LMFDB - Embedded newform 1341.2.a.h.1.12

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:	$12$
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			1.12
Root			$2.68556$ 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)

$f(q)$	$=$	$q+2.68556 q^{2} +5.21223 q^{4} +1.76881 q^{5} -2.90410 q^{7} +8.62665 q^{8} +O(q^{10})$	\(q+2.68556 q^{2} +5.21223 q^{4} +1.76881 q^{5} -2.90410 q^{7} +8.62665 q^{8} +4.75026 q^{10} +4.04066 q^{11} +1.74185 q^{13} -7.79913 q^{14} +12.7429 q^{16} -6.68541 q^{17} +3.89813 q^{19} +9.21947 q^{20} +10.8514 q^{22} -9.43378 q^{23} -1.87130 q^{25} +4.67786 q^{26} -15.1368 q^{28} +2.15036 q^{29} -0.231859 q^{31} +16.9686 q^{32} -17.9541 q^{34} -5.13681 q^{35} -2.01334 q^{37} +10.4687 q^{38} +15.2589 q^{40} +3.61371 q^{41} +5.40723 q^{43} +21.0609 q^{44} -25.3350 q^{46} +3.59321 q^{47} +1.43378 q^{49} -5.02548 q^{50} +9.07895 q^{52} -3.65166 q^{53} +7.14718 q^{55} -25.0526 q^{56} +5.77491 q^{58} +15.1098 q^{59} -12.2543 q^{61} -0.622672 q^{62} +20.0843 q^{64} +3.08102 q^{65} -10.3881 q^{67} -34.8459 q^{68} -13.7952 q^{70} -9.62471 q^{71} +15.9006 q^{73} -5.40695 q^{74} +20.3180 q^{76} -11.7345 q^{77} -13.1448 q^{79} +22.5398 q^{80} +9.70483 q^{82} +2.59849 q^{83} -11.8252 q^{85} +14.5214 q^{86} +34.8574 q^{88} +8.20872 q^{89} -5.05852 q^{91} -49.1711 q^{92} +9.64979 q^{94} +6.89507 q^{95} -2.58237 q^{97} +3.85050 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.68556	1.89898	0.949489	−	0.313801i	$-0.101602\pi$
$2$	2.68556	1.89898	0.949489	+	0.313801i	$0.101602\pi$
$3$	0	0
$3$	0	0
$4$	5.21223	2.60612
$5$	1.76881	0.791038	0.395519	−	0.918458i	$-0.370565\pi$
$5$	1.76881	0.791038	0.395519	+	0.918458i	$0.370565\pi$
$6$	0	0
$7$	−2.90410	−1.09765	−0.548823	−	0.835939i	$-0.684924\pi$
$7$	−2.90410	−1.09765	−0.548823	+	0.835939i	$0.684924\pi$
$8$	8.62665	3.04998
$9$	0	0
$10$	4.75026	1.50216
$11$	4.04066	1.21831	0.609153	−	0.793053i	$-0.291510\pi$
$11$	4.04066	1.21831	0.609153	+	0.793053i	$0.291510\pi$
$12$	0	0
$13$	1.74185	0.483104	0.241552	−	0.970388i	$-0.422344\pi$
$13$	1.74185	0.483104	0.241552	+	0.970388i	$0.422344\pi$
$14$	−7.79913	−2.08440
$15$	0	0
$16$	12.7429	3.18573
$17$	−6.68541	−1.62145	−0.810725	−	0.585427i	$-0.800927\pi$
$17$	−6.68541	−1.62145	−0.810725	+	0.585427i	$0.800927\pi$
$18$	0	0
$19$	3.89813	0.894293	0.447147	−	0.894461i	$-0.352440\pi$
$19$	3.89813	0.894293	0.447147	+	0.894461i	$0.352440\pi$
$20$	9.21947	2.06154
$21$	0	0
$22$	10.8514	2.31354
$23$	−9.43378	−1.96708	−0.983540	−	0.180693i	$-0.942166\pi$
$23$	−9.43378	−1.96708	−0.983540	+	0.180693i	$0.942166\pi$
$24$	0	0
$25$	−1.87130	−0.374259
$26$	4.67786	0.917403
$27$	0	0
$28$	−15.1368	−2.86059
$29$	2.15036	0.399311	0.199656	−	0.979866i	$-0.436018\pi$
$29$	2.15036	0.399311	0.199656	+	0.979866i	$0.436018\pi$
$30$	0	0
$31$	−0.231859	−0.0416431	−0.0208216	−	0.999783i	$-0.506628\pi$
$31$	−0.231859	−0.0416431	−0.0208216	+	0.999783i	$0.506628\pi$
$32$	16.9686	2.99965
$33$	0	0
$34$	−17.9541	−3.07910
$35$	−5.13681	−0.868279
$36$	0	0
$37$	−2.01334	−0.330991	−0.165496	−	0.986211i	$-0.552922\pi$
$37$	−2.01334	−0.330991	−0.165496	+	0.986211i	$0.552922\pi$
$38$	10.4687	1.69824
$39$	0	0
$40$	15.2589	2.41265
$41$	3.61371	0.564366	0.282183	−	0.959361i	$-0.408941\pi$
$41$	3.61371	0.564366	0.282183	+	0.959361i	$0.408941\pi$
$42$	0	0
$43$	5.40723	0.824595	0.412298	−	0.911049i	$-0.364726\pi$
$43$	5.40723	0.824595	0.412298	+	0.911049i	$0.364726\pi$
$44$	21.0609	3.17505
$45$	0	0
$46$	−25.3350	−3.73544
$47$	3.59321	0.524124	0.262062	−	0.965051i	$-0.415598\pi$
$47$	3.59321	0.524124	0.262062	+	0.965051i	$0.415598\pi$
$48$	0	0
$49$	1.43378	0.204826
$50$	−5.02548	−0.710710
$51$	0	0
$52$	9.07895	1.25902
$53$	−3.65166	−0.501595	−0.250797	−	0.968040i	$-0.580693\pi$
$53$	−3.65166	−0.501595	−0.250797	+	0.968040i	$0.580693\pi$
$54$	0	0
$55$	7.14718	0.963726
$56$	−25.0526	−3.34780
$57$	0	0
$58$	5.77491	0.758283
$59$	15.1098	1.96712	0.983561	−	0.180576i	$-0.0577961\pi$
$59$	15.1098	1.96712	0.983561	+	0.180576i	$0.0577961\pi$
$60$	0	0
$61$	−12.2543	−1.56900	−0.784502	−	0.620126i	$-0.787081\pi$
$61$	−12.2543	−1.56900	−0.784502	+	0.620126i	$0.787081\pi$
$62$	−0.622672	−0.0790794
$63$	0	0
$64$	20.0843	2.51053
$65$	3.08102	0.382153
$66$	0	0
$67$	−10.3881	−1.26911	−0.634556	−	0.772877i	$-0.718817\pi$
$67$	−10.3881	−1.26911	−0.634556	+	0.772877i	$0.718817\pi$
$68$	−34.8459	−4.22569
$69$	0	0
$70$	−13.7952	−1.64884
$71$	−9.62471	−1.14224	−0.571122	−	0.820865i	$-0.693492\pi$
$71$	−9.62471	−1.14224	−0.571122	+	0.820865i	$0.693492\pi$
$72$	0	0
$73$	15.9006	1.86103	0.930513	−	0.366258i	$-0.119361\pi$
$73$	15.9006	1.86103	0.930513	+	0.366258i	$0.119361\pi$
$74$	−5.40695	−0.628545
$75$	0	0
$76$	20.3180	2.33063
$77$	−11.7345	−1.33727
$78$	0	0
$79$	−13.1448	−1.47891	−0.739454	−	0.673207i	$-0.764916\pi$
$79$	−13.1448	−1.47891	−0.739454	+	0.673207i	$0.764916\pi$
$80$	22.5398	2.52003
$81$	0	0
$82$	9.70483	1.07172
$83$	2.59849	0.285222	0.142611	−	0.989779i	$-0.454450\pi$
$83$	2.59849	0.285222	0.142611	+	0.989779i	$0.454450\pi$
$84$	0	0
$85$	−11.8252	−1.28263
$86$	14.5214	1.56589
$87$	0	0
$88$	34.8574	3.71581
$89$	8.20872	0.870123	0.435061	−	0.900401i	$-0.356727\pi$
$89$	8.20872	0.870123	0.435061	+	0.900401i	$0.356727\pi$
$90$	0	0
$91$	−5.05852	−0.530277
$92$	−49.1711	−5.12644
$93$	0	0
$94$	9.64979	0.995300
$95$	6.89507	0.707420
$96$	0	0
$97$	−2.58237	−0.262200	−0.131100	−	0.991369i	$-0.541851\pi$
$97$	−2.58237	−0.262200	−0.131100	+	0.991369i	$0.541851\pi$
$98$	3.85050	0.388960
$99$	0	0
$100$	−9.75363	−0.975363
$101$	−7.71863	−0.768033	−0.384016	−	0.923326i	$-0.625459\pi$
$101$	−7.71863	−0.768033	−0.384016	+	0.923326i	$0.625459\pi$
$102$	0	0
$103$	0.704599	0.0694262	0.0347131	−	0.999397i	$-0.488948\pi$
$103$	0.704599	0.0694262	0.0347131	+	0.999397i	$0.488948\pi$
$104$	15.0264	1.47346
$105$	0	0
$106$	−9.80676	−0.952517
$107$	14.7750	1.42836	0.714179	−	0.699963i	$-0.246800\pi$
$107$	14.7750	1.42836	0.714179	+	0.699963i	$0.246800\pi$
$108$	0	0
$109$	−4.98255	−0.477241	−0.238621	−	0.971113i	$-0.576695\pi$
$109$	−4.98255	−0.477241	−0.238621	+	0.971113i	$0.576695\pi$
$110$	19.1942	1.83009
$111$	0	0
$112$	−37.0066	−3.49680
$113$	−7.43793	−0.699702	−0.349851	−	0.936805i	$-0.613768\pi$
$113$	−7.43793	−0.699702	−0.349851	+	0.936805i	$0.613768\pi$
$114$	0	0
$115$	−16.6866	−1.55603
$116$	11.2082	1.04065
$117$	0	0
$118$	40.5781	3.73552
$119$	19.4151	1.77978
$120$	0	0
$121$	5.32695	0.484268
$122$	−32.9097	−2.97950
$123$	0	0
$124$	−1.20850	−0.108527
$125$	−12.1540	−1.08709
$126$	0	0
$127$	−16.2461	−1.44161	−0.720804	−	0.693139i	$-0.756227\pi$
$127$	−16.2461	−1.44161	−0.720804	+	0.693139i	$0.756227\pi$
$128$	20.0004	1.76780
$129$	0	0
$130$	8.27426	0.725700
$131$	−7.75915	−0.677920	−0.338960	−	0.940801i	$-0.610075\pi$
$131$	−7.75915	−0.677920	−0.338960	+	0.940801i	$0.610075\pi$
$132$	0	0
$133$	−11.3206	−0.981617
$134$	−27.8980	−2.41002
$135$	0	0
$136$	−57.6727	−4.94539
$137$	−4.50800	−0.385145	−0.192572	−	0.981283i	$-0.561683\pi$
$137$	−4.50800	−0.385145	−0.192572	+	0.981283i	$0.561683\pi$
$138$	0	0
$139$	−8.43753	−0.715662	−0.357831	−	0.933786i	$-0.616484\pi$
$139$	−8.43753	−0.715662	−0.357831	+	0.933786i	$0.616484\pi$
$140$	−26.7742	−2.26284
$141$	0	0
$142$	−25.8477	−2.16909
$143$	7.03825	0.588568
$144$	0	0
$145$	3.80358	0.315870
$146$	42.7021	3.53405
$147$	0	0
$148$	−10.4940	−0.862602
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	19.1521	1.55857	0.779286	−	0.626668i	$-0.215582\pi$
$151$	19.1521	1.55857	0.779286	+	0.626668i	$0.215582\pi$
$152$	33.6278	2.72758
$153$	0	0
$154$	−31.5136	−2.53944
$155$	−0.410116	−0.0329413
$156$	0	0
$157$	−7.29178	−0.581948	−0.290974	−	0.956731i	$-0.593979\pi$
$157$	−7.29178	−0.581948	−0.290974	+	0.956731i	$0.593979\pi$
$158$	−35.3012	−2.80841
$159$	0	0
$160$	30.0142	2.37283
$161$	27.3966	2.15916
$162$	0	0
$163$	−20.4073	−1.59842	−0.799211	−	0.601051i	$-0.794749\pi$
$163$	−20.4073	−1.59842	−0.799211	+	0.601051i	$0.794749\pi$
$164$	18.8355	1.47080
$165$	0	0
$166$	6.97841	0.541629
$167$	15.7803	1.22112	0.610560	−	0.791970i	$-0.290944\pi$
$167$	15.7803	1.22112	0.610560	+	0.791970i	$0.290944\pi$
$168$	0	0
$169$	−9.96594	−0.766611
$170$	−31.7574	−2.43568
$171$	0	0
$172$	28.1838	2.14899
$173$	−1.25495	−0.0954124	−0.0477062	−	0.998861i	$-0.515191\pi$
$173$	−1.25495	−0.0954124	−0.0477062	+	0.998861i	$0.515191\pi$
$174$	0	0
$175$	5.43443	0.410804
$176$	51.4898	3.88119
$177$	0	0
$178$	22.0450	1.65234
$179$	23.4683	1.75410	0.877052	−	0.480396i	$-0.159507\pi$
$179$	23.4683	1.75410	0.877052	+	0.480396i	$0.159507\pi$
$180$	0	0
$181$	2.32099	0.172518	0.0862588	−	0.996273i	$-0.472509\pi$
$181$	2.32099	0.172518	0.0862588	+	0.996273i	$0.472509\pi$
$182$	−13.5849	−1.00698
$183$	0	0
$184$	−81.3819	−5.99955
$185$	−3.56123	−0.261827
$186$	0	0
$187$	−27.0135	−1.97542
$188$	18.7287	1.36593
$189$	0	0
$190$	18.5171	1.34337
$191$	15.6265	1.13069	0.565345	−	0.824854i	$-0.308743\pi$
$191$	15.6265	1.13069	0.565345	+	0.824854i	$0.308743\pi$
$192$	0	0
$193$	−7.35511	−0.529432	−0.264716	−	0.964326i	$-0.585278\pi$
$193$	−7.35511	−0.529432	−0.264716	+	0.964326i	$0.585278\pi$
$194$	−6.93510	−0.497911
$195$	0	0
$196$	7.47320	0.533800
$197$	1.48023	0.105462	0.0527310	−	0.998609i	$-0.483207\pi$
$197$	1.48023	0.105462	0.0527310	+	0.998609i	$0.483207\pi$
$198$	0	0
$199$	19.5568	1.38634	0.693171	−	0.720773i	$-0.256213\pi$
$199$	19.5568	1.38634	0.693171	+	0.720773i	$0.256213\pi$
$200$	−16.1430	−1.14148
$201$	0	0
$202$	−20.7289	−1.45848
$203$	−6.24484	−0.438302
$204$	0	0
$205$	6.39197	0.446435
$206$	1.89224	0.131839
$207$	0	0
$208$	22.1963	1.53904
$209$	15.7510	1.08952
$210$	0	0
$211$	12.4826	0.859336	0.429668	−	0.902987i	$-0.358631\pi$
$211$	12.4826	0.859336	0.429668	+	0.902987i	$0.358631\pi$
$212$	−19.0333	−1.30721
$213$	0	0
$214$	39.6793	2.71242
$215$	9.56439	0.652286
$216$	0	0
$217$	0.673342	0.0457094
$218$	−13.3809	−0.906271
$219$	0	0
$220$	37.2528	2.51158
$221$	−11.6450	−0.783328
$222$	0	0
$223$	11.8832	0.795761	0.397880	−	0.917437i	$-0.369746\pi$
$223$	11.8832	0.795761	0.397880	+	0.917437i	$0.369746\pi$
$224$	−49.2783	−3.29255
$225$	0	0
$226$	−19.9750	−1.32872
$227$	−8.99453	−0.596988	−0.298494	−	0.954412i	$-0.596484\pi$
$227$	−8.99453	−0.596988	−0.298494	+	0.954412i	$0.596484\pi$
$228$	0	0
$229$	17.0037	1.12363	0.561816	−	0.827262i	$-0.310103\pi$
$229$	17.0037	1.12363	0.561816	+	0.827262i	$0.310103\pi$
$230$	−44.8129	−2.95487
$231$	0	0
$232$	18.5504	1.21789
$233$	−2.97757	−0.195067	−0.0975334	−	0.995232i	$-0.531095\pi$
$233$	−2.97757	−0.195067	−0.0975334	+	0.995232i	$0.531095\pi$
$234$	0	0
$235$	6.35573	0.414602
$236$	78.7555	5.12655
$237$	0	0
$238$	52.1404	3.37976
$239$	7.96606	0.515281	0.257641	−	0.966241i	$-0.417055\pi$
$239$	7.96606	0.515281	0.257641	+	0.966241i	$0.417055\pi$
$240$	0	0
$241$	15.5057	0.998809	0.499404	−	0.866369i	$-0.333552\pi$
$241$	15.5057	0.998809	0.499404	+	0.866369i	$0.333552\pi$
$242$	14.3059	0.919615
$243$	0	0
$244$	−63.8723	−4.08901
$245$	2.53609	0.162025
$246$	0	0
$247$	6.78998	0.432036
$248$	−2.00017	−0.127011
$249$	0	0
$250$	−32.6404	−2.06436
$251$	6.48159	0.409114	0.204557	−	0.978855i	$-0.434425\pi$
$251$	6.48159	0.409114	0.204557	+	0.978855i	$0.434425\pi$
$252$	0	0
$253$	−38.1187	−2.39650
$254$	−43.6299	−2.73758
$255$	0	0
$256$	13.5437	0.846480
$257$	14.6277	0.912451	0.456225	−	0.889864i	$-0.349201\pi$
$257$	14.6277	0.912451	0.456225	+	0.889864i	$0.349201\pi$
$258$	0	0
$259$	5.84694	0.363311
$260$	16.0590	0.995936
$261$	0	0
$262$	−20.8377	−1.28736
$263$	14.1757	0.874110	0.437055	−	0.899435i	$-0.356022\pi$
$263$	14.1757	0.874110	0.437055	+	0.899435i	$0.356022\pi$
$264$	0	0
$265$	−6.45912	−0.396780
$266$	−30.4020	−1.86407
$267$	0	0
$268$	−54.1454	−3.30746
$269$	19.6653	1.19901	0.599506	−	0.800370i	$-0.295364\pi$
$269$	19.6653	1.19901	0.599506	+	0.800370i	$0.295364\pi$
$270$	0	0
$271$	−16.7431	−1.01707	−0.508536	−	0.861040i	$-0.669813\pi$
$271$	−16.7431	−1.01707	−0.508536	+	0.861040i	$0.669813\pi$
$272$	−85.1916	−5.16550
$273$	0	0
$274$	−12.1065	−0.731381
$275$	−7.56128	−0.455962
$276$	0	0
$277$	24.2532	1.45723	0.728617	−	0.684921i	$-0.240163\pi$
$277$	24.2532	1.45723	0.728617	+	0.684921i	$0.240163\pi$
$278$	−22.6595	−1.35903
$279$	0	0
$280$	−44.3134	−2.64823
$281$	−15.0407	−0.897255	−0.448628	−	0.893719i	$-0.648087\pi$
$281$	−15.0407	−0.897255	−0.448628	+	0.893719i	$0.648087\pi$
$282$	0	0
$283$	10.0820	0.599314	0.299657	−	0.954047i	$-0.403128\pi$
$283$	10.0820	0.599314	0.299657	+	0.954047i	$0.403128\pi$
$284$	−50.1663	−2.97682
$285$	0	0
$286$	18.9016	1.11768
$287$	−10.4946	−0.619474
$288$	0	0
$289$	27.6947	1.62910
$290$	10.2147	0.599830
$291$	0	0
$292$	82.8777	4.85005
$293$	−22.2835	−1.30181	−0.650907	−	0.759157i	$-0.725611\pi$
$293$	−22.2835	−1.30181	−0.650907	+	0.759157i	$0.725611\pi$
$294$	0	0
$295$	26.7263	1.55607
$296$	−17.3684	−1.00952
$297$	0	0
$298$	2.68556	0.155570
$299$	−16.4323	−0.950303
$300$	0	0
$301$	−15.7031	−0.905113
$302$	51.4340	2.95970
$303$	0	0
$304$	49.6736	2.84897
$305$	−21.6756	−1.24114
$306$	0	0
$307$	3.58488	0.204600	0.102300	−	0.994754i	$-0.467380\pi$
$307$	3.58488	0.204600	0.102300	+	0.994754i	$0.467380\pi$
$308$	−61.1628	−3.48508
$309$	0	0
$310$	−1.10139	−0.0625548
$311$	14.8449	0.841776	0.420888	−	0.907113i	$-0.361719\pi$
$311$	14.8449	0.841776	0.420888	+	0.907113i	$0.361719\pi$
$312$	0	0
$313$	−32.9702	−1.86359	−0.931793	−	0.362991i	$-0.881756\pi$
$313$	−32.9702	−1.86359	−0.931793	+	0.362991i	$0.881756\pi$
$314$	−19.5825	−1.10511
$315$	0	0
$316$	−68.5139	−3.85421
$317$	−17.6725	−0.992588	−0.496294	−	0.868155i	$-0.665306\pi$
$317$	−17.6725	−0.992588	−0.496294	+	0.868155i	$0.665306\pi$
$318$	0	0
$319$	8.68886	0.486483
$320$	35.5253	1.98593
$321$	0	0
$322$	73.5753	4.10019
$323$	−26.0606	−1.45005
$324$	0	0
$325$	−3.25953	−0.180806
$326$	−54.8050	−3.03537
$327$	0	0
$328$	31.1742	1.72131
$329$	−10.4350	−0.575302
$330$	0	0
$331$	11.8263	0.650033	0.325016	−	0.945708i	$-0.394630\pi$
$331$	11.8263	0.650033	0.325016	+	0.945708i	$0.394630\pi$
$332$	13.5439	0.743321
$333$	0	0
$334$	42.3791	2.31888
$335$	−18.3747	−1.00392
$336$	0	0
$337$	−4.85739	−0.264599	−0.132299	−	0.991210i	$-0.542236\pi$
$337$	−4.85739	−0.264599	−0.132299	+	0.991210i	$0.542236\pi$
$338$	−26.7641	−1.45578
$339$	0	0
$340$	−61.6360	−3.34268
$341$	−0.936865	−0.0507341
$342$	0	0
$343$	16.1648	0.872819
$344$	46.6463	2.51500
$345$	0	0
$346$	−3.37025	−0.181186
$347$	35.0510	1.88164	0.940818	−	0.338913i	$-0.110059\pi$
$347$	35.0510	1.88164	0.940818	+	0.338913i	$0.110059\pi$
$348$	0	0
$349$	−19.0260	−1.01844	−0.509219	−	0.860637i	$-0.670066\pi$
$349$	−19.0260	−1.01844	−0.509219	+	0.860637i	$0.670066\pi$
$350$	14.5945	0.780108
$351$	0	0
$352$	68.5642	3.65448
$353$	−3.21022	−0.170863	−0.0854313	−	0.996344i	$-0.527227\pi$
$353$	−3.21022	−0.170863	−0.0854313	+	0.996344i	$0.527227\pi$
$354$	0	0
$355$	−17.0243	−0.903558
$356$	42.7858	2.26764
$357$	0	0
$358$	63.0256	3.33100
$359$	−9.83688	−0.519171	−0.259585	−	0.965720i	$-0.583586\pi$
$359$	−9.83688	−0.519171	−0.259585	+	0.965720i	$0.583586\pi$
$360$	0	0
$361$	−3.80455	−0.200240
$362$	6.23315	0.327607
$363$	0	0
$364$	−26.3662	−1.38196
$365$	28.1252	1.47214
$366$	0	0
$367$	−18.5258	−0.967039	−0.483520	−	0.875334i	$-0.660642\pi$
$367$	−18.5258	−0.967039	−0.483520	+	0.875334i	$0.660642\pi$
$368$	−120.214	−6.26658
$369$	0	0
$370$	−9.56389	−0.497203
$371$	10.6048	0.550573
$372$	0	0
$373$	−10.6617	−0.552043	−0.276021	−	0.961151i	$-0.589016\pi$
$373$	−10.6617	−0.552043	−0.276021	+	0.961151i	$0.589016\pi$
$374$	−72.5463	−3.75128
$375$	0	0
$376$	30.9974	1.59857
$377$	3.74561	0.192909
$378$	0	0
$379$	−34.8002	−1.78757	−0.893784	−	0.448499i	$-0.851959\pi$
$379$	−34.8002	−1.78757	−0.893784	+	0.448499i	$0.851959\pi$
$380$	35.9387	1.84362
$381$	0	0
$382$	41.9658	2.14716
$383$	28.1747	1.43966	0.719831	−	0.694150i	$-0.244219\pi$
$383$	28.1747	1.43966	0.719831	+	0.694150i	$0.244219\pi$
$384$	0	0
$385$	−20.7561	−1.05783
$386$	−19.7526	−1.00538
$387$	0	0
$388$	−13.4599	−0.683323
$389$	3.20235	0.162366	0.0811829	−	0.996699i	$-0.474130\pi$
$389$	3.20235	0.162366	0.0811829	+	0.996699i	$0.474130\pi$
$390$	0	0
$391$	63.0687	3.18952
$392$	12.3687	0.624714
$393$	0	0
$394$	3.97525	0.200270
$395$	−23.2508	−1.16987
$396$	0	0
$397$	−8.96815	−0.450098	−0.225049	−	0.974347i	$-0.572254\pi$
$397$	−8.96815	−0.450098	−0.225049	+	0.974347i	$0.572254\pi$
$398$	52.5209	2.63263
$399$	0	0
$400$	−23.8458	−1.19229
$401$	21.8654	1.09190	0.545952	−	0.837816i	$-0.316168\pi$
$401$	21.8654	1.09190	0.545952	+	0.837816i	$0.316168\pi$
$402$	0	0
$403$	−0.403865	−0.0201180
$404$	−40.2313	−2.00158
$405$	0	0
$406$	−16.7709	−0.832326
$407$	−8.13523	−0.403248
$408$	0	0
$409$	−27.6036	−1.36491	−0.682456	−	0.730927i	$-0.739088\pi$
$409$	−27.6036	−1.36491	−0.682456	+	0.730927i	$0.739088\pi$
$410$	17.1660	0.847770
$411$	0	0
$412$	3.67253	0.180933
$413$	−43.8802	−2.15920
$414$	0	0
$415$	4.59625	0.225621
$416$	29.5568	1.44914
$417$	0	0
$418$	42.3004	2.06898
$419$	28.8312	1.40850	0.704249	−	0.709953i	$-0.251284\pi$
$419$	28.8312	1.40850	0.704249	+	0.709953i	$0.251284\pi$
$420$	0	0
$421$	−38.5283	−1.87776	−0.938878	−	0.344250i	$-0.888133\pi$
$421$	−38.5283	−1.87776	−0.938878	+	0.344250i	$0.888133\pi$
$422$	33.5227	1.63186
$423$	0	0
$424$	−31.5016	−1.52985
$425$	12.5104	0.606843
$426$	0	0
$427$	35.5877	1.72221
$428$	77.0110	3.72247
$429$	0	0
$430$	25.6857	1.23868
$431$	10.5127	0.506378	0.253189	−	0.967417i	$-0.418521\pi$
$431$	10.5127	0.506378	0.253189	+	0.967417i	$0.418521\pi$
$432$	0	0
$433$	16.1615	0.776672	0.388336	−	0.921518i	$-0.373050\pi$
$433$	16.1615	0.776672	0.388336	+	0.921518i	$0.373050\pi$
$434$	1.80830	0.0868012
$435$	0	0
$436$	−25.9702	−1.24375
$437$	−36.7741	−1.75915
$438$	0	0
$439$	35.4030	1.68969	0.844845	−	0.535011i	$-0.179692\pi$
$439$	35.4030	1.68969	0.844845	+	0.535011i	$0.179692\pi$
$440$	61.6562	2.93934
$441$	0	0
$442$	−31.2734	−1.48752
$443$	6.82614	0.324320	0.162160	−	0.986764i	$-0.448154\pi$
$443$	6.82614	0.324320	0.162160	+	0.986764i	$0.448154\pi$
$444$	0	0
$445$	14.5197	0.688300
$446$	31.9132	1.51113
$447$	0	0
$448$	−58.3266	−2.75567
$449$	0.851723	0.0401953	0.0200977	−	0.999798i	$-0.493602\pi$
$449$	0.851723	0.0401953	0.0200977	+	0.999798i	$0.493602\pi$
$450$	0	0
$451$	14.6018	0.687570
$452$	−38.7682	−1.82350
$453$	0	0
$454$	−24.1553	−1.13367
$455$	−8.94757	−0.419469
$456$	0	0
$457$	19.3669	0.905947	0.452973	−	0.891524i	$-0.350363\pi$
$457$	19.3669	0.905947	0.452973	+	0.891524i	$0.350363\pi$
$458$	45.6643	2.13375
$459$	0	0
$460$	−86.9745	−4.05521
$461$	−5.26819	−0.245364	−0.122682	−	0.992446i	$-0.539149\pi$
$461$	−5.26819	−0.245364	−0.122682	+	0.992446i	$0.539149\pi$
$462$	0	0
$463$	21.4096	0.994988	0.497494	−	0.867468i	$-0.334254\pi$
$463$	21.4096	0.994988	0.497494	+	0.867468i	$0.334254\pi$
$464$	27.4018	1.27210
$465$	0	0
$466$	−7.99643	−0.370428
$467$	37.7230	1.74561	0.872805	−	0.488069i	$-0.162299\pi$
$467$	37.7230	1.74561	0.872805	+	0.488069i	$0.162299\pi$
$468$	0	0
$469$	30.1682	1.39304
$470$	17.0687	0.787320
$471$	0	0
$472$	130.346	5.99968
$473$	21.8488	1.00461
$474$	0	0
$475$	−7.29456	−0.334698
$476$	101.196	4.63831
$477$	0	0
$478$	21.3933	0.978508
$479$	−34.8584	−1.59272	−0.796362	−	0.604821i	$-0.793245\pi$
$479$	−34.8584	−1.59272	−0.796362	+	0.604821i	$0.793245\pi$
$480$	0	0
$481$	−3.50695	−0.159903
$482$	41.6414	1.89672
$483$	0	0
$484$	27.7653	1.26206
$485$	−4.56773	−0.207410
$486$	0	0
$487$	−23.1380	−1.04848	−0.524242	−	0.851569i	$-0.675651\pi$
$487$	−23.1380	−1.04848	−0.524242	+	0.851569i	$0.675651\pi$
$488$	−105.714	−4.78543
$489$	0	0
$490$	6.81082	0.307682
$491$	16.4050	0.740347	0.370173	−	0.928963i	$-0.379298\pi$
$491$	16.4050	0.740347	0.370173	+	0.928963i	$0.379298\pi$
$492$	0	0
$493$	−14.3760	−0.647463
$494$	18.2349	0.820427
$495$	0	0
$496$	−2.95456	−0.132664
$497$	27.9511	1.25378
$498$	0	0
$499$	21.6857	0.970786	0.485393	−	0.874296i	$-0.338677\pi$
$499$	21.6857	0.970786	0.485393	+	0.874296i	$0.338677\pi$
$500$	−63.3497	−2.83309
$501$	0	0
$502$	17.4067	0.776899
$503$	1.58797	0.0708042	0.0354021	−	0.999373i	$-0.488729\pi$
$503$	1.58797	0.0708042	0.0354021	+	0.999373i	$0.488729\pi$
$504$	0	0
$505$	−13.6528	−0.607543
$506$	−102.370	−4.55091
$507$	0	0
$508$	−84.6784	−3.75700
$509$	0.282252	0.0125106	0.00625529	−	0.999980i	$-0.498009\pi$
$509$	0.282252	0.0125106	0.00625529	+	0.999980i	$0.498009\pi$
$510$	0	0
$511$	−46.1769	−2.04275
$512$	−3.62836	−0.160352
$513$	0	0
$514$	39.2836	1.73272
$515$	1.24630	0.0549187
$516$	0	0
$517$	14.5190	0.638543
$518$	15.7023	0.689920
$519$	0	0
$520$	26.5788	1.16556
$521$	17.4689	0.765326	0.382663	−	0.923888i	$-0.375007\pi$
$521$	17.4689	0.765326	0.382663	+	0.923888i	$0.375007\pi$
$522$	0	0
$523$	−21.2884	−0.930877	−0.465438	−	0.885080i	$-0.654103\pi$
$523$	−21.2884	−0.930877	−0.465438	+	0.885080i	$0.654103\pi$
$524$	−40.4425	−1.76674
$525$	0	0
$526$	38.0696	1.65992
$527$	1.55007	0.0675223
$528$	0	0
$529$	65.9962	2.86940
$530$	−17.3463	−0.753477
$531$	0	0
$532$	−59.0054	−2.55821
$533$	6.29455	0.272647
$534$	0	0
$535$	26.1343	1.12988
$536$	−89.6148	−3.87077
$537$	0	0
$538$	52.8123	2.27690
$539$	5.79342	0.249540
$540$	0	0
$541$	30.0776	1.29314	0.646569	−	0.762855i	$-0.276203\pi$
$541$	30.0776	1.29314	0.646569	+	0.762855i	$0.276203\pi$
$542$	−44.9647	−1.93140
$543$	0	0
$544$	−113.442	−4.86378
$545$	−8.81320	−0.377516
$546$	0	0
$547$	−11.3854	−0.486804	−0.243402	−	0.969925i	$-0.578263\pi$
$547$	−11.3854	−0.486804	−0.243402	+	0.969925i	$0.578263\pi$
$548$	−23.4968	−1.00373
$549$	0	0
$550$	−20.3063	−0.865862
$551$	8.38237	0.357101
$552$	0	0
$553$	38.1739	1.62332
$554$	65.1335	2.76726
$555$	0	0
$556$	−43.9784	−1.86510
$557$	24.9245	1.05608	0.528042	−	0.849218i	$-0.322926\pi$
$557$	24.9245	1.05608	0.528042	+	0.849218i	$0.322926\pi$
$558$	0	0
$559$	9.41861	0.398365
$560$	−65.4579	−2.76610
$561$	0	0
$562$	−40.3928	−1.70387
$563$	3.05643	0.128813	0.0644065	−	0.997924i	$-0.479485\pi$
$563$	3.05643	0.128813	0.0644065	+	0.997924i	$0.479485\pi$
$564$	0	0
$565$	−13.1563	−0.553490
$566$	27.0759	1.13808
$567$	0	0
$568$	−83.0290	−3.48382
$569$	23.4970	0.985044	0.492522	−	0.870300i	$-0.336075\pi$
$569$	23.4970	0.985044	0.492522	+	0.870300i	$0.336075\pi$
$570$	0	0
$571$	35.0923	1.46857	0.734283	−	0.678844i	$-0.237519\pi$
$571$	35.0923	1.46857	0.734283	+	0.678844i	$0.237519\pi$
$572$	36.6850	1.53388
$573$	0	0
$574$	−28.1838	−1.17637
$575$	17.6534	0.736198
$576$	0	0
$577$	35.7948	1.49016	0.745078	−	0.666977i	$-0.232412\pi$
$577$	35.7948	1.49016	0.745078	+	0.666977i	$0.232412\pi$
$578$	74.3758	3.09363
$579$	0	0
$580$	19.8251	0.823194
$581$	−7.54627	−0.313072
$582$	0	0
$583$	−14.7551	−0.611096
$584$	137.169	5.67609
$585$	0	0
$586$	−59.8436	−2.47212
$587$	17.7088	0.730918	0.365459	−	0.930827i	$-0.380912\pi$
$587$	17.7088	0.730918	0.365459	+	0.930827i	$0.380912\pi$
$588$	0	0
$589$	−0.903818	−0.0372412
$590$	71.7752	2.95494
$591$	0	0
$592$	−25.6558	−1.05445
$593$	−30.0897	−1.23563	−0.617817	−	0.786322i	$-0.711983\pi$
$593$	−30.0897	−1.23563	−0.617817	+	0.786322i	$0.711983\pi$
$594$	0	0
$595$	34.3417	1.40787
$596$	5.21223	0.213501
$597$	0	0
$598$	−44.1299	−1.80460
$599$	12.8578	0.525356	0.262678	−	0.964884i	$-0.415394\pi$
$599$	12.8578	0.525356	0.262678	+	0.964884i	$0.415394\pi$
$600$	0	0
$601$	3.29618	0.134454	0.0672269	−	0.997738i	$-0.478585\pi$
$601$	3.29618	0.134454	0.0672269	+	0.997738i	$0.478585\pi$
$602$	−42.1717	−1.71879
$603$	0	0
$604$	99.8250	4.06182
$605$	9.42239	0.383075
$606$	0	0
$607$	−12.3417	−0.500933	−0.250466	−	0.968125i	$-0.580584\pi$
$607$	−12.3417	−0.500933	−0.250466	+	0.968125i	$0.580584\pi$
$608$	66.1457	2.68256
$609$	0	0
$610$	−58.2111	−2.35690
$611$	6.25885	0.253206
$612$	0	0
$613$	27.8459	1.12469	0.562343	−	0.826904i	$-0.309900\pi$
$613$	27.8459	1.12469	0.562343	+	0.826904i	$0.309900\pi$
$614$	9.62740	0.388530
$615$	0	0
$616$	−101.229	−4.07864
$617$	17.2549	0.694657	0.347329	−	0.937743i	$-0.387089\pi$
$617$	17.2549	0.694657	0.347329	+	0.937743i	$0.387089\pi$
$618$	0	0
$619$	−0.899155	−0.0361401	−0.0180700	−	0.999837i	$-0.505752\pi$
$619$	−0.899155	−0.0361401	−0.0180700	+	0.999837i	$0.505752\pi$
$620$	−2.13762	−0.0858489
$621$	0	0
$622$	39.8668	1.59851
$623$	−23.8389	−0.955087
$624$	0	0
$625$	−12.1418	−0.485671
$626$	−88.5434	−3.53891
$627$	0	0
$628$	−38.0065	−1.51662
$629$	13.4600	0.536686
$630$	0	0
$631$	−36.4878	−1.45256	−0.726278	−	0.687401i	$-0.758752\pi$
$631$	−36.4878	−1.45256	−0.726278	+	0.687401i	$0.758752\pi$
$632$	−113.396	−4.51064
$633$	0	0
$634$	−47.4606	−1.88490
$635$	−28.7363	−1.14037
$636$	0	0
$637$	2.49744	0.0989521
$638$	23.3345	0.923820
$639$	0	0
$640$	35.3769	1.39840
$641$	−49.4454	−1.95297	−0.976487	−	0.215575i	$-0.930838\pi$
$641$	−49.4454	−1.95297	−0.976487	+	0.215575i	$0.930838\pi$
$642$	0	0
$643$	−9.49010	−0.374253	−0.187127	−	0.982336i	$-0.559917\pi$
$643$	−9.49010	−0.374253	−0.187127	+	0.982336i	$0.559917\pi$
$644$	142.798	5.62701
$645$	0	0
$646$	−69.9874	−2.75362
$647$	−33.4556	−1.31528	−0.657638	−	0.753334i	$-0.728444\pi$
$647$	−33.4556	−1.31528	−0.657638	+	0.753334i	$0.728444\pi$
$648$	0	0
$649$	61.0534	2.39656
$650$	−8.75366	−0.343347
$651$	0	0
$652$	−106.367	−4.16567
$653$	4.68545	0.183356	0.0916780	−	0.995789i	$-0.470777\pi$
$653$	4.68545	0.183356	0.0916780	+	0.995789i	$0.470777\pi$
$654$	0	0
$655$	−13.7245	−0.536260
$656$	46.0491	1.79792
$657$	0	0
$658$	−28.0239	−1.09249
$659$	−19.6838	−0.766773	−0.383387	−	0.923588i	$-0.625242\pi$
$659$	−19.6838	−0.766773	−0.383387	+	0.923588i	$0.625242\pi$
$660$	0	0
$661$	−13.5550	−0.527230	−0.263615	−	0.964628i	$-0.584915\pi$
$661$	−13.5550	−0.527230	−0.263615	+	0.964628i	$0.584915\pi$
$662$	31.7603	1.23440
$663$	0	0
$664$	22.4163	0.869920
$665$	−20.0240	−0.776496
$666$	0	0
$667$	−20.2860	−0.785476
$668$	82.2509	3.18238
$669$	0	0
$670$	−49.3463	−1.90641
$671$	−49.5155	−1.91153
$672$	0	0
$673$	−19.1529	−0.738291	−0.369146	−	0.929372i	$-0.620350\pi$
$673$	−19.1529	−0.738291	−0.369146	+	0.929372i	$0.620350\pi$
$674$	−13.0448	−0.502467
$675$	0	0
$676$	−51.9448	−1.99788
$677$	3.42970	0.131814	0.0659070	−	0.997826i	$-0.479006\pi$
$677$	3.42970	0.131814	0.0659070	+	0.997826i	$0.479006\pi$
$678$	0	0
$679$	7.49945	0.287802
$680$	−102.012	−3.91199
$681$	0	0
$682$	−2.51601	−0.0963429
$683$	−24.6099	−0.941672	−0.470836	−	0.882221i	$-0.656048\pi$
$683$	−24.6099	−0.941672	−0.470836	+	0.882221i	$0.656048\pi$
$684$	0	0
$685$	−7.97382	−0.304664
$686$	43.4117	1.65746
$687$	0	0
$688$	68.9039	2.62693
$689$	−6.36067	−0.242322
$690$	0	0
$691$	−19.3746	−0.737044	−0.368522	−	0.929619i	$-0.620136\pi$
$691$	−19.3746	−0.737044	−0.368522	+	0.929619i	$0.620136\pi$
$692$	−6.54111	−0.248656
$693$	0	0
$694$	94.1315	3.57318
$695$	−14.9244	−0.566116
$696$	0	0
$697$	−24.1591	−0.915092
$698$	−51.0955	−1.93399
$699$	0	0
$700$	28.3255	1.07060
$701$	25.7779	0.973619	0.486810	−	0.873508i	$-0.338161\pi$
$701$	25.7779	0.973619	0.486810	+	0.873508i	$0.338161\pi$
$702$	0	0
$703$	−7.84827	−0.296003
$704$	81.1537	3.05860
$705$	0	0
$706$	−8.62123	−0.324464
$707$	22.4157	0.843028
$708$	0	0
$709$	30.0969	1.13031	0.565157	−	0.824983i	$-0.308816\pi$
$709$	30.0969	1.13031	0.565157	+	0.824983i	$0.308816\pi$
$710$	−45.7199	−1.71584
$711$	0	0
$712$	70.8137	2.65386
$713$	2.18731	0.0819154
$714$	0	0
$715$	12.4494	0.465579
$716$	122.322	4.57140
$717$	0	0
$718$	−26.4175	−0.985894
$719$	−12.7020	−0.473703	−0.236852	−	0.971546i	$-0.576115\pi$
$719$	−12.7020	−0.473703	−0.236852	+	0.971546i	$0.576115\pi$
$720$	0	0
$721$	−2.04622	−0.0762053
$722$	−10.2174	−0.380251
$723$	0	0
$724$	12.0975	0.449601
$725$	−4.02395	−0.149446
$726$	0	0
$727$	41.1007	1.52434	0.762170	−	0.647377i	$-0.224134\pi$
$727$	41.1007	1.52434	0.762170	+	0.647377i	$0.224134\pi$
$728$	−43.6380	−1.61733
$729$	0	0
$730$	75.5320	2.79557
$731$	−36.1496	−1.33704
$732$	0	0
$733$	18.2824	0.675277	0.337639	−	0.941276i	$-0.390372\pi$
$733$	18.2824	0.675277	0.337639	+	0.941276i	$0.390372\pi$
$734$	−49.7522	−1.83639
$735$	0	0
$736$	−160.078	−5.90054
$737$	−41.9750	−1.54617
$738$	0	0
$739$	−43.0008	−1.58181	−0.790906	−	0.611938i	$-0.790390\pi$
$739$	−43.0008	−1.58181	−0.790906	+	0.611938i	$0.790390\pi$
$740$	−18.5619	−0.682351
$741$	0	0
$742$	28.4798	1.04553
$743$	−20.0968	−0.737278	−0.368639	−	0.929573i	$-0.620176\pi$
$743$	−20.0968	−0.737278	−0.368639	+	0.929573i	$0.620176\pi$
$744$	0	0
$745$	1.76881	0.0648043
$746$	−28.6327	−1.04832
$747$	0	0
$748$	−140.801	−5.14818
$749$	−42.9082	−1.56783
$750$	0	0
$751$	7.99317	0.291675	0.145837	−	0.989309i	$-0.453412\pi$
$751$	7.99317	0.291675	0.145837	+	0.989309i	$0.453412\pi$
$752$	45.7880	1.66972
$753$	0	0
$754$	10.0591	0.366329
$755$	33.8764	1.23289
$756$	0	0
$757$	−11.3916	−0.414033	−0.207017	−	0.978337i	$-0.566375\pi$
$757$	−11.3916	−0.414033	−0.207017	+	0.978337i	$0.566375\pi$
$758$	−93.4581	−3.39455
$759$	0	0
$760$	59.4814	2.15762
$761$	23.3694	0.847140	0.423570	−	0.905863i	$-0.360777\pi$
$761$	23.3694	0.847140	0.423570	+	0.905863i	$0.360777\pi$
$762$	0	0
$763$	14.4698	0.523842
$764$	81.4487	2.94671
$765$	0	0
$766$	75.6649	2.73389
$767$	26.3190	0.950324
$768$	0	0
$769$	27.9815	1.00904	0.504519	−	0.863401i	$-0.331670\pi$
$769$	27.9815	1.00904	0.504519	+	0.863401i	$0.331670\pi$
$770$	−55.7418	−2.00879
$771$	0	0
$772$	−38.3365	−1.37976
$773$	−33.6033	−1.20863	−0.604314	−	0.796746i	$-0.706553\pi$
$773$	−33.6033	−1.20863	−0.604314	+	0.796746i	$0.706553\pi$
$774$	0	0
$775$	0.433877	0.0155853
$776$	−22.2772	−0.799704
$777$	0	0
$778$	8.60011	0.308329
$779$	14.0867	0.504709
$780$	0	0
$781$	−38.8902	−1.39160
$782$	169.375	6.05683
$783$	0	0
$784$	18.2705	0.652519
$785$	−12.8978	−0.460343
$786$	0	0
$787$	30.5571	1.08924	0.544621	−	0.838682i	$-0.316673\pi$
$787$	30.5571	1.08924	0.544621	+	0.838682i	$0.316673\pi$
$788$	7.71530	0.274846
$789$	0	0
$790$	−62.4413	−2.22156
$791$	21.6005	0.768024
$792$	0	0
$793$	−21.3452	−0.757991
$794$	−24.0845	−0.854727
$795$	0	0
$796$	101.934	3.61297
$797$	12.5354	0.444026	0.222013	−	0.975044i	$-0.428737\pi$
$797$	12.5354	0.444026	0.222013	+	0.975044i	$0.428737\pi$
$798$	0	0
$799$	−24.0221	−0.849841
$800$	−31.7532	−1.12265
$801$	0	0
$802$	58.7207	2.07350
$803$	64.2490	2.26730
$804$	0	0
$805$	48.4595	1.70797
$806$	−1.08460	−0.0382035
$807$	0	0
$808$	−66.5859	−2.34248
$809$	−23.3115	−0.819587	−0.409793	−	0.912178i	$-0.634399\pi$
$809$	−23.3115	−0.819587	−0.409793	+	0.912178i	$0.634399\pi$
$810$	0	0
$811$	−29.7339	−1.04410	−0.522049	−	0.852916i	$-0.674832\pi$
$811$	−29.7339	−1.04410	−0.522049	+	0.852916i	$0.674832\pi$
$812$	−32.5496	−1.14227
$813$	0	0
$814$	−21.8477	−0.765760
$815$	−36.0967	−1.26441
$816$	0	0
$817$	21.0781	0.737430
$818$	−74.1312	−2.59194
$819$	0	0
$820$	33.3165	1.16346
$821$	19.5920	0.683764	0.341882	−	0.939743i	$-0.388936\pi$
$821$	19.5920	0.683764	0.341882	+	0.939743i	$0.388936\pi$
$822$	0	0
$823$	−19.9205	−0.694385	−0.347192	−	0.937794i	$-0.612865\pi$
$823$	−19.9205	−0.694385	−0.347192	+	0.937794i	$0.612865\pi$
$824$	6.07832	0.211748
$825$	0	0
$826$	−117.843	−4.10028
$827$	−10.7269	−0.373011	−0.186505	−	0.982454i	$-0.559716\pi$
$827$	−10.7269	−0.373011	−0.186505	+	0.982454i	$0.559716\pi$
$828$	0	0
$829$	43.5908	1.51397	0.756985	−	0.653432i	$-0.226672\pi$
$829$	43.5908	1.51397	0.756985	+	0.653432i	$0.226672\pi$
$830$	12.3435	0.428449
$831$	0	0
$832$	34.9839	1.21285
$833$	−9.58541	−0.332115
$834$	0	0
$835$	27.9125	0.965952
$836$	82.0981	2.83942
$837$	0	0
$838$	77.4280	2.67470
$839$	32.8029	1.13248	0.566240	−	0.824240i	$-0.308397\pi$
$839$	32.8029	1.13248	0.566240	+	0.824240i	$0.308397\pi$
$840$	0	0
$841$	−24.3760	−0.840551
$842$	−103.470	−3.56582
$843$	0	0
$844$	65.0621	2.23953
$845$	−17.6279	−0.606418
$846$	0	0
$847$	−15.4700	−0.531555
$848$	−46.5328	−1.59794
$849$	0	0
$850$	33.5974	1.15238
$851$	18.9934	0.651086
$852$	0	0
$853$	−29.0956	−0.996213	−0.498106	−	0.867116i	$-0.665971\pi$
$853$	−29.0956	−0.996213	−0.498106	+	0.867116i	$0.665971\pi$
$854$	95.5729	3.27044
$855$	0	0
$856$	127.459	4.35646
$857$	−38.2314	−1.30596	−0.652980	−	0.757375i	$-0.726481\pi$
$857$	−38.2314	−1.30596	−0.652980	+	0.757375i	$0.726481\pi$
$858$	0	0
$859$	22.8229	0.778707	0.389353	−	0.921088i	$-0.372699\pi$
$859$	22.8229	0.778707	0.389353	+	0.921088i	$0.372699\pi$
$860$	49.8518	1.69993
$861$	0	0
$862$	28.2324	0.961600
$863$	34.3185	1.16822	0.584108	−	0.811676i	$-0.301444\pi$
$863$	34.3185	1.16822	0.584108	+	0.811676i	$0.301444\pi$
$864$	0	0
$865$	−2.21978	−0.0754748
$866$	43.4026	1.47488
$867$	0	0
$868$	3.50961	0.119124
$869$	−53.1138	−1.80176
$870$	0	0
$871$	−18.0946	−0.613113
$872$	−42.9827	−1.45558
$873$	0	0
$874$	−98.7591	−3.34058
$875$	35.2965	1.19324
$876$	0	0
$877$	23.8061	0.803875	0.401938	−	0.915667i	$-0.368337\pi$
$877$	23.8061	0.803875	0.401938	+	0.915667i	$0.368337\pi$
$878$	95.0768	3.20868
$879$	0	0
$880$	91.0759	3.07017
$881$	−34.8966	−1.17570	−0.587848	−	0.808972i	$-0.700025\pi$
$881$	−34.8966	−1.17570	−0.587848	+	0.808972i	$0.700025\pi$
$882$	0	0
$883$	−47.3179	−1.59238	−0.796188	−	0.605050i	$-0.793153\pi$
$883$	−47.3179	−1.59238	−0.796188	+	0.605050i	$0.793153\pi$
$884$	−60.6965	−2.04145
$885$	0	0
$886$	18.3320	0.615876
$887$	26.3284	0.884023	0.442011	−	0.897009i	$-0.354265\pi$
$887$	26.3284	0.884023	0.442011	+	0.897009i	$0.354265\pi$
$888$	0	0
$889$	47.1802	1.58237
$890$	38.9935	1.30707
$891$	0	0
$892$	61.9382	2.07385
$893$	14.0068	0.468720
$894$	0	0
$895$	41.5111	1.38756
$896$	−58.0830	−1.94042
$897$	0	0
$898$	2.28735	0.0763300
$899$	−0.498580	−0.0166286
$900$	0	0
$901$	24.4129	0.813311
$902$	39.2139	1.30568
$903$	0	0
$904$	−64.1644	−2.13408
$905$	4.10539	0.136468
$906$	0	0
$907$	−28.8601	−0.958283	−0.479141	−	0.877738i	$-0.659052\pi$
$907$	−28.8601	−0.958283	−0.479141	+	0.877738i	$0.659052\pi$
$908$	−46.8816	−1.55582
$909$	0	0
$910$	−24.0292	−0.796562
$911$	−16.3776	−0.542615	−0.271308	−	0.962493i	$-0.587456\pi$
$911$	−16.3776	−0.542615	−0.271308	+	0.962493i	$0.587456\pi$
$912$	0	0
$913$	10.4996	0.347487
$914$	52.0111	1.72037
$915$	0	0
$916$	88.6270	2.92832
$917$	22.5333	0.744116
$918$	0	0
$919$	−36.8615	−1.21595	−0.607975	−	0.793957i	$-0.708018\pi$
$919$	−36.8615	−1.21595	−0.607975	+	0.793957i	$0.708018\pi$
$920$	−143.949	−4.74587
$921$	0	0
$922$	−14.1480	−0.465940
$923$	−16.7649	−0.551822
$924$	0	0
$925$	3.76756	0.123877
$926$	57.4967	1.88946
$927$	0	0
$928$	36.4884	1.19779
$929$	43.9733	1.44272	0.721359	−	0.692561i	$-0.243518\pi$
$929$	43.9733	1.44272	0.721359	+	0.692561i	$0.243518\pi$
$930$	0	0
$931$	5.58907	0.183174
$932$	−15.5198	−0.508367
$933$	0	0
$934$	101.307	3.31487
$935$	−47.7818	−1.56263
$936$	0	0
$937$	−43.4006	−1.41784	−0.708919	−	0.705290i	$-0.750817\pi$
$937$	−43.4006	−1.41784	−0.708919	+	0.705290i	$0.750817\pi$
$938$	81.0184	2.64534
$939$	0	0
$940$	33.1275	1.08050
$941$	−33.0047	−1.07592	−0.537962	−	0.842969i	$-0.680805\pi$
$941$	−33.0047	−1.07592	−0.537962	+	0.842969i	$0.680805\pi$
$942$	0	0
$943$	−34.0909	−1.11015
$944$	192.542	6.26671
$945$	0	0
$946$	58.6763	1.90773
$947$	55.8412	1.81459	0.907297	−	0.420491i	$-0.138142\pi$
$947$	55.8412	1.81459	0.907297	+	0.420491i	$0.138142\pi$
$948$	0	0
$949$	27.6966	0.899069
$950$	−19.5900	−0.635583
$951$	0	0
$952$	167.487	5.42829
$953$	−5.08446	−0.164702	−0.0823508	−	0.996603i	$-0.526243\pi$
$953$	−5.08446	−0.164702	−0.0823508	+	0.996603i	$0.526243\pi$
$954$	0	0
$955$	27.6403	0.894419
$956$	41.5209	1.34288
$957$	0	0
$958$	−93.6145	−3.02455
$959$	13.0917	0.422752
$960$	0	0
$961$	−30.9462	−0.998266
$962$	−9.41812	−0.303652
$963$	0	0
$964$	80.8192	2.60301
$965$	−13.0098	−0.418801
$966$	0	0
$967$	−6.27998	−0.201951	−0.100975	−	0.994889i	$-0.532196\pi$
$967$	−6.27998	−0.201951	−0.100975	+	0.994889i	$0.532196\pi$
$968$	45.9537	1.47701
$969$	0	0
$970$	−12.2669	−0.393867
$971$	4.63886	0.148868	0.0744341	−	0.997226i	$-0.476285\pi$
$971$	4.63886	0.148868	0.0744341	+	0.997226i	$0.476285\pi$
$972$	0	0
$973$	24.5034	0.785544
$974$	−62.1386	−1.99105
$975$	0	0
$976$	−156.156	−4.99842
$977$	3.33961	0.106843	0.0534217	−	0.998572i	$-0.482987\pi$
$977$	3.33961	0.106843	0.0534217	+	0.998572i	$0.482987\pi$
$978$	0	0
$979$	33.1687	1.06008
$980$	13.2187	0.422256
$981$	0	0
$982$	44.0566	1.40590
$983$	−25.6065	−0.816722	−0.408361	−	0.912821i	$-0.633899\pi$
$983$	−25.6065	−0.816722	−0.408361	+	0.912821i	$0.633899\pi$
$984$	0	0
$985$	2.61825	0.0834244
$986$	−38.6076	−1.22952
$987$	0	0
$988$	35.3910	1.12594
$989$	−51.0106	−1.62204
$990$	0	0
$991$	2.71945	0.0863863	0.0431931	−	0.999067i	$-0.486247\pi$
$991$	2.71945	0.0863863	0.0431931	+	0.999067i	$0.486247\pi$
$992$	−3.93432	−0.124915
$993$	0	0
$994$	75.0644	2.38090
$995$	34.5923	1.09665
$996$	0	0
$997$	−27.9988	−0.886732	−0.443366	−	0.896341i	$-0.646216\pi$
$997$	−27.9988	−0.886732	−0.443366	+	0.896341i	$0.646216\pi$
$998$	58.2383	1.84350
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1341.2.a.h.1.12	yes	12
3.2	odd	2		1341.2.a.g.1.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1341.2.a.g.1.1	✓	12	3.2	odd	2
1341.2.a.h.1.12	yes	12	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.68556 q^{2} +5.21223 q^{4} +1.76881 q^{5} -2.90410 q^{7} +8.62665 q^{8} +O(q^{10})\)	\(q+2.68556 q^{2} +5.21223 q^{4} +1.76881 q^{5} -2.90410 q^{7} +8.62665 q^{8} +4.75026 q^{10} +4.04066 q^{11} +1.74185 q^{13} -7.79913 q^{14} +12.7429 q^{16} -6.68541 q^{17} +3.89813 q^{19} +9.21947 q^{20} +10.8514 q^{22} -9.43378 q^{23} -1.87130 q^{25} +4.67786 q^{26} -15.1368 q^{28} +2.15036 q^{29} -0.231859 q^{31} +16.9686 q^{32} -17.9541 q^{34} -5.13681 q^{35} -2.01334 q^{37} +10.4687 q^{38} +15.2589 q^{40} +3.61371 q^{41} +5.40723 q^{43} +21.0609 q^{44} -25.3350 q^{46} +3.59321 q^{47} +1.43378 q^{49} -5.02548 q^{50} +9.07895 q^{52} -3.65166 q^{53} +7.14718 q^{55} -25.0526 q^{56} +5.77491 q^{58} +15.1098 q^{59} -12.2543 q^{61} -0.622672 q^{62} +20.0843 q^{64} +3.08102 q^{65} -10.3881 q^{67} -34.8459 q^{68} -13.7952 q^{70} -9.62471 q^{71} +15.9006 q^{73} -5.40695 q^{74} +20.3180 q^{76} -11.7345 q^{77} -13.1448 q^{79} +22.5398 q^{80} +9.70483 q^{82} +2.59849 q^{83} -11.8252 q^{85} +14.5214 q^{86} +34.8574 q^{88} +8.20872 q^{89} -5.05852 q^{91} -49.1711 q^{92} +9.64979 q^{94} +6.89507 q^{95} -2.58237 q^{97} +3.85050 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{2} + 9 q^{4} + 8 q^{5} - 2 q^{7} + 9 q^{8}+O(q^{10}) \) 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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