LMFDB - Embedded newform 256.11.c.k.255.7

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			255.7
Root			$66.5960 - 1.41421i$ of defining polynomial
Character	$\chi$	$=$	256.255
Dual form			256.11.c.k.255.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+141.216i q^{3} +2390.30 q^{5} +14099.9i q^{7} +39107.0 q^{9} +O(q^{10})$	$q+141.216i q^{3} +2390.30 q^{5} +14099.9i q^{7} +39107.0 q^{9} +195559. i q^{11} -450735. q^{13} +337548. i q^{15} +1.62849e6 q^{17} -234694. i q^{19} -1.99113e6 q^{21} +9.12181e6i q^{23} -4.05211e6 q^{25} +1.38612e7i q^{27} +5.34591e6 q^{29} -4.04720e6i q^{31} -2.76161e7 q^{33} +3.37029e7i q^{35} -6.90047e7 q^{37} -6.36510e7i q^{39} -1.80477e8 q^{41} +3.98178e6i q^{43} +9.34774e7 q^{45} +2.35895e7i q^{47} +8.36687e7 q^{49} +2.29968e8i q^{51} +3.40010e8 q^{53} +4.67445e8i q^{55} +3.31426e7 q^{57} -6.12444e8i q^{59} +1.06549e8 q^{61} +5.51405e8i q^{63} -1.07739e9 q^{65} -1.52228e9i q^{67} -1.28814e9 q^{69} +2.90544e9i q^{71} -6.23479e7 q^{73} -5.72223e8i q^{75} -2.75736e9 q^{77} -2.35466e9i q^{79} +3.51808e8 q^{81} +7.35873e9i q^{83} +3.89256e9 q^{85} +7.54928e8i q^{87} +6.48010e9 q^{89} -6.35531e9i q^{91} +5.71529e8 q^{93} -5.60989e8i q^{95} +1.17007e10 q^{97} +7.64775e9i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	141.216i	0.581136i	0.956854	+	0.290568i	$0.0938442\pi$
$3$	141.216i	0.581136i	−0.956854	+	0.290568i	$0.906156\pi$
$4$	0	0
$5$	2390.30	0.764895	0.382447	−	0.923977i	$-0.375081\pi$
$5$	2390.30	0.764895	0.382447	+	0.923977i	$0.375081\pi$
$6$	0	0
$7$	14099.9i	0.838929i	0.907772	+	0.419464i	$0.137782\pi$
$7$	14099.9i	0.838929i	−0.907772	+	0.419464i	$0.862218\pi$
$8$	0	0
$9$	39107.0	0.662281
$10$	0	0
$11$	195559.i	1.21427i	0.794599	+	0.607135i	$0.207681\pi$
$11$	195559.i	1.21427i	−0.794599	+	0.607135i	$0.792319\pi$
$12$	0	0
$13$	−450735.	−1.21396	−0.606980	−	0.794717i	$-0.707619\pi$
$13$	−450735.	−1.21396	−0.606980	+	0.794717i	$0.707619\pi$
$14$	0	0
$15$	337548.i	0.444508i
$16$	0	0
$17$	1.62849e6	1.14694	0.573468	−	0.819228i	$-0.305598\pi$
$17$	1.62849e6	1.14694	0.573468	+	0.819228i	$0.305598\pi$
$18$	0	0
$19$	− 234694.i	− 0.0947839i	−0.998876	−	0.0473920i	$-0.984909\pi$
$19$	− 234694.i	− 0.0947839i	0.998876	−	0.0473920i	$-0.0150910\pi$
$20$	0	0
$21$	−1.99113e6	−0.487532
$22$	0	0
$23$	9.12181e6i	1.41723i	0.705593	+	0.708617i	$0.250681\pi$
$23$	9.12181e6i	1.41723i	−0.705593	+	0.708617i	$0.749319\pi$
$24$	0	0
$25$	−4.05211e6	−0.414936
$26$	0	0
$27$	1.38612e7i	0.966011i
$28$	0	0
$29$	5.34591e6	0.260635	0.130317	−	0.991472i	$-0.458400\pi$
$29$	5.34591e6	0.260635	0.130317	+	0.991472i	$0.458400\pi$
$30$	0	0
$31$	− 4.04720e6i	− 0.141366i	−0.997499	−	0.0706832i	$-0.977482\pi$
$31$	− 4.04720e6i	− 0.141366i	0.997499	−	0.0706832i	$-0.0225179\pi$
$32$	0	0
$33$	−2.76161e7	−0.705656
$34$	0	0
$35$	3.37029e7i	0.641692i
$36$	0	0
$37$	−6.90047e7	−0.995108	−0.497554	−	0.867433i	$-0.665768\pi$
$37$	−6.90047e7	−0.995108	−0.497554	+	0.867433i	$0.665768\pi$
$38$	0	0
$39$	− 6.36510e7i	− 0.705476i
$40$	0	0
$41$	−1.80477e8	−1.55777	−0.778885	−	0.627166i	$-0.784215\pi$
$41$	−1.80477e8	−1.55777	−0.778885	+	0.627166i	$0.784215\pi$
$42$	0	0
$43$	3.98178e6i	0.0270854i	0.999908	+	0.0135427i	$0.00431090\pi$
$43$	3.98178e6i	0.0270854i	−0.999908	+	0.0135427i	$0.995689\pi$
$44$	0	0
$45$	9.34774e7	0.506575
$46$	0	0
$47$	2.35895e7i	0.102856i	0.998677	+	0.0514279i	$0.0163772\pi$
$47$	2.35895e7i	0.102856i	−0.998677	+	0.0514279i	$0.983623\pi$
$48$	0	0
$49$	8.36687e7	0.296198
$50$	0	0
$51$	2.29968e8i	0.666526i
$52$	0	0
$53$	3.40010e8	0.813042	0.406521	−	0.913641i	$-0.366742\pi$
$53$	3.40010e8	0.813042	0.406521	+	0.913641i	$0.366742\pi$
$54$	0	0
$55$	4.67445e8i	0.928788i
$56$	0	0
$57$	3.31426e7	0.0550823
$58$	0	0
$59$	− 6.12444e8i	− 0.856655i	−0.903624	−	0.428328i	$-0.859103\pi$
$59$	− 6.12444e8i	− 0.856655i	0.903624	−	0.428328i	$-0.140897\pi$
$60$	0	0
$61$	1.06549e8	0.126153	0.0630767	−	0.998009i	$-0.479909\pi$
$61$	1.06549e8	0.126153	0.0630767	+	0.998009i	$0.479909\pi$
$62$	0	0
$63$	5.51405e8i	0.555607i
$64$	0	0
$65$	−1.07739e9	−0.928552
$66$	0	0
$67$	− 1.52228e9i	− 1.12751i	−0.825942	−	0.563756i	$-0.809356\pi$
$67$	− 1.52228e9i	− 1.12751i	0.825942	−	0.563756i	$-0.190644\pi$
$68$	0	0
$69$	−1.28814e9	−0.823606
$70$	0	0
$71$	2.90544e9i	1.61035i	0.593039	+	0.805174i	$0.297928\pi$
$71$	2.90544e9i	1.61035i	−0.593039	+	0.805174i	$0.702072\pi$
$72$	0	0
$73$	−6.23479e7	−0.0300751	−0.0150376	−	0.999887i	$-0.504787\pi$
$73$	−6.23479e7	−0.0300751	−0.0150376	+	0.999887i	$0.504787\pi$
$74$	0	0
$75$	− 5.72223e8i	− 0.241134i
$76$	0	0
$77$	−2.75736e9	−1.01869
$78$	0	0
$79$	− 2.35466e9i	− 0.765230i	−0.923908	−	0.382615i	$-0.875024\pi$
$79$	− 2.35466e9i	− 0.765230i	0.923908	−	0.382615i	$-0.124976\pi$
$80$	0	0
$81$	3.51808e8	0.100898
$82$	0	0
$83$	7.35873e9i	1.86815i	0.357073	+	0.934077i	$0.383775\pi$
$83$	7.35873e9i	1.86815i	−0.357073	+	0.934077i	$0.616225\pi$
$84$	0	0
$85$	3.89256e9	0.877286
$86$	0	0
$87$	7.54928e8i	0.151464i
$88$	0	0
$89$	6.48010e9	1.16046	0.580232	−	0.814451i	$-0.302962\pi$
$89$	6.48010e9	1.16046	0.580232	+	0.814451i	$0.302962\pi$
$90$	0	0
$91$	− 6.35531e9i	− 1.01843i
$92$	0	0
$93$	5.71529e8	0.0821531
$94$	0	0
$95$	− 5.60989e8i	− 0.0724997i
$96$	0	0
$97$	1.17007e10	1.36256	0.681278	−	0.732025i	$-0.261425\pi$
$97$	1.17007e10	1.36256	0.681278	+	0.732025i	$0.261425\pi$
$98$	0	0
$99$	7.64775e9i	0.804188i
$100$	0	0
$101$	−6.01091e9	−0.571918	−0.285959	−	0.958242i	$-0.592312\pi$
$101$	−6.01091e9	−0.571918	−0.285959	+	0.958242i	$0.592312\pi$
$102$	0	0
$103$	2.00198e10i	1.72692i	0.504415	+	0.863462i	$0.331708\pi$
$103$	2.00198e10i	1.72692i	−0.504415	+	0.863462i	$0.668292\pi$
$104$	0	0
$105$	−4.75939e9	−0.372910
$106$	0	0
$107$	− 6.67324e9i	− 0.475793i	−0.971291	−	0.237896i	$-0.923542\pi$
$107$	− 6.67324e9i	− 0.475793i	0.971291	−	0.237896i	$-0.0764579\pi$
$108$	0	0
$109$	1.14149e10	0.741891	0.370945	−	0.928655i	$-0.379034\pi$
$109$	1.14149e10	0.741891	0.370945	+	0.928655i	$0.379034\pi$
$110$	0	0
$111$	− 9.74457e9i	− 0.578293i
$112$	0	0
$113$	−1.54730e10	−0.839813	−0.419906	−	0.907567i	$-0.637937\pi$
$113$	−1.54730e10	−0.839813	−0.419906	+	0.907567i	$0.637937\pi$
$114$	0	0
$115$	2.18038e10i	1.08403i
$116$	0	0
$117$	−1.76269e10	−0.803983
$118$	0	0
$119$	2.29615e10i	0.962198i
$120$	0	0
$121$	−1.23060e10	−0.474451
$122$	0	0
$123$	− 2.54863e10i	− 0.905276i
$124$	0	0
$125$	−3.30285e10	−1.08228
$126$	0	0
$127$	− 5.28056e10i	− 1.59831i	−0.601125	−	0.799155i	$-0.705281\pi$
$127$	− 5.28056e10i	− 1.59831i	0.601125	−	0.799155i	$-0.294719\pi$
$128$	0	0
$129$	−5.62291e8	−0.0157403
$130$	0	0
$131$	8.05495e9i	0.208788i	0.994536	+	0.104394i	$0.0332904\pi$
$131$	8.05495e9i	0.208788i	−0.994536	+	0.104394i	$0.966710\pi$
$132$	0	0
$133$	3.30916e9	0.0795170
$134$	0	0
$135$	3.31324e10i	0.738897i
$136$	0	0
$137$	−3.22621e10	−0.668482	−0.334241	−	0.942488i	$-0.608480\pi$
$137$	−3.22621e10	−0.668482	−0.334241	+	0.942488i	$0.608480\pi$
$138$	0	0
$139$	− 7.30560e10i	− 1.40793i	−0.710233	−	0.703967i	$-0.751410\pi$
$139$	− 7.30560e10i	− 1.40793i	0.710233	−	0.703967i	$-0.248590\pi$
$140$	0	0
$141$	−3.33121e9	−0.0597732
$142$	0	0
$143$	− 8.81455e10i	− 1.47408i
$144$	0	0
$145$	1.27783e10	0.199358
$146$	0	0
$147$	1.18154e10i	0.172131i
$148$	0	0
$149$	−5.74043e10	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$149$	−5.74043e10	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$150$	0	0
$151$	− 1.18153e11i	− 1.50508i	−0.658546	−	0.752541i	$-0.728828\pi$
$151$	− 1.18153e11i	− 1.50508i	0.658546	−	0.752541i	$-0.271172\pi$
$152$	0	0
$153$	6.36853e10	0.759594
$154$	0	0
$155$	− 9.67400e9i	− 0.108130i
$156$	0	0
$157$	−1.60999e11	−1.68782	−0.843908	−	0.536488i	$-0.819751\pi$
$157$	−1.60999e11	−1.68782	−0.843908	+	0.536488i	$0.819751\pi$
$158$	0	0
$159$	4.80149e10i	0.472488i
$160$	0	0
$161$	−1.28616e11	−1.18896
$162$	0	0
$163$	1.70734e11i	1.48383i	0.670496	+	0.741913i	$0.266081\pi$
$163$	1.70734e11i	1.48383i	−0.670496	+	0.741913i	$0.733919\pi$
$164$	0	0
$165$	−6.60107e10	−0.539752
$166$	0	0
$167$	1.52124e11i	1.17116i	0.810616	+	0.585578i	$0.199132\pi$
$167$	1.52124e11i	1.17116i	−0.810616	+	0.585578i	$0.800868\pi$
$168$	0	0
$169$	6.53037e10	0.473701
$170$	0	0
$171$	− 9.17820e9i	− 0.0627736i
$172$	0	0
$173$	3.66088e10	0.236241	0.118120	−	0.992999i	$-0.462313\pi$
$173$	3.66088e10	0.236241	0.118120	+	0.992999i	$0.462313\pi$
$174$	0	0
$175$	− 5.71343e10i	− 0.348102i
$176$	0	0
$177$	8.64868e10	0.497833
$178$	0	0
$179$	− 1.44913e11i	− 0.788572i	−0.918988	−	0.394286i	$-0.870992\pi$
$179$	− 1.44913e11i	− 0.788572i	0.918988	−	0.394286i	$-0.129008\pi$
$180$	0	0
$181$	1.71278e10	0.0881674	0.0440837	−	0.999028i	$-0.485963\pi$
$181$	1.71278e10	0.0881674	0.0440837	+	0.999028i	$0.485963\pi$
$182$	0	0
$183$	1.50464e10i	0.0733123i
$184$	0	0
$185$	−1.64942e11	−0.761153
$186$	0	0
$187$	3.18466e11i	1.39269i
$188$	0	0
$189$	−1.95441e11	−0.810415
$190$	0	0
$191$	5.63176e10i	0.221553i	0.993845	+	0.110776i	$0.0353337\pi$
$191$	5.63176e10i	0.221553i	−0.993845	+	0.110776i	$0.964666\pi$
$192$	0	0
$193$	3.09683e11	1.15646	0.578231	−	0.815873i	$-0.303743\pi$
$193$	3.09683e11	1.15646	0.578231	+	0.815873i	$0.303743\pi$
$194$	0	0
$195$	− 1.52145e11i	− 0.539615i
$196$	0	0
$197$	−5.78934e11	−1.95118	−0.975592	−	0.219593i	$-0.929527\pi$
$197$	−5.78934e11	−1.95118	−0.975592	+	0.219593i	$0.929527\pi$
$198$	0	0
$199$	− 4.41355e11i	− 1.41424i	−0.707094	−	0.707120i	$-0.749994\pi$
$199$	− 4.41355e11i	− 1.41424i	0.707094	−	0.707120i	$-0.250006\pi$
$200$	0	0
$201$	2.14971e11	0.655237
$202$	0	0
$203$	7.53767e10i	0.218654i
$204$	0	0
$205$	−4.31394e11	−1.19153
$206$	0	0
$207$	3.56727e11i	0.938608i
$208$	0	0
$209$	4.58967e10	0.115093
$210$	0	0
$211$	− 2.13171e11i	− 0.509702i	−0.966980	−	0.254851i	$-0.917974\pi$
$211$	− 2.13171e11i	− 0.509702i	0.966980	−	0.254851i	$-0.0820264\pi$
$212$	0	0
$213$	−4.10294e11	−0.935831
$214$	0	0
$215$	9.51763e9i	0.0207175i
$216$	0	0
$217$	5.70650e10	0.118596
$218$	0	0
$219$	− 8.80452e9i	− 0.0174777i
$220$	0	0
$221$	−7.34016e11	−1.39234
$222$	0	0
$223$	4.36203e11i	0.790979i	0.918471	+	0.395489i	$0.129425\pi$
$223$	4.36203e11i	0.790979i	−0.918471	+	0.395489i	$0.870575\pi$
$224$	0	0
$225$	−1.58466e11	−0.274804
$226$	0	0
$227$	− 1.09792e12i	− 1.82155i	−0.412900	−	0.910776i	$-0.635484\pi$
$227$	− 1.09792e12i	− 1.82155i	0.412900	−	0.910776i	$-0.364516\pi$
$228$	0	0
$229$	−1.07743e12	−1.71085	−0.855425	−	0.517926i	$-0.826704\pi$
$229$	−1.07743e12	−1.71085	−0.855425	+	0.517926i	$0.826704\pi$
$230$	0	0
$231$	− 3.89384e11i	− 0.591995i
$232$	0	0
$233$	−2.90646e11	−0.423238	−0.211619	−	0.977352i	$-0.567874\pi$
$233$	−2.90646e11	−0.423238	−0.211619	+	0.977352i	$0.567874\pi$
$234$	0	0
$235$	5.63858e10i	0.0786738i
$236$	0	0
$237$	3.32515e11	0.444703
$238$	0	0
$239$	1.30743e12i	1.67659i	0.545213	+	0.838297i	$0.316449\pi$
$239$	1.30743e12i	1.67659i	−0.545213	+	0.838297i	$0.683551\pi$
$240$	0	0
$241$	−7.86902e11	−0.967912	−0.483956	−	0.875092i	$-0.660800\pi$
$241$	−7.86902e11	−0.967912	−0.483956	+	0.875092i	$0.660800\pi$
$242$	0	0
$243$	8.68171e11i	1.02465i
$244$	0	0
$245$	1.99993e11	0.226560
$246$	0	0
$247$	1.05785e11i	0.115064i
$248$	0	0
$249$	−1.03917e12	−1.08565
$250$	0	0
$251$	1.26583e12i	1.27059i	0.772268	+	0.635296i	$0.219122\pi$
$251$	1.26583e12i	1.27059i	−0.772268	+	0.635296i	$0.780878\pi$
$252$	0	0
$253$	−1.78385e12	−1.72090
$254$	0	0
$255$	5.49692e11i	0.509822i
$256$	0	0
$257$	−5.31051e11	−0.473664	−0.236832	−	0.971551i	$-0.576109\pi$
$257$	−5.31051e11	−0.473664	−0.236832	+	0.971551i	$0.576109\pi$
$258$	0	0
$259$	− 9.72959e11i	− 0.834825i
$260$	0	0
$261$	2.09063e11	0.172613
$262$	0	0
$263$	− 1.07957e11i	− 0.0857971i	−0.999079	−	0.0428985i	$-0.986341\pi$
$263$	− 1.07957e11i	− 0.0857971i	0.999079	−	0.0428985i	$-0.0136592\pi$
$264$	0	0
$265$	8.12726e11	0.621891
$266$	0	0
$267$	9.15093e11i	0.674387i
$268$	0	0
$269$	−2.34220e12	−1.66289	−0.831444	−	0.555608i	$-0.812485\pi$
$269$	−2.34220e12	−1.66289	−0.831444	+	0.555608i	$0.812485\pi$
$270$	0	0
$271$	− 7.16998e11i	− 0.490537i	−0.969455	−	0.245268i	$-0.921124\pi$
$271$	− 7.16998e11i	− 0.490537i	0.969455	−	0.245268i	$-0.0788761\pi$
$272$	0	0
$273$	8.97472e11	0.591844
$274$	0	0
$275$	− 7.92428e11i	− 0.503844i
$276$	0	0
$277$	−3.04925e12	−1.86979	−0.934897	−	0.354918i	$-0.884509\pi$
$277$	−3.04925e12	−1.86979	−0.934897	+	0.354918i	$0.884509\pi$
$278$	0	0
$279$	− 1.58274e11i	− 0.0936243i
$280$	0	0
$281$	2.41188e12	1.37665	0.688326	−	0.725402i	$-0.258346\pi$
$281$	2.41188e12	1.37665	0.688326	+	0.725402i	$0.258346\pi$
$282$	0	0
$283$	2.96691e11i	0.163445i	0.996655	+	0.0817226i	$0.0260421\pi$
$283$	2.96691e11i	0.163445i	−0.996655	+	0.0817226i	$0.973958\pi$
$284$	0	0
$285$	7.92206e10	0.0421322
$286$	0	0
$287$	− 2.54471e12i	− 1.30686i
$288$	0	0
$289$	6.35972e11	0.315463
$290$	0	0
$291$	1.65233e12i	0.791830i
$292$	0	0
$293$	−1.28558e12	−0.595337	−0.297668	−	0.954669i	$-0.596209\pi$
$293$	−1.28558e12	−0.595337	−0.297668	+	0.954669i	$0.596209\pi$
$294$	0	0
$295$	− 1.46392e12i	− 0.655251i
$296$	0	0
$297$	−2.71069e12	−1.17300
$298$	0	0
$299$	− 4.11152e12i	− 1.72047i
$300$	0	0
$301$	−5.61426e10	−0.0227227
$302$	0	0
$303$	− 8.48837e11i	− 0.332362i
$304$	0	0
$305$	2.54683e11	0.0964941
$306$	0	0
$307$	3.17537e12i	1.16440i	0.813045	+	0.582201i	$0.197808\pi$
$307$	3.17537e12i	1.16440i	−0.813045	+	0.582201i	$0.802192\pi$
$308$	0	0
$309$	−2.82711e12	−1.00358
$310$	0	0
$311$	1.07044e12i	0.367925i	0.982933	+	0.183963i	$0.0588926\pi$
$311$	1.07044e12i	0.367925i	−0.982933	+	0.183963i	$0.941107\pi$
$312$	0	0
$313$	1.27783e12	0.425356	0.212678	−	0.977122i	$-0.431781\pi$
$313$	1.27783e12	0.425356	0.212678	+	0.977122i	$0.431781\pi$
$314$	0	0
$315$	1.31802e12i	0.424981i
$316$	0	0
$317$	−1.05881e12	−0.330767	−0.165383	−	0.986229i	$-0.552886\pi$
$317$	−1.05881e12	−0.330767	−0.165383	+	0.986229i	$0.552886\pi$
$318$	0	0
$319$	1.04544e12i	0.316481i
$320$	0	0
$321$	9.42368e11	0.276500
$322$	0	0
$323$	− 3.82196e11i	− 0.108711i
$324$	0	0
$325$	1.82643e12	0.503716
$326$	0	0
$327$	1.61197e12i	0.431139i
$328$	0	0
$329$	−3.32608e11	−0.0862887
$330$	0	0
$331$	− 5.36335e12i	− 1.34988i	−0.737872	−	0.674941i	$-0.764169\pi$
$331$	− 5.36335e12i	− 1.34988i	0.737872	−	0.674941i	$-0.235831\pi$
$332$	0	0
$333$	−2.69857e12	−0.659041
$334$	0	0
$335$	− 3.63870e12i	− 0.862428i
$336$	0	0
$337$	4.64297e11	0.106819	0.0534093	−	0.998573i	$-0.482991\pi$
$337$	4.64297e11	0.106819	0.0534093	+	0.998573i	$0.482991\pi$
$338$	0	0
$339$	− 2.18504e12i	− 0.488045i
$340$	0	0
$341$	7.91468e11	0.171657
$342$	0	0
$343$	5.16258e12i	1.08742i
$344$	0	0
$345$	−3.07905e12	−0.629971
$346$	0	0
$347$	− 4.61318e12i	− 0.916966i	−0.888703	−	0.458483i	$-0.848393\pi$
$347$	− 4.61318e12i	− 0.916966i	0.888703	−	0.458483i	$-0.151607\pi$
$348$	0	0
$349$	4.43738e12	0.857038	0.428519	−	0.903533i	$-0.359036\pi$
$349$	4.43738e12	0.857038	0.428519	+	0.903533i	$0.359036\pi$
$350$	0	0
$351$	− 6.24773e12i	− 1.17270i
$352$	0	0
$353$	3.32177e12	0.606033	0.303016	−	0.952985i	$-0.402006\pi$
$353$	3.32177e12	0.606033	0.303016	+	0.952985i	$0.402006\pi$
$354$	0	0
$355$	6.94485e12i	1.23175i
$356$	0	0
$357$	−3.24252e12	−0.559168
$358$	0	0
$359$	3.05100e12i	0.511646i	0.966724	+	0.255823i	$0.0823464\pi$
$359$	3.05100e12i	0.511646i	−0.966724	+	0.255823i	$0.917654\pi$
$360$	0	0
$361$	6.07598e12	0.991016
$362$	0	0
$363$	− 1.73781e12i	− 0.275720i
$364$	0	0
$365$	−1.49030e11	−0.0230043
$366$	0	0
$367$	− 4.42848e12i	− 0.665157i	−0.943076	−	0.332579i	$-0.892081\pi$
$367$	− 4.42848e12i	− 0.665157i	0.943076	−	0.332579i	$-0.107919\pi$
$368$	0	0
$369$	−7.05794e12	−1.03168
$370$	0	0
$371$	4.79411e12i	0.682084i
$372$	0	0
$373$	1.00283e13	1.38894	0.694472	−	0.719519i	$-0.255638\pi$
$373$	1.00283e13	1.38894	0.694472	+	0.719519i	$0.255638\pi$
$374$	0	0
$375$	− 4.66415e12i	− 0.628950i
$376$	0	0
$377$	−2.40959e12	−0.316400
$378$	0	0
$379$	1.21959e13i	1.55962i	0.626015	+	0.779811i	$0.284685\pi$
$379$	1.21959e13i	1.55962i	−0.626015	+	0.779811i	$0.715315\pi$
$380$	0	0
$381$	7.45699e12	0.928835
$382$	0	0
$383$	2.22978e12i	0.270563i	0.990807	+	0.135281i	$0.0431939\pi$
$383$	2.22978e12i	0.270563i	−0.990807	+	0.135281i	$0.956806\pi$
$384$	0	0
$385$	−6.59091e12	−0.779187
$386$	0	0
$387$	1.55716e11i	0.0179381i
$388$	0	0
$389$	5.22537e12	0.586636	0.293318	−	0.956015i	$-0.405240\pi$
$389$	5.22537e12	0.586636	0.293318	+	0.956015i	$0.405240\pi$
$390$	0	0
$391$	1.48547e13i	1.62548i
$392$	0	0
$393$	−1.13749e12	−0.121334
$394$	0	0
$395$	− 5.62832e12i	− 0.585320i
$396$	0	0
$397$	−6.97467e12	−0.707247	−0.353623	−	0.935388i	$-0.615051\pi$
$397$	−6.97467e12	−0.707247	−0.353623	+	0.935388i	$0.615051\pi$
$398$	0	0
$399$	4.67307e11i	0.0462102i
$400$	0	0
$401$	−9.07619e12	−0.875350	−0.437675	−	0.899133i	$-0.644198\pi$
$401$	−9.07619e12	−0.875350	−0.437675	+	0.899133i	$0.644198\pi$
$402$	0	0
$403$	1.82422e12i	0.171613i
$404$	0	0
$405$	8.40926e11	0.0771760
$406$	0	0
$407$	− 1.34945e13i	− 1.20833i
$408$	0	0
$409$	−4.28568e12	−0.374458	−0.187229	−	0.982316i	$-0.559951\pi$
$409$	−4.28568e12	−0.374458	−0.187229	+	0.982316i	$0.559951\pi$
$410$	0	0
$411$	− 4.55592e12i	− 0.388479i
$412$	0	0
$413$	8.63538e12	0.718673
$414$	0	0
$415$	1.75895e13i	1.42894i
$416$	0	0
$417$	1.03167e13	0.818200
$418$	0	0
$419$	− 6.03988e12i	− 0.467690i	−0.972274	−	0.233845i	$-0.924869\pi$
$419$	− 6.03988e12i	− 0.467690i	0.972274	−	0.233845i	$-0.0751308\pi$
$420$	0	0
$421$	1.04217e13	0.788005	0.394003	−	0.919109i	$-0.371090\pi$
$421$	1.04217e13	0.788005	0.394003	+	0.919109i	$0.371090\pi$
$422$	0	0
$423$	9.22514e11i	0.0681194i
$424$	0	0
$425$	−6.59881e12	−0.475905
$426$	0	0
$427$	1.50232e12i	0.105834i
$428$	0	0
$429$	1.24475e13	0.856638
$430$	0	0
$431$	2.31869e13i	1.55904i	0.626379	+	0.779519i	$0.284536\pi$
$431$	2.31869e13i	1.55904i	−0.626379	+	0.779519i	$0.715464\pi$
$432$	0	0
$433$	2.76660e13	1.81763	0.908817	−	0.417195i	$-0.136987\pi$
$433$	2.76660e13	1.81763	0.908817	+	0.417195i	$0.136987\pi$
$434$	0	0
$435$	1.80450e12i	0.115854i
$436$	0	0
$437$	2.14084e12	0.134331
$438$	0	0
$439$	1.25407e13i	0.769130i	0.923098	+	0.384565i	$0.125649\pi$
$439$	1.25407e13i	0.769130i	−0.923098	+	0.384565i	$0.874351\pi$
$440$	0	0
$441$	3.27203e12	0.196166
$442$	0	0
$443$	5.69127e12i	0.333573i	0.985993	+	0.166786i	$0.0533390\pi$
$443$	5.69127e12i	0.333573i	−0.985993	+	0.166786i	$0.946661\pi$
$444$	0	0
$445$	1.54893e13	0.887632
$446$	0	0
$447$	− 8.10641e12i	− 0.454246i
$448$	0	0
$449$	−1.56305e13	−0.856529	−0.428265	−	0.903653i	$-0.640875\pi$
$449$	−1.56305e13	−0.856529	−0.428265	+	0.903653i	$0.640875\pi$
$450$	0	0
$451$	− 3.52940e13i	− 1.89155i
$452$	0	0
$453$	1.66851e13	0.874657
$454$	0	0
$455$	− 1.51911e13i	− 0.778989i
$456$	0	0
$457$	3.13478e13	1.57263	0.786314	−	0.617828i	$-0.211987\pi$
$457$	3.13478e13	1.57263	0.786314	+	0.617828i	$0.211987\pi$
$458$	0	0
$459$	2.25728e13i	1.10795i
$460$	0	0
$461$	3.82856e13	1.83878	0.919392	−	0.393344i	$-0.128682\pi$
$461$	3.82856e13	1.83878	0.919392	+	0.393344i	$0.128682\pi$
$462$	0	0
$463$	− 1.18367e13i	− 0.556320i	−0.960535	−	0.278160i	$-0.910275\pi$
$463$	− 1.18367e13i	− 0.556320i	0.960535	−	0.278160i	$-0.0897245\pi$
$464$	0	0
$465$	1.36612e12	0.0628384
$466$	0	0
$467$	− 1.15025e13i	− 0.517857i	−0.965897	−	0.258928i	$-0.916631\pi$
$467$	− 1.15025e13i	− 0.517857i	0.965897	−	0.258928i	$-0.0833694\pi$
$468$	0	0
$469$	2.14640e13	0.945902
$470$	0	0
$471$	− 2.27356e13i	− 0.980850i
$472$	0	0
$473$	−7.78674e11	−0.0328890
$474$	0	0
$475$	9.51008e11i	0.0393293i
$476$	0	0
$477$	1.32968e13	0.538462
$478$	0	0
$479$	1.10497e13i	0.438202i	0.975702	+	0.219101i	$0.0703124\pi$
$479$	1.10497e13i	0.438202i	−0.975702	+	0.219101i	$0.929688\pi$
$480$	0	0
$481$	3.11029e13	1.20802
$482$	0	0
$483$	− 1.81627e13i	− 0.690947i
$484$	0	0
$485$	2.79682e13	1.04221
$486$	0	0
$487$	− 5.28952e13i	− 1.93095i	−0.260494	−	0.965475i	$-0.583886\pi$
$487$	− 5.28952e13i	− 1.93095i	0.260494	−	0.965475i	$-0.416114\pi$
$488$	0	0
$489$	−2.41104e13	−0.862304
$490$	0	0
$491$	2.91937e13i	1.02301i	0.859279	+	0.511507i	$0.170913\pi$
$491$	2.91937e13i	1.02301i	−0.859279	+	0.511507i	$0.829087\pi$
$492$	0	0
$493$	8.70574e12	0.298931
$494$	0	0
$495$	1.82804e13i	0.615119i
$496$	0	0
$497$	−4.09663e13	−1.35097
$498$	0	0
$499$	2.41620e12i	0.0780964i	0.999237	+	0.0390482i	$0.0124326\pi$
$499$	2.41620e12i	0.0780964i	−0.999237	+	0.0390482i	$0.987567\pi$
$500$	0	0
$501$	−2.14823e13	−0.680600
$502$	0	0
$503$	1.71848e13i	0.533709i	0.963737	+	0.266855i	$0.0859844\pi$
$503$	1.71848e13i	0.533709i	−0.963737	+	0.266855i	$0.914016\pi$
$504$	0	0
$505$	−1.43679e13	−0.437457
$506$	0	0
$507$	9.22192e12i	0.275284i
$508$	0	0
$509$	−4.00990e13	−1.17367	−0.586833	−	0.809708i	$-0.699625\pi$
$509$	−4.00990e13	−1.17367	−0.586833	+	0.809708i	$0.699625\pi$
$510$	0	0
$511$	− 8.79097e11i	− 0.0252309i
$512$	0	0
$513$	3.25315e12	0.0915623
$514$	0	0
$515$	4.78532e13i	1.32091i
$516$	0	0
$517$	−4.61314e12	−0.124895
$518$	0	0
$519$	5.16975e12i	0.137288i
$520$	0	0
$521$	−4.33911e12	−0.113035	−0.0565173	−	0.998402i	$-0.518000\pi$
$521$	−4.33911e12	−0.113035	−0.0565173	+	0.998402i	$0.518000\pi$
$522$	0	0
$523$	− 1.58967e13i	− 0.406254i	−0.979152	−	0.203127i	$-0.934890\pi$
$523$	− 1.58967e13i	− 0.406254i	0.979152	−	0.203127i	$-0.0651104\pi$
$524$	0	0
$525$	8.06827e12	0.202295
$526$	0	0
$527$	− 6.59081e12i	− 0.162138i
$528$	0	0
$529$	−4.17808e13	−1.00855
$530$	0	0
$531$	− 2.39509e13i	− 0.567347i
$532$	0	0
$533$	8.13475e13	1.89107
$534$	0	0
$535$	− 1.59510e13i	− 0.363931i
$536$	0	0
$537$	2.04640e13	0.458267
$538$	0	0
$539$	1.63622e13i	0.359664i
$540$	0	0
$541$	7.32282e12	0.158013	0.0790063	−	0.996874i	$-0.474825\pi$
$541$	7.32282e12	0.158013	0.0790063	+	0.996874i	$0.474825\pi$
$542$	0	0
$543$	2.41872e12i	0.0512373i
$544$	0	0
$545$	2.72850e13	0.567468
$546$	0	0
$547$	3.28228e13i	0.670254i	0.942173	+	0.335127i	$0.108779\pi$
$547$	3.28228e13i	0.670254i	−0.942173	+	0.335127i	$0.891221\pi$
$548$	0	0
$549$	4.16680e12	0.0835490
$550$	0	0
$551$	− 1.25466e12i	− 0.0247040i
$552$	0	0
$553$	3.32004e13	0.641974
$554$	0	0
$555$	− 2.32924e13i	− 0.442333i
$556$	0	0
$557$	4.56379e13	0.851236	0.425618	−	0.904903i	$-0.360057\pi$
$557$	4.56379e13	0.851236	0.425618	+	0.904903i	$0.360057\pi$
$558$	0	0
$559$	− 1.79473e12i	− 0.0328806i
$560$	0	0
$561$	−4.49724e13	−0.809342
$562$	0	0
$563$	− 5.63834e13i	− 0.996803i	−0.866946	−	0.498402i	$-0.833921\pi$
$563$	− 5.63834e13i	− 0.996803i	0.866946	−	0.498402i	$-0.166079\pi$
$564$	0	0
$565$	−3.69851e13	−0.642368
$566$	0	0
$567$	4.96045e12i	0.0846459i
$568$	0	0
$569$	−2.19473e13	−0.367976	−0.183988	−	0.982928i	$-0.558901\pi$
$569$	−2.19473e13	−0.367976	−0.183988	+	0.982928i	$0.558901\pi$
$570$	0	0
$571$	− 7.54189e13i	− 1.24251i	−0.783609	−	0.621254i	$-0.786623\pi$
$571$	− 7.54189e13i	− 1.24251i	0.783609	−	0.621254i	$-0.213377\pi$
$572$	0	0
$573$	−7.95295e12	−0.128752
$574$	0	0
$575$	− 3.69626e13i	− 0.588062i
$576$	0	0
$577$	5.09544e13	0.796714	0.398357	−	0.917230i	$-0.369581\pi$
$577$	5.09544e13	0.796714	0.398357	+	0.917230i	$0.369581\pi$
$578$	0	0
$579$	4.37323e13i	0.672062i
$580$	0	0
$581$	−1.03757e14	−1.56725
$582$	0	0
$583$	6.64922e13i	0.987252i
$584$	0	0
$585$	−4.21335e13	−0.614963
$586$	0	0
$587$	5.02225e13i	0.720622i	0.932832	+	0.360311i	$0.117329\pi$
$587$	5.02225e13i	0.720622i	−0.932832	+	0.360311i	$0.882671\pi$
$588$	0	0
$589$	−9.49855e11	−0.0133993
$590$	0	0
$591$	− 8.17548e13i	− 1.13390i
$592$	0	0
$593$	3.10208e13	0.423038	0.211519	−	0.977374i	$-0.432159\pi$
$593$	3.10208e13	0.423038	0.211519	+	0.977374i	$0.432159\pi$
$594$	0	0
$595$	5.48847e13i	0.735980i
$596$	0	0
$597$	6.23264e13	0.821865
$598$	0	0
$599$	1.11474e14i	1.44557i	0.691074	+	0.722784i	$0.257138\pi$
$599$	1.11474e14i	1.44557i	−0.691074	+	0.722784i	$0.742862\pi$
$600$	0	0
$601$	−2.70844e13	−0.345420	−0.172710	−	0.984973i	$-0.555252\pi$
$601$	−2.70844e13	−0.345420	−0.172710	+	0.984973i	$0.555252\pi$
$602$	0	0
$603$	− 5.95319e13i	− 0.746730i
$604$	0	0
$605$	−2.94151e13	−0.362905
$606$	0	0
$607$	− 1.27275e14i	− 1.54454i	−0.635293	−	0.772271i	$-0.719121\pi$
$607$	− 1.27275e14i	− 1.54454i	0.635293	−	0.772271i	$-0.280879\pi$
$608$	0	0
$609$	−1.06444e13	−0.127068
$610$	0	0
$611$	− 1.06326e13i	− 0.124863i
$612$	0	0
$613$	−3.61511e13	−0.417656	−0.208828	−	0.977952i	$-0.566965\pi$
$613$	−3.61511e13	−0.417656	−0.208828	+	0.977952i	$0.566965\pi$
$614$	0	0
$615$	− 6.09198e13i	− 0.692441i
$616$	0	0
$617$	−2.93353e12	−0.0328069	−0.0164035	−	0.999865i	$-0.505222\pi$
$617$	−2.93353e12	−0.0328069	−0.0164035	+	0.999865i	$0.505222\pi$
$618$	0	0
$619$	− 7.96221e13i	− 0.876153i	−0.898938	−	0.438077i	$-0.855660\pi$
$619$	− 7.96221e13i	− 0.876153i	0.898938	−	0.438077i	$-0.144340\pi$
$620$	0	0
$621$	−1.26439e14	−1.36906
$622$	0	0
$623$	9.13686e13i	0.973546i
$624$	0	0
$625$	−3.93764e13	−0.412892
$626$	0	0
$627$	6.48134e12i	0.0668848i
$628$	0	0
$629$	−1.12373e14	−1.14133
$630$	0	0
$631$	1.14047e14i	1.14008i	0.821617	+	0.570040i	$0.193072\pi$
$631$	1.14047e14i	1.14008i	−0.821617	+	0.570040i	$0.806928\pi$
$632$	0	0
$633$	3.01032e13	0.296206
$634$	0	0
$635$	− 1.26221e14i	− 1.22254i
$636$	0	0
$637$	−3.77124e13	−0.359573
$638$	0	0
$639$	1.13623e14i	1.06650i
$640$	0	0
$641$	1.73595e14	1.60416	0.802078	−	0.597219i	$-0.203728\pi$
$641$	1.73595e14	1.60416	0.802078	+	0.597219i	$0.203728\pi$
$642$	0	0
$643$	1.19143e14i	1.08396i	0.840391	+	0.541981i	$0.182325\pi$
$643$	1.19143e14i	1.08396i	−0.840391	+	0.541981i	$0.817675\pi$
$644$	0	0
$645$	−1.34404e12	−0.0120397
$646$	0	0
$647$	1.33937e14i	1.18135i	0.806909	+	0.590675i	$0.201139\pi$
$647$	1.33937e14i	1.18135i	−0.806909	+	0.590675i	$0.798861\pi$
$648$	0	0
$649$	1.19769e14	1.04021
$650$	0	0
$651$	8.05849e12i	0.0689206i
$652$	0	0
$653$	−2.95366e13	−0.248768	−0.124384	−	0.992234i	$-0.539695\pi$
$653$	−2.95366e13	−0.248768	−0.124384	+	0.992234i	$0.539695\pi$
$654$	0	0
$655$	1.92537e13i	0.159701i
$656$	0	0
$657$	−2.43824e12	−0.0199182
$658$	0	0
$659$	1.56857e14i	1.26205i	0.775763	+	0.631024i	$0.217365\pi$
$659$	1.56857e14i	1.26205i	−0.775763	+	0.631024i	$0.782635\pi$
$660$	0	0
$661$	−1.35298e14	−1.07222	−0.536111	−	0.844147i	$-0.680107\pi$
$661$	−1.35298e14	−1.07222	−0.536111	+	0.844147i	$0.680107\pi$
$662$	0	0
$663$	− 1.03655e14i	− 0.809136i
$664$	0	0
$665$	7.90987e12	0.0608221
$666$	0	0
$667$	4.87644e13i	0.369380i
$668$	0	0
$669$	−6.15989e13	−0.459666
$670$	0	0
$671$	2.08366e13i	0.153184i
$672$	0	0
$673$	−5.56400e13	−0.403006	−0.201503	−	0.979488i	$-0.564583\pi$
$673$	−5.56400e13	−0.403006	−0.201503	+	0.979488i	$0.564583\pi$
$674$	0	0
$675$	− 5.61671e13i	− 0.400833i
$676$	0	0
$677$	−1.32278e14	−0.930132	−0.465066	−	0.885276i	$-0.653969\pi$
$677$	−1.32278e14	−0.930132	−0.465066	+	0.885276i	$0.653969\pi$
$678$	0	0
$679$	1.64979e14i	1.14309i
$680$	0	0
$681$	1.55044e14	1.05857
$682$	0	0
$683$	2.69808e14i	1.81531i	0.419717	+	0.907655i	$0.362129\pi$
$683$	2.69808e14i	1.81531i	−0.419717	+	0.907655i	$0.637871\pi$
$684$	0	0
$685$	−7.71159e13	−0.511318
$686$	0	0
$687$	− 1.52151e14i	− 0.994237i
$688$	0	0
$689$	−1.53255e14	−0.987001
$690$	0	0
$691$	− 1.59626e13i	− 0.101324i	−0.998716	−	0.0506620i	$-0.983867\pi$
$691$	− 1.59626e13i	− 0.101324i	0.998716	−	0.0506620i	$-0.0161331\pi$
$692$	0	0
$693$	−1.07832e14	−0.674657
$694$	0	0
$695$	− 1.74626e14i	− 1.07692i
$696$	0	0
$697$	−2.93905e14	−1.78666
$698$	0	0
$699$	− 4.10439e13i	− 0.245959i
$700$	0	0
$701$	−3.06444e14	−1.81034	−0.905172	−	0.425046i	$-0.860258\pi$
$701$	−3.06444e14	−1.81034	−0.905172	+	0.425046i	$0.860258\pi$
$702$	0	0
$703$	1.61950e13i	0.0943202i
$704$	0	0
$705$	−7.96257e12	−0.0457202
$706$	0	0
$707$	− 8.47532e13i	− 0.479798i
$708$	0	0
$709$	1.57649e14	0.879957	0.439978	−	0.898008i	$-0.354986\pi$
$709$	1.57649e14	0.879957	0.439978	+	0.898008i	$0.354986\pi$
$710$	0	0
$711$	− 9.20836e13i	− 0.506797i
$712$	0	0
$713$	3.69178e13	0.200349
$714$	0	0
$715$	− 2.10694e14i	− 1.12751i
$716$	0	0
$717$	−1.84630e14	−0.974329
$718$	0	0
$719$	− 1.18481e14i	− 0.616603i	−0.951289	−	0.308302i	$-0.900239\pi$
$719$	− 1.18481e14i	− 0.616603i	0.951289	−	0.308302i	$-0.0997606\pi$
$720$	0	0
$721$	−2.82276e14	−1.44877
$722$	0	0
$723$	− 1.11123e14i	− 0.562488i
$724$	0	0
$725$	−2.16622e13	−0.108147
$726$	0	0
$727$	− 6.70029e13i	− 0.329930i	−0.986299	−	0.164965i	$-0.947249\pi$
$727$	− 6.70029e13i	− 0.329930i	0.986299	−	0.164965i	$-0.0527511\pi$
$728$	0	0
$729$	−1.01826e14	−0.494561
$730$	0	0
$731$	6.48427e12i	0.0310652i
$732$	0	0
$733$	1.89850e14	0.897204	0.448602	−	0.893732i	$-0.351922\pi$
$733$	1.89850e14	0.897204	0.448602	+	0.893732i	$0.351922\pi$
$734$	0	0
$735$	2.82422e13i	0.131662i
$736$	0	0
$737$	2.97696e14	1.36910
$738$	0	0
$739$	− 2.22565e14i	− 1.00980i	−0.863178	−	0.504899i	$-0.831530\pi$
$739$	− 2.22565e14i	− 1.00980i	0.863178	−	0.504899i	$-0.168470\pi$
$740$	0	0
$741$	−1.49385e13	−0.0668678
$742$	0	0
$743$	− 1.73922e14i	− 0.768086i	−0.923315	−	0.384043i	$-0.874531\pi$
$743$	− 1.73922e14i	− 0.768086i	0.923315	−	0.384043i	$-0.125469\pi$
$744$	0	0
$745$	−1.37213e14	−0.597881
$746$	0	0
$747$	2.87778e14i	1.23724i
$748$	0	0
$749$	9.40919e13	0.399156
$750$	0	0
$751$	1.86979e14i	0.782696i	0.920243	+	0.391348i	$0.127991\pi$
$751$	1.86979e14i	0.782696i	−0.920243	+	0.391348i	$0.872009\pi$
$752$	0	0
$753$	−1.78755e14	−0.738387
$754$	0	0
$755$	− 2.82421e14i	− 1.15123i
$756$	0	0
$757$	−3.73827e14	−1.50381	−0.751903	−	0.659274i	$-0.770864\pi$
$757$	−3.73827e14	−1.50381	−0.751903	+	0.659274i	$0.770864\pi$
$758$	0	0
$759$	− 2.51909e14i	− 1.00008i
$760$	0	0
$761$	3.10737e13	0.121750	0.0608751	−	0.998145i	$-0.480611\pi$
$761$	3.10737e13	0.121750	0.0608751	+	0.998145i	$0.480611\pi$
$762$	0	0
$763$	1.60949e14i	0.622394i
$764$	0	0
$765$	1.52227e14	0.581010
$766$	0	0
$767$	2.76050e14i	1.03995i
$768$	0	0
$769$	5.13424e14	1.90917	0.954584	−	0.297942i	$-0.0963003\pi$
$769$	5.13424e14	1.90917	0.954584	+	0.297942i	$0.0963003\pi$
$770$	0	0
$771$	− 7.49929e13i	− 0.275263i
$772$	0	0
$773$	−8.30807e13	−0.301025	−0.150512	−	0.988608i	$-0.548092\pi$
$773$	−8.30807e13	−0.301025	−0.150512	+	0.988608i	$0.548092\pi$
$774$	0	0
$775$	1.63997e13i	0.0586580i
$776$	0	0
$777$	1.37397e14	0.485147
$778$	0	0
$779$	4.23570e13i	0.147652i
$780$	0	0
$781$	−5.68185e14	−1.95540
$782$	0	0
$783$	7.41008e13i	0.251776i
$784$	0	0
$785$	−3.84835e14	−1.29100
$786$	0	0
$787$	2.63702e14i	0.873452i	0.899594	+	0.436726i	$0.143862\pi$
$787$	2.63702e14i	0.873452i	−0.899594	+	0.436726i	$0.856138\pi$
$788$	0	0
$789$	1.52453e13	0.0498598
$790$	0	0
$791$	− 2.18168e14i	− 0.704543i
$792$	0	0
$793$	−4.80252e13	−0.153145
$794$	0	0
$795$	1.14770e14i	0.361403i
$796$	0	0
$797$	3.15728e14	0.981797	0.490898	−	0.871217i	$-0.336669\pi$
$797$	3.15728e14	0.981797	0.490898	+	0.871217i	$0.336669\pi$
$798$	0	0
$799$	3.84151e13i	0.117969i
$800$	0	0
$801$	2.53417e14	0.768553
$802$	0	0
$803$	− 1.21927e13i	− 0.0365193i
$804$	0	0
$805$	−3.07431e14	−0.909428
$806$	0	0
$807$	− 3.30756e14i	− 0.966364i
$808$	0	0
$809$	4.13318e14	1.19273	0.596365	−	0.802714i	$-0.296611\pi$
$809$	4.13318e14	1.19273	0.596365	+	0.802714i	$0.296611\pi$
$810$	0	0
$811$	− 3.64163e14i	− 1.03799i	−0.854779	−	0.518993i	$-0.826307\pi$
$811$	− 3.64163e14i	− 1.03799i	0.854779	−	0.518993i	$-0.173693\pi$
$812$	0	0
$813$	1.01252e14	0.285069
$814$	0	0
$815$	4.08106e14i	1.13497i
$816$	0	0
$817$	9.34501e11	0.00256726
$818$	0	0
$819$	− 2.48537e14i	− 0.674485i
$820$	0	0
$821$	3.38706e14	0.908044	0.454022	−	0.890990i	$-0.349989\pi$
$821$	3.38706e14	0.908044	0.454022	+	0.890990i	$0.349989\pi$
$822$	0	0
$823$	5.34579e14i	1.41584i	0.706294	+	0.707918i	$0.250366\pi$
$823$	5.34579e14i	1.41584i	−0.706294	+	0.707918i	$0.749634\pi$
$824$	0	0
$825$	1.11904e14	0.292802
$826$	0	0
$827$	3.88434e14i	1.00413i	0.864830	+	0.502065i	$0.167426\pi$
$827$	3.88434e14i	1.00413i	−0.864830	+	0.502065i	$0.832574\pi$
$828$	0	0
$829$	−2.52125e14	−0.643936	−0.321968	−	0.946751i	$-0.604344\pi$
$829$	−2.52125e14	−0.643936	−0.321968	+	0.946751i	$0.604344\pi$
$830$	0	0
$831$	− 4.30603e14i	− 1.08660i
$832$	0	0
$833$	1.36253e14	0.339721
$834$	0	0
$835$	3.63621e14i	0.895810i
$836$	0	0
$837$	5.60991e13	0.136562
$838$	0	0
$839$	− 4.92796e14i	− 1.18538i	−0.805431	−	0.592690i	$-0.798066\pi$
$839$	− 4.92796e14i	− 1.18538i	0.805431	−	0.592690i	$-0.201934\pi$
$840$	0	0
$841$	−3.92128e14	−0.932070
$842$	0	0
$843$	3.40596e14i	0.800021i
$844$	0	0
$845$	1.56095e14	0.362331
$846$	0	0
$847$	− 1.73514e14i	− 0.398030i
$848$	0	0
$849$	−4.18975e13	−0.0949838
$850$	0	0
$851$	− 6.29448e14i	− 1.41030i
$852$	0	0
$853$	6.70341e14	1.48440	0.742200	−	0.670179i	$-0.233783\pi$
$853$	6.70341e14	1.48440	0.742200	+	0.670179i	$0.233783\pi$
$854$	0	0
$855$	− 2.19386e13i	− 0.0480152i
$856$	0	0
$857$	−5.17726e14	−1.11994	−0.559971	−	0.828512i	$-0.689188\pi$
$857$	−5.17726e14	−1.11994	−0.559971	+	0.828512i	$0.689188\pi$
$858$	0	0
$859$	3.91219e14i	0.836477i	0.908337	+	0.418238i	$0.137352\pi$
$859$	3.91219e14i	0.836477i	−0.908337	+	0.418238i	$0.862648\pi$
$860$	0	0
$861$	3.59354e14	0.759463
$862$	0	0
$863$	7.51394e13i	0.156969i	0.996915	+	0.0784844i	$0.0250081\pi$
$863$	7.51394e13i	0.156969i	−0.996915	+	0.0784844i	$0.974992\pi$
$864$	0	0
$865$	8.75059e13	0.180699
$866$	0	0
$867$	8.98094e13i	0.183327i
$868$	0	0
$869$	4.60475e14	0.929196
$870$	0	0
$871$	6.86146e14i	1.36875i
$872$	0	0
$873$	4.57581e14	0.902395
$874$	0	0
$875$	− 4.65698e14i	− 0.907954i
$876$	0	0
$877$	7.53135e14	1.45169	0.725847	−	0.687857i	$-0.241448\pi$
$877$	7.53135e14	1.45169	0.725847	+	0.687857i	$0.241448\pi$
$878$	0	0
$879$	− 1.81545e14i	− 0.345971i
$880$	0	0
$881$	−5.70675e14	−1.07525	−0.537625	−	0.843184i	$-0.680678\pi$
$881$	−5.70675e14	−1.07525	−0.537625	+	0.843184i	$0.680678\pi$
$882$	0	0
$883$	7.47133e14i	1.39186i	0.718112	+	0.695928i	$0.245007\pi$
$883$	7.47133e14i	1.39186i	−0.718112	+	0.695928i	$0.754993\pi$
$884$	0	0
$885$	2.06729e14	0.380790
$886$	0	0
$887$	2.25550e14i	0.410795i	0.978679	+	0.205398i	$0.0658488\pi$
$887$	2.25550e14i	0.410795i	−0.978679	+	0.205398i	$0.934151\pi$
$888$	0	0
$889$	7.44552e14	1.34087
$890$	0	0
$891$	6.87994e13i	0.122517i
$892$	0	0
$893$	5.53631e12	0.00974907
$894$	0	0
$895$	− 3.46384e14i	− 0.603174i
$896$	0	0
$897$	5.80612e14	0.999825
$898$	0	0
$899$	− 2.16360e13i	− 0.0368450i
$900$	0	0
$901$	5.53702e14	0.932507
$902$	0	0
$903$	− 7.92824e12i	− 0.0132050i
$904$	0	0
$905$	4.09405e13	0.0674388
$906$	0	0
$907$	− 9.30436e14i	− 1.51583i	−0.652354	−	0.757914i	$-0.726219\pi$
$907$	− 9.30436e14i	− 1.51583i	0.652354	−	0.757914i	$-0.273781\pi$
$908$	0	0
$909$	−2.35069e14	−0.378770
$910$	0	0
$911$	8.55426e14i	1.36330i	0.731680	+	0.681648i	$0.238737\pi$
$911$	8.55426e14i	1.36330i	−0.731680	+	0.681648i	$0.761263\pi$
$912$	0	0
$913$	−1.43907e15	−2.26844
$914$	0	0
$915$	3.59653e13i	0.0560761i
$916$	0	0
$917$	−1.13574e14	−0.175159
$918$	0	0
$919$	9.13684e14i	1.39386i	0.717140	+	0.696929i	$0.245451\pi$
$919$	9.13684e14i	1.39386i	−0.717140	+	0.696929i	$0.754549\pi$
$920$	0	0
$921$	−4.48414e14	−0.676676
$922$	0	0
$923$	− 1.30958e15i	− 1.95490i
$924$	0	0
$925$	2.79615e14	0.412906
$926$	0	0
$927$	7.82914e14i	1.14371i
$928$	0	0
$929$	1.01535e15	1.46736	0.733679	−	0.679496i	$-0.237802\pi$
$929$	1.01535e15	1.46736	0.733679	+	0.679496i	$0.237802\pi$
$930$	0	0
$931$	− 1.96366e13i	− 0.0280748i
$932$	0	0
$933$	−1.51163e14	−0.213815
$934$	0	0
$935$	7.61227e14i	1.06526i
$936$	0	0
$937$	8.42490e13	0.116645	0.0583226	−	0.998298i	$-0.481425\pi$
$937$	8.42490e13	0.116645	0.0583226	+	0.998298i	$0.481425\pi$
$938$	0	0
$939$	1.80451e14i	0.247190i
$940$	0	0
$941$	1.10674e15	1.50002	0.750011	−	0.661426i	$-0.230048\pi$
$941$	1.10674e15	1.50002	0.750011	+	0.661426i	$0.230048\pi$
$942$	0	0
$943$	− 1.64628e15i	− 2.20773i
$944$	0	0
$945$	−4.67162e14	−0.619882
$946$	0	0
$947$	9.93001e14i	1.30377i	0.758320	+	0.651883i	$0.226021\pi$
$947$	9.93001e14i	1.30377i	−0.758320	+	0.651883i	$0.773979\pi$
$948$	0	0
$949$	2.81024e13	0.0365100
$950$	0	0
$951$	− 1.49521e14i	− 0.192220i
$952$	0	0
$953$	−1.09464e15	−1.39254	−0.696268	−	0.717782i	$-0.745158\pi$
$953$	−1.09464e15	−1.39254	−0.696268	+	0.717782i	$0.745158\pi$
$954$	0	0
$955$	1.34616e14i	0.169465i
$956$	0	0
$957$	−1.47633e14	−0.183918
$958$	0	0
$959$	− 4.54892e14i	− 0.560809i
$960$	0	0
$961$	8.03248e14	0.980016
$962$	0	0
$963$	− 2.60971e14i	− 0.315109i
$964$	0	0
$965$	7.40235e14	0.884572
$966$	0	0
$967$	8.79811e13i	0.104054i	0.998646	+	0.0520268i	$0.0165681\pi$
$967$	8.79811e13i	0.104054i	−0.998646	+	0.0520268i	$0.983432\pi$
$968$	0	0
$969$	5.39722e13	0.0631759
$970$	0	0
$971$	− 3.50335e14i	− 0.405871i	−0.979192	−	0.202935i	$-0.934952\pi$
$971$	− 3.50335e14i	− 0.405871i	0.979192	−	0.202935i	$-0.0650482\pi$
$972$	0	0
$973$	1.03008e15	1.18116
$974$	0	0
$975$	2.57921e14i	0.292728i
$976$	0	0
$977$	−4.22669e14	−0.474818	−0.237409	−	0.971410i	$-0.576298\pi$
$977$	−4.22669e14	−0.474818	−0.237409	+	0.971410i	$0.576298\pi$
$978$	0	0
$979$	1.26724e15i	1.40912i
$980$	0	0
$981$	4.46403e14	0.491340
$982$	0	0
$983$	5.96699e13i	0.0650112i	0.999472	+	0.0325056i	$0.0103487\pi$
$983$	5.96699e13i	0.0650112i	−0.999472	+	0.0325056i	$0.989651\pi$
$984$	0	0
$985$	−1.38382e15	−1.49245
$986$	0	0
$987$	− 4.69696e13i	− 0.0501454i
$988$	0	0
$989$	−3.63210e13	−0.0383863
$990$	0	0
$991$	− 1.22609e15i	− 1.28279i	−0.767211	−	0.641395i	$-0.778356\pi$
$991$	− 1.22609e15i	− 1.28279i	0.767211	−	0.641395i	$-0.221644\pi$
$992$	0	0
$993$	7.57390e14	0.784465
$994$	0	0
$995$	− 1.05497e15i	− 1.08174i
$996$	0	0
$997$	7.33333e14	0.744432	0.372216	−	0.928146i	$-0.378598\pi$
$997$	7.33333e14	0.744432	0.372216	+	0.928146i	$0.378598\pi$
$998$	0	0
$999$	− 9.56489e14i	− 0.961286i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.11.c.k.255.7		10
4.3	odd	2	inner	256.11.c.k.255.4		10
8.3	odd	2		256.11.c.j.255.7		10
8.5	even	2		256.11.c.j.255.4		10
16.3	odd	4		128.11.d.f.63.13	yes	20
16.5	even	4		128.11.d.f.63.14	yes	20
16.11	odd	4		128.11.d.f.63.8	yes	20
16.13	even	4		128.11.d.f.63.7	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.11.d.f.63.7	✓	20	16.13	even	4
128.11.d.f.63.8	yes	20	16.11	odd	4
128.11.d.f.63.13	yes	20	16.3	odd	4
128.11.d.f.63.14	yes	20	16.5	even	4
256.11.c.j.255.4		10	8.5	even	2
256.11.c.j.255.7		10	8.3	odd	2
256.11.c.k.255.4		10	4.3	odd	2	inner
256.11.c.k.255.7		10	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)

\(f(q)\)	\(=\)	\(q+141.216i q^{3} +2390.30 q^{5} +14099.9i q^{7} +39107.0 q^{9} +O(q^{10})\)	\(q+141.216i q^{3} +2390.30 q^{5} +14099.9i q^{7} +39107.0 q^{9} +195559. i q^{11} -450735. q^{13} +337548. i q^{15} +1.62849e6 q^{17} -234694. i q^{19} -1.99113e6 q^{21} +9.12181e6i q^{23} -4.05211e6 q^{25} +1.38612e7i q^{27} +5.34591e6 q^{29} -4.04720e6i q^{31} -2.76161e7 q^{33} +3.37029e7i q^{35} -6.90047e7 q^{37} -6.36510e7i q^{39} -1.80477e8 q^{41} +3.98178e6i q^{43} +9.34774e7 q^{45} +2.35895e7i q^{47} +8.36687e7 q^{49} +2.29968e8i q^{51} +3.40010e8 q^{53} +4.67445e8i q^{55} +3.31426e7 q^{57} -6.12444e8i q^{59} +1.06549e8 q^{61} +5.51405e8i q^{63} -1.07739e9 q^{65} -1.52228e9i q^{67} -1.28814e9 q^{69} +2.90544e9i q^{71} -6.23479e7 q^{73} -5.72223e8i q^{75} -2.75736e9 q^{77} -2.35466e9i q^{79} +3.51808e8 q^{81} +7.35873e9i q^{83} +3.89256e9 q^{85} +7.54928e8i q^{87} +6.48010e9 q^{89} -6.35531e9i q^{91} +5.71529e8 q^{93} -5.60989e8i q^{95} +1.17007e10 q^{97} +7.64775e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 6232 q^{5} - 218038 q^{9}+O(q^{10}) \) 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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