LMFDB - Embedded newform 2151.2.a.h.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,2,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, 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("2151.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.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:	$17.1758214748$
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} - 15 x^{10} + 47 x^{9} + 75 x^{8} - 256 x^{7} - 134 x^{6} + 571 x^{5} + 23 x^{4} + \cdots - 31 $ x^12 - 3x^11 - 15x^10 + 47x^9 + 75x^8 - 256x^7 - 134x^6 + 571x^5 + 23x^4 - 479x^3 + 129x^2 + 88*x - 31
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-1.40336$ of defining polynomial
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.40336 q^{2} -0.0305942 q^{4} +0.939648 q^{5} +3.20775 q^{7} -2.84964 q^{8} +O(q^{10})$	\(q+1.40336 q^{2} -0.0305942 q^{4} +0.939648 q^{5} +3.20775 q^{7} -2.84964 q^{8} +1.31866 q^{10} +2.93519 q^{11} -0.838130 q^{13} +4.50162 q^{14} -3.93788 q^{16} +1.31188 q^{17} +1.38487 q^{19} -0.0287478 q^{20} +4.11911 q^{22} +5.00490 q^{23} -4.11706 q^{25} -1.17619 q^{26} -0.0981386 q^{28} -4.89770 q^{29} +10.1686 q^{31} +0.173052 q^{32} +1.84104 q^{34} +3.01416 q^{35} +5.39200 q^{37} +1.94347 q^{38} -2.67766 q^{40} -7.45582 q^{41} +10.8774 q^{43} -0.0897996 q^{44} +7.02366 q^{46} +11.3243 q^{47} +3.28968 q^{49} -5.77770 q^{50} +0.0256419 q^{52} -13.5400 q^{53} +2.75804 q^{55} -9.14096 q^{56} -6.87321 q^{58} -7.10285 q^{59} -10.2728 q^{61} +14.2702 q^{62} +8.11860 q^{64} -0.787547 q^{65} +8.90850 q^{67} -0.0401360 q^{68} +4.22993 q^{70} +2.66967 q^{71} +11.8616 q^{73} +7.56689 q^{74} -0.0423690 q^{76} +9.41535 q^{77} -4.33447 q^{79} -3.70022 q^{80} -10.4632 q^{82} +5.27371 q^{83} +1.23271 q^{85} +15.2648 q^{86} -8.36424 q^{88} +12.0397 q^{89} -2.68851 q^{91} -0.153121 q^{92} +15.8921 q^{94} +1.30129 q^{95} -3.41637 q^{97} +4.61659 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 15 q^{4} + q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 + 15 * q^4 + q^5 + 11 * q^7 - 9 * q^8	$ 12 q - 3 q^{2} + 15 q^{4} + q^{5} + 11 q^{7} - 9 q^{8} - 15 q^{11} + 7 q^{13} + 6 q^{14} + 21 q^{16} + 3 q^{17} + 10 q^{19} + 4 q^{20} + 23 q^{22} - 20 q^{23} + 19 q^{25} + 10 q^{26} + 34 q^{28} - 2 q^{29} + 10 q^{31} - 26 q^{32} + 12 q^{34} - 7 q^{35} + 30 q^{37} + 3 q^{38} + 25 q^{40} + 28 q^{41} + 48 q^{43} - 25 q^{44} + 22 q^{46} - 13 q^{47} + 19 q^{49} - 12 q^{50} + 24 q^{52} + 2 q^{53} + 8 q^{55} + 7 q^{56} + 42 q^{58} + 14 q^{59} + 14 q^{61} - 8 q^{62} + 9 q^{64} + 35 q^{65} + 52 q^{67} - 3 q^{68} - 33 q^{70} + 7 q^{71} + 14 q^{73} + 13 q^{74} - 12 q^{76} + 6 q^{77} + 15 q^{79} + 8 q^{80} - 61 q^{82} - 29 q^{83} + 8 q^{85} + 9 q^{86} + 11 q^{88} + 71 q^{89} + 13 q^{91} - 2 q^{92} - 22 q^{94} - 2 q^{95} - 2 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 + 15 * q^4 + q^5 + 11 * q^7 - 9 * q^8 - 15 * q^11 + 7 * q^13 + 6 * q^14 + 21 * q^16 + 3 * q^17 + 10 * q^19 + 4 * q^20 + 23 * q^22 - 20 * q^23 + 19 * q^25 + 10 * q^26 + 34 * q^28 - 2 * q^29 + 10 * q^31 - 26 * q^32 + 12 * q^34 - 7 * q^35 + 30 * q^37 + 3 * q^38 + 25 * q^40 + 28 * q^41 + 48 * q^43 - 25 * q^44 + 22 * q^46 - 13 * q^47 + 19 * q^49 - 12 * q^50 + 24 * q^52 + 2 * q^53 + 8 * q^55 + 7 * q^56 + 42 * q^58 + 14 * q^59 + 14 * q^61 - 8 * q^62 + 9 * q^64 + 35 * q^65 + 52 * q^67 - 3 * q^68 - 33 * q^70 + 7 * q^71 + 14 * q^73 + 13 * q^74 - 12 * q^76 + 6 * q^77 + 15 * q^79 + 8 * q^80 - 61 * q^82 - 29 * q^83 + 8 * q^85 + 9 * q^86 + 11 * q^88 + 71 * q^89 + 13 * q^91 - 2 * q^92 - 22 * q^94 - 2 * q^95 - 2 * 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 $

$2$	1.40336	0.992322	0.496161	−	0.868231i	$-0.334743\pi$
$2$	1.40336	0.992322	0.496161	+	0.868231i	$0.334743\pi$
$3$	0	0
$3$	0	0
$4$	−0.0305942	−0.0152971
$5$	0.939648	0.420223	0.210112	−	0.977677i	$-0.432617\pi$
$5$	0.939648	0.420223	0.210112	+	0.977677i	$0.432617\pi$
$6$	0	0
$7$	3.20775	1.21242	0.606208	−	0.795306i	$-0.292690\pi$
$7$	3.20775	1.21242	0.606208	+	0.795306i	$0.292690\pi$
$8$	−2.84964	−1.00750
$9$	0	0
$10$	1.31866	0.416997
$11$	2.93519	0.884992	0.442496	−	0.896771i	$-0.354093\pi$
$11$	2.93519	0.884992	0.442496	+	0.896771i	$0.354093\pi$
$12$	0	0
$13$	−0.838130	−0.232455	−0.116228	−	0.993223i	$-0.537080\pi$
$13$	−0.838130	−0.232455	−0.116228	+	0.993223i	$0.537080\pi$
$14$	4.50162	1.20311
$15$	0	0
$16$	−3.93788	−0.984469
$17$	1.31188	0.318178	0.159089	−	0.987264i	$-0.449144\pi$
$17$	1.31188	0.318178	0.159089	+	0.987264i	$0.449144\pi$
$18$	0	0
$19$	1.38487	0.317711	0.158856	−	0.987302i	$-0.449220\pi$
$19$	1.38487	0.317711	0.158856	+	0.987302i	$0.449220\pi$
$20$	−0.0287478	−0.00642820
$21$	0	0
$22$	4.11911	0.878197
$23$	5.00490	1.04359	0.521797	−	0.853069i	$-0.325262\pi$
$23$	5.00490	1.04359	0.521797	+	0.853069i	$0.325262\pi$
$24$	0	0
$25$	−4.11706	−0.823412
$26$	−1.17619	−0.230671
$27$	0	0
$28$	−0.0981386	−0.0185465
$29$	−4.89770	−0.909480	−0.454740	−	0.890624i	$-0.650268\pi$
$29$	−4.89770	−0.909480	−0.454740	+	0.890624i	$0.650268\pi$
$30$	0	0
$31$	10.1686	1.82634	0.913171	−	0.407577i	$-0.133626\pi$
$31$	10.1686	1.82634	0.913171	+	0.407577i	$0.133626\pi$
$32$	0.173052	0.0305915
$33$	0	0
$34$	1.84104	0.315735
$35$	3.01416	0.509486
$36$	0	0
$37$	5.39200	0.886440	0.443220	−	0.896413i	$-0.353836\pi$
$37$	5.39200	0.886440	0.443220	+	0.896413i	$0.353836\pi$
$38$	1.94347	0.315272
$39$	0	0
$40$	−2.67766	−0.423376
$41$	−7.45582	−1.16440	−0.582202	−	0.813044i	$-0.697809\pi$
$41$	−7.45582	−1.16440	−0.582202	+	0.813044i	$0.697809\pi$
$42$	0	0
$43$	10.8774	1.65878	0.829391	−	0.558668i	$-0.188687\pi$
$43$	10.8774	1.65878	0.829391	+	0.558668i	$0.188687\pi$
$44$	−0.0897996	−0.0135378
$45$	0	0
$46$	7.02366	1.03558
$47$	11.3243	1.65182	0.825912	−	0.563798i	$-0.190661\pi$
$47$	11.3243	1.65182	0.825912	+	0.563798i	$0.190661\pi$
$48$	0	0
$49$	3.28968	0.469954
$50$	−5.77770	−0.817090
$51$	0	0
$52$	0.0256419	0.00355589
$53$	−13.5400	−1.85986	−0.929932	−	0.367733i	$-0.880134\pi$
$53$	−13.5400	−1.85986	−0.929932	+	0.367733i	$0.880134\pi$
$54$	0	0
$55$	2.75804	0.371894
$56$	−9.14096	−1.22151
$57$	0	0
$58$	−6.87321	−0.902497
$59$	−7.10285	−0.924712	−0.462356	−	0.886694i	$-0.652996\pi$
$59$	−7.10285	−0.924712	−0.462356	+	0.886694i	$0.652996\pi$
$60$	0	0
$61$	−10.2728	−1.31530	−0.657648	−	0.753326i	$-0.728448\pi$
$61$	−10.2728	−1.31530	−0.657648	+	0.753326i	$0.728448\pi$
$62$	14.2702	1.81232
$63$	0	0
$64$	8.11860	1.01483
$65$	−0.787547	−0.0976832
$66$	0	0
$67$	8.90850	1.08835	0.544173	−	0.838973i	$-0.316843\pi$
$67$	8.90850	1.08835	0.544173	+	0.838973i	$0.316843\pi$
$68$	−0.0401360	−0.00486720
$69$	0	0
$70$	4.22993	0.505574
$71$	2.66967	0.316831	0.158416	−	0.987373i	$-0.449361\pi$
$71$	2.66967	0.316831	0.158416	+	0.987373i	$0.449361\pi$
$72$	0	0
$73$	11.8616	1.38830	0.694148	−	0.719832i	$-0.255781\pi$
$73$	11.8616	1.38830	0.694148	+	0.719832i	$0.255781\pi$
$74$	7.56689	0.879634
$75$	0	0
$76$	−0.0423690	−0.00486006
$77$	9.41535	1.07298
$78$	0	0
$79$	−4.33447	−0.487666	−0.243833	−	0.969817i	$-0.578405\pi$
$79$	−4.33447	−0.487666	−0.243833	+	0.969817i	$0.578405\pi$
$80$	−3.70022	−0.413697
$81$	0	0
$82$	−10.4632	−1.15546
$83$	5.27371	0.578865	0.289433	−	0.957198i	$-0.406533\pi$
$83$	5.27371	0.578865	0.289433	+	0.957198i	$0.406533\pi$
$84$	0	0
$85$	1.23271	0.133706
$86$	15.2648	1.64605
$87$	0	0
$88$	−8.36424	−0.891631
$89$	12.0397	1.27621	0.638105	−	0.769949i	$-0.279719\pi$
$89$	12.0397	1.27621	0.638105	+	0.769949i	$0.279719\pi$
$90$	0	0
$91$	−2.68851	−0.281833
$92$	−0.153121	−0.0159640
$93$	0	0
$94$	15.8921	1.63914
$95$	1.30129	0.133510
$96$	0	0
$97$	−3.41637	−0.346880	−0.173440	−	0.984844i	$-0.555488\pi$
$97$	−3.41637	−0.346880	−0.173440	+	0.984844i	$0.555488\pi$
$98$	4.61659	0.466346
$99$	0	0
$100$	0.125958	0.0125958
$101$	−18.2139	−1.81235	−0.906175	−	0.422904i	$-0.861011\pi$
$101$	−18.2139	−1.81235	−0.906175	+	0.422904i	$0.861011\pi$
$102$	0	0
$103$	14.2114	1.40029	0.700144	−	0.714002i	$-0.253119\pi$
$103$	14.2114	1.40029	0.700144	+	0.714002i	$0.253119\pi$
$104$	2.38837	0.234199
$105$	0	0
$106$	−19.0014	−1.84558
$107$	−14.8243	−1.43312	−0.716561	−	0.697524i	$-0.754285\pi$
$107$	−14.8243	−1.43312	−0.716561	+	0.697524i	$0.754285\pi$
$108$	0	0
$109$	12.2953	1.17768	0.588839	−	0.808251i	$-0.299585\pi$
$109$	12.2953	1.17768	0.588839	+	0.808251i	$0.299585\pi$
$110$	3.87051	0.369039
$111$	0	0
$112$	−12.6317	−1.19359
$113$	10.3791	0.976383	0.488192	−	0.872736i	$-0.337657\pi$
$113$	10.3791	0.976383	0.488192	+	0.872736i	$0.337657\pi$
$114$	0	0
$115$	4.70285	0.438543
$116$	0.149841	0.0139124
$117$	0	0
$118$	−9.96782	−0.917612
$119$	4.20820	0.385765
$120$	0	0
$121$	−2.38469	−0.216790
$122$	−14.4164	−1.30520
$123$	0	0
$124$	−0.311101	−0.0279377
$125$	−8.56683	−0.766240
$126$	0	0
$127$	20.7026	1.83706	0.918528	−	0.395355i	$-0.129378\pi$
$127$	20.7026	1.83706	0.918528	+	0.395355i	$0.129378\pi$
$128$	11.0472	0.976442
$129$	0	0
$130$	−1.10521	−0.0969331
$131$	−20.9010	−1.82613	−0.913066	−	0.407812i	$-0.866292\pi$
$131$	−20.9010	−1.82613	−0.913066	+	0.407812i	$0.866292\pi$
$132$	0	0
$133$	4.44232	0.385198
$134$	12.5018	1.07999
$135$	0	0
$136$	−3.73840	−0.320565
$137$	9.83824	0.840537	0.420269	−	0.907400i	$-0.361936\pi$
$137$	9.83824	0.840537	0.420269	+	0.907400i	$0.361936\pi$
$138$	0	0
$139$	−8.34335	−0.707674	−0.353837	−	0.935307i	$-0.615123\pi$
$139$	−8.34335	−0.707674	−0.353837	+	0.935307i	$0.615123\pi$
$140$	−0.0922158	−0.00779365
$141$	0	0
$142$	3.74649	0.314398
$143$	−2.46007	−0.205721
$144$	0	0
$145$	−4.60211	−0.382185
$146$	16.6461	1.37764
$147$	0	0
$148$	−0.164964	−0.0135600
$149$	10.0225	0.821072	0.410536	−	0.911844i	$-0.365342\pi$
$149$	10.0225	0.821072	0.410536	+	0.911844i	$0.365342\pi$
$150$	0	0
$151$	1.37301	0.111734	0.0558670	−	0.998438i	$-0.482208\pi$
$151$	1.37301	0.111734	0.0558670	+	0.998438i	$0.482208\pi$
$152$	−3.94639	−0.320094
$153$	0	0
$154$	13.2131	1.06474
$155$	9.55494	0.767471
$156$	0	0
$157$	2.17348	0.173463	0.0867315	−	0.996232i	$-0.472358\pi$
$157$	2.17348	0.173463	0.0867315	+	0.996232i	$0.472358\pi$
$158$	−6.08280	−0.483921
$159$	0	0
$160$	0.162608	0.0128553
$161$	16.0545	1.26527
$162$	0	0
$163$	−8.74108	−0.684654	−0.342327	−	0.939581i	$-0.611215\pi$
$163$	−8.74108	−0.684654	−0.342327	+	0.939581i	$0.611215\pi$
$164$	0.228105	0.0178120
$165$	0	0
$166$	7.40089	0.574421
$167$	−17.5772	−1.36016	−0.680081	−	0.733137i	$-0.738056\pi$
$167$	−17.5772	−1.36016	−0.680081	+	0.733137i	$0.738056\pi$
$168$	0	0
$169$	−12.2975	−0.945965
$170$	1.72993	0.132679
$171$	0	0
$172$	−0.332784	−0.0253746
$173$	−1.46876	−0.111668	−0.0558338	−	0.998440i	$-0.517782\pi$
$173$	−1.46876	−0.111668	−0.0558338	+	0.998440i	$0.517782\pi$
$174$	0	0
$175$	−13.2065	−0.998319
$176$	−11.5584	−0.871247
$177$	0	0
$178$	16.8960	1.26641
$179$	3.41431	0.255197	0.127599	−	0.991826i	$-0.459273\pi$
$179$	3.41431	0.255197	0.127599	+	0.991826i	$0.459273\pi$
$180$	0	0
$181$	10.0143	0.744354	0.372177	−	0.928162i	$-0.378612\pi$
$181$	10.0143	0.744354	0.372177	+	0.928162i	$0.378612\pi$
$182$	−3.77294	−0.279669
$183$	0	0
$184$	−14.2622	−1.05142
$185$	5.06658	0.372503
$186$	0	0
$187$	3.85062	0.281585
$188$	−0.346459	−0.0252681
$189$	0	0
$190$	1.82617	0.132485
$191$	−9.87283	−0.714373	−0.357187	−	0.934033i	$-0.616264\pi$
$191$	−9.87283	−0.714373	−0.357187	+	0.934033i	$0.616264\pi$
$192$	0	0
$193$	−26.3269	−1.89505	−0.947525	−	0.319682i	$-0.896424\pi$
$193$	−26.3269	−1.89505	−0.947525	+	0.319682i	$0.896424\pi$
$194$	−4.79438	−0.344216
$195$	0	0
$196$	−0.100645	−0.00718894
$197$	2.18190	0.155454	0.0777269	−	0.996975i	$-0.475234\pi$
$197$	2.18190	0.155454	0.0777269	+	0.996975i	$0.475234\pi$
$198$	0	0
$199$	−10.8159	−0.766718	−0.383359	−	0.923599i	$-0.625233\pi$
$199$	−10.8159	−0.766718	−0.383359	+	0.923599i	$0.625233\pi$
$200$	11.7322	0.829589
$201$	0	0
$202$	−25.5605	−1.79843
$203$	−15.7106	−1.10267
$204$	0	0
$205$	−7.00585	−0.489310
$206$	19.9436	1.38954
$207$	0	0
$208$	3.30045	0.228845
$209$	4.06485	0.281172
$210$	0	0
$211$	−13.6548	−0.940036	−0.470018	−	0.882657i	$-0.655753\pi$
$211$	−13.6548	−0.940036	−0.470018	+	0.882657i	$0.655753\pi$
$212$	0.414246	0.0284505
$213$	0	0
$214$	−20.8038	−1.42212
$215$	10.2209	0.697059
$216$	0	0
$217$	32.6185	2.21429
$218$	17.2547	1.16864
$219$	0	0
$220$	−0.0843800	−0.00568890
$221$	−1.09953	−0.0739622
$222$	0	0
$223$	−2.54431	−0.170379	−0.0851897	−	0.996365i	$-0.527150\pi$
$223$	−2.54431	−0.170379	−0.0851897	+	0.996365i	$0.527150\pi$
$224$	0.555107	0.0370897
$225$	0	0
$226$	14.5656	0.968887
$227$	−5.41207	−0.359212	−0.179606	−	0.983739i	$-0.557482\pi$
$227$	−5.41207	−0.359212	−0.179606	+	0.983739i	$0.557482\pi$
$228$	0	0
$229$	−14.2408	−0.941057	−0.470529	−	0.882385i	$-0.655937\pi$
$229$	−14.2408	−0.941057	−0.470529	+	0.882385i	$0.655937\pi$
$230$	6.59977	0.435176
$231$	0	0
$232$	13.9567	0.916302
$233$	−19.0118	−1.24550	−0.622752	−	0.782419i	$-0.713986\pi$
$233$	−19.0118	−1.24550	−0.622752	+	0.782419i	$0.713986\pi$
$234$	0	0
$235$	10.6409	0.694135
$236$	0.217306	0.0141454
$237$	0	0
$238$	5.90559	0.382803
$239$	−1.00000	−0.0646846
$239$	−1.00000	−0.0646846
$240$	0	0
$241$	−1.25112	−0.0805917	−0.0402959	−	0.999188i	$-0.512830\pi$
$241$	−1.25112	−0.0805917	−0.0402959	+	0.999188i	$0.512830\pi$
$242$	−3.34656	−0.215125
$243$	0	0
$244$	0.314288	0.0201202
$245$	3.09114	0.197486
$246$	0	0
$247$	−1.16070	−0.0738537
$248$	−28.9770	−1.84004
$249$	0	0
$250$	−12.0223	−0.760357
$251$	−0.644421	−0.0406755	−0.0203377	−	0.999793i	$-0.506474\pi$
$251$	−0.644421	−0.0406755	−0.0203377	+	0.999793i	$0.506474\pi$
$252$	0	0
$253$	14.6903	0.923573
$254$	29.0531	1.82295
$255$	0	0
$256$	−0.734088	−0.0458805
$257$	5.61298	0.350128	0.175064	−	0.984557i	$-0.443987\pi$
$257$	5.61298	0.350128	0.175064	+	0.984557i	$0.443987\pi$
$258$	0	0
$259$	17.2962	1.07473
$260$	0.0240944	0.00149427
$261$	0	0
$262$	−29.3316	−1.81211
$263$	−18.5914	−1.14640	−0.573199	−	0.819416i	$-0.694298\pi$
$263$	−18.5914	−1.14640	−0.573199	+	0.819416i	$0.694298\pi$
$264$	0	0
$265$	−12.7228	−0.781558
$266$	6.23416	0.382241
$267$	0	0
$268$	−0.272548	−0.0166485
$269$	16.0975	0.981479	0.490740	−	0.871306i	$-0.336727\pi$
$269$	16.0975	0.981479	0.490740	+	0.871306i	$0.336727\pi$
$270$	0	0
$271$	−21.0179	−1.27675	−0.638374	−	0.769726i	$-0.720393\pi$
$271$	−21.0179	−1.27675	−0.638374	+	0.769726i	$0.720393\pi$
$272$	−5.16603	−0.313237
$273$	0	0
$274$	13.8065	0.834084
$275$	−12.0843	−0.728713
$276$	0	0
$277$	8.98776	0.540022	0.270011	−	0.962857i	$-0.412973\pi$
$277$	8.98776	0.540022	0.270011	+	0.962857i	$0.412973\pi$
$278$	−11.7087	−0.702240
$279$	0	0
$280$	−8.58928	−0.513308
$281$	−12.1245	−0.723287	−0.361643	−	0.932317i	$-0.617784\pi$
$281$	−12.1245	−0.723287	−0.361643	+	0.932317i	$0.617784\pi$
$282$	0	0
$283$	−3.12036	−0.185486	−0.0927430	−	0.995690i	$-0.529564\pi$
$283$	−3.12036	−0.185486	−0.0927430	+	0.995690i	$0.529564\pi$
$284$	−0.0816763	−0.00484660
$285$	0	0
$286$	−3.45235	−0.204142
$287$	−23.9164	−1.41174
$288$	0	0
$289$	−15.2790	−0.898763
$290$	−6.45840	−0.379250
$291$	0	0
$292$	−0.362896	−0.0212369
$293$	−19.3663	−1.13139	−0.565695	−	0.824614i	$-0.691392\pi$
$293$	−19.3663	−1.13139	−0.565695	+	0.824614i	$0.691392\pi$
$294$	0	0
$295$	−6.67418	−0.388586
$296$	−15.3653	−0.893089
$297$	0	0
$298$	14.0651	0.814768
$299$	−4.19476	−0.242589
$300$	0	0
$301$	34.8919	2.01114
$302$	1.92682	0.110876
$303$	0	0
$304$	−5.45345	−0.312777
$305$	−9.65280	−0.552718
$306$	0	0
$307$	29.9019	1.70659	0.853296	−	0.521427i	$-0.174600\pi$
$307$	29.9019	1.70659	0.853296	+	0.521427i	$0.174600\pi$
$308$	−0.288055	−0.0164135
$309$	0	0
$310$	13.4090	0.761579
$311$	26.9938	1.53068	0.765338	−	0.643629i	$-0.222572\pi$
$311$	26.9938	1.53068	0.765338	+	0.643629i	$0.222572\pi$
$312$	0	0
$313$	−8.22182	−0.464725	−0.232362	−	0.972629i	$-0.574646\pi$
$313$	−8.22182	−0.464725	−0.232362	+	0.972629i	$0.574646\pi$
$314$	3.05017	0.172131
$315$	0	0
$316$	0.132610	0.00745987
$317$	−14.4734	−0.812907	−0.406454	−	0.913671i	$-0.633235\pi$
$317$	−14.4734	−0.812907	−0.406454	+	0.913671i	$0.633235\pi$
$318$	0	0
$319$	−14.3757	−0.804882
$320$	7.62863	0.426453
$321$	0	0
$322$	22.5302	1.25556
$323$	1.81679	0.101089
$324$	0	0
$325$	3.45063	0.191407
$326$	−12.2668	−0.679397
$327$	0	0
$328$	21.2464	1.17314
$329$	36.3257	2.00270
$330$	0	0
$331$	14.7969	0.813312	0.406656	−	0.913581i	$-0.366695\pi$
$331$	14.7969	0.813312	0.406656	+	0.913581i	$0.366695\pi$
$332$	−0.161345	−0.00885496
$333$	0	0
$334$	−24.6670	−1.34972
$335$	8.37085	0.457348
$336$	0	0
$337$	8.58079	0.467425	0.233713	−	0.972306i	$-0.424913\pi$
$337$	8.58079	0.467425	0.233713	+	0.972306i	$0.424913\pi$
$338$	−17.2578	−0.938701
$339$	0	0
$340$	−0.0377137	−0.00204531
$341$	29.8468	1.61630
$342$	0	0
$343$	−11.9018	−0.642636
$344$	−30.9966	−1.67123
$345$	0	0
$346$	−2.06119	−0.110810
$347$	−11.3694	−0.610341	−0.305171	−	0.952298i	$-0.598714\pi$
$347$	−11.3694	−0.610341	−0.305171	+	0.952298i	$0.598714\pi$
$348$	0	0
$349$	−7.02566	−0.376075	−0.188037	−	0.982162i	$-0.560213\pi$
$349$	−7.02566	−0.376075	−0.188037	+	0.982162i	$0.560213\pi$
$350$	−18.5334	−0.990654
$351$	0	0
$352$	0.507939	0.0270732
$353$	−16.2597	−0.865416	−0.432708	−	0.901534i	$-0.642442\pi$
$353$	−16.2597	−0.865416	−0.432708	+	0.901534i	$0.642442\pi$
$354$	0	0
$355$	2.50855	0.133140
$356$	−0.368346	−0.0195223
$357$	0	0
$358$	4.79148	0.253238
$359$	−17.7636	−0.937528	−0.468764	−	0.883323i	$-0.655301\pi$
$359$	−17.7636	−0.937528	−0.468764	+	0.883323i	$0.655301\pi$
$360$	0	0
$361$	−17.0821	−0.899060
$362$	14.0536	0.738638
$363$	0	0
$364$	0.0822529	0.00431122
$365$	11.1457	0.583394
$366$	0	0
$367$	7.42835	0.387756	0.193878	−	0.981026i	$-0.437893\pi$
$367$	7.42835	0.387756	0.193878	+	0.981026i	$0.437893\pi$
$368$	−19.7087	−1.02739
$369$	0	0
$370$	7.11022	0.369643
$371$	−43.4330	−2.25493
$372$	0	0
$373$	−22.6175	−1.17109	−0.585544	−	0.810641i	$-0.699119\pi$
$373$	−22.6175	−1.17109	−0.585544	+	0.810641i	$0.699119\pi$
$374$	5.40379	0.279423
$375$	0	0
$376$	−32.2703	−1.66422
$377$	4.10491	0.211413
$378$	0	0
$379$	16.2148	0.832898	0.416449	−	0.909159i	$-0.363274\pi$
$379$	16.2148	0.832898	0.416449	+	0.909159i	$0.363274\pi$
$380$	−0.0398120	−0.00204231
$381$	0	0
$382$	−13.8551	−0.708888
$383$	−1.76726	−0.0903025	−0.0451513	−	0.998980i	$-0.514377\pi$
$383$	−1.76726	−0.0903025	−0.0451513	+	0.998980i	$0.514377\pi$
$384$	0	0
$385$	8.84711	0.450891
$386$	−36.9460	−1.88050
$387$	0	0
$388$	0.104521	0.00530625
$389$	8.89925	0.451210	0.225605	−	0.974219i	$-0.427564\pi$
$389$	8.89925	0.451210	0.225605	+	0.974219i	$0.427564\pi$
$390$	0	0
$391$	6.56585	0.332049
$392$	−9.37442	−0.473480
$393$	0	0
$394$	3.06198	0.154260
$395$	−4.07287	−0.204928
$396$	0	0
$397$	−35.1550	−1.76438	−0.882190	−	0.470894i	$-0.843932\pi$
$397$	−35.1550	−1.76438	−0.882190	+	0.470894i	$0.843932\pi$
$398$	−15.1785	−0.760831
$399$	0	0
$400$	16.2125	0.810624
$401$	−17.2452	−0.861182	−0.430591	−	0.902547i	$-0.641695\pi$
$401$	−17.2452	−0.861182	−0.430591	+	0.902547i	$0.641695\pi$
$402$	0	0
$403$	−8.52264	−0.424543
$404$	0.557239	0.0277237
$405$	0	0
$406$	−22.0476	−1.09420
$407$	15.8265	0.784492
$408$	0	0
$409$	−34.6072	−1.71121	−0.855607	−	0.517626i	$-0.826816\pi$
$409$	−34.6072	−1.71121	−0.855607	+	0.517626i	$0.826816\pi$
$410$	−9.83169	−0.485553
$411$	0	0
$412$	−0.434785	−0.0214203
$413$	−22.7842	−1.12114
$414$	0	0
$415$	4.95543	0.243253
$416$	−0.145040	−0.00711116
$417$	0	0
$418$	5.70443	0.279013
$419$	25.9962	1.27000	0.634998	−	0.772514i	$-0.281001\pi$
$419$	25.9962	1.27000	0.634998	+	0.772514i	$0.281001\pi$
$420$	0	0
$421$	−30.1409	−1.46898	−0.734490	−	0.678620i	$-0.762578\pi$
$421$	−30.1409	−1.46898	−0.734490	+	0.678620i	$0.762578\pi$
$422$	−19.1625	−0.932818
$423$	0	0
$424$	38.5842	1.87382
$425$	−5.40110	−0.261992
$426$	0	0
$427$	−32.9526	−1.59469
$428$	0.453538	0.0219226
$429$	0	0
$430$	14.3435	0.691707
$431$	−5.14534	−0.247842	−0.123921	−	0.992292i	$-0.539547\pi$
$431$	−5.14534	−0.247842	−0.123921	+	0.992292i	$0.539547\pi$
$432$	0	0
$433$	−7.62316	−0.366346	−0.183173	−	0.983081i	$-0.558637\pi$
$433$	−7.62316	−0.366346	−0.183173	+	0.983081i	$0.558637\pi$
$434$	45.7753	2.19729
$435$	0	0
$436$	−0.376165	−0.0180150
$437$	6.93115	0.331562
$438$	0	0
$439$	16.4721	0.786172	0.393086	−	0.919502i	$-0.371408\pi$
$439$	16.4721	0.786172	0.393086	+	0.919502i	$0.371408\pi$
$440$	−7.85944	−0.374684
$441$	0	0
$442$	−1.54303	−0.0733944
$443$	17.3698	0.825263	0.412632	−	0.910898i	$-0.364610\pi$
$443$	17.3698	0.825263	0.412632	+	0.910898i	$0.364610\pi$
$444$	0	0
$445$	11.3131	0.536293
$446$	−3.57057	−0.169071
$447$	0	0
$448$	26.0425	1.23039
$449$	13.5485	0.639395	0.319698	−	0.947520i	$-0.396419\pi$
$449$	13.5485	0.639395	0.319698	+	0.947520i	$0.396419\pi$
$450$	0	0
$451$	−21.8842	−1.03049
$452$	−0.317540	−0.0149358
$453$	0	0
$454$	−7.59506	−0.356454
$455$	−2.52626	−0.118433
$456$	0	0
$457$	12.2693	0.573932	0.286966	−	0.957941i	$-0.407353\pi$
$457$	12.2693	0.573932	0.286966	+	0.957941i	$0.407353\pi$
$458$	−19.9849	−0.933832
$459$	0	0
$460$	−0.143880	−0.00670843
$461$	4.98122	0.231999	0.115999	−	0.993249i	$-0.462993\pi$
$461$	4.98122	0.231999	0.115999	+	0.993249i	$0.462993\pi$
$462$	0	0
$463$	22.6666	1.05341	0.526703	−	0.850050i	$-0.323428\pi$
$463$	22.6666	1.05341	0.526703	+	0.850050i	$0.323428\pi$
$464$	19.2865	0.895354
$465$	0	0
$466$	−26.6803	−1.23594
$467$	−12.2896	−0.568696	−0.284348	−	0.958721i	$-0.591777\pi$
$467$	−12.2896	−0.568696	−0.284348	+	0.958721i	$0.591777\pi$
$468$	0	0
$469$	28.5763	1.31953
$470$	14.9330	0.688806
$471$	0	0
$472$	20.2406	0.931649
$473$	31.9271	1.46801
$474$	0	0
$475$	−5.70160	−0.261607
$476$	−0.128746	−0.00590108
$477$	0	0
$478$	−1.40336	−0.0641880
$479$	24.3903	1.11442	0.557211	−	0.830371i	$-0.311872\pi$
$479$	24.3903	1.11442	0.557211	+	0.830371i	$0.311872\pi$
$480$	0	0
$481$	−4.51920	−0.206058
$482$	−1.75577	−0.0799729
$483$	0	0
$484$	0.0729576	0.00331625
$485$	−3.21018	−0.145767
$486$	0	0
$487$	28.1307	1.27472	0.637362	−	0.770564i	$-0.280025\pi$
$487$	28.1307	1.27472	0.637362	+	0.770564i	$0.280025\pi$
$488$	29.2738	1.32516
$489$	0	0
$490$	4.33797	0.195969
$491$	−34.2017	−1.54350	−0.771751	−	0.635925i	$-0.780619\pi$
$491$	−34.2017	−1.54350	−0.771751	+	0.635925i	$0.780619\pi$
$492$	0	0
$493$	−6.42520	−0.289377
$494$	−1.62888	−0.0732866
$495$	0	0
$496$	−40.0428	−1.79798
$497$	8.56363	0.384131
$498$	0	0
$499$	−17.4803	−0.782525	−0.391262	−	0.920279i	$-0.627962\pi$
$499$	−17.4803	−0.782525	−0.391262	+	0.920279i	$0.627962\pi$
$500$	0.262095	0.0117213
$501$	0	0
$502$	−0.904351	−0.0403632
$503$	8.80957	0.392800	0.196400	−	0.980524i	$-0.437075\pi$
$503$	8.80957	0.392800	0.196400	+	0.980524i	$0.437075\pi$
$504$	0	0
$505$	−17.1146	−0.761591
$506$	20.6157	0.916481
$507$	0	0
$508$	−0.633379	−0.0281016
$509$	12.5598	0.556705	0.278352	−	0.960479i	$-0.410212\pi$
$509$	12.5598	0.556705	0.278352	+	0.960479i	$0.410212\pi$
$510$	0	0
$511$	38.0491	1.68319
$512$	−23.1246	−1.02197
$513$	0	0
$514$	7.87701	0.347440
$515$	13.3537	0.588433
$516$	0	0
$517$	33.2390	1.46185
$518$	24.2727	1.06648
$519$	0	0
$520$	2.24423	0.0984159
$521$	17.2622	0.756272	0.378136	−	0.925750i	$-0.376565\pi$
$521$	17.2622	0.756272	0.378136	+	0.925750i	$0.376565\pi$
$522$	0	0
$523$	26.5415	1.16058	0.580289	−	0.814411i	$-0.302940\pi$
$523$	26.5415	1.16058	0.580289	+	0.814411i	$0.302940\pi$
$524$	0.639450	0.0279345
$525$	0	0
$526$	−26.0904	−1.13760
$527$	13.3401	0.581102
$528$	0	0
$529$	2.04907	0.0890898
$530$	−17.8547	−0.775557
$531$	0	0
$532$	−0.135909	−0.00589242
$533$	6.24895	0.270672
$534$	0	0
$535$	−13.9296	−0.602231
$536$	−25.3861	−1.09651
$537$	0	0
$538$	22.5905	0.973944
$539$	9.65582	0.415906
$540$	0	0
$541$	7.80543	0.335582	0.167791	−	0.985823i	$-0.446337\pi$
$541$	7.80543	0.335582	0.167791	+	0.985823i	$0.446337\pi$
$542$	−29.4956	−1.26695
$543$	0	0
$544$	0.227023	0.00973355
$545$	11.5533	0.494887
$546$	0	0
$547$	−19.9714	−0.853915	−0.426958	−	0.904272i	$-0.640415\pi$
$547$	−19.9714	−0.853915	−0.426958	+	0.904272i	$0.640415\pi$
$548$	−0.300993	−0.0128578
$549$	0	0
$550$	−16.9586	−0.723118
$551$	−6.78268	−0.288952
$552$	0	0
$553$	−13.9039	−0.591254
$554$	12.6130	0.535876
$555$	0	0
$556$	0.255258	0.0108254
$557$	13.7018	0.580562	0.290281	−	0.956941i	$-0.406251\pi$
$557$	13.7018	0.580562	0.290281	+	0.956941i	$0.406251\pi$
$558$	0	0
$559$	−9.11664	−0.385593
$560$	−11.8694	−0.501573
$561$	0	0
$562$	−17.0150	−0.717733
$563$	−9.11279	−0.384058	−0.192029	−	0.981389i	$-0.561507\pi$
$563$	−9.11279	−0.384058	−0.192029	+	0.981389i	$0.561507\pi$
$564$	0	0
$565$	9.75270	0.410299
$566$	−4.37897	−0.184062
$567$	0	0
$568$	−7.60760	−0.319208
$569$	15.8220	0.663294	0.331647	−	0.943403i	$-0.392396\pi$
$569$	15.8220	0.663294	0.331647	+	0.943403i	$0.392396\pi$
$570$	0	0
$571$	26.0759	1.09124	0.545621	−	0.838032i	$-0.316294\pi$
$571$	26.0759	1.09124	0.545621	+	0.838032i	$0.316294\pi$
$572$	0.0752638	0.00314694
$573$	0	0
$574$	−33.5633	−1.40090
$575$	−20.6055	−0.859309
$576$	0	0
$577$	10.8491	0.451656	0.225828	−	0.974167i	$-0.427491\pi$
$577$	10.8491	0.451656	0.225828	+	0.974167i	$0.427491\pi$
$578$	−21.4418	−0.891862
$579$	0	0
$580$	0.140798	0.00584632
$581$	16.9168	0.701826
$582$	0	0
$583$	−39.7424	−1.64596
$584$	−33.8014	−1.39871
$585$	0	0
$586$	−27.1778	−1.12270
$587$	16.6335	0.686538	0.343269	−	0.939237i	$-0.388466\pi$
$587$	16.6335	0.686538	0.343269	+	0.939237i	$0.388466\pi$
$588$	0	0
$589$	14.0823	0.580249
$590$	−9.36624	−0.385602
$591$	0	0
$592$	−21.2330	−0.872672
$593$	−15.4406	−0.634070	−0.317035	−	0.948414i	$-0.602687\pi$
$593$	−15.4406	−0.634070	−0.317035	+	0.948414i	$0.602687\pi$
$594$	0	0
$595$	3.95422	0.162107
$596$	−0.306629	−0.0125600
$597$	0	0
$598$	−5.88674	−0.240727
$599$	30.5212	1.24706	0.623531	−	0.781798i	$-0.285697\pi$
$599$	30.5212	1.24706	0.623531	+	0.781798i	$0.285697\pi$
$600$	0	0
$601$	36.6292	1.49414	0.747068	−	0.664747i	$-0.231461\pi$
$601$	36.6292	1.49414	0.747068	+	0.664747i	$0.231461\pi$
$602$	48.9657	1.99569
$603$	0	0
$604$	−0.0420062	−0.00170921
$605$	−2.24077	−0.0911001
$606$	0	0
$607$	−26.0511	−1.05738	−0.528690	−	0.848815i	$-0.677317\pi$
$607$	−26.0511	−1.05738	−0.528690	+	0.848815i	$0.677317\pi$
$608$	0.239654	0.00971926
$609$	0	0
$610$	−13.5463	−0.548474
$611$	−9.49127	−0.383976
$612$	0	0
$613$	−38.9810	−1.57443	−0.787214	−	0.616680i	$-0.788477\pi$
$613$	−38.9810	−1.57443	−0.787214	+	0.616680i	$0.788477\pi$
$614$	41.9630	1.69349
$615$	0	0
$616$	−26.8304	−1.08103
$617$	36.9977	1.48947	0.744737	−	0.667359i	$-0.232575\pi$
$617$	36.9977	1.48947	0.744737	+	0.667359i	$0.232575\pi$
$618$	0	0
$619$	−0.303585	−0.0122021	−0.00610105	−	0.999981i	$-0.501942\pi$
$619$	−0.303585	−0.0122021	−0.00610105	+	0.999981i	$0.501942\pi$
$620$	−0.292326	−0.0117401
$621$	0	0
$622$	37.8818	1.51892
$623$	38.6205	1.54730
$624$	0	0
$625$	12.5355	0.501420
$626$	−11.5381	−0.461157
$627$	0	0
$628$	−0.0664960	−0.00265348
$629$	7.07367	0.282046
$630$	0	0
$631$	31.1549	1.24026	0.620128	−	0.784501i	$-0.287081\pi$
$631$	31.1549	1.24026	0.620128	+	0.784501i	$0.287081\pi$
$632$	12.3517	0.491324
$633$	0	0
$634$	−20.3113	−0.806666
$635$	19.4531	0.771974
$636$	0	0
$637$	−2.75718	−0.109243
$638$	−20.1741	−0.798702
$639$	0	0
$640$	10.3805	0.410324
$641$	3.28825	0.129878	0.0649391	−	0.997889i	$-0.479315\pi$
$641$	3.28825	0.129878	0.0649391	+	0.997889i	$0.479315\pi$
$642$	0	0
$643$	−9.27348	−0.365710	−0.182855	−	0.983140i	$-0.558534\pi$
$643$	−9.27348	−0.365710	−0.182855	+	0.983140i	$0.558534\pi$
$644$	−0.491175	−0.0193550
$645$	0	0
$646$	2.54960	0.100313
$647$	−27.5359	−1.08255	−0.541274	−	0.840846i	$-0.682058\pi$
$647$	−27.5359	−1.08255	−0.541274	+	0.840846i	$0.682058\pi$
$648$	0	0
$649$	−20.8482	−0.818363
$650$	4.84246	0.189937
$651$	0	0
$652$	0.267426	0.0104732
$653$	−7.57614	−0.296477	−0.148239	−	0.988952i	$-0.547360\pi$
$653$	−7.57614	−0.296477	−0.148239	+	0.988952i	$0.547360\pi$
$654$	0	0
$655$	−19.6396	−0.767383
$656$	29.3601	1.14632
$657$	0	0
$658$	50.9778	1.98732
$659$	−6.29023	−0.245033	−0.122516	−	0.992467i	$-0.539096\pi$
$659$	−6.29023	−0.245033	−0.122516	+	0.992467i	$0.539096\pi$
$660$	0	0
$661$	−15.3692	−0.597793	−0.298897	−	0.954285i	$-0.596619\pi$
$661$	−15.3692	−0.597793	−0.298897	+	0.954285i	$0.596619\pi$
$662$	20.7653	0.807067
$663$	0	0
$664$	−15.0282	−0.583208
$665$	4.17422	0.161869
$666$	0	0
$667$	−24.5125	−0.949128
$668$	0.537760	0.0208065
$669$	0	0
$670$	11.7473	0.453837
$671$	−30.1525	−1.16403
$672$	0	0
$673$	−0.135586	−0.00522645	−0.00261322	−	0.999997i	$-0.500832\pi$
$673$	−0.135586	−0.00522645	−0.00261322	+	0.999997i	$0.500832\pi$
$674$	12.0419	0.463836
$675$	0	0
$676$	0.376233	0.0144705
$677$	49.9138	1.91834	0.959171	−	0.282826i	$-0.0912718\pi$
$677$	49.9138	1.91834	0.959171	+	0.282826i	$0.0912718\pi$
$678$	0	0
$679$	−10.9589	−0.420563
$680$	−3.51278	−0.134709
$681$	0	0
$682$	41.8857	1.60389
$683$	19.9786	0.764460	0.382230	−	0.924067i	$-0.375156\pi$
$683$	19.9786	0.764460	0.382230	+	0.924067i	$0.375156\pi$
$684$	0	0
$685$	9.24448	0.353213
$686$	−16.7024	−0.637702
$687$	0	0
$688$	−42.8337	−1.63302
$689$	11.3483	0.432335
$690$	0	0
$691$	−23.2354	−0.883918	−0.441959	−	0.897035i	$-0.645716\pi$
$691$	−23.2354	−0.883918	−0.441959	+	0.897035i	$0.645716\pi$
$692$	0.0449355	0.00170819
$693$	0	0
$694$	−15.9553	−0.605655
$695$	−7.83981	−0.297381
$696$	0	0
$697$	−9.78116	−0.370488
$698$	−9.85950	−0.373187
$699$	0	0
$700$	0.404043	0.0152714
$701$	−18.1262	−0.684619	−0.342309	−	0.939587i	$-0.611209\pi$
$701$	−18.1262	−0.684619	−0.342309	+	0.939587i	$0.611209\pi$
$702$	0	0
$703$	7.46723	0.281632
$704$	23.8296	0.898112
$705$	0	0
$706$	−22.8181	−0.858771
$707$	−58.4256	−2.19732
$708$	0	0
$709$	−6.77354	−0.254386	−0.127193	−	0.991878i	$-0.540597\pi$
$709$	−6.77354	−0.254386	−0.127193	+	0.991878i	$0.540597\pi$
$710$	3.52038	0.132118
$711$	0	0
$712$	−34.3090	−1.28578
$713$	50.8931	1.90596
$714$	0	0
$715$	−2.31160	−0.0864488
$716$	−0.104458	−0.00390378
$717$	0	0
$718$	−24.9287	−0.930330
$719$	−33.6950	−1.25661	−0.628305	−	0.777967i	$-0.716251\pi$
$719$	−33.6950	−1.25661	−0.628305	+	0.777967i	$0.716251\pi$
$720$	0	0
$721$	45.5865	1.69773
$722$	−23.9723	−0.892157
$723$	0	0
$724$	−0.306378	−0.0113865
$725$	20.1641	0.748877
$726$	0	0
$727$	4.34434	0.161123	0.0805613	−	0.996750i	$-0.474329\pi$
$727$	4.34434	0.161123	0.0805613	+	0.996750i	$0.474329\pi$
$728$	7.66131	0.283947
$729$	0	0
$730$	15.6414	0.578915
$731$	14.2698	0.527789
$732$	0	0
$733$	33.0420	1.22043	0.610217	−	0.792235i	$-0.291082\pi$
$733$	33.0420	1.22043	0.610217	+	0.792235i	$0.291082\pi$
$734$	10.4246	0.384779
$735$	0	0
$736$	0.866107	0.0319251
$737$	26.1481	0.963177
$738$	0	0
$739$	50.7274	1.86604	0.933019	−	0.359826i	$-0.117164\pi$
$739$	50.7274	1.86604	0.933019	+	0.359826i	$0.117164\pi$
$740$	−0.155008	−0.00569821
$741$	0	0
$742$	−60.9519	−2.23762
$743$	−13.0892	−0.480198	−0.240099	−	0.970748i	$-0.577180\pi$
$743$	−13.0892	−0.480198	−0.240099	+	0.970748i	$0.577180\pi$
$744$	0	0
$745$	9.41759	0.345034
$746$	−31.7403	−1.16210
$747$	0	0
$748$	−0.117807	−0.00430744
$749$	−47.5528	−1.73754
$750$	0	0
$751$	26.6096	0.970998	0.485499	−	0.874237i	$-0.338638\pi$
$751$	26.6096	0.970998	0.485499	+	0.874237i	$0.338638\pi$
$752$	−44.5938	−1.62617
$753$	0	0
$754$	5.76064	0.209790
$755$	1.29015	0.0469533
$756$	0	0
$757$	18.8005	0.683316	0.341658	−	0.939824i	$-0.389012\pi$
$757$	18.8005	0.683316	0.341658	+	0.939824i	$0.389012\pi$
$758$	22.7551	0.826503
$759$	0	0
$760$	−3.70822	−0.134511
$761$	−37.2721	−1.35111	−0.675557	−	0.737308i	$-0.736097\pi$
$761$	−37.2721	−1.35111	−0.675557	+	0.737308i	$0.736097\pi$
$762$	0	0
$763$	39.4403	1.42784
$764$	0.302051	0.0109278
$765$	0	0
$766$	−2.48009	−0.0896092
$767$	5.95311	0.214954
$768$	0	0
$769$	−40.6267	−1.46503	−0.732517	−	0.680748i	$-0.761655\pi$
$769$	−40.6267	−1.46503	−0.732517	+	0.680748i	$0.761655\pi$
$770$	12.4156	0.447429
$771$	0	0
$772$	0.805450	0.0289888
$773$	15.9586	0.573991	0.286995	−	0.957932i	$-0.407344\pi$
$773$	15.9586	0.573991	0.286995	+	0.957932i	$0.407344\pi$
$774$	0	0
$775$	−41.8649	−1.50383
$776$	9.73544	0.349482
$777$	0	0
$778$	12.4888	0.447745
$779$	−10.3253	−0.369944
$780$	0	0
$781$	7.83596	0.280393
$782$	9.21421	0.329500
$783$	0	0
$784$	−12.9544	−0.462656
$785$	2.04231	0.0728932
$786$	0	0
$787$	−20.6280	−0.735308	−0.367654	−	0.929963i	$-0.619839\pi$
$787$	−20.6280	−0.735308	−0.367654	+	0.929963i	$0.619839\pi$
$788$	−0.0667534	−0.00237799
$789$	0	0
$790$	−5.71569	−0.203355
$791$	33.2936	1.18378
$792$	0	0
$793$	8.60993	0.305747
$794$	−49.3350	−1.75083
$795$	0	0
$796$	0.330903	0.0117286
$797$	4.94682	0.175225	0.0876126	−	0.996155i	$-0.472076\pi$
$797$	4.94682	0.175225	0.0876126	+	0.996155i	$0.472076\pi$
$798$	0	0
$799$	14.8562	0.525575
$800$	−0.712464	−0.0251894
$801$	0	0
$802$	−24.2011	−0.854570
$803$	34.8160	1.22863
$804$	0	0
$805$	15.0856	0.531697
$806$	−11.9603	−0.421283
$807$	0	0
$808$	51.9031	1.82594
$809$	23.4704	0.825175	0.412588	−	0.910918i	$-0.364625\pi$
$809$	23.4704	0.825175	0.412588	+	0.910918i	$0.364625\pi$
$810$	0	0
$811$	−36.8392	−1.29360	−0.646799	−	0.762661i	$-0.723893\pi$
$811$	−36.8392	−1.29360	−0.646799	+	0.762661i	$0.723893\pi$
$812$	0.480653	0.0168676
$813$	0	0
$814$	22.2102	0.778468
$815$	−8.21354	−0.287708
$816$	0	0
$817$	15.0637	0.527014
$818$	−48.5661	−1.69808
$819$	0	0
$820$	0.214338	0.00748502
$821$	11.6446	0.406400	0.203200	−	0.979137i	$-0.434866\pi$
$821$	11.6446	0.406400	0.203200	+	0.979137i	$0.434866\pi$
$822$	0	0
$823$	15.0982	0.526289	0.263145	−	0.964756i	$-0.415240\pi$
$823$	15.0982	0.526289	0.263145	+	0.964756i	$0.415240\pi$
$824$	−40.4973	−1.41079
$825$	0	0
$826$	−31.9743	−1.11253
$827$	−9.80403	−0.340919	−0.170460	−	0.985365i	$-0.554525\pi$
$827$	−9.80403	−0.340919	−0.170460	+	0.985365i	$0.554525\pi$
$828$	0	0
$829$	28.4389	0.987723	0.493862	−	0.869541i	$-0.335585\pi$
$829$	28.4389	0.987723	0.493862	+	0.869541i	$0.335585\pi$
$830$	6.95423	0.241385
$831$	0	0
$832$	−6.80444	−0.235902
$833$	4.31567	0.149529
$834$	0	0
$835$	−16.5164	−0.571572
$836$	−0.124361	−0.00430111
$837$	0	0
$838$	36.4819	1.26024
$839$	46.2364	1.59626	0.798129	−	0.602486i	$-0.205823\pi$
$839$	46.2364	1.59626	0.798129	+	0.602486i	$0.205823\pi$
$840$	0	0
$841$	−5.01256	−0.172847
$842$	−42.2984	−1.45770
$843$	0	0
$844$	0.417758	0.0143798
$845$	−11.5554	−0.397516
$846$	0	0
$847$	−7.64949	−0.262840
$848$	53.3189	1.83098
$849$	0	0
$850$	−7.57966	−0.259980
$851$	26.9865	0.925084
$852$	0	0
$853$	−25.8850	−0.886284	−0.443142	−	0.896451i	$-0.646136\pi$
$853$	−25.8850	−0.886284	−0.443142	+	0.896451i	$0.646136\pi$
$854$	−46.2441	−1.58244
$855$	0	0
$856$	42.2441	1.44387
$857$	−34.7029	−1.18543	−0.592714	−	0.805413i	$-0.701943\pi$
$857$	−34.7029	−1.18543	−0.592714	+	0.805413i	$0.701943\pi$
$858$	0	0
$859$	2.62446	0.0895453	0.0447726	−	0.998997i	$-0.485744\pi$
$859$	2.62446	0.0895453	0.0447726	+	0.998997i	$0.485744\pi$
$860$	−0.312700	−0.0106630
$861$	0	0
$862$	−7.22074	−0.245939
$863$	−42.1802	−1.43583	−0.717916	−	0.696130i	$-0.754904\pi$
$863$	−42.1802	−1.43583	−0.717916	+	0.696130i	$0.754904\pi$
$864$	0	0
$865$	−1.38012	−0.0469254
$866$	−10.6980	−0.363533
$867$	0	0
$868$	−0.997936	−0.0338722
$869$	−12.7225	−0.431580
$870$	0	0
$871$	−7.46648	−0.252992
$872$	−35.0373	−1.18651
$873$	0	0
$874$	9.72686	0.329016
$875$	−27.4803	−0.929003
$876$	0	0
$877$	42.7019	1.44194	0.720971	−	0.692965i	$-0.243696\pi$
$877$	42.7019	1.44194	0.720971	+	0.692965i	$0.243696\pi$
$878$	23.1162	0.780135
$879$	0	0
$880$	−10.8608	−0.366118
$881$	−40.3119	−1.35814	−0.679071	−	0.734072i	$-0.737617\pi$
$881$	−40.3119	−1.35814	−0.679071	+	0.734072i	$0.737617\pi$
$882$	0	0
$883$	5.05156	0.169998	0.0849992	−	0.996381i	$-0.472911\pi$
$883$	5.05156	0.169998	0.0849992	+	0.996381i	$0.472911\pi$
$884$	0.0336392	0.00113141
$885$	0	0
$886$	24.3760	0.818927
$887$	52.7120	1.76990	0.884949	−	0.465689i	$-0.154193\pi$
$887$	52.7120	1.76990	0.884949	+	0.465689i	$0.154193\pi$
$888$	0	0
$889$	66.4088	2.22728
$890$	15.8763	0.532175
$891$	0	0
$892$	0.0778410	0.00260631
$893$	15.6827	0.524803
$894$	0	0
$895$	3.20825	0.107240
$896$	35.4366	1.18385
$897$	0	0
$898$	19.0134	0.634486
$899$	−49.8029	−1.66102
$900$	0	0
$901$	−17.7629	−0.591768
$902$	−30.7113	−1.02258
$903$	0	0
$904$	−29.5767	−0.983708
$905$	9.40987	0.312795
$906$	0	0
$907$	13.2100	0.438632	0.219316	−	0.975654i	$-0.429617\pi$
$907$	13.2100	0.438632	0.219316	+	0.975654i	$0.429617\pi$
$908$	0.165578	0.00549490
$909$	0	0
$910$	−3.54523	−0.117523
$911$	−7.43159	−0.246220	−0.123110	−	0.992393i	$-0.539287\pi$
$911$	−7.43159	−0.246220	−0.123110	+	0.992393i	$0.539287\pi$
$912$	0	0
$913$	15.4793	0.512291
$914$	17.2181	0.569526
$915$	0	0
$916$	0.435685	0.0143954
$917$	−67.0454	−2.21403
$918$	0	0
$919$	−29.3675	−0.968744	−0.484372	−	0.874862i	$-0.660952\pi$
$919$	−29.3675	−0.968744	−0.484372	+	0.874862i	$0.660952\pi$
$920$	−13.4014	−0.441833
$921$	0	0
$922$	6.99042	0.230217
$923$	−2.23753	−0.0736491
$924$	0	0
$925$	−22.1992	−0.729905
$926$	31.8093	1.04532
$927$	0	0
$928$	−0.847555	−0.0278223
$929$	23.5470	0.772551	0.386275	−	0.922384i	$-0.373761\pi$
$929$	23.5470	0.772551	0.386275	+	0.922384i	$0.373761\pi$
$930$	0	0
$931$	4.55578	0.149310
$932$	0.581651	0.0190526
$933$	0	0
$934$	−17.2467	−0.564329
$935$	3.61823	0.118329
$936$	0	0
$937$	−22.3298	−0.729481	−0.364741	−	0.931109i	$-0.618842\pi$
$937$	−22.3298	−0.729481	−0.364741	+	0.931109i	$0.618842\pi$
$938$	40.1027	1.30940
$939$	0	0
$940$	−0.325550	−0.0106183
$941$	−16.5695	−0.540150	−0.270075	−	0.962839i	$-0.587048\pi$
$941$	−16.5695	−0.540150	−0.270075	+	0.962839i	$0.587048\pi$
$942$	0	0
$943$	−37.3157	−1.21517
$944$	27.9701	0.910351
$945$	0	0
$946$	44.8050	1.45674
$947$	−23.2580	−0.755782	−0.377891	−	0.925850i	$-0.623351\pi$
$947$	−23.2580	−0.755782	−0.377891	+	0.925850i	$0.623351\pi$
$948$	0	0
$949$	−9.94157	−0.322717
$950$	−8.00137	−0.259599
$951$	0	0
$952$	−11.9919	−0.388658
$953$	−44.5110	−1.44185	−0.720926	−	0.693012i	$-0.756283\pi$
$953$	−44.5110	−1.44185	−0.720926	+	0.693012i	$0.756283\pi$
$954$	0	0
$955$	−9.27699	−0.300196
$956$	0.0305942	0.000989487 0
$957$	0	0
$958$	34.2283	1.10586
$959$	31.5586	1.01908
$960$	0	0
$961$	72.4012	2.33552
$962$	−6.34204	−0.204476
$963$	0	0
$964$	0.0382770	0.00123282
$965$	−24.7380	−0.796344
$966$	0	0
$967$	5.99683	0.192845	0.0964226	−	0.995340i	$-0.469260\pi$
$967$	5.99683	0.192845	0.0964226	+	0.995340i	$0.469260\pi$
$968$	6.79551	0.218416
$969$	0	0
$970$	−4.50503	−0.144648
$971$	57.0523	1.83090	0.915448	−	0.402437i	$-0.131837\pi$
$971$	57.0523	1.83090	0.915448	+	0.402437i	$0.131837\pi$
$972$	0	0
$973$	−26.7634	−0.857996
$974$	39.4774	1.26494
$975$	0	0
$976$	40.4529	1.29487
$977$	25.7079	0.822470	0.411235	−	0.911529i	$-0.365098\pi$
$977$	25.7079	0.822470	0.411235	+	0.911529i	$0.365098\pi$
$978$	0	0
$979$	35.3389	1.12944
$980$	−0.0945710	−0.00302096
$981$	0	0
$982$	−47.9971	−1.53165
$983$	−2.87126	−0.0915788	−0.0457894	−	0.998951i	$-0.514580\pi$
$983$	−2.87126	−0.0915788	−0.0457894	+	0.998951i	$0.514580\pi$
$984$	0	0
$985$	2.05021	0.0653253
$986$	−9.01684	−0.287155
$987$	0	0
$988$	0.0355107	0.00112975
$989$	54.4402	1.73110
$990$	0	0
$991$	−19.0724	−0.605853	−0.302927	−	0.953014i	$-0.597964\pi$
$991$	−19.0724	−0.605853	−0.302927	+	0.953014i	$0.597964\pi$
$992$	1.75970	0.0558705
$993$	0	0
$994$	12.0178	0.381182
$995$	−10.1631	−0.322193
$996$	0	0
$997$	17.0298	0.539338	0.269669	−	0.962953i	$-0.413086\pi$
$997$	17.0298	0.539338	0.269669	+	0.962953i	$0.413086\pi$
$998$	−24.5310	−0.776517
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.h.1.9		12
3.2	odd	2		717.2.a.g.1.4	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.2.a.g.1.4	✓	12	3.2	odd	2
2151.2.a.h.1.9		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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+1.40336 q^{2} -0.0305942 q^{4} +0.939648 q^{5} +3.20775 q^{7} -2.84964 q^{8} +O(q^{10})\)	\(q+1.40336 q^{2} -0.0305942 q^{4} +0.939648 q^{5} +3.20775 q^{7} -2.84964 q^{8} +1.31866 q^{10} +2.93519 q^{11} -0.838130 q^{13} +4.50162 q^{14} -3.93788 q^{16} +1.31188 q^{17} +1.38487 q^{19} -0.0287478 q^{20} +4.11911 q^{22} +5.00490 q^{23} -4.11706 q^{25} -1.17619 q^{26} -0.0981386 q^{28} -4.89770 q^{29} +10.1686 q^{31} +0.173052 q^{32} +1.84104 q^{34} +3.01416 q^{35} +5.39200 q^{37} +1.94347 q^{38} -2.67766 q^{40} -7.45582 q^{41} +10.8774 q^{43} -0.0897996 q^{44} +7.02366 q^{46} +11.3243 q^{47} +3.28968 q^{49} -5.77770 q^{50} +0.0256419 q^{52} -13.5400 q^{53} +2.75804 q^{55} -9.14096 q^{56} -6.87321 q^{58} -7.10285 q^{59} -10.2728 q^{61} +14.2702 q^{62} +8.11860 q^{64} -0.787547 q^{65} +8.90850 q^{67} -0.0401360 q^{68} +4.22993 q^{70} +2.66967 q^{71} +11.8616 q^{73} +7.56689 q^{74} -0.0423690 q^{76} +9.41535 q^{77} -4.33447 q^{79} -3.70022 q^{80} -10.4632 q^{82} +5.27371 q^{83} +1.23271 q^{85} +15.2648 q^{86} -8.36424 q^{88} +12.0397 q^{89} -2.68851 q^{91} -0.153121 q^{92} +15.8921 q^{94} +1.30129 q^{95} -3.41637 q^{97} +4.61659 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 15 q^{4} + q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) \) 12 * q - 3 * q^2 + 15 * q^4 + q^5 + 11 * q^7 - 9 * q^8	\( 12 q - 3 q^{2} + 15 q^{4} + q^{5} + 11 q^{7} - 9 q^{8} - 15 q^{11} + 7 q^{13} + 6 q^{14} + 21 q^{16} + 3 q^{17} + 10 q^{19} + 4 q^{20} + 23 q^{22} - 20 q^{23} + 19 q^{25} + 10 q^{26} + 34 q^{28} - 2 q^{29} + 10 q^{31} - 26 q^{32} + 12 q^{34} - 7 q^{35} + 30 q^{37} + 3 q^{38} + 25 q^{40} + 28 q^{41} + 48 q^{43} - 25 q^{44} + 22 q^{46} - 13 q^{47} + 19 q^{49} - 12 q^{50} + 24 q^{52} + 2 q^{53} + 8 q^{55} + 7 q^{56} + 42 q^{58} + 14 q^{59} + 14 q^{61} - 8 q^{62} + 9 q^{64} + 35 q^{65} + 52 q^{67} - 3 q^{68} - 33 q^{70} + 7 q^{71} + 14 q^{73} + 13 q^{74} - 12 q^{76} + 6 q^{77} + 15 q^{79} + 8 q^{80} - 61 q^{82} - 29 q^{83} + 8 q^{85} + 9 q^{86} + 11 q^{88} + 71 q^{89} + 13 q^{91} - 2 q^{92} - 22 q^{94} - 2 q^{95} - 2 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 + 15 * q^4 + q^5 + 11 * q^7 - 9 * q^8 - 15 * q^11 + 7 * q^13 + 6 * q^14 + 21 * q^16 + 3 * q^17 + 10 * q^19 + 4 * q^20 + 23 * q^22 - 20 * q^23 + 19 * q^25 + 10 * q^26 + 34 * q^28 - 2 * q^29 + 10 * q^31 - 26 * q^32 + 12 * q^34 - 7 * q^35 + 30 * q^37 + 3 * q^38 + 25 * q^40 + 28 * q^41 + 48 * q^43 - 25 * q^44 + 22 * q^46 - 13 * q^47 + 19 * q^49 - 12 * q^50 + 24 * q^52 + 2 * q^53 + 8 * q^55 + 7 * q^56 + 42 * q^58 + 14 * q^59 + 14 * q^61 - 8 * q^62 + 9 * q^64 + 35 * q^65 + 52 * q^67 - 3 * q^68 - 33 * q^70 + 7 * q^71 + 14 * q^73 + 13 * q^74 - 12 * q^76 + 6 * q^77 + 15 * q^79 + 8 * q^80 - 61 * q^82 - 29 * q^83 + 8 * q^85 + 9 * q^86 + 11 * q^88 + 71 * q^89 + 13 * q^91 - 2 * q^92 - 22 * q^94 - 2 * q^95 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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