LMFDB - Embedded newform 256.11.c.l.255.11

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5593 x^{10} + 23492833 x^{8} - 43148127888 x^{6} + 59505890201856 x^{4} + \cdots + 42\!\cdots\!00 $ x^12 - 5593x^10 + 23492833x^8 - 43148127888x^6 + 59505890201856x^4 - 1615088885760000*x^2 + 42998169600000000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{96}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 64)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			255.11
Root			$42.9174 - 24.7784i$ of defining polynomial
Character	$\chi$	$=$	256.255
Dual form			256.11.c.l.255.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+374.516i q^{3} -5150.99 q^{5} +20446.3i q^{7} -81213.1 q^{9} +O(q^{10})$	$q+374.516i q^{3} -5150.99 q^{5} +20446.3i q^{7} -81213.1 q^{9} -270999. i q^{11} -159348. q^{13} -1.92913e6i q^{15} -2.36784e6 q^{17} +301086. i q^{19} -7.65745e6 q^{21} -509168. i q^{23} +1.67671e7 q^{25} -8.30081e6i q^{27} +1.59080e6 q^{29} -2.19284e7i q^{31} +1.01493e8 q^{33} -1.05319e8i q^{35} +6.44987e7 q^{37} -5.96783e7i q^{39} -9.46251e7 q^{41} -1.15266e7i q^{43} +4.18328e8 q^{45} +1.22204e8i q^{47} -1.35575e8 q^{49} -8.86795e8i q^{51} -3.36435e8 q^{53} +1.39591e9i q^{55} -1.12761e8 q^{57} -5.71022e8i q^{59} +1.47791e9 q^{61} -1.66051e9i q^{63} +8.20800e8 q^{65} -8.73820e8i q^{67} +1.90692e8 q^{69} -2.55947e9i q^{71} -2.99781e9 q^{73} +6.27954e9i q^{75} +5.54092e9 q^{77} +1.87095e9i q^{79} -1.68677e9 q^{81} +5.53569e8i q^{83} +1.21967e10 q^{85} +5.95779e8i q^{87} -2.47230e8 q^{89} -3.25807e9i q^{91} +8.21252e9 q^{93} -1.55089e9i q^{95} +1.05732e10 q^{97} +2.20087e10i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 375204 q^{9}+O(q^{10}) $ 12 * q - 375204 * q^9	$ 12 q - 375204 q^{9} - 1639752 q^{17} + 28703892 q^{25} + 358214544 q^{33} - 496702488 q^{41} + 1385670732 q^{49} - 5733526800 q^{57} + 8895346752 q^{65} - 12161297592 q^{73} - 7708157604 q^{81} + 5871785160 q^{89} + 30465806904 q^{97}+O(q^{100}) $ 12 * q - 375204 * q^9 - 1639752 * q^17 + 28703892 * q^25 + 358214544 * q^33 - 496702488 * q^41 + 1385670732 * q^49 - 5733526800 * q^57 + 8895346752 * q^65 - 12161297592 * q^73 - 7708157604 * q^81 + 5871785160 * q^89 + 30465806904 * 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
$3$	374.516i	1.54122i	0.637309	+	0.770609i	$0.280048\pi$
$3$	374.516i	1.54122i	−0.637309	+	0.770609i	$0.719952\pi$
$4$	0	0
$5$	−5150.99	−1.64832	−0.824159	−	0.566359i	$-0.808352\pi$
$5$	−5150.99	−1.64832	−0.824159	+	0.566359i	$0.808352\pi$
$6$	0	0
$7$	20446.3i	1.21653i	0.793733	+	0.608267i	$0.208135\pi$
$7$	20446.3i	1.21653i	−0.793733	+	0.608267i	$0.791865\pi$
$8$	0	0
$9$	−81213.1	−1.37535
$10$	0	0
$11$	− 270999.i	− 1.68269i	−0.540499	−	0.841344i	$-0.681765\pi$
$11$	− 270999.i	− 1.68269i	0.540499	−	0.841344i	$-0.318235\pi$
$12$	0	0
$13$	−159348.	−0.429170	−0.214585	−	0.976705i	$-0.568840\pi$
$13$	−159348.	−0.429170	−0.214585	+	0.976705i	$0.568840\pi$
$14$	0	0
$15$	− 1.92913e6i	− 2.54042i
$16$	0	0
$17$	−2.36784e6	−1.66766	−0.833831	−	0.552019i	$-0.813858\pi$
$17$	−2.36784e6	−1.66766	−0.833831	+	0.552019i	$0.813858\pi$
$18$	0	0
$19$	301086.i	0.121597i	0.998150	+	0.0607984i	$0.0193647\pi$
$19$	301086.i	0.121597i	−0.998150	+	0.0607984i	$0.980635\pi$
$20$	0	0
$21$	−7.65745e6	−1.87494
$22$	0	0
$23$	− 509168.i	− 0.0791083i	−0.999217	−	0.0395542i	$-0.987406\pi$
$23$	− 509168.i	− 0.0791083i	0.999217	−	0.0395542i	$-0.0125938\pi$
$24$	0	0
$25$	1.67671e7	1.71695
$26$	0	0
$27$	− 8.30081e6i	− 0.578498i
$28$	0	0
$29$	1.59080e6	0.0775577	0.0387788	−	0.999248i	$-0.487653\pi$
$29$	1.59080e6	0.0775577	0.0387788	+	0.999248i	$0.487653\pi$
$30$	0	0
$31$	− 2.19284e7i	− 0.765946i	−0.923760	−	0.382973i	$-0.874900\pi$
$31$	− 2.19284e7i	− 0.765946i	0.923760	−	0.382973i	$-0.125100\pi$
$32$	0	0
$33$	1.01493e8	2.59339
$34$	0	0
$35$	− 1.05319e8i	− 2.00523i
$36$	0	0
$37$	6.44987e7	0.930127	0.465063	−	0.885277i	$-0.346032\pi$
$37$	6.44987e7	0.930127	0.465063	+	0.885277i	$0.346032\pi$
$38$	0	0
$39$	− 5.96783e7i	− 0.661445i
$40$	0	0
$41$	−9.46251e7	−0.816746	−0.408373	−	0.912815i	$-0.633904\pi$
$41$	−9.46251e7	−0.816746	−0.408373	+	0.912815i	$0.633904\pi$
$42$	0	0
$43$	− 1.15266e7i	− 0.0784074i	−0.999231	−	0.0392037i	$-0.987518\pi$
$43$	− 1.15266e7i	− 0.0784074i	0.999231	−	0.0392037i	$-0.0124821\pi$
$44$	0	0
$45$	4.18328e8	2.26701
$46$	0	0
$47$	1.22204e8i	0.532841i	0.963857	+	0.266420i	$0.0858410\pi$
$47$	1.22204e8i	0.532841i	−0.963857	+	0.266420i	$0.914159\pi$
$48$	0	0
$49$	−1.35575e8	−0.479954
$50$	0	0
$51$	− 8.86795e8i	− 2.57023i
$52$	0	0
$53$	−3.36435e8	−0.804491	−0.402246	−	0.915532i	$-0.631770\pi$
$53$	−3.36435e8	−0.804491	−0.402246	+	0.915532i	$0.631770\pi$
$54$	0	0
$55$	1.39591e9i	2.77360i
$56$	0	0
$57$	−1.12761e8	−0.187407
$58$	0	0
$59$	− 5.71022e8i	− 0.798717i	−0.916795	−	0.399358i	$-0.869233\pi$
$59$	− 5.71022e8i	− 0.798717i	0.916795	−	0.399358i	$-0.130767\pi$
$60$	0	0
$61$	1.47791e9	1.74984	0.874919	−	0.484270i	$-0.160915\pi$
$61$	1.47791e9	1.74984	0.874919	+	0.484270i	$0.160915\pi$
$62$	0	0
$63$	− 1.66051e9i	− 1.67316i
$64$	0	0
$65$	8.20800e8	0.707409
$66$	0	0
$67$	− 8.73820e8i	− 0.647214i	−0.946192	−	0.323607i	$-0.895104\pi$
$67$	− 8.73820e8i	− 0.647214i	0.946192	−	0.323607i	$-0.104896\pi$
$68$	0	0
$69$	1.90692e8	0.121923
$70$	0	0
$71$	− 2.55947e9i	− 1.41860i	−0.704909	−	0.709298i	$-0.749012\pi$
$71$	− 2.55947e9i	− 1.41860i	0.704909	−	0.709298i	$-0.250988\pi$
$72$	0	0
$73$	−2.99781e9	−1.44607	−0.723037	−	0.690809i	$-0.757254\pi$
$73$	−2.99781e9	−1.44607	−0.723037	+	0.690809i	$0.757254\pi$
$74$	0	0
$75$	6.27954e9i	2.64619i
$76$	0	0
$77$	5.54092e9	2.04705
$78$	0	0
$79$	1.87095e9i	0.608032i	0.952667	+	0.304016i	$0.0983277\pi$
$79$	1.87095e9i	0.608032i	−0.952667	+	0.304016i	$0.901672\pi$
$80$	0	0
$81$	−1.68677e9	−0.483760
$82$	0	0
$83$	5.53569e8i	0.140534i	0.997528	+	0.0702670i	$0.0223851\pi$
$83$	5.53569e8i	0.140534i	−0.997528	+	0.0702670i	$0.977615\pi$
$84$	0	0
$85$	1.21967e10	2.74884
$86$	0	0
$87$	5.95779e8i	0.119533i
$88$	0	0
$89$	−2.47230e8	−0.0442742	−0.0221371	−	0.999755i	$-0.507047\pi$
$89$	−2.47230e8	−0.0442742	−0.0221371	+	0.999755i	$0.507047\pi$
$90$	0	0
$91$	− 3.25807e9i	− 0.522100i
$92$	0	0
$93$	8.21252e9	1.18049
$94$	0	0
$95$	− 1.55089e9i	− 0.200430i
$96$	0	0
$97$	1.05732e10	1.23126	0.615630	−	0.788035i	$-0.288902\pi$
$97$	1.05732e10	1.23126	0.615630	+	0.788035i	$0.288902\pi$
$98$	0	0
$99$	2.20087e10i	2.31429i
$100$	0	0
$101$	1.31438e9	0.125059	0.0625295	−	0.998043i	$-0.480083\pi$
$101$	1.31438e9	0.125059	0.0625295	+	0.998043i	$0.480083\pi$
$102$	0	0
$103$	1.90481e10i	1.64310i	0.570133	+	0.821552i	$0.306892\pi$
$103$	1.90481e10i	1.64310i	−0.570133	+	0.821552i	$0.693108\pi$
$104$	0	0
$105$	3.94435e10	3.09050
$106$	0	0
$107$	− 2.68758e7i	− 0.00191621i	−1.00000		0.000958104i	$-0.999695\pi$
$107$	− 2.68758e7i	− 0.00191621i	1.00000		0.000958104i	$-0.000304974\pi$
$108$	0	0
$109$	−1.86047e10	−1.20918	−0.604589	−	0.796537i	$-0.706663\pi$
$109$	−1.86047e10	−1.20918	−0.604589	+	0.796537i	$0.706663\pi$
$110$	0	0
$111$	2.41558e10i	1.43353i
$112$	0	0
$113$	−3.04022e10	−1.65011	−0.825055	−	0.565053i	$-0.808856\pi$
$113$	−3.04022e10	−1.65011	−0.825055	+	0.565053i	$0.808856\pi$
$114$	0	0
$115$	2.62272e9i	0.130396i
$116$	0	0
$117$	1.29411e10	0.590260
$118$	0	0
$119$	− 4.84136e10i	− 2.02877i
$120$	0	0
$121$	−4.75029e10	−1.83144
$122$	0	0
$123$	− 3.54386e10i	− 1.25878i
$124$	0	0
$125$	−3.60644e10	−1.18176
$126$	0	0
$127$	7.80454e9i	0.236227i	0.993000	+	0.118113i	$0.0376846\pi$
$127$	7.80454e9i	0.236227i	−0.993000	+	0.118113i	$0.962315\pi$
$128$	0	0
$129$	4.31688e9	0.120843
$130$	0	0
$131$	− 3.72434e10i	− 0.965368i	−0.875795	−	0.482684i	$-0.839662\pi$
$131$	− 3.72434e10i	− 0.965368i	0.875795	−	0.482684i	$-0.160338\pi$
$132$	0	0
$133$	−6.15609e9	−0.147927
$134$	0	0
$135$	4.27574e10i	0.953548i
$136$	0	0
$137$	−6.15033e10	−1.27437	−0.637186	−	0.770710i	$-0.719902\pi$
$137$	−6.15033e10	−1.27437	−0.637186	+	0.770710i	$0.719902\pi$
$138$	0	0
$139$	− 6.75552e10i	− 1.30192i	−0.759111	−	0.650961i	$-0.774366\pi$
$139$	− 6.75552e10i	− 1.30192i	0.759111	−	0.650961i	$-0.225634\pi$
$140$	0	0
$141$	−4.57675e10	−0.821224
$142$	0	0
$143$	4.31831e10i	0.722160i
$144$	0	0
$145$	−8.19418e9	−0.127840
$146$	0	0
$147$	− 5.07750e10i	− 0.739713i
$148$	0	0
$149$	−3.50063e10	−0.476667	−0.238334	−	0.971183i	$-0.576601\pi$
$149$	−3.50063e10	−0.476667	−0.238334	+	0.971183i	$0.576601\pi$
$150$	0	0
$151$	4.03952e10i	0.514570i	0.966336	+	0.257285i	$0.0828279\pi$
$151$	4.03952e10i	0.514570i	−0.966336	+	0.257285i	$0.917172\pi$
$152$	0	0
$153$	1.92300e11	2.29362
$154$	0	0
$155$	1.12953e11i	1.26252i
$156$	0	0
$157$	8.76561e10	0.918934	0.459467	−	0.888195i	$-0.348041\pi$
$157$	8.76561e10	0.918934	0.459467	+	0.888195i	$0.348041\pi$
$158$	0	0
$159$	− 1.26000e11i	− 1.23990i
$160$	0	0
$161$	1.04106e10	0.0962379
$162$	0	0
$163$	− 1.11748e10i	− 0.0971180i	−0.998820	−	0.0485590i	$-0.984537\pi$
$163$	− 1.11748e10i	− 0.0971180i	0.998820	−	0.0485590i	$-0.0154629\pi$
$164$	0	0
$165$	−5.22791e11	−4.27473
$166$	0	0
$167$	1.75627e11i	1.35210i	0.736855	+	0.676051i	$0.236310\pi$
$167$	1.75627e11i	1.35210i	−0.736855	+	0.676051i	$0.763690\pi$
$168$	0	0
$169$	−1.12467e11	−0.815813
$170$	0	0
$171$	− 2.44521e10i	− 0.167238i
$172$	0	0
$173$	−2.39587e11	−1.54608	−0.773042	−	0.634355i	$-0.781266\pi$
$173$	−2.39587e11	−1.54608	−0.773042	+	0.634355i	$0.781266\pi$
$174$	0	0
$175$	3.42824e11i	2.08873i
$176$	0	0
$177$	2.13857e11	1.23100
$178$	0	0
$179$	− 3.90933e10i	− 0.212734i	−0.994327	−	0.106367i	$-0.966078\pi$
$179$	− 3.90933e10i	− 0.212734i	0.994327	−	0.106367i	$-0.0339218\pi$
$180$	0	0
$181$	3.40074e11	1.75057	0.875286	−	0.483606i	$-0.160673\pi$
$181$	3.40074e11	1.75057	0.875286	+	0.483606i	$0.160673\pi$
$182$	0	0
$183$	5.53499e11i	2.69688i
$184$	0	0
$185$	−3.32232e11	−1.53314
$186$	0	0
$187$	6.41682e11i	2.80616i
$188$	0	0
$189$	1.69721e11	0.703762
$190$	0	0
$191$	2.64887e11i	1.04206i	0.853537	+	0.521032i	$0.174453\pi$
$191$	2.64887e11i	1.04206i	−0.853537	+	0.521032i	$0.825547\pi$
$192$	0	0
$193$	4.49917e10	0.168014	0.0840072	−	0.996465i	$-0.473228\pi$
$193$	4.49917e10	0.168014	0.0840072	+	0.996465i	$0.473228\pi$
$194$	0	0
$195$	3.07403e11i	1.09027i
$196$	0	0
$197$	−1.40237e11	−0.472641	−0.236321	−	0.971675i	$-0.575942\pi$
$197$	−1.40237e11	−0.472641	−0.236321	+	0.971675i	$0.575942\pi$
$198$	0	0
$199$	− 4.74503e11i	− 1.52046i	−0.649656	−	0.760228i	$-0.725087\pi$
$199$	− 4.74503e11i	− 1.52046i	0.649656	−	0.760228i	$-0.274913\pi$
$200$	0	0
$201$	3.27259e11	0.997498
$202$	0	0
$203$	3.25259e10i	0.0943515i
$204$	0	0
$205$	4.87413e11	1.34626
$206$	0	0
$207$	4.13511e10i	0.108802i
$208$	0	0
$209$	8.15939e10	0.204610
$210$	0	0
$211$	− 2.35906e11i	− 0.564061i	−0.959405	−	0.282031i	$-0.908992\pi$
$211$	− 2.35906e11i	− 0.564061i	0.959405	−	0.282031i	$-0.0910080\pi$
$212$	0	0
$213$	9.58563e11	2.18637
$214$	0	0
$215$	5.93732e10i	0.129240i
$216$	0	0
$217$	4.48354e11	0.931799
$218$	0	0
$219$	− 1.12273e12i	− 2.22871i
$220$	0	0
$221$	3.77311e11	0.715711
$222$	0	0
$223$	3.71192e11i	0.673092i	0.941667	+	0.336546i	$0.109259\pi$
$223$	3.71192e11i	0.673092i	−0.941667	+	0.336546i	$0.890741\pi$
$224$	0	0
$225$	−1.36171e12	−2.36141
$226$	0	0
$227$	8.63202e11i	1.43213i	0.698032	+	0.716066i	$0.254059\pi$
$227$	8.63202e11i	1.43213i	−0.698032	+	0.716066i	$0.745941\pi$
$228$	0	0
$229$	5.44725e11	0.864968	0.432484	−	0.901642i	$-0.357637\pi$
$229$	5.44725e11	0.864968	0.432484	+	0.901642i	$0.357637\pi$
$230$	0	0
$231$	2.07516e12i	3.15495i
$232$	0	0
$233$	−7.11875e11	−1.03663	−0.518315	−	0.855190i	$-0.673441\pi$
$233$	−7.11875e11	−1.03663	−0.518315	+	0.855190i	$0.673441\pi$
$234$	0	0
$235$	− 6.29474e11i	− 0.878291i
$236$	0	0
$237$	−7.00700e11	−0.937110
$238$	0	0
$239$	5.91014e11i	0.757894i	0.925418	+	0.378947i	$0.123714\pi$
$239$	5.91014e11i	0.757894i	−0.925418	+	0.378947i	$0.876286\pi$
$240$	0	0
$241$	−8.30444e10	−0.102147	−0.0510734	−	0.998695i	$-0.516264\pi$
$241$	−8.30444e10	−0.102147	−0.0510734	+	0.998695i	$0.516264\pi$
$242$	0	0
$243$	− 1.12188e12i	− 1.32408i
$244$	0	0
$245$	6.98346e11	0.791116
$246$	0	0
$247$	− 4.79774e10i	− 0.0521858i
$248$	0	0
$249$	−2.07320e11	−0.216594
$250$	0	0
$251$	− 5.07965e11i	− 0.509877i	−0.966957	−	0.254938i	$-0.917945\pi$
$251$	− 5.07965e11i	− 0.509877i	0.966957	−	0.254938i	$-0.0820552\pi$
$252$	0	0
$253$	−1.37984e11	−0.133115
$254$	0	0
$255$	4.56787e12i	4.23656i
$256$	0	0
$257$	8.74675e11	0.780155	0.390078	−	0.920782i	$-0.372448\pi$
$257$	8.74675e11	0.780155	0.390078	+	0.920782i	$0.372448\pi$
$258$	0	0
$259$	1.31876e12i	1.13153i
$260$	0	0
$261$	−1.29194e11	−0.106669
$262$	0	0
$263$	− 2.67960e11i	− 0.212957i	−0.994315	−	0.106478i	$-0.966042\pi$
$263$	− 2.67960e11i	− 0.212957i	0.994315	−	0.106478i	$-0.0339575\pi$
$264$	0	0
$265$	1.73297e12	1.32606
$266$	0	0
$267$	− 9.25914e10i	− 0.0682361i
$268$	0	0
$269$	1.92015e11	0.136324	0.0681621	−	0.997674i	$-0.478286\pi$
$269$	1.92015e11	0.136324	0.0681621	+	0.997674i	$0.478286\pi$
$270$	0	0
$271$	− 9.56574e11i	− 0.654443i	−0.944948	−	0.327222i	$-0.893888\pi$
$271$	− 9.56574e11i	− 0.654443i	0.944948	−	0.327222i	$-0.106112\pi$
$272$	0	0
$273$	1.22020e12	0.804670
$274$	0	0
$275$	− 4.54386e12i	− 2.88909i
$276$	0	0
$277$	5.30273e11	0.325163	0.162581	−	0.986695i	$-0.448018\pi$
$277$	5.30273e11	0.325163	0.162581	+	0.986695i	$0.448018\pi$
$278$	0	0
$279$	1.78087e12i	1.05344i
$280$	0	0
$281$	7.39601e11	0.422149	0.211075	−	0.977470i	$-0.432304\pi$
$281$	7.39601e11	0.422149	0.211075	+	0.977470i	$0.432304\pi$
$282$	0	0
$283$	1.52426e12i	0.839704i	0.907592	+	0.419852i	$0.137918\pi$
$283$	1.52426e12i	0.839704i	−0.907592	+	0.419852i	$0.862082\pi$
$284$	0	0
$285$	5.80833e11	0.308907
$286$	0	0
$287$	− 1.93473e12i	− 0.993599i
$288$	0	0
$289$	3.59069e12	1.78110
$290$	0	0
$291$	3.95985e12i	1.89764i
$292$	0	0
$293$	−2.37453e12	−1.09961	−0.549806	−	0.835292i	$-0.685298\pi$
$293$	−2.37453e12	−1.09961	−0.549806	+	0.835292i	$0.685298\pi$
$294$	0	0
$295$	2.94133e12i	1.31654i
$296$	0	0
$297$	−2.24951e12	−0.973432
$298$	0	0
$299$	8.11349e10i	0.0339509i
$300$	0	0
$301$	2.35675e11	0.0953852
$302$	0	0
$303$	4.92257e11i	0.192743i
$304$	0	0
$305$	−7.61268e12	−2.88429
$306$	0	0
$307$	2.72818e12i	1.00042i	0.865905	+	0.500208i	$0.166743\pi$
$307$	2.72818e12i	1.00042i	−0.865905	+	0.500208i	$0.833257\pi$
$308$	0	0
$309$	−7.13381e12	−2.53238
$310$	0	0
$311$	4.70819e12i	1.61828i	0.587619	+	0.809138i	$0.300065\pi$
$311$	4.70819e12i	1.61828i	−0.587619	+	0.809138i	$0.699935\pi$
$312$	0	0
$313$	−1.78006e12	−0.592534	−0.296267	−	0.955105i	$-0.595742\pi$
$313$	−1.78006e12	−0.592534	−0.296267	+	0.955105i	$0.595742\pi$
$314$	0	0
$315$	8.55325e12i	2.75790i
$316$	0	0
$317$	−5.75825e11	−0.179885	−0.0899423	−	0.995947i	$-0.528668\pi$
$317$	−5.75825e11	−0.179885	−0.0899423	+	0.995947i	$0.528668\pi$
$318$	0	0
$319$	− 4.31104e11i	− 0.130505i
$320$	0	0
$321$	1.00654e10	0.00295329
$322$	0	0
$323$	− 7.12924e11i	− 0.202783i
$324$	0	0
$325$	−2.67180e12	−0.736864
$326$	0	0
$327$	− 6.96776e12i	− 1.86361i
$328$	0	0
$329$	−2.49863e12	−0.648219
$330$	0	0
$331$	− 4.71906e11i	− 0.118772i	−0.998235	−	0.0593862i	$-0.981086\pi$
$331$	− 4.71906e11i	− 0.118772i	0.998235	−	0.0593862i	$-0.0189143\pi$
$332$	0	0
$333$	−5.23814e12	−1.27925
$334$	0	0
$335$	4.50104e12i	1.06681i
$336$	0	0
$337$	3.70156e12	0.851599	0.425799	−	0.904818i	$-0.359993\pi$
$337$	3.70156e12	0.851599	0.425799	+	0.904818i	$0.359993\pi$
$338$	0	0
$339$	− 1.13861e13i	− 2.54318i
$340$	0	0
$341$	−5.94256e12	−1.28885
$342$	0	0
$343$	3.00356e12i	0.632654i
$344$	0	0
$345$	−9.82251e11	−0.200968
$346$	0	0
$347$	− 1.44216e11i	− 0.0286659i	−0.999897	−	0.0143330i	$-0.995438\pi$
$347$	− 1.44216e11i	− 0.0286659i	0.999897	−	0.0143330i	$-0.00456248\pi$
$348$	0	0
$349$	−8.92335e12	−1.72346	−0.861729	−	0.507369i	$-0.830618\pi$
$349$	−8.92335e12	−1.72346	−0.861729	+	0.507369i	$0.830618\pi$
$350$	0	0
$351$	1.32272e12i	0.248274i
$352$	0	0
$353$	4.48777e12	0.818760	0.409380	−	0.912364i	$-0.365745\pi$
$353$	4.48777e12	0.818760	0.409380	+	0.912364i	$0.365745\pi$
$354$	0	0
$355$	1.31838e13i	2.33830i
$356$	0	0
$357$	1.81317e13	3.12677
$358$	0	0
$359$	4.78315e12i	0.802124i	0.916051	+	0.401062i	$0.131359\pi$
$359$	4.78315e12i	0.802124i	−0.916051	+	0.401062i	$0.868641\pi$
$360$	0	0
$361$	6.04041e12	0.985214
$362$	0	0
$363$	− 1.77906e13i	− 2.82265i
$364$	0	0
$365$	1.54417e13	2.38359
$366$	0	0
$367$	2.85663e12i	0.429065i	0.976717	+	0.214532i	$0.0688228\pi$
$367$	2.85663e12i	0.429065i	−0.976717	+	0.214532i	$0.931177\pi$
$368$	0	0
$369$	7.68480e12	1.12331
$370$	0	0
$371$	− 6.87884e12i	− 0.978691i
$372$	0	0
$373$	−2.40343e12	−0.332879	−0.166440	−	0.986052i	$-0.553227\pi$
$373$	−2.40343e12	−0.332879	−0.166440	+	0.986052i	$0.553227\pi$
$374$	0	0
$375$	− 1.35067e13i	− 1.82135i
$376$	0	0
$377$	−2.53490e11	−0.0332854
$378$	0	0
$379$	4.09475e12i	0.523638i	0.965117	+	0.261819i	$0.0843223\pi$
$379$	4.09475e12i	0.523638i	−0.965117	+	0.261819i	$0.915678\pi$
$380$	0	0
$381$	−2.92292e12	−0.364077
$382$	0	0
$383$	− 1.38243e13i	− 1.67745i	−0.544556	−	0.838725i	$-0.683302\pi$
$383$	− 1.38243e13i	− 1.67745i	0.544556	−	0.838725i	$-0.316698\pi$
$384$	0	0
$385$	−2.85412e13	−3.37418
$386$	0	0
$387$	9.36107e11i	0.107838i
$388$	0	0
$389$	1.29581e13	1.45477	0.727384	−	0.686230i	$-0.240736\pi$
$389$	1.29581e13	1.45477	0.727384	+	0.686230i	$0.240736\pi$
$390$	0	0
$391$	1.20563e12i	0.131926i
$392$	0	0
$393$	1.39482e13	1.48784
$394$	0	0
$395$	− 9.63725e12i	− 1.00223i
$396$	0	0
$397$	3.66407e12	0.371545	0.185772	−	0.982593i	$-0.440521\pi$
$397$	3.66407e12	0.371545	0.185772	+	0.982593i	$0.440521\pi$
$398$	0	0
$399$	− 2.30555e12i	− 0.227987i
$400$	0	0
$401$	−5.46715e12	−0.527277	−0.263639	−	0.964622i	$-0.584923\pi$
$401$	−5.46715e12	−0.527277	−0.263639	+	0.964622i	$0.584923\pi$
$402$	0	0
$403$	3.49424e12i	0.328721i
$404$	0	0
$405$	8.68852e12	0.797390
$406$	0	0
$407$	− 1.74791e13i	− 1.56511i
$408$	0	0
$409$	1.35143e13	1.18080	0.590401	−	0.807110i	$-0.298970\pi$
$409$	1.35143e13	1.18080	0.590401	+	0.807110i	$0.298970\pi$
$410$	0	0
$411$	− 2.30340e13i	− 1.96408i
$412$	0	0
$413$	1.16753e13	0.971665
$414$	0	0
$415$	− 2.85143e12i	− 0.231645i
$416$	0	0
$417$	2.53005e13	2.00654
$418$	0	0
$419$	2.82096e12i	0.218437i	0.994018	+	0.109219i	$0.0348348\pi$
$419$	2.82096e12i	0.218437i	−0.994018	+	0.109219i	$0.965165\pi$
$420$	0	0
$421$	5.70718e12	0.431530	0.215765	−	0.976445i	$-0.430776\pi$
$421$	5.70718e12	0.431530	0.215765	+	0.976445i	$0.430776\pi$
$422$	0	0
$423$	− 9.92460e12i	− 0.732843i
$424$	0	0
$425$	−3.97018e13	−2.86329
$426$	0	0
$427$	3.02177e13i	2.12874i
$428$	0	0
$429$	−1.61728e13	−1.11301
$430$	0	0
$431$	− 2.37871e13i	− 1.59939i	−0.600406	−	0.799695i	$-0.704994\pi$
$431$	− 2.37871e13i	− 1.59939i	0.600406	−	0.799695i	$-0.295006\pi$
$432$	0	0
$433$	1.42751e13	0.937861	0.468930	−	0.883235i	$-0.344640\pi$
$433$	1.42751e13	0.937861	0.468930	+	0.883235i	$0.344640\pi$
$434$	0	0
$435$	− 3.06885e12i	− 0.197029i
$436$	0	0
$437$	1.53303e11	0.00961933
$438$	0	0
$439$	− 2.82092e13i	− 1.73009i	−0.501698	−	0.865043i	$-0.667291\pi$
$439$	− 2.82092e13i	− 1.73009i	0.501698	−	0.865043i	$-0.332709\pi$
$440$	0	0
$441$	1.10105e13	0.660105
$442$	0	0
$443$	2.45483e13i	1.43881i	0.694592	+	0.719404i	$0.255585\pi$
$443$	2.45483e13i	1.43881i	−0.694592	+	0.719404i	$0.744415\pi$
$444$	0	0
$445$	1.27348e12	0.0729779
$446$	0	0
$447$	− 1.31104e13i	− 0.734648i
$448$	0	0
$449$	−1.95821e12	−0.107307	−0.0536533	−	0.998560i	$-0.517087\pi$
$449$	−1.95821e12	−0.107307	−0.0536533	+	0.998560i	$0.517087\pi$
$450$	0	0
$451$	2.56433e13i	1.37433i
$452$	0	0
$453$	−1.51286e13	−0.793064
$454$	0	0
$455$	1.67823e13i	0.860586i
$456$	0	0
$457$	−1.81409e13	−0.910076	−0.455038	−	0.890472i	$-0.650374\pi$
$457$	−1.81409e13	−0.910076	−0.455038	+	0.890472i	$0.650374\pi$
$458$	0	0
$459$	1.96550e13i	0.964740i
$460$	0	0
$461$	3.13673e13	1.50651	0.753256	−	0.657727i	$-0.228482\pi$
$461$	3.13673e13	1.50651	0.753256	+	0.657727i	$0.228482\pi$
$462$	0	0
$463$	3.21781e13i	1.51236i	0.654363	+	0.756181i	$0.272937\pi$
$463$	3.21781e13i	1.51236i	−0.654363	+	0.756181i	$0.727063\pi$
$464$	0	0
$465$	−4.23026e13	−1.94582
$466$	0	0
$467$	− 4.07623e13i	− 1.83516i	−0.397551	−	0.917580i	$-0.630140\pi$
$467$	− 4.07623e13i	− 1.83516i	0.397551	−	0.917580i	$-0.369860\pi$
$468$	0	0
$469$	1.78664e13	0.787358
$470$	0	0
$471$	3.28286e13i	1.41628i
$472$	0	0
$473$	−3.12368e12	−0.131935
$474$	0	0
$475$	5.04833e12i	0.208776i
$476$	0	0
$477$	2.73229e13	1.10646
$478$	0	0
$479$	− 1.10677e13i	− 0.438914i	−0.975622	−	0.219457i	$-0.929571\pi$
$479$	− 1.10677e13i	− 0.438914i	0.975622	−	0.219457i	$-0.0704285\pi$
$480$	0	0
$481$	−1.02777e13	−0.399183
$482$	0	0
$483$	3.89893e12i	0.148324i
$484$	0	0
$485$	−5.44627e13	−2.02951
$486$	0	0
$487$	− 4.47870e13i	− 1.63496i	−0.575958	−	0.817479i	$-0.695371\pi$
$487$	− 4.47870e13i	− 1.63496i	0.575958	−	0.817479i	$-0.304629\pi$
$488$	0	0
$489$	4.18512e12	0.149680
$490$	0	0
$491$	1.03354e13i	0.362176i	0.983467	+	0.181088i	$0.0579618\pi$
$491$	1.03354e13i	0.362176i	−0.983467	+	0.181088i	$0.942038\pi$
$492$	0	0
$493$	−3.76676e12	−0.129340
$494$	0	0
$495$	− 1.13366e14i	− 3.81468i
$496$	0	0
$497$	5.23317e13	1.72577
$498$	0	0
$499$	5.19922e13i	1.68049i	0.542207	+	0.840245i	$0.317589\pi$
$499$	5.19922e13i	1.68049i	−0.542207	+	0.840245i	$0.682411\pi$
$500$	0	0
$501$	−6.57751e13	−2.08388
$502$	0	0
$503$	1.42773e13i	0.443410i	0.975114	+	0.221705i	$0.0711622\pi$
$503$	1.42773e13i	0.443410i	−0.975114	+	0.221705i	$0.928838\pi$
$504$	0	0
$505$	−6.77037e12	−0.206137
$506$	0	0
$507$	− 4.21206e13i	− 1.25734i
$508$	0	0
$509$	−3.93883e13	−1.15286	−0.576432	−	0.817145i	$-0.695556\pi$
$509$	−3.93883e13	−1.15286	−0.576432	+	0.817145i	$0.695556\pi$
$510$	0	0
$511$	− 6.12942e13i	− 1.75920i
$512$	0	0
$513$	2.49926e12	0.0703435
$514$	0	0
$515$	− 9.81165e13i	− 2.70836i
$516$	0	0
$517$	3.31172e13	0.896606
$518$	0	0
$519$	− 8.97292e13i	− 2.38285i
$520$	0	0
$521$	6.05579e13	1.57755	0.788774	−	0.614684i	$-0.210716\pi$
$521$	6.05579e13	1.57755	0.788774	+	0.614684i	$0.210716\pi$
$522$	0	0
$523$	− 4.55041e12i	− 0.116290i	−0.998308	−	0.0581450i	$-0.981481\pi$
$523$	− 4.55041e12i	− 0.116290i	0.998308	−	0.0581450i	$-0.0185186\pi$
$524$	0	0
$525$	−1.28393e14	−3.21918
$526$	0	0
$527$	5.19229e13i	1.27734i
$528$	0	0
$529$	4.11673e13	0.993742
$530$	0	0
$531$	4.63745e13i	1.09852i
$532$	0	0
$533$	1.50783e13	0.350523
$534$	0	0
$535$	1.38437e11i	0.00315852i
$536$	0	0
$537$	1.46410e13	0.327869
$538$	0	0
$539$	3.67407e13i	0.807613i
$540$	0	0
$541$	9.44945e12	0.203901	0.101951	−	0.994789i	$-0.467492\pi$
$541$	9.44945e12	0.203901	0.101951	+	0.994789i	$0.467492\pi$
$542$	0	0
$543$	1.27363e14i	2.69801i
$544$	0	0
$545$	9.58327e13	1.99311
$546$	0	0
$547$	− 6.37468e13i	− 1.30173i	−0.759193	−	0.650866i	$-0.774406\pi$
$547$	− 6.37468e13i	− 1.30173i	0.759193	−	0.650866i	$-0.225594\pi$
$548$	0	0
$549$	−1.20025e14	−2.40664
$550$	0	0
$551$	4.78967e11i	0.00943077i
$552$	0	0
$553$	−3.82540e13	−0.739692
$554$	0	0
$555$	− 1.24426e14i	− 2.36291i
$556$	0	0
$557$	4.42628e13	0.825587	0.412793	−	0.910825i	$-0.364553\pi$
$557$	4.42628e13	0.825587	0.412793	+	0.910825i	$0.364553\pi$
$558$	0	0
$559$	1.83673e12i	0.0336501i
$560$	0	0
$561$	−2.40320e14	−4.32490
$562$	0	0
$563$	− 8.52830e12i	− 0.150772i	−0.997154	−	0.0753859i	$-0.975981\pi$
$563$	− 8.52830e12i	− 0.150772i	0.997154	−	0.0753859i	$-0.0240189\pi$
$564$	0	0
$565$	1.56601e14	2.71990
$566$	0	0
$567$	− 3.44881e13i	− 0.588510i
$568$	0	0
$569$	−2.18979e13	−0.367148	−0.183574	−	0.983006i	$-0.558767\pi$
$569$	−2.18979e13	−0.367148	−0.183574	+	0.983006i	$0.558767\pi$
$570$	0	0
$571$	− 9.09565e13i	− 1.49849i	−0.662294	−	0.749244i	$-0.730417\pi$
$571$	− 9.09565e13i	− 1.49849i	0.662294	−	0.749244i	$-0.269583\pi$
$572$	0	0
$573$	−9.92046e13	−1.60605
$574$	0	0
$575$	− 8.53727e12i	− 0.135825i
$576$	0	0
$577$	−2.64834e13	−0.414090	−0.207045	−	0.978331i	$-0.566385\pi$
$577$	−2.64834e13	−0.414090	−0.207045	+	0.978331i	$0.566385\pi$
$578$	0	0
$579$	1.68501e13i	0.258947i
$580$	0	0
$581$	−1.13184e13	−0.170964
$582$	0	0
$583$	9.11734e13i	1.35371i
$584$	0	0
$585$	−6.66597e13	−0.972936
$586$	0	0
$587$	3.55943e13i	0.510728i	0.966845	+	0.255364i	$0.0821954\pi$
$587$	3.55943e13i	0.510728i	−0.966845	+	0.255364i	$0.917805\pi$
$588$	0	0
$589$	6.60233e12	0.0931366
$590$	0	0
$591$	− 5.25210e13i	− 0.728443i
$592$	0	0
$593$	8.96007e12	0.122191	0.0610953	−	0.998132i	$-0.480541\pi$
$593$	8.96007e12	0.122191	0.0610953	+	0.998132i	$0.480541\pi$
$594$	0	0
$595$	2.49378e14i	3.34405i
$596$	0	0
$597$	1.77709e14	2.34335
$598$	0	0
$599$	3.33910e13i	0.433007i	0.976282	+	0.216504i	$0.0694653\pi$
$599$	3.33910e13i	0.433007i	−0.976282	+	0.216504i	$0.930535\pi$
$600$	0	0
$601$	1.17180e14	1.49445	0.747226	−	0.664570i	$-0.231385\pi$
$601$	1.17180e14	1.49445	0.747226	+	0.664570i	$0.231385\pi$
$602$	0	0
$603$	7.09656e13i	0.890147i
$604$	0	0
$605$	2.44687e14	3.01880
$606$	0	0
$607$	5.38900e13i	0.653981i	0.945028	+	0.326990i	$0.106034\pi$
$607$	5.38900e13i	0.653981i	−0.945028	+	0.326990i	$0.893966\pi$
$608$	0	0
$609$	−1.21815e13	−0.145416
$610$	0	0
$611$	− 1.94730e13i	− 0.228680i
$612$	0	0
$613$	5.59154e13	0.645995	0.322997	−	0.946400i	$-0.395309\pi$
$613$	5.59154e13	0.645995	0.322997	+	0.946400i	$0.395309\pi$
$614$	0	0
$615$	1.82544e14i	2.07487i
$616$	0	0
$617$	7.55299e12	0.0844682	0.0422341	−	0.999108i	$-0.486552\pi$
$617$	7.55299e12	0.0844682	0.0422341	+	0.999108i	$0.486552\pi$
$618$	0	0
$619$	1.25334e14i	1.37917i	0.724206	+	0.689584i	$0.242207\pi$
$619$	1.25334e14i	1.37917i	−0.724206	+	0.689584i	$0.757793\pi$
$620$	0	0
$621$	−4.22651e12	−0.0457640
$622$	0	0
$623$	− 5.05492e12i	− 0.0538610i
$624$	0	0
$625$	2.20266e13	0.230965
$626$	0	0
$627$	3.05582e13i	0.315348i
$628$	0	0
$629$	−1.52723e14	−1.55114
$630$	0	0
$631$	− 1.34112e14i	− 1.34067i	−0.742058	−	0.670335i	$-0.766150\pi$
$631$	− 1.34112e14i	− 1.34067i	0.742058	−	0.670335i	$-0.233850\pi$
$632$	0	0
$633$	8.83505e13	0.869341
$634$	0	0
$635$	− 4.02011e13i	− 0.389376i
$636$	0	0
$637$	2.16036e13	0.205982
$638$	0	0
$639$	2.07863e14i	1.95107i
$640$	0	0
$641$	1.03084e14	0.952576	0.476288	−	0.879289i	$-0.341982\pi$
$641$	1.03084e14	0.952576	0.476288	+	0.879289i	$0.341982\pi$
$642$	0	0
$643$	− 1.57862e14i	− 1.43623i	−0.695927	−	0.718113i	$-0.745006\pi$
$643$	− 1.57862e14i	− 1.43623i	0.695927	−	0.718113i	$-0.254994\pi$
$644$	0	0
$645$	−2.22362e13	−0.199187
$646$	0	0
$647$	− 8.58163e13i	− 0.756917i	−0.925618	−	0.378459i	$-0.876454\pi$
$647$	− 8.58163e13i	− 0.756917i	0.925618	−	0.378459i	$-0.123546\pi$
$648$	0	0
$649$	−1.54746e14	−1.34399
$650$	0	0
$651$	1.67916e14i	1.43610i
$652$	0	0
$653$	6.27358e13	0.528384	0.264192	−	0.964470i	$-0.414895\pi$
$653$	6.27358e13	0.528384	0.264192	+	0.964470i	$0.414895\pi$
$654$	0	0
$655$	1.91840e14i	1.59123i
$656$	0	0
$657$	2.43462e14	1.98886
$658$	0	0
$659$	− 2.22348e13i	− 0.178899i	−0.995991	−	0.0894493i	$-0.971489\pi$
$659$	− 2.22348e13i	− 0.178899i	0.995991	−	0.0894493i	$-0.0285107\pi$
$660$	0	0
$661$	7.77564e13	0.616210	0.308105	−	0.951352i	$-0.400305\pi$
$661$	7.77564e13	0.616210	0.308105	+	0.951352i	$0.400305\pi$
$662$	0	0
$663$	1.41309e14i	1.10307i
$664$	0	0
$665$	3.17099e13	0.243830
$666$	0	0
$667$	− 8.09983e11i	− 0.00613546i
$668$	0	0
$669$	−1.39017e14	−1.03738
$670$	0	0
$671$	− 4.00511e14i	− 2.94443i
$672$	0	0
$673$	−1.91072e14	−1.38396	−0.691979	−	0.721918i	$-0.743261\pi$
$673$	−1.91072e14	−1.38396	−0.691979	+	0.721918i	$0.743261\pi$
$674$	0	0
$675$	− 1.39180e14i	− 0.993252i
$676$	0	0
$677$	2.03836e14	1.43330	0.716650	−	0.697433i	$-0.245674\pi$
$677$	2.03836e14	1.43330	0.716650	+	0.697433i	$0.245674\pi$
$678$	0	0
$679$	2.16184e14i	1.49787i
$680$	0	0
$681$	−3.23283e14	−2.20723
$682$	0	0
$683$	2.03261e14i	1.36757i	0.729681	+	0.683787i	$0.239668\pi$
$683$	2.03261e14i	1.36757i	−0.729681	+	0.683787i	$0.760332\pi$
$684$	0	0
$685$	3.16803e14	2.10057
$686$	0	0
$687$	2.04008e14i	1.33310i
$688$	0	0
$689$	5.36102e13	0.345264
$690$	0	0
$691$	− 7.18947e12i	− 0.0456359i	−0.999740	−	0.0228179i	$-0.992736\pi$
$691$	− 7.18947e12i	− 0.0456359i	0.999740	−	0.0228179i	$-0.00726381\pi$
$692$	0	0
$693$	−4.49995e14	−2.81541
$694$	0	0
$695$	3.47976e14i	2.14598i
$696$	0	0
$697$	2.24057e14	1.36206
$698$	0	0
$699$	− 2.66608e14i	− 1.59767i
$700$	0	0
$701$	1.17495e14	0.694109	0.347054	−	0.937845i	$-0.387182\pi$
$701$	1.17495e14	0.694109	0.347054	+	0.937845i	$0.387182\pi$
$702$	0	0
$703$	1.94196e13i	0.113101i
$704$	0	0
$705$	2.35748e14	1.35364
$706$	0	0
$707$	2.68742e13i	0.152138i
$708$	0	0
$709$	3.20799e13	0.179061	0.0895307	−	0.995984i	$-0.471463\pi$
$709$	3.20799e13	0.179061	0.0895307	+	0.995984i	$0.471463\pi$
$710$	0	0
$711$	− 1.51946e14i	− 0.836258i
$712$	0	0
$713$	−1.11652e13	−0.0605927
$714$	0	0
$715$	− 2.22436e14i	− 1.19035i
$716$	0	0
$717$	−2.21344e14	−1.16808
$718$	0	0
$719$	1.75781e14i	0.914802i	0.889261	+	0.457401i	$0.151220\pi$
$719$	1.75781e14i	0.914802i	−0.889261	+	0.457401i	$0.848780\pi$
$720$	0	0
$721$	−3.89462e14	−1.99889
$722$	0	0
$723$	− 3.11014e13i	− 0.157431i
$724$	0	0
$725$	2.66730e13	0.133163
$726$	0	0
$727$	2.15850e14i	1.06287i	0.847100	+	0.531434i	$0.178347\pi$
$727$	2.15850e14i	1.06287i	−0.847100	+	0.531434i	$0.821653\pi$
$728$	0	0
$729$	3.20558e14	1.55693
$730$	0	0
$731$	2.72931e13i	0.130757i
$732$	0	0
$733$	7.80602e13	0.368901	0.184451	−	0.982842i	$-0.440949\pi$
$733$	7.80602e13	0.368901	0.184451	+	0.982842i	$0.440949\pi$
$734$	0	0
$735$	2.61542e14i	1.21928i
$736$	0	0
$737$	−2.36804e14	−1.08906
$738$	0	0
$739$	− 2.10794e14i	− 0.956394i	−0.878252	−	0.478197i	$-0.841290\pi$
$739$	− 2.10794e14i	− 0.956394i	0.878252	−	0.478197i	$-0.158710\pi$
$740$	0	0
$741$	1.79683e13	0.0804296
$742$	0	0
$743$	3.01822e14i	1.33293i	0.745537	+	0.666464i	$0.232193\pi$
$743$	3.01822e14i	1.33293i	−0.745537	+	0.666464i	$0.767807\pi$
$744$	0	0
$745$	1.80317e14	0.785699
$746$	0	0
$747$	− 4.49571e13i	− 0.193284i
$748$	0	0
$749$	5.49510e11	0.00233113
$750$	0	0
$751$	1.64525e14i	0.688705i	0.938840	+	0.344353i	$0.111902\pi$
$751$	1.64525e14i	0.688705i	−0.938840	+	0.344353i	$0.888098\pi$
$752$	0	0
$753$	1.90241e14	0.785831
$754$	0	0
$755$	− 2.08075e14i	− 0.848175i
$756$	0	0
$757$	−4.65566e14	−1.87285	−0.936423	−	0.350874i	$-0.885884\pi$
$757$	−4.65566e14	−1.87285	−0.936423	+	0.350874i	$0.885884\pi$
$758$	0	0
$759$	− 5.16772e13i	− 0.205159i
$760$	0	0
$761$	−3.79607e14	−1.48734	−0.743671	−	0.668545i	$-0.766917\pi$
$761$	−3.79607e14	−1.48734	−0.743671	+	0.668545i	$0.766917\pi$
$762$	0	0
$763$	− 3.80397e14i	− 1.47101i
$764$	0	0
$765$	−9.90535e14	−3.78062
$766$	0	0
$767$	9.09912e13i	0.342785i
$768$	0	0
$769$	2.33786e14	0.869333	0.434666	−	0.900592i	$-0.356866\pi$
$769$	2.33786e14	0.869333	0.434666	+	0.900592i	$0.356866\pi$
$770$	0	0
$771$	3.27580e14i	1.20239i
$772$	0	0
$773$	2.47200e14	0.895676	0.447838	−	0.894115i	$-0.352194\pi$
$773$	2.47200e14	0.895676	0.447838	+	0.894115i	$0.352194\pi$
$774$	0	0
$775$	− 3.67675e14i	− 1.31509i
$776$	0	0
$777$	−4.93896e14	−1.74393
$778$	0	0
$779$	− 2.84903e13i	− 0.0993138i
$780$	0	0
$781$	−6.93614e14	−2.38706
$782$	0	0
$783$	− 1.32049e13i	− 0.0448669i
$784$	0	0
$785$	−4.51516e14	−1.51469
$786$	0	0
$787$	2.21837e14i	0.734786i	0.930066	+	0.367393i	$0.119750\pi$
$787$	2.21837e14i	0.734786i	−0.930066	+	0.367393i	$0.880250\pi$
$788$	0	0
$789$	1.00355e14	0.328213
$790$	0	0
$791$	− 6.21612e14i	− 2.00741i
$792$	0	0
$793$	−2.35501e14	−0.750978
$794$	0	0
$795$	6.49026e14i	2.04374i
$796$	0	0
$797$	1.57195e14	0.488817	0.244409	−	0.969672i	$-0.421406\pi$
$797$	1.57195e14	0.488817	0.244409	+	0.969672i	$0.421406\pi$
$798$	0	0
$799$	− 2.89361e14i	− 0.888599i
$800$	0	0
$801$	2.00783e13	0.0608925
$802$	0	0
$803$	8.12404e14i	2.43329i
$804$	0	0
$805$	−5.36249e13	−0.158631
$806$	0	0
$807$	7.19125e13i	0.210105i
$808$	0	0
$809$	9.74390e13	0.281184	0.140592	−	0.990068i	$-0.455099\pi$
$809$	9.74390e13	0.281184	0.140592	+	0.990068i	$0.455099\pi$
$810$	0	0
$811$	− 5.40638e14i	− 1.54100i	−0.637440	−	0.770500i	$-0.720007\pi$
$811$	− 5.40638e14i	− 1.54100i	0.637440	−	0.770500i	$-0.279993\pi$
$812$	0	0
$813$	3.58252e14	1.00864
$814$	0	0
$815$	5.75611e13i	0.160081i
$816$	0	0
$817$	3.47048e12	0.00953410
$818$	0	0
$819$	2.64598e14i	0.718071i
$820$	0	0
$821$	5.35880e12	0.0143665	0.00718326	−	0.999974i	$-0.497713\pi$
$821$	5.35880e12	0.0143665	0.00718326	+	0.999974i	$0.497713\pi$
$822$	0	0
$823$	− 1.20214e14i	− 0.318388i	−0.987247	−	0.159194i	$-0.949111\pi$
$823$	− 1.20214e14i	− 0.318388i	0.987247	−	0.159194i	$-0.0508895\pi$
$824$	0	0
$825$	1.70175e15	4.45272
$826$	0	0
$827$	4.40750e14i	1.13937i	0.821863	+	0.569685i	$0.192935\pi$
$827$	4.40750e14i	1.13937i	−0.821863	+	0.569685i	$0.807065\pi$
$828$	0	0
$829$	−8.50403e13	−0.217196	−0.108598	−	0.994086i	$-0.534636\pi$
$829$	−8.50403e13	−0.217196	−0.108598	+	0.994086i	$0.534636\pi$
$830$	0	0
$831$	1.98596e14i	0.501146i
$832$	0	0
$833$	3.21020e14	0.800401
$834$	0	0
$835$	− 9.04654e14i	− 2.22869i
$836$	0	0
$837$	−1.82023e14	−0.443098
$838$	0	0
$839$	3.07345e14i	0.739293i	0.929172	+	0.369647i	$0.120521\pi$
$839$	3.07345e14i	0.739293i	−0.929172	+	0.369647i	$0.879479\pi$
$840$	0	0
$841$	−4.18177e14	−0.993985
$842$	0	0
$843$	2.76992e14i	0.650623i
$844$	0	0
$845$	5.79315e14	1.34472
$846$	0	0
$847$	− 9.71257e14i	− 2.22801i
$848$	0	0
$849$	−5.70859e14	−1.29417
$850$	0	0
$851$	− 3.28407e13i	− 0.0735808i
$852$	0	0
$853$	7.42460e14	1.64410	0.822049	−	0.569416i	$-0.192831\pi$
$853$	7.42460e14	1.64410	0.822049	+	0.569416i	$0.192831\pi$
$854$	0	0
$855$	1.25953e14i	0.275662i
$856$	0	0
$857$	7.73055e14	1.67227	0.836135	−	0.548524i	$-0.184810\pi$
$857$	7.73055e14	1.67227	0.836135	+	0.548524i	$0.184810\pi$
$858$	0	0
$859$	2.47371e14i	0.528913i	0.964398	+	0.264456i	$0.0851925\pi$
$859$	2.47371e14i	0.528913i	−0.964398	+	0.264456i	$0.914808\pi$
$860$	0	0
$861$	7.24587e14	1.53135
$862$	0	0
$863$	6.10391e14i	1.27513i	0.770397	+	0.637565i	$0.220058\pi$
$863$	6.10391e14i	1.27513i	−0.770397	+	0.637565i	$0.779942\pi$
$864$	0	0
$865$	1.23411e15	2.54844
$866$	0	0
$867$	1.34477e15i	2.74506i
$868$	0	0
$869$	5.07025e14	1.02313
$870$	0	0
$871$	1.39241e14i	0.277765i
$872$	0	0
$873$	−8.58686e14	−1.69341
$874$	0	0
$875$	− 7.37384e14i	− 1.43765i
$876$	0	0
$877$	6.53496e14	1.25964	0.629818	−	0.776742i	$-0.283129\pi$
$877$	6.53496e14	1.25964	0.629818	+	0.776742i	$0.283129\pi$
$878$	0	0
$879$	− 8.89299e14i	− 1.69474i
$880$	0	0
$881$	−3.47612e14	−0.654961	−0.327481	−	0.944858i	$-0.606200\pi$
$881$	−3.47612e14	−0.654961	−0.327481	+	0.944858i	$0.606200\pi$
$882$	0	0
$883$	− 7.21047e14i	− 1.34326i	−0.740887	−	0.671630i	$-0.765595\pi$
$883$	− 7.21047e14i	− 1.34326i	0.740887	−	0.671630i	$-0.234405\pi$
$884$	0	0
$885$	−1.10157e15	−2.02907
$886$	0	0
$887$	6.76078e13i	0.123134i	0.998103	+	0.0615671i	$0.0196098\pi$
$887$	6.76078e13i	0.123134i	−0.998103	+	0.0615671i	$0.980390\pi$
$888$	0	0
$889$	−1.59574e14	−0.287378
$890$	0	0
$891$	4.57112e14i	0.814018i
$892$	0	0
$893$	−3.67940e13	−0.0647918
$894$	0	0
$895$	2.01369e14i	0.350653i
$896$	0	0
$897$	−3.03863e13	−0.0523258
$898$	0	0
$899$	− 3.48836e13i	− 0.0594050i
$900$	0	0
$901$	7.96625e14	1.34162
$902$	0	0
$903$	8.82640e13i	0.147009i
$904$	0	0
$905$	−1.75172e15	−2.88550
$906$	0	0
$907$	4.98921e14i	0.812822i	0.913690	+	0.406411i	$0.133220\pi$
$907$	4.98921e14i	0.812822i	−0.913690	+	0.406411i	$0.866780\pi$
$908$	0	0
$909$	−1.06745e14	−0.172000
$910$	0	0
$911$	5.23086e13i	0.0833646i	0.999131	+	0.0416823i	$0.0132717\pi$
$911$	5.23086e13i	0.0833646i	−0.999131	+	0.0416823i	$0.986728\pi$
$912$	0	0
$913$	1.50017e14	0.236475
$914$	0	0
$915$	− 2.85107e15i	− 4.44531i
$916$	0	0
$917$	7.61489e14	1.17440
$918$	0	0
$919$	− 5.15980e14i	− 0.787146i	−0.919293	−	0.393573i	$-0.871239\pi$
$919$	− 5.15980e14i	− 0.787146i	0.919293	−	0.393573i	$-0.128761\pi$
$920$	0	0
$921$	−1.02175e15	−1.54186
$922$	0	0
$923$	4.07847e14i	0.608819i
$924$	0	0
$925$	1.08145e15	1.59698
$926$	0	0
$927$	− 1.54695e15i	− 2.25985i
$928$	0	0
$929$	−3.72498e14	−0.538326	−0.269163	−	0.963095i	$-0.586747\pi$
$929$	−3.72498e14	−0.538326	−0.269163	+	0.963095i	$0.586747\pi$
$930$	0	0
$931$	− 4.08197e13i	− 0.0583609i
$932$	0	0
$933$	−1.76329e15	−2.49411
$934$	0	0
$935$	− 3.30530e15i	− 4.62544i
$936$	0	0
$937$	8.69584e14	1.20396	0.601982	−	0.798509i	$-0.294378\pi$
$937$	8.69584e14	1.20396	0.601982	+	0.798509i	$0.294378\pi$
$938$	0	0
$939$	− 6.66661e14i	− 0.913224i
$940$	0	0
$941$	−1.12583e15	−1.52589	−0.762947	−	0.646461i	$-0.776248\pi$
$941$	−1.12583e15	−1.52589	−0.762947	+	0.646461i	$0.776248\pi$
$942$	0	0
$943$	4.81801e13i	0.0646114i
$944$	0	0
$945$	−8.74230e14	−1.16002
$946$	0	0
$947$	1.62162e14i	0.212911i	0.994317	+	0.106456i	$0.0339502\pi$
$947$	1.62162e14i	0.212911i	−0.994317	+	0.106456i	$0.966050\pi$
$948$	0	0
$949$	4.77696e14	0.620612
$950$	0	0
$951$	− 2.15655e14i	− 0.277241i
$952$	0	0
$953$	1.06606e15	1.35618	0.678091	−	0.734978i	$-0.262807\pi$
$953$	1.06606e15	1.35618	0.678091	+	0.734978i	$0.262807\pi$
$954$	0	0
$955$	− 1.36443e15i	− 1.71765i
$956$	0	0
$957$	1.61455e14	0.201137
$958$	0	0
$959$	− 1.25751e15i	− 1.55032i
$960$	0	0
$961$	3.38775e14	0.413327
$962$	0	0
$963$	2.18267e12i	0.00263546i
$964$	0	0
$965$	−2.31752e14	−0.276941
$966$	0	0
$967$	1.09471e15i	1.29469i	0.762198	+	0.647344i	$0.224120\pi$
$967$	1.09471e15i	1.29469i	−0.762198	+	0.647344i	$0.775880\pi$
$968$	0	0
$969$	2.67001e14	0.312532
$970$	0	0
$971$	3.03445e14i	0.351547i	0.984431	+	0.175773i	$0.0562426\pi$
$971$	3.03445e14i	0.351547i	−0.984431	+	0.175773i	$0.943757\pi$
$972$	0	0
$973$	1.38125e15	1.58383
$974$	0	0
$975$	− 1.00063e15i	− 1.13567i
$976$	0	0
$977$	9.84333e14	1.10578	0.552890	−	0.833254i	$-0.313525\pi$
$977$	9.84333e14	1.10578	0.552890	+	0.833254i	$0.313525\pi$
$978$	0	0
$979$	6.69989e13i	0.0744996i
$980$	0	0
$981$	1.51095e15	1.66305
$982$	0	0
$983$	1.43872e15i	1.56751i	0.621071	+	0.783755i	$0.286698\pi$
$983$	1.43872e15i	1.56751i	−0.621071	+	0.783755i	$0.713302\pi$
$984$	0	0
$985$	7.22359e14	0.779062
$986$	0	0
$987$	− 9.35775e14i	− 0.999046i
$988$	0	0
$989$	−5.86895e12	−0.00620268
$990$	0	0
$991$	1.59630e15i	1.67011i	0.550164	+	0.835057i	$0.314565\pi$
$991$	1.59630e15i	1.67011i	−0.550164	+	0.835057i	$0.685435\pi$
$992$	0	0
$993$	1.76736e14	0.183054
$994$	0	0
$995$	2.44416e15i	2.50619i
$996$	0	0
$997$	−1.45534e15	−1.47737	−0.738684	−	0.674051i	$-0.764553\pi$
$997$	−1.45534e15	−1.47737	−0.738684	+	0.674051i	$0.764553\pi$
$998$	0	0
$999$	− 5.35391e14i	− 0.538076i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.11.c.l.255.11		12
4.3	odd	2	inner	256.11.c.l.255.1		12
8.3	odd	2	inner	256.11.c.l.255.12		12
8.5	even	2	inner	256.11.c.l.255.2		12
16.3	odd	4		64.11.d.b.31.12	yes	12
16.5	even	4		64.11.d.b.31.11	yes	12
16.11	odd	4		64.11.d.b.31.1	✓	12
16.13	even	4		64.11.d.b.31.2	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.11.d.b.31.1	✓	12	16.11	odd	4
64.11.d.b.31.2	yes	12	16.13	even	4
64.11.d.b.31.11	yes	12	16.5	even	4
64.11.d.b.31.12	yes	12	16.3	odd	4
256.11.c.l.255.1		12	4.3	odd	2	inner
256.11.c.l.255.2		12	8.5	even	2	inner
256.11.c.l.255.11		12	1.1	even	1	trivial
256.11.c.l.255.12		12	8.3	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q+374.516i q^{3} -5150.99 q^{5} +20446.3i q^{7} -81213.1 q^{9} +O(q^{10})\)	\(q+374.516i q^{3} -5150.99 q^{5} +20446.3i q^{7} -81213.1 q^{9} -270999. i q^{11} -159348. q^{13} -1.92913e6i q^{15} -2.36784e6 q^{17} +301086. i q^{19} -7.65745e6 q^{21} -509168. i q^{23} +1.67671e7 q^{25} -8.30081e6i q^{27} +1.59080e6 q^{29} -2.19284e7i q^{31} +1.01493e8 q^{33} -1.05319e8i q^{35} +6.44987e7 q^{37} -5.96783e7i q^{39} -9.46251e7 q^{41} -1.15266e7i q^{43} +4.18328e8 q^{45} +1.22204e8i q^{47} -1.35575e8 q^{49} -8.86795e8i q^{51} -3.36435e8 q^{53} +1.39591e9i q^{55} -1.12761e8 q^{57} -5.71022e8i q^{59} +1.47791e9 q^{61} -1.66051e9i q^{63} +8.20800e8 q^{65} -8.73820e8i q^{67} +1.90692e8 q^{69} -2.55947e9i q^{71} -2.99781e9 q^{73} +6.27954e9i q^{75} +5.54092e9 q^{77} +1.87095e9i q^{79} -1.68677e9 q^{81} +5.53569e8i q^{83} +1.21967e10 q^{85} +5.95779e8i q^{87} -2.47230e8 q^{89} -3.25807e9i q^{91} +8.21252e9 q^{93} -1.55089e9i q^{95} +1.05732e10 q^{97} +2.20087e10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 375204 q^{9}+O(q^{10}) \) 12 * q - 375204 * q^9	\( 12 q - 375204 q^{9} - 1639752 q^{17} + 28703892 q^{25} + 358214544 q^{33} - 496702488 q^{41} + 1385670732 q^{49} - 5733526800 q^{57} + 8895346752 q^{65} - 12161297592 q^{73} - 7708157604 q^{81} + 5871785160 q^{89} + 30465806904 q^{97}+O(q^{100}) \) 12 * q - 375204 * q^9 - 1639752 * q^17 + 28703892 * q^25 + 358214544 * q^33 - 496702488 * q^41 + 1385670732 * q^49 - 5733526800 * q^57 + 8895346752 * q^65 - 12161297592 * q^73 - 7708157604 * q^81 + 5871785160 * q^89 + 30465806904 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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