LMFDB - Embedded newform 256.10.b.i.129.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,10,Mod(129,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("256.129");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	256.b (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:	$131.849174058$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 8)
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			256.10.b.i.129.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+60.0000i q^{3} -2074.00i q^{5} +4344.00 q^{7} +16083.0 q^{9} +O(q^{10})$	$q+60.0000i q^{3} -2074.00i q^{5} +4344.00 q^{7} +16083.0 q^{9} +93644.0i q^{11} +12242.0i q^{13} +124440. q^{15} -319598. q^{17} +553516. i q^{19} +260640. i q^{21} +712936. q^{23} -2.34835e6 q^{25} +2.14596e6i q^{27} -2.07584e6i q^{29} -6.42045e6 q^{31} -5.61864e6 q^{33} -9.00946e6i q^{35} -1.81978e7i q^{37} -734520. q^{39} -9.03383e6 q^{41} +1.95947e7i q^{43} -3.33561e7i q^{45} -1.84842e7 q^{47} -2.14833e7 q^{49} -1.91759e7i q^{51} +1.02558e7i q^{53} +1.94218e8 q^{55} -3.32110e7 q^{57} +1.21667e8i q^{59} +4.59490e7i q^{61} +6.98646e7 q^{63} +2.53899e7 q^{65} -5.05354e7i q^{67} +4.27762e7i q^{69} -2.67045e8 q^{71} +1.76213e8 q^{73} -1.40901e8i q^{75} +4.06790e8i q^{77} -2.69686e8 q^{79} +1.87804e8 q^{81} +2.27033e8i q^{83} +6.62846e8i q^{85} +1.24550e8 q^{87} -7.21416e7 q^{89} +5.31792e7i q^{91} -3.85227e8i q^{93} +1.14799e9 q^{95} +2.28777e8 q^{97} +1.50608e9i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8688 q^{7} + 32166 q^{9}+O(q^{10}) $ 2 * q + 8688 * q^7 + 32166 * q^9	$ 2 q + 8688 q^{7} + 32166 q^{9} + 248880 q^{15} - 639196 q^{17} + 1425872 q^{23} - 4696702 q^{25} - 12840896 q^{31} - 11237280 q^{33} - 1469040 q^{39} - 18067668 q^{41} - 36968352 q^{47} - 42966542 q^{49} + 388435312 q^{55} - 66421920 q^{57} + 139729104 q^{63} + 50779816 q^{65} - 534089360 q^{71} + 352426732 q^{73} - 539371360 q^{79} + 375608178 q^{81} + 249100560 q^{87} - 144283188 q^{89} + 2295984368 q^{95} + 457553092 q^{97}+O(q^{100}) $ 2 * q + 8688 * q^7 + 32166 * q^9 + 248880 * q^15 - 639196 * q^17 + 1425872 * q^23 - 4696702 * q^25 - 12840896 * q^31 - 11237280 * q^33 - 1469040 * q^39 - 18067668 * q^41 - 36968352 * q^47 - 42966542 * q^49 + 388435312 * q^55 - 66421920 * q^57 + 139729104 * q^63 + 50779816 * q^65 - 534089360 * q^71 + 352426732 * q^73 - 539371360 * q^79 + 375608178 * q^81 + 249100560 * q^87 - 144283188 * q^89 + 2295984368 * q^95 + 457553092 * q^97

Display 100 coefficients 10 coefficients

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
$3$	60.0000i	0.427667i	0.976870	+	0.213833i	$0.0685950\pi$
$3$	60.0000i	0.427667i	−0.976870	+	0.213833i	$0.931405\pi$
$4$	0	0
$5$	− 2074.00i	− 1.48403i	−0.670381	−	0.742017i	$-0.733869\pi$
$5$	− 2074.00i	− 1.48403i	0.670381	−	0.742017i	$-0.266131\pi$
$6$	0	0
$7$	4344.00	0.683831	0.341915	−	0.939731i	$-0.388924\pi$
$7$	4344.00	0.683831	0.341915	+	0.939731i	$0.388924\pi$
$8$	0	0
$9$	16083.0	0.817101
$10$	0	0
$11$	93644.0i	1.92847i	0.265049	+	0.964235i	$0.414612\pi$
$11$	93644.0i	1.92847i	−0.265049	+	0.964235i	$0.585388\pi$
$12$	0	0
$13$	12242.0i	0.118880i	0.998232	+	0.0594398i	$0.0189314\pi$
$13$	12242.0i	0.118880i	−0.998232	+	0.0594398i	$0.981069\pi$
$14$	0	0
$15$	124440.	0.634672
$16$	0	0
$17$	−319598.	−0.928077	−0.464038	−	0.885815i	$-0.653600\pi$
$17$	−319598.	−0.928077	−0.464038	+	0.885815i	$0.653600\pi$
$18$	0	0
$19$	553516.i	0.974404i	0.873289	+	0.487202i	$0.161982\pi$
$19$	553516.i	0.974404i	−0.873289	+	0.487202i	$0.838018\pi$
$20$	0	0
$21$	260640.i	0.292452i
$22$	0	0
$23$	712936.	0.531221	0.265611	−	0.964080i	$-0.414426\pi$
$23$	712936.	0.531221	0.265611	+	0.964080i	$0.414426\pi$
$24$	0	0
$25$	−2.34835e6	−1.20236
$26$	0	0
$27$	2.14596e6i	0.777114i
$28$	0	0
$29$	− 2.07584e6i	− 0.545007i	−0.962155	−	0.272504i	$-0.912148\pi$
$29$	− 2.07584e6i	− 0.545007i	0.962155	−	0.272504i	$-0.0878517\pi$
$30$	0	0
$31$	−6.42045e6	−1.24864	−0.624321	−	0.781168i	$-0.714624\pi$
$31$	−6.42045e6	−1.24864	−0.624321	+	0.781168i	$0.714624\pi$
$32$	0	0
$33$	−5.61864e6	−0.824743
$34$	0	0
$35$	− 9.00946e6i	− 1.01483i
$36$	0	0
$37$	− 1.81978e7i	− 1.59628i	−0.602470	−	0.798142i	$-0.705817\pi$
$37$	− 1.81978e7i	− 1.59628i	0.602470	−	0.798142i	$-0.294183\pi$
$38$	0	0
$39$	−734520.	−0.0508409
$40$	0	0
$41$	−9.03383e6	−0.499281	−0.249640	−	0.968339i	$-0.580312\pi$
$41$	−9.03383e6	−0.499281	−0.249640	+	0.968339i	$0.580312\pi$
$42$	0	0
$43$	1.95947e7i	0.874040i	0.899452	+	0.437020i	$0.143966\pi$
$43$	1.95947e7i	0.874040i	−0.899452	+	0.437020i	$0.856034\pi$
$44$	0	0
$45$	− 3.33561e7i	− 1.21261i
$46$	0	0
$47$	−1.84842e7	−0.552535	−0.276267	−	0.961081i	$-0.589098\pi$
$47$	−1.84842e7	−0.552535	−0.276267	+	0.961081i	$0.589098\pi$
$48$	0	0
$49$	−2.14833e7	−0.532375
$50$	0	0
$51$	− 1.91759e7i	− 0.396908i
$52$	0	0
$53$	1.02558e7i	0.178536i	0.996008	+	0.0892682i	$0.0284528\pi$
$53$	1.02558e7i	0.178536i	−0.996008	+	0.0892682i	$0.971547\pi$
$54$	0	0
$55$	1.94218e8	2.86191
$56$	0	0
$57$	−3.32110e7	−0.416720
$58$	0	0
$59$	1.21667e8i	1.30719i	0.756847	+	0.653593i	$0.226739\pi$
$59$	1.21667e8i	1.30719i	−0.756847	+	0.653593i	$0.773261\pi$
$60$	0	0
$61$	4.59490e7i	0.424905i	0.977171	+	0.212452i	$0.0681450\pi$
$61$	4.59490e7i	0.424905i	−0.977171	+	0.212452i	$0.931855\pi$
$62$	0	0
$63$	6.98646e7	0.558759
$64$	0	0
$65$	2.53899e7	0.176421
$66$	0	0
$67$	− 5.05354e7i	− 0.306379i	−0.988197	−	0.153190i	$-0.951045\pi$
$67$	− 5.05354e7i	− 0.306379i	0.988197	−	0.153190i	$-0.0489545\pi$
$68$	0	0
$69$	4.27762e7i	0.227186i
$70$	0	0
$71$	−2.67045e8	−1.24716	−0.623579	−	0.781760i	$-0.714322\pi$
$71$	−2.67045e8	−1.24716	−0.623579	+	0.781760i	$0.714322\pi$
$72$	0	0
$73$	1.76213e8	0.726250	0.363125	−	0.931740i	$-0.381710\pi$
$73$	1.76213e8	0.726250	0.363125	+	0.931740i	$0.381710\pi$
$74$	0	0
$75$	− 1.40901e8i	− 0.514208i
$76$	0	0
$77$	4.06790e8i	1.31875i
$78$	0	0
$79$	−2.69686e8	−0.778997	−0.389499	−	0.921027i	$-0.627352\pi$
$79$	−2.69686e8	−0.778997	−0.389499	+	0.921027i	$0.627352\pi$
$80$	0	0
$81$	1.87804e8	0.484755
$82$	0	0
$83$	2.27033e8i	0.525094i	0.964919	+	0.262547i	$0.0845624\pi$
$83$	2.27033e8i	0.525094i	−0.964919	+	0.262547i	$0.915438\pi$
$84$	0	0
$85$	6.62846e8i	1.37730i
$86$	0	0
$87$	1.24550e8	0.233082
$88$	0	0
$89$	−7.21416e7	−0.121880	−0.0609398	−	0.998141i	$-0.519410\pi$
$89$	−7.21416e7	−0.121880	−0.0609398	+	0.998141i	$0.519410\pi$
$90$	0	0
$91$	5.31792e7i	0.0812935i
$92$	0	0
$93$	− 3.85227e8i	− 0.534003i
$94$	0	0
$95$	1.14799e9	1.44605
$96$	0	0
$97$	2.28777e8	0.262385	0.131192	−	0.991357i	$-0.458119\pi$
$97$	2.28777e8	0.262385	0.131192	+	0.991357i	$0.458119\pi$
$98$	0	0
$99$	1.50608e9i	1.57575i
$100$	0	0
$101$	− 8.03256e8i	− 0.768082i	−0.923316	−	0.384041i	$-0.874532\pi$
$101$	− 8.03256e8i	− 0.768082i	0.923316	−	0.384041i	$-0.125468\pi$
$102$	0	0
$103$	−7.81726e8	−0.684363	−0.342182	−	0.939634i	$-0.611166\pi$
$103$	−7.81726e8	−0.684363	−0.342182	+	0.939634i	$0.611166\pi$
$104$	0	0
$105$	5.40567e8	0.434008
$106$	0	0
$107$	− 1.00756e9i	− 0.743093i	−0.928414	−	0.371546i	$-0.878828\pi$
$107$	− 1.00756e9i	− 0.743093i	0.928414	−	0.371546i	$-0.121172\pi$
$108$	0	0
$109$	4.80692e8i	0.326173i	0.986612	+	0.163086i	$0.0521449\pi$
$109$	4.80692e8i	0.326173i	−0.986612	+	0.163086i	$0.947855\pi$
$110$	0	0
$111$	1.09187e9	0.682678
$112$	0	0
$113$	−2.89781e9	−1.67193	−0.835963	−	0.548786i	$-0.815090\pi$
$113$	−2.89781e9	−1.67193	−0.835963	+	0.548786i	$0.815090\pi$
$114$	0	0
$115$	− 1.47863e9i	− 0.788350i
$116$	0	0
$117$	1.96888e8i	0.0971366i
$118$	0	0
$119$	−1.38833e9	−0.634647
$120$	0	0
$121$	−6.41125e9	−2.71900
$122$	0	0
$123$	− 5.42030e8i	− 0.213526i
$124$	0	0
$125$	8.19699e8i	0.300303i
$126$	0	0
$127$	−4.24330e9	−1.44740	−0.723698	−	0.690117i	$-0.757559\pi$
$127$	−4.24330e9	−1.44740	−0.723698	+	0.690117i	$0.757559\pi$
$128$	0	0
$129$	−1.17568e9	−0.373798
$130$	0	0
$131$	− 2.89728e9i	− 0.859546i	−0.902937	−	0.429773i	$-0.858594\pi$
$131$	− 2.89728e9i	− 0.859546i	0.902937	−	0.429773i	$-0.141406\pi$
$132$	0	0
$133$	2.40447e9i	0.666327i
$134$	0	0
$135$	4.45072e9	1.15326
$136$	0	0
$137$	2.35617e9	0.571432	0.285716	−	0.958314i	$-0.407769\pi$
$137$	2.35617e9	0.571432	0.285716	+	0.958314i	$0.407769\pi$
$138$	0	0
$139$	− 2.71527e9i	− 0.616946i	−0.951233	−	0.308473i	$-0.900182\pi$
$139$	− 2.71527e9i	− 0.616946i	0.951233	−	0.308473i	$-0.0998179\pi$
$140$	0	0
$141$	− 1.10905e9i	− 0.236301i
$142$	0	0
$143$	−1.14639e9	−0.229256
$144$	0	0
$145$	−4.30529e9	−0.808809
$146$	0	0
$147$	− 1.28900e9i	− 0.227679i
$148$	0	0
$149$	1.67402e9i	0.278242i	0.990275	+	0.139121i	$0.0444276\pi$
$149$	1.67402e9i	0.278242i	−0.990275	+	0.139121i	$0.955572\pi$
$150$	0	0
$151$	−5.32709e9	−0.833860	−0.416930	−	0.908938i	$-0.636894\pi$
$151$	−5.32709e9	−0.833860	−0.416930	+	0.908938i	$0.636894\pi$
$152$	0	0
$153$	−5.14009e9	−0.758333
$154$	0	0
$155$	1.33160e10i	1.85303i
$156$	0	0
$157$	1.15835e10i	1.52156i	0.649008	+	0.760782i	$0.275184\pi$
$157$	1.15835e10i	1.52156i	−0.649008	+	0.760782i	$0.724816\pi$
$158$	0	0
$159$	−6.15346e8	−0.0763541
$160$	0	0
$161$	3.09699e9	0.363265
$162$	0	0
$163$	− 9.48418e8i	− 0.105234i	−0.998615	−	0.0526169i	$-0.983244\pi$
$163$	− 9.48418e8i	− 0.105234i	0.998615	−	0.0526169i	$-0.0167562\pi$
$164$	0	0
$165$	1.16531e10i	1.22395i
$166$	0	0
$167$	1.44718e10	1.43978	0.719891	−	0.694087i	$-0.244192\pi$
$167$	1.44718e10	1.43978	0.719891	+	0.694087i	$0.244192\pi$
$168$	0	0
$169$	1.04546e10	0.985868
$170$	0	0
$171$	8.90220e9i	0.796186i
$172$	0	0
$173$	1.39886e10i	1.18732i	0.804717	+	0.593658i	$0.202317\pi$
$173$	1.39886e10i	1.18732i	−0.804717	+	0.593658i	$0.797683\pi$
$174$	0	0
$175$	−1.02012e10	−0.822208
$176$	0	0
$177$	−7.29999e9	−0.559040
$178$	0	0
$179$	4.54924e9i	0.331207i	0.986192	+	0.165604i	$0.0529573\pi$
$179$	4.54924e9i	0.331207i	−0.986192	+	0.165604i	$0.947043\pi$
$180$	0	0
$181$	1.56484e10i	1.08372i	0.840469	+	0.541859i	$0.182279\pi$
$181$	1.56484e10i	1.08372i	−0.840469	+	0.541859i	$0.817721\pi$
$182$	0	0
$183$	−2.75694e9	−0.181718
$184$	0	0
$185$	−3.77421e10	−2.36894
$186$	0	0
$187$	− 2.99284e10i	− 1.78977i
$188$	0	0
$189$	9.32205e9i	0.531414i
$190$	0	0
$191$	2.02052e10	1.09853	0.549267	−	0.835647i	$-0.314907\pi$
$191$	2.02052e10	1.09853	0.549267	+	0.835647i	$0.314907\pi$
$192$	0	0
$193$	−7.10827e9	−0.368770	−0.184385	−	0.982854i	$-0.559029\pi$
$193$	−7.10827e9	−0.368770	−0.184385	+	0.982854i	$0.559029\pi$
$194$	0	0
$195$	1.52339e9i	0.0754495i
$196$	0	0
$197$	2.25924e10i	1.06872i	0.845257	+	0.534359i	$0.179447\pi$
$197$	2.25924e10i	1.06872i	−0.845257	+	0.534359i	$0.820553\pi$
$198$	0	0
$199$	−3.55506e10	−1.60697	−0.803485	−	0.595325i	$-0.797023\pi$
$199$	−3.55506e10	−1.60697	−0.803485	+	0.595325i	$0.797023\pi$
$200$	0	0
$201$	3.03213e9	0.131028
$202$	0	0
$203$	− 9.01744e9i	− 0.372693i
$204$	0	0
$205$	1.87362e10i	0.740949i
$206$	0	0
$207$	1.14661e10	0.434061
$208$	0	0
$209$	−5.18335e10	−1.87911
$210$	0	0
$211$	5.58480e9i	0.193971i	0.995286	+	0.0969854i	$0.0309200\pi$
$211$	5.58480e9i	0.193971i	−0.995286	+	0.0969854i	$0.969080\pi$
$212$	0	0
$213$	− 1.60227e10i	− 0.533368i
$214$	0	0
$215$	4.06395e10	1.29710
$216$	0	0
$217$	−2.78904e10	−0.853859
$218$	0	0
$219$	1.05728e10i	0.310593i
$220$	0	0
$221$	− 3.91252e9i	− 0.110329i
$222$	0	0
$223$	4.74713e10	1.28546	0.642731	−	0.766092i	$-0.277801\pi$
$223$	4.74713e10	1.28546	0.642731	+	0.766092i	$0.277801\pi$
$224$	0	0
$225$	−3.77685e10	−0.982446
$226$	0	0
$227$	3.37702e10i	0.844146i	0.906562	+	0.422073i	$0.138697\pi$
$227$	3.37702e10i	0.844146i	−0.906562	+	0.422073i	$0.861303\pi$
$228$	0	0
$229$	7.28989e9i	0.175171i	0.996157	+	0.0875854i	$0.0279151\pi$
$229$	7.28989e9i	0.175171i	−0.996157	+	0.0875854i	$0.972085\pi$
$230$	0	0
$231$	−2.44074e10	−0.563984
$232$	0	0
$233$	−6.79739e10	−1.51092	−0.755458	−	0.655197i	$-0.772586\pi$
$233$	−6.79739e10	−1.51092	−0.755458	+	0.655197i	$0.772586\pi$
$234$	0	0
$235$	3.83362e10i	0.819980i
$236$	0	0
$237$	− 1.61811e10i	− 0.333151i
$238$	0	0
$239$	−3.11283e10	−0.617114	−0.308557	−	0.951206i	$-0.599846\pi$
$239$	−3.11283e10	−0.617114	−0.308557	+	0.951206i	$0.599846\pi$
$240$	0	0
$241$	1.42372e10	0.271861	0.135931	−	0.990718i	$-0.456598\pi$
$241$	1.42372e10	0.271861	0.135931	+	0.990718i	$0.456598\pi$
$242$	0	0
$243$	5.35072e10i	0.984428i
$244$	0	0
$245$	4.45563e10i	0.790063i
$246$	0	0
$247$	−6.77614e9	−0.115837
$248$	0	0
$249$	−1.36220e10	−0.224565
$250$	0	0
$251$	− 5.78389e10i	− 0.919789i	−0.887974	−	0.459894i	$-0.847887\pi$
$251$	− 5.78389e10i	− 0.919789i	0.887974	−	0.459894i	$-0.152113\pi$
$252$	0	0
$253$	6.67622e10i	1.02444i
$254$	0	0
$255$	−3.97708e10	−0.589024
$256$	0	0
$257$	−1.87176e10	−0.267641	−0.133820	−	0.991006i	$-0.542724\pi$
$257$	−1.87176e10	−0.267641	−0.133820	+	0.991006i	$0.542724\pi$
$258$	0	0
$259$	− 7.90510e10i	− 1.09159i
$260$	0	0
$261$	− 3.33857e10i	− 0.445326i
$262$	0	0
$263$	2.80437e10	0.361439	0.180719	−	0.983535i	$-0.442157\pi$
$263$	2.80437e10	0.361439	0.180719	+	0.983535i	$0.442157\pi$
$264$	0	0
$265$	2.12705e10	0.264954
$266$	0	0
$267$	− 4.32850e9i	− 0.0521238i
$268$	0	0
$269$	4.46600e10i	0.520036i	0.965604	+	0.260018i	$0.0837285\pi$
$269$	4.46600e10i	0.520036i	−0.965604	+	0.260018i	$0.916271\pi$
$270$	0	0
$271$	−1.03375e11	−1.16427	−0.582137	−	0.813090i	$-0.697783\pi$
$271$	−1.03375e11	−1.16427	−0.582137	+	0.813090i	$0.697783\pi$
$272$	0	0
$273$	−3.19075e9	−0.0347665
$274$	0	0
$275$	− 2.19909e11i	− 2.31871i
$276$	0	0
$277$	1.81403e11i	1.85133i	0.378341	+	0.925666i	$0.376495\pi$
$277$	1.81403e11i	1.85133i	−0.378341	+	0.925666i	$0.623505\pi$
$278$	0	0
$279$	−1.03260e11	−1.02027
$280$	0	0
$281$	1.25487e11	1.20066	0.600332	−	0.799751i	$-0.295035\pi$
$281$	1.25487e11	1.20066	0.600332	+	0.799751i	$0.295035\pi$
$282$	0	0
$283$	1.33561e11i	1.23777i	0.785481	+	0.618886i	$0.212416\pi$
$283$	1.33561e11i	1.23777i	−0.785481	+	0.618886i	$0.787584\pi$
$284$	0	0
$285$	6.88795e10i	0.618427i
$286$	0	0
$287$	−3.92430e10	−0.341423
$288$	0	0
$289$	−1.64450e10	−0.138673
$290$	0	0
$291$	1.37266e10i	0.112213i
$292$	0	0
$293$	− 3.50635e9i	− 0.0277940i	−0.999903	−	0.0138970i	$-0.995576\pi$
$293$	− 3.50635e9i	− 0.0277940i	0.999903	−	0.0138970i	$-0.00442369\pi$
$294$	0	0
$295$	2.52336e11	1.93991
$296$	0	0
$297$	−2.00956e11	−1.49864
$298$	0	0
$299$	8.72776e9i	0.0631513i
$300$	0	0
$301$	8.51195e10i	0.597695i
$302$	0	0
$303$	4.81954e10	0.328483
$304$	0	0
$305$	9.52981e10	0.630573
$306$	0	0
$307$	2.94357e11i	1.89126i	0.325246	+	0.945629i	$0.394553\pi$
$307$	2.94357e11i	1.89126i	−0.325246	+	0.945629i	$0.605447\pi$
$308$	0	0
$309$	− 4.69035e10i	− 0.292680i
$310$	0	0
$311$	−2.40305e10	−0.145660	−0.0728301	−	0.997344i	$-0.523203\pi$
$311$	−2.40305e10	−0.145660	−0.0728301	+	0.997344i	$0.523203\pi$
$312$	0	0
$313$	2.55229e11	1.50308	0.751539	−	0.659689i	$-0.229312\pi$
$313$	2.55229e11	1.50308	0.751539	+	0.659689i	$0.229312\pi$
$314$	0	0
$315$	− 1.44899e11i	− 0.829217i
$316$	0	0
$317$	− 2.30255e11i	− 1.28069i	−0.768089	−	0.640343i	$-0.778792\pi$
$317$	− 2.30255e11i	− 1.28069i	0.768089	−	0.640343i	$-0.221208\pi$
$318$	0	0
$319$	1.94390e11	1.05103
$320$	0	0
$321$	6.04535e10	0.317796
$322$	0	0
$323$	− 1.76903e11i	− 0.904322i
$324$	0	0
$325$	− 2.87485e10i	− 0.142936i
$326$	0	0
$327$	−2.88415e10	−0.139493
$328$	0	0
$329$	−8.02953e10	−0.377840
$330$	0	0
$331$	− 1.21212e11i	− 0.555035i	−0.960721	−	0.277518i	$-0.910488\pi$
$331$	− 1.21212e11i	− 0.555035i	0.960721	−	0.277518i	$-0.0895117\pi$
$332$	0	0
$333$	− 2.92674e11i	− 1.30432i
$334$	0	0
$335$	−1.04810e11	−0.454677
$336$	0	0
$337$	2.52249e11	1.06536	0.532678	−	0.846318i	$-0.321186\pi$
$337$	2.52249e11	1.06536	0.532678	+	0.846318i	$0.321186\pi$
$338$	0	0
$339$	− 1.73869e11i	− 0.715027i
$340$	0	0
$341$	− 6.01236e11i	− 2.40797i
$342$	0	0
$343$	−2.68619e11	−1.04789
$344$	0	0
$345$	8.87178e10	0.337151
$346$	0	0
$347$	2.99996e11i	1.11079i	0.831585	+	0.555397i	$0.187434\pi$
$347$	2.99996e11i	1.11079i	−0.831585	+	0.555397i	$0.812566\pi$
$348$	0	0
$349$	1.25625e11i	0.453275i	0.973979	+	0.226638i	$0.0727733\pi$
$349$	1.25625e11i	0.453275i	−0.973979	+	0.226638i	$0.927227\pi$
$350$	0	0
$351$	−2.62708e10	−0.0923830
$352$	0	0
$353$	−4.31672e11	−1.47968	−0.739841	−	0.672782i	$-0.765099\pi$
$353$	−4.31672e11	−1.47968	−0.739841	+	0.672782i	$0.765099\pi$
$354$	0	0
$355$	5.53851e11i	1.85082i
$356$	0	0
$357$	− 8.33000e10i	− 0.271418i
$358$	0	0
$359$	−1.83615e11	−0.583421	−0.291711	−	0.956507i	$-0.594224\pi$
$359$	−1.83615e11	−0.583421	−0.291711	+	0.956507i	$0.594224\pi$
$360$	0	0
$361$	1.63077e10	0.0505372
$362$	0	0
$363$	− 3.84675e11i	− 1.16282i
$364$	0	0
$365$	− 3.65467e11i	− 1.07778i
$366$	0	0
$367$	3.77185e11	1.08532	0.542659	−	0.839953i	$-0.317418\pi$
$367$	3.77185e11	1.08532	0.542659	+	0.839953i	$0.317418\pi$
$368$	0	0
$369$	−1.45291e11	−0.407963
$370$	0	0
$371$	4.45510e10i	0.122089i
$372$	0	0
$373$	2.69400e11i	0.720623i	0.932832	+	0.360312i	$0.117330\pi$
$373$	2.69400e11i	0.720623i	−0.932832	+	0.360312i	$0.882670\pi$
$374$	0	0
$375$	−4.91819e10	−0.128430
$376$	0	0
$377$	2.54124e10	0.0647903
$378$	0	0
$379$	− 2.04102e11i	− 0.508124i	−0.967188	−	0.254062i	$-0.918233\pi$
$379$	− 2.04102e11i	− 0.508124i	0.967188	−	0.254062i	$-0.0817668\pi$
$380$	0	0
$381$	− 2.54598e11i	− 0.619003i
$382$	0	0
$383$	−4.10631e10	−0.0975117	−0.0487559	−	0.998811i	$-0.515526\pi$
$383$	−4.10631e10	−0.0975117	−0.0487559	+	0.998811i	$0.515526\pi$
$384$	0	0
$385$	8.43681e11	1.95706
$386$	0	0
$387$	3.15142e11i	0.714179i
$388$	0	0
$389$	− 2.86342e11i	− 0.634032i	−0.948420	−	0.317016i	$-0.897319\pi$
$389$	− 2.86342e11i	− 0.634032i	0.948420	−	0.317016i	$-0.102681\pi$
$390$	0	0
$391$	−2.27853e11	−0.493014
$392$	0	0
$393$	1.73837e11	0.367599
$394$	0	0
$395$	5.59328e11i	1.15606i
$396$	0	0
$397$	− 3.73016e11i	− 0.753651i	−0.926284	−	0.376826i	$-0.877016\pi$
$397$	− 3.73016e11i	− 0.753651i	0.926284	−	0.376826i	$-0.122984\pi$
$398$	0	0
$399$	−1.44268e11	−0.284966
$400$	0	0
$401$	4.70676e11	0.909018	0.454509	−	0.890742i	$-0.349815\pi$
$401$	4.70676e11	0.909018	0.454509	+	0.890742i	$0.349815\pi$
$402$	0	0
$403$	− 7.85991e10i	− 0.148438i
$404$	0	0
$405$	− 3.89506e11i	− 0.719393i
$406$	0	0
$407$	1.70411e12	3.07838
$408$	0	0
$409$	8.60520e11	1.52057	0.760284	−	0.649590i	$-0.225060\pi$
$409$	8.60520e11	1.52057	0.760284	+	0.649590i	$0.225060\pi$
$410$	0	0
$411$	1.41370e11i	0.244382i
$412$	0	0
$413$	5.28520e11i	0.893894i
$414$	0	0
$415$	4.70866e11	0.779257
$416$	0	0
$417$	1.62916e11	0.263847
$418$	0	0
$419$	− 8.46565e11i	− 1.34183i	−0.741535	−	0.670914i	$-0.765902\pi$
$419$	− 8.46565e11i	− 1.34183i	0.741535	−	0.670914i	$-0.234098\pi$
$420$	0	0
$421$	− 2.27835e11i	− 0.353468i	−0.984259	−	0.176734i	$-0.943447\pi$
$421$	− 2.27835e11i	− 0.353468i	0.984259	−	0.176734i	$-0.0565532\pi$
$422$	0	0
$423$	−2.97281e11	−0.451477
$424$	0	0
$425$	7.50528e11	1.11588
$426$	0	0
$427$	1.99602e11i	0.290563i
$428$	0	0
$429$	− 6.87834e10i	− 0.0980451i
$430$	0	0
$431$	−6.47351e11	−0.903633	−0.451817	−	0.892111i	$-0.649224\pi$
$431$	−6.47351e11	−0.903633	−0.451817	+	0.892111i	$0.649224\pi$
$432$	0	0
$433$	5.69898e11	0.779114	0.389557	−	0.921002i	$-0.372628\pi$
$433$	5.69898e11	0.779114	0.389557	+	0.921002i	$0.372628\pi$
$434$	0	0
$435$	− 2.58317e11i	− 0.345901i
$436$	0	0
$437$	3.94621e11i	0.517624i
$438$	0	0
$439$	−5.98042e11	−0.768496	−0.384248	−	0.923230i	$-0.625539\pi$
$439$	−5.98042e11	−0.768496	−0.384248	+	0.923230i	$0.625539\pi$
$440$	0	0
$441$	−3.45515e11	−0.435005
$442$	0	0
$443$	3.10867e11i	0.383494i	0.981444	+	0.191747i	$0.0614152\pi$
$443$	3.10867e11i	0.383494i	−0.981444	+	0.191747i	$0.938585\pi$
$444$	0	0
$445$	1.49622e11i	0.180873i
$446$	0	0
$447$	−1.00441e11	−0.118995
$448$	0	0
$449$	7.47114e11	0.867517	0.433759	−	0.901029i	$-0.357187\pi$
$449$	7.47114e11	0.867517	0.433759	+	0.901029i	$0.357187\pi$
$450$	0	0
$451$	− 8.45964e11i	− 0.962848i
$452$	0	0
$453$	− 3.19625e11i	− 0.356615i
$454$	0	0
$455$	1.10294e11	0.120642
$456$	0	0
$457$	1.54275e12	1.65452	0.827260	−	0.561819i	$-0.189898\pi$
$457$	1.54275e12	1.65452	0.827260	+	0.561819i	$0.189898\pi$
$458$	0	0
$459$	− 6.85845e11i	− 0.721221i
$460$	0	0
$461$	− 1.62766e12i	− 1.67846i	−0.543777	−	0.839230i	$-0.683006\pi$
$461$	− 1.62766e12i	− 1.67846i	0.543777	−	0.839230i	$-0.316994\pi$
$462$	0	0
$463$	−1.11591e12	−1.12854	−0.564268	−	0.825592i	$-0.690842\pi$
$463$	−1.11591e12	−1.12854	−0.564268	+	0.825592i	$0.690842\pi$
$464$	0	0
$465$	−7.98961e11	−0.792478
$466$	0	0
$467$	5.30194e11i	0.515832i	0.966167	+	0.257916i	$0.0830358\pi$
$467$	5.30194e11i	0.515832i	−0.966167	+	0.257916i	$0.916964\pi$
$468$	0	0
$469$	− 2.19526e11i	− 0.209512i
$470$	0	0
$471$	−6.95008e11	−0.650722
$472$	0	0
$473$	−1.83493e12	−1.68556
$474$	0	0
$475$	− 1.29985e12i	− 1.17158i
$476$	0	0
$477$	1.64943e11i	0.145882i
$478$	0	0
$479$	−2.10019e12	−1.82284	−0.911422	−	0.411473i	$-0.865014\pi$
$479$	−2.10019e12	−1.82284	−0.911422	+	0.411473i	$0.865014\pi$
$480$	0	0
$481$	2.22777e11	0.189766
$482$	0	0
$483$	1.85820e11i	0.155357i
$484$	0	0
$485$	− 4.74483e11i	− 0.389388i
$486$	0	0
$487$	1.05307e12	0.848351	0.424176	−	0.905580i	$-0.360564\pi$
$487$	1.05307e12	0.848351	0.424176	+	0.905580i	$0.360564\pi$
$488$	0	0
$489$	5.69051e10	0.0450050
$490$	0	0
$491$	2.10556e12i	1.63494i	0.575971	+	0.817470i	$0.304624\pi$
$491$	2.10556e12i	1.63494i	−0.575971	+	0.817470i	$0.695376\pi$
$492$	0	0
$493$	6.63434e11i	0.505809i
$494$	0	0
$495$	3.12360e12	2.33847
$496$	0	0
$497$	−1.16004e12	−0.852845
$498$	0	0
$499$	− 2.88807e11i	− 0.208523i	−0.994550	−	0.104262i	$-0.966752\pi$
$499$	− 2.88807e11i	− 0.208523i	0.994550	−	0.104262i	$-0.0332479\pi$
$500$	0	0
$501$	8.68305e11i	0.615747i
$502$	0	0
$503$	−5.17681e11	−0.360584	−0.180292	−	0.983613i	$-0.557704\pi$
$503$	−5.17681e11	−0.360584	−0.180292	+	0.983613i	$0.557704\pi$
$504$	0	0
$505$	−1.66595e12	−1.13986
$506$	0	0
$507$	6.27278e11i	0.421623i
$508$	0	0
$509$	− 6.01747e11i	− 0.397360i	−0.980064	−	0.198680i	$-0.936335\pi$
$509$	− 6.01747e11i	− 0.397360i	0.980064	−	0.198680i	$-0.0636654\pi$
$510$	0	0
$511$	7.65471e11	0.496632
$512$	0	0
$513$	−1.18782e12	−0.757223
$514$	0	0
$515$	1.62130e12i	1.01562i
$516$	0	0
$517$	− 1.73093e12i	− 1.06555i
$518$	0	0
$519$	−8.39316e11	−0.507776
$520$	0	0
$521$	1.67285e11	0.0994691	0.0497345	−	0.998762i	$-0.484162\pi$
$521$	1.67285e11	0.0994691	0.0497345	+	0.998762i	$0.484162\pi$
$522$	0	0
$523$	− 1.57966e12i	− 0.923222i	−0.887083	−	0.461611i	$-0.847272\pi$
$523$	− 1.57966e12i	− 0.923222i	0.887083	−	0.461611i	$-0.152728\pi$
$524$	0	0
$525$	− 6.12074e11i	− 0.351631i
$526$	0	0
$527$	2.05196e12	1.15884
$528$	0	0
$529$	−1.29287e12	−0.717804
$530$	0	0
$531$	1.95676e12i	1.06810i
$532$	0	0
$533$	− 1.10592e11i	− 0.0593543i
$534$	0	0
$535$	−2.08968e12	−1.10277
$536$	0	0
$537$	−2.72954e11	−0.141646
$538$	0	0
$539$	− 2.01178e12i	− 1.02667i
$540$	0	0
$541$	3.08736e12i	1.54953i	0.632250	+	0.774765i	$0.282132\pi$
$541$	3.08736e12i	1.54953i	−0.632250	+	0.774765i	$0.717868\pi$
$542$	0	0
$543$	−9.38904e11	−0.463470
$544$	0	0
$545$	9.96955e11	0.484051
$546$	0	0
$547$	− 2.62136e11i	− 0.125194i	−0.998039	−	0.0625969i	$-0.980062\pi$
$547$	− 2.62136e11i	− 0.125194i	0.998039	−	0.0625969i	$-0.0199383\pi$
$548$	0	0
$549$	7.38997e11i	0.347190i
$550$	0	0
$551$	1.14901e12	0.531057
$552$	0	0
$553$	−1.17151e12	−0.532702
$554$	0	0
$555$	− 2.26453e12i	− 1.01312i
$556$	0	0
$557$	− 3.64238e11i	− 0.160338i	−0.996781	−	0.0801691i	$-0.974454\pi$
$557$	− 3.64238e11i	− 0.160338i	0.996781	−	0.0801691i	$-0.0255460\pi$
$558$	0	0
$559$	−2.39879e11	−0.103906
$560$	0	0
$561$	1.79571e12	0.765425
$562$	0	0
$563$	− 3.04052e12i	− 1.27544i	−0.770267	−	0.637721i	$-0.779877\pi$
$563$	− 3.04052e12i	− 1.27544i	0.770267	−	0.637721i	$-0.220123\pi$
$564$	0	0
$565$	6.01006e12i	2.48119i
$566$	0	0
$567$	8.15821e11	0.331491
$568$	0	0
$569$	7.35845e11	0.294294	0.147147	−	0.989115i	$-0.452991\pi$
$569$	7.35845e11	0.294294	0.147147	+	0.989115i	$0.452991\pi$
$570$	0	0
$571$	1.44618e12i	0.569324i	0.958628	+	0.284662i	$0.0918814\pi$
$571$	1.44618e12i	0.569324i	−0.958628	+	0.284662i	$0.908119\pi$
$572$	0	0
$573$	1.21231e12i	0.469806i
$574$	0	0
$575$	−1.67422e12	−0.638717
$576$	0	0
$577$	−2.26945e12	−0.852371	−0.426186	−	0.904636i	$-0.640143\pi$
$577$	−2.26945e12	−0.852371	−0.426186	+	0.904636i	$0.640143\pi$
$578$	0	0
$579$	− 4.26496e11i	− 0.157711i
$580$	0	0
$581$	9.86229e11i	0.359075i
$582$	0	0
$583$	−9.60391e11	−0.344302
$584$	0	0
$585$	4.08346e11	0.144154
$586$	0	0
$587$	3.41977e12i	1.18885i	0.804153	+	0.594423i	$0.202619\pi$
$587$	3.41977e12i	1.18885i	−0.804153	+	0.594423i	$0.797381\pi$
$588$	0	0
$589$	− 3.55382e12i	− 1.21668i
$590$	0	0
$591$	−1.35554e12	−0.457056
$592$	0	0
$593$	−1.32482e12	−0.439959	−0.219979	−	0.975505i	$-0.570599\pi$
$593$	−1.32482e12	−0.439959	−0.219979	+	0.975505i	$0.570599\pi$
$594$	0	0
$595$	2.87940e12i	0.941838i
$596$	0	0
$597$	− 2.13303e12i	− 0.687248i
$598$	0	0
$599$	4.19936e12	1.33279	0.666395	−	0.745599i	$-0.267836\pi$
$599$	4.19936e12	1.33279	0.666395	+	0.745599i	$0.267836\pi$
$600$	0	0
$601$	−1.05682e12	−0.330418	−0.165209	−	0.986259i	$-0.552830\pi$
$601$	−1.05682e12	−0.330418	−0.165209	+	0.986259i	$0.552830\pi$
$602$	0	0
$603$	− 8.12761e11i	− 0.250343i
$604$	0	0
$605$	1.32969e13i	4.03508i
$606$	0	0
$607$	5.97096e12	1.78523	0.892617	−	0.450816i	$-0.148867\pi$
$607$	5.97096e12	1.78523	0.892617	+	0.450816i	$0.148867\pi$
$608$	0	0
$609$	5.41046e11	0.159388
$610$	0	0
$611$	− 2.26283e11i	− 0.0656851i
$612$	0	0
$613$	2.80650e12i	0.802774i	0.915909	+	0.401387i	$0.131472\pi$
$613$	2.80650e12i	0.802774i	−0.915909	+	0.401387i	$0.868528\pi$
$614$	0	0
$615$	−1.12417e12	−0.316879
$616$	0	0
$617$	−1.48302e12	−0.411968	−0.205984	−	0.978555i	$-0.566039\pi$
$617$	−1.48302e12	−0.411968	−0.205984	+	0.978555i	$0.566039\pi$
$618$	0	0
$619$	1.53469e12i	0.420158i	0.977684	+	0.210079i	$0.0673721\pi$
$619$	1.53469e12i	0.420158i	−0.977684	+	0.210079i	$0.932628\pi$
$620$	0	0
$621$	1.52993e12i	0.412819i
$622$	0	0
$623$	−3.13383e11	−0.0833450
$624$	0	0
$625$	−2.88657e12	−0.756696
$626$	0	0
$627$	− 3.11001e12i	− 0.803632i
$628$	0	0
$629$	5.81597e12i	1.48147i
$630$	0	0
$631$	4.43498e12	1.11368	0.556839	−	0.830620i	$-0.312014\pi$
$631$	4.43498e12	1.11368	0.556839	+	0.830620i	$0.312014\pi$
$632$	0	0
$633$	−3.35088e11	−0.0829549
$634$	0	0
$635$	8.80061e12i	2.14798i
$636$	0	0
$637$	− 2.62998e11i	− 0.0632886i
$638$	0	0
$639$	−4.29488e12	−1.01905
$640$	0	0
$641$	−4.56257e12	−1.06745	−0.533725	−	0.845658i	$-0.679208\pi$
$641$	−4.56257e12	−1.06745	−0.533725	+	0.845658i	$0.679208\pi$
$642$	0	0
$643$	3.32818e12i	0.767818i	0.923371	+	0.383909i	$0.125422\pi$
$643$	3.32818e12i	0.767818i	−0.923371	+	0.383909i	$0.874578\pi$
$644$	0	0
$645$	2.43837e12i	0.554729i
$646$	0	0
$647$	−2.31374e12	−0.519093	−0.259547	−	0.965731i	$-0.583573\pi$
$647$	−2.31374e12	−0.519093	−0.259547	+	0.965731i	$0.583573\pi$
$648$	0	0
$649$	−1.13933e13	−2.52087
$650$	0	0
$651$	− 1.67343e12i	− 0.365167i
$652$	0	0
$653$	− 7.03697e12i	− 1.51453i	−0.653110	−	0.757263i	$-0.726536\pi$
$653$	− 7.03697e12i	− 1.51453i	0.653110	−	0.757263i	$-0.273464\pi$
$654$	0	0
$655$	−6.00895e12	−1.27559
$656$	0	0
$657$	2.83404e12	0.593419
$658$	0	0
$659$	− 2.20320e12i	− 0.455060i	−0.973771	−	0.227530i	$-0.926935\pi$
$659$	− 2.20320e12i	− 0.455060i	0.973771	−	0.227530i	$-0.0730650\pi$
$660$	0	0
$661$	− 7.29570e12i	− 1.48648i	−0.669022	−	0.743242i	$-0.733287\pi$
$661$	− 7.29570e12i	− 1.48648i	0.669022	−	0.743242i	$-0.266713\pi$
$662$	0	0
$663$	2.34751e11	0.0471842
$664$	0	0
$665$	4.98688e12	0.988852
$666$	0	0
$667$	− 1.47994e12i	− 0.289519i
$668$	0	0
$669$	2.84828e12i	0.549749i
$670$	0	0
$671$	−4.30284e12	−0.819416
$672$	0	0
$673$	−4.47079e12	−0.840073	−0.420036	−	0.907507i	$-0.637983\pi$
$673$	−4.47079e12	−0.840073	−0.420036	+	0.907507i	$0.637983\pi$
$674$	0	0
$675$	− 5.03947e12i	− 0.934367i
$676$	0	0
$677$	− 3.42095e12i	− 0.625890i	−0.949771	−	0.312945i	$-0.898684\pi$
$677$	− 3.42095e12i	− 0.625890i	0.949771	−	0.312945i	$-0.101316\pi$
$678$	0	0
$679$	9.93805e11	0.179427
$680$	0	0
$681$	−2.02621e12	−0.361013
$682$	0	0
$683$	− 9.18730e12i	− 1.61546i	−0.589556	−	0.807728i	$-0.700697\pi$
$683$	− 9.18730e12i	− 1.61546i	0.589556	−	0.807728i	$-0.299303\pi$
$684$	0	0
$685$	− 4.88670e12i	− 0.848024i
$686$	0	0
$687$	−4.37394e11	−0.0749147
$688$	0	0
$689$	−1.25551e11	−0.0212243
$690$	0	0
$691$	1.88811e12i	0.315047i	0.987515	+	0.157524i	$0.0503511\pi$
$691$	1.88811e12i	0.315047i	−0.987515	+	0.157524i	$0.949649\pi$
$692$	0	0
$693$	6.54240e12i	1.07755i
$694$	0	0
$695$	−5.63148e12	−0.915568
$696$	0	0
$697$	2.88720e12	0.463371
$698$	0	0
$699$	− 4.07843e12i	− 0.646169i
$700$	0	0
$701$	− 1.61907e12i	− 0.253242i	−0.991951	−	0.126621i	$-0.959587\pi$
$701$	− 1.61907e12i	− 0.253242i	0.991951	−	0.126621i	$-0.0404131\pi$
$702$	0	0
$703$	1.00727e13	1.55542
$704$	0	0
$705$	−2.30017e12	−0.350678
$706$	0	0
$707$	− 3.48934e12i	− 0.525238i
$708$	0	0
$709$	− 1.06375e13i	− 1.58099i	−0.612466	−	0.790497i	$-0.709822\pi$
$709$	− 1.06375e13i	− 1.58099i	0.612466	−	0.790497i	$-0.290178\pi$
$710$	0	0
$711$	−4.33735e12	−0.636520
$712$	0	0
$713$	−4.57737e12	−0.663305
$714$	0	0
$715$	2.37761e12i	0.340223i
$716$	0	0
$717$	− 1.86770e12i	− 0.263919i
$718$	0	0
$719$	1.32770e13	1.85276	0.926380	−	0.376589i	$-0.122903\pi$
$719$	1.32770e13	1.85276	0.926380	+	0.376589i	$0.122903\pi$
$720$	0	0
$721$	−3.39582e12	−0.467989
$722$	0	0
$723$	8.54230e11i	0.116266i
$724$	0	0
$725$	4.87480e12i	0.655293i
$726$	0	0
$727$	−2.60017e12	−0.345221	−0.172611	−	0.984990i	$-0.555220\pi$
$727$	−2.60017e12	−0.345221	−0.172611	+	0.984990i	$0.555220\pi$
$728$	0	0
$729$	4.86117e11	0.0637481
$730$	0	0
$731$	− 6.26244e12i	− 0.811176i
$732$	0	0
$733$	1.14818e13i	1.46906i	0.678574	+	0.734532i	$0.262598\pi$
$733$	1.14818e13i	1.46906i	−0.678574	+	0.734532i	$0.737402\pi$
$734$	0	0
$735$	−2.67338e12	−0.337884
$736$	0	0
$737$	4.73234e12	0.590843
$738$	0	0
$739$	7.75984e12i	0.957090i	0.878063	+	0.478545i	$0.158836\pi$
$739$	7.75984e12i	0.957090i	−0.878063	+	0.478545i	$0.841164\pi$
$740$	0	0
$741$	− 4.06569e11i	− 0.0495395i
$742$	0	0
$743$	2.58115e12	0.310717	0.155358	−	0.987858i	$-0.450347\pi$
$743$	2.58115e12	0.310717	0.155358	+	0.987858i	$0.450347\pi$
$744$	0	0
$745$	3.47191e12	0.412920
$746$	0	0
$747$	3.65136e12i	0.429055i
$748$	0	0
$749$	− 4.37683e12i	− 0.508150i
$750$	0	0
$751$	−8.39208e12	−0.962697	−0.481349	−	0.876529i	$-0.659853\pi$
$751$	−8.39208e12	−0.962697	−0.481349	+	0.876529i	$0.659853\pi$
$752$	0	0
$753$	3.47033e12	0.393363
$754$	0	0
$755$	1.10484e13i	1.23748i
$756$	0	0
$757$	8.15875e12i	0.903009i	0.892269	+	0.451505i	$0.149113\pi$
$757$	8.15875e12i	0.903009i	−0.892269	+	0.451505i	$0.850887\pi$
$758$	0	0
$759$	−4.00573e12	−0.438121
$760$	0	0
$761$	6.27433e12	0.678167	0.339083	−	0.940756i	$-0.389883\pi$
$761$	6.27433e12	0.678167	0.339083	+	0.940756i	$0.389883\pi$
$762$	0	0
$763$	2.08813e12i	0.223047i
$764$	0	0
$765$	1.06606e13i	1.12539i
$766$	0	0
$767$	−1.48944e12	−0.155398
$768$	0	0
$769$	6.12027e12	0.631106	0.315553	−	0.948908i	$-0.397810\pi$
$769$	6.12027e12	0.631106	0.315553	+	0.948908i	$0.397810\pi$
$770$	0	0
$771$	− 1.12306e12i	− 0.114461i
$772$	0	0
$773$	− 6.62875e12i	− 0.667765i	−0.942615	−	0.333883i	$-0.891641\pi$
$773$	− 6.62875e12i	− 0.667765i	0.942615	−	0.333883i	$-0.108359\pi$
$774$	0	0
$775$	1.50775e13	1.50131
$776$	0	0
$777$	4.74306e12	0.466836
$778$	0	0
$779$	− 5.00037e12i	− 0.486501i
$780$	0	0
$781$	− 2.50071e13i	− 2.40511i
$782$	0	0
$783$	4.45467e12	0.423533
$784$	0	0
$785$	2.40241e13	2.25805
$786$	0	0
$787$	− 1.19503e13i	− 1.11043i	−0.831706	−	0.555216i	$-0.812636\pi$
$787$	− 1.19503e13i	− 1.11043i	0.831706	−	0.555216i	$-0.187364\pi$
$788$	0	0
$789$	1.68262e12i	0.154575i
$790$	0	0
$791$	−1.25881e13	−1.14331
$792$	0	0
$793$	−5.62507e11	−0.0505125
$794$	0	0
$795$	1.27623e12i	0.113312i
$796$	0	0
$797$	1.18887e13i	1.04369i	0.853041	+	0.521844i	$0.174756\pi$
$797$	1.18887e13i	1.04369i	−0.853041	+	0.521844i	$0.825244\pi$
$798$	0	0
$799$	5.90751e12	0.512795
$800$	0	0
$801$	−1.16025e12	−0.0995879
$802$	0	0
$803$	1.65013e13i	1.40055i
$804$	0	0
$805$	− 6.42317e12i	− 0.539098i
$806$	0	0
$807$	−2.67960e12	−0.222402
$808$	0	0
$809$	1.68063e13	1.37944	0.689722	−	0.724074i	$-0.257733\pi$
$809$	1.68063e13	1.37944	0.689722	+	0.724074i	$0.257733\pi$
$810$	0	0
$811$	1.98473e13i	1.61104i	0.592567	+	0.805521i	$0.298114\pi$
$811$	1.98473e13i	1.61104i	−0.592567	+	0.805521i	$0.701886\pi$
$812$	0	0
$813$	− 6.20252e12i	− 0.497922i
$814$	0	0
$815$	−1.96702e12	−0.156171
$816$	0	0
$817$	−1.08460e13	−0.851668
$818$	0	0
$819$	8.55282e11i	0.0664250i
$820$	0	0
$821$	4.83992e12i	0.371787i	0.982570	+	0.185893i	$0.0595179\pi$
$821$	4.83992e12i	0.371787i	−0.982570	+	0.185893i	$0.940482\pi$
$822$	0	0
$823$	−8.41664e12	−0.639499	−0.319749	−	0.947502i	$-0.603599\pi$
$823$	−8.41664e12	−0.639499	−0.319749	+	0.947502i	$0.603599\pi$
$824$	0	0
$825$	1.31945e13	0.991634
$826$	0	0
$827$	− 2.16658e13i	− 1.61064i	−0.592839	−	0.805321i	$-0.701993\pi$
$827$	− 2.16658e13i	− 1.61064i	0.592839	−	0.805321i	$-0.298007\pi$
$828$	0	0
$829$	3.67734e12i	0.270420i	0.990817	+	0.135210i	$0.0431709\pi$
$829$	3.67734e12i	0.270420i	−0.990817	+	0.135210i	$0.956829\pi$
$830$	0	0
$831$	−1.08842e13	−0.791754
$832$	0	0
$833$	6.86601e12	0.494085
$834$	0	0
$835$	− 3.00144e13i	− 2.13669i
$836$	0	0
$837$	− 1.37780e13i	− 0.970337i
$838$	0	0
$839$	1.46942e13	1.02381	0.511903	−	0.859043i	$-0.328941\pi$
$839$	1.46942e13	1.02381	0.511903	+	0.859043i	$0.328941\pi$
$840$	0	0
$841$	1.01980e13	0.702967
$842$	0	0
$843$	7.52924e12i	0.513484i
$844$	0	0
$845$	− 2.16829e13i	− 1.46306i
$846$	0	0
$847$	−2.78505e13	−1.85933
$848$	0	0
$849$	−8.01366e12	−0.529354
$850$	0	0
$851$	− 1.29738e13i	− 0.847979i
$852$	0	0
$853$	− 1.99845e13i	− 1.29248i	−0.763136	−	0.646238i	$-0.776341\pi$
$853$	− 1.99845e13i	− 1.29248i	0.763136	−	0.646238i	$-0.223659\pi$
$854$	0	0
$855$	1.84632e13	1.18157
$856$	0	0
$857$	−2.11989e13	−1.34245	−0.671226	−	0.741252i	$-0.734232\pi$
$857$	−2.11989e13	−1.34245	−0.671226	+	0.741252i	$0.734232\pi$
$858$	0	0
$859$	2.51809e13i	1.57798i	0.614404	+	0.788992i	$0.289397\pi$
$859$	2.51809e13i	1.57798i	−0.614404	+	0.788992i	$0.710603\pi$
$860$	0	0
$861$	− 2.35458e12i	− 0.146016i
$862$	0	0
$863$	2.15905e13	1.32500	0.662498	−	0.749064i	$-0.269496\pi$
$863$	2.15905e13	1.32500	0.662498	+	0.749064i	$0.269496\pi$
$864$	0	0
$865$	2.90123e13	1.76202
$866$	0	0
$867$	− 9.86700e11i	− 0.0593061i
$868$	0	0
$869$	− 2.52544e13i	− 1.50227i
$870$	0	0
$871$	6.18655e11	0.0364222
$872$	0	0
$873$	3.67941e12	0.214395
$874$	0	0
$875$	3.56077e12i	0.205356i
$876$	0	0
$877$	− 2.63182e13i	− 1.50230i	−0.660129	−	0.751152i	$-0.729498\pi$
$877$	− 2.63182e13i	− 1.50230i	0.660129	−	0.751152i	$-0.270502\pi$
$878$	0	0
$879$	2.10381e11	0.0118866
$880$	0	0
$881$	4.23513e12	0.236851	0.118425	−	0.992963i	$-0.462215\pi$
$881$	4.23513e12	0.236851	0.118425	+	0.992963i	$0.462215\pi$
$882$	0	0
$883$	2.56557e13i	1.42024i	0.704081	+	0.710119i	$0.251359\pi$
$883$	2.56557e13i	1.42024i	−0.704081	+	0.710119i	$0.748641\pi$
$884$	0	0
$885$	1.51402e13i	0.829634i
$886$	0	0
$887$	3.14044e13	1.70347	0.851735	−	0.523973i	$-0.175551\pi$
$887$	3.14044e13	1.70347	0.851735	+	0.523973i	$0.175551\pi$
$888$	0	0
$889$	−1.84329e13	−0.989774
$890$	0	0
$891$	1.75867e13i	0.934836i
$892$	0	0
$893$	− 1.02313e13i	− 0.538392i
$894$	0	0
$895$	9.43512e12	0.491523
$896$	0	0
$897$	−5.23666e11	−0.0270077
$898$	0	0
$899$	1.33278e13i	0.680519i
$900$	0	0
$901$	− 3.27772e12i	− 0.165695i
$902$	0	0
$903$	−5.10717e12	−0.255615
$904$	0	0
$905$	3.24548e13	1.60827
$906$	0	0
$907$	− 3.87001e13i	− 1.89880i	−0.314070	−	0.949400i	$-0.601693\pi$
$907$	− 3.87001e13i	− 1.89880i	0.314070	−	0.949400i	$-0.398307\pi$
$908$	0	0
$909$	− 1.29188e13i	− 0.627601i
$910$	0	0
$911$	1.73436e13	0.834269	0.417135	−	0.908845i	$-0.363034\pi$
$911$	1.73436e13	0.834269	0.417135	+	0.908845i	$0.363034\pi$
$912$	0	0
$913$	−2.12602e13	−1.01263
$914$	0	0
$915$	5.71789e12i	0.269675i
$916$	0	0
$917$	− 1.25858e13i	− 0.587784i
$918$	0	0
$919$	1.75232e12	0.0810387	0.0405194	−	0.999179i	$-0.487099\pi$
$919$	1.75232e12	0.0810387	0.0405194	+	0.999179i	$0.487099\pi$
$920$	0	0
$921$	−1.76614e13	−0.808829
$922$	0	0
$923$	− 3.26916e12i	− 0.148262i
$924$	0	0
$925$	4.27347e13i	1.91930i
$926$	0	0
$927$	−1.25725e13	−0.559194
$928$	0	0
$929$	−1.99977e13	−0.880865	−0.440432	−	0.897786i	$-0.645175\pi$
$929$	−1.99977e13	−0.880865	−0.440432	+	0.897786i	$0.645175\pi$
$930$	0	0
$931$	− 1.18913e13i	− 0.518749i
$932$	0	0
$933$	− 1.44183e12i	− 0.0622940i
$934$	0	0
$935$	−6.20716e13	−2.65608
$936$	0	0
$937$	−4.62137e12	−0.195859	−0.0979293	−	0.995193i	$-0.531222\pi$
$937$	−4.62137e12	−0.195859	−0.0979293	+	0.995193i	$0.531222\pi$
$938$	0	0
$939$	1.53138e13i	0.642816i
$940$	0	0
$941$	− 1.64959e13i	− 0.685838i	−0.939365	−	0.342919i	$-0.888584\pi$
$941$	− 1.64959e13i	− 0.685838i	0.939365	−	0.342919i	$-0.111416\pi$
$942$	0	0
$943$	−6.44055e12	−0.265228
$944$	0	0
$945$	1.93339e13	0.788637
$946$	0	0
$947$	− 1.99606e13i	− 0.806490i	−0.915092	−	0.403245i	$-0.867882\pi$
$947$	− 1.99606e13i	− 0.806490i	0.915092	−	0.403245i	$-0.132118\pi$
$948$	0	0
$949$	2.15720e12i	0.0863363i
$950$	0	0
$951$	1.38153e13	0.547707
$952$	0	0
$953$	8.83087e12	0.346805	0.173402	−	0.984851i	$-0.444524\pi$
$953$	8.83087e12	0.346805	0.173402	+	0.984851i	$0.444524\pi$
$954$	0	0
$955$	− 4.19056e13i	− 1.63026i
$956$	0	0
$957$	1.16634e13i	0.449491i
$958$	0	0
$959$	1.02352e13	0.390763
$960$	0	0
$961$	1.47825e13	0.559105
$962$	0	0
$963$	− 1.62046e13i	− 0.607182i
$964$	0	0
$965$	1.47426e13i	0.547268i
$966$	0	0
$967$	−2.03562e13	−0.748647	−0.374323	−	0.927298i	$-0.622125\pi$
$967$	−2.03562e13	−0.748647	−0.374323	+	0.927298i	$0.622125\pi$
$968$	0	0
$969$	1.06142e13	0.386748
$970$	0	0
$971$	7.66888e12i	0.276851i	0.990373	+	0.138425i	$0.0442041\pi$
$971$	7.66888e12i	0.276851i	−0.990373	+	0.138425i	$0.955796\pi$
$972$	0	0
$973$	− 1.17951e13i	− 0.421886i
$974$	0	0
$975$	1.72491e12	0.0611288
$976$	0	0
$977$	6.23017e12	0.218763	0.109382	−	0.994000i	$-0.465113\pi$
$977$	6.23017e12	0.218763	0.109382	+	0.994000i	$0.465113\pi$
$978$	0	0
$979$	− 6.75563e12i	− 0.235041i
$980$	0	0
$981$	7.73097e12i	0.266516i
$982$	0	0
$983$	−5.22800e13	−1.78585	−0.892924	−	0.450207i	$-0.851350\pi$
$983$	−5.22800e13	−1.78585	−0.892924	+	0.450207i	$0.851350\pi$
$984$	0	0
$985$	4.68565e13	1.58601
$986$	0	0
$987$	− 4.81772e12i	− 0.161590i
$988$	0	0
$989$	1.39698e13i	0.464308i
$990$	0	0
$991$	−3.73672e13	−1.23072	−0.615359	−	0.788247i	$-0.710989\pi$
$991$	−3.73672e13	−1.23072	−0.615359	+	0.788247i	$0.710989\pi$
$992$	0	0
$993$	7.27273e12	0.237370
$994$	0	0
$995$	7.37319e13i	2.38480i
$996$	0	0
$997$	3.66230e13i	1.17389i	0.809628	+	0.586943i	$0.199669\pi$
$997$	3.66230e13i	1.17389i	−0.809628	+	0.586943i	$0.800331\pi$
$998$	0	0
$999$	3.90517e13	1.24049

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.10.b.i.129.2		2
4.3	odd	2		256.10.b.c.129.1		2
8.3	odd	2		256.10.b.c.129.2		2
8.5	even	2	inner	256.10.b.i.129.1		2
16.3	odd	4		16.10.a.c.1.1		1
16.5	even	4		64.10.a.f.1.1		1
16.11	odd	4		64.10.a.d.1.1		1
16.13	even	4		8.10.a.a.1.1	✓	1
48.29	odd	4		72.10.a.e.1.1		1
48.35	even	4		144.10.a.n.1.1		1
80.3	even	4		400.10.c.g.49.2		2
80.13	odd	4		200.10.c.b.49.1		2
80.19	odd	4		400.10.a.d.1.1		1
80.29	even	4		200.10.a.b.1.1		1
80.67	even	4		400.10.c.g.49.1		2
80.77	odd	4		200.10.c.b.49.2		2
112.13	odd	4		392.10.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.10.a.a.1.1	✓	1	16.13	even	4
16.10.a.c.1.1		1	16.3	odd	4
64.10.a.d.1.1		1	16.11	odd	4
64.10.a.f.1.1		1	16.5	even	4
72.10.a.e.1.1		1	48.29	odd	4
144.10.a.n.1.1		1	48.35	even	4
200.10.a.b.1.1		1	80.29	even	4
200.10.c.b.49.1		2	80.13	odd	4
200.10.c.b.49.2		2	80.77	odd	4
256.10.b.c.129.1		2	4.3	odd	2
256.10.b.c.129.2		2	8.3	odd	2
256.10.b.i.129.1		2	8.5	even	2	inner
256.10.b.i.129.2		2	1.1	even	1	trivial
392.10.a.b.1.1		1	112.13	odd	4
400.10.a.d.1.1		1	80.19	odd	4
400.10.c.g.49.1		2	80.67	even	4
400.10.c.g.49.2		2	80.3	even	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:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	256.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+60.0000i q^{3} -2074.00i q^{5} +4344.00 q^{7} +16083.0 q^{9} +O(q^{10})\)	\(q+60.0000i q^{3} -2074.00i q^{5} +4344.00 q^{7} +16083.0 q^{9} +93644.0i q^{11} +12242.0i q^{13} +124440. q^{15} -319598. q^{17} +553516. i q^{19} +260640. i q^{21} +712936. q^{23} -2.34835e6 q^{25} +2.14596e6i q^{27} -2.07584e6i q^{29} -6.42045e6 q^{31} -5.61864e6 q^{33} -9.00946e6i q^{35} -1.81978e7i q^{37} -734520. q^{39} -9.03383e6 q^{41} +1.95947e7i q^{43} -3.33561e7i q^{45} -1.84842e7 q^{47} -2.14833e7 q^{49} -1.91759e7i q^{51} +1.02558e7i q^{53} +1.94218e8 q^{55} -3.32110e7 q^{57} +1.21667e8i q^{59} +4.59490e7i q^{61} +6.98646e7 q^{63} +2.53899e7 q^{65} -5.05354e7i q^{67} +4.27762e7i q^{69} -2.67045e8 q^{71} +1.76213e8 q^{73} -1.40901e8i q^{75} +4.06790e8i q^{77} -2.69686e8 q^{79} +1.87804e8 q^{81} +2.27033e8i q^{83} +6.62846e8i q^{85} +1.24550e8 q^{87} -7.21416e7 q^{89} +5.31792e7i q^{91} -3.85227e8i q^{93} +1.14799e9 q^{95} +2.28777e8 q^{97} +1.50608e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8688 q^{7} + 32166 q^{9}+O(q^{10}) \) 2 * q + 8688 * q^7 + 32166 * q^9	\( 2 q + 8688 q^{7} + 32166 q^{9} + 248880 q^{15} - 639196 q^{17} + 1425872 q^{23} - 4696702 q^{25} - 12840896 q^{31} - 11237280 q^{33} - 1469040 q^{39} - 18067668 q^{41} - 36968352 q^{47} - 42966542 q^{49} + 388435312 q^{55} - 66421920 q^{57} + 139729104 q^{63} + 50779816 q^{65} - 534089360 q^{71} + 352426732 q^{73} - 539371360 q^{79} + 375608178 q^{81} + 249100560 q^{87} - 144283188 q^{89} + 2295984368 q^{95} + 457553092 q^{97}+O(q^{100}) \) 2 * q + 8688 * q^7 + 32166 * q^9 + 248880 * q^15 - 639196 * q^17 + 1425872 * q^23 - 4696702 * q^25 - 12840896 * q^31 - 11237280 * q^33 - 1469040 * q^39 - 18067668 * q^41 - 36968352 * q^47 - 42966542 * q^49 + 388435312 * q^55 - 66421920 * q^57 + 139729104 * q^63 + 50779816 * q^65 - 534089360 * q^71 + 352426732 * q^73 - 539371360 * q^79 + 375608178 * q^81 + 249100560 * q^87 - 144283188 * q^89 + 2295984368 * q^95 + 457553092 * q^97

Introduction

L-functions

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