LMFDB - Embedded newform 256.6.b.c.129.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [256,6,Mod(129,256)] 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("256.129"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(256, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	256.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,-176] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$41.0582578721$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	256.129
Dual form			256.6.b.c.129.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+12.0000i q^{3} -54.0000i q^{5} -88.0000 q^{7} +99.0000 q^{9} +540.000i q^{11} -418.000i q^{13} +648.000 q^{15} +594.000 q^{17} -836.000i q^{19} -1056.00i q^{21} -4104.00 q^{23} +209.000 q^{25} +4104.00i q^{27} -594.000i q^{29} -4256.00 q^{31} -6480.00 q^{33} +4752.00i q^{35} +298.000i q^{37} +5016.00 q^{39} -17226.0 q^{41} -12100.0i q^{43} -5346.00i q^{45} +1296.00 q^{47} -9063.00 q^{49} +7128.00i q^{51} -19494.0i q^{53} +29160.0 q^{55} +10032.0 q^{57} -7668.00i q^{59} -34738.0i q^{61} -8712.00 q^{63} -22572.0 q^{65} -21812.0i q^{67} -49248.0i q^{69} -46872.0 q^{71} -67562.0 q^{73} +2508.00i q^{75} -47520.0i q^{77} +76912.0 q^{79} -25191.0 q^{81} -67716.0i q^{83} -32076.0i q^{85} +7128.00 q^{87} -29754.0 q^{89} +36784.0i q^{91} -51072.0i q^{93} -45144.0 q^{95} -122398. q^{97} +53460.0i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 176 q^{7} + 198 q^{9} + 1296 q^{15} + 1188 q^{17} - 8208 q^{23} + 418 q^{25} - 8512 q^{31} - 12960 q^{33} + 10032 q^{39} - 34452 q^{41} + 2592 q^{47} - 18126 q^{49} + 58320 q^{55} + 20064 q^{57} - 17424 q^{63}+ \cdots - 244796 q^{97}+O(q^{100}) $ 2 * q - 176 * q^7 + 198 * q^9 + 1296 * q^15 + 1188 * q^17 - 8208 * q^23 + 418 * q^25 - 8512 * q^31 - 12960 * q^33 + 10032 * q^39 - 34452 * q^41 + 2592 * q^47 - 18126 * q^49 + 58320 * q^55 + 20064 * q^57 - 17424 * q^63 - 45144 * q^65 - 93744 * q^71 - 135124 * q^73 + 153824 * q^79 - 50382 * q^81 + 14256 * q^87 - 59508 * q^89 - 90288 * q^95 - 244796 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$.

$n$	$5$	$255$
$\chi(n)$	$-1$	$1$

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
$3$	12.0000i	0.769800i	0.922958	+	0.384900i	$0.125764\pi$
$3$	12.0000i	0.769800i	−0.922958	+	0.384900i	$0.874236\pi$
$4$	0	0
$5$	− 54.0000i	− 0.965981i	−0.875625	−	0.482991i	$-0.839550\pi$
$5$	− 54.0000i	− 0.965981i	0.875625	−	0.482991i	$-0.160450\pi$
$6$	0	0
$7$	−88.0000	−0.678793	−0.339397	−	0.940643i	$-0.610223\pi$
$7$	−88.0000	−0.678793	−0.339397	+	0.940643i	$0.610223\pi$
$8$	0	0
$9$	99.0000	0.407407
$10$	0	0
$11$	540.000i	1.34559i	0.739830	+	0.672794i	$0.234906\pi$
$11$	540.000i	1.34559i	−0.739830	+	0.672794i	$0.765094\pi$
$12$	0	0
$13$	− 418.000i	− 0.685990i	−0.939337	−	0.342995i	$-0.888559\pi$
$13$	− 418.000i	− 0.685990i	0.939337	−	0.342995i	$-0.111441\pi$
$14$	0	0
$15$	648.000	0.743613
$16$	0	0
$17$	594.000	0.498499	0.249249	−	0.968439i	$-0.419816\pi$
$17$	594.000	0.498499	0.249249	+	0.968439i	$0.419816\pi$
$18$	0	0
$19$	− 836.000i	− 0.531279i	−0.964072	−	0.265639i	$-0.914417\pi$
$19$	− 836.000i	− 0.531279i	0.964072	−	0.265639i	$-0.0855830\pi$
$20$	0	0
$21$	− 1056.00i	− 0.522535i
$22$	0	0
$23$	−4104.00	−1.61766	−0.808831	−	0.588041i	$-0.799899\pi$
$23$	−4104.00	−1.61766	−0.808831	+	0.588041i	$0.799899\pi$
$24$	0	0
$25$	209.000	0.0668800
$26$	0	0
$27$	4104.00i	1.08342i
$28$	0	0
$29$	− 594.000i	− 0.131157i	−0.997847	−	0.0655785i	$-0.979111\pi$
$29$	− 594.000i	− 0.131157i	0.997847	−	0.0655785i	$-0.0208893\pi$
$30$	0	0
$31$	−4256.00	−0.795422	−0.397711	−	0.917511i	$-0.630195\pi$
$31$	−4256.00	−0.795422	−0.397711	+	0.917511i	$0.630195\pi$
$32$	0	0
$33$	−6480.00	−1.03583
$34$	0	0
$35$	4752.00i	0.655702i
$36$	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
$39$	5016.00	0.528075
$40$	0	0
$41$	−17226.0	−1.60039	−0.800193	−	0.599742i	$-0.795270\pi$
$41$	−17226.0	−1.60039	−0.800193	+	0.599742i	$0.795270\pi$
$42$	0	0
$43$	− 12100.0i	− 0.997963i	−0.866613	−	0.498981i	$-0.833708\pi$
$43$	− 12100.0i	− 0.997963i	0.866613	−	0.498981i	$-0.166292\pi$
$44$	0	0
$45$	− 5346.00i	− 0.393548i
$46$	0	0
$47$	1296.00	0.0855777	0.0427888	−	0.999084i	$-0.486376\pi$
$47$	1296.00	0.0855777	0.0427888	+	0.999084i	$0.486376\pi$
$48$	0	0
$49$	−9063.00	−0.539240
$50$	0	0
$51$	7128.00i	0.383745i
$52$	0	0
$53$	− 19494.0i	− 0.953260i	−0.879104	−	0.476630i	$-0.841858\pi$
$53$	− 19494.0i	− 0.953260i	0.879104	−	0.476630i	$-0.158142\pi$
$54$	0	0
$55$	29160.0	1.29981
$56$	0	0
$57$	10032.0	0.408978
$58$	0	0
$59$	− 7668.00i	− 0.286782i	−0.989666	−	0.143391i	$-0.954199\pi$
$59$	− 7668.00i	− 0.286782i	0.989666	−	0.143391i	$-0.0458007\pi$
$60$	0	0
$61$	− 34738.0i	− 1.19531i	−0.801754	−	0.597655i	$-0.796099\pi$
$61$	− 34738.0i	− 1.19531i	0.801754	−	0.597655i	$-0.203901\pi$
$62$	0	0
$63$	−8712.00	−0.276545
$64$	0	0
$65$	−22572.0	−0.662654
$66$	0	0
$67$	− 21812.0i	− 0.593620i	−0.954937	−	0.296810i	$-0.904077\pi$
$67$	− 21812.0i	− 0.593620i	0.954937	−	0.296810i	$-0.0959228\pi$
$68$	0	0
$69$	− 49248.0i	− 1.24528i
$70$	0	0
$71$	−46872.0	−1.10349	−0.551744	−	0.834014i	$-0.686037\pi$
$71$	−46872.0	−1.10349	−0.551744	+	0.834014i	$0.686037\pi$
$72$	0	0
$73$	−67562.0	−1.48387	−0.741934	−	0.670473i	$-0.766091\pi$
$73$	−67562.0	−1.48387	−0.741934	+	0.670473i	$0.766091\pi$
$74$	0	0
$75$	2508.00i	0.0514842i
$76$	0	0
$77$	− 47520.0i	− 0.913376i
$78$	0	0
$79$	76912.0	1.38652	0.693260	−	0.720687i	$-0.256174\pi$
$79$	76912.0	1.38652	0.693260	+	0.720687i	$0.256174\pi$
$80$	0	0
$81$	−25191.0	−0.426612
$82$	0	0
$83$	− 67716.0i	− 1.07894i	−0.842006	−	0.539468i	$-0.818625\pi$
$83$	− 67716.0i	− 1.07894i	0.842006	−	0.539468i	$-0.181375\pi$
$84$	0	0
$85$	− 32076.0i	− 0.481541i
$86$	0	0
$87$	7128.00	0.100965
$88$	0	0
$89$	−29754.0	−0.398172	−0.199086	−	0.979982i	$-0.563797\pi$
$89$	−29754.0	−0.398172	−0.199086	+	0.979982i	$0.563797\pi$
$90$	0	0
$91$	36784.0i	0.465646i
$92$	0	0
$93$	− 51072.0i	− 0.612316i
$94$	0	0
$95$	−45144.0	−0.513205
$96$	0	0
$97$	−122398.	−1.32082	−0.660412	−	0.750903i	$-0.729618\pi$
$97$	−122398.	−1.32082	−0.660412	+	0.750903i	$0.729618\pi$
$98$	0	0
$99$	53460.0i	0.548202i
$100$	0	0
$101$	− 11286.0i	− 0.110087i	−0.998484	−	0.0550436i	$-0.982470\pi$
$101$	− 11286.0i	− 0.110087i	0.998484	−	0.0550436i	$-0.0175298\pi$
$102$	0	0
$103$	−27256.0	−0.253145	−0.126572	−	0.991957i	$-0.540398\pi$
$103$	−27256.0	−0.253145	−0.126572	+	0.991957i	$0.540398\pi$
$104$	0	0
$105$	−57024.0	−0.504759
$106$	0	0
$107$	122364.i	1.03322i	0.856220	+	0.516612i	$0.172807\pi$
$107$	122364.i	1.03322i	−0.856220	+	0.516612i	$0.827193\pi$
$108$	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
$111$	−3576.00	−0.0275480
$112$	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
$117$	− 41382.0i	− 0.279477i
$118$	0	0
$119$	−52272.0	−0.338378
$120$	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.i	− 0.514111i	−0.966397	−	0.257056i	$-0.917248\pi$
$131$	− 100980.i	− 0.514111i	0.966397	−	0.257056i	$-0.0827524\pi$
$132$	0	0
$133$	73568.0i	0.360628i
$134$	0	0
$135$	221616.	1.04657
$136$	0	0
$137$	317142.	1.44362	0.721809	−	0.692092i	$-0.243311\pi$
$137$	317142.	1.44362	0.721809	+	0.692092i	$0.243311\pi$
$138$	0	0
$139$	− 148324.i	− 0.651140i	−0.945518	−	0.325570i	$-0.894444\pi$
$139$	− 148324.i	− 0.651140i	0.945518	−	0.325570i	$-0.105556\pi$
$140$	0	0
$141$	15552.0i	0.0658777i
$142$	0	0
$143$	225720.	0.923060
$144$	0	0
$145$	−32076.0	−0.126695
$146$	0	0
$147$	− 108756.i	− 0.415107i
$148$	0	0
$149$	− 196614.i	− 0.725519i	−0.931883	−	0.362759i	$-0.881835\pi$
$149$	− 196614.i	− 0.725519i	0.931883	−	0.362759i	$-0.118165\pi$
$150$	0	0
$151$	74360.0	0.265398	0.132699	−	0.991156i	$-0.457636\pi$
$151$	74360.0	0.265398	0.132699	+	0.991156i	$0.457636\pi$
$152$	0	0
$153$	58806.0	0.203092
$154$	0	0
$155$	229824.i	0.768362i
$156$	0	0
$157$	120878.i	0.391380i	0.980666	+	0.195690i	$0.0626946\pi$
$157$	120878.i	0.391380i	−0.980666	+	0.195690i	$0.937305\pi$
$158$	0	0
$159$	233928.	0.733820
$160$	0	0
$161$	361152.	1.09806
$162$	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.i	1.00060i
$166$	0	0
$167$	−491832.	−1.36466	−0.682332	−	0.731043i	$-0.739034\pi$
$167$	−491832.	−1.36466	−0.682332	+	0.731043i	$0.739034\pi$
$168$	0	0
$169$	196569.	0.529417
$170$	0	0
$171$	− 82764.0i	− 0.216447i
$172$	0	0
$173$	707454.i	1.79714i	0.438826	+	0.898572i	$0.355395\pi$
$173$	707454.i	1.79714i	−0.438826	+	0.898572i	$0.644605\pi$
$174$	0	0
$175$	−18392.0	−0.0453977
$176$	0	0
$177$	92016.0	0.220765
$178$	0	0
$179$	− 493668.i	− 1.15160i	−0.817590	−	0.575801i	$-0.804690\pi$
$179$	− 493668.i	− 1.15160i	0.817590	−	0.575801i	$-0.195310\pi$
$180$	0	0
$181$	559450.i	1.26930i	0.772799	+	0.634651i	$0.218856\pi$
$181$	559450.i	1.26930i	−0.772799	+	0.634651i	$0.781144\pi$
$182$	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	$-0.245043\pi$
$191$	724032.	1.43607	0.718033	+	0.696009i	$0.245043\pi$
$192$	0	0
$193$	7106.00	0.0137319	0.00686597	−	0.999976i	$-0.497814\pi$
$193$	7106.00	0.0137319	0.00686597	+	0.999976i	$0.497814\pi$
$194$	0	0
$195$	− 270864.i	− 0.510111i
$196$	0	0
$197$	530442.i	0.973806i	0.873456	+	0.486903i	$0.161873\pi$
$197$	530442.i	0.973806i	−0.873456	+	0.486903i	$0.838127\pi$
$198$	0	0
$199$	56168.0	0.100544	0.0502720	−	0.998736i	$-0.483991\pi$
$199$	56168.0	0.100544	0.0502720	+	0.998736i	$0.483991\pi$
$200$	0	0
$201$	261744.	0.456969
$202$	0	0
$203$	52272.0i	0.0890285i
$204$	0	0
$205$	930204.i	1.54594i
$206$	0	0
$207$	−406296.	−0.659047
$208$	0	0
$209$	451440.	0.714882
$210$	0	0
$211$	339196.i	0.524499i	0.965000	+	0.262249i	$0.0844643\pi$
$211$	339196.i	0.524499i	−0.965000	+	0.262249i	$0.915536\pi$
$212$	0	0
$213$	− 562464.i	− 0.849465i
$214$	0	0
$215$	−653400.	−0.964013
$216$	0	0
$217$	374528.	0.539927
$218$	0	0
$219$	− 810744.i	− 1.14228i
$220$	0	0
$221$	− 248292.i	− 0.341965i
$222$	0	0
$223$	−779360.	−1.04948	−0.524742	−	0.851261i	$-0.675838\pi$
$223$	−779360.	−1.04948	−0.524742	+	0.851261i	$0.675838\pi$
$224$	0	0
$225$	20691.0	0.0272474
$226$	0	0
$227$	744876.i	0.959443i	0.877421	+	0.479722i	$0.159262\pi$
$227$	744876.i	0.959443i	−0.877421	+	0.479722i	$0.840738\pi$
$228$	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
$235$	− 69984.0i	− 0.0826664i
$236$	0	0
$237$	922944.i	1.06734i
$238$	0	0
$239$	−1.15474e6	−1.30764	−0.653820	−	0.756650i	$-0.726834\pi$
$239$	−1.15474e6	−1.30764	−0.653820	+	0.756650i	$0.726834\pi$
$240$	0	0
$241$	657074.	0.728738	0.364369	−	0.931255i	$-0.381285\pi$
$241$	657074.	0.728738	0.364369	+	0.931255i	$0.381285\pi$
$242$	0	0
$243$	694980.i	0.755017i
$244$	0	0
$245$	489402.i	0.520895i
$246$	0	0
$247$	−349448.	−0.364452
$248$	0	0
$249$	812592.	0.830566
$250$	0	0
$251$	1.34190e6i	1.34442i	0.740359	+	0.672211i	$0.234655\pi$
$251$	1.34190e6i	1.34442i	−0.740359	+	0.672211i	$0.765345\pi$
$252$	0	0
$253$	− 2.21616e6i	− 2.17671i
$254$	0	0
$255$	384912.	0.370690
$256$	0	0
$257$	132354.	0.124998	0.0624992	−	0.998045i	$-0.480093\pi$
$257$	132354.	0.124998	0.0624992	+	0.998045i	$0.480093\pi$
$258$	0	0
$259$	− 26224.0i	− 0.0242912i
$260$	0	0
$261$	− 58806.0i	− 0.0534343i
$262$	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
$265$	−1.05268e6	−0.920831
$266$	0	0
$267$	− 357048.i	− 0.306513i
$268$	0	0
$269$	967518.i	0.815227i	0.913155	+	0.407613i	$0.133639\pi$
$269$	967518.i	0.815227i	−0.913155	+	0.407613i	$0.866361\pi$
$270$	0	0
$271$	518320.	0.428721	0.214360	−	0.976755i	$-0.431233\pi$
$271$	518320.	0.428721	0.214360	+	0.976755i	$0.431233\pi$
$272$	0	0
$273$	−441408.	−0.358454
$274$	0	0
$275$	112860.i	0.0899929i
$276$	0	0
$277$	− 2.22273e6i	− 1.74055i	−0.492566	−	0.870275i	$-0.663941\pi$
$277$	− 2.22273e6i	− 1.74055i	0.492566	−	0.870275i	$-0.336059\pi$
$278$	0	0
$279$	−421344.	−0.324061
$280$	0	0
$281$	196614.	0.148542	0.0742709	−	0.997238i	$-0.476337\pi$
$281$	196614.	0.148542	0.0742709	+	0.997238i	$0.476337\pi$
$282$	0	0
$283$	− 1.55228e6i	− 1.15213i	−0.817403	−	0.576067i	$-0.804587\pi$
$283$	− 1.55228e6i	− 1.15213i	0.817403	−	0.576067i	$-0.195413\pi$
$284$	0	0
$285$	− 541728.i	− 0.395066i
$286$	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
$297$	−2.21616e6	−1.45784
$298$	0	0
$299$	1.71547e6i	1.10970i
$300$	0	0
$301$	1.06480e6i	0.677410i
$302$	0	0
$303$	135432.	0.0847451
$304$	0	0
$305$	−1.87585e6	−1.15465
$306$	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.i	− 0.194871i
$310$	0	0
$311$	−730728.	−0.428405	−0.214203	−	0.976789i	$-0.568715\pi$
$311$	−730728.	−0.428405	−0.214203	+	0.976789i	$0.568715\pi$
$312$	0	0
$313$	−584858.	−0.337435	−0.168717	−	0.985664i	$-0.553962\pi$
$313$	−584858.	−0.337435	−0.168717	+	0.985664i	$0.553962\pi$
$314$	0	0
$315$	470448.i	0.267138i
$316$	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.	0.176483
$320$	0	0
$321$	−1.46837e6	−0.795376
$322$	0	0
$323$	− 496584.i	− 0.264842i
$324$	0	0
$325$	− 87362.0i	− 0.0458790i
$326$	0	0
$327$	−1.19882e6	−0.619992
$328$	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	$-0.451017\pi$
$337$	639122.	0.306555	0.153278	+	0.988183i	$0.451017\pi$
$338$	0	0
$339$	− 355752.i	− 0.168131i
$340$	0	0
$341$	− 2.29824e6i	− 1.07031i
$342$	0	0
$343$	2.27656e6	1.04483
$344$	0	0
$345$	−2.65939e6	−1.20291
$346$	0	0
$347$	− 2.90466e6i	− 1.29501i	−0.762063	−	0.647503i	$-0.775813\pi$
$347$	− 2.90466e6i	− 1.29501i	0.762063	−	0.647503i	$-0.224187\pi$
$348$	0	0
$349$	− 3.99157e6i	− 1.75420i	−0.480304	−	0.877102i	$-0.659474\pi$
$349$	− 3.99157e6i	− 1.75420i	0.480304	−	0.877102i	$-0.340526\pi$
$350$	0	0
$351$	1.71547e6	0.743217
$352$	0	0
$353$	1.42922e6	0.610466	0.305233	−	0.952278i	$-0.401266\pi$
$353$	1.42922e6	0.610466	0.305233	+	0.952278i	$0.401266\pi$
$354$	0	0
$355$	2.53109e6i	1.06595i
$356$	0	0
$357$	− 627264.i	− 0.260483i
$358$	0	0
$359$	1.16186e6	0.475794	0.237897	−	0.971290i	$-0.423542\pi$
$359$	1.16186e6	0.475794	0.237897	+	0.971290i	$0.423542\pi$
$360$	0	0
$361$	1.77720e6	0.717743
$362$	0	0
$363$	− 1.56659e6i	− 0.624005i
$364$	0	0
$365$	3.64835e6i	1.43339i
$366$	0	0
$367$	1.08923e6	0.422139	0.211069	−	0.977471i	$-0.432305\pi$
$367$	1.08923e6	0.422139	0.211069	+	0.977471i	$0.432305\pi$
$368$	0	0
$369$	−1.70537e6	−0.652009
$370$	0	0
$371$	1.71547e6i	0.647066i
$372$	0	0
$373$	− 3.50577e6i	− 1.30470i	−0.757918	−	0.652350i	$-0.773783\pi$
$373$	− 3.50577e6i	− 1.30470i	0.757918	−	0.652350i	$-0.226217\pi$
$374$	0	0
$375$	2.16043e6	0.793346
$376$	0	0
$377$	−248292.	−0.0899724
$378$	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
$381$	− 4.03814e6i	− 1.42518i
$382$	0	0
$383$	−5.18746e6	−1.80700	−0.903499	−	0.428591i	$-0.859010\pi$
$383$	−5.18746e6	−1.80700	−0.903499	+	0.428591i	$0.859010\pi$
$384$	0	0
$385$	−2.56608e6	−0.882304
$386$	0	0
$387$	− 1.19790e6i	− 0.406577i
$388$	0	0
$389$	950346.i	0.318425i	0.987244	+	0.159213i	$0.0508956\pi$
$389$	950346.i	0.318425i	−0.987244	+	0.159213i	$0.949104\pi$
$390$	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	$-0.973578\pi$
$397$	− 520738.i	− 0.165822i	0.996557	−	0.0829112i	$-0.0264218\pi$
$398$	0	0
$399$	−882816.	−0.277612
$400$	0	0
$401$	764370.	0.237379	0.118690	−	0.992931i	$-0.462131\pi$
$401$	764370.	0.237379	0.118690	+	0.992931i	$0.462131\pi$
$402$	0	0
$403$	1.77901e6i	0.545651i
$404$	0	0
$405$	1.36031e6i	0.412099i
$406$	0	0
$407$	−160920.	−0.0481531
$408$	0	0
$409$	−2.64051e6	−0.780511	−0.390255	−	0.920707i	$-0.627613\pi$
$409$	−2.64051e6	−0.780511	−0.390255	+	0.920707i	$0.627613\pi$
$410$	0	0
$411$	3.80570e6i	1.11130i
$412$	0	0
$413$	674784.i	0.194666i
$414$	0	0
$415$	−3.65666e6	−1.04223
$416$	0	0
$417$	1.77989e6	0.501248
$418$	0	0
$419$	4.98020e6i	1.38584i	0.721016	+	0.692918i	$0.243675\pi$
$419$	4.98020e6i	1.38584i	−0.721016	+	0.692918i	$0.756325\pi$
$420$	0	0
$421$	237994.i	0.0654426i	0.999465	+	0.0327213i	$0.0104174\pi$
$421$	237994.i	0.0654426i	−0.999465	+	0.0327213i	$0.989583\pi$
$422$	0	0
$423$	128304.	0.0348650
$424$	0	0
$425$	124146.	0.0333396
$426$	0	0
$427$	3.05694e6i	0.811368i
$428$	0	0
$429$	2.70864e6i	0.710572i
$430$	0	0
$431$	3.88238e6	1.00671	0.503356	−	0.864079i	$-0.332098\pi$
$431$	3.88238e6	1.00671	0.503356	+	0.864079i	$0.332098\pi$
$432$	0	0
$433$	−66958.0	−0.0171626	−0.00858129	−	0.999963i	$-0.502732\pi$
$433$	−66958.0	−0.0171626	−0.00858129	+	0.999963i	$0.502732\pi$
$434$	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	$-0.797851\pi$
$439$	−6.50135e6	−1.61006	−0.805031	+	0.593233i	$0.797851\pi$
$440$	0	0
$441$	−897237.	−0.219690
$442$	0	0
$443$	− 4.60760e6i	− 1.11549i	−0.830012	−	0.557745i	$-0.811667\pi$
$443$	− 4.60760e6i	− 1.11549i	0.830012	−	0.557745i	$-0.188333\pi$
$444$	0	0
$445$	1.60672e6i	0.384626i
$446$	0	0
$447$	2.35937e6	0.558505
$448$	0	0
$449$	3.77671e6	0.884092	0.442046	−	0.896992i	$-0.354253\pi$
$449$	3.77671e6	0.884092	0.442046	+	0.896992i	$0.354253\pi$
$450$	0	0
$451$	− 9.30204e6i	− 2.15346i
$452$	0	0
$453$	892320.i	0.204303i
$454$	0	0
$455$	1.98634e6	0.449805
$456$	0	0
$457$	3.18069e6	0.712412	0.356206	−	0.934407i	$-0.384070\pi$
$457$	3.18069e6	0.712412	0.356206	+	0.934407i	$0.384070\pi$
$458$	0	0
$459$	2.43778e6i	0.540085i
$460$	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
$463$	4.35122e6	0.943318	0.471659	−	0.881781i	$-0.343655\pi$
$463$	4.35122e6	0.943318	0.471659	+	0.881781i	$0.343655\pi$
$464$	0	0
$465$	−2.75789e6	−0.591486
$466$	0	0
$467$	− 7.07994e6i	− 1.50223i	−0.660170	−	0.751117i	$-0.729516\pi$
$467$	− 7.07994e6i	− 1.50223i	0.660170	−	0.751117i	$-0.270484\pi$
$468$	0	0
$469$	1.91946e6i	0.402945i
$470$	0	0
$471$	−1.45054e6	−0.301284
$472$	0	0
$473$	6.53400e6	1.34285
$474$	0	0
$475$	− 174724.i	− 0.0355319i
$476$	0	0
$477$	− 1.92991e6i	− 0.388365i
$478$	0	0
$479$	−3.22186e6	−0.641604	−0.320802	−	0.947146i	$-0.603952\pi$
$479$	−3.22186e6	−0.641604	−0.320802	+	0.947146i	$0.603952\pi$
$480$	0	0
$481$	124564.	0.0245488
$482$	0	0
$483$	4.33382e6i	0.845286i
$484$	0	0
$485$	6.60949e6i	1.27589i
$486$	0	0
$487$	2.29710e6	0.438891	0.219446	−	0.975625i	$-0.429575\pi$
$487$	2.29710e6	0.438891	0.219446	+	0.975625i	$0.429575\pi$
$488$	0	0
$489$	−1.33608e6	−0.252674
$490$	0	0
$491$	2.82150e6i	0.528173i	0.964499	+	0.264087i	$0.0850705\pi$
$491$	2.82150e6i	0.528173i	−0.964499	+	0.264087i	$0.914930\pi$
$492$	0	0
$493$	− 352836.i	− 0.0653816i
$494$	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	$0.121265\pi$
$499$	4.13628e6i	0.743634i	−0.928306	+	0.371817i	$0.878735\pi$
$500$	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.	−0.106342
$506$	0	0
$507$	2.35883e6i	0.407546i
$508$	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
$513$	3.43094e6	0.575599
$514$	0	0
$515$	1.47182e6i	0.244533i
$516$	0	0
$517$	699840.i	0.115152i
$518$	0	0
$519$	−8.48945e6	−1.38344
$520$	0	0
$521$	6.74185e6	1.08814	0.544070	−	0.839040i	$-0.316883\pi$
$521$	6.74185e6	1.08814	0.544070	+	0.839040i	$0.316883\pi$
$522$	0	0
$523$	− 7.72196e6i	− 1.23445i	−0.786787	−	0.617224i	$-0.788257\pi$
$523$	− 7.72196e6i	− 1.23445i	0.786787	−	0.617224i	$-0.211743\pi$
$524$	0	0
$525$	− 220704.i	− 0.0349472i
$526$	0	0
$527$	−2.52806e6	−0.396517
$528$	0	0
$529$	1.04065e7	1.61683
$530$	0	0
$531$	− 759132.i	− 0.116837i
$532$	0	0
$533$	7.20047e6i	1.09785i
$534$	0	0
$535$	6.60766e6	0.998075
$536$	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	$-0.984047\pi$
$541$	− 682066.i	− 0.100192i	0.998744	−	0.0500960i	$-0.0159527\pi$
$542$	0	0
$543$	−6.71340e6	−0.977109
$544$	0	0
$545$	5.39471e6	0.777995
$546$	0	0
$547$	− 2.15772e6i	− 0.308337i	−0.988045	−	0.154169i	$-0.950730\pi$
$547$	− 2.15772e6i	− 0.308337i	0.988045	−	0.154169i	$-0.0492699\pi$
$548$	0	0
$549$	− 3.43906e6i	− 0.486978i
$550$	0	0
$551$	−496584.	−0.0696809
$552$	0	0
$553$	−6.76826e6	−0.941161
$554$	0	0
$555$	193104.i	0.0266109i
$556$	0	0
$557$	− 2.67597e6i	− 0.365463i	−0.983163	−	0.182731i	$-0.941506\pi$
$557$	− 2.67597e6i	− 0.365463i	0.983163	−	0.182731i	$-0.0584939\pi$
$558$	0	0
$559$	−5.05780e6	−0.684592
$560$	0	0
$561$	−3.84912e6	−0.516362
$562$	0	0
$563$	3.55331e6i	0.472457i	0.971698	+	0.236228i	$0.0759113\pi$
$563$	3.55331e6i	0.472457i	−0.971698	+	0.236228i	$0.924089\pi$
$564$	0	0
$565$	1.60088e6i	0.210979i
$566$	0	0
$567$	2.21681e6	0.289581
$568$	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
$571$	− 6.08357e6i	− 0.780851i	−0.920634	−	0.390426i	$-0.872328\pi$
$571$	− 6.08357e6i	− 0.780851i	0.920634	−	0.390426i	$-0.127672\pi$
$572$	0	0
$573$	8.68838e6i	1.10548i
$574$	0	0
$575$	−857736.	−0.108189
$576$	0	0
$577$	−1.58241e7	−1.97869	−0.989347	−	0.145579i	$-0.953495\pi$
$577$	−1.58241e7	−1.97869	−0.989347	+	0.145579i	$0.953495\pi$
$578$	0	0
$579$	85272.0i	0.0105709i
$580$	0	0
$581$	5.95901e6i	0.732375i
$582$	0	0
$583$	1.05268e7	1.28269
$584$	0	0
$585$	−2.23463e6	−0.269970
$586$	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
$589$	3.55802e6i	0.422590i
$590$	0	0
$591$	−6.36530e6	−0.749636
$592$	0	0
$593$	8.61122e6	1.00561	0.502803	−	0.864401i	$-0.332302\pi$
$593$	8.61122e6	1.00561	0.502803	+	0.864401i	$0.332302\pi$
$594$	0	0
$595$	2.82269e6i	0.326867i
$596$	0	0
$597$	674016.i	0.0773988i
$598$	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	$-0.694796\pi$
$601$	−1.01740e7	−1.14896	−0.574481	+	0.818518i	$0.694796\pi$
$602$	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	$-0.315193\pi$
$607$	9.95843e6	1.09703	0.548516	+	0.836140i	$0.315193\pi$
$608$	0	0
$609$	−627264.	−0.0685342
$610$	0	0
$611$	− 541728.i	− 0.0587054i
$612$	0	0
$613$	− 4.19586e6i	− 0.450993i	−0.974244	−	0.225497i	$-0.927600\pi$
$613$	− 4.19586e6i	− 0.450993i	0.974244	−	0.225497i	$-0.0724005\pi$
$614$	0	0
$615$	−1.11624e7	−1.19007
$616$	0	0
$617$	−9.12551e6	−0.965038	−0.482519	−	0.875885i	$-0.660278\pi$
$617$	−9.12551e6	−0.965038	−0.482519	+	0.875885i	$0.660278\pi$
$618$	0	0
$619$	6.45734e6i	0.677372i	0.940900	+	0.338686i	$0.109982\pi$
$619$	6.45734e6i	0.677372i	−0.940900	+	0.338686i	$0.890018\pi$
$620$	0	0
$621$	− 1.68428e7i	− 1.75261i
$622$	0	0
$623$	2.61835e6	0.270276
$624$	0	0
$625$	−9.06882e6	−0.928647
$626$	0	0
$627$	5.41728e6i	0.550316i
$628$	0	0
$629$	177012.i	0.0178392i
$630$	0	0
$631$	−1.40514e7	−1.40490	−0.702450	−	0.711733i	$-0.747910\pi$
$631$	−1.40514e7	−1.40490	−0.702450	+	0.711733i	$0.747910\pi$
$632$	0	0
$633$	−4.07035e6	−0.403759
$634$	0	0
$635$	1.81716e7i	1.78838i
$636$	0	0
$637$	3.78833e6i	0.369913i
$638$	0	0
$639$	−4.64033e6	−0.449569
$640$	0	0
$641$	8.47168e6	0.814375	0.407188	−	0.913345i	$-0.366510\pi$
$641$	8.47168e6	0.814375	0.407188	+	0.913345i	$0.366510\pi$
$642$	0	0
$643$	− 488564.i	− 0.0466009i	−0.999729	−	0.0233004i	$-0.992583\pi$
$643$	− 488564.i	− 0.0466009i	0.999729	−	0.0233004i	$-0.00741743\pi$
$644$	0	0
$645$	− 7.84080e6i	− 0.742098i
$646$	0	0
$647$	2.48119e6	0.233023	0.116512	−	0.993189i	$-0.462829\pi$
$647$	2.48119e6	0.233023	0.116512	+	0.993189i	$0.462829\pi$
$648$	0	0
$649$	4.14072e6	0.385891
$650$	0	0
$651$	4.49434e6i	0.415636i
$652$	0	0
$653$	− 5.29130e6i	− 0.485601i	−0.970076	−	0.242800i	$-0.921934\pi$
$653$	− 5.29130e6i	− 0.485601i	0.970076	−	0.242800i	$-0.0780660\pi$
$654$	0	0
$655$	−5.45292e6	−0.496622
$656$	0	0
$657$	−6.68864e6	−0.604539
$658$	0	0
$659$	− 4.72468e6i	− 0.423798i	−0.977292	−	0.211899i	$-0.932035\pi$
$659$	− 4.72468e6i	− 0.423798i	0.977292	−	0.211899i	$-0.0679647\pi$
$660$	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
$663$	2.97950e6	0.263245
$664$	0	0
$665$	3.97267e6	0.348360
$666$	0	0
$667$	2.43778e6i	0.212168i
$668$	0	0
$669$	− 9.35232e6i	− 0.807893i
$670$	0	0
$671$	1.87585e7	1.60839
$672$	0	0
$673$	−9.40925e6	−0.800787	−0.400394	−	0.916343i	$-0.631127\pi$
$673$	−9.40925e6	−0.800787	−0.400394	+	0.916343i	$0.631127\pi$
$674$	0	0
$675$	857736.i	0.0724593i
$676$	0	0
$677$	− 1.50086e7i	− 1.25854i	−0.777185	−	0.629272i	$-0.783353\pi$
$677$	− 1.50086e7i	− 1.25854i	0.777185	−	0.629272i	$-0.216647\pi$
$678$	0	0
$679$	1.07710e7	0.896567
$680$	0	0
$681$	−8.93851e6	−0.738580
$682$	0	0
$683$	− 1.29707e7i	− 1.06393i	−0.846768	−	0.531963i	$-0.821455\pi$
$683$	− 1.29707e7i	− 1.06393i	0.846768	−	0.531963i	$-0.178545\pi$
$684$	0	0
$685$	− 1.71257e7i	− 1.39451i
$686$	0	0
$687$	−3.27295e6	−0.264574
$688$	0	0
$689$	−8.14849e6	−0.653927
$690$	0	0
$691$	− 2.26556e7i	− 1.80501i	−0.430677	−	0.902506i	$-0.641725\pi$
$691$	− 2.26556e7i	− 1.80501i	0.430677	−	0.902506i	$-0.358275\pi$
$692$	0	0
$693$	− 4.70448e6i	− 0.372116i
$694$	0	0
$695$	−8.00950e6	−0.628989
$696$	0	0
$697$	−1.02322e7	−0.797791
$698$	0	0
$699$	1.84615e6i	0.142914i
$700$	0	0
$701$	1.90169e7i	1.46166i	0.682562	+	0.730828i	$0.260866\pi$
$701$	1.90169e7i	1.46166i	−0.682562	+	0.730828i	$0.739134\pi$
$702$	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
$713$	1.74666e7	1.28672
$714$	0	0
$715$	− 1.21889e7i	− 0.891659i
$716$	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
$721$	2.39853e6	0.171833
$722$	0	0
$723$	7.88489e6i	0.560983i
$724$	0	0
$725$	− 124146.i	− 0.00877178i
$726$	0	0
$727$	−7.41230e6	−0.520136	−0.260068	−	0.965590i	$-0.583745\pi$
$727$	−7.41230e6	−0.520136	−0.260068	+	0.965590i	$0.583745\pi$
$728$	0	0
$729$	−1.44612e7	−1.00782
$730$	0	0
$731$	− 7.18740e6i	− 0.497483i
$732$	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
$735$	−5.87282e6	−0.400985
$736$	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
$741$	− 4.19338e6i	− 0.280555i
$742$	0	0
$743$	4.46926e6	0.297005	0.148502	−	0.988912i	$-0.452555\pi$
$743$	4.46926e6	0.297005	0.148502	+	0.988912i	$0.452555\pi$
$744$	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
$755$	− 4.01544e6i	− 0.256369i
$756$	0	0
$757$	− 9.60868e6i	− 0.609430i	−0.952444	−	0.304715i	$-0.901439\pi$
$757$	− 9.60868e6i	− 0.609430i	0.952444	−	0.304715i	$-0.0985612\pi$
$758$	0	0
$759$	2.65939e7	1.67563
$760$	0	0
$761$	−4.54588e6	−0.284549	−0.142274	−	0.989827i	$-0.545442\pi$
$761$	−4.54588e6	−0.284549	−0.142274	+	0.989827i	$0.545442\pi$
$762$	0	0
$763$	− 8.79138e6i	− 0.546696i
$764$	0	0
$765$	− 3.17552e6i	− 0.196183i
$766$	0	0
$767$	−3.20522e6	−0.196730
$768$	0	0
$769$	−2.15923e7	−1.31669	−0.658345	−	0.752716i	$-0.728743\pi$
$769$	−2.15923e7	−1.31669	−0.658345	+	0.752716i	$0.728743\pi$
$770$	0	0
$771$	1.58825e6i	0.0962238i
$772$	0	0
$773$	1.48400e7i	0.893276i	0.894715	+	0.446638i	$0.147379\pi$
$773$	1.48400e7i	0.893276i	−0.894715	+	0.446638i	$0.852621\pi$
$774$	0	0
$775$	−889504.	−0.0531978
$776$	0	0
$777$	314688.	0.0186994
$778$	0	0
$779$	1.44009e7i	0.850251i
$780$	0	0
$781$	− 2.53109e7i	− 1.48484i
$782$	0	0
$783$	2.43778e6	0.142098
$784$	0	0
$785$	6.52741e6	0.378065
$786$	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
$789$	1.13193e7i	0.647330i
$790$	0	0
$791$	2.60885e6	0.148254
$792$	0	0
$793$	−1.45205e7	−0.819970
$794$	0	0
$795$	− 1.26321e7i	− 0.708856i
$796$	0	0
$797$	3.16080e7i	1.76259i	0.472568	+	0.881294i	$0.343327\pi$
$797$	3.16080e7i	1.76259i	−0.472568	+	0.881294i	$0.656673\pi$
$798$	0	0
$799$	769824.	0.0426604
$800$	0	0
$801$	−2.94565e6	−0.162218
$802$	0	0
$803$	− 3.64835e7i	− 1.99668i
$804$	0	0
$805$	− 1.95022e7i	− 1.06070i
$806$	0	0
$807$	−1.16102e7	−0.627562
$808$	0	0
$809$	3.10009e6	0.166534	0.0832669	−	0.996527i	$-0.473465\pi$
$809$	3.10009e6	0.166534	0.0832669	+	0.996527i	$0.473465\pi$
$810$	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
$817$	−1.01156e7	−0.530196
$818$	0	0
$819$	3.64162e6i	0.189707i
$820$	0	0
$821$	2.00184e7i	1.03650i	0.855228	+	0.518252i	$0.173417\pi$
$821$	2.00184e7i	1.03650i	−0.855228	+	0.518252i	$0.826583\pi$
$822$	0	0
$823$	1.53118e7	0.787999	0.394000	−	0.919111i	$-0.371091\pi$
$823$	1.53118e7	0.787999	0.394000	+	0.919111i	$0.371091\pi$
$824$	0	0
$825$	−1.35432e6	−0.0692766
$826$	0	0
$827$	9.59310e6i	0.487748i	0.969807	+	0.243874i	$0.0784183\pi$
$827$	9.59310e6i	0.487748i	−0.969807	+	0.243874i	$0.921582\pi$
$828$	0	0
$829$	2.52209e7i	1.27460i	0.770615	+	0.637302i	$0.219949\pi$
$829$	2.52209e7i	1.27460i	−0.770615	+	0.637302i	$0.780051\pi$
$830$	0	0
$831$	2.66727e7	1.33988
$832$	0	0
$833$	−5.38342e6	−0.268810
$834$	0	0
$835$	2.65589e7i	1.31824i
$836$	0	0
$837$	− 1.74666e7i	− 0.861778i
$838$	0	0
$839$	−1.77623e7	−0.871154	−0.435577	−	0.900151i	$-0.643456\pi$
$839$	−1.77623e7	−0.871154	−0.435577	+	0.900151i	$0.643456\pi$
$840$	0	0
$841$	2.01583e7	0.982798
$842$	0	0
$843$	2.35937e6i	0.114348i
$844$	0	0
$845$	− 1.06147e7i	− 0.511407i
$846$	0	0
$847$	1.14883e7	0.550234
$848$	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
$859$	2.23538e7i	1.03364i	0.856094	+	0.516820i	$0.172884\pi$
$859$	2.23538e7i	1.03364i	−0.856094	+	0.516820i	$0.827116\pi$
$860$	0	0
$861$	1.81907e7i	0.836258i
$862$	0	0
$863$	−1.85838e7	−0.849390	−0.424695	−	0.905337i	$-0.639619\pi$
$863$	−1.85838e7	−0.849390	−0.424695	+	0.905337i	$0.639619\pi$
$864$	0	0
$865$	3.82025e7	1.73601
$866$	0	0
$867$	− 1.28043e7i	− 0.578504i
$868$	0	0
$869$	4.15325e7i	1.86569i
$870$	0	0
$871$	−9.11742e6	−0.407217
$872$	0	0
$873$	−1.21174e7	−0.538114
$874$	0	0
$875$	1.58432e7i	0.699555i
$876$	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
$879$	−1.28660e7	−0.561659
$880$	0	0
$881$	−3.14696e6	−0.136600	−0.0683001	−	0.997665i	$-0.521758\pi$
$881$	−3.14696e6	−0.136600	−0.0683001	+	0.997665i	$0.521758\pi$
$882$	0	0
$883$	− 1.59995e7i	− 0.690566i	−0.938499	−	0.345283i	$-0.887783\pi$
$883$	− 1.59995e7i	− 0.690566i	0.938499	−	0.345283i	$-0.112217\pi$
$884$	0	0
$885$	− 4.96886e6i	− 0.213255i
$886$	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
$889$	2.96131e7	1.25669
$890$	0	0
$891$	− 1.36031e7i	− 0.574044i
$892$	0	0
$893$	− 1.08346e6i	− 0.0454656i
$894$	0	0
$895$	−2.66581e7	−1.11243
$896$	0	0
$897$	−2.05857e7	−0.854248
$898$	0	0
$899$	2.52806e6i	0.104325i
$900$	0	0
$901$	− 1.15794e7i	− 0.475199i
$902$	0	0
$903$	−1.27776e7	−0.521471
$904$	0	0
$905$	3.02103e7	1.22612
$906$	0	0
$907$	1.74396e7i	0.703914i	0.936016	+	0.351957i	$0.114484\pi$
$907$	1.74396e7i	0.703914i	−0.936016	+	0.351957i	$0.885516\pi$
$908$	0	0
$909$	− 1.11731e6i	− 0.0448503i
$910$	0	0
$911$	2.59589e6	0.103631	0.0518155	−	0.998657i	$-0.483499\pi$
$911$	2.59589e6	0.103631	0.0518155	+	0.998657i	$0.483499\pi$
$912$	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
$921$	1.90307e7	0.739275
$922$	0	0
$923$	1.95925e7i	0.756982i
$924$	0	0
$925$	62282.0i	0.00239336i
$926$	0	0
$927$	−2.69834e6	−0.103133
$928$	0	0
$929$	3.96785e7	1.50840	0.754199	−	0.656646i	$-0.228025\pi$
$929$	3.96785e7	1.50840	0.754199	+	0.656646i	$0.228025\pi$
$930$	0	0
$931$	7.57667e6i	0.286486i
$932$	0	0
$933$	− 8.76874e6i	− 0.329787i
$934$	0	0
$935$	1.73210e7	0.647955
$936$	0	0
$937$	−3.93413e7	−1.46386	−0.731930	−	0.681380i	$-0.761380\pi$
$937$	−3.93413e7	−1.46386	−0.731930	+	0.681380i	$0.761380\pi$
$938$	0	0
$939$	− 7.01830e6i	− 0.259757i
$940$	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
$943$	7.06955e7	2.58888
$944$	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
$949$	2.82409e7i	1.01792i
$950$	0	0
$951$	2.97944e7	1.06828
$952$	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
$963$	1.21140e7i	0.420943i
$964$	0	0
$965$	− 383724.i	− 0.0132648i
$966$	0	0
$967$	4.09790e7	1.40927	0.704637	−	0.709568i	$-0.251110\pi$
$967$	4.09790e7	1.40927	0.704637	+	0.709568i	$0.251110\pi$
$968$	0	0
$969$	5.95901e6	0.203875
$970$	0	0
$971$	− 2.72034e7i	− 0.925922i	−0.886379	−	0.462961i	$-0.846787\pi$
$971$	− 2.72034e7i	− 0.925922i	0.886379	−	0.462961i	$-0.153213\pi$
$972$	0	0
$973$	1.30525e7i	0.441990i
$974$	0	0
$975$	1.04834e6	0.0353177
$976$	0	0
$977$	2.53555e7	0.849839	0.424919	−	0.905231i	$-0.360302\pi$
$977$	2.53555e7	0.849839	0.424919	+	0.905231i	$0.360302\pi$
$978$	0	0
$979$	− 1.60672e7i	− 0.535775i
$980$	0	0
$981$	9.89030e6i	0.328123i
$982$	0	0
$983$	1.19139e7	0.393252	0.196626	−	0.980479i	$-0.437002\pi$
$983$	1.19139e7	0.393252	0.196626	+	0.980479i	$0.437002\pi$
$984$	0	0
$985$	2.86439e7	0.940678
$986$	0	0
$987$	− 1.36858e6i	− 0.0447173i
$988$	0	0
$989$	4.96584e7i	1.61437i
$990$	0	0
$991$	−2.91931e7	−0.944268	−0.472134	−	0.881527i	$-0.656516\pi$
$991$	−2.91931e7	−0.944268	−0.472134	+	0.881527i	$0.656516\pi$
$992$	0	0
$993$	−4.53538e6	−0.145962
$994$	0	0
$995$	− 3.03307e6i	− 0.0971237i
$996$	0	0
$997$	1.73001e7i	0.551201i	0.961272	+	0.275601i	$0.0888767\pi$
$997$	1.73001e7i	0.551201i	−0.961272	+	0.275601i	$0.911123\pi$
$998$	0	0
$999$	−1.22299e6	−0.0387713

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.6.b.c.129.2		2
4.3	odd	2		256.6.b.g.129.1		2
8.3	odd	2		256.6.b.g.129.2		2
8.5	even	2	inner	256.6.b.c.129.1		2
16.3	odd	4		64.6.a.f.1.1		1
16.5	even	4		16.6.a.b.1.1		1
16.11	odd	4		4.6.a.a.1.1	✓	1
16.13	even	4		64.6.a.b.1.1		1
48.5	odd	4		144.6.a.c.1.1		1
48.11	even	4		36.6.a.a.1.1		1
48.29	odd	4		576.6.a.bd.1.1		1
48.35	even	4		576.6.a.bc.1.1		1
80.27	even	4		100.6.c.b.49.2		2
80.37	odd	4		400.6.c.f.49.1		2
80.43	even	4		100.6.c.b.49.1		2
80.53	odd	4		400.6.c.f.49.2		2
80.59	odd	4		100.6.a.b.1.1		1
80.69	even	4		400.6.a.d.1.1		1
112.11	odd	12		196.6.e.g.177.1		2
112.27	even	4		196.6.a.e.1.1		1
112.59	even	12		196.6.e.d.177.1		2
112.69	odd	4		784.6.a.d.1.1		1
112.75	even	12		196.6.e.d.165.1		2
112.107	odd	12		196.6.e.g.165.1		2
144.11	even	12		324.6.e.d.109.1		2
144.43	odd	12		324.6.e.a.109.1		2
144.59	even	12		324.6.e.d.217.1		2
144.139	odd	12		324.6.e.a.217.1		2
176.43	even	4		484.6.a.a.1.1		1
208.155	odd	4		676.6.a.a.1.1		1
208.187	even	4		676.6.d.a.337.1		2
208.203	even	4		676.6.d.a.337.2		2
240.59	even	4		900.6.a.h.1.1		1
240.107	odd	4		900.6.d.a.649.1		2
240.203	odd	4		900.6.d.a.649.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.6.a.a.1.1	✓	1	16.11	odd	4
16.6.a.b.1.1		1	16.5	even	4
36.6.a.a.1.1		1	48.11	even	4
64.6.a.b.1.1		1	16.13	even	4
64.6.a.f.1.1		1	16.3	odd	4
100.6.a.b.1.1		1	80.59	odd	4
100.6.c.b.49.1		2	80.43	even	4
100.6.c.b.49.2		2	80.27	even	4
144.6.a.c.1.1		1	48.5	odd	4
196.6.a.e.1.1		1	112.27	even	4
196.6.e.d.165.1		2	112.75	even	12
196.6.e.d.177.1		2	112.59	even	12
196.6.e.g.165.1		2	112.107	odd	12
196.6.e.g.177.1		2	112.11	odd	12
256.6.b.c.129.1		2	8.5	even	2	inner
256.6.b.c.129.2		2	1.1	even	1	trivial
256.6.b.g.129.1		2	4.3	odd	2
256.6.b.g.129.2		2	8.3	odd	2
324.6.e.a.109.1		2	144.43	odd	12
324.6.e.a.217.1		2	144.139	odd	12
324.6.e.d.109.1		2	144.11	even	12
324.6.e.d.217.1		2	144.59	even	12
400.6.a.d.1.1		1	80.69	even	4
400.6.c.f.49.1		2	80.37	odd	4
400.6.c.f.49.2		2	80.53	odd	4
484.6.a.a.1.1		1	176.43	even	4
576.6.a.bc.1.1		1	48.35	even	4
576.6.a.bd.1.1		1	48.29	odd	4
676.6.a.a.1.1		1	208.155	odd	4
676.6.d.a.337.1		2	208.187	even	4
676.6.d.a.337.2		2	208.203	even	4
784.6.a.d.1.1		1	112.69	odd	4
900.6.a.h.1.1		1	240.59	even	4
900.6.d.a.649.1		2	240.107	odd	4
900.6.d.a.649.2		2	240.203	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	256.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+12.0000i q^{3} -54.0000i q^{5} -88.0000 q^{7} +99.0000 q^{9} +540.000i q^{11} -418.000i q^{13} +648.000 q^{15} +594.000 q^{17} -836.000i q^{19} -1056.00i q^{21} -4104.00 q^{23} +209.000 q^{25} +4104.00i q^{27} -594.000i q^{29} -4256.00 q^{31} -6480.00 q^{33} +4752.00i q^{35} +298.000i q^{37} +5016.00 q^{39} -17226.0 q^{41} -12100.0i q^{43} -5346.00i q^{45} +1296.00 q^{47} -9063.00 q^{49} +7128.00i q^{51} -19494.0i q^{53} +29160.0 q^{55} +10032.0 q^{57} -7668.00i q^{59} -34738.0i q^{61} -8712.00 q^{63} -22572.0 q^{65} -21812.0i q^{67} -49248.0i q^{69} -46872.0 q^{71} -67562.0 q^{73} +2508.00i q^{75} -47520.0i q^{77} +76912.0 q^{79} -25191.0 q^{81} -67716.0i q^{83} -32076.0i q^{85} +7128.00 q^{87} -29754.0 q^{89} +36784.0i q^{91} -51072.0i q^{93} -45144.0 q^{95} -122398. q^{97} +53460.0i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 176 q^{7} + 198 q^{9} + 1296 q^{15} + 1188 q^{17} - 8208 q^{23} + 418 q^{25} - 8512 q^{31} - 12960 q^{33} + 10032 q^{39} - 34452 q^{41} + 2592 q^{47} - 18126 q^{49} + 58320 q^{55} + 20064 q^{57} - 17424 q^{63}+ \cdots - 244796 q^{97}+O(q^{100}) \) 2 * q - 176 * q^7 + 198 * q^9 + 1296 * q^15 + 1188 * q^17 - 8208 * q^23 + 418 * q^25 - 8512 * q^31 - 12960 * q^33 + 10032 * q^39 - 34452 * q^41 + 2592 * q^47 - 18126 * q^49 + 58320 * q^55 + 20064 * q^57 - 17424 * q^63 - 45144 * q^65 - 93744 * q^71 - 135124 * q^73 + 153824 * q^79 - 50382 * q^81 + 14256 * q^87 - 59508 * q^89 - 90288 * q^95 - 244796 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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