LMFDB - Embedded newform 75.18.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,18,Mod(1,75)]

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, 0]))

N = Newforms(chi, 18, names="a")

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	75.a (trivial)

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:	yes
Analytic conductor:	$137.416565508$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	75.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-204.000 q^{2} +6561.00 q^{3} -89456.0 q^{4} -1.33844e6 q^{6} +2.08466e7 q^{7} +4.49877e7 q^{8} +4.30467e7 q^{9} +O(q^{10})$

\(q-204.000 q^{2} +6561.00 q^{3} -89456.0 q^{4} -1.33844e6 q^{6} +2.08466e7 q^{7} +4.49877e7 q^{8} +4.30467e7 q^{9} +8.17372e8 q^{11} -5.86921e8 q^{12} -2.99590e8 q^{13} -4.25270e9 q^{14} +2.54768e9 q^{16} +4.47756e10 q^{17} -8.78153e9 q^{18} +7.87487e10 q^{19} +1.36774e11 q^{21} -1.66744e11 q^{22} +7.04672e11 q^{23} +2.95164e11 q^{24} +6.11163e10 q^{26} +2.82430e11 q^{27} -1.86485e12 q^{28} -1.63794e11 q^{29} +1.04986e12 q^{31} -6.41636e12 q^{32} +5.36278e12 q^{33} -9.13422e12 q^{34} -3.85079e12 q^{36} +1.98057e13 q^{37} -1.60647e13 q^{38} -1.96561e12 q^{39} +1.46600e13 q^{41} -2.79020e13 q^{42} -1.16039e14 q^{43} -7.31189e13 q^{44} -1.43753e14 q^{46} +1.76607e14 q^{47} +1.67154e13 q^{48} +2.01949e14 q^{49} +2.93773e14 q^{51} +2.68001e13 q^{52} -1.52863e14 q^{53} -5.76156e13 q^{54} +9.37839e14 q^{56} +5.16670e14 q^{57} +3.34139e13 q^{58} -2.62797e14 q^{59} -1.35855e15 q^{61} -2.14172e14 q^{62} +8.97376e14 q^{63} +9.75007e14 q^{64} -1.09401e15 q^{66} -4.44864e14 q^{67} -4.00545e15 q^{68} +4.62335e15 q^{69} -4.00327e15 q^{71} +1.93657e15 q^{72} -9.24833e14 q^{73} -4.04037e15 q^{74} -7.04454e15 q^{76} +1.70394e16 q^{77} +4.00984e14 q^{78} +1.47473e16 q^{79} +1.85302e15 q^{81} -2.99065e15 q^{82} -2.64230e16 q^{83} -1.22353e16 q^{84} +2.36719e16 q^{86} -1.07465e15 q^{87} +3.67717e16 q^{88} -3.88837e16 q^{89} -6.24542e15 q^{91} -6.30371e16 q^{92} +6.88814e15 q^{93} -3.60277e16 q^{94} -4.20977e16 q^{96} +2.53744e16 q^{97} -4.11975e16 q^{98} +3.51852e16 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 $

$2$	−204.000	−0.563476	−0.281738	−	0.959491i	$-0.590911\pi$
$2$	−204.000	−0.563476	−0.281738	+	0.959491i	$0.590911\pi$
$3$	6561.00	0.577350
$3$	6561.00	0.577350
$4$	−89456.0	−0.682495
$5$	0	0
$5$	0	0
$6$	−1.33844e6	−0.325323
$7$	2.08466e7	1.36679	0.683394	−	0.730050i	$-0.260503\pi$
$7$	2.08466e7	1.36679	0.683394	+	0.730050i	$0.260503\pi$
$8$	4.49877e7	0.948045
$9$	4.30467e7	0.333333
$10$	0	0
$11$	8.17372e8	1.14969	0.574847	−	0.818261i	$-0.305062\pi$
$11$	8.17372e8	1.14969	0.574847	+	0.818261i	$0.305062\pi$
$12$	−5.86921e8	−0.394039
$13$	−2.99590e8	−0.101861	−0.0509306	−	0.998702i	$-0.516219\pi$
$13$	−2.99590e8	−0.101861	−0.0509306	+	0.998702i	$0.516219\pi$
$14$	−4.25270e9	−0.770152
$15$	0	0
$16$	2.54768e9	0.148295
$17$	4.47756e10	1.55677	0.778387	−	0.627785i	$-0.216038\pi$
$17$	4.47756e10	1.55677	0.778387	+	0.627785i	$0.216038\pi$
$18$	−8.78153e9	−0.187825
$19$	7.87487e10	1.06374	0.531872	−	0.846824i	$-0.321489\pi$
$19$	7.87487e10	1.06374	0.531872	+	0.846824i	$0.321489\pi$
$20$	0	0
$21$	1.36774e11	0.789115
$22$	−1.66744e11	−0.647824
$23$	7.04672e11	1.87629	0.938146	−	0.346240i	$-0.112542\pi$
$23$	7.04672e11	1.87629	0.938146	+	0.346240i	$0.112542\pi$
$24$	2.95164e11	0.547354
$25$	0	0
$26$	6.11163e10	0.0573963
$27$	2.82430e11	0.192450
$28$	−1.86485e12	−0.932826
$29$	−1.63794e11	−0.0608015	−0.0304008	−	0.999538i	$-0.509678\pi$
$29$	−1.63794e11	−0.0608015	−0.0304008	+	0.999538i	$0.509678\pi$
$30$	0	0
$31$	1.04986e12	0.221084	0.110542	−	0.993871i	$-0.464741\pi$
$31$	1.04986e12	0.221084	0.110542	+	0.993871i	$0.464741\pi$
$32$	−6.41636e12	−1.03161
$33$	5.36278e12	0.663776
$34$	−9.13422e12	−0.877204
$35$	0	0
$36$	−3.85079e12	−0.227498
$37$	1.98057e13	0.926993	0.463496	−	0.886099i	$-0.346595\pi$
$37$	1.98057e13	0.926993	0.463496	+	0.886099i	$0.346595\pi$
$38$	−1.60647e13	−0.599394
$39$	−1.96561e12	−0.0588095
$40$	0	0
$41$	1.46600e13	0.286729	0.143365	−	0.989670i	$-0.454208\pi$
$41$	1.46600e13	0.286729	0.143365	+	0.989670i	$0.454208\pi$
$42$	−2.79020e13	−0.444647
$43$	−1.16039e14	−1.51399	−0.756993	−	0.653424i	$-0.773332\pi$
$43$	−1.16039e14	−1.51399	−0.756993	+	0.653424i	$0.773332\pi$
$44$	−7.31189e13	−0.784660
$45$	0	0
$46$	−1.43753e14	−1.05724
$47$	1.76607e14	1.08187	0.540935	−	0.841064i	$-0.318070\pi$
$47$	1.76607e14	1.08187	0.540935	+	0.841064i	$0.318070\pi$
$48$	1.67154e13	0.0856180
$49$	2.01949e14	0.868109
$50$	0	0
$51$	2.93773e14	0.898804
$52$	2.68001e13	0.0695197
$53$	−1.52863e14	−0.337255	−0.168628	−	0.985680i	$-0.553934\pi$
$53$	−1.52863e14	−0.337255	−0.168628	+	0.985680i	$0.553934\pi$
$54$	−5.76156e13	−0.108441
$55$	0	0
$56$	9.37839e14	1.29578
$57$	5.16670e14	0.614153
$58$	3.34139e13	0.0342602
$59$	−2.62797e14	−0.233012	−0.116506	−	0.993190i	$-0.537169\pi$
$59$	−2.62797e14	−0.233012	−0.116506	+	0.993190i	$0.537169\pi$
$60$	0	0
$61$	−1.35855e15	−0.907346	−0.453673	−	0.891168i	$-0.649887\pi$
$61$	−1.35855e15	−0.907346	−0.453673	+	0.891168i	$0.649887\pi$
$62$	−2.14172e14	−0.124575
$63$	8.97376e14	0.455596
$64$	9.75007e14	0.432990
$65$	0	0
$66$	−1.09401e15	−0.374022
$67$	−4.44864e14	−0.133842	−0.0669208	−	0.997758i	$-0.521317\pi$
$67$	−4.44864e14	−0.133842	−0.0669208	+	0.997758i	$0.521317\pi$
$68$	−4.00545e15	−1.06249
$69$	4.62335e15	1.08328
$70$	0	0
$71$	−4.00327e15	−0.735731	−0.367865	−	0.929879i	$-0.619911\pi$
$71$	−4.00327e15	−0.735731	−0.367865	+	0.929879i	$0.619911\pi$
$72$	1.93657e15	0.316015
$73$	−9.24833e14	−0.134220	−0.0671102	−	0.997746i	$-0.521378\pi$
$73$	−9.24833e14	−0.134220	−0.0671102	+	0.997746i	$0.521378\pi$
$74$	−4.04037e15	−0.522338
$75$	0	0
$76$	−7.04454e15	−0.726001
$77$	1.70394e16	1.57139
$78$	4.00984e14	0.0331378
$79$	1.47473e16	1.09366	0.546830	−	0.837244i	$-0.315834\pi$
$79$	1.47473e16	1.09366	0.546830	+	0.837244i	$0.315834\pi$
$80$	0	0
$81$	1.85302e15	0.111111
$82$	−2.99065e15	−0.161565
$83$	−2.64230e16	−1.28771	−0.643855	−	0.765148i	$-0.722666\pi$
$83$	−2.64230e16	−1.28771	−0.643855	+	0.765148i	$0.722666\pi$
$84$	−1.22353e16	−0.538567
$85$	0	0
$86$	2.36719e16	0.853094
$87$	−1.07465e15	−0.0351038
$88$	3.67717e16	1.08996
$89$	−3.88837e16	−1.04702	−0.523508	−	0.852021i	$-0.675377\pi$
$89$	−3.88837e16	−1.04702	−0.523508	+	0.852021i	$0.675377\pi$
$90$	0	0
$91$	−6.24542e15	−0.139223
$92$	−6.30371e16	−1.28056
$93$	6.88814e15	0.127643
$94$	−3.60277e16	−0.609608
$95$	0	0
$96$	−4.20977e16	−0.595598
$97$	2.53744e16	0.328727	0.164364	−	0.986400i	$-0.447443\pi$
$97$	2.53744e16	0.328727	0.164364	+	0.986400i	$0.447443\pi$
$98$	−4.11975e16	−0.489158
$99$	3.51852e16	0.383231
$100$	0	0
$101$	−7.44216e16	−0.683860	−0.341930	−	0.939725i	$-0.611081\pi$
$101$	−7.44216e16	−0.683860	−0.341930	+	0.939725i	$0.611081\pi$
$102$	−5.99296e16	−0.506454
$103$	−2.06558e17	−1.60667	−0.803333	−	0.595530i	$-0.796942\pi$
$103$	−2.06558e17	−1.60667	−0.803333	+	0.595530i	$0.796942\pi$
$104$	−1.34779e16	−0.0965689
$105$	0	0
$106$	3.11842e16	0.190035
$107$	2.23373e17	1.25681	0.628405	−	0.777887i	$-0.283708\pi$
$107$	2.23373e17	1.25681	0.628405	+	0.777887i	$0.283708\pi$
$108$	−2.52650e16	−0.131346
$109$	−2.66379e17	−1.28049	−0.640244	−	0.768172i	$-0.721167\pi$
$109$	−2.66379e17	−1.28049	−0.640244	+	0.768172i	$0.721167\pi$
$110$	0	0
$111$	1.29945e17	0.535200
$112$	5.31104e16	0.202687
$113$	6.75157e16	0.238912	0.119456	−	0.992839i	$-0.461885\pi$
$113$	6.75157e16	0.238912	0.119456	+	0.992839i	$0.461885\pi$
$114$	−1.05401e17	−0.346061
$115$	0	0
$116$	1.46523e16	0.0414967
$117$	−1.28964e16	−0.0339537
$118$	5.36106e16	0.131297
$119$	9.33417e17	2.12778
$120$	0	0
$121$	1.62651e17	0.321795
$122$	2.77145e17	0.511267
$123$	9.61845e16	0.165543
$124$	−9.39164e16	−0.150889
$125$	0	0
$126$	−1.83065e17	−0.256717
$127$	2.11177e17	0.276896	0.138448	−	0.990370i	$-0.455789\pi$
$127$	2.11177e17	0.276896	0.138448	+	0.990370i	$0.455789\pi$
$128$	6.42103e17	0.787626
$129$	−7.61331e17	−0.874100
$130$	0	0
$131$	1.02202e18	1.02957	0.514783	−	0.857321i	$-0.327873\pi$
$131$	1.02202e18	1.02957	0.514783	+	0.857321i	$0.327873\pi$
$132$	−4.79733e17	−0.453024
$133$	1.64164e18	1.45391
$134$	9.07522e16	0.0754165
$135$	0	0
$136$	2.01435e18	1.47589
$137$	1.70778e18	1.17573	0.587865	−	0.808959i	$-0.299969\pi$
$137$	1.70778e18	1.17573	0.587865	+	0.808959i	$0.299969\pi$
$138$	−9.43164e17	−0.610401
$139$	1.05313e18	0.640994	0.320497	−	0.947249i	$-0.396150\pi$
$139$	1.05313e18	0.640994	0.320497	+	0.947249i	$0.396150\pi$
$140$	0	0
$141$	1.15872e18	0.624618
$142$	8.16667e17	0.414567
$143$	−2.44876e17	−0.117109
$144$	1.09669e17	0.0494316
$145$	0	0
$146$	1.88666e17	0.0756300
$147$	1.32498e18	0.501203
$148$	−1.77174e18	−0.632668
$149$	−1.41948e18	−0.478681	−0.239340	−	0.970936i	$-0.576931\pi$
$149$	−1.41948e18	−0.478681	−0.239340	+	0.970936i	$0.576931\pi$
$150$	0	0
$151$	−8.64830e17	−0.260391	−0.130196	−	0.991488i	$-0.541561\pi$
$151$	−8.64830e17	−0.260391	−0.130196	+	0.991488i	$0.541561\pi$
$152$	3.54272e18	1.00848
$153$	1.92744e18	0.518925
$154$	−3.47604e18	−0.885438
$155$	0	0
$156$	1.75835e17	0.0401372
$157$	−4.23286e18	−0.915138	−0.457569	−	0.889174i	$-0.651280\pi$
$157$	−4.23286e18	−0.915138	−0.457569	+	0.889174i	$0.651280\pi$
$158$	−3.00845e18	−0.616251
$159$	−1.00294e18	−0.194714
$160$	0	0
$161$	1.46900e19	2.56449
$162$	−3.78016e17	−0.0626084
$163$	4.49946e18	0.707238	0.353619	−	0.935390i	$-0.384951\pi$
$163$	4.49946e18	0.707238	0.353619	+	0.935390i	$0.384951\pi$
$164$	−1.31143e18	−0.195691
$165$	0	0
$166$	5.39028e18	0.725593
$167$	2.89131e18	0.369832	0.184916	−	0.982754i	$-0.440799\pi$
$167$	2.89131e18	0.369832	0.184916	+	0.982754i	$0.440799\pi$
$168$	6.15316e18	0.748117
$169$	−8.56066e18	−0.989624
$170$	0	0
$171$	3.38987e18	0.354582
$172$	1.03804e19	1.03329
$173$	5.47551e18	0.518839	0.259420	−	0.965765i	$-0.416469\pi$
$173$	5.47551e18	0.518839	0.259420	+	0.965765i	$0.416469\pi$
$174$	2.19229e17	0.0197801
$175$	0	0
$176$	2.08241e18	0.170493
$177$	−1.72421e18	−0.134530
$178$	7.93228e18	0.589968
$179$	−2.46059e19	−1.74497	−0.872485	−	0.488642i	$-0.837493\pi$
$179$	−2.46059e19	−1.74497	−0.872485	+	0.488642i	$0.837493\pi$
$180$	0	0
$181$	2.09528e19	1.35199	0.675997	−	0.736904i	$-0.263713\pi$
$181$	2.09528e19	1.35199	0.675997	+	0.736904i	$0.263713\pi$
$182$	1.27406e18	0.0784485
$183$	−8.91346e18	−0.523856
$184$	3.17016e19	1.77881
$185$	0	0
$186$	−1.40518e18	−0.0719237
$187$	3.65983e19	1.78981
$188$	−1.57985e19	−0.738371
$189$	5.88768e18	0.263038
$190$	0	0
$191$	−6.03799e18	−0.246666	−0.123333	−	0.992365i	$-0.539358\pi$
$191$	−6.03799e18	−0.246666	−0.123333	+	0.992365i	$0.539358\pi$
$192$	6.39702e18	0.249987
$193$	−1.32605e19	−0.495817	−0.247909	−	0.968783i	$-0.579743\pi$
$193$	−1.32605e19	−0.495817	−0.247909	+	0.968783i	$0.579743\pi$
$194$	−5.17638e18	−0.185230
$195$	0	0
$196$	−1.80655e19	−0.592480
$197$	2.62791e19	0.825368	0.412684	−	0.910874i	$-0.364591\pi$
$197$	2.62791e19	0.825368	0.412684	+	0.910874i	$0.364591\pi$
$198$	−7.17778e18	−0.215941
$199$	−2.06774e19	−0.595998	−0.297999	−	0.954566i	$-0.596319\pi$
$199$	−2.06774e19	−0.595998	−0.297999	+	0.954566i	$0.596319\pi$
$200$	0	0
$201$	−2.91875e18	−0.0772735
$202$	1.51820e19	0.385338
$203$	−3.41454e18	−0.0831028
$204$	−2.62797e19	−0.613429
$205$	0	0
$206$	4.21378e19	0.905317
$207$	3.03338e19	0.625431
$208$	−7.63260e17	−0.0151055
$209$	6.43670e19	1.22298
$210$	0	0
$211$	1.03175e20	1.80790	0.903948	−	0.427643i	$-0.140656\pi$
$211$	1.03175e20	1.80790	0.903948	+	0.427643i	$0.140656\pi$
$212$	1.36746e19	0.230175
$213$	−2.62655e19	−0.424774
$214$	−4.55682e19	−0.708181
$215$	0	0
$216$	1.27059e19	0.182451
$217$	2.18860e19	0.302175
$218$	5.43414e19	0.721523
$219$	−6.06783e18	−0.0774922
$220$	0	0
$221$	−1.34143e19	−0.158575
$222$	−2.65089e19	−0.301572
$223$	7.25219e19	0.794105	0.397053	−	0.917796i	$-0.370033\pi$
$223$	7.25219e19	0.794105	0.397053	+	0.917796i	$0.370033\pi$
$224$	−1.33759e20	−1.40999
$225$	0	0
$226$	−1.37732e19	−0.134621
$227$	−9.29766e18	−0.0875294	−0.0437647	−	0.999042i	$-0.513935\pi$
$227$	−9.29766e18	−0.0875294	−0.0437647	+	0.999042i	$0.513935\pi$
$228$	−4.62192e19	−0.419157
$229$	9.87595e19	0.862933	0.431467	−	0.902129i	$-0.357996\pi$
$229$	9.87595e19	0.862933	0.431467	+	0.902129i	$0.357996\pi$
$230$	0	0
$231$	1.11796e20	0.907241
$232$	−7.36871e18	−0.0576426
$233$	−3.53284e19	−0.266439	−0.133220	−	0.991087i	$-0.542532\pi$
$233$	−3.53284e19	−0.266439	−0.133220	+	0.991087i	$0.542532\pi$
$234$	2.63086e18	0.0191321
$235$	0	0
$236$	2.35088e19	0.159030
$237$	9.67571e19	0.631425
$238$	−1.90417e20	−1.19895
$239$	−4.68557e19	−0.284695	−0.142348	−	0.989817i	$-0.545465\pi$
$239$	−4.68557e19	−0.284695	−0.142348	+	0.989817i	$0.545465\pi$
$240$	0	0
$241$	−2.55944e20	−1.44877	−0.724385	−	0.689396i	$-0.757876\pi$
$241$	−2.55944e20	−1.44877	−0.724385	+	0.689396i	$0.757876\pi$
$242$	−3.31807e19	−0.181324
$243$	1.21577e19	0.0641500
$244$	1.21531e20	0.619259
$245$	0	0
$246$	−1.96216e19	−0.0932796
$247$	−2.35923e19	−0.108354
$248$	4.72308e19	0.209598
$249$	−1.73361e20	−0.743459
$250$	0	0
$251$	2.06944e20	0.829135	0.414567	−	0.910019i	$-0.363933\pi$
$251$	2.06944e20	0.829135	0.414567	+	0.910019i	$0.363933\pi$
$252$	−8.02757e19	−0.310942
$253$	5.75979e20	2.15716
$254$	−4.30802e19	−0.156024
$255$	0	0
$256$	−2.58785e20	−0.876798
$257$	−4.67477e20	−1.53225	−0.766124	−	0.642693i	$-0.777817\pi$
$257$	−4.67477e20	−1.53225	−0.766124	+	0.642693i	$0.777817\pi$
$258$	1.55312e20	0.492534
$259$	4.12881e20	1.26700
$260$	0	0
$261$	−7.05079e18	−0.0202672
$262$	−2.08492e20	−0.580135
$263$	−1.26367e20	−0.340416	−0.170208	−	0.985408i	$-0.554444\pi$
$263$	−1.26367e20	−0.340416	−0.170208	+	0.985408i	$0.554444\pi$
$264$	2.41259e20	0.629290
$265$	0	0
$266$	−3.34894e20	−0.819245
$267$	−2.55116e20	−0.604495
$268$	3.97957e19	0.0913462
$269$	−1.18589e20	−0.263724	−0.131862	−	0.991268i	$-0.542096\pi$
$269$	−1.18589e20	−0.263724	−0.131862	+	0.991268i	$0.542096\pi$
$270$	0	0
$271$	4.33191e20	0.904567	0.452284	−	0.891874i	$-0.350610\pi$
$271$	4.33191e20	0.904567	0.452284	+	0.891874i	$0.350610\pi$
$272$	1.14074e20	0.230861
$273$	−4.09762e19	−0.0803802
$274$	−3.48388e20	−0.662495
$275$	0	0
$276$	−4.13587e20	−0.739332
$277$	−4.26563e20	−0.739444	−0.369722	−	0.929142i	$-0.620547\pi$
$277$	−4.26563e20	−0.739444	−0.369722	+	0.929142i	$0.620547\pi$
$278$	−2.14838e20	−0.361185
$279$	4.51931e19	0.0736947
$280$	0	0
$281$	−9.97107e20	−1.53016	−0.765082	−	0.643933i	$-0.777301\pi$
$281$	−9.97107e20	−1.53016	−0.765082	+	0.643933i	$0.777301\pi$
$282$	−2.36378e20	−0.351957
$283$	4.85988e20	0.702168	0.351084	−	0.936344i	$-0.385813\pi$
$283$	4.85988e20	0.702168	0.351084	+	0.936344i	$0.385813\pi$
$284$	3.58117e20	0.502133
$285$	0	0
$286$	4.99548e19	0.0659881
$287$	3.05611e20	0.391898
$288$	−2.76203e20	−0.343869
$289$	1.17761e21	1.42355
$290$	0	0
$291$	1.66481e20	0.189791
$292$	8.27318e19	0.0916048
$293$	−1.48194e21	−1.59388	−0.796938	−	0.604061i	$-0.793548\pi$
$293$	−1.48194e21	−1.59388	−0.796938	+	0.604061i	$0.793548\pi$
$294$	−2.70297e20	−0.282416
$295$	0	0
$296$	8.91015e20	0.878831
$297$	2.30850e20	0.221259
$298$	2.89574e20	0.269725
$299$	−2.11113e20	−0.191121
$300$	0	0
$301$	−2.41901e21	−2.06930
$302$	1.76425e20	0.146724
$303$	−4.88280e20	−0.394827
$304$	2.00627e20	0.157748
$305$	0	0
$306$	−3.93198e20	−0.292401
$307$	−3.29698e18	−0.00238473	−0.00119237	−	0.999999i	$-0.500380\pi$
$307$	−3.29698e18	−0.00238473	−0.00119237	+	0.999999i	$0.500380\pi$
$308$	−1.52428e21	−1.07246
$309$	−1.35523e21	−0.927609
$310$	0	0
$311$	−2.79172e21	−1.80887	−0.904437	−	0.426607i	$-0.859709\pi$
$311$	−2.79172e21	−1.80887	−0.904437	+	0.426607i	$0.859709\pi$
$312$	−8.84282e19	−0.0557541
$313$	−1.83126e21	−1.12363	−0.561815	−	0.827263i	$-0.689897\pi$
$313$	−1.83126e21	−1.12363	−0.561815	+	0.827263i	$0.689897\pi$
$314$	8.63503e20	0.515658
$315$	0	0
$316$	−1.31924e21	−0.746418
$317$	−1.74407e21	−0.960639	−0.480319	−	0.877094i	$-0.659479\pi$
$317$	−1.74407e21	−0.960639	−0.480319	+	0.877094i	$0.659479\pi$
$318$	2.04599e20	0.109717
$319$	−1.33881e20	−0.0699031
$320$	0	0
$321$	1.46555e21	0.725619
$322$	−2.99676e21	−1.44503
$323$	3.52602e21	1.65601
$324$	−1.65764e20	−0.0758328
$325$	0	0
$326$	−9.17889e20	−0.398511
$327$	−1.74771e21	−0.739290
$328$	6.59521e20	0.271832
$329$	3.68164e21	1.47869
$330$	0	0
$331$	2.03545e21	0.776468	0.388234	−	0.921561i	$-0.373085\pi$
$331$	2.03545e21	0.776468	0.388234	+	0.921561i	$0.373085\pi$
$332$	2.36369e21	0.878855
$333$	8.52572e20	0.308998
$334$	−5.89828e20	−0.208392
$335$	0	0
$336$	3.48458e20	0.117022
$337$	3.47485e21	1.13784	0.568921	−	0.822392i	$-0.307361\pi$
$337$	3.47485e21	1.13784	0.568921	+	0.822392i	$0.307361\pi$
$338$	1.74638e21	0.557629
$339$	4.42971e20	0.137936
$340$	0	0
$341$	8.58127e20	0.254179
$342$	−6.91534e20	−0.199798
$343$	−6.39613e20	−0.180268
$344$	−5.22032e21	−1.43533
$345$	0	0
$346$	−1.11700e21	−0.292353
$347$	1.59958e21	0.408513	0.204256	−	0.978917i	$-0.434522\pi$
$347$	1.59958e21	0.408513	0.204256	+	0.978917i	$0.434522\pi$
$348$	9.61340e19	0.0239582
$349$	1.65498e21	0.402509	0.201255	−	0.979539i	$-0.435498\pi$
$349$	1.65498e21	0.402509	0.201255	+	0.979539i	$0.435498\pi$
$350$	0	0
$351$	−8.46130e19	−0.0196032
$352$	−5.24455e21	−1.18603
$353$	−8.34568e21	−1.84237	−0.921185	−	0.389126i	$-0.872777\pi$
$353$	−8.34568e21	−1.84237	−0.921185	+	0.389126i	$0.872777\pi$
$354$	3.51739e20	0.0758042
$355$	0	0
$356$	3.47838e21	0.714583
$357$	6.12415e21	1.22847
$358$	5.01960e21	0.983248
$359$	−3.48877e20	−0.0667375	−0.0333688	−	0.999443i	$-0.510624\pi$
$359$	−3.48877e20	−0.0667375	−0.0333688	+	0.999443i	$0.510624\pi$
$360$	0	0
$361$	7.20963e20	0.131553
$362$	−4.27438e21	−0.761816
$363$	1.06715e21	0.185789
$364$	5.58690e20	0.0950187
$365$	0	0
$366$	1.81835e21	0.295180
$367$	6.36726e21	1.00993	0.504965	−	0.863140i	$-0.331505\pi$
$367$	6.36726e21	1.00993	0.504965	+	0.863140i	$0.331505\pi$
$368$	1.79528e21	0.278244
$369$	6.31066e20	0.0955765
$370$	0	0
$371$	−3.18668e21	−0.460957
$372$	−6.16185e20	−0.0871157
$373$	7.78865e21	1.07631	0.538155	−	0.842846i	$-0.319122\pi$
$373$	7.78865e21	1.07631	0.538155	+	0.842846i	$0.319122\pi$
$374$	−7.46606e21	−1.00852
$375$	0	0
$376$	7.94513e21	1.02566
$377$	4.90709e19	0.00619331
$378$	−1.20109e21	−0.148216
$379$	−3.17769e21	−0.383423	−0.191711	−	0.981451i	$-0.561404\pi$
$379$	−3.17769e21	−0.383423	−0.191711	+	0.981451i	$0.561404\pi$
$380$	0	0
$381$	1.38554e21	0.159866
$382$	1.23175e21	0.138990
$383$	−6.69797e21	−0.739186	−0.369593	−	0.929194i	$-0.620503\pi$
$383$	−6.69797e21	−0.739186	−0.369593	+	0.929194i	$0.620503\pi$
$384$	4.21284e21	0.454736
$385$	0	0
$386$	2.70514e21	0.279381
$387$	−4.99509e21	−0.504662
$388$	−2.26989e21	−0.224355
$389$	−1.10700e22	−1.07048	−0.535238	−	0.844701i	$-0.679778\pi$
$389$	−1.10700e22	−1.07048	−0.535238	+	0.844701i	$0.679778\pi$
$390$	0	0
$391$	3.15521e22	2.92096
$392$	9.08520e21	0.823006
$393$	6.70549e21	0.594420
$394$	−5.36094e21	−0.465075
$395$	0	0
$396$	−3.14753e21	−0.261553
$397$	1.43915e22	1.17054	0.585270	−	0.810839i	$-0.300989\pi$
$397$	1.43915e22	1.17054	0.585270	+	0.810839i	$0.300989\pi$
$398$	4.21819e21	0.335831
$399$	1.07708e22	0.839417
$400$	0	0
$401$	8.63300e21	0.644814	0.322407	−	0.946601i	$-0.395508\pi$
$401$	8.63300e21	0.644814	0.322407	+	0.946601i	$0.395508\pi$
$402$	5.95425e20	0.0435417
$403$	−3.14528e20	−0.0225199
$404$	6.65746e21	0.466731
$405$	0	0
$406$	6.96566e20	0.0468264
$407$	1.61887e22	1.06576
$408$	1.32162e22	0.852107
$409$	1.62508e21	0.102619	0.0513094	−	0.998683i	$-0.483661\pi$
$409$	1.62508e21	0.102619	0.0513094	+	0.998683i	$0.483661\pi$
$410$	0	0
$411$	1.12048e22	0.678808
$412$	1.84778e22	1.09654
$413$	−5.47842e21	−0.318478
$414$	−6.18810e21	−0.352415
$415$	0	0
$416$	1.92227e21	0.105081
$417$	6.90955e21	0.370078
$418$	−1.31309e22	−0.689120
$419$	1.93761e22	0.996433	0.498216	−	0.867053i	$-0.333988\pi$
$419$	1.93761e22	0.996433	0.498216	+	0.867053i	$0.333988\pi$
$420$	0	0
$421$	−2.73783e22	−1.35210	−0.676051	−	0.736855i	$-0.736310\pi$
$421$	−2.73783e22	−1.35210	−0.676051	+	0.736855i	$0.736310\pi$
$422$	−2.10477e22	−1.01871
$423$	7.60233e21	0.360623
$424$	−6.87698e21	−0.319733
$425$	0	0
$426$	5.35815e21	0.239350
$427$	−2.83211e22	−1.24015
$428$	−1.99821e22	−0.857766
$429$	−1.60663e21	−0.0676130
$430$	0	0
$431$	−3.54083e22	−1.43235	−0.716174	−	0.697922i	$-0.754108\pi$
$431$	−3.54083e22	−1.43235	−0.716174	+	0.697922i	$0.754108\pi$
$432$	7.19541e20	0.0285393
$433$	−1.25200e22	−0.486920	−0.243460	−	0.969911i	$-0.578282\pi$
$433$	−1.25200e22	−0.486920	−0.243460	+	0.969911i	$0.578282\pi$
$434$	−4.46474e21	−0.170268
$435$	0	0
$436$	2.38292e22	0.873926
$437$	5.54920e22	1.99590
$438$	1.23784e21	0.0436650
$439$	−3.05896e22	−1.05834	−0.529170	−	0.848516i	$-0.677497\pi$
$439$	−3.05896e22	−1.05834	−0.529170	+	0.848516i	$0.677497\pi$
$440$	0	0
$441$	8.69322e21	0.289370
$442$	2.73652e21	0.0893530
$443$	1.49267e22	0.478114	0.239057	−	0.971005i	$-0.423162\pi$
$443$	1.49267e22	0.478114	0.239057	+	0.971005i	$0.423162\pi$
$444$	−1.16244e22	−0.365271
$445$	0	0
$446$	−1.47945e22	−0.447459
$447$	−9.31320e21	−0.276366
$448$	2.03255e22	0.591805
$449$	4.93173e21	0.140898	0.0704491	−	0.997515i	$-0.477557\pi$
$449$	4.93173e21	0.140898	0.0704491	+	0.997515i	$0.477557\pi$
$450$	0	0
$451$	1.19827e22	0.329651
$452$	−6.03968e21	−0.163056
$453$	−5.67415e21	−0.150337
$454$	1.89672e21	0.0493207
$455$	0	0
$456$	2.32438e22	0.582245
$457$	−6.67847e22	−1.64206	−0.821031	−	0.570884i	$-0.806600\pi$
$457$	−6.67847e22	−1.64206	−0.821031	+	0.570884i	$0.806600\pi$
$458$	−2.01469e22	−0.486242
$459$	1.26460e22	0.299601
$460$	0	0
$461$	−3.58593e22	−0.818736	−0.409368	−	0.912369i	$-0.634251\pi$
$461$	−3.58593e22	−0.818736	−0.409368	+	0.912369i	$0.634251\pi$
$462$	−2.28063e22	−0.511208
$463$	2.17954e22	0.479652	0.239826	−	0.970816i	$-0.422910\pi$
$463$	2.17954e22	0.479652	0.239826	+	0.970816i	$0.422910\pi$
$464$	−4.17295e20	−0.00901654
$465$	0	0
$466$	7.20699e21	0.150132
$467$	1.01862e22	0.208362	0.104181	−	0.994558i	$-0.466778\pi$
$467$	1.01862e22	0.208362	0.104181	+	0.994558i	$0.466778\pi$
$468$	1.15366e21	0.0231732
$469$	−9.27388e21	−0.182933
$470$	0	0
$471$	−2.77718e22	−0.528355
$472$	−1.18226e22	−0.220906
$473$	−9.48470e22	−1.74062
$474$	−1.97384e22	−0.355793
$475$	0	0
$476$	−8.34998e22	−1.45220
$477$	−6.58027e21	−0.112418
$478$	9.55856e21	0.160419
$479$	6.74899e22	1.11272	0.556361	−	0.830940i	$-0.312197\pi$
$479$	6.74899e22	1.11272	0.556361	+	0.830940i	$0.312197\pi$
$480$	0	0
$481$	−5.93360e21	−0.0944245
$482$	5.22125e22	0.816347
$483$	9.63810e22	1.48061
$484$	−1.45501e22	−0.219624
$485$	0	0
$486$	−2.48016e21	−0.0361470
$487$	−2.15075e22	−0.308031	−0.154016	−	0.988068i	$-0.549221\pi$
$487$	−2.15075e22	−0.308031	−0.154016	+	0.988068i	$0.549221\pi$
$488$	−6.11182e22	−0.860205
$489$	2.95209e22	0.408324
$490$	0	0
$491$	1.36833e22	0.182809	0.0914044	−	0.995814i	$-0.470864\pi$
$491$	1.36833e22	0.182809	0.0914044	+	0.995814i	$0.470864\pi$
$492$	−8.60428e21	−0.112982
$493$	−7.33397e21	−0.0946542
$494$	4.81283e21	0.0610550
$495$	0	0
$496$	2.67471e21	0.0327856
$497$	−8.34544e22	−1.00559
$498$	3.53657e22	0.418921
$499$	−1.36348e22	−0.158779	−0.0793894	−	0.996844i	$-0.525297\pi$
$499$	−1.36348e22	−0.158779	−0.0793894	+	0.996844i	$0.525297\pi$
$500$	0	0
$501$	1.89699e22	0.213523
$502$	−4.22165e22	−0.467197
$503$	−6.17808e22	−0.672242	−0.336121	−	0.941819i	$-0.609115\pi$
$503$	−6.17808e22	−0.672242	−0.336121	+	0.941819i	$0.609115\pi$
$504$	4.03709e22	0.431925
$505$	0	0
$506$	−1.17500e23	−1.21551
$507$	−5.61665e22	−0.571360
$508$	−1.88911e22	−0.188980
$509$	−9.19553e22	−0.904639	−0.452320	−	0.891856i	$-0.649403\pi$
$509$	−9.19553e22	−0.904639	−0.452320	+	0.891856i	$0.649403\pi$
$510$	0	0
$511$	−1.92796e22	−0.183451
$512$	−3.13696e22	−0.293572
$513$	2.22409e22	0.204718
$514$	9.53654e22	0.863384
$515$	0	0
$516$	6.81056e22	0.596569
$517$	1.44353e23	1.24382
$518$	−8.42278e22	−0.713925
$519$	3.59248e22	0.299552
$520$	0	0
$521$	−1.76716e23	−1.42611	−0.713057	−	0.701106i	$-0.752690\pi$
$521$	−1.76716e23	−1.42611	−0.713057	+	0.701106i	$0.752690\pi$
$522$	1.43836e21	0.0114201
$523$	2.39601e23	1.87165	0.935823	−	0.352470i	$-0.114658\pi$
$523$	2.39601e23	1.87165	0.935823	+	0.352470i	$0.114658\pi$
$524$	−9.14260e22	−0.702674
$525$	0	0
$526$	2.57789e22	0.191816
$527$	4.70082e22	0.344178
$528$	1.36627e22	0.0984345
$529$	3.55513e23	2.52047
$530$	0	0
$531$	−1.13126e22	−0.0776708
$532$	−1.46854e23	−0.992289
$533$	−4.39200e21	−0.0292066
$534$	5.20437e22	0.340618
$535$	0	0
$536$	−2.00134e22	−0.126888
$537$	−1.61439e23	−1.00746
$538$	2.41921e22	0.148602
$539$	1.65067e23	0.998059
$540$	0	0
$541$	1.61766e22	0.0947784	0.0473892	−	0.998877i	$-0.484910\pi$
$541$	1.61766e22	0.0947784	0.0473892	+	0.998877i	$0.484910\pi$
$542$	−8.83710e22	−0.509702
$543$	1.37472e23	0.780575
$544$	−2.87296e23	−1.60598
$545$	0	0
$546$	8.35914e21	0.0452923
$547$	−2.46037e23	−1.31253	−0.656263	−	0.754532i	$-0.727864\pi$
$547$	−2.46037e23	−1.31253	−0.656263	+	0.754532i	$0.727864\pi$
$548$	−1.52771e23	−0.802430
$549$	−5.84812e22	−0.302449
$550$	0	0
$551$	−1.28985e22	−0.0646773
$552$	2.07994e23	1.02700
$553$	3.07431e23	1.49480
$554$	8.70189e22	0.416659
$555$	0	0
$556$	−9.42084e22	−0.437475
$557$	8.61414e22	0.393951	0.196976	−	0.980408i	$-0.436888\pi$
$557$	8.61414e22	0.393951	0.196976	+	0.980408i	$0.436888\pi$
$558$	−9.21939e21	−0.0415252
$559$	3.47641e22	0.154216
$560$	0	0
$561$	2.40122e23	1.03335
$562$	2.03410e23	0.862210
$563$	−2.78426e23	−1.16249	−0.581244	−	0.813730i	$-0.697434\pi$
$563$	−2.78426e23	−1.16249	−0.581244	+	0.813730i	$0.697434\pi$
$564$	−1.03654e23	−0.426299
$565$	0	0
$566$	−9.91416e22	−0.395655
$567$	3.86291e22	0.151865
$568$	−1.80098e23	−0.697506
$569$	−4.52461e23	−1.72634	−0.863172	−	0.504910i	$-0.831526\pi$
$569$	−4.52461e23	−1.72634	−0.863172	+	0.504910i	$0.831526\pi$
$570$	0	0
$571$	7.63957e22	0.282919	0.141459	−	0.989944i	$-0.454821\pi$
$571$	7.63957e22	0.282919	0.141459	+	0.989944i	$0.454821\pi$
$572$	2.19057e22	0.0799264
$573$	−3.96152e22	−0.142412
$574$	−6.23447e22	−0.220825
$575$	0	0
$576$	4.19708e22	0.144330
$577$	−2.40302e23	−0.814261	−0.407131	−	0.913370i	$-0.633471\pi$
$577$	−2.40302e23	−0.814261	−0.407131	+	0.913370i	$0.633471\pi$
$578$	−2.40233e23	−0.802134
$579$	−8.70019e22	−0.286260
$580$	0	0
$581$	−5.50828e23	−1.76002
$582$	−3.39622e22	−0.106943
$583$	−1.24946e23	−0.387740
$584$	−4.16061e22	−0.127247
$585$	0	0
$586$	3.02315e23	0.898111
$587$	1.19663e23	0.350377	0.175188	−	0.984535i	$-0.443947\pi$
$587$	1.19663e23	0.350377	0.175188	+	0.984535i	$0.443947\pi$
$588$	−1.18528e23	−0.342068
$589$	8.26751e22	0.235177
$590$	0	0
$591$	1.72417e23	0.476527
$592$	5.04587e22	0.137468
$593$	2.98534e23	0.801731	0.400866	−	0.916137i	$-0.368709\pi$
$593$	2.98534e23	0.801731	0.400866	+	0.916137i	$0.368709\pi$
$594$	−4.70934e22	−0.124674
$595$	0	0
$596$	1.26981e23	0.326697
$597$	−1.35665e23	−0.344100
$598$	4.30670e22	0.107692
$599$	−2.69094e23	−0.663401	−0.331700	−	0.943385i	$-0.607622\pi$
$599$	−2.69094e23	−0.663401	−0.331700	+	0.943385i	$0.607622\pi$
$600$	0	0
$601$	3.04922e22	0.0730728	0.0365364	−	0.999332i	$-0.488368\pi$
$601$	3.04922e22	0.0730728	0.0365364	+	0.999332i	$0.488368\pi$
$602$	4.93478e23	1.16600
$603$	−1.91499e22	−0.0446139
$604$	7.73642e22	0.177716
$605$	0	0
$606$	9.96091e22	0.222475
$607$	5.58818e23	1.23074	0.615371	−	0.788238i	$-0.289006\pi$
$607$	5.58818e23	1.23074	0.615371	+	0.788238i	$0.289006\pi$
$608$	−5.05279e23	−1.09737
$609$	−2.24028e22	−0.0479794
$610$	0	0
$611$	−5.29095e22	−0.110201
$612$	−1.72421e23	−0.354164
$613$	6.34890e23	1.28613	0.643064	−	0.765812i	$-0.277663\pi$
$613$	6.34890e23	1.28613	0.643064	+	0.765812i	$0.277663\pi$
$614$	6.72584e20	0.00134374
$615$	0	0
$616$	7.66564e23	1.48975
$617$	−2.36658e23	−0.453625	−0.226813	−	0.973938i	$-0.572830\pi$
$617$	−2.36658e23	−0.453625	−0.226813	+	0.973938i	$0.572830\pi$
$618$	2.76466e23	0.522685
$619$	3.23770e23	0.603763	0.301882	−	0.953345i	$-0.402385\pi$
$619$	3.23770e23	0.603763	0.301882	+	0.953345i	$0.402385\pi$
$620$	0	0
$621$	1.99020e23	0.361093
$622$	5.69511e23	1.01926
$623$	−8.10592e23	−1.43105
$624$	−5.00775e21	−0.00872114
$625$	0	0
$626$	3.73577e23	0.633138
$627$	4.22312e23	0.706088
$628$	3.78654e23	0.624577
$629$	8.86814e23	1.44312
$630$	0	0
$631$	−1.43355e23	−0.227073	−0.113536	−	0.993534i	$-0.536218\pi$
$631$	−1.43355e23	−0.227073	−0.113536	+	0.993534i	$0.536218\pi$
$632$	6.63448e23	1.03684
$633$	6.76931e23	1.04379
$634$	3.55790e23	0.541297
$635$	0	0
$636$	8.97188e22	0.132892
$637$	−6.05017e22	−0.0884265
$638$	2.73116e22	0.0393887
$639$	−1.72328e23	−0.245244
$640$	0	0
$641$	4.66875e23	0.647004	0.323502	−	0.946227i	$-0.395140\pi$
$641$	4.66875e23	0.647004	0.323502	+	0.946227i	$0.395140\pi$
$642$	−2.98973e23	−0.408869
$643$	4.89782e23	0.661013	0.330506	−	0.943804i	$-0.392780\pi$
$643$	4.89782e23	0.661013	0.330506	+	0.943804i	$0.392780\pi$
$644$	−1.31411e24	−1.75025
$645$	0	0
$646$	−7.19308e23	−0.933122
$647$	1.09424e24	1.40096	0.700482	−	0.713670i	$-0.252968\pi$
$647$	1.09424e24	1.40096	0.700482	+	0.713670i	$0.252968\pi$
$648$	8.33631e22	0.105338
$649$	−2.14803e23	−0.267893
$650$	0	0
$651$	1.43594e23	0.174461
$652$	−4.02503e23	−0.482686
$653$	−5.18955e23	−0.614282	−0.307141	−	0.951664i	$-0.599372\pi$
$653$	−5.18955e23	−0.614282	−0.307141	+	0.951664i	$0.599372\pi$
$654$	3.56534e23	0.416572
$655$	0	0
$656$	3.73491e22	0.0425205
$657$	−3.98110e22	−0.0447402
$658$	−7.51055e23	−0.833204
$659$	4.01367e23	0.439557	0.219779	−	0.975550i	$-0.429466\pi$
$659$	4.01367e23	0.439557	0.219779	+	0.975550i	$0.429466\pi$
$660$	0	0
$661$	1.72655e24	1.84275	0.921375	−	0.388674i	$-0.127067\pi$
$661$	1.72655e24	1.84275	0.921375	+	0.388674i	$0.127067\pi$
$662$	−4.15233e23	−0.437521
$663$	−8.80113e22	−0.0915532
$664$	−1.18871e24	−1.22081
$665$	0	0
$666$	−1.73925e23	−0.174113
$667$	−1.15421e23	−0.114081
$668$	−2.58645e23	−0.252409
$669$	4.75816e23	0.458477
$670$	0	0
$671$	−1.11044e24	−1.04317
$672$	−8.77593e23	−0.814056
$673$	9.61197e23	0.880409	0.440205	−	0.897898i	$-0.354906\pi$
$673$	9.61197e23	0.880409	0.440205	+	0.897898i	$0.354906\pi$
$674$	−7.08869e23	−0.641146
$675$	0	0
$676$	7.65803e23	0.675414
$677$	6.20817e23	0.540705	0.270352	−	0.962761i	$-0.412860\pi$
$677$	6.20817e23	0.540705	0.270352	+	0.962761i	$0.412860\pi$
$678$	−9.03660e22	−0.0777235
$679$	5.28969e23	0.449301
$680$	0	0
$681$	−6.10020e22	−0.0505351
$682$	−1.75058e23	−0.143224
$683$	2.70132e23	0.218273	0.109137	−	0.994027i	$-0.465191\pi$
$683$	2.70132e23	0.218273	0.109137	+	0.994027i	$0.465191\pi$
$684$	−3.03244e23	−0.242000
$685$	0	0
$686$	1.30481e23	0.101576
$687$	6.47961e23	0.498215
$688$	−2.95630e23	−0.224516
$689$	4.57963e22	0.0343532
$690$	0	0
$691$	−5.16537e23	−0.378040	−0.189020	−	0.981973i	$-0.560531\pi$
$691$	−5.16537e23	−0.378040	−0.189020	+	0.981973i	$0.560531\pi$
$692$	−4.89817e23	−0.354105
$693$	7.33490e23	0.523796
$694$	−3.26315e23	−0.230187
$695$	0	0
$696$	−4.83461e22	−0.0332800
$697$	6.56412e23	0.446373
$698$	−3.37615e23	−0.226804
$699$	−2.31789e23	−0.153829
$700$	0	0
$701$	5.95182e23	0.385520	0.192760	−	0.981246i	$-0.438256\pi$
$701$	5.95182e23	0.385520	0.192760	+	0.981246i	$0.438256\pi$
$702$	1.72611e22	0.0110459
$703$	1.55967e24	0.986084
$704$	7.96944e23	0.497806
$705$	0	0
$706$	1.70252e24	1.03813
$707$	−1.55143e24	−0.934691
$708$	1.54241e23	0.0918159
$709$	1.04141e24	0.612533	0.306266	−	0.951946i	$-0.400920\pi$
$709$	1.04141e24	0.612533	0.306266	+	0.951946i	$0.400920\pi$
$710$	0	0
$711$	6.34823e23	0.364553
$712$	−1.74929e24	−0.992618
$713$	7.39808e23	0.414818
$714$	−1.24933e24	−0.692215
$715$	0	0
$716$	2.20114e24	1.19093
$717$	−3.07420e23	−0.164369
$718$	7.11710e22	0.0376050
$719$	1.57223e24	0.820958	0.410479	−	0.911870i	$-0.365362\pi$
$719$	1.57223e24	0.820958	0.410479	+	0.911870i	$0.365362\pi$
$720$	0	0
$721$	−4.30602e24	−2.19597
$722$	−1.47077e23	−0.0741271
$723$	−1.67925e24	−0.836448
$724$	−1.87436e24	−0.922730
$725$	0	0
$726$	−2.17699e23	−0.104687
$727$	−3.53143e24	−1.67845	−0.839224	−	0.543785i	$-0.816991\pi$
$727$	−3.53143e24	−1.67845	−0.839224	+	0.543785i	$0.816991\pi$
$728$	−2.80967e23	−0.131989
$729$	7.97664e22	0.0370370
$730$	0	0
$731$	−5.19571e24	−2.35693
$732$	7.97363e23	0.357529
$733$	−2.94036e24	−1.30322	−0.651609	−	0.758555i	$-0.725906\pi$
$733$	−2.94036e24	−1.30322	−0.651609	+	0.758555i	$0.725906\pi$
$734$	−1.29892e24	−0.569071
$735$	0	0
$736$	−4.52143e24	−1.93559
$737$	−3.63619e23	−0.153877
$738$	−1.28738e23	−0.0538550
$739$	9.01116e23	0.372651	0.186326	−	0.982488i	$-0.440342\pi$
$739$	9.01116e23	0.372651	0.186326	+	0.982488i	$0.440342\pi$
$740$	0	0
$741$	−1.54789e23	−0.0625584
$742$	6.50082e23	0.259738
$743$	3.55572e24	1.40450	0.702252	−	0.711929i	$-0.252178\pi$
$743$	3.55572e24	1.40450	0.702252	+	0.711929i	$0.252178\pi$
$744$	3.09882e23	0.121011
$745$	0	0
$746$	−1.58888e24	−0.606474
$747$	−1.13742e24	−0.429236
$748$	−3.27394e24	−1.22154
$749$	4.65657e24	1.71779
$750$	0	0
$751$	1.88476e24	0.679697	0.339849	−	0.940480i	$-0.389624\pi$
$751$	1.88476e24	0.679697	0.339849	+	0.940480i	$0.389624\pi$
$752$	4.49938e23	0.160436
$753$	1.35776e24	0.478701
$754$	−1.00105e22	−0.00348978
$755$	0	0
$756$	−5.26689e23	−0.179522
$757$	1.47031e24	0.495558	0.247779	−	0.968817i	$-0.420299\pi$
$757$	1.47031e24	0.495558	0.247779	+	0.968817i	$0.420299\pi$
$758$	6.48248e23	0.216049
$759$	3.77900e24	1.24544
$760$	0	0
$761$	−1.01146e24	−0.325972	−0.162986	−	0.986628i	$-0.552112\pi$
$761$	−1.01146e24	−0.325972	−0.162986	+	0.986628i	$0.552112\pi$
$762$	−2.82649e23	−0.0900804
$763$	−5.55309e24	−1.75015
$764$	5.40134e23	0.168348
$765$	0	0
$766$	1.36639e24	0.416513
$767$	7.87314e22	0.0237349
$768$	−1.69789e24	−0.506220
$769$	−4.17674e24	−1.23158	−0.615791	−	0.787910i	$-0.711163\pi$
$769$	−4.17674e24	−1.23158	−0.615791	+	0.787910i	$0.711163\pi$
$770$	0	0
$771$	−3.06712e24	−0.884644
$772$	1.18623e24	0.338393
$773$	3.39452e24	0.957750	0.478875	−	0.877883i	$-0.341045\pi$
$773$	3.39452e24	0.957750	0.478875	+	0.877883i	$0.341045\pi$
$774$	1.01900e24	0.284365
$775$	0	0
$776$	1.14154e24	0.311648
$777$	2.70892e24	0.731504
$778$	2.25828e24	0.603187
$779$	1.15446e24	0.305007
$780$	0	0
$781$	−3.27216e24	−0.845865
$782$	−6.43663e24	−1.64589
$783$	−4.62602e22	−0.0117013
$784$	5.14501e23	0.128736
$785$	0	0
$786$	−1.36792e24	−0.334941
$787$	7.12654e24	1.72621	0.863105	−	0.505025i	$-0.168517\pi$
$787$	7.12654e24	1.72621	0.863105	+	0.505025i	$0.168517\pi$
$788$	−2.35082e24	−0.563310
$789$	−8.29094e23	−0.196539
$790$	0	0
$791$	1.40747e24	0.326542
$792$	1.58290e24	0.363320
$793$	4.07008e23	0.0924232
$794$	−2.93586e24	−0.659571
$795$	0	0
$796$	1.84972e24	0.406766
$797$	−5.98812e24	−1.30285	−0.651426	−	0.758712i	$-0.725829\pi$
$797$	−5.98812e24	−1.30285	−0.651426	+	0.758712i	$0.725829\pi$
$798$	−2.19724e24	−0.472991
$799$	7.90767e24	1.68423
$800$	0	0
$801$	−1.67382e24	−0.349005
$802$	−1.76113e24	−0.363337
$803$	−7.55933e23	−0.154312
$804$	2.61100e23	0.0527388
$805$	0	0
$806$	6.41636e22	0.0126894
$807$	−7.78061e23	−0.152261
$808$	−3.34806e24	−0.648330
$809$	−2.16052e24	−0.413996	−0.206998	−	0.978341i	$-0.566369\pi$
$809$	−2.16052e24	−0.413996	−0.206998	+	0.978341i	$0.566369\pi$
$810$	0	0
$811$	−7.22229e23	−0.135518	−0.0677591	−	0.997702i	$-0.521585\pi$
$811$	−7.22229e23	−0.135518	−0.0677591	+	0.997702i	$0.521585\pi$
$812$	3.05451e23	0.0567172
$813$	2.84217e24	0.522252
$814$	−3.30249e24	−0.600529
$815$	0	0
$816$	7.48440e23	0.133288
$817$	−9.13790e24	−1.61049
$818$	−3.31516e23	−0.0578232
$819$	−2.68845e23	−0.0464075
$820$	0	0
$821$	1.07145e25	1.81157	0.905786	−	0.423736i	$-0.139281\pi$
$821$	1.07145e25	1.81157	0.905786	+	0.423736i	$0.139281\pi$
$822$	−2.28577e24	−0.382492
$823$	8.71705e24	1.44368	0.721840	−	0.692060i	$-0.243297\pi$
$823$	8.71705e24	1.44368	0.721840	+	0.692060i	$0.243297\pi$
$824$	−9.29257e24	−1.52319
$825$	0	0
$826$	1.11760e24	0.179455
$827$	3.38564e24	0.538077	0.269038	−	0.963129i	$-0.413294\pi$
$827$	3.38564e24	0.538077	0.269038	+	0.963129i	$0.413294\pi$
$828$	−2.71354e24	−0.426853
$829$	1.77079e24	0.275711	0.137855	−	0.990452i	$-0.455979\pi$
$829$	1.77079e24	0.275711	0.137855	+	0.990452i	$0.455979\pi$
$830$	0	0
$831$	−2.79868e24	−0.426918
$832$	−2.92102e23	−0.0441048
$833$	9.04237e24	1.35145
$834$	−1.40955e24	−0.208530
$835$	0	0
$836$	−5.75801e24	−0.834678
$837$	2.96512e23	0.0425476
$838$	−3.95273e24	−0.561466
$839$	−6.16728e23	−0.0867196	−0.0433598	−	0.999060i	$-0.513806\pi$
$839$	−6.16728e23	−0.0867196	−0.0433598	+	0.999060i	$0.513806\pi$
$840$	0	0
$841$	−7.23032e24	−0.996303
$842$	5.58518e24	0.761876
$843$	−6.54202e24	−0.883440
$844$	−9.22962e24	−1.23388
$845$	0	0
$846$	−1.55088e24	−0.203203
$847$	3.39070e24	0.439826
$848$	−3.89448e23	−0.0500132
$849$	3.18857e24	0.405397
$850$	0	0
$851$	1.39565e25	1.73931
$852$	2.34960e24	0.289906
$853$	−8.04241e24	−0.982470	−0.491235	−	0.871027i	$-0.663454\pi$
$853$	−8.04241e24	−0.982470	−0.491235	+	0.871027i	$0.663454\pi$
$854$	5.77751e24	0.698794
$855$	0	0
$856$	1.00491e25	1.19151
$857$	−1.71827e23	−0.0201722	−0.0100861	−	0.999949i	$-0.503211\pi$
$857$	−1.71827e23	−0.0201722	−0.0100861	+	0.999949i	$0.503211\pi$
$858$	3.27753e23	0.0380983
$859$	5.30613e24	0.610711	0.305356	−	0.952238i	$-0.401225\pi$
$859$	5.30613e24	0.610711	0.305356	+	0.952238i	$0.401225\pi$
$860$	0	0
$861$	2.00512e24	0.226263
$862$	7.22330e24	0.807093
$863$	1.56974e25	1.73675	0.868375	−	0.495909i	$-0.165165\pi$
$863$	1.56974e25	1.73675	0.868375	+	0.495909i	$0.165165\pi$
$864$	−1.81217e24	−0.198533
$865$	0	0
$866$	2.55408e24	0.274367
$867$	7.72633e24	0.821885
$868$	−1.95783e24	−0.206233
$869$	1.20540e25	1.25737
$870$	0	0
$871$	1.33277e23	0.0136333
$872$	−1.19838e25	−1.21396
$873$	1.09228e24	0.109576
$874$	−1.13204e25	−1.12464
$875$	0	0
$876$	5.42803e23	0.0528881
$877$	5.17245e24	0.499114	0.249557	−	0.968360i	$-0.419715\pi$
$877$	5.17245e24	0.499114	0.249557	+	0.968360i	$0.419715\pi$
$878$	6.24027e24	0.596349
$879$	−9.72298e24	−0.920225
$880$	0	0
$881$	1.81020e25	1.68048	0.840238	−	0.542217i	$-0.182415\pi$
$881$	1.81020e25	1.68048	0.840238	+	0.542217i	$0.182415\pi$
$882$	−1.77342e24	−0.163053
$883$	1.22106e25	1.11192	0.555959	−	0.831210i	$-0.312351\pi$
$883$	1.22106e25	1.11192	0.555959	+	0.831210i	$0.312351\pi$
$884$	1.19999e24	0.108227
$885$	0	0
$886$	−3.04504e24	−0.269406
$887$	1.59175e25	1.39484	0.697420	−	0.716663i	$-0.254331\pi$
$887$	1.59175e25	1.39484	0.697420	+	0.716663i	$0.254331\pi$
$888$	5.84595e24	0.507393
$889$	4.40232e24	0.378457
$890$	0	0
$891$	1.51461e24	0.127744
$892$	−6.48752e24	−0.541973
$893$	1.39075e25	1.15083
$894$	1.89989e24	0.155726
$895$	0	0
$896$	1.33856e25	1.07652
$897$	−1.38511e24	−0.110344
$898$	−1.00607e24	−0.0793927
$899$	−1.71961e23	−0.0134422
$900$	0	0
$901$	−6.84456e24	−0.525031
$902$	−2.44447e24	−0.185750
$903$	−1.58711e25	−1.19471
$904$	3.03738e24	0.226499
$905$	0	0
$906$	1.15753e24	0.0847113
$907$	5.23010e24	0.379182	0.189591	−	0.981863i	$-0.439284\pi$
$907$	5.23010e24	0.379182	0.189591	+	0.981863i	$0.439284\pi$
$908$	8.31732e23	0.0597384
$909$	−3.20360e24	−0.227953
$910$	0	0
$911$	1.46926e25	1.02611	0.513054	−	0.858356i	$-0.328514\pi$
$911$	1.46926e25	1.02611	0.513054	+	0.858356i	$0.328514\pi$
$912$	1.31631e24	0.0910757
$913$	−2.15974e25	−1.48047
$914$	1.36241e25	0.925262
$915$	0	0
$916$	−8.83463e24	−0.588948
$917$	2.13056e25	1.40720
$918$	−2.57977e24	−0.168818
$919$	3.02532e24	0.196151	0.0980753	−	0.995179i	$-0.468731\pi$
$919$	3.02532e24	0.196151	0.0980753	+	0.995179i	$0.468731\pi$
$920$	0	0
$921$	−2.16315e22	−0.00137683
$922$	7.31530e24	0.461338
$923$	1.19934e24	0.0749424
$924$	−1.00008e25	−0.619187
$925$	0	0
$926$	−4.44626e24	−0.270272
$927$	−8.89164e24	−0.535555
$928$	1.05096e24	0.0627232
$929$	−3.31849e25	−1.96249	−0.981244	−	0.192771i	$-0.938253\pi$
$929$	−3.31849e25	−1.96249	−0.981244	+	0.192771i	$0.938253\pi$
$930$	0	0
$931$	1.59032e25	0.923446
$932$	3.16033e24	0.181843
$933$	−1.83165e25	−1.04435
$934$	−2.07798e24	−0.117407
$935$	0	0
$936$	−5.80178e23	−0.0321896
$937$	4.73059e24	0.260093	0.130047	−	0.991508i	$-0.458487\pi$
$937$	4.73059e24	0.260093	0.130047	+	0.991508i	$0.458487\pi$
$938$	1.89187e24	0.103078
$939$	−1.20149e25	−0.648728
$940$	0	0
$941$	−2.93059e25	−1.55397	−0.776986	−	0.629518i	$-0.783252\pi$
$941$	−2.93059e25	−1.55397	−0.776986	+	0.629518i	$0.783252\pi$
$942$	5.66544e24	0.297715
$943$	1.03305e25	0.537988
$944$	−6.69524e23	−0.0345545
$945$	0	0
$946$	1.93488e25	0.980797
$947$	−2.12220e25	−1.06613	−0.533067	−	0.846073i	$-0.678960\pi$
$947$	−2.12220e25	−1.06613	−0.533067	+	0.846073i	$0.678960\pi$
$948$	−8.65550e24	−0.430945
$949$	2.77070e23	0.0136718
$950$	0	0
$951$	−1.14428e25	−0.554625
$952$	4.19923e25	2.01723
$953$	−3.62846e25	−1.72756	−0.863779	−	0.503870i	$-0.831909\pi$
$953$	−3.62846e25	−1.72756	−0.863779	+	0.503870i	$0.831909\pi$
$954$	1.34238e24	0.0633451
$955$	0	0
$956$	4.19152e24	0.194303
$957$	−8.78390e23	−0.0403586
$958$	−1.37679e25	−0.626992
$959$	3.56014e25	1.60697
$960$	0	0
$961$	−2.14479e25	−0.951122
$962$	1.21045e24	0.0532059
$963$	9.61549e24	0.418936
$964$	2.28957e25	0.988778
$965$	0	0
$966$	−1.96617e25	−0.834288
$967$	−2.27508e25	−0.956910	−0.478455	−	0.878112i	$-0.658803\pi$
$967$	−2.27508e25	−0.956910	−0.478455	+	0.878112i	$0.658803\pi$
$968$	7.31728e24	0.305077
$969$	2.31342e25	0.956098
$970$	0	0
$971$	2.55385e25	1.03713	0.518563	−	0.855039i	$-0.326467\pi$
$971$	2.55385e25	1.03713	0.518563	+	0.855039i	$0.326467\pi$
$972$	−1.08758e24	−0.0437821
$973$	2.19540e25	0.876103
$974$	4.38754e24	0.173568
$975$	0	0
$976$	−3.46116e24	−0.134555
$977$	−3.25195e25	−1.25326	−0.626629	−	0.779318i	$-0.715566\pi$
$977$	−3.25195e25	−1.25326	−0.626629	+	0.779318i	$0.715566\pi$
$978$	−6.02227e24	−0.230081
$979$	−3.17825e25	−1.20375
$980$	0	0
$981$	−1.14668e25	−0.426829
$982$	−2.79139e24	−0.103008
$983$	1.00301e25	0.366946	0.183473	−	0.983025i	$-0.441266\pi$
$983$	1.00301e25	0.366946	0.183473	+	0.983025i	$0.441266\pi$
$984$	4.32712e24	0.156943
$985$	0	0
$986$	1.49613e24	0.0533354
$987$	2.41552e25	0.853720
$988$	2.11047e24	0.0739513
$989$	−8.17693e25	−2.84068
$990$	0	0
$991$	−3.21765e25	−1.09878	−0.549392	−	0.835564i	$-0.685141\pi$
$991$	−3.21765e25	−1.09878	−0.549392	+	0.835564i	$0.685141\pi$
$992$	−6.73628e24	−0.228072
$993$	1.33546e25	0.448294
$994$	1.70247e25	0.566624
$995$	0	0
$996$	1.55082e25	0.507407
$997$	−2.48213e25	−0.805222	−0.402611	−	0.915371i	$-0.631897\pi$
$997$	−2.48213e25	−0.805222	−0.402611	+	0.915371i	$0.631897\pi$
$998$	2.78149e24	0.0894680
$999$	5.59372e24	0.178400

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.18.a.a.1.1		1
5.2	odd	4		75.18.b.a.49.1		2
5.3	odd	4		75.18.b.a.49.2		2
5.4	even	2		3.18.a.a.1.1	✓	1
15.14	odd	2		9.18.a.a.1.1		1
20.19	odd	2		48.18.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.18.a.a.1.1	✓	1	5.4	even	2
9.18.a.a.1.1		1	15.14	odd	2
48.18.a.e.1.1		1	20.19	odd	2
75.18.a.a.1.1		1	1.1	even	1	trivial
75.18.b.a.49.1		2	5.2	odd	4
75.18.b.a.49.2		2	5.3	odd	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:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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