LMFDB - Embedded newform 256.7.c.l.255.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,7,Mod(255,256)]

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([1, 0]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("256.255");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	256.c (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:	$58.8938454067$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 11x^{6} + 516x^{4} - 2816x^{2} + 65536 $ x^8 - 11x^6 + 516x^4 - 2816*x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{42} $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			255.3
Root			$2.84502 - 2.81174i$ of defining polynomial
Character	$\chi$	$=$	256.255
Dual form			256.7.c.l.255.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.49390i q^{3} -59.7107 q^{5} -483.584i q^{7} +656.854 q^{9} +O(q^{10})$	$q-8.49390i q^{3} -59.7107 q^{5} -483.584i q^{7} +656.854 q^{9} +1412.15i q^{11} -3450.70 q^{13} +507.177i q^{15} -3056.78 q^{17} -968.104i q^{19} -4107.51 q^{21} -3314.31i q^{23} -12059.6 q^{25} -11771.3i q^{27} +26351.6 q^{29} -27104.3i q^{31} +11994.7 q^{33} +28875.1i q^{35} +36097.0 q^{37} +29309.9i q^{39} +6860.73 q^{41} +92831.6i q^{43} -39221.2 q^{45} +159323. i q^{47} -116204. q^{49} +25964.0i q^{51} -86612.3 q^{53} -84320.6i q^{55} -8222.98 q^{57} +128806. i q^{59} -189486. q^{61} -317644. i q^{63} +206043. q^{65} +319835. i q^{67} -28151.4 q^{69} +196890. i q^{71} +63957.9 q^{73} +102433. i q^{75} +682894. q^{77} +164678. i q^{79} +378862. q^{81} +802946. i q^{83} +182522. q^{85} -223828. i q^{87} +54145.3 q^{89} +1.66870e6i q^{91} -230221. q^{93} +57806.1i q^{95} -1.10670e6 q^{97} +927577. i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 1320 q^{9}+O(q^{10}) $ 8 * q + 1320 * q^9	$ 8 q + 1320 q^{9} + 8336 q^{17} + 47800 q^{25} + 168096 q^{33} + 235888 q^{41} + 4232 q^{49} - 253344 q^{57} - 410880 q^{65} - 887824 q^{73} - 1136088 q^{81} - 1522448 q^{89} - 1853552 q^{97}+O(q^{100}) $ 8 * q + 1320 * q^9 + 8336 * q^17 + 47800 * q^25 + 168096 * q^33 + 235888 * q^41 + 4232 * q^49 - 253344 * q^57 - 410880 * q^65 - 887824 * q^73 - 1136088 * q^81 - 1522448 * q^89 - 1853552 * 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
$3$	− 8.49390i	− 0.314589i	−0.987552	−	0.157294i	$-0.949723\pi$
$3$	− 8.49390i	− 0.314589i	0.987552	−	0.157294i	$-0.0502772\pi$
$4$	0	0
$5$	−59.7107	−0.477686	−0.238843	−	0.971058i	$-0.576768\pi$
$5$	−59.7107	−0.477686	−0.238843	+	0.971058i	$0.576768\pi$
$6$	0	0
$7$	− 483.584i	− 1.40987i	−0.709274	−	0.704933i	$-0.750977\pi$
$7$	− 483.584i	− 1.40987i	0.709274	−	0.704933i	$-0.249023\pi$
$8$	0	0
$9$	656.854	0.901034
$10$	0	0
$11$	1412.15i	1.06097i	0.847694	+	0.530485i	$0.177990\pi$
$11$	1412.15i	1.06097i	−0.847694	+	0.530485i	$0.822010\pi$
$12$	0	0
$13$	−3450.70	−1.57064	−0.785320	−	0.619090i	$-0.787501\pi$
$13$	−3450.70	−1.57064	−0.785320	+	0.619090i	$0.787501\pi$
$14$	0	0
$15$	507.177i	0.150275i
$16$	0	0
$17$	−3056.78	−0.622182	−0.311091	−	0.950380i	$-0.600694\pi$
$17$	−3056.78	−0.622182	−0.311091	+	0.950380i	$0.600694\pi$
$18$	0	0
$19$	− 968.104i	− 0.141144i	−0.997507	−	0.0705718i	$-0.977518\pi$
$19$	− 968.104i	− 0.141144i	0.997507	−	0.0705718i	$-0.0224824\pi$
$20$	0	0
$21$	−4107.51	−0.443528
$22$	0	0
$23$	− 3314.31i	− 0.272401i	−0.990681	−	0.136201i	$-0.956511\pi$
$23$	− 3314.31i	− 0.272401i	0.990681	−	0.136201i	$-0.0434892\pi$
$24$	0	0
$25$	−12059.6	−0.771817
$26$	0	0
$27$	− 11771.3i	− 0.598044i
$28$	0	0
$29$	26351.6	1.08047	0.540236	−	0.841513i	$-0.318335\pi$
$29$	26351.6	1.08047	0.540236	+	0.841513i	$0.318335\pi$
$30$	0	0
$31$	− 27104.3i	− 0.909815i	−0.890539	−	0.454907i	$-0.849672\pi$
$31$	− 27104.3i	− 0.909815i	0.890539	−	0.454907i	$-0.150328\pi$
$32$	0	0
$33$	11994.7	0.333770
$34$	0	0
$35$	28875.1i	0.673472i
$36$	0	0
$37$	36097.0	0.712633	0.356317	−	0.934365i	$-0.384032\pi$
$37$	36097.0	0.712633	0.356317	+	0.934365i	$0.384032\pi$
$38$	0	0
$39$	29309.9i	0.494106i
$40$	0	0
$41$	6860.73	0.0995449	0.0497724	−	0.998761i	$-0.484150\pi$
$41$	6860.73	0.0995449	0.0497724	+	0.998761i	$0.484150\pi$
$42$	0	0
$43$	92831.6i	1.16759i	0.811901	+	0.583795i	$0.198433\pi$
$43$	92831.6i	1.16759i	−0.811901	+	0.583795i	$0.801567\pi$
$44$	0	0
$45$	−39221.2	−0.430411
$46$	0	0
$47$	159323.i	1.53456i	0.641309	+	0.767282i	$0.278392\pi$
$47$	159323.i	1.53456i	−0.641309	+	0.767282i	$0.721608\pi$
$48$	0	0
$49$	−116204.	−0.987720
$50$	0	0
$51$	25964.0i	0.195732i
$52$	0	0
$53$	−86612.3	−0.581771	−0.290885	−	0.956758i	$-0.593950\pi$
$53$	−86612.3	−0.581771	−0.290885	+	0.956758i	$0.593950\pi$
$54$	0	0
$55$	− 84320.6i	− 0.506810i
$56$	0	0
$57$	−8222.98	−0.0444022
$58$	0	0
$59$	128806.i	0.627165i	0.949561	+	0.313582i	$0.101529\pi$
$59$	128806.i	0.627165i	−0.949561	+	0.313582i	$0.898471\pi$
$60$	0	0
$61$	−189486.	−0.834810	−0.417405	−	0.908721i	$-0.637060\pi$
$61$	−189486.	−0.834810	−0.417405	+	0.908721i	$0.637060\pi$
$62$	0	0
$63$	− 317644.i	− 1.27034i
$64$	0	0
$65$	206043.	0.750272
$66$	0	0
$67$	319835.i	1.06341i	0.846929	+	0.531706i	$0.178449\pi$
$67$	319835.i	1.06341i	−0.846929	+	0.531706i	$0.821551\pi$
$68$	0	0
$69$	−28151.4	−0.0856945
$70$	0	0
$71$	196890.i	0.550108i	0.961429	+	0.275054i	$0.0886957\pi$
$71$	196890.i	0.550108i	−0.961429	+	0.275054i	$0.911304\pi$
$72$	0	0
$73$	63957.9	0.164409	0.0822046	−	0.996615i	$-0.473804\pi$
$73$	63957.9	0.164409	0.0822046	+	0.996615i	$0.473804\pi$
$74$	0	0
$75$	102433.i	0.242805i
$76$	0	0
$77$	682894.	1.49583
$78$	0	0
$79$	164678.i	0.334006i	0.985956	+	0.167003i	$0.0534090\pi$
$79$	164678.i	0.334006i	−0.985956	+	0.167003i	$0.946591\pi$
$80$	0	0
$81$	378862.	0.712896
$82$	0	0
$83$	802946.i	1.40428i	0.712041	+	0.702138i	$0.247771\pi$
$83$	802946.i	1.40428i	−0.712041	+	0.702138i	$0.752229\pi$
$84$	0	0
$85$	182522.	0.297207
$86$	0	0
$87$	− 223828.i	− 0.339905i
$88$	0	0
$89$	54145.3	0.0768052	0.0384026	−	0.999262i	$-0.487773\pi$
$89$	54145.3	0.0768052	0.0384026	+	0.999262i	$0.487773\pi$
$90$	0	0
$91$	1.66870e6i	2.21439i
$92$	0	0
$93$	−230221.	−0.286218
$94$	0	0
$95$	57806.1i	0.0674222i
$96$	0	0
$97$	−1.10670e6	−1.21259	−0.606297	−	0.795238i	$-0.707346\pi$
$97$	−1.10670e6	−1.21259	−0.606297	+	0.795238i	$0.707346\pi$
$98$	0	0
$99$	927577.i	0.955971i
$100$	0	0
$101$	809583.	0.785773	0.392887	−	0.919587i	$-0.371476\pi$
$101$	809583.	0.785773	0.392887	+	0.919587i	$0.371476\pi$
$102$	0	0
$103$	− 619448.i	− 0.566883i	−0.958990	−	0.283442i	$-0.908524\pi$
$103$	− 619448.i	− 0.566883i	0.958990	−	0.283442i	$-0.0914762\pi$
$104$	0	0
$105$	245262.	0.211867
$106$	0	0
$107$	1.08619e6i	0.886657i	0.896359	+	0.443328i	$0.146202\pi$
$107$	1.08619e6i	0.886657i	−0.896359	+	0.443328i	$0.853798\pi$
$108$	0	0
$109$	−1.71692e6	−1.32578	−0.662890	−	0.748717i	$-0.730670\pi$
$109$	−1.71692e6	−1.32578	−0.662890	+	0.748717i	$0.730670\pi$
$110$	0	0
$111$	− 306604.i	− 0.224187i
$112$	0	0
$113$	645423.	0.447311	0.223655	−	0.974668i	$-0.428201\pi$
$113$	645423.	0.447311	0.223655	+	0.974668i	$0.428201\pi$
$114$	0	0
$115$	197900.i	0.130122i
$116$	0	0
$117$	−2.26660e6	−1.41520
$118$	0	0
$119$	1.47821e6i	0.877193i
$120$	0	0
$121$	−222613.	−0.125659
$122$	0	0
$123$	− 58274.4i	− 0.0313157i
$124$	0	0
$125$	1.65307e6	0.846371
$126$	0	0
$127$	2.51504e6i	1.22781i	0.789378	+	0.613907i	$0.210403\pi$
$127$	2.51504e6i	1.22781i	−0.789378	+	0.613907i	$0.789597\pi$
$128$	0	0
$129$	788502.	0.367311
$130$	0	0
$131$	3.77470e6i	1.67907i	0.543305	+	0.839536i	$0.317173\pi$
$131$	3.77470e6i	1.67907i	−0.543305	+	0.839536i	$0.682827\pi$
$132$	0	0
$133$	−468159.	−0.198993
$134$	0	0
$135$	702873.i	0.285677i
$136$	0	0
$137$	−1.40637e6	−0.546939	−0.273470	−	0.961881i	$-0.588171\pi$
$137$	−1.40637e6	−0.546939	−0.273470	+	0.961881i	$0.588171\pi$
$138$	0	0
$139$	2.31561e6i	0.862227i	0.902298	+	0.431114i	$0.141879\pi$
$139$	2.31561e6i	0.862227i	−0.902298	+	0.431114i	$0.858121\pi$
$140$	0	0
$141$	1.35327e6	0.482757
$142$	0	0
$143$	− 4.87291e6i	− 1.66640i
$144$	0	0
$145$	−1.57347e6	−0.516126
$146$	0	0
$147$	987028.i	0.310726i
$148$	0	0
$149$	4.74866e6	1.43553	0.717765	−	0.696285i	$-0.245165\pi$
$149$	4.74866e6	1.43553	0.717765	+	0.696285i	$0.245165\pi$
$150$	0	0
$151$	− 1.11787e6i	− 0.324684i	−0.986735	−	0.162342i	$-0.948095\pi$
$151$	− 1.11787e6i	− 0.324684i	0.986735	−	0.162342i	$-0.0519047\pi$
$152$	0	0
$153$	−2.00786e6	−0.560607
$154$	0	0
$155$	1.61842e6i	0.434605i
$156$	0	0
$157$	−1.00407e6	−0.259458	−0.129729	−	0.991550i	$-0.541411\pi$
$157$	−1.00407e6	−0.259458	−0.129729	+	0.991550i	$0.541411\pi$
$158$	0	0
$159$	735676.i	0.183019i
$160$	0	0
$161$	−1.60275e6	−0.384049
$162$	0	0
$163$	− 1.47315e6i	− 0.340161i	−0.985430	−	0.170080i	$-0.945597\pi$
$163$	− 1.47315e6i	− 0.340161i	0.985430	−	0.170080i	$-0.0544027\pi$
$164$	0	0
$165$	−716211.	−0.159437
$166$	0	0
$167$	− 5.16350e6i	− 1.10865i	−0.832300	−	0.554326i	$-0.812976\pi$
$167$	− 5.16350e6i	− 1.10865i	0.832300	−	0.554326i	$-0.187024\pi$
$168$	0	0
$169$	7.08049e6	1.46691
$170$	0	0
$171$	− 635902.i	− 0.127175i
$172$	0	0
$173$	600245.	0.115928	0.0579642	−	0.998319i	$-0.481539\pi$
$173$	600245.	0.115928	0.0579642	+	0.998319i	$0.481539\pi$
$174$	0	0
$175$	5.83184e6i	1.08816i
$176$	0	0
$177$	1.09407e6	0.197299
$178$	0	0
$179$	− 6.34654e6i	− 1.10657i	−0.832993	−	0.553284i	$-0.813375\pi$
$179$	− 6.34654e6i	− 1.10657i	0.832993	−	0.553284i	$-0.186625\pi$
$180$	0	0
$181$	−1.35031e6	−0.227719	−0.113859	−	0.993497i	$-0.536321\pi$
$181$	−1.35031e6	−0.227719	−0.113859	+	0.993497i	$0.536321\pi$
$182$	0	0
$183$	1.60947e6i	0.262622i
$184$	0	0
$185$	−2.15538e6	−0.340415
$186$	0	0
$187$	− 4.31664e6i	− 0.660117i
$188$	0	0
$189$	−5.69241e6	−0.843162
$190$	0	0
$191$	− 3.32043e6i	− 0.476534i	−0.971200	−	0.238267i	$-0.923421\pi$
$191$	− 3.32043e6i	− 0.476534i	0.971200	−	0.238267i	$-0.0765794\pi$
$192$	0	0
$193$	5.59202e6	0.777852	0.388926	−	0.921269i	$-0.372846\pi$
$193$	5.59202e6	0.777852	0.388926	+	0.921269i	$0.372846\pi$
$194$	0	0
$195$	− 1.75011e6i	− 0.236027i
$196$	0	0
$197$	−7.19799e6	−0.941483	−0.470742	−	0.882271i	$-0.656014\pi$
$197$	−7.19799e6	−0.941483	−0.470742	+	0.882271i	$0.656014\pi$
$198$	0	0
$199$	1.15838e7i	1.46991i	0.678116	+	0.734955i	$0.262797\pi$
$199$	1.15838e7i	1.46991i	−0.678116	+	0.734955i	$0.737203\pi$
$200$	0	0
$201$	2.71665e6	0.334538
$202$	0	0
$203$	− 1.27432e7i	− 1.52332i
$204$	0	0
$205$	−409659.	−0.0475511
$206$	0	0
$207$	− 2.17701e6i	− 0.245443i
$208$	0	0
$209$	1.36711e6	0.149749
$210$	0	0
$211$	639344.i	0.0680592i	0.999421	+	0.0340296i	$0.0108341\pi$
$211$	639344.i	0.0680592i	−0.999421	+	0.0340296i	$0.989166\pi$
$212$	0	0
$213$	1.67236e6	0.173058
$214$	0	0
$215$	− 5.54304e6i	− 0.557741i
$216$	0	0
$217$	−1.31072e7	−1.28272
$218$	0	0
$219$	− 543252.i	− 0.0517213i
$220$	0	0
$221$	1.05480e7	0.977224
$222$	0	0
$223$	− 9.51144e6i	− 0.857693i	−0.903377	−	0.428846i	$-0.858920\pi$
$223$	− 9.51144e6i	− 0.857693i	0.903377	−	0.428846i	$-0.141080\pi$
$224$	0	0
$225$	−7.92141e6	−0.695433
$226$	0	0
$227$	− 1.27875e7i	− 1.09322i	−0.837387	−	0.546610i	$-0.815918\pi$
$227$	− 1.27875e7i	− 1.09322i	0.837387	−	0.546610i	$-0.184082\pi$
$228$	0	0
$229$	4.61930e6	0.384653	0.192327	−	0.981331i	$-0.438397\pi$
$229$	4.61930e6	0.384653	0.192327	+	0.981331i	$0.438397\pi$
$230$	0	0
$231$	− 5.80043e6i	− 0.470570i
$232$	0	0
$233$	9.41375e6	0.744209	0.372105	−	0.928191i	$-0.378636\pi$
$233$	9.41375e6	0.744209	0.372105	+	0.928191i	$0.378636\pi$
$234$	0	0
$235$	− 9.51329e6i	− 0.733039i
$236$	0	0
$237$	1.39876e6	0.105075
$238$	0	0
$239$	2.57043e7i	1.88283i	0.337247	+	0.941416i	$0.390504\pi$
$239$	2.57043e7i	1.88283i	−0.337247	+	0.941416i	$0.609496\pi$
$240$	0	0
$241$	8.16838e6	0.583559	0.291779	−	0.956486i	$-0.405753\pi$
$241$	8.16838e6	0.583559	0.291779	+	0.956486i	$0.405753\pi$
$242$	0	0
$243$	− 1.17993e7i	− 0.822313i
$244$	0	0
$245$	6.93864e6	0.471820
$246$	0	0
$247$	3.34063e6i	0.221686i
$248$	0	0
$249$	6.82015e6	0.441769
$250$	0	0
$251$	− 8.44540e6i	− 0.534071i	−0.963687	−	0.267035i	$-0.913956\pi$
$251$	− 8.44540e6i	− 0.534071i	0.963687	−	0.267035i	$-0.0860441\pi$
$252$	0	0
$253$	4.68031e6	0.289010
$254$	0	0
$255$	− 1.55033e6i	− 0.0934981i
$256$	0	0
$257$	−1.84481e7	−1.08681	−0.543403	−	0.839472i	$-0.682864\pi$
$257$	−1.84481e7	−1.08681	−0.543403	+	0.839472i	$0.682864\pi$
$258$	0	0
$259$	− 1.74559e7i	− 1.00472i
$260$	0	0
$261$	1.73092e7	0.973542
$262$	0	0
$263$	3.84095e6i	0.211140i	0.994412	+	0.105570i	$0.0336668\pi$
$263$	3.84095e6i	0.211140i	−0.994412	+	0.105570i	$0.966333\pi$
$264$	0	0
$265$	5.17168e6	0.277903
$266$	0	0
$267$	− 459904.i	− 0.0241621i
$268$	0	0
$269$	−3.10430e7	−1.59480	−0.797400	−	0.603451i	$-0.793792\pi$
$269$	−3.10430e7	−1.59480	−0.797400	+	0.603451i	$0.793792\pi$
$270$	0	0
$271$	− 2.35560e7i	− 1.18357i	−0.806097	−	0.591784i	$-0.798424\pi$
$271$	− 2.35560e7i	− 1.18357i	0.806097	−	0.591784i	$-0.201576\pi$
$272$	0	0
$273$	1.41738e7	0.696623
$274$	0	0
$275$	− 1.70300e7i	− 0.818875i
$276$	0	0
$277$	−2.32189e7	−1.09245	−0.546225	−	0.837638i	$-0.683936\pi$
$277$	−2.32189e7	−1.09245	−0.546225	+	0.837638i	$0.683936\pi$
$278$	0	0
$279$	− 1.78035e7i	− 0.819774i
$280$	0	0
$281$	−2.99519e7	−1.34991	−0.674955	−	0.737859i	$-0.735837\pi$
$281$	−2.99519e7	−1.34991	−0.674955	+	0.737859i	$0.735837\pi$
$282$	0	0
$283$	− 3.68036e7i	− 1.62380i	−0.583800	−	0.811898i	$-0.698435\pi$
$283$	− 3.68036e7i	− 1.62380i	0.583800	−	0.811898i	$-0.301565\pi$
$284$	0	0
$285$	491000.	0.0212103
$286$	0	0
$287$	− 3.31774e6i	− 0.140345i
$288$	0	0
$289$	−1.47937e7	−0.612890
$290$	0	0
$291$	9.40021e6i	0.381469i
$292$	0	0
$293$	−3.36919e7	−1.33944	−0.669718	−	0.742615i	$-0.733585\pi$
$293$	−3.36919e7	−1.33944	−0.669718	+	0.742615i	$0.733585\pi$
$294$	0	0
$295$	− 7.69112e6i	− 0.299588i
$296$	0	0
$297$	1.66229e7	0.634508
$298$	0	0
$299$	1.14367e7i	0.427844i
$300$	0	0
$301$	4.48918e7	1.64614
$302$	0	0
$303$	− 6.87652e6i	− 0.247196i
$304$	0	0
$305$	1.13143e7	0.398776
$306$	0	0
$307$	3.69782e7i	1.27800i	0.769207	+	0.639000i	$0.220651\pi$
$307$	3.69782e7i	1.27800i	−0.769207	+	0.639000i	$0.779349\pi$
$308$	0	0
$309$	−5.26153e6	−0.178335
$310$	0	0
$311$	3.65356e7i	1.21461i	0.794470	+	0.607303i	$0.207749\pi$
$311$	3.65356e7i	1.21461i	−0.794470	+	0.607303i	$0.792251\pi$
$312$	0	0
$313$	−5.50169e7	−1.79417	−0.897083	−	0.441861i	$-0.854318\pi$
$313$	−5.50169e7	−1.79417	−0.897083	+	0.441861i	$0.854318\pi$
$314$	0	0
$315$	1.89667e7i	0.606821i
$316$	0	0
$317$	−4.12495e7	−1.29491	−0.647457	−	0.762102i	$-0.724168\pi$
$317$	−4.12495e7	−1.29491	−0.647457	+	0.762102i	$0.724168\pi$
$318$	0	0
$319$	3.72125e7i	1.14635i
$320$	0	0
$321$	9.22601e6	0.278932
$322$	0	0
$323$	2.95928e6i	0.0878170i
$324$	0	0
$325$	4.16141e7	1.21225
$326$	0	0
$327$	1.45834e7i	0.417076i
$328$	0	0
$329$	7.70461e7	2.16353
$330$	0	0
$331$	2.72892e7i	0.752502i	0.926518	+	0.376251i	$0.122787\pi$
$331$	2.72892e7i	0.752502i	−0.926518	+	0.376251i	$0.877213\pi$
$332$	0	0
$333$	2.37104e7	0.642107
$334$	0	0
$335$	− 1.90976e7i	− 0.507977i
$336$	0	0
$337$	−1.36843e6	−0.0357547	−0.0178774	−	0.999840i	$-0.505691\pi$
$337$	−1.36843e6	−0.0357547	−0.0178774	+	0.999840i	$0.505691\pi$
$338$	0	0
$339$	− 5.48216e6i	− 0.140719i
$340$	0	0
$341$	3.82754e7	0.965287
$342$	0	0
$343$	− 698651.i	− 0.0173132i
$344$	0	0
$345$	1.68094e6	0.0409350
$346$	0	0
$347$	1.07803e7i	0.258013i	0.991644	+	0.129006i	$0.0411788\pi$
$347$	1.07803e7i	0.258013i	−0.991644	+	0.129006i	$0.958821\pi$
$348$	0	0
$349$	1.33186e6	0.0313316	0.0156658	−	0.999877i	$-0.495013\pi$
$349$	1.33186e6	0.0313316	0.0156658	+	0.999877i	$0.495013\pi$
$350$	0	0
$351$	4.06192e7i	0.939312i
$352$	0	0
$353$	798852.	0.0181611	0.00908055	−	0.999959i	$-0.497110\pi$
$353$	798852.	0.0181611	0.00908055	+	0.999959i	$0.497110\pi$
$354$	0	0
$355$	− 1.17564e7i	− 0.262779i
$356$	0	0
$357$	1.25558e7	0.275955
$358$	0	0
$359$	− 5.67611e7i	− 1.22678i	−0.789779	−	0.613392i	$-0.789805\pi$
$359$	− 5.67611e7i	− 1.22678i	0.789779	−	0.613392i	$-0.210195\pi$
$360$	0	0
$361$	4.61087e7	0.980078
$362$	0	0
$363$	1.89086e6i	0.0395311i
$364$	0	0
$365$	−3.81897e6	−0.0785359
$366$	0	0
$367$	− 1.80379e7i	− 0.364911i	−0.983214	−	0.182456i	$-0.941595\pi$
$367$	− 1.80379e7i	− 0.364911i	0.983214	−	0.182456i	$-0.0584046\pi$
$368$	0	0
$369$	4.50650e6	0.0896933
$370$	0	0
$371$	4.18843e7i	0.820218i
$372$	0	0
$373$	−3.58446e7	−0.690712	−0.345356	−	0.938472i	$-0.612242\pi$
$373$	−3.58446e7	−0.690712	−0.345356	+	0.938472i	$0.612242\pi$
$374$	0	0
$375$	− 1.40410e7i	− 0.266259i
$376$	0	0
$377$	−9.09315e7	−1.69703
$378$	0	0
$379$	9.70544e7i	1.78278i	0.453238	+	0.891390i	$0.350269\pi$
$379$	9.70544e7i	1.78278i	−0.453238	+	0.891390i	$0.649731\pi$
$380$	0	0
$381$	2.13625e7	0.386257
$382$	0	0
$383$	− 3.16741e7i	− 0.563779i	−0.959447	−	0.281889i	$-0.909039\pi$
$383$	− 3.16741e7i	− 0.563779i	0.959447	−	0.281889i	$-0.0909611\pi$
$384$	0	0
$385$	−4.07761e7	−0.714534
$386$	0	0
$387$	6.09768e7i	1.05204i
$388$	0	0
$389$	−5.50526e7	−0.935252	−0.467626	−	0.883926i	$-0.654891\pi$
$389$	−5.50526e7	−0.935252	−0.467626	+	0.883926i	$0.654891\pi$
$390$	0	0
$391$	1.01311e7i	0.169483i
$392$	0	0
$393$	3.20620e7	0.528217
$394$	0	0
$395$	− 9.83305e6i	− 0.159550i
$396$	0	0
$397$	−8.33093e7	−1.33144	−0.665721	−	0.746201i	$-0.731876\pi$
$397$	−8.33093e7	−1.33144	−0.665721	+	0.746201i	$0.731876\pi$
$398$	0	0
$399$	3.97650e6i	0.0626011i
$400$	0	0
$401$	4.24726e7	0.658682	0.329341	−	0.944211i	$-0.393174\pi$
$401$	4.24726e7	0.658682	0.329341	+	0.944211i	$0.393174\pi$
$402$	0	0
$403$	9.35286e7i	1.42899i
$404$	0	0
$405$	−2.26221e7	−0.340540
$406$	0	0
$407$	5.09745e7i	0.756083i
$408$	0	0
$409$	−4.61036e7	−0.673853	−0.336926	−	0.941531i	$-0.609387\pi$
$409$	−4.61036e7	−0.673853	−0.336926	+	0.941531i	$0.609387\pi$
$410$	0	0
$411$	1.19456e7i	0.172061i
$412$	0	0
$413$	6.22887e7	0.884218
$414$	0	0
$415$	− 4.79445e7i	− 0.670802i
$416$	0	0
$417$	1.96686e7	0.271247
$418$	0	0
$419$	3.86614e7i	0.525575i	0.964854	+	0.262788i	$0.0846419\pi$
$419$	3.86614e7i	0.525575i	−0.964854	+	0.262788i	$0.915358\pi$
$420$	0	0
$421$	2.58211e7	0.346042	0.173021	−	0.984918i	$-0.444647\pi$
$421$	2.58211e7	0.346042	0.173021	+	0.984918i	$0.444647\pi$
$422$	0	0
$423$	1.04652e8i	1.38269i
$424$	0	0
$425$	3.68636e7	0.480210
$426$	0	0
$427$	9.16323e7i	1.17697i
$428$	0	0
$429$	−4.13900e7	−0.524232
$430$	0	0
$431$	3.09589e7i	0.386681i	0.981132	+	0.193341i	$0.0619322\pi$
$431$	3.09589e7i	0.386681i	−0.981132	+	0.193341i	$0.938068\pi$
$432$	0	0
$433$	4.64302e7	0.571921	0.285961	−	0.958241i	$-0.407687\pi$
$433$	4.64302e7	0.571921	0.285961	+	0.958241i	$0.407687\pi$
$434$	0	0
$435$	1.33649e7i	0.162368i
$436$	0	0
$437$	−3.20859e6	−0.0384477
$438$	0	0
$439$	− 1.33692e8i	− 1.58020i	−0.612979	−	0.790099i	$-0.710029\pi$
$439$	− 1.33692e8i	− 1.58020i	0.612979	−	0.790099i	$-0.289971\pi$
$440$	0	0
$441$	−7.63292e7	−0.889969
$442$	0	0
$443$	− 5.29536e6i	− 0.0609094i	−0.999536	−	0.0304547i	$-0.990304\pi$
$443$	− 5.29536e6i	− 0.0609094i	0.999536	−	0.0304547i	$-0.00969552\pi$
$444$	0	0
$445$	−3.23305e6	−0.0366887
$446$	0	0
$447$	− 4.03347e7i	− 0.451602i
$448$	0	0
$449$	−1.20011e8	−1.32582	−0.662908	−	0.748701i	$-0.730678\pi$
$449$	−1.20011e8	−1.32582	−0.662908	+	0.748701i	$0.730678\pi$
$450$	0	0
$451$	9.68840e6i	0.105614i
$452$	0	0
$453$	−9.49508e6	−0.102142
$454$	0	0
$455$	− 9.96392e7i	− 1.05778i
$456$	0	0
$457$	1.84689e8	1.93505	0.967527	−	0.252767i	$-0.0813405\pi$
$457$	1.84689e8	1.93505	0.967527	+	0.252767i	$0.0813405\pi$
$458$	0	0
$459$	3.59823e7i	0.372092i
$460$	0	0
$461$	1.46730e8	1.49767	0.748836	−	0.662755i	$-0.230613\pi$
$461$	1.46730e8	1.49767	0.748836	+	0.662755i	$0.230613\pi$
$462$	0	0
$463$	1.55124e8i	1.56292i	0.623956	+	0.781459i	$0.285524\pi$
$463$	1.55124e8i	1.56292i	−0.623956	+	0.781459i	$0.714476\pi$
$464$	0	0
$465$	1.37467e7	0.136722
$466$	0	0
$467$	− 1.07539e8i	− 1.05589i	−0.849280	−	0.527943i	$-0.822963\pi$
$467$	− 1.07539e8i	− 1.05589i	0.849280	−	0.527943i	$-0.177037\pi$
$468$	0	0
$469$	1.54667e8	1.49927
$470$	0	0
$471$	8.52850e6i	0.0816225i
$472$	0	0
$473$	−1.31092e8	−1.23878
$474$	0	0
$475$	1.16750e7i	0.108937i
$476$	0	0
$477$	−5.68916e7	−0.524195
$478$	0	0
$479$	− 1.48881e7i	− 0.135467i	−0.997703	−	0.0677333i	$-0.978423\pi$
$479$	− 1.48881e7i	− 0.135467i	0.997703	−	0.0677333i	$-0.0215767\pi$
$480$	0	0
$481$	−1.24560e8	−1.11929
$482$	0	0
$483$	1.36136e7i	0.120818i
$484$	0	0
$485$	6.60819e7	0.579239
$486$	0	0
$487$	9.69346e7i	0.839251i	0.907697	+	0.419625i	$0.137839\pi$
$487$	9.69346e7i	0.839251i	−0.907697	+	0.419625i	$0.862161\pi$
$488$	0	0
$489$	−1.25128e7	−0.107011
$490$	0	0
$491$	− 6.45564e7i	− 0.545374i	−0.962103	−	0.272687i	$-0.912088\pi$
$491$	− 6.45564e7i	− 0.545374i	0.962103	−	0.272687i	$-0.0879124\pi$
$492$	0	0
$493$	−8.05512e7	−0.672250
$494$	0	0
$495$	− 5.53863e7i	− 0.456653i
$496$	0	0
$497$	9.52126e7	0.775578
$498$	0	0
$499$	− 8.09040e7i	− 0.651131i	−0.945520	−	0.325565i	$-0.894445\pi$
$499$	− 8.09040e7i	− 0.651131i	0.945520	−	0.325565i	$-0.105555\pi$
$500$	0	0
$501$	−4.38583e7	−0.348770
$502$	0	0
$503$	1.32342e8i	1.03990i	0.854196	+	0.519951i	$0.174050\pi$
$503$	1.32342e8i	1.03990i	−0.854196	+	0.519951i	$0.825950\pi$
$504$	0	0
$505$	−4.83408e7	−0.375352
$506$	0	0
$507$	− 6.01410e7i	− 0.461473i
$508$	0	0
$509$	−1.39233e7	−0.105582	−0.0527909	−	0.998606i	$-0.516812\pi$
$509$	−1.39233e7	−0.105582	−0.0527909	+	0.998606i	$0.516812\pi$
$510$	0	0
$511$	− 3.09290e7i	− 0.231795i
$512$	0	0
$513$	−1.13958e7	−0.0844101
$514$	0	0
$515$	3.69877e7i	0.270792i
$516$	0	0
$517$	−2.24989e8	−1.62813
$518$	0	0
$519$	− 5.09842e6i	− 0.0364698i
$520$	0	0
$521$	9.25151e7	0.654183	0.327092	−	0.944993i	$-0.393931\pi$
$521$	9.25151e7	0.654183	0.327092	+	0.944993i	$0.393931\pi$
$522$	0	0
$523$	− 5.17905e7i	− 0.362030i	−0.983480	−	0.181015i	$-0.942062\pi$
$523$	− 5.17905e7i	− 0.362030i	0.983480	−	0.181015i	$-0.0579383\pi$
$524$	0	0
$525$	4.95351e7	0.342322
$526$	0	0
$527$	8.28518e7i	0.566070i
$528$	0	0
$529$	1.37051e8	0.925797
$530$	0	0
$531$	8.46070e7i	0.565097i
$532$	0	0
$533$	−2.36743e7	−0.156349
$534$	0	0
$535$	− 6.48573e7i	− 0.423543i
$536$	0	0
$537$	−5.39069e7	−0.348114
$538$	0	0
$539$	− 1.64098e8i	− 1.04794i
$540$	0	0
$541$	−2.48317e7	−0.156825	−0.0784125	−	0.996921i	$-0.524985\pi$
$541$	−2.48317e7	−0.156825	−0.0784125	+	0.996921i	$0.524985\pi$
$542$	0	0
$543$	1.14694e7i	0.0716378i
$544$	0	0
$545$	1.02519e8	0.633306
$546$	0	0
$547$	− 8.23167e7i	− 0.502951i	−0.967864	−	0.251476i	$-0.919084\pi$
$547$	− 8.23167e7i	− 0.502951i	0.967864	−	0.251476i	$-0.0809159\pi$
$548$	0	0
$549$	−1.24465e8	−0.752192
$550$	0	0
$551$	− 2.55111e7i	− 0.152502i
$552$	0	0
$553$	7.96357e7	0.470904
$554$	0	0
$555$	1.83076e7i	0.107091i
$556$	0	0
$557$	1.99169e7	0.115254	0.0576271	−	0.998338i	$-0.481647\pi$
$557$	1.99169e7	0.115254	0.0576271	+	0.998338i	$0.481647\pi$
$558$	0	0
$559$	− 3.20333e8i	− 1.83386i
$560$	0	0
$561$	−3.66651e7	−0.207666
$562$	0	0
$563$	− 3.05819e8i	− 1.71372i	−0.515550	−	0.856859i	$-0.672412\pi$
$563$	− 3.05819e8i	− 1.71372i	0.515550	−	0.856859i	$-0.327588\pi$
$564$	0	0
$565$	−3.85387e7	−0.213674
$566$	0	0
$567$	− 1.83212e8i	− 1.00509i
$568$	0	0
$569$	2.25936e6	0.0122644	0.00613222	−	0.999981i	$-0.498048\pi$
$569$	2.25936e6	0.0122644	0.00613222	+	0.999981i	$0.498048\pi$
$570$	0	0
$571$	− 6.55464e7i	− 0.352079i	−0.984383	−	0.176040i	$-0.943671\pi$
$571$	− 6.55464e7i	− 0.352079i	0.984383	−	0.176040i	$-0.0563287\pi$
$572$	0	0
$573$	−2.82034e7	−0.149912
$574$	0	0
$575$	3.99693e7i	0.210244i
$576$	0	0
$577$	5.43876e7	0.283121	0.141561	−	0.989930i	$-0.454788\pi$
$577$	5.43876e7	0.283121	0.141561	+	0.989930i	$0.454788\pi$
$578$	0	0
$579$	− 4.74981e7i	− 0.244703i
$580$	0	0
$581$	3.88292e8	1.97984
$582$	0	0
$583$	− 1.22310e8i	− 0.617242i
$584$	0	0
$585$	1.35340e8	0.676020
$586$	0	0
$587$	− 3.85369e8i	− 1.90529i	−0.304078	−	0.952647i	$-0.598348\pi$
$587$	− 3.85369e8i	− 1.90529i	0.304078	−	0.952647i	$-0.401652\pi$
$588$	0	0
$589$	−2.62398e7	−0.128414
$590$	0	0
$591$	6.11390e7i	0.296180i
$592$	0	0
$593$	−3.00090e8	−1.43909	−0.719545	−	0.694446i	$-0.755650\pi$
$593$	−3.00090e8	−1.43909	−0.719545	+	0.694446i	$0.755650\pi$
$594$	0	0
$595$	− 8.82649e7i	− 0.419022i
$596$	0	0
$597$	9.83914e7	0.462417
$598$	0	0
$599$	3.11371e8i	1.44876i	0.689400	+	0.724381i	$0.257874\pi$
$599$	3.11371e8i	1.44876i	−0.689400	+	0.724381i	$0.742126\pi$
$600$	0	0
$601$	−2.10925e8	−0.971637	−0.485818	−	0.874060i	$-0.661478\pi$
$601$	−2.10925e8	−0.971637	−0.485818	+	0.874060i	$0.661478\pi$
$602$	0	0
$603$	2.10085e8i	0.958171i
$604$	0	0
$605$	1.32924e7	0.0600257
$606$	0	0
$607$	3.28208e8i	1.46752i	0.679410	+	0.733759i	$0.262236\pi$
$607$	3.28208e8i	1.46752i	−0.679410	+	0.733759i	$0.737764\pi$
$608$	0	0
$609$	−1.08240e8	−0.479220
$610$	0	0
$611$	− 5.49776e8i	− 2.41025i
$612$	0	0
$613$	1.43677e8	0.623743	0.311872	−	0.950124i	$-0.399044\pi$
$613$	1.43677e8	0.623743	0.311872	+	0.950124i	$0.399044\pi$
$614$	0	0
$615$	3.47960e6i	0.0149591i
$616$	0	0
$617$	1.50375e8	0.640206	0.320103	−	0.947383i	$-0.396283\pi$
$617$	1.50375e8	0.640206	0.320103	+	0.947383i	$0.396283\pi$
$618$	0	0
$619$	− 2.07480e8i	− 0.874793i	−0.899269	−	0.437396i	$-0.855901\pi$
$619$	− 2.07480e8i	− 0.874793i	0.899269	−	0.437396i	$-0.144099\pi$
$620$	0	0
$621$	−3.90137e7	−0.162908
$622$	0	0
$623$	− 2.61838e7i	− 0.108285i
$624$	0	0
$625$	8.97259e7	0.367517
$626$	0	0
$627$	− 1.16121e7i	− 0.0471094i
$628$	0	0
$629$	−1.10341e8	−0.443388
$630$	0	0
$631$	9.08069e7i	0.361436i	0.983535	+	0.180718i	$0.0578421\pi$
$631$	9.08069e7i	0.361436i	−0.983535	+	0.180718i	$0.942158\pi$
$632$	0	0
$633$	5.43052e6	0.0214107
$634$	0	0
$635$	− 1.50174e8i	− 0.586509i
$636$	0	0
$637$	4.00986e8	1.55135
$638$	0	0
$639$	1.29328e8i	0.495666i
$640$	0	0
$641$	1.95456e8	0.742121	0.371061	−	0.928609i	$-0.378994\pi$
$641$	1.95456e8	0.742121	0.371061	+	0.928609i	$0.378994\pi$
$642$	0	0
$643$	3.72026e8i	1.39939i	0.714440	+	0.699697i	$0.246682\pi$
$643$	3.72026e8i	1.39939i	−0.714440	+	0.699697i	$0.753318\pi$
$644$	0	0
$645$	−4.70820e7	−0.175459
$646$	0	0
$647$	− 2.47447e8i	− 0.913627i	−0.889563	−	0.456813i	$-0.848991\pi$
$647$	− 2.47447e8i	− 0.913627i	0.889563	−	0.456813i	$-0.151009\pi$
$648$	0	0
$649$	−1.81894e8	−0.665404
$650$	0	0
$651$	1.11331e8i	0.403528i
$652$	0	0
$653$	−1.98187e8	−0.711764	−0.355882	−	0.934531i	$-0.615820\pi$
$653$	−1.98187e8	−0.711764	−0.355882	+	0.934531i	$0.615820\pi$
$654$	0	0
$655$	− 2.25390e8i	− 0.802068i
$656$	0	0
$657$	4.20110e7	0.148138
$658$	0	0
$659$	5.71625e7i	0.199735i	0.995001	+	0.0998677i	$0.0318419\pi$
$659$	5.71625e7i	0.199735i	−0.995001	+	0.0998677i	$0.968158\pi$
$660$	0	0
$661$	−5.04948e8	−1.74841	−0.874203	−	0.485561i	$-0.838615\pi$
$661$	−5.04948e8	−1.74841	−0.874203	+	0.485561i	$0.838615\pi$
$662$	0	0
$663$	− 8.95938e7i	− 0.307424i
$664$	0	0
$665$	2.79541e7	0.0950563
$666$	0	0
$667$	− 8.73374e7i	− 0.294322i
$668$	0	0
$669$	−8.07893e7	−0.269821
$670$	0	0
$671$	− 2.67583e8i	− 0.885709i
$672$	0	0
$673$	−1.59423e8	−0.523004	−0.261502	−	0.965203i	$-0.584218\pi$
$673$	−1.59423e8	−0.523004	−0.261502	+	0.965203i	$0.584218\pi$
$674$	0	0
$675$	1.41958e8i	0.461580i
$676$	0	0
$677$	−1.89358e8	−0.610265	−0.305132	−	0.952310i	$-0.598701\pi$
$677$	−1.89358e8	−0.610265	−0.305132	+	0.952310i	$0.598701\pi$
$678$	0	0
$679$	5.35183e8i	1.70959i
$680$	0	0
$681$	−1.08616e8	−0.343915
$682$	0	0
$683$	6.31662e7i	0.198254i	0.995075	+	0.0991272i	$0.0316051\pi$
$683$	6.31662e7i	0.198254i	−0.995075	+	0.0991272i	$0.968395\pi$
$684$	0	0
$685$	8.39755e7	0.261265
$686$	0	0
$687$	− 3.92358e7i	− 0.121008i
$688$	0	0
$689$	2.98872e8	0.913752
$690$	0	0
$691$	1.37266e8i	0.416033i	0.978125	+	0.208016i	$0.0667007\pi$
$691$	1.37266e8i	0.416033i	−0.978125	+	0.208016i	$0.933299\pi$
$692$	0	0
$693$	4.48561e8	1.34779
$694$	0	0
$695$	− 1.38267e8i	− 0.411873i
$696$	0	0
$697$	−2.09718e7	−0.0619350
$698$	0	0
$699$	− 7.99595e7i	− 0.234120i
$700$	0	0
$701$	−6.02955e8	−1.75037	−0.875187	−	0.483784i	$-0.839262\pi$
$701$	−6.02955e8	−1.75037	−0.875187	+	0.483784i	$0.839262\pi$
$702$	0	0
$703$	− 3.49456e7i	− 0.100584i
$704$	0	0
$705$	−8.08050e7	−0.230606
$706$	0	0
$707$	− 3.91501e8i	− 1.10783i
$708$	0	0
$709$	4.41370e7	0.123841	0.0619205	−	0.998081i	$-0.480277\pi$
$709$	4.41370e7	0.123841	0.0619205	+	0.998081i	$0.480277\pi$
$710$	0	0
$711$	1.08169e8i	0.300951i
$712$	0	0
$713$	−8.98319e7	−0.247835
$714$	0	0
$715$	2.90965e8i	0.796017i
$716$	0	0
$717$	2.18330e8	0.592318
$718$	0	0
$719$	3.59817e8i	0.968045i	0.875056	+	0.484022i	$0.160825\pi$
$719$	3.59817e8i	0.968045i	−0.875056	+	0.484022i	$0.839175\pi$
$720$	0	0
$721$	−2.99555e8	−0.799229
$722$	0	0
$723$	− 6.93814e7i	− 0.183581i
$724$	0	0
$725$	−3.17791e8	−0.833926
$726$	0	0
$727$	− 1.23355e7i	− 0.0321035i	−0.999871	−	0.0160518i	$-0.994890\pi$
$727$	− 1.23355e7i	− 0.0321035i	0.999871	−	0.0160518i	$-0.00510965\pi$
$728$	0	0
$729$	1.75968e8	0.454205
$730$	0	0
$731$	− 2.83766e8i	− 0.726453i
$732$	0	0
$733$	−3.54832e7	−0.0900972	−0.0450486	−	0.998985i	$-0.514344\pi$
$733$	−3.54832e7	−0.0900972	−0.0450486	+	0.998985i	$0.514344\pi$
$734$	0	0
$735$	− 5.89361e7i	− 0.148429i
$736$	0	0
$737$	−4.51656e8	−1.12825
$738$	0	0
$739$	4.32139e8i	1.07076i	0.844612	+	0.535378i	$0.179831\pi$
$739$	4.32139e8i	1.07076i	−0.844612	+	0.535378i	$0.820169\pi$
$740$	0	0
$741$	2.83750e7	0.0697399
$742$	0	0
$743$	− 7.42345e8i	− 1.80984i	−0.425586	−	0.904918i	$-0.639932\pi$
$743$	− 7.42345e8i	− 1.80984i	0.425586	−	0.904918i	$-0.360068\pi$
$744$	0	0
$745$	−2.83546e8	−0.685732
$746$	0	0
$747$	5.27418e8i	1.26530i
$748$	0	0
$749$	5.25265e8	1.25007
$750$	0	0
$751$	6.22745e8i	1.47025i	0.677933	+	0.735124i	$0.262876\pi$
$751$	6.22745e8i	1.47025i	−0.677933	+	0.735124i	$0.737124\pi$
$752$	0	0
$753$	−7.17344e7	−0.168013
$754$	0	0
$755$	6.67488e7i	0.155097i
$756$	0	0
$757$	8.38267e8	1.93239	0.966194	−	0.257815i	$-0.0830024\pi$
$757$	8.38267e8	1.93239	0.966194	+	0.257815i	$0.0830024\pi$
$758$	0	0
$759$	− 3.97541e7i	− 0.0909193i
$760$	0	0
$761$	6.44538e8	1.46250	0.731248	−	0.682112i	$-0.238938\pi$
$761$	6.44538e8	1.46250	0.731248	+	0.682112i	$0.238938\pi$
$762$	0	0
$763$	8.30277e8i	1.86917i
$764$	0	0
$765$	1.19891e8	0.267794
$766$	0	0
$767$	− 4.44472e8i	− 0.985050i
$768$	0	0
$769$	7.83458e8	1.72281	0.861404	−	0.507921i	$-0.169586\pi$
$769$	7.83458e8	1.72281	0.861404	+	0.507921i	$0.169586\pi$
$770$	0	0
$771$	1.56696e8i	0.341897i
$772$	0	0
$773$	−7.79311e7	−0.168722	−0.0843611	−	0.996435i	$-0.526885\pi$
$773$	−7.79311e7	−0.168722	−0.0843611	+	0.996435i	$0.526885\pi$
$774$	0	0
$775$	3.26868e8i	0.702210i
$776$	0	0
$777$	−1.48269e8	−0.316073
$778$	0	0
$779$	− 6.64190e6i	− 0.0140501i
$780$	0	0
$781$	−2.78038e8	−0.583649
$782$	0	0
$783$	− 3.10193e8i	− 0.646170i
$784$	0	0
$785$	5.99539e7	0.123939
$786$	0	0
$787$	4.76683e8i	0.977924i	0.872305	+	0.488962i	$0.162624\pi$
$787$	4.76683e8i	0.977924i	−0.872305	+	0.488962i	$0.837376\pi$
$788$	0	0
$789$	3.26246e7	0.0664224
$790$	0	0
$791$	− 3.12116e8i	− 0.630648i
$792$	0	0
$793$	6.53858e8	1.31118
$794$	0	0
$795$	− 4.39277e7i	− 0.0874253i
$796$	0	0
$797$	2.49479e7	0.0492788	0.0246394	−	0.999696i	$-0.492156\pi$
$797$	2.49479e7	0.0492788	0.0246394	+	0.999696i	$0.492156\pi$
$798$	0	0
$799$	− 4.87016e8i	− 0.954779i
$800$	0	0
$801$	3.55655e7	0.0692040
$802$	0	0
$803$	9.03184e7i	0.174433i
$804$	0	0
$805$	9.57010e7	0.183455
$806$	0	0
$807$	2.63676e8i	0.501707i
$808$	0	0
$809$	−6.82458e8	−1.28893	−0.644467	−	0.764632i	$-0.722921\pi$
$809$	−6.82458e8	−1.28893	−0.644467	+	0.764632i	$0.722921\pi$
$810$	0	0
$811$	3.51060e8i	0.658140i	0.944306	+	0.329070i	$0.106735\pi$
$811$	3.51060e8i	0.658140i	−0.944306	+	0.329070i	$0.893265\pi$
$812$	0	0
$813$	−2.00082e8	−0.372337
$814$	0	0
$815$	8.79628e7i	0.162490i
$816$	0	0
$817$	8.98706e7	0.164798
$818$	0	0
$819$	1.09609e9i	1.99524i
$820$	0	0
$821$	−6.38664e8	−1.15410	−0.577049	−	0.816709i	$-0.695796\pi$
$821$	−6.38664e8	−1.15410	−0.577049	+	0.816709i	$0.695796\pi$
$822$	0	0
$823$	1.77607e8i	0.318611i	0.987229	+	0.159305i	$0.0509254\pi$
$823$	1.77607e8i	0.318611i	−0.987229	+	0.159305i	$0.949075\pi$
$824$	0	0
$825$	−1.44651e8	−0.257609
$826$	0	0
$827$	2.12047e7i	0.0374900i	0.999824	+	0.0187450i	$0.00596707\pi$
$827$	2.12047e7i	0.0374900i	−0.999824	+	0.0187450i	$0.994033\pi$
$828$	0	0
$829$	6.30661e8	1.10696	0.553480	−	0.832862i	$-0.313299\pi$
$829$	6.30661e8	1.10696	0.553480	+	0.832862i	$0.313299\pi$
$830$	0	0
$831$	1.97219e8i	0.343673i
$832$	0	0
$833$	3.55211e8	0.614542
$834$	0	0
$835$	3.08316e8i	0.529587i
$836$	0	0
$837$	−3.19053e8	−0.544109
$838$	0	0
$839$	9.45472e8i	1.60089i	0.599403	+	0.800447i	$0.295405\pi$
$839$	9.45472e8i	1.60089i	−0.599403	+	0.800447i	$0.704595\pi$
$840$	0	0
$841$	9.95855e7	0.167420
$842$	0	0
$843$	2.54408e8i	0.424667i
$844$	0	0
$845$	−4.22781e8	−0.700721
$846$	0	0
$847$	1.07652e8i	0.177163i
$848$	0	0
$849$	−3.12606e8	−0.510828
$850$	0	0
$851$	− 1.19637e8i	− 0.194122i
$852$	0	0
$853$	−3.08597e8	−0.497216	−0.248608	−	0.968604i	$-0.579973\pi$
$853$	−3.08597e8	−0.497216	−0.248608	+	0.968604i	$0.579973\pi$
$854$	0	0
$855$	3.79702e7i	0.0607497i
$856$	0	0
$857$	−2.45043e8	−0.389314	−0.194657	−	0.980871i	$-0.562359\pi$
$857$	−2.45043e8	−0.389314	−0.194657	+	0.980871i	$0.562359\pi$
$858$	0	0
$859$	6.90100e8i	1.08876i	0.838839	+	0.544380i	$0.183235\pi$
$859$	6.90100e8i	1.08876i	−0.838839	+	0.544380i	$0.816765\pi$
$860$	0	0
$861$	−2.81805e7	−0.0441509
$862$	0	0
$863$	6.89145e8i	1.07221i	0.844153	+	0.536103i	$0.180104\pi$
$863$	6.89145e8i	1.07221i	−0.844153	+	0.536103i	$0.819896\pi$
$864$	0	0
$865$	−3.58410e7	−0.0553773
$866$	0	0
$867$	1.25656e8i	0.192808i
$868$	0	0
$869$	−2.32551e8	−0.354371
$870$	0	0
$871$	− 1.10365e9i	− 1.67024i
$872$	0	0
$873$	−7.26941e8	−1.09259
$874$	0	0
$875$	− 7.99397e8i	− 1.19327i
$876$	0	0
$877$	−8.03085e8	−1.19059	−0.595296	−	0.803507i	$-0.702965\pi$
$877$	−8.03085e8	−1.19059	−0.595296	+	0.803507i	$0.702965\pi$
$878$	0	0
$879$	2.86175e8i	0.421372i
$880$	0	0
$881$	−4.58820e8	−0.670988	−0.335494	−	0.942042i	$-0.608903\pi$
$881$	−4.58820e8	−0.670988	−0.335494	+	0.942042i	$0.608903\pi$
$882$	0	0
$883$	− 1.80291e8i	− 0.261874i	−0.991391	−	0.130937i	$-0.958201\pi$
$883$	− 1.80291e8i	− 0.261874i	0.991391	−	0.130937i	$-0.0417986\pi$
$884$	0	0
$885$	−6.53277e7	−0.0942469
$886$	0	0
$887$	6.63542e8i	0.950818i	0.879765	+	0.475409i	$0.157700\pi$
$887$	6.63542e8i	0.950818i	−0.879765	+	0.475409i	$0.842300\pi$
$888$	0	0
$889$	1.21623e9	1.73105
$890$	0	0
$891$	5.35011e8i	0.756362i
$892$	0	0
$893$	1.54241e8	0.216594
$894$	0	0
$895$	3.78956e8i	0.528591i
$896$	0	0
$897$	9.71419e7	0.134595
$898$	0	0
$899$	− 7.14242e8i	− 0.983029i
$900$	0	0
$901$	2.64755e8	0.361967
$902$	0	0
$903$	− 3.81307e8i	− 0.517859i
$904$	0	0
$905$	8.06281e7	0.108778
$906$	0	0
$907$	2.51825e8i	0.337503i	0.985659	+	0.168751i	$0.0539735\pi$
$907$	2.51825e8i	0.337503i	−0.985659	+	0.168751i	$0.946027\pi$
$908$	0	0
$909$	5.31777e8	0.708008
$910$	0	0
$911$	− 1.33414e9i	− 1.76459i	−0.470694	−	0.882297i	$-0.655996\pi$
$911$	− 1.33414e9i	− 1.76459i	0.470694	−	0.882297i	$-0.344004\pi$
$912$	0	0
$913$	−1.13388e9	−1.48990
$914$	0	0
$915$	− 9.61028e7i	− 0.125451i
$916$	0	0
$917$	1.82539e9	2.36726
$918$	0	0
$919$	− 9.68344e8i	− 1.24762i	−0.781575	−	0.623811i	$-0.785583\pi$
$919$	− 9.68344e8i	− 1.24762i	0.781575	−	0.623811i	$-0.214417\pi$
$920$	0	0
$921$	3.14089e8	0.402044
$922$	0	0
$923$	− 6.79406e8i	− 0.864021i
$924$	0	0
$925$	−4.35317e8	−0.550022
$926$	0	0
$927$	− 4.06887e8i	− 0.510781i
$928$	0	0
$929$	9.60464e8	1.19794	0.598968	−	0.800773i	$-0.295577\pi$
$929$	9.60464e8	1.19794	0.598968	+	0.800773i	$0.295577\pi$
$930$	0	0
$931$	1.12498e8i	0.139410i
$932$	0	0
$933$	3.10330e8	0.382102
$934$	0	0
$935$	2.57750e8i	0.315328i
$936$	0	0
$937$	−9.88309e8	−1.20136	−0.600681	−	0.799489i	$-0.705104\pi$
$937$	−9.88309e8	−1.20136	−0.600681	+	0.799489i	$0.705104\pi$
$938$	0	0
$939$	4.67308e8i	0.564425i
$940$	0	0
$941$	1.34529e8	0.161454	0.0807269	−	0.996736i	$-0.474276\pi$
$941$	1.34529e8	0.161454	0.0807269	+	0.996736i	$0.474276\pi$
$942$	0	0
$943$	− 2.27386e7i	− 0.0271162i
$944$	0	0
$945$	3.39898e8	0.402766
$946$	0	0
$947$	− 4.96808e8i	− 0.584976i	−0.956269	−	0.292488i	$-0.905517\pi$
$947$	− 4.96808e8i	− 0.584976i	0.956269	−	0.292488i	$-0.0944832\pi$
$948$	0	0
$949$	−2.20699e8	−0.258227
$950$	0	0
$951$	3.50369e8i	0.407366i
$952$	0	0
$953$	4.73453e8	0.547014	0.273507	−	0.961870i	$-0.411816\pi$
$953$	4.73453e8	0.547014	0.273507	+	0.961870i	$0.411816\pi$
$954$	0	0
$955$	1.98265e8i	0.227633i
$956$	0	0
$957$	3.16080e8	0.360629
$958$	0	0
$959$	6.80099e8i	0.771110i
$960$	0	0
$961$	1.52861e8	0.172238
$962$	0	0
$963$	7.13470e8i	0.798908i
$964$	0	0
$965$	−3.33903e8	−0.371568
$966$	0	0
$967$	− 3.26806e8i	− 0.361418i	−0.983537	−	0.180709i	$-0.942161\pi$
$967$	− 3.26806e8i	− 0.361418i	0.983537	−	0.180709i	$-0.0578393\pi$
$968$	0	0
$969$	2.51358e7	0.0276263
$970$	0	0
$971$	− 7.38503e8i	− 0.806667i	−0.915053	−	0.403333i	$-0.867851\pi$
$971$	− 7.38503e8i	− 0.806667i	0.915053	−	0.403333i	$-0.132149\pi$
$972$	0	0
$973$	1.11979e9	1.21562
$974$	0	0
$975$	− 3.53466e8i	− 0.381359i
$976$	0	0
$977$	−8.98069e8	−0.962999	−0.481500	−	0.876446i	$-0.659908\pi$
$977$	−8.98069e8	−0.962999	−0.481500	+	0.876446i	$0.659908\pi$
$978$	0	0
$979$	7.64614e7i	0.0814880i
$980$	0	0
$981$	−1.12777e9	−1.19457
$982$	0	0
$983$	5.63791e8i	0.593551i	0.954947	+	0.296775i	$0.0959112\pi$
$983$	5.63791e8i	0.593551i	−0.954947	+	0.296775i	$0.904089\pi$
$984$	0	0
$985$	4.29797e8	0.449733
$986$	0	0
$987$	− 6.54422e8i	− 0.680622i
$988$	0	0
$989$	3.07672e8	0.318053
$990$	0	0
$991$	4.94832e8i	0.508437i	0.967147	+	0.254218i	$0.0818182\pi$
$991$	4.94832e8i	0.508437i	−0.967147	+	0.254218i	$0.918182\pi$
$992$	0	0
$993$	2.31792e8	0.236729
$994$	0	0
$995$	− 6.91675e8i	− 0.702155i
$996$	0	0
$997$	−2.35279e8	−0.237409	−0.118705	−	0.992930i	$-0.537874\pi$
$997$	−2.35279e8	−0.237409	−0.118705	+	0.992930i	$0.537874\pi$
$998$	0	0
$999$	− 4.24909e8i	− 0.426186i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.7.c.l.255.3		8
4.3	odd	2	inner	256.7.c.l.255.5		8
8.3	odd	2	inner	256.7.c.l.255.4		8
8.5	even	2	inner	256.7.c.l.255.6		8
16.3	odd	4		32.7.d.b.15.2		4
16.5	even	4		32.7.d.b.15.1		4
16.11	odd	4		8.7.d.b.3.4	yes	4
16.13	even	4		8.7.d.b.3.3	✓	4
48.5	odd	4		288.7.b.b.271.3		4
48.11	even	4		72.7.b.b.19.1		4
48.29	odd	4		72.7.b.b.19.2		4
48.35	even	4		288.7.b.b.271.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.7.d.b.3.3	✓	4	16.13	even	4
8.7.d.b.3.4	yes	4	16.11	odd	4
32.7.d.b.15.1		4	16.5	even	4
32.7.d.b.15.2		4	16.3	odd	4
72.7.b.b.19.1		4	48.11	even	4
72.7.b.b.19.2		4	48.29	odd	4
256.7.c.l.255.3		8	1.1	even	1	trivial
256.7.c.l.255.4		8	8.3	odd	2	inner
256.7.c.l.255.5		8	4.3	odd	2	inner
256.7.c.l.255.6		8	8.5	even	2	inner
288.7.b.b.271.2		4	48.35	even	4
288.7.b.b.271.3		4	48.5	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	256.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-8.49390i q^{3} -59.7107 q^{5} -483.584i q^{7} +656.854 q^{9} +O(q^{10})\)	\(q-8.49390i q^{3} -59.7107 q^{5} -483.584i q^{7} +656.854 q^{9} +1412.15i q^{11} -3450.70 q^{13} +507.177i q^{15} -3056.78 q^{17} -968.104i q^{19} -4107.51 q^{21} -3314.31i q^{23} -12059.6 q^{25} -11771.3i q^{27} +26351.6 q^{29} -27104.3i q^{31} +11994.7 q^{33} +28875.1i q^{35} +36097.0 q^{37} +29309.9i q^{39} +6860.73 q^{41} +92831.6i q^{43} -39221.2 q^{45} +159323. i q^{47} -116204. q^{49} +25964.0i q^{51} -86612.3 q^{53} -84320.6i q^{55} -8222.98 q^{57} +128806. i q^{59} -189486. q^{61} -317644. i q^{63} +206043. q^{65} +319835. i q^{67} -28151.4 q^{69} +196890. i q^{71} +63957.9 q^{73} +102433. i q^{75} +682894. q^{77} +164678. i q^{79} +378862. q^{81} +802946. i q^{83} +182522. q^{85} -223828. i q^{87} +54145.3 q^{89} +1.66870e6i q^{91} -230221. q^{93} +57806.1i q^{95} -1.10670e6 q^{97} +927577. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 1320 q^{9}+O(q^{10}) \) 8 * q + 1320 * q^9	\( 8 q + 1320 q^{9} + 8336 q^{17} + 47800 q^{25} + 168096 q^{33} + 235888 q^{41} + 4232 q^{49} - 253344 q^{57} - 410880 q^{65} - 887824 q^{73} - 1136088 q^{81} - 1522448 q^{89} - 1853552 q^{97}+O(q^{100}) \) 8 * q + 1320 * q^9 + 8336 * q^17 + 47800 * q^25 + 168096 * q^33 + 235888 * q^41 + 4232 * q^49 - 253344 * q^57 - 410880 * q^65 - 887824 * q^73 - 1136088 * q^81 - 1522448 * q^89 - 1853552 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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