LMFDB - Embedded newform 1341.2.a.g.1.6

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			1.6
Root			$0.423108$ 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)

$f(q)$	$=$	$q-0.423108 q^{2} -1.82098 q^{4} -4.20441 q^{5} +2.99566 q^{7} +1.61669 q^{8} +O(q^{10})$	\(q-0.423108 q^{2} -1.82098 q^{4} -4.20441 q^{5} +2.99566 q^{7} +1.61669 q^{8} +1.77892 q^{10} -2.55358 q^{11} +2.04378 q^{13} -1.26749 q^{14} +2.95793 q^{16} +6.75895 q^{17} +1.19722 q^{19} +7.65615 q^{20} +1.08044 q^{22} -1.18306 q^{23} +12.6771 q^{25} -0.864741 q^{26} -5.45504 q^{28} -7.94943 q^{29} -5.64721 q^{31} -4.48490 q^{32} -2.85977 q^{34} -12.5950 q^{35} -8.25046 q^{37} -0.506553 q^{38} -6.79722 q^{40} +3.91138 q^{41} -9.54389 q^{43} +4.65002 q^{44} +0.500563 q^{46} +1.09410 q^{47} +1.97399 q^{49} -5.36378 q^{50} -3.72169 q^{52} +5.08335 q^{53} +10.7363 q^{55} +4.84305 q^{56} +3.36347 q^{58} -11.5707 q^{59} +15.2312 q^{61} +2.38938 q^{62} -4.01826 q^{64} -8.59291 q^{65} -4.66090 q^{67} -12.3079 q^{68} +5.32905 q^{70} -8.01473 q^{71} -8.75750 q^{73} +3.49084 q^{74} -2.18011 q^{76} -7.64966 q^{77} +14.9010 q^{79} -12.4363 q^{80} -1.65494 q^{82} -6.19956 q^{83} -28.4174 q^{85} +4.03810 q^{86} -4.12834 q^{88} -8.51933 q^{89} +6.12248 q^{91} +2.15433 q^{92} -0.462923 q^{94} -5.03361 q^{95} -5.18102 q^{97} -0.835209 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.423108	−0.299183	−0.149591	−	0.988748i	$-0.547796\pi$
$2$	−0.423108	−0.299183	−0.149591	+	0.988748i	$0.547796\pi$
$3$	0	0
$3$	0	0
$4$	−1.82098	−0.910490
$5$	−4.20441	−1.88027	−0.940136	−	0.340801i	$-0.889302\pi$
$5$	−4.20441	−1.88027	−0.940136	+	0.340801i	$0.889302\pi$
$6$	0	0
$7$	2.99566	1.13225	0.566127	−	0.824318i	$-0.308441\pi$
$7$	2.99566	1.13225	0.566127	+	0.824318i	$0.308441\pi$
$8$	1.61669	0.571585
$9$	0	0
$10$	1.77892	0.562544
$11$	−2.55358	−0.769934	−0.384967	−	0.922930i	$-0.625787\pi$
$11$	−2.55358	−0.769934	−0.384967	+	0.922930i	$0.625787\pi$
$12$	0	0
$13$	2.04378	0.566844	0.283422	−	0.958995i	$-0.408530\pi$
$13$	2.04378	0.566844	0.283422	+	0.958995i	$0.408530\pi$
$14$	−1.26749	−0.338751
$15$	0	0
$16$	2.95793	0.739481
$17$	6.75895	1.63929	0.819644	−	0.572874i	$-0.194171\pi$
$17$	6.75895	1.63929	0.819644	+	0.572874i	$0.194171\pi$
$18$	0	0
$19$	1.19722	0.274661	0.137331	−	0.990525i	$-0.456148\pi$
$19$	1.19722	0.274661	0.137331	+	0.990525i	$0.456148\pi$
$20$	7.65615	1.71197
$21$	0	0
$22$	1.08044	0.230351
$23$	−1.18306	−0.246686	−0.123343	−	0.992364i	$-0.539361\pi$
$23$	−1.18306	−0.246686	−0.123343	+	0.992364i	$0.539361\pi$
$24$	0	0
$25$	12.6771	2.53542
$26$	−0.864741	−0.169590
$27$	0	0
$28$	−5.45504	−1.03091
$29$	−7.94943	−1.47617	−0.738086	−	0.674706i	$-0.764270\pi$
$29$	−7.94943	−1.47617	−0.738086	+	0.674706i	$0.764270\pi$
$30$	0	0
$31$	−5.64721	−1.01427	−0.507134	−	0.861867i	$-0.669295\pi$
$31$	−5.64721	−1.01427	−0.507134	+	0.861867i	$0.669295\pi$
$32$	−4.48490	−0.792825
$33$	0	0
$34$	−2.85977	−0.490446
$35$	−12.5950	−2.12894
$36$	0	0
$37$	−8.25046	−1.35637	−0.678184	−	0.734893i	$-0.737233\pi$
$37$	−8.25046	−1.35637	−0.678184	+	0.734893i	$0.737233\pi$
$38$	−0.506553	−0.0821738
$39$	0	0
$40$	−6.79722	−1.07474
$41$	3.91138	0.610855	0.305428	−	0.952215i	$-0.401201\pi$
$41$	3.91138	0.610855	0.305428	+	0.952215i	$0.401201\pi$
$42$	0	0
$43$	−9.54389	−1.45543	−0.727715	−	0.685880i	$-0.759418\pi$
$43$	−9.54389	−1.45543	−0.727715	+	0.685880i	$0.759418\pi$
$44$	4.65002	0.701017
$45$	0	0
$46$	0.500563	0.0738040
$47$	1.09410	0.159591	0.0797955	−	0.996811i	$-0.474573\pi$
$47$	1.09410	0.159591	0.0797955	+	0.996811i	$0.474573\pi$
$48$	0	0
$49$	1.97399	0.281998
$50$	−5.36378	−0.758553
$51$	0	0
$52$	−3.72169	−0.516105
$53$	5.08335	0.698252	0.349126	−	0.937076i	$-0.386479\pi$
$53$	5.08335	0.698252	0.349126	+	0.937076i	$0.386479\pi$
$54$	0	0
$55$	10.7363	1.44768
$56$	4.84305	0.647179
$57$	0	0
$58$	3.36347	0.441645
$59$	−11.5707	−1.50638	−0.753188	−	0.657805i	$-0.771485\pi$
$59$	−11.5707	−1.50638	−0.753188	+	0.657805i	$0.771485\pi$
$60$	0	0
$61$	15.2312	1.95015	0.975076	−	0.221871i	$-0.0712165\pi$
$61$	15.2312	1.95015	0.975076	+	0.221871i	$0.0712165\pi$
$62$	2.38938	0.303451
$63$	0	0
$64$	−4.01826	−0.502282
$65$	−8.59291	−1.06582
$66$	0	0
$67$	−4.66090	−0.569420	−0.284710	−	0.958614i	$-0.591897\pi$
$67$	−4.66090	−0.569420	−0.284710	+	0.958614i	$0.591897\pi$
$68$	−12.3079	−1.49255
$69$	0	0
$70$	5.32905	0.636943
$71$	−8.01473	−0.951174	−0.475587	−	0.879669i	$-0.657764\pi$
$71$	−8.01473	−0.951174	−0.475587	+	0.879669i	$0.657764\pi$
$72$	0	0
$73$	−8.75750	−1.02499	−0.512494	−	0.858691i	$-0.671278\pi$
$73$	−8.75750	−1.02499	−0.512494	+	0.858691i	$0.671278\pi$
$74$	3.49084	0.405801
$75$	0	0
$76$	−2.18011	−0.250076
$77$	−7.64966	−0.871760
$78$	0	0
$79$	14.9010	1.67649	0.838245	−	0.545294i	$-0.183582\pi$
$79$	14.9010	1.67649	0.838245	+	0.545294i	$0.183582\pi$
$80$	−12.4363	−1.39043
$81$	0	0
$82$	−1.65494	−0.182757
$83$	−6.19956	−0.680490	−0.340245	−	0.940337i	$-0.610510\pi$
$83$	−6.19956	−0.680490	−0.340245	+	0.940337i	$0.610510\pi$
$84$	0	0
$85$	−28.4174	−3.08230
$86$	4.03810	0.435439
$87$	0	0
$88$	−4.12834	−0.440083
$89$	−8.51933	−0.903047	−0.451524	−	0.892259i	$-0.649119\pi$
$89$	−8.51933	−0.903047	−0.451524	+	0.892259i	$0.649119\pi$
$90$	0	0
$91$	6.12248	0.641811
$92$	2.15433	0.224605
$93$	0	0
$94$	−0.462923	−0.0477469
$95$	−5.03361	−0.516437
$96$	0	0
$97$	−5.18102	−0.526053	−0.263026	−	0.964789i	$-0.584721\pi$
$97$	−5.18102	−0.526053	−0.263026	+	0.964789i	$0.584721\pi$
$98$	−0.835209	−0.0843689
$99$	0	0
$100$	−23.0847	−2.30847
$101$	7.56601	0.752846	0.376423	−	0.926448i	$-0.377154\pi$
$101$	7.56601	0.752846	0.376423	+	0.926448i	$0.377154\pi$
$102$	0	0
$103$	8.30450	0.818267	0.409133	−	0.912475i	$-0.365831\pi$
$103$	8.30450	0.818267	0.409133	+	0.912475i	$0.365831\pi$
$104$	3.30416	0.323999
$105$	0	0
$106$	−2.15081	−0.208905
$107$	−2.06293	−0.199431	−0.0997156	−	0.995016i	$-0.531793\pi$
$107$	−2.06293	−0.199431	−0.0997156	+	0.995016i	$0.531793\pi$
$108$	0	0
$109$	0.640887	0.0613858	0.0306929	−	0.999529i	$-0.490229\pi$
$109$	0.640887	0.0613858	0.0306929	+	0.999529i	$0.490229\pi$
$110$	−4.54262	−0.433122
$111$	0	0
$112$	8.86094	0.837280
$113$	−10.1052	−0.950616	−0.475308	−	0.879819i	$-0.657663\pi$
$113$	−10.1052	−0.950616	−0.475308	+	0.879819i	$0.657663\pi$
$114$	0	0
$115$	4.97408	0.463836
$116$	14.4758	1.34404
$117$	0	0
$118$	4.89565	0.450682
$119$	20.2475	1.85609
$120$	0	0
$121$	−4.47923	−0.407202
$122$	−6.44443	−0.583451
$123$	0	0
$124$	10.2834	0.923481
$125$	−32.2777	−2.88701
$126$	0	0
$127$	−5.15242	−0.457204	−0.228602	−	0.973520i	$-0.573415\pi$
$127$	−5.15242	−0.457204	−0.228602	+	0.973520i	$0.573415\pi$
$128$	10.6699	0.943099
$129$	0	0
$130$	3.63573	0.318875
$131$	8.49912	0.742572	0.371286	−	0.928519i	$-0.378917\pi$
$131$	8.49912	0.742572	0.371286	+	0.928519i	$0.378917\pi$
$132$	0	0
$133$	3.58646	0.310986
$134$	1.97207	0.170361
$135$	0	0
$136$	10.9271	0.936992
$137$	−4.74675	−0.405542	−0.202771	−	0.979226i	$-0.564995\pi$
$137$	−4.74675	−0.405542	−0.202771	+	0.979226i	$0.564995\pi$
$138$	0	0
$139$	−6.87968	−0.583526	−0.291763	−	0.956491i	$-0.594242\pi$
$139$	−6.87968	−0.583526	−0.291763	+	0.956491i	$0.594242\pi$
$140$	22.9352	1.93838
$141$	0	0
$142$	3.39110	0.284575
$143$	−5.21897	−0.436432
$144$	0	0
$145$	33.4227	2.77560
$146$	3.70537	0.306659
$147$	0	0
$148$	15.0239	1.23496
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−9.86374	−0.802700	−0.401350	−	0.915925i	$-0.631459\pi$
$151$	−9.86374	−0.802700	−0.401350	+	0.915925i	$0.631459\pi$
$152$	1.93553	0.156992
$153$	0	0
$154$	3.23663	0.260815
$155$	23.7432	1.90710
$156$	0	0
$157$	−15.6107	−1.24587	−0.622935	−	0.782273i	$-0.714060\pi$
$157$	−15.6107	−1.24587	−0.622935	+	0.782273i	$0.714060\pi$
$158$	−6.30472	−0.501577
$159$	0	0
$160$	18.8564	1.49073
$161$	−3.54405	−0.279311
$162$	0	0
$163$	−20.1593	−1.57899	−0.789497	−	0.613754i	$-0.789659\pi$
$163$	−20.1593	−1.57899	−0.789497	+	0.613754i	$0.789659\pi$
$164$	−7.12254	−0.556177
$165$	0	0
$166$	2.62308	0.203591
$167$	0.127884	0.00989596	0.00494798	−	0.999988i	$-0.498425\pi$
$167$	0.127884	0.00989596	0.00494798	+	0.999988i	$0.498425\pi$
$168$	0	0
$169$	−8.82295	−0.678688
$170$	12.0237	0.922172
$171$	0	0
$172$	17.3792	1.32515
$173$	−7.51672	−0.571486	−0.285743	−	0.958306i	$-0.592240\pi$
$173$	−7.51672	−0.571486	−0.285743	+	0.958306i	$0.592240\pi$
$174$	0	0
$175$	37.9763	2.87074
$176$	−7.55330	−0.569352
$177$	0	0
$178$	3.60460	0.270176
$179$	−1.39670	−0.104394	−0.0521970	−	0.998637i	$-0.516622\pi$
$179$	−1.39670	−0.104394	−0.0521970	+	0.998637i	$0.516622\pi$
$180$	0	0
$181$	−21.9325	−1.63023	−0.815115	−	0.579299i	$-0.803326\pi$
$181$	−21.9325	−1.63023	−0.815115	+	0.579299i	$0.803326\pi$
$182$	−2.59047	−0.192019
$183$	0	0
$184$	−1.91264	−0.141002
$185$	34.6883	2.55034
$186$	0	0
$187$	−17.2595	−1.26214
$188$	−1.99234	−0.145306
$189$	0	0
$190$	2.12976	0.154509
$191$	18.2239	1.31863	0.659316	−	0.751866i	$-0.270846\pi$
$191$	18.2239	1.31863	0.659316	+	0.751866i	$0.270846\pi$
$192$	0	0
$193$	−6.40158	−0.460796	−0.230398	−	0.973097i	$-0.574003\pi$
$193$	−6.40158	−0.460796	−0.230398	+	0.973097i	$0.574003\pi$
$194$	2.19213	0.157386
$195$	0	0
$196$	−3.59459	−0.256756
$197$	23.4449	1.67038	0.835190	−	0.549962i	$-0.185358\pi$
$197$	23.4449	1.67038	0.835190	+	0.549962i	$0.185358\pi$
$198$	0	0
$199$	17.1088	1.21281	0.606406	−	0.795155i	$-0.292611\pi$
$199$	17.1088	1.21281	0.606406	+	0.795155i	$0.292611\pi$
$200$	20.4949	1.44921
$201$	0	0
$202$	−3.20124	−0.225238
$203$	−23.8138	−1.67140
$204$	0	0
$205$	−16.4451	−1.14857
$206$	−3.51370	−0.244811
$207$	0	0
$208$	6.04536	0.419170
$209$	−3.05720	−0.211471
$210$	0	0
$211$	3.12717	0.215283	0.107641	−	0.994190i	$-0.465670\pi$
$211$	3.12717	0.215283	0.107641	+	0.994190i	$0.465670\pi$
$212$	−9.25667	−0.635751
$213$	0	0
$214$	0.872843	0.0596663
$215$	40.1265	2.73660
$216$	0	0
$217$	−16.9171	−1.14841
$218$	−0.271164	−0.0183656
$219$	0	0
$220$	−19.5506	−1.31810
$221$	13.8138	0.929220
$222$	0	0
$223$	−13.3821	−0.896130	−0.448065	−	0.894001i	$-0.647887\pi$
$223$	−13.3821	−0.896130	−0.448065	+	0.894001i	$0.647887\pi$
$224$	−13.4352	−0.897679
$225$	0	0
$226$	4.27559	0.284408
$227$	11.6389	0.772500	0.386250	−	0.922394i	$-0.373770\pi$
$227$	11.6389	0.772500	0.386250	+	0.922394i	$0.373770\pi$
$228$	0	0
$229$	−1.03770	−0.0685730	−0.0342865	−	0.999412i	$-0.510916\pi$
$229$	−1.03770	−0.0685730	−0.0342865	+	0.999412i	$0.510916\pi$
$230$	−2.10457	−0.138772
$231$	0	0
$232$	−12.8517	−0.843758
$233$	20.9009	1.36926	0.684632	−	0.728889i	$-0.259963\pi$
$233$	20.9009	1.36926	0.684632	+	0.728889i	$0.259963\pi$
$234$	0	0
$235$	−4.60005	−0.300074
$236$	21.0700	1.37154
$237$	0	0
$238$	−8.56690	−0.555309
$239$	−27.1069	−1.75340	−0.876701	−	0.481037i	$-0.840260\pi$
$239$	−27.1069	−1.75340	−0.876701	+	0.481037i	$0.840260\pi$
$240$	0	0
$241$	−16.7277	−1.07753	−0.538764	−	0.842457i	$-0.681109\pi$
$241$	−16.7277	−1.07753	−0.538764	+	0.842457i	$0.681109\pi$
$242$	1.89520	0.121828
$243$	0	0
$244$	−27.7357	−1.77559
$245$	−8.29946	−0.530233
$246$	0	0
$247$	2.44686	0.155690
$248$	−9.12977	−0.579741
$249$	0	0
$250$	13.6570	0.863742
$251$	−29.7067	−1.87507	−0.937535	−	0.347892i	$-0.886898\pi$
$251$	−29.7067	−1.87507	−0.937535	+	0.347892i	$0.886898\pi$
$252$	0	0
$253$	3.02104	0.189931
$254$	2.18003	0.136787
$255$	0	0
$256$	3.52197	0.220123
$257$	−4.69106	−0.292621	−0.146310	−	0.989239i	$-0.546740\pi$
$257$	−4.69106	−0.292621	−0.146310	+	0.989239i	$0.546740\pi$
$258$	0	0
$259$	−24.7156	−1.53575
$260$	15.6475	0.970418
$261$	0	0
$262$	−3.59605	−0.222164
$263$	−22.4803	−1.38619	−0.693096	−	0.720845i	$-0.743754\pi$
$263$	−22.4803	−1.38619	−0.693096	+	0.720845i	$0.743754\pi$
$264$	0	0
$265$	−21.3725	−1.31290
$266$	−1.51746	−0.0930416
$267$	0	0
$268$	8.48741	0.518451
$269$	13.5875	0.828445	0.414223	−	0.910176i	$-0.364053\pi$
$269$	13.5875	0.828445	0.414223	+	0.910176i	$0.364053\pi$
$270$	0	0
$271$	32.5994	1.98027	0.990136	−	0.140106i	$-0.0447444\pi$
$271$	32.5994	1.98027	0.990136	+	0.140106i	$0.0447444\pi$
$272$	19.9925	1.21222
$273$	0	0
$274$	2.00839	0.121331
$275$	−32.3720	−1.95210
$276$	0	0
$277$	6.40829	0.385037	0.192518	−	0.981293i	$-0.438335\pi$
$277$	6.40829	0.385037	0.192518	+	0.981293i	$0.438335\pi$
$278$	2.91085	0.174581
$279$	0	0
$280$	−20.3622	−1.21687
$281$	13.3102	0.794020	0.397010	−	0.917814i	$-0.370048\pi$
$281$	13.3102	0.794020	0.397010	+	0.917814i	$0.370048\pi$
$282$	0	0
$283$	−10.2679	−0.610362	−0.305181	−	0.952294i	$-0.598717\pi$
$283$	−10.2679	−0.610362	−0.305181	+	0.952294i	$0.598717\pi$
$284$	14.5947	0.866034
$285$	0	0
$286$	2.20819	0.130573
$287$	11.7172	0.691643
$288$	0	0
$289$	28.6835	1.68726
$290$	−14.1414	−0.830413
$291$	0	0
$292$	15.9472	0.933241
$293$	−0.797096	−0.0465668	−0.0232834	−	0.999729i	$-0.507412\pi$
$293$	−0.797096	−0.0465668	−0.0232834	+	0.999729i	$0.507412\pi$
$294$	0	0
$295$	48.6480	2.83240
$296$	−13.3384	−0.775279
$297$	0	0
$298$	0.423108	0.0245100
$299$	−2.41792	−0.139832
$300$	0	0
$301$	−28.5903	−1.64792
$302$	4.17343	0.240154
$303$	0	0
$304$	3.54129	0.203107
$305$	−64.0382	−3.66681
$306$	0	0
$307$	−2.70410	−0.154331	−0.0771655	−	0.997018i	$-0.524587\pi$
$307$	−2.70410	−0.154331	−0.0771655	+	0.997018i	$0.524587\pi$
$308$	13.9299	0.793729
$309$	0	0
$310$	−10.0459	−0.570571
$311$	24.0221	1.36217	0.681085	−	0.732204i	$-0.261508\pi$
$311$	24.0221	1.36217	0.681085	+	0.732204i	$0.261508\pi$
$312$	0	0
$313$	−11.1883	−0.632402	−0.316201	−	0.948692i	$-0.602407\pi$
$313$	−11.1883	−0.632402	−0.316201	+	0.948692i	$0.602407\pi$
$314$	6.60502	0.372743
$315$	0	0
$316$	−27.1344	−1.52643
$317$	−12.7911	−0.718422	−0.359211	−	0.933256i	$-0.616954\pi$
$317$	−12.7911	−0.718422	−0.359211	+	0.933256i	$0.616954\pi$
$318$	0	0
$319$	20.2995	1.13655
$320$	16.8944	0.944426
$321$	0	0
$322$	1.49952	0.0835648
$323$	8.09195	0.450248
$324$	0	0
$325$	25.9093	1.43719
$326$	8.52954	0.472408
$327$	0	0
$328$	6.32348	0.349156
$329$	3.27756	0.180697
$330$	0	0
$331$	−27.0194	−1.48512	−0.742559	−	0.669781i	$-0.766388\pi$
$331$	−27.0194	−1.48512	−0.742559	+	0.669781i	$0.766388\pi$
$332$	11.2893	0.619579
$333$	0	0
$334$	−0.0541087	−0.00296070
$335$	19.5964	1.07066
$336$	0	0
$337$	−23.9778	−1.30616	−0.653078	−	0.757291i	$-0.726523\pi$
$337$	−23.9778	−1.30616	−0.653078	+	0.757291i	$0.726523\pi$
$338$	3.73306	0.203052
$339$	0	0
$340$	51.7476	2.80641
$341$	14.4206	0.780919
$342$	0	0
$343$	−15.0562	−0.812960
$344$	−15.4295	−0.831902
$345$	0	0
$346$	3.18039	0.170979
$347$	27.7797	1.49129	0.745647	−	0.666342i	$-0.232141\pi$
$347$	27.7797	1.49129	0.745647	+	0.666342i	$0.232141\pi$
$348$	0	0
$349$	2.56458	0.137279	0.0686393	−	0.997642i	$-0.478134\pi$
$349$	2.56458	0.137279	0.0686393	+	0.997642i	$0.478134\pi$
$350$	−16.0681	−0.858875
$351$	0	0
$352$	11.4525	0.610423
$353$	−5.96155	−0.317301	−0.158651	−	0.987335i	$-0.550714\pi$
$353$	−5.96155	−0.317301	−0.158651	+	0.987335i	$0.550714\pi$
$354$	0	0
$355$	33.6972	1.78846
$356$	15.5135	0.822215
$357$	0	0
$358$	0.590953	0.0312328
$359$	−17.1429	−0.904766	−0.452383	−	0.891824i	$-0.649426\pi$
$359$	−17.1429	−0.904766	−0.452383	+	0.891824i	$0.649426\pi$
$360$	0	0
$361$	−17.5667	−0.924561
$362$	9.27982	0.487736
$363$	0	0
$364$	−11.1489	−0.584362
$365$	36.8202	1.92726
$366$	0	0
$367$	14.2487	0.743777	0.371888	−	0.928278i	$-0.378710\pi$
$367$	14.2487	0.743777	0.371888	+	0.928278i	$0.378710\pi$
$368$	−3.49941	−0.182419
$369$	0	0
$370$	−14.6769	−0.763017
$371$	15.2280	0.790598
$372$	0	0
$373$	28.2921	1.46491	0.732453	−	0.680817i	$-0.238375\pi$
$373$	28.2921	1.46491	0.732453	+	0.680817i	$0.238375\pi$
$374$	7.30265	0.377611
$375$	0	0
$376$	1.76882	0.0912199
$377$	−16.2469	−0.836759
$378$	0	0
$379$	11.4114	0.586165	0.293082	−	0.956087i	$-0.405319\pi$
$379$	11.4114	0.586165	0.293082	+	0.956087i	$0.405319\pi$
$380$	9.16610	0.470211
$381$	0	0
$382$	−7.71066	−0.394512
$383$	−32.8780	−1.67999	−0.839994	−	0.542595i	$-0.817442\pi$
$383$	−32.8780	−1.67999	−0.839994	+	0.542595i	$0.817442\pi$
$384$	0	0
$385$	32.1624	1.63915
$386$	2.70856	0.137862
$387$	0	0
$388$	9.43453	0.478965
$389$	−2.96547	−0.150356	−0.0751778	−	0.997170i	$-0.523952\pi$
$389$	−2.96547	−0.150356	−0.0751778	+	0.997170i	$0.523952\pi$
$390$	0	0
$391$	−7.99626	−0.404388
$392$	3.19132	0.161186
$393$	0	0
$394$	−9.91973	−0.499749
$395$	−62.6498	−3.15226
$396$	0	0
$397$	17.1216	0.859307	0.429653	−	0.902994i	$-0.358636\pi$
$397$	17.1216	0.859307	0.429653	+	0.902994i	$0.358636\pi$
$398$	−7.23888	−0.362852
$399$	0	0
$400$	37.4979	1.87490
$401$	−1.46187	−0.0730021	−0.0365010	−	0.999334i	$-0.511621\pi$
$401$	−1.46187	−0.0730021	−0.0365010	+	0.999334i	$0.511621\pi$
$402$	0	0
$403$	−11.5417	−0.574931
$404$	−13.7775	−0.685458
$405$	0	0
$406$	10.0758	0.500054
$407$	21.0682	1.04431
$408$	0	0
$409$	6.69539	0.331066	0.165533	−	0.986204i	$-0.447066\pi$
$409$	6.69539	0.331066	0.165533	+	0.986204i	$0.447066\pi$
$410$	6.95804	0.343633
$411$	0	0
$412$	−15.1223	−0.745024
$413$	−34.6619	−1.70560
$414$	0	0
$415$	26.0655	1.27951
$416$	−9.16616	−0.449408
$417$	0	0
$418$	1.29352	0.0632684
$419$	17.1609	0.838367	0.419184	−	0.907901i	$-0.362316\pi$
$419$	17.1609	0.838367	0.419184	+	0.907901i	$0.362316\pi$
$420$	0	0
$421$	−28.7755	−1.40243	−0.701215	−	0.712950i	$-0.747359\pi$
$421$	−28.7755	−1.40243	−0.701215	+	0.712950i	$0.747359\pi$
$422$	−1.32313	−0.0644089
$423$	0	0
$424$	8.21819	0.399110
$425$	85.6839	4.15628
$426$	0	0
$427$	45.6274	2.20807
$428$	3.75656	0.181580
$429$	0	0
$430$	−16.9778	−0.818744
$431$	−14.8156	−0.713643	−0.356821	−	0.934173i	$-0.616139\pi$
$431$	−14.8156	−0.713643	−0.356821	+	0.934173i	$0.616139\pi$
$432$	0	0
$433$	−21.0658	−1.01236	−0.506180	−	0.862428i	$-0.668943\pi$
$433$	−21.0658	−1.01236	−0.506180	+	0.862428i	$0.668943\pi$
$434$	7.15777	0.343584
$435$	0	0
$436$	−1.16704	−0.0558911
$437$	−1.41639	−0.0677549
$438$	0	0
$439$	−35.1208	−1.67622	−0.838111	−	0.545499i	$-0.816340\pi$
$439$	−35.1208	−1.67622	−0.838111	+	0.545499i	$0.816340\pi$
$440$	17.3573	0.827475
$441$	0	0
$442$	−5.84475	−0.278006
$443$	−5.32280	−0.252894	−0.126447	−	0.991973i	$-0.540357\pi$
$443$	−5.32280	−0.252894	−0.126447	+	0.991973i	$0.540357\pi$
$444$	0	0
$445$	35.8188	1.69797
$446$	5.66206	0.268106
$447$	0	0
$448$	−12.0373	−0.568710
$449$	38.4202	1.81316	0.906581	−	0.422032i	$-0.138683\pi$
$449$	38.4202	1.81316	0.906581	+	0.422032i	$0.138683\pi$
$450$	0	0
$451$	−9.98803	−0.470318
$452$	18.4013	0.865526
$453$	0	0
$454$	−4.92451	−0.231119
$455$	−25.7415	−1.20678
$456$	0	0
$457$	33.2547	1.55559	0.777795	−	0.628519i	$-0.216338\pi$
$457$	33.2547	1.55559	0.777795	+	0.628519i	$0.216338\pi$
$458$	0.439058	0.0205158
$459$	0	0
$460$	−9.05770	−0.422318
$461$	21.0851	0.982032	0.491016	−	0.871151i	$-0.336626\pi$
$461$	21.0851	0.982032	0.491016	+	0.871151i	$0.336626\pi$
$462$	0	0
$463$	16.9621	0.788295	0.394148	−	0.919047i	$-0.371040\pi$
$463$	16.9621	0.788295	0.394148	+	0.919047i	$0.371040\pi$
$464$	−23.5138	−1.09160
$465$	0	0
$466$	−8.84335	−0.409660
$467$	9.98785	0.462183	0.231091	−	0.972932i	$-0.425770\pi$
$467$	9.98785	0.462183	0.231091	+	0.972932i	$0.425770\pi$
$468$	0	0
$469$	−13.9625	−0.644728
$470$	1.94632	0.0897770
$471$	0	0
$472$	−18.7062	−0.861022
$473$	24.3711	1.12058
$474$	0	0
$475$	15.1773	0.696381
$476$	−36.8704	−1.68995
$477$	0	0
$478$	11.4692	0.524587
$479$	−41.0580	−1.87599	−0.937995	−	0.346650i	$-0.887319\pi$
$479$	−41.0580	−1.87599	−0.937995	+	0.346650i	$0.887319\pi$
$480$	0	0
$481$	−16.8622	−0.768848
$482$	7.07764	0.322378
$483$	0	0
$484$	8.15658	0.370754
$485$	21.7831	0.989122
$486$	0	0
$487$	32.4577	1.47080	0.735400	−	0.677633i	$-0.236994\pi$
$487$	32.4577	1.47080	0.735400	+	0.677633i	$0.236994\pi$
$488$	24.6240	1.11468
$489$	0	0
$490$	3.51157	0.158636
$491$	12.4943	0.563858	0.281929	−	0.959435i	$-0.409026\pi$
$491$	12.4943	0.563858	0.281929	+	0.959435i	$0.409026\pi$
$492$	0	0
$493$	−53.7299	−2.41987
$494$	−1.03529	−0.0465797
$495$	0	0
$496$	−16.7040	−0.750033
$497$	−24.0094	−1.07697
$498$	0	0
$499$	23.9507	1.07218	0.536091	−	0.844160i	$-0.319901\pi$
$499$	23.9507	1.07218	0.536091	+	0.844160i	$0.319901\pi$
$500$	58.7770	2.62859
$501$	0	0
$502$	12.5691	0.560988
$503$	1.27880	0.0570190	0.0285095	−	0.999594i	$-0.490924\pi$
$503$	1.27880	0.0570190	0.0285095	+	0.999594i	$0.490924\pi$
$504$	0	0
$505$	−31.8106	−1.41555
$506$	−1.27823	−0.0568242
$507$	0	0
$508$	9.38246	0.416279
$509$	18.8636	0.836113	0.418056	−	0.908421i	$-0.362711\pi$
$509$	18.8636	0.836113	0.418056	+	0.908421i	$0.362711\pi$
$510$	0	0
$511$	−26.2345	−1.16055
$512$	−22.8301	−1.00896
$513$	0	0
$514$	1.98483	0.0875470
$515$	−34.9156	−1.53856
$516$	0	0
$517$	−2.79388	−0.122874
$518$	10.4574	0.459470
$519$	0	0
$520$	−13.8921	−0.609207
$521$	5.18952	0.227357	0.113678	−	0.993518i	$-0.463737\pi$
$521$	5.18952	0.227357	0.113678	+	0.993518i	$0.463737\pi$
$522$	0	0
$523$	−27.5993	−1.20683	−0.603417	−	0.797426i	$-0.706195\pi$
$523$	−27.5993	−1.20683	−0.603417	+	0.797426i	$0.706195\pi$
$524$	−15.4767	−0.676104
$525$	0	0
$526$	9.51158	0.414725
$527$	−38.1692	−1.66268
$528$	0	0
$529$	−21.6004	−0.939146
$530$	9.04288	0.392798
$531$	0	0
$532$	−6.53088	−0.283150
$533$	7.99402	0.346259
$534$	0	0
$535$	8.67342	0.374985
$536$	−7.53522	−0.325472
$537$	0	0
$538$	−5.74899	−0.247856
$539$	−5.04073	−0.217120
$540$	0	0
$541$	−4.76375	−0.204810	−0.102405	−	0.994743i	$-0.532654\pi$
$541$	−4.76375	−0.204810	−0.102405	+	0.994743i	$0.532654\pi$
$542$	−13.7931	−0.592463
$543$	0	0
$544$	−30.3132	−1.29967
$545$	−2.69455	−0.115422
$546$	0	0
$547$	−23.2588	−0.994474	−0.497237	−	0.867615i	$-0.665652\pi$
$547$	−23.2588	−0.994474	−0.497237	+	0.867615i	$0.665652\pi$
$548$	8.64373	0.369242
$549$	0	0
$550$	13.6969	0.584036
$551$	−9.51722	−0.405447
$552$	0	0
$553$	44.6383	1.89821
$554$	−2.71140	−0.115196
$555$	0	0
$556$	12.5277	0.531295
$557$	−28.1978	−1.19478	−0.597389	−	0.801951i	$-0.703795\pi$
$557$	−28.1978	−1.19478	−0.597389	+	0.801951i	$0.703795\pi$
$558$	0	0
$559$	−19.5057	−0.825001
$560$	−37.2551	−1.57431
$561$	0	0
$562$	−5.63165	−0.237557
$563$	−34.1122	−1.43766	−0.718830	−	0.695186i	$-0.755322\pi$
$563$	−34.1122	−1.43766	−0.718830	+	0.695186i	$0.755322\pi$
$564$	0	0
$565$	42.4864	1.78742
$566$	4.34442	0.182610
$567$	0	0
$568$	−12.9573	−0.543677
$569$	−15.0040	−0.629001	−0.314500	−	0.949257i	$-0.601837\pi$
$569$	−15.0040	−0.629001	−0.314500	+	0.949257i	$0.601837\pi$
$570$	0	0
$571$	−1.55661	−0.0651419	−0.0325709	−	0.999469i	$-0.510369\pi$
$571$	−1.55661	−0.0651419	−0.0325709	+	0.999469i	$0.510369\pi$
$572$	9.50363	0.397367
$573$	0	0
$574$	−4.95763	−0.206927
$575$	−14.9978	−0.625451
$576$	0	0
$577$	−13.4240	−0.558848	−0.279424	−	0.960168i	$-0.590143\pi$
$577$	−13.4240	−0.558848	−0.279424	+	0.960168i	$0.590143\pi$
$578$	−12.1362	−0.504800
$579$	0	0
$580$	−60.8621	−2.52716
$581$	−18.5718	−0.770487
$582$	0	0
$583$	−12.9807	−0.537607
$584$	−14.1581	−0.585868
$585$	0	0
$586$	0.337258	0.0139320
$587$	39.6560	1.63678	0.818388	−	0.574665i	$-0.194868\pi$
$587$	39.6560	1.63678	0.818388	+	0.574665i	$0.194868\pi$
$588$	0	0
$589$	−6.76095	−0.278580
$590$	−20.5834	−0.847404
$591$	0	0
$592$	−24.4042	−1.00301
$593$	23.0642	0.947132	0.473566	−	0.880758i	$-0.342967\pi$
$593$	23.0642	0.947132	0.473566	+	0.880758i	$0.342967\pi$
$594$	0	0
$595$	−85.1290	−3.48995
$596$	1.82098	0.0745902
$597$	0	0
$598$	1.02304	0.0418353
$599$	−5.78018	−0.236172	−0.118086	−	0.993003i	$-0.537676\pi$
$599$	−5.78018	−0.236172	−0.118086	+	0.993003i	$0.537676\pi$
$600$	0	0
$601$	−2.97912	−0.121521	−0.0607603	−	0.998152i	$-0.519353\pi$
$601$	−2.97912	−0.121521	−0.0607603	+	0.998152i	$0.519353\pi$
$602$	12.0968	0.493028
$603$	0	0
$604$	17.9617	0.730850
$605$	18.8325	0.765651
$606$	0	0
$607$	10.6225	0.431154	0.215577	−	0.976487i	$-0.430837\pi$
$607$	10.6225	0.431154	0.215577	+	0.976487i	$0.430837\pi$
$608$	−5.36941	−0.217758
$609$	0	0
$610$	27.0951	1.09705
$611$	2.23611	0.0904632
$612$	0	0
$613$	24.9485	1.00766	0.503830	−	0.863803i	$-0.331924\pi$
$613$	24.9485	1.00766	0.503830	+	0.863803i	$0.331924\pi$
$614$	1.14413	0.0461732
$615$	0	0
$616$	−12.3671	−0.498285
$617$	11.5084	0.463311	0.231656	−	0.972798i	$-0.425586\pi$
$617$	11.5084	0.463311	0.231656	+	0.972798i	$0.425586\pi$
$618$	0	0
$619$	−7.35049	−0.295441	−0.147721	−	0.989029i	$-0.547194\pi$
$619$	−7.35049	−0.295441	−0.147721	+	0.989029i	$0.547194\pi$
$620$	−43.2359	−1.73639
$621$	0	0
$622$	−10.1640	−0.407538
$623$	−25.5210	−1.02248
$624$	0	0
$625$	72.3234	2.89293
$626$	4.73388	0.189204
$627$	0	0
$628$	28.4268	1.13435
$629$	−55.7645	−2.22348
$630$	0	0
$631$	6.10638	0.243091	0.121546	−	0.992586i	$-0.461215\pi$
$631$	6.10638	0.243091	0.121546	+	0.992586i	$0.461215\pi$
$632$	24.0902	0.958257
$633$	0	0
$634$	5.41203	0.214939
$635$	21.6629	0.859667
$636$	0	0
$637$	4.03440	0.159849
$638$	−8.58889	−0.340037
$639$	0	0
$640$	−44.8609	−1.77328
$641$	23.1526	0.914472	0.457236	−	0.889345i	$-0.348839\pi$
$641$	23.1526	0.914472	0.457236	+	0.889345i	$0.348839\pi$
$642$	0	0
$643$	19.0417	0.750930	0.375465	−	0.926837i	$-0.377483\pi$
$643$	19.0417	0.750930	0.375465	+	0.926837i	$0.377483\pi$
$644$	6.45365	0.254309
$645$	0	0
$646$	−3.42377	−0.134706
$647$	−29.6694	−1.16642	−0.583212	−	0.812320i	$-0.698204\pi$
$647$	−29.6694	−1.16642	−0.583212	+	0.812320i	$0.698204\pi$
$648$	0	0
$649$	29.5467	1.15981
$650$	−10.9624	−0.429981
$651$	0	0
$652$	36.7096	1.43766
$653$	−15.2880	−0.598266	−0.299133	−	0.954211i	$-0.596697\pi$
$653$	−15.2880	−0.598266	−0.299133	+	0.954211i	$0.596697\pi$
$654$	0	0
$655$	−35.7338	−1.39624
$656$	11.5696	0.451716
$657$	0	0
$658$	−1.38676	−0.0540615
$659$	−25.7440	−1.00285	−0.501423	−	0.865203i	$-0.667190\pi$
$659$	−25.7440	−1.00285	−0.501423	+	0.865203i	$0.667190\pi$
$660$	0	0
$661$	5.62933	0.218956	0.109478	−	0.993989i	$-0.465082\pi$
$661$	5.62933	0.218956	0.109478	+	0.993989i	$0.465082\pi$
$662$	11.4321	0.444321
$663$	0	0
$664$	−10.0227	−0.388958
$665$	−15.0790	−0.584738
$666$	0	0
$667$	9.40467	0.364150
$668$	−0.232874	−0.00901017
$669$	0	0
$670$	−8.29138	−0.320324
$671$	−38.8940	−1.50149
$672$	0	0
$673$	−31.1491	−1.20071	−0.600354	−	0.799735i	$-0.704974\pi$
$673$	−31.1491	−1.20071	−0.600354	+	0.799735i	$0.704974\pi$
$674$	10.1452	0.390779
$675$	0	0
$676$	16.0664	0.617939
$677$	25.6839	0.987114	0.493557	−	0.869713i	$-0.335696\pi$
$677$	25.6839	0.987114	0.493557	+	0.869713i	$0.335696\pi$
$678$	0	0
$679$	−15.5206	−0.595625
$680$	−45.9421	−1.76180
$681$	0	0
$682$	−6.10147	−0.233637
$683$	−25.4542	−0.973977	−0.486988	−	0.873408i	$-0.661905\pi$
$683$	−25.4542	−0.973977	−0.486988	+	0.873408i	$0.661905\pi$
$684$	0	0
$685$	19.9573	0.762529
$686$	6.37041	0.243224
$687$	0	0
$688$	−28.2301	−1.07626
$689$	10.3893	0.395800
$690$	0	0
$691$	14.4737	0.550604	0.275302	−	0.961358i	$-0.411222\pi$
$691$	14.4737	0.550604	0.275302	+	0.961358i	$0.411222\pi$
$692$	13.6878	0.520332
$693$	0	0
$694$	−11.7538	−0.446169
$695$	28.9250	1.09719
$696$	0	0
$697$	26.4368	1.00137
$698$	−1.08509	−0.0410714
$699$	0	0
$700$	−69.1541	−2.61378
$701$	36.2028	1.36736	0.683680	−	0.729782i	$-0.260379\pi$
$701$	36.2028	1.36736	0.683680	+	0.729782i	$0.260379\pi$
$702$	0	0
$703$	−9.87761	−0.372541
$704$	10.2609	0.386724
$705$	0	0
$706$	2.52238	0.0949309
$707$	22.6652	0.852412
$708$	0	0
$709$	26.3530	0.989709	0.494854	−	0.868976i	$-0.335221\pi$
$709$	26.3530	0.989709	0.494854	+	0.868976i	$0.335221\pi$
$710$	−14.2576	−0.535077
$711$	0	0
$712$	−13.7731	−0.516168
$713$	6.68100	0.250205
$714$	0	0
$715$	21.9427	0.820610
$716$	2.54335	0.0950496
$717$	0	0
$718$	7.25329	0.270690
$719$	−7.55870	−0.281892	−0.140946	−	0.990017i	$-0.545014\pi$
$719$	−7.55870	−0.281892	−0.140946	+	0.990017i	$0.545014\pi$
$720$	0	0
$721$	24.8775	0.926486
$722$	7.43260	0.276613
$723$	0	0
$724$	39.9386	1.48431
$725$	−100.776	−3.74272
$726$	0	0
$727$	26.7419	0.991802	0.495901	−	0.868379i	$-0.334838\pi$
$727$	26.7419	0.991802	0.495901	+	0.868379i	$0.334838\pi$
$728$	9.89814	0.366850
$729$	0	0
$730$	−15.5789	−0.576601
$731$	−64.5067	−2.38587
$732$	0	0
$733$	−31.3605	−1.15833	−0.579163	−	0.815212i	$-0.696620\pi$
$733$	−31.3605	−1.15833	−0.579163	+	0.815212i	$0.696620\pi$
$734$	−6.02874	−0.222525
$735$	0	0
$736$	5.30591	0.195579
$737$	11.9020	0.438415
$738$	0	0
$739$	−7.37243	−0.271199	−0.135600	−	0.990764i	$-0.543296\pi$
$739$	−7.37243	−0.271199	−0.135600	+	0.990764i	$0.543296\pi$
$740$	−63.1668	−2.32206
$741$	0	0
$742$	−6.44309	−0.236533
$743$	28.0885	1.03047	0.515233	−	0.857050i	$-0.327706\pi$
$743$	28.0885	1.03047	0.515233	+	0.857050i	$0.327706\pi$
$744$	0	0
$745$	4.20441	0.154038
$746$	−11.9706	−0.438275
$747$	0	0
$748$	31.4293	1.14917
$749$	−6.17984	−0.225807
$750$	0	0
$751$	19.4411	0.709415	0.354707	−	0.934977i	$-0.384581\pi$
$751$	19.4411	0.709415	0.354707	+	0.934977i	$0.384581\pi$
$752$	3.23627	0.118015
$753$	0	0
$754$	6.87420	0.250344
$755$	41.4712	1.50929
$756$	0	0
$757$	41.5403	1.50981	0.754904	−	0.655835i	$-0.227683\pi$
$757$	41.5403	1.50981	0.754904	+	0.655835i	$0.227683\pi$
$758$	−4.82826	−0.175370
$759$	0	0
$760$	−8.13777	−0.295188
$761$	3.35453	0.121602	0.0608008	−	0.998150i	$-0.480635\pi$
$761$	3.35453	0.121602	0.0608008	+	0.998150i	$0.480635\pi$
$762$	0	0
$763$	1.91988	0.0695043
$764$	−33.1853	−1.20060
$765$	0	0
$766$	13.9110	0.502623
$767$	−23.6480	−0.853880
$768$	0	0
$769$	−44.8270	−1.61650	−0.808251	−	0.588838i	$-0.799586\pi$
$769$	−44.8270	−1.61650	−0.808251	+	0.588838i	$0.799586\pi$
$770$	−13.6082	−0.490404
$771$	0	0
$772$	11.6571	0.419550
$773$	−11.3087	−0.406744	−0.203372	−	0.979102i	$-0.565190\pi$
$773$	−11.3087	−0.406744	−0.203372	+	0.979102i	$0.565190\pi$
$774$	0	0
$775$	−71.5902	−2.57160
$776$	−8.37608	−0.300684
$777$	0	0
$778$	1.25472	0.0449837
$779$	4.68278	0.167778
$780$	0	0
$781$	20.4663	0.732340
$782$	3.38328	0.120986
$783$	0	0
$784$	5.83890	0.208532
$785$	65.6339	2.34258
$786$	0	0
$787$	−16.5410	−0.589622	−0.294811	−	0.955556i	$-0.595257\pi$
$787$	−16.5410	−0.589622	−0.294811	+	0.955556i	$0.595257\pi$
$788$	−42.6927	−1.52086
$789$	0	0
$790$	26.5077	0.943100
$791$	−30.2717	−1.07634
$792$	0	0
$793$	31.1292	1.10543
$794$	−7.24427	−0.257090
$795$	0	0
$796$	−31.1548	−1.10425
$797$	−38.9469	−1.37957	−0.689785	−	0.724014i	$-0.742295\pi$
$797$	−38.9469	−1.37957	−0.689785	+	0.724014i	$0.742295\pi$
$798$	0	0
$799$	7.39498	0.261616
$800$	−56.8555	−2.01014
$801$	0	0
$802$	0.618527	0.0218409
$803$	22.3630	0.789173
$804$	0	0
$805$	14.9007	0.525180
$806$	4.88337	0.172009
$807$	0	0
$808$	12.2319	0.430316
$809$	40.0092	1.40665	0.703325	−	0.710869i	$-0.251698\pi$
$809$	40.0092	1.40665	0.703325	+	0.710869i	$0.251698\pi$
$810$	0	0
$811$	−22.0952	−0.775867	−0.387934	−	0.921687i	$-0.626811\pi$
$811$	−22.0952	−0.775867	−0.387934	+	0.921687i	$0.626811\pi$
$812$	43.3645	1.52179
$813$	0	0
$814$	−8.91413	−0.312440
$815$	84.7579	2.96894
$816$	0	0
$817$	−11.4261	−0.399750
$818$	−2.83287	−0.0990491
$819$	0	0
$820$	29.9461	1.04576
$821$	12.3639	0.431502	0.215751	−	0.976448i	$-0.430780\pi$
$821$	12.3639	0.431502	0.215751	+	0.976448i	$0.430780\pi$
$822$	0	0
$823$	7.49704	0.261330	0.130665	−	0.991427i	$-0.458289\pi$
$823$	7.49704	0.261330	0.130665	+	0.991427i	$0.458289\pi$
$824$	13.4258	0.467709
$825$	0	0
$826$	14.6657	0.510286
$827$	55.9269	1.94477	0.972384	−	0.233385i	$-0.0749804\pi$
$827$	55.9269	1.94477	0.972384	+	0.233385i	$0.0749804\pi$
$828$	0	0
$829$	28.4117	0.986781	0.493390	−	0.869808i	$-0.335757\pi$
$829$	28.4117	0.986781	0.493390	+	0.869808i	$0.335757\pi$
$830$	−11.0285	−0.382806
$831$	0	0
$832$	−8.21245	−0.284715
$833$	13.3421	0.462276
$834$	0	0
$835$	−0.537677	−0.0186071
$836$	5.56709	0.192542
$837$	0	0
$838$	−7.26094	−0.250825
$839$	−10.0821	−0.348072	−0.174036	−	0.984739i	$-0.555681\pi$
$839$	−10.0821	−0.348072	−0.174036	+	0.984739i	$0.555681\pi$
$840$	0	0
$841$	34.1935	1.17909
$842$	12.1751	0.419583
$843$	0	0
$844$	−5.69450	−0.196013
$845$	37.0953	1.27612
$846$	0	0
$847$	−13.4182	−0.461056
$848$	15.0362	0.516344
$849$	0	0
$850$	−36.2536	−1.24349
$851$	9.76081	0.334596
$852$	0	0
$853$	21.1030	0.722554	0.361277	−	0.932459i	$-0.382341\pi$
$853$	21.1030	0.722554	0.361277	+	0.932459i	$0.382341\pi$
$854$	−19.3053	−0.660615
$855$	0	0
$856$	−3.33512	−0.113992
$857$	15.8624	0.541849	0.270924	−	0.962601i	$-0.412671\pi$
$857$	15.8624	0.541849	0.270924	+	0.962601i	$0.412671\pi$
$858$	0	0
$859$	−20.9410	−0.714497	−0.357249	−	0.934009i	$-0.616285\pi$
$859$	−20.9410	−0.714497	−0.357249	+	0.934009i	$0.616285\pi$
$860$	−73.0695	−2.49165
$861$	0	0
$862$	6.26860	0.213509
$863$	−31.8010	−1.08252	−0.541260	−	0.840856i	$-0.682052\pi$
$863$	−31.8010	−1.08252	−0.541260	+	0.840856i	$0.682052\pi$
$864$	0	0
$865$	31.6034	1.07455
$866$	8.91312	0.302880
$867$	0	0
$868$	30.8057	1.04561
$869$	−38.0508	−1.29079
$870$	0	0
$871$	−9.52588	−0.322772
$872$	1.03611	0.0350872
$873$	0	0
$874$	0.599284	0.0202711
$875$	−96.6931	−3.26882
$876$	0	0
$877$	−8.35720	−0.282203	−0.141101	−	0.989995i	$-0.545064\pi$
$877$	−8.35720	−0.282203	−0.141101	+	0.989995i	$0.545064\pi$
$878$	14.8599	0.501497
$879$	0	0
$880$	31.7572	1.07054
$881$	−41.5872	−1.40111	−0.700555	−	0.713599i	$-0.747064\pi$
$881$	−41.5872	−1.40111	−0.700555	+	0.713599i	$0.747064\pi$
$882$	0	0
$883$	13.4253	0.451796	0.225898	−	0.974151i	$-0.427468\pi$
$883$	13.4253	0.451796	0.225898	+	0.974151i	$0.427468\pi$
$884$	−25.1547	−0.846045
$885$	0	0
$886$	2.25212	0.0756614
$887$	34.5350	1.15957	0.579786	−	0.814769i	$-0.303136\pi$
$887$	34.5350	1.15957	0.579786	+	0.814769i	$0.303136\pi$
$888$	0	0
$889$	−15.4349	−0.517670
$890$	−15.1552	−0.508004
$891$	0	0
$892$	24.3685	0.815917
$893$	1.30988	0.0438334
$894$	0	0
$895$	5.87229	0.196289
$896$	31.9636	1.06783
$897$	0	0
$898$	−16.2559	−0.542467
$899$	44.8921	1.49723
$900$	0	0
$901$	34.3581	1.14464
$902$	4.22601	0.140711
$903$	0	0
$904$	−16.3369	−0.543358
$905$	92.2133	3.06527
$906$	0	0
$907$	7.68731	0.255253	0.127626	−	0.991822i	$-0.459264\pi$
$907$	7.68731	0.255253	0.127626	+	0.991822i	$0.459264\pi$
$908$	−21.1942	−0.703354
$909$	0	0
$910$	10.8914	0.361047
$911$	−28.7198	−0.951530	−0.475765	−	0.879572i	$-0.657829\pi$
$911$	−28.7198	−0.951530	−0.475765	+	0.879572i	$0.657829\pi$
$912$	0	0
$913$	15.8311	0.523932
$914$	−14.0703	−0.465405
$915$	0	0
$916$	1.88963	0.0624350
$917$	25.4605	0.840779
$918$	0	0
$919$	−23.1866	−0.764856	−0.382428	−	0.923985i	$-0.624912\pi$
$919$	−23.1866	−0.764856	−0.382428	+	0.923985i	$0.624912\pi$
$920$	8.04154	0.265122
$921$	0	0
$922$	−8.92128	−0.293807
$923$	−16.3804	−0.539167
$924$	0	0
$925$	−104.592	−3.43896
$926$	−7.17680	−0.235844
$927$	0	0
$928$	35.6524	1.17035
$929$	4.39598	0.144227	0.0721137	−	0.997396i	$-0.477026\pi$
$929$	4.39598	0.144227	0.0721137	+	0.997396i	$0.477026\pi$
$930$	0	0
$931$	2.36330	0.0774539
$932$	−38.0601	−1.24670
$933$	0	0
$934$	−4.22594	−0.138277
$935$	72.5662	2.37317
$936$	0	0
$937$	−7.42639	−0.242610	−0.121305	−	0.992615i	$-0.538708\pi$
$937$	−7.42639	−0.242610	−0.121305	+	0.992615i	$0.538708\pi$
$938$	5.90764	0.192891
$939$	0	0
$940$	8.37660	0.273215
$941$	−27.3295	−0.890916	−0.445458	−	0.895303i	$-0.646959\pi$
$941$	−27.3295	−0.890916	−0.445458	+	0.895303i	$0.646959\pi$
$942$	0	0
$943$	−4.62741	−0.150689
$944$	−34.2253	−1.11394
$945$	0	0
$946$	−10.3116	−0.335259
$947$	0.771762	0.0250789	0.0125395	−	0.999921i	$-0.496008\pi$
$947$	0.771762	0.0250789	0.0125395	+	0.999921i	$0.496008\pi$
$948$	0	0
$949$	−17.8984	−0.581008
$950$	−6.42163	−0.208345
$951$	0	0
$952$	32.7339	1.06091
$953$	−42.5845	−1.37945	−0.689723	−	0.724073i	$-0.742268\pi$
$953$	−42.5845	−1.37945	−0.689723	+	0.724073i	$0.742268\pi$
$954$	0	0
$955$	−76.6207	−2.47939
$956$	49.3612	1.59645
$957$	0	0
$958$	17.3720	0.561263
$959$	−14.2196	−0.459176
$960$	0	0
$961$	0.890941	0.0287400
$962$	7.13451	0.230026
$963$	0	0
$964$	30.4609	0.981078
$965$	26.9149	0.866421
$966$	0	0
$967$	−34.5903	−1.11235	−0.556174	−	0.831066i	$-0.687731\pi$
$967$	−34.5903	−1.11235	−0.556174	+	0.831066i	$0.687731\pi$
$968$	−7.24151	−0.232751
$969$	0	0
$970$	−9.21662	−0.295928
$971$	−31.3938	−1.00748	−0.503738	−	0.863857i	$-0.668042\pi$
$971$	−31.3938	−1.00748	−0.503738	+	0.863857i	$0.668042\pi$
$972$	0	0
$973$	−20.6092	−0.660700
$974$	−13.7331	−0.440038
$975$	0	0
$976$	45.0527	1.44210
$977$	−15.2981	−0.489431	−0.244715	−	0.969595i	$-0.578695\pi$
$977$	−15.2981	−0.489431	−0.244715	+	0.969595i	$0.578695\pi$
$978$	0	0
$979$	21.7548	0.695286
$980$	15.1131	0.482772
$981$	0	0
$982$	−5.28642	−0.168696
$983$	−49.3979	−1.57555	−0.787775	−	0.615963i	$-0.788767\pi$
$983$	−49.3979	−1.57555	−0.787775	+	0.615963i	$0.788767\pi$
$984$	0	0
$985$	−98.5721	−3.14077
$986$	22.7335	0.723983
$987$	0	0
$988$	−4.45568	−0.141754
$989$	11.2910	0.359034
$990$	0	0
$991$	53.2570	1.69176	0.845882	−	0.533370i	$-0.179075\pi$
$991$	53.2570	1.69176	0.845882	+	0.533370i	$0.179075\pi$
$992$	25.3271	0.804137
$993$	0	0
$994$	10.1586	0.322211
$995$	−71.9326	−2.28042
$996$	0	0
$997$	13.1338	0.415952	0.207976	−	0.978134i	$-0.433312\pi$
$997$	13.1338	0.415952	0.207976	+	0.978134i	$0.433312\pi$
$998$	−10.1337	−0.320778
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1341.2.a.g.1.6	✓	12
3.2	odd	2		1341.2.a.h.1.7	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1341.2.a.g.1.6	✓	12	1.1	even	1	trivial
1341.2.a.h.1.7	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)

\(f(q)\)	\(=\)	\(q-0.423108 q^{2} -1.82098 q^{4} -4.20441 q^{5} +2.99566 q^{7} +1.61669 q^{8} +O(q^{10})\)	\(q-0.423108 q^{2} -1.82098 q^{4} -4.20441 q^{5} +2.99566 q^{7} +1.61669 q^{8} +1.77892 q^{10} -2.55358 q^{11} +2.04378 q^{13} -1.26749 q^{14} +2.95793 q^{16} +6.75895 q^{17} +1.19722 q^{19} +7.65615 q^{20} +1.08044 q^{22} -1.18306 q^{23} +12.6771 q^{25} -0.864741 q^{26} -5.45504 q^{28} -7.94943 q^{29} -5.64721 q^{31} -4.48490 q^{32} -2.85977 q^{34} -12.5950 q^{35} -8.25046 q^{37} -0.506553 q^{38} -6.79722 q^{40} +3.91138 q^{41} -9.54389 q^{43} +4.65002 q^{44} +0.500563 q^{46} +1.09410 q^{47} +1.97399 q^{49} -5.36378 q^{50} -3.72169 q^{52} +5.08335 q^{53} +10.7363 q^{55} +4.84305 q^{56} +3.36347 q^{58} -11.5707 q^{59} +15.2312 q^{61} +2.38938 q^{62} -4.01826 q^{64} -8.59291 q^{65} -4.66090 q^{67} -12.3079 q^{68} +5.32905 q^{70} -8.01473 q^{71} -8.75750 q^{73} +3.49084 q^{74} -2.18011 q^{76} -7.64966 q^{77} +14.9010 q^{79} -12.4363 q^{80} -1.65494 q^{82} -6.19956 q^{83} -28.4174 q^{85} +4.03810 q^{86} -4.12834 q^{88} -8.51933 q^{89} +6.12248 q^{91} +2.15433 q^{92} -0.462923 q^{94} -5.03361 q^{95} -5.18102 q^{97} -0.835209 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 9 q^{4} - 8 q^{5} - 2 q^{7} - 9 q^{8}+O(q^{10}) \) 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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