LMFDB - Embedded newform 3234.2.a.bl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3234,2,Mod(1,3234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3234, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3234.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3234.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:	$25.8236200137$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.4352.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 4x + 2 $ x^4 - 6x^2 - 4x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.68554$ of defining polynomial
Character	$\chi$	$=$	3234.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.79793 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -3.79793 q^{10} +1.00000 q^{11} -1.00000 q^{12} -0.361009 q^{13} +3.79793 q^{15} +1.00000 q^{16} +4.11582 q^{17} +1.00000 q^{18} -4.15894 q^{19} -3.79793 q^{20} +1.00000 q^{22} +0.542661 q^{23} -1.00000 q^{24} +9.42429 q^{25} -0.361009 q^{26} -1.00000 q^{27} +0.767438 q^{29} +3.79793 q^{30} +8.80801 q^{31} +1.00000 q^{32} -1.00000 q^{33} +4.11582 q^{34} +1.00000 q^{36} +2.28577 q^{37} -4.15894 q^{38} +0.361009 q^{39} -3.79793 q^{40} -9.13690 q^{41} -10.1995 q^{43} +1.00000 q^{44} -3.79793 q^{45} +0.542661 q^{46} +2.98737 q^{47} -1.00000 q^{48} +9.42429 q^{50} -4.11582 q^{51} -0.361009 q^{52} -12.4564 q^{53} -1.00000 q^{54} -3.79793 q^{55} +4.15894 q^{57} +0.767438 q^{58} -2.25689 q^{59} +3.79793 q^{60} -12.2748 q^{61} +8.80801 q^{62} +1.00000 q^{64} +1.37109 q^{65} -1.00000 q^{66} +0.603650 q^{67} +4.11582 q^{68} -0.542661 q^{69} -6.87385 q^{71} +1.00000 q^{72} -9.45479 q^{73} +2.28577 q^{74} -9.42429 q^{75} -4.15894 q^{76} +0.361009 q^{78} +0.0321169 q^{79} -3.79793 q^{80} +1.00000 q^{81} -9.13690 q^{82} -10.8690 q^{83} -15.6316 q^{85} -10.1995 q^{86} -0.767438 q^{87} +1.00000 q^{88} +1.29585 q^{89} -3.79793 q^{90} +0.542661 q^{92} -8.80801 q^{93} +2.98737 q^{94} +15.7954 q^{95} -1.00000 q^{96} +18.2628 q^{97} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} - 4 q^{6} + 4 q^{8} + 4 q^{9} - 4 q^{10} + 4 q^{11} - 4 q^{12} - 8 q^{13} + 4 q^{15} + 4 q^{16} - 4 q^{17} + 4 q^{18} - 12 q^{19} - 4 q^{20} + 4 q^{22} - 8 q^{23}+ \cdots + 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^5 - 4 * q^6 + 4 * q^8 + 4 * q^9 - 4 * q^10 + 4 * q^11 - 4 * q^12 - 8 * q^13 + 4 * q^15 + 4 * q^16 - 4 * q^17 + 4 * q^18 - 12 * q^19 - 4 * q^20 + 4 * q^22 - 8 * q^23 - 4 * q^24 + 4 * q^25 - 8 * q^26 - 4 * q^27 - 8 * q^29 + 4 * q^30 - 4 * q^31 + 4 * q^32 - 4 * q^33 - 4 * q^34 + 4 * q^36 + 8 * q^37 - 12 * q^38 + 8 * q^39 - 4 * q^40 - 12 * q^41 - 8 * q^43 + 4 * q^44 - 4 * q^45 - 8 * q^46 - 4 * q^47 - 4 * q^48 + 4 * q^50 + 4 * q^51 - 8 * q^52 - 8 * q^53 - 4 * q^54 - 4 * q^55 + 12 * q^57 - 8 * q^58 + 4 * q^60 - 24 * q^61 - 4 * q^62 + 4 * q^64 - 16 * q^65 - 4 * q^66 - 8 * q^67 - 4 * q^68 + 8 * q^69 + 8 * q^71 + 4 * q^72 - 4 * q^73 + 8 * q^74 - 4 * q^75 - 12 * q^76 + 8 * q^78 - 8 * q^79 - 4 * q^80 + 4 * q^81 - 12 * q^82 - 4 * q^83 - 8 * q^85 - 8 * q^86 + 8 * q^87 + 4 * q^88 - 24 * q^89 - 4 * q^90 - 8 * q^92 + 4 * q^93 - 4 * q^94 + 8 * q^95 - 4 * q^96 + 4 * q^99

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−3.79793	−1.69849	−0.849244	−	0.528001i	$-0.822942\pi$
$5$	−3.79793	−1.69849	−0.849244	+	0.528001i	$0.822942\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−3.79793	−1.20101
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.00000	−0.288675
$13$	−0.361009	−0.100126	−0.0500629	−	0.998746i	$-0.515942\pi$
$13$	−0.361009	−0.100126	−0.0500629	+	0.998746i	$0.515942\pi$
$14$	0	0
$15$	3.79793	0.980622
$16$	1.00000	0.250000
$17$	4.11582	0.998232	0.499116	−	0.866535i	$-0.333658\pi$
$17$	4.11582	0.998232	0.499116	+	0.866535i	$0.333658\pi$
$18$	1.00000	0.235702
$19$	−4.15894	−0.954127	−0.477063	−	0.878869i	$-0.658299\pi$
$19$	−4.15894	−0.954127	−0.477063	+	0.878869i	$0.658299\pi$
$20$	−3.79793	−0.849244
$21$	0	0
$22$	1.00000	0.213201
$23$	0.542661	0.113153	0.0565763	−	0.998398i	$-0.481982\pi$
$23$	0.542661	0.113153	0.0565763	+	0.998398i	$0.481982\pi$
$24$	−1.00000	−0.204124
$25$	9.42429	1.88486
$26$	−0.361009	−0.0707997
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0.767438	0.142510	0.0712548	−	0.997458i	$-0.477300\pi$
$29$	0.767438	0.142510	0.0712548	+	0.997458i	$0.477300\pi$
$30$	3.79793	0.693404
$31$	8.80801	1.58197	0.790983	−	0.611838i	$-0.209570\pi$
$31$	8.80801	1.58197	0.790983	+	0.611838i	$0.209570\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	4.11582	0.705857
$35$	0	0
$36$	1.00000	0.166667
$37$	2.28577	0.375778	0.187889	−	0.982190i	$-0.439836\pi$
$37$	2.28577	0.375778	0.187889	+	0.982190i	$0.439836\pi$
$38$	−4.15894	−0.674669
$39$	0.361009	0.0578077
$40$	−3.79793	−0.600506
$41$	−9.13690	−1.42694	−0.713472	−	0.700683i	$-0.752879\pi$
$41$	−9.13690	−1.42694	−0.713472	+	0.700683i	$0.752879\pi$
$42$	0	0
$43$	−10.1995	−1.55541	−0.777706	−	0.628629i	$-0.783617\pi$
$43$	−10.1995	−1.55541	−0.777706	+	0.628629i	$0.783617\pi$
$44$	1.00000	0.150756
$45$	−3.79793	−0.566162
$46$	0.542661	0.0800110
$47$	2.98737	0.435753	0.217876	−	0.975976i	$-0.430087\pi$
$47$	2.98737	0.435753	0.217876	+	0.975976i	$0.430087\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	9.42429	1.33280
$51$	−4.11582	−0.576330
$52$	−0.361009	−0.0500629
$53$	−12.4564	−1.71102	−0.855510	−	0.517787i	$-0.826756\pi$
$53$	−12.4564	−1.71102	−0.855510	+	0.517787i	$0.826756\pi$
$54$	−1.00000	−0.136083
$55$	−3.79793	−0.512113
$56$	0	0
$57$	4.15894	0.550865
$58$	0.767438	0.100770
$59$	−2.25689	−0.293823	−0.146911	−	0.989150i	$-0.546933\pi$
$59$	−2.25689	−0.293823	−0.146911	+	0.989150i	$0.546933\pi$
$60$	3.79793	0.490311
$61$	−12.2748	−1.57162	−0.785811	−	0.618467i	$-0.787754\pi$
$61$	−12.2748	−1.57162	−0.785811	+	0.618467i	$0.787754\pi$
$62$	8.80801	1.11862
$63$	0	0
$64$	1.00000	0.125000
$65$	1.37109	0.170063
$66$	−1.00000	−0.123091
$67$	0.603650	0.0737475	0.0368738	−	0.999320i	$-0.488260\pi$
$67$	0.603650	0.0737475	0.0368738	+	0.999320i	$0.488260\pi$
$68$	4.11582	0.499116
$69$	−0.542661	−0.0653287
$70$	0	0
$71$	−6.87385	−0.815776	−0.407888	−	0.913032i	$-0.633735\pi$
$71$	−6.87385	−0.815776	−0.407888	+	0.913032i	$0.633735\pi$
$72$	1.00000	0.117851
$73$	−9.45479	−1.10660	−0.553300	−	0.832982i	$-0.686632\pi$
$73$	−9.45479	−1.10660	−0.553300	+	0.832982i	$0.686632\pi$
$74$	2.28577	0.265715
$75$	−9.42429	−1.08822
$76$	−4.15894	−0.477063
$77$	0	0
$78$	0.361009	0.0408762
$79$	0.0321169	0.00361344	0.00180672	−	0.999998i	$-0.499425\pi$
$79$	0.0321169	0.00361344	0.00180672	+	0.999998i	$0.499425\pi$
$80$	−3.79793	−0.424622
$81$	1.00000	0.111111
$82$	−9.13690	−1.00900
$83$	−10.8690	−1.19303	−0.596514	−	0.802603i	$-0.703448\pi$
$83$	−10.8690	−1.19303	−0.596514	+	0.802603i	$0.703448\pi$
$84$	0	0
$85$	−15.6316	−1.69548
$86$	−10.1995	−1.09984
$87$	−0.767438	−0.0822780
$88$	1.00000	0.106600
$89$	1.29585	0.137359	0.0686797	−	0.997639i	$-0.478121\pi$
$89$	1.29585	0.137359	0.0686797	+	0.997639i	$0.478121\pi$
$90$	−3.79793	−0.400337
$91$	0	0
$92$	0.542661	0.0565763
$93$	−8.80801	−0.913348
$94$	2.98737	0.308124
$95$	15.7954	1.62057
$96$	−1.00000	−0.102062
$97$	18.2628	1.85431	0.927153	−	0.374683i	$-0.122248\pi$
$97$	18.2628	1.85431	0.927153	+	0.374683i	$0.122248\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	9.42429	0.942429
$101$	−2.26790	−0.225665	−0.112832	−	0.993614i	$-0.535992\pi$
$101$	−2.26790	−0.225665	−0.112832	+	0.993614i	$0.535992\pi$
$102$	−4.11582	−0.407527
$103$	13.4438	1.32465	0.662327	−	0.749215i	$-0.269569\pi$
$103$	13.4438	1.32465	0.662327	+	0.749215i	$0.269569\pi$
$104$	−0.361009	−0.0353998
$105$	0	0
$106$	−12.4564	−1.20987
$107$	−6.79956	−0.657338	−0.328669	−	0.944445i	$-0.606600\pi$
$107$	−6.79956	−0.657338	−0.328669	+	0.944445i	$0.606600\pi$
$108$	−1.00000	−0.0962250
$109$	3.20730	0.307204	0.153602	−	0.988133i	$-0.450913\pi$
$109$	3.20730	0.307204	0.153602	+	0.988133i	$0.450913\pi$
$110$	−3.79793	−0.362119
$111$	−2.28577	−0.216955
$112$	0	0
$113$	−6.37887	−0.600074	−0.300037	−	0.953928i	$-0.596999\pi$
$113$	−6.37887	−0.600074	−0.300037	+	0.953928i	$0.596999\pi$
$114$	4.15894	0.389521
$115$	−2.06099	−0.192188
$116$	0.767438	0.0712548
$117$	−0.361009	−0.0333753
$118$	−2.25689	−0.207764
$119$	0	0
$120$	3.79793	0.346702
$121$	1.00000	0.0909091
$122$	−12.2748	−1.11130
$123$	9.13690	0.823847
$124$	8.80801	0.790983
$125$	−16.8032	−1.50292
$126$	0	0
$127$	−10.7101	−0.950364	−0.475182	−	0.879888i	$-0.657618\pi$
$127$	−10.7101	−0.950364	−0.475182	+	0.879888i	$0.657618\pi$
$128$	1.00000	0.0883883
$129$	10.1995	0.898017
$130$	1.37109	0.120252
$131$	−11.8732	−1.03736	−0.518682	−	0.854967i	$-0.673577\pi$
$131$	−11.8732	−1.03736	−0.518682	+	0.854967i	$0.673577\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	0.603650	0.0521474
$135$	3.79793	0.326874
$136$	4.11582	0.352928
$137$	−3.21699	−0.274846	−0.137423	−	0.990512i	$-0.543882\pi$
$137$	−3.21699	−0.274846	−0.137423	+	0.990512i	$0.543882\pi$
$138$	−0.542661	−0.0461943
$139$	−0.608497	−0.0516120	−0.0258060	−	0.999667i	$-0.508215\pi$
$139$	−0.608497	−0.0516120	−0.0258060	+	0.999667i	$0.508215\pi$
$140$	0	0
$141$	−2.98737	−0.251582
$142$	−6.87385	−0.576840
$143$	−0.361009	−0.0301891
$144$	1.00000	0.0833333
$145$	−2.91468	−0.242051
$146$	−9.45479	−0.782484
$147$	0	0
$148$	2.28577	0.187889
$149$	−14.1064	−1.15564	−0.577821	−	0.816163i	$-0.696097\pi$
$149$	−14.1064	−1.15564	−0.577821	+	0.816163i	$0.696097\pi$
$150$	−9.42429	−0.769490
$151$	−3.42847	−0.279005	−0.139502	−	0.990222i	$-0.544550\pi$
$151$	−3.42847	−0.279005	−0.139502	+	0.990222i	$0.544550\pi$
$152$	−4.15894	−0.337335
$153$	4.11582	0.332744
$154$	0	0
$155$	−33.4522	−2.68695
$156$	0.361009	0.0289039
$157$	16.8833	1.34743	0.673715	−	0.738991i	$-0.264697\pi$
$157$	16.8833	1.34743	0.673715	+	0.738991i	$0.264697\pi$
$158$	0.0321169	0.00255508
$159$	12.4564	0.987858
$160$	−3.79793	−0.300253
$161$	0	0
$162$	1.00000	0.0785674
$163$	−15.8275	−1.23971	−0.619853	−	0.784718i	$-0.712808\pi$
$163$	−15.8275	−1.23971	−0.619853	+	0.784718i	$0.712808\pi$
$164$	−9.13690	−0.713472
$165$	3.79793	0.295669
$166$	−10.8690	−0.843598
$167$	−25.3981	−1.96536	−0.982682	−	0.185300i	$-0.940674\pi$
$167$	−25.3981	−1.96536	−0.982682	+	0.185300i	$0.940674\pi$
$168$	0	0
$169$	−12.8697	−0.989975
$170$	−15.6316	−1.19889
$171$	−4.15894	−0.318042
$172$	−10.1995	−0.777706
$173$	−12.1207	−0.921517	−0.460758	−	0.887526i	$-0.652423\pi$
$173$	−12.1207	−0.921517	−0.460758	+	0.887526i	$0.652423\pi$
$174$	−0.767438	−0.0581793
$175$	0	0
$176$	1.00000	0.0753778
$177$	2.25689	0.169639
$178$	1.29585	0.0971277
$179$	22.5376	1.68454	0.842268	−	0.539059i	$-0.181220\pi$
$179$	22.5376	1.68454	0.842268	+	0.539059i	$0.181220\pi$
$180$	−3.79793	−0.283081
$181$	21.4905	1.59738	0.798689	−	0.601745i	$-0.205527\pi$
$181$	21.4905	1.59738	0.798689	+	0.601745i	$0.205527\pi$
$182$	0	0
$183$	12.2748	0.907376
$184$	0.542661	0.0400055
$185$	−8.68119	−0.638254
$186$	−8.80801	−0.645835
$187$	4.11582	0.300978
$188$	2.98737	0.217876
$189$	0	0
$190$	15.7954	1.14592
$191$	−5.80049	−0.419708	−0.209854	−	0.977733i	$-0.567299\pi$
$191$	−5.80049	−0.419708	−0.209854	+	0.977733i	$0.567299\pi$
$192$	−1.00000	−0.0721688
$193$	26.1623	1.88320	0.941602	−	0.336729i	$-0.109321\pi$
$193$	26.1623	1.88320	0.941602	+	0.336729i	$0.109321\pi$
$194$	18.2628	1.31119
$195$	−1.37109	−0.0981856
$196$	0	0
$197$	−16.9958	−1.21090	−0.605451	−	0.795882i	$-0.707007\pi$
$197$	−16.9958	−1.21090	−0.605451	+	0.795882i	$0.707007\pi$
$198$	1.00000	0.0710669
$199$	−25.9745	−1.84128	−0.920641	−	0.390410i	$-0.872333\pi$
$199$	−25.9745	−1.84128	−0.920641	+	0.390410i	$0.872333\pi$
$200$	9.42429	0.666398
$201$	−0.603650	−0.0425782
$202$	−2.26790	−0.159569
$203$	0	0
$204$	−4.11582	−0.288165
$205$	34.7013	2.42365
$206$	13.4438	0.936672
$207$	0.542661	0.0377175
$208$	−0.361009	−0.0250315
$209$	−4.15894	−0.287680
$210$	0	0
$211$	12.7555	0.878123	0.439062	−	0.898457i	$-0.355311\pi$
$211$	12.7555	0.878123	0.439062	+	0.898457i	$0.355311\pi$
$212$	−12.4564	−0.855510
$213$	6.87385	0.470988
$214$	−6.79956	−0.464808
$215$	38.7371	2.64185
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	3.20730	0.217226
$219$	9.45479	0.638895
$220$	−3.79793	−0.256057
$221$	−1.48585	−0.0999489
$222$	−2.28577	−0.153411
$223$	10.0658	0.674058	0.337029	−	0.941494i	$-0.390578\pi$
$223$	10.0658	0.674058	0.337029	+	0.941494i	$0.390578\pi$
$224$	0	0
$225$	9.42429	0.628286
$226$	−6.37887	−0.424316
$227$	1.58055	0.104905	0.0524525	−	0.998623i	$-0.483296\pi$
$227$	1.58055	0.104905	0.0524525	+	0.998623i	$0.483296\pi$
$228$	4.15894	0.275433
$229$	−23.5969	−1.55933	−0.779664	−	0.626198i	$-0.784610\pi$
$229$	−23.5969	−1.55933	−0.779664	+	0.626198i	$0.784610\pi$
$230$	−2.06099	−0.135898
$231$	0	0
$232$	0.767438	0.0503848
$233$	−6.75774	−0.442715	−0.221357	−	0.975193i	$-0.571049\pi$
$233$	−6.75774	−0.442715	−0.221357	+	0.975193i	$0.571049\pi$
$234$	−0.361009	−0.0235999
$235$	−11.3458	−0.740120
$236$	−2.25689	−0.146911
$237$	−0.0321169	−0.00208622
$238$	0	0
$239$	−10.8642	−0.702744	−0.351372	−	0.936236i	$-0.614285\pi$
$239$	−10.8642	−0.702744	−0.351372	+	0.936236i	$0.614285\pi$
$240$	3.79793	0.245155
$241$	−10.8988	−0.702055	−0.351027	−	0.936365i	$-0.614168\pi$
$241$	−10.8988	−0.702055	−0.351027	+	0.936365i	$0.614168\pi$
$242$	1.00000	0.0642824
$243$	−1.00000	−0.0641500
$244$	−12.2748	−0.785811
$245$	0	0
$246$	9.13690	0.582548
$247$	1.50142	0.0955328
$248$	8.80801	0.559309
$249$	10.8690	0.688795
$250$	−16.8032	−1.06273
$251$	14.4885	0.914508	0.457254	−	0.889336i	$-0.348833\pi$
$251$	14.4885	0.914508	0.457254	+	0.889336i	$0.348833\pi$
$252$	0	0
$253$	0.542661	0.0341168
$254$	−10.7101	−0.672009
$255$	15.6316	0.978888
$256$	1.00000	0.0625000
$257$	−21.6591	−1.35106	−0.675530	−	0.737332i	$-0.736085\pi$
$257$	−21.6591	−1.35106	−0.675530	+	0.737332i	$0.736085\pi$
$258$	10.1995	0.634994
$259$	0	0
$260$	1.37109	0.0850313
$261$	0.767438	0.0475032
$262$	−11.8732	−0.733527
$263$	−13.6890	−0.844098	−0.422049	−	0.906573i	$-0.638689\pi$
$263$	−13.6890	−0.844098	−0.422049	+	0.906573i	$0.638689\pi$
$264$	−1.00000	−0.0615457
$265$	47.3086	2.90614
$266$	0	0
$267$	−1.29585	−0.0793044
$268$	0.603650	0.0368738
$269$	−9.45479	−0.576469	−0.288234	−	0.957560i	$-0.593068\pi$
$269$	−9.45479	−0.576469	−0.288234	+	0.957560i	$0.593068\pi$
$270$	3.79793	0.231135
$271$	−0.363303	−0.0220691	−0.0110346	−	0.999939i	$-0.503512\pi$
$271$	−0.363303	−0.0220691	−0.0110346	+	0.999939i	$0.503512\pi$
$272$	4.11582	0.249558
$273$	0	0
$274$	−3.21699	−0.194346
$275$	9.42429	0.568306
$276$	−0.542661	−0.0326643
$277$	20.4445	1.22839	0.614194	−	0.789155i	$-0.289481\pi$
$277$	20.4445	1.22839	0.614194	+	0.789155i	$0.289481\pi$
$278$	−0.608497	−0.0364952
$279$	8.80801	0.527322
$280$	0	0
$281$	−9.55045	−0.569732	−0.284866	−	0.958567i	$-0.591949\pi$
$281$	−9.55045	−0.569732	−0.284866	+	0.958567i	$0.591949\pi$
$282$	−2.98737	−0.177895
$283$	1.73465	0.103114	0.0515571	−	0.998670i	$-0.483582\pi$
$283$	1.73465	0.103114	0.0515571	+	0.998670i	$0.483582\pi$
$284$	−6.87385	−0.407888
$285$	−15.7954	−0.935638
$286$	−0.361009	−0.0213469
$287$	0	0
$288$	1.00000	0.0589256
$289$	−0.0600590	−0.00353288
$290$	−2.91468	−0.171156
$291$	−18.2628	−1.07058
$292$	−9.45479	−0.553300
$293$	28.7922	1.68206	0.841028	−	0.540992i	$-0.181951\pi$
$293$	28.7922	1.68206	0.841028	+	0.540992i	$0.181951\pi$
$294$	0	0
$295$	8.57153	0.499054
$296$	2.28577	0.132857
$297$	−1.00000	−0.0580259
$298$	−14.1064	−0.817162
$299$	−0.195905	−0.0113295
$300$	−9.42429	−0.544112
$301$	0	0
$302$	−3.42847	−0.197286
$303$	2.26790	0.130288
$304$	−4.15894	−0.238532
$305$	46.6187	2.66938
$306$	4.11582	0.235286
$307$	21.4066	1.22174	0.610868	−	0.791732i	$-0.290820\pi$
$307$	21.4066	1.22174	0.610868	+	0.791732i	$0.290820\pi$
$308$	0	0
$309$	−13.4438	−0.764790
$310$	−33.4522	−1.89996
$311$	8.88096	0.503593	0.251797	−	0.967780i	$-0.418979\pi$
$311$	8.88096	0.503593	0.251797	+	0.967780i	$0.418979\pi$
$312$	0.361009	0.0204381
$313$	−17.1100	−0.967117	−0.483558	−	0.875312i	$-0.660656\pi$
$313$	−17.1100	−0.967117	−0.483558	+	0.875312i	$0.660656\pi$
$314$	16.8833	0.952777
$315$	0	0
$316$	0.0321169	0.00180672
$317$	25.9408	1.45698	0.728489	−	0.685057i	$-0.240223\pi$
$317$	25.9408	1.45698	0.728489	+	0.685057i	$0.240223\pi$
$318$	12.4564	0.698521
$319$	0.767438	0.0429683
$320$	−3.79793	−0.212311
$321$	6.79956	0.379514
$322$	0	0
$323$	−17.1174	−0.952440
$324$	1.00000	0.0555556
$325$	−3.40225	−0.188723
$326$	−15.8275	−0.876604
$327$	−3.20730	−0.177364
$328$	−9.13690	−0.504501
$329$	0	0
$330$	3.79793	0.209069
$331$	−26.0880	−1.43393	−0.716963	−	0.697111i	$-0.754468\pi$
$331$	−26.0880	−1.43393	−0.716963	+	0.697111i	$0.754468\pi$
$332$	−10.8690	−0.596514
$333$	2.28577	0.125259
$334$	−25.3981	−1.38972
$335$	−2.29262	−0.125259
$336$	0	0
$337$	19.8206	1.07970	0.539850	−	0.841761i	$-0.318481\pi$
$337$	19.8206	1.07970	0.539850	+	0.841761i	$0.318481\pi$
$338$	−12.8697	−0.700018
$339$	6.37887	0.346453
$340$	−15.6316	−0.847742
$341$	8.80801	0.476981
$342$	−4.15894	−0.224890
$343$	0	0
$344$	−10.1995	−0.549921
$345$	2.06099	0.110960
$346$	−12.1207	−0.651611
$347$	−25.3050	−1.35844	−0.679222	−	0.733933i	$-0.737682\pi$
$347$	−25.3050	−1.35844	−0.679222	+	0.733933i	$0.737682\pi$
$348$	−0.767438	−0.0411390
$349$	−0.556914	−0.0298109	−0.0149055	−	0.999889i	$-0.504745\pi$
$349$	−0.556914	−0.0298109	−0.0149055	+	0.999889i	$0.504745\pi$
$350$	0	0
$351$	0.361009	0.0192692
$352$	1.00000	0.0533002
$353$	−8.84629	−0.470841	−0.235420	−	0.971894i	$-0.575647\pi$
$353$	−8.84629	−0.470841	−0.235420	+	0.971894i	$0.575647\pi$
$354$	2.25689	0.119953
$355$	26.1064	1.38558
$356$	1.29585	0.0686797
$357$	0	0
$358$	22.5376	1.19115
$359$	2.49724	0.131799	0.0658997	−	0.997826i	$-0.479008\pi$
$359$	2.49724	0.131799	0.0658997	+	0.997826i	$0.479008\pi$
$360$	−3.79793	−0.200169
$361$	−1.70320	−0.0896423
$362$	21.4905	1.12952
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	35.9086	1.87954
$366$	12.2748	0.641612
$367$	−5.25757	−0.274443	−0.137221	−	0.990540i	$-0.543817\pi$
$367$	−5.25757	−0.274443	−0.137221	+	0.990540i	$0.543817\pi$
$368$	0.542661	0.0282881
$369$	−9.13690	−0.475648
$370$	−8.68119	−0.451313
$371$	0	0
$372$	−8.80801	−0.456674
$373$	−6.95365	−0.360046	−0.180023	−	0.983662i	$-0.557617\pi$
$373$	−6.95365	−0.360046	−0.180023	+	0.983662i	$0.557617\pi$
$374$	4.11582	0.212824
$375$	16.8032	0.867712
$376$	2.98737	0.154062
$377$	−0.277052	−0.0142689
$378$	0	0
$379$	36.7981	1.89019	0.945095	−	0.326797i	$-0.105969\pi$
$379$	36.7981	1.89019	0.945095	+	0.326797i	$0.105969\pi$
$380$	15.7954	0.810286
$381$	10.7101	0.548693
$382$	−5.80049	−0.296779
$383$	23.1190	1.18133	0.590664	−	0.806918i	$-0.298866\pi$
$383$	23.1190	1.18133	0.590664	+	0.806918i	$0.298866\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	26.1623	1.33163
$387$	−10.1995	−0.518470
$388$	18.2628	0.927153
$389$	−14.5715	−0.738806	−0.369403	−	0.929269i	$-0.620438\pi$
$389$	−14.5715	−0.738806	−0.369403	+	0.929269i	$0.620438\pi$
$390$	−1.37109	−0.0694277
$391$	2.23349	0.112953
$392$	0	0
$393$	11.8732	0.598922
$394$	−16.9958	−0.856237
$395$	−0.121978	−0.00613737
$396$	1.00000	0.0502519
$397$	−14.7075	−0.738149	−0.369074	−	0.929400i	$-0.620325\pi$
$397$	−14.7075	−0.738149	−0.369074	+	0.929400i	$0.620325\pi$
$398$	−25.9745	−1.30198
$399$	0	0
$400$	9.42429	0.471215
$401$	−4.09672	−0.204580	−0.102290	−	0.994755i	$-0.532617\pi$
$401$	−4.09672	−0.204580	−0.102290	+	0.994755i	$0.532617\pi$
$402$	−0.603650	−0.0301073
$403$	−3.17977	−0.158396
$404$	−2.26790	−0.112832
$405$	−3.79793	−0.188721
$406$	0	0
$407$	2.28577	0.113301
$408$	−4.11582	−0.203763
$409$	5.41396	0.267703	0.133851	−	0.991001i	$-0.457266\pi$
$409$	5.41396	0.267703	0.133851	+	0.991001i	$0.457266\pi$
$410$	34.7013	1.71378
$411$	3.21699	0.158683
$412$	13.4438	0.662327
$413$	0	0
$414$	0.542661	0.0266703
$415$	41.2797	2.02634
$416$	−0.361009	−0.0176999
$417$	0.608497	0.0297982
$418$	−4.15894	−0.203420
$419$	20.1097	0.982421	0.491210	−	0.871041i	$-0.336555\pi$
$419$	20.1097	0.982421	0.491210	+	0.871041i	$0.336555\pi$
$420$	0	0
$421$	33.0192	1.60926	0.804629	−	0.593777i	$-0.202364\pi$
$421$	33.0192	1.60926	0.804629	+	0.593777i	$0.202364\pi$
$422$	12.7555	0.620927
$423$	2.98737	0.145251
$424$	−12.4564	−0.604937
$425$	38.7887	1.88153
$426$	6.87385	0.333039
$427$	0	0
$428$	−6.79956	−0.328669
$429$	0.361009	0.0174297
$430$	38.7371	1.86807
$431$	26.1385	1.25905	0.629524	−	0.776981i	$-0.283250\pi$
$431$	26.1385	1.25905	0.629524	+	0.776981i	$0.283250\pi$
$432$	−1.00000	−0.0481125
$433$	−34.7124	−1.66817	−0.834085	−	0.551637i	$-0.814004\pi$
$433$	−34.7124	−1.66817	−0.834085	+	0.551637i	$0.814004\pi$
$434$	0	0
$435$	2.91468	0.139748
$436$	3.20730	0.153602
$437$	−2.25689	−0.107962
$438$	9.45479	0.451767
$439$	−15.4000	−0.735000	−0.367500	−	0.930024i	$-0.619786\pi$
$439$	−15.4000	−0.735000	−0.367500	+	0.930024i	$0.619786\pi$
$440$	−3.79793	−0.181059
$441$	0	0
$442$	−1.48585	−0.0706745
$443$	1.75321	0.0832973	0.0416486	−	0.999132i	$-0.486739\pi$
$443$	1.75321	0.0832973	0.0416486	+	0.999132i	$0.486739\pi$
$444$	−2.28577	−0.108478
$445$	−4.92153	−0.233303
$446$	10.0658	0.476631
$447$	14.1064	0.667210
$448$	0	0
$449$	0.549515	0.0259332	0.0129666	−	0.999916i	$-0.495872\pi$
$449$	0.549515	0.0259332	0.0129666	+	0.999916i	$0.495872\pi$
$450$	9.42429	0.444265
$451$	−9.13690	−0.430240
$452$	−6.37887	−0.300037
$453$	3.42847	0.161083
$454$	1.58055	0.0741791
$455$	0	0
$456$	4.15894	0.194760
$457$	−11.9339	−0.558245	−0.279122	−	0.960256i	$-0.590043\pi$
$457$	−11.9339	−0.558245	−0.279122	+	0.960256i	$0.590043\pi$
$458$	−23.5969	−1.10261
$459$	−4.11582	−0.192110
$460$	−2.06099	−0.0960941
$461$	−29.3091	−1.36506	−0.682532	−	0.730856i	$-0.739121\pi$
$461$	−29.3091	−1.36506	−0.682532	+	0.730856i	$0.739121\pi$
$462$	0	0
$463$	4.75774	0.221111	0.110556	−	0.993870i	$-0.464737\pi$
$463$	4.75774	0.221111	0.110556	+	0.993870i	$0.464737\pi$
$464$	0.767438	0.0356274
$465$	33.4522	1.55131
$466$	−6.75774	−0.313046
$467$	33.4201	1.54650	0.773249	−	0.634102i	$-0.218630\pi$
$467$	33.4201	1.54650	0.773249	+	0.634102i	$0.218630\pi$
$468$	−0.361009	−0.0166876
$469$	0	0
$470$	−11.3458	−0.523344
$471$	−16.8833	−0.777939
$472$	−2.25689	−0.103882
$473$	−10.1995	−0.468974
$474$	−0.0321169	−0.00147518
$475$	−39.1951	−1.79839
$476$	0	0
$477$	−12.4564	−0.570340
$478$	−10.8642	−0.496915
$479$	39.8087	1.81891	0.909453	−	0.415808i	$-0.136501\pi$
$479$	39.8087	1.81891	0.909453	+	0.415808i	$0.136501\pi$
$480$	3.79793	0.173351
$481$	−0.825182	−0.0376251
$482$	−10.8988	−0.496428
$483$	0	0
$484$	1.00000	0.0454545
$485$	−69.3609	−3.14952
$486$	−1.00000	−0.0453609
$487$	2.97206	0.134677	0.0673384	−	0.997730i	$-0.478549\pi$
$487$	2.97206	0.134677	0.0673384	+	0.997730i	$0.478549\pi$
$488$	−12.2748	−0.555652
$489$	15.8275	0.715744
$490$	0	0
$491$	−37.6972	−1.70125	−0.850625	−	0.525773i	$-0.823776\pi$
$491$	−37.6972	−1.70125	−0.850625	+	0.525773i	$0.823776\pi$
$492$	9.13690	0.411923
$493$	3.15863	0.142258
$494$	1.50142	0.0675519
$495$	−3.79793	−0.170704
$496$	8.80801	0.395491
$497$	0	0
$498$	10.8690	0.487052
$499$	44.5292	1.99340	0.996701	−	0.0811669i	$-0.0258647\pi$
$499$	44.5292	1.99340	0.996701	+	0.0811669i	$0.0258647\pi$
$500$	−16.8032	−0.751460
$501$	25.3981	1.13470
$502$	14.4885	0.646655
$503$	22.3477	0.996436	0.498218	−	0.867052i	$-0.333988\pi$
$503$	22.3477	0.996436	0.498218	+	0.867052i	$0.333988\pi$
$504$	0	0
$505$	8.61334	0.383289
$506$	0.542661	0.0241242
$507$	12.8697	0.571562
$508$	−10.7101	−0.475182
$509$	−2.35390	−0.104335	−0.0521673	−	0.998638i	$-0.516613\pi$
$509$	−2.35390	−0.104335	−0.0521673	+	0.998638i	$0.516613\pi$
$510$	15.6316	0.692179
$511$	0	0
$512$	1.00000	0.0441942
$513$	4.15894	0.183622
$514$	−21.6591	−0.955344
$515$	−51.0586	−2.24991
$516$	10.1995	0.449009
$517$	2.98737	0.131384
$518$	0	0
$519$	12.1207	0.532038
$520$	1.37109	0.0601262
$521$	2.58940	0.113443	0.0567217	−	0.998390i	$-0.481935\pi$
$521$	2.58940	0.113443	0.0567217	+	0.998390i	$0.481935\pi$
$522$	0.767438	0.0335899
$523$	20.4671	0.894965	0.447483	−	0.894293i	$-0.352321\pi$
$523$	20.4671	0.894965	0.447483	+	0.894293i	$0.352321\pi$
$524$	−11.8732	−0.518682
$525$	0	0
$526$	−13.6890	−0.596868
$527$	36.2522	1.57917
$528$	−1.00000	−0.0435194
$529$	−22.7055	−0.987196
$530$	47.3086	2.05495
$531$	−2.25689	−0.0979409
$532$	0	0
$533$	3.29850	0.142874
$534$	−1.29585	−0.0560767
$535$	25.8243	1.11648
$536$	0.603650	0.0260737
$537$	−22.5376	−0.972567
$538$	−9.45479	−0.407625
$539$	0	0
$540$	3.79793	0.163437
$541$	6.31788	0.271627	0.135814	−	0.990734i	$-0.456635\pi$
$541$	6.31788	0.271627	0.135814	+	0.990734i	$0.456635\pi$
$542$	−0.363303	−0.0156052
$543$	−21.4905	−0.922246
$544$	4.11582	0.176464
$545$	−12.1811	−0.521781
$546$	0	0
$547$	−10.6371	−0.454810	−0.227405	−	0.973800i	$-0.573024\pi$
$547$	−10.6371	−0.454810	−0.227405	+	0.973800i	$0.573024\pi$
$548$	−3.21699	−0.137423
$549$	−12.2748	−0.523874
$550$	9.42429	0.401853
$551$	−3.19173	−0.135972
$552$	−0.542661	−0.0230972
$553$	0	0
$554$	20.4445	0.868601
$555$	8.68119	0.368496
$556$	−0.608497	−0.0258060
$557$	−13.6316	−0.577589	−0.288795	−	0.957391i	$-0.593254\pi$
$557$	−13.6316	−0.577589	−0.288795	+	0.957391i	$0.593254\pi$
$558$	8.80801	0.372873
$559$	3.68212	0.155737
$560$	0	0
$561$	−4.11582	−0.173770
$562$	−9.55045	−0.402861
$563$	37.8882	1.59680	0.798399	−	0.602128i	$-0.205681\pi$
$563$	37.8882	1.59680	0.798399	+	0.602128i	$0.205681\pi$
$564$	−2.98737	−0.125791
$565$	24.2265	1.01922
$566$	1.73465	0.0729127
$567$	0	0
$568$	−6.87385	−0.288420
$569$	0.799920	0.0335344	0.0167672	−	0.999859i	$-0.494663\pi$
$569$	0.799920	0.0335344	0.0167672	+	0.999859i	$0.494663\pi$
$570$	−15.7954	−0.661596
$571$	−31.3949	−1.31383	−0.656917	−	0.753963i	$-0.728140\pi$
$571$	−31.3949	−1.31383	−0.656917	+	0.753963i	$0.728140\pi$
$572$	−0.361009	−0.0150945
$573$	5.80049	0.242319
$574$	0	0
$575$	5.11419	0.213277
$576$	1.00000	0.0416667
$577$	−35.6375	−1.48361	−0.741804	−	0.670617i	$-0.766030\pi$
$577$	−35.6375	−1.48361	−0.741804	+	0.670617i	$0.766030\pi$
$578$	−0.0600590	−0.00249813
$579$	−26.1623	−1.08727
$580$	−2.91468	−0.121025
$581$	0	0
$582$	−18.2628	−0.757017
$583$	−12.4564	−0.515892
$584$	−9.45479	−0.391242
$585$	1.37109	0.0566875
$586$	28.7922	1.18939
$587$	−28.0811	−1.15903	−0.579516	−	0.814961i	$-0.696759\pi$
$587$	−28.0811	−1.15903	−0.579516	+	0.814961i	$0.696759\pi$
$588$	0	0
$589$	−36.6320	−1.50940
$590$	8.57153	0.352884
$591$	16.9958	0.699115
$592$	2.28577	0.0939444
$593$	−10.3370	−0.424489	−0.212245	−	0.977217i	$-0.568077\pi$
$593$	−10.3370	−0.424489	−0.212245	+	0.977217i	$0.568077\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	−14.1064	−0.577821
$597$	25.9745	1.06306
$598$	−0.195905	−0.00801117
$599$	36.5949	1.49523	0.747614	−	0.664133i	$-0.231199\pi$
$599$	36.5949	1.49523	0.747614	+	0.664133i	$0.231199\pi$
$600$	−9.42429	−0.384745
$601$	−34.9800	−1.42686	−0.713431	−	0.700725i	$-0.752860\pi$
$601$	−34.9800	−1.42686	−0.713431	+	0.700725i	$0.752860\pi$
$602$	0	0
$603$	0.603650	0.0245825
$604$	−3.42847	−0.139502
$605$	−3.79793	−0.154408
$606$	2.26790	0.0921273
$607$	−0.0642338	−0.00260717	−0.00130359	−	0.999999i	$-0.500415\pi$
$607$	−0.0642338	−0.00260717	−0.00130359	+	0.999999i	$0.500415\pi$
$608$	−4.15894	−0.168667
$609$	0	0
$610$	46.6187	1.88754
$611$	−1.07847	−0.0436301
$612$	4.11582	0.166372
$613$	1.17157	0.0473194	0.0236597	−	0.999720i	$-0.492468\pi$
$613$	1.17157	0.0473194	0.0236597	+	0.999720i	$0.492468\pi$
$614$	21.4066	0.863898
$615$	−34.7013	−1.39929
$616$	0	0
$617$	−3.40965	−0.137267	−0.0686337	−	0.997642i	$-0.521864\pi$
$617$	−3.40965	−0.137267	−0.0686337	+	0.997642i	$0.521864\pi$
$618$	−13.4438	−0.540788
$619$	−2.33804	−0.0939738	−0.0469869	−	0.998896i	$-0.514962\pi$
$619$	−2.33804	−0.0939738	−0.0469869	+	0.998896i	$0.514962\pi$
$620$	−33.4522	−1.34347
$621$	−0.542661	−0.0217762
$622$	8.88096	0.356094
$623$	0	0
$624$	0.361009	0.0144519
$625$	16.6958	0.667833
$626$	−17.1100	−0.683855
$627$	4.15894	0.166092
$628$	16.8833	0.673715
$629$	9.40779	0.375113
$630$	0	0
$631$	11.0124	0.438396	0.219198	−	0.975680i	$-0.429656\pi$
$631$	11.0124	0.438396	0.219198	+	0.975680i	$0.429656\pi$
$632$	0.0321169	0.00127754
$633$	−12.7555	−0.506985
$634$	25.9408	1.03024
$635$	40.6761	1.61418
$636$	12.4564	0.493929
$637$	0	0
$638$	0.767438	0.0303832
$639$	−6.87385	−0.271925
$640$	−3.79793	−0.150126
$641$	−7.83260	−0.309369	−0.154685	−	0.987964i	$-0.549436\pi$
$641$	−7.83260	−0.309369	−0.154685	+	0.987964i	$0.549436\pi$
$642$	6.79956	0.268357
$643$	17.4486	0.688107	0.344053	−	0.938950i	$-0.388200\pi$
$643$	17.4486	0.688107	0.344053	+	0.938950i	$0.388200\pi$
$644$	0	0
$645$	−38.7371	−1.52527
$646$	−17.1174	−0.673477
$647$	43.7782	1.72110	0.860550	−	0.509367i	$-0.170120\pi$
$647$	43.7782	1.72110	0.860550	+	0.509367i	$0.170120\pi$
$648$	1.00000	0.0392837
$649$	−2.25689	−0.0885909
$650$	−3.40225	−0.133447
$651$	0	0
$652$	−15.8275	−0.619853
$653$	−43.8119	−1.71449	−0.857247	−	0.514906i	$-0.827827\pi$
$653$	−43.8119	−1.71449	−0.857247	+	0.514906i	$0.827827\pi$
$654$	−3.20730	−0.125415
$655$	45.0935	1.76195
$656$	−9.13690	−0.356736
$657$	−9.45479	−0.368866
$658$	0	0
$659$	−39.0838	−1.52249	−0.761245	−	0.648465i	$-0.775411\pi$
$659$	−39.0838	−1.52249	−0.761245	+	0.648465i	$0.775411\pi$
$660$	3.79793	0.147834
$661$	12.2832	0.477762	0.238881	−	0.971049i	$-0.423219\pi$
$661$	12.2832	0.477762	0.238881	+	0.971049i	$0.423219\pi$
$662$	−26.0880	−1.01394
$663$	1.48585	0.0577055
$664$	−10.8690	−0.421799
$665$	0	0
$666$	2.28577	0.0885716
$667$	0.416459	0.0161253
$668$	−25.3981	−0.982682
$669$	−10.0658	−0.389168
$670$	−2.29262	−0.0885717
$671$	−12.2748	−0.473862
$672$	0	0
$673$	−4.69315	−0.180907	−0.0904537	−	0.995901i	$-0.528832\pi$
$673$	−4.69315	−0.180907	−0.0904537	+	0.995901i	$0.528832\pi$
$674$	19.8206	0.763463
$675$	−9.42429	−0.362741
$676$	−12.8697	−0.494987
$677$	43.7114	1.67997	0.839983	−	0.542612i	$-0.182565\pi$
$677$	43.7114	1.67997	0.839983	+	0.542612i	$0.182565\pi$
$678$	6.37887	0.244979
$679$	0	0
$680$	−15.6316	−0.599444
$681$	−1.58055	−0.0605670
$682$	8.80801	0.337276
$683$	−29.7156	−1.13703	−0.568517	−	0.822671i	$-0.692483\pi$
$683$	−29.7156	−1.13703	−0.568517	+	0.822671i	$0.692483\pi$
$684$	−4.15894	−0.159021
$685$	12.2179	0.466823
$686$	0	0
$687$	23.5969	0.900279
$688$	−10.1995	−0.388853
$689$	4.49688	0.171317
$690$	2.06099	0.0784605
$691$	−9.23070	−0.351152	−0.175576	−	0.984466i	$-0.556179\pi$
$691$	−9.23070	−0.351152	−0.175576	+	0.984466i	$0.556179\pi$
$692$	−12.1207	−0.460758
$693$	0	0
$694$	−25.3050	−0.960564
$695$	2.31103	0.0876623
$696$	−0.767438	−0.0290897
$697$	−37.6058	−1.42442
$698$	−0.556914	−0.0210795
$699$	6.75774	0.255601
$700$	0	0
$701$	−15.3559	−0.579984	−0.289992	−	0.957029i	$-0.593653\pi$
$701$	−15.3559	−0.579984	−0.289992	+	0.957029i	$0.593653\pi$
$702$	0.361009	0.0136254
$703$	−9.50637	−0.358539
$704$	1.00000	0.0376889
$705$	11.3458	0.427309
$706$	−8.84629	−0.332935
$707$	0	0
$708$	2.25689	0.0848193
$709$	−23.4425	−0.880403	−0.440202	−	0.897899i	$-0.645093\pi$
$709$	−23.4425	−0.880403	−0.440202	+	0.897899i	$0.645093\pi$
$710$	26.1064	0.979756
$711$	0.0321169	0.00120448
$712$	1.29585	0.0485639
$713$	4.77976	0.179003
$714$	0	0
$715$	1.37109	0.0512758
$716$	22.5376	0.842268
$717$	10.8642	0.405729
$718$	2.49724	0.0931962
$719$	22.1791	0.827141	0.413570	−	0.910472i	$-0.364282\pi$
$719$	22.1791	0.827141	0.413570	+	0.910472i	$0.364282\pi$
$720$	−3.79793	−0.141541
$721$	0	0
$722$	−1.70320	−0.0633867
$723$	10.8988	0.405332
$724$	21.4905	0.798689
$725$	7.23256	0.268611
$726$	−1.00000	−0.0371135
$727$	−26.8736	−0.996686	−0.498343	−	0.866980i	$-0.666058\pi$
$727$	−26.8736	−0.996686	−0.498343	+	0.866980i	$0.666058\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	35.9086	1.32904
$731$	−41.9793	−1.55266
$732$	12.2748	0.453688
$733$	−42.2729	−1.56139	−0.780693	−	0.624915i	$-0.785133\pi$
$733$	−42.2729	−1.56139	−0.780693	+	0.624915i	$0.785133\pi$
$734$	−5.25757	−0.194060
$735$	0	0
$736$	0.542661	0.0200027
$737$	0.603650	0.0222357
$738$	−9.13690	−0.336334
$739$	−18.5463	−0.682236	−0.341118	−	0.940021i	$-0.610806\pi$
$739$	−18.5463	−0.682236	−0.341118	+	0.940021i	$0.610806\pi$
$740$	−8.68119	−0.319127
$741$	−1.50142	−0.0551559
$742$	0	0
$743$	26.6917	0.979222	0.489611	−	0.871941i	$-0.337139\pi$
$743$	26.6917	0.979222	0.489611	+	0.871941i	$0.337139\pi$
$744$	−8.80801	−0.322917
$745$	53.5752	1.96284
$746$	−6.95365	−0.254591
$747$	−10.8690	−0.397676
$748$	4.11582	0.150489
$749$	0	0
$750$	16.8032	0.613565
$751$	−14.1638	−0.516844	−0.258422	−	0.966032i	$-0.583202\pi$
$751$	−14.1638	−0.516844	−0.258422	+	0.966032i	$0.583202\pi$
$752$	2.98737	0.108938
$753$	−14.4885	−0.527991
$754$	−0.277052	−0.0100896
$755$	13.0211	0.473886
$756$	0	0
$757$	23.3069	0.847102	0.423551	−	0.905872i	$-0.360783\pi$
$757$	23.3069	0.847102	0.423551	+	0.905872i	$0.360783\pi$
$758$	36.7981	1.33657
$759$	−0.542661	−0.0196973
$760$	15.7954	0.572959
$761$	−39.2524	−1.42290	−0.711450	−	0.702737i	$-0.751961\pi$
$761$	−39.2524	−1.42290	−0.711450	+	0.702737i	$0.751961\pi$
$762$	10.7101	0.387984
$763$	0	0
$764$	−5.80049	−0.209854
$765$	−15.6316	−0.565161
$766$	23.1190	0.835325
$767$	0.814759	0.0294192
$768$	−1.00000	−0.0360844
$769$	35.6372	1.28511	0.642556	−	0.766239i	$-0.277874\pi$
$769$	35.6372	1.28511	0.642556	+	0.766239i	$0.277874\pi$
$770$	0	0
$771$	21.6591	0.780035
$772$	26.1623	0.941602
$773$	−32.6368	−1.17386	−0.586932	−	0.809636i	$-0.699665\pi$
$773$	−32.6368	−1.17386	−0.586932	+	0.809636i	$0.699665\pi$
$774$	−10.1995	−0.366614
$775$	83.0093	2.98178
$776$	18.2628	0.655596
$777$	0	0
$778$	−14.5715	−0.522415
$779$	37.9998	1.36149
$780$	−1.37109	−0.0490928
$781$	−6.87385	−0.245966
$782$	2.23349	0.0798695
$783$	−0.767438	−0.0274260
$784$	0	0
$785$	−64.1215	−2.28859
$786$	11.8732	0.423502
$787$	−14.0832	−0.502010	−0.251005	−	0.967986i	$-0.580761\pi$
$787$	−14.0832	−0.502010	−0.251005	+	0.967986i	$0.580761\pi$
$788$	−16.9958	−0.605451
$789$	13.6890	0.487340
$790$	−0.121978	−0.00433978
$791$	0	0
$792$	1.00000	0.0355335
$793$	4.43130	0.157360
$794$	−14.7075	−0.521950
$795$	−47.3086	−1.67786
$796$	−25.9745	−0.920641
$797$	21.3873	0.757579	0.378789	−	0.925483i	$-0.376340\pi$
$797$	21.3873	0.757579	0.378789	+	0.925483i	$0.376340\pi$
$798$	0	0
$799$	12.2955	0.434982
$800$	9.42429	0.333199
$801$	1.29585	0.0457864
$802$	−4.09672	−0.144660
$803$	−9.45479	−0.333652
$804$	−0.603650	−0.0212891
$805$	0	0
$806$	−3.17977	−0.112003
$807$	9.45479	0.332824
$808$	−2.26790	−0.0797846
$809$	−8.69315	−0.305635	−0.152817	−	0.988254i	$-0.548835\pi$
$809$	−8.69315	−0.305635	−0.152817	+	0.988254i	$0.548835\pi$
$810$	−3.79793	−0.133446
$811$	44.5886	1.56572	0.782859	−	0.622200i	$-0.213761\pi$
$811$	44.5886	1.56572	0.782859	+	0.622200i	$0.213761\pi$
$812$	0	0
$813$	0.363303	0.0127416
$814$	2.28577	0.0801161
$815$	60.1118	2.10562
$816$	−4.11582	−0.144082
$817$	42.4192	1.48406
$818$	5.41396	0.189295
$819$	0	0
$820$	34.7013	1.21182
$821$	49.5824	1.73044	0.865219	−	0.501394i	$-0.167179\pi$
$821$	49.5824	1.73044	0.865219	+	0.501394i	$0.167179\pi$
$822$	3.21699	0.112206
$823$	−24.9132	−0.868419	−0.434209	−	0.900812i	$-0.642972\pi$
$823$	−24.9132	−0.868419	−0.434209	+	0.900812i	$0.642972\pi$
$824$	13.4438	0.468336
$825$	−9.42429	−0.328112
$826$	0	0
$827$	42.7183	1.48546	0.742730	−	0.669591i	$-0.233531\pi$
$827$	42.7183	1.48546	0.742730	+	0.669591i	$0.233531\pi$
$828$	0.542661	0.0188588
$829$	9.29198	0.322724	0.161362	−	0.986895i	$-0.448411\pi$
$829$	9.29198	0.322724	0.161362	+	0.986895i	$0.448411\pi$
$830$	41.2797	1.43284
$831$	−20.4445	−0.709210
$832$	−0.361009	−0.0125157
$833$	0	0
$834$	0.608497	0.0210705
$835$	96.4603	3.33815
$836$	−4.15894	−0.143840
$837$	−8.80801	−0.304449
$838$	20.1097	0.694676
$839$	14.4061	0.497355	0.248678	−	0.968586i	$-0.420004\pi$
$839$	14.4061	0.497355	0.248678	+	0.968586i	$0.420004\pi$
$840$	0	0
$841$	−28.4110	−0.979691
$842$	33.0192	1.13792
$843$	9.55045	0.328935
$844$	12.7555	0.439062
$845$	48.8781	1.68146
$846$	2.98737	0.102708
$847$	0	0
$848$	−12.4564	−0.427755
$849$	−1.73465	−0.0595330
$850$	38.7887	1.33044
$851$	1.24040	0.0425202
$852$	6.87385	0.235494
$853$	32.5161	1.11333	0.556665	−	0.830737i	$-0.312081\pi$
$853$	32.5161	1.11333	0.556665	+	0.830737i	$0.312081\pi$
$854$	0	0
$855$	15.7954	0.540191
$856$	−6.79956	−0.232404
$857$	−49.1364	−1.67847	−0.839234	−	0.543771i	$-0.816996\pi$
$857$	−49.1364	−1.67847	−0.839234	+	0.543771i	$0.816996\pi$
$858$	0.361009	0.0123246
$859$	−13.2430	−0.451846	−0.225923	−	0.974145i	$-0.572540\pi$
$859$	−13.2430	−0.451846	−0.225923	+	0.974145i	$0.572540\pi$
$860$	38.7371	1.32092
$861$	0	0
$862$	26.1385	0.890282
$863$	−49.8617	−1.69731	−0.848656	−	0.528944i	$-0.822588\pi$
$863$	−49.8617	−1.69731	−0.848656	+	0.528944i	$0.822588\pi$
$864$	−1.00000	−0.0340207
$865$	46.0335	1.56518
$866$	−34.7124	−1.17957
$867$	0.0600590	0.00203971
$868$	0	0
$869$	0.0321169	0.00108949
$870$	2.91468	0.0988169
$871$	−0.217923	−0.00738404
$872$	3.20730	0.108613
$873$	18.2628	0.618102
$874$	−2.25689	−0.0763406
$875$	0	0
$876$	9.45479	0.319448
$877$	48.5769	1.64033	0.820163	−	0.572131i	$-0.193883\pi$
$877$	48.5769	1.64033	0.820163	+	0.572131i	$0.193883\pi$
$878$	−15.4000	−0.519723
$879$	−28.7922	−0.971135
$880$	−3.79793	−0.128028
$881$	−16.1788	−0.545078	−0.272539	−	0.962145i	$-0.587863\pi$
$881$	−16.1788	−0.545078	−0.272539	+	0.962145i	$0.587863\pi$
$882$	0	0
$883$	−14.4340	−0.485742	−0.242871	−	0.970059i	$-0.578089\pi$
$883$	−14.4340	−0.485742	−0.242871	+	0.970059i	$0.578089\pi$
$884$	−1.48585	−0.0499744
$885$	−8.57153	−0.288129
$886$	1.75321	0.0589001
$887$	−6.14672	−0.206387	−0.103193	−	0.994661i	$-0.532906\pi$
$887$	−6.14672	−0.206387	−0.103193	+	0.994661i	$0.532906\pi$
$888$	−2.28577	−0.0767053
$889$	0	0
$890$	−4.92153	−0.164970
$891$	1.00000	0.0335013
$892$	10.0658	0.337029
$893$	−12.4243	−0.415763
$894$	14.1064	0.471789
$895$	−85.5961	−2.86116
$896$	0	0
$897$	0.195905	0.00654109
$898$	0.549515	0.0183376
$899$	6.75960	0.225445
$900$	9.42429	0.314143
$901$	−51.2683	−1.70799
$902$	−9.13690	−0.304226
$903$	0	0
$904$	−6.37887	−0.212158
$905$	−81.6195	−2.71312
$906$	3.42847	0.113903
$907$	23.7882	0.789873	0.394936	−	0.918708i	$-0.370767\pi$
$907$	23.7882	0.789873	0.394936	+	0.918708i	$0.370767\pi$
$908$	1.58055	0.0524525
$909$	−2.26790	−0.0752216
$910$	0	0
$911$	38.4526	1.27399	0.636997	−	0.770867i	$-0.280177\pi$
$911$	38.4526	1.27399	0.636997	+	0.770867i	$0.280177\pi$
$912$	4.15894	0.137716
$913$	−10.8690	−0.359711
$914$	−11.9339	−0.394739
$915$	−46.6187	−1.54117
$916$	−23.5969	−0.779664
$917$	0	0
$918$	−4.11582	−0.135842
$919$	16.3276	0.538597	0.269299	−	0.963057i	$-0.413208\pi$
$919$	16.3276	0.538597	0.269299	+	0.963057i	$0.413208\pi$
$920$	−2.06099	−0.0679488
$921$	−21.4066	−0.705370
$922$	−29.3091	−0.965245
$923$	2.48152	0.0816802
$924$	0	0
$925$	21.5417	0.708288
$926$	4.75774	0.156349
$927$	13.4438	0.441552
$928$	0.767438	0.0251924
$929$	−46.0270	−1.51010	−0.755049	−	0.655668i	$-0.772387\pi$
$929$	−46.0270	−1.51010	−0.755049	+	0.655668i	$0.772387\pi$
$930$	33.4522	1.09694
$931$	0	0
$932$	−6.75774	−0.221357
$933$	−8.88096	−0.290750
$934$	33.4201	1.09354
$935$	−15.6316	−0.511208
$936$	−0.361009	−0.0117999
$937$	19.6326	0.641371	0.320685	−	0.947186i	$-0.396087\pi$
$937$	19.6326	0.641371	0.320685	+	0.947186i	$0.396087\pi$
$938$	0	0
$939$	17.1100	0.558365
$940$	−11.3458	−0.370060
$941$	12.1279	0.395358	0.197679	−	0.980267i	$-0.436660\pi$
$941$	12.1279	0.395358	0.197679	+	0.980267i	$0.436660\pi$
$942$	−16.8833	−0.550086
$943$	−4.95824	−0.161462
$944$	−2.25689	−0.0734557
$945$	0	0
$946$	−10.1995	−0.331615
$947$	0.0403155	0.00131008	0.000655038	−	1.00000i	$-0.499791\pi$
$947$	0.0403155	0.00131008	0.000655038		1.00000i	$0.499791\pi$
$948$	−0.0321169	−0.00104311
$949$	3.41326	0.110799
$950$	−39.1951	−1.27166
$951$	−25.9408	−0.841187
$952$	0	0
$953$	45.7949	1.48344	0.741721	−	0.670709i	$-0.234010\pi$
$953$	45.7949	1.48344	0.741721	+	0.670709i	$0.234010\pi$
$954$	−12.4564	−0.403291
$955$	22.0299	0.712869
$956$	−10.8642	−0.351372
$957$	−0.767438	−0.0248078
$958$	39.8087	1.28616
$959$	0	0
$960$	3.79793	0.122578
$961$	46.5811	1.50262
$962$	−0.825182	−0.0266049
$963$	−6.79956	−0.219113
$964$	−10.8988	−0.351027
$965$	−99.3626	−3.19860
$966$	0	0
$967$	19.1340	0.615308	0.307654	−	0.951498i	$-0.400456\pi$
$967$	19.1340	0.615308	0.307654	+	0.951498i	$0.400456\pi$
$968$	1.00000	0.0321412
$969$	17.1174	0.549891
$970$	−69.3609	−2.22704
$971$	−58.3278	−1.87183	−0.935915	−	0.352227i	$-0.885425\pi$
$971$	−58.3278	−1.87183	−0.935915	+	0.352227i	$0.885425\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	2.97206	0.0952309
$975$	3.40225	0.108959
$976$	−12.2748	−0.392905
$977$	−32.1169	−1.02751	−0.513755	−	0.857937i	$-0.671746\pi$
$977$	−32.1169	−1.02751	−0.513755	+	0.857937i	$0.671746\pi$
$978$	15.8275	0.506108
$979$	1.29585	0.0414154
$980$	0	0
$981$	3.20730	0.102401
$982$	−37.6972	−1.20297
$983$	−14.1388	−0.450957	−0.225479	−	0.974248i	$-0.572395\pi$
$983$	−14.1388	−0.450957	−0.225479	+	0.974248i	$0.572395\pi$
$984$	9.13690	0.291274
$985$	64.5490	2.05670
$986$	3.15863	0.100591
$987$	0	0
$988$	1.50142	0.0477664
$989$	−5.53488	−0.175999
$990$	−3.79793	−0.120706
$991$	31.0458	0.986203	0.493102	−	0.869972i	$-0.335863\pi$
$991$	31.0458	0.986203	0.493102	+	0.869972i	$0.335863\pi$
$992$	8.80801	0.279655
$993$	26.0880	0.827878
$994$	0	0
$995$	98.6493	3.12739
$996$	10.8690	0.344397
$997$	−12.9422	−0.409885	−0.204942	−	0.978774i	$-0.565701\pi$
$997$	−12.9422	−0.409885	−0.204942	+	0.978774i	$0.565701\pi$
$998$	44.5292	1.40955
$999$	−2.28577	−0.0723184

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3234.2.a.bl.1.1	✓	4
3.2	odd	2		9702.2.a.ea.1.4		4
7.6	odd	2		3234.2.a.bm.1.4	yes	4
21.20	even	2		9702.2.a.dz.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3234.2.a.bl.1.1	✓	4	1.1	even	1	trivial
3234.2.a.bm.1.4	yes	4	7.6	odd	2
9702.2.a.dz.1.1		4	21.20	even	2
9702.2.a.ea.1.4		4	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:	\( N \)	\(=\)	\( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3234.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.79793 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -3.79793 q^{10} +1.00000 q^{11} -1.00000 q^{12} -0.361009 q^{13} +3.79793 q^{15} +1.00000 q^{16} +4.11582 q^{17} +1.00000 q^{18} -4.15894 q^{19} -3.79793 q^{20} +1.00000 q^{22} +0.542661 q^{23} -1.00000 q^{24} +9.42429 q^{25} -0.361009 q^{26} -1.00000 q^{27} +0.767438 q^{29} +3.79793 q^{30} +8.80801 q^{31} +1.00000 q^{32} -1.00000 q^{33} +4.11582 q^{34} +1.00000 q^{36} +2.28577 q^{37} -4.15894 q^{38} +0.361009 q^{39} -3.79793 q^{40} -9.13690 q^{41} -10.1995 q^{43} +1.00000 q^{44} -3.79793 q^{45} +0.542661 q^{46} +2.98737 q^{47} -1.00000 q^{48} +9.42429 q^{50} -4.11582 q^{51} -0.361009 q^{52} -12.4564 q^{53} -1.00000 q^{54} -3.79793 q^{55} +4.15894 q^{57} +0.767438 q^{58} -2.25689 q^{59} +3.79793 q^{60} -12.2748 q^{61} +8.80801 q^{62} +1.00000 q^{64} +1.37109 q^{65} -1.00000 q^{66} +0.603650 q^{67} +4.11582 q^{68} -0.542661 q^{69} -6.87385 q^{71} +1.00000 q^{72} -9.45479 q^{73} +2.28577 q^{74} -9.42429 q^{75} -4.15894 q^{76} +0.361009 q^{78} +0.0321169 q^{79} -3.79793 q^{80} +1.00000 q^{81} -9.13690 q^{82} -10.8690 q^{83} -15.6316 q^{85} -10.1995 q^{86} -0.767438 q^{87} +1.00000 q^{88} +1.29585 q^{89} -3.79793 q^{90} +0.542661 q^{92} -8.80801 q^{93} +2.98737 q^{94} +15.7954 q^{95} -1.00000 q^{96} +18.2628 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} - 4 q^{6} + 4 q^{8} + 4 q^{9} - 4 q^{10} + 4 q^{11} - 4 q^{12} - 8 q^{13} + 4 q^{15} + 4 q^{16} - 4 q^{17} + 4 q^{18} - 12 q^{19} - 4 q^{20} + 4 q^{22} - 8 q^{23}+ \cdots + 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^5 - 4 * q^6 + 4 * q^8 + 4 * q^9 - 4 * q^10 + 4 * q^11 - 4 * q^12 - 8 * q^13 + 4 * q^15 + 4 * q^16 - 4 * q^17 + 4 * q^18 - 12 * q^19 - 4 * q^20 + 4 * q^22 - 8 * q^23 - 4 * q^24 + 4 * q^25 - 8 * q^26 - 4 * q^27 - 8 * q^29 + 4 * q^30 - 4 * q^31 + 4 * q^32 - 4 * q^33 - 4 * q^34 + 4 * q^36 + 8 * q^37 - 12 * q^38 + 8 * q^39 - 4 * q^40 - 12 * q^41 - 8 * q^43 + 4 * q^44 - 4 * q^45 - 8 * q^46 - 4 * q^47 - 4 * q^48 + 4 * q^50 + 4 * q^51 - 8 * q^52 - 8 * q^53 - 4 * q^54 - 4 * q^55 + 12 * q^57 - 8 * q^58 + 4 * q^60 - 24 * q^61 - 4 * q^62 + 4 * q^64 - 16 * q^65 - 4 * q^66 - 8 * q^67 - 4 * q^68 + 8 * q^69 + 8 * q^71 + 4 * q^72 - 4 * q^73 + 8 * q^74 - 4 * q^75 - 12 * q^76 + 8 * q^78 - 8 * q^79 - 4 * q^80 + 4 * q^81 - 12 * q^82 - 4 * q^83 - 8 * q^85 - 8 * q^86 + 8 * q^87 + 4 * q^88 - 24 * q^89 - 4 * q^90 - 8 * q^92 + 4 * q^93 - 4 * q^94 + 8 * q^95 - 4 * q^96 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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