LMFDB - Embedded newform 256.8.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("256.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$79.9705665239$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{46}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 46 $ x^2 - 46
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 128)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.78233$ of defining polynomial
Character	$\chi$	$=$	256.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-54.2586 q^{3} -180.000 q^{5} -217.035 q^{7} +757.000 q^{9} +O(q^{10})$	$q-54.2586 q^{3} -180.000 q^{5} -217.035 q^{7} +757.000 q^{9} +2658.67 q^{11} +3820.00 q^{13} +9766.56 q^{15} -15842.0 q^{17} -35865.0 q^{19} +11776.0 q^{21} -103960. q^{23} -45725.0 q^{25} +77589.9 q^{27} -97748.0 q^{29} -123276. q^{31} -144256. q^{33} +39066.2 q^{35} +238748. q^{37} -207268. q^{39} +455622. q^{41} -401677. q^{43} -136260. q^{45} -1.35213e6 q^{47} -776439. q^{49} +859565. q^{51} +1.28436e6 q^{53} -478561. q^{55} +1.94598e6 q^{57} -109548. q^{59} -1.60256e6 q^{61} -164295. q^{63} -687600. q^{65} -2.34772e6 q^{67} +5.64070e6 q^{69} -1.54463e6 q^{71} +2.74276e6 q^{73} +2.48098e6 q^{75} -577024. q^{77} +6.27186e6 q^{79} -5.86548e6 q^{81} -549586. q^{83} +2.85156e6 q^{85} +5.30367e6 q^{87} +346458. q^{89} -829072. q^{91} +6.68877e6 q^{93} +6.45569e6 q^{95} -6.84130e6 q^{97} +2.01262e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 360 q^{5} + 1514 q^{9}+O(q^{10}) $ 2 * q - 360 * q^5 + 1514 * q^9	$ 2 q - 360 q^{5} + 1514 q^{9} + 7640 q^{13} - 31684 q^{17} + 23552 q^{21} - 91450 q^{25} - 195496 q^{29} - 288512 q^{33} + 477496 q^{37} + 911244 q^{41} - 272520 q^{45} - 1552878 q^{49} + 2568728 q^{53} + 3891968 q^{57} - 3205128 q^{61} - 1375200 q^{65} + 11281408 q^{69} + 5485524 q^{73} - 1154048 q^{77} - 11730958 q^{81} + 5703120 q^{85} + 692916 q^{89} + 13377536 q^{93} - 13682596 q^{97}+O(q^{100}) $ 2 * q - 360 * q^5 + 1514 * q^9 + 7640 * q^13 - 31684 * q^17 + 23552 * q^21 - 91450 * q^25 - 195496 * q^29 - 288512 * q^33 + 477496 * q^37 + 911244 * q^41 - 272520 * q^45 - 1552878 * q^49 + 2568728 * q^53 + 3891968 * q^57 - 3205128 * q^61 - 1375200 * q^65 + 11281408 * q^69 + 5485524 * q^73 - 1154048 * q^77 - 11730958 * q^81 + 5703120 * q^85 + 692916 * q^89 + 13377536 * q^93 - 13682596 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−54.2586	−1.16023	−0.580116	−	0.814534i	$-0.696993\pi$
$3$	−54.2586	−1.16023	−0.580116	+	0.814534i	$0.696993\pi$
$4$	0	0
$5$	−180.000	−0.643988	−0.321994	−	0.946742i	$-0.604353\pi$
$5$	−180.000	−0.643988	−0.321994	+	0.946742i	$0.604353\pi$
$6$	0	0
$7$	−217.035	−0.239158	−0.119579	−	0.992825i	$-0.538155\pi$
$7$	−217.035	−0.239158	−0.119579	+	0.992825i	$0.538155\pi$
$8$	0	0
$9$	757.000	0.346136
$10$	0	0
$11$	2658.67	0.602269	0.301135	−	0.953582i	$-0.402635\pi$
$11$	2658.67	0.602269	0.301135	+	0.953582i	$0.402635\pi$
$12$	0	0
$13$	3820.00	0.482238	0.241119	−	0.970496i	$-0.422486\pi$
$13$	3820.00	0.482238	0.241119	+	0.970496i	$0.422486\pi$
$14$	0	0
$15$	9766.56	0.747174
$16$	0	0
$17$	−15842.0	−0.782058	−0.391029	−	0.920378i	$-0.627881\pi$
$17$	−15842.0	−0.782058	−0.391029	+	0.920378i	$0.627881\pi$
$18$	0	0
$19$	−35865.0	−1.19959	−0.599795	−	0.800154i	$-0.704751\pi$
$19$	−35865.0	−1.19959	−0.599795	+	0.800154i	$0.704751\pi$
$20$	0	0
$21$	11776.0	0.277479
$22$	0	0
$23$	−103960.	−1.78163	−0.890814	−	0.454368i	$-0.849865\pi$
$23$	−103960.	−1.78163	−0.890814	+	0.454368i	$0.849865\pi$
$24$	0	0
$25$	−45725.0	−0.585280
$26$	0	0
$27$	77589.9	0.758633
$28$	0	0
$29$	−97748.0	−0.744243	−0.372122	−	0.928184i	$-0.621370\pi$
$29$	−97748.0	−0.744243	−0.372122	+	0.928184i	$0.621370\pi$
$30$	0	0
$31$	−123276.	−0.743210	−0.371605	−	0.928391i	$-0.621192\pi$
$31$	−123276.	−0.743210	−0.371605	+	0.928391i	$0.621192\pi$
$32$	0	0
$33$	−144256.	−0.698771
$34$	0	0
$35$	39066.2	0.154015
$36$	0	0
$37$	238748.	0.774879	0.387439	−	0.921895i	$-0.373360\pi$
$37$	238748.	0.774879	0.387439	+	0.921895i	$0.373360\pi$
$38$	0	0
$39$	−207268.	−0.559508
$40$	0	0
$41$	455622.	1.03243	0.516216	−	0.856459i	$-0.327340\pi$
$41$	455622.	1.03243	0.516216	+	0.856459i	$0.327340\pi$
$42$	0	0
$43$	−401677.	−0.770437	−0.385218	−	0.922825i	$-0.625874\pi$
$43$	−401677.	−0.770437	−0.385218	+	0.922825i	$0.625874\pi$
$44$	0	0
$45$	−136260.	−0.222907
$46$	0	0
$47$	−1.35213e6	−1.89965	−0.949827	−	0.312776i	$-0.898741\pi$
$47$	−1.35213e6	−1.89965	−0.949827	+	0.312776i	$0.898741\pi$
$48$	0	0
$49$	−776439.	−0.942803
$50$	0	0
$51$	859565.	0.907368
$52$	0	0
$53$	1.28436e6	1.18501	0.592506	−	0.805566i	$-0.298139\pi$
$53$	1.28436e6	1.18501	0.592506	+	0.805566i	$0.298139\pi$
$54$	0	0
$55$	−478561.	−0.387854
$56$	0	0
$57$	1.94598e6	1.39180
$58$	0	0
$59$	−109548.	−0.0694422	−0.0347211	−	0.999397i	$-0.511054\pi$
$59$	−109548.	−0.0694422	−0.0347211	+	0.999397i	$0.511054\pi$
$60$	0	0
$61$	−1.60256e6	−0.903984	−0.451992	−	0.892022i	$-0.649287\pi$
$61$	−1.60256e6	−0.903984	−0.451992	+	0.892022i	$0.649287\pi$
$62$	0	0
$63$	−164295.	−0.0827814
$64$	0	0
$65$	−687600.	−0.310555
$66$	0	0
$67$	−2.34772e6	−0.953639	−0.476819	−	0.879001i	$-0.658210\pi$
$67$	−2.34772e6	−0.953639	−0.476819	+	0.879001i	$0.658210\pi$
$68$	0	0
$69$	5.64070e6	2.06710
$70$	0	0
$71$	−1.54463e6	−0.512179	−0.256089	−	0.966653i	$-0.582434\pi$
$71$	−1.54463e6	−0.512179	−0.256089	+	0.966653i	$0.582434\pi$
$72$	0	0
$73$	2.74276e6	0.825198	0.412599	−	0.910913i	$-0.364621\pi$
$73$	2.74276e6	0.825198	0.412599	+	0.910913i	$0.364621\pi$
$74$	0	0
$75$	2.48098e6	0.679060
$76$	0	0
$77$	−577024.	−0.144038
$78$	0	0
$79$	6.27186e6	1.43121	0.715603	−	0.698508i	$-0.246152\pi$
$79$	6.27186e6	1.43121	0.715603	+	0.698508i	$0.246152\pi$
$80$	0	0
$81$	−5.86548e6	−1.22633
$82$	0	0
$83$	−549586.	−0.105502	−0.0527512	−	0.998608i	$-0.516799\pi$
$83$	−549586.	−0.105502	−0.0527512	+	0.998608i	$0.516799\pi$
$84$	0	0
$85$	2.85156e6	0.503635
$86$	0	0
$87$	5.30367e6	0.863494
$88$	0	0
$89$	346458.	0.0520937	0.0260469	−	0.999661i	$-0.491708\pi$
$89$	346458.	0.0520937	0.0260469	+	0.999661i	$0.491708\pi$
$90$	0	0
$91$	−829072.	−0.115331
$92$	0	0
$93$	6.68877e6	0.862295
$94$	0	0
$95$	6.45569e6	0.772521
$96$	0	0
$97$	−6.84130e6	−0.761092	−0.380546	−	0.924762i	$-0.624264\pi$
$97$	−6.84130e6	−0.761092	−0.380546	+	0.924762i	$0.624264\pi$
$98$	0	0
$99$	2.01262e6	0.208467
$100$	0	0
$101$	1.81944e7	1.75717	0.878585	−	0.477587i	$-0.158488\pi$
$101$	1.81944e7	1.75717	0.878585	+	0.477587i	$0.158488\pi$
$102$	0	0
$103$	1.20912e7	1.09028	0.545142	−	0.838344i	$-0.316476\pi$
$103$	1.20912e7	1.09028	0.545142	+	0.838344i	$0.316476\pi$
$104$	0	0
$105$	−2.11968e6	−0.178693
$106$	0	0
$107$	−712253.	−0.0562071	−0.0281035	−	0.999605i	$-0.508947\pi$
$107$	−712253.	−0.0562071	−0.0281035	+	0.999605i	$0.508947\pi$
$108$	0	0
$109$	2.08543e7	1.54242	0.771210	−	0.636580i	$-0.219652\pi$
$109$	2.08543e7	1.54242	0.771210	+	0.636580i	$0.219652\pi$
$110$	0	0
$111$	−1.29541e7	−0.899038
$112$	0	0
$113$	1.86270e7	1.21442	0.607208	−	0.794543i	$-0.292289\pi$
$113$	1.86270e7	1.21442	0.607208	+	0.794543i	$0.292289\pi$
$114$	0	0
$115$	1.87127e7	1.14735
$116$	0	0
$117$	2.89174e6	0.166920
$118$	0	0
$119$	3.43826e6	0.187036
$120$	0	0
$121$	−1.24186e7	−0.637272
$122$	0	0
$123$	−2.47214e7	−1.19786
$124$	0	0
$125$	2.22930e7	1.02090
$126$	0	0
$127$	2.92823e6	0.126851	0.0634253	−	0.997987i	$-0.479798\pi$
$127$	2.92823e6	0.126851	0.0634253	+	0.997987i	$0.479798\pi$
$128$	0	0
$129$	2.17944e7	0.893885
$130$	0	0
$131$	−3.95817e7	−1.53831	−0.769157	−	0.639059i	$-0.779324\pi$
$131$	−3.95817e7	−1.53831	−0.769157	+	0.639059i	$0.779324\pi$
$132$	0	0
$133$	7.78394e6	0.286892
$134$	0	0
$135$	−1.39662e7	−0.488550
$136$	0	0
$137$	−1.03205e7	−0.342908	−0.171454	−	0.985192i	$-0.554846\pi$
$137$	−1.03205e7	−0.342908	−0.171454	+	0.985192i	$0.554846\pi$
$138$	0	0
$139$	3.81107e7	1.20363	0.601817	−	0.798634i	$-0.294443\pi$
$139$	3.81107e7	1.20363	0.601817	+	0.798634i	$0.294443\pi$
$140$	0	0
$141$	7.33645e7	2.20404
$142$	0	0
$143$	1.01561e7	0.290437
$144$	0	0
$145$	1.75946e7	0.479283
$146$	0	0
$147$	4.21285e7	1.09387
$148$	0	0
$149$	2.43731e6	0.0603613	0.0301806	−	0.999544i	$-0.490392\pi$
$149$	2.43731e6	0.0603613	0.0301806	+	0.999544i	$0.490392\pi$
$150$	0	0
$151$	−6.59470e7	−1.55875	−0.779374	−	0.626559i	$-0.784463\pi$
$151$	−6.59470e7	−1.55875	−0.779374	+	0.626559i	$0.784463\pi$
$152$	0	0
$153$	−1.19924e7	−0.270699
$154$	0	0
$155$	2.21896e7	0.478618
$156$	0	0
$157$	7.55101e7	1.55724	0.778622	−	0.627494i	$-0.215919\pi$
$157$	7.55101e7	1.55724	0.778622	+	0.627494i	$0.215919\pi$
$158$	0	0
$159$	−6.96878e7	−1.37489
$160$	0	0
$161$	2.25628e7	0.426091
$162$	0	0
$163$	−9.30710e7	−1.68329	−0.841643	−	0.540035i	$-0.818411\pi$
$163$	−9.30710e7	−1.68329	−0.841643	+	0.540035i	$0.818411\pi$
$164$	0	0
$165$	2.59661e7	0.450000
$166$	0	0
$167$	−5.10615e7	−0.848371	−0.424186	−	0.905575i	$-0.639440\pi$
$167$	−5.10615e7	−0.848371	−0.424186	+	0.905575i	$0.639440\pi$
$168$	0	0
$169$	−4.81561e7	−0.767446
$170$	0	0
$171$	−2.71498e7	−0.415222
$172$	0	0
$173$	1.42642e7	0.209452	0.104726	−	0.994501i	$-0.466603\pi$
$173$	1.42642e7	0.209452	0.104726	+	0.994501i	$0.466603\pi$
$174$	0	0
$175$	9.92391e6	0.139975
$176$	0	0
$177$	5.94394e6	0.0805689
$178$	0	0
$179$	9.64245e7	1.25661	0.628307	−	0.777966i	$-0.283748\pi$
$179$	9.64245e7	1.25661	0.628307	+	0.777966i	$0.283748\pi$
$180$	0	0
$181$	5.35944e7	0.671807	0.335904	−	0.941896i	$-0.390958\pi$
$181$	5.35944e7	0.671807	0.335904	+	0.941896i	$0.390958\pi$
$182$	0	0
$183$	8.69529e7	1.04883
$184$	0	0
$185$	−4.29746e7	−0.499012
$186$	0	0
$187$	−4.21187e7	−0.471009
$188$	0	0
$189$	−1.68397e7	−0.181434
$190$	0	0
$191$	2.81442e7	0.292262	0.146131	−	0.989265i	$-0.453318\pi$
$191$	2.81442e7	0.292262	0.146131	+	0.989265i	$0.453318\pi$
$192$	0	0
$193$	−7.65870e7	−0.766839	−0.383420	−	0.923574i	$-0.625254\pi$
$193$	−7.65870e7	−0.766839	−0.383420	+	0.923574i	$0.625254\pi$
$194$	0	0
$195$	3.73082e7	0.360316
$196$	0	0
$197$	1.63881e7	0.152720	0.0763600	−	0.997080i	$-0.475670\pi$
$197$	1.63881e7	0.152720	0.0763600	+	0.997080i	$0.475670\pi$
$198$	0	0
$199$	1.07038e8	0.962834	0.481417	−	0.876492i	$-0.340122\pi$
$199$	1.07038e8	0.962834	0.481417	+	0.876492i	$0.340122\pi$
$200$	0	0
$201$	1.27384e8	1.10644
$202$	0	0
$203$	2.12147e7	0.177992
$204$	0	0
$205$	−8.20120e7	−0.664873
$206$	0	0
$207$	−7.86974e7	−0.616686
$208$	0	0
$209$	−9.53532e7	−0.722476
$210$	0	0
$211$	5.66316e7	0.415021	0.207511	−	0.978233i	$-0.433464\pi$
$211$	5.66316e7	0.415021	0.207511	+	0.978233i	$0.433464\pi$
$212$	0	0
$213$	8.38098e7	0.594246
$214$	0	0
$215$	7.23018e7	0.496152
$216$	0	0
$217$	2.67551e7	0.177745
$218$	0	0
$219$	−1.48819e8	−0.957420
$220$	0	0
$221$	−6.05164e7	−0.377138
$222$	0	0
$223$	2.19794e8	1.32724	0.663618	−	0.748072i	$-0.269020\pi$
$223$	2.19794e8	1.32724	0.663618	+	0.748072i	$0.269020\pi$
$224$	0	0
$225$	−3.46138e7	−0.202587
$226$	0	0
$227$	−2.78334e8	−1.57934	−0.789670	−	0.613532i	$-0.789748\pi$
$227$	−2.78334e8	−1.57934	−0.789670	+	0.613532i	$0.789748\pi$
$228$	0	0
$229$	−1.11568e8	−0.613923	−0.306962	−	0.951722i	$-0.599312\pi$
$229$	−1.11568e8	−0.613923	−0.306962	+	0.951722i	$0.599312\pi$
$230$	0	0
$231$	3.13085e7	0.167117
$232$	0	0
$233$	3.70241e8	1.91751	0.958757	−	0.284225i	$-0.0917364\pi$
$233$	3.70241e8	1.91751	0.958757	+	0.284225i	$0.0917364\pi$
$234$	0	0
$235$	2.43383e8	1.22335
$236$	0	0
$237$	−3.40303e8	−1.66053
$238$	0	0
$239$	2.74632e8	1.30124	0.650621	−	0.759403i	$-0.274509\pi$
$239$	2.74632e8	1.30124	0.650621	+	0.759403i	$0.274509\pi$
$240$	0	0
$241$	−4.14029e6	−0.0190533	−0.00952667	−	0.999955i	$-0.503032\pi$
$241$	−4.14029e6	−0.0190533	−0.00952667	+	0.999955i	$0.503032\pi$
$242$	0	0
$243$	1.48564e8	0.664188
$244$	0	0
$245$	1.39759e8	0.607154
$246$	0	0
$247$	−1.37004e8	−0.578488
$248$	0	0
$249$	2.98198e7	0.122407
$250$	0	0
$251$	9.61262e7	0.383693	0.191846	−	0.981425i	$-0.438552\pi$
$251$	9.61262e7	0.383693	0.191846	+	0.981425i	$0.438552\pi$
$252$	0	0
$253$	−2.76394e8	−1.07302
$254$	0	0
$255$	−1.54722e8	−0.584334
$256$	0	0
$257$	−1.34766e8	−0.495237	−0.247618	−	0.968858i	$-0.579648\pi$
$257$	−1.34766e8	−0.495237	−0.247618	+	0.968858i	$0.579648\pi$
$258$	0	0
$259$	−5.18166e7	−0.185319
$260$	0	0
$261$	−7.39952e7	−0.257610
$262$	0	0
$263$	−3.85690e7	−0.130735	−0.0653677	−	0.997861i	$-0.520822\pi$
$263$	−3.85690e7	−0.130735	−0.0653677	+	0.997861i	$0.520822\pi$
$264$	0	0
$265$	−2.31186e8	−0.763133
$266$	0	0
$267$	−1.87983e7	−0.0604408
$268$	0	0
$269$	−1.67903e8	−0.525927	−0.262964	−	0.964806i	$-0.584700\pi$
$269$	−1.67903e8	−0.525927	−0.262964	+	0.964806i	$0.584700\pi$
$270$	0	0
$271$	−5.56602e7	−0.169884	−0.0849420	−	0.996386i	$-0.527071\pi$
$271$	−5.56602e7	−0.169884	−0.0849420	+	0.996386i	$0.527071\pi$
$272$	0	0
$273$	4.49843e7	0.133811
$274$	0	0
$275$	−1.21568e8	−0.352496
$276$	0	0
$277$	−4.17551e8	−1.18040	−0.590201	−	0.807256i	$-0.700952\pi$
$277$	−4.17551e8	−1.18040	−0.590201	+	0.807256i	$0.700952\pi$
$278$	0	0
$279$	−9.33197e7	−0.257252
$280$	0	0
$281$	−7.19842e7	−0.193537	−0.0967687	−	0.995307i	$-0.530851\pi$
$281$	−7.19842e7	−0.193537	−0.0967687	+	0.995307i	$0.530851\pi$
$282$	0	0
$283$	−7.18753e7	−0.188507	−0.0942534	−	0.995548i	$-0.530046\pi$
$283$	−7.18753e7	−0.188507	−0.0942534	+	0.995548i	$0.530046\pi$
$284$	0	0
$285$	−3.50277e8	−0.896303
$286$	0	0
$287$	−9.88857e7	−0.246915
$288$	0	0
$289$	−1.59370e8	−0.388386
$290$	0	0
$291$	3.71200e8	0.883043
$292$	0	0
$293$	−6.78092e8	−1.57490	−0.787449	−	0.616380i	$-0.788598\pi$
$293$	−6.78092e8	−1.57490	−0.787449	+	0.616380i	$0.788598\pi$
$294$	0	0
$295$	1.97187e7	0.0447199
$296$	0	0
$297$	2.06286e8	0.456901
$298$	0	0
$299$	−3.97125e8	−0.859169
$300$	0	0
$301$	8.71777e7	0.184256
$302$	0	0
$303$	−9.87205e8	−2.03872
$304$	0	0
$305$	2.88462e8	0.582155
$306$	0	0
$307$	−9.75677e7	−0.192452	−0.0962259	−	0.995360i	$-0.530677\pi$
$307$	−9.75677e7	−0.192452	−0.0962259	+	0.995360i	$0.530677\pi$
$308$	0	0
$309$	−6.56053e8	−1.26498
$310$	0	0
$311$	1.38783e8	0.261622	0.130811	−	0.991407i	$-0.458242\pi$
$311$	1.38783e8	0.261622	0.130811	+	0.991407i	$0.458242\pi$
$312$	0	0
$313$	−9.77285e8	−1.80142	−0.900712	−	0.434417i	$-0.856955\pi$
$313$	−9.77285e8	−1.80142	−0.900712	+	0.434417i	$0.856955\pi$
$314$	0	0
$315$	2.95731e7	0.0533102
$316$	0	0
$317$	8.52052e6	0.0150231	0.00751154	−	0.999972i	$-0.497609\pi$
$317$	8.52052e6	0.0150231	0.00751154	+	0.999972i	$0.497609\pi$
$318$	0	0
$319$	−2.59880e8	−0.448235
$320$	0	0
$321$	3.86459e7	0.0652132
$322$	0	0
$323$	5.68173e8	0.938148
$324$	0	0
$325$	−1.74670e8	−0.282244
$326$	0	0
$327$	−1.13153e9	−1.78956
$328$	0	0
$329$	2.93458e8	0.454318
$330$	0	0
$331$	−4.69843e8	−0.712122	−0.356061	−	0.934463i	$-0.615881\pi$
$331$	−4.69843e8	−0.712122	−0.356061	+	0.934463i	$0.615881\pi$
$332$	0	0
$333$	1.80732e8	0.268214
$334$	0	0
$335$	4.22589e8	0.614132
$336$	0	0
$337$	1.01378e9	1.44291	0.721453	−	0.692464i	$-0.243475\pi$
$337$	1.01378e9	1.44291	0.721453	+	0.692464i	$0.243475\pi$
$338$	0	0
$339$	−1.01067e9	−1.40900
$340$	0	0
$341$	−3.27750e8	−0.447612
$342$	0	0
$343$	3.47251e8	0.464638
$344$	0	0
$345$	−1.01533e9	−1.33119
$346$	0	0
$347$	1.12846e9	1.44989	0.724943	−	0.688808i	$-0.241866\pi$
$347$	1.12846e9	1.44989	0.724943	+	0.688808i	$0.241866\pi$
$348$	0	0
$349$	6.48422e8	0.816523	0.408262	−	0.912865i	$-0.366135\pi$
$349$	6.48422e8	0.816523	0.408262	+	0.912865i	$0.366135\pi$
$350$	0	0
$351$	2.96393e8	0.365842
$352$	0	0
$353$	3.46140e8	0.418832	0.209416	−	0.977827i	$-0.432844\pi$
$353$	3.46140e8	0.418832	0.209416	+	0.977827i	$0.432844\pi$
$354$	0	0
$355$	2.78034e8	0.329837
$356$	0	0
$357$	−1.86555e8	−0.217005
$358$	0	0
$359$	1.10066e9	1.25552	0.627759	−	0.778408i	$-0.283972\pi$
$359$	1.10066e9	1.25552	0.627759	+	0.778408i	$0.283972\pi$
$360$	0	0
$361$	3.92424e8	0.439016
$362$	0	0
$363$	6.73818e8	0.739383
$364$	0	0
$365$	−4.93697e8	−0.531417
$366$	0	0
$367$	1.66008e9	1.75307	0.876534	−	0.481340i	$-0.159850\pi$
$367$	1.66008e9	1.75307	0.876534	+	0.481340i	$0.159850\pi$
$368$	0	0
$369$	3.44906e8	0.357362
$370$	0	0
$371$	−2.78751e8	−0.283406
$372$	0	0
$373$	−1.18885e9	−1.18617	−0.593083	−	0.805141i	$-0.702089\pi$
$373$	−1.18885e9	−1.18617	−0.593083	+	0.805141i	$0.702089\pi$
$374$	0	0
$375$	−1.20959e9	−1.18448
$376$	0	0
$377$	−3.73397e8	−0.358903
$378$	0	0
$379$	−3.90447e8	−0.368404	−0.184202	−	0.982888i	$-0.558970\pi$
$379$	−3.90447e8	−0.368404	−0.184202	+	0.982888i	$0.558970\pi$
$380$	0	0
$381$	−1.58882e8	−0.147176
$382$	0	0
$383$	3.11693e8	0.283486	0.141743	−	0.989904i	$-0.454729\pi$
$383$	3.11693e8	0.283486	0.141743	+	0.989904i	$0.454729\pi$
$384$	0	0
$385$	1.03864e8	0.0927585
$386$	0	0
$387$	−3.04069e8	−0.266676
$388$	0	0
$389$	1.80781e9	1.55715	0.778573	−	0.627553i	$-0.215944\pi$
$389$	1.80781e9	1.55715	0.778573	+	0.627553i	$0.215944\pi$
$390$	0	0
$391$	1.64693e9	1.39334
$392$	0	0
$393$	2.14765e9	1.78480
$394$	0	0
$395$	−1.12894e9	−0.921678
$396$	0	0
$397$	1.90692e9	1.52956	0.764779	−	0.644293i	$-0.222848\pi$
$397$	1.90692e9	1.52956	0.764779	+	0.644293i	$0.222848\pi$
$398$	0	0
$399$	−4.22346e8	−0.332861
$400$	0	0
$401$	−7.73206e8	−0.598811	−0.299405	−	0.954126i	$-0.596788\pi$
$401$	−7.73206e8	−0.598811	−0.299405	+	0.954126i	$0.596788\pi$
$402$	0	0
$403$	−4.70913e8	−0.358404
$404$	0	0
$405$	1.05579e9	0.789739
$406$	0	0
$407$	6.34753e8	0.466685
$408$	0	0
$409$	−1.38156e9	−0.998480	−0.499240	−	0.866464i	$-0.666387\pi$
$409$	−1.38156e9	−0.998480	−0.499240	+	0.866464i	$0.666387\pi$
$410$	0	0
$411$	5.59975e8	0.397853
$412$	0	0
$413$	2.37757e7	0.0166077
$414$	0	0
$415$	9.89254e7	0.0679422
$416$	0	0
$417$	−2.06783e9	−1.39649
$418$	0	0
$419$	2.82274e9	1.87466	0.937328	−	0.348447i	$-0.113291\pi$
$419$	2.82274e9	1.87466	0.937328	+	0.348447i	$0.113291\pi$
$420$	0	0
$421$	5.59886e7	0.0365689	0.0182845	−	0.999833i	$-0.494180\pi$
$421$	5.59886e7	0.0365689	0.0182845	+	0.999833i	$0.494180\pi$
$422$	0	0
$423$	−1.02356e9	−0.657539
$424$	0	0
$425$	7.24375e8	0.457723
$426$	0	0
$427$	3.47812e8	0.216196
$428$	0	0
$429$	−5.51058e8	−0.336974
$430$	0	0
$431$	−7.29079e8	−0.438636	−0.219318	−	0.975653i	$-0.570383\pi$
$431$	−7.29079e8	−0.438636	−0.219318	+	0.975653i	$0.570383\pi$
$432$	0	0
$433$	7.51244e8	0.444706	0.222353	−	0.974966i	$-0.428626\pi$
$433$	7.51244e8	0.444706	0.222353	+	0.974966i	$0.428626\pi$
$434$	0	0
$435$	−9.54661e8	−0.556080
$436$	0	0
$437$	3.72851e9	2.13722
$438$	0	0
$439$	3.14004e9	1.77137	0.885685	−	0.464286i	$-0.153689\pi$
$439$	3.14004e9	1.77137	0.885685	+	0.464286i	$0.153689\pi$
$440$	0	0
$441$	−5.87764e8	−0.326338
$442$	0	0
$443$	−1.89553e9	−1.03590	−0.517949	−	0.855411i	$-0.673304\pi$
$443$	−1.89553e9	−1.03590	−0.517949	+	0.855411i	$0.673304\pi$
$444$	0	0
$445$	−6.23624e7	−0.0335477
$446$	0	0
$447$	−1.32245e8	−0.0700330
$448$	0	0
$449$	−2.27689e9	−1.18708	−0.593540	−	0.804804i	$-0.702270\pi$
$449$	−2.27689e9	−1.18708	−0.593540	+	0.804804i	$0.702270\pi$
$450$	0	0
$451$	1.21135e9	0.621802
$452$	0	0
$453$	3.57820e9	1.80851
$454$	0	0
$455$	1.49233e8	0.0742720
$456$	0	0
$457$	2.29182e9	1.12324	0.561621	−	0.827395i	$-0.310178\pi$
$457$	2.29182e9	1.12324	0.561621	+	0.827395i	$0.310178\pi$
$458$	0	0
$459$	−1.22918e9	−0.593295
$460$	0	0
$461$	2.14064e8	0.101763	0.0508816	−	0.998705i	$-0.483797\pi$
$461$	2.14064e8	0.101763	0.0508816	+	0.998705i	$0.483797\pi$
$462$	0	0
$463$	−1.03421e9	−0.484258	−0.242129	−	0.970244i	$-0.577846\pi$
$463$	−1.03421e9	−0.484258	−0.242129	+	0.970244i	$0.577846\pi$
$464$	0	0
$465$	−1.20398e9	−0.555307
$466$	0	0
$467$	−2.93319e9	−1.33269	−0.666347	−	0.745642i	$-0.732143\pi$
$467$	−2.93319e9	−1.33269	−0.666347	+	0.745642i	$0.732143\pi$
$468$	0	0
$469$	5.09536e8	0.228071
$470$	0	0
$471$	−4.09708e9	−1.80676
$472$	0	0
$473$	−1.06793e9	−0.464010
$474$	0	0
$475$	1.63993e9	0.702096
$476$	0	0
$477$	9.72264e8	0.410175
$478$	0	0
$479$	−4.96997e8	−0.206623	−0.103312	−	0.994649i	$-0.532944\pi$
$479$	−4.96997e8	−0.206623	−0.103312	+	0.994649i	$0.532944\pi$
$480$	0	0
$481$	9.12017e8	0.373676
$482$	0	0
$483$	−1.22423e9	−0.494364
$484$	0	0
$485$	1.23143e9	0.490134
$486$	0	0
$487$	−3.67557e9	−1.44203	−0.721013	−	0.692921i	$-0.756323\pi$
$487$	−3.67557e9	−1.44203	−0.721013	+	0.692921i	$0.756323\pi$
$488$	0	0
$489$	5.04991e9	1.95300
$490$	0	0
$491$	3.00322e9	1.14499	0.572494	−	0.819909i	$-0.305976\pi$
$491$	3.00322e9	1.14499	0.572494	+	0.819909i	$0.305976\pi$
$492$	0	0
$493$	1.54852e9	0.582041
$494$	0	0
$495$	−3.62271e8	−0.134250
$496$	0	0
$497$	3.35239e8	0.122492
$498$	0	0
$499$	3.95565e9	1.42517	0.712583	−	0.701587i	$-0.247525\pi$
$499$	3.95565e9	1.42517	0.712583	+	0.701587i	$0.247525\pi$
$500$	0	0
$501$	2.77053e9	0.984307
$502$	0	0
$503$	−8.20917e8	−0.287615	−0.143807	−	0.989606i	$-0.545935\pi$
$503$	−8.20917e8	−0.287615	−0.143807	+	0.989606i	$0.545935\pi$
$504$	0	0
$505$	−3.27500e9	−1.13160
$506$	0	0
$507$	2.61289e9	0.890415
$508$	0	0
$509$	1.25294e9	0.421130	0.210565	−	0.977580i	$-0.432470\pi$
$509$	1.25294e9	0.421130	0.210565	+	0.977580i	$0.432470\pi$
$510$	0	0
$511$	−5.95274e8	−0.197353
$512$	0	0
$513$	−2.78276e9	−0.910048
$514$	0	0
$515$	−2.17642e9	−0.702129
$516$	0	0
$517$	−3.59486e9	−1.14410
$518$	0	0
$519$	−7.73955e8	−0.243013
$520$	0	0
$521$	4.46501e9	1.38322	0.691609	−	0.722272i	$-0.256902\pi$
$521$	4.46501e9	1.38322	0.691609	+	0.722272i	$0.256902\pi$
$522$	0	0
$523$	−3.16654e9	−0.967898	−0.483949	−	0.875096i	$-0.660798\pi$
$523$	−3.16654e9	−0.967898	−0.483949	+	0.875096i	$0.660798\pi$
$524$	0	0
$525$	−5.38458e8	−0.162403
$526$	0	0
$527$	1.95293e9	0.581233
$528$	0	0
$529$	7.40276e9	2.17420
$530$	0	0
$531$	−8.29280e7	−0.0240364
$532$	0	0
$533$	1.74048e9	0.497878
$534$	0	0
$535$	1.28206e8	0.0361967
$536$	0	0
$537$	−5.23186e9	−1.45796
$538$	0	0
$539$	−2.06430e9	−0.567821
$540$	0	0
$541$	3.93146e9	1.06749	0.533745	−	0.845646i	$-0.320784\pi$
$541$	3.93146e9	1.06749	0.533745	+	0.845646i	$0.320784\pi$
$542$	0	0
$543$	−2.90796e9	−0.779452
$544$	0	0
$545$	−3.75377e9	−0.993300
$546$	0	0
$547$	−6.36970e9	−1.66404	−0.832019	−	0.554747i	$-0.812815\pi$
$547$	−6.36970e9	−1.66404	−0.832019	+	0.554747i	$0.812815\pi$
$548$	0	0
$549$	−1.21314e9	−0.312902
$550$	0	0
$551$	3.50573e9	0.892787
$552$	0	0
$553$	−1.36121e9	−0.342285
$554$	0	0
$555$	2.33175e9	0.578970
$556$	0	0
$557$	−4.19132e8	−0.102768	−0.0513840	−	0.998679i	$-0.516363\pi$
$557$	−4.19132e8	−0.102768	−0.0513840	+	0.998679i	$0.516363\pi$
$558$	0	0
$559$	−1.53441e9	−0.371534
$560$	0	0
$561$	2.28530e9	0.546480
$562$	0	0
$563$	7.94827e9	1.87712	0.938562	−	0.345109i	$-0.112158\pi$
$563$	7.94827e9	1.87712	0.938562	+	0.345109i	$0.112158\pi$
$564$	0	0
$565$	−3.35285e9	−0.782069
$566$	0	0
$567$	1.27301e9	0.293286
$568$	0	0
$569$	−3.46382e9	−0.788247	−0.394123	−	0.919058i	$-0.628952\pi$
$569$	−3.46382e9	−0.788247	−0.394123	+	0.919058i	$0.628952\pi$
$570$	0	0
$571$	−4.37465e9	−0.983370	−0.491685	−	0.870773i	$-0.663619\pi$
$571$	−4.37465e9	−0.983370	−0.491685	+	0.870773i	$0.663619\pi$
$572$	0	0
$573$	−1.52706e9	−0.339091
$574$	0	0
$575$	4.75355e9	1.04275
$576$	0	0
$577$	6.14946e9	1.33267	0.666333	−	0.745654i	$-0.267863\pi$
$577$	6.14946e9	1.33267	0.666333	+	0.745654i	$0.267863\pi$
$578$	0	0
$579$	4.15551e9	0.889711
$580$	0	0
$581$	1.19279e8	0.0252318
$582$	0	0
$583$	3.41470e9	0.713696
$584$	0	0
$585$	−5.20513e8	−0.107494
$586$	0	0
$587$	−1.09870e9	−0.224206	−0.112103	−	0.993697i	$-0.535759\pi$
$587$	−1.09870e9	−0.224206	−0.112103	+	0.993697i	$0.535759\pi$
$588$	0	0
$589$	4.42128e9	0.891547
$590$	0	0
$591$	−8.89195e8	−0.177191
$592$	0	0
$593$	4.28953e8	0.0844731	0.0422366	−	0.999108i	$-0.486552\pi$
$593$	4.28953e8	0.0844731	0.0422366	+	0.999108i	$0.486552\pi$
$594$	0	0
$595$	−6.18887e8	−0.120449
$596$	0	0
$597$	−5.80772e9	−1.11711
$598$	0	0
$599$	7.24934e9	1.37817	0.689087	−	0.724678i	$-0.258012\pi$
$599$	7.24934e9	1.37817	0.689087	+	0.724678i	$0.258012\pi$
$600$	0	0
$601$	−9.22234e9	−1.73293	−0.866464	−	0.499240i	$-0.833612\pi$
$601$	−9.22234e9	−1.73293	−0.866464	+	0.499240i	$0.833612\pi$
$602$	0	0
$603$	−1.77722e9	−0.330089
$604$	0	0
$605$	2.23535e9	0.410395
$606$	0	0
$607$	5.75921e9	1.04521	0.522604	−	0.852576i	$-0.324961\pi$
$607$	5.75921e9	1.04521	0.522604	+	0.852576i	$0.324961\pi$
$608$	0	0
$609$	−1.15108e9	−0.206512
$610$	0	0
$611$	−5.16512e9	−0.916086
$612$	0	0
$613$	−1.07539e10	−1.88563	−0.942813	−	0.333322i	$-0.891830\pi$
$613$	−1.07539e10	−1.88563	−0.942813	+	0.333322i	$0.891830\pi$
$614$	0	0
$615$	4.44986e9	0.771406
$616$	0	0
$617$	5.82077e9	0.997660	0.498830	−	0.866700i	$-0.333763\pi$
$617$	5.82077e9	0.997660	0.498830	+	0.866700i	$0.333763\pi$
$618$	0	0
$619$	5.98336e9	1.01398	0.506988	−	0.861953i	$-0.330759\pi$
$619$	5.98336e9	1.01398	0.506988	+	0.861953i	$0.330759\pi$
$620$	0	0
$621$	−8.06621e9	−1.35160
$622$	0	0
$623$	−7.51934e7	−0.0124587
$624$	0	0
$625$	−4.40474e8	−0.0721673
$626$	0	0
$627$	5.17374e9	0.838239
$628$	0	0
$629$	−3.78225e9	−0.606000
$630$	0	0
$631$	−6.68543e9	−1.05932	−0.529659	−	0.848211i	$-0.677680\pi$
$631$	−6.68543e9	−1.05932	−0.529659	+	0.848211i	$0.677680\pi$
$632$	0	0
$633$	−3.07276e9	−0.481521
$634$	0	0
$635$	−5.27081e8	−0.0816901
$636$	0	0
$637$	−2.96600e9	−0.454656
$638$	0	0
$639$	−1.16929e9	−0.177284
$640$	0	0
$641$	−5.46365e9	−0.819370	−0.409685	−	0.912227i	$-0.634361\pi$
$641$	−5.46365e9	−0.819370	−0.409685	+	0.912227i	$0.634361\pi$
$642$	0	0
$643$	−3.42325e9	−0.507808	−0.253904	−	0.967229i	$-0.581715\pi$
$643$	−3.42325e9	−0.507808	−0.253904	+	0.967229i	$0.581715\pi$
$644$	0	0
$645$	−3.92300e9	−0.575651
$646$	0	0
$647$	−1.04327e10	−1.51437	−0.757185	−	0.653201i	$-0.773426\pi$
$647$	−1.04327e10	−1.51437	−0.757185	+	0.653201i	$0.773426\pi$
$648$	0	0
$649$	−2.91253e8	−0.0418229
$650$	0	0
$651$	−1.45169e9	−0.206225
$652$	0	0
$653$	−1.21062e9	−0.170142	−0.0850712	−	0.996375i	$-0.527112\pi$
$653$	−1.21062e9	−0.170142	−0.0850712	+	0.996375i	$0.527112\pi$
$654$	0	0
$655$	7.12471e9	0.990656
$656$	0	0
$657$	2.07627e9	0.285631
$658$	0	0
$659$	−7.92203e9	−1.07830	−0.539148	−	0.842211i	$-0.681254\pi$
$659$	−7.92203e9	−1.07830	−0.539148	+	0.842211i	$0.681254\pi$
$660$	0	0
$661$	−4.62495e9	−0.622876	−0.311438	−	0.950267i	$-0.600811\pi$
$661$	−4.62495e9	−0.622876	−0.311438	+	0.950267i	$0.600811\pi$
$662$	0	0
$663$	3.28354e9	0.437567
$664$	0	0
$665$	−1.40111e9	−0.184755
$666$	0	0
$667$	1.01618e10	1.32596
$668$	0	0
$669$	−1.19257e10	−1.53990
$670$	0	0
$671$	−4.26069e9	−0.544442
$672$	0	0
$673$	7.57112e9	0.957430	0.478715	−	0.877970i	$-0.341103\pi$
$673$	7.57112e9	0.957430	0.478715	+	0.877970i	$0.341103\pi$
$674$	0	0
$675$	−3.54780e9	−0.444013
$676$	0	0
$677$	8.36067e9	1.03557	0.517787	−	0.855510i	$-0.326756\pi$
$677$	8.36067e9	1.03557	0.517787	+	0.855510i	$0.326756\pi$
$678$	0	0
$679$	1.48480e9	0.182022
$680$	0	0
$681$	1.51020e10	1.83240
$682$	0	0
$683$	7.57602e9	0.909847	0.454924	−	0.890530i	$-0.349667\pi$
$683$	7.57602e9	0.909847	0.454924	+	0.890530i	$0.349667\pi$
$684$	0	0
$685$	1.85769e9	0.220829
$686$	0	0
$687$	6.05351e9	0.712293
$688$	0	0
$689$	4.90627e9	0.571458
$690$	0	0
$691$	1.16658e10	1.34506	0.672531	−	0.740069i	$-0.265207\pi$
$691$	1.16658e10	1.34506	0.672531	+	0.740069i	$0.265207\pi$
$692$	0	0
$693$	−4.36807e8	−0.0498567
$694$	0	0
$695$	−6.85992e9	−0.775126
$696$	0	0
$697$	−7.21796e9	−0.807421
$698$	0	0
$699$	−2.00888e10	−2.22476
$700$	0	0
$701$	−8.61744e9	−0.944855	−0.472427	−	0.881370i	$-0.656622\pi$
$701$	−8.61744e9	−0.944855	−0.472427	+	0.881370i	$0.656622\pi$
$702$	0	0
$703$	−8.56269e9	−0.929537
$704$	0	0
$705$	−1.32056e10	−1.41937
$706$	0	0
$707$	−3.94882e9	−0.420242
$708$	0	0
$709$	−5.63807e9	−0.594112	−0.297056	−	0.954860i	$-0.596005\pi$
$709$	−5.63807e9	−0.594112	−0.297056	+	0.954860i	$0.596005\pi$
$710$	0	0
$711$	4.74780e9	0.495392
$712$	0	0
$713$	1.28157e10	1.32412
$714$	0	0
$715$	−1.82810e9	−0.187038
$716$	0	0
$717$	−1.49011e10	−1.50974
$718$	0	0
$719$	−7.95313e9	−0.797971	−0.398986	−	0.916957i	$-0.630638\pi$
$719$	−7.95313e9	−0.797971	−0.398986	+	0.916957i	$0.630638\pi$
$720$	0	0
$721$	−2.62421e9	−0.260751
$722$	0	0
$723$	2.24647e8	0.0221063
$724$	0	0
$725$	4.46953e9	0.435591
$726$	0	0
$727$	−6.10429e9	−0.589202	−0.294601	−	0.955620i	$-0.595187\pi$
$727$	−6.10429e9	−0.589202	−0.294601	+	0.955620i	$0.595187\pi$
$728$	0	0
$729$	4.76693e9	0.455714
$730$	0	0
$731$	6.36336e9	0.602526
$732$	0	0
$733$	−9.35527e9	−0.877389	−0.438695	−	0.898636i	$-0.644559\pi$
$733$	−9.35527e9	−0.877389	−0.438695	+	0.898636i	$0.644559\pi$
$734$	0	0
$735$	−7.58313e9	−0.704438
$736$	0	0
$737$	−6.24181e9	−0.574347
$738$	0	0
$739$	1.08015e9	0.0984528	0.0492264	−	0.998788i	$-0.484324\pi$
$739$	1.08015e9	0.0984528	0.0492264	+	0.998788i	$0.484324\pi$
$740$	0	0
$741$	7.43366e9	0.671180
$742$	0	0
$743$	−1.64759e10	−1.47363	−0.736813	−	0.676097i	$-0.763670\pi$
$743$	−1.64759e10	−1.47363	−0.736813	+	0.676097i	$0.763670\pi$
$744$	0	0
$745$	−4.38715e8	−0.0388719
$746$	0	0
$747$	−4.16036e8	−0.0365182
$748$	0	0
$749$	1.54584e8	0.0134424
$750$	0	0
$751$	−1.57676e10	−1.35839	−0.679196	−	0.733957i	$-0.737671\pi$
$751$	−1.57676e10	−1.35839	−0.679196	+	0.733957i	$0.737671\pi$
$752$	0	0
$753$	−5.21568e9	−0.445172
$754$	0	0
$755$	1.18705e10	1.00381
$756$	0	0
$757$	−1.54352e10	−1.29323	−0.646614	−	0.762817i	$-0.723816\pi$
$757$	−1.54352e10	−1.29323	−0.646614	+	0.762817i	$0.723816\pi$
$758$	0	0
$759$	1.49968e10	1.24495
$760$	0	0
$761$	7.00289e9	0.576012	0.288006	−	0.957629i	$-0.407008\pi$
$761$	7.00289e9	0.576012	0.288006	+	0.957629i	$0.407008\pi$
$762$	0	0
$763$	−4.52610e9	−0.368883
$764$	0	0
$765$	2.15863e9	0.174326
$766$	0	0
$767$	−4.18474e8	−0.0334877
$768$	0	0
$769$	−7.66268e9	−0.607629	−0.303814	−	0.952731i	$-0.598260\pi$
$769$	−7.66268e9	−0.607629	−0.303814	+	0.952731i	$0.598260\pi$
$770$	0	0
$771$	7.31220e9	0.574589
$772$	0	0
$773$	1.92438e10	1.49852	0.749261	−	0.662275i	$-0.230409\pi$
$773$	1.92438e10	1.49852	0.749261	+	0.662275i	$0.230409\pi$
$774$	0	0
$775$	5.63678e9	0.434986
$776$	0	0
$777$	2.81150e9	0.215013
$778$	0	0
$779$	−1.63409e10	−1.23849
$780$	0	0
$781$	−4.10668e9	−0.308469
$782$	0	0
$783$	−7.58425e9	−0.564608
$784$	0	0
$785$	−1.35918e10	−1.00285
$786$	0	0
$787$	−2.37916e10	−1.73985	−0.869925	−	0.493184i	$-0.835833\pi$
$787$	−2.37916e10	−1.73985	−0.869925	+	0.493184i	$0.835833\pi$
$788$	0	0
$789$	2.09270e9	0.151683
$790$	0	0
$791$	−4.04269e9	−0.290438
$792$	0	0
$793$	−6.12179e9	−0.435936
$794$	0	0
$795$	1.25438e10	0.885410
$796$	0	0
$797$	8.06044e9	0.563968	0.281984	−	0.959419i	$-0.409007\pi$
$797$	8.06044e9	0.563968	0.281984	+	0.959419i	$0.409007\pi$
$798$	0	0
$799$	2.14204e10	1.48564
$800$	0	0
$801$	2.62269e8	0.0180315
$802$	0	0
$803$	7.29211e9	0.496991
$804$	0	0
$805$	−4.06131e9	−0.274398
$806$	0	0
$807$	9.11019e9	0.610197
$808$	0	0
$809$	−1.19696e10	−0.794805	−0.397403	−	0.917644i	$-0.630088\pi$
$809$	−1.19696e10	−0.794805	−0.397403	+	0.917644i	$0.630088\pi$
$810$	0	0
$811$	1.76446e9	0.116156	0.0580778	−	0.998312i	$-0.481503\pi$
$811$	1.76446e9	0.116156	0.0580778	+	0.998312i	$0.481503\pi$
$812$	0	0
$813$	3.02005e9	0.197105
$814$	0	0
$815$	1.67528e10	1.08401
$816$	0	0
$817$	1.44061e10	0.924208
$818$	0	0
$819$	−6.27608e8	−0.0399204
$820$	0	0
$821$	2.29211e10	1.44556	0.722778	−	0.691080i	$-0.242865\pi$
$821$	2.29211e10	1.44556	0.722778	+	0.691080i	$0.242865\pi$
$822$	0	0
$823$	−1.36475e10	−0.853404	−0.426702	−	0.904392i	$-0.640325\pi$
$823$	−1.36475e10	−0.853404	−0.426702	+	0.904392i	$0.640325\pi$
$824$	0	0
$825$	6.59611e9	0.408977
$826$	0	0
$827$	1.92551e10	1.18379	0.591897	−	0.806014i	$-0.298379\pi$
$827$	1.92551e10	1.18379	0.591897	+	0.806014i	$0.298379\pi$
$828$	0	0
$829$	8.10248e9	0.493943	0.246972	−	0.969023i	$-0.420565\pi$
$829$	8.10248e9	0.493943	0.246972	+	0.969023i	$0.420565\pi$
$830$	0	0
$831$	2.26557e10	1.36954
$832$	0	0
$833$	1.23003e10	0.737327
$834$	0	0
$835$	9.19107e9	0.546341
$836$	0	0
$837$	−9.56494e9	−0.563823
$838$	0	0
$839$	1.36587e10	0.798438	0.399219	−	0.916856i	$-0.369281\pi$
$839$	1.36587e10	0.798438	0.399219	+	0.916856i	$0.369281\pi$
$840$	0	0
$841$	−7.69520e9	−0.446102
$842$	0	0
$843$	3.90576e9	0.224548
$844$	0	0
$845$	8.66810e9	0.494226
$846$	0	0
$847$	2.69527e9	0.152409
$848$	0	0
$849$	3.89985e9	0.218711
$850$	0	0
$851$	−2.48201e10	−1.38055
$852$	0	0
$853$	3.83840e9	0.211753	0.105876	−	0.994379i	$-0.466235\pi$
$853$	3.83840e9	0.211753	0.105876	+	0.994379i	$0.466235\pi$
$854$	0	0
$855$	4.88696e9	0.267398
$856$	0	0
$857$	−6.18270e9	−0.335541	−0.167770	−	0.985826i	$-0.553657\pi$
$857$	−6.18270e9	−0.335541	−0.167770	+	0.985826i	$0.553657\pi$
$858$	0	0
$859$	7.81827e9	0.420857	0.210429	−	0.977609i	$-0.432514\pi$
$859$	7.81827e9	0.420857	0.210429	+	0.977609i	$0.432514\pi$
$860$	0	0
$861$	5.36540e9	0.286478
$862$	0	0
$863$	−1.74642e10	−0.924934	−0.462467	−	0.886636i	$-0.653036\pi$
$863$	−1.74642e10	−0.924934	−0.462467	+	0.886636i	$0.653036\pi$
$864$	0	0
$865$	−2.56755e9	−0.134885
$866$	0	0
$867$	8.64718e9	0.450617
$868$	0	0
$869$	1.66748e10	0.861971
$870$	0	0
$871$	−8.96828e9	−0.459881
$872$	0	0
$873$	−5.17886e9	−0.263442
$874$	0	0
$875$	−4.83835e9	−0.244157
$876$	0	0
$877$	3.68179e10	1.84315	0.921574	−	0.388203i	$-0.126904\pi$
$877$	3.68179e10	1.84315	0.921574	+	0.388203i	$0.126904\pi$
$878$	0	0
$879$	3.67924e10	1.82724
$880$	0	0
$881$	1.15013e10	0.566671	0.283336	−	0.959021i	$-0.408559\pi$
$881$	1.15013e10	0.566671	0.283336	+	0.959021i	$0.408559\pi$
$882$	0	0
$883$	−1.54418e10	−0.754806	−0.377403	−	0.926049i	$-0.623183\pi$
$883$	−1.54418e10	−0.754806	−0.377403	+	0.926049i	$0.623183\pi$
$884$	0	0
$885$	−1.06991e9	−0.0518854
$886$	0	0
$887$	1.97963e10	0.952471	0.476235	−	0.879318i	$-0.342001\pi$
$887$	1.97963e10	0.952471	0.476235	+	0.879318i	$0.342001\pi$
$888$	0	0
$889$	−6.35527e8	−0.0303374
$890$	0	0
$891$	−1.55944e10	−0.738578
$892$	0	0
$893$	4.84939e10	2.27881
$894$	0	0
$895$	−1.73564e10	−0.809244
$896$	0	0
$897$	2.15475e10	0.996835
$898$	0	0
$899$	1.20499e10	0.553129
$900$	0	0
$901$	−2.03469e10	−0.926747
$902$	0	0
$903$	−4.73014e9	−0.213780
$904$	0	0
$905$	−9.64700e9	−0.432635
$906$	0	0
$907$	−3.23910e10	−1.44145	−0.720724	−	0.693222i	$-0.756191\pi$
$907$	−3.23910e10	−1.44145	−0.720724	+	0.693222i	$0.756191\pi$
$908$	0	0
$909$	1.37732e10	0.608220
$910$	0	0
$911$	−9.19954e9	−0.403136	−0.201568	−	0.979475i	$-0.564604\pi$
$911$	−9.19954e9	−0.403136	−0.201568	+	0.979475i	$0.564604\pi$
$912$	0	0
$913$	−1.46117e9	−0.0635408
$914$	0	0
$915$	−1.56515e10	−0.675434
$916$	0	0
$917$	8.59060e9	0.367901
$918$	0	0
$919$	1.73860e10	0.738917	0.369458	−	0.929247i	$-0.379543\pi$
$919$	1.73860e10	0.738917	0.369458	+	0.929247i	$0.379543\pi$
$920$	0	0
$921$	5.29389e9	0.223289
$922$	0	0
$923$	−5.90051e9	−0.246992
$924$	0	0
$925$	−1.09168e10	−0.453521
$926$	0	0
$927$	9.15305e9	0.377387
$928$	0	0
$929$	2.41112e10	0.986653	0.493326	−	0.869844i	$-0.335781\pi$
$929$	2.41112e10	0.986653	0.493326	+	0.869844i	$0.335781\pi$
$930$	0	0
$931$	2.78470e10	1.13098
$932$	0	0
$933$	−7.53015e9	−0.303541
$934$	0	0
$935$	7.58137e9	0.303324
$936$	0	0
$937$	−1.58105e10	−0.627850	−0.313925	−	0.949448i	$-0.601644\pi$
$937$	−1.58105e10	−0.627850	−0.313925	+	0.949448i	$0.601644\pi$
$938$	0	0
$939$	5.30261e10	2.09007
$940$	0	0
$941$	1.50751e10	0.589788	0.294894	−	0.955530i	$-0.404716\pi$
$941$	1.50751e10	0.589788	0.294894	+	0.955530i	$0.404716\pi$
$942$	0	0
$943$	−4.73663e10	−1.83941
$944$	0	0
$945$	3.03114e9	0.116841
$946$	0	0
$947$	3.51432e10	1.34467	0.672337	−	0.740245i	$-0.265291\pi$
$947$	3.51432e10	1.34467	0.672337	+	0.740245i	$0.265291\pi$
$948$	0	0
$949$	1.04774e10	0.397942
$950$	0	0
$951$	−4.62312e8	−0.0174302
$952$	0	0
$953$	1.61528e10	0.604538	0.302269	−	0.953223i	$-0.402256\pi$
$953$	1.61528e10	0.604538	0.302269	+	0.953223i	$0.402256\pi$
$954$	0	0
$955$	−5.06595e9	−0.188213
$956$	0	0
$957$	1.41007e10	0.520056
$958$	0	0
$959$	2.23990e9	0.0820094
$960$	0	0
$961$	−1.23157e10	−0.447640
$962$	0	0
$963$	−5.39176e8	−0.0194553
$964$	0	0
$965$	1.37857e10	0.493835
$966$	0	0
$967$	−2.27594e10	−0.809408	−0.404704	−	0.914448i	$-0.632625\pi$
$967$	−2.27594e10	−0.809408	−0.404704	+	0.914448i	$0.632625\pi$
$968$	0	0
$969$	−3.08283e10	−1.08847
$970$	0	0
$971$	−3.40071e10	−1.19207	−0.596036	−	0.802958i	$-0.703258\pi$
$971$	−3.40071e10	−1.19207	−0.596036	+	0.802958i	$0.703258\pi$
$972$	0	0
$973$	−8.27133e9	−0.287859
$974$	0	0
$975$	9.47733e9	0.327469
$976$	0	0
$977$	−8.64935e9	−0.296724	−0.148362	−	0.988933i	$-0.547400\pi$
$977$	−8.64935e9	−0.296724	−0.148362	+	0.988933i	$0.547400\pi$
$978$	0	0
$979$	9.21119e8	0.0313745
$980$	0	0
$981$	1.57867e10	0.533888
$982$	0	0
$983$	−4.00601e10	−1.34516	−0.672581	−	0.740024i	$-0.734814\pi$
$983$	−4.00601e10	−1.34516	−0.672581	+	0.740024i	$0.734814\pi$
$984$	0	0
$985$	−2.94985e9	−0.0983498
$986$	0	0
$987$	−1.59226e10	−0.527114
$988$	0	0
$989$	4.17581e10	1.37263
$990$	0	0
$991$	−2.06848e10	−0.675139	−0.337570	−	0.941301i	$-0.609605\pi$
$991$	−2.06848e10	−0.675139	−0.337570	+	0.941301i	$0.609605\pi$
$992$	0	0
$993$	2.54930e10	0.826227
$994$	0	0
$995$	−1.92668e10	−0.620053
$996$	0	0
$997$	1.01849e10	0.325480	0.162740	−	0.986669i	$-0.447967\pi$
$997$	1.01849e10	0.325480	0.162740	+	0.986669i	$0.447967\pi$
$998$	0	0
$999$	1.85244e10	0.587849

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.8.a.f.1.1		2
4.3	odd	2	inner	256.8.a.f.1.2		2
8.3	odd	2		256.8.a.h.1.1		2
8.5	even	2		256.8.a.h.1.2		2
16.3	odd	4		128.8.b.e.65.4	yes	4
16.5	even	4		128.8.b.e.65.3	yes	4
16.11	odd	4		128.8.b.e.65.1	✓	4
16.13	even	4		128.8.b.e.65.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.8.b.e.65.1	✓	4	16.11	odd	4
128.8.b.e.65.2	yes	4	16.13	even	4
128.8.b.e.65.3	yes	4	16.5	even	4
128.8.b.e.65.4	yes	4	16.3	odd	4
256.8.a.f.1.1		2	1.1	even	1	trivial
256.8.a.f.1.2		2	4.3	odd	2	inner
256.8.a.h.1.1		2	8.3	odd	2
256.8.a.h.1.2		2	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-54.2586 q^{3} -180.000 q^{5} -217.035 q^{7} +757.000 q^{9} +O(q^{10})\)	\(q-54.2586 q^{3} -180.000 q^{5} -217.035 q^{7} +757.000 q^{9} +2658.67 q^{11} +3820.00 q^{13} +9766.56 q^{15} -15842.0 q^{17} -35865.0 q^{19} +11776.0 q^{21} -103960. q^{23} -45725.0 q^{25} +77589.9 q^{27} -97748.0 q^{29} -123276. q^{31} -144256. q^{33} +39066.2 q^{35} +238748. q^{37} -207268. q^{39} +455622. q^{41} -401677. q^{43} -136260. q^{45} -1.35213e6 q^{47} -776439. q^{49} +859565. q^{51} +1.28436e6 q^{53} -478561. q^{55} +1.94598e6 q^{57} -109548. q^{59} -1.60256e6 q^{61} -164295. q^{63} -687600. q^{65} -2.34772e6 q^{67} +5.64070e6 q^{69} -1.54463e6 q^{71} +2.74276e6 q^{73} +2.48098e6 q^{75} -577024. q^{77} +6.27186e6 q^{79} -5.86548e6 q^{81} -549586. q^{83} +2.85156e6 q^{85} +5.30367e6 q^{87} +346458. q^{89} -829072. q^{91} +6.68877e6 q^{93} +6.45569e6 q^{95} -6.84130e6 q^{97} +2.01262e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 360 q^{5} + 1514 q^{9}+O(q^{10}) \) 2 * q - 360 * q^5 + 1514 * q^9	\( 2 q - 360 q^{5} + 1514 q^{9} + 7640 q^{13} - 31684 q^{17} + 23552 q^{21} - 91450 q^{25} - 195496 q^{29} - 288512 q^{33} + 477496 q^{37} + 911244 q^{41} - 272520 q^{45} - 1552878 q^{49} + 2568728 q^{53} + 3891968 q^{57} - 3205128 q^{61} - 1375200 q^{65} + 11281408 q^{69} + 5485524 q^{73} - 1154048 q^{77} - 11730958 q^{81} + 5703120 q^{85} + 692916 q^{89} + 13377536 q^{93} - 13682596 q^{97}+O(q^{100}) \) 2 * q - 360 * q^5 + 1514 * q^9 + 7640 * q^13 - 31684 * q^17 + 23552 * q^21 - 91450 * q^25 - 195496 * q^29 - 288512 * q^33 + 477496 * q^37 + 911244 * q^41 - 272520 * q^45 - 1552878 * q^49 + 2568728 * q^53 + 3891968 * q^57 - 3205128 * q^61 - 1375200 * q^65 + 11281408 * q^69 + 5485524 * q^73 - 1154048 * q^77 - 11730958 * q^81 + 5703120 * q^85 + 692916 * q^89 + 13377536 * q^93 - 13682596 * q^97

Introduction

L-functions

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