LMFDB - Embedded newform 256.11.c.i.255.6

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 15338x^{6} + 68362851x^{4} - 73193774690x^{2} + 22797234876025 $ x^8 - 15338x^6 + 68362851x^4 - 73193774690*x^2 + 22797234876025
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{40}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 64)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			255.6
Root			$-26.1371 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	256.255
Dual form			256.11.c.i.255.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+104.549i q^{3} +4603.82 q^{5} +23924.8i q^{7} +48118.6 q^{9} +O(q^{10})$	$q+104.549i q^{3} +4603.82 q^{5} +23924.8i q^{7} +48118.6 q^{9} -216099. i q^{11} -641501. q^{13} +481323. i q^{15} +924243. q^{17} -2.63967e6i q^{19} -2.50130e6 q^{21} +2.37147e6i q^{23} +1.14296e7 q^{25} +1.12042e7i q^{27} -6.16297e6 q^{29} +2.62621e7i q^{31} +2.25929e7 q^{33} +1.10146e8i q^{35} +2.49705e7 q^{37} -6.70680e7i q^{39} +2.23974e8 q^{41} +2.18054e8i q^{43} +2.21529e8 q^{45} +1.78203e8i q^{47} -2.89921e8 q^{49} +9.66283e7i q^{51} +1.65786e8 q^{53} -9.94882e8i q^{55} +2.75974e8 q^{57} +2.04996e8i q^{59} -1.00749e9 q^{61} +1.15123e9i q^{63} -2.95336e9 q^{65} +1.16117e9i q^{67} -2.47934e8 q^{69} -1.04167e8i q^{71} +5.20565e8 q^{73} +1.19494e9i q^{75} +5.17013e9 q^{77} +4.06922e9i q^{79} +1.66997e9 q^{81} -4.48866e9i q^{83} +4.25505e9 q^{85} -6.44330e8i q^{87} +4.01435e9 q^{89} -1.53478e10i q^{91} -2.74566e9 q^{93} -1.21526e10i q^{95} +7.96410e9 q^{97} -1.03984e10i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 18456 q^{9}+O(q^{10}) $ 8 * q - 18456 * q^9	$ 8 q - 18456 q^{9} - 1077552 q^{17} + 14789560 q^{25} - 194826912 q^{33} + 826044912 q^{41} - 2838950520 q^{49} + 6691637664 q^{57} - 10336279680 q^{65} + 13655425840 q^{73} - 8599581720 q^{81} - 14080263120 q^{89} + 47348700112 q^{97}+O(q^{100}) $ 8 * q - 18456 * q^9 - 1077552 * q^17 + 14789560 * q^25 - 194826912 * q^33 + 826044912 * q^41 - 2838950520 * q^49 + 6691637664 * q^57 - 10336279680 * q^65 + 13655425840 * q^73 - 8599581720 * q^81 - 14080263120 * q^89 + 47348700112 * 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
$3$	104.549i	0.430241i	0.976588	+	0.215121i	$0.0690144\pi$
$3$	104.549i	0.430241i	−0.976588	+	0.215121i	$0.930986\pi$
$4$	0	0
$5$	4603.82	1.47322	0.736612	−	0.676316i	$-0.236425\pi$
$5$	4603.82	1.47322	0.736612	+	0.676316i	$0.236425\pi$
$6$	0	0
$7$	23924.8i	1.42350i	0.702432	+	0.711751i	$0.252098\pi$
$7$	23924.8i	1.42350i	−0.702432	+	0.711751i	$0.747902\pi$
$8$	0	0
$9$	48118.6	0.814893
$10$	0	0
$11$	− 216099.i	− 1.34181i	−0.741545	−	0.670903i	$-0.765907\pi$
$11$	− 216099.i	− 1.34181i	0.741545	−	0.670903i	$-0.234093\pi$
$12$	0	0
$13$	−641501.	−1.72775	−0.863874	−	0.503707i	$-0.831969\pi$
$13$	−641501.	−1.72775	−0.863874	+	0.503707i	$0.831969\pi$
$14$	0	0
$15$	481323.i	0.633841i
$16$	0	0
$17$	924243.	0.650941	0.325471	−	0.945552i	$-0.394477\pi$
$17$	924243.	0.650941	0.325471	+	0.945552i	$0.394477\pi$
$18$	0	0
$19$	− 2.63967e6i	− 1.06606i	−0.846096	−	0.533031i	$-0.821053\pi$
$19$	− 2.63967e6i	− 1.06606i	0.846096	−	0.533031i	$-0.178947\pi$
$20$	0	0
$21$	−2.50130e6	−0.612449
$22$	0	0
$23$	2.37147e6i	0.368450i	0.982884	+	0.184225i	$0.0589776\pi$
$23$	2.37147e6i	0.368450i	−0.982884	+	0.184225i	$0.941022\pi$
$24$	0	0
$25$	1.14296e7	1.17039
$26$	0	0
$27$	1.12042e7i	0.780841i
$28$	0	0
$29$	−6.16297e6	−0.300469	−0.150235	−	0.988650i	$-0.548003\pi$
$29$	−6.16297e6	−0.300469	−0.150235	+	0.988650i	$0.548003\pi$
$30$	0	0
$31$	2.62621e7i	0.917319i	0.888612	+	0.458660i	$0.151670\pi$
$31$	2.62621e7i	0.917319i	−0.888612	+	0.458660i	$0.848330\pi$
$32$	0	0
$33$	2.25929e7	0.577300
$34$	0	0
$35$	1.10146e8i	2.09714i
$36$	0	0
$37$	2.49705e7	0.360096	0.180048	−	0.983658i	$-0.442375\pi$
$37$	2.49705e7	0.360096	0.180048	+	0.983658i	$0.442375\pi$
$38$	0	0
$39$	− 6.70680e7i	− 0.743348i
$40$	0	0
$41$	2.23974e8	1.93321	0.966606	−	0.256269i	$-0.0824933\pi$
$41$	2.23974e8	1.93321	0.966606	+	0.256269i	$0.0824933\pi$
$42$	0	0
$43$	2.18054e8i	1.48328i	0.670800	+	0.741638i	$0.265951\pi$
$43$	2.18054e8i	1.48328i	−0.670800	+	0.741638i	$0.734049\pi$
$44$	0	0
$45$	2.21529e8	1.20052
$46$	0	0
$47$	1.78203e8i	0.777007i	0.921447	+	0.388503i	$0.127008\pi$
$47$	1.78203e8i	0.777007i	−0.921447	+	0.388503i	$0.872992\pi$
$48$	0	0
$49$	−2.89921e8	−1.02636
$50$	0	0
$51$	9.66283e7i	0.280062i
$52$	0	0
$53$	1.65786e8	0.396432	0.198216	−	0.980158i	$-0.436485\pi$
$53$	1.65786e8	0.396432	0.198216	+	0.980158i	$0.436485\pi$
$54$	0	0
$55$	− 9.94882e8i	− 1.97678i
$56$	0	0
$57$	2.75974e8	0.458664
$58$	0	0
$59$	2.04996e8i	0.286738i	0.989669	+	0.143369i	$0.0457935\pi$
$59$	2.04996e8i	0.286738i	−0.989669	+	0.143369i	$0.954206\pi$
$60$	0	0
$61$	−1.00749e9	−1.19286	−0.596431	−	0.802664i	$-0.703415\pi$
$61$	−1.00749e9	−1.19286	−0.596431	+	0.802664i	$0.703415\pi$
$62$	0	0
$63$	1.15123e9i	1.16000i
$64$	0	0
$65$	−2.95336e9	−2.54536
$66$	0	0
$67$	1.16117e9i	0.860045i	0.902818	+	0.430023i	$0.141494\pi$
$67$	1.16117e9i	0.860045i	−0.902818	+	0.430023i	$0.858506\pi$
$68$	0	0
$69$	−2.47934e8	−0.158522
$70$	0	0
$71$	− 1.04167e8i	− 0.0577347i	−0.999583	−	0.0288673i	$-0.990810\pi$
$71$	− 1.04167e8i	− 0.0577347i	0.999583	−	0.0288673i	$-0.00919003\pi$
$72$	0	0
$73$	5.20565e8	0.251108	0.125554	−	0.992087i	$-0.459929\pi$
$73$	5.20565e8	0.251108	0.125554	+	0.992087i	$0.459929\pi$
$74$	0	0
$75$	1.19494e9i	0.503548i
$76$	0	0
$77$	5.17013e9	1.91006
$78$	0	0
$79$	4.06922e9i	1.32244i	0.750193	+	0.661219i	$0.229961\pi$
$79$	4.06922e9i	1.32244i	−0.750193	+	0.661219i	$0.770039\pi$
$80$	0	0
$81$	1.66997e9	0.478943
$82$	0	0
$83$	− 4.48866e9i	− 1.13953i	−0.821808	−	0.569765i	$-0.807034\pi$
$83$	− 4.48866e9i	− 1.13953i	0.821808	−	0.569765i	$-0.192966\pi$
$84$	0	0
$85$	4.25505e9	0.958982
$86$	0	0
$87$	− 6.44330e8i	− 0.129274i
$88$	0	0
$89$	4.01435e9	0.718895	0.359448	−	0.933165i	$-0.382965\pi$
$89$	4.01435e9	0.718895	0.359448	+	0.933165i	$0.382965\pi$
$90$	0	0
$91$	− 1.53478e10i	− 2.45945i
$92$	0	0
$93$	−2.74566e9	−0.394668
$94$	0	0
$95$	− 1.21526e10i	− 1.57055i
$96$	0	0
$97$	7.96410e9	0.927424	0.463712	−	0.885986i	$-0.346517\pi$
$97$	7.96410e9	0.927424	0.463712	+	0.885986i	$0.346517\pi$
$98$	0	0
$99$	− 1.03984e10i	− 1.09343i
$100$	0	0
$101$	4.06315e9	0.386595	0.193298	−	0.981140i	$-0.438082\pi$
$101$	4.06315e9	0.386595	0.193298	+	0.981140i	$0.438082\pi$
$102$	0	0
$103$	1.19709e10i	1.03262i	0.856402	+	0.516310i	$0.172695\pi$
$103$	1.19709e10i	1.03262i	−0.856402	+	0.516310i	$0.827305\pi$
$104$	0	0
$105$	−1.15156e10	−0.902274
$106$	0	0
$107$	1.26411e10i	0.901296i	0.892702	+	0.450648i	$0.148807\pi$
$107$	1.26411e10i	0.901296i	−0.892702	+	0.450648i	$0.851193\pi$
$108$	0	0
$109$	−2.92461e9	−0.190079	−0.0950397	−	0.995473i	$-0.530298\pi$
$109$	−2.92461e9	−0.190079	−0.0950397	+	0.995473i	$0.530298\pi$
$110$	0	0
$111$	2.61063e9i	0.154928i
$112$	0	0
$113$	2.62948e10	1.42717	0.713587	−	0.700566i	$-0.247069\pi$
$113$	2.62948e10	1.42717	0.713587	+	0.700566i	$0.247069\pi$
$114$	0	0
$115$	1.09178e10i	0.542810i
$116$	0	0
$117$	−3.08681e10	−1.40793
$118$	0	0
$119$	2.21123e10i	0.926616i
$120$	0	0
$121$	−2.07614e10	−0.800443
$122$	0	0
$123$	2.34162e10i	0.831747i
$124$	0	0
$125$	7.66045e9	0.251018
$126$	0	0
$127$	− 4.01237e10i	− 1.21446i	−0.794527	−	0.607228i	$-0.792281\pi$
$127$	− 4.01237e10i	− 1.21446i	0.794527	−	0.607228i	$-0.207719\pi$
$128$	0	0
$129$	−2.27973e10	−0.638167
$130$	0	0
$131$	5.94283e10i	1.54041i	0.637795	+	0.770206i	$0.279847\pi$
$131$	5.94283e10i	1.54041i	−0.637795	+	0.770206i	$0.720153\pi$
$132$	0	0
$133$	6.31537e10	1.51754
$134$	0	0
$135$	5.15822e10i	1.15035i
$136$	0	0
$137$	−4.33980e10	−0.899221	−0.449610	−	0.893225i	$-0.648437\pi$
$137$	−4.33980e10	−0.899221	−0.449610	+	0.893225i	$0.648437\pi$
$138$	0	0
$139$	− 2.54752e10i	− 0.490958i	−0.969402	−	0.245479i	$-0.921055\pi$
$139$	− 2.54752e10i	− 0.490958i	0.969402	−	0.245479i	$-0.0789452\pi$
$140$	0	0
$141$	−1.86308e10	−0.334300
$142$	0	0
$143$	1.38628e11i	2.31830i
$144$	0	0
$145$	−2.83732e10	−0.442658
$146$	0	0
$147$	− 3.03108e10i	− 0.441581i
$148$	0	0
$149$	−1.15719e11	−1.57570	−0.787850	−	0.615867i	$-0.788806\pi$
$149$	−1.15719e11	−1.57570	−0.787850	+	0.615867i	$0.788806\pi$
$150$	0	0
$151$	− 2.64000e10i	− 0.336294i	−0.985762	−	0.168147i	$-0.946222\pi$
$151$	− 2.64000e10i	− 0.336294i	0.985762	−	0.168147i	$-0.0537784\pi$
$152$	0	0
$153$	4.44733e10	0.530447
$154$	0	0
$155$	1.20906e11i	1.35142i
$156$	0	0
$157$	−7.49235e9	−0.0785452	−0.0392726	−	0.999229i	$-0.512504\pi$
$157$	−7.49235e9	−0.0785452	−0.0392726	+	0.999229i	$0.512504\pi$
$158$	0	0
$159$	1.73327e10i	0.170561i
$160$	0	0
$161$	−5.67370e10	−0.524490
$162$	0	0
$163$	− 6.38082e10i	− 0.554547i	−0.960791	−	0.277274i	$-0.910569\pi$
$163$	− 6.38082e10i	− 0.554547i	0.960791	−	0.277274i	$-0.0894309\pi$
$164$	0	0
$165$	1.04014e11	0.850492
$166$	0	0
$167$	2.81872e9i	0.0217005i	0.999941	+	0.0108503i	$0.00345381\pi$
$167$	2.81872e9i	0.0217005i	−0.999941	+	0.0108503i	$0.996546\pi$
$168$	0	0
$169$	2.73665e11	1.98512
$170$	0	0
$171$	− 1.27017e11i	− 0.868726i
$172$	0	0
$173$	−6.05906e10	−0.390998	−0.195499	−	0.980704i	$-0.562633\pi$
$173$	−6.05906e10	−0.390998	−0.195499	+	0.980704i	$0.562633\pi$
$174$	0	0
$175$	2.73450e11i	1.66605i
$176$	0	0
$177$	−2.14320e10	−0.123366
$178$	0	0
$179$	1.33712e11i	0.727623i	0.931473	+	0.363812i	$0.118525\pi$
$179$	1.33712e11i	0.727623i	−0.931473	+	0.363812i	$0.881475\pi$
$180$	0	0
$181$	−2.46597e11	−1.26939	−0.634695	−	0.772763i	$-0.718874\pi$
$181$	−2.46597e11	−1.26939	−0.634695	+	0.772763i	$0.718874\pi$
$182$	0	0
$183$	− 1.05331e11i	− 0.513219i
$184$	0	0
$185$	1.14960e11	0.530502
$186$	0	0
$187$	− 1.99728e11i	− 0.873437i
$188$	0	0
$189$	−2.68059e11	−1.11153
$190$	0	0
$191$	3.23796e11i	1.27381i	0.770942	+	0.636905i	$0.219786\pi$
$191$	3.23796e11i	1.27381i	−0.770942	+	0.636905i	$0.780214\pi$
$192$	0	0
$193$	1.68797e11	0.630344	0.315172	−	0.949035i	$-0.397938\pi$
$193$	1.68797e11	0.630344	0.315172	+	0.949035i	$0.397938\pi$
$194$	0	0
$195$	− 3.08769e11i	− 1.09512i
$196$	0	0
$197$	2.54348e11	0.857229	0.428615	−	0.903487i	$-0.359002\pi$
$197$	2.54348e11	0.857229	0.428615	+	0.903487i	$0.359002\pi$
$198$	0	0
$199$	4.94164e11i	1.58345i	0.610875	+	0.791727i	$0.290818\pi$
$199$	4.94164e11i	1.58345i	−0.610875	+	0.791727i	$0.709182\pi$
$200$	0	0
$201$	−1.21399e11	−0.370027
$202$	0	0
$203$	− 1.47448e11i	− 0.427719i
$204$	0	0
$205$	1.03114e12	2.84805
$206$	0	0
$207$	1.14112e11i	0.300248i
$208$	0	0
$209$	−5.70432e11	−1.43045
$210$	0	0
$211$	− 6.05486e11i	− 1.44774i	−0.689935	−	0.723872i	$-0.742361\pi$
$211$	− 6.05486e11i	− 1.44774i	0.689935	−	0.723872i	$-0.257639\pi$
$212$	0	0
$213$	1.08905e10	0.0248398
$214$	0	0
$215$	1.00388e12i	2.18520i
$216$	0	0
$217$	−6.28315e11	−1.30581
$218$	0	0
$219$	5.44244e10i	0.108037i
$220$	0	0
$221$	−5.92903e11	−1.12466
$222$	0	0
$223$	6.86349e11i	1.24457i	0.782789	+	0.622287i	$0.213796\pi$
$223$	6.86349e11i	1.24457i	−0.782789	+	0.622287i	$0.786204\pi$
$224$	0	0
$225$	5.49974e11	0.953739
$226$	0	0
$227$	1.08236e12i	1.79573i	0.440267	+	0.897867i	$0.354884\pi$
$227$	1.08236e12i	1.79573i	−0.440267	+	0.897867i	$0.645116\pi$
$228$	0	0
$229$	9.34449e11	1.48381	0.741905	−	0.670505i	$-0.233923\pi$
$229$	9.34449e11	1.48381	0.741905	+	0.670505i	$0.233923\pi$
$230$	0	0
$231$	5.40530e11i	0.821788i
$232$	0	0
$233$	−9.49512e11	−1.38268	−0.691339	−	0.722531i	$-0.742979\pi$
$233$	−9.49512e11	−1.38268	−0.691339	+	0.722531i	$0.742979\pi$
$234$	0	0
$235$	8.20413e11i	1.14470i
$236$	0	0
$237$	−4.25431e11	−0.568968
$238$	0	0
$239$	− 5.24272e11i	− 0.672306i	−0.941807	−	0.336153i	$-0.890874\pi$
$239$	− 5.24272e11i	− 0.672306i	0.941807	−	0.336153i	$-0.109126\pi$
$240$	0	0
$241$	9.10526e11	1.11997	0.559986	−	0.828502i	$-0.310806\pi$
$241$	9.10526e11	1.11997	0.559986	+	0.828502i	$0.310806\pi$
$242$	0	0
$243$	8.36191e11i	0.986902i
$244$	0	0
$245$	−1.33474e12	−1.51205
$246$	0	0
$247$	1.69335e12i	1.84189i
$248$	0	0
$249$	4.69283e11	0.490273
$250$	0	0
$251$	7.59962e11i	0.762822i	0.924405	+	0.381411i	$0.124562\pi$
$251$	7.59962e11i	0.762822i	−0.924405	+	0.381411i	$0.875438\pi$
$252$	0	0
$253$	5.12473e11	0.494389
$254$	0	0
$255$	4.44860e11i	0.412593i
$256$	0	0
$257$	−3.75645e11	−0.335052	−0.167526	−	0.985868i	$-0.553578\pi$
$257$	−3.75645e11	−0.335052	−0.167526	+	0.985868i	$0.553578\pi$
$258$	0	0
$259$	5.97414e11i	0.512597i
$260$	0	0
$261$	−2.96554e11	−0.244850
$262$	0	0
$263$	8.37394e10i	0.0665505i	0.999446	+	0.0332752i	$0.0105938\pi$
$263$	8.37394e10i	0.0665505i	−0.999446	+	0.0332752i	$0.989406\pi$
$264$	0	0
$265$	7.63249e11	0.584033
$266$	0	0
$267$	4.19695e11i	0.309298i
$268$	0	0
$269$	6.03960e11	0.428792	0.214396	−	0.976747i	$-0.431222\pi$
$269$	6.03960e11	0.428792	0.214396	+	0.976747i	$0.431222\pi$
$270$	0	0
$271$	9.82561e11i	0.672222i	0.941822	+	0.336111i	$0.109112\pi$
$271$	9.82561e11i	0.672222i	−0.941822	+	0.336111i	$0.890888\pi$
$272$	0	0
$273$	1.60459e12	1.05816
$274$	0	0
$275$	− 2.46992e12i	− 1.57043i
$276$	0	0
$277$	1.37282e12	0.841809	0.420904	−	0.907105i	$-0.361713\pi$
$277$	1.37282e12	0.841809	0.420904	+	0.907105i	$0.361713\pi$
$278$	0	0
$279$	1.26369e12i	0.747517i
$280$	0	0
$281$	1.20339e11	0.0686869	0.0343435	−	0.999410i	$-0.489066\pi$
$281$	1.20339e11	0.0686869	0.0343435	+	0.999410i	$0.489066\pi$
$282$	0	0
$283$	− 3.18064e12i	− 1.75220i	−0.482133	−	0.876098i	$-0.660138\pi$
$283$	− 3.18064e12i	− 1.75220i	0.482133	−	0.876098i	$-0.339862\pi$
$284$	0	0
$285$	1.27054e12	0.675714
$286$	0	0
$287$	5.35854e12i	2.75193i
$288$	0	0
$289$	−1.16177e12	−0.576276
$290$	0	0
$291$	8.32636e11i	0.399016i
$292$	0	0
$293$	1.06170e12	0.491660	0.245830	−	0.969313i	$-0.420939\pi$
$293$	1.06170e12	0.491660	0.245830	+	0.969313i	$0.420939\pi$
$294$	0	0
$295$	9.43764e11i	0.422429i
$296$	0	0
$297$	2.42122e12	1.04774
$298$	0	0
$299$	− 1.52130e12i	− 0.636590i
$300$	0	0
$301$	−5.21690e12	−2.11145
$302$	0	0
$303$	4.24797e11i	0.166329i
$304$	0	0
$305$	−4.63829e12	−1.75735
$306$	0	0
$307$	− 1.59620e12i	− 0.585324i	−0.956216	−	0.292662i	$-0.905459\pi$
$307$	− 1.59620e12i	− 0.585324i	0.956216	−	0.292662i	$-0.0945410\pi$
$308$	0	0
$309$	−1.25154e12	−0.444276
$310$	0	0
$311$	− 2.26639e12i	− 0.778992i	−0.921028	−	0.389496i	$-0.872649\pi$
$311$	− 2.26639e12i	− 0.778992i	0.921028	−	0.389496i	$-0.127351\pi$
$312$	0	0
$313$	2.06372e12	0.686955	0.343477	−	0.939161i	$-0.388395\pi$
$313$	2.06372e12	0.686955	0.343477	+	0.939161i	$0.388395\pi$
$314$	0	0
$315$	5.30005e12i	1.70894i
$316$	0	0
$317$	1.88686e12	0.589444	0.294722	−	0.955583i	$-0.404773\pi$
$317$	1.88686e12	0.589444	0.294722	+	0.955583i	$0.404773\pi$
$318$	0	0
$319$	1.33181e12i	0.403172i
$320$	0	0
$321$	−1.32161e12	−0.387775
$322$	0	0
$323$	− 2.43970e12i	− 0.693944i
$324$	0	0
$325$	−7.33207e12	−2.02213
$326$	0	0
$327$	− 3.05764e11i	− 0.0817800i
$328$	0	0
$329$	−4.26346e12	−1.10607
$330$	0	0
$331$	− 5.19423e12i	− 1.30732i	−0.756789	−	0.653659i	$-0.773233\pi$
$331$	− 5.19423e12i	− 1.30732i	0.756789	−	0.653659i	$-0.226767\pi$
$332$	0	0
$333$	1.20154e12	0.293440
$334$	0	0
$335$	5.34581e12i	1.26704i
$336$	0	0
$337$	1.18939e12	0.273637	0.136819	−	0.990596i	$-0.456312\pi$
$337$	1.18939e12	0.273637	0.136819	+	0.990596i	$0.456312\pi$
$338$	0	0
$339$	2.74908e12i	0.614029i
$340$	0	0
$341$	5.67521e12	1.23086
$342$	0	0
$343$	− 1.78130e11i	− 0.0375203i
$344$	0	0
$345$	−1.14144e12	−0.233539
$346$	0	0
$347$	− 7.17866e11i	− 0.142691i	−0.997452	−	0.0713454i	$-0.977271\pi$
$347$	− 7.17866e11i	− 0.142691i	0.997452	−	0.0713454i	$-0.0227293\pi$
$348$	0	0
$349$	−4.23235e12	−0.817438	−0.408719	−	0.912660i	$-0.634024\pi$
$349$	−4.23235e12	−0.817438	−0.408719	+	0.912660i	$0.634024\pi$
$350$	0	0
$351$	− 7.18752e12i	− 1.34910i
$352$	0	0
$353$	−6.41088e12	−1.16962	−0.584810	−	0.811171i	$-0.698831\pi$
$353$	−6.41088e12	−1.16962	−0.584810	+	0.811171i	$0.698831\pi$
$354$	0	0
$355$	− 4.79564e11i	− 0.0850560i
$356$	0	0
$357$	−2.31181e12	−0.398668
$358$	0	0
$359$	1.39654e12i	0.234196i	0.993120	+	0.117098i	$0.0373592\pi$
$359$	1.39654e12i	0.234196i	−0.993120	+	0.117098i	$0.962641\pi$
$360$	0	0
$361$	−8.36815e11	−0.136488
$362$	0	0
$363$	− 2.17058e12i	− 0.344383i
$364$	0	0
$365$	2.39659e12	0.369938
$366$	0	0
$367$	− 7.81335e12i	− 1.17357i	−0.809745	−	0.586783i	$-0.800394\pi$
$367$	− 7.81335e12i	− 1.17357i	0.809745	−	0.586783i	$-0.199606\pi$
$368$	0	0
$369$	1.07773e13	1.57536
$370$	0	0
$371$	3.96640e12i	0.564321i
$372$	0	0
$373$	8.92815e12	1.23657	0.618283	−	0.785955i	$-0.287828\pi$
$373$	8.92815e12	1.23657	0.618283	+	0.785955i	$0.287828\pi$
$374$	0	0
$375$	8.00889e11i	0.107998i
$376$	0	0
$377$	3.95355e12	0.519135
$378$	0	0
$379$	9.71433e12i	1.24227i	0.783703	+	0.621136i	$0.213329\pi$
$379$	9.71433e12i	1.24227i	−0.783703	+	0.621136i	$0.786671\pi$
$380$	0	0
$381$	4.19487e12	0.522509
$382$	0	0
$383$	− 1.13101e13i	− 1.37237i	−0.727427	−	0.686185i	$-0.759284\pi$
$383$	− 1.13101e13i	− 1.37237i	0.727427	−	0.686185i	$-0.240716\pi$
$384$	0	0
$385$	2.38024e13	2.81395
$386$	0	0
$387$	1.04925e13i	1.20871i
$388$	0	0
$389$	−5.54283e12	−0.622277	−0.311139	−	0.950365i	$-0.600710\pi$
$389$	−5.54283e12	−0.622277	−0.311139	+	0.950365i	$0.600710\pi$
$390$	0	0
$391$	2.19182e12i	0.239840i
$392$	0	0
$393$	−6.21315e12	−0.662749
$394$	0	0
$395$	1.87340e13i	1.94825i
$396$	0	0
$397$	6.88396e12	0.698048	0.349024	−	0.937114i	$-0.386513\pi$
$397$	6.88396e12	0.698048	0.349024	+	0.937114i	$0.386513\pi$
$398$	0	0
$399$	6.60263e12i	0.652908i
$400$	0	0
$401$	−1.02992e13	−0.993298	−0.496649	−	0.867951i	$-0.665436\pi$
$401$	−1.02992e13	−0.993298	−0.496649	+	0.867951i	$0.665436\pi$
$402$	0	0
$403$	− 1.68471e13i	− 1.58490i
$404$	0	0
$405$	7.68824e12	0.705589
$406$	0	0
$407$	− 5.39610e12i	− 0.483179i
$408$	0	0
$409$	−7.24683e12	−0.633186	−0.316593	−	0.948561i	$-0.602539\pi$
$409$	−7.24683e12	−0.633186	−0.316593	+	0.948561i	$0.602539\pi$
$410$	0	0
$411$	− 4.53719e12i	− 0.386882i
$412$	0	0
$413$	−4.90448e12	−0.408172
$414$	0	0
$415$	− 2.06650e13i	− 1.67878i
$416$	0	0
$417$	2.66340e12	0.211230
$418$	0	0
$419$	2.36280e13i	1.82960i	0.403904	+	0.914801i	$0.367653\pi$
$419$	2.36280e13i	1.82960i	−0.403904	+	0.914801i	$0.632347\pi$
$420$	0	0
$421$	−8.45398e12	−0.639220	−0.319610	−	0.947549i	$-0.603552\pi$
$421$	−8.45398e12	−0.639220	−0.319610	+	0.947549i	$0.603552\pi$
$422$	0	0
$423$	8.57486e12i	0.633177i
$424$	0	0
$425$	1.05637e13	0.761853
$426$	0	0
$427$	− 2.41039e13i	− 1.69804i
$428$	0	0
$429$	−1.44933e13	−0.997429
$430$	0	0
$431$	− 2.82283e12i	− 0.189801i	−0.995487	−	0.0949006i	$-0.969747\pi$
$431$	− 2.82283e12i	− 0.189801i	0.995487	−	0.0949006i	$-0.0302533\pi$
$432$	0	0
$433$	−1.24364e13	−0.817064	−0.408532	−	0.912744i	$-0.633959\pi$
$433$	−1.24364e13	−0.817064	−0.408532	+	0.912744i	$0.633959\pi$
$434$	0	0
$435$	− 2.96638e12i	− 0.190450i
$436$	0	0
$437$	6.25992e12	0.392791
$438$	0	0
$439$	− 2.35706e13i	− 1.44560i	−0.691057	−	0.722800i	$-0.742855\pi$
$439$	− 2.35706e13i	− 1.44560i	0.691057	−	0.722800i	$-0.257145\pi$
$440$	0	0
$441$	−1.39506e13	−0.836371
$442$	0	0
$443$	− 1.62108e13i	− 0.950138i	−0.879948	−	0.475069i	$-0.842423\pi$
$443$	− 1.62108e13i	− 0.950138i	0.879948	−	0.475069i	$-0.157577\pi$
$444$	0	0
$445$	1.84814e13	1.05909
$446$	0	0
$447$	− 1.20983e13i	− 0.677931i
$448$	0	0
$449$	−1.74665e13	−0.957136	−0.478568	−	0.878051i	$-0.658844\pi$
$449$	−1.74665e13	−0.957136	−0.478568	+	0.878051i	$0.658844\pi$
$450$	0	0
$451$	− 4.84007e13i	− 2.59399i
$452$	0	0
$453$	2.76009e12	0.144688
$454$	0	0
$455$	− 7.06585e13i	− 3.62332i
$456$	0	0
$457$	−1.00232e13	−0.502835	−0.251417	−	0.967879i	$-0.580897\pi$
$457$	−1.00232e13	−0.502835	−0.251417	+	0.967879i	$0.580897\pi$
$458$	0	0
$459$	1.03554e13i	0.508282i
$460$	0	0
$461$	−5.97818e11	−0.0287120	−0.0143560	−	0.999897i	$-0.504570\pi$
$461$	−5.97818e11	−0.0287120	−0.0143560	+	0.999897i	$0.504570\pi$
$462$	0	0
$463$	− 1.07126e13i	− 0.503491i	−0.967794	−	0.251745i	$-0.918995\pi$
$463$	− 1.07126e13i	− 0.503491i	0.967794	−	0.251745i	$-0.0810045\pi$
$464$	0	0
$465$	−1.26405e13	−0.581435
$466$	0	0
$467$	− 5.18390e11i	− 0.0233385i	−0.999932	−	0.0116692i	$-0.996285\pi$
$467$	− 5.18390e11i	− 0.0233385i	0.999932	−	0.0116692i	$-0.00371452\pi$
$468$	0	0
$469$	−2.77807e13	−1.22428
$470$	0	0
$471$	− 7.83315e11i	− 0.0337934i
$472$	0	0
$473$	4.71213e13	1.99027
$474$	0	0
$475$	− 3.01703e13i	− 1.24770i
$476$	0	0
$477$	7.97739e12	0.323049
$478$	0	0
$479$	3.00914e13i	1.19334i	0.802486	+	0.596671i	$0.203510\pi$
$479$	3.00914e13i	1.19334i	−0.802486	+	0.596671i	$0.796490\pi$
$480$	0	0
$481$	−1.60186e13	−0.622155
$482$	0	0
$483$	− 5.93177e12i	− 0.225657i
$484$	0	0
$485$	3.66653e13	1.36630
$486$	0	0
$487$	1.28195e13i	0.467977i	0.972239	+	0.233989i	$0.0751779\pi$
$487$	1.28195e13i	0.467977i	−0.972239	+	0.233989i	$0.924822\pi$
$488$	0	0
$489$	6.67106e12	0.238589
$490$	0	0
$491$	3.99853e13i	1.40118i	0.713565	+	0.700589i	$0.247079\pi$
$491$	3.99853e13i	1.40118i	−0.713565	+	0.700589i	$0.752921\pi$
$492$	0	0
$493$	−5.69609e12	−0.195588
$494$	0	0
$495$	− 4.78723e13i	− 1.61086i
$496$	0	0
$497$	2.49216e12	0.0821854
$498$	0	0
$499$	2.58812e13i	0.836530i	0.908325	+	0.418265i	$0.137362\pi$
$499$	2.58812e13i	0.836530i	−0.908325	+	0.418265i	$0.862638\pi$
$500$	0	0
$501$	−2.94693e11	−0.00933645
$502$	0	0
$503$	− 6.16237e13i	− 1.91385i	−0.290334	−	0.956925i	$-0.593766\pi$
$503$	− 6.16237e13i	− 1.91385i	0.290334	−	0.956925i	$-0.406234\pi$
$504$	0	0
$505$	1.87060e13	0.569541
$506$	0	0
$507$	2.86113e13i	0.854078i
$508$	0	0
$509$	−1.82690e13	−0.534719	−0.267359	−	0.963597i	$-0.586151\pi$
$509$	−1.82690e13	−0.534719	−0.267359	+	0.963597i	$0.586151\pi$
$510$	0	0
$511$	1.24544e13i	0.357453i
$512$	0	0
$513$	2.95755e13	0.832425
$514$	0	0
$515$	5.51119e13i	1.52128i
$516$	0	0
$517$	3.85094e13	1.04259
$518$	0	0
$519$	− 6.33466e12i	− 0.168224i
$520$	0	0
$521$	7.50446e12	0.195493	0.0977464	−	0.995211i	$-0.468837\pi$
$521$	7.50446e12	0.195493	0.0977464	+	0.995211i	$0.468837\pi$
$522$	0	0
$523$	− 4.61985e13i	− 1.18065i	−0.807167	−	0.590323i	$-0.799000\pi$
$523$	− 4.61985e13i	− 1.18065i	0.807167	−	0.590323i	$-0.201000\pi$
$524$	0	0
$525$	−2.85888e13	−0.716802
$526$	0	0
$527$	2.42725e13i	0.597121i
$528$	0	0
$529$	3.58026e13	0.864244
$530$	0	0
$531$	9.86411e12i	0.233660i
$532$	0	0
$533$	−1.43680e14	−3.34010
$534$	0	0
$535$	5.81976e13i	1.32781i
$536$	0	0
$537$	−1.39794e13	−0.313053
$538$	0	0
$539$	6.26516e13i	1.37717i
$540$	0	0
$541$	−4.13820e13	−0.892945	−0.446473	−	0.894797i	$-0.647320\pi$
$541$	−4.13820e13	−0.892945	−0.446473	+	0.894797i	$0.647320\pi$
$542$	0	0
$543$	− 2.57814e13i	− 0.546144i
$544$	0	0
$545$	−1.34644e13	−0.280029
$546$	0	0
$547$	− 4.19636e13i	− 0.856911i	−0.903563	−	0.428455i	$-0.859058\pi$
$547$	− 4.19636e13i	− 0.856911i	0.903563	−	0.428455i	$-0.140942\pi$
$548$	0	0
$549$	−4.84789e13	−0.972055
$550$	0	0
$551$	1.62682e13i	0.320319i
$552$	0	0
$553$	−9.73552e13	−1.88249
$554$	0	0
$555$	1.20189e13i	0.228244i
$556$	0	0
$557$	1.00972e14	1.88332	0.941662	−	0.336561i	$-0.109264\pi$
$557$	1.00972e14	1.88332	0.941662	+	0.336561i	$0.109264\pi$
$558$	0	0
$559$	− 1.39882e14i	− 2.56273i
$560$	0	0
$561$	2.08813e13	0.375788
$562$	0	0
$563$	− 1.16086e13i	− 0.205228i	−0.994721	−	0.102614i	$-0.967279\pi$
$563$	− 1.16086e13i	− 0.205228i	0.994721	−	0.102614i	$-0.0327206\pi$
$564$	0	0
$565$	1.21056e14	2.10255
$566$	0	0
$567$	3.99537e13i	0.681776i
$568$	0	0
$569$	8.82509e13	1.47965	0.739823	−	0.672801i	$-0.234909\pi$
$569$	8.82509e13	1.47965	0.739823	+	0.672801i	$0.234909\pi$
$570$	0	0
$571$	− 2.61816e13i	− 0.431336i	−0.976467	−	0.215668i	$-0.930807\pi$
$571$	− 2.61816e13i	− 0.431336i	0.976467	−	0.215668i	$-0.0691928\pi$
$572$	0	0
$573$	−3.38524e13	−0.548045
$574$	0	0
$575$	2.71049e13i	0.431229i
$576$	0	0
$577$	4.73483e13	0.740330	0.370165	−	0.928966i	$-0.379301\pi$
$577$	4.73483e13	0.740330	0.370165	+	0.928966i	$0.379301\pi$
$578$	0	0
$579$	1.76475e13i	0.271200i
$580$	0	0
$581$	1.07390e14	1.62212
$582$	0	0
$583$	− 3.58262e13i	− 0.531935i
$584$	0	0
$585$	−1.42111e14	−2.07419
$586$	0	0
$587$	− 4.82284e13i	− 0.692009i	−0.938233	−	0.346005i	$-0.887538\pi$
$587$	− 4.82284e13i	− 0.692009i	0.938233	−	0.346005i	$-0.112462\pi$
$588$	0	0
$589$	6.93233e13	0.977919
$590$	0	0
$591$	2.65917e13i	0.368815i
$592$	0	0
$593$	−3.90864e13	−0.533031	−0.266516	−	0.963831i	$-0.585872\pi$
$593$	−3.90864e13	−0.533031	−0.266516	+	0.963831i	$0.585872\pi$
$594$	0	0
$595$	1.01801e14i	1.36511i
$596$	0	0
$597$	−5.16641e13	−0.681267
$598$	0	0
$599$	5.37750e13i	0.697343i	0.937245	+	0.348672i	$0.113367\pi$
$599$	5.37750e13i	0.697343i	−0.937245	+	0.348672i	$0.886633\pi$
$600$	0	0
$601$	−1.19066e14	−1.51850	−0.759251	−	0.650798i	$-0.774434\pi$
$601$	−1.19066e14	−1.51850	−0.759251	+	0.650798i	$0.774434\pi$
$602$	0	0
$603$	5.58738e13i	0.700844i
$604$	0	0
$605$	−9.55819e13	−1.17923
$606$	0	0
$607$	2.69720e13i	0.327318i	0.986517	+	0.163659i	$0.0523297\pi$
$607$	2.69720e13i	0.327318i	−0.986517	+	0.163659i	$0.947670\pi$
$608$	0	0
$609$	1.54155e13	0.184022
$610$	0	0
$611$	− 1.14317e14i	− 1.34247i
$612$	0	0
$613$	1.44696e12	0.0167169	0.00835844	−	0.999965i	$-0.497339\pi$
$613$	1.44696e12	0.0167169	0.00835844	+	0.999965i	$0.497339\pi$
$614$	0	0
$615$	1.07804e14i	1.22535i
$616$	0	0
$617$	1.06522e14	1.19127	0.595637	−	0.803254i	$-0.296900\pi$
$617$	1.06522e14	1.19127	0.595637	+	0.803254i	$0.296900\pi$
$618$	0	0
$619$	4.85778e13i	0.534546i	0.963621	+	0.267273i	$0.0861225\pi$
$619$	4.85778e13i	0.534546i	−0.963621	+	0.267273i	$0.913877\pi$
$620$	0	0
$621$	−2.65705e13	−0.287701
$622$	0	0
$623$	9.60426e13i	1.02335i
$624$	0	0
$625$	−7.63494e13	−0.800582
$626$	0	0
$627$	− 5.96378e13i	− 0.615438i
$628$	0	0
$629$	2.30788e13	0.234401
$630$	0	0
$631$	7.46155e13i	0.745903i	0.927851	+	0.372952i	$0.121654\pi$
$631$	7.46155e13i	0.745903i	−0.927851	+	0.372952i	$0.878346\pi$
$632$	0	0
$633$	6.33027e13	0.622879
$634$	0	0
$635$	− 1.84722e14i	− 1.78917i
$636$	0	0
$637$	1.85984e14	1.77329
$638$	0	0
$639$	− 5.01235e12i	− 0.0470475i
$640$	0	0
$641$	−4.63888e13	−0.428670	−0.214335	−	0.976760i	$-0.568758\pi$
$641$	−4.63888e13	−0.428670	−0.214335	+	0.976760i	$0.568758\pi$
$642$	0	0
$643$	− 1.11537e14i	− 1.01476i	−0.861722	−	0.507380i	$-0.830614\pi$
$643$	− 1.11537e14i	− 1.01476i	0.861722	−	0.507380i	$-0.169386\pi$
$644$	0	0
$645$	−1.04955e14	−0.940162
$646$	0	0
$647$	− 7.06443e13i	− 0.623097i	−0.950230	−	0.311549i	$-0.899152\pi$
$647$	− 7.06443e13i	− 0.623097i	0.950230	−	0.311549i	$-0.100848\pi$
$648$	0	0
$649$	4.42994e13	0.384746
$650$	0	0
$651$	− 6.56894e13i	− 0.561811i
$652$	0	0
$653$	−8.26908e12	−0.0696452	−0.0348226	−	0.999394i	$-0.511087\pi$
$653$	−8.26908e12	−0.0696452	−0.0348226	+	0.999394i	$0.511087\pi$
$654$	0	0
$655$	2.73598e14i	2.26937i
$656$	0	0
$657$	2.50489e13	0.204626
$658$	0	0
$659$	− 5.96139e13i	− 0.479645i	−0.970817	−	0.239823i	$-0.922911\pi$
$659$	− 5.96139e13i	− 0.479645i	0.970817	−	0.239823i	$-0.0770893\pi$
$660$	0	0
$661$	−1.01869e14	−0.807302	−0.403651	−	0.914913i	$-0.632259\pi$
$661$	−1.01869e14	−0.807302	−0.403651	+	0.914913i	$0.632259\pi$
$662$	0	0
$663$	− 6.19872e13i	− 0.483876i
$664$	0	0
$665$	2.90748e14	2.23568
$666$	0	0
$667$	− 1.46153e13i	− 0.110708i
$668$	0	0
$669$	−7.17568e13	−0.535467
$670$	0	0
$671$	2.17717e14i	1.60059i
$672$	0	0
$673$	1.35652e14	0.982545	0.491273	−	0.871006i	$-0.336532\pi$
$673$	1.35652e14	0.982545	0.491273	+	0.871006i	$0.336532\pi$
$674$	0	0
$675$	1.28059e14i	0.913886i
$676$	0	0
$677$	1.77090e14	1.24524	0.622618	−	0.782526i	$-0.286069\pi$
$677$	1.77090e14	1.24524	0.622618	+	0.782526i	$0.286069\pi$
$678$	0	0
$679$	1.90540e14i	1.32019i
$680$	0	0
$681$	−1.13159e14	−0.772599
$682$	0	0
$683$	1.56085e13i	0.105017i	0.998620	+	0.0525083i	$0.0167216\pi$
$683$	1.56085e13i	0.105017i	−0.998620	+	0.0525083i	$0.983278\pi$
$684$	0	0
$685$	−1.99796e14	−1.32475
$686$	0	0
$687$	9.76953e13i	0.638396i
$688$	0	0
$689$	−1.06352e14	−0.684934
$690$	0	0
$691$	1.15000e14i	0.729974i	0.931013	+	0.364987i	$0.118927\pi$
$691$	1.15000e14i	0.729974i	−0.931013	+	0.364987i	$0.881073\pi$
$692$	0	0
$693$	2.48779e14	1.55650
$694$	0	0
$695$	− 1.17283e14i	− 0.723290i
$696$	0	0
$697$	2.07007e14	1.25841
$698$	0	0
$699$	− 9.92701e13i	− 0.594885i
$700$	0	0
$701$	2.50440e14	1.47949	0.739747	−	0.672885i	$-0.234945\pi$
$701$	2.50440e14	1.47949	0.739747	+	0.672885i	$0.234945\pi$
$702$	0	0
$703$	− 6.59139e13i	− 0.383885i
$704$	0	0
$705$	−8.57731e13	−0.492499
$706$	0	0
$707$	9.72101e13i	0.550319i
$708$	0	0
$709$	−1.99396e14	−1.11298	−0.556488	−	0.830856i	$-0.687851\pi$
$709$	−1.99396e14	−1.11298	−0.556488	+	0.830856i	$0.687851\pi$
$710$	0	0
$711$	1.95805e14i	1.07765i
$712$	0	0
$713$	−6.22798e13	−0.337987
$714$	0	0
$715$	6.38218e14i	3.41538i
$716$	0	0
$717$	5.48119e13	0.289254
$718$	0	0
$719$	2.70554e14i	1.40802i	0.710188	+	0.704012i	$0.248610\pi$
$719$	2.70554e14i	1.40802i	−0.710188	+	0.704012i	$0.751390\pi$
$720$	0	0
$721$	−2.86401e14	−1.46994
$722$	0	0
$723$	9.51943e13i	0.481858i
$724$	0	0
$725$	−7.04400e13	−0.351665
$726$	0	0
$727$	9.73122e13i	0.479176i	0.970875	+	0.239588i	$0.0770124\pi$
$727$	9.73122e13i	0.479176i	−0.970875	+	0.239588i	$0.922988\pi$
$728$	0	0
$729$	1.11874e13	0.0543367
$730$	0	0
$731$	2.01535e14i	0.965526i
$732$	0	0
$733$	−3.65427e13	−0.172695	−0.0863477	−	0.996265i	$-0.527520\pi$
$733$	−3.65427e13	−0.172695	−0.0863477	+	0.996265i	$0.527520\pi$
$734$	0	0
$735$	− 1.39546e14i	− 0.650548i
$736$	0	0
$737$	2.50928e14	1.15401
$738$	0	0
$739$	5.41934e13i	0.245881i	0.992414	+	0.122940i	$0.0392324\pi$
$739$	5.41934e13i	0.245881i	−0.992414	+	0.122940i	$0.960768\pi$
$740$	0	0
$741$	−1.77038e14	−0.792455
$742$	0	0
$743$	− 3.84093e13i	− 0.169626i	−0.996397	−	0.0848130i	$-0.972971\pi$
$743$	− 3.84093e13i	− 0.169626i	0.996397	−	0.0848130i	$-0.0270293\pi$
$744$	0	0
$745$	−5.32750e14	−2.32136
$746$	0	0
$747$	− 2.15988e14i	− 0.928595i
$748$	0	0
$749$	−3.02437e14	−1.28300
$750$	0	0
$751$	− 5.22355e13i	− 0.218658i	−0.994006	−	0.109329i	$-0.965130\pi$
$751$	− 5.22355e13i	− 0.218658i	0.994006	−	0.109329i	$-0.0348703\pi$
$752$	0	0
$753$	−7.94530e13	−0.328198
$754$	0	0
$755$	− 1.21541e14i	− 0.495437i
$756$	0	0
$757$	2.86392e14	1.15208	0.576039	−	0.817422i	$-0.304598\pi$
$757$	2.86392e14	1.15208	0.576039	+	0.817422i	$0.304598\pi$
$758$	0	0
$759$	5.35784e13i	0.212706i
$760$	0	0
$761$	−2.90918e14	−1.13985	−0.569925	−	0.821697i	$-0.693028\pi$
$761$	−2.90918e14	−1.13985	−0.569925	+	0.821697i	$0.693028\pi$
$762$	0	0
$763$	− 6.99706e13i	− 0.270578i
$764$	0	0
$765$	2.04747e14	0.781467
$766$	0	0
$767$	− 1.31505e14i	− 0.495411i
$768$	0	0
$769$	2.54977e14	0.948131	0.474066	−	0.880490i	$-0.342786\pi$
$769$	2.54977e14	0.948131	0.474066	+	0.880490i	$0.342786\pi$
$770$	0	0
$771$	− 3.92731e13i	− 0.144153i
$772$	0	0
$773$	−2.89814e14	−1.05008	−0.525040	−	0.851078i	$-0.675949\pi$
$773$	−2.89814e14	−1.05008	−0.525040	+	0.851078i	$0.675949\pi$
$774$	0	0
$775$	3.00164e14i	1.07362i
$776$	0	0
$777$	−6.24587e13	−0.220540
$778$	0	0
$779$	− 5.91220e14i	− 2.06092i
$780$	0	0
$781$	−2.25103e13	−0.0774687
$782$	0	0
$783$	− 6.90513e13i	− 0.234619i
$784$	0	0
$785$	−3.44935e13	−0.115715
$786$	0	0
$787$	1.80246e14i	0.597024i	0.954406	+	0.298512i	$0.0964903\pi$
$787$	1.80246e14i	0.597024i	−0.954406	+	0.298512i	$0.903510\pi$
$788$	0	0
$789$	−8.75484e12	−0.0286327
$790$	0	0
$791$	6.29097e14i	2.03159i
$792$	0	0
$793$	6.46304e14	2.06097
$794$	0	0
$795$	7.97966e13i	0.251275i
$796$	0	0
$797$	−4.40616e14	−1.37015	−0.685075	−	0.728472i	$-0.740231\pi$
$797$	−4.40616e14	−1.37015	−0.685075	+	0.728472i	$0.740231\pi$
$798$	0	0
$799$	1.64703e14i	0.505786i
$800$	0	0
$801$	1.93165e14	0.585822
$802$	0	0
$803$	− 1.12494e14i	− 0.336938i
$804$	0	0
$805$	−2.61207e14	−0.772691
$806$	0	0
$807$	6.31432e13i	0.184484i
$808$	0	0
$809$	−5.61205e14	−1.61949	−0.809747	−	0.586779i	$-0.800396\pi$
$809$	−5.61205e14	−1.61949	−0.809747	+	0.586779i	$0.800396\pi$
$810$	0	0
$811$	− 2.56130e14i	− 0.730055i	−0.930997	−	0.365027i	$-0.881060\pi$
$811$	− 2.56130e14i	− 0.730055i	0.930997	−	0.365027i	$-0.118940\pi$
$812$	0	0
$813$	−1.02725e14	−0.289218
$814$	0	0
$815$	− 2.93762e14i	− 0.816972i
$816$	0	0
$817$	5.75592e14	1.58126
$818$	0	0
$819$	− 7.38514e14i	− 2.00419i
$820$	0	0
$821$	2.95104e14	0.791150	0.395575	−	0.918434i	$-0.370545\pi$
$821$	2.95104e14	0.791150	0.395575	+	0.918434i	$0.370545\pi$
$822$	0	0
$823$	− 2.11141e14i	− 0.559208i	−0.960115	−	0.279604i	$-0.909797\pi$
$823$	− 2.11141e14i	− 0.559208i	0.960115	−	0.279604i	$-0.0902032\pi$
$824$	0	0
$825$	2.58226e14	0.675664
$826$	0	0
$827$	4.56935e14i	1.18121i	0.806961	+	0.590605i	$0.201111\pi$
$827$	4.56935e14i	1.18121i	−0.806961	+	0.590605i	$0.798889\pi$
$828$	0	0
$829$	1.75989e14	0.449484	0.224742	−	0.974418i	$-0.427846\pi$
$829$	1.75989e14	0.449484	0.224742	+	0.974418i	$0.427846\pi$
$830$	0	0
$831$	1.43526e14i	0.362181i
$832$	0	0
$833$	−2.67957e14	−0.668099
$834$	0	0
$835$	1.29769e13i	0.0319697i
$836$	0	0
$837$	−2.94246e14	−0.716281
$838$	0	0
$839$	− 4.23502e14i	− 1.01870i	−0.860560	−	0.509350i	$-0.829886\pi$
$839$	− 4.23502e14i	− 1.01870i	0.860560	−	0.509350i	$-0.170114\pi$
$840$	0	0
$841$	−3.82725e14	−0.909718
$842$	0	0
$843$	1.25813e13i	0.0295519i
$844$	0	0
$845$	1.25990e15	2.92452
$846$	0	0
$847$	− 4.96713e14i	− 1.13943i
$848$	0	0
$849$	3.32532e14	0.753867
$850$	0	0
$851$	5.92168e13i	0.132677i
$852$	0	0
$853$	3.49003e14	0.772831	0.386415	−	0.922325i	$-0.373713\pi$
$853$	3.49003e14	0.772831	0.386415	+	0.922325i	$0.373713\pi$
$854$	0	0
$855$	− 5.84766e14i	− 1.27983i
$856$	0	0
$857$	6.07945e14	1.31510	0.657552	−	0.753410i	$-0.271592\pi$
$857$	6.07945e14	1.31510	0.657552	+	0.753410i	$0.271592\pi$
$858$	0	0
$859$	− 2.82319e14i	− 0.603635i	−0.953366	−	0.301817i	$-0.902407\pi$
$859$	− 2.82319e14i	− 0.603635i	0.953366	−	0.301817i	$-0.0975933\pi$
$860$	0	0
$861$	−5.60228e14	−1.18399
$862$	0	0
$863$	− 3.31638e14i	− 0.692805i	−0.938086	−	0.346402i	$-0.887403\pi$
$863$	− 3.31638e14i	− 0.692805i	0.938086	−	0.346402i	$-0.112597\pi$
$864$	0	0
$865$	−2.78948e14	−0.576028
$866$	0	0
$867$	− 1.21461e14i	− 0.247937i
$868$	0	0
$869$	8.79355e14	1.77446
$870$	0	0
$871$	− 7.44891e14i	− 1.48594i
$872$	0	0
$873$	3.83221e14	0.755751
$874$	0	0
$875$	1.83275e14i	0.357324i
$876$	0	0
$877$	−7.21866e14	−1.39142	−0.695711	−	0.718322i	$-0.744911\pi$
$877$	−7.21866e14	−1.39142	−0.695711	+	0.718322i	$0.744911\pi$
$878$	0	0
$879$	1.11000e14i	0.211532i
$880$	0	0
$881$	−8.88764e14	−1.67458	−0.837292	−	0.546756i	$-0.815862\pi$
$881$	−8.88764e14	−1.67458	−0.837292	+	0.546756i	$0.815862\pi$
$882$	0	0
$883$	9.47565e14i	1.76525i	0.470080	+	0.882624i	$0.344225\pi$
$883$	9.47565e14i	1.76525i	−0.470080	+	0.882624i	$0.655775\pi$
$884$	0	0
$885$	−9.86692e13	−0.181746
$886$	0	0
$887$	2.37839e14i	0.433176i	0.976263	+	0.216588i	$0.0694929\pi$
$887$	2.37839e14i	0.433176i	−0.976263	+	0.216588i	$0.930507\pi$
$888$	0	0
$889$	9.59951e14	1.72878
$890$	0	0
$891$	− 3.60879e14i	− 0.642648i
$892$	0	0
$893$	4.70397e14	0.828337
$894$	0	0
$895$	6.15588e14i	1.07195i
$896$	0	0
$897$	1.59050e14	0.273887
$898$	0	0
$899$	− 1.61852e14i	− 0.275626i
$900$	0	0
$901$	1.53227e14	0.258054
$902$	0	0
$903$	− 5.45420e14i	− 0.908431i
$904$	0	0
$905$	−1.13529e15	−1.87010
$906$	0	0
$907$	1.75418e14i	0.285784i	0.989738	+	0.142892i	$0.0456402\pi$
$907$	1.75418e14i	0.285784i	−0.989738	+	0.142892i	$0.954360\pi$
$908$	0	0
$909$	1.95513e14	0.315034
$910$	0	0
$911$	− 1.11663e15i	− 1.77958i	−0.456371	−	0.889790i	$-0.650851\pi$
$911$	− 1.11663e15i	− 1.77958i	0.456371	−	0.889790i	$-0.349149\pi$
$912$	0	0
$913$	−9.69995e14	−1.52903
$914$	0	0
$915$	− 4.84927e14i	− 0.756086i
$916$	0	0
$917$	−1.42181e15	−2.19278
$918$	0	0
$919$	− 1.08673e15i	− 1.65784i	−0.559365	−	0.828922i	$-0.688955\pi$
$919$	− 1.08673e15i	− 1.65784i	0.559365	−	0.828922i	$-0.311045\pi$
$920$	0	0
$921$	1.66881e14	0.251830
$922$	0	0
$923$	6.68229e13i	0.0997510i
$924$	0	0
$925$	2.85402e14	0.421452
$926$	0	0
$927$	5.76023e14i	0.841475i
$928$	0	0
$929$	5.26060e14	0.760251	0.380126	−	0.924935i	$-0.375881\pi$
$929$	5.26060e14	0.760251	0.380126	+	0.924935i	$0.375881\pi$
$930$	0	0
$931$	7.65296e14i	1.09416i
$932$	0	0
$933$	2.36948e14	0.335154
$934$	0	0
$935$	− 9.19513e14i	− 1.28677i
$936$	0	0
$937$	−4.20234e14	−0.581826	−0.290913	−	0.956749i	$-0.593959\pi$
$937$	−4.20234e14	−0.581826	−0.290913	+	0.956749i	$0.593959\pi$
$938$	0	0
$939$	2.15759e14i	0.295556i
$940$	0	0
$941$	−4.49342e14	−0.609017	−0.304508	−	0.952510i	$-0.598492\pi$
$941$	−4.49342e14	−0.609017	−0.304508	+	0.952510i	$0.598492\pi$
$942$	0	0
$943$	5.31149e14i	0.712292i
$944$	0	0
$945$	−1.23409e15	−1.63753
$946$	0	0
$947$	− 1.78285e14i	− 0.234080i	−0.993127	−	0.117040i	$-0.962659\pi$
$947$	− 1.78285e14i	− 0.234080i	0.993127	−	0.117040i	$-0.0373406\pi$
$948$	0	0
$949$	−3.33943e14	−0.433852
$950$	0	0
$951$	1.97268e14i	0.253603i
$952$	0	0
$953$	2.56243e14	0.325977	0.162989	−	0.986628i	$-0.447887\pi$
$953$	2.56243e14	0.325977	0.162989	+	0.986628i	$0.447887\pi$
$954$	0	0
$955$	1.49070e15i	1.87661i
$956$	0	0
$957$	−1.39239e14	−0.173461
$958$	0	0
$959$	− 1.03829e15i	− 1.28004i
$960$	0	0
$961$	1.29932e14	0.158525
$962$	0	0
$963$	6.08274e14i	0.734459i
$964$	0	0
$965$	7.77111e14	0.928638
$966$	0	0
$967$	5.98475e14i	0.707805i	0.935282	+	0.353902i	$0.115145\pi$
$967$	5.98475e14i	0.707805i	−0.935282	+	0.353902i	$0.884855\pi$
$968$	0	0
$969$	2.55067e14	0.298563
$970$	0	0
$971$	2.89179e13i	0.0335020i	0.999860	+	0.0167510i	$0.00533225\pi$
$971$	2.89179e13i	0.0335020i	−0.999860	+	0.0167510i	$0.994668\pi$
$972$	0	0
$973$	6.09490e14	0.698879
$974$	0	0
$975$	− 7.66558e14i	− 0.870005i
$976$	0	0
$977$	7.15159e14	0.803397	0.401698	−	0.915772i	$-0.368420\pi$
$977$	7.15159e14	0.803397	0.401698	+	0.915772i	$0.368420\pi$
$978$	0	0
$979$	− 8.67498e14i	− 0.964618i
$980$	0	0
$981$	−1.40728e14	−0.154894
$982$	0	0
$983$	− 4.43034e14i	− 0.482691i	−0.970439	−	0.241345i	$-0.922411\pi$
$983$	− 4.43034e14i	− 0.482691i	0.970439	−	0.241345i	$-0.0775887\pi$
$984$	0	0
$985$	1.17097e15	1.26289
$986$	0	0
$987$	− 4.45739e14i	− 0.475877i
$988$	0	0
$989$	−5.17110e14	−0.546514
$990$	0	0
$991$	− 1.41297e15i	− 1.47830i	−0.673539	−	0.739152i	$-0.735227\pi$
$991$	− 1.41297e15i	− 1.47830i	0.673539	−	0.739152i	$-0.264773\pi$
$992$	0	0
$993$	5.43050e14	0.562462
$994$	0	0
$995$	2.27504e15i	2.33278i
$996$	0	0
$997$	1.40666e15	1.42795	0.713974	−	0.700172i	$-0.246893\pi$
$997$	1.40666e15	1.42795	0.713974	+	0.700172i	$0.246893\pi$
$998$	0	0
$999$	2.79775e14i	0.281178i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.11.c.i.255.6		8
4.3	odd	2	inner	256.11.c.i.255.4		8
8.3	odd	2	inner	256.11.c.i.255.5		8
8.5	even	2	inner	256.11.c.i.255.3		8
16.3	odd	4		64.11.d.a.31.5	yes	8
16.5	even	4		64.11.d.a.31.6	yes	8
16.11	odd	4		64.11.d.a.31.4	yes	8
16.13	even	4		64.11.d.a.31.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.11.d.a.31.3	✓	8	16.13	even	4
64.11.d.a.31.4	yes	8	16.11	odd	4
64.11.d.a.31.5	yes	8	16.3	odd	4
64.11.d.a.31.6	yes	8	16.5	even	4
256.11.c.i.255.3		8	8.5	even	2	inner
256.11.c.i.255.4		8	4.3	odd	2	inner
256.11.c.i.255.5		8	8.3	odd	2	inner
256.11.c.i.255.6		8	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+104.549i q^{3} +4603.82 q^{5} +23924.8i q^{7} +48118.6 q^{9} +O(q^{10})\)	\(q+104.549i q^{3} +4603.82 q^{5} +23924.8i q^{7} +48118.6 q^{9} -216099. i q^{11} -641501. q^{13} +481323. i q^{15} +924243. q^{17} -2.63967e6i q^{19} -2.50130e6 q^{21} +2.37147e6i q^{23} +1.14296e7 q^{25} +1.12042e7i q^{27} -6.16297e6 q^{29} +2.62621e7i q^{31} +2.25929e7 q^{33} +1.10146e8i q^{35} +2.49705e7 q^{37} -6.70680e7i q^{39} +2.23974e8 q^{41} +2.18054e8i q^{43} +2.21529e8 q^{45} +1.78203e8i q^{47} -2.89921e8 q^{49} +9.66283e7i q^{51} +1.65786e8 q^{53} -9.94882e8i q^{55} +2.75974e8 q^{57} +2.04996e8i q^{59} -1.00749e9 q^{61} +1.15123e9i q^{63} -2.95336e9 q^{65} +1.16117e9i q^{67} -2.47934e8 q^{69} -1.04167e8i q^{71} +5.20565e8 q^{73} +1.19494e9i q^{75} +5.17013e9 q^{77} +4.06922e9i q^{79} +1.66997e9 q^{81} -4.48866e9i q^{83} +4.25505e9 q^{85} -6.44330e8i q^{87} +4.01435e9 q^{89} -1.53478e10i q^{91} -2.74566e9 q^{93} -1.21526e10i q^{95} +7.96410e9 q^{97} -1.03984e10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 18456 q^{9}+O(q^{10}) \) 8 * q - 18456 * q^9	\( 8 q - 18456 q^{9} - 1077552 q^{17} + 14789560 q^{25} - 194826912 q^{33} + 826044912 q^{41} - 2838950520 q^{49} + 6691637664 q^{57} - 10336279680 q^{65} + 13655425840 q^{73} - 8599581720 q^{81} - 14080263120 q^{89} + 47348700112 q^{97}+O(q^{100}) \) 8 * q - 18456 * q^9 - 1077552 * q^17 + 14789560 * q^25 - 194826912 * q^33 + 826044912 * q^41 - 2838950520 * q^49 + 6691637664 * q^57 - 10336279680 * q^65 + 13655425840 * q^73 - 8599581720 * q^81 - 14080263120 * q^89 + 47348700112 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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