− 0.104325i
$900$	0	0
$901$	−1.15794e7	−0.475199
$902$	0	0
$903$	− 1.27776e7i	− 0.521471i
$904$	0	0
$905$	3.02103e7i	1.22612i
$906$	0	0
$907$	−1.74396e7	−0.703914	−0.351957	−	0.936016i	$-0.614484\pi$
$907$	−1.74396e7	−0.703914	−0.351957	+	0.936016i	$0.614484\pi$
$908$	0	0
$909$	1.11731e6	0.0448503
$910$	0	0
$911$	−2.59589e6	−0.103631	−0.0518155	−	0.998657i	$-0.516501\pi$
$911$	−2.59589e6	−0.103631	−0.0518155	+	0.998657i	$0.516501\pi$
$912$	0	0
$913$	3.65666e7	1.45180
$914$	0	0
$915$	2.25102e7i	0.888847i
$916$	0	0
$917$	8.88624e6i	0.348975i
$918$	0	0
$919$	−1.76411e7	−0.689028	−0.344514	−	0.938781i	$-0.611956\pi$
$919$	−1.76411e7	−0.689028	−0.344514	+	0.938781i	$0.611956\pi$
$920$	0	0
$921$	1.90307e7i	0.739275i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	62282.0i	0.00239336i
$926$	0	0
$927$	−2.69834e6	−0.103133
$928$	0	0
$929$	3.96785e7i	1.50840i	0.656646	+	0.754199i	$0.271975\pi$
$929$	3.96785e7i	1.50840i	−0.656646	+	0.754199i	$0.728025\pi$
$930$	0	0
$931$	7.57667e6i	0.286486i
$932$	0	0
$933$	−8.76874e6	−0.329787
$934$	0	0
$935$	−1.73210e7	−0.647955
$936$	0	0
$937$	3.93413e7	1.46386	0.731930	−	0.681380i	$-0.238620\pi$
$937$	3.93413e7	1.46386	0.731930	+	0.681380i	$0.238620\pi$
$938$	0	0
$939$	−7.01830e6	−0.259757
$940$	0	0
$941$	4.62506e7i	1.70272i	0.524581	+	0.851361i	$0.324222\pi$
$941$	4.62506e7i	1.70272i	−0.524581	+	0.851361i	$0.675778\pi$
$942$	0	0
$943$	7.06955e7i	2.58888i
$944$	0	0
$945$	−1.95022e7	−0.710402
$946$	0	0
$947$	3.79025e7i	1.37339i	0.726947	+	0.686693i	$0.240938\pi$
$947$	3.79025e7i	1.37339i	−0.726947	+	0.686693i	$0.759062\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	2.97944e7i	1.06828i
$952$	0	0
$953$	2.66462e7	0.950394	0.475197	−	0.879879i	$-0.342377\pi$
$953$	2.66462e7	0.950394	0.475197	+	0.879879i	$0.342377\pi$
$954$	0	0
$955$	− 3.90977e7i	− 1.38721i
$956$	0	0
$957$	− 3.84912e6i	− 0.135857i
$958$	0	0
$959$	−2.79085e7	−0.979918
$960$	0	0
$961$	1.05156e7	0.367304
$962$	0	0
$963$	−1.21140e7	−0.420943
$964$	0	0
$965$	383724.	0.0132648
$966$	0	0
$967$	4.09790e7i	1.40927i	0.709568	+	0.704637i	$0.248890\pi$
$967$	4.09790e7i	1.40927i	−0.709568	+	0.704637i	$0.751110\pi$
$968$	0	0
$969$	5.95901e6i	0.203875i
$970$	0	0
$971$	−2.72034e7	−0.925922	−0.462961	−	0.886379i	$-0.653213\pi$
$971$	−2.72034e7	−0.925922	−0.462961	+	0.886379i	$0.653213\pi$
$972$	0	0
$973$	− 1.30525e7i	− 0.441990i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.53555e7i	0.849839i	0.905231	+	0.424919i	$0.139698\pi$
$977$	2.53555e7i	0.849839i	−0.905231	+	0.424919i	$0.860302\pi$
$978$	0	0
$979$	−1.60672e7	−0.535775
$980$	0	0
$981$	− 9.89030e6i	− 0.328123i
$982$	0	0
$983$	− 1.19139e7i	− 0.393252i	−0.980479	−	0.196626i	$-0.937002\pi$
$983$	− 1.19139e7i	− 0.393252i	0.980479	−	0.196626i	$-0.0629984\pi$
$984$	0	0
$985$	2.86439e7	0.940678
$986$	0	0
$987$	1.36858e6	0.0447173
$988$	0	0
$989$	4.96584e7	1.61437
$990$	0	0
$991$	2.91931e7	0.944268	0.472134	−	0.881527i	$-0.343484\pi$
$991$	2.91931e7	0.944268	0.472134	+	0.881527i	$0.343484\pi$
$992$	0	0
$993$	− 4.53538e6i	− 0.145962i
$994$	0	0
$995$	− 3.03307e6i	− 0.0971237i
$996$	0	0
$997$	−1.73001e7	−0.551201	−0.275601	−	0.961272i	$-0.588877\pi$
$997$	−1.73001e7	−0.551201	−0.275601	+	0.961272i	$0.588877\pi$
$998$	0	0
$999$	1.22299e6i	0.0387713i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	676.6.d.a.337.2		2
13.5	odd	4		4.6.a.a.1.1	✓	1
13.8	odd	4		676.6.a.a.1.1		1
13.12	even	2	inner	676.6.d.a.337.1		2
39.5	even	4		36.6.a.a.1.1		1
52.31	even	4		16.6.a.b.1.1		1
65.18	even	4		100.6.c.b.49.1		2
65.44	odd	4		100.6.a.b.1.1		1
65.57	even	4		100.6.c.b.49.2		2
91.5	even	12		196.6.e.d.165.1		2
91.18	odd	12		196.6.e.g.177.1		2
91.31	even	12		196.6.e.d.177.1		2
91.44	odd	12		196.6.e.g.165.1		2
91.83	even	4		196.6.a.e.1.1		1
104.5	odd	4		64.6.a.f.1.1		1
104.83	even	4		64.6.a.b.1.1		1
117.5	even	12		324.6.e.d.217.1		2
117.31	odd	12		324.6.e.a.217.1		2
117.70	odd	12		324.6.e.a.109.1		2
117.83	even	12		324.6.e.d.109.1		2
143.109	even	4		484.6.a.a.1.1		1
156.83	odd	4		144.6.a.c.1.1		1
195.44	even	4		900.6.a.h.1.1		1
195.83	odd	4		900.6.d.a.649.2		2
195.122	odd	4		900.6.d.a.649.1		2
208.5	odd	4		256.6.b.g.129.2		2
208.83	even	4		256.6.b.c.129.2		2
208.109	odd	4		256.6.b.g.129.1		2
208.187	even	4		256.6.b.c.129.1		2
260.83	odd	4		400.6.c.f.49.2		2
260.187	odd	4		400.6.c.f.49.1		2
260.239	even	4		400.6.a.d.1.1		1
312.5	even	4		576.6.a.bc.1.1		1
312.83	odd	4		576.6.a.bd.1.1		1
364.83	odd	4		784.6.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.6.a.a.1.1	✓	1	13.5	odd	4
16.6.a.b.1.1		1	52.31	even	4
36.6.a.a.1.1		1	39.5	even	4
64.6.a.b.1.1		1	104.83	even	4
64.6.a.f.1.1		1	104.5	odd	4
100.6.a.b.1.1		1	65.44	odd	4
100.6.c.b.49.1		2	65.18	even	4
100.6.c.b.49.2		2	65.57	even	4
144.6.a.c.1.1		1	156.83	odd	4
196.6.a.e.1.1		1	91.83	even	4
196.6.e.d.165.1		2	91.5	even	12
196.6.e.d.177.1		2	91.31	even	12
196.6.e.g.165.1		2	91.44	odd	12
196.6.e.g.177.1		2	91.18	odd	12
256.6.b.c.129.1		2	208.187	even	4
256.6.b.c.129.2		2	208.83	even	4
256.6.b.g.129.1		2	208.109	odd	4
256.6.b.g.129.2		2	208.5	odd	4
324.6.e.a.109.1		2	117.70	odd	12
324.6.e.a.217.1		2	117.31	odd	12
324.6.e.d.109.1		2	117.83	even	12
324.6.e.d.217.1		2	117.5	even	12
400.6.a.d.1.1		1	260.239	even	4
400.6.c.f.49.1		2	260.187	odd	4
400.6.c.f.49.2		2	260.83	odd	4
484.6.a.a.1.1		1	143.109	even	4
576.6.a.bc.1.1		1	312.5	even	4
576.6.a.bd.1.1		1	312.83	odd	4
676.6.a.a.1.1		1	13.8	odd	4
676.6.d.a.337.1		2	13.12	even	2	inner
676.6.d.a.337.2		2	1.1	even	1	trivial
784.6.a.d.1.1		1	364.83	odd	4
900.6.a.h.1.1		1	195.44	even	4
900.6.d.a.649.1		2	195.122	odd	4
900.6.d.a.649.2		2	195.83	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 676 = 2^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	676.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-12.0000 q^{3} +54.0000i q^{5} +88.0000i q^{7} -99.0000 q^{9} +O(q^{10})\)	\(q-12.0000 q^{3} +54.0000i q^{5} +88.0000i q^{7} -99.0000 q^{9} -540.000i q^{11} -648.000i q^{15} -594.000 q^{17} +836.000i q^{19} -1056.00i q^{21} +4104.00 q^{23} +209.000 q^{25} +4104.00 q^{27} -594.000 q^{29} +4256.00i q^{31} +6480.00i q^{33} -4752.00 q^{35} +298.000i q^{37} +17226.0i q^{41} +12100.0 q^{43} -5346.00i q^{45} +1296.00i q^{47} +9063.00 q^{49} +7128.00 q^{51} +19494.0 q^{53} +29160.0 q^{55} -10032.0i q^{57} +7668.00i q^{59} -34738.0 q^{61} -8712.00i q^{63} +21812.0i q^{67} -49248.0 q^{69} -46872.0i q^{71} -67562.0i q^{73} -2508.00 q^{75} +47520.0 q^{77} -76912.0 q^{79} -25191.0 q^{81} +67716.0i q^{83} -32076.0i q^{85} +7128.00 q^{87} -29754.0i q^{89} -51072.0i q^{93} -45144.0 q^{95} -122398. i q^{97} +53460.0i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 24 q^{3} - 198 q^{9}+O(q^{10}) \) 2 * q - 24 * q^3 - 198 * q^9	\( 2 q - 24 q^{3} - 198 q^{9} - 1188 q^{17} + 8208 q^{23} + 418 q^{25} + 8208 q^{27} - 1188 q^{29} - 9504 q^{35} + 24200 q^{43} + 18126 q^{49} + 14256 q^{51} + 38988 q^{53} + 58320 q^{55} - 69476 q^{61} - 98496 q^{69} - 5016 q^{75} + 95040 q^{77} - 153824 q^{79} - 50382 q^{81} + 14256 q^{87} - 90288 q^{95}+O(q^{100}) \) 2 * q - 24 * q^3 - 198 * q^9 - 1188 * q^17 + 8208 * q^23 + 418 * q^25 + 8208 * q^27 - 1188 * q^29 - 9504 * q^35 + 24200 * q^43 + 18126 * q^49 + 14256 * q^51 + 38988 * q^53 + 58320 * q^55 - 69476 * q^61 - 98496 * q^69 - 5016 * q^75 + 95040 * q^77 - 153824 * q^79 - 50382 * q^81 + 14256 * q^87 - 90288 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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