LMFDB - Embedded newform 1341.2.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1341,2,Mod(1,1341)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1341, 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("1341.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1341 = 3^{2} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1341.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:	$10.7079389111$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 12 x^{10} + 38 x^{9} + 46 x^{8} - 162 x^{7} - 59 x^{6} + 280 x^{5} - 14 x^{4} + \cdots - 3 $ x^12 - 3x^11 - 12x^10 + 38x^9 + 46x^8 - 162x^7 - 59x^6 + 280x^5 - 14x^4 - 173x^3 + 49x^2 + 11*x - 3
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			1.3
Root			$2.26491$ of defining polynomial
Character	$\chi$	$=$	1341.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.26491 q^{2} +3.12982 q^{4} -3.26427 q^{5} +2.33298 q^{7} -2.55895 q^{8} +O(q^{10})$	\(q-2.26491 q^{2} +3.12982 q^{4} -3.26427 q^{5} +2.33298 q^{7} -2.55895 q^{8} +7.39329 q^{10} +1.17074 q^{11} -0.357254 q^{13} -5.28399 q^{14} -0.463856 q^{16} -1.97482 q^{17} -4.92302 q^{19} -10.2166 q^{20} -2.65162 q^{22} -0.133314 q^{23} +5.65548 q^{25} +0.809148 q^{26} +7.30181 q^{28} +6.96702 q^{29} +2.15664 q^{31} +6.16849 q^{32} +4.47279 q^{34} -7.61548 q^{35} -1.96058 q^{37} +11.1502 q^{38} +8.35311 q^{40} +4.35387 q^{41} +2.67465 q^{43} +3.66421 q^{44} +0.301944 q^{46} +0.574948 q^{47} -1.55721 q^{49} -12.8092 q^{50} -1.11814 q^{52} -5.50183 q^{53} -3.82162 q^{55} -5.96997 q^{56} -15.7797 q^{58} +2.14099 q^{59} -11.6639 q^{61} -4.88460 q^{62} -13.0434 q^{64} +1.16617 q^{65} +15.4105 q^{67} -6.18083 q^{68} +17.2484 q^{70} -8.39677 q^{71} -1.93636 q^{73} +4.44055 q^{74} -15.4082 q^{76} +2.73131 q^{77} +1.22658 q^{79} +1.51415 q^{80} -9.86112 q^{82} -12.4131 q^{83} +6.44634 q^{85} -6.05784 q^{86} -2.99586 q^{88} -6.22010 q^{89} -0.833465 q^{91} -0.417249 q^{92} -1.30221 q^{94} +16.0701 q^{95} -5.70333 q^{97} +3.52694 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 9 q^{4} - 8 q^{5} - 2 q^{7} - 9 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 + 9 * q^4 - 8 * q^5 - 2 * q^7 - 9 * q^8	$ 12 q - 3 q^{2} + 9 q^{4} - 8 q^{5} - 2 q^{7} - 9 q^{8} - 22 q^{11} - 2 q^{13} - 12 q^{14} + 11 q^{16} - 8 q^{17} - 6 q^{19} - 24 q^{20} + 4 q^{22} - 12 q^{23} + 8 q^{25} - 26 q^{26} - 20 q^{28} - 16 q^{29} - 2 q^{31} - 21 q^{32} - 6 q^{34} - 32 q^{35} - 4 q^{37} - 15 q^{38} + 6 q^{40} - 24 q^{41} + 12 q^{43} - 37 q^{44} - 22 q^{46} - 26 q^{47} + 24 q^{49} + 5 q^{50} - 10 q^{52} - 20 q^{53} + 6 q^{55} + 13 q^{56} + 10 q^{58} - 72 q^{59} - 8 q^{61} + 2 q^{62} + q^{64} + 4 q^{65} - 14 q^{67} + 14 q^{68} + 18 q^{70} - 38 q^{71} + 27 q^{74} - 2 q^{76} - 6 q^{77} + 10 q^{79} - 56 q^{80} + 6 q^{82} - 42 q^{83} - 32 q^{85} + 14 q^{86} + 22 q^{88} - 44 q^{89} + 50 q^{92} + 2 q^{94} - 4 q^{95} - 22 q^{97} - 20 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 + 9 * q^4 - 8 * q^5 - 2 * q^7 - 9 * q^8 - 22 * q^11 - 2 * q^13 - 12 * q^14 + 11 * q^16 - 8 * q^17 - 6 * q^19 - 24 * q^20 + 4 * q^22 - 12 * q^23 + 8 * q^25 - 26 * q^26 - 20 * q^28 - 16 * q^29 - 2 * q^31 - 21 * q^32 - 6 * q^34 - 32 * q^35 - 4 * q^37 - 15 * q^38 + 6 * q^40 - 24 * q^41 + 12 * q^43 - 37 * q^44 - 22 * q^46 - 26 * q^47 + 24 * q^49 + 5 * q^50 - 10 * q^52 - 20 * q^53 + 6 * q^55 + 13 * q^56 + 10 * q^58 - 72 * q^59 - 8 * q^61 + 2 * q^62 + q^64 + 4 * q^65 - 14 * q^67 + 14 * q^68 + 18 * q^70 - 38 * q^71 + 27 * q^74 - 2 * q^76 - 6 * q^77 + 10 * q^79 - 56 * q^80 + 6 * q^82 - 42 * q^83 - 32 * q^85 + 14 * q^86 + 22 * q^88 - 44 * q^89 + 50 * q^92 + 2 * q^94 - 4 * q^95 - 22 * q^97 - 20 * 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 $

$2$	−2.26491	−1.60153	−0.800767	−	0.598976i	$-0.795575\pi$
$2$	−2.26491	−1.60153	−0.800767	+	0.598976i	$0.795575\pi$
$3$	0	0
$3$	0	0
$4$	3.12982	1.56491
$5$	−3.26427	−1.45983	−0.729914	−	0.683539i	$-0.760440\pi$
$5$	−3.26427	−1.45983	−0.729914	+	0.683539i	$0.760440\pi$
$6$	0	0
$7$	2.33298	0.881783	0.440892	−	0.897560i	$-0.354662\pi$
$7$	2.33298	0.881783	0.440892	+	0.897560i	$0.354662\pi$
$8$	−2.55895	−0.904725
$9$	0	0
$10$	7.39329	2.33796
$11$	1.17074	0.352992	0.176496	−	0.984301i	$-0.443524\pi$
$11$	1.17074	0.352992	0.176496	+	0.984301i	$0.443524\pi$
$12$	0	0
$13$	−0.357254	−0.0990843	−0.0495422	−	0.998772i	$-0.515776\pi$
$13$	−0.357254	−0.0990843	−0.0495422	+	0.998772i	$0.515776\pi$
$14$	−5.28399	−1.41221
$15$	0	0
$16$	−0.463856	−0.115964
$17$	−1.97482	−0.478963	−0.239482	−	0.970901i	$-0.576978\pi$
$17$	−1.97482	−0.478963	−0.239482	+	0.970901i	$0.576978\pi$
$18$	0	0
$19$	−4.92302	−1.12942	−0.564709	−	0.825290i	$-0.691011\pi$
$19$	−4.92302	−1.12942	−0.564709	+	0.825290i	$0.691011\pi$
$20$	−10.2166	−2.28450
$21$	0	0
$22$	−2.65162	−0.565328
$23$	−0.133314	−0.0277979	−0.0138989	−	0.999903i	$-0.504424\pi$
$23$	−0.133314	−0.0277979	−0.0138989	+	0.999903i	$0.504424\pi$
$24$	0	0
$25$	5.65548	1.13110
$26$	0.809148	0.158687
$27$	0	0
$28$	7.30181	1.37991
$29$	6.96702	1.29374	0.646872	−	0.762599i	$-0.276077\pi$
$29$	6.96702	1.29374	0.646872	+	0.762599i	$0.276077\pi$
$30$	0	0
$31$	2.15664	0.387344	0.193672	−	0.981066i	$-0.437960\pi$
$31$	2.15664	0.387344	0.193672	+	0.981066i	$0.437960\pi$
$32$	6.16849	1.09044
$33$	0	0
$34$	4.47279	0.767076
$35$	−7.61548	−1.28725
$36$	0	0
$37$	−1.96058	−0.322318	−0.161159	−	0.986928i	$-0.551523\pi$
$37$	−1.96058	−0.322318	−0.161159	+	0.986928i	$0.551523\pi$
$38$	11.1502	1.80880
$39$	0	0
$40$	8.35311	1.32074
$41$	4.35387	0.679960	0.339980	−	0.940433i	$-0.389580\pi$
$41$	4.35387	0.679960	0.339980	+	0.940433i	$0.389580\pi$
$42$	0	0
$43$	2.67465	0.407880	0.203940	−	0.978983i	$-0.434625\pi$
$43$	2.67465	0.407880	0.203940	+	0.978983i	$0.434625\pi$
$44$	3.66421	0.552401
$45$	0	0
$46$	0.301944	0.0445192
$47$	0.574948	0.0838648	0.0419324	−	0.999120i	$-0.486649\pi$
$47$	0.574948	0.0838648	0.0419324	+	0.999120i	$0.486649\pi$
$48$	0	0
$49$	−1.55721	−0.222458
$50$	−12.8092	−1.81149
$51$	0	0
$52$	−1.11814	−0.155058
$53$	−5.50183	−0.755734	−0.377867	−	0.925860i	$-0.623342\pi$
$53$	−5.50183	−0.755734	−0.377867	+	0.925860i	$0.623342\pi$
$54$	0	0
$55$	−3.82162	−0.515307
$56$	−5.96997	−0.797771
$57$	0	0
$58$	−15.7797	−2.07197
$59$	2.14099	0.278733	0.139366	−	0.990241i	$-0.455493\pi$
$59$	2.14099	0.278733	0.139366	+	0.990241i	$0.455493\pi$
$60$	0	0
$61$	−11.6639	−1.49341	−0.746705	−	0.665155i	$-0.768365\pi$
$61$	−11.6639	−1.49341	−0.746705	+	0.665155i	$0.768365\pi$
$62$	−4.88460	−0.620345
$63$	0	0
$64$	−13.0434	−1.63042
$65$	1.16617	0.144646
$66$	0	0
$67$	15.4105	1.88269	0.941345	−	0.337446i	$-0.109563\pi$
$67$	15.4105	1.88269	0.941345	+	0.337446i	$0.109563\pi$
$68$	−6.18083	−0.749535
$69$	0	0
$70$	17.2484	2.06158
$71$	−8.39677	−0.996514	−0.498257	−	0.867029i	$-0.666026\pi$
$71$	−8.39677	−0.996514	−0.498257	+	0.867029i	$0.666026\pi$
$72$	0	0
$73$	−1.93636	−0.226634	−0.113317	−	0.993559i	$-0.536148\pi$
$73$	−1.93636	−0.226634	−0.113317	+	0.993559i	$0.536148\pi$
$74$	4.44055	0.516203
$75$	0	0
$76$	−15.4082	−1.76744
$77$	2.73131	0.311262
$78$	0	0
$79$	1.22658	0.138002	0.0690008	−	0.997617i	$-0.478019\pi$
$79$	1.22658	0.138002	0.0690008	+	0.997617i	$0.478019\pi$
$80$	1.51415	0.169287
$81$	0	0
$82$	−9.86112	−1.08898
$83$	−12.4131	−1.36252	−0.681258	−	0.732043i	$-0.738567\pi$
$83$	−12.4131	−1.36252	−0.681258	+	0.732043i	$0.738567\pi$
$84$	0	0
$85$	6.44634	0.699204
$86$	−6.05784	−0.653234
$87$	0	0
$88$	−2.99586	−0.319360
$89$	−6.22010	−0.659329	−0.329664	−	0.944098i	$-0.606936\pi$
$89$	−6.22010	−0.659329	−0.329664	+	0.944098i	$0.606936\pi$
$90$	0	0
$91$	−0.833465	−0.0873709
$92$	−0.417249	−0.0435012
$93$	0	0
$94$	−1.30221	−0.134312
$95$	16.0701	1.64875
$96$	0	0
$97$	−5.70333	−0.579086	−0.289543	−	0.957165i	$-0.593503\pi$
$97$	−5.70333	−0.579086	−0.289543	+	0.957165i	$0.593503\pi$
$98$	3.52694	0.356274
$99$	0	0
$100$	17.7007	1.77007
$101$	−7.80201	−0.776329	−0.388164	−	0.921590i	$-0.626891\pi$
$101$	−7.80201	−0.776329	−0.388164	+	0.921590i	$0.626891\pi$
$102$	0	0
$103$	−2.93312	−0.289009	−0.144505	−	0.989504i	$-0.546159\pi$
$103$	−2.93312	−0.289009	−0.144505	+	0.989504i	$0.546159\pi$
$104$	0.914193	0.0896440
$105$	0	0
$106$	12.4611	1.21033
$107$	−9.39664	−0.908408	−0.454204	−	0.890898i	$-0.650076\pi$
$107$	−9.39664	−0.908408	−0.454204	+	0.890898i	$0.650076\pi$
$108$	0	0
$109$	−9.25768	−0.886725	−0.443362	−	0.896342i	$-0.646215\pi$
$109$	−9.25768	−0.886725	−0.443362	+	0.896342i	$0.646215\pi$
$110$	8.65562	0.825281
$111$	0	0
$112$	−1.08217	−0.102255
$113$	8.10929	0.762858	0.381429	−	0.924398i	$-0.375432\pi$
$113$	8.10929	0.762858	0.381429	+	0.924398i	$0.375432\pi$
$114$	0	0
$115$	0.435173	0.0405801
$116$	21.8055	2.02459
$117$	0	0
$118$	−4.84914	−0.446400
$119$	−4.60721	−0.422342
$120$	0	0
$121$	−9.62937	−0.875397
$122$	26.4177	2.39175
$123$	0	0
$124$	6.74991	0.606160
$125$	−2.13967	−0.191378
$126$	0	0
$127$	17.5754	1.55956	0.779780	−	0.626053i	$-0.215331\pi$
$127$	17.5754	1.55956	0.779780	+	0.626053i	$0.215331\pi$
$128$	17.2051	1.52073
$129$	0	0
$130$	−2.64128	−0.231656
$131$	−6.82793	−0.596559	−0.298280	−	0.954479i	$-0.596413\pi$
$131$	−6.82793	−0.596559	−0.298280	+	0.954479i	$0.596413\pi$
$132$	0	0
$133$	−11.4853	−0.995901
$134$	−34.9034	−3.01519
$135$	0	0
$136$	5.05345	0.433330
$137$	−18.1980	−1.55476	−0.777379	−	0.629033i	$-0.783451\pi$
$137$	−18.1980	−1.55476	−0.777379	+	0.629033i	$0.783451\pi$
$138$	0	0
$139$	−6.94358	−0.588946	−0.294473	−	0.955660i	$-0.595144\pi$
$139$	−6.94358	−0.588946	−0.294473	+	0.955660i	$0.595144\pi$
$140$	−23.8351	−2.01443
$141$	0	0
$142$	19.0179	1.59595
$143$	−0.418251	−0.0349759
$144$	0	0
$145$	−22.7423	−1.88864
$146$	4.38568	0.362962
$147$	0	0
$148$	−6.13628	−0.504399
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−19.4161	−1.58006	−0.790031	−	0.613067i	$-0.789935\pi$
$151$	−19.4161	−1.58006	−0.790031	+	0.613067i	$0.789935\pi$
$152$	12.5977	1.02181
$153$	0	0
$154$	−6.18618	−0.498497
$155$	−7.03987	−0.565456
$156$	0	0
$157$	−13.1169	−1.04685	−0.523423	−	0.852073i	$-0.675345\pi$
$157$	−13.1169	−1.04685	−0.523423	+	0.852073i	$0.675345\pi$
$158$	−2.77811	−0.221014
$159$	0	0
$160$	−20.1356	−1.59186
$161$	−0.311018	−0.0245117
$162$	0	0
$163$	−14.6617	−1.14839	−0.574196	−	0.818718i	$-0.694685\pi$
$163$	−14.6617	−1.14839	−0.574196	+	0.818718i	$0.694685\pi$
$164$	13.6268	1.06408
$165$	0	0
$166$	28.1146	2.18212
$167$	9.88027	0.764558	0.382279	−	0.924047i	$-0.375139\pi$
$167$	9.88027	0.764558	0.382279	+	0.924047i	$0.375139\pi$
$168$	0	0
$169$	−12.8724	−0.990182
$170$	−14.6004	−1.11980
$171$	0	0
$172$	8.37118	0.638296
$173$	−9.58585	−0.728798	−0.364399	−	0.931243i	$-0.618726\pi$
$173$	−9.58585	−0.728798	−0.364399	+	0.931243i	$0.618726\pi$
$174$	0	0
$175$	13.1941	0.997382
$176$	−0.543055	−0.0409343
$177$	0	0
$178$	14.0880	1.05594
$179$	9.63389	0.720071	0.360035	−	0.932939i	$-0.382765\pi$
$179$	9.63389	0.720071	0.360035	+	0.932939i	$0.382765\pi$
$180$	0	0
$181$	5.77372	0.429158	0.214579	−	0.976707i	$-0.431162\pi$
$181$	5.77372	0.429158	0.214579	+	0.976707i	$0.431162\pi$
$182$	1.88772	0.139927
$183$	0	0
$184$	0.341143	0.0251494
$185$	6.39988	0.470528
$186$	0	0
$187$	−2.31200	−0.169070
$188$	1.79949	0.131241
$189$	0	0
$190$	−36.3973	−2.64054
$191$	−5.51956	−0.399381	−0.199691	−	0.979859i	$-0.563994\pi$
$191$	−5.51956	−0.399381	−0.199691	+	0.979859i	$0.563994\pi$
$192$	0	0
$193$	−2.45399	−0.176642	−0.0883211	−	0.996092i	$-0.528150\pi$
$193$	−2.45399	−0.176642	−0.0883211	+	0.996092i	$0.528150\pi$
$194$	12.9175	0.927426
$195$	0	0
$196$	−4.87378	−0.348127
$197$	−25.1966	−1.79519	−0.897593	−	0.440826i	$-0.854686\pi$
$197$	−25.1966	−1.79519	−0.897593	+	0.440826i	$0.854686\pi$
$198$	0	0
$199$	6.06213	0.429733	0.214867	−	0.976643i	$-0.431068\pi$
$199$	6.06213	0.429733	0.214867	+	0.976643i	$0.431068\pi$
$200$	−14.4721	−1.02333
$201$	0	0
$202$	17.6709	1.24332
$203$	16.2539	1.14080
$204$	0	0
$205$	−14.2122	−0.992624
$206$	6.64326	0.462858
$207$	0	0
$208$	0.165714	0.0114902
$209$	−5.76358	−0.398675
$210$	0	0
$211$	1.16149	0.0799599	0.0399800	−	0.999200i	$-0.487271\pi$
$211$	1.16149	0.0799599	0.0399800	+	0.999200i	$0.487271\pi$
$212$	−17.2197	−1.18266
$213$	0	0
$214$	21.2826	1.45485
$215$	−8.73078	−0.595435
$216$	0	0
$217$	5.03140	0.341554
$218$	20.9678	1.42012
$219$	0	0
$220$	−11.9610	−0.806409
$221$	0.705511	0.0474578
$222$	0	0
$223$	9.79450	0.655888	0.327944	−	0.944697i	$-0.393644\pi$
$223$	9.79450	0.655888	0.327944	+	0.944697i	$0.393644\pi$
$224$	14.3910	0.961536
$225$	0	0
$226$	−18.3668	−1.22174
$227$	−17.9319	−1.19018	−0.595090	−	0.803659i	$-0.702884\pi$
$227$	−17.9319	−1.19018	−0.595090	+	0.803659i	$0.702884\pi$
$228$	0	0
$229$	−0.618753	−0.0408883	−0.0204442	−	0.999791i	$-0.506508\pi$
$229$	−0.618753	−0.0408883	−0.0204442	+	0.999791i	$0.506508\pi$
$230$	−0.985628	−0.0649904
$231$	0	0
$232$	−17.8282	−1.17048
$233$	24.7617	1.62219	0.811097	−	0.584912i	$-0.198871\pi$
$233$	24.7617	1.62219	0.811097	+	0.584912i	$0.198871\pi$
$234$	0	0
$235$	−1.87679	−0.122428
$236$	6.70091	0.436192
$237$	0	0
$238$	10.4349	0.676395
$239$	13.2395	0.856390	0.428195	−	0.903686i	$-0.359150\pi$
$239$	13.2395	0.856390	0.428195	+	0.903686i	$0.359150\pi$
$240$	0	0
$241$	−18.9376	−1.21988	−0.609939	−	0.792448i	$-0.708806\pi$
$241$	−18.9376	−1.21988	−0.609939	+	0.792448i	$0.708806\pi$
$242$	21.8097	1.40198
$243$	0	0
$244$	−36.5060	−2.33706
$245$	5.08315	0.324751
$246$	0	0
$247$	1.75877	0.111908
$248$	−5.51874	−0.350440
$249$	0	0
$250$	4.84616	0.306498
$251$	−11.9087	−0.751668	−0.375834	−	0.926687i	$-0.622644\pi$
$251$	−11.9087	−0.751668	−0.375834	+	0.926687i	$0.622644\pi$
$252$	0	0
$253$	−0.156076	−0.00981241
$254$	−39.8066	−2.49769
$255$	0	0
$256$	−12.8813	−0.805079
$257$	0.783242	0.0488573	0.0244286	−	0.999702i	$-0.492223\pi$
$257$	0.783242	0.0488573	0.0244286	+	0.999702i	$0.492223\pi$
$258$	0	0
$259$	−4.57400	−0.284215
$260$	3.64992	0.226358
$261$	0	0
$262$	15.4647	0.955410
$263$	29.2618	1.80436	0.902180	−	0.431360i	$-0.141966\pi$
$263$	29.2618	1.80436	0.902180	+	0.431360i	$0.141966\pi$
$264$	0	0
$265$	17.9595	1.10324
$266$	26.0132	1.59497
$267$	0	0
$268$	48.2321	2.94624
$269$	−18.1401	−1.10602	−0.553010	−	0.833175i	$-0.686521\pi$
$269$	−18.1401	−1.10602	−0.553010	+	0.833175i	$0.686521\pi$
$270$	0	0
$271$	−30.8644	−1.87488	−0.937439	−	0.348150i	$-0.886810\pi$
$271$	−30.8644	−1.87488	−0.937439	+	0.348150i	$0.886810\pi$
$272$	0.916030	0.0555425
$273$	0	0
$274$	41.2168	2.49000
$275$	6.62110	0.399267
$276$	0	0
$277$	16.0148	0.962239	0.481119	−	0.876655i	$-0.340230\pi$
$277$	16.0148	0.962239	0.481119	+	0.876655i	$0.340230\pi$
$278$	15.7266	0.943218
$279$	0	0
$280$	19.4876	1.16461
$281$	19.6562	1.17259	0.586296	−	0.810097i	$-0.300585\pi$
$281$	19.6562	1.17259	0.586296	+	0.810097i	$0.300585\pi$
$282$	0	0
$283$	22.5962	1.34320	0.671601	−	0.740913i	$-0.265607\pi$
$283$	22.5962	1.34320	0.671601	+	0.740913i	$0.265607\pi$
$284$	−26.2804	−1.55946
$285$	0	0
$286$	0.947302	0.0560152
$287$	10.1575	0.599577
$288$	0	0
$289$	−13.1001	−0.770594
$290$	51.5092	3.02473
$291$	0	0
$292$	−6.06046	−0.354662
$293$	−30.8700	−1.80344	−0.901721	−	0.432318i	$-0.857696\pi$
$293$	−30.8700	−1.80344	−0.901721	+	0.432318i	$0.857696\pi$
$294$	0	0
$295$	−6.98876	−0.406902
$296$	5.01703	0.291609
$297$	0	0
$298$	2.26491	0.131203
$299$	0.0476269	0.00275433
$300$	0	0
$301$	6.23990	0.359662
$302$	43.9758	2.53052
$303$	0	0
$304$	2.28357	0.130972
$305$	38.0742	2.18012
$306$	0	0
$307$	−17.1756	−0.980261	−0.490131	−	0.871649i	$-0.663051\pi$
$307$	−17.1756	−0.980261	−0.490131	+	0.871649i	$0.663051\pi$
$308$	8.54853	0.487098
$309$	0	0
$310$	15.9447	0.905597
$311$	−8.23714	−0.467085	−0.233543	−	0.972347i	$-0.575032\pi$
$311$	−8.23714	−0.467085	−0.233543	+	0.972347i	$0.575032\pi$
$312$	0	0
$313$	8.65580	0.489255	0.244627	−	0.969617i	$-0.421334\pi$
$313$	8.65580	0.489255	0.244627	+	0.969617i	$0.421334\pi$
$314$	29.7087	1.67656
$315$	0	0
$316$	3.83899	0.215960
$317$	0.743587	0.0417640	0.0208820	−	0.999782i	$-0.493353\pi$
$317$	0.743587	0.0417640	0.0208820	+	0.999782i	$0.493353\pi$
$318$	0	0
$319$	8.15658	0.456681
$320$	42.5771	2.38013
$321$	0	0
$322$	0.704429	0.0392563
$323$	9.72206	0.540950
$324$	0	0
$325$	−2.02044	−0.112074
$326$	33.2074	1.83919
$327$	0	0
$328$	−11.1413	−0.615176
$329$	1.34134	0.0739506
$330$	0	0
$331$	13.0338	0.716402	0.358201	−	0.933644i	$-0.383390\pi$
$331$	13.0338	0.716402	0.358201	+	0.933644i	$0.383390\pi$
$332$	−38.8508	−2.13222
$333$	0	0
$334$	−22.3779	−1.22447
$335$	−50.3040	−2.74840
$336$	0	0
$337$	16.4567	0.896454	0.448227	−	0.893920i	$-0.352056\pi$
$337$	16.4567	0.896454	0.448227	+	0.893920i	$0.352056\pi$
$338$	29.1548	1.58581
$339$	0	0
$340$	20.1759	1.09419
$341$	2.52487	0.136729
$342$	0	0
$343$	−19.9638	−1.07794
$344$	−6.84429	−0.369019
$345$	0	0
$346$	21.7111	1.16720
$347$	−6.57789	−0.353120	−0.176560	−	0.984290i	$-0.556497\pi$
$347$	−6.57789	−0.353120	−0.176560	+	0.984290i	$0.556497\pi$
$348$	0	0
$349$	8.57120	0.458806	0.229403	−	0.973332i	$-0.426323\pi$
$349$	8.57120	0.458806	0.229403	+	0.973332i	$0.426323\pi$
$350$	−29.8835	−1.59734
$351$	0	0
$352$	7.22170	0.384918
$353$	35.2138	1.87424	0.937121	−	0.349004i	$-0.113480\pi$
$353$	35.2138	1.87424	0.937121	+	0.349004i	$0.113480\pi$
$354$	0	0
$355$	27.4094	1.45474
$356$	−19.4678	−1.03179
$357$	0	0
$358$	−21.8199	−1.15322
$359$	−16.5477	−0.873356	−0.436678	−	0.899618i	$-0.643845\pi$
$359$	−16.5477	−0.873356	−0.436678	+	0.899618i	$0.643845\pi$
$360$	0	0
$361$	5.23609	0.275584
$362$	−13.0770	−0.687310
$363$	0	0
$364$	−2.60860	−0.136728
$365$	6.32081	0.330846
$366$	0	0
$367$	32.9577	1.72038	0.860190	−	0.509973i	$-0.170345\pi$
$367$	32.9577	1.72038	0.860190	+	0.509973i	$0.170345\pi$
$368$	0.0618384	0.00322355
$369$	0	0
$370$	−14.4952	−0.753567
$371$	−12.8356	−0.666394
$372$	0	0
$373$	4.41491	0.228596	0.114298	−	0.993447i	$-0.463538\pi$
$373$	4.41491	0.228596	0.114298	+	0.993447i	$0.463538\pi$
$374$	5.23647	0.270772
$375$	0	0
$376$	−1.47126	−0.0758746
$377$	−2.48899	−0.128190
$378$	0	0
$379$	−22.9982	−1.18134	−0.590670	−	0.806913i	$-0.701137\pi$
$379$	−22.9982	−1.18134	−0.590670	+	0.806913i	$0.701137\pi$
$380$	50.2965	2.58015
$381$	0	0
$382$	12.5013	0.639623
$383$	5.63111	0.287736	0.143868	−	0.989597i	$-0.454046\pi$
$383$	5.63111	0.287736	0.143868	+	0.989597i	$0.454046\pi$
$384$	0	0
$385$	−8.91576	−0.454389
$386$	5.55807	0.282899
$387$	0	0
$388$	−17.8504	−0.906218
$389$	22.8668	1.15939	0.579697	−	0.814832i	$-0.303171\pi$
$389$	22.8668	1.15939	0.579697	+	0.814832i	$0.303171\pi$
$390$	0	0
$391$	0.263271	0.0133142
$392$	3.98481	0.201263
$393$	0	0
$394$	57.0681	2.87505
$395$	−4.00391	−0.201458
$396$	0	0
$397$	10.2066	0.512254	0.256127	−	0.966643i	$-0.417553\pi$
$397$	10.2066	0.512254	0.256127	+	0.966643i	$0.417553\pi$
$398$	−13.7302	−0.688232
$399$	0	0
$400$	−2.62333	−0.131166
$401$	−22.2499	−1.11111	−0.555553	−	0.831481i	$-0.687493\pi$
$401$	−22.2499	−1.11111	−0.555553	+	0.831481i	$0.687493\pi$
$402$	0	0
$403$	−0.770468	−0.0383798
$404$	−24.4189	−1.21489
$405$	0	0
$406$	−36.8137	−1.82703
$407$	−2.29533	−0.113776
$408$	0	0
$409$	−0.866253	−0.0428335	−0.0214167	−	0.999771i	$-0.506818\pi$
$409$	−0.866253	−0.0428335	−0.0214167	+	0.999771i	$0.506818\pi$
$410$	32.1894	1.58972
$411$	0	0
$412$	−9.18015	−0.452274
$413$	4.99488	0.245782
$414$	0	0
$415$	40.5198	1.98904
$416$	−2.20371	−0.108046
$417$	0	0
$418$	13.0540	0.638491
$419$	−22.0356	−1.07651	−0.538254	−	0.842783i	$-0.680916\pi$
$419$	−22.0356	−1.07651	−0.538254	+	0.842783i	$0.680916\pi$
$420$	0	0
$421$	8.55526	0.416958	0.208479	−	0.978027i	$-0.433149\pi$
$421$	8.55526	0.416958	0.208479	+	0.978027i	$0.433149\pi$
$422$	−2.63066	−0.128059
$423$	0	0
$424$	14.0789	0.683731
$425$	−11.1685	−0.541754
$426$	0	0
$427$	−27.2117	−1.31686
$428$	−29.4098	−1.42158
$429$	0	0
$430$	19.7745	0.953609
$431$	−35.7807	−1.72350	−0.861748	−	0.507337i	$-0.830630\pi$
$431$	−35.7807	−1.72350	−0.861748	+	0.507337i	$0.830630\pi$
$432$	0	0
$433$	32.0576	1.54059	0.770296	−	0.637687i	$-0.220109\pi$
$433$	32.0576	1.54059	0.770296	+	0.637687i	$0.220109\pi$
$434$	−11.3957	−0.547010
$435$	0	0
$436$	−28.9749	−1.38765
$437$	0.656306	0.0313954
$438$	0	0
$439$	4.26005	0.203321	0.101661	−	0.994819i	$-0.467584\pi$
$439$	4.26005	0.203321	0.101661	+	0.994819i	$0.467584\pi$
$440$	9.77932	0.466211
$441$	0	0
$442$	−1.59792	−0.0760052
$443$	−5.26019	−0.249919	−0.124960	−	0.992162i	$-0.539880\pi$
$443$	−5.26019	−0.249919	−0.124960	+	0.992162i	$0.539880\pi$
$444$	0	0
$445$	20.3041	0.962506
$446$	−22.1837	−1.05043
$447$	0	0
$448$	−30.4299	−1.43768
$449$	−10.0886	−0.476112	−0.238056	−	0.971251i	$-0.576510\pi$
$449$	−10.0886	−0.476112	−0.238056	+	0.971251i	$0.576510\pi$
$450$	0	0
$451$	5.09725	0.240020
$452$	25.3806	1.19380
$453$	0	0
$454$	40.6141	1.90611
$455$	2.72066	0.127546
$456$	0	0
$457$	35.6597	1.66809	0.834044	−	0.551697i	$-0.186020\pi$
$457$	35.6597	1.66809	0.834044	+	0.551697i	$0.186020\pi$
$458$	1.40142	0.0654840
$459$	0	0
$460$	1.36201	0.0635042
$461$	−0.679517	−0.0316483	−0.0158241	−	0.999875i	$-0.505037\pi$
$461$	−0.679517	−0.0316483	−0.0158241	+	0.999875i	$0.505037\pi$
$462$	0	0
$463$	−4.92866	−0.229054	−0.114527	−	0.993420i	$-0.536535\pi$
$463$	−4.92866	−0.229054	−0.114527	+	0.993420i	$0.536535\pi$
$464$	−3.23169	−0.150028
$465$	0	0
$466$	−56.0831	−2.59800
$467$	−9.61515	−0.444936	−0.222468	−	0.974940i	$-0.571411\pi$
$467$	−9.61515	−0.444936	−0.222468	+	0.974940i	$0.571411\pi$
$468$	0	0
$469$	35.9523	1.66012
$470$	4.25076	0.196073
$471$	0	0
$472$	−5.47867	−0.252176
$473$	3.13132	0.143978
$474$	0	0
$475$	−27.8420	−1.27748
$476$	−14.4197	−0.660928
$477$	0	0
$478$	−29.9862	−1.37154
$479$	20.2302	0.924341	0.462171	−	0.886791i	$-0.347071\pi$
$479$	20.2302	0.924341	0.462171	+	0.886791i	$0.347071\pi$
$480$	0	0
$481$	0.700425	0.0319366
$482$	42.8920	1.95368
$483$	0	0
$484$	−30.1382	−1.36992
$485$	18.6172	0.845366
$486$	0	0
$487$	38.3846	1.73937	0.869686	−	0.493605i	$-0.164321\pi$
$487$	38.3846	1.73937	0.869686	+	0.493605i	$0.164321\pi$
$488$	29.8473	1.35113
$489$	0	0
$490$	−11.5129	−0.520099
$491$	−39.4565	−1.78065	−0.890323	−	0.455330i	$-0.849521\pi$
$491$	−39.4565	−1.78065	−0.890323	+	0.455330i	$0.849521\pi$
$492$	0	0
$493$	−13.7586	−0.619656
$494$	−3.98345	−0.179224
$495$	0	0
$496$	−1.00037	−0.0449180
$497$	−19.5895	−0.878709
$498$	0	0
$499$	26.9298	1.20554	0.602771	−	0.797914i	$-0.294063\pi$
$499$	26.9298	1.20554	0.602771	+	0.797914i	$0.294063\pi$
$500$	−6.69678	−0.299489
$501$	0	0
$502$	26.9721	1.20382
$503$	1.82391	0.0813241	0.0406621	−	0.999173i	$-0.487053\pi$
$503$	1.82391	0.0813241	0.0406621	+	0.999173i	$0.487053\pi$
$504$	0	0
$505$	25.4679	1.13331
$506$	0.353498	0.0157149
$507$	0	0
$508$	55.0077	2.44057
$509$	−18.3663	−0.814070	−0.407035	−	0.913413i	$-0.633437\pi$
$509$	−18.3663	−0.814070	−0.407035	+	0.913413i	$0.633437\pi$
$510$	0	0
$511$	−4.51749	−0.199842
$512$	−5.23525	−0.231368
$513$	0	0
$514$	−1.77397	−0.0782466
$515$	9.57451	0.421903
$516$	0	0
$517$	0.673115	0.0296036
$518$	10.3597	0.455179
$519$	0	0
$520$	−2.98418	−0.130865
$521$	13.2275	0.579508	0.289754	−	0.957101i	$-0.406427\pi$
$521$	13.2275	0.579508	0.289754	+	0.957101i	$0.406427\pi$
$522$	0	0
$523$	11.5350	0.504390	0.252195	−	0.967676i	$-0.418848\pi$
$523$	11.5350	0.504390	0.252195	+	0.967676i	$0.418848\pi$
$524$	−21.3702	−0.933562
$525$	0	0
$526$	−66.2754	−2.88974
$527$	−4.25897	−0.185524
$528$	0	0
$529$	−22.9822	−0.999227
$530$	−40.6766	−1.76688
$531$	0	0
$532$	−35.9469	−1.55850
$533$	−1.55543	−0.0673734
$534$	0	0
$535$	30.6732	1.32612
$536$	−39.4346	−1.70332
$537$	0	0
$538$	41.0856	1.77133
$539$	−1.82309	−0.0785259
$540$	0	0
$541$	−30.6376	−1.31722	−0.658608	−	0.752487i	$-0.728854\pi$
$541$	−30.6376	−1.31722	−0.658608	+	0.752487i	$0.728854\pi$
$542$	69.9051	3.00268
$543$	0	0
$544$	−12.1816	−0.522283
$545$	30.2196	1.29447
$546$	0	0
$547$	−29.0744	−1.24313	−0.621565	−	0.783362i	$-0.713503\pi$
$547$	−29.0744	−1.24313	−0.621565	+	0.783362i	$0.713503\pi$
$548$	−56.9564	−2.43306
$549$	0	0
$550$	−14.9962	−0.639440
$551$	−34.2988	−1.46118
$552$	0	0
$553$	2.86160	0.121687
$554$	−36.2722	−1.54106
$555$	0	0
$556$	−21.7322	−0.921649
$557$	13.3798	0.566922	0.283461	−	0.958984i	$-0.408517\pi$
$557$	13.3798	0.566922	0.283461	+	0.958984i	$0.408517\pi$
$558$	0	0
$559$	−0.955528	−0.0404145
$560$	3.53248	0.149275
$561$	0	0
$562$	−44.5196	−1.87795
$563$	−1.59393	−0.0671761	−0.0335880	−	0.999436i	$-0.510693\pi$
$563$	−1.59393	−0.0671761	−0.0335880	+	0.999436i	$0.510693\pi$
$564$	0	0
$565$	−26.4709	−1.11364
$566$	−51.1783	−2.15118
$567$	0	0
$568$	21.4869	0.901570
$569$	32.3304	1.35536	0.677681	−	0.735356i	$-0.262985\pi$
$569$	32.3304	1.35536	0.677681	+	0.735356i	$0.262985\pi$
$570$	0	0
$571$	−18.3742	−0.768938	−0.384469	−	0.923138i	$-0.625615\pi$
$571$	−18.3742	−0.768938	−0.384469	+	0.923138i	$0.625615\pi$
$572$	−1.30905	−0.0547342
$573$	0	0
$574$	−23.0058	−0.960243
$575$	−0.753954	−0.0314421
$576$	0	0
$577$	−22.8091	−0.949555	−0.474777	−	0.880106i	$-0.657471\pi$
$577$	−22.8091	−0.949555	−0.474777	+	0.880106i	$0.657471\pi$
$578$	29.6706	1.23413
$579$	0	0
$580$	−71.1793	−2.95556
$581$	−28.9595	−1.20144
$582$	0	0
$583$	−6.44121	−0.266768
$584$	4.95504	0.205041
$585$	0	0
$586$	69.9177	2.88827
$587$	−12.1654	−0.502122	−0.251061	−	0.967971i	$-0.580779\pi$
$587$	−12.1654	−0.502122	−0.251061	+	0.967971i	$0.580779\pi$
$588$	0	0
$589$	−10.6172	−0.437474
$590$	15.8289	0.651667
$591$	0	0
$592$	0.909428	0.0373772
$593$	−30.9918	−1.27268	−0.636340	−	0.771409i	$-0.719552\pi$
$593$	−30.9918	−1.27268	−0.636340	+	0.771409i	$0.719552\pi$
$594$	0	0
$595$	15.0392	0.616546
$596$	−3.12982	−0.128203
$597$	0	0
$598$	−0.107871	−0.00441116
$599$	−15.6053	−0.637617	−0.318808	−	0.947819i	$-0.603283\pi$
$599$	−15.6053	−0.637617	−0.318808	+	0.947819i	$0.603283\pi$
$600$	0	0
$601$	−9.04197	−0.368830	−0.184415	−	0.982849i	$-0.559039\pi$
$601$	−9.04197	−0.368830	−0.184415	+	0.982849i	$0.559039\pi$
$602$	−14.1328	−0.576011
$603$	0	0
$604$	−60.7690	−2.47266
$605$	31.4329	1.27793
$606$	0	0
$607$	−2.50687	−0.101751	−0.0508754	−	0.998705i	$-0.516201\pi$
$607$	−2.50687	−0.101751	−0.0508754	+	0.998705i	$0.516201\pi$
$608$	−30.3676	−1.23157
$609$	0	0
$610$	−86.2347	−3.49154
$611$	−0.205402	−0.00830969
$612$	0	0
$613$	−28.7999	−1.16322	−0.581608	−	0.813469i	$-0.697576\pi$
$613$	−28.7999	−1.16322	−0.581608	+	0.813469i	$0.697576\pi$
$614$	38.9011	1.56992
$615$	0	0
$616$	−6.98929	−0.281607
$617$	2.84587	0.114570	0.0572852	−	0.998358i	$-0.481756\pi$
$617$	2.84587	0.114570	0.0572852	+	0.998358i	$0.481756\pi$
$618$	0	0
$619$	−4.37429	−0.175818	−0.0879088	−	0.996129i	$-0.528018\pi$
$619$	−4.37429	−0.175818	−0.0879088	+	0.996129i	$0.528018\pi$
$620$	−22.0335	−0.884889
$621$	0	0
$622$	18.6564	0.748053
$623$	−14.5114	−0.581385
$624$	0	0
$625$	−21.2929	−0.851718
$626$	−19.6046	−0.783558
$627$	0	0
$628$	−41.0537	−1.63822
$629$	3.87179	0.154378
$630$	0	0
$631$	−6.24126	−0.248461	−0.124230	−	0.992253i	$-0.539646\pi$
$631$	−6.24126	−0.248461	−0.124230	+	0.992253i	$0.539646\pi$
$632$	−3.13877	−0.124853
$633$	0	0
$634$	−1.68416	−0.0668865
$635$	−57.3708	−2.27669
$636$	0	0
$637$	0.556318	0.0220421
$638$	−18.4739	−0.731390
$639$	0	0
$640$	−56.1621	−2.22000
$641$	0.313130	0.0123679	0.00618395	−	0.999981i	$-0.498032\pi$
$641$	0.313130	0.0123679	0.00618395	+	0.999981i	$0.498032\pi$
$642$	0	0
$643$	23.0109	0.907463	0.453731	−	0.891138i	$-0.350093\pi$
$643$	23.0109	0.907463	0.453731	+	0.891138i	$0.350093\pi$
$644$	−0.973433	−0.0383586
$645$	0	0
$646$	−22.0196	−0.866349
$647$	−33.0819	−1.30058	−0.650292	−	0.759684i	$-0.725353\pi$
$647$	−33.0819	−1.30058	−0.650292	+	0.759684i	$0.725353\pi$
$648$	0	0
$649$	2.50654	0.0983903
$650$	4.57612	0.179490
$651$	0	0
$652$	−45.8885	−1.79713
$653$	1.44202	0.0564304	0.0282152	−	0.999602i	$-0.491018\pi$
$653$	1.44202	0.0564304	0.0282152	+	0.999602i	$0.491018\pi$
$654$	0	0
$655$	22.2882	0.870873
$656$	−2.01957	−0.0788508
$657$	0	0
$658$	−3.03802	−0.118434
$659$	34.4043	1.34020	0.670100	−	0.742271i	$-0.266251\pi$
$659$	34.4043	1.34020	0.670100	+	0.742271i	$0.266251\pi$
$660$	0	0
$661$	43.4259	1.68907	0.844536	−	0.535499i	$-0.179876\pi$
$661$	43.4259	1.68907	0.844536	+	0.535499i	$0.179876\pi$
$662$	−29.5204	−1.14734
$663$	0	0
$664$	31.7645	1.23270
$665$	37.4911	1.45384
$666$	0	0
$667$	−0.928801	−0.0359633
$668$	30.9235	1.19647
$669$	0	0
$670$	113.934	4.40166
$671$	−13.6554	−0.527161
$672$	0	0
$673$	−38.1075	−1.46894	−0.734468	−	0.678643i	$-0.762568\pi$
$673$	−38.1075	−1.46894	−0.734468	+	0.678643i	$0.762568\pi$
$674$	−37.2730	−1.43570
$675$	0	0
$676$	−40.2882	−1.54955
$677$	46.6608	1.79332	0.896659	−	0.442721i	$-0.145987\pi$
$677$	46.6608	1.79332	0.896659	+	0.442721i	$0.145987\pi$
$678$	0	0
$679$	−13.3058	−0.510628
$680$	−16.4959	−0.632587
$681$	0	0
$682$	−5.71861	−0.218977
$683$	−27.9517	−1.06954	−0.534771	−	0.844997i	$-0.679602\pi$
$683$	−27.9517	−1.06954	−0.534771	+	0.844997i	$0.679602\pi$
$684$	0	0
$685$	59.4032	2.26968
$686$	45.2162	1.72636
$687$	0	0
$688$	−1.24065	−0.0472994
$689$	1.96555	0.0748814
$690$	0	0
$691$	−46.3617	−1.76368	−0.881841	−	0.471547i	$-0.843696\pi$
$691$	−46.3617	−1.76368	−0.881841	+	0.471547i	$0.843696\pi$
$692$	−30.0020	−1.14051
$693$	0	0
$694$	14.8983	0.565533
$695$	22.6657	0.859760
$696$	0	0
$697$	−8.59809	−0.325676
$698$	−19.4130	−0.734793
$699$	0	0
$700$	41.2953	1.56081
$701$	42.4123	1.60189	0.800945	−	0.598738i	$-0.204331\pi$
$701$	42.4123	1.60189	0.800945	+	0.598738i	$0.204331\pi$
$702$	0	0
$703$	9.65198	0.364031
$704$	−15.2704	−0.575525
$705$	0	0
$706$	−79.7562	−3.00166
$707$	−18.2019	−0.684554
$708$	0	0
$709$	44.6000	1.67499	0.837494	−	0.546446i	$-0.184020\pi$
$709$	44.6000	1.67499	0.837494	+	0.546446i	$0.184020\pi$
$710$	−62.0798	−2.32981
$711$	0	0
$712$	15.9169	0.596511
$713$	−0.287510	−0.0107673
$714$	0	0
$715$	1.36529	0.0510588
$716$	30.1524	1.12685
$717$	0	0
$718$	37.4791	1.39871
$719$	27.2904	1.01776	0.508880	−	0.860837i	$-0.330060\pi$
$719$	27.2904	1.01776	0.508880	+	0.860837i	$0.330060\pi$
$720$	0	0
$721$	−6.84291	−0.254843
$722$	−11.8593	−0.441357
$723$	0	0
$724$	18.0707	0.671594
$725$	39.4019	1.46335
$726$	0	0
$727$	45.4346	1.68507	0.842537	−	0.538638i	$-0.181061\pi$
$727$	45.4346	1.68507	0.842537	+	0.538638i	$0.181061\pi$
$728$	2.13279	0.0790466
$729$	0	0
$730$	−14.3161	−0.529862
$731$	−5.28194	−0.195360
$732$	0	0
$733$	19.6689	0.726488	0.363244	−	0.931694i	$-0.381669\pi$
$733$	19.6689	0.726488	0.363244	+	0.931694i	$0.381669\pi$
$734$	−74.6464	−2.75525
$735$	0	0
$736$	−0.822345	−0.0303120
$737$	18.0417	0.664574
$738$	0	0
$739$	13.2769	0.488400	0.244200	−	0.969725i	$-0.421475\pi$
$739$	13.2769	0.488400	0.244200	+	0.969725i	$0.421475\pi$
$740$	20.0305	0.736335
$741$	0	0
$742$	29.0716	1.06725
$743$	29.1597	1.06977	0.534883	−	0.844926i	$-0.320356\pi$
$743$	29.1597	1.06977	0.534883	+	0.844926i	$0.320356\pi$
$744$	0	0
$745$	3.26427	0.119594
$746$	−9.99939	−0.366104
$747$	0	0
$748$	−7.23615	−0.264580
$749$	−21.9222	−0.801019
$750$	0	0
$751$	32.0879	1.17091	0.585453	−	0.810706i	$-0.300917\pi$
$751$	32.0879	1.17091	0.585453	+	0.810706i	$0.300917\pi$
$752$	−0.266693	−0.00972530
$753$	0	0
$754$	5.63735	0.205300
$755$	63.3795	2.30662
$756$	0	0
$757$	−4.22696	−0.153631	−0.0768157	−	0.997045i	$-0.524475\pi$
$757$	−4.22696	−0.153631	−0.0768157	+	0.997045i	$0.524475\pi$
$758$	52.0890	1.89196
$759$	0	0
$760$	−41.1225	−1.49167
$761$	−12.4904	−0.452776	−0.226388	−	0.974037i	$-0.572692\pi$
$761$	−12.4904	−0.452776	−0.226388	+	0.974037i	$0.572692\pi$
$762$	0	0
$763$	−21.5980	−0.781899
$764$	−17.2752	−0.624997
$765$	0	0
$766$	−12.7540	−0.460819
$767$	−0.764875	−0.0276180
$768$	0	0
$769$	−1.17544	−0.0423873	−0.0211937	−	0.999775i	$-0.506747\pi$
$769$	−1.17544	−0.0423873	−0.0211937	+	0.999775i	$0.506747\pi$
$770$	20.1934	0.727719
$771$	0	0
$772$	−7.68056	−0.276429
$773$	1.49430	0.0537464	0.0268732	−	0.999639i	$-0.491445\pi$
$773$	1.49430	0.0537464	0.0268732	+	0.999639i	$0.491445\pi$
$774$	0	0
$775$	12.1969	0.438124
$776$	14.5945	0.523913
$777$	0	0
$778$	−51.7913	−1.85681
$779$	−21.4342	−0.767958
$780$	0	0
$781$	−9.83044	−0.351761
$782$	−0.596284	−0.0213231
$783$	0	0
$784$	0.722320	0.0257971
$785$	42.8173	1.52821
$786$	0	0
$787$	−0.462163	−0.0164743	−0.00823717	−	0.999966i	$-0.502622\pi$
$787$	−0.462163	−0.0164743	−0.00823717	+	0.999966i	$0.502622\pi$
$788$	−78.8610	−2.80931
$789$	0	0
$790$	9.06850	0.322643
$791$	18.9188	0.672675
$792$	0	0
$793$	4.16697	0.147974
$794$	−23.1170	−0.820393
$795$	0	0
$796$	18.9734	0.672494
$797$	24.2182	0.857854	0.428927	−	0.903339i	$-0.358892\pi$
$797$	24.2182	0.857854	0.428927	+	0.903339i	$0.358892\pi$
$798$	0	0
$799$	−1.13542	−0.0401682
$800$	34.8858	1.23340
$801$	0	0
$802$	50.3940	1.77947
$803$	−2.26698	−0.0799998
$804$	0	0
$805$	1.01525	0.0357828
$806$	1.74504	0.0614665
$807$	0	0
$808$	19.9649	0.702364
$809$	30.4313	1.06991	0.534954	−	0.844881i	$-0.320329\pi$
$809$	30.4313	1.06991	0.534954	+	0.844881i	$0.320329\pi$
$810$	0	0
$811$	−46.7185	−1.64051	−0.820254	−	0.571999i	$-0.806168\pi$
$811$	−46.7185	−1.64051	−0.820254	+	0.571999i	$0.806168\pi$
$812$	50.8719	1.78525
$813$	0	0
$814$	5.19873	0.182215
$815$	47.8598	1.67646
$816$	0	0
$817$	−13.1673	−0.460667
$818$	1.96199	0.0685993
$819$	0	0
$820$	−44.4817	−1.55337
$821$	25.0629	0.874701	0.437350	−	0.899291i	$-0.355917\pi$
$821$	25.0629	0.874701	0.437350	+	0.899291i	$0.355917\pi$
$822$	0	0
$823$	33.5855	1.17072	0.585359	−	0.810774i	$-0.300954\pi$
$823$	33.5855	1.17072	0.585359	+	0.810774i	$0.300954\pi$
$824$	7.50570	0.261474
$825$	0	0
$826$	−11.3129	−0.393628
$827$	6.48144	0.225382	0.112691	−	0.993630i	$-0.464053\pi$
$827$	6.48144	0.225382	0.112691	+	0.993630i	$0.464053\pi$
$828$	0	0
$829$	21.9229	0.761414	0.380707	−	0.924696i	$-0.375681\pi$
$829$	21.9229	0.761414	0.380707	+	0.924696i	$0.375681\pi$
$830$	−91.7737	−3.18551
$831$	0	0
$832$	4.65979	0.161549
$833$	3.07520	0.106549
$834$	0	0
$835$	−32.2519	−1.11612
$836$	−18.0390	−0.623891
$837$	0	0
$838$	49.9086	1.72406
$839$	−9.19078	−0.317301	−0.158650	−	0.987335i	$-0.550714\pi$
$839$	−9.19078	−0.317301	−0.158650	+	0.987335i	$0.550714\pi$
$840$	0	0
$841$	19.5394	0.673773
$842$	−19.3769	−0.667772
$843$	0	0
$844$	3.63524	0.125130
$845$	42.0189	1.44550
$846$	0	0
$847$	−22.4651	−0.771910
$848$	2.55205	0.0876379
$849$	0	0
$850$	25.2958	0.867637
$851$	0.261373	0.00895975
$852$	0	0
$853$	−12.8148	−0.438771	−0.219386	−	0.975638i	$-0.570405\pi$
$853$	−12.8148	−0.438771	−0.219386	+	0.975638i	$0.570405\pi$
$854$	61.6320	2.10900
$855$	0	0
$856$	24.0455	0.821859
$857$	−13.5255	−0.462021	−0.231011	−	0.972951i	$-0.574203\pi$
$857$	−13.5255	−0.462021	−0.231011	+	0.972951i	$0.574203\pi$
$858$	0	0
$859$	1.14410	0.0390361	0.0195181	−	0.999810i	$-0.493787\pi$
$859$	1.14410	0.0390361	0.0195181	+	0.999810i	$0.493787\pi$
$860$	−27.3258	−0.931802
$861$	0	0
$862$	81.0401	2.76024
$863$	10.0654	0.342632	0.171316	−	0.985216i	$-0.445198\pi$
$863$	10.0654	0.342632	0.171316	+	0.985216i	$0.445198\pi$
$864$	0	0
$865$	31.2908	1.06392
$866$	−72.6077	−2.46731
$867$	0	0
$868$	15.7474	0.534502
$869$	1.43601	0.0487134
$870$	0	0
$871$	−5.50545	−0.186545
$872$	23.6899	0.802242
$873$	0	0
$874$	−1.48648	−0.0502808
$875$	−4.99180	−0.168754
$876$	0	0
$877$	42.7977	1.44518	0.722588	−	0.691279i	$-0.242952\pi$
$877$	42.7977	1.44518	0.722588	+	0.691279i	$0.242952\pi$
$878$	−9.64864	−0.325626
$879$	0	0
$880$	1.77268	0.0597570
$881$	−37.9521	−1.27864	−0.639319	−	0.768942i	$-0.720784\pi$
$881$	−37.9521	−1.27864	−0.639319	+	0.768942i	$0.720784\pi$
$882$	0	0
$883$	−47.6152	−1.60238	−0.801190	−	0.598410i	$-0.795799\pi$
$883$	−47.6152	−1.60238	−0.801190	+	0.598410i	$0.795799\pi$
$884$	2.20812	0.0742672
$885$	0	0
$886$	11.9139	0.400254
$887$	−23.4672	−0.787953	−0.393976	−	0.919121i	$-0.628901\pi$
$887$	−23.4672	−0.787953	−0.393976	+	0.919121i	$0.628901\pi$
$888$	0	0
$889$	41.0029	1.37519
$890$	−45.9870	−1.54149
$891$	0	0
$892$	30.6550	1.02641
$893$	−2.83048	−0.0947184
$894$	0	0
$895$	−31.4477	−1.05118
$896$	40.1391	1.34095
$897$	0	0
$898$	22.8498	0.762509
$899$	15.0254	0.501124
$900$	0	0
$901$	10.8651	0.361969
$902$	−11.5448	−0.384400
$903$	0	0
$904$	−20.7513	−0.690176
$905$	−18.8470	−0.626496
$906$	0	0
$907$	−56.1065	−1.86299	−0.931493	−	0.363759i	$-0.881493\pi$
$907$	−56.1065	−1.86299	−0.931493	+	0.363759i	$0.881493\pi$
$908$	−56.1236	−1.86253
$909$	0	0
$910$	−6.16205	−0.204270
$911$	6.04044	0.200129	0.100064	−	0.994981i	$-0.468095\pi$
$911$	6.04044	0.200129	0.100064	+	0.994981i	$0.468095\pi$
$912$	0	0
$913$	−14.5325	−0.480957
$914$	−80.7660	−2.67150
$915$	0	0
$916$	−1.93659	−0.0639866
$917$	−15.9294	−0.526036
$918$	0	0
$919$	31.5361	1.04028	0.520139	−	0.854081i	$-0.325880\pi$
$919$	31.5361	1.04028	0.520139	+	0.854081i	$0.325880\pi$
$920$	−1.11358	−0.0367138
$921$	0	0
$922$	1.53905	0.0506858
$923$	2.99978	0.0987389
$924$	0	0
$925$	−11.0880	−0.364572
$926$	11.1630	0.366838
$927$	0	0
$928$	42.9760	1.41076
$929$	−28.5997	−0.938326	−0.469163	−	0.883111i	$-0.655444\pi$
$929$	−28.5997	−0.938326	−0.469163	+	0.883111i	$0.655444\pi$
$930$	0	0
$931$	7.66616	0.251248
$932$	77.4997	2.53859
$933$	0	0
$934$	21.7775	0.712580
$935$	7.54700	0.246813
$936$	0	0
$937$	54.3167	1.77445	0.887225	−	0.461336i	$-0.152630\pi$
$937$	54.3167	1.77445	0.887225	+	0.461336i	$0.152630\pi$
$938$	−81.4288	−2.65875
$939$	0	0
$940$	−5.87402	−0.191589
$941$	−33.2964	−1.08543	−0.542716	−	0.839916i	$-0.682604\pi$
$941$	−33.2964	−1.08543	−0.542716	+	0.839916i	$0.682604\pi$
$942$	0	0
$943$	−0.580431	−0.0189014
$944$	−0.993109	−0.0323229
$945$	0	0
$946$	−7.09216	−0.230586
$947$	29.0438	0.943797	0.471899	−	0.881653i	$-0.343569\pi$
$947$	29.0438	0.943797	0.471899	+	0.881653i	$0.343569\pi$
$948$	0	0
$949$	0.691772	0.0224559
$950$	63.0597	2.04593
$951$	0	0
$952$	11.7896	0.382103
$953$	−16.2657	−0.526897	−0.263448	−	0.964673i	$-0.584860\pi$
$953$	−16.2657	−0.526897	−0.263448	+	0.964673i	$0.584860\pi$
$954$	0	0
$955$	18.0174	0.583028
$956$	41.4372	1.34017
$957$	0	0
$958$	−45.8196	−1.48036
$959$	−42.4555	−1.37096
$960$	0	0
$961$	−26.3489	−0.849964
$962$	−1.58640	−0.0511476
$963$	0	0
$964$	−59.2714	−1.90900
$965$	8.01050	0.257867
$966$	0	0
$967$	−20.1217	−0.647069	−0.323534	−	0.946216i	$-0.604871\pi$
$967$	−20.1217	−0.647069	−0.323534	+	0.946216i	$0.604871\pi$
$968$	24.6410	0.791993
$969$	0	0
$970$	−42.1664	−1.35388
$971$	1.66253	0.0533532	0.0266766	−	0.999644i	$-0.491508\pi$
$971$	1.66253	0.0533532	0.0266766	+	0.999644i	$0.491508\pi$
$972$	0	0
$973$	−16.1992	−0.519323
$974$	−86.9377	−2.78566
$975$	0	0
$976$	5.41037	0.173182
$977$	−7.63971	−0.244416	−0.122208	−	0.992505i	$-0.538997\pi$
$977$	−7.63971	−0.244416	−0.122208	+	0.992505i	$0.538997\pi$
$978$	0	0
$979$	−7.28212	−0.232738
$980$	15.9094	0.508206
$981$	0	0
$982$	89.3654	2.85176
$983$	−12.7105	−0.405401	−0.202701	−	0.979241i	$-0.564972\pi$
$983$	−12.7105	−0.405401	−0.202701	+	0.979241i	$0.564972\pi$
$984$	0	0
$985$	82.2487	2.62066
$986$	31.1620	0.992400
$987$	0	0
$988$	5.50462	0.175125
$989$	−0.356568	−0.0113382
$990$	0	0
$991$	−37.0267	−1.17619	−0.588096	−	0.808791i	$-0.700122\pi$
$991$	−37.0267	−1.17619	−0.588096	+	0.808791i	$0.700122\pi$
$992$	13.3032	0.422378
$993$	0	0
$994$	44.3685	1.40728
$995$	−19.7885	−0.627336
$996$	0	0
$997$	−21.0433	−0.666448	−0.333224	−	0.942848i	$-0.608137\pi$
$997$	−21.0433	−0.666448	−0.333224	+	0.942848i	$0.608137\pi$
$998$	−60.9935	−1.93072
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1341.2.a.g.1.3	✓	12
3.2	odd	2		1341.2.a.h.1.10	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1341.2.a.g.1.3	✓	12	1.1	even	1	trivial
1341.2.a.h.1.10	yes	12	3.2	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:	\( N \)	\(=\)	\( 1341 = 3^{2} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1341.a (trivial)

\(f(q)\)	\(=\)	\(q-2.26491 q^{2} +3.12982 q^{4} -3.26427 q^{5} +2.33298 q^{7} -2.55895 q^{8} +O(q^{10})\)	\(q-2.26491 q^{2} +3.12982 q^{4} -3.26427 q^{5} +2.33298 q^{7} -2.55895 q^{8} +7.39329 q^{10} +1.17074 q^{11} -0.357254 q^{13} -5.28399 q^{14} -0.463856 q^{16} -1.97482 q^{17} -4.92302 q^{19} -10.2166 q^{20} -2.65162 q^{22} -0.133314 q^{23} +5.65548 q^{25} +0.809148 q^{26} +7.30181 q^{28} +6.96702 q^{29} +2.15664 q^{31} +6.16849 q^{32} +4.47279 q^{34} -7.61548 q^{35} -1.96058 q^{37} +11.1502 q^{38} +8.35311 q^{40} +4.35387 q^{41} +2.67465 q^{43} +3.66421 q^{44} +0.301944 q^{46} +0.574948 q^{47} -1.55721 q^{49} -12.8092 q^{50} -1.11814 q^{52} -5.50183 q^{53} -3.82162 q^{55} -5.96997 q^{56} -15.7797 q^{58} +2.14099 q^{59} -11.6639 q^{61} -4.88460 q^{62} -13.0434 q^{64} +1.16617 q^{65} +15.4105 q^{67} -6.18083 q^{68} +17.2484 q^{70} -8.39677 q^{71} -1.93636 q^{73} +4.44055 q^{74} -15.4082 q^{76} +2.73131 q^{77} +1.22658 q^{79} +1.51415 q^{80} -9.86112 q^{82} -12.4131 q^{83} +6.44634 q^{85} -6.05784 q^{86} -2.99586 q^{88} -6.22010 q^{89} -0.833465 q^{91} -0.417249 q^{92} -1.30221 q^{94} +16.0701 q^{95} -5.70333 q^{97} +3.52694 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 9 q^{4} - 8 q^{5} - 2 q^{7} - 9 q^{8}+O(q^{10}) \) 12 * q - 3 * q^2 + 9 * q^4 - 8 * q^5 - 2 * q^7 - 9 * q^8	\( 12 q - 3 q^{2} + 9 q^{4} - 8 q^{5} - 2 q^{7} - 9 q^{8} - 22 q^{11} - 2 q^{13} - 12 q^{14} + 11 q^{16} - 8 q^{17} - 6 q^{19} - 24 q^{20} + 4 q^{22} - 12 q^{23} + 8 q^{25} - 26 q^{26} - 20 q^{28} - 16 q^{29} - 2 q^{31} - 21 q^{32} - 6 q^{34} - 32 q^{35} - 4 q^{37} - 15 q^{38} + 6 q^{40} - 24 q^{41} + 12 q^{43} - 37 q^{44} - 22 q^{46} - 26 q^{47} + 24 q^{49} + 5 q^{50} - 10 q^{52} - 20 q^{53} + 6 q^{55} + 13 q^{56} + 10 q^{58} - 72 q^{59} - 8 q^{61} + 2 q^{62} + q^{64} + 4 q^{65} - 14 q^{67} + 14 q^{68} + 18 q^{70} - 38 q^{71} + 27 q^{74} - 2 q^{76} - 6 q^{77} + 10 q^{79} - 56 q^{80} + 6 q^{82} - 42 q^{83} - 32 q^{85} + 14 q^{86} + 22 q^{88} - 44 q^{89} + 50 q^{92} + 2 q^{94} - 4 q^{95} - 22 q^{97} - 20 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 + 9 * q^4 - 8 * q^5 - 2 * q^7 - 9 * q^8 - 22 * q^11 - 2 * q^13 - 12 * q^14 + 11 * q^16 - 8 * q^17 - 6 * q^19 - 24 * q^20 + 4 * q^22 - 12 * q^23 + 8 * q^25 - 26 * q^26 - 20 * q^28 - 16 * q^29 - 2 * q^31 - 21 * q^32 - 6 * q^34 - 32 * q^35 - 4 * q^37 - 15 * q^38 + 6 * q^40 - 24 * q^41 + 12 * q^43 - 37 * q^44 - 22 * q^46 - 26 * q^47 + 24 * q^49 + 5 * q^50 - 10 * q^52 - 20 * q^53 + 6 * q^55 + 13 * q^56 + 10 * q^58 - 72 * q^59 - 8 * q^61 + 2 * q^62 + q^64 + 4 * q^65 - 14 * q^67 + 14 * q^68 + 18 * q^70 - 38 * q^71 + 27 * q^74 - 2 * q^76 - 6 * q^77 + 10 * q^79 - 56 * q^80 + 6 * q^82 - 42 * q^83 - 32 * q^85 + 14 * q^86 + 22 * q^88 - 44 * q^89 + 50 * q^92 + 2 * q^94 - 4 * q^95 - 22 * q^97 - 20 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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