LMFDB - Embedded newform 855.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,2,Mod(1,855)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("855.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	855.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:	$6.82720937282$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	855.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.311108 q^{2} -1.90321 q^{4} +1.00000 q^{5} +2.21432 q^{7} -1.21432 q^{8} +O(q^{10})$	\(q+0.311108 q^{2} -1.90321 q^{4} +1.00000 q^{5} +2.21432 q^{7} -1.21432 q^{8} +0.311108 q^{10} +1.31111 q^{11} -1.59210 q^{13} +0.688892 q^{14} +3.42864 q^{16} +0.474572 q^{17} -1.00000 q^{19} -1.90321 q^{20} +0.407896 q^{22} +7.33185 q^{23} +1.00000 q^{25} -0.495316 q^{26} -4.21432 q^{28} +0.260253 q^{29} +3.95407 q^{31} +3.49532 q^{32} +0.147643 q^{34} +2.21432 q^{35} +0.541249 q^{37} -0.311108 q^{38} -1.21432 q^{40} +12.1684 q^{41} -6.02074 q^{43} -2.49532 q^{44} +2.28100 q^{46} +4.28100 q^{47} -2.09679 q^{49} +0.311108 q^{50} +3.03011 q^{52} +7.52543 q^{53} +1.31111 q^{55} -2.68889 q^{56} +0.0809666 q^{58} +7.93978 q^{59} -2.76986 q^{61} +1.23014 q^{62} -5.76986 q^{64} -1.59210 q^{65} +5.18421 q^{67} -0.903212 q^{68} +0.688892 q^{70} +9.47949 q^{71} -5.67307 q^{73} +0.168387 q^{74} +1.90321 q^{76} +2.90321 q^{77} -3.86665 q^{79} +3.42864 q^{80} +3.78568 q^{82} -7.33185 q^{83} +0.474572 q^{85} -1.87310 q^{86} -1.59210 q^{88} +2.68889 q^{89} -3.52543 q^{91} -13.9541 q^{92} +1.33185 q^{94} -1.00000 q^{95} +15.8272 q^{97} -0.652327 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + q^{4} + 3 q^{5} + 3 q^{8}+O(q^{10}) $ 3 * q + q^2 + q^4 + 3 * q^5 + 3 * q^8	$ 3 q + q^{2} + q^{4} + 3 q^{5} + 3 q^{8} + q^{10} + 4 q^{11} + 2 q^{13} + 2 q^{14} - 3 q^{16} + 8 q^{17} - 3 q^{19} + q^{20} + 8 q^{22} + 2 q^{23} + 3 q^{25} + 12 q^{26} - 6 q^{28} + 14 q^{29} - 8 q^{31} - 3 q^{32} - 6 q^{34} + 8 q^{37} - q^{38} + 3 q^{40} + 10 q^{41} + 2 q^{43} + 6 q^{44} + 6 q^{47} - 13 q^{49} + q^{50} + 16 q^{52} + 16 q^{53} + 4 q^{55} - 8 q^{56} - 6 q^{58} + 10 q^{59} - 2 q^{61} + 10 q^{62} - 11 q^{64} + 2 q^{65} + 2 q^{67} + 4 q^{68} + 2 q^{70} + 2 q^{71} - 4 q^{73} - 26 q^{74} - q^{76} + 2 q^{77} - 12 q^{79} - 3 q^{80} + 18 q^{82} - 2 q^{83} + 8 q^{85} + 8 q^{86} + 2 q^{88} + 8 q^{89} - 4 q^{91} - 22 q^{92} - 16 q^{94} - 3 q^{95} + 14 q^{97} - 9 q^{98}+O(q^{100}) $ 3 * q + q^2 + q^4 + 3 * q^5 + 3 * q^8 + q^10 + 4 * q^11 + 2 * q^13 + 2 * q^14 - 3 * q^16 + 8 * q^17 - 3 * q^19 + q^20 + 8 * q^22 + 2 * q^23 + 3 * q^25 + 12 * q^26 - 6 * q^28 + 14 * q^29 - 8 * q^31 - 3 * q^32 - 6 * q^34 + 8 * q^37 - q^38 + 3 * q^40 + 10 * q^41 + 2 * q^43 + 6 * q^44 + 6 * q^47 - 13 * q^49 + q^50 + 16 * q^52 + 16 * q^53 + 4 * q^55 - 8 * q^56 - 6 * q^58 + 10 * q^59 - 2 * q^61 + 10 * q^62 - 11 * q^64 + 2 * q^65 + 2 * q^67 + 4 * q^68 + 2 * q^70 + 2 * q^71 - 4 * q^73 - 26 * q^74 - q^76 + 2 * q^77 - 12 * q^79 - 3 * q^80 + 18 * q^82 - 2 * q^83 + 8 * q^85 + 8 * q^86 + 2 * q^88 + 8 * q^89 - 4 * q^91 - 22 * q^92 - 16 * q^94 - 3 * q^95 + 14 * q^97 - 9 * q^98

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 $

$2$	0.311108	0.219986	0.109993	−	0.993932i	$-0.464917\pi$
$2$	0.311108	0.219986	0.109993	+	0.993932i	$0.464917\pi$
$3$	0	0
$3$	0	0
$4$	−1.90321	−0.951606
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.21432	0.836934	0.418467	−	0.908232i	$-0.362568\pi$
$7$	2.21432	0.836934	0.418467	+	0.908232i	$0.362568\pi$
$8$	−1.21432	−0.429327
$9$	0	0
$10$	0.311108	0.0983809
$11$	1.31111	0.395314	0.197657	−	0.980271i	$-0.436667\pi$
$11$	1.31111	0.395314	0.197657	+	0.980271i	$0.436667\pi$
$12$	0	0
$13$	−1.59210	−0.441570	−0.220785	−	0.975322i	$-0.570862\pi$
$13$	−1.59210	−0.441570	−0.220785	+	0.975322i	$0.570862\pi$
$14$	0.688892	0.184114
$15$	0	0
$16$	3.42864	0.857160
$17$	0.474572	0.115101	0.0575504	−	0.998343i	$-0.481671\pi$
$17$	0.474572	0.115101	0.0575504	+	0.998343i	$0.481671\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−1.90321	−0.425571
$21$	0	0
$22$	0.407896	0.0869637
$23$	7.33185	1.52880	0.764398	−	0.644744i	$-0.223036\pi$
$23$	7.33185	1.52880	0.764398	+	0.644744i	$0.223036\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.495316	−0.0971395
$27$	0	0
$28$	−4.21432	−0.796432
$29$	0.260253	0.0483277	0.0241639	−	0.999708i	$-0.492308\pi$
$29$	0.260253	0.0483277	0.0241639	+	0.999708i	$0.492308\pi$
$30$	0	0
$31$	3.95407	0.710171	0.355086	−	0.934834i	$-0.384452\pi$
$31$	3.95407	0.710171	0.355086	+	0.934834i	$0.384452\pi$
$32$	3.49532	0.617890
$33$	0	0
$34$	0.147643	0.0253206
$35$	2.21432	0.374288
$36$	0	0
$37$	0.541249	0.0889808	0.0444904	−	0.999010i	$-0.485834\pi$
$37$	0.541249	0.0889808	0.0444904	+	0.999010i	$0.485834\pi$
$38$	−0.311108	−0.0504684
$39$	0	0
$40$	−1.21432	−0.192001
$41$	12.1684	1.90038	0.950191	−	0.311667i	$-0.100887\pi$
$41$	12.1684	1.90038	0.950191	+	0.311667i	$0.100887\pi$
$42$	0	0
$43$	−6.02074	−0.918155	−0.459077	−	0.888396i	$-0.651820\pi$
$43$	−6.02074	−0.918155	−0.459077	+	0.888396i	$0.651820\pi$
$44$	−2.49532	−0.376183
$45$	0	0
$46$	2.28100	0.336315
$47$	4.28100	0.624447	0.312224	−	0.950009i	$-0.398926\pi$
$47$	4.28100	0.624447	0.312224	+	0.950009i	$0.398926\pi$
$48$	0	0
$49$	−2.09679	−0.299541
$50$	0.311108	0.0439973
$51$	0	0
$52$	3.03011	0.420201
$53$	7.52543	1.03370	0.516848	−	0.856077i	$-0.327105\pi$
$53$	7.52543	1.03370	0.516848	+	0.856077i	$0.327105\pi$
$54$	0	0
$55$	1.31111	0.176790
$56$	−2.68889	−0.359318
$57$	0	0
$58$	0.0809666	0.0106314
$59$	7.93978	1.03367	0.516835	−	0.856085i	$-0.327110\pi$
$59$	7.93978	1.03367	0.516835	+	0.856085i	$0.327110\pi$
$60$	0	0
$61$	−2.76986	−0.354644	−0.177322	−	0.984153i	$-0.556743\pi$
$61$	−2.76986	−0.354644	−0.177322	+	0.984153i	$0.556743\pi$
$62$	1.23014	0.156228
$63$	0	0
$64$	−5.76986	−0.721232
$65$	−1.59210	−0.197476
$66$	0	0
$67$	5.18421	0.633352	0.316676	−	0.948534i	$-0.397433\pi$
$67$	5.18421	0.633352	0.316676	+	0.948534i	$0.397433\pi$
$68$	−0.903212	−0.109531
$69$	0	0
$70$	0.688892	0.0823384
$71$	9.47949	1.12501	0.562505	−	0.826794i	$-0.309838\pi$
$71$	9.47949	1.12501	0.562505	+	0.826794i	$0.309838\pi$
$72$	0	0
$73$	−5.67307	−0.663983	−0.331991	−	0.943282i	$-0.607720\pi$
$73$	−5.67307	−0.663983	−0.331991	+	0.943282i	$0.607720\pi$
$74$	0.168387	0.0195746
$75$	0	0
$76$	1.90321	0.218313
$77$	2.90321	0.330852
$78$	0	0
$79$	−3.86665	−0.435032	−0.217516	−	0.976057i	$-0.569795\pi$
$79$	−3.86665	−0.435032	−0.217516	+	0.976057i	$0.569795\pi$
$80$	3.42864	0.383334
$81$	0	0
$82$	3.78568	0.418058
$83$	−7.33185	−0.804775	−0.402388	−	0.915469i	$-0.631820\pi$
$83$	−7.33185	−0.804775	−0.402388	+	0.915469i	$0.631820\pi$
$84$	0	0
$85$	0.474572	0.0514746
$86$	−1.87310	−0.201982
$87$	0	0
$88$	−1.59210	−0.169719
$89$	2.68889	0.285022	0.142511	−	0.989793i	$-0.454482\pi$
$89$	2.68889	0.285022	0.142511	+	0.989793i	$0.454482\pi$
$90$	0	0
$91$	−3.52543	−0.369565
$92$	−13.9541	−1.45481
$93$	0	0
$94$	1.33185	0.137370
$95$	−1.00000	−0.102598
$96$	0	0
$97$	15.8272	1.60701	0.803503	−	0.595301i	$-0.202967\pi$
$97$	15.8272	1.60701	0.803503	+	0.595301i	$0.202967\pi$
$98$	−0.652327	−0.0658950
$99$	0	0
$100$	−1.90321	−0.190321
$101$	−15.8479	−1.57693	−0.788463	−	0.615082i	$-0.789123\pi$
$101$	−15.8479	−1.57693	−0.788463	+	0.615082i	$0.789123\pi$
$102$	0	0
$103$	−6.10171	−0.601219	−0.300610	−	0.953747i	$-0.597190\pi$
$103$	−6.10171	−0.601219	−0.300610	+	0.953747i	$0.597190\pi$
$104$	1.93332	0.189578
$105$	0	0
$106$	2.34122	0.227399
$107$	−15.0923	−1.45903	−0.729516	−	0.683964i	$-0.760255\pi$
$107$	−15.0923	−1.45903	−0.729516	+	0.683964i	$0.760255\pi$
$108$	0	0
$109$	1.47949	0.141710	0.0708549	−	0.997487i	$-0.477427\pi$
$109$	1.47949	0.141710	0.0708549	+	0.997487i	$0.477427\pi$
$110$	0.407896	0.0388913
$111$	0	0
$112$	7.59210	0.717386
$113$	−8.94470	−0.841447	−0.420723	−	0.907189i	$-0.638224\pi$
$113$	−8.94470	−0.841447	−0.420723	+	0.907189i	$0.638224\pi$
$114$	0	0
$115$	7.33185	0.683699
$116$	−0.495316	−0.0459889
$117$	0	0
$118$	2.47013	0.227394
$119$	1.05086	0.0963317
$120$	0	0
$121$	−9.28100	−0.843727
$122$	−0.861725	−0.0780169
$123$	0	0
$124$	−7.52543	−0.675803
$125$	1.00000	0.0894427
$126$	0	0
$127$	3.37778	0.299730	0.149865	−	0.988706i	$-0.452116\pi$
$127$	3.37778	0.299730	0.149865	+	0.988706i	$0.452116\pi$
$128$	−8.78568	−0.776552
$129$	0	0
$130$	−0.495316	−0.0434421
$131$	−8.30174	−0.725326	−0.362663	−	0.931920i	$-0.618132\pi$
$131$	−8.30174	−0.725326	−0.362663	+	0.931920i	$0.618132\pi$
$132$	0	0
$133$	−2.21432	−0.192006
$134$	1.61285	0.139329
$135$	0	0
$136$	−0.576283	−0.0494158
$137$	2.99063	0.255507	0.127753	−	0.991806i	$-0.459223\pi$
$137$	2.99063	0.255507	0.127753	+	0.991806i	$0.459223\pi$
$138$	0	0
$139$	−17.8479	−1.51384	−0.756920	−	0.653508i	$-0.773297\pi$
$139$	−17.8479	−1.51384	−0.756920	+	0.653508i	$0.773297\pi$
$140$	−4.21432	−0.356175
$141$	0	0
$142$	2.94914	0.247487
$143$	−2.08742	−0.174559
$144$	0	0
$145$	0.260253	0.0216128
$146$	−1.76494	−0.146067
$147$	0	0
$148$	−1.03011	−0.0846746
$149$	−14.1017	−1.15526	−0.577629	−	0.816300i	$-0.696022\pi$
$149$	−14.1017	−1.15526	−0.577629	+	0.816300i	$0.696022\pi$
$150$	0	0
$151$	−16.3827	−1.33321	−0.666603	−	0.745413i	$-0.732252\pi$
$151$	−16.3827	−1.33321	−0.666603	+	0.745413i	$0.732252\pi$
$152$	1.21432	0.0984943
$153$	0	0
$154$	0.903212	0.0727829
$155$	3.95407	0.317598
$156$	0	0
$157$	15.3461	1.22476	0.612378	−	0.790565i	$-0.290213\pi$
$157$	15.3461	1.22476	0.612378	+	0.790565i	$0.290213\pi$
$158$	−1.20294	−0.0957011
$159$	0	0
$160$	3.49532	0.276329
$161$	16.2351	1.27950
$162$	0	0
$163$	−1.78568	−0.139865	−0.0699326	−	0.997552i	$-0.522278\pi$
$163$	−1.78568	−0.139865	−0.0699326	+	0.997552i	$0.522278\pi$
$164$	−23.1590	−1.80842
$165$	0	0
$166$	−2.28100	−0.177040
$167$	8.70964	0.673972	0.336986	−	0.941510i	$-0.390593\pi$
$167$	8.70964	0.673972	0.336986	+	0.941510i	$0.390593\pi$
$168$	0	0
$169$	−10.4652	−0.805016
$170$	0.147643	0.0113237
$171$	0	0
$172$	11.4588	0.873722
$173$	16.3827	1.24555	0.622777	−	0.782399i	$-0.286004\pi$
$173$	16.3827	1.24555	0.622777	+	0.782399i	$0.286004\pi$
$174$	0	0
$175$	2.21432	0.167387
$176$	4.49532	0.338847
$177$	0	0
$178$	0.836535	0.0627010
$179$	−4.62222	−0.345481	−0.172740	−	0.984967i	$-0.555262\pi$
$179$	−4.62222	−0.345481	−0.172740	+	0.984967i	$0.555262\pi$
$180$	0	0
$181$	−7.67307	−0.570335	−0.285167	−	0.958478i	$-0.592049\pi$
$181$	−7.67307	−0.570335	−0.285167	+	0.958478i	$0.592049\pi$
$182$	−1.09679	−0.0812993
$183$	0	0
$184$	−8.90321	−0.656353
$185$	0.541249	0.0397934
$186$	0	0
$187$	0.622216	0.0455009
$188$	−8.14764	−0.594228
$189$	0	0
$190$	−0.311108	−0.0225701
$191$	−12.0667	−0.873114	−0.436557	−	0.899677i	$-0.643802\pi$
$191$	−12.0667	−0.873114	−0.436557	+	0.899677i	$0.643802\pi$
$192$	0	0
$193$	19.3067	1.38972	0.694862	−	0.719143i	$-0.255465\pi$
$193$	19.3067	1.38972	0.694862	+	0.719143i	$0.255465\pi$
$194$	4.92396	0.353519
$195$	0	0
$196$	3.99063	0.285045
$197$	8.38271	0.597243	0.298622	−	0.954372i	$-0.403473\pi$
$197$	8.38271	0.597243	0.298622	+	0.954372i	$0.403473\pi$
$198$	0	0
$199$	0.990632	0.0702240	0.0351120	−	0.999383i	$-0.488821\pi$
$199$	0.990632	0.0702240	0.0351120	+	0.999383i	$0.488821\pi$
$200$	−1.21432	−0.0858654
$201$	0	0
$202$	−4.93041	−0.346902
$203$	0.576283	0.0404471
$204$	0	0
$205$	12.1684	0.849877
$206$	−1.89829	−0.132260
$207$	0	0
$208$	−5.45875	−0.378496
$209$	−1.31111	−0.0906912
$210$	0	0
$211$	−0.133353	−0.00918041	−0.00459020	−	0.999989i	$-0.501461\pi$
$211$	−0.133353	−0.00918041	−0.00459020	+	0.999989i	$0.501461\pi$
$212$	−14.3225	−0.983672
$213$	0	0
$214$	−4.69535	−0.320967
$215$	−6.02074	−0.410611
$216$	0	0
$217$	8.75557	0.594367
$218$	0.460282	0.0311743
$219$	0	0
$220$	−2.49532	−0.168234
$221$	−0.755569	−0.0508251
$222$	0	0
$223$	7.34614	0.491934	0.245967	−	0.969278i	$-0.420895\pi$
$223$	7.34614	0.491934	0.245967	+	0.969278i	$0.420895\pi$
$224$	7.73975	0.517134
$225$	0	0
$226$	−2.78277	−0.185107
$227$	−22.5161	−1.49444	−0.747222	−	0.664575i	$-0.768613\pi$
$227$	−22.5161	−1.49444	−0.747222	+	0.664575i	$0.768613\pi$
$228$	0	0
$229$	−6.22077	−0.411080	−0.205540	−	0.978649i	$-0.565895\pi$
$229$	−6.22077	−0.411080	−0.205540	+	0.978649i	$0.565895\pi$
$230$	2.28100	0.150404
$231$	0	0
$232$	−0.316030	−0.0207484
$233$	13.3176	0.872462	0.436231	−	0.899835i	$-0.356313\pi$
$233$	13.3176	0.872462	0.436231	+	0.899835i	$0.356313\pi$
$234$	0	0
$235$	4.28100	0.279261
$236$	−15.1111	−0.983647
$237$	0	0
$238$	0.326929	0.0211917
$239$	7.05731	0.456499	0.228250	−	0.973603i	$-0.426700\pi$
$239$	7.05731	0.456499	0.228250	+	0.973603i	$0.426700\pi$
$240$	0	0
$241$	−8.36842	−0.539057	−0.269529	−	0.962992i	$-0.586868\pi$
$241$	−8.36842	−0.539057	−0.269529	+	0.962992i	$0.586868\pi$
$242$	−2.88739	−0.185608
$243$	0	0
$244$	5.27163	0.337481
$245$	−2.09679	−0.133959
$246$	0	0
$247$	1.59210	0.101303
$248$	−4.80150	−0.304896
$249$	0	0
$250$	0.311108	0.0196762
$251$	7.35260	0.464092	0.232046	−	0.972705i	$-0.425458\pi$
$251$	7.35260	0.464092	0.232046	+	0.972705i	$0.425458\pi$
$252$	0	0
$253$	9.61285	0.604355
$254$	1.05086	0.0659365
$255$	0	0
$256$	8.80642	0.550401
$257$	10.2395	0.638723	0.319362	−	0.947633i	$-0.396532\pi$
$257$	10.2395	0.638723	0.319362	+	0.947633i	$0.396532\pi$
$258$	0	0
$259$	1.19850	0.0744711
$260$	3.03011	0.187920
$261$	0	0
$262$	−2.58274	−0.159562
$263$	−27.4652	−1.69358	−0.846789	−	0.531930i	$-0.821467\pi$
$263$	−27.4652	−1.69358	−0.846789	+	0.531930i	$0.821467\pi$
$264$	0	0
$265$	7.52543	0.462283
$266$	−0.688892	−0.0422387
$267$	0	0
$268$	−9.86665	−0.602701
$269$	3.07604	0.187550	0.0937749	−	0.995593i	$-0.470107\pi$
$269$	3.07604	0.187550	0.0937749	+	0.995593i	$0.470107\pi$
$270$	0	0
$271$	11.7748	0.715267	0.357634	−	0.933862i	$-0.383584\pi$
$271$	11.7748	0.715267	0.357634	+	0.933862i	$0.383584\pi$
$272$	1.62714	0.0986597
$273$	0	0
$274$	0.930409	0.0562081
$275$	1.31111	0.0790628
$276$	0	0
$277$	2.94914	0.177197	0.0885985	−	0.996067i	$-0.471761\pi$
$277$	2.94914	0.177197	0.0885985	+	0.996067i	$0.471761\pi$
$278$	−5.55262	−0.333024
$279$	0	0
$280$	−2.68889	−0.160692
$281$	10.2603	0.612075	0.306038	−	0.952019i	$-0.400997\pi$
$281$	10.2603	0.612075	0.306038	+	0.952019i	$0.400997\pi$
$282$	0	0
$283$	24.6113	1.46299	0.731495	−	0.681846i	$-0.238823\pi$
$283$	24.6113	1.46299	0.731495	+	0.681846i	$0.238823\pi$
$284$	−18.0415	−1.07057
$285$	0	0
$286$	−0.649413	−0.0384006
$287$	26.9447	1.59050
$288$	0	0
$289$	−16.7748	−0.986752
$290$	0.0809666	0.00475453
$291$	0	0
$292$	10.7971	0.631850
$293$	3.39207	0.198167	0.0990836	−	0.995079i	$-0.468409\pi$
$293$	3.39207	0.198167	0.0990836	+	0.995079i	$0.468409\pi$
$294$	0	0
$295$	7.93978	0.462272
$296$	−0.657249	−0.0382018
$297$	0	0
$298$	−4.38715	−0.254141
$299$	−11.6731	−0.675071
$300$	0	0
$301$	−13.3319	−0.768435
$302$	−5.09679	−0.293287
$303$	0	0
$304$	−3.42864	−0.196646
$305$	−2.76986	−0.158602
$306$	0	0
$307$	−24.8069	−1.41580	−0.707902	−	0.706310i	$-0.750358\pi$
$307$	−24.8069	−1.41580	−0.707902	+	0.706310i	$0.750358\pi$
$308$	−5.52543	−0.314840
$309$	0	0
$310$	1.23014	0.0698673
$311$	−25.0672	−1.42143	−0.710714	−	0.703481i	$-0.751628\pi$
$311$	−25.0672	−1.42143	−0.710714	+	0.703481i	$0.751628\pi$
$312$	0	0
$313$	−15.9081	−0.899181	−0.449590	−	0.893235i	$-0.648430\pi$
$313$	−15.9081	−0.899181	−0.449590	+	0.893235i	$0.648430\pi$
$314$	4.77430	0.269430
$315$	0	0
$316$	7.35905	0.413979
$317$	−22.3225	−1.25376	−0.626878	−	0.779118i	$-0.715667\pi$
$317$	−22.3225	−1.25376	−0.626878	+	0.779118i	$0.715667\pi$
$318$	0	0
$319$	0.341219	0.0191046
$320$	−5.76986	−0.322545
$321$	0	0
$322$	5.05086	0.281473
$323$	−0.474572	−0.0264059
$324$	0	0
$325$	−1.59210	−0.0883140
$326$	−0.555539	−0.0307685
$327$	0	0
$328$	−14.7763	−0.815885
$329$	9.47949	0.522621
$330$	0	0
$331$	−3.25872	−0.179116	−0.0895578	−	0.995982i	$-0.528545\pi$
$331$	−3.25872	−0.179116	−0.0895578	+	0.995982i	$0.528545\pi$
$332$	13.9541	0.765829
$333$	0	0
$334$	2.70964	0.148265
$335$	5.18421	0.283244
$336$	0	0
$337$	−16.6242	−0.905579	−0.452790	−	0.891617i	$-0.649571\pi$
$337$	−16.6242	−0.905579	−0.452790	+	0.891617i	$0.649571\pi$
$338$	−3.25581	−0.177093
$339$	0	0
$340$	−0.903212	−0.0489835
$341$	5.18421	0.280741
$342$	0	0
$343$	−20.1432	−1.08763
$344$	7.31111	0.394189
$345$	0	0
$346$	5.09679	0.274005
$347$	10.2079	0.547987	0.273993	−	0.961732i	$-0.411655\pi$
$347$	10.2079	0.547987	0.273993	+	0.961732i	$0.411655\pi$
$348$	0	0
$349$	−19.5526	−1.04663	−0.523314	−	0.852140i	$-0.675305\pi$
$349$	−19.5526	−1.04663	−0.523314	+	0.852140i	$0.675305\pi$
$350$	0.688892	0.0368228
$351$	0	0
$352$	4.58274	0.244261
$353$	23.7146	1.26220	0.631099	−	0.775702i	$-0.282604\pi$
$353$	23.7146	1.26220	0.631099	+	0.775702i	$0.282604\pi$
$354$	0	0
$355$	9.47949	0.503119
$356$	−5.11753	−0.271229
$357$	0	0
$358$	−1.43801	−0.0760011
$359$	−14.4351	−0.761855	−0.380928	−	0.924605i	$-0.624395\pi$
$359$	−14.4351	−0.761855	−0.380928	+	0.924605i	$0.624395\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−2.38715	−0.125466
$363$	0	0
$364$	6.70964	0.351680
$365$	−5.67307	−0.296942
$366$	0	0
$367$	14.9195	0.778792	0.389396	−	0.921070i	$-0.372684\pi$
$367$	14.9195	0.778792	0.389396	+	0.921070i	$0.372684\pi$
$368$	25.1383	1.31042
$369$	0	0
$370$	0.168387	0.00875401
$371$	16.6637	0.865136
$372$	0	0
$373$	33.0716	1.71238	0.856191	−	0.516659i	$-0.172825\pi$
$373$	33.0716	1.71238	0.856191	+	0.516659i	$0.172825\pi$
$374$	0.193576	0.0100096
$375$	0	0
$376$	−5.19850	−0.268092
$377$	−0.414349	−0.0213401
$378$	0	0
$379$	23.3131	1.19751	0.598757	−	0.800931i	$-0.295661\pi$
$379$	23.3131	1.19751	0.598757	+	0.800931i	$0.295661\pi$
$380$	1.90321	0.0976327
$381$	0	0
$382$	−3.75404	−0.192073
$383$	8.94914	0.457280	0.228640	−	0.973511i	$-0.426572\pi$
$383$	8.94914	0.457280	0.228640	+	0.973511i	$0.426572\pi$
$384$	0	0
$385$	2.90321	0.147961
$386$	6.00645	0.305720
$387$	0	0
$388$	−30.1225	−1.52924
$389$	30.2034	1.53137	0.765687	−	0.643213i	$-0.222399\pi$
$389$	30.2034	1.53137	0.765687	+	0.643213i	$0.222399\pi$
$390$	0	0
$391$	3.47949	0.175966
$392$	2.54617	0.128601
$393$	0	0
$394$	2.60793	0.131385
$395$	−3.86665	−0.194552
$396$	0	0
$397$	18.7654	0.941809	0.470905	−	0.882184i	$-0.343928\pi$
$397$	18.7654	0.941809	0.470905	+	0.882184i	$0.343928\pi$
$398$	0.308193	0.0154483
$399$	0	0
$400$	3.42864	0.171432
$401$	39.3526	1.96517	0.982587	−	0.185801i	$-0.0594880\pi$
$401$	39.3526	1.96517	0.982587	+	0.185801i	$0.0594880\pi$
$402$	0	0
$403$	−6.29529	−0.313591
$404$	30.1619	1.50061
$405$	0	0
$406$	0.179286	0.00889782
$407$	0.709636	0.0351753
$408$	0	0
$409$	−26.2034	−1.29568	−0.647838	−	0.761778i	$-0.724327\pi$
$409$	−26.2034	−1.29568	−0.647838	+	0.761778i	$0.724327\pi$
$410$	3.78568	0.186961
$411$	0	0
$412$	11.6128	0.572124
$413$	17.5812	0.865115
$414$	0	0
$415$	−7.33185	−0.359906
$416$	−5.56491	−0.272842
$417$	0	0
$418$	−0.407896	−0.0199508
$419$	−21.9432	−1.07199	−0.535997	−	0.844220i	$-0.680064\pi$
$419$	−21.9432	−1.07199	−0.535997	+	0.844220i	$0.680064\pi$
$420$	0	0
$421$	3.84791	0.187536	0.0937679	−	0.995594i	$-0.470109\pi$
$421$	3.84791	0.187536	0.0937679	+	0.995594i	$0.470109\pi$
$422$	−0.0414872	−0.00201956
$423$	0	0
$424$	−9.13828	−0.443794
$425$	0.474572	0.0230201
$426$	0	0
$427$	−6.13335	−0.296814
$428$	28.7239	1.38842
$429$	0	0
$430$	−1.87310	−0.0903289
$431$	−14.9175	−0.718551	−0.359275	−	0.933232i	$-0.616976\pi$
$431$	−14.9175	−0.718551	−0.359275	+	0.933232i	$0.616976\pi$
$432$	0	0
$433$	−3.23659	−0.155541	−0.0777704	−	0.996971i	$-0.524780\pi$
$433$	−3.23659	−0.155541	−0.0777704	+	0.996971i	$0.524780\pi$
$434$	2.72393	0.130753
$435$	0	0
$436$	−2.81579	−0.134852
$437$	−7.33185	−0.350730
$438$	0	0
$439$	−30.1847	−1.44064	−0.720318	−	0.693644i	$-0.756004\pi$
$439$	−30.1847	−1.44064	−0.720318	+	0.693644i	$0.756004\pi$
$440$	−1.59210	−0.0759006
$441$	0	0
$442$	−0.235063	−0.0111808
$443$	−35.9639	−1.70870	−0.854348	−	0.519701i	$-0.826044\pi$
$443$	−35.9639	−1.70870	−0.854348	+	0.519701i	$0.826044\pi$
$444$	0	0
$445$	2.68889	0.127466
$446$	2.28544	0.108219
$447$	0	0
$448$	−12.7763	−0.603624
$449$	28.9052	1.36412	0.682061	−	0.731295i	$-0.261084\pi$
$449$	28.9052	1.36412	0.682061	+	0.731295i	$0.261084\pi$
$450$	0	0
$451$	15.9541	0.751248
$452$	17.0237	0.800726
$453$	0	0
$454$	−7.00492	−0.328757
$455$	−3.52543	−0.165275
$456$	0	0
$457$	30.7467	1.43827	0.719134	−	0.694871i	$-0.244539\pi$
$457$	30.7467	1.43827	0.719134	+	0.694871i	$0.244539\pi$
$458$	−1.93533	−0.0904321
$459$	0	0
$460$	−13.9541	−0.650612
$461$	20.7368	0.965811	0.482905	−	0.875673i	$-0.339582\pi$
$461$	20.7368	0.965811	0.482905	+	0.875673i	$0.339582\pi$
$462$	0	0
$463$	22.5412	1.04758	0.523790	−	0.851847i	$-0.324518\pi$
$463$	22.5412	1.04758	0.523790	+	0.851847i	$0.324518\pi$
$464$	0.892313	0.0414246
$465$	0	0
$466$	4.14320	0.191930
$467$	17.8622	0.826564	0.413282	−	0.910603i	$-0.364382\pi$
$467$	17.8622	0.826564	0.413282	+	0.910603i	$0.364382\pi$
$468$	0	0
$469$	11.4795	0.530074
$470$	1.33185	0.0614337
$471$	0	0
$472$	−9.64143	−0.443783
$473$	−7.89384	−0.362959
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	−2.00000	−0.0916698
$477$	0	0
$478$	2.19558	0.100424
$479$	−28.5970	−1.30663	−0.653316	−	0.757086i	$-0.726623\pi$
$479$	−28.5970	−1.30663	−0.653316	+	0.757086i	$0.726623\pi$
$480$	0	0
$481$	−0.861725	−0.0392913
$482$	−2.60348	−0.118585
$483$	0	0
$484$	17.6637	0.802896
$485$	15.8272	0.718675
$486$	0	0
$487$	−30.3497	−1.37528	−0.687638	−	0.726054i	$-0.741352\pi$
$487$	−30.3497	−1.37528	−0.687638	+	0.726054i	$0.741352\pi$
$488$	3.36349	0.152258
$489$	0	0
$490$	−0.652327	−0.0294691
$491$	−8.45383	−0.381516	−0.190758	−	0.981637i	$-0.561095\pi$
$491$	−8.45383	−0.381516	−0.190758	+	0.981637i	$0.561095\pi$
$492$	0	0
$493$	0.123509	0.00556255
$494$	0.495316	0.0222853
$495$	0	0
$496$	13.5571	0.608730
$497$	20.9906	0.941559
$498$	0	0
$499$	12.7239	0.569601	0.284801	−	0.958587i	$-0.408073\pi$
$499$	12.7239	0.569601	0.284801	+	0.958587i	$0.408073\pi$
$500$	−1.90321	−0.0851142
$501$	0	0
$502$	2.28745	0.102094
$503$	22.1289	0.986679	0.493340	−	0.869837i	$-0.335776\pi$
$503$	22.1289	0.986679	0.493340	+	0.869837i	$0.335776\pi$
$504$	0	0
$505$	−15.8479	−0.705223
$506$	2.99063	0.132950
$507$	0	0
$508$	−6.42864	−0.285225
$509$	16.9556	0.751544	0.375772	−	0.926712i	$-0.377378\pi$
$509$	16.9556	0.751544	0.375772	+	0.926712i	$0.377378\pi$
$510$	0	0
$511$	−12.5620	−0.555710
$512$	20.3111	0.897633
$513$	0	0
$514$	3.18559	0.140510
$515$	−6.10171	−0.268873
$516$	0	0
$517$	5.61285	0.246853
$518$	0.372862	0.0163826
$519$	0	0
$520$	1.93332	0.0847818
$521$	−31.6291	−1.38570	−0.692849	−	0.721083i	$-0.743645\pi$
$521$	−31.6291	−1.38570	−0.692849	+	0.721083i	$0.743645\pi$
$522$	0	0
$523$	9.14272	0.399783	0.199892	−	0.979818i	$-0.435941\pi$
$523$	9.14272	0.399783	0.199892	+	0.979818i	$0.435941\pi$
$524$	15.8000	0.690225
$525$	0	0
$526$	−8.54464	−0.372564
$527$	1.87649	0.0817412
$528$	0	0
$529$	30.7560	1.33722
$530$	2.34122	0.101696
$531$	0	0
$532$	4.21432	0.182714
$533$	−19.3733	−0.839152
$534$	0	0
$535$	−15.0923	−0.652499
$536$	−6.29529	−0.271915
$537$	0	0
$538$	0.956981	0.0412584
$539$	−2.74912	−0.118413
$540$	0	0
$541$	−0.488863	−0.0210178	−0.0105089	−	0.999945i	$-0.503345\pi$
$541$	−0.488863	−0.0210178	−0.0105089	+	0.999945i	$0.503345\pi$
$542$	3.66323	0.157349
$543$	0	0
$544$	1.65878	0.0711196
$545$	1.47949	0.0633746
$546$	0	0
$547$	−8.42864	−0.360383	−0.180191	−	0.983632i	$-0.557672\pi$
$547$	−8.42864	−0.360383	−0.180191	+	0.983632i	$0.557672\pi$
$548$	−5.69181	−0.243142
$549$	0	0
$550$	0.407896	0.0173927
$551$	−0.260253	−0.0110871
$552$	0	0
$553$	−8.56199	−0.364093
$554$	0.917502	0.0389809
$555$	0	0
$556$	33.9684	1.44058
$557$	−5.57136	−0.236066	−0.118033	−	0.993010i	$-0.537659\pi$
$557$	−5.57136	−0.236066	−0.118033	+	0.993010i	$0.537659\pi$
$558$	0	0
$559$	9.58565	0.405430
$560$	7.59210	0.320825
$561$	0	0
$562$	3.19204	0.134648
$563$	9.68736	0.408274	0.204137	−	0.978942i	$-0.434561\pi$
$563$	9.68736	0.408274	0.204137	+	0.978942i	$0.434561\pi$
$564$	0	0
$565$	−8.94470	−0.376306
$566$	7.65677	0.321838
$567$	0	0
$568$	−11.5111	−0.482997
$569$	16.3718	0.686342	0.343171	−	0.939273i	$-0.388499\pi$
$569$	16.3718	0.686342	0.343171	+	0.939273i	$0.388499\pi$
$570$	0	0
$571$	−12.9491	−0.541905	−0.270952	−	0.962593i	$-0.587339\pi$
$571$	−12.9491	−0.541905	−0.270952	+	0.962593i	$0.587339\pi$
$572$	3.97280	0.166111
$573$	0	0
$574$	8.38271	0.349887
$575$	7.33185	0.305759
$576$	0	0
$577$	0.266706	0.0111031	0.00555156	−	0.999985i	$-0.498233\pi$
$577$	0.266706	0.0111031	0.00555156	+	0.999985i	$0.498233\pi$
$578$	−5.21877	−0.217072
$579$	0	0
$580$	−0.495316	−0.0205669
$581$	−16.2351	−0.673544
$582$	0	0
$583$	9.86665	0.408635
$584$	6.88892	0.285066
$585$	0	0
$586$	1.05530	0.0435941
$587$	−23.1985	−0.957504	−0.478752	−	0.877950i	$-0.658911\pi$
$587$	−23.1985	−0.957504	−0.478752	+	0.877950i	$0.658911\pi$
$588$	0	0
$589$	−3.95407	−0.162924
$590$	2.47013	0.101694
$591$	0	0
$592$	1.85575	0.0762708
$593$	−19.1655	−0.787032	−0.393516	−	0.919318i	$-0.628741\pi$
$593$	−19.1655	−0.787032	−0.393516	+	0.919318i	$0.628741\pi$
$594$	0	0
$595$	1.05086	0.0430809
$596$	26.8385	1.09935
$597$	0	0
$598$	−3.63158	−0.148506
$599$	29.1052	1.18921	0.594604	−	0.804019i	$-0.297309\pi$
$599$	29.1052	1.18921	0.594604	+	0.804019i	$0.297309\pi$
$600$	0	0
$601$	−29.9210	−1.22050	−0.610252	−	0.792207i	$-0.708932\pi$
$601$	−29.9210	−1.22050	−0.610252	+	0.792207i	$0.708932\pi$
$602$	−4.14764	−0.169045
$603$	0	0
$604$	31.1798	1.26869
$605$	−9.28100	−0.377326
$606$	0	0
$607$	−41.3461	−1.67819	−0.839094	−	0.543986i	$-0.816914\pi$
$607$	−41.3461	−1.67819	−0.839094	+	0.543986i	$0.816914\pi$
$608$	−3.49532	−0.141754
$609$	0	0
$610$	−0.861725	−0.0348902
$611$	−6.81579	−0.275737
$612$	0	0
$613$	−7.58427	−0.306326	−0.153163	−	0.988201i	$-0.548946\pi$
$613$	−7.58427	−0.306326	−0.153163	+	0.988201i	$0.548946\pi$
$614$	−7.71762	−0.311458
$615$	0	0
$616$	−3.52543	−0.142044
$617$	19.6173	0.789762	0.394881	−	0.918732i	$-0.370786\pi$
$617$	19.6173	0.789762	0.394881	+	0.918732i	$0.370786\pi$
$618$	0	0
$619$	24.1748	0.971669	0.485834	−	0.874051i	$-0.338516\pi$
$619$	24.1748	0.971669	0.485834	+	0.874051i	$0.338516\pi$
$620$	−7.52543	−0.302228
$621$	0	0
$622$	−7.79859	−0.312695
$623$	5.95407	0.238545
$624$	0	0
$625$	1.00000	0.0400000
$626$	−4.94914	−0.197808
$627$	0	0
$628$	−29.2070	−1.16548
$629$	0.256862	0.0102418
$630$	0	0
$631$	46.9719	1.86992	0.934961	−	0.354751i	$-0.115434\pi$
$631$	46.9719	1.86992	0.934961	+	0.354751i	$0.115434\pi$
$632$	4.69535	0.186771
$633$	0	0
$634$	−6.94470	−0.275809
$635$	3.37778	0.134043
$636$	0	0
$637$	3.33830	0.132268
$638$	0.106156	0.00420276
$639$	0	0
$640$	−8.78568	−0.347285
$641$	−42.5466	−1.68049	−0.840246	−	0.542206i	$-0.817589\pi$
$641$	−42.5466	−1.68049	−0.840246	+	0.542206i	$0.817589\pi$
$642$	0	0
$643$	27.5605	1.08688	0.543439	−	0.839449i	$-0.317122\pi$
$643$	27.5605	1.08688	0.543439	+	0.839449i	$0.317122\pi$
$644$	−30.8988	−1.21758
$645$	0	0
$646$	−0.147643	−0.00580894
$647$	−47.8149	−1.87980	−0.939899	−	0.341454i	$-0.889081\pi$
$647$	−47.8149	−1.87980	−0.939899	+	0.341454i	$0.889081\pi$
$648$	0	0
$649$	10.4099	0.408624
$650$	−0.495316	−0.0194279
$651$	0	0
$652$	3.39853	0.133097
$653$	−18.5477	−0.725828	−0.362914	−	0.931823i	$-0.618218\pi$
$653$	−18.5477	−0.725828	−0.362914	+	0.931823i	$0.618218\pi$
$654$	0	0
$655$	−8.30174	−0.324376
$656$	41.7210	1.62893
$657$	0	0
$658$	2.94914	0.114970
$659$	−35.8608	−1.39694	−0.698470	−	0.715640i	$-0.746135\pi$
$659$	−35.8608	−1.39694	−0.698470	+	0.715640i	$0.746135\pi$
$660$	0	0
$661$	−18.5620	−0.721978	−0.360989	−	0.932570i	$-0.617561\pi$
$661$	−18.5620	−0.721978	−0.360989	+	0.932570i	$0.617561\pi$
$662$	−1.01381	−0.0394030
$663$	0	0
$664$	8.90321	0.345512
$665$	−2.21432	−0.0858676
$666$	0	0
$667$	1.90813	0.0738832
$668$	−16.5763	−0.641356
$669$	0	0
$670$	1.61285	0.0623097
$671$	−3.63158	−0.140196
$672$	0	0
$673$	1.82717	0.0704321	0.0352161	−	0.999380i	$-0.488788\pi$
$673$	1.82717	0.0704321	0.0352161	+	0.999380i	$0.488788\pi$
$674$	−5.17193	−0.199215
$675$	0	0
$676$	19.9175	0.766058
$677$	31.7703	1.22103	0.610517	−	0.792003i	$-0.290962\pi$
$677$	31.7703	1.22103	0.610517	+	0.792003i	$0.290962\pi$
$678$	0	0
$679$	35.0464	1.34496
$680$	−0.576283	−0.0220994
$681$	0	0
$682$	1.61285	0.0617591
$683$	−7.39652	−0.283020	−0.141510	−	0.989937i	$-0.545196\pi$
$683$	−7.39652	−0.283020	−0.141510	+	0.989937i	$0.545196\pi$
$684$	0	0
$685$	2.99063	0.114266
$686$	−6.26671	−0.239264
$687$	0	0
$688$	−20.6430	−0.787005
$689$	−11.9813	−0.456450
$690$	0	0
$691$	−9.33630	−0.355169	−0.177585	−	0.984106i	$-0.556828\pi$
$691$	−9.33630	−0.355169	−0.177585	+	0.984106i	$0.556828\pi$
$692$	−31.1798	−1.18528
$693$	0	0
$694$	3.17575	0.120550
$695$	−17.8479	−0.677010
$696$	0	0
$697$	5.77478	0.218735
$698$	−6.08297	−0.230244
$699$	0	0
$700$	−4.21432	−0.159286
$701$	−9.92687	−0.374933	−0.187466	−	0.982271i	$-0.560028\pi$
$701$	−9.92687	−0.374933	−0.187466	+	0.982271i	$0.560028\pi$
$702$	0	0
$703$	−0.541249	−0.0204136
$704$	−7.56491	−0.285113
$705$	0	0
$706$	7.37778	0.277667
$707$	−35.0923	−1.31978
$708$	0	0
$709$	−51.0563	−1.91746	−0.958729	−	0.284322i	$-0.908232\pi$
$709$	−51.0563	−1.91746	−0.958729	+	0.284322i	$0.908232\pi$
$710$	2.94914	0.110679
$711$	0	0
$712$	−3.26517	−0.122368
$713$	28.9906	1.08571
$714$	0	0
$715$	−2.08742	−0.0780651
$716$	8.79706	0.328761
$717$	0	0
$718$	−4.49087	−0.167598
$719$	−40.5555	−1.51247	−0.756233	−	0.654302i	$-0.772962\pi$
$719$	−40.5555	−1.51247	−0.756233	+	0.654302i	$0.772962\pi$
$720$	0	0
$721$	−13.5111	−0.503181
$722$	0.311108	0.0115782
$723$	0	0
$724$	14.6035	0.542734
$725$	0.260253	0.00966554
$726$	0	0
$727$	6.25581	0.232015	0.116008	−	0.993248i	$-0.462990\pi$
$727$	6.25581	0.232015	0.116008	+	0.993248i	$0.462990\pi$
$728$	4.28100	0.158664
$729$	0	0
$730$	−1.76494	−0.0653232
$731$	−2.85728	−0.105680
$732$	0	0
$733$	23.0638	0.851879	0.425940	−	0.904752i	$-0.359944\pi$
$733$	23.0638	0.851879	0.425940	+	0.904752i	$0.359944\pi$
$734$	4.64158	0.171324
$735$	0	0
$736$	25.6271	0.944629
$737$	6.79706	0.250373
$738$	0	0
$739$	−14.7052	−0.540939	−0.270470	−	0.962729i	$-0.587179\pi$
$739$	−14.7052	−0.540939	−0.270470	+	0.962729i	$0.587179\pi$
$740$	−1.03011	−0.0378677
$741$	0	0
$742$	5.18421	0.190318
$743$	46.6004	1.70960	0.854802	−	0.518955i	$-0.173679\pi$
$743$	46.6004	1.70960	0.854802	+	0.518955i	$0.173679\pi$
$744$	0	0
$745$	−14.1017	−0.516647
$746$	10.2888	0.376701
$747$	0	0
$748$	−1.18421	−0.0432989
$749$	−33.4193	−1.22111
$750$	0	0
$751$	−44.2908	−1.61620	−0.808098	−	0.589048i	$-0.799503\pi$
$751$	−44.2908	−1.61620	−0.808098	+	0.589048i	$0.799503\pi$
$752$	14.6780	0.535251
$753$	0	0
$754$	−0.128907	−0.00469453
$755$	−16.3827	−0.596228
$756$	0	0
$757$	−42.3912	−1.54073	−0.770367	−	0.637601i	$-0.779927\pi$
$757$	−42.3912	−1.54073	−0.770367	+	0.637601i	$0.779927\pi$
$758$	7.25289	0.263437
$759$	0	0
$760$	1.21432	0.0440480
$761$	20.0098	0.725356	0.362678	−	0.931914i	$-0.381862\pi$
$761$	20.0098	0.725356	0.362678	+	0.931914i	$0.381862\pi$
$762$	0	0
$763$	3.27607	0.118602
$764$	22.9654	0.830861
$765$	0	0
$766$	2.78415	0.100595
$767$	−12.6410	−0.456438
$768$	0	0
$769$	23.4479	0.845551	0.422776	−	0.906234i	$-0.361056\pi$
$769$	23.4479	0.845551	0.422776	+	0.906234i	$0.361056\pi$
$770$	0.903212	0.0325495
$771$	0	0
$772$	−36.7447	−1.32247
$773$	−40.9733	−1.47371	−0.736853	−	0.676053i	$-0.763689\pi$
$773$	−40.9733	−1.47371	−0.736853	+	0.676053i	$0.763689\pi$
$774$	0	0
$775$	3.95407	0.142034
$776$	−19.2192	−0.689931
$777$	0	0
$778$	9.39652	0.336882
$779$	−12.1684	−0.435978
$780$	0	0
$781$	12.4286	0.444732
$782$	1.08250	0.0387100
$783$	0	0
$784$	−7.18913	−0.256755
$785$	15.3461	0.547727
$786$	0	0
$787$	−36.1017	−1.28689	−0.643443	−	0.765494i	$-0.722495\pi$
$787$	−36.1017	−1.28689	−0.643443	+	0.765494i	$0.722495\pi$
$788$	−15.9541	−0.568340
$789$	0	0
$790$	−1.20294	−0.0427988
$791$	−19.8064	−0.704235
$792$	0	0
$793$	4.40990	0.156600
$794$	5.83807	0.207185
$795$	0	0
$796$	−1.88538	−0.0668256
$797$	−3.77923	−0.133867	−0.0669335	−	0.997757i	$-0.521322\pi$
$797$	−3.77923	−0.133867	−0.0669335	+	0.997757i	$0.521322\pi$
$798$	0	0
$799$	2.03164	0.0718744
$800$	3.49532	0.123578
$801$	0	0
$802$	12.2429	0.432312
$803$	−7.43801	−0.262482
$804$	0	0
$805$	16.2351	0.572211
$806$	−1.95851	−0.0689857
$807$	0	0
$808$	19.2444	0.677017
$809$	−19.6128	−0.689551	−0.344776	−	0.938685i	$-0.612045\pi$
$809$	−19.6128	−0.689551	−0.344776	+	0.938685i	$0.612045\pi$
$810$	0	0
$811$	37.4479	1.31497	0.657486	−	0.753467i	$-0.271620\pi$
$811$	37.4479	1.31497	0.657486	+	0.753467i	$0.271620\pi$
$812$	−1.09679	−0.0384897
$813$	0	0
$814$	0.220773	0.00773810
$815$	−1.78568	−0.0625497
$816$	0	0
$817$	6.02074	0.210639
$818$	−8.15209	−0.285031
$819$	0	0
$820$	−23.1590	−0.808748
$821$	−38.8702	−1.35658	−0.678289	−	0.734795i	$-0.737278\pi$
$821$	−38.8702	−1.35658	−0.678289	+	0.734795i	$0.737278\pi$
$822$	0	0
$823$	51.9605	1.81123	0.905615	−	0.424101i	$-0.139410\pi$
$823$	51.9605	1.81123	0.905615	+	0.424101i	$0.139410\pi$
$824$	7.40943	0.258120
$825$	0	0
$826$	5.46965	0.190313
$827$	18.5161	0.643866	0.321933	−	0.946762i	$-0.395667\pi$
$827$	18.5161	0.643866	0.321933	+	0.946762i	$0.395667\pi$
$828$	0	0
$829$	41.0134	1.42445	0.712227	−	0.701949i	$-0.247687\pi$
$829$	41.0134	1.42445	0.712227	+	0.701949i	$0.247687\pi$
$830$	−2.28100	−0.0791745
$831$	0	0
$832$	9.18622	0.318475
$833$	−0.995078	−0.0344774
$834$	0	0
$835$	8.70964	0.301409
$836$	2.49532	0.0863023
$837$	0	0
$838$	−6.82669	−0.235824
$839$	28.8256	0.995171	0.497586	−	0.867415i	$-0.334220\pi$
$839$	28.8256	0.995171	0.497586	+	0.867415i	$0.334220\pi$
$840$	0	0
$841$	−28.9323	−0.997664
$842$	1.19712	0.0412553
$843$	0	0
$844$	0.253799	0.00873613
$845$	−10.4652	−0.360014
$846$	0	0
$847$	−20.5511	−0.706144
$848$	25.8020	0.886044
$849$	0	0
$850$	0.147643	0.00506412
$851$	3.96836	0.136034
$852$	0	0
$853$	11.0825	0.379458	0.189729	−	0.981837i	$-0.439239\pi$
$853$	11.0825	0.379458	0.189729	+	0.981837i	$0.439239\pi$
$854$	−1.90813	−0.0652950
$855$	0	0
$856$	18.3269	0.626402
$857$	−10.3640	−0.354026	−0.177013	−	0.984208i	$-0.556644\pi$
$857$	−10.3640	−0.354026	−0.177013	+	0.984208i	$0.556644\pi$
$858$	0	0
$859$	11.8381	0.403910	0.201955	−	0.979395i	$-0.435271\pi$
$859$	11.8381	0.403910	0.201955	+	0.979395i	$0.435271\pi$
$860$	11.4588	0.390740
$861$	0	0
$862$	−4.64095	−0.158071
$863$	−25.9684	−0.883973	−0.441987	−	0.897022i	$-0.645726\pi$
$863$	−25.9684	−0.883973	−0.441987	+	0.897022i	$0.645726\pi$
$864$	0	0
$865$	16.3827	0.557029
$866$	−1.00693	−0.0342169
$867$	0	0
$868$	−16.6637	−0.565603
$869$	−5.06959	−0.171974
$870$	0	0
$871$	−8.25380	−0.279669
$872$	−1.79658	−0.0608399
$873$	0	0
$874$	−2.28100	−0.0771559
$875$	2.21432	0.0748577
$876$	0	0
$877$	−56.0533	−1.89279	−0.946393	−	0.323016i	$-0.895303\pi$
$877$	−56.0533	−1.89279	−0.946393	+	0.323016i	$0.895303\pi$
$878$	−9.39069	−0.316920
$879$	0	0
$880$	4.49532	0.151537
$881$	36.2449	1.22112	0.610561	−	0.791969i	$-0.290944\pi$
$881$	36.2449	1.22112	0.610561	+	0.791969i	$0.290944\pi$
$882$	0	0
$883$	−9.61084	−0.323431	−0.161715	−	0.986837i	$-0.551703\pi$
$883$	−9.61084	−0.323431	−0.161715	+	0.986837i	$0.551703\pi$
$884$	1.43801	0.0483654
$885$	0	0
$886$	−11.1887	−0.375890
$887$	−2.22216	−0.0746127	−0.0373064	−	0.999304i	$-0.511878\pi$
$887$	−2.22216	−0.0746127	−0.0373064	+	0.999304i	$0.511878\pi$
$888$	0	0
$889$	7.47949	0.250854
$890$	0.836535	0.0280407
$891$	0	0
$892$	−13.9813	−0.468127
$893$	−4.28100	−0.143258
$894$	0	0
$895$	−4.62222	−0.154504
$896$	−19.4543	−0.649923
$897$	0	0
$898$	8.99264	0.300088
$899$	1.02906	0.0343210
$900$	0	0
$901$	3.57136	0.118979
$902$	4.96343	0.165264
$903$	0	0
$904$	10.8617	0.361256
$905$	−7.67307	−0.255062
$906$	0	0
$907$	−46.6735	−1.54977	−0.774885	−	0.632102i	$-0.782192\pi$
$907$	−46.6735	−1.54977	−0.774885	+	0.632102i	$0.782192\pi$
$908$	42.8528	1.42212
$909$	0	0
$910$	−1.09679	−0.0363582
$911$	−23.6128	−0.782329	−0.391164	−	0.920321i	$-0.627928\pi$
$911$	−23.6128	−0.782329	−0.391164	+	0.920321i	$0.627928\pi$
$912$	0	0
$913$	−9.61285	−0.318139
$914$	9.56553	0.316400
$915$	0	0
$916$	11.8394	0.391186
$917$	−18.3827	−0.607050
$918$	0	0
$919$	−9.75605	−0.321822	−0.160911	−	0.986969i	$-0.551443\pi$
$919$	−9.75605	−0.321822	−0.160911	+	0.986969i	$0.551443\pi$
$920$	−8.90321	−0.293530
$921$	0	0
$922$	6.45139	0.212465
$923$	−15.0923	−0.496770
$924$	0	0
$925$	0.541249	0.0177962
$926$	7.01276	0.230454
$927$	0	0
$928$	0.909665	0.0298612
$929$	42.0415	1.37934	0.689668	−	0.724125i	$-0.257756\pi$
$929$	42.0415	1.37934	0.689668	+	0.724125i	$0.257756\pi$
$930$	0	0
$931$	2.09679	0.0687195
$932$	−25.3461	−0.830240
$933$	0	0
$934$	5.55707	0.181833
$935$	0.622216	0.0203486
$936$	0	0
$937$	−5.00937	−0.163649	−0.0818245	−	0.996647i	$-0.526075\pi$
$937$	−5.00937	−0.163649	−0.0818245	+	0.996647i	$0.526075\pi$
$938$	3.57136	0.116609
$939$	0	0
$940$	−8.14764	−0.265747
$941$	58.8834	1.91954	0.959772	−	0.280779i	$-0.0905929\pi$
$941$	58.8834	1.91954	0.959772	+	0.280779i	$0.0905929\pi$
$942$	0	0
$943$	89.2168	2.90530
$944$	27.2226	0.886021
$945$	0	0
$946$	−2.45584	−0.0798461
$947$	−50.1990	−1.63125	−0.815624	−	0.578583i	$-0.803606\pi$
$947$	−50.1990	−1.63125	−0.815624	+	0.578583i	$0.803606\pi$
$948$	0	0
$949$	9.03212	0.293195
$950$	−0.311108	−0.0100937
$951$	0	0
$952$	−1.27607	−0.0413578
$953$	4.17929	0.135380	0.0676902	−	0.997706i	$-0.478437\pi$
$953$	4.17929	0.135380	0.0676902	+	0.997706i	$0.478437\pi$
$954$	0	0
$955$	−12.0667	−0.390468
$956$	−13.4316	−0.434407
$957$	0	0
$958$	−8.89676	−0.287441
$959$	6.62222	0.213842
$960$	0	0
$961$	−15.3654	−0.495657
$962$	−0.268089	−0.00864355
$963$	0	0
$964$	15.9269	0.512970
$965$	19.3067	0.621503
$966$	0	0
$967$	−46.1225	−1.48320	−0.741599	−	0.670843i	$-0.765932\pi$
$967$	−46.1225	−1.48320	−0.741599	+	0.670843i	$0.765932\pi$
$968$	11.2701	0.362235
$969$	0	0
$970$	4.92396	0.158099
$971$	13.9170	0.446619	0.223309	−	0.974748i	$-0.428314\pi$
$971$	13.9170	0.446619	0.223309	+	0.974748i	$0.428314\pi$
$972$	0	0
$973$	−39.5210	−1.26698
$974$	−9.44202	−0.302542
$975$	0	0
$976$	−9.49685	−0.303987
$977$	−12.5161	−0.400424	−0.200212	−	0.979753i	$-0.564163\pi$
$977$	−12.5161	−0.400424	−0.200212	+	0.979753i	$0.564163\pi$
$978$	0	0
$979$	3.52543	0.112673
$980$	3.99063	0.127476
$981$	0	0
$982$	−2.63005	−0.0839283
$983$	−56.7926	−1.81140	−0.905701	−	0.423916i	$-0.860655\pi$
$983$	−56.7926	−1.81140	−0.905701	+	0.423916i	$0.860655\pi$
$984$	0	0
$985$	8.38271	0.267095
$986$	0.0384245	0.00122369
$987$	0	0
$988$	−3.03011	−0.0964007
$989$	−44.1432	−1.40367
$990$	0	0
$991$	19.8163	0.629485	0.314742	−	0.949177i	$-0.398082\pi$
$991$	19.8163	0.629485	0.314742	+	0.949177i	$0.398082\pi$
$992$	13.8207	0.438808
$993$	0	0
$994$	6.53035	0.207130
$995$	0.990632	0.0314051
$996$	0	0
$997$	−7.52098	−0.238192	−0.119096	−	0.992883i	$-0.538000\pi$
$997$	−7.52098	−0.238192	−0.119096	+	0.992883i	$0.538000\pi$
$998$	3.95851	0.125305
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.a.l.1.2	yes	3
3.2	odd	2		855.2.a.h.1.2	✓	3
5.4	even	2		4275.2.a.bb.1.2		3
15.14	odd	2		4275.2.a.bj.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
855.2.a.h.1.2	✓	3	3.2	odd	2
855.2.a.l.1.2	yes	3	1.1	even	1	trivial
4275.2.a.bb.1.2		3	5.4	even	2
4275.2.a.bj.1.2		3	15.14	odd	2

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:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.a (trivial)

\(f(q)\)	\(=\)	\(q+0.311108 q^{2} -1.90321 q^{4} +1.00000 q^{5} +2.21432 q^{7} -1.21432 q^{8} +O(q^{10})\)	\(q+0.311108 q^{2} -1.90321 q^{4} +1.00000 q^{5} +2.21432 q^{7} -1.21432 q^{8} +0.311108 q^{10} +1.31111 q^{11} -1.59210 q^{13} +0.688892 q^{14} +3.42864 q^{16} +0.474572 q^{17} -1.00000 q^{19} -1.90321 q^{20} +0.407896 q^{22} +7.33185 q^{23} +1.00000 q^{25} -0.495316 q^{26} -4.21432 q^{28} +0.260253 q^{29} +3.95407 q^{31} +3.49532 q^{32} +0.147643 q^{34} +2.21432 q^{35} +0.541249 q^{37} -0.311108 q^{38} -1.21432 q^{40} +12.1684 q^{41} -6.02074 q^{43} -2.49532 q^{44} +2.28100 q^{46} +4.28100 q^{47} -2.09679 q^{49} +0.311108 q^{50} +3.03011 q^{52} +7.52543 q^{53} +1.31111 q^{55} -2.68889 q^{56} +0.0809666 q^{58} +7.93978 q^{59} -2.76986 q^{61} +1.23014 q^{62} -5.76986 q^{64} -1.59210 q^{65} +5.18421 q^{67} -0.903212 q^{68} +0.688892 q^{70} +9.47949 q^{71} -5.67307 q^{73} +0.168387 q^{74} +1.90321 q^{76} +2.90321 q^{77} -3.86665 q^{79} +3.42864 q^{80} +3.78568 q^{82} -7.33185 q^{83} +0.474572 q^{85} -1.87310 q^{86} -1.59210 q^{88} +2.68889 q^{89} -3.52543 q^{91} -13.9541 q^{92} +1.33185 q^{94} -1.00000 q^{95} +15.8272 q^{97} -0.652327 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + q^{4} + 3 q^{5} + 3 q^{8}+O(q^{10}) \) 3 * q + q^2 + q^4 + 3 * q^5 + 3 * q^8	\( 3 q + q^{2} + q^{4} + 3 q^{5} + 3 q^{8} + q^{10} + 4 q^{11} + 2 q^{13} + 2 q^{14} - 3 q^{16} + 8 q^{17} - 3 q^{19} + q^{20} + 8 q^{22} + 2 q^{23} + 3 q^{25} + 12 q^{26} - 6 q^{28} + 14 q^{29} - 8 q^{31} - 3 q^{32} - 6 q^{34} + 8 q^{37} - q^{38} + 3 q^{40} + 10 q^{41} + 2 q^{43} + 6 q^{44} + 6 q^{47} - 13 q^{49} + q^{50} + 16 q^{52} + 16 q^{53} + 4 q^{55} - 8 q^{56} - 6 q^{58} + 10 q^{59} - 2 q^{61} + 10 q^{62} - 11 q^{64} + 2 q^{65} + 2 q^{67} + 4 q^{68} + 2 q^{70} + 2 q^{71} - 4 q^{73} - 26 q^{74} - q^{76} + 2 q^{77} - 12 q^{79} - 3 q^{80} + 18 q^{82} - 2 q^{83} + 8 q^{85} + 8 q^{86} + 2 q^{88} + 8 q^{89} - 4 q^{91} - 22 q^{92} - 16 q^{94} - 3 q^{95} + 14 q^{97} - 9 q^{98}+O(q^{100}) \) 3 * q + q^2 + q^4 + 3 * q^5 + 3 * q^8 + q^10 + 4 * q^11 + 2 * q^13 + 2 * q^14 - 3 * q^16 + 8 * q^17 - 3 * q^19 + q^20 + 8 * q^22 + 2 * q^23 + 3 * q^25 + 12 * q^26 - 6 * q^28 + 14 * q^29 - 8 * q^31 - 3 * q^32 - 6 * q^34 + 8 * q^37 - q^38 + 3 * q^40 + 10 * q^41 + 2 * q^43 + 6 * q^44 + 6 * q^47 - 13 * q^49 + q^50 + 16 * q^52 + 16 * q^53 + 4 * q^55 - 8 * q^56 - 6 * q^58 + 10 * q^59 - 2 * q^61 + 10 * q^62 - 11 * q^64 + 2 * q^65 + 2 * q^67 + 4 * q^68 + 2 * q^70 + 2 * q^71 - 4 * q^73 - 26 * q^74 - q^76 + 2 * q^77 - 12 * q^79 - 3 * q^80 + 18 * q^82 - 2 * q^83 + 8 * q^85 + 8 * q^86 + 2 * q^88 + 8 * q^89 - 4 * q^91 - 22 * q^92 - 16 * q^94 - 3 * q^95 + 14 * q^97 - 9 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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