LMFDB - Embedded newform 128.8.b.c.65.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [128,8,Mod(65,128)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(128, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("128.65");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$N$	$=$	$128 = 2^{7}$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	128.b (of order $2$ , degree $1$ , 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:	$39.9852832620$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 4x^{2} + 9$ x^4 + 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{9}\cdot 13^{2}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.4
Root			$-0.707107 - 1.58114i$ of defining polynomial
Character	$\chi$	$=$	128.65
Dual form			128.8.b.c.65.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+82.2192i q^{3} +232.551i q^{5} +1606.55 q^{7} -4573.00 q^{9} +3370.99i q^{11} +10929.9i q^{13} -19120.2 q^{15} +23110.0 q^{17} +24748.0i q^{19} +132089. i q^{21} +43829.3 q^{23} +24045.0 q^{25} -196175. i q^{27} +79765.0i q^{29} -108612. q^{31} -277160. q^{33} +373604. i q^{35} -173716. i q^{37} -898648. q^{39} +43498.0 q^{41} -609984. i q^{43} -1.06346e6i q^{45} +31814.1 q^{47} +1.75745e6 q^{49} +1.90009e6i q^{51} -1.98854e6i q^{53} -783927. q^{55} -2.03476e6 q^{57} -1.92878e6i q^{59} +1.63786e6i q^{61} -7.34674e6 q^{63} -2.54176e6 q^{65} -1.97614e6i q^{67} +3.60361e6i q^{69} -3.01137e6 q^{71} +2.39083e6 q^{73} +1.97696e6i q^{75} +5.41565e6i q^{77} +2.37384e6 q^{79} +6.12821e6 q^{81} -2.95060e6i q^{83} +5.37426e6i q^{85} -6.55822e6 q^{87} -7.18279e6 q^{89} +1.75594e7i q^{91} -8.92996e6i q^{93} -5.75517e6 q^{95} +1.49150e7 q^{97} -1.54155e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 18292 q^{9} + 92440 q^{17} + 96180 q^{25} - 1108640 q^{33} + 173992 q^{41} + 7029796 q^{49} - 8139040 q^{57} - 10167040 q^{65} + 9563320 q^{73} + 24512836 q^{81} - 28731144 q^{89} + 59660120 q^{97}+O(q^{100})$ 4 * q - 18292 * q^9 + 92440 * q^17 + 96180 * q^25 - 1108640 * q^33 + 173992 * q^41 + 7029796 * q^49 - 8139040 * q^57 - 10167040 * q^65 + 9563320 * q^73 + 24512836 * q^81 - 28731144 * q^89 + 59660120 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/128\mathbb{Z}\right)^\times$ .

$n$	$5$	$127$
$\chi(n)$	$-1$	$1$

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
$3$	82.2192i	1.75812i	0.476709	+	0.879061i	$0.341829\pi$
$3$	82.2192i	1.75812i	−0.476709	+	0.879061i	$0.658171\pi$
$4$	0	0
$5$	232.551i	0.832000i	0.909365	+	0.416000i	$0.136568\pi$
$5$	232.551i	0.832000i	−0.909365	+	0.416000i	$0.863432\pi$
$6$	0	0
$7$	1606.55	1.77031	0.885157	−	0.465293i	$-0.154051\pi$
$7$	1606.55	1.77031	0.885157	+	0.465293i	$0.154051\pi$
$8$	0	0
$9$	−4573.00	−2.09099
$10$	0	0
$11$	3370.99i	0.763630i	0.924239	+	0.381815i	$0.124701\pi$
$11$	3370.99i	0.763630i	−0.924239	+	0.381815i	$0.875299\pi$
$12$	0	0
$13$	10929.9i	1.37979i	0.723907	+	0.689897i	$0.242344\pi$
$13$	10929.9i	1.37979i	−0.723907	+	0.689897i	$0.757656\pi$
$14$	0	0
$15$	−19120.2	−1.46276
$16$	0	0
$17$	23110.0	1.14085	0.570425	−	0.821350i	$-0.306778\pi$
$17$	23110.0	1.14085	0.570425	+	0.821350i	$0.306778\pi$
$18$	0	0
$19$	24748.0i	0.827756i	0.910332	+	0.413878i	$0.135826\pi$
$19$	24748.0i	0.827756i	−0.910332	+	0.413878i	$0.864174\pi$
$20$	0	0
$21$	132089.i	3.11243i
$22$	0	0
$23$	43829.3	0.751134	0.375567	−	0.926795i	$-0.377448\pi$
$23$	43829.3	0.751134	0.375567	+	0.926795i	$0.377448\pi$
$24$	0	0
$25$	24045.0	0.307776
$26$	0	0
$27$	− 196175.i	− 1.91810i
$28$	0	0
$29$	79765.0i	0.607323i	0.952780	+	0.303661i	$0.0982091\pi$
$29$	79765.0i	0.607323i	−0.952780	+	0.303661i	$0.901791\pi$
$30$	0	0
$31$	−108612.	−0.654802	−0.327401	−	0.944885i	$-0.606173\pi$
$31$	−108612.	−0.654802	−0.327401	+	0.944885i	$0.606173\pi$
$32$	0	0
$33$	−277160.	−1.34255
$34$	0	0
$35$	373604.i	1.47290i
$36$	0	0
$37$	− 173716.i	− 0.563810i	−0.959442	−	0.281905i	$-0.909034\pi$
$37$	− 173716.i	− 0.563810i	0.959442	−	0.281905i	$-0.0909663\pi$
$38$	0	0
$39$	−898648.	−2.42585
$40$	0	0
$41$	43498.0	0.0985657	0.0492828	−	0.998785i	$-0.484306\pi$
$41$	43498.0	0.0985657	0.0492828	+	0.998785i	$0.484306\pi$
$42$	0	0
$43$	− 609984.i	− 1.16998i	−0.811040	−	0.584991i	$-0.801098\pi$
$43$	− 609984.i	− 1.16998i	0.811040	−	0.584991i	$-0.198902\pi$
$44$	0	0
$45$	− 1.06346e6i	− 1.73971i
$46$	0	0
$47$	31814.1	0.0446969	0.0223485	−	0.999750i	$-0.492886\pi$
$47$	31814.1	0.0446969	0.0223485	+	0.999750i	$0.492886\pi$
$48$	0	0
$49$	1.75745e6	2.13401
$50$	0	0
$51$	1.90009e6i	2.00575i
$52$	0	0
$53$	− 1.98854e6i	− 1.83472i	−0.398059	−	0.917360i	$-0.630316\pi$
$53$	− 1.98854e6i	− 1.83472i	0.398059	−	0.917360i	$-0.369684\pi$
$54$	0	0
$55$	−783927.	−0.635340
$56$	0	0
$57$	−2.03476e6	−1.45530
$58$	0	0
$59$	− 1.92878e6i	− 1.22265i	−0.791381	−	0.611323i	$-0.790638\pi$
$59$	− 1.92878e6i	− 1.22265i	0.791381	−	0.611323i	$-0.209362\pi$
$60$	0	0
$61$	1.63786e6i	0.923893i	0.886908	+	0.461946i	$0.152849\pi$
$61$	1.63786e6i	0.923893i	−0.886908	+	0.461946i	$0.847151\pi$
$62$	0	0
$63$	−7.34674e6	−3.70171
$64$	0	0
$65$	−2.54176e6	−1.14799
$66$	0	0
$67$	− 1.97614e6i	− 0.802704i	−0.915924	−	0.401352i	$-0.868540\pi$
$67$	− 1.97614e6i	− 0.802704i	0.915924	−	0.401352i	$-0.131460\pi$
$68$	0	0
$69$	3.60361e6i	1.32058i
$70$	0	0
$71$	−3.01137e6	−0.998527	−0.499264	−	0.866450i	$-0.666396\pi$
$71$	−3.01137e6	−0.998527	−0.499264	+	0.866450i	$0.666396\pi$
$72$	0	0
$73$	2.39083e6	0.719314	0.359657	−	0.933085i	$-0.382894\pi$
$73$	2.39083e6	0.719314	0.359657	+	0.933085i	$0.382894\pi$
$74$	0	0
$75$	1.97696e6i	0.541108i
$76$	0	0
$77$	5.41565e6i	1.35186i
$78$	0	0
$79$	2.37384e6	0.541698	0.270849	−	0.962622i	$-0.412696\pi$
$79$	2.37384e6	0.541698	0.270849	+	0.962622i	$0.412696\pi$
$80$	0	0
$81$	6.12821e6	1.28126
$82$	0	0
$83$	− 2.95060e6i	− 0.566418i	−0.959058	−	0.283209i	$-0.908601\pi$
$83$	− 2.95060e6i	− 0.566418i	0.959058	−	0.283209i	$-0.0913990\pi$
$84$	0	0
$85$	5.37426e6i	0.949188i
$86$	0	0
$87$	−6.55822e6	−1.06775
$88$	0	0
$89$	−7.18279e6	−1.08001	−0.540005	−	0.841662i	$-0.681578\pi$
$89$	−7.18279e6	−1.08001	−0.540005	+	0.841662i	$0.681578\pi$
$90$	0	0
$91$	1.75594e7i	2.44267i
$92$	0	0
$93$	− 8.92996e6i	− 1.15122i
$94$	0	0
$95$	−5.75517e6	−0.688693
$96$	0	0
$97$	1.49150e7	1.65929	0.829646	−	0.558289i	$-0.188542\pi$
$97$	1.49150e7	1.65929	0.829646	+	0.558289i	$0.188542\pi$
$98$	0	0
$99$	− 1.54155e7i	− 1.59674i
$100$	0	0
$101$	538821.i	0.0520379i	0.999661	+	0.0260189i	$0.00828302\pi$
$101$	538821.i	0.0520379i	−0.999661	+	0.0260189i	$0.991717\pi$
$102$	0	0
$103$	−9.22151e6	−0.831518	−0.415759	−	0.909475i	$-0.636484\pi$
$103$	−9.22151e6	−0.831518	−0.415759	+	0.909475i	$0.636484\pi$
$104$	0	0
$105$	−3.07174e7	−2.58954
$106$	0	0
$107$	− 6.74584e6i	− 0.532345i	−0.963925	−	0.266172i	$-0.914241\pi$
$107$	− 6.74584e6i	− 0.532345i	0.963925	−	0.266172i	$-0.0857590\pi$
$108$	0	0
$109$	− 1.86257e7i	− 1.37759i	−0.724956	−	0.688795i	$-0.758140\pi$
$109$	− 1.86257e7i	− 1.37759i	0.724956	−	0.688795i	$-0.241860\pi$
$110$	0	0
$111$	1.42828e7	0.991247
$112$	0	0
$113$	395570.	0.0257898	0.0128949	−	0.999917i	$-0.495895\pi$
$113$	395570.	0.0257898	0.0128949	+	0.999917i	$0.495895\pi$
$114$	0	0
$115$	1.01926e7i	0.624943i
$116$	0	0
$117$	− 4.99824e7i	− 2.88514i
$118$	0	0
$119$	3.71273e7	2.01966
$120$	0	0
$121$	8.12361e6	0.416870
$122$	0	0
$123$	3.57637e6i	0.173290i
$124$	0	0
$125$	2.37597e7i	1.08807i
$126$	0	0
$127$	1.98061e7	0.857999	0.429000	−	0.903305i	$-0.358866\pi$
$127$	1.98061e7	0.857999	0.429000	+	0.903305i	$0.358866\pi$
$128$	0	0
$129$	5.01524e7	2.05697
$130$	0	0
$131$	− 3.41160e7i	− 1.32589i	−0.748667	−	0.662946i	$-0.769306\pi$
$131$	− 3.41160e7i	− 1.32589i	0.748667	−	0.662946i	$-0.230694\pi$
$132$	0	0
$133$	3.97588e7i	1.46539i
$134$	0	0
$135$	4.56207e7	1.59586
$136$	0	0
$137$	−6.25391e6	−0.207792	−0.103896	−	0.994588i	$-0.533131\pi$
$137$	−6.25391e6	−0.207792	−0.103896	+	0.994588i	$0.533131\pi$
$138$	0	0
$139$	− 2.70701e7i	− 0.854945i	−0.904028	−	0.427472i	$-0.859404\pi$
$139$	− 2.70701e7i	− 0.854945i	0.904028	−	0.427472i	$-0.140596\pi$
$140$	0	0
$141$	2.61573e6i	0.0785827i
$142$	0	0
$143$	−3.68446e7	−1.05365
$144$	0	0
$145$	−1.85494e7	−0.505292
$146$	0	0
$147$	1.44496e8i	3.75185i
$148$	0	0
$149$	6.04123e7i	1.49615i	0.663617	+	0.748073i	$0.269021\pi$
$149$	6.04123e7i	1.49615i	−0.663617	+	0.748073i	$0.730979\pi$
$150$	0	0
$151$	−3.17723e6	−0.0750981	−0.0375491	−	0.999295i	$-0.511955\pi$
$151$	−3.17723e6	−0.0750981	−0.0375491	+	0.999295i	$0.511955\pi$
$152$	0	0
$153$	−1.05682e8	−2.38551
$154$	0	0
$155$	− 2.52577e7i	− 0.544796i
$156$	0	0
$157$	7.56710e7i	1.56056i	0.625430	+	0.780280i	$0.284923\pi$
$157$	7.56710e7i	1.56056i	−0.625430	+	0.780280i	$0.715077\pi$
$158$	0	0
$159$	1.63497e8	3.22566
$160$	0	0
$161$	7.04138e7	1.32974
$162$	0	0
$163$	− 4.80232e7i	− 0.868549i	−0.900781	−	0.434274i	$-0.857005\pi$
$163$	− 4.80232e7i	− 0.868549i	0.900781	−	0.434274i	$-0.142995\pi$
$164$	0	0
$165$	− 6.44539e7i	− 1.11700i
$166$	0	0
$167$	4.59703e7	0.763782	0.381891	−	0.924207i	$-0.375273\pi$
$167$	4.59703e7	0.763782	0.381891	+	0.924207i	$0.375273\pi$
$168$	0	0
$169$	−5.67142e7	−0.903833
$170$	0	0
$171$	− 1.13173e8i	− 1.73083i
$172$	0	0
$173$	8.15580e6i	0.119758i	0.998206	+	0.0598791i	$0.0190715\pi$
$173$	8.15580e6i	0.119758i	−0.998206	+	0.0598791i	$0.980928\pi$
$174$	0	0
$175$	3.86294e7	0.544860
$176$	0	0
$177$	1.58583e8	2.14956
$178$	0	0
$179$	7.41717e7i	0.966613i	0.875451	+	0.483306i	$0.160564\pi$
$179$	7.41717e7i	0.966613i	−0.875451	+	0.483306i	$0.839436\pi$
$180$	0	0
$181$	6.18746e7i	0.775600i	0.921744	+	0.387800i	$0.126765\pi$
$181$	6.18746e7i	0.775600i	−0.921744	+	0.387800i	$0.873235\pi$
$182$	0	0
$183$	−1.34663e8	−1.62432
$184$	0	0
$185$	4.03978e7	0.469090
$186$	0	0
$187$	7.79035e7i	0.871187i
$188$	0	0
$189$	− 3.15164e8i	− 3.39563i
$190$	0	0
$191$	−1.68405e8	−1.74879	−0.874394	−	0.485217i	$-0.838741\pi$
$191$	−1.68405e8	−1.74879	−0.874394	+	0.485217i	$0.838741\pi$
$192$	0	0
$193$	−1.72770e8	−1.72988	−0.864942	−	0.501873i	$-0.832645\pi$
$193$	−1.72770e8	−1.72988	−0.864942	+	0.501873i	$0.832645\pi$
$194$	0	0
$195$	− 2.08982e8i	− 2.01830i
$196$	0	0
$197$	− 5.80203e7i	− 0.540690i	−0.962764	−	0.270345i	$-0.912862\pi$
$197$	− 5.80203e7i	− 0.540690i	0.962764	−	0.270345i	$-0.0871378\pi$
$198$	0	0
$199$	4.62046e7	0.415623	0.207812	−	0.978169i	$-0.433366\pi$
$199$	4.62046e7	0.415623	0.207812	+	0.978169i	$0.433366\pi$
$200$	0	0
$201$	1.62477e8	1.41125
$202$	0	0
$203$	1.28146e8i	1.07515i
$204$	0	0
$205$	1.01155e7i	0.0820067i
$206$	0	0
$207$	−2.00431e8	−1.57061
$208$	0	0
$209$	−8.34252e7	−0.632099
$210$	0	0
$211$	6.11363e7i	0.448034i	0.974585	+	0.224017i	$0.0719171\pi$
$211$	6.11363e7i	0.448034i	−0.974585	+	0.224017i	$0.928083\pi$
$212$	0	0
$213$	− 2.47592e8i	− 1.75553i
$214$	0	0
$215$	1.41853e8	0.973425
$216$	0	0
$217$	−1.74490e8	−1.15921
$218$	0	0
$219$	1.96572e8i	1.26464i
$220$	0	0
$221$	2.52590e8i	1.57414i
$222$	0	0
$223$	−5.16836e7	−0.312094	−0.156047	−	0.987750i	$-0.549875\pi$
$223$	−5.16836e7	−0.312094	−0.156047	+	0.987750i	$0.549875\pi$
$224$	0	0
$225$	−1.09958e8	−0.643557
$226$	0	0
$227$	− 2.46371e8i	− 1.39797i	−0.715135	−	0.698986i	$-0.753635\pi$
$227$	− 2.46371e8i	− 1.39797i	0.715135	−	0.698986i	$-0.246365\pi$
$228$	0	0
$229$	1.74927e8i	0.962571i	0.876564	+	0.481285i	$0.159830\pi$
$229$	1.74927e8i	0.962571i	−0.876564	+	0.481285i	$0.840170\pi$
$230$	0	0
$231$	−4.45270e8	−2.37674
$232$	0	0
$233$	2.18403e7	0.113113	0.0565565	−	0.998399i	$-0.481988\pi$
$233$	2.18403e7	0.113113	0.0565565	+	0.998399i	$0.481988\pi$
$234$	0	0
$235$	7.39841e6i	0.0371879i
$236$	0	0
$237$	1.95175e8i	0.952371i
$238$	0	0
$239$	−5.17693e6	−0.0245290	−0.0122645	−	0.999925i	$-0.503904\pi$
$239$	−5.17693e6	−0.0245290	−0.0122645	+	0.999925i	$0.503904\pi$
$240$	0	0
$241$	3.13874e8	1.44443	0.722213	−	0.691671i	$-0.243125\pi$
$241$	3.13874e8	1.44443	0.722213	+	0.691671i	$0.243125\pi$
$242$	0	0
$243$	7.48217e7i	0.334507i
$244$	0	0
$245$	4.08697e8i	1.77550i
$246$	0	0
$247$	−2.70493e8	−1.14213
$248$	0	0
$249$	2.42596e8	0.995832
$250$	0	0
$251$	1.98948e8i	0.794111i	0.917795	+	0.397056i	$0.129968\pi$
$251$	1.98948e8i	0.794111i	−0.917795	+	0.397056i	$0.870032\pi$
$252$	0	0
$253$	1.47748e8i	0.573588i
$254$	0	0
$255$	−4.41867e8	−1.66879
$256$	0	0
$257$	5.54400e7	0.203731	0.101865	−	0.994798i	$-0.467519\pi$
$257$	5.54400e7	0.203731	0.101865	+	0.994798i	$0.467519\pi$
$258$	0	0
$259$	− 2.79082e8i	− 0.998121i
$260$	0	0
$261$	− 3.64765e8i	− 1.26991i
$262$	0	0
$263$	−2.64976e8	−0.898177	−0.449089	−	0.893487i	$-0.648251\pi$
$263$	−2.64976e8	−0.898177	−0.449089	+	0.893487i	$0.648251\pi$
$264$	0	0
$265$	4.62438e8	1.52649
$266$	0	0
$267$	− 5.90563e8i	− 1.89879i
$268$	0	0
$269$	1.72913e7i	0.0541621i	0.999633	+	0.0270811i	$0.00862122\pi$
$269$	1.72913e7i	0.0541621i	−0.999633	+	0.0270811i	$0.991379\pi$
$270$	0	0
$271$	5.96568e8	1.82082	0.910412	−	0.413704i	$-0.135765\pi$
$271$	5.96568e8	1.82082	0.910412	+	0.413704i	$0.135765\pi$
$272$	0	0
$273$	−1.44372e9	−4.29451
$274$	0	0
$275$	8.10554e7i	0.235027i
$276$	0	0
$277$	4.38547e8i	1.23976i	0.784697	+	0.619879i	$0.212818\pi$
$277$	4.38547e8i	1.23976i	−0.784697	+	0.619879i	$0.787182\pi$
$278$	0	0
$279$	4.96681e8	1.36919
$280$	0	0
$281$	−9.85846e7	−0.265056	−0.132528	−	0.991179i	$-0.542309\pi$
$281$	−9.85846e7	−0.265056	−0.132528	+	0.991179i	$0.542309\pi$
$282$	0	0
$283$	− 7.70834e7i	− 0.202166i	−0.994878	−	0.101083i	$-0.967769\pi$
$283$	− 7.70834e7i	− 0.202166i	0.994878	−	0.101083i	$-0.0322308\pi$
$284$	0	0
$285$	− 4.73186e8i	− 1.21081i
$286$	0	0
$287$	6.98816e7	0.174492
$288$	0	0
$289$	1.23733e8	0.301540
$290$	0	0
$291$	1.22630e9i	2.91724i
$292$	0	0
$293$	− 3.83106e8i	− 0.889780i	−0.895585	−	0.444890i	$-0.853243\pi$
$293$	− 3.83106e8i	− 0.889780i	0.895585	−	0.444890i	$-0.146757\pi$
$294$	0	0
$295$	4.48540e8	1.01724
$296$	0	0
$297$	6.61304e8	1.46472
$298$	0	0
$299$	4.79050e8i	1.03641i
$300$	0	0
$301$	− 9.79968e8i	− 2.07123i
$302$	0	0
$303$	−4.43014e7	−0.0914889
$304$	0	0
$305$	−3.80885e8	−0.768679
$306$	0	0
$307$	− 2.19599e8i	− 0.433158i	−0.976265	−	0.216579i	$-0.930510\pi$
$307$	− 2.19599e8i	− 0.433158i	0.976265	−	0.216579i	$-0.0694899\pi$
$308$	0	0
$309$	− 7.58185e8i	− 1.46191i
$310$	0	0
$311$	−4.17348e8	−0.786750	−0.393375	−	0.919378i	$-0.628693\pi$
$311$	−4.17348e8	−0.786750	−0.393375	+	0.919378i	$0.628693\pi$
$312$	0	0
$313$	4.56920e8	0.842239	0.421120	−	0.907005i	$-0.361637\pi$
$313$	4.56920e8	0.842239	0.421120	+	0.907005i	$0.361637\pi$
$314$	0	0
$315$	− 1.70849e9i	− 3.07982i
$316$	0	0
$317$	1.87499e8i	0.330592i	0.986244	+	0.165296i	$0.0528579\pi$
$317$	1.87499e8i	0.330592i	−0.986244	+	0.165296i	$0.947142\pi$
$318$	0	0
$319$	−2.68887e8	−0.463770
$320$	0	0
$321$	5.54638e8	0.935927
$322$	0	0
$323$	5.71926e8i	0.944346i
$324$	0	0
$325$	2.62809e8i	0.424668i
$326$	0	0
$327$	1.53139e9	2.42197
$328$	0	0
$329$	5.11109e7	0.0791276
$330$	0	0
$331$	3.21982e8i	0.488016i	0.969773	+	0.244008i	$0.0784623\pi$
$331$	3.21982e8i	0.488016i	−0.969773	+	0.244008i	$0.921538\pi$
$332$	0	0
$333$	7.94402e8i	1.17892i
$334$	0	0
$335$	4.59553e8	0.667850
$336$	0	0
$337$	−3.23755e7	−0.0460799	−0.0230399	−	0.999735i	$-0.507334\pi$
$337$	−3.23755e7	−0.0460799	−0.0230399	+	0.999735i	$0.507334\pi$
$338$	0	0
$339$	3.25235e7i	0.0453417i
$340$	0	0
$341$	− 3.66128e8i	− 0.500027i
$342$	0	0
$343$	1.50036e9	2.00755
$344$	0	0
$345$	−8.38024e8	−1.09873
$346$	0	0
$347$	2.06409e8i	0.265201i	0.991170	+	0.132600i	$0.0423327\pi$
$347$	2.06409e8i	0.265201i	−0.991170	+	0.132600i	$0.957667\pi$
$348$	0	0
$349$	2.00076e8i	0.251945i	0.992034	+	0.125972i	$0.0402050\pi$
$349$	2.00076e8i	0.251945i	−0.992034	+	0.125972i	$0.959795\pi$
$350$	0	0
$351$	2.14417e9	2.64658
$352$	0	0
$353$	−4.23810e8	−0.512814	−0.256407	−	0.966569i	$-0.582539\pi$
$353$	−4.23810e8	−0.512814	−0.256407	+	0.966569i	$0.582539\pi$
$354$	0	0
$355$	− 7.00297e8i	− 0.830775i
$356$	0	0
$357$	3.05258e9i	3.55081i
$358$	0	0
$359$	−1.36046e9	−1.55187	−0.775936	−	0.630812i	$-0.782722\pi$
$359$	−1.36046e9	−1.55187	−0.775936	+	0.630812i	$0.782722\pi$
$360$	0	0
$361$	2.81409e8	0.314820
$362$	0	0
$363$	6.67917e8i	0.732908i
$364$	0	0
$365$	5.55990e8i	0.598469i
$366$	0	0
$367$	1.09202e8	0.115319	0.0576593	−	0.998336i	$-0.481636\pi$
$367$	1.09202e8	0.115319	0.0576593	+	0.998336i	$0.481636\pi$
$368$	0	0
$369$	−1.98916e8	−0.206100
$370$	0	0
$371$	− 3.19469e9i	− 3.24803i
$372$	0	0
$373$	− 1.24855e9i	− 1.24573i	−0.782330	−	0.622865i	$-0.785969\pi$
$373$	− 1.24855e9i	− 1.24573i	0.782330	−	0.622865i	$-0.214031\pi$
$374$	0	0
$375$	−1.95351e9	−1.91296
$376$	0	0
$377$	−8.71824e8	−0.837981
$378$	0	0
$379$	1.62778e9i	1.53588i	0.640520	+	0.767942i	$0.278719\pi$
$379$	1.62778e9i	1.53588i	−0.640520	+	0.767942i	$0.721281\pi$
$380$	0	0
$381$	1.62845e9i	1.50847i
$382$	0	0
$383$	8.33436e8	0.758013	0.379006	−	0.925394i	$-0.376266\pi$
$383$	8.33436e8	0.758013	0.379006	+	0.925394i	$0.376266\pi$
$384$	0	0
$385$	−1.25942e9	−1.12475
$386$	0	0
$387$	2.78946e9i	2.44642i
$388$	0	0
$389$	− 2.13085e9i	− 1.83540i	−0.397278	−	0.917698i	$-0.630045\pi$
$389$	− 2.13085e9i	− 1.83540i	0.397278	−	0.917698i	$-0.369955\pi$
$390$	0	0
$391$	1.01290e9	0.856931
$392$	0	0
$393$	2.80499e9	2.33108
$394$	0	0
$395$	5.52040e8i	0.450693i
$396$	0	0
$397$	5.03042e8i	0.403494i	0.979438	+	0.201747i	$0.0646620\pi$
$397$	5.03042e8i	0.403494i	−0.979438	+	0.201747i	$0.935338\pi$
$398$	0	0
$399$	−3.26894e9	−2.57633
$400$	0	0
$401$	2.25217e9	1.74420	0.872101	−	0.489327i	$-0.162757\pi$
$401$	2.25217e9	1.74420	0.872101	+	0.489327i	$0.162757\pi$
$402$	0	0
$403$	− 1.18711e9i	− 0.903493i
$404$	0	0
$405$	1.42512e9i	1.06601i
$406$	0	0
$407$	5.85593e8	0.430542
$408$	0	0
$409$	−1.95602e9	−1.41365	−0.706826	−	0.707388i	$-0.749873\pi$
$409$	−1.95602e9	−1.41365	−0.706826	+	0.707388i	$0.749873\pi$
$410$	0	0
$411$	− 5.14192e8i	− 0.365324i
$412$	0	0
$413$	− 3.09868e9i	− 2.16447i
$414$	0	0
$415$	6.86165e8	0.471260
$416$	0	0
$417$	2.22568e9	1.50310
$418$	0	0
$419$	− 4.76013e8i	− 0.316133i	−0.987428	−	0.158066i	$-0.949474\pi$
$419$	− 4.76013e8i	− 0.316133i	0.987428	−	0.158066i	$-0.0505260\pi$
$420$	0	0
$421$	− 1.26889e9i	− 0.828775i	−0.910101	−	0.414387i	$-0.863996\pi$
$421$	− 1.26889e9i	− 0.828775i	0.910101	−	0.414387i	$-0.136004\pi$
$422$	0	0
$423$	−1.45486e8	−0.0934610
$424$	0	0
$425$	5.55680e8	0.351126
$426$	0	0
$427$	2.63129e9i	1.63558i
$428$	0	0
$429$	− 3.02933e9i	− 1.85245i
$430$	0	0
$431$	−1.43299e9	−0.862127	−0.431063	−	0.902322i	$-0.641861\pi$
$431$	−1.43299e9	−0.862127	−0.431063	+	0.902322i	$0.641861\pi$
$432$	0	0
$433$	−5.93740e8	−0.351470	−0.175735	−	0.984438i	$-0.556230\pi$
$433$	−5.93740e8	−0.351470	−0.175735	+	0.984438i	$0.556230\pi$
$434$	0	0
$435$	− 1.52512e9i	− 0.888366i
$436$	0	0
$437$	1.08469e9i	0.621755i
$438$	0	0
$439$	1.14709e9	0.647103	0.323551	−	0.946211i	$-0.395123\pi$
$439$	1.14709e9	0.647103	0.323551	+	0.946211i	$0.395123\pi$
$440$	0	0
$441$	−8.03681e9	−4.46220
$442$	0	0
$443$	− 1.87714e9i	− 1.02585i	−0.858434	−	0.512924i	$-0.828562\pi$
$443$	− 1.87714e9i	− 1.02585i	0.858434	−	0.512924i	$-0.171438\pi$
$444$	0	0
$445$	− 1.67036e9i	− 0.898569i
$446$	0	0
$447$	−4.96706e9	−2.63041
$448$	0	0
$449$	2.11344e9	1.10186	0.550931	−	0.834551i	$-0.314273\pi$
$449$	2.11344e9	1.10186	0.550931	+	0.834551i	$0.314273\pi$
$450$	0	0
$451$	1.46631e8i	0.0752677i
$452$	0	0
$453$	− 2.61229e8i	− 0.132032i
$454$	0	0
$455$	−4.08346e9	−2.03230
$456$	0	0
$457$	−1.16152e9	−0.569271	−0.284635	−	0.958636i	$-0.591873\pi$
$457$	−1.16152e9	−0.569271	−0.284635	+	0.958636i	$0.591873\pi$
$458$	0	0
$459$	− 4.53361e9i	− 2.18826i
$460$	0	0
$461$	− 5.90214e8i	− 0.280580i	−0.990110	−	0.140290i	$-0.955197\pi$
$461$	− 5.90214e8i	− 0.280580i	0.990110	−	0.140290i	$-0.0448034\pi$
$462$	0	0
$463$	2.34116e9	1.09622	0.548109	−	0.836407i	$-0.315348\pi$
$463$	2.34116e9	1.09622	0.548109	+	0.836407i	$0.315348\pi$
$464$	0	0
$465$	2.07667e9	0.957817
$466$	0	0
$467$	1.58376e9i	0.719581i	0.933033	+	0.359791i	$0.117152\pi$
$467$	1.58376e9i	0.719581i	−0.933033	+	0.359791i	$0.882848\pi$
$468$	0	0
$469$	− 3.17476e9i	− 1.42104i
$470$	0	0
$471$	−6.22161e9	−2.74365
$472$	0	0
$473$	2.05625e9	0.893433
$474$	0	0
$475$	5.95065e8i	0.254763i
$476$	0	0
$477$	9.09361e9i	3.83638i
$478$	0	0
$479$	−2.79309e9	−1.16121	−0.580605	−	0.814185i	$-0.697184\pi$
$479$	−2.79309e9	−1.16121	−0.580605	+	0.814185i	$0.697184\pi$
$480$	0	0
$481$	1.89869e9	0.777942
$482$	0	0
$483$	5.78937e9i	2.33785i
$484$	0	0
$485$	3.46851e9i	1.38053i
$486$	0	0
$487$	2.33046e9	0.914305	0.457152	−	0.889388i	$-0.348869\pi$
$487$	2.33046e9	0.914305	0.457152	+	0.889388i	$0.348869\pi$
$488$	0	0
$489$	3.94843e9	1.52701
$490$	0	0
$491$	2.02375e9i	0.771564i	0.922590	+	0.385782i	$0.126068\pi$
$491$	2.02375e9i	0.771564i	−0.922590	+	0.385782i	$0.873932\pi$
$492$	0	0
$493$	1.84337e9i	0.692864i
$494$	0	0
$495$	3.58490e9	1.32849
$496$	0	0
$497$	−4.83791e9	−1.76771
$498$	0	0
$499$	5.37651e8i	0.193708i	0.995299	+	0.0968542i	$0.0308780\pi$
$499$	5.37651e8i	0.193708i	−0.995299	+	0.0968542i	$0.969122\pi$
$500$	0	0
$501$	3.77964e9i	1.34282i
$502$	0	0
$503$	1.86741e9	0.654262	0.327131	−	0.944979i	$-0.393918\pi$
$503$	1.86741e9	0.654262	0.327131	+	0.944979i	$0.393918\pi$
$504$	0	0
$505$	−1.25303e8	−0.0432955
$506$	0	0
$507$	− 4.66300e9i	− 1.58905i
$508$	0	0
$509$	− 4.22019e9i	− 1.41847i	−0.704973	−	0.709234i	$-0.749041\pi$
$509$	− 4.22019e9i	− 1.41847i	0.704973	−	0.709234i	$-0.250959\pi$
$510$	0	0
$511$	3.84098e9	1.27341
$512$	0	0
$513$	4.85494e9	1.58772
$514$	0	0
$515$	− 2.14447e9i	− 0.691823i
$516$	0	0
$517$	1.07245e8i	0.0341319i
$518$	0	0
$519$	−6.70563e8	−0.210549
$520$	0	0
$521$	−3.86723e9	−1.19803	−0.599016	−	0.800737i	$-0.704441\pi$
$521$	−3.86723e9	−1.19803	−0.599016	+	0.800737i	$0.704441\pi$
$522$	0	0
$523$	− 5.30082e9i	− 1.62027i	−0.586244	−	0.810134i	$-0.699394\pi$
$523$	− 5.30082e9i	− 1.62027i	0.586244	−	0.810134i	$-0.300606\pi$
$524$	0	0
$525$	3.17608e9i	0.957930i
$526$	0	0
$527$	−2.51001e9	−0.747032
$528$	0	0
$529$	−1.48382e9	−0.435798
$530$	0	0
$531$	8.82031e9i	2.55654i
$532$	0	0
$533$	4.75429e8i	0.136000i
$534$	0	0
$535$	1.56875e9	0.442911
$536$	0	0
$537$	−6.09834e9	−1.69942
$538$	0	0
$539$	5.92434e9i	1.62959i
$540$	0	0
$541$	− 6.07700e8i	− 0.165006i	−0.996591	−	0.0825028i	$-0.973709\pi$
$541$	− 6.07700e8i	− 0.165006i	0.996591	−	0.0825028i	$-0.0262913\pi$
$542$	0	0
$543$	−5.08728e9	−1.36360
$544$	0	0
$545$	4.33143e9	1.14616
$546$	0	0
$547$	− 6.34461e9i	− 1.65748i	−0.559631	−	0.828742i	$-0.689057\pi$
$547$	− 6.34461e9i	− 1.65748i	0.559631	−	0.828742i	$-0.310943\pi$
$548$	0	0
$549$	− 7.48992e9i	− 1.93185i
$550$	0	0
$551$	−1.97402e9	−0.502715
$552$	0	0
$553$	3.81369e9	0.958975
$554$	0	0
$555$	3.32147e9i	0.824717i
$556$	0	0
$557$	3.46471e9i	0.849520i	0.905306	+	0.424760i	$0.139642\pi$
$557$	3.46471e9i	0.849520i	−0.905306	+	0.424760i	$0.860358\pi$
$558$	0	0
$559$	6.66707e9	1.61433
$560$	0	0
$561$	−6.40517e9	−1.53165
$562$	0	0
$563$	− 3.62033e9i	− 0.855006i	−0.904014	−	0.427503i	$-0.859393\pi$
$563$	− 3.62033e9i	− 0.855006i	0.904014	−	0.427503i	$-0.140607\pi$
$564$	0	0
$565$	9.19902e7i	0.0214572i
$566$	0	0
$567$	9.84525e9	2.26823
$568$	0	0
$569$	−8.30120e9	−1.88907	−0.944535	−	0.328411i	$-0.893487\pi$
$569$	−8.30120e9	−1.88907	−0.944535	+	0.328411i	$0.893487\pi$
$570$	0	0
$571$	− 1.75351e9i	− 0.394169i	−0.980386	−	0.197085i	$-0.936853\pi$
$571$	− 1.75351e9i	− 0.394169i	0.980386	−	0.197085i	$-0.0631474\pi$
$572$	0	0
$573$	− 1.38461e10i	− 3.07458i
$574$	0	0
$575$	1.05388e9	0.231181
$576$	0	0
$577$	4.61658e9	1.00047	0.500236	−	0.865889i	$-0.333247\pi$
$577$	4.61658e9	1.00047	0.500236	+	0.865889i	$0.333247\pi$
$578$	0	0
$579$	− 1.42050e10i	− 3.04135i
$580$	0	0
$581$	− 4.74028e9i	− 1.00274i
$582$	0	0
$583$	6.70336e9	1.40105
$584$	0	0
$585$	1.16235e10	2.40044
$586$	0	0
$587$	− 7.58842e6i	− 0.00154852i	−1.00000		0.000774262i	$-0.999754\pi$
$587$	− 7.58842e6i	− 0.00154852i	1.00000		0.000774262i	$-0.000246455\pi$
$588$	0	0
$589$	− 2.68792e9i	− 0.542017i
$590$	0	0
$591$	4.77039e9	0.950599
$592$	0	0
$593$	3.39305e9	0.668189	0.334094	−	0.942540i	$-0.391570\pi$
$593$	3.39305e9	0.668189	0.334094	+	0.942540i	$0.391570\pi$
$594$	0	0
$595$	8.63399e9i	1.68036i
$596$	0	0
$597$	3.79891e9i	0.730716i
$598$	0	0
$599$	6.24638e9	1.18750	0.593752	−	0.804648i	$-0.297646\pi$
$599$	6.24638e9	1.18750	0.593752	+	0.804648i	$0.297646\pi$
$600$	0	0
$601$	6.30382e9	1.18452	0.592261	−	0.805746i	$-0.298235\pi$
$601$	6.30382e9	1.18452	0.592261	+	0.805746i	$0.298235\pi$
$602$	0	0
$603$	9.03688e9i	1.67845i
$604$	0	0
$605$	1.88915e9i	0.346836i
$606$	0	0
$607$	7.33337e9	1.33089	0.665447	−	0.746445i	$-0.268241\pi$
$607$	7.33337e9	1.33089	0.665447	+	0.746445i	$0.268241\pi$
$608$	0	0
$609$	−1.05361e10	−1.89025
$610$	0	0
$611$	3.47725e8i	0.0616726i
$612$	0	0
$613$	2.12088e9i	0.371882i	0.982561	+	0.185941i	$0.0595333\pi$
$613$	2.12088e9i	0.371882i	−0.982561	+	0.185941i	$0.940467\pi$
$614$	0	0
$615$	−8.31689e8	−0.144178
$616$	0	0
$617$	8.12132e9	1.39197	0.695983	−	0.718058i	$-0.254969\pi$
$617$	8.12132e9	1.39197	0.695983	+	0.718058i	$0.254969\pi$
$618$	0	0
$619$	8.07704e9i	1.36878i	0.729114	+	0.684392i	$0.239932\pi$
$619$	8.07704e9i	1.36878i	−0.729114	+	0.684392i	$0.760068\pi$
$620$	0	0
$621$	− 8.59822e9i	− 1.44075i
$622$	0	0
$623$	−1.15395e10	−1.91196
$624$	0	0
$625$	−3.64684e9	−0.597498
$626$	0	0
$627$	− 6.85915e9i	− 1.11131i
$628$	0	0
$629$	− 4.01457e9i	− 0.643223i
$630$	0	0
$631$	−6.61772e9	−1.04859	−0.524295	−	0.851537i	$-0.675671\pi$
$631$	−6.61772e9	−1.04859	−0.524295	+	0.851537i	$0.675671\pi$
$632$	0	0
$633$	−5.02658e9	−0.787698
$634$	0	0
$635$	4.60594e9i	0.713855i
$636$	0	0
$637$	1.92087e10i	2.94450i
$638$	0	0
$639$	1.37710e10	2.08791
$640$	0	0
$641$	−1.11366e10	−1.67013	−0.835063	−	0.550155i	$-0.814569\pi$
$641$	−1.11366e10	−1.67013	−0.835063	+	0.550155i	$0.814569\pi$
$642$	0	0
$643$	− 2.96548e9i	− 0.439903i	−0.975511	−	0.219952i	$-0.929410\pi$
$643$	− 2.96548e9i	− 0.439903i	0.975511	−	0.219952i	$-0.0705899\pi$
$644$	0	0
$645$	1.16630e10i	1.71140i
$646$	0	0
$647$	−5.69850e9	−0.827171	−0.413586	−	0.910465i	$-0.635724\pi$
$647$	−5.69850e9	−0.827171	−0.413586	+	0.910465i	$0.635724\pi$
$648$	0	0
$649$	6.50190e9	0.933649
$650$	0	0
$651$	− 1.43464e10i	− 2.03802i
$652$	0	0
$653$	6.52934e9i	0.917642i	0.888529	+	0.458821i	$0.151728\pi$
$653$	6.52934e9i	0.917642i	−0.888529	+	0.458821i	$0.848272\pi$
$654$	0	0
$655$	7.93370e9	1.10314
$656$	0	0
$657$	−1.09333e10	−1.50408
$658$	0	0
$659$	1.34359e10i	1.82881i	0.404802	+	0.914404i	$0.367340\pi$
$659$	1.34359e10i	1.82881i	−0.404802	+	0.914404i	$0.632660\pi$
$660$	0	0
$661$	7.88705e9i	1.06221i	0.847307	+	0.531104i	$0.178223\pi$
$661$	7.88705e9i	1.06221i	−0.847307	+	0.531104i	$0.821777\pi$
$662$	0	0
$663$	−2.07678e10	−2.76753
$664$	0	0
$665$	−9.24595e9	−1.21920
$666$	0	0
$667$	3.49605e9i	0.456180i
$668$	0	0
$669$	− 4.24938e9i	− 0.548699i
$670$	0	0
$671$	−5.52120e9	−0.705512
$672$	0	0
$673$	−1.66540e9	−0.210603	−0.105302	−	0.994440i	$-0.533581\pi$
$673$	−1.66540e9	−0.210603	−0.105302	+	0.994440i	$0.533581\pi$
$674$	0	0
$675$	− 4.71703e9i	− 0.590344i
$676$	0	0
$677$	1.16341e10i	1.44103i	0.693438	+	0.720516i	$0.256095\pi$
$677$	1.16341e10i	1.44103i	−0.693438	+	0.720516i	$0.743905\pi$
$678$	0	0
$679$	2.39617e10	2.93747
$680$	0	0
$681$	2.02564e10	2.45781
$682$	0	0
$683$	− 9.34980e9i	− 1.12287i	−0.827521	−	0.561435i	$-0.810249\pi$
$683$	− 9.34980e9i	− 1.12287i	0.827521	−	0.561435i	$-0.189751\pi$
$684$	0	0
$685$	− 1.45435e9i	− 0.172883i
$686$	0	0
$687$	−1.43824e10	−1.69232
$688$	0	0
$689$	2.17346e10	2.53154
$690$	0	0
$691$	− 6.10881e9i	− 0.704341i	−0.935936	−	0.352171i	$-0.885444\pi$
$691$	− 6.10881e9i	− 0.704341i	0.935936	−	0.352171i	$-0.114556\pi$
$692$	0	0
$693$	− 2.47658e10i	− 2.82674i
$694$	0	0
$695$	6.29518e9	0.711314
$696$	0	0
$697$	1.00524e9	0.112449
$698$	0	0
$699$	1.79569e9i	0.198866i
$700$	0	0
$701$	1.61502e10i	1.77078i	0.464845	+	0.885392i	$0.346110\pi$
$701$	1.61502e10i	1.77078i	−0.464845	+	0.885392i	$0.653890\pi$
$702$	0	0
$703$	4.29911e9	0.466697
$704$	0	0
$705$	−6.08292e8	−0.0653808
$706$	0	0
$707$	8.65641e8i	0.0921234i
$708$	0	0
$709$	− 9.31508e9i	− 0.981578i	−0.871279	−	0.490789i	$-0.836709\pi$
$709$	− 9.31508e9i	− 0.981578i	0.871279	−	0.490789i	$-0.163291\pi$
$710$	0	0
$711$	−1.08556e10	−1.13269
$712$	0	0
$713$	−4.76037e9	−0.491844
$714$	0	0
$715$	− 8.56824e9i	− 0.876639i
$716$	0	0
$717$	− 4.25643e8i	− 0.0431249i
$718$	0	0
$719$	−1.84156e10	−1.84772	−0.923859	−	0.382733i	$-0.874983\pi$
$719$	−1.84156e10	−1.84772	−0.923859	+	0.382733i	$0.874983\pi$
$720$	0	0
$721$	−1.48148e10	−1.47205
$722$	0	0
$723$	2.58064e10i	2.53948i
$724$	0	0
$725$	1.91795e9i	0.186919i
$726$	0	0
$727$	4.46970e8	0.0431427	0.0215714	−	0.999767i	$-0.493133\pi$
$727$	4.46970e8	0.0431427	0.0215714	+	0.999767i	$0.493133\pi$
$728$	0	0
$729$	7.25061e9	0.693152
$730$	0	0
$731$	− 1.40967e10i	− 1.33477i
$732$	0	0
$733$	− 1.43625e10i	− 1.34699i	−0.739190	−	0.673497i	$-0.764791\pi$
$733$	− 1.43625e10i	− 1.34699i	0.739190	−	0.673497i	$-0.235209\pi$
$734$	0	0
$735$	−3.36027e10	−3.12154
$736$	0	0
$737$	6.66154e9	0.612969
$738$	0	0
$739$	7.82928e9i	0.713618i	0.934177	+	0.356809i	$0.116135\pi$
$739$	7.82928e9i	0.713618i	−0.934177	+	0.356809i	$0.883865\pi$
$740$	0	0
$741$	− 2.22397e10i	− 2.00801i
$742$	0	0
$743$	−1.85028e10	−1.65492	−0.827460	−	0.561525i	$-0.810215\pi$
$743$	−1.85028e10	−1.65492	−0.827460	+	0.561525i	$0.810215\pi$
$744$	0	0
$745$	−1.40490e10	−1.24479
$746$	0	0
$747$	1.34931e10i	1.18438i
$748$	0	0
$749$	− 1.08375e10i	− 0.942417i
$750$	0	0
$751$	−2.06945e9	−0.178285	−0.0891424	−	0.996019i	$-0.528413\pi$
$751$	−2.06945e9	−0.178285	−0.0891424	+	0.996019i	$0.528413\pi$
$752$	0	0
$753$	−1.63573e10	−1.39614
$754$	0	0
$755$	− 7.38868e8i	− 0.0624817i
$756$	0	0
$757$	2.15642e10i	1.80675i	0.428849	+	0.903376i	$0.358919\pi$
$757$	2.15642e10i	1.80675i	−0.428849	+	0.903376i	$0.641081\pi$
$758$	0	0
$759$	−1.21477e10	−1.00844
$760$	0	0
$761$	1.43334e10	1.17897	0.589484	−	0.807780i	$-0.299331\pi$
$761$	1.43334e10	1.17897	0.589484	+	0.807780i	$0.299331\pi$
$762$	0	0
$763$	− 2.99231e10i	− 2.43877i
$764$	0	0
$765$	− 2.45765e10i	− 1.98474i
$766$	0	0
$767$	2.10814e10	1.68700
$768$	0	0
$769$	−1.52724e10	−1.21106	−0.605530	−	0.795822i	$-0.707039\pi$
$769$	−1.52724e10	−1.21106	−0.605530	+	0.795822i	$0.707039\pi$
$770$	0	0
$771$	4.55823e9i	0.358184i
$772$	0	0
$773$	1.21986e10i	0.949911i	0.880010	+	0.474956i	$0.157536\pi$
$773$	1.21986e10i	0.949911i	−0.880010	+	0.474956i	$0.842464\pi$
$774$	0	0
$775$	−2.61157e9	−0.201532
$776$	0	0
$777$	2.29459e10	1.75482
$778$	0	0
$779$	1.07649e9i	0.0815883i
$780$	0	0
$781$	− 1.01513e10i	− 0.762505i
$782$	0	0
$783$	1.56479e10	1.16490
$784$	0	0
$785$	−1.75974e10	−1.29839
$786$	0	0
$787$	7.73131e9i	0.565381i	0.959211	+	0.282691i	$0.0912270\pi$
$787$	7.73131e9i	0.565381i	−0.959211	+	0.282691i	$0.908773\pi$
$788$	0	0
$789$	− 2.17862e10i	− 1.57911i
$790$	0	0
$791$	6.35502e8	0.0456561
$792$	0	0
$793$	−1.79016e10	−1.27478
$794$	0	0
$795$	3.80213e10i	2.68375i
$796$	0	0
$797$	− 1.27527e9i	− 0.0892275i	−0.999004	−	0.0446138i	$-0.985794\pi$
$797$	− 1.27527e9i	− 0.0892275i	0.999004	−	0.0446138i	$-0.0142057\pi$
$798$	0	0
$799$	7.35225e8	0.0509925
$800$	0	0
$801$	3.28469e10	2.25829
$802$	0	0
$803$	8.05946e9i	0.549290i
$804$	0	0
$805$	1.63748e10i	1.10635i
$806$	0	0
$807$	−1.42168e9	−0.0952236
$808$	0	0
$809$	−1.50809e10	−1.00140	−0.500701	−	0.865621i	$-0.666924\pi$
$809$	−1.50809e10	−1.00140	−0.500701	+	0.865621i	$0.666924\pi$
$810$	0	0
$811$	2.20340e10i	1.45051i	0.688483	+	0.725253i	$0.258277\pi$
$811$	2.20340e10i	1.45051i	−0.688483	+	0.725253i	$0.741723\pi$
$812$	0	0
$813$	4.90494e10i	3.20123i
$814$	0	0
$815$	1.11678e10	0.722633
$816$	0	0
$817$	1.50959e10	0.968459
$818$	0	0
$819$	− 8.02991e10i	− 5.10760i
$820$	0	0
$821$	1.55711e9i	0.0982015i	0.998794	+	0.0491008i	$0.0156355\pi$
$821$	1.55711e9i	0.0982015i	−0.998794	+	0.0491008i	$0.984364\pi$
$822$	0	0
$823$	1.46343e10	0.915109	0.457554	−	0.889182i	$-0.348726\pi$
$823$	1.46343e10	0.915109	0.457554	+	0.889182i	$0.348726\pi$
$824$	0	0
$825$	−6.66431e9	−0.413206
$826$	0	0
$827$	− 2.98290e10i	− 1.83388i	−0.399029	−	0.916938i	$-0.630653\pi$
$827$	− 2.98290e10i	− 1.83388i	0.399029	−	0.916938i	$-0.369347\pi$
$828$	0	0
$829$	2.83611e10i	1.72895i	0.502677	+	0.864474i	$0.332348\pi$
$829$	2.83611e10i	1.72895i	−0.502677	+	0.864474i	$0.667652\pi$
$830$	0	0
$831$	−3.60570e10	−2.17965
$832$	0	0
$833$	4.06146e10	2.43459
$834$	0	0
$835$	1.06904e10i	0.635467i
$836$	0	0
$837$	2.13069e10i	1.25597i
$838$	0	0
$839$	−3.34789e10	−1.95706	−0.978529	−	0.206107i	$-0.933920\pi$
$839$	−3.34789e10	−1.95706	−0.978529	+	0.206107i	$0.933920\pi$
$840$	0	0
$841$	1.08874e10	0.631159
$842$	0	0
$843$	− 8.10555e9i	− 0.466000i
$844$	0	0
$845$	− 1.31889e10i	− 0.751989i
$846$	0	0
$847$	1.30510e10	0.737990
$848$	0	0
$849$	6.33774e9	0.355433
$850$	0	0
$851$	− 7.61384e9i	− 0.423497i
$852$	0	0
$853$	− 2.58519e9i	− 0.142617i	−0.997454	−	0.0713083i	$-0.977283\pi$
$853$	− 2.58519e9i	− 0.142617i	0.997454	−	0.0713083i	$-0.0227174\pi$
$854$	0	0
$855$	2.63184e10	1.44005
$856$	0	0
$857$	−2.64139e10	−1.43351	−0.716754	−	0.697326i	$-0.754373\pi$
$857$	−2.64139e10	−1.43351	−0.716754	+	0.697326i	$0.754373\pi$
$858$	0	0
$859$	2.11238e10i	1.13710i	0.822650	+	0.568548i	$0.192494\pi$
$859$	2.11238e10i	1.13710i	−0.822650	+	0.568548i	$0.807506\pi$
$860$	0	0
$861$	5.74561e9i	0.306779i
$862$	0	0
$863$	1.92964e10	1.02197	0.510987	−	0.859589i	$-0.329280\pi$
$863$	1.92964e10	1.02197	0.510987	+	0.859589i	$0.329280\pi$
$864$	0	0
$865$	−1.89664e9	−0.0996388
$866$	0	0
$867$	1.01733e10i	0.530144i
$868$	0	0
$869$	8.00219e9i	0.413657i
$870$	0	0
$871$	2.15990e10	1.10757
$872$	0	0
$873$	−6.82064e10	−3.46957
$874$	0	0
$875$	3.81711e10i	1.92622i
$876$	0	0
$877$	− 2.07991e10i	− 1.04123i	−0.853792	−	0.520615i	$-0.825703\pi$
$877$	− 2.07991e10i	− 1.04123i	0.853792	−	0.520615i	$-0.174297\pi$
$878$	0	0
$879$	3.14987e10	1.56434
$880$	0	0
$881$	8.49220e9	0.418412	0.209206	−	0.977872i	$-0.432912\pi$
$881$	8.49220e9	0.418412	0.209206	+	0.977872i	$0.432912\pi$
$882$	0	0
$883$	2.47905e9i	0.121178i	0.998163	+	0.0605888i	$0.0192978\pi$
$883$	2.47905e9i	0.121178i	−0.998163	+	0.0605888i	$0.980702\pi$
$884$	0	0
$885$	3.68786e10i	1.78843i
$886$	0	0
$887$	8.57742e9	0.412690	0.206345	−	0.978479i	$-0.433843\pi$
$887$	8.57742e9	0.412690	0.206345	+	0.978479i	$0.433843\pi$
$888$	0	0
$889$	3.18195e10	1.51893
$890$	0	0
$891$	2.06581e10i	0.978405i
$892$	0	0
$893$	7.87336e8i	0.0369982i
$894$	0	0
$895$	−1.72487e10	−0.804222
$896$	0	0
$897$	−3.93871e10	−1.82214
$898$	0	0
$899$	− 8.66341e9i	− 0.397676i
$900$	0	0
$901$	− 4.59553e10i	− 2.09314i
$902$	0	0
$903$	8.05722e10	3.64148
$904$	0	0
$905$	−1.43890e10	−0.645299
$906$	0	0
$907$	− 1.62696e10i	− 0.724023i	−0.932174	−	0.362011i	$-0.882090\pi$
$907$	− 1.62696e10i	− 0.724023i	0.932174	−	0.362011i	$-0.117910\pi$
$908$	0	0
$909$	− 2.46403e9i	− 0.108811i
$910$	0	0
$911$	−5.39206e9	−0.236287	−0.118144	−	0.992997i	$-0.537694\pi$
$911$	−5.39206e9	−0.236287	−0.118144	+	0.992997i	$0.537694\pi$
$912$	0	0
$913$	9.94644e9	0.432534
$914$	0	0
$915$	− 3.13161e10i	− 1.35143i
$916$	0	0
$917$	− 5.48089e10i	− 2.34724i
$918$	0	0
$919$	−1.78439e9	−0.0758376	−0.0379188	−	0.999281i	$-0.512073\pi$
$919$	−1.78439e9	−0.0758376	−0.0379188	+	0.999281i	$0.512073\pi$
$920$	0	0
$921$	1.80553e10	0.761545
$922$	0	0
$923$	− 3.29140e10i	− 1.37776i
$924$	0	0
$925$	− 4.17699e9i	− 0.173527i
$926$	0	0
$927$	4.21700e10	1.73870
$928$	0	0
$929$	5.70081e9	0.233282	0.116641	−	0.993174i	$-0.462787\pi$
$929$	5.70081e9	0.233282	0.116641	+	0.993174i	$0.462787\pi$
$930$	0	0
$931$	4.34933e10i	1.76644i
$932$	0	0
$933$	− 3.43140e10i	− 1.38320i
$934$	0	0
$935$	−1.81165e10	−0.724828
$936$	0	0
$937$	4.18534e10	1.66204	0.831022	−	0.556240i	$-0.187756\pi$
$937$	4.18534e10	1.66204	0.831022	+	0.556240i	$0.187756\pi$
$938$	0	0
$939$	3.75676e10i	1.48076i
$940$	0	0
$941$	1.59230e10i	0.622961i	0.950253	+	0.311480i	$0.100825\pi$
$941$	1.59230e10i	0.622961i	−0.950253	+	0.311480i	$0.899175\pi$
$942$	0	0
$943$	1.90649e9	0.0740360
$944$	0	0
$945$	7.32918e10	2.82517
$946$	0	0
$947$	1.00392e10i	0.384127i	0.981383	+	0.192063i	$0.0615179\pi$
$947$	1.00392e10i	0.384127i	−0.981383	+	0.192063i	$0.938482\pi$
$948$	0	0
$949$	2.61315e10i	0.992506i
$950$	0	0
$951$	−1.54160e10	−0.581220
$952$	0	0
$953$	−3.04755e10	−1.14058	−0.570290	−	0.821443i	$-0.693169\pi$
$953$	−3.04755e10	−1.14058	−0.570290	+	0.821443i	$0.693169\pi$
$954$	0	0
$955$	− 3.91627e10i	− 1.45499i
$956$	0	0
$957$	− 2.21077e10i	− 0.815363i
$958$	0	0
$959$	−1.00472e10	−0.367858
$960$	0	0
$961$	−1.57161e10	−0.571234
$962$	0	0
$963$	3.08487e10i	1.11313i
$964$	0	0
$965$	− 4.01778e10i	− 1.43926i
$966$	0	0
$967$	−6.74850e9	−0.240002	−0.120001	−	0.992774i	$-0.538290\pi$
$967$	−6.74850e9	−0.240002	−0.120001	+	0.992774i	$0.538290\pi$
$968$	0	0
$969$	−4.70233e10	−1.66027
$970$	0	0
$971$	− 3.79338e10i	− 1.32972i	−0.746969	−	0.664859i	$-0.768492\pi$
$971$	− 3.79338e10i	− 1.32972i	0.746969	−	0.664859i	$-0.231508\pi$
$972$	0	0
$973$	− 4.34894e10i	− 1.51352i
$974$	0	0
$975$	−2.16080e10	−0.746618
$976$	0	0
$977$	3.54073e10	1.21468	0.607340	−	0.794442i	$-0.292237\pi$
$977$	3.54073e10	1.21468	0.607340	+	0.794442i	$0.292237\pi$
$978$	0	0
$979$	− 2.42131e10i	− 0.824728i
$980$	0	0
$981$	8.51754e10i	2.88053i
$982$	0	0
$983$	3.38292e8	0.0113594	0.00567970	−	0.999984i	$-0.498192\pi$
$983$	3.38292e8	0.0113594	0.00567970	+	0.999984i	$0.498192\pi$
$984$	0	0
$985$	1.34927e10	0.449854
$986$	0	0
$987$	4.20230e9i	0.139116i
$988$	0	0
$989$	− 2.67352e10i	− 0.878813i
$990$	0	0
$991$	2.44058e10	0.796590	0.398295	−	0.917257i	$-0.369602\pi$
$991$	2.44058e10	0.796590	0.398295	+	0.917257i	$0.369602\pi$
$992$	0	0
$993$	−2.64731e10	−0.857992
$994$	0	0
$995$	1.07449e10i	0.345798i
$996$	0	0
$997$	− 2.79556e10i	− 0.893380i	−0.894689	−	0.446690i	$-0.852603\pi$
$997$	− 2.79556e10i	− 0.893380i	0.894689	−	0.446690i	$-0.147397\pi$
$998$	0	0
$999$	−3.40787e10	−1.08144

Display

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a_n

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a_n

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a_n

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a_n

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)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.8.b.c.65.4	yes	4
4.3	odd	2	inner	128.8.b.c.65.2	yes	4
8.3	odd	2	inner	128.8.b.c.65.3	yes	4
8.5	even	2	inner	128.8.b.c.65.1	✓	4
16.3	odd	4		256.8.a.n.1.4		4
16.5	even	4		256.8.a.n.1.3		4
16.11	odd	4		256.8.a.n.1.1		4
16.13	even	4		256.8.a.n.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.8.b.c.65.1	✓	4	8.5	even	2	inner
128.8.b.c.65.2	yes	4	4.3	odd	2	inner
128.8.b.c.65.3	yes	4	8.3	odd	2	inner
128.8.b.c.65.4	yes	4	1.1	even	1	trivial
256.8.a.n.1.1		4	16.11	odd	4
256.8.a.n.1.2		4	16.13	even	4
256.8.a.n.1.3		4	16.5	even	4
256.8.a.n.1.4		4	16.3	odd	4

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

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Modular forms

Varieties

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Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	128.8.b.c.65.4

Level	$128$
Weight	$8$
Character	128.65
Analytic conductor	$39.985$
Analytic rank	$0$
Dimension	$4$
Inner twists	$4$