LMFDB - Embedded newform 256.8.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,8,Mod(1,256)]

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, 0]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("256.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	256.a (trivial)

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:	yes
Analytic conductor:	$79.9705665239$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 163x^{4} + 4820x^{2} - 15296 $ x^6 - 163x^4 + 4820x^2 - 15296
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{27} $
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.89807$ of defining polynomial
Character	$\chi$	$=$	256.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-76.9497 q^{3} -338.443 q^{5} +438.996 q^{7} +3734.25 q^{9} +O(q^{10})$	$q-76.9497 q^{3} -338.443 q^{5} +438.996 q^{7} +3734.25 q^{9} -1966.58 q^{11} -2210.98 q^{13} +26043.0 q^{15} -12114.9 q^{17} -32872.2 q^{19} -33780.6 q^{21} -19605.1 q^{23} +36418.4 q^{25} -119060. q^{27} -160689. q^{29} -229270. q^{31} +151328. q^{33} -148575. q^{35} -496284. q^{37} +170134. q^{39} -599971. q^{41} +88346.0 q^{43} -1.26383e6 q^{45} +820344. q^{47} -630825. q^{49} +932236. q^{51} -1.53717e6 q^{53} +665574. q^{55} +2.52950e6 q^{57} +1.82480e6 q^{59} +484582. q^{61} +1.63932e6 q^{63} +748290. q^{65} +79878.2 q^{67} +1.50860e6 q^{69} -1.27078e6 q^{71} -3.70820e6 q^{73} -2.80238e6 q^{75} -863321. q^{77} -2.55846e6 q^{79} +994857. q^{81} -1.53414e6 q^{83} +4.10019e6 q^{85} +1.23650e7 q^{87} -1.99492e6 q^{89} -970612. q^{91} +1.76423e7 q^{93} +1.11253e7 q^{95} -28917.7 q^{97} -7.34370e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 688 q^{7} + 2918 q^{9}+O(q^{10}) $ 6 * q + 688 * q^7 + 2918 * q^9	$ 6 q + 688 q^{7} + 2918 q^{9} + 17872 q^{15} + 1452 q^{17} + 1296 q^{23} + 39314 q^{25} - 89280 q^{31} + 53880 q^{33} + 328208 q^{39} - 521244 q^{41} + 1566432 q^{47} - 511050 q^{49} + 3270256 q^{55} + 1889896 q^{57} + 5776816 q^{63} + 1416480 q^{65} + 7597104 q^{71} - 2089564 q^{73} + 16015904 q^{79} - 723058 q^{81} + 37453776 q^{87} - 2169084 q^{89} + 48537936 q^{95} - 1088308 q^{97}+O(q^{100}) $ 6 * q + 688 * q^7 + 2918 * q^9 + 17872 * q^15 + 1452 * q^17 + 1296 * q^23 + 39314 * q^25 - 89280 * q^31 + 53880 * q^33 + 328208 * q^39 - 521244 * q^41 + 1566432 * q^47 - 511050 * q^49 + 3270256 * q^55 + 1889896 * q^57 + 5776816 * q^63 + 1416480 * q^65 + 7597104 * q^71 - 2089564 * q^73 + 16015904 * q^79 - 723058 * q^81 + 37453776 * q^87 - 2169084 * q^89 + 48537936 * q^95 - 1088308 * q^97

Display 100 coefficients 10 coefficients

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
$3$	−76.9497	−1.64544	−0.822721	−	0.568446i	$-0.807545\pi$
$3$	−76.9497	−1.64544	−0.822721	+	0.568446i	$0.807545\pi$
$4$	0	0
$5$	−338.443	−1.21085	−0.605425	−	0.795903i	$-0.706997\pi$
$5$	−338.443	−1.21085	−0.605425	+	0.795903i	$0.706997\pi$
$6$	0	0
$7$	438.996	0.483746	0.241873	−	0.970308i	$-0.422238\pi$
$7$	438.996	0.483746	0.241873	+	0.970308i	$0.422238\pi$
$8$	0	0
$9$	3734.25	1.70748
$10$	0	0
$11$	−1966.58	−0.445489	−0.222744	−	0.974877i	$-0.571502\pi$
$11$	−1966.58	−0.445489	−0.222744	+	0.974877i	$0.571502\pi$
$12$	0	0
$13$	−2210.98	−0.279115	−0.139557	−	0.990214i	$-0.544568\pi$
$13$	−2210.98	−0.279115	−0.139557	+	0.990214i	$0.544568\pi$
$14$	0	0
$15$	26043.0	1.99238
$16$	0	0
$17$	−12114.9	−0.598064	−0.299032	−	0.954243i	$-0.596664\pi$
$17$	−12114.9	−0.598064	−0.299032	+	0.954243i	$0.596664\pi$
$18$	0	0
$19$	−32872.2	−1.09949	−0.549744	−	0.835333i	$-0.685275\pi$
$19$	−32872.2	−1.09949	−0.549744	+	0.835333i	$0.685275\pi$
$20$	0	0
$21$	−33780.6	−0.795976
$22$	0	0
$23$	−19605.1	−0.335986	−0.167993	−	0.985788i	$-0.553729\pi$
$23$	−19605.1	−0.335986	−0.167993	+	0.985788i	$0.553729\pi$
$24$	0	0
$25$	36418.4	0.466155
$26$	0	0
$27$	−119060.	−1.16411
$28$	0	0
$29$	−160689.	−1.22347	−0.611735	−	0.791063i	$-0.709528\pi$
$29$	−160689.	−1.22347	−0.611735	+	0.791063i	$0.709528\pi$
$30$	0	0
$31$	−229270.	−1.38224	−0.691118	−	0.722742i	$-0.742881\pi$
$31$	−229270.	−1.38224	−0.691118	+	0.722742i	$0.742881\pi$
$32$	0	0
$33$	151328.	0.733026
$34$	0	0
$35$	−148575.	−0.585744
$36$	0	0
$37$	−496284.	−1.61074	−0.805368	−	0.592775i	$-0.798032\pi$
$37$	−496284.	−1.61074	−0.805368	+	0.592775i	$0.798032\pi$
$38$	0	0
$39$	170134.	0.459267
$40$	0	0
$41$	−599971.	−1.35952	−0.679762	−	0.733433i	$-0.737917\pi$
$41$	−599971.	−1.35952	−0.679762	+	0.733433i	$0.737917\pi$
$42$	0	0
$43$	88346.0	0.169452	0.0847262	−	0.996404i	$-0.472998\pi$
$43$	88346.0	0.169452	0.0847262	+	0.996404i	$0.472998\pi$
$44$	0	0
$45$	−1.26383e6	−2.06750
$46$	0	0
$47$	820344.	1.15253	0.576267	−	0.817262i	$-0.304509\pi$
$47$	820344.	1.15253	0.576267	+	0.817262i	$0.304509\pi$
$48$	0	0
$49$	−630825.	−0.765989
$50$	0	0
$51$	932236.	0.984080
$52$	0	0
$53$	−1.53717e6	−1.41826	−0.709131	−	0.705077i	$-0.750913\pi$
$53$	−1.53717e6	−1.41826	−0.709131	+	0.705077i	$0.750913\pi$
$54$	0	0
$55$	665574.	0.539420
$56$	0	0
$57$	2.52950e6	1.80914
$58$	0	0
$59$	1.82480e6	1.15673	0.578367	−	0.815776i	$-0.303690\pi$
$59$	1.82480e6	1.15673	0.578367	+	0.815776i	$0.303690\pi$
$60$	0	0
$61$	484582.	0.273346	0.136673	−	0.990616i	$-0.456359\pi$
$61$	484582.	0.273346	0.136673	+	0.990616i	$0.456359\pi$
$62$	0	0
$63$	1.63932e6	0.825986
$64$	0	0
$65$	748290.	0.337966
$66$	0	0
$67$	79878.2	0.0324464	0.0162232	−	0.999868i	$-0.494836\pi$
$67$	79878.2	0.0324464	0.0162232	+	0.999868i	$0.494836\pi$
$68$	0	0
$69$	1.50860e6	0.552845
$70$	0	0
$71$	−1.27078e6	−0.421373	−0.210686	−	0.977554i	$-0.567570\pi$
$71$	−1.27078e6	−0.421373	−0.210686	+	0.977554i	$0.567570\pi$
$72$	0	0
$73$	−3.70820e6	−1.11566	−0.557832	−	0.829954i	$-0.688367\pi$
$73$	−3.70820e6	−1.11566	−0.557832	+	0.829954i	$0.688367\pi$
$74$	0	0
$75$	−2.80238e6	−0.767031
$76$	0	0
$77$	−863321.	−0.215504
$78$	0	0
$79$	−2.55846e6	−0.583827	−0.291914	−	0.956445i	$-0.594292\pi$
$79$	−2.55846e6	−0.583827	−0.291914	+	0.956445i	$0.594292\pi$
$80$	0	0
$81$	994857.	0.208000
$82$	0	0
$83$	−1.53414e6	−0.294505	−0.147252	−	0.989099i	$-0.547043\pi$
$83$	−1.53414e6	−0.294505	−0.147252	+	0.989099i	$0.547043\pi$
$84$	0	0
$85$	4.10019e6	0.724166
$86$	0	0
$87$	1.23650e7	2.01315
$88$	0	0
$89$	−1.99492e6	−0.299958	−0.149979	−	0.988689i	$-0.547921\pi$
$89$	−1.99492e6	−0.299958	−0.149979	+	0.988689i	$0.547921\pi$
$90$	0	0
$91$	−970612.	−0.135021
$92$	0	0
$93$	1.76423e7	2.27439
$94$	0	0
$95$	1.11253e7	1.33131
$96$	0	0
$97$	−28917.7	−0.00321708	−0.00160854	−	0.999999i	$-0.500512\pi$
$97$	−28917.7	−0.00321708	−0.00160854	+	0.999999i	$0.500512\pi$
$98$	0	0
$99$	−7.34370e6	−0.760662
$100$	0	0
$101$	1.68077e7	1.62324	0.811622	−	0.584182i	$-0.198585\pi$
$101$	1.68077e7	1.62324	0.811622	+	0.584182i	$0.198585\pi$
$102$	0	0
$103$	1.27746e7	1.15191	0.575953	−	0.817483i	$-0.304631\pi$
$103$	1.27746e7	1.15191	0.575953	+	0.817483i	$0.304631\pi$
$104$	0	0
$105$	1.14328e7	0.963807
$106$	0	0
$107$	−1.35610e7	−1.07016	−0.535078	−	0.844803i	$-0.679718\pi$
$107$	−1.35610e7	−1.07016	−0.535078	+	0.844803i	$0.679718\pi$
$108$	0	0
$109$	−4.74206e6	−0.350731	−0.175366	−	0.984503i	$-0.556111\pi$
$109$	−4.74206e6	−0.350731	−0.175366	+	0.984503i	$0.556111\pi$
$110$	0	0
$111$	3.81889e7	2.65037
$112$	0	0
$113$	−8.06832e6	−0.526028	−0.263014	−	0.964792i	$-0.584717\pi$
$113$	−8.06832e6	−0.526028	−0.263014	+	0.964792i	$0.584717\pi$
$114$	0	0
$115$	6.63520e6	0.406828
$116$	0	0
$117$	−8.25635e6	−0.476582
$118$	0	0
$119$	−5.31839e6	−0.289311
$120$	0	0
$121$	−1.56197e7	−0.801540
$122$	0	0
$123$	4.61676e7	2.23702
$124$	0	0
$125$	1.41153e7	0.646405
$126$	0	0
$127$	−1.12410e7	−0.486960	−0.243480	−	0.969906i	$-0.578289\pi$
$127$	−1.12410e7	−0.486960	−0.243480	+	0.969906i	$0.578289\pi$
$128$	0	0
$129$	−6.79820e6	−0.278824
$130$	0	0
$131$	8.81527e6	0.342599	0.171299	−	0.985219i	$-0.445203\pi$
$131$	8.81527e6	0.342599	0.171299	+	0.985219i	$0.445203\pi$
$132$	0	0
$133$	−1.44308e7	−0.531874
$134$	0	0
$135$	4.02951e7	1.40956
$136$	0	0
$137$	3.33729e7	1.10885	0.554424	−	0.832234i	$-0.312939\pi$
$137$	3.33729e7	1.10885	0.554424	+	0.832234i	$0.312939\pi$
$138$	0	0
$139$	−4.68161e7	−1.47857	−0.739287	−	0.673391i	$-0.764837\pi$
$139$	−4.68161e7	−1.47857	−0.739287	+	0.673391i	$0.764837\pi$
$140$	0	0
$141$	−6.31252e7	−1.89643
$142$	0	0
$143$	4.34806e6	0.124343
$144$	0	0
$145$	5.43840e7	1.48144
$146$	0	0
$147$	4.85418e7	1.26039
$148$	0	0
$149$	−3.83709e7	−0.950277	−0.475139	−	0.879911i	$-0.657602\pi$
$149$	−3.83709e7	−0.950277	−0.475139	+	0.879911i	$0.657602\pi$
$150$	0	0
$151$	7.17648e7	1.69626	0.848130	−	0.529788i	$-0.177729\pi$
$151$	7.17648e7	1.69626	0.848130	+	0.529788i	$0.177729\pi$
$152$	0	0
$153$	−4.52400e7	−1.02118
$154$	0	0
$155$	7.75948e7	1.67368
$156$	0	0
$157$	−4.03778e7	−0.832710	−0.416355	−	0.909202i	$-0.636693\pi$
$157$	−4.03778e7	−0.832710	−0.416355	+	0.909202i	$0.636693\pi$
$158$	0	0
$159$	1.18285e8	2.33367
$160$	0	0
$161$	−8.60656e6	−0.162532
$162$	0	0
$163$	−9.84512e7	−1.78059	−0.890296	−	0.455383i	$-0.849502\pi$
$163$	−9.84512e7	−1.78059	−0.890296	+	0.455383i	$0.849502\pi$
$164$	0	0
$165$	−5.12157e7	−0.887584
$166$	0	0
$167$	4.24811e7	0.705810	0.352905	−	0.935659i	$-0.385194\pi$
$167$	4.24811e7	0.705810	0.352905	+	0.935659i	$0.385194\pi$
$168$	0	0
$169$	−5.78601e7	−0.922095
$170$	0	0
$171$	−1.22753e8	−1.87735
$172$	0	0
$173$	2.65257e7	0.389498	0.194749	−	0.980853i	$-0.437611\pi$
$173$	2.65257e7	0.389498	0.194749	+	0.980853i	$0.437611\pi$
$174$	0	0
$175$	1.59875e7	0.225501
$176$	0	0
$177$	−1.40418e8	−1.90334
$178$	0	0
$179$	2.42148e7	0.315570	0.157785	−	0.987474i	$-0.449565\pi$
$179$	2.42148e7	0.315570	0.157785	+	0.987474i	$0.449565\pi$
$180$	0	0
$181$	2.78961e7	0.349679	0.174839	−	0.984597i	$-0.444059\pi$
$181$	2.78961e7	0.349679	0.174839	+	0.984597i	$0.444059\pi$
$182$	0	0
$183$	−3.72884e7	−0.449775
$184$	0	0
$185$	1.67964e8	1.95036
$186$	0	0
$187$	2.38249e7	0.266431
$188$	0	0
$189$	−5.22671e7	−0.563135
$190$	0	0
$191$	1.67596e7	0.174039	0.0870195	−	0.996207i	$-0.472266\pi$
$191$	1.67596e7	0.174039	0.0870195	+	0.996207i	$0.472266\pi$
$192$	0	0
$193$	8.75008e7	0.876116	0.438058	−	0.898947i	$-0.355667\pi$
$193$	8.75008e7	0.876116	0.438058	+	0.898947i	$0.355667\pi$
$194$	0	0
$195$	−5.75806e7	−0.556103
$196$	0	0
$197$	2.56239e7	0.238789	0.119394	−	0.992847i	$-0.461905\pi$
$197$	2.56239e7	0.238789	0.119394	+	0.992847i	$0.461905\pi$
$198$	0	0
$199$	−5.31884e7	−0.478444	−0.239222	−	0.970965i	$-0.576892\pi$
$199$	−5.31884e7	−0.478444	−0.239222	+	0.970965i	$0.576892\pi$
$200$	0	0
$201$	−6.14660e6	−0.0533886
$202$	0	0
$203$	−7.05419e7	−0.591849
$204$	0	0
$205$	2.03056e8	1.64618
$206$	0	0
$207$	−7.32103e7	−0.573688
$208$	0	0
$209$	6.46457e7	0.489810
$210$	0	0
$211$	2.01165e7	0.147423	0.0737114	−	0.997280i	$-0.476516\pi$
$211$	2.01165e7	0.147423	0.0737114	+	0.997280i	$0.476516\pi$
$212$	0	0
$213$	9.77862e7	0.693344
$214$	0	0
$215$	−2.99001e7	−0.205181
$216$	0	0
$217$	−1.00649e8	−0.668651
$218$	0	0
$219$	2.85345e8	1.83576
$220$	0	0
$221$	2.67858e7	0.166929
$222$	0	0
$223$	−1.67012e8	−1.00851	−0.504254	−	0.863555i	$-0.668232\pi$
$223$	−1.67012e8	−1.00851	−0.504254	+	0.863555i	$0.668232\pi$
$224$	0	0
$225$	1.35995e8	0.795949
$226$	0	0
$227$	1.37308e8	0.779122	0.389561	−	0.921001i	$-0.372627\pi$
$227$	1.37308e8	0.779122	0.389561	+	0.921001i	$0.372627\pi$
$228$	0	0
$229$	2.67935e8	1.47436	0.737182	−	0.675694i	$-0.236156\pi$
$229$	2.67935e8	1.47436	0.737182	+	0.675694i	$0.236156\pi$
$230$	0	0
$231$	6.64322e7	0.354599
$232$	0	0
$233$	−1.98032e8	−1.02563	−0.512815	−	0.858499i	$-0.671397\pi$
$233$	−1.98032e8	−1.02563	−0.512815	+	0.858499i	$0.671397\pi$
$234$	0	0
$235$	−2.77639e8	−1.39554
$236$	0	0
$237$	1.96873e8	0.960654
$238$	0	0
$239$	8.22277e7	0.389606	0.194803	−	0.980842i	$-0.437593\pi$
$239$	8.22277e7	0.389606	0.194803	+	0.980842i	$0.437593\pi$
$240$	0	0
$241$	−2.70650e8	−1.24551	−0.622757	−	0.782415i	$-0.713988\pi$
$241$	−2.70650e8	−1.24551	−0.622757	+	0.782415i	$0.713988\pi$
$242$	0	0
$243$	1.83831e8	0.821860
$244$	0	0
$245$	2.13498e8	0.927498
$246$	0	0
$247$	7.26797e7	0.306884
$248$	0	0
$249$	1.18052e8	0.484591
$250$	0	0
$251$	2.90747e8	1.16053	0.580266	−	0.814427i	$-0.302949\pi$
$251$	2.90747e8	1.16053	0.580266	+	0.814427i	$0.302949\pi$
$252$	0	0
$253$	3.85549e7	0.149678
$254$	0	0
$255$	−3.15508e8	−1.19157
$256$	0	0
$257$	4.36047e7	0.160239	0.0801193	−	0.996785i	$-0.474470\pi$
$257$	4.36047e7	0.160239	0.0801193	+	0.996785i	$0.474470\pi$
$258$	0	0
$259$	−2.17867e8	−0.779188
$260$	0	0
$261$	−6.00053e8	−2.08905
$262$	0	0
$263$	−4.27678e8	−1.44968	−0.724840	−	0.688917i	$-0.758086\pi$
$263$	−4.27678e8	−1.44968	−0.724840	+	0.688917i	$0.758086\pi$
$264$	0	0
$265$	5.20244e8	1.71730
$266$	0	0
$267$	1.53509e8	0.493564
$268$	0	0
$269$	−8.26134e7	−0.258772	−0.129386	−	0.991594i	$-0.541301\pi$
$269$	−8.26134e7	−0.258772	−0.129386	+	0.991594i	$0.541301\pi$
$270$	0	0
$271$	6.15189e7	0.187766	0.0938829	−	0.995583i	$-0.470072\pi$
$271$	6.15189e7	0.187766	0.0938829	+	0.995583i	$0.470072\pi$
$272$	0	0
$273$	7.46883e7	0.222169
$274$	0	0
$275$	−7.16196e7	−0.207667
$276$	0	0
$277$	−4.39237e7	−0.124171	−0.0620854	−	0.998071i	$-0.519775\pi$
$277$	−4.39237e7	−0.124171	−0.0620854	+	0.998071i	$0.519775\pi$
$278$	0	0
$279$	−8.56153e8	−2.36013
$280$	0	0
$281$	−5.80931e8	−1.56190	−0.780948	−	0.624596i	$-0.785264\pi$
$281$	−5.80931e8	−1.56190	−0.780948	+	0.624596i	$0.785264\pi$
$282$	0	0
$283$	−6.03790e8	−1.58356	−0.791778	−	0.610810i	$-0.790844\pi$
$283$	−6.03790e8	−1.58356	−0.791778	+	0.610810i	$0.790844\pi$
$284$	0	0
$285$	−8.56091e8	−2.19060
$286$	0	0
$287$	−2.63385e8	−0.657665
$288$	0	0
$289$	−2.63568e8	−0.642319
$290$	0	0
$291$	2.22520e6	0.00529352
$292$	0	0
$293$	1.10504e8	0.256649	0.128325	−	0.991732i	$-0.459040\pi$
$293$	1.10504e8	0.256649	0.128325	+	0.991732i	$0.459040\pi$
$294$	0	0
$295$	−6.17591e8	−1.40063
$296$	0	0
$297$	2.34142e8	0.518599
$298$	0	0
$299$	4.33464e7	0.0937787
$300$	0	0
$301$	3.87836e7	0.0819719
$302$	0	0
$303$	−1.29335e9	−2.67095
$304$	0	0
$305$	−1.64003e8	−0.330981
$306$	0	0
$307$	−7.03386e8	−1.38742	−0.693712	−	0.720252i	$-0.744026\pi$
$307$	−7.03386e8	−1.38742	−0.693712	+	0.720252i	$0.744026\pi$
$308$	0	0
$309$	−9.83002e8	−1.89539
$310$	0	0
$311$	8.61240e8	1.62354	0.811769	−	0.583978i	$-0.198505\pi$
$311$	8.61240e8	1.62354	0.811769	+	0.583978i	$0.198505\pi$
$312$	0	0
$313$	−2.42056e8	−0.446181	−0.223090	−	0.974798i	$-0.571615\pi$
$313$	−2.42056e8	−0.446181	−0.223090	+	0.974798i	$0.571615\pi$
$314$	0	0
$315$	−5.54817e8	−1.00014
$316$	0	0
$317$	5.38362e8	0.949221	0.474610	−	0.880196i	$-0.342589\pi$
$317$	5.38362e8	0.949221	0.474610	+	0.880196i	$0.342589\pi$
$318$	0	0
$319$	3.16007e8	0.545042
$320$	0	0
$321$	1.04351e9	1.76088
$322$	0	0
$323$	3.98242e8	0.657565
$324$	0	0
$325$	−8.05203e7	−0.130111
$326$	0	0
$327$	3.64900e8	0.577108
$328$	0	0
$329$	3.60128e8	0.557534
$330$	0	0
$331$	1.05054e9	1.59226	0.796131	−	0.605124i	$-0.206877\pi$
$331$	1.05054e9	1.59226	0.796131	+	0.605124i	$0.206877\pi$
$332$	0	0
$333$	−1.85325e9	−2.75029
$334$	0	0
$335$	−2.70342e7	−0.0392877
$336$	0	0
$337$	−2.04579e8	−0.291177	−0.145589	−	0.989345i	$-0.546508\pi$
$337$	−2.04579e8	−0.291177	−0.145589	+	0.989345i	$0.546508\pi$
$338$	0	0
$339$	6.20855e8	0.865548
$340$	0	0
$341$	4.50878e8	0.615770
$342$	0	0
$343$	−6.38462e8	−0.854291
$344$	0	0
$345$	−5.10576e8	−0.669412
$346$	0	0
$347$	−6.28852e8	−0.807970	−0.403985	−	0.914766i	$-0.632375\pi$
$347$	−6.28852e8	−0.807970	−0.403985	+	0.914766i	$0.632375\pi$
$348$	0	0
$349$	9.86717e8	1.24252	0.621260	−	0.783604i	$-0.286621\pi$
$349$	9.86717e8	1.24252	0.621260	+	0.783604i	$0.286621\pi$
$350$	0	0
$351$	2.63240e8	0.324921
$352$	0	0
$353$	5.59732e8	0.677281	0.338641	−	0.940916i	$-0.390033\pi$
$353$	5.59732e8	0.677281	0.338641	+	0.940916i	$0.390033\pi$
$354$	0	0
$355$	4.30087e8	0.510219
$356$	0	0
$357$	4.09248e8	0.476045
$358$	0	0
$359$	−1.42390e8	−0.162424	−0.0812119	−	0.996697i	$-0.525879\pi$
$359$	−1.42390e8	−0.162424	−0.0812119	+	0.996697i	$0.525879\pi$
$360$	0	0
$361$	1.86707e8	0.208875
$362$	0	0
$363$	1.20193e9	1.31889
$364$	0	0
$365$	1.25501e9	1.35090
$366$	0	0
$367$	7.13452e8	0.753414	0.376707	−	0.926333i	$-0.377056\pi$
$367$	7.13452e8	0.753414	0.376707	+	0.926333i	$0.377056\pi$
$368$	0	0
$369$	−2.24044e9	−2.32136
$370$	0	0
$371$	−6.74812e8	−0.686079
$372$	0	0
$373$	4.14729e8	0.413794	0.206897	−	0.978363i	$-0.433664\pi$
$373$	4.14729e8	0.413794	0.206897	+	0.978363i	$0.433664\pi$
$374$	0	0
$375$	−1.08617e9	−1.06362
$376$	0	0
$377$	3.55280e8	0.341489
$378$	0	0
$379$	−1.23625e8	−0.116646	−0.0583229	−	0.998298i	$-0.518575\pi$
$379$	−1.23625e8	−0.116646	−0.0583229	+	0.998298i	$0.518575\pi$
$380$	0	0
$381$	8.64994e8	0.801264
$382$	0	0
$383$	−1.35784e9	−1.23496	−0.617480	−	0.786586i	$-0.711846\pi$
$383$	−1.35784e9	−1.23496	−0.617480	+	0.786586i	$0.711846\pi$
$384$	0	0
$385$	2.92184e8	0.260942
$386$	0	0
$387$	3.29906e8	0.289336
$388$	0	0
$389$	1.00573e9	0.866281	0.433141	−	0.901326i	$-0.357405\pi$
$389$	1.00573e9	0.866281	0.433141	+	0.901326i	$0.357405\pi$
$390$	0	0
$391$	2.37513e8	0.200941
$392$	0	0
$393$	−6.78332e8	−0.563726
$394$	0	0
$395$	8.65893e8	0.706927
$396$	0	0
$397$	2.30080e8	0.184549	0.0922747	−	0.995734i	$-0.470586\pi$
$397$	2.30080e8	0.184549	0.0922747	+	0.995734i	$0.470586\pi$
$398$	0	0
$399$	1.11044e9	0.875167
$400$	0	0
$401$	−1.24791e9	−0.966446	−0.483223	−	0.875497i	$-0.660534\pi$
$401$	−1.24791e9	−0.966446	−0.483223	+	0.875497i	$0.660534\pi$
$402$	0	0
$403$	5.06912e8	0.385802
$404$	0	0
$405$	−3.36702e8	−0.251857
$406$	0	0
$407$	9.75982e8	0.717565
$408$	0	0
$409$	1.79923e9	1.30033	0.650166	−	0.759792i	$-0.274699\pi$
$409$	1.79923e9	1.30033	0.650166	+	0.759792i	$0.274699\pi$
$410$	0	0
$411$	−2.56803e9	−1.82454
$412$	0	0
$413$	8.01081e8	0.559566
$414$	0	0
$415$	5.19219e8	0.356601
$416$	0	0
$417$	3.60248e9	2.43291
$418$	0	0
$419$	2.66870e8	0.177235	0.0886177	−	0.996066i	$-0.471755\pi$
$419$	2.66870e8	0.177235	0.0886177	+	0.996066i	$0.471755\pi$
$420$	0	0
$421$	2.70575e9	1.76726	0.883630	−	0.468185i	$-0.155092\pi$
$421$	2.70575e9	1.76726	0.883630	+	0.468185i	$0.155092\pi$
$422$	0	0
$423$	3.06337e9	1.96792
$424$	0	0
$425$	−4.41204e8	−0.278791
$426$	0	0
$427$	2.12730e8	0.132230
$428$	0	0
$429$	−3.34582e8	−0.204598
$430$	0	0
$431$	1.78455e9	1.07364	0.536820	−	0.843697i	$-0.319626\pi$
$431$	1.78455e9	1.07364	0.536820	+	0.843697i	$0.319626\pi$
$432$	0	0
$433$	1.21276e9	0.717905	0.358953	−	0.933356i	$-0.383134\pi$
$433$	1.21276e9	0.717905	0.358953	+	0.933356i	$0.383134\pi$
$434$	0	0
$435$	−4.18483e9	−2.43762
$436$	0	0
$437$	6.44461e8	0.369413
$438$	0	0
$439$	1.76141e9	0.993654	0.496827	−	0.867850i	$-0.334498\pi$
$439$	1.76141e9	0.993654	0.496827	+	0.867850i	$0.334498\pi$
$440$	0	0
$441$	−2.35566e9	−1.30791
$442$	0	0
$443$	−8.06208e8	−0.440590	−0.220295	−	0.975433i	$-0.570702\pi$
$443$	−8.06208e8	−0.440590	−0.220295	+	0.975433i	$0.570702\pi$
$444$	0	0
$445$	6.75166e8	0.363204
$446$	0	0
$447$	2.95263e9	1.56363
$448$	0	0
$449$	5.85913e8	0.305472	0.152736	−	0.988267i	$-0.451192\pi$
$449$	5.85913e8	0.305472	0.152736	+	0.988267i	$0.451192\pi$
$450$	0	0
$451$	1.17989e9	0.605653
$452$	0	0
$453$	−5.52228e9	−2.79110
$454$	0	0
$455$	3.28496e8	0.163490
$456$	0	0
$457$	−5.52640e8	−0.270854	−0.135427	−	0.990787i	$-0.543241\pi$
$457$	−5.52640e8	−0.270854	−0.135427	+	0.990787i	$0.543241\pi$
$458$	0	0
$459$	1.44240e9	0.696213
$460$	0	0
$461$	−1.74101e9	−0.827652	−0.413826	−	0.910356i	$-0.635808\pi$
$461$	−1.74101e9	−0.827652	−0.413826	+	0.910356i	$0.635808\pi$
$462$	0	0
$463$	−2.84431e9	−1.33181	−0.665906	−	0.746035i	$-0.731955\pi$
$463$	−2.84431e9	−1.33181	−0.665906	+	0.746035i	$0.731955\pi$
$464$	0	0
$465$	−5.97090e9	−2.75394
$466$	0	0
$467$	1.63130e9	0.741183	0.370592	−	0.928796i	$-0.379155\pi$
$467$	1.63130e9	0.741183	0.370592	+	0.928796i	$0.379155\pi$
$468$	0	0
$469$	3.50662e7	0.0156958
$470$	0	0
$471$	3.10706e9	1.37018
$472$	0	0
$473$	−1.73739e8	−0.0754891
$474$	0	0
$475$	−1.19715e9	−0.512532
$476$	0	0
$477$	−5.74018e9	−2.42165
$478$	0	0
$479$	3.69345e9	1.53553	0.767765	−	0.640732i	$-0.221369\pi$
$479$	3.69345e9	1.53553	0.767765	+	0.640732i	$0.221369\pi$
$480$	0	0
$481$	1.09727e9	0.449580
$482$	0	0
$483$	6.62272e8	0.267437
$484$	0	0
$485$	9.78697e6	0.00389540
$486$	0	0
$487$	−1.57153e9	−0.616554	−0.308277	−	0.951297i	$-0.599752\pi$
$487$	−1.57153e9	−0.616554	−0.308277	+	0.951297i	$0.599752\pi$
$488$	0	0
$489$	7.57578e9	2.92986
$490$	0	0
$491$	−1.63493e9	−0.623323	−0.311662	−	0.950193i	$-0.600886\pi$
$491$	−1.63493e9	−0.623323	−0.311662	+	0.950193i	$0.600886\pi$
$492$	0	0
$493$	1.94673e9	0.731713
$494$	0	0
$495$	2.48542e9	0.921047
$496$	0	0
$497$	−5.57868e8	−0.203838
$498$	0	0
$499$	6.86870e8	0.247470	0.123735	−	0.992315i	$-0.460513\pi$
$499$	6.86870e8	0.247470	0.123735	+	0.992315i	$0.460513\pi$
$500$	0	0
$501$	−3.26890e9	−1.16137
$502$	0	0
$503$	−2.33472e9	−0.817990	−0.408995	−	0.912537i	$-0.634121\pi$
$503$	−2.33472e9	−0.817990	−0.408995	+	0.912537i	$0.634121\pi$
$504$	0	0
$505$	−5.68845e9	−1.96550
$506$	0	0
$507$	4.45231e9	1.51725
$508$	0	0
$509$	6.21342e8	0.208842	0.104421	−	0.994533i	$-0.466701\pi$
$509$	6.21342e8	0.208842	0.104421	+	0.994533i	$0.466701\pi$
$510$	0	0
$511$	−1.62789e9	−0.539699
$512$	0	0
$513$	3.91378e9	1.27993
$514$	0	0
$515$	−4.32347e9	−1.39478
$516$	0	0
$517$	−1.61327e9	−0.513441
$518$	0	0
$519$	−2.04114e9	−0.640896
$520$	0	0
$521$	1.65562e9	0.512897	0.256448	−	0.966558i	$-0.417448\pi$
$521$	1.65562e9	0.512897	0.256448	+	0.966558i	$0.417448\pi$
$522$	0	0
$523$	4.42671e9	1.35309	0.676543	−	0.736403i	$-0.263477\pi$
$523$	4.42671e9	1.35309	0.676543	+	0.736403i	$0.263477\pi$
$524$	0	0
$525$	−1.23024e9	−0.371049
$526$	0	0
$527$	2.77758e9	0.826665
$528$	0	0
$529$	−3.02047e9	−0.887113
$530$	0	0
$531$	6.81427e9	1.97510
$532$	0	0
$533$	1.32652e9	0.379463
$534$	0	0
$535$	4.58961e9	1.29580
$536$	0	0
$537$	−1.86332e9	−0.519252
$538$	0	0
$539$	1.24057e9	0.341240
$540$	0	0
$541$	−3.71878e9	−1.00974	−0.504871	−	0.863195i	$-0.668460\pi$
$541$	−3.71878e9	−1.00974	−0.504871	+	0.863195i	$0.668460\pi$
$542$	0	0
$543$	−2.14660e9	−0.575376
$544$	0	0
$545$	1.60492e9	0.424683
$546$	0	0
$547$	3.20749e9	0.837934	0.418967	−	0.908002i	$-0.362392\pi$
$547$	3.20749e9	0.837934	0.418967	+	0.908002i	$0.362392\pi$
$548$	0	0
$549$	1.80955e9	0.466732
$550$	0	0
$551$	5.28219e9	1.34519
$552$	0	0
$553$	−1.12316e9	−0.282424
$554$	0	0
$555$	−1.29247e10	−3.20920
$556$	0	0
$557$	4.80739e9	1.17873	0.589367	−	0.807865i	$-0.299377\pi$
$557$	4.80739e9	1.17873	0.589367	+	0.807865i	$0.299377\pi$
$558$	0	0
$559$	−1.95331e8	−0.0472967
$560$	0	0
$561$	−1.83332e9	−0.438397
$562$	0	0
$563$	−3.77127e9	−0.890653	−0.445326	−	0.895368i	$-0.646912\pi$
$563$	−3.77127e9	−0.890653	−0.445326	+	0.895368i	$0.646912\pi$
$564$	0	0
$565$	2.73066e9	0.636940
$566$	0	0
$567$	4.36739e8	0.100619
$568$	0	0
$569$	−2.09341e9	−0.476388	−0.238194	−	0.971218i	$-0.576555\pi$
$569$	−2.09341e9	−0.476388	−0.238194	+	0.971218i	$0.576555\pi$
$570$	0	0
$571$	6.86085e9	1.54224	0.771119	−	0.636691i	$-0.219697\pi$
$571$	6.86085e9	1.54224	0.771119	+	0.636691i	$0.219697\pi$
$572$	0	0
$573$	−1.28965e9	−0.286371
$574$	0	0
$575$	−7.13986e8	−0.156622
$576$	0	0
$577$	5.70742e9	1.23687	0.618435	−	0.785836i	$-0.287767\pi$
$577$	5.70742e9	1.23687	0.618435	+	0.785836i	$0.287767\pi$
$578$	0	0
$579$	−6.73316e9	−1.44160
$580$	0	0
$581$	−6.73483e8	−0.142466
$582$	0	0
$583$	3.02297e9	0.631820
$584$	0	0
$585$	2.79430e9	0.577069
$586$	0	0
$587$	−1.56510e9	−0.319380	−0.159690	−	0.987167i	$-0.551049\pi$
$587$	−1.56510e9	−0.319380	−0.159690	+	0.987167i	$0.551049\pi$
$588$	0	0
$589$	7.53661e9	1.51975
$590$	0	0
$591$	−1.97175e9	−0.392913
$592$	0	0
$593$	−2.78410e9	−0.548268	−0.274134	−	0.961692i	$-0.588391\pi$
$593$	−2.78410e9	−0.548268	−0.274134	+	0.961692i	$0.588391\pi$
$594$	0	0
$595$	1.79997e9	0.350312
$596$	0	0
$597$	4.09283e9	0.787251
$598$	0	0
$599$	−6.18885e9	−1.17657	−0.588283	−	0.808655i	$-0.700196\pi$
$599$	−6.18885e9	−1.17657	−0.588283	+	0.808655i	$0.700196\pi$
$600$	0	0
$601$	2.42206e9	0.455118	0.227559	−	0.973764i	$-0.426926\pi$
$601$	2.42206e9	0.455118	0.227559	+	0.973764i	$0.426926\pi$
$602$	0	0
$603$	2.98285e8	0.0554014
$604$	0	0
$605$	5.28639e9	0.970544
$606$	0	0
$607$	−4.62465e9	−0.839302	−0.419651	−	0.907686i	$-0.637848\pi$
$607$	−4.62465e9	−0.839302	−0.419651	+	0.907686i	$0.637848\pi$
$608$	0	0
$609$	5.42817e9	0.973852
$610$	0	0
$611$	−1.81376e9	−0.321689
$612$	0	0
$613$	5.45433e9	0.956378	0.478189	−	0.878257i	$-0.341293\pi$
$613$	5.45433e9	0.956378	0.478189	+	0.878257i	$0.341293\pi$
$614$	0	0
$615$	−1.56251e10	−2.70869
$616$	0	0
$617$	2.32837e9	0.399074	0.199537	−	0.979890i	$-0.436056\pi$
$617$	2.32837e9	0.399074	0.199537	+	0.979890i	$0.436056\pi$
$618$	0	0
$619$	−9.58626e9	−1.62455	−0.812273	−	0.583278i	$-0.801770\pi$
$619$	−9.58626e9	−1.62455	−0.812273	+	0.583278i	$0.801770\pi$
$620$	0	0
$621$	2.33419e9	0.391125
$622$	0	0
$623$	−8.75763e8	−0.145104
$624$	0	0
$625$	−7.62240e9	−1.24885
$626$	0	0
$627$	−4.97446e9	−0.805953
$628$	0	0
$629$	6.01242e9	0.963324
$630$	0	0
$631$	−1.18616e10	−1.87949	−0.939747	−	0.341870i	$-0.888940\pi$
$631$	−1.18616e10	−1.87949	−0.939747	+	0.341870i	$0.888940\pi$
$632$	0	0
$633$	−1.54796e9	−0.242576
$634$	0	0
$635$	3.80445e9	0.589635
$636$	0	0
$637$	1.39474e9	0.213799
$638$	0	0
$639$	−4.74542e9	−0.719484
$640$	0	0
$641$	−1.06130e8	−0.0159161	−0.00795805	−	0.999968i	$-0.502533\pi$
$641$	−1.06130e8	−0.0159161	−0.00795805	+	0.999968i	$0.502533\pi$
$642$	0	0
$643$	2.19289e9	0.325296	0.162648	−	0.986684i	$-0.447996\pi$
$643$	2.19289e9	0.325296	0.162648	+	0.986684i	$0.447996\pi$
$644$	0	0
$645$	2.30080e9	0.337614
$646$	0	0
$647$	4.23914e9	0.615337	0.307668	−	0.951494i	$-0.400451\pi$
$647$	4.23914e9	0.615337	0.307668	+	0.951494i	$0.400451\pi$
$648$	0	0
$649$	−3.58862e9	−0.515312
$650$	0	0
$651$	7.74489e9	1.10023
$652$	0	0
$653$	−9.93257e9	−1.39594	−0.697968	−	0.716129i	$-0.745912\pi$
$653$	−9.93257e9	−1.39594	−0.697968	+	0.716129i	$0.745912\pi$
$654$	0	0
$655$	−2.98346e9	−0.414836
$656$	0	0
$657$	−1.38474e10	−1.90497
$658$	0	0
$659$	1.36634e10	1.85977	0.929884	−	0.367852i	$-0.119907\pi$
$659$	1.36634e10	1.85977	0.929884	+	0.367852i	$0.119907\pi$
$660$	0	0
$661$	−1.03765e10	−1.39747	−0.698737	−	0.715378i	$-0.746254\pi$
$661$	−1.03765e10	−1.39747	−0.698737	+	0.715378i	$0.746254\pi$
$662$	0	0
$663$	−2.06115e9	−0.274671
$664$	0	0
$665$	4.88398e9	0.644019
$666$	0	0
$667$	3.15032e9	0.411069
$668$	0	0
$669$	1.28515e10	1.65944
$670$	0	0
$671$	−9.52968e8	−0.121773
$672$	0	0
$673$	4.70776e9	0.595336	0.297668	−	0.954669i	$-0.403791\pi$
$673$	4.70776e9	0.595336	0.297668	+	0.954669i	$0.403791\pi$
$674$	0	0
$675$	−4.33599e9	−0.542657
$676$	0	0
$677$	−9.55050e9	−1.18295	−0.591474	−	0.806324i	$-0.701454\pi$
$677$	−9.55050e9	−1.18295	−0.591474	+	0.806324i	$0.701454\pi$
$678$	0	0
$679$	−1.26947e7	−0.00155625
$680$	0	0
$681$	−1.05658e10	−1.28200
$682$	0	0
$683$	−1.06442e10	−1.27832	−0.639161	−	0.769073i	$-0.720718\pi$
$683$	−1.06442e10	−1.27832	−0.639161	+	0.769073i	$0.720718\pi$
$684$	0	0
$685$	−1.12948e10	−1.34265
$686$	0	0
$687$	−2.06175e10	−2.42598
$688$	0	0
$689$	3.39865e9	0.395858
$690$	0	0
$691$	−8.41537e9	−0.970287	−0.485143	−	0.874435i	$-0.661233\pi$
$691$	−8.41537e9	−0.970287	−0.485143	+	0.874435i	$0.661233\pi$
$692$	0	0
$693$	−3.22386e9	−0.367967
$694$	0	0
$695$	1.58445e10	1.79033
$696$	0	0
$697$	7.26858e9	0.813083
$698$	0	0
$699$	1.52385e10	1.68761
$700$	0	0
$701$	−8.95355e9	−0.981707	−0.490854	−	0.871242i	$-0.663315\pi$
$701$	−8.95355e9	−0.981707	−0.490854	+	0.871242i	$0.663315\pi$
$702$	0	0
$703$	1.63139e10	1.77099
$704$	0	0
$705$	2.13643e10	2.29629
$706$	0	0
$707$	7.37853e9	0.785239
$708$	0	0
$709$	8.11796e9	0.855431	0.427716	−	0.903913i	$-0.359318\pi$
$709$	8.11796e9	0.855431	0.427716	+	0.903913i	$0.359318\pi$
$710$	0	0
$711$	−9.55395e9	−0.996872
$712$	0	0
$713$	4.49486e9	0.464412
$714$	0	0
$715$	−1.47157e9	−0.150560
$716$	0	0
$717$	−6.32739e9	−0.641073
$718$	0	0
$719$	9.54339e8	0.0957528	0.0478764	−	0.998853i	$-0.484755\pi$
$719$	9.54339e8	0.0957528	0.0478764	+	0.998853i	$0.484755\pi$
$720$	0	0
$721$	5.60800e9	0.557231
$722$	0	0
$723$	2.08264e10	2.04942
$724$	0	0
$725$	−5.85203e9	−0.570327
$726$	0	0
$727$	−1.75084e10	−1.68996	−0.844979	−	0.534799i	$-0.820387\pi$
$727$	−1.75084e10	−1.68996	−0.844979	+	0.534799i	$0.820387\pi$
$728$	0	0
$729$	−1.63215e10	−1.56032
$730$	0	0
$731$	−1.07030e9	−0.101343
$732$	0	0
$733$	−1.16062e10	−1.08849	−0.544247	−	0.838925i	$-0.683185\pi$
$733$	−1.16062e10	−1.08849	−0.544247	+	0.838925i	$0.683185\pi$
$734$	0	0
$735$	−1.64286e10	−1.52614
$736$	0	0
$737$	−1.57087e8	−0.0144545
$738$	0	0
$739$	−4.74800e9	−0.432768	−0.216384	−	0.976308i	$-0.569426\pi$
$739$	−4.74800e9	−0.432768	−0.216384	+	0.976308i	$0.569426\pi$
$740$	0	0
$741$	−5.59268e9	−0.504959
$742$	0	0
$743$	4.35857e9	0.389837	0.194918	−	0.980819i	$-0.437556\pi$
$743$	4.35857e9	0.389837	0.194918	+	0.980819i	$0.437556\pi$
$744$	0	0
$745$	1.29864e10	1.15064
$746$	0	0
$747$	−5.72888e9	−0.502860
$748$	0	0
$749$	−5.95321e9	−0.517684
$750$	0	0
$751$	−6.10987e9	−0.526371	−0.263186	−	0.964745i	$-0.584773\pi$
$751$	−6.10987e9	−0.526371	−0.263186	+	0.964745i	$0.584773\pi$
$752$	0	0
$753$	−2.23729e10	−1.90959
$754$	0	0
$755$	−2.42883e10	−2.05391
$756$	0	0
$757$	−2.29472e10	−1.92262	−0.961312	−	0.275460i	$-0.911170\pi$
$757$	−2.29472e10	−1.92262	−0.961312	+	0.275460i	$0.911170\pi$
$758$	0	0
$759$	−2.96679e9	−0.246286
$760$	0	0
$761$	−7.15151e9	−0.588236	−0.294118	−	0.955769i	$-0.595026\pi$
$761$	−7.15151e9	−0.588236	−0.294118	+	0.955769i	$0.595026\pi$
$762$	0	0
$763$	−2.08175e9	−0.169665
$764$	0	0
$765$	1.53111e10	1.23650
$766$	0	0
$767$	−4.03460e9	−0.322862
$768$	0	0
$769$	−1.85381e10	−1.47002	−0.735011	−	0.678055i	$-0.762823\pi$
$769$	−1.85381e10	−1.47002	−0.735011	+	0.678055i	$0.762823\pi$
$770$	0	0
$771$	−3.35537e9	−0.263663
$772$	0	0
$773$	−8.25535e9	−0.642846	−0.321423	−	0.946936i	$-0.604161\pi$
$773$	−8.25535e9	−0.642846	−0.321423	+	0.946936i	$0.604161\pi$
$774$	0	0
$775$	−8.34966e9	−0.644336
$776$	0	0
$777$	1.67648e10	1.28211
$778$	0	0
$779$	1.97223e10	1.49478
$780$	0	0
$781$	2.49909e9	0.187717
$782$	0	0
$783$	1.91317e10	1.42425
$784$	0	0
$785$	1.36656e10	1.00829
$786$	0	0
$787$	−1.53185e9	−0.112023	−0.0560113	−	0.998430i	$-0.517838\pi$
$787$	−1.53185e9	−0.112023	−0.0560113	+	0.998430i	$0.517838\pi$
$788$	0	0
$789$	3.29097e10	2.38536
$790$	0	0
$791$	−3.54196e9	−0.254464
$792$	0	0
$793$	−1.07140e9	−0.0762950
$794$	0	0
$795$	−4.00326e10	−2.82572
$796$	0	0
$797$	2.78553e9	0.194896	0.0974480	−	0.995241i	$-0.468932\pi$
$797$	2.78553e9	0.194896	0.0974480	+	0.995241i	$0.468932\pi$
$798$	0	0
$799$	−9.93837e9	−0.689289
$800$	0	0
$801$	−7.44954e9	−0.512172
$802$	0	0
$803$	7.29247e9	0.497016
$804$	0	0
$805$	2.91283e9	0.196802
$806$	0	0
$807$	6.35708e9	0.425794
$808$	0	0
$809$	−1.17657e10	−0.781267	−0.390634	−	0.920546i	$-0.627744\pi$
$809$	−1.17657e10	−0.781267	−0.390634	+	0.920546i	$0.627744\pi$
$810$	0	0
$811$	−6.29491e9	−0.414397	−0.207198	−	0.978299i	$-0.566435\pi$
$811$	−6.29491e9	−0.414397	−0.207198	+	0.978299i	$0.566435\pi$
$812$	0	0
$813$	−4.73386e9	−0.308957
$814$	0	0
$815$	3.33201e10	2.15603
$816$	0	0
$817$	−2.90413e9	−0.186311
$818$	0	0
$819$	−3.62451e9	−0.230545
$820$	0	0
$821$	4.27400e9	0.269546	0.134773	−	0.990876i	$-0.456969\pi$
$821$	4.27400e9	0.269546	0.134773	+	0.990876i	$0.456969\pi$
$822$	0	0
$823$	3.16411e9	0.197858	0.0989288	−	0.995095i	$-0.468458\pi$
$823$	3.16411e9	0.197858	0.0989288	+	0.995095i	$0.468458\pi$
$824$	0	0
$825$	5.51111e9	0.341704
$826$	0	0
$827$	3.47251e9	0.213488	0.106744	−	0.994287i	$-0.465957\pi$
$827$	3.47251e9	0.213488	0.106744	+	0.994287i	$0.465957\pi$
$828$	0	0
$829$	5.21388e9	0.317848	0.158924	−	0.987291i	$-0.449197\pi$
$829$	5.21388e9	0.317848	0.158924	+	0.987291i	$0.449197\pi$
$830$	0	0
$831$	3.37991e9	0.204316
$832$	0	0
$833$	7.64237e9	0.458111
$834$	0	0
$835$	−1.43774e10	−0.854629
$836$	0	0
$837$	2.72970e10	1.60908
$838$	0	0
$839$	2.48652e10	1.45353	0.726766	−	0.686885i	$-0.241023\pi$
$839$	2.48652e10	1.45353	0.726766	+	0.686885i	$0.241023\pi$
$840$	0	0
$841$	8.57107e9	0.496877
$842$	0	0
$843$	4.47024e10	2.57001
$844$	0	0
$845$	1.95823e10	1.11652
$846$	0	0
$847$	−6.85701e9	−0.387742
$848$	0	0
$849$	4.64614e10	2.60565
$850$	0	0
$851$	9.72969e9	0.541185
$852$	0	0
$853$	−3.01930e10	−1.66565	−0.832826	−	0.553536i	$-0.813278\pi$
$853$	−3.01930e10	−1.66565	−0.832826	+	0.553536i	$0.813278\pi$
$854$	0	0
$855$	4.15448e10	2.27319
$856$	0	0
$857$	−2.57761e10	−1.39889	−0.699446	−	0.714685i	$-0.746570\pi$
$857$	−2.57761e10	−1.39889	−0.699446	+	0.714685i	$0.746570\pi$
$858$	0	0
$859$	−2.77848e10	−1.49565	−0.747827	−	0.663894i	$-0.768903\pi$
$859$	−2.77848e10	−1.49565	−0.747827	+	0.663894i	$0.768903\pi$
$860$	0	0
$861$	2.02674e10	1.08215
$862$	0	0
$863$	3.02990e10	1.60469	0.802344	−	0.596862i	$-0.203586\pi$
$863$	3.02990e10	1.60469	0.802344	+	0.596862i	$0.203586\pi$
$864$	0	0
$865$	−8.97743e9	−0.471624
$866$	0	0
$867$	2.02815e10	1.05690
$868$	0	0
$869$	5.03142e9	0.260089
$870$	0	0
$871$	−1.76609e8	−0.00905627
$872$	0	0
$873$	−1.07986e8	−0.00549309
$874$	0	0
$875$	6.19656e9	0.312696
$876$	0	0
$877$	2.37410e9	0.118850	0.0594252	−	0.998233i	$-0.481073\pi$
$877$	2.37410e9	0.118850	0.0594252	+	0.998233i	$0.481073\pi$
$878$	0	0
$879$	−8.50323e9	−0.422302
$880$	0	0
$881$	−3.28058e10	−1.61635	−0.808175	−	0.588942i	$-0.799544\pi$
$881$	−3.28058e10	−1.61635	−0.808175	+	0.588942i	$0.799544\pi$
$882$	0	0
$883$	−2.14967e10	−1.05078	−0.525388	−	0.850863i	$-0.676080\pi$
$883$	−2.14967e10	−1.05078	−0.525388	+	0.850863i	$0.676080\pi$
$884$	0	0
$885$	4.75234e10	2.30466
$886$	0	0
$887$	1.50080e10	0.722089	0.361045	−	0.932549i	$-0.382420\pi$
$887$	1.50080e10	0.722089	0.361045	+	0.932549i	$0.382420\pi$
$888$	0	0
$889$	−4.93477e9	−0.235565
$890$	0	0
$891$	−1.95647e9	−0.0926617
$892$	0	0
$893$	−2.69665e10	−1.26720
$894$	0	0
$895$	−8.19532e9	−0.382107
$896$	0	0
$897$	−3.33549e9	−0.154307
$898$	0	0
$899$	3.68412e10	1.69112
$900$	0	0
$901$	1.86226e10	0.848212
$902$	0	0
$903$	−2.98438e9	−0.134880
$904$	0	0
$905$	−9.44124e9	−0.423408
$906$	0	0
$907$	1.57906e10	0.702704	0.351352	−	0.936243i	$-0.385722\pi$
$907$	1.57906e10	0.702704	0.351352	+	0.936243i	$0.385722\pi$
$908$	0	0
$909$	6.27643e10	2.77165
$910$	0	0
$911$	−6.82128e9	−0.298917	−0.149459	−	0.988768i	$-0.547753\pi$
$911$	−6.82128e9	−0.298917	−0.149459	+	0.988768i	$0.547753\pi$
$912$	0	0
$913$	3.01701e9	0.131199
$914$	0	0
$915$	1.26200e10	0.544609
$916$	0	0
$917$	3.86987e9	0.165731
$918$	0	0
$919$	1.08588e10	0.461506	0.230753	−	0.973012i	$-0.425881\pi$
$919$	1.08588e10	0.461506	0.230753	+	0.973012i	$0.425881\pi$
$920$	0	0
$921$	5.41253e10	2.28293
$922$	0	0
$923$	2.80967e9	0.117611
$924$	0	0
$925$	−1.80739e10	−0.750853
$926$	0	0
$927$	4.77036e10	1.96685
$928$	0	0
$929$	−2.99248e9	−0.122455	−0.0612275	−	0.998124i	$-0.519502\pi$
$929$	−2.99248e9	−0.122455	−0.0612275	+	0.998124i	$0.519502\pi$
$930$	0	0
$931$	2.07366e10	0.842197
$932$	0	0
$933$	−6.62721e10	−2.67144
$934$	0	0
$935$	−8.06335e9	−0.322608
$936$	0	0
$937$	−2.39333e10	−0.950417	−0.475208	−	0.879873i	$-0.657627\pi$
$937$	−2.39333e10	−0.950417	−0.475208	+	0.879873i	$0.657627\pi$
$938$	0	0
$939$	1.86261e10	0.734164
$940$	0	0
$941$	−3.17291e10	−1.24135	−0.620675	−	0.784068i	$-0.713141\pi$
$941$	−3.17291e10	−1.24135	−0.620675	+	0.784068i	$0.713141\pi$
$942$	0	0
$943$	1.17625e10	0.456781
$944$	0	0
$945$	1.76894e10	0.681871
$946$	0	0
$947$	−4.13161e10	−1.58086	−0.790431	−	0.612551i	$-0.790143\pi$
$947$	−4.13161e10	−1.58086	−0.790431	+	0.612551i	$0.790143\pi$
$948$	0	0
$949$	8.19876e9	0.311399
$950$	0	0
$951$	−4.14268e10	−1.56189
$952$	0	0
$953$	4.31993e10	1.61678	0.808390	−	0.588647i	$-0.200339\pi$
$953$	4.31993e10	1.61678	0.808390	+	0.588647i	$0.200339\pi$
$954$	0	0
$955$	−5.67216e9	−0.210735
$956$	0	0
$957$	−2.43167e10	−0.896835
$958$	0	0
$959$	1.46506e10	0.536401
$960$	0	0
$961$	2.50523e10	0.910574
$962$	0	0
$963$	−5.06400e10	−1.82727
$964$	0	0
$965$	−2.96140e10	−1.06084
$966$	0	0
$967$	−6.99318e9	−0.248703	−0.124352	−	0.992238i	$-0.539685\pi$
$967$	−6.99318e9	−0.248703	−0.124352	+	0.992238i	$0.539685\pi$
$968$	0	0
$969$	−3.06446e10	−1.08198
$970$	0	0
$971$	−8.48357e9	−0.297380	−0.148690	−	0.988884i	$-0.547506\pi$
$971$	−8.48357e9	−0.297380	−0.148690	+	0.988884i	$0.547506\pi$
$972$	0	0
$973$	−2.05521e10	−0.715255
$974$	0	0
$975$	6.19601e9	0.214090
$976$	0	0
$977$	−1.72197e10	−0.590739	−0.295370	−	0.955383i	$-0.595443\pi$
$977$	−1.72197e10	−0.590739	−0.295370	+	0.955383i	$0.595443\pi$
$978$	0	0
$979$	3.92317e9	0.133628
$980$	0	0
$981$	−1.77081e10	−0.598866
$982$	0	0
$983$	3.76267e10	1.26345	0.631725	−	0.775192i	$-0.282347\pi$
$983$	3.76267e10	1.26345	0.631725	+	0.775192i	$0.282347\pi$
$984$	0	0
$985$	−8.67222e9	−0.289137
$986$	0	0
$987$	−2.77117e10	−0.917389
$988$	0	0
$989$	−1.73203e9	−0.0569336
$990$	0	0
$991$	−3.13036e9	−0.102173	−0.0510866	−	0.998694i	$-0.516268\pi$
$991$	−3.13036e9	−0.102173	−0.0510866	+	0.998694i	$0.516268\pi$
$992$	0	0
$993$	−8.08387e10	−2.61997
$994$	0	0
$995$	1.80012e10	0.579323
$996$	0	0
$997$	1.67605e9	0.0535617	0.0267808	−	0.999641i	$-0.491474\pi$
$997$	1.67605e9	0.0535617	0.0267808	+	0.999641i	$0.491474\pi$
$998$	0	0
$999$	5.90878e10	1.87508

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.8.a.r.1.1		6
4.3	odd	2		256.8.a.q.1.6		6
8.3	odd	2		256.8.a.q.1.1		6
8.5	even	2	inner	256.8.a.r.1.6		6
16.3	odd	4		32.8.b.a.17.6		6
16.5	even	4		8.8.b.a.5.5	✓	6
16.11	odd	4		32.8.b.a.17.1		6
16.13	even	4		8.8.b.a.5.6	yes	6
48.5	odd	4		72.8.d.b.37.2		6
48.11	even	4		288.8.d.b.145.6		6
48.29	odd	4		72.8.d.b.37.1		6
48.35	even	4		288.8.d.b.145.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.8.b.a.5.5	✓	6	16.5	even	4
8.8.b.a.5.6	yes	6	16.13	even	4
32.8.b.a.17.1		6	16.11	odd	4
32.8.b.a.17.6		6	16.3	odd	4
72.8.d.b.37.1		6	48.29	odd	4
72.8.d.b.37.2		6	48.5	odd	4
256.8.a.q.1.1		6	8.3	odd	2
256.8.a.q.1.6		6	4.3	odd	2
256.8.a.r.1.1		6	1.1	even	1	trivial
256.8.a.r.1.6		6	8.5	even	2	inner
288.8.d.b.145.1		6	48.35	even	4
288.8.d.b.145.6		6	48.11	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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	256.a (trivial)

\(f(q)\)	\(=\)	\(q-76.9497 q^{3} -338.443 q^{5} +438.996 q^{7} +3734.25 q^{9} +O(q^{10})\)	\(q-76.9497 q^{3} -338.443 q^{5} +438.996 q^{7} +3734.25 q^{9} -1966.58 q^{11} -2210.98 q^{13} +26043.0 q^{15} -12114.9 q^{17} -32872.2 q^{19} -33780.6 q^{21} -19605.1 q^{23} +36418.4 q^{25} -119060. q^{27} -160689. q^{29} -229270. q^{31} +151328. q^{33} -148575. q^{35} -496284. q^{37} +170134. q^{39} -599971. q^{41} +88346.0 q^{43} -1.26383e6 q^{45} +820344. q^{47} -630825. q^{49} +932236. q^{51} -1.53717e6 q^{53} +665574. q^{55} +2.52950e6 q^{57} +1.82480e6 q^{59} +484582. q^{61} +1.63932e6 q^{63} +748290. q^{65} +79878.2 q^{67} +1.50860e6 q^{69} -1.27078e6 q^{71} -3.70820e6 q^{73} -2.80238e6 q^{75} -863321. q^{77} -2.55846e6 q^{79} +994857. q^{81} -1.53414e6 q^{83} +4.10019e6 q^{85} +1.23650e7 q^{87} -1.99492e6 q^{89} -970612. q^{91} +1.76423e7 q^{93} +1.11253e7 q^{95} -28917.7 q^{97} -7.34370e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 688 q^{7} + 2918 q^{9}+O(q^{10}) \) 6 * q + 688 * q^7 + 2918 * q^9	\( 6 q + 688 q^{7} + 2918 q^{9} + 17872 q^{15} + 1452 q^{17} + 1296 q^{23} + 39314 q^{25} - 89280 q^{31} + 53880 q^{33} + 328208 q^{39} - 521244 q^{41} + 1566432 q^{47} - 511050 q^{49} + 3270256 q^{55} + 1889896 q^{57} + 5776816 q^{63} + 1416480 q^{65} + 7597104 q^{71} - 2089564 q^{73} + 16015904 q^{79} - 723058 q^{81} + 37453776 q^{87} - 2169084 q^{89} + 48537936 q^{95} - 1088308 q^{97}+O(q^{100}) \) 6 * q + 688 * q^7 + 2918 * q^9 + 17872 * q^15 + 1452 * q^17 + 1296 * q^23 + 39314 * q^25 - 89280 * q^31 + 53880 * q^33 + 328208 * q^39 - 521244 * q^41 + 1566432 * q^47 - 511050 * q^49 + 3270256 * q^55 + 1889896 * q^57 + 5776816 * q^63 + 1416480 * q^65 + 7597104 * q^71 - 2089564 * q^73 + 16015904 * q^79 - 723058 * q^81 + 37453776 * q^87 - 2169084 * q^89 + 48537936 * q^95 - 1088308 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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