LMFDB - Embedded newform 1338.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1338,2,Mod(1,1338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1338, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1338.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1338 = 2 \cdot 3 \cdot 223$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1338.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:	$10.6839837904$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.10273.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 2x^{3} - 5x^{2} + x + 2$ x^4 - 2x^3 - 5x^2 + x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.673533$ of defining polynomial
Character	$\chi$	$=$	1338.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.326467 q^{5} +1.00000 q^{6} -1.59754 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.326467 q^{10} -2.21989 q^{11} -1.00000 q^{12} +2.49096 q^{13} +1.59754 q^{14} -0.326467 q^{15} +1.00000 q^{16} +6.46037 q^{17} -1.00000 q^{18} -7.66776 q^{19} +0.326467 q^{20} +1.59754 q^{21} +2.21989 q^{22} +2.91402 q^{23} +1.00000 q^{24} -4.89342 q^{25} -2.49096 q^{26} -1.00000 q^{27} -1.59754 q^{28} -4.18930 q^{29} +0.326467 q^{30} +5.66776 q^{31} -1.00000 q^{32} +2.21989 q^{33} -6.46037 q^{34} -0.521543 q^{35} +1.00000 q^{36} -2.05540 q^{37} +7.66776 q^{38} -2.49096 q^{39} -0.326467 q^{40} +12.2901 q^{41} -1.59754 q^{42} -8.89342 q^{43} -2.21989 q^{44} +0.326467 q^{45} -2.91402 q^{46} -8.26530 q^{47} -1.00000 q^{48} -4.44787 q^{49} +4.89342 q^{50} -6.46037 q^{51} +2.49096 q^{52} -6.20990 q^{53} +1.00000 q^{54} -0.724719 q^{55} +1.59754 q^{56} +7.66776 q^{57} +4.18930 q^{58} -0.283381 q^{59} -0.326467 q^{60} -2.94460 q^{61} -5.66776 q^{62} -1.59754 q^{63} +1.00000 q^{64} +0.813215 q^{65} -2.21989 q^{66} -5.67353 q^{67} +6.46037 q^{68} -2.91402 q^{69} +0.521543 q^{70} +4.12141 q^{71} -1.00000 q^{72} +7.69257 q^{73} +2.05540 q^{74} +4.89342 q^{75} -7.66776 q^{76} +3.54635 q^{77} +2.49096 q^{78} -0.116565 q^{79} +0.326467 q^{80} +1.00000 q^{81} -12.2901 q^{82} -7.08850 q^{83} +1.59754 q^{84} +2.10910 q^{85} +8.89342 q^{86} +4.18930 q^{87} +2.21989 q^{88} +0.513252 q^{89} -0.326467 q^{90} -3.97940 q^{91} +2.91402 q^{92} -5.66776 q^{93} +8.26530 q^{94} -2.50327 q^{95} +1.00000 q^{96} -6.38186 q^{97} +4.44787 q^{98} -2.21989 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} - 5 q^{7} - 4 q^{8} + 4 q^{9} - 2 q^{10} + 4 q^{11} - 4 q^{12} - 5 q^{13} + 5 q^{14} - 2 q^{15} + 4 q^{16} - 2 q^{17} - 4 q^{18} - 7 q^{19} + 2 q^{20}+ \cdots + 4 q^{99}+O(q^{100})$ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^5 + 4 * q^6 - 5 * q^7 - 4 * q^8 + 4 * q^9 - 2 * q^10 + 4 * q^11 - 4 * q^12 - 5 * q^13 + 5 * q^14 - 2 * q^15 + 4 * q^16 - 2 * q^17 - 4 * q^18 - 7 * q^19 + 2 * q^20 + 5 * q^21 - 4 * q^22 - 4 * q^23 + 4 * q^24 - 6 * q^25 + 5 * q^26 - 4 * q^27 - 5 * q^28 + 9 * q^29 + 2 * q^30 - q^31 - 4 * q^32 - 4 * q^33 + 2 * q^34 + 4 * q^36 - 11 * q^37 + 7 * q^38 + 5 * q^39 - 2 * q^40 + 14 * q^41 - 5 * q^42 - 22 * q^43 + 4 * q^44 + 2 * q^45 + 4 * q^46 - 8 * q^47 - 4 * q^48 - 7 * q^49 + 6 * q^50 + 2 * q^51 - 5 * q^52 + 3 * q^53 + 4 * q^54 - 13 * q^55 + 5 * q^56 + 7 * q^57 - 9 * q^58 - 6 * q^59 - 2 * q^60 - 9 * q^61 + q^62 - 5 * q^63 + 4 * q^64 - 3 * q^65 + 4 * q^66 - 22 * q^67 - 2 * q^68 + 4 * q^69 + 5 * q^71 - 4 * q^72 - 3 * q^73 + 11 * q^74 + 6 * q^75 - 7 * q^76 + 2 * q^77 - 5 * q^78 - 29 * q^79 + 2 * q^80 + 4 * q^81 - 14 * q^82 - 12 * q^83 + 5 * q^84 - 10 * q^85 + 22 * q^86 - 9 * q^87 - 4 * q^88 + 9 * q^89 - 2 * q^90 - 18 * q^91 - 4 * q^92 + q^93 + 8 * q^94 - 2 * q^95 + 4 * q^96 - 29 * q^97 + 7 * q^98 + 4 * q^99

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0.326467	0.146000	0.0730002	−	0.997332i	$-0.476743\pi$
$5$	0.326467	0.146000	0.0730002	+	0.997332i	$0.476743\pi$
$6$	1.00000	0.408248
$7$	−1.59754	−0.603813	−0.301906	−	0.953338i	$-0.597623\pi$
$7$	−1.59754	−0.603813	−0.301906	+	0.953338i	$0.597623\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−0.326467	−0.103238
$11$	−2.21989	−0.669321	−0.334660	−	0.942339i	$-0.608622\pi$
$11$	−2.21989	−0.669321	−0.334660	+	0.942339i	$0.608622\pi$
$12$	−1.00000	−0.288675
$13$	2.49096	0.690867	0.345434	−	0.938443i	$-0.387732\pi$
$13$	2.49096	0.690867	0.345434	+	0.938443i	$0.387732\pi$
$14$	1.59754	0.426960
$15$	−0.326467	−0.0842933
$16$	1.00000	0.250000
$17$	6.46037	1.56687	0.783435	−	0.621473i	$-0.213466\pi$
$17$	6.46037	1.56687	0.783435	+	0.621473i	$0.213466\pi$
$18$	−1.00000	−0.235702
$19$	−7.66776	−1.75910	−0.879552	−	0.475802i	$-0.842158\pi$
$19$	−7.66776	−1.75910	−0.879552	+	0.475802i	$0.842158\pi$
$20$	0.326467	0.0730002
$21$	1.59754	0.348611
$22$	2.21989	0.473281
$23$	2.91402	0.607615	0.303808	−	0.952733i	$-0.401742\pi$
$23$	2.91402	0.607615	0.303808	+	0.952733i	$0.401742\pi$
$24$	1.00000	0.204124
$25$	−4.89342	−0.978684
$26$	−2.49096	−0.488517
$27$	−1.00000	−0.192450
$28$	−1.59754	−0.301906
$29$	−4.18930	−0.777934	−0.388967	−	0.921252i	$-0.627168\pi$
$29$	−4.18930	−0.777934	−0.388967	+	0.921252i	$0.627168\pi$
$30$	0.326467	0.0596044
$31$	5.66776	1.01796	0.508980	−	0.860779i	$-0.330023\pi$
$31$	5.66776	1.01796	0.508980	+	0.860779i	$0.330023\pi$
$32$	−1.00000	−0.176777
$33$	2.21989	0.386433
$34$	−6.46037	−1.10794
$35$	−0.521543	−0.0881568
$36$	1.00000	0.166667
$37$	−2.05540	−0.337905	−0.168952	−	0.985624i	$-0.554038\pi$
$37$	−2.05540	−0.337905	−0.168952	+	0.985624i	$0.554038\pi$
$38$	7.66776	1.24387
$39$	−2.49096	−0.398872
$40$	−0.326467	−0.0516189
$41$	12.2901	1.91939	0.959696	−	0.281040i	$-0.0906794\pi$
$41$	12.2901	1.91939	0.959696	+	0.281040i	$0.0906794\pi$
$42$	−1.59754	−0.246505
$43$	−8.89342	−1.35623	−0.678117	−	0.734954i	$-0.737204\pi$
$43$	−8.89342	−1.35623	−0.678117	+	0.734954i	$0.737204\pi$
$44$	−2.21989	−0.334660
$45$	0.326467	0.0486668
$46$	−2.91402	−0.429649
$47$	−8.26530	−1.20562	−0.602809	−	0.797886i	$-0.705952\pi$
$47$	−8.26530	−1.20562	−0.602809	+	0.797886i	$0.705952\pi$
$48$	−1.00000	−0.144338
$49$	−4.44787	−0.635410
$50$	4.89342	0.692034
$51$	−6.46037	−0.904633
$52$	2.49096	0.345434
$53$	−6.20990	−0.852996	−0.426498	−	0.904489i	$-0.640253\pi$
$53$	−6.20990	−0.852996	−0.426498	+	0.904489i	$0.640253\pi$
$54$	1.00000	0.136083
$55$	−0.724719	−0.0977211
$56$	1.59754	0.213480
$57$	7.66776	1.01562
$58$	4.18930	0.550082
$59$	−0.283381	−0.0368930	−0.0184465	−	0.999830i	$-0.505872\pi$
$59$	−0.283381	−0.0368930	−0.0184465	+	0.999830i	$0.505872\pi$
$60$	−0.326467	−0.0421467
$61$	−2.94460	−0.377018	−0.188509	−	0.982071i	$-0.560365\pi$
$61$	−2.94460	−0.377018	−0.188509	+	0.982071i	$0.560365\pi$
$62$	−5.66776	−0.719806
$63$	−1.59754	−0.201271
$64$	1.00000	0.125000
$65$	0.813215	0.100867
$66$	−2.21989	−0.273249
$67$	−5.67353	−0.693132	−0.346566	−	0.938026i	$-0.612652\pi$
$67$	−5.67353	−0.693132	−0.346566	+	0.938026i	$0.612652\pi$
$68$	6.46037	0.783435
$69$	−2.91402	−0.350807
$70$	0.521543	0.0623363
$71$	4.12141	0.489121	0.244560	−	0.969634i	$-0.421356\pi$
$71$	4.12141	0.489121	0.244560	+	0.969634i	$0.421356\pi$
$72$	−1.00000	−0.117851
$73$	7.69257	0.900347	0.450173	−	0.892941i	$-0.351362\pi$
$73$	7.69257	0.900347	0.450173	+	0.892941i	$0.351362\pi$
$74$	2.05540	0.238935
$75$	4.89342	0.565043
$76$	−7.66776	−0.879552
$77$	3.54635	0.404144
$78$	2.49096	0.282045
$79$	−0.116565	−0.0131146	−0.00655732	−	0.999979i	$-0.502087\pi$
$79$	−0.116565	−0.0131146	−0.00655732	+	0.999979i	$0.502087\pi$
$80$	0.326467	0.0365001
$81$	1.00000	0.111111
$82$	−12.2901	−1.35722
$83$	−7.08850	−0.778063	−0.389032	−	0.921224i	$-0.627190\pi$
$83$	−7.08850	−0.778063	−0.389032	+	0.921224i	$0.627190\pi$
$84$	1.59754	0.174306
$85$	2.10910	0.228764
$86$	8.89342	0.959002
$87$	4.18930	0.449140
$88$	2.21989	0.236641
$89$	0.513252	0.0544046	0.0272023	−	0.999630i	$-0.491340\pi$
$89$	0.513252	0.0544046	0.0272023	+	0.999630i	$0.491340\pi$
$90$	−0.326467	−0.0344126
$91$	−3.97940	−0.417154
$92$	2.91402	0.303808
$93$	−5.66776	−0.587719
$94$	8.26530	0.852500
$95$	−2.50327	−0.256830
$96$	1.00000	0.102062
$97$	−6.38186	−0.647980	−0.323990	−	0.946061i	$-0.605024\pi$
$97$	−6.38186	−0.647980	−0.323990	+	0.946061i	$0.605024\pi$
$98$	4.44787	0.449303
$99$	−2.21989	−0.223107
$100$	−4.89342	−0.489342
$101$	−12.2801	−1.22192	−0.610959	−	0.791662i	$-0.709216\pi$
$101$	−12.2801	−1.22192	−0.610959	+	0.791662i	$0.709216\pi$
$102$	6.46037	0.639672
$103$	−19.4504	−1.91650	−0.958252	−	0.285926i	$-0.907699\pi$
$103$	−19.4504	−1.91650	−0.958252	+	0.285926i	$0.907699\pi$
$104$	−2.49096	−0.244258
$105$	0.521543	0.0508974
$106$	6.20990	0.603159
$107$	−6.16028	−0.595537	−0.297768	−	0.954638i	$-0.596242\pi$
$107$	−6.16028	−0.595537	−0.297768	+	0.954638i	$0.596242\pi$
$108$	−1.00000	−0.0962250
$109$	−9.68026	−0.927201	−0.463600	−	0.886044i	$-0.653443\pi$
$109$	−9.68026	−0.927201	−0.463600	+	0.886044i	$0.653443\pi$
$110$	0.724719	0.0690992
$111$	2.05540	0.195089
$112$	−1.59754	−0.150953
$113$	3.79915	0.357394	0.178697	−	0.983904i	$-0.442812\pi$
$113$	3.79915	0.357394	0.178697	+	0.983904i	$0.442812\pi$
$114$	−7.66776	−0.718151
$115$	0.951330	0.0887120
$116$	−4.18930	−0.388967
$117$	2.49096	0.230289
$118$	0.283381	0.0260873
$119$	−10.3207	−0.946096
$120$	0.326467	0.0298022
$121$	−6.07211	−0.552010
$122$	2.94460	0.266592
$123$	−12.2901	−1.10816
$124$	5.66776	0.508980
$125$	−3.22987	−0.288888
$126$	1.59754	0.142320
$127$	−17.9861	−1.59601	−0.798005	−	0.602650i	$-0.794111\pi$
$127$	−17.9861	−1.59601	−0.798005	+	0.602650i	$0.794111\pi$
$128$	−1.00000	−0.0883883
$129$	8.89342	0.783022
$130$	−0.813215	−0.0713236
$131$	7.32898	0.640336	0.320168	−	0.947361i	$-0.396261\pi$
$131$	7.32898	0.640336	0.320168	+	0.947361i	$0.396261\pi$
$132$	2.21989	0.193216
$133$	12.2495	1.06217
$134$	5.67353	0.490119
$135$	−0.326467	−0.0280978
$136$	−6.46037	−0.553972
$137$	−8.66776	−0.740537	−0.370268	−	0.928925i	$-0.620734\pi$
$137$	−8.66776	−0.740537	−0.370268	+	0.928925i	$0.620734\pi$
$138$	2.91402	0.248058
$139$	5.85706	0.496789	0.248395	−	0.968659i	$-0.420097\pi$
$139$	5.85706	0.496789	0.248395	+	0.968659i	$0.420097\pi$
$140$	−0.521543	−0.0440784
$141$	8.26530	0.696064
$142$	−4.12141	−0.345861
$143$	−5.52964	−0.462412
$144$	1.00000	0.0833333
$145$	−1.36767	−0.113579
$146$	−7.69257	−0.636641
$147$	4.44787	0.366854
$148$	−2.05540	−0.168952
$149$	−15.7586	−1.29099	−0.645497	−	0.763763i	$-0.723350\pi$
$149$	−15.7586	−1.29099	−0.645497	+	0.763763i	$0.723350\pi$
$150$	−4.89342	−0.399546
$151$	−7.78263	−0.633341	−0.316671	−	0.948536i	$-0.602565\pi$
$151$	−7.78263	−0.633341	−0.316671	+	0.948536i	$0.602565\pi$
$152$	7.66776	0.621937
$153$	6.46037	0.522290
$154$	−3.54635	−0.285773
$155$	1.85033	0.148622
$156$	−2.49096	−0.199436
$157$	3.52575	0.281386	0.140693	−	0.990053i	$-0.455067\pi$
$157$	3.52575	0.281386	0.140693	+	0.990053i	$0.455067\pi$
$158$	0.116565	0.00927345
$159$	6.20990	0.492477
$160$	−0.326467	−0.0258095
$161$	−4.65526	−0.366886
$162$	−1.00000	−0.0785674
$163$	0.0621204	0.00486564	0.00243282	−	0.999997i	$-0.499226\pi$
$163$	0.0621204	0.00486564	0.00243282	+	0.999997i	$0.499226\pi$
$164$	12.2901	0.959696
$165$	0.724719	0.0564193
$166$	7.08850	0.550174
$167$	2.08850	0.161613	0.0808063	−	0.996730i	$-0.474250\pi$
$167$	2.08850	0.161613	0.0808063	+	0.996730i	$0.474250\pi$
$168$	−1.59754	−0.123253
$169$	−6.79513	−0.522702
$170$	−2.10910	−0.161760
$171$	−7.66776	−0.586368
$172$	−8.89342	−0.678117
$173$	−14.2597	−1.08415	−0.542073	−	0.840331i	$-0.682360\pi$
$173$	−14.2597	−1.08415	−0.542073	+	0.840331i	$0.682360\pi$
$174$	−4.18930	−0.317590
$175$	7.81742	0.590942
$176$	−2.21989	−0.167330
$177$	0.283381	0.0213002
$178$	−0.513252	−0.0384699
$179$	24.5192	1.83265	0.916327	−	0.400431i	$-0.131140\pi$
$179$	24.5192	1.83265	0.916327	+	0.400431i	$0.131140\pi$
$180$	0.326467	0.0243334
$181$	−18.2125	−1.35373	−0.676864	−	0.736108i	$-0.736661\pi$
$181$	−18.2125	−1.35373	−0.676864	+	0.736108i	$0.736661\pi$
$182$	3.97940	0.294973
$183$	2.94460	0.217671
$184$	−2.91402	−0.214824
$185$	−0.671018	−0.0493342
$186$	5.66776	0.415580
$187$	−14.3413	−1.04874
$188$	−8.26530	−0.602809
$189$	1.59754	0.116204
$190$	2.50327	0.181606
$191$	21.0644	1.52417	0.762085	−	0.647477i	$-0.224176\pi$
$191$	21.0644	1.52417	0.762085	+	0.647477i	$0.224176\pi$
$192$	−1.00000	−0.0721688
$193$	−27.0985	−1.95059	−0.975296	−	0.220901i	$-0.929100\pi$
$193$	−27.0985	−1.95059	−0.975296	+	0.220901i	$0.929100\pi$
$194$	6.38186	0.458191
$195$	−0.813215	−0.0582355
$196$	−4.44787	−0.317705
$197$	17.7547	1.26497	0.632485	−	0.774573i	$-0.282035\pi$
$197$	17.7547	1.26497	0.632485	+	0.774573i	$0.282035\pi$
$198$	2.21989	0.157760
$199$	20.2190	1.43328	0.716642	−	0.697442i	$-0.245678\pi$
$199$	20.2190	1.43328	0.716642	+	0.697442i	$0.245678\pi$
$200$	4.89342	0.346017
$201$	5.67353	0.400180
$202$	12.2801	0.864026
$203$	6.69257	0.469726
$204$	−6.46037	−0.452317
$205$	4.01231	0.280232
$206$	19.4504	1.35517
$207$	2.91402	0.202538
$208$	2.49096	0.172717
$209$	17.0216	1.17741
$210$	−0.521543	−0.0359899
$211$	−11.3049	−0.778264	−0.389132	−	0.921182i	$-0.627225\pi$
$211$	−11.3049	−0.778264	−0.389132	+	0.921182i	$0.627225\pi$
$212$	−6.20990	−0.426498
$213$	−4.12141	−0.282394
$214$	6.16028	0.421108
$215$	−2.90340	−0.198011
$216$	1.00000	0.0680414
$217$	−9.05446	−0.614657
$218$	9.68026	0.655630
$219$	−7.69257	−0.519816
$220$	−0.724719	−0.0488605
$221$	16.0925	1.08250
$222$	−2.05540	−0.137949
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	1.59754	0.106740
$225$	−4.89342	−0.326228
$226$	−3.79915	−0.252716
$227$	18.8653	1.25214	0.626069	−	0.779768i	$-0.284663\pi$
$227$	18.8653	1.25214	0.626069	+	0.779768i	$0.284663\pi$
$228$	7.66776	0.507810
$229$	25.1506	1.66200	0.831000	−	0.556273i	$-0.187769\pi$
$229$	25.1506	1.66200	0.831000	+	0.556273i	$0.187769\pi$
$230$	−0.951330	−0.0627289
$231$	−3.54635	−0.233333
$232$	4.18930	0.275041
$233$	−2.13736	−0.140023	−0.0700115	−	0.997546i	$-0.522304\pi$
$233$	−2.13736	−0.140023	−0.0700115	+	0.997546i	$0.522304\pi$
$234$	−2.49096	−0.162839
$235$	−2.69834	−0.176021
$236$	−0.283381	−0.0184465
$237$	0.116565	0.00757174
$238$	10.3207	0.668991
$239$	0.112547	0.00728005	0.00364003	−	0.999993i	$-0.498841\pi$
$239$	0.112547	0.00728005	0.00364003	+	0.999993i	$0.498841\pi$
$240$	−0.326467	−0.0210733
$241$	13.8957	0.895104	0.447552	−	0.894258i	$-0.352296\pi$
$241$	13.8957	0.895104	0.447552	+	0.894258i	$0.352296\pi$
$242$	6.07211	0.390330
$243$	−1.00000	−0.0641500
$244$	−2.94460	−0.188509
$245$	−1.45208	−0.0927701
$246$	12.2901	0.783589
$247$	−19.1001	−1.21531
$248$	−5.66776	−0.359903
$249$	7.08850	0.449215
$250$	3.22987	0.204275
$251$	1.18698	0.0749213	0.0374607	−	0.999298i	$-0.488073\pi$
$251$	1.18698	0.0749213	0.0374607	+	0.999298i	$0.488073\pi$
$252$	−1.59754	−0.100635
$253$	−6.46879	−0.406689
$254$	17.9861	1.12855
$255$	−2.10910	−0.132077
$256$	1.00000	0.0625000
$257$	−9.49749	−0.592437	−0.296219	−	0.955120i	$-0.595726\pi$
$257$	−9.49749	−0.592437	−0.296219	+	0.955120i	$0.595726\pi$
$258$	−8.89342	−0.553680
$259$	3.28357	0.204031
$260$	0.813215	0.0504334
$261$	−4.18930	−0.259311
$262$	−7.32898	−0.452786
$263$	−20.2440	−1.24830	−0.624148	−	0.781306i	$-0.714554\pi$
$263$	−20.2440	−1.24830	−0.624148	+	0.781306i	$0.714554\pi$
$264$	−2.21989	−0.136625
$265$	−2.02733	−0.124538
$266$	−12.2495	−0.751067
$267$	−0.513252	−0.0314105
$268$	−5.67353	−0.346566
$269$	−9.55885	−0.582814	−0.291407	−	0.956599i	$-0.594123\pi$
$269$	−9.55885	−0.582814	−0.291407	+	0.956599i	$0.594123\pi$
$270$	0.326467	0.0198681
$271$	24.0547	1.46122	0.730608	−	0.682797i	$-0.239237\pi$
$271$	24.0547	1.46122	0.730608	+	0.682797i	$0.239237\pi$
$272$	6.46037	0.391718
$273$	3.97940	0.240844
$274$	8.66776	0.523638
$275$	10.8628	0.655054
$276$	−2.91402	−0.175403
$277$	30.8702	1.85481	0.927405	−	0.374059i	$-0.122034\pi$
$277$	30.8702	1.85481	0.927405	+	0.374059i	$0.122034\pi$
$278$	−5.85706	−0.351283
$279$	5.66776	0.339320
$280$	0.521543	0.0311681
$281$	6.78665	0.404857	0.202429	−	0.979297i	$-0.435117\pi$
$281$	6.78665	0.404857	0.202429	+	0.979297i	$0.435117\pi$
$282$	−8.26530	−0.492191
$283$	12.2466	0.727984	0.363992	−	0.931402i	$-0.381414\pi$
$283$	12.2466	0.727984	0.363992	+	0.931402i	$0.381414\pi$
$284$	4.12141	0.244560
$285$	2.50327	0.148281
$286$	5.52964	0.326975
$287$	−19.6339	−1.15895
$288$	−1.00000	−0.0589256
$289$	24.7364	1.45508
$290$	1.36767	0.0803122
$291$	6.38186	0.374111
$292$	7.69257	0.450173
$293$	−10.7176	−0.626127	−0.313064	−	0.949732i	$-0.601355\pi$
$293$	−10.7176	−0.626127	−0.313064	+	0.949732i	$0.601355\pi$
$294$	−4.44787	−0.259405
$295$	−0.0925144	−0.00538640
$296$	2.05540	0.119467
$297$	2.21989	0.128811
$298$	15.7586	0.912870
$299$	7.25870	0.419781
$300$	4.89342	0.282522
$301$	14.2076	0.818911
$302$	7.78263	0.447840
$303$	12.2801	0.705475
$304$	−7.66776	−0.439776
$305$	−0.961315	−0.0550447
$306$	−6.46037	−0.369315
$307$	6.39669	0.365078	0.182539	−	0.983199i	$-0.441568\pi$
$307$	6.39669	0.365078	0.182539	+	0.983199i	$0.441568\pi$
$308$	3.54635	0.202072
$309$	19.4504	1.10649
$310$	−1.85033	−0.105092
$311$	−6.95535	−0.394402	−0.197201	−	0.980363i	$-0.563185\pi$
$311$	−6.95535	−0.394402	−0.197201	+	0.980363i	$0.563185\pi$
$312$	2.49096	0.141023
$313$	6.45365	0.364782	0.182391	−	0.983226i	$-0.441616\pi$
$313$	6.45365	0.364782	0.182391	+	0.983226i	$0.441616\pi$
$314$	−3.52575	−0.198970
$315$	−0.521543	−0.0293856
$316$	−0.116565	−0.00655732
$317$	5.56695	0.312671	0.156336	−	0.987704i	$-0.450032\pi$
$317$	5.56695	0.312671	0.156336	+	0.987704i	$0.450032\pi$
$318$	−6.20990	−0.348234
$319$	9.29977	0.520687
$320$	0.326467	0.0182500
$321$	6.16028	0.343833
$322$	4.65526	0.259427
$323$	−49.5366	−2.75629
$324$	1.00000	0.0555556
$325$	−12.1893	−0.676141
$326$	−0.0621204	−0.00344053
$327$	9.68026	0.535320
$328$	−12.2901	−0.678608
$329$	13.2041	0.727967
$330$	−0.724719	−0.0398945
$331$	−24.9131	−1.36935	−0.684673	−	0.728850i	$-0.740055\pi$
$331$	−24.9131	−1.36935	−0.684673	+	0.728850i	$0.740055\pi$
$332$	−7.08850	−0.389032
$333$	−2.05540	−0.112635
$334$	−2.08850	−0.114277
$335$	−1.85222	−0.101198
$336$	1.59754	0.0871528
$337$	12.2978	0.669902	0.334951	−	0.942236i	$-0.391280\pi$
$337$	12.2978	0.669902	0.334951	+	0.942236i	$0.391280\pi$
$338$	6.79513	0.369606
$339$	−3.79915	−0.206341
$340$	2.10910	0.114382
$341$	−12.5818	−0.681341
$342$	7.66776	0.414625
$343$	18.2884	0.987481
$344$	8.89342	0.479501
$345$	−0.951330	−0.0512179
$346$	14.2597	0.766607
$347$	−2.80103	−0.150367	−0.0751837	−	0.997170i	$-0.523954\pi$
$347$	−2.80103	−0.150367	−0.0751837	+	0.997170i	$0.523954\pi$
$348$	4.18930	0.224570
$349$	13.6847	0.732523	0.366262	−	0.930512i	$-0.380638\pi$
$349$	13.6847	0.732523	0.366262	+	0.930512i	$0.380638\pi$
$350$	−7.81742	−0.417859
$351$	−2.49096	−0.132957
$352$	2.21989	0.118320
$353$	19.7430	1.05081	0.525407	−	0.850851i	$-0.323913\pi$
$353$	19.7430	1.05081	0.525407	+	0.850851i	$0.323913\pi$
$354$	−0.283381	−0.0150615
$355$	1.34550	0.0714118
$356$	0.513252	0.0272023
$357$	10.3207	0.546229
$358$	−24.5192	−1.29588
$359$	9.56526	0.504835	0.252418	−	0.967618i	$-0.418774\pi$
$359$	9.56526	0.504835	0.252418	+	0.967618i	$0.418774\pi$
$360$	−0.326467	−0.0172063
$361$	39.7945	2.09445
$362$	18.2125	0.957230
$363$	6.07211	0.318703
$364$	−3.97940	−0.208577
$365$	2.51137	0.131451
$366$	−2.94460	−0.153917
$367$	−24.7755	−1.29327	−0.646635	−	0.762800i	$-0.723824\pi$
$367$	−24.7755	−1.29327	−0.646635	+	0.762800i	$0.723824\pi$
$368$	2.91402	0.151904
$369$	12.2901	0.639797
$370$	0.671018	0.0348846
$371$	9.92055	0.515049
$372$	−5.66776	−0.293860
$373$	35.7002	1.84849	0.924244	−	0.381802i	$-0.124696\pi$
$373$	35.7002	1.84849	0.924244	+	0.381802i	$0.124696\pi$
$374$	14.3413	0.741571
$375$	3.22987	0.166790
$376$	8.26530	0.426250
$377$	−10.4354	−0.537449
$378$	−1.59754	−0.0821685
$379$	−14.3802	−0.738660	−0.369330	−	0.929298i	$-0.620413\pi$
$379$	−14.3802	−0.738660	−0.369330	+	0.929298i	$0.620413\pi$
$380$	−2.50327	−0.128415
$381$	17.9861	0.921457
$382$	−21.0644	−1.07775
$383$	−30.9671	−1.58235	−0.791173	−	0.611593i	$-0.790529\pi$
$383$	−30.9671	−1.58235	−0.791173	+	0.611593i	$0.790529\pi$
$384$	1.00000	0.0510310
$385$	1.15777	0.0590052
$386$	27.0985	1.37928
$387$	−8.89342	−0.452078
$388$	−6.38186	−0.323990
$389$	8.50767	0.431356	0.215678	−	0.976465i	$-0.430804\pi$
$389$	8.50767	0.431356	0.215678	+	0.976465i	$0.430804\pi$
$390$	0.813215	0.0411787
$391$	18.8257	0.952054
$392$	4.44787	0.224651
$393$	−7.32898	−0.369698
$394$	−17.7547	−0.894468
$395$	−0.0380547	−0.00191474
$396$	−2.21989	−0.111553
$397$	−14.2651	−0.715945	−0.357973	−	0.933732i	$-0.616532\pi$
$397$	−14.2651	−0.715945	−0.357973	+	0.933732i	$0.616532\pi$
$398$	−20.2190	−1.01348
$399$	−12.2495	−0.613244
$400$	−4.89342	−0.244671
$401$	31.5109	1.57358	0.786791	−	0.617220i	$-0.211741\pi$
$401$	31.5109	1.57358	0.786791	+	0.617220i	$0.211741\pi$
$402$	−5.67353	−0.282970
$403$	14.1181	0.703275
$404$	−12.2801	−0.610959
$405$	0.326467	0.0162223
$406$	−6.69257	−0.332147
$407$	4.56274	0.226167
$408$	6.46037	0.319836
$409$	−9.16293	−0.453078	−0.226539	−	0.974002i	$-0.572741\pi$
$409$	−9.16293	−0.453078	−0.226539	+	0.974002i	$0.572741\pi$
$410$	−4.01231	−0.198154
$411$	8.66776	0.427549
$412$	−19.4504	−0.958252
$413$	0.452712	0.0222765
$414$	−2.91402	−0.143216
$415$	−2.31416	−0.113598
$416$	−2.49096	−0.122129
$417$	−5.85706	−0.286821
$418$	−17.0216	−0.832551
$419$	−29.4201	−1.43727	−0.718634	−	0.695389i	$-0.755232\pi$
$419$	−29.4201	−1.43727	−0.718634	+	0.695389i	$0.755232\pi$
$420$	0.521543	0.0254487
$421$	28.8396	1.40556	0.702778	−	0.711409i	$-0.251943\pi$
$421$	28.8396	1.40556	0.702778	+	0.711409i	$0.251943\pi$
$422$	11.3049	0.550315
$423$	−8.26530	−0.401872
$424$	6.20990	0.301579
$425$	−31.6133	−1.53347
$426$	4.12141	0.199683
$427$	4.70412	0.227648
$428$	−6.16028	−0.297768
$429$	5.52964	0.266974
$430$	2.90340	0.140015
$431$	29.9517	1.44272	0.721360	−	0.692560i	$-0.243517\pi$
$431$	29.9517	1.44272	0.721360	+	0.692560i	$0.243517\pi$
$432$	−1.00000	−0.0481125
$433$	−11.1833	−0.537437	−0.268718	−	0.963219i	$-0.586600\pi$
$433$	−11.1833	−0.537437	−0.268718	+	0.963219i	$0.586600\pi$
$434$	9.05446	0.434628
$435$	1.36767	0.0655746
$436$	−9.68026	−0.463600
$437$	−22.3440	−1.06886
$438$	7.69257	0.367565
$439$	−30.8644	−1.47308	−0.736539	−	0.676395i	$-0.763541\pi$
$439$	−30.8644	−1.47308	−0.736539	+	0.676395i	$0.763541\pi$
$440$	0.724719	0.0345496
$441$	−4.44787	−0.211803
$442$	−16.0925	−0.765443
$443$	−35.8542	−1.70349	−0.851743	−	0.523960i	$-0.824454\pi$
$443$	−35.8542	−1.70349	−0.851743	+	0.523960i	$0.824454\pi$
$444$	2.05540	0.0975447
$445$	0.167560	0.00794309
$446$	1.00000	0.0473514
$447$	15.7586	0.745355
$448$	−1.59754	−0.0754766
$449$	−15.7066	−0.741242	−0.370621	−	0.928784i	$-0.620855\pi$
$449$	−15.7066	−0.741242	−0.370621	+	0.928784i	$0.620855\pi$
$450$	4.89342	0.230678
$451$	−27.2826	−1.28469
$452$	3.79915	0.178697
$453$	7.78263	0.365660
$454$	−18.8653	−0.885395
$455$	−1.29914	−0.0609047
$456$	−7.66776	−0.359076
$457$	−29.0459	−1.35871	−0.679355	−	0.733809i	$-0.737741\pi$
$457$	−29.0459	−1.35871	−0.679355	+	0.733809i	$0.737741\pi$
$458$	−25.1506	−1.17521
$459$	−6.46037	−0.301544
$460$	0.951330	0.0443560
$461$	21.5743	1.00482	0.502408	−	0.864631i	$-0.332448\pi$
$461$	21.5743	1.00482	0.502408	+	0.864631i	$0.332448\pi$
$462$	3.54635	0.164991
$463$	24.3686	1.13251	0.566253	−	0.824232i	$-0.308393\pi$
$463$	24.3686	1.13251	0.566253	+	0.824232i	$0.308393\pi$
$464$	−4.18930	−0.194483
$465$	−1.85033	−0.0858072
$466$	2.13736	0.0990111
$467$	−32.4960	−1.50374	−0.751868	−	0.659314i	$-0.770847\pi$
$467$	−32.4960	−1.50374	−0.751868	+	0.659314i	$0.770847\pi$
$468$	2.49096	0.115145
$469$	9.06369	0.418522
$470$	2.69834	0.124465
$471$	−3.52575	−0.162458
$472$	0.283381	0.0130437
$473$	19.7424	0.907756
$474$	−0.116565	−0.00535403
$475$	37.5216	1.72161
$476$	−10.3207	−0.473048
$477$	−6.20990	−0.284332
$478$	−0.112547	−0.00514778
$479$	34.9175	1.59542	0.797710	−	0.603041i	$-0.206044\pi$
$479$	34.9175	1.59542	0.797710	+	0.603041i	$0.206044\pi$
$480$	0.326467	0.0149011
$481$	−5.11990	−0.233447
$482$	−13.8957	−0.632934
$483$	4.65526	0.211822
$484$	−6.07211	−0.276005
$485$	−2.08346	−0.0946053
$486$	1.00000	0.0453609
$487$	−6.46037	−0.292747	−0.146374	−	0.989229i	$-0.546760\pi$
$487$	−6.46037	−0.292747	−0.146374	+	0.989229i	$0.546760\pi$
$488$	2.94460	0.133296
$489$	−0.0621204	−0.00280918
$490$	1.45208	0.0655984
$491$	−0.708520	−0.0319750	−0.0159875	−	0.999872i	$-0.505089\pi$
$491$	−0.708520	−0.0319750	−0.0159875	+	0.999872i	$0.505089\pi$
$492$	−12.2901	−0.554081
$493$	−27.0644	−1.21892
$494$	19.1001	0.859352
$495$	−0.724719	−0.0325737
$496$	5.66776	0.254490
$497$	−6.58410	−0.295337
$498$	−7.08850	−0.317643
$499$	20.2230	0.905304	0.452652	−	0.891687i	$-0.350478\pi$
$499$	20.2230	0.905304	0.452652	+	0.891687i	$0.350478\pi$
$500$	−3.22987	−0.144444
$501$	−2.08850	−0.0933071
$502$	−1.18698	−0.0529774
$503$	−36.3380	−1.62023	−0.810116	−	0.586269i	$-0.800596\pi$
$503$	−36.3380	−1.62023	−0.810116	+	0.586269i	$0.800596\pi$
$504$	1.59754	0.0711600
$505$	−4.00905	−0.178400
$506$	6.46879	0.287573
$507$	6.79513	0.301782
$508$	−17.9861	−0.798005
$509$	4.16857	0.184769	0.0923844	−	0.995723i	$-0.470551\pi$
$509$	4.16857	0.184769	0.0923844	+	0.995723i	$0.470551\pi$
$510$	2.10910	0.0933923
$511$	−12.2892	−0.543641
$512$	−1.00000	−0.0441942
$513$	7.66776	0.338540
$514$	9.49749	0.418916
$515$	−6.34990	−0.279810
$516$	8.89342	0.391511
$517$	18.3480	0.806945
$518$	−3.28357	−0.144272
$519$	14.2597	0.625932
$520$	−0.813215	−0.0356618
$521$	−16.3684	−0.717114	−0.358557	−	0.933508i	$-0.616731\pi$
$521$	−16.3684	−0.717114	−0.358557	+	0.933508i	$0.616731\pi$
$522$	4.18930	0.183361
$523$	31.9673	1.39783	0.698916	−	0.715204i	$-0.253666\pi$
$523$	31.9673	1.39783	0.698916	+	0.715204i	$0.253666\pi$
$524$	7.32898	0.320168
$525$	−7.81742	−0.341180
$526$	20.2440	0.882678
$527$	36.6158	1.59501
$528$	2.21989	0.0966081
$529$	−14.5085	−0.630804
$530$	2.02733	0.0880614
$531$	−0.283381	−0.0122977
$532$	12.2495	0.531085
$533$	30.6141	1.32605
$534$	0.513252	0.0222106
$535$	−2.01113	−0.0869486
$536$	5.67353	0.245059
$537$	−24.5192	−1.05808
$538$	9.55885	0.412111
$539$	9.87377	0.425293
$540$	−0.326467	−0.0140489
$541$	31.0977	1.33700	0.668498	−	0.743714i	$-0.266937\pi$
$541$	31.0977	1.33700	0.668498	+	0.743714i	$0.266937\pi$
$542$	−24.0547	−1.03324
$543$	18.2125	0.781575
$544$	−6.46037	−0.276986
$545$	−3.16028	−0.135372
$546$	−3.97940	−0.170303
$547$	−21.5025	−0.919381	−0.459691	−	0.888079i	$-0.652040\pi$
$547$	−21.5025	−0.919381	−0.459691	+	0.888079i	$0.652040\pi$
$548$	−8.66776	−0.370268
$549$	−2.94460	−0.125673
$550$	−10.8628	−0.463193
$551$	32.1225	1.36847
$552$	2.91402	0.124029
$553$	0.186218	0.00791879
$554$	−30.8702	−1.31155
$555$	0.671018	0.0284831
$556$	5.85706	0.248395
$557$	0.144826	0.00613649	0.00306824	−	0.999995i	$-0.499023\pi$
$557$	0.144826	0.00613649	0.00306824	+	0.999995i	$0.499023\pi$
$558$	−5.66776	−0.239935
$559$	−22.1531	−0.936978
$560$	−0.521543	−0.0220392
$561$	14.3413	0.605490
$562$	−6.78665	−0.286277
$563$	26.8290	1.13071	0.565354	−	0.824849i	$-0.308740\pi$
$563$	26.8290	1.13071	0.565354	+	0.824849i	$0.308740\pi$
$564$	8.26530	0.348032
$565$	1.24030	0.0521796
$566$	−12.2466	−0.514762
$567$	−1.59754	−0.0670903
$568$	−4.12141	−0.172930
$569$	−25.5083	−1.06936	−0.534682	−	0.845054i	$-0.679569\pi$
$569$	−25.5083	−1.06936	−0.534682	+	0.845054i	$0.679569\pi$
$570$	−2.50327	−0.104850
$571$	1.66059	0.0694937	0.0347468	−	0.999396i	$-0.488938\pi$
$571$	1.66059	0.0694937	0.0347468	+	0.999396i	$0.488938\pi$
$572$	−5.52964	−0.231206
$573$	−21.0644	−0.879980
$574$	19.6339	0.819504
$575$	−14.2595	−0.594663
$576$	1.00000	0.0416667
$577$	−9.53888	−0.397109	−0.198554	−	0.980090i	$-0.563625\pi$
$577$	−9.53888	−0.397109	−0.198554	+	0.980090i	$0.563625\pi$
$578$	−24.7364	−1.02890
$579$	27.0985	1.12618
$580$	−1.36767	−0.0567893
$581$	11.3241	0.469805
$582$	−6.38186	−0.264537
$583$	13.7853	0.570928
$584$	−7.69257	−0.318321
$585$	0.813215	0.0336223
$586$	10.7176	0.442739
$587$	13.8126	0.570107	0.285053	−	0.958512i	$-0.407989\pi$
$587$	13.8126	0.570107	0.285053	+	0.958512i	$0.407989\pi$
$588$	4.44787	0.183427
$589$	−43.4590	−1.79070
$590$	0.0925144	0.00380876
$591$	−17.7547	−0.730330
$592$	−2.05540	−0.0844762
$593$	−9.72780	−0.399473	−0.199736	−	0.979850i	$-0.564009\pi$
$593$	−9.72780	−0.399473	−0.199736	+	0.979850i	$0.564009\pi$
$594$	−2.21989	−0.0910830
$595$	−3.36936	−0.138130
$596$	−15.7586	−0.645497
$597$	−20.2190	−0.827507
$598$	−7.25870	−0.296830
$599$	36.0854	1.47441	0.737204	−	0.675670i	$-0.236146\pi$
$599$	36.0854	1.47441	0.737204	+	0.675670i	$0.236146\pi$
$600$	−4.89342	−0.199773
$601$	11.5725	0.472054	0.236027	−	0.971747i	$-0.424155\pi$
$601$	11.5725	0.472054	0.236027	+	0.971747i	$0.424155\pi$
$602$	−14.2076	−0.579058
$603$	−5.67353	−0.231044
$604$	−7.78263	−0.316671
$605$	−1.98234	−0.0805936
$606$	−12.2801	−0.498846
$607$	13.4321	0.545193	0.272596	−	0.962129i	$-0.412118\pi$
$607$	13.4321	0.545193	0.272596	+	0.962129i	$0.412118\pi$
$608$	7.66776	0.310969
$609$	−6.69257	−0.271197
$610$	0.961315	0.0389225
$611$	−20.5885	−0.832922
$612$	6.46037	0.261145
$613$	−11.7622	−0.475072	−0.237536	−	0.971379i	$-0.576340\pi$
$613$	−11.7622	−0.475072	−0.237536	+	0.971379i	$0.576340\pi$
$614$	−6.39669	−0.258149
$615$	−4.01231	−0.161792
$616$	−3.54635	−0.142887
$617$	−42.4730	−1.70990	−0.854950	−	0.518711i	$-0.826412\pi$
$617$	−42.4730	−1.70990	−0.854950	+	0.518711i	$0.826412\pi$
$618$	−19.4504	−0.782409
$619$	−43.7701	−1.75927	−0.879635	−	0.475649i	$-0.842213\pi$
$619$	−43.7701	−1.75927	−0.879635	+	0.475649i	$0.842213\pi$
$620$	1.85033	0.0743112
$621$	−2.91402	−0.116936
$622$	6.95535	0.278884
$623$	−0.819940	−0.0328502
$624$	−2.49096	−0.0997181
$625$	23.4127	0.936506
$626$	−6.45365	−0.257940
$627$	−17.0216	−0.679775
$628$	3.52575	0.140693
$629$	−13.2786	−0.529453
$630$	0.521543	0.0207788
$631$	−38.5593	−1.53502	−0.767511	−	0.641036i	$-0.778505\pi$
$631$	−38.5593	−1.53502	−0.767511	+	0.641036i	$0.778505\pi$
$632$	0.116565	0.00463673
$633$	11.3049	0.449331
$634$	−5.56695	−0.221092
$635$	−5.87187	−0.233018
$636$	6.20990	0.246239
$637$	−11.0795	−0.438984
$638$	−9.29977	−0.368181
$639$	4.12141	0.163040
$640$	−0.326467	−0.0129047
$641$	47.9344	1.89330	0.946648	−	0.322268i	$-0.104445\pi$
$641$	47.9344	1.89330	0.946648	+	0.322268i	$0.104445\pi$
$642$	−6.16028	−0.243127
$643$	−33.1025	−1.30544	−0.652718	−	0.757601i	$-0.726371\pi$
$643$	−33.1025	−1.30544	−0.652718	+	0.757601i	$0.726371\pi$
$644$	−4.65526	−0.183443
$645$	2.90340	0.114321
$646$	49.5366	1.94899
$647$	21.7062	0.853359	0.426680	−	0.904403i	$-0.359683\pi$
$647$	21.7062	0.853359	0.426680	+	0.904403i	$0.359683\pi$
$648$	−1.00000	−0.0392837
$649$	0.629073	0.0246933
$650$	12.1893	0.478104
$651$	9.05446	0.354872
$652$	0.0621204	0.00243282
$653$	−32.0324	−1.25352	−0.626761	−	0.779211i	$-0.715620\pi$
$653$	−32.0324	−1.25352	−0.626761	+	0.779211i	$0.715620\pi$
$654$	−9.68026	−0.378528
$655$	2.39267	0.0934893
$656$	12.2901	0.479848
$657$	7.69257	0.300116
$658$	−13.2041	−0.514750
$659$	−21.7155	−0.845915	−0.422958	−	0.906149i	$-0.639008\pi$
$659$	−21.7155	−0.845915	−0.422958	+	0.906149i	$0.639008\pi$
$660$	0.724719	0.0282096
$661$	17.2593	0.671310	0.335655	−	0.941985i	$-0.391042\pi$
$661$	17.2593	0.671310	0.335655	+	0.941985i	$0.391042\pi$
$662$	24.9131	0.968275
$663$	−16.0925	−0.624981
$664$	7.08850	0.275087
$665$	3.99906	0.155077
$666$	2.05540	0.0796449
$667$	−12.2077	−0.472684
$668$	2.08850	0.0808063
$669$	1.00000	0.0386622
$670$	1.85222	0.0715575
$671$	6.53669	0.252346
$672$	−1.59754	−0.0616264
$673$	22.1687	0.854541	0.427270	−	0.904124i	$-0.359475\pi$
$673$	22.1687	0.854541	0.427270	+	0.904124i	$0.359475\pi$
$674$	−12.2978	−0.473692
$675$	4.89342	0.188348
$676$	−6.79513	−0.261351
$677$	4.75877	0.182894	0.0914472	−	0.995810i	$-0.470851\pi$
$677$	4.75877	0.182894	0.0914472	+	0.995810i	$0.470851\pi$
$678$	3.79915	0.145905
$679$	10.1953	0.391258
$680$	−2.10910	−0.0808801
$681$	−18.8653	−0.722922
$682$	12.5818	0.481781
$683$	18.3973	0.703954	0.351977	−	0.936009i	$-0.385510\pi$
$683$	18.3973	0.703954	0.351977	+	0.936009i	$0.385510\pi$
$684$	−7.66776	−0.293184
$685$	−2.82973	−0.108119
$686$	−18.2884	−0.698255
$687$	−25.1506	−0.959556
$688$	−8.89342	−0.339058
$689$	−15.4686	−0.589307
$690$	0.951330	0.0362165
$691$	28.6842	1.09120	0.545598	−	0.838047i	$-0.316302\pi$
$691$	28.6842	1.09120	0.545598	+	0.838047i	$0.316302\pi$
$692$	−14.2597	−0.542073
$693$	3.54635	0.134715
$694$	2.80103	0.106326
$695$	1.91213	0.0725314
$696$	−4.18930	−0.158795
$697$	79.3987	3.00744
$698$	−13.6847	−0.517972
$699$	2.13736	0.0808423
$700$	7.81742	0.295471
$701$	−49.5192	−1.87032	−0.935158	−	0.354231i	$-0.884743\pi$
$701$	−49.5192	−1.87032	−0.935158	+	0.354231i	$0.884743\pi$
$702$	2.49096	0.0940151
$703$	15.7603	0.594410
$704$	−2.21989	−0.0836651
$705$	2.69834	0.101625
$706$	−19.7430	−0.743038
$707$	19.6180	0.737809
$708$	0.283381	0.0106501
$709$	−11.5396	−0.433380	−0.216690	−	0.976240i	$-0.569526\pi$
$709$	−11.5396	−0.433380	−0.216690	+	0.976240i	$0.569526\pi$
$710$	−1.34550	−0.0504958
$711$	−0.116565	−0.00437155
$712$	−0.513252	−0.0192349
$713$	16.5160	0.618528
$714$	−10.3207	−0.386242
$715$	−1.80524	−0.0675123
$716$	24.5192	0.916327
$717$	−0.112547	−0.00420314
$718$	−9.56526	−0.356972
$719$	−4.75550	−0.177350	−0.0886750	−	0.996061i	$-0.528263\pi$
$719$	−4.75550	−0.177350	−0.0886750	+	0.996061i	$0.528263\pi$
$720$	0.326467	0.0121667
$721$	31.0727	1.15721
$722$	−39.7945	−1.48100
$723$	−13.8957	−0.516788
$724$	−18.2125	−0.676864
$725$	20.5000	0.761351
$726$	−6.07211	−0.225357
$727$	11.4025	0.422894	0.211447	−	0.977389i	$-0.432182\pi$
$727$	11.4025	0.422894	0.211447	+	0.977389i	$0.432182\pi$
$728$	3.97940	0.147486
$729$	1.00000	0.0370370
$730$	−2.51137	−0.0929499
$731$	−57.4548	−2.12504
$732$	2.94460	0.108836
$733$	4.38123	0.161824	0.0809122	−	0.996721i	$-0.474217\pi$
$733$	4.38123	0.161824	0.0809122	+	0.996721i	$0.474217\pi$
$734$	24.7755	0.914480
$735$	1.45208	0.0535608
$736$	−2.91402	−0.107412
$737$	12.5946	0.463928
$738$	−12.2901	−0.452405
$739$	19.2143	0.706810	0.353405	−	0.935471i	$-0.385024\pi$
$739$	19.2143	0.706810	0.353405	+	0.935471i	$0.385024\pi$
$740$	−0.671018	−0.0246671
$741$	19.1001	0.701658
$742$	−9.92055	−0.364195
$743$	11.5536	0.423862	0.211931	−	0.977285i	$-0.432025\pi$
$743$	11.5536	0.423862	0.211931	+	0.977285i	$0.432025\pi$
$744$	5.66776	0.207790
$745$	−5.14465	−0.188485
$746$	−35.7002	−1.30708
$747$	−7.08850	−0.259354
$748$	−14.3413	−0.524370
$749$	9.84128	0.359593
$750$	−3.22987	−0.117938
$751$	−28.3814	−1.03565	−0.517827	−	0.855486i	$-0.673259\pi$
$751$	−28.3814	−1.03565	−0.517827	+	0.855486i	$0.673259\pi$
$752$	−8.26530	−0.301404
$753$	−1.18698	−0.0432558
$754$	10.4354	0.380034
$755$	−2.54077	−0.0924680
$756$	1.59754	0.0581019
$757$	−2.52311	−0.0917039	−0.0458520	−	0.998948i	$-0.514600\pi$
$757$	−2.52311	−0.0917039	−0.0458520	+	0.998948i	$0.514600\pi$
$758$	14.3802	0.522311
$759$	6.46879	0.234802
$760$	2.50327	0.0908030
$761$	−38.2137	−1.38525	−0.692623	−	0.721300i	$-0.743545\pi$
$761$	−38.2137	−1.38525	−0.692623	+	0.721300i	$0.743545\pi$
$762$	−17.9861	−0.651569
$763$	15.4646	0.559855
$764$	21.0644	0.762085
$765$	2.10910	0.0762545
$766$	30.9671	1.11889
$767$	−0.705890	−0.0254882
$768$	−1.00000	−0.0360844
$769$	−18.2995	−0.659898	−0.329949	−	0.943999i	$-0.607032\pi$
$769$	−18.2995	−0.659898	−0.329949	+	0.943999i	$0.607032\pi$
$770$	−1.15777	−0.0417230
$771$	9.49749	0.342044
$772$	−27.0985	−0.975296
$773$	−14.7714	−0.531290	−0.265645	−	0.964071i	$-0.585585\pi$
$773$	−14.7714	−0.531290	−0.265645	+	0.964071i	$0.585585\pi$
$774$	8.89342	0.319667
$775$	−27.7347	−0.996260
$776$	6.38186	0.229095
$777$	−3.28357	−0.117797
$778$	−8.50767	−0.305015
$779$	−94.2376	−3.37641
$780$	−0.813215	−0.0291178
$781$	−9.14905	−0.327379
$782$	−18.8257	−0.673204
$783$	4.18930	0.149713
$784$	−4.44787	−0.158853
$785$	1.15104	0.0410824
$786$	7.32898	0.261416
$787$	−3.28144	−0.116971	−0.0584853	−	0.998288i	$-0.518627\pi$
$787$	−3.28144	−0.116971	−0.0584853	+	0.998288i	$0.518627\pi$
$788$	17.7547	0.632485
$789$	20.2440	0.720704
$790$	0.0380547	0.00135393
$791$	−6.06928	−0.215799
$792$	2.21989	0.0788802
$793$	−7.33489	−0.260469
$794$	14.2651	0.506250
$795$	2.02733	0.0719018
$796$	20.2190	0.716642
$797$	15.6573	0.554612	0.277306	−	0.960782i	$-0.410558\pi$
$797$	15.6573	0.554612	0.277306	+	0.960782i	$0.410558\pi$
$798$	12.2495	0.433629
$799$	−53.3969	−1.88905
$800$	4.89342	0.173009
$801$	0.513252	0.0181349
$802$	−31.5109	−1.11269
$803$	−17.0766	−0.602621
$804$	5.67353	0.200090
$805$	−1.51979	−0.0535654
$806$	−14.1181	−0.497290
$807$	9.55885	0.336488
$808$	12.2801	0.432013
$809$	−31.0320	−1.09103	−0.545513	−	0.838102i	$-0.683665\pi$
$809$	−31.0320	−1.09103	−0.545513	+	0.838102i	$0.683665\pi$
$810$	−0.326467	−0.0114709
$811$	22.1470	0.777688	0.388844	−	0.921304i	$-0.372875\pi$
$811$	22.1470	0.777688	0.388844	+	0.921304i	$0.372875\pi$
$812$	6.69257	0.234863
$813$	−24.0547	−0.843633
$814$	−4.56274	−0.159924
$815$	0.0202802	0.000710385 0
$816$	−6.46037	−0.226158
$817$	68.1926	2.38576
$818$	9.16293	0.320374
$819$	−3.97940	−0.139051
$820$	4.01231	0.140116
$821$	−7.24105	−0.252715	−0.126357	−	0.991985i	$-0.540329\pi$
$821$	−7.24105	−0.252715	−0.126357	+	0.991985i	$0.540329\pi$
$822$	−8.66776	−0.302323
$823$	12.0581	0.420319	0.210160	−	0.977667i	$-0.432602\pi$
$823$	12.0581	0.420319	0.210160	+	0.977667i	$0.432602\pi$
$824$	19.4504	0.677586
$825$	−10.8628	−0.378195
$826$	−0.452712	−0.0157519
$827$	28.5104	0.991403	0.495701	−	0.868493i	$-0.334911\pi$
$827$	28.5104	0.991403	0.495701	+	0.868493i	$0.334911\pi$
$828$	2.91402	0.101269
$829$	9.20256	0.319618	0.159809	−	0.987148i	$-0.448912\pi$
$829$	9.20256	0.319618	0.159809	+	0.987148i	$0.448912\pi$
$830$	2.31416	0.0803256
$831$	−30.8702	−1.07088
$832$	2.49096	0.0863584
$833$	−28.7349	−0.995606
$834$	5.85706	0.202813
$835$	0.681824	0.0235955
$836$	17.0216	0.588703
$837$	−5.66776	−0.195906
$838$	29.4201	1.01630
$839$	−16.9775	−0.586129	−0.293064	−	0.956093i	$-0.594675\pi$
$839$	−16.9775	−0.586129	−0.293064	+	0.956093i	$0.594675\pi$
$840$	−0.521543	−0.0179949
$841$	−11.4498	−0.394819
$842$	−28.8396	−0.993878
$843$	−6.78665	−0.233745
$844$	−11.3049	−0.389132
$845$	−2.21838	−0.0763147
$846$	8.26530	0.284167
$847$	9.70042	0.333310
$848$	−6.20990	−0.213249
$849$	−12.2466	−0.420302
$850$	31.6133	1.08433
$851$	−5.98946	−0.205316
$852$	−4.12141	−0.141197
$853$	26.0908	0.893333	0.446666	−	0.894701i	$-0.352611\pi$
$853$	26.0908	0.893333	0.446666	+	0.894701i	$0.352611\pi$
$854$	−4.70412	−0.160972
$855$	−2.50327	−0.0856099
$856$	6.16028	0.210554
$857$	−14.7870	−0.505115	−0.252558	−	0.967582i	$-0.581272\pi$
$857$	−14.7870	−0.505115	−0.252558	+	0.967582i	$0.581272\pi$
$858$	−5.52964	−0.188779
$859$	25.5670	0.872334	0.436167	−	0.899866i	$-0.356336\pi$
$859$	25.5670	0.872334	0.436167	+	0.899866i	$0.356336\pi$
$860$	−2.90340	−0.0990053
$861$	19.6339	0.669122
$862$	−29.9517	−1.02016
$863$	44.4272	1.51232	0.756159	−	0.654388i	$-0.227074\pi$
$863$	44.4272	1.51232	0.756159	+	0.654388i	$0.227074\pi$
$864$	1.00000	0.0340207
$865$	−4.65532	−0.158286
$866$	11.1833	0.380025
$867$	−24.7364	−0.840093
$868$	−9.05446	−0.307328
$869$	0.258762	0.00877790
$870$	−1.36767	−0.0463683
$871$	−14.1325	−0.478863
$872$	9.68026	0.327815
$873$	−6.38186	−0.215993
$874$	22.3440	0.755797
$875$	5.15984	0.174435
$876$	−7.69257	−0.259908
$877$	−11.9340	−0.402983	−0.201491	−	0.979490i	$-0.564579\pi$
$877$	−11.9340	−0.402983	−0.201491	+	0.979490i	$0.564579\pi$
$878$	30.8644	1.04162
$879$	10.7176	0.361495
$880$	−0.724719	−0.0244303
$881$	34.8986	1.17576	0.587882	−	0.808947i	$-0.299962\pi$
$881$	34.8986	1.17576	0.587882	+	0.808947i	$0.299962\pi$
$882$	4.44787	0.149768
$883$	−37.8330	−1.27318	−0.636591	−	0.771202i	$-0.719656\pi$
$883$	−37.8330	−1.27318	−0.636591	+	0.771202i	$0.719656\pi$
$884$	16.0925	0.541250
$885$	0.0925144	0.00310984
$886$	35.8542	1.20455
$887$	−41.1789	−1.38265	−0.691326	−	0.722543i	$-0.742973\pi$
$887$	−41.1789	−1.38265	−0.691326	+	0.722543i	$0.742973\pi$
$888$	−2.05540	−0.0689745
$889$	28.7335	0.963691
$890$	−0.167560	−0.00561661
$891$	−2.21989	−0.0743690
$892$	−1.00000	−0.0334825
$893$	63.3763	2.12081
$894$	−15.7586	−0.527046
$895$	8.00471	0.267568
$896$	1.59754	0.0533700
$897$	−7.25870	−0.242361
$898$	15.7066	0.524137
$899$	−23.7439	−0.791905
$900$	−4.89342	−0.163114
$901$	−40.1183	−1.33653
$902$	27.2826	0.908412
$903$	−14.2076	−0.472799
$904$	−3.79915	−0.126358
$905$	−5.94579	−0.197645
$906$	−7.78263	−0.258561
$907$	−53.0797	−1.76248	−0.881241	−	0.472668i	$-0.843291\pi$
$907$	−53.0797	−1.76248	−0.881241	+	0.472668i	$0.843291\pi$
$908$	18.8653	0.626069
$909$	−12.2801	−0.407306
$910$	1.29914	0.0430661
$911$	40.1408	1.32992	0.664961	−	0.746878i	$-0.268448\pi$
$911$	40.1408	1.32992	0.664961	+	0.746878i	$0.268448\pi$
$912$	7.66776	0.253905
$913$	15.7357	0.520774
$914$	29.0459	0.960754
$915$	0.961315	0.0317801
$916$	25.1506	0.831000
$917$	−11.7083	−0.386643
$918$	6.46037	0.213224
$919$	−23.1072	−0.762237	−0.381118	−	0.924526i	$-0.624461\pi$
$919$	−23.1072	−0.762237	−0.381118	+	0.924526i	$0.624461\pi$
$920$	−0.951330	−0.0313644
$921$	−6.39669	−0.210778
$922$	−21.5743	−0.710512
$923$	10.2662	0.337918
$924$	−3.54635	−0.116666
$925$	10.0579	0.330702
$926$	−24.3686	−0.800802
$927$	−19.4504	−0.638835
$928$	4.18930	0.137521
$929$	−1.79556	−0.0589105	−0.0294553	−	0.999566i	$-0.509377\pi$
$929$	−1.79556	−0.0589105	−0.0294553	+	0.999566i	$0.509377\pi$
$930$	1.85033	0.0606748
$931$	34.1052	1.11775
$932$	−2.13736	−0.0700115
$933$	6.95535	0.227708
$934$	32.4960	1.06330
$935$	−4.68195	−0.153116
$936$	−2.49096	−0.0814195
$937$	−9.00535	−0.294192	−0.147096	−	0.989122i	$-0.546993\pi$
$937$	−9.00535	−0.294192	−0.147096	+	0.989122i	$0.546993\pi$
$938$	−9.06369	−0.295940
$939$	−6.45365	−0.210607
$940$	−2.69834	−0.0880103
$941$	−13.2698	−0.432584	−0.216292	−	0.976329i	$-0.569396\pi$
$941$	−13.2698	−0.432584	−0.216292	+	0.976329i	$0.569396\pi$
$942$	3.52575	0.114875
$943$	35.8136	1.16625
$944$	−0.283381	−0.00922326
$945$	0.521543	0.0169658
$946$	−19.7424	−0.641880
$947$	38.7970	1.26073	0.630366	−	0.776298i	$-0.282905\pi$
$947$	38.7970	1.26073	0.630366	+	0.776298i	$0.282905\pi$
$948$	0.116565	0.00378587
$949$	19.1619	0.622020
$950$	−37.5216	−1.21736
$951$	−5.56695	−0.180521
$952$	10.3207	0.334496
$953$	57.9423	1.87694	0.938468	−	0.345367i	$-0.112246\pi$
$953$	57.9423	1.87694	0.938468	+	0.345367i	$0.112246\pi$
$954$	6.20990	0.201053
$955$	6.87684	0.222529
$956$	0.112547	0.00364003
$957$	−9.29977	−0.300619
$958$	−34.9175	−1.12813
$959$	13.8471	0.447145
$960$	−0.326467	−0.0105367
$961$	1.12348	0.0362414
$962$	5.11990	0.165072
$963$	−6.16028	−0.198512
$964$	13.8957	0.447552
$965$	−8.84675	−0.284787
$966$	−4.65526	−0.149780
$967$	35.0066	1.12574	0.562868	−	0.826547i	$-0.309698\pi$
$967$	35.0066	1.12574	0.562868	+	0.826547i	$0.309698\pi$
$968$	6.07211	0.195165
$969$	49.5366	1.59134
$970$	2.08346	0.0668960
$971$	31.1405	0.999345	0.499672	−	0.866214i	$-0.333454\pi$
$971$	31.1405	0.999345	0.499672	+	0.866214i	$0.333454\pi$
$972$	−1.00000	−0.0320750
$973$	−9.35688	−0.299968
$974$	6.46037	0.207004
$975$	12.1893	0.390370
$976$	−2.94460	−0.0942545
$977$	−9.44587	−0.302200	−0.151100	−	0.988518i	$-0.548282\pi$
$977$	−9.44587	−0.302200	−0.151100	+	0.988518i	$0.548282\pi$
$978$	0.0621204	0.00198639
$979$	−1.13936	−0.0364141
$980$	−1.45208	−0.0463850
$981$	−9.68026	−0.309067
$982$	0.708520	0.0226098
$983$	−20.5690	−0.656051	−0.328025	−	0.944669i	$-0.606383\pi$
$983$	−20.5690	−0.656051	−0.328025	+	0.944669i	$0.606383\pi$
$984$	12.2901	0.391794
$985$	5.79631	0.184686
$986$	27.0644	0.861908
$987$	−13.2041	−0.420292
$988$	−19.1001	−0.607654
$989$	−25.9156	−0.824068
$990$	0.724719	0.0230331
$991$	−7.89877	−0.250913	−0.125456	−	0.992099i	$-0.540040\pi$
$991$	−7.89877	−0.250913	−0.125456	+	0.992099i	$0.540040\pi$
$992$	−5.66776	−0.179951
$993$	24.9131	0.790593
$994$	6.58410	0.208835
$995$	6.60081	0.209260
$996$	7.08850	0.224608
$997$	6.14766	0.194698	0.0973492	−	0.995250i	$-0.468964\pi$
$997$	6.14766	0.194698	0.0973492	+	0.995250i	$0.468964\pi$
$998$	−20.2230	−0.640147
$999$	2.05540	0.0650298

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1338.2.a.g.1.2	✓	4
3.2	odd	2		4014.2.a.q.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1338.2.a.g.1.2	✓	4	1.1	even	1	trivial
4014.2.a.q.1.3		4	3.2	odd	2

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Classical	Maass
Hilbert	Bianchi

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

Introduction

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Modular forms

Varieties

Fields

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Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

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

Label	1338.2.a.g.1.2

Level	$1338$
Weight	$2$
Character	1338.1
Self dual	yes
Analytic conductor	$10.684$
Analytic rank	$1$
Dimension	$4$
CM	no
Inner twists	$1$