LMFDB - Embedded newform 256.11.c.j.255.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,11,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, 11, names="a")

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

chi := DirichletCharacter("256.255");

S:= CuspForms(chi, 11);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 11 $
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:	$162.651456684$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4 x^{9} - 9850 x^{8} - 22678 x^{7} + 31760900 x^{6} + 262382084 x^{5} - 36066825359 x^{4} + \cdots + 11\!\cdots\!28 $ x^10 - 4x^9 - 9850x^8 - 22678x^7 + 31760900x^6 + 262382084x^5 - 36066825359x^4 - 492294015528x^3 + 12219836291940x^2 + 252889346123568*x + 1185399433622628
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{85}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			255.2
Root			$-26.8622 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	256.255
Dual form			256.11.c.j.255.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-357.190i q^{3} +1631.23 q^{5} -3378.22i q^{7} -68535.8 q^{9} +O(q^{10})$	$q-357.190i q^{3} +1631.23 q^{5} -3378.22i q^{7} -68535.8 q^{9} +221936. i q^{11} -374761. q^{13} -582660. i q^{15} -430286. q^{17} -303528. i q^{19} -1.20667e6 q^{21} -8.29165e6i q^{23} -7.10471e6 q^{25} +3.38860e6i q^{27} -3.72645e7 q^{29} +7.54540e6i q^{31} +7.92733e7 q^{33} -5.51065e6i q^{35} +7.50121e7 q^{37} +1.33861e8i q^{39} -6.61543e7 q^{41} +2.22796e7i q^{43} -1.11798e8 q^{45} +1.88846e8i q^{47} +2.71063e8 q^{49} +1.53694e8i q^{51} +5.80175e8 q^{53} +3.62029e8i q^{55} -1.08417e8 q^{57} -3.07323e7i q^{59} +5.40106e8 q^{61} +2.31529e8i q^{63} -6.11322e8 q^{65} -2.14996e9i q^{67} -2.96170e9 q^{69} +2.89668e9i q^{71} +3.51296e9 q^{73} +2.53773e9i q^{75} +7.49747e8 q^{77} +4.62994e9i q^{79} -2.83660e9 q^{81} +2.00858e9i q^{83} -7.01896e8 q^{85} +1.33105e10i q^{87} -5.96540e9 q^{89} +1.26602e9i q^{91} +2.69514e9 q^{93} -4.95125e8i q^{95} -5.56137e9 q^{97} -1.52105e10i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 6232 q^{5} - 218038 q^{9}+O(q^{10}) $ 10 * q - 6232 * q^5 - 218038 * q^9	$ 10 q - 6232 q^{5} - 218038 q^{9} - 485560 q^{13} - 1468772 q^{17} - 15545024 q^{21} + 4383462 q^{25} - 25263800 q^{29} - 122737264 q^{33} - 178255288 q^{37} + 91656876 q^{41} + 718346408 q^{45} - 842454934 q^{49} + 798458664 q^{53} - 1612102544 q^{57} - 102636184 q^{61} + 376325920 q^{65} - 207224128 q^{69} + 2854265572 q^{73} - 9037740608 q^{77} + 11990017466 q^{81} + 5473132400 q^{85} - 11790814556 q^{89} + 24576098304 q^{93} - 9363277860 q^{97}+O(q^{100}) $ 10 * q - 6232 * q^5 - 218038 * q^9 - 485560 * q^13 - 1468772 * q^17 - 15545024 * q^21 + 4383462 * q^25 - 25263800 * q^29 - 122737264 * q^33 - 178255288 * q^37 + 91656876 * q^41 + 718346408 * q^45 - 842454934 * q^49 + 798458664 * q^53 - 1612102544 * q^57 - 102636184 * q^61 + 376325920 * q^65 - 207224128 * q^69 + 2854265572 * q^73 - 9037740608 * q^77 + 11990017466 * q^81 + 5473132400 * q^85 - 11790814556 * q^89 + 24576098304 * q^93 - 9363277860 * 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
$3$	− 357.190i	− 1.46992i	−0.678111	−	0.734959i	$-0.737201\pi$
$3$	− 357.190i	− 1.46992i	0.678111	−	0.734959i	$-0.262799\pi$
$4$	0	0
$5$	1631.23	0.521994	0.260997	−	0.965340i	$-0.415949\pi$
$5$	1631.23	0.521994	0.260997	+	0.965340i	$0.415949\pi$
$6$	0	0
$7$	− 3378.22i	− 0.201000i	−0.994937	−	0.100500i	$-0.967956\pi$
$7$	− 3378.22i	− 0.201000i	0.994937	−	0.100500i	$-0.0320443\pi$
$8$	0	0
$9$	−68535.8	−1.16066
$10$	0	0
$11$	221936.i	1.37805i	0.724739	+	0.689023i	$0.241960\pi$
$11$	221936.i	1.37805i	−0.724739	+	0.689023i	$0.758040\pi$
$12$	0	0
$13$	−374761.	−1.00934	−0.504670	−	0.863313i	$-0.668386\pi$
$13$	−374761.	−1.00934	−0.504670	+	0.863313i	$0.668386\pi$
$14$	0	0
$15$	− 582660.i	− 0.767289i
$16$	0	0
$17$	−430286.	−0.303049	−0.151524	−	0.988454i	$-0.548418\pi$
$17$	−430286.	−0.303049	−0.151524	+	0.988454i	$0.548418\pi$
$18$	0	0
$19$	− 303528.i	− 0.122583i	−0.998120	−	0.0612916i	$-0.980478\pi$
$19$	− 303528.i	− 0.122583i	0.998120	−	0.0612916i	$-0.0195220\pi$
$20$	0	0
$21$	−1.20667e6	−0.295454
$22$	0	0
$23$	− 8.29165e6i	− 1.28826i	−0.764918	−	0.644128i	$-0.777221\pi$
$23$	− 8.29165e6i	− 1.28826i	0.764918	−	0.644128i	$-0.222779\pi$
$24$	0	0
$25$	−7.10471e6	−0.727522
$26$	0	0
$27$	3.38860e6i	0.236157i
$28$	0	0
$29$	−3.72645e7	−1.81679	−0.908397	−	0.418110i	$-0.862693\pi$
$29$	−3.72645e7	−1.81679	−0.908397	+	0.418110i	$0.862693\pi$
$30$	0	0
$31$	7.54540e6i	0.263556i	0.991279	+	0.131778i	$0.0420687\pi$
$31$	7.54540e6i	0.263556i	−0.991279	+	0.131778i	$0.957931\pi$
$32$	0	0
$33$	7.92733e7	2.02562
$34$	0	0
$35$	− 5.51065e6i	− 0.104921i
$36$	0	0
$37$	7.50121e7	1.08174	0.540870	−	0.841106i	$-0.318095\pi$
$37$	7.50121e7	1.08174	0.540870	+	0.841106i	$0.318095\pi$
$38$	0	0
$39$	1.33861e8i	1.48365i
$40$	0	0
$41$	−6.61543e7	−0.571004	−0.285502	−	0.958378i	$-0.592160\pi$
$41$	−6.61543e7	−0.571004	−0.285502	+	0.958378i	$0.592160\pi$
$42$	0	0
$43$	2.22796e7i	0.151553i	0.997125	+	0.0757767i	$0.0241436\pi$
$43$	2.22796e7i	0.151553i	−0.997125	+	0.0757767i	$0.975856\pi$
$44$	0	0
$45$	−1.11798e8	−0.605858
$46$	0	0
$47$	1.88846e8i	0.823413i	0.911317	+	0.411706i	$0.135067\pi$
$47$	1.88846e8i	0.823413i	−0.911317	+	0.411706i	$0.864933\pi$
$48$	0	0
$49$	2.71063e8	0.959599
$50$	0	0
$51$	1.53694e8i	0.445457i
$52$	0	0
$53$	5.80175e8	1.38733	0.693665	−	0.720298i	$-0.255995\pi$
$53$	5.80175e8	1.38733	0.693665	+	0.720298i	$0.255995\pi$
$54$	0	0
$55$	3.62029e8i	0.719332i
$56$	0	0
$57$	−1.08417e8	−0.180187
$58$	0	0
$59$	− 3.07323e7i	− 0.0429868i	−0.999769	−	0.0214934i	$-0.993158\pi$
$59$	− 3.07323e7i	− 0.0429868i	0.999769	−	0.0214934i	$-0.00684209\pi$
$60$	0	0
$61$	5.40106e8	0.639484	0.319742	−	0.947505i	$-0.396404\pi$
$61$	5.40106e8	0.639484	0.319742	+	0.947505i	$0.396404\pi$
$62$	0	0
$63$	2.31529e8i	0.233293i
$64$	0	0
$65$	−6.11322e8	−0.526869
$66$	0	0
$67$	− 2.14996e9i	− 1.59242i	−0.605023	−	0.796208i	$-0.706836\pi$
$67$	− 2.14996e9i	− 1.59242i	0.605023	−	0.796208i	$-0.293164\pi$
$68$	0	0
$69$	−2.96170e9	−1.89363
$70$	0	0
$71$	2.89668e9i	1.60549i	0.596320	+	0.802747i	$0.296629\pi$
$71$	2.89668e9i	1.60549i	−0.596320	+	0.802747i	$0.703371\pi$
$72$	0	0
$73$	3.51296e9	1.69457	0.847284	−	0.531140i	$-0.178236\pi$
$73$	3.51296e9	1.69457	0.847284	+	0.531140i	$0.178236\pi$
$74$	0	0
$75$	2.53773e9i	1.06940i
$76$	0	0
$77$	7.49747e8	0.276988
$78$	0	0
$79$	4.62994e9i	1.50466i	0.658784	+	0.752332i	$0.271071\pi$
$79$	4.62994e9i	1.50466i	−0.658784	+	0.752332i	$0.728929\pi$
$80$	0	0
$81$	−2.83660e9	−0.813528
$82$	0	0
$83$	2.00858e9i	0.509917i	0.966952	+	0.254958i	$0.0820618\pi$
$83$	2.00858e9i	0.509917i	−0.966952	+	0.254958i	$0.917938\pi$
$84$	0	0
$85$	−7.01896e8	−0.158190
$86$	0	0
$87$	1.33105e10i	2.67054i
$88$	0	0
$89$	−5.96540e9	−1.06829	−0.534146	−	0.845392i	$-0.679367\pi$
$89$	−5.96540e9	−1.06829	−0.534146	+	0.845392i	$0.679367\pi$
$90$	0	0
$91$	1.26602e9i	0.202878i
$92$	0	0
$93$	2.69514e9	0.387406
$94$	0	0
$95$	− 4.95125e8i	− 0.0639878i
$96$	0	0
$97$	−5.56137e9	−0.647624	−0.323812	−	0.946121i	$-0.604965\pi$
$97$	−5.56137e9	−0.647624	−0.323812	+	0.946121i	$0.604965\pi$
$98$	0	0
$99$	− 1.52105e10i	− 1.59944i
$100$	0	0
$101$	1.09675e10	1.04352	0.521760	−	0.853092i	$-0.325276\pi$
$101$	1.09675e10	1.04352	0.521760	+	0.853092i	$0.325276\pi$
$102$	0	0
$103$	1.43593e10i	1.23864i	0.785138	+	0.619321i	$0.212592\pi$
$103$	1.43593e10i	1.23864i	−0.785138	+	0.619321i	$0.787408\pi$
$104$	0	0
$105$	−1.96835e9	−0.154226
$106$	0	0
$107$	1.94881e10i	1.38947i	0.719265	+	0.694736i	$0.244479\pi$
$107$	1.94881e10i	1.38947i	−0.719265	+	0.694736i	$0.755521\pi$
$108$	0	0
$109$	3.05191e8	0.0198353	0.00991766	−	0.999951i	$-0.496843\pi$
$109$	3.05191e8	0.0198353	0.00991766	+	0.999951i	$0.496843\pi$
$110$	0	0
$111$	− 2.67936e10i	− 1.59007i
$112$	0	0
$113$	3.20895e10	1.74169	0.870845	−	0.491558i	$-0.163572\pi$
$113$	3.20895e10	1.74169	0.870845	+	0.491558i	$0.163572\pi$
$114$	0	0
$115$	− 1.35256e10i	− 0.672462i
$116$	0	0
$117$	2.56845e10	1.17150
$118$	0	0
$119$	1.45360e9i	0.0609129i
$120$	0	0
$121$	−2.33180e10	−0.899011
$122$	0	0
$123$	2.36297e10i	0.839329i
$124$	0	0
$125$	−2.75194e10	−0.901757
$126$	0	0
$127$	− 2.30721e10i	− 0.698343i	−0.937059	−	0.349172i	$-0.886463\pi$
$127$	− 2.30721e10i	− 0.698343i	0.937059	−	0.349172i	$-0.113537\pi$
$128$	0	0
$129$	7.95807e9	0.222771
$130$	0	0
$131$	− 2.67038e10i	− 0.692177i	−0.938202	−	0.346088i	$-0.887510\pi$
$131$	− 2.67038e10i	− 0.692177i	0.938202	−	0.346088i	$-0.112490\pi$
$132$	0	0
$133$	−1.02538e9	−0.0246393
$134$	0	0
$135$	5.52759e9i	0.123273i
$136$	0	0
$137$	5.54146e10	1.14821	0.574106	−	0.818781i	$-0.305350\pi$
$137$	5.54146e10	1.14821	0.574106	+	0.818781i	$0.305350\pi$
$138$	0	0
$139$	3.35031e10i	0.645670i	0.946455	+	0.322835i	$0.104636\pi$
$139$	3.35031e10i	0.645670i	−0.946455	+	0.322835i	$0.895364\pi$
$140$	0	0
$141$	6.74538e10	1.21035
$142$	0	0
$143$	− 8.31727e10i	− 1.39092i
$144$	0	0
$145$	−6.07871e10	−0.948356
$146$	0	0
$147$	− 9.68210e10i	− 1.41053i
$148$	0	0
$149$	−5.66453e10	−0.771316	−0.385658	−	0.922642i	$-0.626026\pi$
$149$	−5.66453e10	−0.771316	−0.385658	+	0.922642i	$0.626026\pi$
$150$	0	0
$151$	4.10467e10i	0.522870i	0.965221	+	0.261435i	$0.0841957\pi$
$151$	4.10467e10i	0.522870i	−0.965221	+	0.261435i	$0.915804\pi$
$152$	0	0
$153$	2.94900e10	0.351736
$154$	0	0
$155$	1.23083e10i	0.137575i
$156$	0	0
$157$	9.56421e10	1.00265	0.501327	−	0.865258i	$-0.332845\pi$
$157$	9.56421e10	1.00265	0.501327	+	0.865258i	$0.332845\pi$
$158$	0	0
$159$	− 2.07233e11i	− 2.03926i
$160$	0	0
$161$	−2.80110e10	−0.258940
$162$	0	0
$163$	1.29530e11i	1.12572i	0.826551	+	0.562862i	$0.190300\pi$
$163$	1.29530e11i	1.12572i	−0.826551	+	0.562862i	$0.809700\pi$
$164$	0	0
$165$	1.29313e11	1.05736
$166$	0	0
$167$	− 2.13263e11i	− 1.64185i	−0.571037	−	0.820924i	$-0.693459\pi$
$167$	− 2.13263e11i	− 1.64185i	0.571037	−	0.820924i	$-0.306541\pi$
$168$	0	0
$169$	2.58698e9	0.0187655
$170$	0	0
$171$	2.08026e10i	0.142278i
$172$	0	0
$173$	2.86886e11	1.85131	0.925655	−	0.378369i	$-0.123515\pi$
$173$	2.86886e11	1.85131	0.925655	+	0.378369i	$0.123515\pi$
$174$	0	0
$175$	2.40012e10i	0.146232i
$176$	0	0
$177$	−1.09773e10	−0.0631871
$178$	0	0
$179$	− 1.03817e11i	− 0.564940i	−0.959276	−	0.282470i	$-0.908846\pi$
$179$	− 1.03817e11i	− 0.564940i	0.959276	−	0.282470i	$-0.0911538\pi$
$180$	0	0
$181$	2.69873e11	1.38921	0.694604	−	0.719392i	$-0.255580\pi$
$181$	2.69873e11	1.38921	0.694604	+	0.719392i	$0.255580\pi$
$182$	0	0
$183$	− 1.92921e11i	− 0.939990i
$184$	0	0
$185$	1.22362e11	0.564662
$186$	0	0
$187$	− 9.54958e10i	− 0.417615i
$188$	0	0
$189$	1.14474e10	0.0474677
$190$	0	0
$191$	2.00628e11i	0.789270i	0.918838	+	0.394635i	$0.129129\pi$
$191$	2.00628e11i	0.789270i	−0.918838	+	0.394635i	$0.870871\pi$
$192$	0	0
$193$	5.78528e10	0.216042	0.108021	−	0.994149i	$-0.465549\pi$
$193$	5.78528e10	0.216042	0.108021	+	0.994149i	$0.465549\pi$
$194$	0	0
$195$	2.18358e11i	0.774455i
$196$	0	0
$197$	1.26196e11	0.425318	0.212659	−	0.977126i	$-0.431788\pi$
$197$	1.26196e11	0.425318	0.212659	+	0.977126i	$0.431788\pi$
$198$	0	0
$199$	4.00882e11i	1.28455i	0.766475	+	0.642275i	$0.222009\pi$
$199$	4.00882e11i	1.28455i	−0.766475	+	0.642275i	$0.777991\pi$
$200$	0	0
$201$	−7.67945e11	−2.34072
$202$	0	0
$203$	1.25888e11i	0.365176i
$204$	0	0
$205$	−1.07913e11	−0.298061
$206$	0	0
$207$	5.68275e11i	1.49523i
$208$	0	0
$209$	6.73638e10	0.168925
$210$	0	0
$211$	1.33181e11i	0.318442i	0.987243	+	0.159221i	$0.0508982\pi$
$211$	1.33181e11i	0.318442i	−0.987243	+	0.159221i	$0.949102\pi$
$212$	0	0
$213$	1.03467e12	2.35994
$214$	0	0
$215$	3.63433e10i	0.0791100i
$216$	0	0
$217$	2.54900e10	0.0529750
$218$	0	0
$219$	− 1.25479e12i	− 2.49088i
$220$	0	0
$221$	1.61254e11	0.305879
$222$	0	0
$223$	1.10007e12i	1.99479i	0.0721394	+	0.997395i	$0.477017\pi$
$223$	1.10007e12i	1.99479i	−0.0721394	+	0.997395i	$0.522983\pi$
$224$	0	0
$225$	4.86927e11	0.844406
$226$	0	0
$227$	3.84857e11i	0.638514i	0.947668	+	0.319257i	$0.103433\pi$
$227$	3.84857e11i	0.638514i	−0.947668	+	0.319257i	$0.896567\pi$
$228$	0	0
$229$	−1.44125e11	−0.228856	−0.114428	−	0.993432i	$-0.536503\pi$
$229$	−1.44125e11	−0.228856	−0.114428	+	0.993432i	$0.536503\pi$
$230$	0	0
$231$	− 2.67802e11i	− 0.407150i
$232$	0	0
$233$	7.15459e11	1.04185	0.520925	−	0.853602i	$-0.325587\pi$
$233$	7.15459e11	1.04185	0.520925	+	0.853602i	$0.325587\pi$
$234$	0	0
$235$	3.08051e11i	0.429817i
$236$	0	0
$237$	1.65377e12	2.21173
$238$	0	0
$239$	3.35606e11i	0.430369i	0.976573	+	0.215184i	$0.0690352\pi$
$239$	3.35606e11i	0.430369i	−0.976573	+	0.215184i	$0.930965\pi$
$240$	0	0
$241$	−4.07370e11	−0.501076	−0.250538	−	0.968107i	$-0.580607\pi$
$241$	−4.07370e11	−0.501076	−0.250538	+	0.968107i	$0.580607\pi$
$242$	0	0
$243$	1.21330e12i	1.43198i
$244$	0	0
$245$	4.42167e11	0.500905
$246$	0	0
$247$	1.13750e11i	0.123728i
$248$	0	0
$249$	7.17446e11	0.749536
$250$	0	0
$251$	− 1.09100e12i	− 1.09511i	−0.836771	−	0.547553i	$-0.815560\pi$
$251$	− 1.09100e12i	− 1.09511i	0.836771	−	0.547553i	$-0.184440\pi$
$252$	0	0
$253$	1.84021e12	1.77528
$254$	0	0
$255$	2.50710e11i	0.232526i
$256$	0	0
$257$	−7.23543e11	−0.645355	−0.322678	−	0.946509i	$-0.604583\pi$
$257$	−7.23543e11	−0.645355	−0.322678	+	0.946509i	$0.604583\pi$
$258$	0	0
$259$	− 2.53407e11i	− 0.217430i
$260$	0	0
$261$	2.55395e12	2.10868
$262$	0	0
$263$	− 4.92538e11i	− 0.391436i	−0.980660	−	0.195718i	$-0.937296\pi$
$263$	− 4.92538e11i	− 0.391436i	0.980660	−	0.195718i	$-0.0627038\pi$
$264$	0	0
$265$	9.46401e11	0.724179
$266$	0	0
$267$	2.13078e12i	1.57030i
$268$	0	0
$269$	1.53398e12	1.08908	0.544539	−	0.838736i	$-0.316705\pi$
$269$	1.53398e12	1.08908	0.544539	+	0.838736i	$0.316705\pi$
$270$	0	0
$271$	5.68865e11i	0.389191i	0.980884	+	0.194596i	$0.0623394\pi$
$271$	5.68865e11i	0.389191i	−0.980884	+	0.194596i	$0.937661\pi$
$272$	0	0
$273$	4.52211e11	0.298214
$274$	0	0
$275$	− 1.57679e12i	− 1.00256i
$276$	0	0
$277$	−1.41619e12	−0.868406	−0.434203	−	0.900815i	$-0.642970\pi$
$277$	−1.41619e12	−0.868406	−0.434203	+	0.900815i	$0.642970\pi$
$278$	0	0
$279$	− 5.17130e11i	− 0.305899i
$280$	0	0
$281$	−1.55865e12	−0.889644	−0.444822	−	0.895619i	$-0.646733\pi$
$281$	−1.55865e12	−0.889644	−0.444822	+	0.895619i	$0.646733\pi$
$282$	0	0
$283$	− 3.27132e12i	− 1.80215i	−0.433661	−	0.901076i	$-0.642778\pi$
$283$	− 3.27132e12i	− 1.80215i	0.433661	−	0.901076i	$-0.357222\pi$
$284$	0	0
$285$	−1.76854e11	−0.0940568
$286$	0	0
$287$	2.23483e11i	0.114772i
$288$	0	0
$289$	−1.83085e12	−0.908162
$290$	0	0
$291$	1.98647e12i	0.951954i
$292$	0	0
$293$	3.22504e11	0.149347	0.0746735	−	0.997208i	$-0.476209\pi$
$293$	3.22504e11	0.149347	0.0746735	+	0.997208i	$0.476209\pi$
$294$	0	0
$295$	− 5.01316e10i	− 0.0224389i
$296$	0	0
$297$	−7.52051e11	−0.325436
$298$	0	0
$299$	3.10738e12i	1.30029i
$300$	0	0
$301$	7.52654e10	0.0304623
$302$	0	0
$303$	− 3.91748e12i	− 1.53389i
$304$	0	0
$305$	8.81038e11	0.333807
$306$	0	0
$307$	− 1.67280e12i	− 0.613413i	−0.951804	−	0.306706i	$-0.900773\pi$
$307$	− 1.67280e12i	− 0.613413i	0.951804	−	0.306706i	$-0.0992269\pi$
$308$	0	0
$309$	5.12898e12	1.82070
$310$	0	0
$311$	− 9.45387e11i	− 0.324943i	−0.986713	−	0.162472i	$-0.948053\pi$
$311$	− 9.45387e11i	− 0.324943i	0.986713	−	0.162472i	$-0.0519466\pi$
$312$	0	0
$313$	−2.07906e12	−0.692061	−0.346031	−	0.938223i	$-0.612471\pi$
$313$	−2.07906e12	−0.692061	−0.346031	+	0.938223i	$0.612471\pi$
$314$	0	0
$315$	3.77677e11i	0.121778i
$316$	0	0
$317$	−2.84326e11	−0.0888219	−0.0444109	−	0.999013i	$-0.514141\pi$
$317$	−2.84326e11	−0.0888219	−0.0444109	+	0.999013i	$0.514141\pi$
$318$	0	0
$319$	− 8.27033e12i	− 2.50362i
$320$	0	0
$321$	6.96095e12	2.04241
$322$	0	0
$323$	1.30604e11i	0.0371487i
$324$	0	0
$325$	2.66256e12	0.734316
$326$	0	0
$327$	− 1.09011e11i	− 0.0291563i
$328$	0	0
$329$	6.37961e11	0.165506
$330$	0	0
$331$	6.51688e12i	1.64021i	0.572213	+	0.820105i	$0.306085\pi$
$331$	6.51688e12i	1.64021i	−0.572213	+	0.820105i	$0.693915\pi$
$332$	0	0
$333$	−5.14102e12	−1.25553
$334$	0	0
$335$	− 3.50709e12i	− 0.831232i
$336$	0	0
$337$	−2.58435e12	−0.594569	−0.297284	−	0.954789i	$-0.596081\pi$
$337$	−2.58435e12	−0.594569	−0.297284	+	0.954789i	$0.596081\pi$
$338$	0	0
$339$	− 1.14621e13i	− 2.56014i
$340$	0	0
$341$	−1.67459e12	−0.363193
$342$	0	0
$343$	− 1.86997e12i	− 0.393880i
$344$	0	0
$345$	−4.83122e12	−0.988464
$346$	0	0
$347$	− 6.46207e12i	− 1.28447i	−0.766507	−	0.642236i	$-0.778007\pi$
$347$	− 6.46207e12i	− 1.28447i	0.766507	−	0.642236i	$-0.221993\pi$
$348$	0	0
$349$	8.33885e12	1.61057	0.805284	−	0.592889i	$-0.202013\pi$
$349$	8.33885e12	1.61057	0.805284	+	0.592889i	$0.202013\pi$
$350$	0	0
$351$	− 1.26991e12i	− 0.238363i
$352$	0	0
$353$	−9.00977e12	−1.64377	−0.821884	−	0.569655i	$-0.807077\pi$
$353$	−9.00977e12	−1.64377	−0.821884	+	0.569655i	$0.807077\pi$
$354$	0	0
$355$	4.72516e12i	0.838059i
$356$	0	0
$357$	5.19211e11	0.0895370
$358$	0	0
$359$	5.63848e12i	0.945561i	0.881180	+	0.472781i	$0.156750\pi$
$359$	5.63848e12i	0.945561i	−0.881180	+	0.472781i	$0.843250\pi$
$360$	0	0
$361$	6.03894e12	0.984973
$362$	0	0
$363$	8.32897e12i	1.32147i
$364$	0	0
$365$	5.73045e12	0.884555
$366$	0	0
$367$	4.02140e12i	0.604013i	0.953306	+	0.302007i	$0.0976565\pi$
$367$	4.02140e12i	0.604013i	−0.953306	+	0.302007i	$0.902344\pi$
$368$	0	0
$369$	4.53394e12	0.662741
$370$	0	0
$371$	− 1.95996e12i	− 0.278854i
$372$	0	0
$373$	−6.73592e12	−0.932937	−0.466469	−	0.884538i	$-0.654474\pi$
$373$	−6.73592e12	−0.932937	−0.466469	+	0.884538i	$0.654474\pi$
$374$	0	0
$375$	9.82967e12i	1.32551i
$376$	0	0
$377$	1.39653e13	1.83376
$378$	0	0
$379$	4.18648e12i	0.535369i	0.963507	+	0.267685i	$0.0862585\pi$
$379$	4.18648e12i	0.535369i	−0.963507	+	0.267685i	$0.913742\pi$
$380$	0	0
$381$	−8.24114e12	−1.02651
$382$	0	0
$383$	8.21642e12i	0.996985i	0.866894	+	0.498493i	$0.166113\pi$
$383$	8.21642e12i	0.996985i	−0.866894	+	0.498493i	$0.833887\pi$
$384$	0	0
$385$	1.22301e12	0.144586
$386$	0	0
$387$	− 1.52695e12i	− 0.175902i
$388$	0	0
$389$	−1.57111e13	−1.76383	−0.881917	−	0.471405i	$-0.843747\pi$
$389$	−1.57111e13	−1.76383	−0.881917	+	0.471405i	$0.843747\pi$
$390$	0	0
$391$	3.56778e12i	0.390404i
$392$	0	0
$393$	−9.53835e12	−1.01744
$394$	0	0
$395$	7.55250e12i	0.785426i
$396$	0	0
$397$	−9.20683e12	−0.933593	−0.466796	−	0.884365i	$-0.654592\pi$
$397$	−9.20683e12	−0.933593	−0.466796	+	0.884365i	$0.654592\pi$
$398$	0	0
$399$	3.66257e11i	0.0362178i
$400$	0	0
$401$	−6.09208e12	−0.587549	−0.293774	−	0.955875i	$-0.594911\pi$
$401$	−6.09208e12	−0.587549	−0.293774	+	0.955875i	$0.594911\pi$
$402$	0	0
$403$	− 2.82772e12i	− 0.266018i
$404$	0	0
$405$	−4.62715e12	−0.424657
$406$	0	0
$407$	1.66479e13i	1.49069i
$408$	0	0
$409$	7.26275e12	0.634577	0.317288	−	0.948329i	$-0.397228\pi$
$409$	7.26275e12	0.634577	0.317288	+	0.948329i	$0.397228\pi$
$410$	0	0
$411$	− 1.97936e13i	− 1.68778i
$412$	0	0
$413$	−1.03820e11	−0.00864037
$414$	0	0
$415$	3.27647e12i	0.266174i
$416$	0	0
$417$	1.19670e13	0.949082
$418$	0	0
$419$	7.35633e12i	0.569628i	0.958583	+	0.284814i	$0.0919318\pi$
$419$	7.35633e12i	0.569628i	−0.958583	+	0.284814i	$0.908068\pi$
$420$	0	0
$421$	−8.92453e12	−0.674800	−0.337400	−	0.941361i	$-0.609547\pi$
$421$	−8.92453e12	−0.674800	−0.337400	+	0.941361i	$0.609547\pi$
$422$	0	0
$423$	− 1.29427e13i	− 0.955702i
$424$	0	0
$425$	3.05705e12	0.220474
$426$	0	0
$427$	− 1.82459e12i	− 0.128537i
$428$	0	0
$429$	−2.97085e13	−2.04453
$430$	0	0
$431$	9.87918e12i	0.664255i	0.943235	+	0.332127i	$0.107766\pi$
$431$	9.87918e12i	0.664255i	−0.943235	+	0.332127i	$0.892234\pi$
$432$	0	0
$433$	−1.75826e13	−1.15517	−0.577583	−	0.816332i	$-0.696004\pi$
$433$	−1.75826e13	−1.15517	−0.577583	+	0.816332i	$0.696004\pi$
$434$	0	0
$435$	2.17126e13i	1.39401i
$436$	0	0
$437$	−2.51675e12	−0.157919
$438$	0	0
$439$	− 1.16369e13i	− 0.713700i	−0.934162	−	0.356850i	$-0.883851\pi$
$439$	− 1.16369e13i	− 0.713700i	0.934162	−	0.356850i	$-0.116149\pi$
$440$	0	0
$441$	−1.85775e13	−1.11377
$442$	0	0
$443$	− 1.46689e13i	− 0.859765i	−0.902885	−	0.429883i	$-0.858555\pi$
$443$	− 1.46689e13i	− 0.859765i	0.902885	−	0.429883i	$-0.141445\pi$
$444$	0	0
$445$	−9.73096e12	−0.557642
$446$	0	0
$447$	2.02331e13i	1.13377i
$448$	0	0
$449$	2.66096e13	1.45816	0.729081	−	0.684427i	$-0.239948\pi$
$449$	2.66096e13	1.45816	0.729081	+	0.684427i	$0.239948\pi$
$450$	0	0
$451$	− 1.46820e13i	− 0.786869i
$452$	0	0
$453$	1.46615e13	0.768577
$454$	0	0
$455$	2.06518e12i	0.105901i
$456$	0	0
$457$	−3.79259e13	−1.90263	−0.951316	−	0.308219i	$-0.900267\pi$
$457$	−3.79259e13	−1.90263	−0.951316	+	0.308219i	$0.900267\pi$
$458$	0	0
$459$	− 1.45807e12i	− 0.0715671i
$460$	0	0
$461$	−1.82920e12	−0.0878528	−0.0439264	−	0.999035i	$-0.513987\pi$
$461$	−1.82920e12	−0.0878528	−0.0439264	+	0.999035i	$0.513987\pi$
$462$	0	0
$463$	2.03740e13i	0.957571i	0.877932	+	0.478785i	$0.158923\pi$
$463$	2.03740e13i	0.957571i	−0.877932	+	0.478785i	$0.841077\pi$
$464$	0	0
$465$	4.39640e12	0.202224
$466$	0	0
$467$	1.48747e13i	0.669674i	0.942276	+	0.334837i	$0.108681\pi$
$467$	1.48747e13i	0.669674i	−0.942276	+	0.334837i	$0.891319\pi$
$468$	0	0
$469$	−7.26303e12	−0.320076
$470$	0	0
$471$	− 3.41624e13i	− 1.47382i
$472$	0	0
$473$	−4.94465e12	−0.208848
$474$	0	0
$475$	2.15648e12i	0.0891820i
$476$	0	0
$477$	−3.97628e13	−1.61022
$478$	0	0
$479$	− 1.32955e12i	− 0.0527263i	−0.999652	−	0.0263631i	$-0.991607\pi$
$479$	− 1.32955e12i	− 0.0527263i	0.999652	−	0.0263631i	$-0.00839262\pi$
$480$	0	0
$481$	−2.81116e13	−1.09184
$482$	0	0
$483$	1.00053e13i	0.380621i
$484$	0	0
$485$	−9.07188e12	−0.338056
$486$	0	0
$487$	1.96646e13i	0.717861i	0.933364	+	0.358930i	$0.116858\pi$
$487$	1.96646e13i	0.717861i	−0.933364	+	0.358930i	$0.883142\pi$
$488$	0	0
$489$	4.62668e13	1.65472
$490$	0	0
$491$	− 2.19337e13i	− 0.768607i	−0.923207	−	0.384304i	$-0.874442\pi$
$491$	− 2.19337e13i	− 0.768607i	0.923207	−	0.384304i	$-0.125558\pi$
$492$	0	0
$493$	1.60344e13	0.550577
$494$	0	0
$495$	− 2.48119e13i	− 0.834900i
$496$	0	0
$497$	9.78561e12	0.322705
$498$	0	0
$499$	1.54603e13i	0.499707i	0.968284	+	0.249854i	$0.0803825\pi$
$499$	1.54603e13i	0.499707i	−0.968284	+	0.249854i	$0.919618\pi$
$500$	0	0
$501$	−7.61755e13	−2.41338
$502$	0	0
$503$	− 4.31328e13i	− 1.33958i	−0.742552	−	0.669789i	$-0.766385\pi$
$503$	− 4.31328e13i	− 1.33958i	0.742552	−	0.669789i	$-0.233615\pi$
$504$	0	0
$505$	1.78905e13	0.544711
$506$	0	0
$507$	− 9.24045e11i	− 0.0275838i
$508$	0	0
$509$	−1.98059e13	−0.579704	−0.289852	−	0.957072i	$-0.593606\pi$
$509$	−1.98059e13	−0.579704	−0.289852	+	0.957072i	$0.593606\pi$
$510$	0	0
$511$	− 1.18675e13i	− 0.340609i
$512$	0	0
$513$	1.02854e12	0.0289489
$514$	0	0
$515$	2.34233e13i	0.646564i
$516$	0	0
$517$	−4.19116e13	−1.13470
$518$	0	0
$519$	− 1.02473e14i	− 2.72127i
$520$	0	0
$521$	4.15852e13	1.08330	0.541652	−	0.840603i	$-0.317799\pi$
$521$	4.15852e13	1.08330	0.541652	+	0.840603i	$0.317799\pi$
$522$	0	0
$523$	− 4.71269e13i	− 1.20437i	−0.798356	−	0.602185i	$-0.794297\pi$
$523$	− 4.71269e13i	− 1.20437i	0.798356	−	0.602185i	$-0.205703\pi$
$524$	0	0
$525$	8.57300e12	0.214949
$526$	0	0
$527$	− 3.24668e12i	− 0.0798704i
$528$	0	0
$529$	−2.73250e13	−0.659602
$530$	0	0
$531$	2.10626e12i	0.0498931i
$532$	0	0
$533$	2.47920e13	0.576336
$534$	0	0
$535$	3.17896e13i	0.725297i
$536$	0	0
$537$	−3.70823e13	−0.830416
$538$	0	0
$539$	6.01585e13i	1.32237i
$540$	0	0
$541$	−7.14446e12	−0.154164	−0.0770820	−	0.997025i	$-0.524560\pi$
$541$	−7.14446e12	−0.154164	−0.0770820	+	0.997025i	$0.524560\pi$
$542$	0	0
$543$	− 9.63961e13i	− 2.04202i
$544$	0	0
$545$	4.97838e11	0.0103539
$546$	0	0
$547$	− 1.86732e13i	− 0.381313i	−0.981657	−	0.190657i	$-0.938938\pi$
$547$	− 1.86732e13i	− 0.381313i	0.981657	−	0.190657i	$-0.0610617\pi$
$548$	0	0
$549$	−3.70166e13	−0.742224
$550$	0	0
$551$	1.13108e13i	0.222708i
$552$	0	0
$553$	1.56409e13	0.302438
$554$	0	0
$555$	− 4.37066e13i	− 0.830008i
$556$	0	0
$557$	−1.22729e13	−0.228914	−0.114457	−	0.993428i	$-0.536513\pi$
$557$	−1.22729e13	−0.228914	−0.114457	+	0.993428i	$0.536513\pi$
$558$	0	0
$559$	− 8.34953e12i	− 0.152969i
$560$	0	0
$561$	−3.41101e13	−0.613860
$562$	0	0
$563$	− 5.80649e13i	− 1.02653i	−0.858230	−	0.513265i	$-0.828436\pi$
$563$	− 5.80649e13i	− 1.02653i	0.858230	−	0.513265i	$-0.171564\pi$
$564$	0	0
$565$	5.23455e13	0.909153
$566$	0	0
$567$	9.58264e12i	0.163520i
$568$	0	0
$569$	−2.56734e13	−0.430449	−0.215224	−	0.976565i	$-0.569048\pi$
$569$	−2.56734e13	−0.430449	−0.215224	+	0.976565i	$0.569048\pi$
$570$	0	0
$571$	1.01602e14i	1.67387i	0.547302	+	0.836936i	$0.315655\pi$
$571$	1.01602e14i	1.67387i	−0.547302	+	0.836936i	$0.684345\pi$
$572$	0	0
$573$	7.16625e13	1.16016
$574$	0	0
$575$	5.89098e13i	0.937234i
$576$	0	0
$577$	9.45205e13	1.47791	0.738954	−	0.673756i	$-0.235320\pi$
$577$	9.45205e13	1.47791	0.738954	+	0.673756i	$0.235320\pi$
$578$	0	0
$579$	− 2.06644e13i	− 0.317564i
$580$	0	0
$581$	6.78542e12	0.102493
$582$	0	0
$583$	1.28762e14i	1.91180i
$584$	0	0
$585$	4.18974e13	0.611516
$586$	0	0
$587$	− 1.47399e13i	− 0.211497i	−0.994393	−	0.105749i	$-0.966276\pi$
$587$	− 1.47399e13i	− 0.211497i	0.994393	−	0.105749i	$-0.0337239\pi$
$588$	0	0
$589$	2.29024e12	0.0323076
$590$	0	0
$591$	− 4.50759e13i	− 0.625183i
$592$	0	0
$593$	−3.51156e12	−0.0478880	−0.0239440	−	0.999713i	$-0.507622\pi$
$593$	−3.51156e12	−0.0478880	−0.0239440	+	0.999713i	$0.507622\pi$
$594$	0	0
$595$	2.37116e12i	0.0317962i
$596$	0	0
$597$	1.43191e14	1.88818
$598$	0	0
$599$	3.39041e13i	0.439661i	0.975538	+	0.219831i	$0.0705505\pi$
$599$	3.39041e13i	0.439661i	−0.975538	+	0.219831i	$0.929450\pi$
$600$	0	0
$601$	−8.19479e13	−1.04512	−0.522559	−	0.852603i	$-0.675023\pi$
$601$	−8.19479e13	−1.04512	−0.522559	+	0.852603i	$0.675023\pi$
$602$	0	0
$603$	1.47349e14i	1.84825i
$604$	0	0
$605$	−3.80371e13	−0.469279
$606$	0	0
$607$	1.43545e14i	1.74199i	0.491292	+	0.870995i	$0.336525\pi$
$607$	1.43545e14i	1.74199i	−0.491292	+	0.870995i	$0.663475\pi$
$608$	0	0
$609$	4.49658e13	0.536779
$610$	0	0
$611$	− 7.07719e13i	− 0.831102i
$612$	0	0
$613$	1.13152e14	1.30725	0.653627	−	0.756817i	$-0.273246\pi$
$613$	1.13152e14	1.30725	0.653627	+	0.756817i	$0.273246\pi$
$614$	0	0
$615$	3.85455e13i	0.438125i
$616$	0	0
$617$	−1.07662e14	−1.20403	−0.602017	−	0.798483i	$-0.705636\pi$
$617$	−1.07662e14	−1.20403	−0.602017	+	0.798483i	$0.705636\pi$
$618$	0	0
$619$	− 8.63434e13i	− 0.950114i	−0.879955	−	0.475057i	$-0.842427\pi$
$619$	− 8.63434e13i	− 0.950114i	0.879955	−	0.475057i	$-0.157573\pi$
$620$	0	0
$621$	2.80971e13	0.304231
$622$	0	0
$623$	2.01524e13i	0.214727i
$624$	0	0
$625$	2.44913e13	0.256810
$626$	0	0
$627$	− 2.40617e13i	− 0.248307i
$628$	0	0
$629$	−3.22767e13	−0.327820
$630$	0	0
$631$	1.73312e14i	1.73253i	0.499585	+	0.866265i	$0.333486\pi$
$631$	1.73312e14i	1.73253i	−0.499585	+	0.866265i	$0.666514\pi$
$632$	0	0
$633$	4.75709e13	0.468083
$634$	0	0
$635$	− 3.76360e13i	− 0.364531i
$636$	0	0
$637$	−1.01584e14	−0.968561
$638$	0	0
$639$	− 1.98526e14i	− 1.86343i
$640$	0	0
$641$	−7.35419e12	−0.0679586	−0.0339793	−	0.999423i	$-0.510818\pi$
$641$	−7.35419e12	−0.0679586	−0.0339793	+	0.999423i	$0.510818\pi$
$642$	0	0
$643$	− 1.48182e14i	− 1.34816i	−0.738659	−	0.674080i	$-0.764540\pi$
$643$	− 1.48182e14i	− 1.34816i	0.738659	−	0.674080i	$-0.235460\pi$
$644$	0	0
$645$	1.29815e13	0.116285
$646$	0	0
$647$	1.12723e14i	0.994241i	0.867682	+	0.497120i	$0.165609\pi$
$647$	1.12723e14i	0.994241i	−0.867682	+	0.497120i	$0.834391\pi$
$648$	0	0
$649$	6.82060e12	0.0592378
$650$	0	0
$651$	− 9.10477e12i	− 0.0778689i
$652$	0	0
$653$	1.28640e14	1.08346	0.541728	−	0.840554i	$-0.317770\pi$
$653$	1.28640e14	1.08346	0.541728	+	0.840554i	$0.317770\pi$
$654$	0	0
$655$	− 4.35602e13i	− 0.361313i
$656$	0	0
$657$	−2.40764e14	−1.96682
$658$	0	0
$659$	1.61390e14i	1.29852i	0.760566	+	0.649261i	$0.224922\pi$
$659$	1.61390e14i	1.29852i	−0.760566	+	0.649261i	$0.775078\pi$
$660$	0	0
$661$	−4.61797e13	−0.365968	−0.182984	−	0.983116i	$-0.558576\pi$
$661$	−4.61797e13	−0.365968	−0.182984	+	0.983116i	$0.558576\pi$
$662$	0	0
$663$	− 5.75984e13i	− 0.449617i
$664$	0	0
$665$	−1.67264e12	−0.0128616
$666$	0	0
$667$	3.08984e14i	2.34049i
$668$	0	0
$669$	3.92935e14	2.93218
$670$	0	0
$671$	1.19869e14i	0.881239i
$672$	0	0
$673$	1.10635e14	0.801343	0.400672	−	0.916222i	$-0.368777\pi$
$673$	1.10635e14	0.801343	0.400672	+	0.916222i	$0.368777\pi$
$674$	0	0
$675$	− 2.40750e13i	− 0.171810i
$676$	0	0
$677$	9.19690e13	0.646693	0.323346	−	0.946281i	$-0.395192\pi$
$677$	9.19690e13	0.646693	0.323346	+	0.946281i	$0.395192\pi$
$678$	0	0
$679$	1.87875e13i	0.130173i
$680$	0	0
$681$	1.37467e14	0.938563
$682$	0	0
$683$	2.33078e13i	0.156819i	0.996921	+	0.0784094i	$0.0249841\pi$
$683$	2.33078e13i	0.156819i	−0.996921	+	0.0784094i	$0.975016\pi$
$684$	0	0
$685$	9.03942e13	0.599360
$686$	0	0
$687$	5.14800e13i	0.336399i
$688$	0	0
$689$	−2.17427e14	−1.40029
$690$	0	0
$691$	3.52382e13i	0.223678i	0.993726	+	0.111839i	$0.0356741\pi$
$691$	3.52382e13i	0.223678i	−0.993726	+	0.111839i	$0.964326\pi$
$692$	0	0
$693$	−5.13845e13	−0.321489
$694$	0	0
$695$	5.46513e13i	0.337036i
$696$	0	0
$697$	2.84652e13	0.173042
$698$	0	0
$699$	− 2.55555e14i	− 1.53143i
$700$	0	0
$701$	−6.66933e13	−0.393996	−0.196998	−	0.980404i	$-0.563119\pi$
$701$	−6.66933e13	−0.393996	−0.196998	+	0.980404i	$0.563119\pi$
$702$	0	0
$703$	− 2.27683e13i	− 0.132603i
$704$	0	0
$705$	1.10033e14	0.631796
$706$	0	0
$707$	− 3.70506e13i	− 0.209748i
$708$	0	0
$709$	1.72014e14	0.960133	0.480067	−	0.877232i	$-0.340612\pi$
$709$	1.72014e14	0.960133	0.480067	+	0.877232i	$0.340612\pi$
$710$	0	0
$711$	− 3.17317e14i	− 1.74640i
$712$	0	0
$713$	6.25638e13	0.339528
$714$	0	0
$715$	− 1.35674e14i	− 0.726050i
$716$	0	0
$717$	1.19875e14	0.632607
$718$	0	0
$719$	2.98544e14i	1.55369i	0.629691	+	0.776845i	$0.283181\pi$
$719$	2.98544e14i	1.55369i	−0.629691	+	0.776845i	$0.716819\pi$
$720$	0	0
$721$	4.85086e13	0.248968
$722$	0	0
$723$	1.45508e14i	0.736540i
$724$	0	0
$725$	2.64753e14	1.32176
$726$	0	0
$727$	6.65143e12i	0.0327524i	0.999866	+	0.0163762i	$0.00521294\pi$
$727$	6.65143e12i	0.0327524i	−0.999866	+	0.0163762i	$0.994787\pi$
$728$	0	0
$729$	2.65880e14	1.29136
$730$	0	0
$731$	− 9.58661e12i	− 0.0459280i
$732$	0	0
$733$	2.98324e14	1.40984	0.704919	−	0.709288i	$-0.250983\pi$
$733$	2.98324e14	1.40984	0.704919	+	0.709288i	$0.250983\pi$
$734$	0	0
$735$	− 1.57938e14i	− 0.736290i
$736$	0	0
$737$	4.77153e14	2.19442
$738$	0	0
$739$	− 7.60252e13i	− 0.344933i	−0.985015	−	0.172467i	$-0.944826\pi$
$739$	− 7.60252e13i	− 0.344933i	0.985015	−	0.172467i	$-0.0551737\pi$
$740$	0	0
$741$	4.06305e13	0.181870
$742$	0	0
$743$	− 1.19329e14i	− 0.526988i	−0.964661	−	0.263494i	$-0.915125\pi$
$743$	− 1.19329e14i	− 0.526988i	0.964661	−	0.263494i	$-0.0848749\pi$
$744$	0	0
$745$	−9.24016e13	−0.402623
$746$	0	0
$747$	− 1.37660e14i	− 0.591840i
$748$	0	0
$749$	6.58349e13	0.279285
$750$	0	0
$751$	− 2.91765e14i	− 1.22133i	−0.791888	−	0.610666i	$-0.790902\pi$
$751$	− 2.91765e14i	− 1.22133i	0.791888	−	0.610666i	$-0.209098\pi$
$752$	0	0
$753$	−3.89694e14	−1.60972
$754$	0	0
$755$	6.69568e13i	0.272935i
$756$	0	0
$757$	1.54894e14	0.623094	0.311547	−	0.950231i	$-0.399153\pi$
$757$	1.54894e14	0.623094	0.311547	+	0.950231i	$0.399153\pi$
$758$	0	0
$759$	− 6.57306e14i	− 2.60951i
$760$	0	0
$761$	−4.32948e14	−1.69634	−0.848169	−	0.529726i	$-0.822295\pi$
$761$	−4.32948e14	−1.69634	−0.848169	+	0.529726i	$0.822295\pi$
$762$	0	0
$763$	− 1.03100e12i	− 0.00398691i
$764$	0	0
$765$	4.81050e13	0.183604
$766$	0	0
$767$	1.15173e13i	0.0433883i
$768$	0	0
$769$	−8.32144e13	−0.309433	−0.154717	−	0.987959i	$-0.549446\pi$
$769$	−8.32144e13	−0.309433	−0.154717	+	0.987959i	$0.549446\pi$
$770$	0	0
$771$	2.58443e14i	0.948620i
$772$	0	0
$773$	1.71162e14	0.620168	0.310084	−	0.950709i	$-0.399643\pi$
$773$	1.71162e14	0.620168	0.310084	+	0.950709i	$0.399643\pi$
$774$	0	0
$775$	− 5.36078e13i	− 0.191743i
$776$	0	0
$777$	−9.05146e13	−0.319605
$778$	0	0
$779$	2.00797e13i	0.0699955i
$780$	0	0
$781$	−6.42877e14	−2.21244
$782$	0	0
$783$	− 1.26274e14i	− 0.429049i
$784$	0	0
$785$	1.56015e14	0.523380
$786$	0	0
$787$	− 4.24986e13i	− 0.140767i	−0.997520	−	0.0703836i	$-0.977578\pi$
$787$	− 4.24986e13i	− 0.140767i	0.997520	−	0.0703836i	$-0.0224223\pi$
$788$	0	0
$789$	−1.75930e14	−0.575379
$790$	0	0
$791$	− 1.08405e14i	− 0.350081i
$792$	0	0
$793$	−2.02410e14	−0.645456
$794$	0	0
$795$	− 3.38045e14i	− 1.06448i
$796$	0	0
$797$	−4.77089e14	−1.48357	−0.741785	−	0.670638i	$-0.766021\pi$
$797$	−4.77089e14	−1.48357	−0.741785	+	0.670638i	$0.766021\pi$
$798$	0	0
$799$	− 8.12575e13i	− 0.249534i
$800$	0	0
$801$	4.08844e14	1.23992
$802$	0	0
$803$	7.79651e14i	2.33519i
$804$	0	0
$805$	−4.56924e13	−0.135165
$806$	0	0
$807$	− 5.47923e14i	− 1.60085i
$808$	0	0
$809$	1.50179e14	0.433378	0.216689	−	0.976241i	$-0.430474\pi$
$809$	1.50179e14	0.433378	0.216689	+	0.976241i	$0.430474\pi$
$810$	0	0
$811$	− 5.28650e13i	− 0.150683i	−0.997158	−	0.0753414i	$-0.975995\pi$
$811$	− 5.28650e13i	− 0.150683i	0.997158	−	0.0753414i	$-0.0240047\pi$
$812$	0	0
$813$	2.03193e14	0.572079
$814$	0	0
$815$	2.11293e14i	0.587621i
$816$	0	0
$817$	6.76250e12	0.0185779
$818$	0	0
$819$	− 8.67678e13i	− 0.235472i
$820$	0	0
$821$	−9.70849e13	−0.260277	−0.130139	−	0.991496i	$-0.541542\pi$
$821$	−9.70849e13	−0.260277	−0.130139	+	0.991496i	$0.541542\pi$
$822$	0	0
$823$	2.13357e14i	0.565077i	0.959256	+	0.282538i	$0.0911764\pi$
$823$	2.13357e14i	0.565077i	−0.959256	+	0.282538i	$0.908824\pi$
$824$	0	0
$825$	−5.63213e14	−1.47368
$826$	0	0
$827$	− 6.64118e14i	− 1.71679i	−0.512988	−	0.858396i	$-0.671461\pi$
$827$	− 6.64118e14i	− 1.71679i	0.512988	−	0.858396i	$-0.328539\pi$
$828$	0	0
$829$	3.35800e14	0.857645	0.428822	−	0.903389i	$-0.358929\pi$
$829$	3.35800e14	0.857645	0.428822	+	0.903389i	$0.358929\pi$
$830$	0	0
$831$	5.05849e14i	1.27649i
$832$	0	0
$833$	−1.16634e14	−0.290805
$834$	0	0
$835$	− 3.47882e14i	− 0.857036i
$836$	0	0
$837$	−2.55683e13	−0.0622407
$838$	0	0
$839$	4.65135e14i	1.11884i	0.828883	+	0.559422i	$0.188977\pi$
$839$	4.65135e14i	1.11884i	−0.828883	+	0.559422i	$0.811023\pi$
$840$	0	0
$841$	9.67937e14	2.30074
$842$	0	0
$843$	5.56734e14i	1.30770i
$844$	0	0
$845$	4.21997e12	0.00979549
$846$	0	0
$847$	7.87733e13i	0.180702i
$848$	0	0
$849$	−1.16848e15	−2.64902
$850$	0	0
$851$	− 6.21975e14i	− 1.39356i
$852$	0	0
$853$	−1.00085e14	−0.221627	−0.110813	−	0.993841i	$-0.535346\pi$
$853$	−1.00085e14	−0.221627	−0.110813	+	0.993841i	$0.535346\pi$
$854$	0	0
$855$	3.39338e13i	0.0742681i
$856$	0	0
$857$	−6.57399e13	−0.142208	−0.0711041	−	0.997469i	$-0.522652\pi$
$857$	−6.57399e13	−0.142208	−0.0711041	+	0.997469i	$0.522652\pi$
$858$	0	0
$859$	7.08850e14i	1.51561i	0.652479	+	0.757807i	$0.273729\pi$
$859$	7.08850e14i	1.51561i	−0.652479	+	0.757807i	$0.726271\pi$
$860$	0	0
$861$	7.98261e13	0.168705
$862$	0	0
$863$	5.84183e14i	1.22038i	0.792255	+	0.610190i	$0.208907\pi$
$863$	5.84183e14i	1.22038i	−0.792255	+	0.610190i	$0.791093\pi$
$864$	0	0
$865$	4.67978e14	0.966373
$866$	0	0
$867$	6.53961e14i	1.33492i
$868$	0	0
$869$	−1.02755e15	−2.07350
$870$	0	0
$871$	8.05720e14i	1.60729i
$872$	0	0
$873$	3.81153e14	0.751671
$874$	0	0
$875$	9.29666e13i	0.181254i
$876$	0	0
$877$	6.71506e13	0.129435	0.0647176	−	0.997904i	$-0.479385\pi$
$877$	6.71506e13	0.129435	0.0647176	+	0.997904i	$0.479385\pi$
$878$	0	0
$879$	− 1.15195e14i	− 0.219528i
$880$	0	0
$881$	−6.22156e14	−1.17225	−0.586124	−	0.810221i	$-0.699347\pi$
$881$	−6.22156e14	−1.17225	−0.586124	+	0.810221i	$0.699347\pi$
$882$	0	0
$883$	− 3.41804e14i	− 0.636756i	−0.947964	−	0.318378i	$-0.896862\pi$
$883$	− 3.41804e14i	− 0.636756i	0.947964	−	0.318378i	$-0.103138\pi$
$884$	0	0
$885$	−1.79065e13	−0.0329833
$886$	0	0
$887$	4.17872e14i	0.761071i	0.924766	+	0.380536i	$0.124260\pi$
$887$	4.17872e14i	0.761071i	−0.924766	+	0.380536i	$0.875740\pi$
$888$	0	0
$889$	−7.79426e13	−0.140367
$890$	0	0
$891$	− 6.29542e14i	− 1.12108i
$892$	0	0
$893$	5.73200e13	0.100937
$894$	0	0
$895$	− 1.69349e14i	− 0.294896i
$896$	0	0
$897$	1.10993e15	1.91132
$898$	0	0
$899$	− 2.81176e14i	− 0.478827i
$900$	0	0
$901$	−2.49641e14	−0.420428
$902$	0	0
$903$	− 2.68841e13i	− 0.0447771i
$904$	0	0
$905$	4.40226e14	0.725159
$906$	0	0
$907$	1.12857e15i	1.83862i	0.393534	+	0.919310i	$0.371252\pi$
$907$	1.12857e15i	1.83862i	−0.393534	+	0.919310i	$0.628748\pi$
$908$	0	0
$909$	−7.51666e14	−1.21117
$910$	0	0
$911$	1.29762e14i	0.206803i	0.994640	+	0.103402i	$0.0329727\pi$
$911$	1.29762e14i	0.206803i	−0.994640	+	0.103402i	$0.967027\pi$
$912$	0	0
$913$	−4.45776e14	−0.702689
$914$	0	0
$915$	− 3.14698e14i	− 0.490669i
$916$	0	0
$917$	−9.02113e13	−0.139128
$918$	0	0
$919$	9.03339e14i	1.37808i	0.724725	+	0.689038i	$0.241967\pi$
$919$	9.03339e14i	1.37808i	−0.724725	+	0.689038i	$0.758033\pi$
$920$	0	0
$921$	−5.97509e14	−0.901667
$922$	0	0
$923$	− 1.08556e15i	− 1.62049i
$924$	0	0
$925$	−5.32939e14	−0.786990
$926$	0	0
$927$	− 9.84123e14i	− 1.43764i
$928$	0	0
$929$	−1.77009e14	−0.255810	−0.127905	−	0.991786i	$-0.540825\pi$
$929$	−1.77009e14	−0.255810	−0.127905	+	0.991786i	$0.540825\pi$
$930$	0	0
$931$	− 8.22753e13i	− 0.117631i
$932$	0	0
$933$	−3.37683e14	−0.477640
$934$	0	0
$935$	− 1.55776e14i	− 0.217993i
$936$	0	0
$937$	−8.62032e14	−1.19351	−0.596754	−	0.802424i	$-0.703543\pi$
$937$	−8.62032e14	−1.19351	−0.596754	+	0.802424i	$0.703543\pi$
$938$	0	0
$939$	7.42619e14i	1.01727i
$940$	0	0
$941$	2.00926e14	0.272325	0.136162	−	0.990687i	$-0.456523\pi$
$941$	2.00926e14	0.272325	0.136162	+	0.990687i	$0.456523\pi$
$942$	0	0
$943$	5.48529e14i	0.735598i
$944$	0	0
$945$	1.86734e13	0.0247779
$946$	0	0
$947$	6.34666e14i	0.833288i	0.909070	+	0.416644i	$0.136794\pi$
$947$	6.34666e14i	0.833288i	−0.909070	+	0.416644i	$0.863206\pi$
$948$	0	0
$949$	−1.31652e15	−1.71039
$950$	0	0
$951$	1.01558e14i	0.130561i
$952$	0	0
$953$	8.63241e14	1.09816	0.549082	−	0.835768i	$-0.314977\pi$
$953$	8.63241e14	1.09816	0.549082	+	0.835768i	$0.314977\pi$
$954$	0	0
$955$	3.27272e14i	0.411994i
$956$	0	0
$957$	−2.95408e15	−3.68012
$958$	0	0
$959$	− 1.87203e14i	− 0.230791i
$960$	0	0
$961$	7.62695e14	0.930538
$962$	0	0
$963$	− 1.33563e15i	− 1.61270i
$964$	0	0
$965$	9.43713e13	0.112773
$966$	0	0
$967$	1.22416e15i	1.44780i	0.689907	+	0.723898i	$0.257651\pi$
$967$	1.22416e15i	1.44780i	−0.689907	+	0.723898i	$0.742349\pi$
$968$	0	0
$969$	4.66504e13	0.0546055
$970$	0	0
$971$	1.15812e15i	1.34171i	0.741590	+	0.670854i	$0.234072\pi$
$971$	1.15812e15i	1.34171i	−0.741590	+	0.670854i	$0.765928\pi$
$972$	0	0
$973$	1.13181e14	0.129780
$974$	0	0
$975$	− 9.51041e14i	− 1.07938i
$976$	0	0
$977$	3.65953e14	0.411105	0.205553	−	0.978646i	$-0.434101\pi$
$977$	3.65953e14	0.411105	0.205553	+	0.978646i	$0.434101\pi$
$978$	0	0
$979$	− 1.32394e15i	− 1.47216i
$980$	0	0
$981$	−2.09165e13	−0.0230221
$982$	0	0
$983$	− 1.68339e14i	− 0.183407i	−0.995786	−	0.0917036i	$-0.970769\pi$
$983$	− 1.68339e14i	− 0.183407i	0.995786	−	0.0917036i	$-0.0292312\pi$
$984$	0	0
$985$	2.05855e14	0.222014
$986$	0	0
$987$	− 2.27873e14i	− 0.243281i
$988$	0	0
$989$	1.84735e14	0.195240
$990$	0	0
$991$	− 6.47388e14i	− 0.677323i	−0.940908	−	0.338662i	$-0.890026\pi$
$991$	− 6.47388e14i	− 0.677323i	0.940908	−	0.338662i	$-0.109974\pi$
$992$	0	0
$993$	2.32776e15	2.41098
$994$	0	0
$995$	6.53931e14i	0.670528i
$996$	0	0
$997$	1.46527e15	1.48744	0.743722	−	0.668488i	$-0.233058\pi$
$997$	1.46527e15	1.48744	0.743722	+	0.668488i	$0.233058\pi$
$998$	0	0
$999$	2.54186e14i	0.255461i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.11.c.j.255.2		10
4.3	odd	2	inner	256.11.c.j.255.9		10
8.3	odd	2		256.11.c.k.255.2		10
8.5	even	2		256.11.c.k.255.9		10
16.3	odd	4		128.11.d.f.63.3	✓	20
16.5	even	4		128.11.d.f.63.4	yes	20
16.11	odd	4		128.11.d.f.63.18	yes	20
16.13	even	4		128.11.d.f.63.17	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.11.d.f.63.3	✓	20	16.3	odd	4
128.11.d.f.63.4	yes	20	16.5	even	4
128.11.d.f.63.17	yes	20	16.13	even	4
128.11.d.f.63.18	yes	20	16.11	odd	4
256.11.c.j.255.2		10	1.1	even	1	trivial
256.11.c.j.255.9		10	4.3	odd	2	inner
256.11.c.k.255.2		10	8.3	odd	2
256.11.c.k.255.9		10	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:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	256.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-357.190i q^{3} +1631.23 q^{5} -3378.22i q^{7} -68535.8 q^{9} +O(q^{10})\)	\(q-357.190i q^{3} +1631.23 q^{5} -3378.22i q^{7} -68535.8 q^{9} +221936. i q^{11} -374761. q^{13} -582660. i q^{15} -430286. q^{17} -303528. i q^{19} -1.20667e6 q^{21} -8.29165e6i q^{23} -7.10471e6 q^{25} +3.38860e6i q^{27} -3.72645e7 q^{29} +7.54540e6i q^{31} +7.92733e7 q^{33} -5.51065e6i q^{35} +7.50121e7 q^{37} +1.33861e8i q^{39} -6.61543e7 q^{41} +2.22796e7i q^{43} -1.11798e8 q^{45} +1.88846e8i q^{47} +2.71063e8 q^{49} +1.53694e8i q^{51} +5.80175e8 q^{53} +3.62029e8i q^{55} -1.08417e8 q^{57} -3.07323e7i q^{59} +5.40106e8 q^{61} +2.31529e8i q^{63} -6.11322e8 q^{65} -2.14996e9i q^{67} -2.96170e9 q^{69} +2.89668e9i q^{71} +3.51296e9 q^{73} +2.53773e9i q^{75} +7.49747e8 q^{77} +4.62994e9i q^{79} -2.83660e9 q^{81} +2.00858e9i q^{83} -7.01896e8 q^{85} +1.33105e10i q^{87} -5.96540e9 q^{89} +1.26602e9i q^{91} +2.69514e9 q^{93} -4.95125e8i q^{95} -5.56137e9 q^{97} -1.52105e10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 6232 q^{5} - 218038 q^{9}+O(q^{10}) \) 10 * q - 6232 * q^5 - 218038 * q^9	\( 10 q - 6232 q^{5} - 218038 q^{9} - 485560 q^{13} - 1468772 q^{17} - 15545024 q^{21} + 4383462 q^{25} - 25263800 q^{29} - 122737264 q^{33} - 178255288 q^{37} + 91656876 q^{41} + 718346408 q^{45} - 842454934 q^{49} + 798458664 q^{53} - 1612102544 q^{57} - 102636184 q^{61} + 376325920 q^{65} - 207224128 q^{69} + 2854265572 q^{73} - 9037740608 q^{77} + 11990017466 q^{81} + 5473132400 q^{85} - 11790814556 q^{89} + 24576098304 q^{93} - 9363277860 q^{97}+O(q^{100}) \) 10 * q - 6232 * q^5 - 218038 * q^9 - 485560 * q^13 - 1468772 * q^17 - 15545024 * q^21 + 4383462 * q^25 - 25263800 * q^29 - 122737264 * q^33 - 178255288 * q^37 + 91656876 * q^41 + 718346408 * q^45 - 842454934 * q^49 + 798458664 * q^53 - 1612102544 * q^57 - 102636184 * q^61 + 376325920 * q^65 - 207224128 * q^69 + 2854265572 * q^73 - 9037740608 * q^77 + 11990017466 * q^81 + 5473132400 * q^85 - 11790814556 * q^89 + 24576098304 * q^93 - 9363277860 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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