LMFDB - Embedded newform 75.16.b.b.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,16,Mod(49,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 16, names="a")

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

chi := DirichletCharacter("75.49");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	75.b (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:	$107.020128825$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	75.49
Dual form			75.16.b.b.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-72.0000i q^{2} -2187.00i q^{3} +27584.0 q^{4} -157464. q^{6} -2.14900e6i q^{7} -4.34534e6i q^{8} -4.78297e6 q^{9} +O(q^{10})$	\(q-72.0000i q^{2} -2187.00i q^{3} +27584.0 q^{4} -157464. q^{6} -2.14900e6i q^{7} -4.34534e6i q^{8} -4.78297e6 q^{9} +3.71693e7 q^{11} -6.03262e7i q^{12} +2.79974e8i q^{13} -1.54728e8 q^{14} +5.91008e8 q^{16} +2.49291e9i q^{17} +3.44374e8i q^{18} +4.66978e9 q^{19} -4.69986e9 q^{21} -2.67619e9i q^{22} +1.84679e10i q^{23} -9.50327e9 q^{24} +2.01581e10 q^{26} +1.04604e10i q^{27} -5.92780e10i q^{28} +1.15953e11 q^{29} -5.61870e10 q^{31} -1.84941e11i q^{32} -8.12893e10i q^{33} +1.79490e11 q^{34} -1.31933e11 q^{36} +6.14765e11i q^{37} -3.36224e11i q^{38} +6.12304e11 q^{39} +5.49860e11 q^{41} +3.38390e11i q^{42} +9.82884e11i q^{43} +1.02528e12 q^{44} +1.32969e12 q^{46} +2.07614e12i q^{47} -1.29253e12i q^{48} +1.29361e11 q^{49} +5.45200e12 q^{51} +7.72281e12i q^{52} +1.20484e13i q^{53} +7.53145e11 q^{54} -9.33814e12 q^{56} -1.02128e13i q^{57} -8.34865e12i q^{58} -2.30879e13 q^{59} -8.50581e12 q^{61} +4.04547e12i q^{62} +1.02786e13i q^{63} +6.05040e12 q^{64} -5.85283e12 q^{66} -1.23310e13i q^{67} +6.87645e13i q^{68} +4.03894e13 q^{69} +5.89892e13 q^{71} +2.07836e13i q^{72} +5.60983e12i q^{73} +4.42631e13 q^{74} +1.28811e14 q^{76} -7.98769e13i q^{77} -4.40859e13i q^{78} -1.59919e14 q^{79} +2.28768e13 q^{81} -3.95899e13i q^{82} -5.76759e13i q^{83} -1.29641e14 q^{84} +7.07677e13 q^{86} -2.53590e14i q^{87} -1.61513e14i q^{88} +3.62288e14 q^{89} +6.01665e14 q^{91} +5.09419e14i q^{92} +1.22881e14i q^{93} +1.49482e14 q^{94} -4.04466e14 q^{96} -5.39787e14i q^{97} -9.31396e12i q^{98} -1.77780e14 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 55168 q^{4} - 314928 q^{6} - 9565938 q^{9}+O(q^{10}) $ 2 * q + 55168 * q^4 - 314928 * q^6 - 9565938 * q^9	$ 2 q + 55168 q^{4} - 314928 q^{6} - 9565938 q^{9} + 74338632 q^{11} - 309456000 q^{14} + 1182015488 q^{16} + 9339564488 q^{19} - 9399726000 q^{21} - 19006534656 q^{24} + 40316294304 q^{26} + 231906898836 q^{29} - 112374046400 q^{31} + 358979436576 q^{34} - 263866833792 q^{36} + 1224607439484 q^{39} + 1099719584820 q^{41} + 2050556825088 q^{44} + 2659382409600 q^{46} + 258721019886 q^{49} + 10904000385996 q^{51} + 1506290861232 q^{54} - 18676288512000 q^{56} - 46175811516648 q^{59} - 17011618284884 q^{61} + 12100809785344 q^{64} - 11705658349248 q^{66} + 80778740691600 q^{69} + 117978385384944 q^{71} + 88526149463520 q^{74} + 257622546836992 q^{76} - 319837367653600 q^{79} + 45753584909922 q^{81} - 259282041984000 q^{84} + 141535359940032 q^{86} + 724575220827948 q^{89} + 12\!\cdots\!00 q^{91}+ \cdots - 355559372358408 q^{99}+O(q^{100}) $ 2 * q + 55168 * q^4 - 314928 * q^6 - 9565938 * q^9 + 74338632 * q^11 - 309456000 * q^14 + 1182015488 * q^16 + 9339564488 * q^19 - 9399726000 * q^21 - 19006534656 * q^24 + 40316294304 * q^26 + 231906898836 * q^29 - 112374046400 * q^31 + 358979436576 * q^34 - 263866833792 * q^36 + 1224607439484 * q^39 + 1099719584820 * q^41 + 2050556825088 * q^44 + 2659382409600 * q^46 + 258721019886 * q^49 + 10904000385996 * q^51 + 1506290861232 * q^54 - 18676288512000 * q^56 - 46175811516648 * q^59 - 17011618284884 * q^61 + 12100809785344 * q^64 - 11705658349248 * q^66 + 80778740691600 * q^69 + 117978385384944 * q^71 + 88526149463520 * q^74 + 257622546836992 * q^76 - 319837367653600 * q^79 + 45753584909922 * q^81 - 259282041984000 * q^84 + 141535359940032 * q^86 + 724575220827948 * q^89 + 1203329395268000 * q^91 + 298964782497024 * q^94 - 808931014410240 * q^96 - 355559372358408 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 72.0000i	− 0.397748i	−0.980025	−	0.198874i	$-0.936272\pi$
$2$	− 72.0000i	− 0.397748i	0.980025	−	0.198874i	$-0.0637284\pi$
$3$	− 2187.00i	− 0.577350i
$3$	− 2187.00i	− 0.577350i
$4$	27584.0	0.841797
$5$	0	0
$5$	0	0
$6$	−157464.	−0.229640
$7$	− 2.14900e6i	− 0.986282i	−0.869949	−	0.493141i	$-0.835849\pi$
$7$	− 2.14900e6i	− 0.986282i	0.869949	−	0.493141i	$-0.164151\pi$
$8$	− 4.34534e6i	− 0.732570i
$9$	−4.78297e6	−0.333333
$10$	0	0
$11$	3.71693e7	0.575095	0.287547	−	0.957766i	$-0.407160\pi$
$11$	3.71693e7	0.575095	0.287547	+	0.957766i	$0.407160\pi$
$12$	− 6.03262e7i	− 0.486012i
$13$	2.79974e8i	1.23749i	0.785590	+	0.618747i	$0.212359\pi$
$13$	2.79974e8i	1.23749i	−0.785590	+	0.618747i	$0.787641\pi$
$14$	−1.54728e8	−0.392291
$15$	0	0
$16$	5.91008e8	0.550419
$17$	2.49291e9i	1.47347i	0.676184	+	0.736733i	$0.263633\pi$
$17$	2.49291e9i	1.47347i	−0.676184	+	0.736733i	$0.736367\pi$
$18$	3.44374e8i	0.132583i
$19$	4.66978e9	1.19852	0.599259	−	0.800555i	$-0.295462\pi$
$19$	4.66978e9	1.19852	0.599259	+	0.800555i	$0.295462\pi$
$20$	0	0
$21$	−4.69986e9	−0.569430
$22$	− 2.67619e9i	− 0.228743i
$23$	1.84679e10i	1.13099i	0.824751	+	0.565496i	$0.191315\pi$
$23$	1.84679e10i	1.13099i	−0.824751	+	0.565496i	$0.808685\pi$
$24$	−9.50327e9	−0.422950
$25$	0	0
$26$	2.01581e10	0.492210
$27$	1.04604e10i	0.192450i
$28$	− 5.92780e10i	− 0.830249i
$29$	1.15953e11	1.24824	0.624121	−	0.781328i	$-0.285457\pi$
$29$	1.15953e11	1.24824	0.624121	+	0.781328i	$0.285457\pi$
$30$	0	0
$31$	−5.61870e10	−0.366795	−0.183397	−	0.983039i	$-0.558710\pi$
$31$	−5.61870e10	−0.366795	−0.183397	+	0.983039i	$0.558710\pi$
$32$	− 1.84941e11i	− 0.951498i
$33$	− 8.12893e10i	− 0.332031i
$34$	1.79490e11	0.586068
$35$	0	0
$36$	−1.31933e11	−0.280599
$37$	6.14765e11i	1.06462i	0.846548	+	0.532312i	$0.178677\pi$
$37$	6.14765e11i	1.06462i	−0.846548	+	0.532312i	$0.821323\pi$
$38$	− 3.36224e11i	− 0.476708i
$39$	6.12304e11	0.714467
$40$	0	0
$41$	5.49860e11	0.440933	0.220467	−	0.975394i	$-0.429242\pi$
$41$	5.49860e11	0.440933	0.220467	+	0.975394i	$0.429242\pi$
$42$	3.38390e11i	0.226489i
$43$	9.82884e11i	0.551428i	0.961240	+	0.275714i	$0.0889143\pi$
$43$	9.82884e11i	0.551428i	−0.961240	+	0.275714i	$0.911086\pi$
$44$	1.02528e12	0.484113
$45$	0	0
$46$	1.32969e12	0.449849
$47$	2.07614e12i	0.597756i	0.954291	+	0.298878i	$0.0966123\pi$
$47$	2.07614e12i	0.597756i	−0.954291	+	0.298878i	$0.903388\pi$
$48$	− 1.29253e12i	− 0.317784i
$49$	1.29361e11	0.0272478
$50$	0	0
$51$	5.45200e12	0.850706
$52$	7.72281e12i	1.04172i
$53$	1.20484e13i	1.40883i	0.709787	+	0.704417i	$0.248791\pi$
$53$	1.20484e13i	1.40883i	−0.709787	+	0.704417i	$0.751209\pi$
$54$	7.53145e11	0.0765466
$55$	0	0
$56$	−9.33814e12	−0.722521
$57$	− 1.02128e13i	− 0.691965i
$58$	− 8.34865e12i	− 0.496485i
$59$	−2.30879e13	−1.20780	−0.603899	−	0.797061i	$-0.706387\pi$
$59$	−2.30879e13	−1.20780	−0.603899	+	0.797061i	$0.706387\pi$
$60$	0	0
$61$	−8.50581e12	−0.346531	−0.173265	−	0.984875i	$-0.555432\pi$
$61$	−8.50581e12	−0.346531	−0.173265	+	0.984875i	$0.555432\pi$
$62$	4.04547e12i	0.145892i
$63$	1.02786e13i	0.328761i
$64$	6.05040e12	0.171963
$65$	0	0
$66$	−5.85283e12	−0.132065
$67$	− 1.23310e13i	− 0.248564i	−0.992247	−	0.124282i	$-0.960337\pi$
$67$	− 1.23310e13i	− 0.248564i	0.992247	−	0.124282i	$-0.0396627\pi$
$68$	6.87645e13i	1.24036i
$69$	4.03894e13	0.652979
$70$	0	0
$71$	5.89892e13	0.769724	0.384862	−	0.922974i	$-0.374249\pi$
$71$	5.89892e13	0.769724	0.384862	+	0.922974i	$0.374249\pi$
$72$	2.07836e13i	0.244190i
$73$	5.60983e12i	0.0594331i	0.999558	+	0.0297165i	$0.00946046\pi$
$73$	5.60983e12i	0.0594331i	−0.999558	+	0.0297165i	$0.990540\pi$
$74$	4.42631e13	0.423452
$75$	0	0
$76$	1.28811e14	1.00891
$77$	− 7.98769e13i	− 0.567206i
$78$	− 4.40859e13i	− 0.284178i
$79$	−1.59919e14	−0.936906	−0.468453	−	0.883488i	$-0.655188\pi$
$79$	−1.59919e14	−0.936906	−0.468453	+	0.883488i	$0.655188\pi$
$80$	0	0
$81$	2.28768e13	0.111111
$82$	− 3.95899e13i	− 0.175380i
$83$	− 5.76759e13i	− 0.233297i	−0.993173	−	0.116648i	$-0.962785\pi$
$83$	− 5.76759e13i	− 0.233297i	0.993173	−	0.116648i	$-0.0372151\pi$
$84$	−1.29641e14	−0.479345
$85$	0	0
$86$	7.07677e13	0.219329
$87$	− 2.53590e14i	− 0.720673i
$88$	− 1.61513e14i	− 0.421297i
$89$	3.62288e14	0.868217	0.434109	−	0.900861i	$-0.357063\pi$
$89$	3.62288e14	0.868217	0.434109	+	0.900861i	$0.357063\pi$
$90$	0	0
$91$	6.01665e14	1.22052
$92$	5.09419e14i	0.952066i
$93$	1.22881e14i	0.211769i
$94$	1.49482e14	0.237756
$95$	0	0
$96$	−4.04466e14	−0.549348
$97$	− 5.39787e14i	− 0.678319i	−0.940729	−	0.339160i	$-0.889857\pi$
$97$	− 5.39787e14i	− 0.678319i	0.940729	−	0.339160i	$-0.110143\pi$
$98$	− 9.31396e12i	− 0.0108377i
$99$	−1.77780e14	−0.191698
$100$	0	0
$101$	−4.22221e14	−0.391859	−0.195929	−	0.980618i	$-0.562772\pi$
$101$	−4.22221e14	−0.391859	−0.195929	+	0.980618i	$0.562772\pi$
$102$	− 3.92544e14i	− 0.338366i
$103$	1.45689e14i	0.116720i	0.998296	+	0.0583602i	$0.0185872\pi$
$103$	1.45689e14i	0.116720i	−0.998296	+	0.0583602i	$0.981413\pi$
$104$	1.21658e15	0.906551
$105$	0	0
$106$	8.67483e14	0.560360
$107$	− 4.08696e14i	− 0.246049i	−0.992404	−	0.123024i	$-0.960741\pi$
$107$	− 4.08696e14i	− 0.246049i	0.992404	−	0.123024i	$-0.0392594\pi$
$108$	2.88538e14i	0.162004i
$109$	−1.10134e15	−0.577060	−0.288530	−	0.957471i	$-0.593167\pi$
$109$	−1.10134e15	−0.577060	−0.288530	+	0.957471i	$0.593167\pi$
$110$	0	0
$111$	1.34449e15	0.614661
$112$	− 1.27008e15i	− 0.542868i
$113$	2.13023e15i	0.851802i	0.904770	+	0.425901i	$0.140043\pi$
$113$	2.13023e15i	0.851802i	−0.904770	+	0.425901i	$0.859957\pi$
$114$	−7.35323e14	−0.275227
$115$	0	0
$116$	3.19846e15	1.05077
$117$	− 1.33911e15i	− 0.412498i
$118$	1.66233e15i	0.480399i
$119$	5.35727e15	1.45325
$120$	0	0
$121$	−2.79569e15	−0.669266
$122$	6.12418e14i	0.137832i
$123$	− 1.20254e15i	− 0.254573i
$124$	−1.54986e15	−0.308767
$125$	0	0
$126$	7.40059e14	0.130764
$127$	− 6.25962e15i	− 1.04236i	−0.853445	−	0.521182i	$-0.825491\pi$
$127$	− 6.25962e15i	− 1.04236i	0.853445	−	0.521182i	$-0.174509\pi$
$128$	− 6.49577e15i	− 1.01990i
$129$	2.14957e15	0.318367
$130$	0	0
$131$	1.01059e16	1.33365	0.666825	−	0.745214i	$-0.267653\pi$
$131$	1.01059e16	1.33365	0.666825	+	0.745214i	$0.267653\pi$
$132$	− 2.24228e15i	− 0.279503i
$133$	− 1.00354e16i	− 1.18208i
$134$	−8.87833e14	−0.0988656
$135$	0	0
$136$	1.08326e16	1.07942
$137$	− 9.93739e15i	− 0.937277i	−0.883390	−	0.468638i	$-0.844745\pi$
$137$	− 9.93739e15i	− 0.937277i	0.883390	−	0.468638i	$-0.155255\pi$
$138$	− 2.90803e15i	− 0.259721i
$139$	1.95078e16	1.65043	0.825216	−	0.564817i	$-0.191053\pi$
$139$	1.95078e16	1.65043	0.825216	+	0.564817i	$0.191053\pi$
$140$	0	0
$141$	4.54053e15	0.345114
$142$	− 4.24722e15i	− 0.306156i
$143$	1.04065e16i	0.711676i
$144$	−2.82677e15	−0.183473
$145$	0	0
$146$	4.03908e14	0.0236394
$147$	− 2.82911e14i	− 0.0157315i
$148$	1.69577e16i	0.896197i
$149$	−1.58935e16	−0.798589	−0.399295	−	0.916823i	$-0.630745\pi$
$149$	−1.58935e16	−0.798589	−0.399295	+	0.916823i	$0.630745\pi$
$150$	0	0
$151$	3.10597e16	1.41211	0.706057	−	0.708155i	$-0.250472\pi$
$151$	3.10597e16	1.41211	0.706057	+	0.708155i	$0.250472\pi$
$152$	− 2.02918e16i	− 0.877999i
$153$	− 1.19235e16i	− 0.491155i
$154$	−5.75113e15	−0.225605
$155$	0	0
$156$	1.68898e16	0.601436
$157$	− 5.41372e16i	− 1.83759i	−0.394738	−	0.918794i	$-0.629165\pi$
$157$	− 5.41372e16i	− 1.83759i	0.394738	−	0.918794i	$-0.370835\pi$
$158$	1.15141e16i	0.372652i
$159$	2.63498e16	0.813390
$160$	0	0
$161$	3.96876e16	1.11548
$162$	− 1.64713e15i	− 0.0441942i
$163$	5.54612e16i	1.42096i	0.703717	+	0.710481i	$0.251522\pi$
$163$	5.54612e16i	1.42096i	−0.703717	+	0.710481i	$0.748478\pi$
$164$	1.51673e16	0.371176
$165$	0	0
$166$	−4.15266e15	−0.0927932
$167$	− 5.83552e15i	− 0.124654i	−0.998056	−	0.0623270i	$-0.980148\pi$
$167$	− 5.83552e15i	− 0.124654i	0.998056	−	0.0623270i	$-0.0198522\pi$
$168$	2.04225e16i	0.417148i
$169$	−2.71997e16	−0.531390
$170$	0	0
$171$	−2.23354e16	−0.399506
$172$	2.71119e16i	0.464191i
$173$	− 4.40958e16i	− 0.722855i	−0.932400	−	0.361428i	$-0.882289\pi$
$173$	− 4.40958e16i	− 0.722855i	0.932400	−	0.361428i	$-0.117711\pi$
$174$	−1.82585e16	−0.286646
$175$	0	0
$176$	2.19674e16	0.316543
$177$	5.04932e16i	0.697323i
$178$	− 2.60847e16i	− 0.345331i
$179$	5.10594e16	0.648153	0.324077	−	0.946031i	$-0.394946\pi$
$179$	5.10594e16	0.648153	0.324077	+	0.946031i	$0.394946\pi$
$180$	0	0
$181$	3.16143e16	0.369228	0.184614	−	0.982811i	$-0.440897\pi$
$181$	3.16143e16	0.369228	0.184614	+	0.982811i	$0.440897\pi$
$182$	− 4.33199e16i	− 0.485458i
$183$	1.86022e16i	0.200070i
$184$	8.02495e16	0.828531
$185$	0	0
$186$	8.84743e15	0.0842307
$187$	9.26599e16i	0.847383i
$188$	5.72684e16i	0.503189i
$189$	2.24793e16	0.189810
$190$	0	0
$191$	9.44291e16	0.736810	0.368405	−	0.929665i	$-0.379904\pi$
$191$	9.44291e16	0.736810	0.368405	+	0.929665i	$0.379904\pi$
$192$	− 1.32322e16i	− 0.0992828i
$193$	− 4.17744e16i	− 0.301460i	−0.988575	−	0.150730i	$-0.951838\pi$
$193$	− 4.17744e16i	− 0.301460i	0.988575	−	0.150730i	$-0.0481624\pi$
$194$	−3.88646e16	−0.269800
$195$	0	0
$196$	3.56828e15	0.0229371
$197$	− 1.48383e17i	− 0.918094i	−0.888412	−	0.459047i	$-0.848191\pi$
$197$	− 1.48383e17i	− 0.918094i	0.888412	−	0.459047i	$-0.151809\pi$
$198$	1.28001e16i	0.0762475i
$199$	−1.97555e17	−1.13316	−0.566579	−	0.824007i	$-0.691733\pi$
$199$	−1.97555e17	−1.13316	−0.566579	+	0.824007i	$0.691733\pi$
$200$	0	0
$201$	−2.69679e16	−0.143508
$202$	3.03999e16i	0.155861i
$203$	− 2.49184e17i	− 1.23112i
$204$	1.50388e17	0.716122
$205$	0	0
$206$	1.04896e16	0.0464253
$207$	− 8.83316e16i	− 0.376997i
$208$	1.65467e17i	0.681140i
$209$	1.73573e17	0.689262
$210$	0	0
$211$	−4.48570e17	−1.65849	−0.829244	−	0.558887i	$-0.811228\pi$
$211$	−4.48570e17	−1.65849	−0.829244	+	0.558887i	$0.811228\pi$
$212$	3.32342e17i	1.18595i
$213$	− 1.29009e17i	− 0.444400i
$214$	−2.94261e16	−0.0978654
$215$	0	0
$216$	4.54538e16	0.140983
$217$	1.20746e17i	0.361763i
$218$	7.92962e16i	0.229524i
$219$	1.22687e16	0.0343137
$220$	0	0
$221$	−6.97951e17	−1.82341
$222$	− 9.68033e16i	− 0.244480i
$223$	− 1.59977e17i	− 0.390634i	−0.980740	−	0.195317i	$-0.937426\pi$
$223$	− 1.59977e17i	− 0.390634i	0.980740	−	0.195317i	$-0.0625735\pi$
$224$	−3.97438e17	−0.938445
$225$	0	0
$226$	1.53377e17	0.338802
$227$	1.45803e17i	0.311582i	0.987790	+	0.155791i	$0.0497926\pi$
$227$	1.45803e17i	0.311582i	−0.987790	+	0.155791i	$0.950207\pi$
$228$	− 2.81710e17i	− 0.582494i
$229$	−6.81194e17	−1.36303	−0.681514	−	0.731805i	$-0.738678\pi$
$229$	−6.81194e17	−1.36303	−0.681514	+	0.731805i	$0.738678\pi$
$230$	0	0
$231$	−1.74691e17	−0.327476
$232$	− 5.03858e17i	− 0.914425i
$233$	3.89035e17i	0.683628i	0.939768	+	0.341814i	$0.111041\pi$
$233$	3.89035e17i	0.683628i	−0.939768	+	0.341814i	$0.888959\pi$
$234$	−9.64158e16	−0.164070
$235$	0	0
$236$	−6.36857e17	−1.01672
$237$	3.49742e17i	0.540923i
$238$	− 3.85723e17i	− 0.578028i
$239$	−6.76227e16	−0.0981993	−0.0490996	−	0.998794i	$-0.515635\pi$
$239$	−6.76227e16	−0.0981993	−0.0490996	+	0.998794i	$0.515635\pi$
$240$	0	0
$241$	−9.00528e17	−1.22848	−0.614242	−	0.789118i	$-0.710538\pi$
$241$	−9.00528e17	−1.22848	−0.614242	+	0.789118i	$0.710538\pi$
$242$	2.01290e17i	0.266199i
$243$	− 5.00315e16i	− 0.0641500i
$244$	−2.34624e17	−0.291709
$245$	0	0
$246$	−8.65831e16	−0.101256
$247$	1.30742e18i	1.48316i
$248$	2.44152e17i	0.268703i
$249$	−1.26137e17	−0.134694
$250$	0	0
$251$	1.58929e18	1.59827	0.799134	−	0.601153i	$-0.205292\pi$
$251$	1.58929e18	1.59827	0.799134	+	0.601153i	$0.205292\pi$
$252$	2.83525e17i	0.276750i
$253$	6.86440e17i	0.650428i
$254$	−4.50692e17	−0.414598
$255$	0	0
$256$	−2.69436e17	−0.233698
$257$	2.29738e18i	1.93524i	0.252418	+	0.967618i	$0.418774\pi$
$257$	2.29738e18i	1.93524i	−0.252418	+	0.967618i	$0.581226\pi$
$258$	− 1.54769e17i	− 0.126630i
$259$	1.32113e18	1.05002
$260$	0	0
$261$	−5.54602e17	−0.416081
$262$	− 7.27628e17i	− 0.530456i
$263$	5.05657e17i	0.358252i	0.983826	+	0.179126i	$0.0573269\pi$
$263$	5.05657e17i	0.358252i	−0.983826	+	0.179126i	$0.942673\pi$
$264$	−3.53230e17	−0.243236
$265$	0	0
$266$	−7.22546e17	−0.470168
$267$	− 7.92323e17i	− 0.501265i
$268$	− 3.40139e17i	− 0.209240i
$269$	2.44731e18	1.46402	0.732009	−	0.681295i	$-0.238583\pi$
$269$	2.44731e18	1.46402	0.732009	+	0.681295i	$0.238583\pi$
$270$	0	0
$271$	−1.53807e18	−0.870373	−0.435187	−	0.900340i	$-0.643318\pi$
$271$	−1.53807e18	−0.870373	−0.435187	+	0.900340i	$0.643318\pi$
$272$	1.47333e18i	0.811024i
$273$	− 1.31584e18i	− 0.704666i
$274$	−7.15492e17	−0.372800
$275$	0	0
$276$	1.11410e18	0.549675
$277$	− 7.89959e17i	− 0.379321i	−0.981850	−	0.189660i	$-0.939261\pi$
$277$	− 7.89959e17i	− 0.379321i	0.981850	−	0.189660i	$-0.0607387\pi$
$278$	− 1.40456e18i	− 0.656456i
$279$	2.68741e17	0.122265
$280$	0	0
$281$	−6.15303e17	−0.265333	−0.132667	−	0.991161i	$-0.542354\pi$
$281$	−6.15303e17	−0.265333	−0.132667	+	0.991161i	$0.542354\pi$
$282$	− 3.26918e17i	− 0.137268i
$283$	1.26279e18i	0.516337i	0.966100	+	0.258169i	$0.0831190\pi$
$283$	1.26279e18i	0.516337i	−0.966100	+	0.258169i	$0.916881\pi$
$284$	1.62716e18	0.647951
$285$	0	0
$286$	7.49265e17	0.283067
$287$	− 1.18165e18i	− 0.434885i
$288$	8.84566e17i	0.317166i
$289$	−3.35219e18	−1.17110
$290$	0	0
$291$	−1.18051e18	−0.391628
$292$	1.54742e17i	0.0500306i
$293$	3.49864e18i	1.10253i	0.834329	+	0.551267i	$0.185855\pi$
$293$	3.49864e18i	1.10253i	−0.834329	+	0.551267i	$0.814145\pi$
$294$	−2.03696e16	−0.00625717
$295$	0	0
$296$	2.67137e18	0.779912
$297$	3.88804e17i	0.110677i
$298$	1.14433e18i	0.317637i
$299$	−5.17055e18	−1.39960
$300$	0	0
$301$	2.11222e18	0.543864
$302$	− 2.23630e18i	− 0.561665i
$303$	9.23398e17i	0.226240i
$304$	2.75988e18	0.659687
$305$	0	0
$306$	−8.58494e17	−0.195356
$307$	1.28861e18i	0.286142i	0.989712	+	0.143071i	$0.0456978\pi$
$307$	1.28861e18i	0.286142i	−0.989712	+	0.143071i	$0.954302\pi$
$308$	− 2.20332e18i	− 0.477472i
$309$	3.18621e17	0.0673886
$310$	0	0
$311$	8.43030e18	1.69879	0.849396	−	0.527756i	$-0.176967\pi$
$311$	8.43030e18	1.69879	0.849396	+	0.527756i	$0.176967\pi$
$312$	− 2.66067e18i	− 0.523397i
$313$	− 7.88359e18i	− 1.51405i	−0.653383	−	0.757027i	$-0.726651\pi$
$313$	− 7.88359e18i	− 1.51405i	0.653383	−	0.757027i	$-0.273349\pi$
$314$	−3.89788e18	−0.730896
$315$	0	0
$316$	−4.41120e18	−0.788685
$317$	− 8.71539e18i	− 1.52175i	−0.648900	−	0.760873i	$-0.724771\pi$
$317$	− 8.71539e18i	− 1.52175i	0.648900	−	0.760873i	$-0.275229\pi$
$318$	− 1.89719e18i	− 0.323524i
$319$	4.30991e18	0.717857
$320$	0	0
$321$	−8.93817e17	−0.142056
$322$	− 2.85751e18i	− 0.443678i
$323$	1.16414e19i	1.76598i
$324$	6.31033e17	0.0935330
$325$	0	0
$326$	3.99321e18	0.565184
$327$	2.40862e18i	0.333166i
$328$	− 2.38933e18i	− 0.323015i
$329$	4.46163e18	0.589556
$330$	0	0
$331$	1.26995e19	1.60353	0.801763	−	0.597643i	$-0.203896\pi$
$331$	1.26995e19	1.60353	0.801763	+	0.597643i	$0.203896\pi$
$332$	− 1.59093e18i	− 0.196388i
$333$	− 2.94040e18i	− 0.354875i
$334$	−4.20157e17	−0.0495808
$335$	0	0
$336$	−2.77766e18	−0.313425
$337$	− 6.64250e18i	− 0.733005i	−0.930417	−	0.366503i	$-0.880555\pi$
$337$	− 6.64250e18i	− 0.733005i	0.930417	−	0.366503i	$-0.119445\pi$
$338$	1.95838e18i	0.211359i
$339$	4.65882e18	0.491788
$340$	0	0
$341$	−2.08843e18	−0.210942
$342$	1.60815e18i	0.158903i
$343$	− 1.04805e19i	− 1.01316i
$344$	4.27097e18	0.403960
$345$	0	0
$346$	−3.17490e18	−0.287514
$347$	1.87835e18i	0.166458i	0.996530	+	0.0832289i	$0.0265233\pi$
$347$	1.87835e18i	0.166458i	−0.996530	+	0.0832289i	$0.973477\pi$
$348$	− 6.99503e18i	− 0.606660i
$349$	1.48545e19	1.26086	0.630432	−	0.776245i	$-0.282878\pi$
$349$	1.48545e19	1.26086	0.630432	+	0.776245i	$0.282878\pi$
$350$	0	0
$351$	−2.92863e18	−0.238156
$352$	− 6.87412e18i	− 0.547202i
$353$	8.23380e18i	0.641638i	0.947140	+	0.320819i	$0.103958\pi$
$353$	8.23380e18i	0.641638i	−0.947140	+	0.320819i	$0.896042\pi$
$354$	3.63551e18	0.277358
$355$	0	0
$356$	9.99334e18	0.730863
$357$	− 1.17163e19i	− 0.839036i
$358$	− 3.67627e18i	− 0.257801i
$359$	−1.87417e19	−1.28706	−0.643532	−	0.765419i	$-0.722532\pi$
$359$	−1.87417e19	−1.28706	−0.643532	+	0.765419i	$0.722532\pi$
$360$	0	0
$361$	6.62574e18	0.436446
$362$	− 2.27623e18i	− 0.146859i
$363$	6.11417e18i	0.386401i
$364$	1.65963e19	1.02743
$365$	0	0
$366$	1.33936e18	0.0795772
$367$	8.63117e18i	0.502428i	0.967932	+	0.251214i	$0.0808298\pi$
$367$	8.63117e18i	0.502428i	−0.967932	+	0.251214i	$0.919170\pi$
$368$	1.09147e19i	0.622519i
$369$	−2.62996e18	−0.146978
$370$	0	0
$371$	2.58920e19	1.38951
$372$	3.38955e18i	0.178267i
$373$	9.95975e18i	0.513372i	0.966495	+	0.256686i	$0.0826306\pi$
$373$	9.95975e18i	0.513372i	−0.966495	+	0.256686i	$0.917369\pi$
$374$	6.67151e18	0.337044
$375$	0	0
$376$	9.02156e18	0.437898
$377$	3.24640e19i	1.54469i
$378$	− 1.61851e18i	− 0.0754965i
$379$	−1.00949e19	−0.461645	−0.230823	−	0.972996i	$-0.574142\pi$
$379$	−1.00949e19	−0.461645	−0.230823	+	0.972996i	$0.574142\pi$
$380$	0	0
$381$	−1.36898e19	−0.601810
$382$	− 6.79890e18i	− 0.293065i
$383$	2.71764e19i	1.14869i	0.818614	+	0.574343i	$0.194743\pi$
$383$	2.71764e19i	1.14869i	−0.818614	+	0.574343i	$0.805257\pi$
$384$	−1.42062e19	−0.588837
$385$	0	0
$386$	−3.00775e18	−0.119905
$387$	− 4.70111e18i	− 0.183809i
$388$	− 1.48895e19i	− 0.571007i
$389$	9.05285e18	0.340536	0.170268	−	0.985398i	$-0.445537\pi$
$389$	9.05285e18	0.340536	0.170268	+	0.985398i	$0.445537\pi$
$390$	0	0
$391$	−4.60389e19	−1.66648
$392$	− 5.62116e17i	− 0.0199609i
$393$	− 2.21017e19i	− 0.769984i
$394$	−1.06836e19	−0.365170
$395$	0	0
$396$	−4.90387e18	−0.161371
$397$	− 1.76495e19i	− 0.569908i	−0.958541	−	0.284954i	$-0.908022\pi$
$397$	− 1.76495e19i	− 0.569908i	0.958541	−	0.284954i	$-0.0919783\pi$
$398$	1.42240e19i	0.450711i
$399$	−2.19473e19	−0.682472
$400$	0	0
$401$	2.81135e19	0.842040	0.421020	−	0.907051i	$-0.361672\pi$
$401$	2.81135e19	0.842040	0.421020	+	0.907051i	$0.361672\pi$
$402$	1.94169e18i	0.0570801i
$403$	− 1.57309e19i	− 0.453906i
$404$	−1.16465e19	−0.329866
$405$	0	0
$406$	−1.79412e19	−0.489674
$407$	2.28504e19i	0.612260i
$408$	− 2.36908e19i	− 0.623202i
$409$	−6.42955e19	−1.66056	−0.830282	−	0.557343i	$-0.811821\pi$
$409$	−6.42955e19	−1.66056	−0.830282	+	0.557343i	$0.811821\pi$
$410$	0	0
$411$	−2.17331e19	−0.541137
$412$	4.01868e18i	0.0982549i
$413$	4.96159e19i	1.19123i
$414$	−6.35987e18	−0.149950
$415$	0	0
$416$	5.17787e19	1.17747
$417$	− 4.26636e19i	− 0.952878i
$418$	− 1.24972e19i	− 0.274152i
$419$	2.34474e19	0.505230	0.252615	−	0.967567i	$-0.418709\pi$
$419$	2.34474e19	0.505230	0.252615	+	0.967567i	$0.418709\pi$
$420$	0	0
$421$	−7.28227e19	−1.51409	−0.757044	−	0.653364i	$-0.773357\pi$
$421$	−7.28227e19	−1.51409	−0.757044	+	0.653364i	$0.773357\pi$
$422$	3.22971e19i	0.659659i
$423$	− 9.93013e18i	− 0.199252i
$424$	5.23543e19	1.03207
$425$	0	0
$426$	−9.28867e18	−0.176759
$427$	1.82790e19i	0.341777i
$428$	− 1.12735e19i	− 0.207123i
$429$	2.27589e19	0.410886
$430$	0	0
$431$	−4.71831e19	−0.822635	−0.411317	−	0.911492i	$-0.634931\pi$
$431$	−4.71831e19	−0.822635	−0.411317	+	0.911492i	$0.634931\pi$
$432$	6.18215e18i	0.105928i
$433$	− 3.79099e19i	− 0.638400i	−0.947687	−	0.319200i	$-0.896586\pi$
$433$	− 3.79099e19i	− 0.638400i	0.947687	−	0.319200i	$-0.103414\pi$
$434$	8.69371e18	0.143890
$435$	0	0
$436$	−3.03793e19	−0.485768
$437$	8.62412e19i	1.35551i
$438$	− 8.83346e17i	− 0.0136482i
$439$	2.26832e19	0.344525	0.172262	−	0.985051i	$-0.444892\pi$
$439$	2.26832e19	0.344525	0.172262	+	0.985051i	$0.444892\pi$
$440$	0	0
$441$	−6.18727e17	−0.00908259
$442$	5.02525e19i	0.725255i
$443$	9.94888e19i	1.41171i	0.708355	+	0.705856i	$0.249438\pi$
$443$	9.94888e19i	1.41171i	−0.708355	+	0.705856i	$0.750562\pi$
$444$	3.70864e19	0.517420
$445$	0	0
$446$	−1.15183e19	−0.155374
$447$	3.47592e19i	0.461066i
$448$	− 1.30023e19i	− 0.169604i
$449$	−1.30159e20	−1.66965	−0.834825	−	0.550515i	$-0.814432\pi$
$449$	−1.30159e20	−1.66965	−0.834825	+	0.550515i	$0.814432\pi$
$450$	0	0
$451$	2.04379e19	0.253579
$452$	5.87604e19i	0.717045i
$453$	− 6.79275e19i	− 0.815285i
$454$	1.04978e19	0.123931
$455$	0	0
$456$	−4.43782e19	−0.506913
$457$	− 1.01910e20i	− 1.14510i	−0.819870	−	0.572550i	$-0.805954\pi$
$457$	− 1.01910e20i	− 1.14510i	0.819870	−	0.572550i	$-0.194046\pi$
$458$	4.90460e19i	0.542141i
$459$	−2.60767e19	−0.283569
$460$	0	0
$461$	−1.08142e20	−1.13825	−0.569123	−	0.822253i	$-0.692717\pi$
$461$	−1.08142e20	−1.13825	−0.569123	+	0.822253i	$0.692717\pi$
$462$	1.25777e19i	0.130253i
$463$	− 1.69113e20i	− 1.72314i	−0.507641	−	0.861569i	$-0.669482\pi$
$463$	− 1.69113e20i	− 1.72314i	0.507641	−	0.861569i	$-0.330518\pi$
$464$	6.85294e19	0.687056
$465$	0	0
$466$	2.80105e19	0.271911
$467$	− 4.58418e19i	− 0.437910i	−0.975735	−	0.218955i	$-0.929735\pi$
$467$	− 4.58418e19i	− 0.437910i	0.975735	−	0.218955i	$-0.0702648\pi$
$468$	− 3.69380e19i	− 0.347239i
$469$	−2.64993e19	−0.245154
$470$	0	0
$471$	−1.18398e20	−1.06093
$472$	1.00325e20i	0.884797i
$473$	3.65331e19i	0.317124i
$474$	2.51814e19	0.215151
$475$	0	0
$476$	1.47775e20	1.22334
$477$	− 5.76270e19i	− 0.469611i
$478$	4.86884e18i	0.0390585i
$479$	−1.31513e20	−1.03861	−0.519303	−	0.854590i	$-0.673808\pi$
$479$	−1.31513e20	−1.03861	−0.519303	+	0.854590i	$0.673808\pi$
$480$	0	0
$481$	−1.72118e20	−1.31747
$482$	6.48380e19i	0.488626i
$483$	− 8.67968e19i	− 0.644021i
$484$	−7.71163e19	−0.563386
$485$	0	0
$486$	−3.60227e18	−0.0255155
$487$	6.13671e19i	0.428024i	0.976831	+	0.214012i	$0.0686532\pi$
$487$	6.13671e19i	0.428024i	−0.976831	+	0.214012i	$0.931347\pi$
$488$	3.69607e19i	0.253858i
$489$	1.21294e20	0.820392
$490$	0	0
$491$	2.42057e20	1.58784	0.793918	−	0.608024i	$-0.208038\pi$
$491$	2.42057e20	1.58784	0.793918	+	0.608024i	$0.208038\pi$
$492$	− 3.31710e19i	− 0.214299i
$493$	2.89062e20i	1.83924i
$494$	9.41342e19	0.589923
$495$	0	0
$496$	−3.32070e19	−0.201891
$497$	− 1.26768e20i	− 0.759165i
$498$	9.08188e18i	0.0535742i
$499$	−1.46049e20	−0.848680	−0.424340	−	0.905503i	$-0.639494\pi$
$499$	−1.46049e20	−0.848680	−0.424340	+	0.905503i	$0.639494\pi$
$500$	0	0
$501$	−1.27623e19	−0.0719690
$502$	− 1.14429e20i	− 0.635707i
$503$	1.68041e20i	0.919720i	0.887991	+	0.459860i	$0.152100\pi$
$503$	1.68041e20i	0.919720i	−0.887991	+	0.459860i	$0.847900\pi$
$504$	4.46641e19	0.240840
$505$	0	0
$506$	4.94237e19	0.258706
$507$	5.94857e19i	0.306798i
$508$	− 1.72665e20i	− 0.877459i
$509$	−7.37709e19	−0.369404	−0.184702	−	0.982795i	$-0.559132\pi$
$509$	−7.37709e19	−0.369404	−0.184702	+	0.982795i	$0.559132\pi$
$510$	0	0
$511$	1.20555e19	0.0586178
$512$	− 1.93454e20i	− 0.926943i
$513$	4.88476e19i	0.230655i
$514$	1.65411e20	0.769736
$515$	0	0
$516$	5.92937e19	0.268001
$517$	7.71689e19i	0.343766i
$518$	− 9.51213e19i	− 0.417643i
$519$	−9.64375e19	−0.417341
$520$	0	0
$521$	3.22023e20	1.35396	0.676978	−	0.736003i	$-0.263289\pi$
$521$	3.22023e20	1.35396	0.676978	+	0.736003i	$0.263289\pi$
$522$	3.99313e19i	0.165495i
$523$	1.19064e19i	0.0486429i	0.999704	+	0.0243214i	$0.00774252\pi$
$523$	1.19064e19i	0.0486429i	−0.999704	+	0.0243214i	$0.992257\pi$
$524$	2.78762e20	1.12266
$525$	0	0
$526$	3.64073e19	0.142494
$527$	− 1.40069e20i	− 0.540460i
$528$	− 4.80426e19i	− 0.182756i
$529$	−7.44293e19	−0.279143
$530$	0	0
$531$	1.10429e20	0.402599
$532$	− 2.76815e20i	− 0.995069i
$533$	1.53947e20i	0.545652i
$534$	−5.70473e19	−0.199377
$535$	0	0
$536$	−5.35825e19	−0.182090
$537$	− 1.11667e20i	− 0.374211i
$538$	− 1.76206e20i	− 0.582309i
$539$	4.80824e18	0.0156701
$540$	0	0
$541$	1.59033e20	0.504089	0.252045	−	0.967716i	$-0.418897\pi$
$541$	1.59033e20	0.504089	0.252045	+	0.967716i	$0.418897\pi$
$542$	1.10741e20i	0.346189i
$543$	− 6.91405e19i	− 0.213174i
$544$	4.61041e20	1.40200
$545$	0	0
$546$	−9.47405e19	−0.280279
$547$	− 4.25405e20i	− 1.24136i	−0.784064	−	0.620680i	$-0.786857\pi$
$547$	− 4.25405e20i	− 1.24136i	0.784064	−	0.620680i	$-0.213143\pi$
$548$	− 2.74113e20i	− 0.788997i
$549$	4.06830e19	0.115510
$550$	0	0
$551$	5.41477e20	1.49604
$552$	− 1.75506e20i	− 0.478353i
$553$	3.43665e20i	0.924054i
$554$	−5.68771e19	−0.150874
$555$	0	0
$556$	5.38104e20	1.38933
$557$	2.71579e20i	0.691803i	0.938271	+	0.345901i	$0.112427\pi$
$557$	2.71579e20i	0.691803i	−0.938271	+	0.345901i	$0.887573\pi$
$558$	− 1.93493e19i	− 0.0486306i
$559$	−2.75182e20	−0.682389
$560$	0	0
$561$	2.02647e20	0.489237
$562$	4.43018e19i	0.105536i
$563$	2.68438e20i	0.631002i	0.948925	+	0.315501i	$0.102173\pi$
$563$	2.68438e20i	0.631002i	−0.948925	+	0.315501i	$0.897827\pi$
$564$	1.25246e20	0.290516
$565$	0	0
$566$	9.09210e19	0.205372
$567$	− 4.91622e19i	− 0.109587i
$568$	− 2.56328e20i	− 0.563877i
$569$	1.34893e20	0.292853	0.146426	−	0.989222i	$-0.453223\pi$
$569$	1.34893e20	0.292853	0.146426	+	0.989222i	$0.453223\pi$
$570$	0	0
$571$	−6.81153e20	−1.44037	−0.720185	−	0.693782i	$-0.755943\pi$
$571$	−6.81153e20	−1.44037	−0.720185	+	0.693782i	$0.755943\pi$
$572$	2.87052e20i	0.599087i
$573$	− 2.06517e20i	− 0.425398i
$574$	−8.50787e19	−0.172974
$575$	0	0
$576$	−2.89389e19	−0.0573209
$577$	− 2.42927e20i	− 0.474961i	−0.971392	−	0.237480i	$-0.923678\pi$
$577$	− 2.42927e20i	− 0.474961i	0.971392	−	0.237480i	$-0.0763215\pi$
$578$	2.41358e20i	0.465803i
$579$	−9.13605e19	−0.174048
$580$	0	0
$581$	−1.23945e20	−0.230096
$582$	8.49970e19i	0.155769i
$583$	4.47830e20i	0.810213i
$584$	2.43766e19	0.0435389
$585$	0	0
$586$	2.51902e20	0.438530
$587$	− 1.06241e20i	− 0.182602i	−0.995823	−	0.0913012i	$-0.970897\pi$
$587$	− 1.06241e20i	− 0.182602i	0.995823	−	0.0913012i	$-0.0291026\pi$
$588$	− 7.80383e18i	− 0.0132427i
$589$	−2.62381e20	−0.439610
$590$	0	0
$591$	−3.24513e20	−0.530062
$592$	3.63331e20i	0.585989i
$593$	7.70415e20i	1.22692i	0.789727	+	0.613458i	$0.210222\pi$
$593$	7.70415e20i	1.22692i	−0.789727	+	0.613458i	$0.789778\pi$
$594$	2.79939e19	0.0440215
$595$	0	0
$596$	−4.38407e20	−0.672250
$597$	4.32053e20i	0.654229i
$598$	3.72279e20i	0.556686i
$599$	7.59796e19	0.112201	0.0561004	−	0.998425i	$-0.482133\pi$
$599$	7.59796e19	0.112201	0.0561004	+	0.998425i	$0.482133\pi$
$600$	0	0
$601$	7.78003e20	1.12053	0.560264	−	0.828314i	$-0.310700\pi$
$601$	7.78003e20	1.12053	0.560264	+	0.828314i	$0.310700\pi$
$602$	− 1.52080e20i	− 0.216321i
$603$	5.89788e19i	0.0828546i
$604$	8.56750e20	1.18871
$605$	0	0
$606$	6.64846e19	0.0899863
$607$	3.21867e20i	0.430290i	0.976582	+	0.215145i	$0.0690224\pi$
$607$	3.21867e20i	0.430290i	−0.976582	+	0.215145i	$0.930978\pi$
$608$	− 8.63633e20i	− 1.14039i
$609$	−5.44965e20	−0.710787
$610$	0	0
$611$	−5.81267e20	−0.739719
$612$	− 3.28898e20i	− 0.413453i
$613$	− 6.68867e20i	− 0.830589i	−0.909687	−	0.415294i	$-0.863679\pi$
$613$	− 6.68867e20i	− 0.830589i	0.909687	−	0.415294i	$-0.136321\pi$
$614$	9.27796e19	0.113812
$615$	0	0
$616$	−3.47092e20	−0.415518
$617$	− 8.47904e20i	− 1.00279i	−0.865220	−	0.501393i	$-0.832821\pi$
$617$	− 8.47904e20i	− 1.00279i	0.865220	−	0.501393i	$-0.167179\pi$
$618$	− 2.29407e19i	− 0.0268036i
$619$	3.51027e20	0.405192	0.202596	−	0.979262i	$-0.435062\pi$
$619$	3.51027e20	0.405192	0.202596	+	0.979262i	$0.435062\pi$
$620$	0	0
$621$	−1.93181e20	−0.217660
$622$	− 6.06982e20i	− 0.675690i
$623$	− 7.78556e20i	− 0.856307i
$624$	3.61876e20	0.393256
$625$	0	0
$626$	−5.67619e20	−0.602212
$627$	− 3.79603e20i	− 0.397945i
$628$	− 1.49332e21i	− 1.54688i
$629$	−1.53256e21	−1.56869
$630$	0	0
$631$	−1.30718e21	−1.30652	−0.653260	−	0.757134i	$-0.726599\pi$
$631$	−1.30718e21	−1.30652	−0.653260	+	0.757134i	$0.726599\pi$
$632$	6.94902e20i	0.686349i
$633$	9.81023e20i	0.957528i
$634$	−6.27508e20	−0.605271
$635$	0	0
$636$	7.26833e20	0.684709
$637$	3.62176e19i	0.0337190i
$638$	− 3.10314e20i	− 0.285526i
$639$	−2.82143e20	−0.256575
$640$	0	0
$641$	−1.77975e21	−1.58097	−0.790486	−	0.612480i	$-0.790172\pi$
$641$	−1.77975e21	−1.58097	−0.790486	+	0.612480i	$0.790172\pi$
$642$	6.43548e19i	0.0565026i
$643$	5.18269e19i	0.0449752i	0.999747	+	0.0224876i	$0.00715862\pi$
$643$	5.18269e19i	0.0449752i	−0.999747	+	0.0224876i	$0.992841\pi$
$644$	1.09474e21	0.939005
$645$	0	0
$646$	8.38178e20	0.702413
$647$	1.80268e21i	1.49327i	0.665235	+	0.746634i	$0.268331\pi$
$647$	1.80268e21i	1.49327i	−0.665235	+	0.746634i	$0.731669\pi$
$648$	− 9.94075e19i	− 0.0813967i
$649$	−8.58162e20	−0.694599
$650$	0	0
$651$	2.64071e20	0.208864
$652$	1.52984e21i	1.19616i
$653$	− 6.01117e20i	− 0.464633i	−0.972640	−	0.232317i	$-0.925369\pi$
$653$	− 6.01117e20i	− 0.464633i	0.972640	−	0.232317i	$-0.0746306\pi$
$654$	1.73421e20	0.132516
$655$	0	0
$656$	3.24971e20	0.242698
$657$	− 2.68316e19i	− 0.0198110i
$658$	− 3.21238e20i	− 0.234494i
$659$	1.34492e21	0.970635	0.485318	−	0.874338i	$-0.338704\pi$
$659$	1.34492e21	0.970635	0.485318	+	0.874338i	$0.338704\pi$
$660$	0	0
$661$	3.00301e20	0.211858	0.105929	−	0.994374i	$-0.466218\pi$
$661$	3.00301e20	0.211858	0.105929	+	0.994374i	$0.466218\pi$
$662$	− 9.14362e20i	− 0.637798i
$663$	1.52642e21i	1.05274i
$664$	−2.50622e20	−0.170906
$665$	0	0
$666$	−2.11709e20	−0.141151
$667$	2.14142e21i	1.41175i
$668$	− 1.60967e20i	− 0.104933i
$669$	−3.49869e20	−0.225533
$670$	0	0
$671$	−3.16155e20	−0.199288
$672$	8.69196e20i	0.541812i
$673$	2.48252e21i	1.53031i	0.643845	+	0.765156i	$0.277338\pi$
$673$	2.48252e21i	1.53031i	−0.643845	+	0.765156i	$0.722662\pi$
$674$	−4.78260e20	−0.291551
$675$	0	0
$676$	−7.50276e20	−0.447323
$677$	− 1.84000e21i	− 1.08493i	−0.840077	−	0.542467i	$-0.817491\pi$
$677$	− 1.84000e21i	− 1.08493i	0.840077	−	0.542467i	$-0.182509\pi$
$678$	− 3.35435e20i	− 0.195608i
$679$	−1.16000e21	−0.669014
$680$	0	0
$681$	3.18871e20	0.179892
$682$	1.50367e20i	0.0839016i
$683$	− 2.19144e21i	− 1.20941i	−0.796450	−	0.604705i	$-0.793291\pi$
$683$	− 2.19144e21i	− 1.20941i	0.796450	−	0.604705i	$-0.206709\pi$
$684$	−6.16100e20	−0.336303
$685$	0	0
$686$	−7.54596e20	−0.402980
$687$	1.48977e21i	0.786945i
$688$	5.80892e20i	0.303517i
$689$	−3.37324e21	−1.74342
$690$	0	0
$691$	−5.25387e20	−0.265701	−0.132851	−	0.991136i	$-0.542413\pi$
$691$	−5.25387e20	−0.265701	−0.132851	+	0.991136i	$0.542413\pi$
$692$	− 1.21634e21i	− 0.608497i
$693$	3.82049e20i	0.189069i
$694$	1.35241e20	0.0662082
$695$	0	0
$696$	−1.10194e21	−0.527943
$697$	1.37075e21i	0.649701i
$698$	− 1.06953e21i	− 0.501505i
$699$	8.50820e20	0.394693
$700$	0	0
$701$	−3.78227e21	−1.71739	−0.858693	−	0.512491i	$-0.828723\pi$
$701$	−3.78227e21	−1.71739	−0.858693	+	0.512491i	$0.828723\pi$
$702$	2.10861e20i	0.0947259i
$703$	2.87082e21i	1.27597i
$704$	2.24889e20	0.0988949
$705$	0	0
$706$	5.92834e20	0.255210
$707$	9.07353e20i	0.386483i
$708$	1.39281e21i	0.587004i
$709$	4.24647e21	1.77085	0.885424	−	0.464784i	$-0.153868\pi$
$709$	4.24647e21	1.77085	0.885424	+	0.464784i	$0.153868\pi$
$710$	0	0
$711$	7.64886e20	0.312302
$712$	− 1.57426e21i	− 0.636030i
$713$	− 1.03766e21i	− 0.414842i
$714$	−8.43577e20	−0.333725
$715$	0	0
$716$	1.40842e21	0.545613
$717$	1.47891e20i	0.0566954i
$718$	1.34940e21i	0.511927i
$719$	3.17360e20	0.119148	0.0595738	−	0.998224i	$-0.481026\pi$
$719$	3.17360e20	0.119148	0.0595738	+	0.998224i	$0.481026\pi$
$720$	0	0
$721$	3.13085e20	0.115119
$722$	− 4.77053e20i	− 0.173595i
$723$	1.96946e21i	0.709266i
$724$	8.72049e20	0.310815
$725$	0	0
$726$	4.40221e20	0.153690
$727$	1.25112e21i	0.432307i	0.976359	+	0.216154i	$0.0693512\pi$
$727$	1.25112e21i	0.432307i	−0.976359	+	0.216154i	$0.930649\pi$
$728$	− 2.61444e21i	− 0.894115i
$729$	−1.09419e20	−0.0370370
$730$	0	0
$731$	−2.45025e21	−0.812511
$732$	5.13123e20i	0.168418i
$733$	− 4.26993e21i	− 1.38721i	−0.720357	−	0.693603i	$-0.756022\pi$
$733$	− 4.26993e21i	− 1.38721i	0.720357	−	0.693603i	$-0.243978\pi$
$734$	6.21444e20	0.199840
$735$	0	0
$736$	3.41547e21	1.07614
$737$	− 4.58335e20i	− 0.142948i
$738$	1.89357e20i	0.0584601i
$739$	−1.73302e21	−0.529627	−0.264813	−	0.964300i	$-0.585310\pi$
$739$	−1.73302e21	−0.529627	−0.264813	+	0.964300i	$0.585310\pi$
$740$	0	0
$741$	2.85933e21	0.856302
$742$	− 1.86422e21i	− 0.552673i
$743$	2.63739e21i	0.774032i	0.922073	+	0.387016i	$0.126494\pi$
$743$	2.63739e21i	0.774032i	−0.922073	+	0.387016i	$0.873506\pi$
$744$	5.33960e20	0.155136
$745$	0	0
$746$	7.17102e20	0.204192
$747$	2.75862e20i	0.0777656i
$748$	2.55593e21i	0.713324i
$749$	−8.78287e20	−0.242674
$750$	0	0
$751$	2.27078e21	0.615001	0.307500	−	0.951548i	$-0.400507\pi$
$751$	2.27078e21	0.615001	0.307500	+	0.951548i	$0.400507\pi$
$752$	1.22702e21i	0.329016i
$753$	− 3.47577e21i	− 0.922761i
$754$	2.33741e21	0.614397
$755$	0	0
$756$	6.20069e20	0.159782
$757$	− 1.72959e21i	− 0.441291i	−0.975354	−	0.220645i	$-0.929184\pi$
$757$	− 1.72959e21i	− 0.441291i	0.975354	−	0.220645i	$-0.0708164\pi$
$758$	7.26834e20i	0.183618i
$759$	1.50125e21	0.375525
$760$	0	0
$761$	−2.06739e20	−0.0507035	−0.0253518	−	0.999679i	$-0.508071\pi$
$761$	−2.06739e20	−0.0507035	−0.0253518	+	0.999679i	$0.508071\pi$
$762$	9.85664e20i	0.239368i
$763$	2.36677e21i	0.569144i
$764$	2.60473e21	0.620245
$765$	0	0
$766$	1.95670e21	0.456887
$767$	− 6.46402e21i	− 1.49464i
$768$	5.89256e20i	0.134926i
$769$	−3.01490e21	−0.683636	−0.341818	−	0.939766i	$-0.611043\pi$
$769$	−3.01490e21	−0.683636	−0.341818	+	0.939766i	$0.611043\pi$
$770$	0	0
$771$	5.02437e21	1.11731
$772$	− 1.15230e21i	− 0.253768i
$773$	− 8.99357e21i	− 1.96149i	−0.195293	−	0.980745i	$-0.562566\pi$
$773$	− 8.99357e21i	− 1.96149i	0.195293	−	0.980745i	$-0.437434\pi$
$774$	−3.38480e20	−0.0731098
$775$	0	0
$776$	−2.34556e21	−0.496916
$777$	− 2.88931e21i	− 0.606229i
$778$	− 6.51805e20i	− 0.135447i
$779$	2.56773e21	0.528467
$780$	0	0
$781$	2.19259e21	0.442664
$782$	3.31480e21i	0.662838i
$783$	1.21291e21i	0.240224i
$784$	7.64531e19	0.0149977
$785$	0	0
$786$	−1.59132e21	−0.306259
$787$	− 5.92807e21i	− 1.13006i	−0.825069	−	0.565032i	$-0.808864\pi$
$787$	− 5.92807e21i	− 1.13006i	0.825069	−	0.565032i	$-0.191136\pi$
$788$	− 4.09299e21i	− 0.772849i
$789$	1.10587e21	0.206837
$790$	0	0
$791$	4.57787e21	0.840117
$792$	7.72514e20i	0.140432i
$793$	− 2.38141e21i	− 0.428830i
$794$	−1.27077e21	−0.226680
$795$	0	0
$796$	−5.44936e21	−0.953889
$797$	− 3.83837e21i	− 0.665593i	−0.942999	−	0.332797i	$-0.892008\pi$
$797$	− 3.83837e21i	− 0.665593i	0.942999	−	0.332797i	$-0.107992\pi$
$798$	1.58021e21i	0.271452i
$799$	−5.17565e21	−0.880773
$800$	0	0
$801$	−1.73281e21	−0.289406
$802$	− 2.02417e21i	− 0.334919i
$803$	2.08513e20i	0.0341796i
$804$	−7.43883e20	−0.120805
$805$	0	0
$806$	−1.13263e21	−0.180540
$807$	− 5.35226e21i	− 0.845251i
$808$	1.83470e21i	0.287064i
$809$	6.17948e21	0.957940	0.478970	−	0.877831i	$-0.341010\pi$
$809$	6.17948e21	0.957940	0.478970	+	0.877831i	$0.341010\pi$
$810$	0	0
$811$	1.87862e21	0.285879	0.142939	−	0.989731i	$-0.454345\pi$
$811$	1.87862e21	0.285879	0.142939	+	0.989731i	$0.454345\pi$
$812$	− 6.87349e21i	− 1.03635i
$813$	3.36375e21i	0.502510i
$814$	1.64523e21	0.243525
$815$	0	0
$816$	3.22217e21	0.468245
$817$	4.58986e21i	0.660897i
$818$	4.62927e21i	0.660485i
$819$	−2.87774e21	−0.406839
$820$	0	0
$821$	3.93227e21	0.545845	0.272923	−	0.962036i	$-0.412010\pi$
$821$	3.93227e21	0.545845	0.272923	+	0.962036i	$0.412010\pi$
$822$	1.56478e21i	0.215236i
$823$	7.92927e21i	1.08077i	0.841417	+	0.540386i	$0.181722\pi$
$823$	7.92927e21i	1.08077i	−0.841417	+	0.540386i	$0.818278\pi$
$824$	6.33068e20	0.0855059
$825$	0	0
$826$	3.57235e21	0.473809
$827$	− 1.97038e21i	− 0.258976i	−0.991581	−	0.129488i	$-0.958667\pi$
$827$	− 1.97038e21i	− 0.258976i	0.991581	−	0.129488i	$-0.0413333\pi$
$828$	− 2.43654e21i	− 0.317355i
$829$	−2.43930e21	−0.314851	−0.157426	−	0.987531i	$-0.550319\pi$
$829$	−2.43930e21	−0.314851	−0.157426	+	0.987531i	$0.550319\pi$
$830$	0	0
$831$	−1.72764e21	−0.219001
$832$	1.69396e21i	0.212803i
$833$	3.22484e20i	0.0401487i
$834$	−3.07178e21	−0.379005
$835$	0	0
$836$	4.78783e21	0.580218
$837$	− 5.87736e20i	− 0.0705897i
$838$	− 1.68821e21i	− 0.200954i
$839$	1.35318e22	1.59639	0.798196	−	0.602398i	$-0.205788\pi$
$839$	1.35318e22	1.59639	0.798196	+	0.602398i	$0.205788\pi$
$840$	0	0
$841$	4.81601e21	0.558107
$842$	5.24323e21i	0.602225i
$843$	1.34567e21i	0.153190i
$844$	−1.23734e22	−1.39611
$845$	0	0
$846$	−7.14970e20	−0.0792519
$847$	6.00794e21i	0.660085i
$848$	7.12068e21i	0.775448i
$849$	2.76173e21	0.298107
$850$	0	0
$851$	−1.13534e22	−1.20408
$852$	− 3.55859e21i	− 0.374095i
$853$	7.30038e21i	0.760725i	0.924837	+	0.380363i	$0.124201\pi$
$853$	7.30038e21i	0.760725i	−0.924837	+	0.380363i	$0.875799\pi$
$854$	1.31609e21	0.135941
$855$	0	0
$856$	−1.77592e21	−0.180248
$857$	1.16475e22i	1.17187i	0.810359	+	0.585934i	$0.199272\pi$
$857$	1.16475e22i	1.17187i	−0.810359	+	0.585934i	$0.800728\pi$
$858$	− 1.63864e21i	− 0.163429i
$859$	1.05473e22	1.04278	0.521388	−	0.853320i	$-0.325414\pi$
$859$	1.05473e22	1.04278	0.521388	+	0.853320i	$0.325414\pi$
$860$	0	0
$861$	−2.58427e21	−0.251081
$862$	3.39718e21i	0.327201i
$863$	− 1.25514e22i	− 1.19843i	−0.800588	−	0.599215i	$-0.795480\pi$
$863$	− 1.25514e22i	− 1.19843i	0.800588	−	0.599215i	$-0.204520\pi$
$864$	1.93455e21	0.183116
$865$	0	0
$866$	−2.72951e21	−0.253922
$867$	7.33124e21i	0.676136i
$868$	3.33066e21i	0.304531i
$869$	−5.94407e21	−0.538810
$870$	0	0
$871$	3.45237e21	0.307596
$872$	4.78569e21i	0.422737i
$873$	2.58178e21i	0.226106i
$874$	6.20937e21	0.539153
$875$	0	0
$876$	3.38420e20	0.0288852
$877$	− 1.57294e22i	− 1.33112i	−0.746346	−	0.665558i	$-0.768193\pi$
$877$	− 1.57294e22i	− 1.33112i	0.746346	−	0.665558i	$-0.231807\pi$
$878$	− 1.63319e21i	− 0.137034i
$879$	7.65152e21	0.636548
$880$	0	0
$881$	1.80412e22	1.47553	0.737764	−	0.675059i	$-0.235882\pi$
$881$	1.80412e22	1.47553	0.737764	+	0.675059i	$0.235882\pi$
$882$	4.45484e19i	0.00361258i
$883$	− 2.41866e22i	− 1.94478i	−0.233371	−	0.972388i	$-0.574976\pi$
$883$	− 2.41866e22i	− 1.94478i	0.233371	−	0.972388i	$-0.425024\pi$
$884$	−1.92523e22	−1.53494
$885$	0	0
$886$	7.16320e21	0.561505
$887$	7.05392e21i	0.548282i	0.961690	+	0.274141i	$0.0883934\pi$
$887$	7.05392e21i	0.548282i	−0.961690	+	0.274141i	$0.911607\pi$
$888$	− 5.84228e21i	− 0.450282i
$889$	−1.34519e22	−1.02807
$890$	0	0
$891$	8.50315e20	0.0638994
$892$	− 4.41279e21i	− 0.328834i
$893$	9.69514e21i	0.716421i
$894$	2.50266e21	0.183388
$895$	0	0
$896$	−1.39594e22	−1.00590
$897$	1.13080e22i	0.808057i
$898$	9.37143e21i	0.664100i
$899$	−6.51508e21	−0.457849
$900$	0	0
$901$	−3.00356e22	−2.07587
$902$	− 1.47153e21i	− 0.100860i
$903$	− 4.61942e21i	− 0.314000i
$904$	9.25660e21	0.624005
$905$	0	0
$906$	−4.89078e21	−0.324277
$907$	2.48498e22i	1.63406i	0.576596	+	0.817030i	$0.304381\pi$
$907$	2.48498e22i	1.63406i	−0.576596	+	0.817030i	$0.695619\pi$
$908$	4.02183e21i	0.262289i
$909$	2.01947e21	0.130620
$910$	0	0
$911$	−2.83848e21	−0.180592	−0.0902958	−	0.995915i	$-0.528781\pi$
$911$	−2.83848e21	−0.180592	−0.0902958	+	0.995915i	$0.528781\pi$
$912$	− 6.03585e21i	− 0.380870i
$913$	− 2.14377e21i	− 0.134168i
$914$	−7.33749e21	−0.455461
$915$	0	0
$916$	−1.87901e22	−1.14739
$917$	− 2.17177e22i	− 1.31536i
$918$	1.87753e21i	0.112789i
$919$	1.87762e22	1.11877	0.559387	−	0.828907i	$-0.311037\pi$
$919$	1.87762e22	1.11877	0.559387	+	0.828907i	$0.311037\pi$
$920$	0	0
$921$	2.81818e21	0.165204
$922$	7.78620e21i	0.452734i
$923$	1.65155e22i	0.952528i
$924$	−4.81867e21	−0.275669
$925$	0	0
$926$	−1.21761e22	−0.685374
$927$	− 6.96825e20i	− 0.0389068i
$928$	− 2.14445e22i	− 1.18770i
$929$	2.61729e21	0.143792	0.0718958	−	0.997412i	$-0.477095\pi$
$929$	2.61729e21	0.143792	0.0718958	+	0.997412i	$0.477095\pi$
$930$	0	0
$931$	6.04085e20	0.0326570
$932$	1.07311e22i	0.575476i
$933$	− 1.84371e22i	− 0.980798i
$934$	−3.30061e21	−0.174178
$935$	0	0
$936$	−5.81889e21	−0.302184
$937$	1.24366e22i	0.640701i	0.947299	+	0.320350i	$0.103801\pi$
$937$	1.24366e22i	0.640701i	−0.947299	+	0.320350i	$0.896199\pi$
$938$	1.90795e21i	0.0975094i
$939$	−1.72414e22	−0.874140
$940$	0	0
$941$	1.71266e22	0.854574	0.427287	−	0.904116i	$-0.359469\pi$
$941$	1.71266e22	0.854574	0.427287	+	0.904116i	$0.359469\pi$
$942$	8.52465e21i	0.421983i
$943$	1.01548e22i	0.498692i
$944$	−1.36451e22	−0.664795
$945$	0	0
$946$	2.63039e21	0.126135
$947$	1.26035e21i	0.0599607i	0.999550	+	0.0299804i	$0.00954447\pi$
$947$	1.26035e21i	0.0599607i	−0.999550	+	0.0299804i	$0.990456\pi$
$948$	9.64729e21i	0.455347i
$949$	−1.57061e21	−0.0735480
$950$	0	0
$951$	−1.90606e22	−0.878581
$952$	− 2.32792e22i	− 1.06461i
$953$	2.19887e22i	0.997706i	0.866687	+	0.498853i	$0.166245\pi$
$953$	2.19887e22i	0.997706i	−0.866687	+	0.498853i	$0.833755\pi$
$954$	−4.14915e21	−0.186787
$955$	0	0
$956$	−1.86531e21	−0.0826639
$957$	− 9.42577e21i	− 0.414455i
$958$	9.46890e21i	0.413103i
$959$	−2.13554e22	−0.924419
$960$	0	0
$961$	−2.03083e22	−0.865461
$962$	1.23925e22i	0.524019i
$963$	1.95478e21i	0.0820163i
$964$	−2.48402e22	−1.03413
$965$	0	0
$966$	−6.24937e21	−0.256158
$967$	1.21130e21i	0.0492666i	0.999697	+	0.0246333i	$0.00784181\pi$
$967$	1.21130e21i	0.0492666i	−0.999697	+	0.0246333i	$0.992158\pi$
$968$	1.21482e22i	0.490284i
$969$	2.54597e22	1.01959
$970$	0	0
$971$	1.26825e22	0.500103	0.250051	−	0.968233i	$-0.419552\pi$
$971$	1.26825e22	0.500103	0.250051	+	0.968233i	$0.419552\pi$
$972$	− 1.38007e21i	− 0.0540013i
$973$	− 4.19223e22i	− 1.62779i
$974$	4.41843e21	0.170245
$975$	0	0
$976$	−5.02700e21	−0.190737
$977$	− 3.29367e21i	− 0.124014i	−0.998076	−	0.0620070i	$-0.980250\pi$
$977$	− 3.29367e21i	− 0.124014i	0.998076	−	0.0620070i	$-0.0197501\pi$
$978$	− 8.73314e21i	− 0.326309i
$979$	1.34660e22	0.499307
$980$	0	0
$981$	5.26766e21	0.192353
$982$	− 1.74281e22i	− 0.631558i
$983$	− 3.74166e22i	− 1.34559i	−0.739828	−	0.672796i	$-0.765093\pi$
$983$	− 3.74166e22i	− 1.34559i	0.739828	−	0.672796i	$-0.234907\pi$
$984$	−5.22546e21	−0.186493
$985$	0	0
$986$	2.08125e22	0.731554
$987$	− 9.75759e21i	− 0.340380i
$988$	3.60638e22i	1.24852i
$989$	−1.81518e22	−0.623661
$990$	0	0
$991$	−4.40305e22	−1.49005	−0.745026	−	0.667035i	$-0.767563\pi$
$991$	−4.40305e22	−1.49005	−0.745026	+	0.667035i	$0.767563\pi$
$992$	1.03913e22i	0.349005i
$993$	− 2.77738e22i	− 0.925796i
$994$	−9.12728e21	−0.301956
$995$	0	0
$996$	−3.47937e21	−0.113385
$997$	2.62067e22i	0.847614i	0.905753	+	0.423807i	$0.139306\pi$
$997$	2.62067e22i	0.847614i	−0.905753	+	0.423807i	$0.860694\pi$
$998$	1.05155e22i	0.337560i
$999$	−6.43066e21	−0.204887

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.16.b.b.49.1		2
5.2	odd	4		75.16.a.a.1.1		1
5.3	odd	4		3.16.a.b.1.1	✓	1
5.4	even	2	inner	75.16.b.b.49.2		2
15.8	even	4		9.16.a.c.1.1		1
20.3	even	4		48.16.a.a.1.1		1
35.13	even	4		147.16.a.b.1.1		1
60.23	odd	4		144.16.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.16.a.b.1.1	✓	1	5.3	odd	4
9.16.a.c.1.1		1	15.8	even	4
48.16.a.a.1.1		1	20.3	even	4
75.16.a.a.1.1		1	5.2	odd	4
75.16.b.b.49.1		2	1.1	even	1	trivial
75.16.b.b.49.2		2	5.4	even	2	inner
144.16.a.l.1.1		1	60.23	odd	4
147.16.a.b.1.1		1	35.13	even	4

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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	75.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-72.0000i q^{2} -2187.00i q^{3} +27584.0 q^{4} -157464. q^{6} -2.14900e6i q^{7} -4.34534e6i q^{8} -4.78297e6 q^{9} +O(q^{10})\)	\(q-72.0000i q^{2} -2187.00i q^{3} +27584.0 q^{4} -157464. q^{6} -2.14900e6i q^{7} -4.34534e6i q^{8} -4.78297e6 q^{9} +3.71693e7 q^{11} -6.03262e7i q^{12} +2.79974e8i q^{13} -1.54728e8 q^{14} +5.91008e8 q^{16} +2.49291e9i q^{17} +3.44374e8i q^{18} +4.66978e9 q^{19} -4.69986e9 q^{21} -2.67619e9i q^{22} +1.84679e10i q^{23} -9.50327e9 q^{24} +2.01581e10 q^{26} +1.04604e10i q^{27} -5.92780e10i q^{28} +1.15953e11 q^{29} -5.61870e10 q^{31} -1.84941e11i q^{32} -8.12893e10i q^{33} +1.79490e11 q^{34} -1.31933e11 q^{36} +6.14765e11i q^{37} -3.36224e11i q^{38} +6.12304e11 q^{39} +5.49860e11 q^{41} +3.38390e11i q^{42} +9.82884e11i q^{43} +1.02528e12 q^{44} +1.32969e12 q^{46} +2.07614e12i q^{47} -1.29253e12i q^{48} +1.29361e11 q^{49} +5.45200e12 q^{51} +7.72281e12i q^{52} +1.20484e13i q^{53} +7.53145e11 q^{54} -9.33814e12 q^{56} -1.02128e13i q^{57} -8.34865e12i q^{58} -2.30879e13 q^{59} -8.50581e12 q^{61} +4.04547e12i q^{62} +1.02786e13i q^{63} +6.05040e12 q^{64} -5.85283e12 q^{66} -1.23310e13i q^{67} +6.87645e13i q^{68} +4.03894e13 q^{69} +5.89892e13 q^{71} +2.07836e13i q^{72} +5.60983e12i q^{73} +4.42631e13 q^{74} +1.28811e14 q^{76} -7.98769e13i q^{77} -4.40859e13i q^{78} -1.59919e14 q^{79} +2.28768e13 q^{81} -3.95899e13i q^{82} -5.76759e13i q^{83} -1.29641e14 q^{84} +7.07677e13 q^{86} -2.53590e14i q^{87} -1.61513e14i q^{88} +3.62288e14 q^{89} +6.01665e14 q^{91} +5.09419e14i q^{92} +1.22881e14i q^{93} +1.49482e14 q^{94} -4.04466e14 q^{96} -5.39787e14i q^{97} -9.31396e12i q^{98} -1.77780e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 55168 q^{4} - 314928 q^{6} - 9565938 q^{9}+O(q^{10}) \) 2 * q + 55168 * q^4 - 314928 * q^6 - 9565938 * q^9	\( 2 q + 55168 q^{4} - 314928 q^{6} - 9565938 q^{9} + 74338632 q^{11} - 309456000 q^{14} + 1182015488 q^{16} + 9339564488 q^{19} - 9399726000 q^{21} - 19006534656 q^{24} + 40316294304 q^{26} + 231906898836 q^{29} - 112374046400 q^{31} + 358979436576 q^{34} - 263866833792 q^{36} + 1224607439484 q^{39} + 1099719584820 q^{41} + 2050556825088 q^{44} + 2659382409600 q^{46} + 258721019886 q^{49} + 10904000385996 q^{51} + 1506290861232 q^{54} - 18676288512000 q^{56} - 46175811516648 q^{59} - 17011618284884 q^{61} + 12100809785344 q^{64} - 11705658349248 q^{66} + 80778740691600 q^{69} + 117978385384944 q^{71} + 88526149463520 q^{74} + 257622546836992 q^{76} - 319837367653600 q^{79} + 45753584909922 q^{81} - 259282041984000 q^{84} + 141535359940032 q^{86} + 724575220827948 q^{89} + 12\!\cdots\!00 q^{91}+ \cdots - 355559372358408 q^{99}+O(q^{100}) \) 2 * q + 55168 * q^4 - 314928 * q^6 - 9565938 * q^9 + 74338632 * q^11 - 309456000 * q^14 + 1182015488 * q^16 + 9339564488 * q^19 - 9399726000 * q^21 - 19006534656 * q^24 + 40316294304 * q^26 + 231906898836 * q^29 - 112374046400 * q^31 + 358979436576 * q^34 - 263866833792 * q^36 + 1224607439484 * q^39 + 1099719584820 * q^41 + 2050556825088 * q^44 + 2659382409600 * q^46 + 258721019886 * q^49 + 10904000385996 * q^51 + 1506290861232 * q^54 - 18676288512000 * q^56 - 46175811516648 * q^59 - 17011618284884 * q^61 + 12100809785344 * q^64 - 11705658349248 * q^66 + 80778740691600 * q^69 + 117978385384944 * q^71 + 88526149463520 * q^74 + 257622546836992 * q^76 - 319837367653600 * q^79 + 45753584909922 * q^81 - 259282041984000 * q^84 + 141535359940032 * q^86 + 724575220827948 * q^89 + 1203329395268000 * q^91 + 298964782497024 * q^94 - 808931014410240 * q^96 - 355559372358408 * q^99

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