LMFDB - Embedded newform 256.7.c.l.255.6

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			255.6
Root			$-2.84502 + 2.81174i$ of defining polynomial
Character	$\chi$	$=$	256.255
Dual form			256.7.c.l.255.4

$q$-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	$0.0502772\pi$
$3$	8.49390i	0.314589i	−0.987552	+	0.157294i	$0.949723\pi$
$4$	0	0
$5$	59.7107	0.477686	0.238843	−	0.971058i	$-0.423232\pi$
$5$	59.7107	0.477686	0.238843	+	0.971058i	$0.423232\pi$
$6$	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	$-0.822010\pi$
$11$	− 1412.15i	− 1.06097i	0.847694	−	0.530485i	$-0.177990\pi$
$12$	0	0
$13$	3450.70	1.57064	0.785320	−	0.619090i	$-0.212499\pi$
$13$	3450.70	1.57064	0.785320	+	0.619090i	$0.212499\pi$
$14$	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	$0.0224824\pi$
$19$	968.104i	0.141144i	−0.997507	+	0.0705718i	$0.977518\pi$
$20$	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	$-0.681665\pi$
$29$	−26351.6	−1.08047	−0.540236	+	0.841513i	$0.681665\pi$
$30$	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	$-0.615968\pi$
$37$	−36097.0	−0.712633	−0.356317	+	0.934365i	$0.615968\pi$
$38$	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	$-0.801567\pi$
$43$	− 92831.6i	− 1.16759i	0.811901	−	0.583795i	$-0.198433\pi$
$44$	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	$-0.406050\pi$
$53$	86612.3	0.581771	0.290885	+	0.956758i	$0.406050\pi$
$54$	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	$-0.898471\pi$
$59$	− 128806.i	− 0.627165i	0.949561	−	0.313582i	$-0.101529\pi$
$60$	0	0
$61$	189486.	0.834810	0.417405	−	0.908721i	$-0.362940\pi$
$61$	189486.	0.834810	0.417405	+	0.908721i	$0.362940\pi$
$62$	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	$-0.821551\pi$
$67$	− 319835.i	− 1.06341i	0.846929	−	0.531706i	$-0.178449\pi$
$68$	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	$-0.752229\pi$
$83$	− 802946.i	− 1.40428i	0.712041	−	0.702138i	$-0.247771\pi$
$84$	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	$-0.628524\pi$
$101$	−809583.	−0.785773	−0.392887	+	0.919587i	$0.628524\pi$
$102$	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	$-0.853798\pi$
$107$	− 1.08619e6i	− 0.886657i	0.896359	−	0.443328i	$-0.146202\pi$
$108$	0	0
$109$	1.71692e6	1.32578	0.662890	−	0.748717i	$-0.269330\pi$
$109$	1.71692e6	1.32578	0.662890	+	0.748717i	$0.269330\pi$
$110$	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	$-0.682827\pi$
$131$	− 3.77470e6i	− 1.67907i	0.543305	−	0.839536i	$-0.317173\pi$
$132$	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	$-0.858121\pi$
$139$	− 2.31561e6i	− 0.862227i	0.902298	−	0.431114i	$-0.141879\pi$
$140$	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	$-0.754835\pi$
$149$	−4.74866e6	−1.43553	−0.717765	+	0.696285i	$0.754835\pi$
$150$	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	$-0.458589\pi$
$157$	1.00407e6	0.259458	0.129729	+	0.991550i	$0.458589\pi$
$158$	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	$0.0544027\pi$
$163$	1.47315e6i	0.340161i	−0.985430	+	0.170080i	$0.945597\pi$
$164$	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	$-0.518461\pi$
$173$	−600245.	−0.115928	−0.0579642	+	0.998319i	$0.518461\pi$
$174$	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	$0.186625\pi$
$179$	6.34654e6i	1.10657i	−0.832993	+	0.553284i	$0.813375\pi$
$180$	0	0
$181$	1.35031e6	0.227719	0.113859	−	0.993497i	$-0.463679\pi$
$181$	1.35031e6	0.227719	0.113859	+	0.993497i	$0.463679\pi$
$182$	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	$-0.343986\pi$
$197$	7.19799e6	0.941483	0.470742	+	0.882271i	$0.343986\pi$
$198$	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	$-0.989166\pi$
$211$	− 639344.i	− 0.0680592i	0.999421	−	0.0340296i	$-0.0108341\pi$
$212$	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	$0.184082\pi$
$227$	1.27875e7i	1.09322i	−0.837387	+	0.546610i	$0.815918\pi$
$228$	0	0
$229$	−4.61930e6	−0.384653	−0.192327	−	0.981331i	$-0.561603\pi$
$229$	−4.61930e6	−0.384653	−0.192327	+	0.981331i	$0.561603\pi$
$230$	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	$0.0860441\pi$
$251$	8.44540e6i	0.534071i	−0.963687	+	0.267035i	$0.913956\pi$
$252$	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	$-0.206208\pi$
$269$	3.10430e7	1.59480	0.797400	+	0.603451i	$0.206208\pi$
$270$	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	$-0.316064\pi$
$277$	2.32189e7	1.09245	0.546225	+	0.837638i	$0.316064\pi$
$278$	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	$0.301565\pi$
$283$	3.68036e7i	1.62380i	−0.583800	+	0.811898i	$0.698435\pi$
$284$	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	$-0.266415\pi$
$293$	3.36919e7	1.33944	0.669718	+	0.742615i	$0.266415\pi$
$294$	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	$-0.779349\pi$
$307$	− 3.69782e7i	− 1.27800i	0.769207	−	0.639000i	$-0.220651\pi$
$308$	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	$-0.275832\pi$
$317$	4.12495e7	1.29491	0.647457	+	0.762102i	$0.275832\pi$
$318$	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	$-0.877213\pi$
$331$	− 2.72892e7i	− 0.752502i	0.926518	−	0.376251i	$-0.122787\pi$
$332$	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	$-0.958821\pi$
$347$	− 1.07803e7i	− 0.258013i	0.991644	−	0.129006i	$-0.0411788\pi$
$348$	0	0
$349$	−1.33186e6	−0.0313316	−0.0156658	−	0.999877i	$-0.504987\pi$
$349$	−1.33186e6	−0.0313316	−0.0156658	+	0.999877i	$0.504987\pi$
$350$	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	$-0.387758\pi$
$373$	3.58446e7	0.690712	0.345356	+	0.938472i	$0.387758\pi$
$374$	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	$-0.649731\pi$
$379$	− 9.70544e7i	− 1.78278i	0.453238	−	0.891390i	$-0.350269\pi$
$380$	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	$-0.345109\pi$
$389$	5.50526e7	0.935252	0.467626	+	0.883926i	$0.345109\pi$
$390$	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	$-0.268124\pi$
$397$	8.33093e7	1.33144	0.665721	+	0.746201i	$0.268124\pi$
$398$	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	$-0.915358\pi$
$419$	− 3.86614e7i	− 0.525575i	0.964854	−	0.262788i	$-0.0846419\pi$
$420$	0	0
$421$	−2.58211e7	−0.346042	−0.173021	−	0.984918i	$-0.555353\pi$
$421$	−2.58211e7	−0.346042	−0.173021	+	0.984918i	$0.555353\pi$
$422$	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	$0.00969552\pi$
$443$	5.29536e6i	0.0609094i	−0.999536	+	0.0304547i	$0.990304\pi$
$444$	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	$-0.769387\pi$
$461$	−1.46730e8	−1.49767	−0.748836	+	0.662755i	$0.769387\pi$
$462$	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	$0.177037\pi$
$467$	1.07539e8i	1.05589i	−0.849280	+	0.527943i	$0.822963\pi$
$468$	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	$0.0879124\pi$
$491$	6.45564e7i	0.545374i	−0.962103	+	0.272687i	$0.912088\pi$
$492$	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	$0.105555\pi$
$499$	8.09040e7i	0.651131i	−0.945520	+	0.325565i	$0.894445\pi$
$500$	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	$-0.483188\pi$
$509$	1.39233e7	0.105582	0.0527909	+	0.998606i	$0.483188\pi$
$510$	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	$0.0579383\pi$
$523$	5.17905e7i	0.362030i	−0.983480	+	0.181015i	$0.942062\pi$
$524$	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	$-0.475015\pi$
$541$	2.48317e7	0.156825	0.0784125	+	0.996921i	$0.475015\pi$
$542$	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	$0.0809159\pi$
$547$	8.23167e7i	0.502951i	−0.967864	+	0.251476i	$0.919084\pi$
$548$	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	$-0.518353\pi$
$557$	−1.99169e7	−0.115254	−0.0576271	+	0.998338i	$0.518353\pi$
$558$	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	$0.327588\pi$
$563$	3.05819e8i	1.71372i	−0.515550	+	0.856859i	$0.672412\pi$
$564$	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	$0.0563287\pi$
$571$	6.55464e7i	0.352079i	−0.984383	+	0.176040i	$0.943671\pi$
$572$	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	$0.401652\pi$
$587$	3.85369e8i	1.90529i	−0.304078	+	0.952647i	$0.598348\pi$
$588$	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	$-0.600956\pi$
$613$	−1.43677e8	−0.623743	−0.311872	+	0.950124i	$0.600956\pi$
$614$	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	$0.144099\pi$
$619$	2.07480e8i	0.874793i	−0.899269	+	0.437396i	$0.855901\pi$
$620$	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	$-0.753318\pi$
$643$	− 3.72026e8i	− 1.39939i	0.714440	−	0.699697i	$-0.246682\pi$
$644$	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	$-0.384180\pi$
$653$	1.98187e8	0.711764	0.355882	+	0.934531i	$0.384180\pi$
$654$	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	$-0.968158\pi$
$659$	− 5.71625e7i	− 0.199735i	0.995001	−	0.0998677i	$-0.0318419\pi$
$660$	0	0
$661$	5.04948e8	1.74841	0.874203	−	0.485561i	$-0.161385\pi$
$661$	5.04948e8	1.74841	0.874203	+	0.485561i	$0.161385\pi$
$662$	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	$-0.401299\pi$
$677$	1.89358e8	0.610265	0.305132	+	0.952310i	$0.401299\pi$
$678$	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	$-0.968395\pi$
$683$	− 6.31662e7i	− 0.198254i	0.995075	−	0.0991272i	$-0.0316051\pi$
$684$	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	$-0.933299\pi$
$691$	− 1.37266e8i	− 0.416033i	0.978125	−	0.208016i	$-0.0667007\pi$
$692$	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	$-0.160738\pi$
$701$	6.02955e8	1.75037	0.875187	+	0.483784i	$0.160738\pi$
$702$	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	$-0.519723\pi$
$709$	−4.41370e7	−0.123841	−0.0619205	+	0.998081i	$0.519723\pi$
$710$	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	$-0.485656\pi$
$733$	3.54832e7	0.0900972	0.0450486	+	0.998985i	$0.485656\pi$
$734$	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	$-0.820169\pi$
$739$	− 4.32139e8i	− 1.07076i	0.844612	−	0.535378i	$-0.179831\pi$
$740$	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	$-0.916998\pi$
$757$	−8.38267e8	−1.93239	−0.966194	+	0.257815i	$0.916998\pi$
$758$	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	$-0.473115\pi$
$773$	7.79311e7	0.168722	0.0843611	+	0.996435i	$0.473115\pi$
$774$	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	$-0.837376\pi$
$787$	− 4.76683e8i	− 0.977924i	0.872305	−	0.488962i	$-0.162624\pi$
$788$	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	$-0.507844\pi$
$797$	−2.49479e7	−0.0492788	−0.0246394	+	0.999696i	$0.507844\pi$
$798$	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	$-0.893265\pi$
$811$	− 3.51060e8i	− 0.658140i	0.944306	−	0.329070i	$-0.106735\pi$
$812$	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	$-0.304204\pi$
$821$	6.38664e8	1.15410	0.577049	+	0.816709i	$0.304204\pi$
$822$	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	$-0.994033\pi$
$827$	− 2.12047e7i	− 0.0374900i	0.999824	−	0.0187450i	$-0.00596707\pi$
$828$	0	0
$829$	−6.30661e8	−1.10696	−0.553480	−	0.832862i	$-0.686701\pi$
$829$	−6.30661e8	−1.10696	−0.553480	+	0.832862i	$0.686701\pi$
$830$	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	$-0.420027\pi$
$853$	3.08597e8	0.497216	0.248608	+	0.968604i	$0.420027\pi$
$854$	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	$-0.816765\pi$
$859$	− 6.90100e8i	− 1.08876i	0.838839	−	0.544380i	$-0.183235\pi$
$860$	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	$-0.297035\pi$
$877$	8.03085e8	1.19059	0.595296	+	0.803507i	$0.297035\pi$
$878$	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	$0.0417986\pi$
$883$	1.80291e8i	0.261874i	−0.991391	+	0.130937i	$0.958201\pi$
$884$	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	$-0.946027\pi$
$907$	− 2.51825e8i	− 0.337503i	0.985659	−	0.168751i	$-0.0539735\pi$
$908$	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	$-0.525724\pi$
$941$	−1.34529e8	−0.161454	−0.0807269	+	0.996736i	$0.525724\pi$
$942$	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	$0.0944832\pi$
$947$	4.96808e8i	0.584976i	−0.956269	+	0.292488i	$0.905517\pi$
$948$	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	$0.132149\pi$
$971$	7.38503e8i	0.806667i	−0.915053	+	0.403333i	$0.867851\pi$
$972$	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	$-0.462126\pi$
$997$	2.35279e8	0.237409	0.118705	+	0.992930i	$0.462126\pi$
$998$	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.255.6		8
4.3	odd	2	inner	256.7.c.l.255.4		8
8.3	odd	2	inner	256.7.c.l.255.5		8
8.5	even	2	inner	256.7.c.l.255.3		8
16.3	odd	4		8.7.d.b.3.4	yes	4
16.5	even	4		8.7.d.b.3.3	✓	4
16.11	odd	4		32.7.d.b.15.2		4
16.13	even	4		32.7.d.b.15.1		4
48.5	odd	4		72.7.b.b.19.2		4
48.11	even	4		288.7.b.b.271.2		4
48.29	odd	4		288.7.b.b.271.3		4
48.35	even	4		72.7.b.b.19.1		4

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