LMFDB - Embedded newform 1341.2.a.e.1.5

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:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 15x^{7} + 12x^{6} + 75x^{5} - 48x^{4} - 137x^{3} + 76x^{2} + 68x - 39 $ x^9 - x^8 - 15x^7 + 12x^6 + 75x^5 - 48x^4 - 137x^3 + 76x^2 + 68*x - 39
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 149)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.652876$ 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)

$f(q)$	$=$	$q+0.652876 q^{2} -1.57375 q^{4} -1.28287 q^{5} +3.13476 q^{7} -2.33322 q^{8} +O(q^{10})$	\(q+0.652876 q^{2} -1.57375 q^{4} -1.28287 q^{5} +3.13476 q^{7} -2.33322 q^{8} -0.837555 q^{10} -4.06966 q^{11} +0.561009 q^{13} +2.04661 q^{14} +1.62420 q^{16} +3.36793 q^{17} +4.48355 q^{19} +2.01892 q^{20} -2.65698 q^{22} -2.73895 q^{23} -3.35424 q^{25} +0.366270 q^{26} -4.93334 q^{28} +6.42545 q^{29} +8.47125 q^{31} +5.72684 q^{32} +2.19884 q^{34} -4.02149 q^{35} +9.30394 q^{37} +2.92720 q^{38} +2.99321 q^{40} -7.10347 q^{41} -4.83865 q^{43} +6.40463 q^{44} -1.78820 q^{46} +9.51064 q^{47} +2.82673 q^{49} -2.18991 q^{50} -0.882890 q^{52} +5.96662 q^{53} +5.22084 q^{55} -7.31408 q^{56} +4.19502 q^{58} +0.226831 q^{59} +8.20421 q^{61} +5.53068 q^{62} +0.490507 q^{64} -0.719702 q^{65} +4.46449 q^{67} -5.30030 q^{68} -2.62554 q^{70} +7.93072 q^{71} -8.89197 q^{73} +6.07432 q^{74} -7.05600 q^{76} -12.7574 q^{77} +7.38920 q^{79} -2.08364 q^{80} -4.63768 q^{82} +2.37875 q^{83} -4.32062 q^{85} -3.15904 q^{86} +9.49539 q^{88} -5.37386 q^{89} +1.75863 q^{91} +4.31044 q^{92} +6.20927 q^{94} -5.75181 q^{95} -6.87043 q^{97} +1.84551 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{2} + 13 q^{4} + q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) $ 9 * q + q^2 + 13 * q^4 + q^5 + 3 * q^7 + 6 * q^8	$ 9 q + q^{2} + 13 q^{4} + q^{5} + 3 q^{7} + 6 q^{8} - 6 q^{10} - 5 q^{11} + 7 q^{13} + 8 q^{14} + 13 q^{16} + 5 q^{17} + 30 q^{19} + 10 q^{20} - 7 q^{22} + 4 q^{23} + 6 q^{25} + 15 q^{26} - 12 q^{28} + 16 q^{29} + 22 q^{31} + 38 q^{32} + 9 q^{34} + 11 q^{35} - 7 q^{37} + 18 q^{38} - 7 q^{40} - 6 q^{41} + 4 q^{43} - 6 q^{44} + q^{46} + 6 q^{47} + 14 q^{49} + 16 q^{50} + 50 q^{52} + 2 q^{53} - 2 q^{55} - 7 q^{56} + 2 q^{58} - 43 q^{59} + q^{61} - 33 q^{62} + 18 q^{64} + 20 q^{65} + 33 q^{67} + 16 q^{68} - 3 q^{70} - 15 q^{71} - 11 q^{73} - 33 q^{74} + 59 q^{76} + 30 q^{77} + q^{79} - 65 q^{80} + 5 q^{82} + 4 q^{83} - 34 q^{85} + 7 q^{86} - 37 q^{88} + 19 q^{89} + 62 q^{91} - 17 q^{92} + 17 q^{94} + 21 q^{95} - q^{97} - 36 q^{98}+O(q^{100}) $ 9 * q + q^2 + 13 * q^4 + q^5 + 3 * q^7 + 6 * q^8 - 6 * q^10 - 5 * q^11 + 7 * q^13 + 8 * q^14 + 13 * q^16 + 5 * q^17 + 30 * q^19 + 10 * q^20 - 7 * q^22 + 4 * q^23 + 6 * q^25 + 15 * q^26 - 12 * q^28 + 16 * q^29 + 22 * q^31 + 38 * q^32 + 9 * q^34 + 11 * q^35 - 7 * q^37 + 18 * q^38 - 7 * q^40 - 6 * q^41 + 4 * q^43 - 6 * q^44 + q^46 + 6 * q^47 + 14 * q^49 + 16 * q^50 + 50 * q^52 + 2 * q^53 - 2 * q^55 - 7 * q^56 + 2 * q^58 - 43 * q^59 + q^61 - 33 * q^62 + 18 * q^64 + 20 * q^65 + 33 * q^67 + 16 * q^68 - 3 * q^70 - 15 * q^71 - 11 * q^73 - 33 * q^74 + 59 * q^76 + 30 * q^77 + q^79 - 65 * q^80 + 5 * q^82 + 4 * q^83 - 34 * q^85 + 7 * q^86 - 37 * q^88 + 19 * q^89 + 62 * q^91 - 17 * q^92 + 17 * q^94 + 21 * q^95 - q^97 - 36 * 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 $

$2$	0.652876	0.461653	0.230827	−	0.972995i	$-0.425857\pi$
$2$	0.652876	0.461653	0.230827	+	0.972995i	$0.425857\pi$
$3$	0	0
$3$	0	0
$4$	−1.57375	−0.786876
$5$	−1.28287	−0.573717	−0.286858	−	0.957973i	$-0.592611\pi$
$5$	−1.28287	−0.573717	−0.286858	+	0.957973i	$0.592611\pi$
$6$	0	0
$7$	3.13476	1.18483	0.592414	−	0.805633i	$-0.298175\pi$
$7$	3.13476	1.18483	0.592414	+	0.805633i	$0.298175\pi$
$8$	−2.33322	−0.824917
$9$	0	0
$10$	−0.837555	−0.264858
$11$	−4.06966	−1.22705	−0.613524	−	0.789676i	$-0.710248\pi$
$11$	−4.06966	−1.22705	−0.613524	+	0.789676i	$0.710248\pi$
$12$	0	0
$13$	0.561009	0.155596	0.0777980	−	0.996969i	$-0.475211\pi$
$13$	0.561009	0.155596	0.0777980	+	0.996969i	$0.475211\pi$
$14$	2.04661	0.546980
$15$	0	0
$16$	1.62420	0.406051
$17$	3.36793	0.816844	0.408422	−	0.912793i	$-0.366079\pi$
$17$	3.36793	0.816844	0.408422	+	0.912793i	$0.366079\pi$
$18$	0	0
$19$	4.48355	1.02860	0.514298	−	0.857611i	$-0.328052\pi$
$19$	4.48355	1.02860	0.514298	+	0.857611i	$0.328052\pi$
$20$	2.01892	0.451444
$21$	0	0
$22$	−2.65698	−0.566470
$23$	−2.73895	−0.571111	−0.285556	−	0.958362i	$-0.592178\pi$
$23$	−2.73895	−0.571111	−0.285556	+	0.958362i	$0.592178\pi$
$24$	0	0
$25$	−3.35424	−0.670849
$26$	0.366270	0.0718314
$27$	0	0
$28$	−4.93334	−0.932314
$29$	6.42545	1.19318	0.596588	−	0.802547i	$-0.296523\pi$
$29$	6.42545	1.19318	0.596588	+	0.802547i	$0.296523\pi$
$30$	0	0
$31$	8.47125	1.52148	0.760741	−	0.649056i	$-0.224836\pi$
$31$	8.47125	1.52148	0.760741	+	0.649056i	$0.224836\pi$
$32$	5.72684	1.01237
$33$	0	0
$34$	2.19884	0.377098
$35$	−4.02149	−0.679756
$36$	0	0
$37$	9.30394	1.52956	0.764779	−	0.644292i	$-0.222848\pi$
$37$	9.30394	1.52956	0.764779	+	0.644292i	$0.222848\pi$
$38$	2.92720	0.474855
$39$	0	0
$40$	2.99321	0.473269
$41$	−7.10347	−1.10938	−0.554688	−	0.832059i	$-0.687162\pi$
$41$	−7.10347	−1.10938	−0.554688	+	0.832059i	$0.687162\pi$
$42$	0	0
$43$	−4.83865	−0.737888	−0.368944	−	0.929452i	$-0.620281\pi$
$43$	−4.83865	−0.737888	−0.368944	+	0.929452i	$0.620281\pi$
$44$	6.40463	0.965535
$45$	0	0
$46$	−1.78820	−0.263655
$47$	9.51064	1.38727	0.693634	−	0.720327i	$-0.256008\pi$
$47$	9.51064	1.38727	0.693634	+	0.720327i	$0.256008\pi$
$48$	0	0
$49$	2.82673	0.403819
$50$	−2.18991	−0.309699
$51$	0	0
$52$	−0.882890	−0.122435
$53$	5.96662	0.819579	0.409789	−	0.912180i	$-0.365602\pi$
$53$	5.96662	0.819579	0.409789	+	0.912180i	$0.365602\pi$
$54$	0	0
$55$	5.22084	0.703978
$56$	−7.31408	−0.977385
$57$	0	0
$58$	4.19502	0.550834
$59$	0.226831	0.0295308	0.0147654	−	0.999891i	$-0.495300\pi$
$59$	0.226831	0.0295308	0.0147654	+	0.999891i	$0.495300\pi$
$60$	0	0
$61$	8.20421	1.05044	0.525221	−	0.850966i	$-0.323983\pi$
$61$	8.20421	1.05044	0.525221	+	0.850966i	$0.323983\pi$
$62$	5.53068	0.702397
$63$	0	0
$64$	0.490507	0.0613134
$65$	−0.719702	−0.0892681
$66$	0	0
$67$	4.46449	0.545424	0.272712	−	0.962096i	$-0.412079\pi$
$67$	4.46449	0.545424	0.272712	+	0.962096i	$0.412079\pi$
$68$	−5.30030	−0.642755
$69$	0	0
$70$	−2.62554	−0.313812
$71$	7.93072	0.941204	0.470602	−	0.882346i	$-0.344037\pi$
$71$	7.93072	0.941204	0.470602	+	0.882346i	$0.344037\pi$
$72$	0	0
$73$	−8.89197	−1.04073	−0.520363	−	0.853945i	$-0.674203\pi$
$73$	−8.89197	−1.04073	−0.520363	+	0.853945i	$0.674203\pi$
$74$	6.07432	0.706126
$75$	0	0
$76$	−7.05600	−0.809378
$77$	−12.7574	−1.45384
$78$	0	0
$79$	7.38920	0.831350	0.415675	−	0.909513i	$-0.363545\pi$
$79$	7.38920	0.831350	0.415675	+	0.909513i	$0.363545\pi$
$80$	−2.08364	−0.232958
$81$	0	0
$82$	−4.63768	−0.512147
$83$	2.37875	0.261101	0.130551	−	0.991442i	$-0.458325\pi$
$83$	2.37875	0.261101	0.130551	+	0.991442i	$0.458325\pi$
$84$	0	0
$85$	−4.32062	−0.468637
$86$	−3.15904	−0.340648
$87$	0	0
$88$	9.49539	1.01221
$89$	−5.37386	−0.569628	−0.284814	−	0.958583i	$-0.591932\pi$
$89$	−5.37386	−0.569628	−0.284814	+	0.958583i	$0.591932\pi$
$90$	0	0
$91$	1.75863	0.184355
$92$	4.31044	0.449394
$93$	0	0
$94$	6.20927	0.640437
$95$	−5.75181	−0.590123
$96$	0	0
$97$	−6.87043	−0.697586	−0.348793	−	0.937200i	$-0.613408\pi$
$97$	−6.87043	−0.697586	−0.348793	+	0.937200i	$0.613408\pi$
$98$	1.84551	0.186424
$99$	0	0
$100$	5.27875	0.527875
$101$	−2.26896	−0.225770	−0.112885	−	0.993608i	$-0.536009\pi$
$101$	−2.26896	−0.225770	−0.112885	+	0.993608i	$0.536009\pi$
$102$	0	0
$103$	−5.79288	−0.570789	−0.285395	−	0.958410i	$-0.592125\pi$
$103$	−5.79288	−0.570789	−0.285395	+	0.958410i	$0.592125\pi$
$104$	−1.30896	−0.128354
$105$	0	0
$106$	3.89546	0.378361
$107$	10.7966	1.04374	0.521871	−	0.853024i	$-0.325234\pi$
$107$	10.7966	1.04374	0.521871	+	0.853024i	$0.325234\pi$
$108$	0	0
$109$	19.6999	1.88691	0.943454	−	0.331503i	$-0.107556\pi$
$109$	19.6999	1.88691	0.943454	+	0.331503i	$0.107556\pi$
$110$	3.40856	0.324994
$111$	0	0
$112$	5.09149	0.481101
$113$	12.6967	1.19440	0.597202	−	0.802091i	$-0.296279\pi$
$113$	12.6967	1.19440	0.597202	+	0.802091i	$0.296279\pi$
$114$	0	0
$115$	3.51372	0.327656
$116$	−10.1121	−0.938883
$117$	0	0
$118$	0.148092	0.0136330
$119$	10.5577	0.967820
$120$	0	0
$121$	5.56210	0.505645
$122$	5.35633	0.484940
$123$	0	0
$124$	−13.3317	−1.19722
$125$	10.7174	0.958594
$126$	0	0
$127$	−6.59797	−0.585475	−0.292737	−	0.956193i	$-0.594566\pi$
$127$	−6.59797	−0.585475	−0.292737	+	0.956193i	$0.594566\pi$
$128$	−11.1334	−0.984066
$129$	0	0
$130$	−0.469876	−0.0412109
$131$	−4.85756	−0.424407	−0.212204	−	0.977225i	$-0.568064\pi$
$131$	−4.85756	−0.424407	−0.212204	+	0.977225i	$0.568064\pi$
$132$	0	0
$133$	14.0549	1.21871
$134$	2.91476	0.251797
$135$	0	0
$136$	−7.85812	−0.673828
$137$	22.2148	1.89793	0.948967	−	0.315374i	$-0.102130\pi$
$137$	22.2148	1.89793	0.948967	+	0.315374i	$0.102130\pi$
$138$	0	0
$139$	−16.3964	−1.39073	−0.695363	−	0.718659i	$-0.744756\pi$
$139$	−16.3964	−1.39073	−0.695363	+	0.718659i	$0.744756\pi$
$140$	6.32884	0.534884
$141$	0	0
$142$	5.17778	0.434510
$143$	−2.28311	−0.190924
$144$	0	0
$145$	−8.24302	−0.684546
$146$	−5.80536	−0.480455
$147$	0	0
$148$	−14.6421	−1.20357
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−20.7290	−1.68690	−0.843451	−	0.537206i	$-0.819480\pi$
$151$	−20.7290	−1.68690	−0.843451	+	0.537206i	$0.819480\pi$
$152$	−10.4611	−0.848507
$153$	0	0
$154$	−8.32900	−0.671170
$155$	−10.8675	−0.872900
$156$	0	0
$157$	−6.42049	−0.512411	−0.256206	−	0.966622i	$-0.582472\pi$
$157$	−6.42049	−0.512411	−0.256206	+	0.966622i	$0.582472\pi$
$158$	4.82423	0.383795
$159$	0	0
$160$	−7.34679	−0.580815
$161$	−8.58597	−0.676669
$162$	0	0
$163$	8.63691	0.676495	0.338248	−	0.941057i	$-0.390166\pi$
$163$	8.63691	0.676495	0.338248	+	0.941057i	$0.390166\pi$
$164$	11.1791	0.872941
$165$	0	0
$166$	1.55303	0.120538
$167$	−19.5407	−1.51211	−0.756054	−	0.654510i	$-0.772875\pi$
$167$	−19.5407	−1.51211	−0.756054	+	0.654510i	$0.772875\pi$
$168$	0	0
$169$	−12.6853	−0.975790
$170$	−2.82083	−0.216348
$171$	0	0
$172$	7.61485	0.580627
$173$	−8.57024	−0.651583	−0.325792	−	0.945442i	$-0.605631\pi$
$173$	−8.57024	−0.651583	−0.325792	+	0.945442i	$0.605631\pi$
$174$	0	0
$175$	−10.5148	−0.794841
$176$	−6.60995	−0.498244
$177$	0	0
$178$	−3.50846	−0.262970
$179$	−25.5200	−1.90746	−0.953729	−	0.300669i	$-0.902790\pi$
$179$	−25.5200	−1.90746	−0.953729	+	0.300669i	$0.902790\pi$
$180$	0	0
$181$	16.8967	1.25592	0.627960	−	0.778246i	$-0.283890\pi$
$181$	16.8967	1.25592	0.627960	+	0.778246i	$0.283890\pi$
$182$	1.14817	0.0851079
$183$	0	0
$184$	6.39057	0.471119
$185$	−11.9358	−0.877534
$186$	0	0
$187$	−13.7063	−1.00231
$188$	−14.9674	−1.09161
$189$	0	0
$190$	−3.75522	−0.272432
$191$	11.0286	0.798001	0.399001	−	0.916951i	$-0.369357\pi$
$191$	11.0286	0.798001	0.399001	+	0.916951i	$0.369357\pi$
$192$	0	0
$193$	−18.7180	−1.34735	−0.673675	−	0.739028i	$-0.735285\pi$
$193$	−18.7180	−1.34735	−0.673675	+	0.739028i	$0.735285\pi$
$194$	−4.48554	−0.322043
$195$	0	0
$196$	−4.44858	−0.317756
$197$	−15.3627	−1.09455	−0.547273	−	0.836954i	$-0.684334\pi$
$197$	−15.3627	−1.09455	−0.547273	+	0.836954i	$0.684334\pi$
$198$	0	0
$199$	18.8788	1.33829	0.669143	−	0.743134i	$-0.266662\pi$
$199$	18.8788	1.33829	0.669143	+	0.743134i	$0.266662\pi$
$200$	7.82618	0.553395
$201$	0	0
$202$	−1.48135	−0.104228
$203$	20.1423	1.41371
$204$	0	0
$205$	9.11283	0.636467
$206$	−3.78203	−0.263507
$207$	0	0
$208$	0.911194	0.0631799
$209$	−18.2465	−1.26214
$210$	0	0
$211$	1.80692	0.124393	0.0621967	−	0.998064i	$-0.480189\pi$
$211$	1.80692	0.124393	0.0621967	+	0.998064i	$0.480189\pi$
$212$	−9.38999	−0.644907
$213$	0	0
$214$	7.04881	0.481847
$215$	6.20736	0.423339
$216$	0	0
$217$	26.5554	1.80270
$218$	12.8616	0.871097
$219$	0	0
$220$	−8.21631	−0.553944
$221$	1.88944	0.127098
$222$	0	0
$223$	19.7078	1.31973	0.659866	−	0.751384i	$-0.270613\pi$
$223$	19.7078	1.31973	0.659866	+	0.751384i	$0.270613\pi$
$224$	17.9523	1.19949
$225$	0	0
$226$	8.28937	0.551400
$227$	−7.44860	−0.494381	−0.247190	−	0.968967i	$-0.579507\pi$
$227$	−7.44860	−0.494381	−0.247190	+	0.968967i	$0.579507\pi$
$228$	0	0
$229$	−23.6641	−1.56377	−0.781885	−	0.623423i	$-0.785741\pi$
$229$	−23.6641	−1.56377	−0.781885	+	0.623423i	$0.785741\pi$
$230$	2.29402	0.151263
$231$	0	0
$232$	−14.9920	−0.984272
$233$	−9.11870	−0.597386	−0.298693	−	0.954349i	$-0.596551\pi$
$233$	−9.11870	−0.597386	−0.298693	+	0.954349i	$0.596551\pi$
$234$	0	0
$235$	−12.2009	−0.795900
$236$	−0.356975	−0.0232371
$237$	0	0
$238$	6.89285	0.446797
$239$	3.20427	0.207267	0.103633	−	0.994616i	$-0.466953\pi$
$239$	3.20427	0.207267	0.103633	+	0.994616i	$0.466953\pi$
$240$	0	0
$241$	−11.0256	−0.710219	−0.355109	−	0.934825i	$-0.615556\pi$
$241$	−11.0256	−0.710219	−0.355109	+	0.934825i	$0.615556\pi$
$242$	3.63136	0.233433
$243$	0	0
$244$	−12.9114	−0.826568
$245$	−3.62633	−0.231678
$246$	0	0
$247$	2.51531	0.160045
$248$	−19.7653	−1.25510
$249$	0	0
$250$	6.99714	0.442538
$251$	−0.228411	−0.0144172	−0.00720859	−	0.999974i	$-0.502295\pi$
$251$	−0.228411	−0.0144172	−0.00720859	+	0.999974i	$0.502295\pi$
$252$	0	0
$253$	11.1466	0.700780
$254$	−4.30765	−0.270286
$255$	0	0
$256$	−8.24977	−0.515611
$257$	−16.8677	−1.05218	−0.526090	−	0.850429i	$-0.676342\pi$
$257$	−16.8677	−1.05218	−0.526090	+	0.850429i	$0.676342\pi$
$258$	0	0
$259$	29.1657	1.81227
$260$	1.13263	0.0702429
$261$	0	0
$262$	−3.17139	−0.195929
$263$	14.8077	0.913085	0.456542	−	0.889702i	$-0.349088\pi$
$263$	14.8077	0.913085	0.456542	+	0.889702i	$0.349088\pi$
$264$	0	0
$265$	−7.65440	−0.470206
$266$	9.17608	0.562622
$267$	0	0
$268$	−7.02600	−0.429181
$269$	23.8683	1.45528	0.727638	−	0.685961i	$-0.240618\pi$
$269$	23.8683	1.45528	0.727638	+	0.685961i	$0.240618\pi$
$270$	0	0
$271$	−26.6358	−1.61801	−0.809005	−	0.587802i	$-0.799993\pi$
$271$	−26.6358	−1.61801	−0.809005	+	0.587802i	$0.799993\pi$
$272$	5.47021	0.331680
$273$	0	0
$274$	14.5035	0.876187
$275$	13.6506	0.823163
$276$	0	0
$277$	−2.44125	−0.146680	−0.0733401	−	0.997307i	$-0.523366\pi$
$277$	−2.44125	−0.146680	−0.0733401	+	0.997307i	$0.523366\pi$
$278$	−10.7048	−0.642033
$279$	0	0
$280$	9.38302	0.560743
$281$	14.3792	0.857792	0.428896	−	0.903354i	$-0.358903\pi$
$281$	14.3792	0.857792	0.428896	+	0.903354i	$0.358903\pi$
$282$	0	0
$283$	18.9460	1.12622	0.563111	−	0.826381i	$-0.309604\pi$
$283$	18.9460	1.12622	0.563111	+	0.826381i	$0.309604\pi$
$284$	−12.4810	−0.740611
$285$	0	0
$286$	−1.49059	−0.0881405
$287$	−22.2677	−1.31442
$288$	0	0
$289$	−5.65702	−0.332766
$290$	−5.38167	−0.316023
$291$	0	0
$292$	13.9938	0.818923
$293$	11.6187	0.678773	0.339386	−	0.940647i	$-0.389781\pi$
$293$	11.6187	0.678773	0.339386	+	0.940647i	$0.389781\pi$
$294$	0	0
$295$	−0.290994	−0.0169423
$296$	−21.7081	−1.26176
$297$	0	0
$298$	−0.652876	−0.0378201
$299$	−1.53658	−0.0888626
$300$	0	0
$301$	−15.1680	−0.874271
$302$	−13.5335	−0.778764
$303$	0	0
$304$	7.28220	0.417663
$305$	−10.5249	−0.602656
$306$	0	0
$307$	30.1083	1.71837	0.859185	−	0.511664i	$-0.170971\pi$
$307$	30.1083	1.71837	0.859185	+	0.511664i	$0.170971\pi$
$308$	20.0770	1.14399
$309$	0	0
$310$	−7.09514	−0.402977
$311$	5.55072	0.314752	0.157376	−	0.987539i	$-0.449696\pi$
$311$	5.55072	0.314752	0.157376	+	0.987539i	$0.449696\pi$
$312$	0	0
$313$	25.3761	1.43434	0.717170	−	0.696899i	$-0.245437\pi$
$313$	25.3761	1.43434	0.717170	+	0.696899i	$0.245437\pi$
$314$	−4.19179	−0.236556
$315$	0	0
$316$	−11.6288	−0.654170
$317$	33.0006	1.85350	0.926749	−	0.375680i	$-0.122591\pi$
$317$	33.0006	1.85350	0.926749	+	0.375680i	$0.122591\pi$
$318$	0	0
$319$	−26.1494	−1.46408
$320$	−0.629257	−0.0351765
$321$	0	0
$322$	−5.60557	−0.312386
$323$	15.1003	0.840203
$324$	0	0
$325$	−1.88176	−0.104381
$326$	5.63883	0.312306
$327$	0	0
$328$	16.5739	0.915143
$329$	29.8136	1.64368
$330$	0	0
$331$	−10.2505	−0.563417	−0.281709	−	0.959500i	$-0.590901\pi$
$331$	−10.2505	−0.563417	−0.281709	+	0.959500i	$0.590901\pi$
$332$	−3.74356	−0.205455
$333$	0	0
$334$	−12.7577	−0.698069
$335$	−5.72736	−0.312919
$336$	0	0
$337$	6.36051	0.346479	0.173239	−	0.984880i	$-0.444577\pi$
$337$	6.36051	0.346479	0.173239	+	0.984880i	$0.444577\pi$
$338$	−8.28191	−0.450476
$339$	0	0
$340$	6.79959	0.368760
$341$	−34.4751	−1.86693
$342$	0	0
$343$	−13.0822	−0.706372
$344$	11.2896	0.608696
$345$	0	0
$346$	−5.59530	−0.300805
$347$	24.8591	1.33451	0.667254	−	0.744830i	$-0.267469\pi$
$347$	24.8591	1.33451	0.667254	+	0.744830i	$0.267469\pi$
$348$	0	0
$349$	−10.2447	−0.548387	−0.274194	−	0.961675i	$-0.588411\pi$
$349$	−10.2447	−0.548387	−0.274194	+	0.961675i	$0.588411\pi$
$350$	−6.86483	−0.366941
$351$	0	0
$352$	−23.3063	−1.24223
$353$	9.80888	0.522074	0.261037	−	0.965329i	$-0.415936\pi$
$353$	9.80888	0.522074	0.261037	+	0.965329i	$0.415936\pi$
$354$	0	0
$355$	−10.1741	−0.539985
$356$	8.45712	0.448227
$357$	0	0
$358$	−16.6614	−0.880583
$359$	−5.22566	−0.275800	−0.137900	−	0.990446i	$-0.544035\pi$
$359$	−5.22566	−0.275800	−0.137900	+	0.990446i	$0.544035\pi$
$360$	0	0
$361$	1.10221	0.0580108
$362$	11.0314	0.579799
$363$	0	0
$364$	−2.76765	−0.145064
$365$	11.4072	0.597083
$366$	0	0
$367$	24.1173	1.25891	0.629457	−	0.777035i	$-0.283277\pi$
$367$	24.1173	1.25891	0.629457	+	0.777035i	$0.283277\pi$
$368$	−4.44862	−0.231900
$369$	0	0
$370$	−7.79257	−0.405116
$371$	18.7039	0.971060
$372$	0	0
$373$	−21.5829	−1.11752	−0.558760	−	0.829330i	$-0.688723\pi$
$373$	−21.5829	−1.11752	−0.558760	+	0.829330i	$0.688723\pi$
$374$	−8.94853	−0.462718
$375$	0	0
$376$	−22.1904	−1.14438
$377$	3.60474	0.185653
$378$	0	0
$379$	−24.8950	−1.27877	−0.639385	−	0.768886i	$-0.720811\pi$
$379$	−24.8950	−1.27877	−0.639385	+	0.768886i	$0.720811\pi$
$380$	9.05193	0.464354
$381$	0	0
$382$	7.20031	0.368400
$383$	−1.20565	−0.0616060	−0.0308030	−	0.999525i	$-0.509806\pi$
$383$	−1.20565	−0.0616060	−0.0308030	+	0.999525i	$0.509806\pi$
$384$	0	0
$385$	16.3661	0.834093
$386$	−12.2205	−0.622008
$387$	0	0
$388$	10.8124	0.548914
$389$	24.6195	1.24826	0.624128	−	0.781322i	$-0.285454\pi$
$389$	24.6195	1.24826	0.624128	+	0.781322i	$0.285454\pi$
$390$	0	0
$391$	−9.22461	−0.466509
$392$	−6.59539	−0.333117
$393$	0	0
$394$	−10.0299	−0.505300
$395$	−9.47938	−0.476959
$396$	0	0
$397$	−15.0492	−0.755300	−0.377650	−	0.925948i	$-0.623268\pi$
$397$	−15.0492	−0.755300	−0.377650	+	0.925948i	$0.623268\pi$
$398$	12.3255	0.617823
$399$	0	0
$400$	−5.44798	−0.272399
$401$	−31.8579	−1.59091	−0.795454	−	0.606013i	$-0.792768\pi$
$401$	−31.8579	−1.59091	−0.795454	+	0.606013i	$0.792768\pi$
$402$	0	0
$403$	4.75245	0.236736
$404$	3.57079	0.177653
$405$	0	0
$406$	13.1504	0.652643
$407$	−37.8638	−1.87684
$408$	0	0
$409$	16.0555	0.793891	0.396946	−	0.917842i	$-0.370070\pi$
$409$	16.0555	0.793891	0.396946	+	0.917842i	$0.370070\pi$
$410$	5.94955	0.293827
$411$	0	0
$412$	9.11656	0.449141
$413$	0.711060	0.0349890
$414$	0	0
$415$	−3.05162	−0.149798
$416$	3.21281	0.157521
$417$	0	0
$418$	−11.9127	−0.582669
$419$	−14.8446	−0.725208	−0.362604	−	0.931943i	$-0.618112\pi$
$419$	−14.8446	−0.725208	−0.362604	+	0.931943i	$0.618112\pi$
$420$	0	0
$421$	−17.2582	−0.841114	−0.420557	−	0.907266i	$-0.638165\pi$
$421$	−17.2582	−0.841114	−0.420557	+	0.907266i	$0.638165\pi$
$422$	1.17969	0.0574266
$423$	0	0
$424$	−13.9214	−0.676084
$425$	−11.2969	−0.547979
$426$	0	0
$427$	25.7183	1.24459
$428$	−16.9911	−0.821296
$429$	0	0
$430$	4.05264	0.195436
$431$	−22.0624	−1.06271	−0.531355	−	0.847150i	$-0.678317\pi$
$431$	−22.0624	−1.06271	−0.531355	+	0.847150i	$0.678317\pi$
$432$	0	0
$433$	17.7074	0.850963	0.425482	−	0.904967i	$-0.360105\pi$
$433$	17.7074	0.850963	0.425482	+	0.904967i	$0.360105\pi$
$434$	17.3374	0.832220
$435$	0	0
$436$	−31.0028	−1.48476
$437$	−12.2802	−0.587443
$438$	0	0
$439$	8.66548	0.413581	0.206790	−	0.978385i	$-0.433698\pi$
$439$	8.66548	0.413581	0.206790	+	0.978385i	$0.433698\pi$
$440$	−12.1814	−0.580723
$441$	0	0
$442$	1.23357	0.0586750
$443$	−6.60184	−0.313663	−0.156831	−	0.987625i	$-0.550128\pi$
$443$	−6.60184	−0.313663	−0.156831	+	0.987625i	$0.550128\pi$
$444$	0	0
$445$	6.89396	0.326805
$446$	12.8667	0.609258
$447$	0	0
$448$	1.53762	0.0726459
$449$	2.10312	0.0992523	0.0496262	−	0.998768i	$-0.484197\pi$
$449$	2.10312	0.0992523	0.0496262	+	0.998768i	$0.484197\pi$
$450$	0	0
$451$	28.9087	1.36126
$452$	−19.9815	−0.939849
$453$	0	0
$454$	−4.86301	−0.228232
$455$	−2.25609	−0.105767
$456$	0	0
$457$	−8.27482	−0.387080	−0.193540	−	0.981092i	$-0.561997\pi$
$457$	−8.27482	−0.387080	−0.193540	+	0.981092i	$0.561997\pi$
$458$	−15.4497	−0.721919
$459$	0	0
$460$	−5.52973	−0.257825
$461$	−29.0899	−1.35485	−0.677425	−	0.735592i	$-0.736904\pi$
$461$	−29.0899	−1.35485	−0.677425	+	0.735592i	$0.736904\pi$
$462$	0	0
$463$	30.5334	1.41901	0.709503	−	0.704702i	$-0.248919\pi$
$463$	30.5334	1.41901	0.709503	+	0.704702i	$0.248919\pi$
$464$	10.4362	0.484491
$465$	0	0
$466$	−5.95338	−0.275785
$467$	21.2513	0.983394	0.491697	−	0.870766i	$-0.336377\pi$
$467$	21.2513	0.983394	0.491697	+	0.870766i	$0.336377\pi$
$468$	0	0
$469$	13.9951	0.646234
$470$	−7.96568	−0.367430
$471$	0	0
$472$	−0.529245	−0.0243605
$473$	19.6917	0.905423
$474$	0	0
$475$	−15.0389	−0.690033
$476$	−16.6152	−0.761555
$477$	0	0
$478$	2.09199	0.0956853
$479$	29.7734	1.36038	0.680190	−	0.733035i	$-0.261897\pi$
$479$	29.7734	1.36038	0.680190	+	0.733035i	$0.261897\pi$
$480$	0	0
$481$	5.21960	0.237993
$482$	−7.19832	−0.327875
$483$	0	0
$484$	−8.75337	−0.397880
$485$	8.81387	0.400217
$486$	0	0
$487$	−35.2316	−1.59649	−0.798247	−	0.602330i	$-0.794239\pi$
$487$	−35.2316	−1.59649	−0.798247	+	0.602330i	$0.794239\pi$
$488$	−19.1422	−0.866527
$489$	0	0
$490$	−2.36755	−0.106955
$491$	35.8902	1.61970	0.809851	−	0.586636i	$-0.199548\pi$
$491$	35.8902	1.61970	0.809851	+	0.586636i	$0.199548\pi$
$492$	0	0
$493$	21.6405	0.974639
$494$	1.64219	0.0738855
$495$	0	0
$496$	13.7590	0.617799
$497$	24.8609	1.11517
$498$	0	0
$499$	31.3839	1.40493	0.702467	−	0.711716i	$-0.252082\pi$
$499$	31.3839	1.40493	0.702467	+	0.711716i	$0.252082\pi$
$500$	−16.8666	−0.754295
$501$	0	0
$502$	−0.149124	−0.00665574
$503$	6.34561	0.282937	0.141468	−	0.989943i	$-0.454818\pi$
$503$	6.34561	0.282937	0.141468	+	0.989943i	$0.454818\pi$
$504$	0	0
$505$	2.91079	0.129528
$506$	7.27734	0.323517
$507$	0	0
$508$	10.3836	0.460696
$509$	19.8363	0.879230	0.439615	−	0.898186i	$-0.355115\pi$
$509$	19.8363	0.879230	0.439615	+	0.898186i	$0.355115\pi$
$510$	0	0
$511$	−27.8742	−1.23308
$512$	16.8808	0.746033
$513$	0	0
$514$	−11.0125	−0.485742
$515$	7.43151	0.327471
$516$	0	0
$517$	−38.7050	−1.70224
$518$	19.0416	0.836638
$519$	0	0
$520$	1.67922	0.0736387
$521$	−15.9862	−0.700369	−0.350184	−	0.936681i	$-0.613881\pi$
$521$	−15.9862	−0.700369	−0.350184	+	0.936681i	$0.613881\pi$
$522$	0	0
$523$	−0.185190	−0.00809779	−0.00404890	−	0.999992i	$-0.501289\pi$
$523$	−0.185190	−0.00809779	−0.00404890	+	0.999992i	$0.501289\pi$
$524$	7.64461	0.333956
$525$	0	0
$526$	9.66762	0.421528
$527$	28.5306	1.24281
$528$	0	0
$529$	−15.4981	−0.673832
$530$	−4.99737	−0.217072
$531$	0	0
$532$	−22.1189	−0.958975
$533$	−3.98511	−0.172614
$534$	0	0
$535$	−13.8506	−0.598813
$536$	−10.4166	−0.449929
$537$	0	0
$538$	15.5830	0.671833
$539$	−11.5038	−0.495505
$540$	0	0
$541$	7.74383	0.332933	0.166467	−	0.986047i	$-0.446764\pi$
$541$	7.74383	0.332933	0.166467	+	0.986047i	$0.446764\pi$
$542$	−17.3899	−0.746959
$543$	0	0
$544$	19.2876	0.826950
$545$	−25.2724	−1.08255
$546$	0	0
$547$	−8.10945	−0.346735	−0.173368	−	0.984857i	$-0.555465\pi$
$547$	−8.10945	−0.346735	−0.173368	+	0.984857i	$0.555465\pi$
$548$	−34.9605	−1.49344
$549$	0	0
$550$	8.91216	0.380016
$551$	28.8088	1.22730
$552$	0	0
$553$	23.1634	0.985007
$554$	−1.59383	−0.0677154
$555$	0	0
$556$	25.8039	1.09433
$557$	−11.7009	−0.495782	−0.247891	−	0.968788i	$-0.579737\pi$
$557$	−11.7009	−0.495782	−0.247891	+	0.968788i	$0.579737\pi$
$558$	0	0
$559$	−2.71453	−0.114812
$560$	−6.53173	−0.276016
$561$	0	0
$562$	9.38785	0.396003
$563$	40.3752	1.70161	0.850805	−	0.525481i	$-0.176115\pi$
$563$	40.3752	1.70161	0.850805	+	0.525481i	$0.176115\pi$
$564$	0	0
$565$	−16.2882	−0.685250
$566$	12.3694	0.519924
$567$	0	0
$568$	−18.5041	−0.776415
$569$	−2.42844	−0.101805	−0.0509027	−	0.998704i	$-0.516210\pi$
$569$	−2.42844	−0.101805	−0.0509027	+	0.998704i	$0.516210\pi$
$570$	0	0
$571$	40.9194	1.71242	0.856211	−	0.516626i	$-0.172812\pi$
$571$	40.9194	1.71242	0.856211	+	0.516626i	$0.172812\pi$
$572$	3.59306	0.150233
$573$	0	0
$574$	−14.5380	−0.606806
$575$	9.18712	0.383129
$576$	0	0
$577$	−21.9950	−0.915663	−0.457832	−	0.889039i	$-0.651374\pi$
$577$	−21.9950	−0.915663	−0.457832	+	0.889039i	$0.651374\pi$
$578$	−3.69333	−0.153622
$579$	0	0
$580$	12.9725	0.538653
$581$	7.45681	0.309361
$582$	0	0
$583$	−24.2821	−1.00566
$584$	20.7469	0.858513
$585$	0	0
$586$	7.58558	0.313357
$587$	−9.49236	−0.391792	−0.195896	−	0.980625i	$-0.562761\pi$
$587$	−9.49236	−0.391792	−0.195896	+	0.980625i	$0.562761\pi$
$588$	0	0
$589$	37.9813	1.56499
$590$	−0.189983	−0.00782148
$591$	0	0
$592$	15.1115	0.621079
$593$	−24.6368	−1.01171	−0.505857	−	0.862617i	$-0.668824\pi$
$593$	−24.6368	−1.01171	−0.505857	+	0.862617i	$0.668824\pi$
$594$	0	0
$595$	−13.5441	−0.555255
$596$	1.57375	0.0644634
$597$	0	0
$598$	−1.00320	−0.0410237
$599$	−9.69513	−0.396132	−0.198066	−	0.980189i	$-0.563466\pi$
$599$	−9.69513	−0.396132	−0.198066	+	0.980189i	$0.563466\pi$
$600$	0	0
$601$	−39.3803	−1.60636	−0.803179	−	0.595738i	$-0.796860\pi$
$601$	−39.3803	−1.60636	−0.803179	+	0.595738i	$0.796860\pi$
$602$	−9.90284	−0.403610
$603$	0	0
$604$	32.6223	1.32738
$605$	−7.13545	−0.290097
$606$	0	0
$607$	−13.3232	−0.540773	−0.270386	−	0.962752i	$-0.587151\pi$
$607$	−13.3232	−0.540773	−0.270386	+	0.962752i	$0.587151\pi$
$608$	25.6766	1.04132
$609$	0	0
$610$	−6.87148	−0.278218
$611$	5.33556	0.215853
$612$	0	0
$613$	4.47235	0.180637	0.0903183	−	0.995913i	$-0.471212\pi$
$613$	4.47235	0.180637	0.0903183	+	0.995913i	$0.471212\pi$
$614$	19.6570	0.793291
$615$	0	0
$616$	29.7658	1.19930
$617$	−46.5017	−1.87209	−0.936044	−	0.351882i	$-0.885542\pi$
$617$	−46.5017	−1.87209	−0.936044	+	0.351882i	$0.885542\pi$
$618$	0	0
$619$	23.1348	0.929868	0.464934	−	0.885345i	$-0.346078\pi$
$619$	23.1348	0.929868	0.464934	+	0.885345i	$0.346078\pi$
$620$	17.1028	0.686864
$621$	0	0
$622$	3.62393	0.145306
$623$	−16.8458	−0.674911
$624$	0	0
$625$	3.02218	0.120887
$626$	16.5674	0.662167
$627$	0	0
$628$	10.1043	0.403204
$629$	31.3351	1.24941
$630$	0	0
$631$	4.75459	0.189277	0.0946385	−	0.995512i	$-0.469830\pi$
$631$	4.75459	0.189277	0.0946385	+	0.995512i	$0.469830\pi$
$632$	−17.2406	−0.685794
$633$	0	0
$634$	21.5453	0.855673
$635$	8.46433	0.335897
$636$	0	0
$637$	1.58582	0.0628327
$638$	−17.0723	−0.675899
$639$	0	0
$640$	14.2828	0.564575
$641$	−11.5990	−0.458134	−0.229067	−	0.973411i	$-0.573567\pi$
$641$	−11.5990	−0.458134	−0.229067	+	0.973411i	$0.573567\pi$
$642$	0	0
$643$	26.8460	1.05870	0.529351	−	0.848403i	$-0.322436\pi$
$643$	26.8460	1.05870	0.529351	+	0.848403i	$0.322436\pi$
$644$	13.5122	0.532455
$645$	0	0
$646$	9.85862	0.387882
$647$	36.5612	1.43737	0.718684	−	0.695337i	$-0.244745\pi$
$647$	36.5612	1.43737	0.718684	+	0.695337i	$0.244745\pi$
$648$	0	0
$649$	−0.923122	−0.0362357
$650$	−1.22856	−0.0481880
$651$	0	0
$652$	−13.5924	−0.532318
$653$	6.99409	0.273700	0.136850	−	0.990592i	$-0.456302\pi$
$653$	6.99409	0.273700	0.136850	+	0.990592i	$0.456302\pi$
$654$	0	0
$655$	6.23162	0.243490
$656$	−11.5375	−0.450463
$657$	0	0
$658$	19.4646	0.758808
$659$	−30.2713	−1.17920	−0.589601	−	0.807694i	$-0.700715\pi$
$659$	−30.2713	−1.17920	−0.589601	+	0.807694i	$0.700715\pi$
$660$	0	0
$661$	8.73300	0.339675	0.169837	−	0.985472i	$-0.445676\pi$
$661$	8.73300	0.339675	0.169837	+	0.985472i	$0.445676\pi$
$662$	−6.69229	−0.260103
$663$	0	0
$664$	−5.55014	−0.215387
$665$	−18.0306	−0.699195
$666$	0	0
$667$	−17.5990	−0.681436
$668$	30.7523	1.18984
$669$	0	0
$670$	−3.73925	−0.144460
$671$	−33.3883	−1.28894
$672$	0	0
$673$	−4.17598	−0.160972	−0.0804862	−	0.996756i	$-0.525647\pi$
$673$	−4.17598	−0.160972	−0.0804862	+	0.996756i	$0.525647\pi$
$674$	4.15262	0.159953
$675$	0	0
$676$	19.9635	0.767826
$677$	−14.0549	−0.540172	−0.270086	−	0.962836i	$-0.587052\pi$
$677$	−14.0549	−0.540172	−0.270086	+	0.962836i	$0.587052\pi$
$678$	0	0
$679$	−21.5372	−0.826520
$680$	10.0809	0.386587
$681$	0	0
$682$	−22.5079	−0.861874
$683$	4.36809	0.167140	0.0835702	−	0.996502i	$-0.473368\pi$
$683$	4.36809	0.167140	0.0835702	+	0.996502i	$0.473368\pi$
$684$	0	0
$685$	−28.4986	−1.08888
$686$	−8.54105	−0.326099
$687$	0	0
$688$	−7.85896	−0.299620
$689$	3.34733	0.127523
$690$	0	0
$691$	20.1625	0.767018	0.383509	−	0.923537i	$-0.374716\pi$
$691$	20.1625	0.767018	0.383509	+	0.923537i	$0.374716\pi$
$692$	13.4874	0.512716
$693$	0	0
$694$	16.2299	0.616080
$695$	21.0345	0.797883
$696$	0	0
$697$	−23.9240	−0.906186
$698$	−6.68853	−0.253165
$699$	0	0
$700$	16.5476	0.625442
$701$	5.94273	0.224454	0.112227	−	0.993683i	$-0.464202\pi$
$701$	5.94273	0.224454	0.112227	+	0.993683i	$0.464202\pi$
$702$	0	0
$703$	41.7147	1.57330
$704$	−1.99620	−0.0752344
$705$	0	0
$706$	6.40398	0.241017
$707$	−7.11267	−0.267499
$708$	0	0
$709$	19.2413	0.722622	0.361311	−	0.932445i	$-0.382329\pi$
$709$	19.2413	0.722622	0.361311	+	0.932445i	$0.382329\pi$
$710$	−6.64242	−0.249286
$711$	0	0
$712$	12.5384	0.469895
$713$	−23.2024	−0.868935
$714$	0	0
$715$	2.92894	0.109536
$716$	40.1622	1.50093
$717$	0	0
$718$	−3.41171	−0.127324
$719$	5.26138	0.196217	0.0981083	−	0.995176i	$-0.468721\pi$
$719$	5.26138	0.196217	0.0981083	+	0.995176i	$0.468721\pi$
$720$	0	0
$721$	−18.1593	−0.676287
$722$	0.719603	0.0267809
$723$	0	0
$724$	−26.5912	−0.988254
$725$	−21.5525	−0.800441
$726$	0	0
$727$	−11.8275	−0.438659	−0.219329	−	0.975651i	$-0.570387\pi$
$727$	−11.8275	−0.438659	−0.219329	+	0.975651i	$0.570387\pi$
$728$	−4.10327	−0.152077
$729$	0	0
$730$	7.44752	0.275645
$731$	−16.2963	−0.602739
$732$	0	0
$733$	3.27481	0.120958	0.0604789	−	0.998169i	$-0.480737\pi$
$733$	3.27481	0.120958	0.0604789	+	0.998169i	$0.480737\pi$
$734$	15.7456	0.581182
$735$	0	0
$736$	−15.6855	−0.578177
$737$	−18.1689	−0.669261
$738$	0	0
$739$	34.7670	1.27893	0.639463	−	0.768822i	$-0.279157\pi$
$739$	34.7670	1.27893	0.639463	+	0.768822i	$0.279157\pi$
$740$	18.7839	0.690511
$741$	0	0
$742$	12.2114	0.448293
$743$	−29.9985	−1.10054	−0.550268	−	0.834988i	$-0.685475\pi$
$743$	−29.9985	−1.10054	−0.550268	+	0.834988i	$0.685475\pi$
$744$	0	0
$745$	1.28287	0.0470007
$746$	−14.0910	−0.515906
$747$	0	0
$748$	21.5704	0.788691
$749$	33.8446	1.23666
$750$	0	0
$751$	21.1905	0.773251	0.386625	−	0.922237i	$-0.373641\pi$
$751$	21.1905	0.773251	0.386625	+	0.922237i	$0.373641\pi$
$752$	15.4472	0.563302
$753$	0	0
$754$	2.35345	0.0857075
$755$	26.5926	0.967805
$756$	0	0
$757$	16.6924	0.606694	0.303347	−	0.952880i	$-0.401896\pi$
$757$	16.6924	0.606694	0.303347	+	0.952880i	$0.401896\pi$
$758$	−16.2534	−0.590348
$759$	0	0
$760$	13.4202	0.486803
$761$	−1.40843	−0.0510555	−0.0255277	−	0.999674i	$-0.508127\pi$
$761$	−1.40843	−0.0510555	−0.0255277	+	0.999674i	$0.508127\pi$
$762$	0	0
$763$	61.7545	2.23566
$764$	−17.3563	−0.627928
$765$	0	0
$766$	−0.787142	−0.0284406
$767$	0.127254	0.00459488
$768$	0	0
$769$	31.2989	1.12867	0.564334	−	0.825546i	$-0.309133\pi$
$769$	31.2989	1.12867	0.564334	+	0.825546i	$0.309133\pi$
$770$	10.6850	0.385062
$771$	0	0
$772$	29.4575	1.06020
$773$	−36.1845	−1.30147	−0.650734	−	0.759306i	$-0.725539\pi$
$773$	−36.1845	−1.30147	−0.650734	+	0.759306i	$0.725539\pi$
$774$	0	0
$775$	−28.4146	−1.02068
$776$	16.0302	0.575451
$777$	0	0
$778$	16.0735	0.576261
$779$	−31.8487	−1.14110
$780$	0	0
$781$	−32.2753	−1.15490
$782$	−6.02253	−0.215365
$783$	0	0
$784$	4.59119	0.163971
$785$	8.23666	0.293979
$786$	0	0
$787$	−50.2756	−1.79213	−0.896066	−	0.443922i	$-0.853587\pi$
$787$	−50.2756	−1.79213	−0.896066	+	0.443922i	$0.853587\pi$
$788$	24.1771	0.861272
$789$	0	0
$790$	−6.18886	−0.220190
$791$	39.8011	1.41516
$792$	0	0
$793$	4.60264	0.163445
$794$	−9.82529	−0.348686
$795$	0	0
$796$	−29.7106	−1.05307
$797$	−8.11382	−0.287406	−0.143703	−	0.989621i	$-0.545901\pi$
$797$	−8.11382	−0.287406	−0.143703	+	0.989621i	$0.545901\pi$
$798$	0	0
$799$	32.0312	1.13318
$800$	−19.2092	−0.679148
$801$	0	0
$802$	−20.7993	−0.734448
$803$	36.1873	1.27702
$804$	0	0
$805$	11.0147	0.388216
$806$	3.10276	0.109290
$807$	0	0
$808$	5.29399	0.186242
$809$	−21.3652	−0.751159	−0.375579	−	0.926790i	$-0.622556\pi$
$809$	−21.3652	−0.751159	−0.375579	+	0.926790i	$0.622556\pi$
$810$	0	0
$811$	29.5273	1.03684	0.518422	−	0.855125i	$-0.326520\pi$
$811$	29.5273	1.03684	0.518422	+	0.855125i	$0.326520\pi$
$812$	−31.6989	−1.11242
$813$	0	0
$814$	−24.7204	−0.866449
$815$	−11.0800	−0.388117
$816$	0	0
$817$	−21.6943	−0.758989
$818$	10.4822	0.366502
$819$	0	0
$820$	−14.3413	−0.500821
$821$	3.77958	0.131908	0.0659541	−	0.997823i	$-0.478991\pi$
$821$	3.77958	0.131908	0.0659541	+	0.997823i	$0.478991\pi$
$822$	0	0
$823$	−22.6493	−0.789506	−0.394753	−	0.918787i	$-0.629170\pi$
$823$	−22.6493	−0.789506	−0.394753	+	0.918787i	$0.629170\pi$
$824$	13.5160	0.470854
$825$	0	0
$826$	0.464234	0.0161528
$827$	9.42257	0.327655	0.163827	−	0.986489i	$-0.447616\pi$
$827$	9.42257	0.327655	0.163827	+	0.986489i	$0.447616\pi$
$828$	0	0
$829$	10.9566	0.380538	0.190269	−	0.981732i	$-0.439064\pi$
$829$	10.9566	0.380538	0.190269	+	0.981732i	$0.439064\pi$
$830$	−1.99233	−0.0691549
$831$	0	0
$832$	0.275179	0.00954012
$833$	9.52025	0.329857
$834$	0	0
$835$	25.0682	0.867522
$836$	28.7155	0.993146
$837$	0	0
$838$	−9.69171	−0.334794
$839$	−0.0127848	−0.000441380 0	−0.000220690	−	1.00000i	$-0.500070\pi$
$839$	−0.0127848	−0.000441380 0	−0.000220690		1.00000i	$0.500070\pi$
$840$	0	0
$841$	12.2864	0.423670
$842$	−11.2675	−0.388303
$843$	0	0
$844$	−2.84364	−0.0978823
$845$	16.2736	0.559827
$846$	0	0
$847$	17.4359	0.599103
$848$	9.69101	0.332791
$849$	0	0
$850$	−7.37546	−0.252976
$851$	−25.4831	−0.873548
$852$	0	0
$853$	−15.9380	−0.545707	−0.272854	−	0.962056i	$-0.587967\pi$
$853$	−15.9380	−0.545707	−0.272854	+	0.962056i	$0.587967\pi$
$854$	16.7908	0.574570
$855$	0	0
$856$	−25.1907	−0.861001
$857$	43.1271	1.47319	0.736597	−	0.676332i	$-0.236431\pi$
$857$	43.1271	1.47319	0.736597	+	0.676332i	$0.236431\pi$
$858$	0	0
$859$	−9.93465	−0.338966	−0.169483	−	0.985533i	$-0.554210\pi$
$859$	−9.93465	−0.338966	−0.169483	+	0.985533i	$0.554210\pi$
$860$	−9.76886	−0.333115
$861$	0	0
$862$	−14.4040	−0.490603
$863$	−45.3714	−1.54446	−0.772231	−	0.635342i	$-0.780859\pi$
$863$	−45.3714	−1.54446	−0.772231	+	0.635342i	$0.780859\pi$
$864$	0	0
$865$	10.9945	0.373824
$866$	11.5607	0.392850
$867$	0	0
$868$	−41.7916	−1.41850
$869$	−30.0715	−1.02011
$870$	0	0
$871$	2.50462	0.0848658
$872$	−45.9642	−1.55654
$873$	0	0
$874$	−8.01747	−0.271195
$875$	33.5965	1.13577
$876$	0	0
$877$	−39.2996	−1.32705	−0.663526	−	0.748153i	$-0.730941\pi$
$877$	−39.2996	−1.32705	−0.663526	+	0.748153i	$0.730941\pi$
$878$	5.65749	0.190931
$879$	0	0
$880$	8.47971	0.285851
$881$	−50.5025	−1.70147	−0.850737	−	0.525592i	$-0.823844\pi$
$881$	−50.5025	−1.70147	−0.850737	+	0.525592i	$0.823844\pi$
$882$	0	0
$883$	−22.4683	−0.756119	−0.378059	−	0.925781i	$-0.623408\pi$
$883$	−22.4683	−0.756119	−0.378059	+	0.925781i	$0.623408\pi$
$884$	−2.97352	−0.100010
$885$	0	0
$886$	−4.31018	−0.144803
$887$	−18.6086	−0.624817	−0.312409	−	0.949948i	$-0.601136\pi$
$887$	−18.6086	−0.624817	−0.312409	+	0.949948i	$0.601136\pi$
$888$	0	0
$889$	−20.6831	−0.693687
$890$	4.50090	0.150871
$891$	0	0
$892$	−31.0152	−1.03847
$893$	42.6414	1.42694
$894$	0	0
$895$	32.7389	1.09434
$896$	−34.9007	−1.16595
$897$	0	0
$898$	1.37308	0.0458201
$899$	54.4316	1.81540
$900$	0	0
$901$	20.0952	0.669468
$902$	18.8738	0.628428
$903$	0	0
$904$	−29.6241	−0.985284
$905$	−21.6762	−0.720542
$906$	0	0
$907$	−11.8605	−0.393820	−0.196910	−	0.980422i	$-0.563091\pi$
$907$	−11.8605	−0.393820	−0.196910	+	0.980422i	$0.563091\pi$
$908$	11.7223	0.389017
$909$	0	0
$910$	−1.47295	−0.0488278
$911$	5.02291	0.166416	0.0832082	−	0.996532i	$-0.473483\pi$
$911$	5.02291	0.166416	0.0832082	+	0.996532i	$0.473483\pi$
$912$	0	0
$913$	−9.68068	−0.320384
$914$	−5.40243	−0.178696
$915$	0	0
$916$	37.2415	1.23049
$917$	−15.2273	−0.502850
$918$	0	0
$919$	11.7397	0.387257	0.193629	−	0.981075i	$-0.437974\pi$
$919$	11.7397	0.387257	0.193629	+	0.981075i	$0.437974\pi$
$920$	−8.19827	−0.270289
$921$	0	0
$922$	−18.9921	−0.625471
$923$	4.44921	0.146448
$924$	0	0
$925$	−31.2077	−1.02610
$926$	19.9345	0.655089
$927$	0	0
$928$	36.7975	1.20794
$929$	−41.2552	−1.35354	−0.676769	−	0.736195i	$-0.736620\pi$
$929$	−41.2552	−1.35354	−0.676769	+	0.736195i	$0.736620\pi$
$930$	0	0
$931$	12.6738	0.415367
$932$	14.3506	0.470069
$933$	0	0
$934$	13.8745	0.453987
$935$	17.5834	0.575040
$936$	0	0
$937$	12.4311	0.406106	0.203053	−	0.979168i	$-0.434914\pi$
$937$	12.4311	0.406106	0.203053	+	0.979168i	$0.434914\pi$
$938$	9.13707	0.298336
$939$	0	0
$940$	19.2012	0.626275
$941$	44.8622	1.46247	0.731233	−	0.682128i	$-0.238945\pi$
$941$	44.8622	1.46247	0.731233	+	0.682128i	$0.238945\pi$
$942$	0	0
$943$	19.4561	0.633577
$944$	0.368419	0.0119910
$945$	0	0
$946$	12.8562	0.417991
$947$	8.67052	0.281754	0.140877	−	0.990027i	$-0.455008\pi$
$947$	8.67052	0.281754	0.140877	+	0.990027i	$0.455008\pi$
$948$	0	0
$949$	−4.98848	−0.161933
$950$	−9.81855	−0.318556
$951$	0	0
$952$	−24.6333	−0.798371
$953$	−53.6754	−1.73872	−0.869359	−	0.494181i	$-0.835468\pi$
$953$	−53.6754	−1.73872	−0.869359	+	0.494181i	$0.835468\pi$
$954$	0	0
$955$	−14.1483	−0.457827
$956$	−5.04272	−0.163093
$957$	0	0
$958$	19.4383	0.628024
$959$	69.6380	2.24873
$960$	0	0
$961$	40.7621	1.31491
$962$	3.40775	0.109870
$963$	0	0
$964$	17.3515	0.558854
$965$	24.0127	0.772997
$966$	0	0
$967$	31.4986	1.01293	0.506463	−	0.862262i	$-0.330953\pi$
$967$	31.4986	1.01293	0.506463	+	0.862262i	$0.330953\pi$
$968$	−12.9776	−0.417115
$969$	0	0
$970$	5.75436	0.184761
$971$	2.16055	0.0693353	0.0346677	−	0.999399i	$-0.488963\pi$
$971$	2.16055	0.0693353	0.0346677	+	0.999399i	$0.488963\pi$
$972$	0	0
$973$	−51.3988	−1.64777
$974$	−23.0018	−0.737027
$975$	0	0
$976$	13.3253	0.426533
$977$	−3.57558	−0.114393	−0.0571965	−	0.998363i	$-0.518216\pi$
$977$	−3.57558	−0.114393	−0.0571965	+	0.998363i	$0.518216\pi$
$978$	0	0
$979$	21.8697	0.698960
$980$	5.70695	0.182302
$981$	0	0
$982$	23.4318	0.747740
$983$	22.6442	0.722237	0.361119	−	0.932520i	$-0.382395\pi$
$983$	22.6442	0.722237	0.361119	+	0.932520i	$0.382395\pi$
$984$	0	0
$985$	19.7083	0.627959
$986$	14.1286	0.449945
$987$	0	0
$988$	−3.95848	−0.125936
$989$	13.2528	0.421416
$990$	0	0
$991$	−4.51947	−0.143566	−0.0717829	−	0.997420i	$-0.522869\pi$
$991$	−4.51947	−0.143566	−0.0717829	+	0.997420i	$0.522869\pi$
$992$	48.5135	1.54030
$993$	0	0
$994$	16.2311	0.514820
$995$	−24.2191	−0.767797
$996$	0	0
$997$	−4.73666	−0.150012	−0.0750058	−	0.997183i	$-0.523898\pi$
$997$	−4.73666	−0.150012	−0.0750058	+	0.997183i	$0.523898\pi$
$998$	20.4898	0.648592
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1341.2.a.e.1.5		9
3.2	odd	2		149.2.a.b.1.5	✓	9
12.11	even	2		2384.2.a.j.1.9		9
15.14	odd	2		3725.2.a.c.1.5		9
21.20	even	2		7301.2.a.j.1.5		9
24.5	odd	2		9536.2.a.v.1.9		9
24.11	even	2		9536.2.a.w.1.1		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
149.2.a.b.1.5	✓	9	3.2	odd	2
1341.2.a.e.1.5		9	1.1	even	1	trivial
2384.2.a.j.1.9		9	12.11	even	2
3725.2.a.c.1.5		9	15.14	odd	2
7301.2.a.j.1.5		9	21.20	even	2
9536.2.a.v.1.9		9	24.5	odd	2
9536.2.a.w.1.1		9	24.11	even	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)

\(f(q)\)	\(=\)	\(q+0.652876 q^{2} -1.57375 q^{4} -1.28287 q^{5} +3.13476 q^{7} -2.33322 q^{8} +O(q^{10})\)	\(q+0.652876 q^{2} -1.57375 q^{4} -1.28287 q^{5} +3.13476 q^{7} -2.33322 q^{8} -0.837555 q^{10} -4.06966 q^{11} +0.561009 q^{13} +2.04661 q^{14} +1.62420 q^{16} +3.36793 q^{17} +4.48355 q^{19} +2.01892 q^{20} -2.65698 q^{22} -2.73895 q^{23} -3.35424 q^{25} +0.366270 q^{26} -4.93334 q^{28} +6.42545 q^{29} +8.47125 q^{31} +5.72684 q^{32} +2.19884 q^{34} -4.02149 q^{35} +9.30394 q^{37} +2.92720 q^{38} +2.99321 q^{40} -7.10347 q^{41} -4.83865 q^{43} +6.40463 q^{44} -1.78820 q^{46} +9.51064 q^{47} +2.82673 q^{49} -2.18991 q^{50} -0.882890 q^{52} +5.96662 q^{53} +5.22084 q^{55} -7.31408 q^{56} +4.19502 q^{58} +0.226831 q^{59} +8.20421 q^{61} +5.53068 q^{62} +0.490507 q^{64} -0.719702 q^{65} +4.46449 q^{67} -5.30030 q^{68} -2.62554 q^{70} +7.93072 q^{71} -8.89197 q^{73} +6.07432 q^{74} -7.05600 q^{76} -12.7574 q^{77} +7.38920 q^{79} -2.08364 q^{80} -4.63768 q^{82} +2.37875 q^{83} -4.32062 q^{85} -3.15904 q^{86} +9.49539 q^{88} -5.37386 q^{89} +1.75863 q^{91} +4.31044 q^{92} +6.20927 q^{94} -5.75181 q^{95} -6.87043 q^{97} +1.84551 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{2} + 13 q^{4} + q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) \) 9 * q + q^2 + 13 * q^4 + q^5 + 3 * q^7 + 6 * q^8	\( 9 q + q^{2} + 13 q^{4} + q^{5} + 3 q^{7} + 6 q^{8} - 6 q^{10} - 5 q^{11} + 7 q^{13} + 8 q^{14} + 13 q^{16} + 5 q^{17} + 30 q^{19} + 10 q^{20} - 7 q^{22} + 4 q^{23} + 6 q^{25} + 15 q^{26} - 12 q^{28} + 16 q^{29} + 22 q^{31} + 38 q^{32} + 9 q^{34} + 11 q^{35} - 7 q^{37} + 18 q^{38} - 7 q^{40} - 6 q^{41} + 4 q^{43} - 6 q^{44} + q^{46} + 6 q^{47} + 14 q^{49} + 16 q^{50} + 50 q^{52} + 2 q^{53} - 2 q^{55} - 7 q^{56} + 2 q^{58} - 43 q^{59} + q^{61} - 33 q^{62} + 18 q^{64} + 20 q^{65} + 33 q^{67} + 16 q^{68} - 3 q^{70} - 15 q^{71} - 11 q^{73} - 33 q^{74} + 59 q^{76} + 30 q^{77} + q^{79} - 65 q^{80} + 5 q^{82} + 4 q^{83} - 34 q^{85} + 7 q^{86} - 37 q^{88} + 19 q^{89} + 62 q^{91} - 17 q^{92} + 17 q^{94} + 21 q^{95} - q^{97} - 36 q^{98}+O(q^{100}) \) 9 * q + q^2 + 13 * q^4 + q^5 + 3 * q^7 + 6 * q^8 - 6 * q^10 - 5 * q^11 + 7 * q^13 + 8 * q^14 + 13 * q^16 + 5 * q^17 + 30 * q^19 + 10 * q^20 - 7 * q^22 + 4 * q^23 + 6 * q^25 + 15 * q^26 - 12 * q^28 + 16 * q^29 + 22 * q^31 + 38 * q^32 + 9 * q^34 + 11 * q^35 - 7 * q^37 + 18 * q^38 - 7 * q^40 - 6 * q^41 + 4 * q^43 - 6 * q^44 + q^46 + 6 * q^47 + 14 * q^49 + 16 * q^50 + 50 * q^52 + 2 * q^53 - 2 * q^55 - 7 * q^56 + 2 * q^58 - 43 * q^59 + q^61 - 33 * q^62 + 18 * q^64 + 20 * q^65 + 33 * q^67 + 16 * q^68 - 3 * q^70 - 15 * q^71 - 11 * q^73 - 33 * q^74 + 59 * q^76 + 30 * q^77 + q^79 - 65 * q^80 + 5 * q^82 + 4 * q^83 - 34 * q^85 + 7 * q^86 - 37 * q^88 + 19 * q^89 + 62 * q^91 - 17 * q^92 + 17 * q^94 + 21 * q^95 - q^97 - 36 * q^98

Introduction

L-functions

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