LMFDB - Embedded newform 50.18.b.e.49.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [50,18,Mod(49,50)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("50.49");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$ N $	$=$	$ 50 = 2 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	50.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:	$91.6110436723$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 18031x^{2} + 81270225 $ x^4 + 18031*x^2 + 81270225
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.3
Root			$95.4487i$ of defining polynomial
Character	$\chi$	$=$	50.49
Dual form			50.18.b.e.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+256.000i q^{2} -18345.8i q^{3} -65536.0 q^{4} +4.69652e6 q^{6} -1.60786e7i q^{7} -1.67772e7i q^{8} -2.07428e8 q^{9} +7.88217e8 q^{11} +1.20231e9i q^{12} -2.87409e9i q^{13} +4.11612e9 q^{14} +4.29497e9 q^{16} -2.01570e10i q^{17} -5.31015e10i q^{18} +2.28657e10 q^{19} -2.94975e11 q^{21} +2.01784e11i q^{22} -6.65718e11i q^{23} -3.07791e11 q^{24} +7.35766e11 q^{26} +1.43625e12i q^{27} +1.05373e12i q^{28} +2.42182e12 q^{29} +8.75874e12 q^{31} +1.09951e12i q^{32} -1.44605e13i q^{33} +5.16020e12 q^{34} +1.35940e13 q^{36} -2.75298e13i q^{37} +5.85361e12i q^{38} -5.27274e13 q^{39} +3.91759e13 q^{41} -7.55135e13i q^{42} -1.29981e14i q^{43} -5.16566e13 q^{44} +1.70424e14 q^{46} +3.00620e14i q^{47} -7.87946e13i q^{48} -2.58908e13 q^{49} -3.69796e14 q^{51} +1.88356e14i q^{52} +1.01926e14i q^{53} -3.67679e14 q^{54} -2.69754e14 q^{56} -4.19488e14i q^{57} +6.19987e14i q^{58} +9.67154e14 q^{59} -1.14011e15 q^{61} +2.24224e15i q^{62} +3.33515e15i q^{63} -2.81475e14 q^{64} +3.70188e15 q^{66} +4.04577e15i q^{67} +1.32101e15i q^{68} -1.22131e16 q^{69} -7.35604e15 q^{71} +3.48006e15i q^{72} +6.99946e14i q^{73} +7.04762e15 q^{74} -1.49852e15 q^{76} -1.26734e16i q^{77} -1.34982e16i q^{78} -1.08787e16 q^{79} -4.38165e14 q^{81} +1.00290e16i q^{82} -1.23277e16i q^{83} +1.93315e16 q^{84} +3.32752e16 q^{86} -4.44303e16i q^{87} -1.32241e16i q^{88} +2.21332e16 q^{89} -4.62113e16 q^{91} +4.36285e16i q^{92} -1.60686e17i q^{93} -7.69588e16 q^{94} +2.01714e16 q^{96} -9.41354e16i q^{97} -6.62805e15i q^{98} -1.63498e17 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 262144 q^{4} + 3229696 q^{6} - 446391812 q^{9} + 2378817408 q^{11} + 3350448128 q^{14} + 17179869184 q^{16} + 273409661200 q^{19} - 819503796752 q^{21} - 211661357056 q^{24} + 1033174644736 q^{26}+ \cdots - 33\!\cdots\!24 q^{99}+O(q^{100}) $ 4 * q - 262144 * q^4 + 3229696 * q^6 - 446391812 * q^9 + 2378817408 * q^11 + 3350448128 * q^14 + 17179869184 * q^16 + 273409661200 * q^19 - 819503796752 * q^21 - 211661357056 * q^24 + 1033174644736 * q^26 + 9393086840040 * q^29 + 14241734757488 * q^31 - 9602887391232 * q^34 + 29254733791232 * q^36 - 126067430445424 * q^39 + 19245423761688 * q^41 - 155898177650688 * q^44 + 332407895402496 * q^46 + 231656293774092 * q^49 - 1676437434168912 * q^51 - 1434109657968640 * q^54 - 219574968516608 * q^56 - 1309232143990320 * q^59 + 2658080770325768 * q^61 - 1125899906842624 * q^64 + 4931077776801792 * q^66 - 24823071318707664 * q^69 + 4070263898695248 * q^71 + 5085754694371328 * q^74 - 17918175556403200 * q^76 - 38376855356990240 * q^79 - 37806028031428396 * q^81 + 53707000823939072 * q^84 - 3946719238068224 * q^86 + 34917825832790040 * q^89 - 108749216422866032 * q^91 - 238629158829053952 * q^94 + 13871438696022016 * q^96 - 339648391656292224 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/50\mathbb{Z}\right)^\times$.

$n$	$27$
$\chi(n)$	$-1$

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 $

$2$	256.000i	0.707107i
$2$	256.000i	0.707107i
$3$	− 18345.8i	− 1.61438i	−0.590292	−	0.807190i	$-0.700987\pi$
$3$	− 18345.8i	− 1.61438i	0.590292	−	0.807190i	$-0.299013\pi$
$4$	−65536.0	−0.500000
$5$	0	0
$5$	0	0
$6$	4.69652e6	1.14154
$7$	− 1.60786e7i	− 1.05418i	−0.849809	−	0.527090i	$-0.823283\pi$
$7$	− 1.60786e7i	− 1.05418i	0.849809	−	0.527090i	$-0.176717\pi$
$8$	− 1.67772e7i	− 0.353553i
$9$	−2.07428e8	−1.60622
$10$	0	0
$11$	7.88217e8	1.10868	0.554342	−	0.832289i	$-0.312970\pi$
$11$	7.88217e8	1.10868	0.554342	+	0.832289i	$0.312970\pi$
$12$	1.20231e9i	0.807190i
$13$	− 2.87409e9i	− 0.977195i	−0.872509	−	0.488597i	$-0.837509\pi$
$13$	− 2.87409e9i	− 0.977195i	0.872509	−	0.488597i	$-0.162491\pi$
$14$	4.11612e9	0.745418
$15$	0	0
$16$	4.29497e9	0.250000
$17$	− 2.01570e10i	− 0.700826i	−0.936595	−	0.350413i	$-0.886041\pi$
$17$	− 2.01570e10i	− 0.700826i	0.936595	−	0.350413i	$-0.113959\pi$
$18$	− 5.31015e10i	− 1.13577i
$19$	2.28657e10	0.308872	0.154436	−	0.988003i	$-0.450644\pi$
$19$	2.28657e10	0.308872	0.154436	+	0.988003i	$0.450644\pi$
$20$	0	0
$21$	−2.94975e11	−1.70185
$22$	2.01784e11i	0.783959i
$23$	− 6.65718e11i	− 1.77257i	−0.463140	−	0.886285i	$-0.653277\pi$
$23$	− 6.65718e11i	− 1.77257i	0.463140	−	0.886285i	$-0.346723\pi$
$24$	−3.07791e11	−0.570769
$25$	0	0
$26$	7.35766e11	0.690981
$27$	1.43625e12i	0.978672i
$28$	1.05373e12i	0.527090i
$29$	2.42182e12	0.899000	0.449500	−	0.893280i	$-0.351602\pi$
$29$	2.42182e12	0.899000	0.449500	+	0.893280i	$0.351602\pi$
$30$	0	0
$31$	8.75874e12	1.84445	0.922226	−	0.386651i	$-0.126368\pi$
$31$	8.75874e12	1.84445	0.922226	+	0.386651i	$0.126368\pi$
$32$	1.09951e12i	0.176777i
$33$	− 1.44605e13i	− 1.78984i
$34$	5.16020e12	0.495559
$35$	0	0
$36$	1.35940e13	0.803111
$37$	− 2.75298e13i	− 1.28851i	−0.764810	−	0.644255i	$-0.777167\pi$
$37$	− 2.75298e13i	− 1.28851i	0.764810	−	0.644255i	$-0.222833\pi$
$38$	5.85361e12i	0.218405i
$39$	−5.27274e13	−1.57756
$40$	0	0
$41$	3.91759e13	0.766225	0.383112	−	0.923702i	$-0.374852\pi$
$41$	3.91759e13	0.766225	0.383112	+	0.923702i	$0.374852\pi$
$42$	− 7.55135e13i	− 1.20339i
$43$	− 1.29981e14i	− 1.69589i	−0.530082	−	0.847947i	$-0.677839\pi$
$43$	− 1.29981e14i	− 1.69589i	0.530082	−	0.847947i	$-0.322161\pi$
$44$	−5.16566e13	−0.554342
$45$	0	0
$46$	1.70424e14	1.25340
$47$	3.00620e14i	1.84156i	0.390079	+	0.920781i	$0.372448\pi$
$47$	3.00620e14i	1.84156i	−0.390079	+	0.920781i	$0.627552\pi$
$48$	− 7.87946e13i	− 0.403595i
$49$	−2.58908e13	−0.111296
$50$	0	0
$51$	−3.69796e14	−1.13140
$52$	1.88356e14i	0.488597i
$53$	1.01926e14i	0.224874i	0.993659	+	0.112437i	$0.0358657\pi$
$53$	1.01926e14i	0.224874i	−0.993659	+	0.112437i	$0.964134\pi$
$54$	−3.67679e14	−0.692026
$55$	0	0
$56$	−2.69754e14	−0.372709
$57$	− 4.19488e14i	− 0.498636i
$58$	6.19987e14i	0.635689i
$59$	9.67154e14	0.857538	0.428769	−	0.903414i	$-0.358947\pi$
$59$	9.67154e14	0.857538	0.428769	+	0.903414i	$0.358947\pi$
$60$	0	0
$61$	−1.14011e15	−0.761455	−0.380727	−	0.924687i	$-0.624326\pi$
$61$	−1.14011e15	−0.761455	−0.380727	+	0.924687i	$0.624326\pi$
$62$	2.24224e15i	1.30422i
$63$	3.33515e15i	1.69325i
$64$	−2.81475e14	−0.125000
$65$	0	0
$66$	3.70188e15	1.26561
$67$	4.04577e15i	1.21721i	0.793473	+	0.608605i	$0.208271\pi$
$67$	4.04577e15i	1.21721i	−0.793473	+	0.608605i	$0.791729\pi$
$68$	1.32101e15i	0.350413i
$69$	−1.22131e16	−2.86160
$70$	0	0
$71$	−7.35604e15	−1.35191	−0.675955	−	0.736943i	$-0.736269\pi$
$71$	−7.35604e15	−1.35191	−0.675955	+	0.736943i	$0.736269\pi$
$72$	3.48006e15i	0.567885i
$73$	6.99946e14i	0.101583i	0.998709	+	0.0507914i	$0.0161744\pi$
$73$	6.99946e14i	0.101583i	−0.998709	+	0.0507914i	$0.983826\pi$
$74$	7.04762e15	0.911115
$75$	0	0
$76$	−1.49852e15	−0.154436
$77$	− 1.26734e16i	− 1.16875i
$78$	− 1.34982e16i	− 1.11551i
$79$	−1.08787e16	−0.806761	−0.403381	−	0.915032i	$-0.632165\pi$
$79$	−1.08787e16	−0.806761	−0.403381	+	0.915032i	$0.632165\pi$
$80$	0	0
$81$	−4.38165e14	−0.0262733
$82$	1.00290e16i	0.541803i
$83$	− 1.23277e16i	− 0.600783i	−0.953816	−	0.300391i	$-0.902883\pi$
$83$	− 1.23277e16i	− 0.600783i	0.953816	−	0.300391i	$-0.0971173\pi$
$84$	1.93315e16	0.850924
$85$	0	0
$86$	3.32752e16	1.19918
$87$	− 4.44303e16i	− 1.45133i
$88$	− 1.32241e16i	− 0.391979i
$89$	2.21332e16	0.595976	0.297988	−	0.954570i	$-0.403684\pi$
$89$	2.21332e16	0.595976	0.297988	+	0.954570i	$0.403684\pi$
$90$	0	0
$91$	−4.62113e16	−1.03014
$92$	4.36285e16i	0.886285i
$93$	− 1.60686e17i	− 2.97765i
$94$	−7.69588e16	−1.30218
$95$	0	0
$96$	2.01714e16	0.285385
$97$	− 9.41354e16i	− 1.21953i	−0.792581	−	0.609766i	$-0.791263\pi$
$97$	− 9.41354e16i	− 1.21953i	0.792581	−	0.609766i	$-0.208737\pi$
$98$	− 6.62805e15i	− 0.0786981i
$99$	−1.63498e17	−1.78079
$100$	0	0
$101$	3.77746e16	0.347111	0.173556	−	0.984824i	$-0.444474\pi$
$101$	3.77746e16	0.347111	0.173556	+	0.984824i	$0.444474\pi$
$102$	− 9.46679e16i	− 0.800021i
$103$	1.63051e17i	1.26826i	0.773226	+	0.634130i	$0.218642\pi$
$103$	1.63051e17i	1.26826i	−0.773226	+	0.634130i	$0.781358\pi$
$104$	−4.82192e16	−0.345491
$105$	0	0
$106$	−2.60930e16	−0.159010
$107$	2.33848e17i	1.31574i	0.753131	+	0.657871i	$0.228543\pi$
$107$	2.33848e17i	1.31574i	−0.753131	+	0.657871i	$0.771457\pi$
$108$	− 9.41259e16i	− 0.489336i
$109$	4.47257e16	0.214997	0.107498	−	0.994205i	$-0.465716\pi$
$109$	4.47257e16	0.214997	0.107498	+	0.994205i	$0.465716\pi$
$110$	0	0
$111$	−5.05055e17	−2.08015
$112$	− 6.90571e16i	− 0.263545i
$113$	− 2.41764e17i	− 0.855510i	−0.903895	−	0.427755i	$-0.859305\pi$
$113$	− 2.41764e17i	− 0.855510i	0.903895	−	0.427755i	$-0.140695\pi$
$114$	1.07389e17	0.352589
$115$	0	0
$116$	−1.58717e17	−0.449500
$117$	5.96165e17i	1.56959i
$118$	2.47591e17i	0.606371i
$119$	−3.24097e17	−0.738797
$120$	0	0
$121$	1.15840e17	0.229182
$122$	− 2.91869e17i	− 0.538430i
$123$	− 7.18713e17i	− 1.23698i
$124$	−5.74013e17	−0.922226
$125$	0	0
$126$	−8.53798e17	−1.19731
$127$	− 1.14453e17i	− 0.150070i	−0.997181	−	0.0750352i	$-0.976093\pi$
$127$	− 1.14453e17i	− 0.150070i	0.997181	−	0.0750352i	$-0.0239069\pi$
$128$	− 7.20576e16i	− 0.0883883i
$129$	−2.38461e18	−2.73782
$130$	0	0
$131$	4.93656e16	0.0497300	0.0248650	−	0.999691i	$-0.492084\pi$
$131$	4.93656e16	0.0497300	0.0248650	+	0.999691i	$0.492084\pi$
$132$	9.47681e17i	0.894919i
$133$	− 3.67648e17i	− 0.325606i
$134$	−1.03572e18	−0.860698
$135$	0	0
$136$	−3.38179e17	−0.247780
$137$	4.16706e17i	0.286883i	0.989659	+	0.143442i	$0.0458169\pi$
$137$	4.16706e17i	0.286883i	−0.989659	+	0.143442i	$0.954183\pi$
$138$	− 3.12656e18i	− 2.02346i
$139$	−5.01647e17	−0.305332	−0.152666	−	0.988278i	$-0.548786\pi$
$139$	−5.01647e17	−0.305332	−0.152666	+	0.988278i	$0.548786\pi$
$140$	0	0
$141$	5.51512e18	2.97298
$142$	− 1.88315e18i	− 0.955945i
$143$	− 2.26540e18i	− 1.08340i
$144$	−8.90895e17	−0.401555
$145$	0	0
$146$	−1.79186e17	−0.0718298
$147$	4.74987e17i	0.179674i
$148$	1.80419e18i	0.644255i
$149$	1.16146e18	0.391670	0.195835	−	0.980637i	$-0.437258\pi$
$149$	1.16146e18	0.391670	0.195835	+	0.980637i	$0.437258\pi$
$150$	0	0
$151$	−5.58219e18	−1.68074	−0.840371	−	0.542012i	$-0.817663\pi$
$151$	−5.58219e18	−1.68074	−0.840371	+	0.542012i	$0.817663\pi$
$152$	− 3.83622e17i	− 0.109203i
$153$	4.18113e18i	1.12568i
$154$	3.24440e18	0.826434
$155$	0	0
$156$	3.45554e18	0.788782
$157$	2.25949e18i	0.488499i	0.969712	+	0.244249i	$0.0785415\pi$
$157$	2.25949e18i	0.488499i	−0.969712	+	0.244249i	$0.921458\pi$
$158$	− 2.78494e18i	− 0.570466i
$159$	1.86991e18	0.363033
$160$	0	0
$161$	−1.07038e19	−1.86861
$162$	− 1.12170e17i	− 0.0185780i
$163$	− 2.17080e18i	− 0.341212i	−0.985339	−	0.170606i	$-0.945427\pi$
$163$	− 2.17080e18i	− 0.341212i	0.985339	−	0.170606i	$-0.0545725\pi$
$164$	−2.56743e18	−0.383112
$165$	0	0
$166$	3.15589e18	0.424818
$167$	− 2.02306e18i	− 0.258772i	−0.991594	−	0.129386i	$-0.958699\pi$
$167$	− 2.02306e18i	− 0.258772i	0.991594	−	0.129386i	$-0.0413007\pi$
$168$	4.94885e18i	0.601694i
$169$	3.90047e17	0.0450900
$170$	0	0
$171$	−4.74297e18	−0.496116
$172$	8.51844e18i	0.847947i
$173$	1.22669e19i	1.16237i	0.813773	+	0.581183i	$0.197410\pi$
$173$	1.22669e19i	1.16237i	−0.813773	+	0.581183i	$0.802590\pi$
$174$	1.13741e19	1.02624
$175$	0	0
$176$	3.38537e18	0.277171
$177$	− 1.77432e19i	− 1.38439i
$178$	5.66610e18i	0.421419i
$179$	1.77359e19	1.25778	0.628888	−	0.777496i	$-0.283510\pi$
$179$	1.77359e19	1.25778	0.628888	+	0.777496i	$0.283510\pi$
$180$	0	0
$181$	−1.40303e19	−0.905315	−0.452657	−	0.891684i	$-0.649524\pi$
$181$	−1.40303e19	−0.905315	−0.452657	+	0.891684i	$0.649524\pi$
$182$	− 1.18301e19i	− 0.728419i
$183$	2.09163e19i	1.22928i
$184$	−1.11689e19	−0.626698
$185$	0	0
$186$	4.11356e19	2.10551
$187$	− 1.58881e19i	− 0.776996i
$188$	− 1.97015e19i	− 0.920781i
$189$	2.30928e19	1.03170
$190$	0	0
$191$	1.22487e19	0.500387	0.250194	−	0.968196i	$-0.419506\pi$
$191$	1.22487e19	0.500387	0.250194	+	0.968196i	$0.419506\pi$
$192$	5.16388e18i	0.201797i
$193$	3.98799e19i	1.49114i	0.666430	+	0.745568i	$0.267821\pi$
$193$	3.98799e19i	1.49114i	−0.666430	+	0.745568i	$0.732179\pi$
$194$	2.40987e19	0.862340
$195$	0	0
$196$	1.69678e18	0.0556479
$197$	1.32923e19i	0.417481i	0.977971	+	0.208740i	$0.0669364\pi$
$197$	1.32923e19i	0.417481i	−0.977971	+	0.208740i	$0.933064\pi$
$198$	− 4.18555e19i	− 1.25921i
$199$	−6.95301e18	−0.200411	−0.100206	−	0.994967i	$-0.531950\pi$
$199$	−6.95301e18	−0.200411	−0.100206	+	0.994967i	$0.531950\pi$
$200$	0	0
$201$	7.42229e19	1.96504
$202$	9.67031e18i	0.245445i
$203$	− 3.89395e19i	− 0.947708i
$204$	2.42350e19	0.565700
$205$	0	0
$206$	−4.17412e19	−0.896795
$207$	1.38088e20i	2.84714i
$208$	− 1.23441e19i	− 0.244299i
$209$	1.80231e19	0.342441
$210$	0	0
$211$	−1.70414e19	−0.298611	−0.149305	−	0.988791i	$-0.547704\pi$
$211$	−1.70414e19	−0.298611	−0.149305	+	0.988791i	$0.547704\pi$
$212$	− 6.67982e18i	− 0.112437i
$213$	1.34952e20i	2.18250i
$214$	−5.98650e19	−0.930370
$215$	0	0
$216$	2.40962e19	0.346013
$217$	− 1.40828e20i	− 1.94439i
$218$	1.14498e19i	0.152026i
$219$	1.28411e19	0.163993
$220$	0	0
$221$	−5.79330e19	−0.684844
$222$	− 1.29294e20i	− 1.47088i
$223$	2.89177e19i	0.316645i	0.987387	+	0.158322i	$0.0506086\pi$
$223$	2.89177e19i	0.316645i	−0.987387	+	0.158322i	$0.949391\pi$
$224$	1.76786e19	0.186354
$225$	0	0
$226$	6.18916e19	0.604937
$227$	1.32244e20i	1.24496i	0.782635	+	0.622481i	$0.213875\pi$
$227$	1.32244e20i	1.24496i	−0.782635	+	0.622481i	$0.786125\pi$
$228$	2.74916e19i	0.249318i
$229$	1.21349e19	0.106031	0.0530157	−	0.998594i	$-0.483117\pi$
$229$	1.21349e19	0.106031	0.0530157	+	0.998594i	$0.483117\pi$
$230$	0	0
$231$	−2.32504e20	−1.88681
$232$	− 4.06315e19i	− 0.317844i
$233$	1.12741e20i	0.850271i	0.905130	+	0.425135i	$0.139773\pi$
$233$	1.12741e20i	0.850271i	−0.905130	+	0.425135i	$0.860227\pi$
$234$	−1.52618e20	−1.10987
$235$	0	0
$236$	−6.33834e19	−0.428769
$237$	1.99578e20i	1.30242i
$238$	− 8.29687e19i	− 0.522409i
$239$	1.64296e20	0.998261	0.499131	−	0.866527i	$-0.333653\pi$
$239$	1.64296e20	0.998261	0.499131	+	0.866527i	$0.333653\pi$
$240$	0	0
$241$	1.25647e20	0.711223	0.355612	−	0.934634i	$-0.384273\pi$
$241$	1.25647e20	0.711223	0.355612	+	0.934634i	$0.384273\pi$
$242$	2.96549e19i	0.162056i
$243$	1.93516e20i	1.02109i
$244$	7.47184e19	0.380727
$245$	0	0
$246$	1.83990e20	0.874675
$247$	− 6.57178e19i	− 0.301828i
$248$	− 1.46947e20i	− 0.652112i
$249$	−2.26161e20	−0.969892
$250$	0	0
$251$	−1.37960e20	−0.552746	−0.276373	−	0.961050i	$-0.589133\pi$
$251$	−1.37960e20	−0.552746	−0.276373	+	0.961050i	$0.589133\pi$
$252$	− 2.18572e20i	− 0.846624i
$253$	− 5.24730e20i	− 1.96522i
$254$	2.92999e19	0.106116
$255$	0	0
$256$	1.84467e19	0.0625000
$257$	1.08129e19i	0.0354414i	0.999843	+	0.0177207i	$0.00564097\pi$
$257$	1.08129e19i	0.0354414i	−0.999843	+	0.0177207i	$0.994359\pi$
$258$	− 6.10459e20i	− 1.93593i
$259$	−4.42640e20	−1.35832
$260$	0	0
$261$	−5.02353e20	−1.44399
$262$	1.26376e19i	0.0351644i
$263$	− 7.17651e20i	− 1.93326i	−0.256185	−	0.966628i	$-0.582466\pi$
$263$	− 7.17651e20i	− 1.93326i	0.256185	−	0.966628i	$-0.417534\pi$
$264$	−2.42606e20	−0.632803
$265$	0	0
$266$	9.41178e19	0.230238
$267$	− 4.06051e20i	− 0.962132i
$268$	− 2.65144e20i	− 0.608605i
$269$	5.80280e20	1.29046	0.645228	−	0.763990i	$-0.276762\pi$
$269$	5.80280e20	1.29046	0.645228	+	0.763990i	$0.276762\pi$
$270$	0	0
$271$	1.64836e20	0.344201	0.172101	−	0.985079i	$-0.444945\pi$
$271$	1.64836e20	0.344201	0.172101	+	0.985079i	$0.444945\pi$
$272$	− 8.65737e19i	− 0.175207i
$273$	8.47782e20i	1.66304i
$274$	−1.06677e20	−0.202857
$275$	0	0
$276$	8.00399e20	1.43080
$277$	− 1.52391e20i	− 0.264168i	−0.991239	−	0.132084i	$-0.957833\pi$
$277$	− 1.52391e20i	− 0.264168i	0.991239	−	0.132084i	$-0.0421669\pi$
$278$	− 1.28422e20i	− 0.215902i
$279$	−1.81681e21	−2.96260
$280$	0	0
$281$	1.60032e20	0.245586	0.122793	−	0.992432i	$-0.460815\pi$
$281$	1.60032e20	0.245586	0.122793	+	0.992432i	$0.460815\pi$
$282$	1.41187e21i	2.10222i
$283$	2.34430e20i	0.338711i	0.985555	+	0.169356i	$0.0541686\pi$
$283$	2.34430e20i	0.338711i	−0.985555	+	0.169356i	$0.945831\pi$
$284$	4.82085e20	0.675955
$285$	0	0
$286$	5.79943e20	0.766080
$287$	− 6.29894e20i	− 0.807739i
$288$	− 2.28069e20i	− 0.283943i
$289$	4.20935e20	0.508842
$290$	0	0
$291$	−1.72699e21	−1.96879
$292$	− 4.58716e19i	− 0.0507914i
$293$	1.57885e21i	1.69811i	0.528303	+	0.849056i	$0.322828\pi$
$293$	1.57885e21i	1.69811i	−0.528303	+	0.849056i	$0.677172\pi$
$294$	−1.21597e20	−0.127049
$295$	0	0
$296$	−4.61873e20	−0.455557
$297$	1.13208e21i	1.08504i
$298$	2.97333e20i	0.276952i
$299$	−1.91333e21	−1.73215
$300$	0	0
$301$	−2.08991e21	−1.78778
$302$	− 1.42904e21i	− 1.18846i
$303$	− 6.93005e20i	− 0.560369i
$304$	9.82072e19	0.0772179
$305$	0	0
$306$	−1.07037e21	−0.795978
$307$	2.38391e21i	1.72430i	0.506650	+	0.862152i	$0.330884\pi$
$307$	2.38391e21i	1.72430i	−0.506650	+	0.862152i	$0.669116\pi$
$308$	8.30566e20i	0.584377i
$309$	2.99131e21	2.04745
$310$	0	0
$311$	1.58302e21	1.02570	0.512852	−	0.858477i	$-0.328589\pi$
$311$	1.58302e21	1.02570	0.512852	+	0.858477i	$0.328589\pi$
$312$	8.84618e20i	0.557753i
$313$	− 2.98369e20i	− 0.183074i	−0.995802	−	0.0915370i	$-0.970822\pi$
$313$	− 2.98369e20i	− 0.183074i	0.995802	−	0.0915370i	$-0.0291780\pi$
$314$	−5.78430e20	−0.345421
$315$	0	0
$316$	7.12944e20	0.403381
$317$	1.01873e21i	0.561117i	0.959837	+	0.280559i	$0.0905197\pi$
$317$	1.01873e21i	0.561117i	−0.959837	+	0.280559i	$0.909480\pi$
$318$	4.78697e20i	0.256703i
$319$	1.90892e21	0.996707
$320$	0	0
$321$	4.29012e21	2.12411
$322$	− 2.74017e21i	− 1.32131i
$323$	− 4.60903e20i	− 0.216465i
$324$	2.87156e19	0.0131366
$325$	0	0
$326$	5.55724e20	0.241273
$327$	− 8.20527e20i	− 0.347086i
$328$	− 6.57263e20i	− 0.270901i
$329$	4.83355e21	1.94134
$330$	0	0
$331$	−6.27305e20	−0.239299	−0.119649	−	0.992816i	$-0.538177\pi$
$331$	−6.27305e20	−0.239299	−0.119649	+	0.992816i	$0.538177\pi$
$332$	8.07907e20i	0.300391i
$333$	5.71044e21i	2.06963i
$334$	5.17903e20	0.182980
$335$	0	0
$336$	−1.26691e21	−0.425462
$337$	− 3.61857e21i	− 1.18490i	−0.805606	−	0.592451i	$-0.798160\pi$
$337$	− 3.61857e21i	− 1.18490i	0.805606	−	0.592451i	$-0.201840\pi$
$338$	9.98521e19i	0.0318834i
$339$	−4.43535e21	−1.38112
$340$	0	0
$341$	6.90379e21	2.04492
$342$	− 1.21420e21i	− 0.350807i
$343$	− 3.32408e21i	− 0.936854i
$344$	−2.18072e21	−0.599589
$345$	0	0
$346$	−3.14033e21	−0.821917
$347$	− 7.74896e21i	− 1.97899i	−0.144579	−	0.989493i	$-0.546183\pi$
$347$	− 7.74896e21i	− 1.97899i	0.144579	−	0.989493i	$-0.453817\pi$
$348$	2.91178e21i	0.725663i
$349$	7.13117e21	1.73438	0.867191	−	0.497976i	$-0.165923\pi$
$349$	7.13117e21	1.73438	0.867191	+	0.497976i	$0.165923\pi$
$350$	0	0
$351$	4.12790e21	0.956354
$352$	8.66654e20i	0.195990i
$353$	− 7.70298e21i	− 1.70049i	−0.526388	−	0.850244i	$-0.676454\pi$
$353$	− 7.70298e21i	− 1.70049i	0.526388	−	0.850244i	$-0.323546\pi$
$354$	4.54226e21	0.978913
$355$	0	0
$356$	−1.45052e21	−0.297988
$357$	5.94581e21i	1.19270i
$358$	4.54040e21i	0.889382i
$359$	4.55697e21	0.871712	0.435856	−	0.900016i	$-0.356446\pi$
$359$	4.55697e21	0.871712	0.435856	+	0.900016i	$0.356446\pi$
$360$	0	0
$361$	−4.95755e21	−0.904598
$362$	− 3.59176e21i	− 0.640154i
$363$	− 2.12517e21i	− 0.369987i
$364$	3.02850e21	0.515070
$365$	0	0
$366$	−5.35456e21	−0.869230
$367$	6.69643e19i	0.0106214i	0.999986	+	0.00531070i	$0.00169046\pi$
$367$	6.69643e19i	0.0106214i	−0.999986	+	0.00531070i	$0.998310\pi$
$368$	− 2.85924e21i	− 0.443142i
$369$	−8.12617e21	−1.23073
$370$	0	0
$371$	1.63883e21	0.237058
$372$	1.05307e22i	1.48882i
$373$	5.61068e21i	0.775336i	0.921799	+	0.387668i	$0.126719\pi$
$373$	5.61068e21i	0.775336i	−0.921799	+	0.387668i	$0.873281\pi$
$374$	4.06736e21	0.549419
$375$	0	0
$376$	5.04357e21	0.651091
$377$	− 6.96053e21i	− 0.878498i
$378$	5.91177e21i	0.729520i
$379$	−6.43379e21	−0.776307	−0.388153	−	0.921595i	$-0.626887\pi$
$379$	−6.43379e21	−0.776307	−0.388153	+	0.921595i	$0.626887\pi$
$380$	0	0
$381$	−2.09973e21	−0.242271
$382$	3.13567e21i	0.353827i
$383$	1.11213e22i	1.22735i	0.789560	+	0.613674i	$0.210309\pi$
$383$	1.11213e22i	1.22735i	−0.789560	+	0.613674i	$0.789691\pi$
$384$	−1.32195e21	−0.142692
$385$	0	0
$386$	−1.02093e22	−1.05439
$387$	2.69617e22i	2.72398i
$388$	6.16926e21i	0.609766i
$389$	−6.91791e21	−0.668965	−0.334483	−	0.942402i	$-0.608562\pi$
$389$	−6.91791e21	−0.668965	−0.334483	+	0.942402i	$0.608562\pi$
$390$	0	0
$391$	−1.34189e22	−1.24226
$392$	4.34376e20i	0.0393490i
$393$	− 9.05651e20i	− 0.0802831i
$394$	−3.40282e21	−0.295204
$395$	0	0
$396$	1.07150e22	0.890397
$397$	− 2.17800e22i	− 1.77149i	−0.464171	−	0.885746i	$-0.653648\pi$
$397$	− 2.17800e22i	− 1.77149i	0.464171	−	0.885746i	$-0.346352\pi$
$398$	− 1.77997e21i	− 0.141712i
$399$	−6.74479e21	−0.525652
$400$	0	0
$401$	4.05717e21	0.303037	0.151519	−	0.988454i	$-0.451584\pi$
$401$	4.05717e21	0.303037	0.151519	+	0.988454i	$0.451584\pi$
$402$	1.90011e22i	1.38949i
$403$	− 2.51734e22i	− 1.80239i
$404$	−2.47560e21	−0.173556
$405$	0	0
$406$	9.96852e21	0.670130
$407$	− 2.16994e22i	− 1.42855i
$408$	6.20415e21i	0.400010i
$409$	6.48833e21	0.409717	0.204859	−	0.978792i	$-0.434327\pi$
$409$	6.48833e21	0.409717	0.204859	+	0.978792i	$0.434327\pi$
$410$	0	0
$411$	7.64480e21	0.463138
$412$	− 1.06857e22i	− 0.634130i
$413$	− 1.55505e22i	− 0.904000i
$414$	−3.53506e22	−2.01323
$415$	0	0
$416$	3.16009e21	0.172745
$417$	9.20310e21i	0.492922i
$418$	4.61391e21i	0.242143i
$419$	−2.17453e22	−1.11827	−0.559135	−	0.829077i	$-0.688867\pi$
$419$	−2.17453e22	−1.11827	−0.559135	+	0.829077i	$0.688867\pi$
$420$	0	0
$421$	1.45920e22	0.720638	0.360319	−	0.932829i	$-0.382668\pi$
$421$	1.45920e22	0.720638	0.360319	+	0.932829i	$0.382668\pi$
$422$	− 4.36261e21i	− 0.211150i
$423$	− 6.23570e22i	− 2.95796i
$424$	1.71003e21	0.0795051
$425$	0	0
$426$	−3.45478e22	−1.54326
$427$	1.83314e22i	0.802710i
$428$	− 1.53254e22i	− 0.657871i
$429$	−4.15606e22	−1.74902
$430$	0	0
$431$	−2.61293e22	−1.05699	−0.528495	−	0.848937i	$-0.677243\pi$
$431$	−2.61293e22	−1.05699	−0.528495	+	0.848937i	$0.677243\pi$
$432$	6.16864e21i	0.244668i
$433$	− 1.27697e22i	− 0.496632i	−0.968679	−	0.248316i	$-0.920123\pi$
$433$	− 1.27697e22i	− 0.496632i	0.968679	−	0.248316i	$-0.0798772\pi$
$434$	3.60521e22	1.37489
$435$	0	0
$436$	−2.93114e21	−0.107498
$437$	− 1.52221e22i	− 0.547497i
$438$	3.28731e21i	0.115961i
$439$	2.83409e22	0.980540	0.490270	−	0.871571i	$-0.336898\pi$
$439$	2.83409e22	0.980540	0.490270	+	0.871571i	$0.336898\pi$
$440$	0	0
$441$	5.37047e21	0.178766
$442$	− 1.48308e22i	− 0.484258i
$443$	− 1.45222e22i	− 0.465158i	−0.972578	−	0.232579i	$-0.925284\pi$
$443$	− 1.45222e22i	− 0.465158i	0.972578	−	0.232579i	$-0.0747164\pi$
$444$	3.30993e22	1.04007
$445$	0	0
$446$	−7.40293e21	−0.223902
$447$	− 2.13078e22i	− 0.632303i
$448$	4.52572e21i	0.131773i
$449$	−1.38898e22	−0.396828	−0.198414	−	0.980118i	$-0.563579\pi$
$449$	−1.38898e22	−0.396828	−0.198414	+	0.980118i	$0.563579\pi$
$450$	0	0
$451$	3.08791e22	0.849502
$452$	1.58443e22i	0.427755i
$453$	1.02410e23i	2.71335i
$454$	−3.38544e22	−0.880321
$455$	0	0
$456$	−7.03785e21	−0.176294
$457$	3.72555e22i	0.916016i	0.888948	+	0.458008i	$0.151437\pi$
$457$	3.72555e22i	0.916016i	−0.888948	+	0.458008i	$0.848563\pi$
$458$	3.10653e21i	0.0749755i
$459$	2.89505e22	0.685879
$460$	0	0
$461$	1.60242e22	0.365864	0.182932	−	0.983126i	$-0.441441\pi$
$461$	1.60242e22	0.365864	0.182932	+	0.983126i	$0.441441\pi$
$462$	− 5.95210e22i	− 1.33418i
$463$	6.01797e22i	1.32438i	0.749337	+	0.662189i	$0.230372\pi$
$463$	6.01797e22i	1.32438i	−0.749337	+	0.662189i	$0.769628\pi$
$464$	1.04017e22	0.224750
$465$	0	0
$466$	−2.88617e22	−0.601232
$467$	4.21105e22i	0.861384i	0.902499	+	0.430692i	$0.141730\pi$
$467$	4.21105e22i	0.861384i	−0.902499	+	0.430692i	$0.858270\pi$
$468$	− 3.90703e22i	− 0.784796i
$469$	6.50504e22	1.28316
$470$	0	0
$471$	4.14521e22	0.788622
$472$	− 1.62262e22i	− 0.303186i
$473$	− 1.02453e23i	− 1.88021i
$474$	−5.10919e22	−0.920949
$475$	0	0
$476$	2.12400e22	0.369399
$477$	− 2.11423e22i	− 0.361198i
$478$	4.20597e22i	0.705877i
$479$	4.13412e22	0.681603	0.340801	−	0.940135i	$-0.389302\pi$
$479$	4.13412e22	0.681603	0.340801	+	0.940135i	$0.389302\pi$
$480$	0	0
$481$	−7.91229e22	−1.25913
$482$	3.21655e22i	0.502911i
$483$	1.96370e23i	3.01664i
$484$	−7.59166e21	−0.114591
$485$	0	0
$486$	−4.95400e22	−0.722018
$487$	− 9.82373e22i	− 1.40696i	−0.710717	−	0.703478i	$-0.751630\pi$
$487$	− 9.82373e22i	− 1.40696i	0.710717	−	0.703478i	$-0.248370\pi$
$488$	1.91279e22i	0.269215i
$489$	−3.98250e22	−0.550846
$490$	0	0
$491$	−2.86911e21	−0.0383314	−0.0191657	−	0.999816i	$-0.506101\pi$
$491$	−2.86911e21	−0.0383314	−0.0191657	+	0.999816i	$0.506101\pi$
$492$	4.71016e22i	0.618489i
$493$	− 4.88167e22i	− 0.630043i
$494$	1.68238e22	0.213424
$495$	0	0
$496$	3.76185e22	0.461113
$497$	1.18275e23i	1.42516i
$498$	− 5.78972e22i	− 0.685817i
$499$	8.68275e22	1.01112	0.505560	−	0.862791i	$-0.331286\pi$
$499$	8.68275e22	1.01112	0.505560	+	0.862791i	$0.331286\pi$
$500$	0	0
$501$	−3.71146e22	−0.417757
$502$	− 3.53177e22i	− 0.390851i
$503$	2.63166e22i	0.286353i	0.989697	+	0.143177i	$0.0457317\pi$
$503$	2.63166e22i	0.286353i	−0.989697	+	0.143177i	$0.954268\pi$
$504$	5.59545e22	0.598653
$505$	0	0
$506$	1.34331e23	1.38962
$507$	− 7.15572e21i	− 0.0727924i
$508$	7.50078e21i	0.0750352i
$509$	−2.52122e22	−0.248033	−0.124017	−	0.992280i	$-0.539578\pi$
$509$	−2.52122e22	−0.248033	−0.124017	+	0.992280i	$0.539578\pi$
$510$	0	0
$511$	1.12541e22	0.107087
$512$	4.72237e21i	0.0441942i
$513$	3.28407e22i	0.302284i
$514$	−2.76810e21	−0.0250608
$515$	0	0
$516$	1.56278e23	1.36891
$517$	2.36954e23i	2.04171i
$518$	− 1.13316e23i	− 0.960479i
$519$	2.25046e23	1.87650
$520$	0	0
$521$	8.93522e22	0.721082	0.360541	−	0.932743i	$-0.382592\pi$
$521$	8.93522e22	0.721082	0.360541	+	0.932743i	$0.382592\pi$
$522$	− 1.28602e23i	− 1.02106i
$523$	9.62048e22i	0.751506i	0.926720	+	0.375753i	$0.122616\pi$
$523$	9.62048e22i	0.751506i	−0.926720	+	0.375753i	$0.877384\pi$
$524$	−3.23523e21	−0.0248650
$525$	0	0
$526$	1.83719e23	1.36702
$527$	− 1.76550e23i	− 1.29264i
$528$	− 6.21072e22i	− 0.447460i
$529$	−3.02130e23	−2.14200
$530$	0	0
$531$	−2.00615e23	−1.37740
$532$	2.40942e22i	0.162803i
$533$	− 1.12595e23i	− 0.748751i
$534$	1.03949e23	0.680330
$535$	0	0
$536$	6.78768e22	0.430349
$537$	− 3.25380e23i	− 2.03053i
$538$	1.48552e23i	0.912490i
$539$	−2.04076e22	−0.123392
$540$	0	0
$541$	−1.09224e23	−0.639944	−0.319972	−	0.947427i	$-0.603673\pi$
$541$	−1.09224e23	−0.639944	−0.319972	+	0.947427i	$0.603673\pi$
$542$	4.21979e22i	0.243387i
$543$	2.57397e23i	1.46152i
$544$	2.21629e22	0.123890
$545$	0	0
$546$	−2.17032e23	−1.17594
$547$	2.52518e23i	1.34710i	0.739140	+	0.673551i	$0.235232\pi$
$547$	2.52518e23i	1.34710i	−0.739140	+	0.673551i	$0.764768\pi$
$548$	− 2.73092e22i	− 0.143442i
$549$	2.36491e23	1.22307
$550$	0	0
$551$	5.53766e22	0.277675
$552$	2.04902e23i	1.01173i
$553$	1.74914e23i	0.850472i
$554$	3.90120e22	0.186795
$555$	0	0
$556$	3.28759e22	0.152666
$557$	− 1.44437e23i	− 0.660554i	−0.943884	−	0.330277i	$-0.892858\pi$
$557$	− 1.44437e23i	− 0.660554i	0.943884	−	0.330277i	$-0.107142\pi$
$558$	− 4.65102e23i	− 2.09487i
$559$	−3.73577e23	−1.65722
$560$	0	0
$561$	−2.91480e23	−1.25437
$562$	4.09682e22i	0.173655i
$563$	7.42080e22i	0.309834i	0.987927	+	0.154917i	$0.0495111\pi$
$563$	7.42080e22i	0.309834i	−0.987927	+	0.154917i	$0.950489\pi$
$564$	−3.61439e23	−1.48649
$565$	0	0
$566$	−6.00142e22	−0.239505
$567$	7.04507e21i	0.0276968i
$568$	1.23414e23i	0.477972i
$569$	−1.99386e23	−0.760747	−0.380373	−	0.924833i	$-0.624205\pi$
$569$	−1.99386e23	−0.760747	−0.380373	+	0.924833i	$0.624205\pi$
$570$	0	0
$571$	5.76243e22	0.213402	0.106701	−	0.994291i	$-0.465971\pi$
$571$	5.76243e22	0.213402	0.106701	+	0.994291i	$0.465971\pi$
$572$	1.48466e23i	0.541701i
$573$	− 2.24712e23i	− 0.807815i
$574$	1.61253e23	0.571158
$575$	0	0
$576$	5.83857e22	0.200778
$577$	2.51358e23i	0.851722i	0.904789	+	0.425861i	$0.140029\pi$
$577$	2.51358e23i	0.851722i	−0.904789	+	0.425861i	$0.859971\pi$
$578$	1.07759e23i	0.359806i
$579$	7.31628e23	2.40726
$580$	0	0
$581$	−1.98212e23	−0.633333
$582$	− 4.42109e23i	− 1.39214i
$583$	8.03398e22i	0.249315i
$584$	1.17431e22	0.0359149
$585$	0	0
$586$	−4.04186e23	−1.20075
$587$	− 2.04262e23i	− 0.598087i	−0.954240	−	0.299043i	$-0.903332\pi$
$587$	− 2.04262e23i	− 0.598087i	0.954240	−	0.299043i	$-0.0966675\pi$
$588$	− 3.11288e22i	− 0.0898369i
$589$	2.00274e23	0.569699
$590$	0	0
$591$	2.43857e23	0.673973
$592$	− 1.18239e23i	− 0.322128i
$593$	− 2.40118e23i	− 0.644851i	−0.946595	−	0.322425i	$-0.895502\pi$
$593$	− 2.40118e23i	− 0.644851i	0.946595	−	0.322425i	$-0.104498\pi$
$594$	−2.89811e23	−0.767239
$595$	0	0
$596$	−7.61172e22	−0.195835
$597$	1.27558e23i	0.323539i
$598$	− 4.89812e23i	− 1.22481i
$599$	−6.64163e23	−1.63737	−0.818684	−	0.574244i	$-0.805296\pi$
$599$	−6.64163e23	−1.63737	−0.818684	+	0.574244i	$0.805296\pi$
$600$	0	0
$601$	−5.64984e23	−1.35395	−0.676975	−	0.736006i	$-0.736710\pi$
$601$	−5.64984e23	−1.35395	−0.676975	+	0.736006i	$0.736710\pi$
$602$	− 5.35018e23i	− 1.26415i
$603$	− 8.39206e23i	− 1.95511i
$604$	3.65835e23	0.840371
$605$	0	0
$606$	1.77409e23	0.396241
$607$	2.77451e23i	0.611057i	0.952183	+	0.305529i	$0.0988331\pi$
$607$	2.77451e23i	0.611057i	−0.952183	+	0.305529i	$0.901167\pi$
$608$	2.51411e22i	0.0546013i
$609$	−7.14376e23	−1.52996
$610$	0	0
$611$	8.64009e23	1.79957
$612$	− 2.74014e23i	− 0.562841i
$613$	− 3.66032e23i	− 0.741491i	−0.928735	−	0.370745i	$-0.879102\pi$
$613$	− 3.66032e23i	− 0.741491i	0.928735	−	0.370745i	$-0.120898\pi$
$614$	−6.10281e23	−1.21927
$615$	0	0
$616$	−2.12625e23	−0.413217
$617$	5.54311e23i	1.06250i	0.847215	+	0.531251i	$0.178278\pi$
$617$	5.54311e23i	1.06250i	−0.847215	+	0.531251i	$0.821722\pi$
$618$	7.65775e23i	1.44777i
$619$	−7.96802e23	−1.48587	−0.742934	−	0.669365i	$-0.766566\pi$
$619$	−7.96802e23	−1.48587	−0.742934	+	0.669365i	$0.766566\pi$
$620$	0	0
$621$	9.56135e23	1.73476
$622$	4.05252e23i	0.725282i
$623$	− 3.55871e23i	− 0.628266i
$624$	−2.26462e23	−0.394391
$625$	0	0
$626$	7.63825e22	0.129453
$627$	− 3.30648e23i	− 0.552830i
$628$	− 1.48078e23i	− 0.244249i
$629$	−5.54918e23	−0.903022
$630$	0	0
$631$	4.62634e23	0.732804	0.366402	−	0.930457i	$-0.380590\pi$
$631$	4.62634e23	0.732804	0.366402	+	0.930457i	$0.380590\pi$
$632$	1.82514e23i	0.285233i
$633$	3.12639e23i	0.482071i
$634$	−2.60794e23	−0.396770
$635$	0	0
$636$	−1.22547e23	−0.181516
$637$	7.44124e22i	0.108758i
$638$	4.88684e23i	0.704779i
$639$	1.52585e24	2.17147
$640$	0	0
$641$	2.38553e23	0.330592	0.165296	−	0.986244i	$-0.447142\pi$
$641$	2.38553e23	0.330592	0.165296	+	0.986244i	$0.447142\pi$
$642$	1.09827e24i	1.50197i
$643$	− 1.39064e24i	− 1.87681i	−0.345536	−	0.938405i	$-0.612303\pi$
$643$	− 1.39064e24i	− 1.87681i	0.345536	−	0.938405i	$-0.387697\pi$
$644$	7.01485e23	0.934304
$645$	0	0
$646$	1.17991e23	0.153064
$647$	1.93162e23i	0.247306i	0.992325	+	0.123653i	$0.0394610\pi$
$647$	1.93162e23i	0.247306i	−0.992325	+	0.123653i	$0.960539\pi$
$648$	7.35118e21i	0.00928901i
$649$	7.62328e23	0.950740
$650$	0	0
$651$	−2.58361e24	−3.13898
$652$	1.42265e23i	0.170606i
$653$	1.41423e24i	1.67401i	0.547192	+	0.837007i	$0.315697\pi$
$653$	1.41423e24i	1.67401i	−0.547192	+	0.837007i	$0.684303\pi$
$654$	2.10055e23	0.245427
$655$	0	0
$656$	1.68259e23	0.191556
$657$	− 1.45188e23i	− 0.163164i
$658$	1.23739e24i	1.37273i
$659$	3.77712e23	0.413651	0.206826	−	0.978378i	$-0.433687\pi$
$659$	3.77712e23	0.413651	0.206826	+	0.978378i	$0.433687\pi$
$660$	0	0
$661$	6.30952e23	0.673416	0.336708	−	0.941609i	$-0.390686\pi$
$661$	6.30952e23	0.673416	0.336708	+	0.941609i	$0.390686\pi$
$662$	− 1.60590e23i	− 0.169210i
$663$	1.06283e24i	1.10560i
$664$	−2.06824e23	−0.212409
$665$	0	0
$666$	−1.46187e24	−1.46345
$667$	− 1.61225e24i	− 1.59354i
$668$	1.32583e23i	0.129386i
$669$	5.30518e23	0.511185
$670$	0	0
$671$	−8.98656e23	−0.844213
$672$	− 3.24328e23i	− 0.300847i
$673$	1.46706e24i	1.34375i	0.740663	+	0.671876i	$0.234511\pi$
$673$	1.46706e24i	1.34375i	−0.740663	+	0.671876i	$0.765489\pi$
$674$	9.26353e23	0.837852
$675$	0	0
$676$	−2.55621e22	−0.0225450
$677$	− 8.89634e22i	− 0.0774832i	−0.999249	−	0.0387416i	$-0.987665\pi$
$677$	− 8.89634e22i	− 0.0774832i	0.999249	−	0.0387416i	$-0.0123349\pi$
$678$	− 1.13545e24i	− 0.976598i
$679$	−1.51357e24	−1.28561
$680$	0	0
$681$	2.42612e24	2.00984
$682$	1.76737e24i	1.44597i
$683$	− 8.93594e23i	− 0.722045i	−0.932557	−	0.361022i	$-0.882428\pi$
$683$	− 8.93594e23i	− 0.722045i	0.932557	−	0.361022i	$-0.117572\pi$
$684$	3.10835e23	0.248058
$685$	0	0
$686$	8.50966e23	0.662456
$687$	− 2.22624e23i	− 0.171175i
$688$	− 5.58265e23i	− 0.423973i
$689$	2.92944e23	0.219746
$690$	0	0
$691$	−7.89975e23	−0.578162	−0.289081	−	0.957305i	$-0.593350\pi$
$691$	−7.89975e23	−0.578162	−0.289081	+	0.957305i	$0.593350\pi$
$692$	− 8.03924e23i	− 0.581183i
$693$	2.62882e24i	1.87728i
$694$	1.98373e24	1.39935
$695$	0	0
$696$	−7.45416e23	−0.513121
$697$	− 7.89669e23i	− 0.536991i
$698$	1.82558e24i	1.22639i
$699$	2.06833e24	1.37266
$700$	0	0
$701$	3.61705e23	0.234288	0.117144	−	0.993115i	$-0.462626\pi$
$701$	3.61705e23	0.234288	0.117144	+	0.993115i	$0.462626\pi$
$702$	1.05674e24i	0.676244i
$703$	− 6.29486e23i	− 0.397984i
$704$	−2.21863e23	−0.138586
$705$	0	0
$706$	1.97196e24	1.20243
$707$	− 6.07363e23i	− 0.365918i
$708$	1.16282e24i	0.692196i
$709$	−4.46298e23	−0.262501	−0.131251	−	0.991349i	$-0.541899\pi$
$709$	−4.46298e23	−0.262501	−0.131251	+	0.991349i	$0.541899\pi$
$710$	0	0
$711$	2.25654e24	1.29584
$712$	− 3.71333e23i	− 0.210709i
$713$	− 5.83085e24i	− 3.26942i
$714$	−1.52213e24	−0.843366
$715$	0	0
$716$	−1.16234e24	−0.628888
$717$	− 3.01414e24i	− 1.61157i
$718$	1.16658e24i	0.616393i
$719$	−1.55258e24	−0.810698	−0.405349	−	0.914162i	$-0.632850\pi$
$719$	−1.55258e24	−0.810698	−0.405349	+	0.914162i	$0.632850\pi$
$720$	0	0
$721$	2.62164e24	1.33697
$722$	− 1.26913e24i	− 0.639648i
$723$	− 2.30509e24i	− 1.14818i
$724$	9.19491e23	0.452657
$725$	0	0
$726$	5.44043e23	0.261621
$727$	− 1.38114e23i	− 0.0656439i	−0.999461	−	0.0328219i	$-0.989551\pi$
$727$	− 1.38114e23i	− 0.0656439i	0.999461	−	0.0328219i	$-0.0104494\pi$
$728$	7.75296e23i	0.364209i
$729$	3.49361e24	1.62215
$730$	0	0
$731$	−2.62003e24	−1.18853
$732$	− 1.37077e24i	− 0.614639i
$733$	1.51067e24i	0.669555i	0.942297	+	0.334778i	$0.108661\pi$
$733$	1.51067e24i	0.669555i	−0.942297	+	0.334778i	$0.891339\pi$
$734$	−1.71428e22	−0.00751046
$735$	0	0
$736$	7.31964e23	0.313349
$737$	3.18895e24i	1.34950i
$738$	− 2.08030e24i	− 0.870255i
$739$	3.13836e24	1.29785	0.648926	−	0.760851i	$-0.275218\pi$
$739$	3.13836e24	1.29785	0.648926	+	0.760851i	$0.275218\pi$
$740$	0	0
$741$	−1.20565e24	−0.487265
$742$	4.19540e23i	0.167625i
$743$	− 4.99845e23i	− 0.197438i	−0.995115	−	0.0987190i	$-0.968526\pi$
$743$	− 4.99845e23i	− 0.197438i	0.995115	−	0.0987190i	$-0.0314745\pi$
$744$	−2.69586e24	−1.05276
$745$	0	0
$746$	−1.43633e24	−0.548246
$747$	2.55710e24i	0.964991i
$748$	1.04124e24i	0.388498i
$749$	3.75994e24	1.38703
$750$	0	0
$751$	4.01494e24	1.44790	0.723952	−	0.689851i	$-0.242324\pi$
$751$	4.01494e24	1.44790	0.723952	+	0.689851i	$0.242324\pi$
$752$	1.29115e24i	0.460391i
$753$	2.53098e24i	0.892342i
$754$	1.78190e24	0.621192
$755$	0	0
$756$	−1.51341e24	−0.515848
$757$	3.79032e24i	1.27750i	0.769414	+	0.638750i	$0.220548\pi$
$757$	3.79032e24i	1.27750i	−0.769414	+	0.638750i	$0.779452\pi$
$758$	− 1.64705e24i	− 0.548932i
$759$	−9.62659e24	−3.17261
$760$	0	0
$761$	−1.23826e24	−0.399065	−0.199532	−	0.979891i	$-0.563942\pi$
$761$	−1.23826e24	−0.399065	−0.199532	+	0.979891i	$0.563942\pi$
$762$	− 5.37530e23i	− 0.171311i
$763$	− 7.19126e23i	− 0.226645i
$764$	−8.02731e23	−0.250194
$765$	0	0
$766$	−2.84706e24	−0.867866
$767$	− 2.77968e24i	− 0.837982i
$768$	− 3.38420e23i	− 0.100899i
$769$	−3.58282e24	−1.05646	−0.528228	−	0.849102i	$-0.677143\pi$
$769$	−3.58282e24	−1.05646	−0.528228	+	0.849102i	$0.677143\pi$
$770$	0	0
$771$	1.98371e23	0.0572158
$772$	− 2.61357e24i	− 0.745568i
$773$	1.21994e24i	0.344203i	0.985079	+	0.172101i	$0.0550557\pi$
$773$	1.21994e24i	0.344203i	−0.985079	+	0.172101i	$0.944944\pi$
$774$	−6.90219e24	−1.92615
$775$	0	0
$776$	−1.57933e24	−0.431170
$777$	8.12058e24i	2.19285i
$778$	− 1.77099e24i	− 0.473030i
$779$	8.95783e23	0.236665
$780$	0	0
$781$	−5.79815e24	−1.49884
$782$	− 3.43523e24i	− 0.878413i
$783$	3.47834e24i	0.879826i
$784$	−1.11200e23	−0.0278240
$785$	0	0
$786$	2.31847e23	0.0567688
$787$	1.07097e24i	0.259414i	0.991552	+	0.129707i	$0.0414036\pi$
$787$	1.07097e24i	0.259414i	−0.991552	+	0.129707i	$0.958596\pi$
$788$	− 8.71123e23i	− 0.208740i
$789$	−1.31659e25	−3.12101
$790$	0	0
$791$	−3.88723e24	−0.901862
$792$	2.74304e24i	0.629606i
$793$	3.27678e24i	0.744090i
$794$	5.57569e24	1.25263
$795$	0	0
$796$	4.55672e23	0.100206
$797$	− 5.86608e22i	− 0.0127630i	−0.999980	−	0.00638149i	$-0.997969\pi$
$797$	− 5.86608e22i	− 0.0127630i	0.999980	−	0.00638149i	$-0.00203130\pi$
$798$	− 1.72667e24i	− 0.371692i
$799$	6.05961e24	1.29062
$800$	0	0
$801$	−4.59104e24	−0.957270
$802$	1.03864e24i	0.214280i
$803$	5.51709e23i	0.112623i
$804$	−4.86427e24	−0.982520
$805$	0	0
$806$	6.44439e24	1.27448
$807$	− 1.06457e25i	− 2.08329i
$808$	− 6.33753e23i	− 0.122722i
$809$	−2.47056e24	−0.473406	−0.236703	−	0.971582i	$-0.576067\pi$
$809$	−2.47056e24	−0.473406	−0.236703	+	0.971582i	$0.576067\pi$
$810$	0	0
$811$	9.80457e24	1.83972	0.919859	−	0.392249i	$-0.128303\pi$
$811$	9.80457e24	1.83972	0.919859	+	0.392249i	$0.128303\pi$
$812$	2.55194e24i	0.473854i
$813$	− 3.02404e24i	− 0.555671i
$814$	5.55506e24	1.01014
$815$	0	0
$816$	−1.58826e24	−0.282850
$817$	− 2.97210e24i	− 0.523813i
$818$	1.66101e24i	0.289714i
$819$	9.58550e24	1.65463
$820$	0	0
$821$	3.65428e23	0.0617853	0.0308927	−	0.999523i	$-0.490165\pi$
$821$	3.65428e23	0.0617853	0.0308927	+	0.999523i	$0.490165\pi$
$822$	1.95707e24i	0.327488i
$823$	− 9.20886e24i	− 1.52513i	−0.646911	−	0.762565i	$-0.723940\pi$
$823$	− 9.20886e24i	− 1.52513i	0.646911	−	0.762565i	$-0.276060\pi$
$824$	2.73555e24	0.448398
$825$	0	0
$826$	3.98092e24	0.639224
$827$	− 2.05375e24i	− 0.326400i	−0.986593	−	0.163200i	$-0.947818\pi$
$827$	− 2.05375e24i	− 0.326400i	0.986593	−	0.163200i	$-0.0521816\pi$
$828$	− 9.04975e24i	− 1.42357i
$829$	−2.35707e24	−0.366994	−0.183497	−	0.983020i	$-0.558742\pi$
$829$	−2.35707e24	−0.366994	−0.183497	+	0.983020i	$0.558742\pi$
$830$	0	0
$831$	−2.79573e24	−0.426468
$832$	8.08983e23i	0.122149i
$833$	5.21882e23i	0.0779991i
$834$	−2.35599e24	−0.348548
$835$	0	0
$836$	−1.18116e24	−0.171221
$837$	1.25797e25i	1.80511i
$838$	− 5.56680e24i	− 0.790736i
$839$	−4.69193e24	−0.659743	−0.329872	−	0.944026i	$-0.607006\pi$
$839$	−4.69193e24	−0.659743	−0.329872	+	0.944026i	$0.607006\pi$
$840$	0	0
$841$	−1.39192e24	−0.191800
$842$	3.73555e24i	0.509568i
$843$	− 2.93591e24i	− 0.396469i
$844$	1.11683e24	0.149305
$845$	0	0
$846$	1.59634e25	2.09159
$847$	− 1.86254e24i	− 0.241600i
$848$	4.37769e23i	0.0562186i
$849$	4.30081e24	0.546808
$850$	0	0
$851$	−1.83271e25	−2.28398
$852$	− 8.84423e24i	− 1.09125i
$853$	− 8.89565e24i	− 1.08670i	−0.839505	−	0.543352i	$-0.817155\pi$
$853$	− 8.89565e24i	− 1.08670i	0.839505	−	0.543352i	$-0.182845\pi$
$854$	−4.69284e24	−0.567602
$855$	0	0
$856$	3.92331e24	0.465185
$857$	− 1.23863e25i	− 1.45413i	−0.686567	−	0.727066i	$-0.740883\pi$
$857$	− 1.23863e25i	− 1.45413i	0.686567	−	0.727066i	$-0.259117\pi$
$858$	− 1.06395e25i	− 1.23674i
$859$	−3.17743e24	−0.365708	−0.182854	−	0.983140i	$-0.558534\pi$
$859$	−3.17743e24	−0.365708	−0.182854	+	0.983140i	$0.558534\pi$
$860$	0	0
$861$	−1.15559e25	−1.30400
$862$	− 6.68909e24i	− 0.747404i
$863$	1.42021e25i	1.57130i	0.618669	+	0.785652i	$0.287672\pi$
$863$	1.42021e25i	1.57130i	−0.618669	+	0.785652i	$0.712328\pi$
$864$	−1.57917e24	−0.173006
$865$	0	0
$866$	3.26905e24	0.351172
$867$	− 7.72238e24i	− 0.821465i
$868$	9.22933e24i	0.972193i
$869$	−8.57475e24	−0.894444
$870$	0	0
$871$	1.16279e25	1.18945
$872$	− 7.50372e23i	− 0.0760128i
$873$	1.95263e25i	1.95884i
$874$	3.89685e24	0.387139
$875$	0	0
$876$	−8.41551e23	−0.0819966
$877$	8.28800e24i	0.799748i	0.916570	+	0.399874i	$0.130946\pi$
$877$	8.28800e24i	0.799748i	−0.916570	+	0.399874i	$0.869054\pi$
$878$	7.25527e24i	0.693347i
$879$	2.89653e25	2.74140
$880$	0	0
$881$	7.02937e24	0.652561	0.326281	−	0.945273i	$-0.394205\pi$
$881$	7.02937e24	0.652561	0.326281	+	0.945273i	$0.394205\pi$
$882$	1.37484e24i	0.126407i
$883$	− 3.64222e23i	− 0.0331665i	−0.999862	−	0.0165833i	$-0.994721\pi$
$883$	− 3.64222e23i	− 0.0331665i	0.999862	−	0.0165833i	$-0.00527886\pi$
$884$	3.79670e24	0.342422
$885$	0	0
$886$	3.71768e24	0.328916
$887$	− 1.85309e25i	− 1.62385i	−0.583765	−	0.811923i	$-0.698421\pi$
$887$	− 1.85309e25i	− 1.62385i	0.583765	−	0.811923i	$-0.301579\pi$
$888$	8.47342e24i	0.735442i
$889$	−1.84024e24	−0.158201
$890$	0	0
$891$	−3.45369e23	−0.0291288
$892$	− 1.89515e24i	− 0.158322i
$893$	6.87388e24i	0.568807i
$894$	5.45481e24	0.447106
$895$	0	0
$896$	−1.15859e24	−0.0931772
$897$	3.51015e25i	2.79634i
$898$	− 3.55579e24i	− 0.280600i
$899$	2.12121e25	1.65816
$900$	0	0
$901$	2.05452e24	0.157598
$902$	7.90506e24i	0.600689i
$903$	3.83411e25i	2.88615i
$904$	−4.05613e24	−0.302469
$905$	0	0
$906$	−2.62169e25	−1.91863
$907$	1.87024e25i	1.35592i	0.735097	+	0.677962i	$0.237137\pi$
$907$	1.87024e25i	1.35592i	−0.735097	+	0.677962i	$0.762863\pi$
$908$	− 8.66674e24i	− 0.622481i
$909$	−7.83551e24	−0.557538
$910$	0	0
$911$	−1.41129e25	−0.985620	−0.492810	−	0.870137i	$-0.664030\pi$
$911$	−1.41129e25	−0.985620	−0.492810	+	0.870137i	$0.664030\pi$
$912$	− 1.80169e24i	− 0.124659i
$913$	− 9.71689e24i	− 0.666079i
$914$	−9.53741e24	−0.647721
$915$	0	0
$916$	−7.95273e23	−0.0530157
$917$	− 7.93730e23i	− 0.0524244i
$918$	7.41132e24i	0.484990i
$919$	2.33638e25	1.51482	0.757411	−	0.652938i	$-0.226464\pi$
$919$	2.33638e25	1.51482	0.757411	+	0.652938i	$0.226464\pi$
$920$	0	0
$921$	4.37347e25	2.78368
$922$	4.10221e24i	0.258705i
$923$	2.11419e25i	1.32108i
$924$	1.52374e25	0.943406
$925$	0	0
$926$	−1.54060e25	−0.936476
$927$	− 3.38214e25i	− 2.03711i
$928$	2.66282e24i	0.158922i
$929$	−2.98004e25	−1.76234	−0.881169	−	0.472802i	$-0.843243\pi$
$929$	−2.98004e25	−1.76234	−0.881169	+	0.472802i	$0.843243\pi$
$930$	0	0
$931$	−5.92010e23	−0.0343761
$932$	− 7.38860e24i	− 0.425135i
$933$	− 2.90417e25i	− 1.65588i
$934$	−1.07803e25	−0.609090
$935$	0	0
$936$	1.00020e25	0.554935
$937$	2.48730e25i	1.36755i	0.729694	+	0.683774i	$0.239663\pi$
$937$	2.48730e25i	1.36755i	−0.729694	+	0.683774i	$0.760337\pi$
$938$	1.66529e25i	0.907331i
$939$	−5.47381e24	−0.295551
$940$	0	0
$941$	−3.05795e25	−1.62150	−0.810752	−	0.585390i	$-0.800941\pi$
$941$	−3.05795e25	−1.62150	−0.810752	+	0.585390i	$0.800941\pi$
$942$	1.06117e25i	0.557640i
$943$	− 2.60801e25i	− 1.35819i
$944$	4.15389e24	0.214385
$945$	0	0
$946$	2.62281e25	1.32951
$947$	3.09044e25i	1.55255i	0.630393	+	0.776276i	$0.282894\pi$
$947$	3.09044e25i	1.55255i	−0.630393	+	0.776276i	$0.717106\pi$
$948$	− 1.30795e25i	− 0.651210i
$949$	2.01170e24	0.0992661
$950$	0	0
$951$	1.86893e25	0.905856
$952$	5.43744e24i	0.261204i
$953$	2.50647e24i	0.119337i	0.998218	+	0.0596683i	$0.0190043\pi$
$953$	2.50647e24i	0.119337i	−0.998218	+	0.0596683i	$0.980996\pi$
$954$	5.41242e24	0.255406
$955$	0	0
$956$	−1.07673e25	−0.499131
$957$	− 3.50207e25i	− 1.60906i
$958$	1.05834e25i	0.481966i
$959$	6.70005e24	0.302426
$960$	0	0
$961$	5.41655e25	2.40200
$962$	− 2.02555e25i	− 0.890337i
$963$	− 4.85065e25i	− 2.11337i
$964$	−8.23438e24	−0.355612
$965$	0	0
$966$	−5.02707e25	−2.13309
$967$	1.13278e25i	0.476455i	0.971209	+	0.238228i	$0.0765664\pi$
$967$	1.13278e25i	0.476455i	−0.971209	+	0.238228i	$0.923434\pi$
$968$	− 1.94347e24i	− 0.0810282i
$969$	−8.45564e24	−0.349457
$970$	0	0
$971$	4.00712e25	1.62731	0.813653	−	0.581351i	$-0.197476\pi$
$971$	4.00712e25	1.62731	0.813653	+	0.581351i	$0.197476\pi$
$972$	− 1.26822e25i	− 0.510544i
$973$	8.06578e24i	0.321875i
$974$	2.51488e25	0.994868
$975$	0	0
$976$	−4.89675e24	−0.190364
$977$	− 8.86356e24i	− 0.341589i	−0.985307	−	0.170795i	$-0.945367\pi$
$977$	− 8.86356e24i	− 0.341589i	0.985307	−	0.170795i	$-0.0546335\pi$
$978$	− 1.01952e25i	− 0.389507i
$979$	1.74458e25	0.660750
$980$	0	0
$981$	−9.27734e24	−0.345332
$982$	− 7.34492e23i	− 0.0271044i
$983$	− 1.56885e25i	− 0.573954i	−0.957938	−	0.286977i	$-0.907350\pi$
$983$	− 1.56885e25i	− 0.573954i	0.957938	−	0.286977i	$-0.0926503\pi$
$984$	−1.20580e25	−0.437338
$985$	0	0
$986$	1.24971e25	0.445507
$987$	− 8.86754e25i	− 3.13406i
$988$	4.30688e24i	0.150914i
$989$	−8.65307e25	−3.00609
$990$	0	0
$991$	−2.85830e25	−0.976073	−0.488037	−	0.872823i	$-0.662287\pi$
$991$	−2.85830e25	−0.976073	−0.488037	+	0.872823i	$0.662287\pi$
$992$	9.63034e24i	0.326056i
$993$	1.15084e25i	0.386319i
$994$	−3.02783e25	−1.00774
$995$	0	0
$996$	1.48217e25	0.484946
$997$	6.58977e24i	0.213777i	0.994271	+	0.106889i	$0.0340888\pi$
$997$	6.58977e24i	0.213777i	−0.994271	+	0.106889i	$0.965911\pi$
$998$	2.22278e25i	0.714970i
$999$	3.95396e25	1.26103

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.18.b.e.49.3		4
5.2	odd	4		10.18.a.b.1.1	✓	2
5.3	odd	4		50.18.a.g.1.2		2
5.4	even	2	inner	50.18.b.e.49.2		4
15.2	even	4		90.18.a.n.1.2		2
20.7	even	4		80.18.a.e.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.18.a.b.1.1	✓	2	5.2	odd	4
50.18.a.g.1.2		2	5.3	odd	4
50.18.b.e.49.2		4	5.4	even	2	inner
50.18.b.e.49.3		4	1.1	even	1	trivial
80.18.a.e.1.2		2	20.7	even	4
90.18.a.n.1.2		2	15.2	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:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	50.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+256.000i q^{2} -18345.8i q^{3} -65536.0 q^{4} +4.69652e6 q^{6} -1.60786e7i q^{7} -1.67772e7i q^{8} -2.07428e8 q^{9} +7.88217e8 q^{11} +1.20231e9i q^{12} -2.87409e9i q^{13} +4.11612e9 q^{14} +4.29497e9 q^{16} -2.01570e10i q^{17} -5.31015e10i q^{18} +2.28657e10 q^{19} -2.94975e11 q^{21} +2.01784e11i q^{22} -6.65718e11i q^{23} -3.07791e11 q^{24} +7.35766e11 q^{26} +1.43625e12i q^{27} +1.05373e12i q^{28} +2.42182e12 q^{29} +8.75874e12 q^{31} +1.09951e12i q^{32} -1.44605e13i q^{33} +5.16020e12 q^{34} +1.35940e13 q^{36} -2.75298e13i q^{37} +5.85361e12i q^{38} -5.27274e13 q^{39} +3.91759e13 q^{41} -7.55135e13i q^{42} -1.29981e14i q^{43} -5.16566e13 q^{44} +1.70424e14 q^{46} +3.00620e14i q^{47} -7.87946e13i q^{48} -2.58908e13 q^{49} -3.69796e14 q^{51} +1.88356e14i q^{52} +1.01926e14i q^{53} -3.67679e14 q^{54} -2.69754e14 q^{56} -4.19488e14i q^{57} +6.19987e14i q^{58} +9.67154e14 q^{59} -1.14011e15 q^{61} +2.24224e15i q^{62} +3.33515e15i q^{63} -2.81475e14 q^{64} +3.70188e15 q^{66} +4.04577e15i q^{67} +1.32101e15i q^{68} -1.22131e16 q^{69} -7.35604e15 q^{71} +3.48006e15i q^{72} +6.99946e14i q^{73} +7.04762e15 q^{74} -1.49852e15 q^{76} -1.26734e16i q^{77} -1.34982e16i q^{78} -1.08787e16 q^{79} -4.38165e14 q^{81} +1.00290e16i q^{82} -1.23277e16i q^{83} +1.93315e16 q^{84} +3.32752e16 q^{86} -4.44303e16i q^{87} -1.32241e16i q^{88} +2.21332e16 q^{89} -4.62113e16 q^{91} +4.36285e16i q^{92} -1.60686e17i q^{93} -7.69588e16 q^{94} +2.01714e16 q^{96} -9.41354e16i q^{97} -6.62805e15i q^{98} -1.63498e17 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 262144 q^{4} + 3229696 q^{6} - 446391812 q^{9} + 2378817408 q^{11} + 3350448128 q^{14} + 17179869184 q^{16} + 273409661200 q^{19} - 819503796752 q^{21} - 211661357056 q^{24} + 1033174644736 q^{26}+ \cdots - 33\!\cdots\!24 q^{99}+O(q^{100}) \) 4 * q - 262144 * q^4 + 3229696 * q^6 - 446391812 * q^9 + 2378817408 * q^11 + 3350448128 * q^14 + 17179869184 * q^16 + 273409661200 * q^19 - 819503796752 * q^21 - 211661357056 * q^24 + 1033174644736 * q^26 + 9393086840040 * q^29 + 14241734757488 * q^31 - 9602887391232 * q^34 + 29254733791232 * q^36 - 126067430445424 * q^39 + 19245423761688 * q^41 - 155898177650688 * q^44 + 332407895402496 * q^46 + 231656293774092 * q^49 - 1676437434168912 * q^51 - 1434109657968640 * q^54 - 219574968516608 * q^56 - 1309232143990320 * q^59 + 2658080770325768 * q^61 - 1125899906842624 * q^64 + 4931077776801792 * q^66 - 24823071318707664 * q^69 + 4070263898695248 * q^71 + 5085754694371328 * q^74 - 17918175556403200 * q^76 - 38376855356990240 * q^79 - 37806028031428396 * q^81 + 53707000823939072 * q^84 - 3946719238068224 * q^86 + 34917825832790040 * q^89 - 108749216422866032 * q^91 - 238629158829053952 * q^94 + 13871438696022016 * q^96 - 339648391656292224 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more