LMFDB - Embedded newform 1850.2.a.bd.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(1,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, 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("1850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1850 = 2 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1850.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:	$14.7723243739$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1791440.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 9x^{3} + 13x - 4 $ x^5 - 9x^3 + 13x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 370)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.62545$ of defining polynomial
Character	$\chi$	$=$	1850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.62545 q^{3} +1.00000 q^{4} +2.62545 q^{6} +1.83227 q^{7} -1.00000 q^{8} +3.89300 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.62545 q^{3} +1.00000 q^{4} +2.62545 q^{6} +1.83227 q^{7} -1.00000 q^{8} +3.89300 q^{9} +4.19017 q^{11} -2.62545 q^{12} +0.369454 q^{13} -1.83227 q^{14} +1.00000 q^{16} +5.08317 q^{17} -3.89300 q^{18} +3.55963 q^{19} -4.81053 q^{21} -4.19017 q^{22} -5.62036 q^{23} +2.62545 q^{24} -0.369454 q^{26} -2.34453 q^{27} +1.83227 q^{28} +1.20681 q^{29} +10.1030 q^{31} -1.00000 q^{32} -11.0011 q^{33} -5.08317 q^{34} +3.89300 q^{36} -1.00000 q^{37} -3.55963 q^{38} -0.969984 q^{39} -8.01447 q^{41} +4.81053 q^{42} +2.27264 q^{43} +4.19017 q^{44} +5.62036 q^{46} -10.9154 q^{47} -2.62545 q^{48} -3.64280 q^{49} -13.3456 q^{51} +0.369454 q^{52} +9.94355 q^{53} +2.34453 q^{54} -1.83227 q^{56} -9.34563 q^{57} -1.20681 q^{58} -5.34563 q^{59} -9.79428 q^{61} -10.1030 q^{62} +7.13302 q^{63} +1.00000 q^{64} +11.0011 q^{66} -1.85073 q^{67} +5.08317 q^{68} +14.7560 q^{69} +2.86038 q^{71} -3.89300 q^{72} +8.09942 q^{73} +1.00000 q^{74} +3.55963 q^{76} +7.67751 q^{77} +0.969984 q^{78} +6.06361 q^{79} -5.52355 q^{81} +8.01447 q^{82} +8.93200 q^{83} -4.81053 q^{84} -2.27264 q^{86} -3.16843 q^{87} -4.19017 q^{88} +11.4773 q^{89} +0.676938 q^{91} -5.62036 q^{92} -26.5249 q^{93} +10.9154 q^{94} +2.62545 q^{96} -6.05864 q^{97} +3.64280 q^{98} +16.3123 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9}+O(q^{10}) $ 5 * q - 5 * q^2 + 5 * q^4 + q^7 - 5 * q^8 + 3 * q^9	$ 5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9} + 3 q^{11} + 6 q^{13} - q^{14} + 5 q^{16} - 9 q^{17} - 3 q^{18} + 4 q^{19} + 16 q^{21} - 3 q^{22} - 6 q^{23} - 6 q^{26} + q^{28} + 11 q^{29} + 23 q^{31} - 5 q^{32} - 20 q^{33} + 9 q^{34} + 3 q^{36} - 5 q^{37} - 4 q^{38} + 20 q^{39} - 7 q^{41} - 16 q^{42} + 17 q^{43} + 3 q^{44} + 6 q^{46} - 12 q^{47} + 30 q^{49} - 20 q^{51} + 6 q^{52} + 7 q^{53} - q^{56} - 11 q^{58} + 20 q^{59} - 9 q^{61} - 23 q^{62} + 33 q^{63} + 5 q^{64} + 20 q^{66} - 12 q^{67} - 9 q^{68} + 16 q^{69} + 6 q^{71} - 3 q^{72} + 6 q^{73} + 5 q^{74} + 4 q^{76} + q^{77} - 20 q^{78} + 20 q^{79} - 7 q^{81} + 7 q^{82} - 12 q^{83} + 16 q^{84} - 17 q^{86} + 34 q^{87} - 3 q^{88} + 12 q^{89} + 16 q^{91} - 6 q^{92} + 4 q^{93} + 12 q^{94} - 3 q^{97} - 30 q^{98} - 11 q^{99}+O(q^{100}) $ 5 * q - 5 * q^2 + 5 * q^4 + q^7 - 5 * q^8 + 3 * q^9 + 3 * q^11 + 6 * q^13 - q^14 + 5 * q^16 - 9 * q^17 - 3 * q^18 + 4 * q^19 + 16 * q^21 - 3 * q^22 - 6 * q^23 - 6 * q^26 + q^28 + 11 * q^29 + 23 * q^31 - 5 * q^32 - 20 * q^33 + 9 * q^34 + 3 * q^36 - 5 * q^37 - 4 * q^38 + 20 * q^39 - 7 * q^41 - 16 * q^42 + 17 * q^43 + 3 * q^44 + 6 * q^46 - 12 * q^47 + 30 * q^49 - 20 * q^51 + 6 * q^52 + 7 * q^53 - q^56 - 11 * q^58 + 20 * q^59 - 9 * q^61 - 23 * q^62 + 33 * q^63 + 5 * q^64 + 20 * q^66 - 12 * q^67 - 9 * q^68 + 16 * q^69 + 6 * q^71 - 3 * q^72 + 6 * q^73 + 5 * q^74 + 4 * q^76 + q^77 - 20 * q^78 + 20 * q^79 - 7 * q^81 + 7 * q^82 - 12 * q^83 + 16 * q^84 - 17 * q^86 + 34 * q^87 - 3 * q^88 + 12 * q^89 + 16 * q^91 - 6 * q^92 + 4 * q^93 + 12 * q^94 - 3 * q^97 - 30 * q^98 - 11 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.62545	−1.51581	−0.757903	−	0.652367i	$-0.773776\pi$
$3$	−2.62545	−1.51581	−0.757903	+	0.652367i	$0.773776\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	2.62545	1.07184
$7$	1.83227	0.692532	0.346266	−	0.938136i	$-0.387449\pi$
$7$	1.83227	0.692532	0.346266	+	0.938136i	$0.387449\pi$
$8$	−1.00000	−0.353553
$9$	3.89300	1.29767
$10$	0	0
$11$	4.19017	1.26338	0.631692	−	0.775219i	$-0.282361\pi$
$11$	4.19017	1.26338	0.631692	+	0.775219i	$0.282361\pi$
$12$	−2.62545	−0.757903
$13$	0.369454	0.102468	0.0512340	−	0.998687i	$-0.483685\pi$
$13$	0.369454	0.102468	0.0512340	+	0.998687i	$0.483685\pi$
$14$	−1.83227	−0.489694
$15$	0	0
$16$	1.00000	0.250000
$17$	5.08317	1.23285	0.616425	−	0.787414i	$-0.288580\pi$
$17$	5.08317	1.23285	0.616425	+	0.787414i	$0.288580\pi$
$18$	−3.89300	−0.917589
$19$	3.55963	0.816634	0.408317	−	0.912840i	$-0.366116\pi$
$19$	3.55963	0.816634	0.408317	+	0.912840i	$0.366116\pi$
$20$	0	0
$21$	−4.81053	−1.04974
$22$	−4.19017	−0.893348
$23$	−5.62036	−1.17193	−0.585963	−	0.810338i	$-0.699284\pi$
$23$	−5.62036	−1.17193	−0.585963	+	0.810338i	$0.699284\pi$
$24$	2.62545	0.535918
$25$	0	0
$26$	−0.369454	−0.0724559
$27$	−2.34453	−0.451205
$28$	1.83227	0.346266
$29$	1.20681	0.224100	0.112050	−	0.993703i	$-0.464258\pi$
$29$	1.20681	0.224100	0.112050	+	0.993703i	$0.464258\pi$
$30$	0	0
$31$	10.1030	1.81455	0.907276	−	0.420535i	$-0.138158\pi$
$31$	10.1030	1.81455	0.907276	+	0.420535i	$0.138158\pi$
$32$	−1.00000	−0.176777
$33$	−11.0011	−1.91504
$34$	−5.08317	−0.871757
$35$	0	0
$36$	3.89300	0.648833
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−3.55963	−0.577447
$39$	−0.969984	−0.155322
$40$	0	0
$41$	−8.01447	−1.25165	−0.625825	−	0.779964i	$-0.715238\pi$
$41$	−8.01447	−1.25165	−0.625825	+	0.779964i	$0.715238\pi$
$42$	4.81053	0.742281
$43$	2.27264	0.346575	0.173287	−	0.984871i	$-0.444561\pi$
$43$	2.27264	0.346575	0.173287	+	0.984871i	$0.444561\pi$
$44$	4.19017	0.631692
$45$	0	0
$46$	5.62036	0.828677
$47$	−10.9154	−1.59218	−0.796090	−	0.605178i	$-0.793102\pi$
$47$	−10.9154	−1.59218	−0.796090	+	0.605178i	$0.793102\pi$
$48$	−2.62545	−0.378951
$49$	−3.64280	−0.520400
$50$	0	0
$51$	−13.3456	−1.86876
$52$	0.369454	0.0512340
$53$	9.94355	1.36585	0.682926	−	0.730488i	$-0.260707\pi$
$53$	9.94355	1.36585	0.682926	+	0.730488i	$0.260707\pi$
$54$	2.34453	0.319050
$55$	0	0
$56$	−1.83227	−0.244847
$57$	−9.34563	−1.23786
$58$	−1.20681	−0.158463
$59$	−5.34563	−0.695941	−0.347971	−	0.937505i	$-0.613129\pi$
$59$	−5.34563	−0.695941	−0.347971	+	0.937505i	$0.613129\pi$
$60$	0	0
$61$	−9.79428	−1.25403	−0.627015	−	0.779008i	$-0.715723\pi$
$61$	−9.79428	−1.25403	−0.627015	+	0.779008i	$0.715723\pi$
$62$	−10.1030	−1.28308
$63$	7.13302	0.898676
$64$	1.00000	0.125000
$65$	0	0
$66$	11.0011	1.35414
$67$	−1.85073	−0.226103	−0.113052	−	0.993589i	$-0.536063\pi$
$67$	−1.85073	−0.226103	−0.113052	+	0.993589i	$0.536063\pi$
$68$	5.08317	0.616425
$69$	14.7560	1.77641
$70$	0	0
$71$	2.86038	0.339464	0.169732	−	0.985490i	$-0.445710\pi$
$71$	2.86038	0.339464	0.169732	+	0.985490i	$0.445710\pi$
$72$	−3.89300	−0.458795
$73$	8.09942	0.947966	0.473983	−	0.880534i	$-0.342816\pi$
$73$	8.09942	0.947966	0.473983	+	0.880534i	$0.342816\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	3.55963	0.408317
$77$	7.67751	0.874934
$78$	0.969984	0.109829
$79$	6.06361	0.682209	0.341105	−	0.940025i	$-0.389199\pi$
$79$	6.06361	0.682209	0.341105	+	0.940025i	$0.389199\pi$
$80$	0	0
$81$	−5.52355	−0.613727
$82$	8.01447	0.885050
$83$	8.93200	0.980414	0.490207	−	0.871606i	$-0.336921\pi$
$83$	8.93200	0.980414	0.490207	+	0.871606i	$0.336921\pi$
$84$	−4.81053	−0.524872
$85$	0	0
$86$	−2.27264	−0.245065
$87$	−3.16843	−0.339692
$88$	−4.19017	−0.446674
$89$	11.4773	1.21659	0.608295	−	0.793711i	$-0.291854\pi$
$89$	11.4773	1.21659	0.608295	+	0.793711i	$0.291854\pi$
$90$	0	0
$91$	0.676938	0.0709624
$92$	−5.62036	−0.585963
$93$	−26.5249	−2.75051
$94$	10.9154	1.12584
$95$	0	0
$96$	2.62545	0.267959
$97$	−6.05864	−0.615162	−0.307581	−	0.951522i	$-0.599520\pi$
$97$	−6.05864	−0.615162	−0.307581	+	0.951522i	$0.599520\pi$
$98$	3.64280	0.367978
$99$	16.3123	1.63945
$100$	0	0
$101$	4.96528	0.494064	0.247032	−	0.969007i	$-0.420545\pi$
$101$	4.96528	0.494064	0.247032	+	0.969007i	$0.420545\pi$
$102$	13.3456	1.32141
$103$	7.33338	0.722579	0.361289	−	0.932454i	$-0.382337\pi$
$103$	7.33338	0.722579	0.361289	+	0.932454i	$0.382337\pi$
$104$	−0.369454	−0.0362279
$105$	0	0
$106$	−9.94355	−0.965803
$107$	8.21182	0.793867	0.396933	−	0.917847i	$-0.370074\pi$
$107$	8.21182	0.793867	0.396933	+	0.917847i	$0.370074\pi$
$108$	−2.34453	−0.225603
$109$	−20.5503	−1.96836	−0.984179	−	0.177175i	$-0.943304\pi$
$109$	−20.5503	−1.96836	−0.984179	+	0.177175i	$0.943304\pi$
$110$	0	0
$111$	2.62545	0.249197
$112$	1.83227	0.173133
$113$	0.652984	0.0614276	0.0307138	−	0.999528i	$-0.490222\pi$
$113$	0.652984	0.0614276	0.0307138	+	0.999528i	$0.490222\pi$
$114$	9.34563	0.875298
$115$	0	0
$116$	1.20681	0.112050
$117$	1.43828	0.132969
$118$	5.34563	0.492105
$119$	9.31373	0.853788
$120$	0	0
$121$	6.55754	0.596140
$122$	9.79428	0.886733
$123$	21.0416	1.89726
$124$	10.1030	0.907276
$125$	0	0
$126$	−7.13302	−0.635460
$127$	19.2856	1.71132	0.855660	−	0.517539i	$-0.173152\pi$
$127$	19.2856	1.71132	0.855660	+	0.517539i	$0.173152\pi$
$128$	−1.00000	−0.0883883
$129$	−5.96671	−0.525340
$130$	0	0
$131$	−8.82072	−0.770670	−0.385335	−	0.922777i	$-0.625914\pi$
$131$	−8.82072	−0.770670	−0.385335	+	0.922777i	$0.625914\pi$
$132$	−11.0011	−0.957522
$133$	6.52218	0.565545
$134$	1.85073	0.159879
$135$	0	0
$136$	−5.08317	−0.435878
$137$	−4.66126	−0.398239	−0.199119	−	0.979975i	$-0.563808\pi$
$137$	−4.66126	−0.398239	−0.199119	+	0.979975i	$0.563808\pi$
$138$	−14.7560	−1.25611
$139$	16.4050	1.39145	0.695727	−	0.718306i	$-0.255082\pi$
$139$	16.4050	1.39145	0.695727	+	0.718306i	$0.255082\pi$
$140$	0	0
$141$	28.6580	2.41344
$142$	−2.86038	−0.240037
$143$	1.54808	0.129457
$144$	3.89300	0.324417
$145$	0	0
$146$	−8.09942	−0.670313
$147$	9.56399	0.788825
$148$	−1.00000	−0.0821995
$149$	−5.86259	−0.480282	−0.240141	−	0.970738i	$-0.577194\pi$
$149$	−5.86259	−0.480282	−0.240141	+	0.970738i	$0.577194\pi$
$150$	0	0
$151$	13.1460	1.06981	0.534903	−	0.844913i	$-0.320348\pi$
$151$	13.1460	1.06981	0.534903	+	0.844913i	$0.320348\pi$
$152$	−3.55963	−0.288724
$153$	19.7888	1.59983
$154$	−7.67751	−0.618672
$155$	0	0
$156$	−0.969984	−0.0776609
$157$	7.33629	0.585500	0.292750	−	0.956189i	$-0.405430\pi$
$157$	7.33629	0.585500	0.292750	+	0.956189i	$0.405430\pi$
$158$	−6.06361	−0.482395
$159$	−26.1063	−2.07037
$160$	0	0
$161$	−10.2980	−0.811596
$162$	5.52355	0.433971
$163$	18.7968	1.47228	0.736138	−	0.676831i	$-0.236647\pi$
$163$	18.7968	1.47228	0.736138	+	0.676831i	$0.236647\pi$
$164$	−8.01447	−0.625825
$165$	0	0
$166$	−8.93200	−0.693257
$167$	5.92772	0.458700	0.229350	−	0.973344i	$-0.426340\pi$
$167$	5.92772	0.458700	0.229350	+	0.973344i	$0.426340\pi$
$168$	4.81053	0.371140
$169$	−12.8635	−0.989500
$170$	0	0
$171$	13.8576	1.05972
$172$	2.27264	0.173287
$173$	−5.78242	−0.439629	−0.219815	−	0.975542i	$-0.570545\pi$
$173$	−5.78242	−0.439629	−0.219815	+	0.975542i	$0.570545\pi$
$174$	3.16843	0.240198
$175$	0	0
$176$	4.19017	0.315846
$177$	14.0347	1.05491
$178$	−11.4773	−0.860259
$179$	7.13581	0.533355	0.266678	−	0.963786i	$-0.414074\pi$
$179$	7.13581	0.533355	0.266678	+	0.963786i	$0.414074\pi$
$180$	0	0
$181$	11.5365	0.857503	0.428752	−	0.903422i	$-0.358954\pi$
$181$	11.5365	0.857503	0.428752	+	0.903422i	$0.358954\pi$
$182$	−0.676938	−0.0501780
$183$	25.7144	1.90086
$184$	5.62036	0.414338
$185$	0	0
$186$	26.5249	1.94490
$187$	21.2994	1.55756
$188$	−10.9154	−0.796090
$189$	−4.29581	−0.312474
$190$	0	0
$191$	−20.6048	−1.49091	−0.745456	−	0.666555i	$-0.767768\pi$
$191$	−20.6048	−1.49091	−0.745456	+	0.666555i	$0.767768\pi$
$192$	−2.62545	−0.189476
$193$	26.1663	1.88349	0.941747	−	0.336321i	$-0.109183\pi$
$193$	26.1663	1.88349	0.941747	+	0.336321i	$0.109183\pi$
$194$	6.05864	0.434985
$195$	0	0
$196$	−3.64280	−0.260200
$197$	−14.6067	−1.04069	−0.520343	−	0.853957i	$-0.674196\pi$
$197$	−14.6067	−1.04069	−0.520343	+	0.853957i	$0.674196\pi$
$198$	−16.3123	−1.15927
$199$	5.81275	0.412055	0.206027	−	0.978546i	$-0.433946\pi$
$199$	5.81275	0.412055	0.206027	+	0.978546i	$0.433946\pi$
$200$	0	0
$201$	4.85901	0.342728
$202$	−4.96528	−0.349356
$203$	2.21121	0.155196
$204$	−13.3456	−0.934381
$205$	0	0
$206$	−7.33338	−0.510940
$207$	−21.8801	−1.52077
$208$	0.369454	0.0256170
$209$	14.9154	1.03172
$210$	0	0
$211$	16.0976	1.10821	0.554104	−	0.832448i	$-0.313061\pi$
$211$	16.0976	1.10821	0.554104	+	0.832448i	$0.313061\pi$
$212$	9.94355	0.682926
$213$	−7.50978	−0.514562
$214$	−8.21182	−0.561349
$215$	0	0
$216$	2.34453	0.159525
$217$	18.5114	1.25664
$218$	20.5503	1.39184
$219$	−21.2646	−1.43693
$220$	0	0
$221$	1.87800	0.126328
$222$	−2.62545	−0.176209
$223$	−25.3608	−1.69828	−0.849142	−	0.528164i	$-0.822881\pi$
$223$	−25.3608	−1.69828	−0.849142	+	0.528164i	$0.822881\pi$
$224$	−1.83227	−0.122423
$225$	0	0
$226$	−0.652984	−0.0434359
$227$	5.57339	0.369919	0.184960	−	0.982746i	$-0.440785\pi$
$227$	5.57339	0.369919	0.184960	+	0.982746i	$0.440785\pi$
$228$	−9.34563	−0.618929
$229$	9.02035	0.596081	0.298041	−	0.954553i	$-0.403667\pi$
$229$	9.02035	0.596081	0.298041	+	0.954553i	$0.403667\pi$
$230$	0	0
$231$	−20.1569	−1.32623
$232$	−1.20681	−0.0792313
$233$	−11.8100	−0.773696	−0.386848	−	0.922144i	$-0.626436\pi$
$233$	−11.8100	−0.773696	−0.386848	+	0.922144i	$0.626436\pi$
$234$	−1.43828	−0.0940236
$235$	0	0
$236$	−5.34563	−0.347971
$237$	−15.9197	−1.03410
$238$	−9.31373	−0.603719
$239$	0.907160	0.0586793	0.0293396	−	0.999569i	$-0.490660\pi$
$239$	0.907160	0.0586793	0.0293396	+	0.999569i	$0.490660\pi$
$240$	0	0
$241$	−4.15616	−0.267722	−0.133861	−	0.991000i	$-0.542738\pi$
$241$	−4.15616	−0.267722	−0.133861	+	0.991000i	$0.542738\pi$
$242$	−6.55754	−0.421534
$243$	21.5354	1.38150
$244$	−9.79428	−0.627015
$245$	0	0
$246$	−21.0416	−1.34156
$247$	1.31512	0.0836789
$248$	−10.1030	−0.641541
$249$	−23.4505	−1.48612
$250$	0	0
$251$	9.31888	0.588203	0.294101	−	0.955774i	$-0.404980\pi$
$251$	9.31888	0.588203	0.294101	+	0.955774i	$0.404980\pi$
$252$	7.13302	0.449338
$253$	−23.5503	−1.48059
$254$	−19.2856	−1.21009
$255$	0	0
$256$	1.00000	0.0625000
$257$	13.6312	0.850294	0.425147	−	0.905124i	$-0.360222\pi$
$257$	13.6312	0.850294	0.425147	+	0.905124i	$0.360222\pi$
$258$	5.96671	0.371471
$259$	−1.83227	−0.113852
$260$	0	0
$261$	4.69813	0.290807
$262$	8.82072	0.544946
$263$	−27.2170	−1.67827	−0.839137	−	0.543921i	$-0.816939\pi$
$263$	−27.2170	−1.67827	−0.839137	+	0.543921i	$0.816939\pi$
$264$	11.0011	0.677071
$265$	0	0
$266$	−6.52218	−0.399901
$267$	−30.1331	−1.84411
$268$	−1.85073	−0.113052
$269$	23.0550	1.40569	0.702845	−	0.711343i	$-0.251913\pi$
$269$	23.0550	1.40569	0.702845	+	0.711343i	$0.251913\pi$
$270$	0	0
$271$	22.9320	1.39302	0.696510	−	0.717547i	$-0.254735\pi$
$271$	22.9320	1.39302	0.696510	+	0.717547i	$0.254735\pi$
$272$	5.08317	0.308213
$273$	−1.77727	−0.107565
$274$	4.66126	0.281597
$275$	0	0
$276$	14.7560	0.888206
$277$	11.7534	0.706192	0.353096	−	0.935587i	$-0.385129\pi$
$277$	11.7534	0.706192	0.353096	+	0.935587i	$0.385129\pi$
$278$	−16.4050	−0.983906
$279$	39.3310	2.35468
$280$	0	0
$281$	−29.0962	−1.73573	−0.867865	−	0.496799i	$-0.834508\pi$
$281$	−29.0962	−1.73573	−0.867865	+	0.496799i	$0.834508\pi$
$282$	−28.6580	−1.70656
$283$	10.1829	0.605311	0.302655	−	0.953100i	$-0.402127\pi$
$283$	10.1829	0.605311	0.302655	+	0.953100i	$0.402127\pi$
$284$	2.86038	0.169732
$285$	0	0
$286$	−1.54808	−0.0915396
$287$	−14.6846	−0.866807
$288$	−3.89300	−0.229397
$289$	8.83864	0.519920
$290$	0	0
$291$	15.9067	0.932466
$292$	8.09942	0.473983
$293$	−18.1243	−1.05883	−0.529415	−	0.848363i	$-0.677589\pi$
$293$	−18.1243	−1.05883	−0.529415	+	0.848363i	$0.677589\pi$
$294$	−9.56399	−0.557783
$295$	0	0
$296$	1.00000	0.0581238
$297$	−9.82399	−0.570046
$298$	5.86259	0.339611
$299$	−2.07646	−0.120085
$300$	0	0
$301$	4.16409	0.240014
$302$	−13.1460	−0.756467
$303$	−13.0361	−0.748905
$304$	3.55963	0.204159
$305$	0	0
$306$	−19.7888	−1.13125
$307$	6.15561	0.351319	0.175660	−	0.984451i	$-0.443794\pi$
$307$	6.15561	0.351319	0.175660	+	0.984451i	$0.443794\pi$
$308$	7.67751	0.437467
$309$	−19.2534	−1.09529
$310$	0	0
$311$	−29.4512	−1.67002	−0.835011	−	0.550234i	$-0.814539\pi$
$311$	−29.4512	−1.67002	−0.835011	+	0.550234i	$0.814539\pi$
$312$	0.969984	0.0549145
$313$	−13.0102	−0.735378	−0.367689	−	0.929949i	$-0.619851\pi$
$313$	−13.0102	−0.735378	−0.367689	+	0.929949i	$0.619851\pi$
$314$	−7.33629	−0.414011
$315$	0	0
$316$	6.06361	0.341105
$317$	7.39826	0.415528	0.207764	−	0.978179i	$-0.433381\pi$
$317$	7.39826	0.415528	0.207764	+	0.978179i	$0.433381\pi$
$318$	26.1063	1.46397
$319$	5.05676	0.283124
$320$	0	0
$321$	−21.5598	−1.20335
$322$	10.2980	0.573885
$323$	18.0942	1.00679
$324$	−5.52355	−0.306864
$325$	0	0
$326$	−18.7968	−1.04106
$327$	53.9537	2.98365
$328$	8.01447	0.442525
$329$	−20.0000	−1.10264
$330$	0	0
$331$	33.2545	1.82783	0.913916	−	0.405903i	$-0.133043\pi$
$331$	33.2545	1.82783	0.913916	+	0.405903i	$0.133043\pi$
$332$	8.93200	0.490207
$333$	−3.89300	−0.213335
$334$	−5.92772	−0.324350
$335$	0	0
$336$	−4.81053	−0.262436
$337$	−22.6288	−1.23267	−0.616334	−	0.787485i	$-0.711383\pi$
$337$	−22.6288	−1.23267	−0.616334	+	0.787485i	$0.711383\pi$
$338$	12.8635	0.699682
$339$	−1.71438	−0.0931123
$340$	0	0
$341$	42.3333	2.29248
$342$	−13.8576	−0.749334
$343$	−19.5004	−1.05293
$344$	−2.27264	−0.122533
$345$	0	0
$346$	5.78242	0.310865
$347$	24.1396	1.29588	0.647941	−	0.761691i	$-0.275630\pi$
$347$	24.1396	1.29588	0.647941	+	0.761691i	$0.275630\pi$
$348$	−3.16843	−0.169846
$349$	−10.3702	−0.555102	−0.277551	−	0.960711i	$-0.589523\pi$
$349$	−10.3702	−0.555102	−0.277551	+	0.960711i	$0.589523\pi$
$350$	0	0
$351$	−0.866196	−0.0462341
$352$	−4.19017	−0.223337
$353$	3.92699	0.209013	0.104506	−	0.994524i	$-0.466674\pi$
$353$	3.92699	0.209013	0.104506	+	0.994524i	$0.466674\pi$
$354$	−14.0347	−0.745935
$355$	0	0
$356$	11.4773	0.608295
$357$	−24.4528	−1.29418
$358$	−7.13581	−0.377139
$359$	−9.85179	−0.519957	−0.259979	−	0.965614i	$-0.583716\pi$
$359$	−9.85179	−0.519957	−0.259979	+	0.965614i	$0.583716\pi$
$360$	0	0
$361$	−6.32907	−0.333109
$362$	−11.5365	−0.606346
$363$	−17.2165	−0.903632
$364$	0.676938	0.0354812
$365$	0	0
$366$	−25.7144	−1.34411
$367$	−33.2559	−1.73594	−0.867972	−	0.496614i	$-0.834577\pi$
$367$	−33.2559	−1.73594	−0.867972	+	0.496614i	$0.834577\pi$
$368$	−5.62036	−0.292981
$369$	−31.2003	−1.62422
$370$	0	0
$371$	18.2192	0.945896
$372$	−26.5249	−1.37525
$373$	28.5387	1.47768	0.738839	−	0.673882i	$-0.235375\pi$
$373$	28.5387	1.47768	0.738839	+	0.673882i	$0.235375\pi$
$374$	−21.2994	−1.10136
$375$	0	0
$376$	10.9154	0.562921
$377$	0.445862	0.0229631
$378$	4.29581	0.220953
$379$	0.476554	0.0244789	0.0122395	−	0.999925i	$-0.496104\pi$
$379$	0.476554	0.0244789	0.0122395	+	0.999925i	$0.496104\pi$
$380$	0	0
$381$	−50.6334	−2.59403
$382$	20.6048	1.05423
$383$	−7.13962	−0.364818	−0.182409	−	0.983223i	$-0.558389\pi$
$383$	−7.13962	−0.364818	−0.182409	+	0.983223i	$0.558389\pi$
$384$	2.62545	0.133980
$385$	0	0
$386$	−26.1663	−1.33183
$387$	8.84740	0.449738
$388$	−6.05864	−0.307581
$389$	−24.2060	−1.22729	−0.613646	−	0.789581i	$-0.710298\pi$
$389$	−24.2060	−1.22729	−0.613646	+	0.789581i	$0.710298\pi$
$390$	0	0
$391$	−28.5693	−1.44481
$392$	3.64280	0.183989
$393$	23.1584	1.16819
$394$	14.6067	0.735876
$395$	0	0
$396$	16.3123	0.819726
$397$	−20.7491	−1.04137	−0.520684	−	0.853750i	$-0.674323\pi$
$397$	−20.7491	−1.04137	−0.520684	+	0.853750i	$0.674323\pi$
$398$	−5.81275	−0.291367
$399$	−17.1237	−0.857256
$400$	0	0
$401$	−34.8410	−1.73988	−0.869939	−	0.493159i	$-0.835842\pi$
$401$	−34.8410	−1.73988	−0.869939	+	0.493159i	$0.835842\pi$
$402$	−4.85901	−0.242346
$403$	3.73259	0.185934
$404$	4.96528	0.247032
$405$	0	0
$406$	−2.21121	−0.109740
$407$	−4.19017	−0.207699
$408$	13.3456	0.660707
$409$	25.2132	1.24671	0.623356	−	0.781938i	$-0.285769\pi$
$409$	25.2132	1.24671	0.623356	+	0.781938i	$0.285769\pi$
$410$	0	0
$411$	12.2379	0.603652
$412$	7.33338	0.361289
$413$	−9.79462	−0.481962
$414$	21.8801	1.07535
$415$	0	0
$416$	−0.369454	−0.0181140
$417$	−43.0705	−2.10917
$418$	−14.9154	−0.729538
$419$	5.69695	0.278314	0.139157	−	0.990270i	$-0.455561\pi$
$419$	5.69695	0.278314	0.139157	+	0.990270i	$0.455561\pi$
$420$	0	0
$421$	3.38092	0.164776	0.0823880	−	0.996600i	$-0.473745\pi$
$421$	3.38092	0.164776	0.0823880	+	0.996600i	$0.473745\pi$
$422$	−16.0976	−0.783621
$423$	−42.4938	−2.06612
$424$	−9.94355	−0.482901
$425$	0	0
$426$	7.50978	0.363850
$427$	−17.9457	−0.868455
$428$	8.21182	0.396933
$429$	−4.06440	−0.196231
$430$	0	0
$431$	−0.381731	−0.0183873	−0.00919367	−	0.999958i	$-0.502926\pi$
$431$	−0.381731	−0.0183873	−0.00919367	+	0.999958i	$0.502926\pi$
$432$	−2.34453	−0.112801
$433$	21.6393	1.03992	0.519958	−	0.854192i	$-0.325948\pi$
$433$	21.6393	1.03992	0.519958	+	0.854192i	$0.325948\pi$
$434$	−18.5114	−0.888575
$435$	0	0
$436$	−20.5503	−0.984179
$437$	−20.0064	−0.957035
$438$	21.2646	1.01606
$439$	26.4663	1.26317	0.631583	−	0.775308i	$-0.282405\pi$
$439$	26.4663	1.26317	0.631583	+	0.775308i	$0.282405\pi$
$440$	0	0
$441$	−14.1814	−0.675305
$442$	−1.87800	−0.0893273
$443$	4.65489	0.221161	0.110580	−	0.993867i	$-0.464729\pi$
$443$	4.65489	0.221161	0.110580	+	0.993867i	$0.464729\pi$
$444$	2.62545	0.124598
$445$	0	0
$446$	25.3608	1.20087
$447$	15.3920	0.728015
$448$	1.83227	0.0865665
$449$	−26.5531	−1.25312	−0.626558	−	0.779375i	$-0.715537\pi$
$449$	−26.5531	−1.25312	−0.626558	+	0.779375i	$0.715537\pi$
$450$	0	0
$451$	−33.5820	−1.58131
$452$	0.652984	0.0307138
$453$	−34.5142	−1.62162
$454$	−5.57339	−0.261572
$455$	0	0
$456$	9.34563	0.437649
$457$	−4.27264	−0.199866	−0.0999329	−	0.994994i	$-0.531863\pi$
$457$	−4.27264	−0.199866	−0.0999329	+	0.994994i	$0.531863\pi$
$458$	−9.02035	−0.421493
$459$	−11.9177	−0.556269
$460$	0	0
$461$	32.1873	1.49911	0.749555	−	0.661942i	$-0.230267\pi$
$461$	32.1873	1.49911	0.749555	+	0.661942i	$0.230267\pi$
$462$	20.1569	0.937786
$463$	−18.4014	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$463$	−18.4014	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$464$	1.20681	0.0560250
$465$	0	0
$466$	11.8100	0.547086
$467$	3.06777	0.141959	0.0709797	−	0.997478i	$-0.477387\pi$
$467$	3.06777	0.141959	0.0709797	+	0.997478i	$0.477387\pi$
$468$	1.43828	0.0664847
$469$	−3.39104	−0.156584
$470$	0	0
$471$	−19.2611	−0.887504
$472$	5.34563	0.246052
$473$	9.52276	0.437857
$474$	15.9197	0.731217
$475$	0	0
$476$	9.31373	0.426894
$477$	38.7102	1.77242
$478$	−0.907160	−0.0414925
$479$	8.78798	0.401533	0.200767	−	0.979639i	$-0.435657\pi$
$479$	8.78798	0.401533	0.200767	+	0.979639i	$0.435657\pi$
$480$	0	0
$481$	−0.369454	−0.0168457
$482$	4.15616	0.189308
$483$	27.0369	1.23022
$484$	6.55754	0.298070
$485$	0	0
$486$	−21.5354	−0.976866
$487$	22.4814	1.01873	0.509366	−	0.860550i	$-0.329880\pi$
$487$	22.4814	1.01873	0.509366	+	0.860550i	$0.329880\pi$
$488$	9.79428	0.443366
$489$	−49.3500	−2.23168
$490$	0	0
$491$	−0.0441752	−0.00199360	−0.000996799	−	1.00000i	$-0.500317\pi$
$491$	−0.0441752	−0.00199360	−0.000996799		1.00000i	$0.500317\pi$
$492$	21.0416	0.948629
$493$	6.13445	0.276282
$494$	−1.31512	−0.0591699
$495$	0	0
$496$	10.1030	0.453638
$497$	5.24097	0.235090
$498$	23.4505	1.05084
$499$	28.2445	1.26440	0.632199	−	0.774806i	$-0.282153\pi$
$499$	28.2445	1.26440	0.632199	+	0.774806i	$0.282153\pi$
$500$	0	0
$501$	−15.5629	−0.695301
$502$	−9.31888	−0.415922
$503$	19.7713	0.881560	0.440780	−	0.897615i	$-0.354702\pi$
$503$	19.7713	0.881560	0.440780	+	0.897615i	$0.354702\pi$
$504$	−7.13302	−0.317730
$505$	0	0
$506$	23.5503	1.04694
$507$	33.7725	1.49989
$508$	19.2856	0.855660
$509$	43.4809	1.92726	0.963628	−	0.267247i	$-0.0861141\pi$
$509$	43.4809	1.92726	0.963628	+	0.267247i	$0.0861141\pi$
$510$	0	0
$511$	14.8403	0.656496
$512$	−1.00000	−0.0441942
$513$	−8.34565	−0.368470
$514$	−13.6312	−0.601249
$515$	0	0
$516$	−5.96671	−0.262670
$517$	−45.7376	−2.01154
$518$	1.83227	0.0805052
$519$	15.1815	0.666393
$520$	0	0
$521$	−18.0325	−0.790019	−0.395009	−	0.918677i	$-0.629259\pi$
$521$	−18.0325	−0.790019	−0.395009	+	0.918677i	$0.629259\pi$
$522$	−4.69813	−0.205632
$523$	−31.6788	−1.38522	−0.692610	−	0.721313i	$-0.743539\pi$
$523$	−31.6788	−1.38522	−0.692610	+	0.721313i	$0.743539\pi$
$524$	−8.82072	−0.385335
$525$	0	0
$526$	27.2170	1.18672
$527$	51.3553	2.23707
$528$	−11.0011	−0.478761
$529$	8.58844	0.373410
$530$	0	0
$531$	−20.8105	−0.903100
$532$	6.52218	0.282773
$533$	−2.96098	−0.128254
$534$	30.1331	1.30398
$535$	0	0
$536$	1.85073	0.0799395
$537$	−18.7347	−0.808463
$538$	−23.0550	−0.993973
$539$	−15.2639	−0.657465
$540$	0	0
$541$	9.39442	0.403898	0.201949	−	0.979396i	$-0.435273\pi$
$541$	9.39442	0.403898	0.201949	+	0.979396i	$0.435273\pi$
$542$	−22.9320	−0.985014
$543$	−30.2886	−1.29981
$544$	−5.08317	−0.217939
$545$	0	0
$546$	1.77727	0.0760601
$547$	−24.7824	−1.05962	−0.529809	−	0.848117i	$-0.677737\pi$
$547$	−24.7824	−1.05962	−0.529809	+	0.848117i	$0.677737\pi$
$548$	−4.66126	−0.199119
$549$	−38.1291	−1.62731
$550$	0	0
$551$	4.29581	0.183008
$552$	−14.7560	−0.628056
$553$	11.1102	0.472452
$554$	−11.7534	−0.499353
$555$	0	0
$556$	16.4050	0.695727
$557$	22.1627	0.939062	0.469531	−	0.882916i	$-0.344423\pi$
$557$	22.1627	0.939062	0.469531	+	0.882916i	$0.344423\pi$
$558$	−39.3310	−1.66501
$559$	0.839637	0.0355128
$560$	0	0
$561$	−55.9205	−2.36096
$562$	29.0962	1.22735
$563$	−6.46846	−0.272613	−0.136306	−	0.990667i	$-0.543523\pi$
$563$	−6.46846	−0.272613	−0.136306	+	0.990667i	$0.543523\pi$
$564$	28.6580	1.20672
$565$	0	0
$566$	−10.1829	−0.428019
$567$	−10.1206	−0.425026
$568$	−2.86038	−0.120019
$569$	8.26550	0.346508	0.173254	−	0.984877i	$-0.444572\pi$
$569$	8.26550	0.346508	0.173254	+	0.984877i	$0.444572\pi$
$570$	0	0
$571$	34.5179	1.44453	0.722264	−	0.691618i	$-0.243102\pi$
$571$	34.5179	1.44453	0.722264	+	0.691618i	$0.243102\pi$
$572$	1.54808	0.0647283
$573$	54.0970	2.25993
$574$	14.6846	0.612925
$575$	0	0
$576$	3.89300	0.162208
$577$	3.05347	0.127117	0.0635587	−	0.997978i	$-0.479755\pi$
$577$	3.05347	0.127117	0.0635587	+	0.997978i	$0.479755\pi$
$578$	−8.83864	−0.367639
$579$	−68.6985	−2.85501
$580$	0	0
$581$	16.3658	0.678968
$582$	−15.9067	−0.659353
$583$	41.6652	1.72559
$584$	−8.09942	−0.335156
$585$	0	0
$586$	18.1243	0.748706
$587$	35.1344	1.45015	0.725076	−	0.688668i	$-0.241804\pi$
$587$	35.1344	1.45015	0.725076	+	0.688668i	$0.241804\pi$
$588$	9.56399	0.394412
$589$	35.9629	1.48183
$590$	0	0
$591$	38.3492	1.57748
$592$	−1.00000	−0.0410997
$593$	12.0400	0.494424	0.247212	−	0.968961i	$-0.420486\pi$
$593$	12.0400	0.494424	0.247212	+	0.968961i	$0.420486\pi$
$594$	9.82399	0.403083
$595$	0	0
$596$	−5.86259	−0.240141
$597$	−15.2611	−0.624595
$598$	2.07646	0.0849129
$599$	−17.4701	−0.713810	−0.356905	−	0.934141i	$-0.616168\pi$
$599$	−17.4701	−0.713810	−0.356905	+	0.934141i	$0.616168\pi$
$600$	0	0
$601$	38.7044	1.57878	0.789392	−	0.613890i	$-0.210396\pi$
$601$	38.7044	1.57878	0.789392	+	0.613890i	$0.210396\pi$
$602$	−4.16409	−0.169716
$603$	−7.20491	−0.293406
$604$	13.1460	0.534903
$605$	0	0
$606$	13.0361	0.529556
$607$	28.7903	1.16856	0.584282	−	0.811551i	$-0.301376\pi$
$607$	28.7903	1.16856	0.584282	+	0.811551i	$0.301376\pi$
$608$	−3.55963	−0.144362
$609$	−5.80542	−0.235247
$610$	0	0
$611$	−4.03275	−0.163148
$612$	19.7888	0.799915
$613$	7.66539	0.309602	0.154801	−	0.987946i	$-0.450526\pi$
$613$	7.66539	0.309602	0.154801	+	0.987946i	$0.450526\pi$
$614$	−6.15561	−0.248420
$615$	0	0
$616$	−7.67751	−0.309336
$617$	−36.5388	−1.47100	−0.735498	−	0.677527i	$-0.763052\pi$
$617$	−36.5388	−1.47100	−0.735498	+	0.677527i	$0.763052\pi$
$618$	19.2534	0.774487
$619$	−45.8085	−1.84120	−0.920599	−	0.390510i	$-0.872299\pi$
$619$	−45.8085	−1.84120	−0.920599	+	0.390510i	$0.872299\pi$
$620$	0	0
$621$	13.1771	0.528779
$622$	29.4512	1.18088
$623$	21.0294	0.842527
$624$	−0.969984	−0.0388304
$625$	0	0
$626$	13.0102	0.519991
$627$	−39.1598	−1.56389
$628$	7.33629	0.292750
$629$	−5.08317	−0.202679
$630$	0	0
$631$	3.42682	0.136420	0.0682098	−	0.997671i	$-0.478271\pi$
$631$	3.42682	0.136420	0.0682098	+	0.997671i	$0.478271\pi$
$632$	−6.06361	−0.241197
$633$	−42.2636	−1.67983
$634$	−7.39826	−0.293823
$635$	0	0
$636$	−26.1063	−1.03518
$637$	−1.34585	−0.0533244
$638$	−5.05676	−0.200199
$639$	11.1354	0.440511
$640$	0	0
$641$	38.3241	1.51371	0.756855	−	0.653583i	$-0.226735\pi$
$641$	38.3241	1.51371	0.756855	+	0.653583i	$0.226735\pi$
$642$	21.5598	0.850896
$643$	−1.36660	−0.0538935	−0.0269468	−	0.999637i	$-0.508578\pi$
$643$	−1.36660	−0.0538935	−0.0269468	+	0.999637i	$0.508578\pi$
$644$	−10.2980	−0.405798
$645$	0	0
$646$	−18.0942	−0.711906
$647$	−11.2690	−0.443028	−0.221514	−	0.975157i	$-0.571100\pi$
$647$	−11.2690	−0.443028	−0.221514	+	0.975157i	$0.571100\pi$
$648$	5.52355	0.216985
$649$	−22.3991	−0.879241
$650$	0	0
$651$	−48.6008	−1.90482
$652$	18.7968	0.736138
$653$	29.5439	1.15614	0.578070	−	0.815987i	$-0.303806\pi$
$653$	29.5439	1.15614	0.578070	+	0.815987i	$0.303806\pi$
$654$	−53.9537	−2.10976
$655$	0	0
$656$	−8.01447	−0.312912
$657$	31.5311	1.23014
$658$	20.0000	0.779681
$659$	−49.9126	−1.94432	−0.972159	−	0.234324i	$-0.924712\pi$
$659$	−49.9126	−1.94432	−0.972159	+	0.234324i	$0.924712\pi$
$660$	0	0
$661$	25.9950	1.01109	0.505545	−	0.862800i	$-0.331292\pi$
$661$	25.9950	1.01109	0.505545	+	0.862800i	$0.331292\pi$
$662$	−33.2545	−1.29247
$663$	−4.93059	−0.191488
$664$	−8.93200	−0.346629
$665$	0	0
$666$	3.89300	0.150851
$667$	−6.78273	−0.262628
$668$	5.92772	0.229350
$669$	66.5836	2.57427
$670$	0	0
$671$	−41.0397	−1.58432
$672$	4.81053	0.185570
$673$	−6.91054	−0.266382	−0.133191	−	0.991090i	$-0.542522\pi$
$673$	−6.91054	−0.266382	−0.133191	+	0.991090i	$0.542522\pi$
$674$	22.6288	0.871627
$675$	0	0
$676$	−12.8635	−0.494750
$677$	15.6913	0.603065	0.301532	−	0.953456i	$-0.402502\pi$
$677$	15.6913	0.603065	0.301532	+	0.953456i	$0.402502\pi$
$678$	1.71438	0.0658403
$679$	−11.1011	−0.426019
$680$	0	0
$681$	−14.6327	−0.560725
$682$	−42.3333	−1.62103
$683$	18.2002	0.696412	0.348206	−	0.937418i	$-0.386791\pi$
$683$	18.2002	0.696412	0.348206	+	0.937418i	$0.386791\pi$
$684$	13.8576	0.529859
$685$	0	0
$686$	19.5004	0.744531
$687$	−23.6825	−0.903544
$688$	2.27264	0.0866437
$689$	3.67368	0.139956
$690$	0	0
$691$	−2.58658	−0.0983982	−0.0491991	−	0.998789i	$-0.515667\pi$
$691$	−2.58658	−0.0983982	−0.0491991	+	0.998789i	$0.515667\pi$
$692$	−5.78242	−0.219815
$693$	29.8886	1.13537
$694$	−24.1396	−0.916327
$695$	0	0
$696$	3.16843	0.120099
$697$	−40.7389	−1.54310
$698$	10.3702	0.392516
$699$	31.0065	1.17277
$700$	0	0
$701$	−26.5092	−1.00124	−0.500620	−	0.865667i	$-0.666895\pi$
$701$	−26.5092	−1.00124	−0.500620	+	0.865667i	$0.666895\pi$
$702$	0.866196	0.0326925
$703$	−3.55963	−0.134254
$704$	4.19017	0.157923
$705$	0	0
$706$	−3.92699	−0.147794
$707$	9.09773	0.342155
$708$	14.0347	0.527456
$709$	−25.0474	−0.940674	−0.470337	−	0.882487i	$-0.655868\pi$
$709$	−25.0474	−0.940674	−0.470337	+	0.882487i	$0.655868\pi$
$710$	0	0
$711$	23.6056	0.885281
$712$	−11.4773	−0.430129
$713$	−56.7825	−2.12652
$714$	24.4528	0.915121
$715$	0	0
$716$	7.13581	0.266678
$717$	−2.38171	−0.0889464
$718$	9.85179	0.367665
$719$	24.9551	0.930668	0.465334	−	0.885135i	$-0.345934\pi$
$719$	24.9551	0.930668	0.465334	+	0.885135i	$0.345934\pi$
$720$	0	0
$721$	13.4367	0.500409
$722$	6.32907	0.235544
$723$	10.9118	0.405814
$724$	11.5365	0.428752
$725$	0	0
$726$	17.2165	0.638964
$727$	21.6774	0.803970	0.401985	−	0.915646i	$-0.368320\pi$
$727$	21.6774	0.803970	0.401985	+	0.915646i	$0.368320\pi$
$728$	−0.676938	−0.0250890
$729$	−39.9695	−1.48035
$730$	0	0
$731$	11.5522	0.427275
$732$	25.7144	0.950432
$733$	−6.33267	−0.233903	−0.116951	−	0.993138i	$-0.537312\pi$
$733$	−6.33267	−0.233903	−0.116951	+	0.993138i	$0.537312\pi$
$734$	33.2559	1.22750
$735$	0	0
$736$	5.62036	0.207169
$737$	−7.75489	−0.285655
$738$	31.2003	1.14850
$739$	12.8076	0.471136	0.235568	−	0.971858i	$-0.424305\pi$
$739$	12.8076	0.471136	0.235568	+	0.971858i	$0.424305\pi$
$740$	0	0
$741$	−3.45278	−0.126841
$742$	−18.2192	−0.668849
$743$	−11.0694	−0.406097	−0.203048	−	0.979169i	$-0.565085\pi$
$743$	−11.0694	−0.406097	−0.203048	+	0.979169i	$0.565085\pi$
$744$	26.5249	0.972452
$745$	0	0
$746$	−28.5387	−1.04488
$747$	34.7723	1.27225
$748$	21.2994	0.778782
$749$	15.0463	0.549778
$750$	0	0
$751$	−46.5316	−1.69796	−0.848981	−	0.528423i	$-0.822784\pi$
$751$	−46.5316	−1.69796	−0.848981	+	0.528423i	$0.822784\pi$
$752$	−10.9154	−0.398045
$753$	−24.4663	−0.891601
$754$	−0.445862	−0.0162374
$755$	0	0
$756$	−4.29581	−0.156237
$757$	−32.9872	−1.19894	−0.599470	−	0.800397i	$-0.704622\pi$
$757$	−32.9872	−1.19894	−0.599470	+	0.800397i	$0.704622\pi$
$758$	−0.476554	−0.0173092
$759$	61.8301	2.24429
$760$	0	0
$761$	28.7913	1.04369	0.521843	−	0.853042i	$-0.325245\pi$
$761$	28.7913	1.04369	0.521843	+	0.853042i	$0.325245\pi$
$762$	50.6334	1.83426
$763$	−37.6536	−1.36315
$764$	−20.6048	−0.745456
$765$	0	0
$766$	7.13962	0.257965
$767$	−1.97496	−0.0713118
$768$	−2.62545	−0.0947379
$769$	41.9025	1.51104	0.755521	−	0.655124i	$-0.227384\pi$
$769$	41.9025	1.51104	0.755521	+	0.655124i	$0.227384\pi$
$770$	0	0
$771$	−35.7882	−1.28888
$772$	26.1663	0.941747
$773$	−31.0931	−1.11834	−0.559171	−	0.829052i	$-0.688881\pi$
$773$	−31.0931	−1.11834	−0.559171	+	0.829052i	$0.688881\pi$
$774$	−8.84740	−0.318013
$775$	0	0
$776$	6.05864	0.217493
$777$	4.81053	0.172577
$778$	24.2060	0.867827
$779$	−28.5285	−1.02214
$780$	0	0
$781$	11.9855	0.428874
$782$	28.5693	1.02163
$783$	−2.82941	−0.101115
$784$	−3.64280	−0.130100
$785$	0	0
$786$	−23.1584	−0.826032
$787$	18.2842	0.651760	0.325880	−	0.945411i	$-0.394339\pi$
$787$	18.2842	0.651760	0.325880	+	0.945411i	$0.394339\pi$
$788$	−14.6067	−0.520343
$789$	71.4570	2.54394
$790$	0	0
$791$	1.19644	0.0425406
$792$	−16.3123	−0.579634
$793$	−3.61854	−0.128498
$794$	20.7491	0.736358
$795$	0	0
$796$	5.81275	0.206027
$797$	40.6025	1.43821	0.719106	−	0.694900i	$-0.244551\pi$
$797$	40.6025	1.43821	0.719106	+	0.694900i	$0.244551\pi$
$798$	17.1237	0.606172
$799$	−55.4851	−1.96292
$800$	0	0
$801$	44.6811	1.57873
$802$	34.8410	1.23028
$803$	33.9380	1.19764
$804$	4.85901	0.171364
$805$	0	0
$806$	−3.73259	−0.131475
$807$	−60.5299	−2.13075
$808$	−4.96528	−0.174678
$809$	41.1512	1.44680	0.723400	−	0.690429i	$-0.242578\pi$
$809$	41.1512	1.44680	0.723400	+	0.690429i	$0.242578\pi$
$810$	0	0
$811$	−6.71866	−0.235924	−0.117962	−	0.993018i	$-0.537636\pi$
$811$	−6.71866	−0.235924	−0.117962	+	0.993018i	$0.537636\pi$
$812$	2.21121	0.0775981
$813$	−60.2069	−2.11155
$814$	4.19017	0.146865
$815$	0	0
$816$	−13.3456	−0.467190
$817$	8.08975	0.283025
$818$	−25.2132	−0.881558
$819$	2.63532	0.0920856
$820$	0	0
$821$	−11.5263	−0.402272	−0.201136	−	0.979563i	$-0.564463\pi$
$821$	−11.5263	−0.402272	−0.201136	+	0.979563i	$0.564463\pi$
$822$	−12.2379	−0.426847
$823$	4.69569	0.163681	0.0818407	−	0.996645i	$-0.473920\pi$
$823$	4.69569	0.163681	0.0818407	+	0.996645i	$0.473920\pi$
$824$	−7.33338	−0.255470
$825$	0	0
$826$	9.79462	0.340798
$827$	28.4859	0.990550	0.495275	−	0.868736i	$-0.335067\pi$
$827$	28.4859	0.990550	0.495275	+	0.868736i	$0.335067\pi$
$828$	−21.8801	−0.760385
$829$	−40.2771	−1.39888	−0.699441	−	0.714691i	$-0.746567\pi$
$829$	−40.2771	−1.39888	−0.699441	+	0.714691i	$0.746567\pi$
$830$	0	0
$831$	−30.8579	−1.07045
$832$	0.369454	0.0128085
$833$	−18.5170	−0.641575
$834$	43.0705	1.49141
$835$	0	0
$836$	14.9154	0.515861
$837$	−23.6868	−0.818736
$838$	−5.69695	−0.196798
$839$	−9.99007	−0.344895	−0.172448	−	0.985019i	$-0.555168\pi$
$839$	−9.99007	−0.344895	−0.172448	+	0.985019i	$0.555168\pi$
$840$	0	0
$841$	−27.5436	−0.949779
$842$	−3.38092	−0.116514
$843$	76.3906	2.63103
$844$	16.0976	0.554104
$845$	0	0
$846$	42.4938	1.46097
$847$	12.0152	0.412846
$848$	9.94355	0.341463
$849$	−26.7347	−0.917533
$850$	0	0
$851$	5.62036	0.192663
$852$	−7.50978	−0.257281
$853$	18.0500	0.618019	0.309010	−	0.951059i	$-0.400002\pi$
$853$	18.0500	0.618019	0.309010	+	0.951059i	$0.400002\pi$
$854$	17.9457	0.614091
$855$	0	0
$856$	−8.21182	−0.280674
$857$	−24.3118	−0.830474	−0.415237	−	0.909713i	$-0.636301\pi$
$857$	−24.3118	−0.830474	−0.415237	+	0.909713i	$0.636301\pi$
$858$	4.06440	0.138756
$859$	35.2112	1.20139	0.600696	−	0.799477i	$-0.294890\pi$
$859$	35.2112	1.20139	0.600696	+	0.799477i	$0.294890\pi$
$860$	0	0
$861$	38.5538	1.31391
$862$	0.381731	0.0130018
$863$	−23.2986	−0.793093	−0.396546	−	0.918015i	$-0.629791\pi$
$863$	−23.2986	−0.793093	−0.396546	+	0.918015i	$0.629791\pi$
$864$	2.34453	0.0797626
$865$	0	0
$866$	−21.6393	−0.735332
$867$	−23.2054	−0.788098
$868$	18.5114	0.628318
$869$	25.4076	0.861893
$870$	0	0
$871$	−0.683761	−0.0231684
$872$	20.5503	0.695920
$873$	−23.5863	−0.798275
$874$	20.0064	0.676726
$875$	0	0
$876$	−21.2646	−0.718466
$877$	−47.8713	−1.61650	−0.808250	−	0.588839i	$-0.799585\pi$
$877$	−47.8713	−1.61650	−0.808250	+	0.588839i	$0.799585\pi$
$878$	−26.4663	−0.893194
$879$	47.5844	1.60498
$880$	0	0
$881$	−40.1274	−1.35193	−0.675963	−	0.736936i	$-0.736272\pi$
$881$	−40.1274	−1.35193	−0.675963	+	0.736936i	$0.736272\pi$
$882$	14.1814	0.477513
$883$	−45.6747	−1.53708	−0.768539	−	0.639803i	$-0.779016\pi$
$883$	−45.6747	−1.53708	−0.768539	+	0.639803i	$0.779016\pi$
$884$	1.87800	0.0631639
$885$	0	0
$886$	−4.65489	−0.156384
$887$	17.5417	0.588992	0.294496	−	0.955653i	$-0.404848\pi$
$887$	17.5417	0.588992	0.294496	+	0.955653i	$0.404848\pi$
$888$	−2.62545	−0.0881044
$889$	35.3364	1.18514
$890$	0	0
$891$	−23.1446	−0.775374
$892$	−25.3608	−0.849142
$893$	−38.8549	−1.30023
$894$	−15.3920	−0.514784
$895$	0	0
$896$	−1.83227	−0.0612117
$897$	5.45166	0.182026
$898$	26.5531	0.886087
$899$	12.1924	0.406641
$900$	0	0
$901$	50.5448	1.68389
$902$	33.5820	1.11816
$903$	−10.9326	−0.363815
$904$	−0.652984	−0.0217179
$905$	0	0
$906$	34.5142	1.14666
$907$	2.04294	0.0678347	0.0339173	−	0.999425i	$-0.489202\pi$
$907$	2.04294	0.0678347	0.0339173	+	0.999425i	$0.489202\pi$
$908$	5.57339	0.184960
$909$	19.3299	0.641131
$910$	0	0
$911$	54.3145	1.79952	0.899760	−	0.436385i	$-0.143742\pi$
$911$	54.3145	1.79952	0.899760	+	0.436385i	$0.143742\pi$
$912$	−9.34563	−0.309465
$913$	37.4266	1.23864
$914$	4.27264	0.141326
$915$	0	0
$916$	9.02035	0.298041
$917$	−16.1619	−0.533713
$918$	11.9177	0.393341
$919$	40.9771	1.35171	0.675855	−	0.737034i	$-0.263774\pi$
$919$	40.9771	1.35171	0.675855	+	0.737034i	$0.263774\pi$
$920$	0	0
$921$	−16.1613	−0.532531
$922$	−32.1873	−1.06003
$923$	1.05678	0.0347842
$924$	−20.1569	−0.663115
$925$	0	0
$926$	18.4014	0.604708
$927$	28.5488	0.937667
$928$	−1.20681	−0.0396156
$929$	−22.2458	−0.729860	−0.364930	−	0.931035i	$-0.618907\pi$
$929$	−22.2458	−0.729860	−0.364930	+	0.931035i	$0.618907\pi$
$930$	0	0
$931$	−12.9670	−0.424976
$932$	−11.8100	−0.386848
$933$	77.3226	2.53143
$934$	−3.06777	−0.100380
$935$	0	0
$936$	−1.43828	−0.0470118
$937$	0.989771	0.0323344	0.0161672	−	0.999869i	$-0.494854\pi$
$937$	0.989771	0.0323344	0.0161672	+	0.999869i	$0.494854\pi$
$938$	3.39104	0.110721
$939$	34.1576	1.11469
$940$	0	0
$941$	0.255111	0.00831639	0.00415820	−	0.999991i	$-0.498676\pi$
$941$	0.255111	0.00831639	0.00415820	+	0.999991i	$0.498676\pi$
$942$	19.2611	0.627560
$943$	45.0442	1.46684
$944$	−5.34563	−0.173985
$945$	0	0
$946$	−9.52276	−0.309612
$947$	−58.4983	−1.90094	−0.950469	−	0.310819i	$-0.899397\pi$
$947$	−58.4983	−1.90094	−0.950469	+	0.310819i	$0.899397\pi$
$948$	−15.9197	−0.517048
$949$	2.99236	0.0971362
$950$	0	0
$951$	−19.4238	−0.629860
$952$	−9.31373	−0.301860
$953$	−34.4365	−1.11551	−0.557755	−	0.830006i	$-0.688337\pi$
$953$	−34.4365	−1.11551	−0.557755	+	0.830006i	$0.688337\pi$
$954$	−38.7102	−1.25329
$955$	0	0
$956$	0.907160	0.0293396
$957$	−13.2763	−0.429161
$958$	−8.78798	−0.283927
$959$	−8.54068	−0.275793
$960$	0	0
$961$	71.0706	2.29260
$962$	0.369454	0.0119117
$963$	31.9686	1.03017
$964$	−4.15616	−0.133861
$965$	0	0
$966$	−27.0369	−0.869898
$967$	−19.0310	−0.611997	−0.305998	−	0.952032i	$-0.598990\pi$
$967$	−19.0310	−0.611997	−0.305998	+	0.952032i	$0.598990\pi$
$968$	−6.55754	−0.210767
$969$	−47.5054	−1.52609
$970$	0	0
$971$	50.9856	1.63621	0.818103	−	0.575071i	$-0.195026\pi$
$971$	50.9856	1.63621	0.818103	+	0.575071i	$0.195026\pi$
$972$	21.5354	0.690748
$973$	30.0583	0.963626
$974$	−22.4814	−0.720352
$975$	0	0
$976$	−9.79428	−0.313507
$977$	26.8574	0.859245	0.429622	−	0.903009i	$-0.358647\pi$
$977$	26.8574	0.859245	0.429622	+	0.903009i	$0.358647\pi$
$978$	49.3500	1.57804
$979$	48.0918	1.53702
$980$	0	0
$981$	−80.0022	−2.55427
$982$	0.0441752	0.00140969
$983$	7.09389	0.226260	0.113130	−	0.993580i	$-0.463912\pi$
$983$	7.09389	0.226260	0.113130	+	0.993580i	$0.463912\pi$
$984$	−21.0416	−0.670782
$985$	0	0
$986$	−6.13445	−0.195361
$987$	52.5091	1.67138
$988$	1.31512	0.0418395
$989$	−12.7731	−0.406160
$990$	0	0
$991$	−27.2573	−0.865856	−0.432928	−	0.901429i	$-0.642520\pi$
$991$	−27.2573	−0.865856	−0.432928	+	0.901429i	$0.642520\pi$
$992$	−10.1030	−0.320771
$993$	−87.3081	−2.77064
$994$	−5.24097	−0.166234
$995$	0	0
$996$	−23.4505	−0.743059
$997$	24.6984	0.782207	0.391103	−	0.920347i	$-0.372094\pi$
$997$	24.6984	0.782207	0.391103	+	0.920347i	$0.372094\pi$
$998$	−28.2445	−0.894064
$999$	2.34453	0.0741777

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.a.bd.1.1		5
5.2	odd	4		370.2.b.d.149.5	✓	10
5.3	odd	4		370.2.b.d.149.6	yes	10
5.4	even	2		1850.2.a.be.1.5		5
15.2	even	4		3330.2.d.p.1999.8		10
15.8	even	4		3330.2.d.p.1999.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.b.d.149.5	✓	10	5.2	odd	4
370.2.b.d.149.6	yes	10	5.3	odd	4
1850.2.a.bd.1.1		5	1.1	even	1	trivial
1850.2.a.be.1.5		5	5.4	even	2
3330.2.d.p.1999.3		10	15.8	even	4
3330.2.d.p.1999.8		10	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:	\( N \)	\(=\)	\( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1850.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.62545 q^{3} +1.00000 q^{4} +2.62545 q^{6} +1.83227 q^{7} -1.00000 q^{8} +3.89300 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.62545 q^{3} +1.00000 q^{4} +2.62545 q^{6} +1.83227 q^{7} -1.00000 q^{8} +3.89300 q^{9} +4.19017 q^{11} -2.62545 q^{12} +0.369454 q^{13} -1.83227 q^{14} +1.00000 q^{16} +5.08317 q^{17} -3.89300 q^{18} +3.55963 q^{19} -4.81053 q^{21} -4.19017 q^{22} -5.62036 q^{23} +2.62545 q^{24} -0.369454 q^{26} -2.34453 q^{27} +1.83227 q^{28} +1.20681 q^{29} +10.1030 q^{31} -1.00000 q^{32} -11.0011 q^{33} -5.08317 q^{34} +3.89300 q^{36} -1.00000 q^{37} -3.55963 q^{38} -0.969984 q^{39} -8.01447 q^{41} +4.81053 q^{42} +2.27264 q^{43} +4.19017 q^{44} +5.62036 q^{46} -10.9154 q^{47} -2.62545 q^{48} -3.64280 q^{49} -13.3456 q^{51} +0.369454 q^{52} +9.94355 q^{53} +2.34453 q^{54} -1.83227 q^{56} -9.34563 q^{57} -1.20681 q^{58} -5.34563 q^{59} -9.79428 q^{61} -10.1030 q^{62} +7.13302 q^{63} +1.00000 q^{64} +11.0011 q^{66} -1.85073 q^{67} +5.08317 q^{68} +14.7560 q^{69} +2.86038 q^{71} -3.89300 q^{72} +8.09942 q^{73} +1.00000 q^{74} +3.55963 q^{76} +7.67751 q^{77} +0.969984 q^{78} +6.06361 q^{79} -5.52355 q^{81} +8.01447 q^{82} +8.93200 q^{83} -4.81053 q^{84} -2.27264 q^{86} -3.16843 q^{87} -4.19017 q^{88} +11.4773 q^{89} +0.676938 q^{91} -5.62036 q^{92} -26.5249 q^{93} +10.9154 q^{94} +2.62545 q^{96} -6.05864 q^{97} +3.64280 q^{98} +16.3123 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9}+O(q^{10}) \) 5 * q - 5 * q^2 + 5 * q^4 + q^7 - 5 * q^8 + 3 * q^9	\( 5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9} + 3 q^{11} + 6 q^{13} - q^{14} + 5 q^{16} - 9 q^{17} - 3 q^{18} + 4 q^{19} + 16 q^{21} - 3 q^{22} - 6 q^{23} - 6 q^{26} + q^{28} + 11 q^{29} + 23 q^{31} - 5 q^{32} - 20 q^{33} + 9 q^{34} + 3 q^{36} - 5 q^{37} - 4 q^{38} + 20 q^{39} - 7 q^{41} - 16 q^{42} + 17 q^{43} + 3 q^{44} + 6 q^{46} - 12 q^{47} + 30 q^{49} - 20 q^{51} + 6 q^{52} + 7 q^{53} - q^{56} - 11 q^{58} + 20 q^{59} - 9 q^{61} - 23 q^{62} + 33 q^{63} + 5 q^{64} + 20 q^{66} - 12 q^{67} - 9 q^{68} + 16 q^{69} + 6 q^{71} - 3 q^{72} + 6 q^{73} + 5 q^{74} + 4 q^{76} + q^{77} - 20 q^{78} + 20 q^{79} - 7 q^{81} + 7 q^{82} - 12 q^{83} + 16 q^{84} - 17 q^{86} + 34 q^{87} - 3 q^{88} + 12 q^{89} + 16 q^{91} - 6 q^{92} + 4 q^{93} + 12 q^{94} - 3 q^{97} - 30 q^{98} - 11 q^{99}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^4 + q^7 - 5 * q^8 + 3 * q^9 + 3 * q^11 + 6 * q^13 - q^14 + 5 * q^16 - 9 * q^17 - 3 * q^18 + 4 * q^19 + 16 * q^21 - 3 * q^22 - 6 * q^23 - 6 * q^26 + q^28 + 11 * q^29 + 23 * q^31 - 5 * q^32 - 20 * q^33 + 9 * q^34 + 3 * q^36 - 5 * q^37 - 4 * q^38 + 20 * q^39 - 7 * q^41 - 16 * q^42 + 17 * q^43 + 3 * q^44 + 6 * q^46 - 12 * q^47 + 30 * q^49 - 20 * q^51 + 6 * q^52 + 7 * q^53 - q^56 - 11 * q^58 + 20 * q^59 - 9 * q^61 - 23 * q^62 + 33 * q^63 + 5 * q^64 + 20 * q^66 - 12 * q^67 - 9 * q^68 + 16 * q^69 + 6 * q^71 - 3 * q^72 + 6 * q^73 + 5 * q^74 + 4 * q^76 + q^77 - 20 * q^78 + 20 * q^79 - 7 * q^81 + 7 * q^82 - 12 * q^83 + 16 * q^84 - 17 * q^86 + 34 * q^87 - 3 * q^88 + 12 * q^89 + 16 * q^91 - 6 * q^92 + 4 * q^93 + 12 * q^94 - 3 * q^97 - 30 * q^98 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more