LMFDB - Embedded newform 256.4.b.e.129.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,4,Mod(129,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("256.129");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	256.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$15.1044889615$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 32)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	256.129
Dual form			256.4.b.e.129.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000i q^{3} +10.0000i q^{5} +16.0000 q^{7} -37.0000 q^{9} +O(q^{10})$	$q-8.00000i q^{3} +10.0000i q^{5} +16.0000 q^{7} -37.0000 q^{9} -40.0000i q^{11} -50.0000i q^{13} +80.0000 q^{15} -30.0000 q^{17} -40.0000i q^{19} -128.000i q^{21} +48.0000 q^{23} +25.0000 q^{25} +80.0000i q^{27} -34.0000i q^{29} -320.000 q^{31} -320.000 q^{33} +160.000i q^{35} -310.000i q^{37} -400.000 q^{39} -410.000 q^{41} +152.000i q^{43} -370.000i q^{45} +416.000 q^{47} -87.0000 q^{49} +240.000i q^{51} +410.000i q^{53} +400.000 q^{55} -320.000 q^{57} -200.000i q^{59} +30.0000i q^{61} -592.000 q^{63} +500.000 q^{65} -776.000i q^{67} -384.000i q^{69} +400.000 q^{71} +630.000 q^{73} -200.000i q^{75} -640.000i q^{77} +1120.00 q^{79} -359.000 q^{81} -552.000i q^{83} -300.000i q^{85} -272.000 q^{87} +326.000 q^{89} -800.000i q^{91} +2560.00i q^{93} +400.000 q^{95} -110.000 q^{97} +1480.00i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 32 q^{7} - 74 q^{9}+O(q^{10}) $ 2 * q + 32 * q^7 - 74 * q^9	$ 2 q + 32 q^{7} - 74 q^{9} + 160 q^{15} - 60 q^{17} + 96 q^{23} + 50 q^{25} - 640 q^{31} - 640 q^{33} - 800 q^{39} - 820 q^{41} + 832 q^{47} - 174 q^{49} + 800 q^{55} - 640 q^{57} - 1184 q^{63} + 1000 q^{65} + 800 q^{71} + 1260 q^{73} + 2240 q^{79} - 718 q^{81} - 544 q^{87} + 652 q^{89} + 800 q^{95} - 220 q^{97}+O(q^{100}) $ 2 * q + 32 * q^7 - 74 * q^9 + 160 * q^15 - 60 * q^17 + 96 * q^23 + 50 * q^25 - 640 * q^31 - 640 * q^33 - 800 * q^39 - 820 * q^41 + 832 * q^47 - 174 * q^49 + 800 * q^55 - 640 * q^57 - 1184 * q^63 + 1000 * q^65 + 800 * q^71 + 1260 * q^73 + 2240 * q^79 - 718 * q^81 - 544 * q^87 + 652 * q^89 + 800 * q^95 - 220 * 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
$3$	− 8.00000i	− 1.53960i	−0.638285	−	0.769800i	$-0.720356\pi$
$3$	− 8.00000i	− 1.53960i	0.638285	−	0.769800i	$-0.279644\pi$
$4$	0	0
$5$	10.0000i	0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$	10.0000i	0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	16.0000	0.863919	0.431959	−	0.901893i	$-0.357822\pi$
$7$	16.0000	0.863919	0.431959	+	0.901893i	$0.357822\pi$
$8$	0	0
$9$	−37.0000	−1.37037
$10$	0	0
$11$	− 40.0000i	− 1.09640i	−0.836346	−	0.548202i	$-0.815312\pi$
$11$	− 40.0000i	− 1.09640i	0.836346	−	0.548202i	$-0.184688\pi$
$12$	0	0
$13$	− 50.0000i	− 1.06673i	−0.845885	−	0.533366i	$-0.820927\pi$
$13$	− 50.0000i	− 1.06673i	0.845885	−	0.533366i	$-0.179073\pi$
$14$	0	0
$15$	80.0000	1.37706
$16$	0	0
$17$	−30.0000	−0.428004	−0.214002	−	0.976833i	$-0.568650\pi$
$17$	−30.0000	−0.428004	−0.214002	+	0.976833i	$0.568650\pi$
$18$	0	0
$19$	− 40.0000i	− 0.482980i	−0.970403	−	0.241490i	$-0.922364\pi$
$19$	− 40.0000i	− 0.482980i	0.970403	−	0.241490i	$-0.0776362\pi$
$20$	0	0
$21$	− 128.000i	− 1.33009i
$22$	0	0
$23$	48.0000	0.435161	0.217580	−	0.976042i	$-0.430184\pi$
$23$	48.0000	0.435161	0.217580	+	0.976042i	$0.430184\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	80.0000i	0.570222i
$28$	0	0
$29$	− 34.0000i	− 0.217712i	−0.994058	−	0.108856i	$-0.965281\pi$
$29$	− 34.0000i	− 0.217712i	0.994058	−	0.108856i	$-0.0347187\pi$
$30$	0	0
$31$	−320.000	−1.85399	−0.926995	−	0.375073i	$-0.877617\pi$
$31$	−320.000	−1.85399	−0.926995	+	0.375073i	$0.877617\pi$
$32$	0	0
$33$	−320.000	−1.68803
$34$	0	0
$35$	160.000i	0.772712i
$36$	0	0
$37$	− 310.000i	− 1.37740i	−0.725048	−	0.688698i	$-0.758182\pi$
$37$	− 310.000i	− 1.37740i	0.725048	−	0.688698i	$-0.241818\pi$
$38$	0	0
$39$	−400.000	−1.64234
$40$	0	0
$41$	−410.000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−410.000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	152.000i	0.539065i	0.962991	+	0.269532i	$0.0868691\pi$
$43$	152.000i	0.539065i	−0.962991	+	0.269532i	$0.913131\pi$
$44$	0	0
$45$	− 370.000i	− 1.22570i
$46$	0	0
$47$	416.000	1.29106	0.645530	−	0.763735i	$-0.276636\pi$
$47$	416.000	1.29106	0.645530	+	0.763735i	$0.276636\pi$
$48$	0	0
$49$	−87.0000	−0.253644
$50$	0	0
$51$	240.000i	0.658955i
$52$	0	0
$53$	410.000i	1.06260i	0.847184	+	0.531300i	$0.178296\pi$
$53$	410.000i	1.06260i	−0.847184	+	0.531300i	$0.821704\pi$
$54$	0	0
$55$	400.000	0.980654
$56$	0	0
$57$	−320.000	−0.743597
$58$	0	0
$59$	− 200.000i	− 0.441318i	−0.975351	−	0.220659i	$-0.929179\pi$
$59$	− 200.000i	− 0.441318i	0.975351	−	0.220659i	$-0.0708208\pi$
$60$	0	0
$61$	30.0000i	0.0629690i	0.999504	+	0.0314845i	$0.0100235\pi$
$61$	30.0000i	0.0629690i	−0.999504	+	0.0314845i	$0.989977\pi$
$62$	0	0
$63$	−592.000	−1.18389
$64$	0	0
$65$	500.000	0.954113
$66$	0	0
$67$	− 776.000i	− 1.41498i	−0.706725	−	0.707489i	$-0.749828\pi$
$67$	− 776.000i	− 1.41498i	0.706725	−	0.707489i	$-0.250172\pi$
$68$	0	0
$69$	− 384.000i	− 0.669973i
$70$	0	0
$71$	400.000	0.668609	0.334305	−	0.942465i	$-0.391499\pi$
$71$	400.000	0.668609	0.334305	+	0.942465i	$0.391499\pi$
$72$	0	0
$73$	630.000	1.01008	0.505041	−	0.863096i	$-0.331478\pi$
$73$	630.000	1.01008	0.505041	+	0.863096i	$0.331478\pi$
$74$	0	0
$75$	− 200.000i	− 0.307920i
$76$	0	0
$77$	− 640.000i	− 0.947205i
$78$	0	0
$79$	1120.00	1.59506	0.797531	−	0.603278i	$-0.206139\pi$
$79$	1120.00	1.59506	0.797531	+	0.603278i	$0.206139\pi$
$80$	0	0
$81$	−359.000	−0.492455
$82$	0	0
$83$	− 552.000i	− 0.729998i	−0.931008	−	0.364999i	$-0.881069\pi$
$83$	− 552.000i	− 0.729998i	0.931008	−	0.364999i	$-0.118931\pi$
$84$	0	0
$85$	− 300.000i	− 0.382818i
$86$	0	0
$87$	−272.000	−0.335189
$88$	0	0
$89$	326.000	0.388269	0.194134	−	0.980975i	$-0.437810\pi$
$89$	326.000	0.388269	0.194134	+	0.980975i	$0.437810\pi$
$90$	0	0
$91$	− 800.000i	− 0.921569i
$92$	0	0
$93$	2560.00i	2.85440i
$94$	0	0
$95$	400.000	0.431991
$96$	0	0
$97$	−110.000	−0.115142	−0.0575712	−	0.998341i	$-0.518336\pi$
$97$	−110.000	−0.115142	−0.0575712	+	0.998341i	$0.518336\pi$
$98$	0	0
$99$	1480.00i	1.50248i
$100$	0	0
$101$	1098.00i	1.08173i	0.841108	+	0.540867i	$0.181904\pi$
$101$	1098.00i	1.08173i	−0.841108	+	0.540867i	$0.818096\pi$
$102$	0	0
$103$	−48.0000	−0.0459183	−0.0229591	−	0.999736i	$-0.507309\pi$
$103$	−48.0000	−0.0459183	−0.0229591	+	0.999736i	$0.507309\pi$
$104$	0	0
$105$	1280.00	1.18967
$106$	0	0
$107$	664.000i	0.599919i	0.953952	+	0.299959i	$0.0969731\pi$
$107$	664.000i	0.599919i	−0.953952	+	0.299959i	$0.903027\pi$
$108$	0	0
$109$	− 370.000i	− 0.325134i	−0.986698	−	0.162567i	$-0.948023\pi$
$109$	− 370.000i	− 0.325134i	0.986698	−	0.162567i	$-0.0519773\pi$
$110$	0	0
$111$	−2480.00	−2.12064
$112$	0	0
$113$	1490.00	1.24042	0.620210	−	0.784436i	$-0.287047\pi$
$113$	1490.00	1.24042	0.620210	+	0.784436i	$0.287047\pi$
$114$	0	0
$115$	480.000i	0.389219i
$116$	0	0
$117$	1850.00i	1.46182i
$118$	0	0
$119$	−480.000	−0.369761
$120$	0	0
$121$	−269.000	−0.202104
$122$	0	0
$123$	3280.00i	2.40445i
$124$	0	0
$125$	1500.00i	1.07331i
$126$	0	0
$127$	1024.00	0.715475	0.357737	−	0.933822i	$-0.383548\pi$
$127$	1024.00	0.715475	0.357737	+	0.933822i	$0.383548\pi$
$128$	0	0
$129$	1216.00	0.829944
$130$	0	0
$131$	− 1160.00i	− 0.773662i	−0.922151	−	0.386831i	$-0.873570\pi$
$131$	− 1160.00i	− 0.773662i	0.922151	−	0.386831i	$-0.126430\pi$
$132$	0	0
$133$	− 640.000i	− 0.417256i
$134$	0	0
$135$	−800.000	−0.510022
$136$	0	0
$137$	−570.000	−0.355463	−0.177731	−	0.984079i	$-0.556876\pi$
$137$	−570.000	−0.355463	−0.177731	+	0.984079i	$0.556876\pi$
$138$	0	0
$139$	− 1960.00i	− 1.19601i	−0.801493	−	0.598004i	$-0.795961\pi$
$139$	− 1960.00i	− 1.19601i	0.801493	−	0.598004i	$-0.204039\pi$
$140$	0	0
$141$	− 3328.00i	− 1.98772i
$142$	0	0
$143$	−2000.00	−1.16957
$144$	0	0
$145$	340.000	0.194727
$146$	0	0
$147$	696.000i	0.390511i
$148$	0	0
$149$	2010.00i	1.10514i	0.833467	+	0.552569i	$0.186352\pi$
$149$	2010.00i	1.10514i	−0.833467	+	0.552569i	$0.813648\pi$
$150$	0	0
$151$	−720.000	−0.388032	−0.194016	−	0.980998i	$-0.562151\pi$
$151$	−720.000	−0.388032	−0.194016	+	0.980998i	$0.562151\pi$
$152$	0	0
$153$	1110.00	0.586524
$154$	0	0
$155$	− 3200.00i	− 1.65826i
$156$	0	0
$157$	1790.00i	0.909921i	0.890512	+	0.454960i	$0.150347\pi$
$157$	1790.00i	0.909921i	−0.890512	+	0.454960i	$0.849653\pi$
$158$	0	0
$159$	3280.00	1.63598
$160$	0	0
$161$	768.000	0.375943
$162$	0	0
$163$	1208.00i	0.580478i	0.956954	+	0.290239i	$0.0937348\pi$
$163$	1208.00i	0.580478i	−0.956954	+	0.290239i	$0.906265\pi$
$164$	0	0
$165$	− 3200.00i	− 1.50982i
$166$	0	0
$167$	2896.00	1.34191	0.670956	−	0.741497i	$-0.265884\pi$
$167$	2896.00	1.34191	0.670956	+	0.741497i	$0.265884\pi$
$168$	0	0
$169$	−303.000	−0.137915
$170$	0	0
$171$	1480.00i	0.661862i
$172$	0	0
$173$	750.000i	0.329604i	0.986327	+	0.164802i	$0.0526985\pi$
$173$	750.000i	0.329604i	−0.986327	+	0.164802i	$0.947302\pi$
$174$	0	0
$175$	400.000	0.172784
$176$	0	0
$177$	−1600.00	−0.679454
$178$	0	0
$179$	− 2280.00i	− 0.952040i	−0.879434	−	0.476020i	$-0.842079\pi$
$179$	− 2280.00i	− 0.952040i	0.879434	−	0.476020i	$-0.157921\pi$
$180$	0	0
$181$	442.000i	0.181512i	0.995873	+	0.0907558i	$0.0289283\pi$
$181$	442.000i	0.181512i	−0.995873	+	0.0907558i	$0.971072\pi$
$182$	0	0
$183$	240.000	0.0969471
$184$	0	0
$185$	3100.00	1.23198
$186$	0	0
$187$	1200.00i	0.469266i
$188$	0	0
$189$	1280.00i	0.492626i
$190$	0	0
$191$	1920.00	0.727363	0.363681	−	0.931523i	$-0.381520\pi$
$191$	1920.00	0.727363	0.363681	+	0.931523i	$0.381520\pi$
$192$	0	0
$193$	−5070.00	−1.89091	−0.945457	−	0.325746i	$-0.894385\pi$
$193$	−5070.00	−1.89091	−0.945457	+	0.325746i	$0.894385\pi$
$194$	0	0
$195$	− 4000.00i	− 1.46895i
$196$	0	0
$197$	− 1910.00i	− 0.690771i	−0.938461	−	0.345385i	$-0.887748\pi$
$197$	− 1910.00i	− 0.690771i	0.938461	−	0.345385i	$-0.112252\pi$
$198$	0	0
$199$	2960.00	1.05442	0.527208	−	0.849736i	$-0.323239\pi$
$199$	2960.00	1.05442	0.527208	+	0.849736i	$0.323239\pi$
$200$	0	0
$201$	−6208.00	−2.17850
$202$	0	0
$203$	− 544.000i	− 0.188085i
$204$	0	0
$205$	− 4100.00i	− 1.39686i
$206$	0	0
$207$	−1776.00	−0.596331
$208$	0	0
$209$	−1600.00	−0.529542
$210$	0	0
$211$	− 40.0000i	− 0.0130508i	−0.999979	−	0.00652539i	$-0.997923\pi$
$211$	− 40.0000i	− 0.0130508i	0.999979	−	0.00652539i	$-0.00207711\pi$
$212$	0	0
$213$	− 3200.00i	− 1.02939i
$214$	0	0
$215$	−1520.00	−0.482154
$216$	0	0
$217$	−5120.00	−1.60170
$218$	0	0
$219$	− 5040.00i	− 1.55512i
$220$	0	0
$221$	1500.00i	0.456565i
$222$	0	0
$223$	−4288.00	−1.28765	−0.643824	−	0.765173i	$-0.722653\pi$
$223$	−4288.00	−1.28765	−0.643824	+	0.765173i	$0.722653\pi$
$224$	0	0
$225$	−925.000	−0.274074
$226$	0	0
$227$	6456.00i	1.88766i	0.330425	+	0.943832i	$0.392808\pi$
$227$	6456.00i	1.88766i	−0.330425	+	0.943832i	$0.607192\pi$
$228$	0	0
$229$	1066.00i	0.307613i	0.988101	+	0.153806i	$0.0491532\pi$
$229$	1066.00i	0.307613i	−0.988101	+	0.153806i	$0.950847\pi$
$230$	0	0
$231$	−5120.00	−1.45832
$232$	0	0
$233$	5910.00	1.66170	0.830852	−	0.556494i	$-0.187854\pi$
$233$	5910.00	1.66170	0.830852	+	0.556494i	$0.187854\pi$
$234$	0	0
$235$	4160.00i	1.15476i
$236$	0	0
$237$	− 8960.00i	− 2.45576i
$238$	0	0
$239$	3360.00	0.909374	0.454687	−	0.890651i	$-0.349751\pi$
$239$	3360.00	0.909374	0.454687	+	0.890651i	$0.349751\pi$
$240$	0	0
$241$	3970.00	1.06112	0.530561	−	0.847647i	$-0.321981\pi$
$241$	3970.00	1.06112	0.530561	+	0.847647i	$0.321981\pi$
$242$	0	0
$243$	5032.00i	1.32841i
$244$	0	0
$245$	− 870.000i	− 0.226866i
$246$	0	0
$247$	−2000.00	−0.515210
$248$	0	0
$249$	−4416.00	−1.12391
$250$	0	0
$251$	6840.00i	1.72007i	0.510237	+	0.860034i	$0.329558\pi$
$251$	6840.00i	1.72007i	−0.510237	+	0.860034i	$0.670442\pi$
$252$	0	0
$253$	− 1920.00i	− 0.477112i
$254$	0	0
$255$	−2400.00	−0.589388
$256$	0	0
$257$	4610.00	1.11893	0.559463	−	0.828855i	$-0.311007\pi$
$257$	4610.00	1.11893	0.559463	+	0.828855i	$0.311007\pi$
$258$	0	0
$259$	− 4960.00i	− 1.18996i
$260$	0	0
$261$	1258.00i	0.298346i
$262$	0	0
$263$	−4848.00	−1.13666	−0.568328	−	0.822802i	$-0.692409\pi$
$263$	−4848.00	−1.13666	−0.568328	+	0.822802i	$0.692409\pi$
$264$	0	0
$265$	−4100.00	−0.950419
$266$	0	0
$267$	− 2608.00i	− 0.597779i
$268$	0	0
$269$	5550.00i	1.25795i	0.777424	+	0.628977i	$0.216526\pi$
$269$	5550.00i	1.25795i	−0.777424	+	0.628977i	$0.783474\pi$
$270$	0	0
$271$	480.000	0.107594	0.0537969	−	0.998552i	$-0.482868\pi$
$271$	480.000	0.107594	0.0537969	+	0.998552i	$0.482868\pi$
$272$	0	0
$273$	−6400.00	−1.41885
$274$	0	0
$275$	− 1000.00i	− 0.219281i
$276$	0	0
$277$	− 1030.00i	− 0.223418i	−0.993741	−	0.111709i	$-0.964368\pi$
$277$	− 1030.00i	− 0.223418i	0.993741	−	0.111709i	$-0.0356324\pi$
$278$	0	0
$279$	11840.0	2.54065
$280$	0	0
$281$	3270.00	0.694206	0.347103	−	0.937827i	$-0.387165\pi$
$281$	3270.00	0.694206	0.347103	+	0.937827i	$0.387165\pi$
$282$	0	0
$283$	2168.00i	0.455386i	0.973733	+	0.227693i	$0.0731183\pi$
$283$	2168.00i	0.455386i	−0.973733	+	0.227693i	$0.926882\pi$
$284$	0	0
$285$	− 3200.00i	− 0.665093i
$286$	0	0
$287$	−6560.00	−1.34921
$288$	0	0
$289$	−4013.00	−0.816813
$290$	0	0
$291$	880.000i	0.177273i
$292$	0	0
$293$	− 2070.00i	− 0.412733i	−0.978475	−	0.206366i	$-0.933836\pi$
$293$	− 2070.00i	− 0.412733i	0.978475	−	0.206366i	$-0.0661639\pi$
$294$	0	0
$295$	2000.00	0.394727
$296$	0	0
$297$	3200.00	0.625195
$298$	0	0
$299$	− 2400.00i	− 0.464199i
$300$	0	0
$301$	2432.00i	0.465708i
$302$	0	0
$303$	8784.00	1.66544
$304$	0	0
$305$	−300.000	−0.0563211
$306$	0	0
$307$	− 1896.00i	− 0.352477i	−0.984347	−	0.176238i	$-0.943607\pi$
$307$	− 1896.00i	− 0.352477i	0.984347	−	0.176238i	$-0.0563930\pi$
$308$	0	0
$309$	384.000i	0.0706958i
$310$	0	0
$311$	−1680.00	−0.306315	−0.153158	−	0.988202i	$-0.548944\pi$
$311$	−1680.00	−0.306315	−0.153158	+	0.988202i	$0.548944\pi$
$312$	0	0
$313$	−970.000	−0.175168	−0.0875841	−	0.996157i	$-0.527915\pi$
$313$	−970.000	−0.175168	−0.0875841	+	0.996157i	$0.527915\pi$
$314$	0	0
$315$	− 5920.00i	− 1.05890i
$316$	0	0
$317$	7230.00i	1.28100i	0.767958	+	0.640500i	$0.221273\pi$
$317$	7230.00i	1.28100i	−0.767958	+	0.640500i	$0.778727\pi$
$318$	0	0
$319$	−1360.00	−0.238700
$320$	0	0
$321$	5312.00	0.923635
$322$	0	0
$323$	1200.00i	0.206718i
$324$	0	0
$325$	− 1250.00i	− 0.213346i
$326$	0	0
$327$	−2960.00	−0.500576
$328$	0	0
$329$	6656.00	1.11537
$330$	0	0
$331$	− 5800.00i	− 0.963132i	−0.876410	−	0.481566i	$-0.840068\pi$
$331$	− 5800.00i	− 0.963132i	0.876410	−	0.481566i	$-0.159932\pi$
$332$	0	0
$333$	11470.0i	1.88754i
$334$	0	0
$335$	7760.00	1.26559
$336$	0	0
$337$	−1870.00	−0.302271	−0.151136	−	0.988513i	$-0.548293\pi$
$337$	−1870.00	−0.302271	−0.151136	+	0.988513i	$0.548293\pi$
$338$	0	0
$339$	− 11920.0i	− 1.90975i
$340$	0	0
$341$	12800.0i	2.03272i
$342$	0	0
$343$	−6880.00	−1.08305
$344$	0	0
$345$	3840.00	0.599242
$346$	0	0
$347$	376.000i	0.0581693i	0.999577	+	0.0290846i	$0.00925923\pi$
$347$	376.000i	0.0581693i	−0.999577	+	0.0290846i	$0.990741\pi$
$348$	0	0
$349$	− 7586.00i	− 1.16352i	−0.813360	−	0.581761i	$-0.802364\pi$
$349$	− 7586.00i	− 1.16352i	0.813360	−	0.581761i	$-0.197636\pi$
$350$	0	0
$351$	4000.00	0.608274
$352$	0	0
$353$	2530.00	0.381468	0.190734	−	0.981642i	$-0.438913\pi$
$353$	2530.00	0.381468	0.190734	+	0.981642i	$0.438913\pi$
$354$	0	0
$355$	4000.00i	0.598022i
$356$	0	0
$357$	3840.00i	0.569284i
$358$	0	0
$359$	9680.00	1.42309	0.711547	−	0.702638i	$-0.247995\pi$
$359$	9680.00	1.42309	0.711547	+	0.702638i	$0.247995\pi$
$360$	0	0
$361$	5259.00	0.766730
$362$	0	0
$363$	2152.00i	0.311159i
$364$	0	0
$365$	6300.00i	0.903444i
$366$	0	0
$367$	−2784.00	−0.395977	−0.197989	−	0.980204i	$-0.563441\pi$
$367$	−2784.00	−0.395977	−0.197989	+	0.980204i	$0.563441\pi$
$368$	0	0
$369$	15170.0	2.14016
$370$	0	0
$371$	6560.00i	0.918001i
$372$	0	0
$373$	− 7910.00i	− 1.09803i	−0.835813	−	0.549014i	$-0.815003\pi$
$373$	− 7910.00i	− 1.09803i	0.835813	−	0.549014i	$-0.184997\pi$
$374$	0	0
$375$	12000.0	1.65247
$376$	0	0
$377$	−1700.00	−0.232240
$378$	0	0
$379$	1720.00i	0.233115i	0.993184	+	0.116557i	$0.0371859\pi$
$379$	1720.00i	0.233115i	−0.993184	+	0.116557i	$0.962814\pi$
$380$	0	0
$381$	− 8192.00i	− 1.10155i
$382$	0	0
$383$	−11008.0	−1.46862	−0.734311	−	0.678813i	$-0.762495\pi$
$383$	−11008.0	−1.46862	−0.734311	+	0.678813i	$0.762495\pi$
$384$	0	0
$385$	6400.00	0.847206
$386$	0	0
$387$	− 5624.00i	− 0.738718i
$388$	0	0
$389$	12330.0i	1.60708i	0.595248	+	0.803542i	$0.297054\pi$
$389$	12330.0i	1.60708i	−0.595248	+	0.803542i	$0.702946\pi$
$390$	0	0
$391$	−1440.00	−0.186250
$392$	0	0
$393$	−9280.00	−1.19113
$394$	0	0
$395$	11200.0i	1.42667i
$396$	0	0
$397$	− 4370.00i	− 0.552453i	−0.961093	−	0.276227i	$-0.910916\pi$
$397$	− 4370.00i	− 0.552453i	0.961093	−	0.276227i	$-0.0890841\pi$
$398$	0	0
$399$	−5120.00	−0.642408
$400$	0	0
$401$	3298.00	0.410709	0.205354	−	0.978688i	$-0.434165\pi$
$401$	3298.00	0.410709	0.205354	+	0.978688i	$0.434165\pi$
$402$	0	0
$403$	16000.0i	1.97771i
$404$	0	0
$405$	− 3590.00i	− 0.440466i
$406$	0	0
$407$	−12400.0	−1.51018
$408$	0	0
$409$	9110.00	1.10137	0.550685	−	0.834713i	$-0.314366\pi$
$409$	9110.00	1.10137	0.550685	+	0.834713i	$0.314366\pi$
$410$	0	0
$411$	4560.00i	0.547271i
$412$	0	0
$413$	− 3200.00i	− 0.381263i
$414$	0	0
$415$	5520.00	0.652930
$416$	0	0
$417$	−15680.0	−1.84137
$418$	0	0
$419$	− 7880.00i	− 0.918767i	−0.888238	−	0.459383i	$-0.848070\pi$
$419$	− 7880.00i	− 0.918767i	0.888238	−	0.459383i	$-0.151930\pi$
$420$	0	0
$421$	5290.00i	0.612396i	0.951968	+	0.306198i	$0.0990570\pi$
$421$	5290.00i	0.612396i	−0.951968	+	0.306198i	$0.900943\pi$
$422$	0	0
$423$	−15392.0	−1.76923
$424$	0	0
$425$	−750.000	−0.0856008
$426$	0	0
$427$	480.000i	0.0544001i
$428$	0	0
$429$	16000.0i	1.80067i
$430$	0	0
$431$	−13920.0	−1.55569	−0.777845	−	0.628456i	$-0.783687\pi$
$431$	−13920.0	−1.55569	−0.777845	+	0.628456i	$0.783687\pi$
$432$	0	0
$433$	4930.00	0.547161	0.273580	−	0.961849i	$-0.411792\pi$
$433$	4930.00	0.547161	0.273580	+	0.961849i	$0.411792\pi$
$434$	0	0
$435$	− 2720.00i	− 0.299802i
$436$	0	0
$437$	− 1920.00i	− 0.210174i
$438$	0	0
$439$	−10640.0	−1.15676	−0.578382	−	0.815766i	$-0.696316\pi$
$439$	−10640.0	−1.15676	−0.578382	+	0.815766i	$0.696316\pi$
$440$	0	0
$441$	3219.00	0.347587
$442$	0	0
$443$	− 9288.00i	− 0.996131i	−0.867139	−	0.498066i	$-0.834044\pi$
$443$	− 9288.00i	− 0.996131i	0.867139	−	0.498066i	$-0.165956\pi$
$444$	0	0
$445$	3260.00i	0.347278i
$446$	0	0
$447$	16080.0	1.70147
$448$	0	0
$449$	12850.0	1.35062	0.675311	−	0.737533i	$-0.264010\pi$
$449$	12850.0	1.35062	0.675311	+	0.737533i	$0.264010\pi$
$450$	0	0
$451$	16400.0i	1.71230i
$452$	0	0
$453$	5760.00i	0.597414i
$454$	0	0
$455$	8000.00	0.824276
$456$	0	0
$457$	−10490.0	−1.07375	−0.536873	−	0.843663i	$-0.680394\pi$
$457$	−10490.0	−1.07375	−0.536873	+	0.843663i	$0.680394\pi$
$458$	0	0
$459$	− 2400.00i	− 0.244058i
$460$	0	0
$461$	11118.0i	1.12325i	0.827393	+	0.561624i	$0.189823\pi$
$461$	11118.0i	1.12325i	−0.827393	+	0.561624i	$0.810177\pi$
$462$	0	0
$463$	−5792.00	−0.581376	−0.290688	−	0.956818i	$-0.593884\pi$
$463$	−5792.00	−0.581376	−0.290688	+	0.956818i	$0.593884\pi$
$464$	0	0
$465$	−25600.0	−2.55306
$466$	0	0
$467$	− 2216.00i	− 0.219581i	−0.993955	−	0.109790i	$-0.964982\pi$
$467$	− 2216.00i	− 0.219581i	0.993955	−	0.109790i	$-0.0350180\pi$
$468$	0	0
$469$	− 12416.0i	− 1.22243i
$470$	0	0
$471$	14320.0	1.40091
$472$	0	0
$473$	6080.00	0.591033
$474$	0	0
$475$	− 1000.00i	− 0.0965961i
$476$	0	0
$477$	− 15170.0i	− 1.45616i
$478$	0	0
$479$	10560.0	1.00730	0.503652	−	0.863907i	$-0.331989\pi$
$479$	10560.0	1.00730	0.503652	+	0.863907i	$0.331989\pi$
$480$	0	0
$481$	−15500.0	−1.46931
$482$	0	0
$483$	− 6144.00i	− 0.578803i
$484$	0	0
$485$	− 1100.00i	− 0.102986i
$486$	0	0
$487$	13264.0	1.23419	0.617094	−	0.786890i	$-0.288310\pi$
$487$	13264.0	1.23419	0.617094	+	0.786890i	$0.288310\pi$
$488$	0	0
$489$	9664.00	0.893704
$490$	0	0
$491$	− 4840.00i	− 0.444860i	−0.974949	−	0.222430i	$-0.928601\pi$
$491$	− 4840.00i	− 0.444860i	0.974949	−	0.222430i	$-0.0713988\pi$
$492$	0	0
$493$	1020.00i	0.0931815i
$494$	0	0
$495$	−14800.0	−1.34386
$496$	0	0
$497$	6400.00	0.577624
$498$	0	0
$499$	− 19560.0i	− 1.75476i	−0.479795	−	0.877381i	$-0.659289\pi$
$499$	− 19560.0i	− 1.75476i	0.479795	−	0.877381i	$-0.340711\pi$
$500$	0	0
$501$	− 23168.0i	− 2.06601i
$502$	0	0
$503$	−528.000	−0.0468039	−0.0234019	−	0.999726i	$-0.507450\pi$
$503$	−528.000	−0.0468039	−0.0234019	+	0.999726i	$0.507450\pi$
$504$	0	0
$505$	−10980.0	−0.967532
$506$	0	0
$507$	2424.00i	0.212335i
$508$	0	0
$509$	− 19554.0i	− 1.70278i	−0.524532	−	0.851391i	$-0.675760\pi$
$509$	− 19554.0i	− 1.70278i	0.524532	−	0.851391i	$-0.324240\pi$
$510$	0	0
$511$	10080.0	0.872628
$512$	0	0
$513$	3200.00	0.275406
$514$	0	0
$515$	− 480.000i	− 0.0410705i
$516$	0	0
$517$	− 16640.0i	− 1.41552i
$518$	0	0
$519$	6000.00	0.507458
$520$	0	0
$521$	−15162.0	−1.27497	−0.637485	−	0.770463i	$-0.720025\pi$
$521$	−15162.0	−1.27497	−0.637485	+	0.770463i	$0.720025\pi$
$522$	0	0
$523$	10968.0i	0.917012i	0.888691	+	0.458506i	$0.151615\pi$
$523$	10968.0i	0.917012i	−0.888691	+	0.458506i	$0.848385\pi$
$524$	0	0
$525$	− 3200.00i	− 0.266018i
$526$	0	0
$527$	9600.00	0.793515
$528$	0	0
$529$	−9863.00	−0.810635
$530$	0	0
$531$	7400.00i	0.604770i
$532$	0	0
$533$	20500.0i	1.66595i
$534$	0	0
$535$	−6640.00	−0.536584
$536$	0	0
$537$	−18240.0	−1.46576
$538$	0	0
$539$	3480.00i	0.278097i
$540$	0	0
$541$	− 6722.00i	− 0.534198i	−0.963669	−	0.267099i	$-0.913935\pi$
$541$	− 6722.00i	− 0.534198i	0.963669	−	0.267099i	$-0.0860651\pi$
$542$	0	0
$543$	3536.00	0.279455
$544$	0	0
$545$	3700.00	0.290808
$546$	0	0
$547$	− 20424.0i	− 1.59647i	−0.602348	−	0.798233i	$-0.705768\pi$
$547$	− 20424.0i	− 1.59647i	0.602348	−	0.798233i	$-0.294232\pi$
$548$	0	0
$549$	− 1110.00i	− 0.0862908i
$550$	0	0
$551$	−1360.00	−0.105151
$552$	0	0
$553$	17920.0	1.37800
$554$	0	0
$555$	− 24800.0i	− 1.89676i
$556$	0	0
$557$	− 6610.00i	− 0.502827i	−0.967880	−	0.251414i	$-0.919105\pi$
$557$	− 6610.00i	− 0.502827i	0.967880	−	0.251414i	$-0.0808954\pi$
$558$	0	0
$559$	7600.00	0.575037
$560$	0	0
$561$	9600.00	0.722482
$562$	0	0
$563$	2712.00i	0.203015i	0.994835	+	0.101507i	$0.0323665\pi$
$563$	2712.00i	0.203015i	−0.994835	+	0.101507i	$0.967633\pi$
$564$	0	0
$565$	14900.0i	1.10946i
$566$	0	0
$567$	−5744.00	−0.425441
$568$	0	0
$569$	−3530.00	−0.260080	−0.130040	−	0.991509i	$-0.541511\pi$
$569$	−3530.00	−0.260080	−0.130040	+	0.991509i	$0.541511\pi$
$570$	0	0
$571$	− 13640.0i	− 0.999678i	−0.866118	−	0.499839i	$-0.833392\pi$
$571$	− 13640.0i	− 0.999678i	0.866118	−	0.499839i	$-0.166608\pi$
$572$	0	0
$573$	− 15360.0i	− 1.11985i
$574$	0	0
$575$	1200.00	0.0870321
$576$	0	0
$577$	−6270.00	−0.452380	−0.226190	−	0.974083i	$-0.572627\pi$
$577$	−6270.00	−0.452380	−0.226190	+	0.974083i	$0.572627\pi$
$578$	0	0
$579$	40560.0i	2.91125i
$580$	0	0
$581$	− 8832.00i	− 0.630659i
$582$	0	0
$583$	16400.0	1.16504
$584$	0	0
$585$	−18500.0	−1.30749
$586$	0	0
$587$	− 8616.00i	− 0.605827i	−0.953018	−	0.302913i	$-0.902041\pi$
$587$	− 8616.00i	− 0.605827i	0.953018	−	0.302913i	$-0.0979593\pi$
$588$	0	0
$589$	12800.0i	0.895441i
$590$	0	0
$591$	−15280.0	−1.06351
$592$	0	0
$593$	5490.00	0.380181	0.190090	−	0.981767i	$-0.439122\pi$
$593$	5490.00	0.380181	0.190090	+	0.981767i	$0.439122\pi$
$594$	0	0
$595$	− 4800.00i	− 0.330724i
$596$	0	0
$597$	− 23680.0i	− 1.62338i
$598$	0	0
$599$	−15440.0	−1.05319	−0.526595	−	0.850116i	$-0.676532\pi$
$599$	−15440.0	−1.05319	−0.526595	+	0.850116i	$0.676532\pi$
$600$	0	0
$601$	−8890.00	−0.603379	−0.301689	−	0.953406i	$-0.597551\pi$
$601$	−8890.00	−0.603379	−0.301689	+	0.953406i	$0.597551\pi$
$602$	0	0
$603$	28712.0i	1.93904i
$604$	0	0
$605$	− 2690.00i	− 0.180767i
$606$	0	0
$607$	−23744.0	−1.58771	−0.793854	−	0.608108i	$-0.791929\pi$
$607$	−23744.0	−1.58771	−0.793854	+	0.608108i	$0.791929\pi$
$608$	0	0
$609$	−4352.00	−0.289576
$610$	0	0
$611$	− 20800.0i	− 1.37721i
$612$	0	0
$613$	15210.0i	1.00216i	0.865400	+	0.501082i	$0.167064\pi$
$613$	15210.0i	1.00216i	−0.865400	+	0.501082i	$0.832936\pi$
$614$	0	0
$615$	−32800.0	−2.15061
$616$	0	0
$617$	12630.0	0.824092	0.412046	−	0.911163i	$-0.364814\pi$
$617$	12630.0	0.824092	0.412046	+	0.911163i	$0.364814\pi$
$618$	0	0
$619$	11160.0i	0.724650i	0.932052	+	0.362325i	$0.118017\pi$
$619$	11160.0i	0.724650i	−0.932052	+	0.362325i	$0.881983\pi$
$620$	0	0
$621$	3840.00i	0.248138i
$622$	0	0
$623$	5216.00	0.335433
$624$	0	0
$625$	−11875.0	−0.760000
$626$	0	0
$627$	12800.0i	0.815284i
$628$	0	0
$629$	9300.00i	0.589531i
$630$	0	0
$631$	13040.0	0.822685	0.411342	−	0.911481i	$-0.365060\pi$
$631$	13040.0	0.822685	0.411342	+	0.911481i	$0.365060\pi$
$632$	0	0
$633$	−320.000	−0.0200930
$634$	0	0
$635$	10240.0i	0.639940i
$636$	0	0
$637$	4350.00i	0.270570i
$638$	0	0
$639$	−14800.0	−0.916242
$640$	0	0
$641$	−16910.0	−1.04197	−0.520987	−	0.853565i	$-0.674436\pi$
$641$	−16910.0	−1.04197	−0.520987	+	0.853565i	$0.674436\pi$
$642$	0	0
$643$	− 4488.00i	− 0.275256i	−0.990484	−	0.137628i	$-0.956052\pi$
$643$	− 4488.00i	− 0.275256i	0.990484	−	0.137628i	$-0.0439478\pi$
$644$	0	0
$645$	12160.0i	0.742325i
$646$	0	0
$647$	2064.00	0.125416	0.0627080	−	0.998032i	$-0.480026\pi$
$647$	2064.00	0.125416	0.0627080	+	0.998032i	$0.480026\pi$
$648$	0	0
$649$	−8000.00	−0.483864
$650$	0	0
$651$	40960.0i	2.46597i
$652$	0	0
$653$	4270.00i	0.255893i	0.991781	+	0.127946i	$0.0408386\pi$
$653$	4270.00i	0.255893i	−0.991781	+	0.127946i	$0.959161\pi$
$654$	0	0
$655$	11600.0	0.691984
$656$	0	0
$657$	−23310.0	−1.38419
$658$	0	0
$659$	19800.0i	1.17041i	0.810886	+	0.585204i	$0.198985\pi$
$659$	19800.0i	1.17041i	−0.810886	+	0.585204i	$0.801015\pi$
$660$	0	0
$661$	− 27110.0i	− 1.59524i	−0.603157	−	0.797622i	$-0.706091\pi$
$661$	− 27110.0i	− 1.59524i	0.603157	−	0.797622i	$-0.293909\pi$
$662$	0	0
$663$	12000.0	0.702928
$664$	0	0
$665$	6400.00	0.373205
$666$	0	0
$667$	− 1632.00i	− 0.0947396i
$668$	0	0
$669$	34304.0i	1.98247i
$670$	0	0
$671$	1200.00	0.0690395
$672$	0	0
$673$	32210.0	1.84488	0.922440	−	0.386140i	$-0.126192\pi$
$673$	32210.0	1.84488	0.922440	+	0.386140i	$0.126192\pi$
$674$	0	0
$675$	2000.00i	0.114044i
$676$	0	0
$677$	− 27190.0i	− 1.54357i	−0.635884	−	0.771785i	$-0.719364\pi$
$677$	− 27190.0i	− 1.54357i	0.635884	−	0.771785i	$-0.280636\pi$
$678$	0	0
$679$	−1760.00	−0.0994736
$680$	0	0
$681$	51648.0	2.90625
$682$	0	0
$683$	− 20328.0i	− 1.13884i	−0.822046	−	0.569421i	$-0.807167\pi$
$683$	− 20328.0i	− 1.13884i	0.822046	−	0.569421i	$-0.192833\pi$
$684$	0	0
$685$	− 5700.00i	− 0.317935i
$686$	0	0
$687$	8528.00	0.473600
$688$	0	0
$689$	20500.0	1.13351
$690$	0	0
$691$	− 12520.0i	− 0.689267i	−0.938737	−	0.344633i	$-0.888003\pi$
$691$	− 12520.0i	− 0.689267i	0.938737	−	0.344633i	$-0.111997\pi$
$692$	0	0
$693$	23680.0i	1.29802i
$694$	0	0
$695$	19600.0	1.06974
$696$	0	0
$697$	12300.0	0.668430
$698$	0	0
$699$	− 47280.0i	− 2.55836i
$700$	0	0
$701$	11550.0i	0.622307i	0.950360	+	0.311154i	$0.100715\pi$
$701$	11550.0i	0.622307i	−0.950360	+	0.311154i	$0.899285\pi$
$702$	0	0
$703$	−12400.0	−0.665256
$704$	0	0
$705$	33280.0	1.77787
$706$	0	0
$707$	17568.0i	0.934530i
$708$	0	0
$709$	34154.0i	1.80914i	0.426325	+	0.904570i	$0.359808\pi$
$709$	34154.0i	1.80914i	−0.426325	+	0.904570i	$0.640192\pi$
$710$	0	0
$711$	−41440.0	−2.18582
$712$	0	0
$713$	−15360.0	−0.806783
$714$	0	0
$715$	− 20000.0i	− 1.04609i
$716$	0	0
$717$	− 26880.0i	− 1.40007i
$718$	0	0
$719$	22880.0	1.18676	0.593380	−	0.804923i	$-0.297793\pi$
$719$	22880.0	1.18676	0.593380	+	0.804923i	$0.297793\pi$
$720$	0	0
$721$	−768.000	−0.0396696
$722$	0	0
$723$	− 31760.0i	− 1.63370i
$724$	0	0
$725$	− 850.000i	− 0.0435424i
$726$	0	0
$727$	10416.0	0.531373	0.265686	−	0.964060i	$-0.414401\pi$
$727$	10416.0	0.531373	0.265686	+	0.964060i	$0.414401\pi$
$728$	0	0
$729$	30563.0	1.55276
$730$	0	0
$731$	− 4560.00i	− 0.230722i
$732$	0	0
$733$	14750.0i	0.743252i	0.928383	+	0.371626i	$0.121200\pi$
$733$	14750.0i	0.743252i	−0.928383	+	0.371626i	$0.878800\pi$
$734$	0	0
$735$	−6960.00	−0.349284
$736$	0	0
$737$	−31040.0	−1.55139
$738$	0	0
$739$	2360.00i	0.117475i	0.998273	+	0.0587375i	$0.0187075\pi$
$739$	2360.00i	0.117475i	−0.998273	+	0.0587375i	$0.981293\pi$
$740$	0	0
$741$	16000.0i	0.793218i
$742$	0	0
$743$	32208.0	1.59031	0.795153	−	0.606409i	$-0.207391\pi$
$743$	32208.0	1.59031	0.795153	+	0.606409i	$0.207391\pi$
$744$	0	0
$745$	−20100.0	−0.988466
$746$	0	0
$747$	20424.0i	1.00037i
$748$	0	0
$749$	10624.0i	0.518281i
$750$	0	0
$751$	36640.0	1.78031	0.890155	−	0.455658i	$-0.150596\pi$
$751$	36640.0	1.78031	0.890155	+	0.455658i	$0.150596\pi$
$752$	0	0
$753$	54720.0	2.64822
$754$	0	0
$755$	− 7200.00i	− 0.347066i
$756$	0	0
$757$	12090.0i	0.580474i	0.956955	+	0.290237i	$0.0937341\pi$
$757$	12090.0i	0.580474i	−0.956955	+	0.290237i	$0.906266\pi$
$758$	0	0
$759$	−15360.0	−0.734562
$760$	0	0
$761$	3318.00	0.158052	0.0790259	−	0.996873i	$-0.474819\pi$
$761$	3318.00	0.158052	0.0790259	+	0.996873i	$0.474819\pi$
$762$	0	0
$763$	− 5920.00i	− 0.280889i
$764$	0	0
$765$	11100.0i	0.524603i
$766$	0	0
$767$	−10000.0	−0.470768
$768$	0	0
$769$	11506.0	0.539554	0.269777	−	0.962923i	$-0.413050\pi$
$769$	11506.0	0.539554	0.269777	+	0.962923i	$0.413050\pi$
$770$	0	0
$771$	− 36880.0i	− 1.72270i
$772$	0	0
$773$	− 22230.0i	− 1.03436i	−0.855878	−	0.517178i	$-0.826982\pi$
$773$	− 22230.0i	− 1.03436i	0.855878	−	0.517178i	$-0.173018\pi$
$774$	0	0
$775$	−8000.00	−0.370798
$776$	0	0
$777$	−39680.0	−1.83206
$778$	0	0
$779$	16400.0i	0.754289i
$780$	0	0
$781$	− 16000.0i	− 0.733067i
$782$	0	0
$783$	2720.00	0.124144
$784$	0	0
$785$	−17900.0	−0.813858
$786$	0	0
$787$	21336.0i	0.966387i	0.875514	+	0.483193i	$0.160523\pi$
$787$	21336.0i	0.966387i	−0.875514	+	0.483193i	$0.839477\pi$
$788$	0	0
$789$	38784.0i	1.75000i
$790$	0	0
$791$	23840.0	1.07162
$792$	0	0
$793$	1500.00	0.0671709
$794$	0	0
$795$	32800.0i	1.46327i
$796$	0	0
$797$	− 7170.00i	− 0.318663i	−0.987225	−	0.159332i	$-0.949066\pi$
$797$	− 7170.00i	− 0.318663i	0.987225	−	0.159332i	$-0.0509339\pi$
$798$	0	0
$799$	−12480.0	−0.552579
$800$	0	0
$801$	−12062.0	−0.532072
$802$	0	0
$803$	− 25200.0i	− 1.10746i
$804$	0	0
$805$	7680.00i	0.336254i
$806$	0	0
$807$	44400.0	1.93675
$808$	0	0
$809$	23654.0	1.02797	0.513987	−	0.857798i	$-0.328168\pi$
$809$	23654.0	1.02797	0.513987	+	0.857798i	$0.328168\pi$
$810$	0	0
$811$	− 30440.0i	− 1.31799i	−0.752146	−	0.658997i	$-0.770981\pi$
$811$	− 30440.0i	− 1.31799i	0.752146	−	0.658997i	$-0.229019\pi$
$812$	0	0
$813$	− 3840.00i	− 0.165652i
$814$	0	0
$815$	−12080.0	−0.519195
$816$	0	0
$817$	6080.00	0.260358
$818$	0	0
$819$	29600.0i	1.26289i
$820$	0	0
$821$	19930.0i	0.847213i	0.905846	+	0.423606i	$0.139236\pi$
$821$	19930.0i	0.847213i	−0.905846	+	0.423606i	$0.860764\pi$
$822$	0	0
$823$	−9872.00	−0.418124	−0.209062	−	0.977902i	$-0.567041\pi$
$823$	−9872.00	−0.418124	−0.209062	+	0.977902i	$0.567041\pi$
$824$	0	0
$825$	−8000.00	−0.337605
$826$	0	0
$827$	− 5704.00i	− 0.239840i	−0.992784	−	0.119920i	$-0.961736\pi$
$827$	− 5704.00i	− 0.239840i	0.992784	−	0.119920i	$-0.0382638\pi$
$828$	0	0
$829$	27230.0i	1.14082i	0.821361	+	0.570408i	$0.193215\pi$
$829$	27230.0i	1.14082i	−0.821361	+	0.570408i	$0.806785\pi$
$830$	0	0
$831$	−8240.00	−0.343974
$832$	0	0
$833$	2610.00	0.108561
$834$	0	0
$835$	28960.0i	1.20024i
$836$	0	0
$837$	− 25600.0i	− 1.05719i
$838$	0	0
$839$	−18800.0	−0.773597	−0.386799	−	0.922164i	$-0.626419\pi$
$839$	−18800.0	−0.773597	−0.386799	+	0.922164i	$0.626419\pi$
$840$	0	0
$841$	23233.0	0.952602
$842$	0	0
$843$	− 26160.0i	− 1.06880i
$844$	0	0
$845$	− 3030.00i	− 0.123355i
$846$	0	0
$847$	−4304.00	−0.174601
$848$	0	0
$849$	17344.0	0.701113
$850$	0	0
$851$	− 14880.0i	− 0.599389i
$852$	0	0
$853$	12090.0i	0.485292i	0.970115	+	0.242646i	$0.0780153\pi$
$853$	12090.0i	0.485292i	−0.970115	+	0.242646i	$0.921985\pi$
$854$	0	0
$855$	−14800.0	−0.591988
$856$	0	0
$857$	470.000	0.0187338	0.00936692	−	0.999956i	$-0.497018\pi$
$857$	470.000	0.0187338	0.00936692	+	0.999956i	$0.497018\pi$
$858$	0	0
$859$	24440.0i	0.970759i	0.874304	+	0.485380i	$0.161319\pi$
$859$	24440.0i	0.970759i	−0.874304	+	0.485380i	$0.838681\pi$
$860$	0	0
$861$	52480.0i	2.07725i
$862$	0	0
$863$	22592.0	0.891125	0.445562	−	0.895251i	$-0.353004\pi$
$863$	22592.0	0.891125	0.445562	+	0.895251i	$0.353004\pi$
$864$	0	0
$865$	−7500.00	−0.294807
$866$	0	0
$867$	32104.0i	1.25757i
$868$	0	0
$869$	− 44800.0i	− 1.74883i
$870$	0	0
$871$	−38800.0	−1.50940
$872$	0	0
$873$	4070.00	0.157788
$874$	0	0
$875$	24000.0i	0.927255i
$876$	0	0
$877$	− 17330.0i	− 0.667266i	−0.942703	−	0.333633i	$-0.891725\pi$
$877$	− 17330.0i	− 0.667266i	0.942703	−	0.333633i	$-0.108275\pi$
$878$	0	0
$879$	−16560.0	−0.635444
$880$	0	0
$881$	−31470.0	−1.20346	−0.601732	−	0.798698i	$-0.705522\pi$
$881$	−31470.0	−1.20346	−0.601732	+	0.798698i	$0.705522\pi$
$882$	0	0
$883$	3352.00i	0.127751i	0.997958	+	0.0638753i	$0.0203460\pi$
$883$	3352.00i	0.127751i	−0.997958	+	0.0638753i	$0.979654\pi$
$884$	0	0
$885$	− 16000.0i	− 0.607722i
$886$	0	0
$887$	−48144.0	−1.82245	−0.911227	−	0.411904i	$-0.864864\pi$
$887$	−48144.0	−1.82245	−0.911227	+	0.411904i	$0.864864\pi$
$888$	0	0
$889$	16384.0	0.618112
$890$	0	0
$891$	14360.0i	0.539931i
$892$	0	0
$893$	− 16640.0i	− 0.623557i
$894$	0	0
$895$	22800.0	0.851531
$896$	0	0
$897$	−19200.0	−0.714682
$898$	0	0
$899$	10880.0i	0.403636i
$900$	0	0
$901$	− 12300.0i	− 0.454797i
$902$	0	0
$903$	19456.0	0.717005
$904$	0	0
$905$	−4420.00	−0.162349
$906$	0	0
$907$	16216.0i	0.593653i	0.954931	+	0.296827i	$0.0959283\pi$
$907$	16216.0i	0.593653i	−0.954931	+	0.296827i	$0.904072\pi$
$908$	0	0
$909$	− 40626.0i	− 1.48238i
$910$	0	0
$911$	−49440.0	−1.79805	−0.899023	−	0.437901i	$-0.855722\pi$
$911$	−49440.0	−1.79805	−0.899023	+	0.437901i	$0.855722\pi$
$912$	0	0
$913$	−22080.0	−0.800374
$914$	0	0
$915$	2400.00i	0.0867121i
$916$	0	0
$917$	− 18560.0i	− 0.668381i
$918$	0	0
$919$	−16080.0	−0.577182	−0.288591	−	0.957452i	$-0.593187\pi$
$919$	−16080.0	−0.577182	−0.288591	+	0.957452i	$0.593187\pi$
$920$	0	0
$921$	−15168.0	−0.542674
$922$	0	0
$923$	− 20000.0i	− 0.713226i
$924$	0	0
$925$	− 7750.00i	− 0.275479i
$926$	0	0
$927$	1776.00	0.0629250
$928$	0	0
$929$	−11310.0	−0.399428	−0.199714	−	0.979854i	$-0.564001\pi$
$929$	−11310.0	−0.399428	−0.199714	+	0.979854i	$0.564001\pi$
$930$	0	0
$931$	3480.00i	0.122505i
$932$	0	0
$933$	13440.0i	0.471603i
$934$	0	0
$935$	−12000.0	−0.419724
$936$	0	0
$937$	−25130.0	−0.876159	−0.438080	−	0.898936i	$-0.644341\pi$
$937$	−25130.0	−0.876159	−0.438080	+	0.898936i	$0.644341\pi$
$938$	0	0
$939$	7760.00i	0.269689i
$940$	0	0
$941$	− 22322.0i	− 0.773301i	−0.922226	−	0.386651i	$-0.873632\pi$
$941$	− 22322.0i	− 0.773301i	0.922226	−	0.386651i	$-0.126368\pi$
$942$	0	0
$943$	−19680.0	−0.679607
$944$	0	0
$945$	−12800.0	−0.440618
$946$	0	0
$947$	− 36456.0i	− 1.25096i	−0.780239	−	0.625481i	$-0.784903\pi$
$947$	− 36456.0i	− 1.25096i	0.780239	−	0.625481i	$-0.215097\pi$
$948$	0	0
$949$	− 31500.0i	− 1.07749i
$950$	0	0
$951$	57840.0	1.97223
$952$	0	0
$953$	−40650.0	−1.38172	−0.690862	−	0.722987i	$-0.742769\pi$
$953$	−40650.0	−1.38172	−0.690862	+	0.722987i	$0.742769\pi$
$954$	0	0
$955$	19200.0i	0.650573i
$956$	0	0
$957$	10880.0i	0.367503i
$958$	0	0
$959$	−9120.00	−0.307091
$960$	0	0
$961$	72609.0	2.43728
$962$	0	0
$963$	− 24568.0i	− 0.822111i
$964$	0	0
$965$	− 50700.0i	− 1.69129i
$966$	0	0
$967$	34704.0	1.15409	0.577045	−	0.816712i	$-0.304206\pi$
$967$	34704.0	1.15409	0.577045	+	0.816712i	$0.304206\pi$
$968$	0	0
$969$	9600.00	0.318263
$970$	0	0
$971$	− 30760.0i	− 1.01662i	−0.861175	−	0.508309i	$-0.830271\pi$
$971$	− 30760.0i	− 1.01662i	0.861175	−	0.508309i	$-0.169729\pi$
$972$	0	0
$973$	− 31360.0i	− 1.03325i
$974$	0	0
$975$	−10000.0	−0.328468
$976$	0	0
$977$	−38110.0	−1.24795	−0.623975	−	0.781444i	$-0.714483\pi$
$977$	−38110.0	−1.24795	−0.623975	+	0.781444i	$0.714483\pi$
$978$	0	0
$979$	− 13040.0i	− 0.425700i
$980$	0	0
$981$	13690.0i	0.445554i
$982$	0	0
$983$	19632.0	0.636992	0.318496	−	0.947924i	$-0.396822\pi$
$983$	19632.0	0.636992	0.318496	+	0.947924i	$0.396822\pi$
$984$	0	0
$985$	19100.0	0.617844
$986$	0	0
$987$	− 53248.0i	− 1.71723i
$988$	0	0
$989$	7296.00i	0.234580i
$990$	0	0
$991$	47680.0	1.52836	0.764180	−	0.645003i	$-0.223144\pi$
$991$	47680.0	1.52836	0.764180	+	0.645003i	$0.223144\pi$
$992$	0	0
$993$	−46400.0	−1.48284
$994$	0	0
$995$	29600.0i	0.943099i
$996$	0	0
$997$	39690.0i	1.26078i	0.776280	+	0.630389i	$0.217104\pi$
$997$	39690.0i	1.26078i	−0.776280	+	0.630389i	$0.782896\pi$
$998$	0	0
$999$	24800.0	0.785423

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.4.b.e.129.1		2
4.3	odd	2		256.4.b.c.129.2		2
8.3	odd	2		256.4.b.c.129.1		2
8.5	even	2	inner	256.4.b.e.129.2		2
16.3	odd	4		64.4.a.a.1.1		1
16.5	even	4		32.4.a.a.1.1	✓	1
16.11	odd	4		32.4.a.c.1.1	yes	1
16.13	even	4		64.4.a.e.1.1		1
48.5	odd	4		288.4.a.h.1.1		1
48.11	even	4		288.4.a.i.1.1		1
48.29	odd	4		576.4.a.g.1.1		1
48.35	even	4		576.4.a.h.1.1		1
80.19	odd	4		1600.4.a.bw.1.1		1
80.27	even	4		800.4.c.a.449.1		2
80.29	even	4		1600.4.a.e.1.1		1
80.37	odd	4		800.4.c.b.449.2		2
80.43	even	4		800.4.c.a.449.2		2
80.53	odd	4		800.4.c.b.449.1		2
80.59	odd	4		800.4.a.a.1.1		1
80.69	even	4		800.4.a.k.1.1		1
112.27	even	4		1568.4.a.c.1.1		1
112.69	odd	4		1568.4.a.o.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.4.a.a.1.1	✓	1	16.5	even	4
32.4.a.c.1.1	yes	1	16.11	odd	4
64.4.a.a.1.1		1	16.3	odd	4
64.4.a.e.1.1		1	16.13	even	4
256.4.b.c.129.1		2	8.3	odd	2
256.4.b.c.129.2		2	4.3	odd	2
256.4.b.e.129.1		2	1.1	even	1	trivial
256.4.b.e.129.2		2	8.5	even	2	inner
288.4.a.h.1.1		1	48.5	odd	4
288.4.a.i.1.1		1	48.11	even	4
576.4.a.g.1.1		1	48.29	odd	4
576.4.a.h.1.1		1	48.35	even	4
800.4.a.a.1.1		1	80.59	odd	4
800.4.a.k.1.1		1	80.69	even	4
800.4.c.a.449.1		2	80.27	even	4
800.4.c.a.449.2		2	80.43	even	4
800.4.c.b.449.1		2	80.53	odd	4
800.4.c.b.449.2		2	80.37	odd	4
1568.4.a.c.1.1		1	112.27	even	4
1568.4.a.o.1.1		1	112.69	odd	4
1600.4.a.e.1.1		1	80.29	even	4
1600.4.a.bw.1.1		1	80.19	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	256.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-8.00000i q^{3} +10.0000i q^{5} +16.0000 q^{7} -37.0000 q^{9} +O(q^{10})\)	\(q-8.00000i q^{3} +10.0000i q^{5} +16.0000 q^{7} -37.0000 q^{9} -40.0000i q^{11} -50.0000i q^{13} +80.0000 q^{15} -30.0000 q^{17} -40.0000i q^{19} -128.000i q^{21} +48.0000 q^{23} +25.0000 q^{25} +80.0000i q^{27} -34.0000i q^{29} -320.000 q^{31} -320.000 q^{33} +160.000i q^{35} -310.000i q^{37} -400.000 q^{39} -410.000 q^{41} +152.000i q^{43} -370.000i q^{45} +416.000 q^{47} -87.0000 q^{49} +240.000i q^{51} +410.000i q^{53} +400.000 q^{55} -320.000 q^{57} -200.000i q^{59} +30.0000i q^{61} -592.000 q^{63} +500.000 q^{65} -776.000i q^{67} -384.000i q^{69} +400.000 q^{71} +630.000 q^{73} -200.000i q^{75} -640.000i q^{77} +1120.00 q^{79} -359.000 q^{81} -552.000i q^{83} -300.000i q^{85} -272.000 q^{87} +326.000 q^{89} -800.000i q^{91} +2560.00i q^{93} +400.000 q^{95} -110.000 q^{97} +1480.00i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 32 q^{7} - 74 q^{9}+O(q^{10}) \) 2 * q + 32 * q^7 - 74 * q^9	\( 2 q + 32 q^{7} - 74 q^{9} + 160 q^{15} - 60 q^{17} + 96 q^{23} + 50 q^{25} - 640 q^{31} - 640 q^{33} - 800 q^{39} - 820 q^{41} + 832 q^{47} - 174 q^{49} + 800 q^{55} - 640 q^{57} - 1184 q^{63} + 1000 q^{65} + 800 q^{71} + 1260 q^{73} + 2240 q^{79} - 718 q^{81} - 544 q^{87} + 652 q^{89} + 800 q^{95} - 220 q^{97}+O(q^{100}) \) 2 * q + 32 * q^7 - 74 * q^9 + 160 * q^15 - 60 * q^17 + 96 * q^23 + 50 * q^25 - 640 * q^31 - 640 * q^33 - 800 * q^39 - 820 * q^41 + 832 * q^47 - 174 * q^49 + 800 * q^55 - 640 * q^57 - 1184 * q^63 + 1000 * q^65 + 800 * q^71 + 1260 * q^73 + 2240 * q^79 - 718 * q^81 - 544 * q^87 + 652 * q^89 + 800 * q^95 - 220 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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