LMFDB - Embedded newform 9702.2.a.ea.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9702,2,Mod(1,9702)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9702, 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("9702.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9702.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:	$77.4708600410$
Analytic rank:	$0$
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:	no (minimal twist has level 3234)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.68554$ of defining polynomial
Character	$\chi$	$=$	9702.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +3.79793 q^{5} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +3.79793 q^{5} -1.00000 q^{8} -3.79793 q^{10} -1.00000 q^{11} -0.361009 q^{13} +1.00000 q^{16} -4.11582 q^{17} -4.15894 q^{19} +3.79793 q^{20} +1.00000 q^{22} -0.542661 q^{23} +9.42429 q^{25} +0.361009 q^{26} -0.767438 q^{29} +8.80801 q^{31} -1.00000 q^{32} +4.11582 q^{34} +2.28577 q^{37} +4.15894 q^{38} -3.79793 q^{40} +9.13690 q^{41} -10.1995 q^{43} -1.00000 q^{44} +0.542661 q^{46} -2.98737 q^{47} -9.42429 q^{50} -0.361009 q^{52} +12.4564 q^{53} -3.79793 q^{55} +0.767438 q^{58} +2.25689 q^{59} -12.2748 q^{61} -8.80801 q^{62} +1.00000 q^{64} -1.37109 q^{65} +0.603650 q^{67} -4.11582 q^{68} +6.87385 q^{71} -9.45479 q^{73} -2.28577 q^{74} -4.15894 q^{76} +0.0321169 q^{79} +3.79793 q^{80} -9.13690 q^{82} +10.8690 q^{83} -15.6316 q^{85} +10.1995 q^{86} +1.00000 q^{88} -1.29585 q^{89} -0.542661 q^{92} +2.98737 q^{94} -15.7954 q^{95} +18.2628 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} + 4 q^{5} - 4 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^5 - 4 * q^8	$ 4 q - 4 q^{2} + 4 q^{4} + 4 q^{5} - 4 q^{8} - 4 q^{10} - 4 q^{11} - 8 q^{13} + 4 q^{16} + 4 q^{17} - 12 q^{19} + 4 q^{20} + 4 q^{22} + 8 q^{23} + 4 q^{25} + 8 q^{26} + 8 q^{29} - 4 q^{31} - 4 q^{32} - 4 q^{34} + 8 q^{37} + 12 q^{38} - 4 q^{40} + 12 q^{41} - 8 q^{43} - 4 q^{44} - 8 q^{46} + 4 q^{47} - 4 q^{50} - 8 q^{52} + 8 q^{53} - 4 q^{55} - 8 q^{58} - 24 q^{61} + 4 q^{62} + 4 q^{64} + 16 q^{65} - 8 q^{67} + 4 q^{68} - 8 q^{71} - 4 q^{73} - 8 q^{74} - 12 q^{76} - 8 q^{79} + 4 q^{80} - 12 q^{82} + 4 q^{83} - 8 q^{85} + 8 q^{86} + 4 q^{88} + 24 q^{89} + 8 q^{92} - 4 q^{94} - 8 q^{95}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^5 - 4 * q^8 - 4 * q^10 - 4 * q^11 - 8 * q^13 + 4 * q^16 + 4 * q^17 - 12 * q^19 + 4 * q^20 + 4 * q^22 + 8 * q^23 + 4 * q^25 + 8 * q^26 + 8 * q^29 - 4 * q^31 - 4 * q^32 - 4 * q^34 + 8 * q^37 + 12 * q^38 - 4 * q^40 + 12 * q^41 - 8 * q^43 - 4 * q^44 - 8 * q^46 + 4 * q^47 - 4 * q^50 - 8 * q^52 + 8 * q^53 - 4 * q^55 - 8 * q^58 - 24 * q^61 + 4 * q^62 + 4 * q^64 + 16 * q^65 - 8 * q^67 + 4 * q^68 - 8 * q^71 - 4 * q^73 - 8 * q^74 - 12 * q^76 - 8 * q^79 + 4 * q^80 - 12 * q^82 + 4 * q^83 - 8 * q^85 + 8 * q^86 + 4 * q^88 + 24 * q^89 + 8 * q^92 - 4 * q^94 - 8 * q^95

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.79793	1.69849	0.849244	−	0.528001i	$-0.177058\pi$
$5$	3.79793	1.69849	0.849244	+	0.528001i	$0.177058\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−3.79793	−1.20101
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−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
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.11582	−0.998232	−0.499116	−	0.866535i	$-0.666342\pi$
$17$	−4.11582	−0.998232	−0.499116	+	0.866535i	$0.666342\pi$
$18$	0	0
$19$	−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	$-0.518018\pi$
$23$	−0.542661	−0.113153	−0.0565763	+	0.998398i	$0.518018\pi$
$24$	0	0
$25$	9.42429	1.88486
$26$	0.361009	0.0707997
$27$	0	0
$28$	0	0
$29$	−0.767438	−0.142510	−0.0712548	−	0.997458i	$-0.522700\pi$
$29$	−0.767438	−0.142510	−0.0712548	+	0.997458i	$0.522700\pi$
$30$	0	0
$31$	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
$33$	0	0
$34$	4.11582	0.705857
$35$	0	0
$36$	0	0
$37$	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	0
$40$	−3.79793	−0.600506
$41$	9.13690	1.42694	0.713472	−	0.700683i	$-0.247121\pi$
$41$	9.13690	1.42694	0.713472	+	0.700683i	$0.247121\pi$
$42$	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
$45$	0	0
$46$	0.542661	0.0800110
$47$	−2.98737	−0.435753	−0.217876	−	0.975976i	$-0.569913\pi$
$47$	−2.98737	−0.435753	−0.217876	+	0.975976i	$0.569913\pi$
$48$	0	0
$49$	0	0
$50$	−9.42429	−1.33280
$51$	0	0
$52$	−0.361009	−0.0500629
$53$	12.4564	1.71102	0.855510	−	0.517787i	$-0.173244\pi$
$53$	12.4564	1.71102	0.855510	+	0.517787i	$0.173244\pi$
$54$	0	0
$55$	−3.79793	−0.512113
$56$	0	0
$57$	0	0
$58$	0.767438	0.100770
$59$	2.25689	0.293823	0.146911	−	0.989150i	$-0.453067\pi$
$59$	2.25689	0.293823	0.146911	+	0.989150i	$0.453067\pi$
$60$	0	0
$61$	−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
$66$	0	0
$67$	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	0
$70$	0	0
$71$	6.87385	0.815776	0.407888	−	0.913032i	$-0.366265\pi$
$71$	6.87385	0.815776	0.407888	+	0.913032i	$0.366265\pi$
$72$	0	0
$73$	−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
$75$	0	0
$76$	−4.15894	−0.477063
$77$	0	0
$78$	0	0
$79$	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
$81$	0	0
$82$	−9.13690	−1.00900
$83$	10.8690	1.19303	0.596514	−	0.802603i	$-0.296552\pi$
$83$	10.8690	1.19303	0.596514	+	0.802603i	$0.296552\pi$
$84$	0	0
$85$	−15.6316	−1.69548
$86$	10.1995	1.09984
$87$	0	0
$88$	1.00000	0.106600
$89$	−1.29585	−0.137359	−0.0686797	−	0.997639i	$-0.521879\pi$
$89$	−1.29585	−0.137359	−0.0686797	+	0.997639i	$0.521879\pi$
$90$	0	0
$91$	0	0
$92$	−0.542661	−0.0565763
$93$	0	0
$94$	2.98737	0.308124
$95$	−15.7954	−1.62057
$96$	0	0
$97$	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
$99$	0	0
$100$	9.42429	0.942429
$101$	2.26790	0.225665	0.112832	−	0.993614i	$-0.464008\pi$
$101$	2.26790	0.225665	0.112832	+	0.993614i	$0.464008\pi$
$102$	0	0
$103$	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	$-0.393400\pi$
$107$	6.79956	0.657338	0.328669	+	0.944445i	$0.393400\pi$
$108$	0	0
$109$	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
$111$	0	0
$112$	0	0
$113$	6.37887	0.600074	0.300037	−	0.953928i	$-0.403001\pi$
$113$	6.37887	0.600074	0.300037	+	0.953928i	$0.403001\pi$
$114$	0	0
$115$	−2.06099	−0.192188
$116$	−0.767438	−0.0712548
$117$	0	0
$118$	−2.25689	−0.207764
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	12.2748	1.11130
$123$	0	0
$124$	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
$129$	0	0
$130$	1.37109	0.120252
$131$	11.8732	1.03736	0.518682	−	0.854967i	$-0.326423\pi$
$131$	11.8732	1.03736	0.518682	+	0.854967i	$0.326423\pi$
$132$	0	0
$133$	0	0
$134$	−0.603650	−0.0521474
$135$	0	0
$136$	4.11582	0.352928
$137$	3.21699	0.274846	0.137423	−	0.990512i	$-0.456118\pi$
$137$	3.21699	0.274846	0.137423	+	0.990512i	$0.456118\pi$
$138$	0	0
$139$	−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
$141$	0	0
$142$	−6.87385	−0.576840
$143$	0.361009	0.0301891
$144$	0	0
$145$	−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	$-0.303903\pi$
$149$	14.1064	1.15564	0.577821	+	0.816163i	$0.303903\pi$
$150$	0	0
$151$	−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
$153$	0	0
$154$	0	0
$155$	33.4522	2.68695
$156$	0	0
$157$	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
$159$	0	0
$160$	−3.79793	−0.300253
$161$	0	0
$162$	0	0
$163$	−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
$165$	0	0
$166$	−10.8690	−0.843598
$167$	25.3981	1.96536	0.982682	−	0.185300i	$-0.0593257\pi$
$167$	25.3981	1.96536	0.982682	+	0.185300i	$0.0593257\pi$
$168$	0	0
$169$	−12.8697	−0.989975
$170$	15.6316	1.19889
$171$	0	0
$172$	−10.1995	−0.777706
$173$	12.1207	0.921517	0.460758	−	0.887526i	$-0.347577\pi$
$173$	12.1207	0.921517	0.460758	+	0.887526i	$0.347577\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	1.29585	0.0971277
$179$	−22.5376	−1.68454	−0.842268	−	0.539059i	$-0.818780\pi$
$179$	−22.5376	−1.68454	−0.842268	+	0.539059i	$0.818780\pi$
$180$	0	0
$181$	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
$183$	0	0
$184$	0.542661	0.0400055
$185$	8.68119	0.638254
$186$	0	0
$187$	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	$-0.432701\pi$
$191$	5.80049	0.419708	0.209854	+	0.977733i	$0.432701\pi$
$192$	0	0
$193$	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
$195$	0	0
$196$	0	0
$197$	16.9958	1.21090	0.605451	−	0.795882i	$-0.292993\pi$
$197$	16.9958	1.21090	0.605451	+	0.795882i	$0.292993\pi$
$198$	0	0
$199$	−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	0
$202$	−2.26790	−0.159569
$203$	0	0
$204$	0	0
$205$	34.7013	2.42365
$206$	−13.4438	−0.936672
$207$	0	0
$208$	−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
$213$	0	0
$214$	−6.79956	−0.464808
$215$	−38.7371	−2.64185
$216$	0	0
$217$	0	0
$218$	−3.20730	−0.217226
$219$	0	0
$220$	−3.79793	−0.256057
$221$	1.48585	0.0999489
$222$	0	0
$223$	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
$225$	0	0
$226$	−6.37887	−0.424316
$227$	−1.58055	−0.104905	−0.0524525	−	0.998623i	$-0.516704\pi$
$227$	−1.58055	−0.104905	−0.0524525	+	0.998623i	$0.516704\pi$
$228$	0	0
$229$	−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	$-0.428951\pi$
$233$	6.75774	0.442715	0.221357	+	0.975193i	$0.428951\pi$
$234$	0	0
$235$	−11.3458	−0.740120
$236$	2.25689	0.146911
$237$	0	0
$238$	0	0
$239$	10.8642	0.702744	0.351372	−	0.936236i	$-0.385715\pi$
$239$	10.8642	0.702744	0.351372	+	0.936236i	$0.385715\pi$
$240$	0	0
$241$	−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
$243$	0	0
$244$	−12.2748	−0.785811
$245$	0	0
$246$	0	0
$247$	1.50142	0.0955328
$248$	−8.80801	−0.559309
$249$	0	0
$250$	−16.8032	−1.06273
$251$	−14.4885	−0.914508	−0.457254	−	0.889336i	$-0.651167\pi$
$251$	−14.4885	−0.914508	−0.457254	+	0.889336i	$0.651167\pi$
$252$	0	0
$253$	0.542661	0.0341168
$254$	10.7101	0.672009
$255$	0	0
$256$	1.00000	0.0625000
$257$	21.6591	1.35106	0.675530	−	0.737332i	$-0.263915\pi$
$257$	21.6591	1.35106	0.675530	+	0.737332i	$0.263915\pi$
$258$	0	0
$259$	0	0
$260$	−1.37109	−0.0850313
$261$	0	0
$262$	−11.8732	−0.733527
$263$	13.6890	0.844098	0.422049	−	0.906573i	$-0.361311\pi$
$263$	13.6890	0.844098	0.422049	+	0.906573i	$0.361311\pi$
$264$	0	0
$265$	47.3086	2.90614
$266$	0	0
$267$	0	0
$268$	0.603650	0.0368738
$269$	9.45479	0.576469	0.288234	−	0.957560i	$-0.406932\pi$
$269$	9.45479	0.576469	0.288234	+	0.957560i	$0.406932\pi$
$270$	0	0
$271$	−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	0
$277$	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
$279$	0	0
$280$	0	0
$281$	9.55045	0.569732	0.284866	−	0.958567i	$-0.408051\pi$
$281$	9.55045	0.569732	0.284866	+	0.958567i	$0.408051\pi$
$282$	0	0
$283$	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
$285$	0	0
$286$	−0.361009	−0.0213469
$287$	0	0
$288$	0	0
$289$	−0.0600590	−0.00353288
$290$	2.91468	0.171156
$291$	0	0
$292$	−9.45479	−0.553300
$293$	−28.7922	−1.68206	−0.841028	−	0.540992i	$-0.818049\pi$
$293$	−28.7922	−1.68206	−0.841028	+	0.540992i	$0.818049\pi$
$294$	0	0
$295$	8.57153	0.499054
$296$	−2.28577	−0.132857
$297$	0	0
$298$	−14.1064	−0.817162
$299$	0.195905	0.0113295
$300$	0	0
$301$	0	0
$302$	3.42847	0.197286
$303$	0	0
$304$	−4.15894	−0.238532
$305$	−46.6187	−2.66938
$306$	0	0
$307$	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
$309$	0	0
$310$	−33.4522	−1.89996
$311$	−8.88096	−0.503593	−0.251797	−	0.967780i	$-0.581021\pi$
$311$	−8.88096	−0.503593	−0.251797	+	0.967780i	$0.581021\pi$
$312$	0	0
$313$	−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	$-0.759777\pi$
$317$	−25.9408	−1.45698	−0.728489	+	0.685057i	$0.759777\pi$
$318$	0	0
$319$	0.767438	0.0429683
$320$	3.79793	0.212311
$321$	0	0
$322$	0	0
$323$	17.1174	0.952440
$324$	0	0
$325$	−3.40225	−0.188723
$326$	15.8275	0.876604
$327$	0	0
$328$	−9.13690	−0.504501
$329$	0	0
$330$	0	0
$331$	−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
$333$	0	0
$334$	−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
$339$	0	0
$340$	−15.6316	−0.847742
$341$	−8.80801	−0.476981
$342$	0	0
$343$	0	0
$344$	10.1995	0.549921
$345$	0	0
$346$	−12.1207	−0.651611
$347$	25.3050	1.35844	0.679222	−	0.733933i	$-0.262318\pi$
$347$	25.3050	1.35844	0.679222	+	0.733933i	$0.262318\pi$
$348$	0	0
$349$	−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	0
$352$	1.00000	0.0533002
$353$	8.84629	0.470841	0.235420	−	0.971894i	$-0.424353\pi$
$353$	8.84629	0.470841	0.235420	+	0.971894i	$0.424353\pi$
$354$	0	0
$355$	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	$-0.520992\pi$
$359$	−2.49724	−0.131799	−0.0658997	+	0.997826i	$0.520992\pi$
$360$	0	0
$361$	−1.70320	−0.0896423
$362$	−21.4905	−1.12952
$363$	0	0
$364$	0	0
$365$	−35.9086	−1.87954
$366$	0	0
$367$	−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
$369$	0	0
$370$	−8.68119	−0.451313
$371$	0	0
$372$	0	0
$373$	−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
$375$	0	0
$376$	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
$381$	0	0
$382$	−5.80049	−0.296779
$383$	−23.1190	−1.18133	−0.590664	−	0.806918i	$-0.701134\pi$
$383$	−23.1190	−1.18133	−0.590664	+	0.806918i	$0.701134\pi$
$384$	0	0
$385$	0	0
$386$	−26.1623	−1.33163
$387$	0	0
$388$	18.2628	0.927153
$389$	14.5715	0.738806	0.369403	−	0.929269i	$-0.379562\pi$
$389$	14.5715	0.738806	0.369403	+	0.929269i	$0.379562\pi$
$390$	0	0
$391$	2.23349	0.112953
$392$	0	0
$393$	0	0
$394$	−16.9958	−0.856237
$395$	0.121978	0.00613737
$396$	0	0
$397$	−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	$-0.467383\pi$
$401$	4.09672	0.204580	0.102290	+	0.994755i	$0.467383\pi$
$402$	0	0
$403$	−3.17977	−0.158396
$404$	2.26790	0.112832
$405$	0	0
$406$	0	0
$407$	−2.28577	−0.113301
$408$	0	0
$409$	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
$411$	0	0
$412$	13.4438	0.662327
$413$	0	0
$414$	0	0
$415$	41.2797	2.02634
$416$	0.361009	0.0176999
$417$	0	0
$418$	−4.15894	−0.203420
$419$	−20.1097	−0.982421	−0.491210	−	0.871041i	$-0.663445\pi$
$419$	−20.1097	−0.982421	−0.491210	+	0.871041i	$0.663445\pi$
$420$	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
$423$	0	0
$424$	−12.4564	−0.604937
$425$	−38.7887	−1.88153
$426$	0	0
$427$	0	0
$428$	6.79956	0.328669
$429$	0	0
$430$	38.7371	1.86807
$431$	−26.1385	−1.25905	−0.629524	−	0.776981i	$-0.716750\pi$
$431$	−26.1385	−1.25905	−0.629524	+	0.776981i	$0.716750\pi$
$432$	0	0
$433$	−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
$435$	0	0
$436$	3.20730	0.153602
$437$	2.25689	0.107962
$438$	0	0
$439$	−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	$-0.513261\pi$
$443$	−1.75321	−0.0832973	−0.0416486	+	0.999132i	$0.513261\pi$
$444$	0	0
$445$	−4.92153	−0.233303
$446$	−10.0658	−0.476631
$447$	0	0
$448$	0	0
$449$	−0.549515	−0.0259332	−0.0129666	−	0.999916i	$-0.504128\pi$
$449$	−0.549515	−0.0259332	−0.0129666	+	0.999916i	$0.504128\pi$
$450$	0	0
$451$	−9.13690	−0.430240
$452$	6.37887	0.300037
$453$	0	0
$454$	1.58055	0.0741791
$455$	0	0
$456$	0	0
$457$	−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
$459$	0	0
$460$	−2.06099	−0.0960941
$461$	29.3091	1.36506	0.682532	−	0.730856i	$-0.260879\pi$
$461$	29.3091	1.36506	0.682532	+	0.730856i	$0.260879\pi$
$462$	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
$465$	0	0
$466$	−6.75774	−0.313046
$467$	−33.4201	−1.54650	−0.773249	−	0.634102i	$-0.781370\pi$
$467$	−33.4201	−1.54650	−0.773249	+	0.634102i	$0.781370\pi$
$468$	0	0
$469$	0	0
$470$	11.3458	0.523344
$471$	0	0
$472$	−2.25689	−0.103882
$473$	10.1995	0.468974
$474$	0	0
$475$	−39.1951	−1.79839
$476$	0	0
$477$	0	0
$478$	−10.8642	−0.496915
$479$	−39.8087	−1.81891	−0.909453	−	0.415808i	$-0.863499\pi$
$479$	−39.8087	−1.81891	−0.909453	+	0.415808i	$0.863499\pi$
$480$	0	0
$481$	−0.825182	−0.0376251
$482$	10.8988	0.496428
$483$	0	0
$484$	1.00000	0.0454545
$485$	69.3609	3.14952
$486$	0	0
$487$	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
$489$	0	0
$490$	0	0
$491$	37.6972	1.70125	0.850625	−	0.525773i	$-0.176224\pi$
$491$	37.6972	1.70125	0.850625	+	0.525773i	$0.176224\pi$
$492$	0	0
$493$	3.15863	0.142258
$494$	−1.50142	−0.0675519
$495$	0	0
$496$	8.80801	0.395491
$497$	0	0
$498$	0	0
$499$	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
$501$	0	0
$502$	14.4885	0.646655
$503$	−22.3477	−0.996436	−0.498218	−	0.867052i	$-0.666012\pi$
$503$	−22.3477	−0.996436	−0.498218	+	0.867052i	$0.666012\pi$
$504$	0	0
$505$	8.61334	0.383289
$506$	−0.542661	−0.0241242
$507$	0	0
$508$	−10.7101	−0.475182
$509$	2.35390	0.104335	0.0521673	−	0.998638i	$-0.483387\pi$
$509$	2.35390	0.104335	0.0521673	+	0.998638i	$0.483387\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−21.6591	−0.955344
$515$	51.0586	2.24991
$516$	0	0
$517$	2.98737	0.131384
$518$	0	0
$519$	0	0
$520$	1.37109	0.0601262
$521$	−2.58940	−0.113443	−0.0567217	−	0.998390i	$-0.518065\pi$
$521$	−2.58940	−0.113443	−0.0567217	+	0.998390i	$0.518065\pi$
$522$	0	0
$523$	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
$528$	0	0
$529$	−22.7055	−0.987196
$530$	−47.3086	−2.05495
$531$	0	0
$532$	0	0
$533$	−3.29850	−0.142874
$534$	0	0
$535$	25.8243	1.11648
$536$	−0.603650	−0.0260737
$537$	0	0
$538$	−9.45479	−0.407625
$539$	0	0
$540$	0	0
$541$	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
$543$	0	0
$544$	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
$549$	0	0
$550$	9.42429	0.401853
$551$	3.19173	0.135972
$552$	0	0
$553$	0	0
$554$	−20.4445	−0.868601
$555$	0	0
$556$	−0.608497	−0.0258060
$557$	13.6316	0.577589	0.288795	−	0.957391i	$-0.406746\pi$
$557$	13.6316	0.577589	0.288795	+	0.957391i	$0.406746\pi$
$558$	0	0
$559$	3.68212	0.155737
$560$	0	0
$561$	0	0
$562$	−9.55045	−0.402861
$563$	−37.8882	−1.59680	−0.798399	−	0.602128i	$-0.794319\pi$
$563$	−37.8882	−1.59680	−0.798399	+	0.602128i	$0.794319\pi$
$564$	0	0
$565$	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	$-0.505337\pi$
$569$	−0.799920	−0.0335344	−0.0167672	+	0.999859i	$0.505337\pi$
$570$	0	0
$571$	−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
$573$	0	0
$574$	0	0
$575$	−5.11419	−0.213277
$576$	0	0
$577$	−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
$579$	0	0
$580$	−2.91468	−0.121025
$581$	0	0
$582$	0	0
$583$	−12.4564	−0.515892
$584$	9.45479	0.391242
$585$	0	0
$586$	28.7922	1.18939
$587$	28.0811	1.15903	0.579516	−	0.814961i	$-0.303241\pi$
$587$	28.0811	1.15903	0.579516	+	0.814961i	$0.303241\pi$
$588$	0	0
$589$	−36.6320	−1.50940
$590$	−8.57153	−0.352884
$591$	0	0
$592$	2.28577	0.0939444
$593$	10.3370	0.424489	0.212245	−	0.977217i	$-0.431923\pi$
$593$	10.3370	0.424489	0.212245	+	0.977217i	$0.431923\pi$
$594$	0	0
$595$	0	0
$596$	14.1064	0.577821
$597$	0	0
$598$	−0.195905	−0.00801117
$599$	−36.5949	−1.49523	−0.747614	−	0.664133i	$-0.768801\pi$
$599$	−36.5949	−1.49523	−0.747614	+	0.664133i	$0.768801\pi$
$600$	0	0
$601$	−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	0
$604$	−3.42847	−0.139502
$605$	3.79793	0.154408
$606$	0	0
$607$	−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
$612$	0	0
$613$	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
$615$	0	0
$616$	0	0
$617$	3.40965	0.137267	0.0686337	−	0.997642i	$-0.478136\pi$
$617$	3.40965	0.137267	0.0686337	+	0.997642i	$0.478136\pi$
$618$	0	0
$619$	−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	0
$622$	8.88096	0.356094
$623$	0	0
$624$	0	0
$625$	16.6958	0.667833
$626$	17.1100	0.683855
$627$	0	0
$628$	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
$633$	0	0
$634$	25.9408	1.03024
$635$	−40.6761	−1.61418
$636$	0	0
$637$	0	0
$638$	−0.767438	−0.0303832
$639$	0	0
$640$	−3.79793	−0.150126
$641$	7.83260	0.309369	0.154685	−	0.987964i	$-0.450564\pi$
$641$	7.83260	0.309369	0.154685	+	0.987964i	$0.450564\pi$
$642$	0	0
$643$	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
$645$	0	0
$646$	−17.1174	−0.673477
$647$	−43.7782	−1.72110	−0.860550	−	0.509367i	$-0.829880\pi$
$647$	−43.7782	−1.72110	−0.860550	+	0.509367i	$0.829880\pi$
$648$	0	0
$649$	−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	$-0.172173\pi$
$653$	43.8119	1.71449	0.857247	+	0.514906i	$0.172173\pi$
$654$	0	0
$655$	45.0935	1.76195
$656$	9.13690	0.356736
$657$	0	0
$658$	0	0
$659$	39.0838	1.52249	0.761245	−	0.648465i	$-0.224589\pi$
$659$	39.0838	1.52249	0.761245	+	0.648465i	$0.224589\pi$
$660$	0	0
$661$	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
$663$	0	0
$664$	−10.8690	−0.421799
$665$	0	0
$666$	0	0
$667$	0.416459	0.0161253
$668$	25.3981	0.982682
$669$	0	0
$670$	−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
$675$	0	0
$676$	−12.8697	−0.494987
$677$	−43.7114	−1.67997	−0.839983	−	0.542612i	$-0.817435\pi$
$677$	−43.7114	−1.67997	−0.839983	+	0.542612i	$0.817435\pi$
$678$	0	0
$679$	0	0
$680$	15.6316	0.599444
$681$	0	0
$682$	8.80801	0.337276
$683$	29.7156	1.13703	0.568517	−	0.822671i	$-0.307517\pi$
$683$	29.7156	1.13703	0.568517	+	0.822671i	$0.307517\pi$
$684$	0	0
$685$	12.2179	0.466823
$686$	0	0
$687$	0	0
$688$	−10.1995	−0.388853
$689$	−4.49688	−0.171317
$690$	0	0
$691$	−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	0
$697$	−37.6058	−1.42442
$698$	0.556914	0.0210795
$699$	0	0
$700$	0	0
$701$	15.3559	0.579984	0.289992	−	0.957029i	$-0.406347\pi$
$701$	15.3559	0.579984	0.289992	+	0.957029i	$0.406347\pi$
$702$	0	0
$703$	−9.50637	−0.358539
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−8.84629	−0.332935
$707$	0	0
$708$	0	0
$709$	−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	0
$712$	1.29585	0.0485639
$713$	−4.77976	−0.179003
$714$	0	0
$715$	1.37109	0.0512758
$716$	−22.5376	−0.842268
$717$	0	0
$718$	2.49724	0.0931962
$719$	−22.1791	−0.827141	−0.413570	−	0.910472i	$-0.635718\pi$
$719$	−22.1791	−0.827141	−0.413570	+	0.910472i	$0.635718\pi$
$720$	0	0
$721$	0	0
$722$	1.70320	0.0633867
$723$	0	0
$724$	21.4905	0.798689
$725$	−7.23256	−0.268611
$726$	0	0
$727$	−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
$729$	0	0
$730$	35.9086	1.32904
$731$	41.9793	1.55266
$732$	0	0
$733$	−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
$738$	0	0
$739$	−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
$741$	0	0
$742$	0	0
$743$	−26.6917	−0.979222	−0.489611	−	0.871941i	$-0.662861\pi$
$743$	−26.6917	−0.979222	−0.489611	+	0.871941i	$0.662861\pi$
$744$	0	0
$745$	53.5752	1.96284
$746$	6.95365	0.254591
$747$	0	0
$748$	4.11582	0.150489
$749$	0	0
$750$	0	0
$751$	−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
$753$	0	0
$754$	−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	0
$760$	15.7954	0.572959
$761$	39.2524	1.42290	0.711450	−	0.702737i	$-0.248039\pi$
$761$	39.2524	1.42290	0.711450	+	0.702737i	$0.248039\pi$
$762$	0	0
$763$	0	0
$764$	5.80049	0.209854
$765$	0	0
$766$	23.1190	0.835325
$767$	−0.814759	−0.0294192
$768$	0	0
$769$	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
$771$	0	0
$772$	26.1623	0.941602
$773$	32.6368	1.17386	0.586932	−	0.809636i	$-0.300335\pi$
$773$	32.6368	1.17386	0.586932	+	0.809636i	$0.300335\pi$
$774$	0	0
$775$	83.0093	2.98178
$776$	−18.2628	−0.655596
$777$	0	0
$778$	−14.5715	−0.522415
$779$	−37.9998	−1.36149
$780$	0	0
$781$	−6.87385	−0.245966
$782$	−2.23349	−0.0798695
$783$	0	0
$784$	0	0
$785$	64.1215	2.28859
$786$	0	0
$787$	−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
$789$	0	0
$790$	−0.121978	−0.00433978
$791$	0	0
$792$	0	0
$793$	4.43130	0.157360
$794$	14.7075	0.521950
$795$	0	0
$796$	−25.9745	−0.920641
$797$	−21.3873	−0.757579	−0.378789	−	0.925483i	$-0.623660\pi$
$797$	−21.3873	−0.757579	−0.378789	+	0.925483i	$0.623660\pi$
$798$	0	0
$799$	12.2955	0.434982
$800$	−9.42429	−0.333199
$801$	0	0
$802$	−4.09672	−0.144660
$803$	9.45479	0.333652
$804$	0	0
$805$	0	0
$806$	3.17977	0.112003
$807$	0	0
$808$	−2.26790	−0.0797846
$809$	8.69315	0.305635	0.152817	−	0.988254i	$-0.451165\pi$
$809$	8.69315	0.305635	0.152817	+	0.988254i	$0.451165\pi$
$810$	0	0
$811$	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	0
$814$	2.28577	0.0801161
$815$	−60.1118	−2.10562
$816$	0	0
$817$	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	$-0.832821\pi$
$821$	−49.5824	−1.73044	−0.865219	+	0.501394i	$0.832821\pi$
$822$	0	0
$823$	−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
$825$	0	0
$826$	0	0
$827$	−42.7183	−1.48546	−0.742730	−	0.669591i	$-0.766469\pi$
$827$	−42.7183	−1.48546	−0.742730	+	0.669591i	$0.766469\pi$
$828$	0	0
$829$	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
$831$	0	0
$832$	−0.361009	−0.0125157
$833$	0	0
$834$	0	0
$835$	96.4603	3.33815
$836$	4.15894	0.143840
$837$	0	0
$838$	20.1097	0.694676
$839$	−14.4061	−0.497355	−0.248678	−	0.968586i	$-0.579996\pi$
$839$	−14.4061	−0.497355	−0.248678	+	0.968586i	$0.579996\pi$
$840$	0	0
$841$	−28.4110	−0.979691
$842$	−33.0192	−1.13792
$843$	0	0
$844$	12.7555	0.439062
$845$	−48.8781	−1.68146
$846$	0	0
$847$	0	0
$848$	12.4564	0.427755
$849$	0	0
$850$	38.7887	1.33044
$851$	−1.24040	−0.0425202
$852$	0	0
$853$	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
$855$	0	0
$856$	−6.79956	−0.232404
$857$	49.1364	1.67847	0.839234	−	0.543771i	$-0.183004\pi$
$857$	49.1364	1.67847	0.839234	+	0.543771i	$0.183004\pi$
$858$	0	0
$859$	−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	$-0.177412\pi$
$863$	49.8617	1.69731	0.848656	+	0.528944i	$0.177412\pi$
$864$	0	0
$865$	46.0335	1.56518
$866$	34.7124	1.17957
$867$	0	0
$868$	0	0
$869$	−0.0321169	−0.00108949
$870$	0	0
$871$	−0.217923	−0.00738404
$872$	−3.20730	−0.108613
$873$	0	0
$874$	−2.25689	−0.0763406
$875$	0	0
$876$	0	0
$877$	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
$879$	0	0
$880$	−3.79793	−0.128028
$881$	16.1788	0.545078	0.272539	−	0.962145i	$-0.412137\pi$
$881$	16.1788	0.545078	0.272539	+	0.962145i	$0.412137\pi$
$882$	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
$885$	0	0
$886$	1.75321	0.0589001
$887$	6.14672	0.206387	0.103193	−	0.994661i	$-0.467094\pi$
$887$	6.14672	0.206387	0.103193	+	0.994661i	$0.467094\pi$
$888$	0	0
$889$	0	0
$890$	4.92153	0.164970
$891$	0	0
$892$	10.0658	0.337029
$893$	12.4243	0.415763
$894$	0	0
$895$	−85.5961	−2.86116
$896$	0	0
$897$	0	0
$898$	0.549515	0.0183376
$899$	−6.75960	−0.225445
$900$	0	0
$901$	−51.2683	−1.70799
$902$	9.13690	0.304226
$903$	0	0
$904$	−6.37887	−0.212158
$905$	81.6195	2.71312
$906$	0	0
$907$	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
$909$	0	0
$910$	0	0
$911$	−38.4526	−1.27399	−0.636997	−	0.770867i	$-0.719823\pi$
$911$	−38.4526	−1.27399	−0.636997	+	0.770867i	$0.719823\pi$
$912$	0	0
$913$	−10.8690	−0.359711
$914$	11.9339	0.394739
$915$	0	0
$916$	−23.5969	−0.779664
$917$	0	0
$918$	0	0
$919$	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
$921$	0	0
$922$	−29.3091	−0.965245
$923$	−2.48152	−0.0816802
$924$	0	0
$925$	21.5417	0.708288
$926$	−4.75774	−0.156349
$927$	0	0
$928$	0.767438	0.0251924
$929$	46.0270	1.51010	0.755049	−	0.655668i	$-0.227613\pi$
$929$	46.0270	1.51010	0.755049	+	0.655668i	$0.227613\pi$
$930$	0	0
$931$	0	0
$932$	6.75774	0.221357
$933$	0	0
$934$	33.4201	1.09354
$935$	15.6316	0.511208
$936$	0	0
$937$	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
$939$	0	0
$940$	−11.3458	−0.370060
$941$	−12.1279	−0.395358	−0.197679	−	0.980267i	$-0.563340\pi$
$941$	−12.1279	−0.395358	−0.197679	+	0.980267i	$0.563340\pi$
$942$	0	0
$943$	−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	$-0.500209\pi$
$947$	−0.0403155	−0.00131008	−0.000655038		1.00000i	$0.500209\pi$
$948$	0	0
$949$	3.41326	0.110799
$950$	39.1951	1.27166
$951$	0	0
$952$	0	0
$953$	−45.7949	−1.48344	−0.741721	−	0.670709i	$-0.765990\pi$
$953$	−45.7949	−1.48344	−0.741721	+	0.670709i	$0.765990\pi$
$954$	0	0
$955$	22.0299	0.712869
$956$	10.8642	0.351372
$957$	0	0
$958$	39.8087	1.28616
$959$	0	0
$960$	0	0
$961$	46.5811	1.50262
$962$	0.825182	0.0266049
$963$	0	0
$964$	−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
$969$	0	0
$970$	−69.3609	−2.22704
$971$	58.3278	1.87183	0.935915	−	0.352227i	$-0.114575\pi$
$971$	58.3278	1.87183	0.935915	+	0.352227i	$0.114575\pi$
$972$	0	0
$973$	0	0
$974$	−2.97206	−0.0952309
$975$	0	0
$976$	−12.2748	−0.392905
$977$	32.1169	1.02751	0.513755	−	0.857937i	$-0.328254\pi$
$977$	32.1169	1.02751	0.513755	+	0.857937i	$0.328254\pi$
$978$	0	0
$979$	1.29585	0.0414154
$980$	0	0
$981$	0	0
$982$	−37.6972	−1.20297
$983$	14.1388	0.450957	0.225479	−	0.974248i	$-0.427605\pi$
$983$	14.1388	0.450957	0.225479	+	0.974248i	$0.427605\pi$
$984$	0	0
$985$	64.5490	2.05670
$986$	−3.15863	−0.100591
$987$	0	0
$988$	1.50142	0.0477664
$989$	5.53488	0.175999
$990$	0	0
$991$	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
$993$	0	0
$994$	0	0
$995$	−98.6493	−3.12739
$996$	0	0
$997$	−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
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3234.2.a.bl.1.1	✓	4	3.2	odd	2
3234.2.a.bm.1.4	yes	4	21.20	even	2
9702.2.a.dz.1.1		4	7.6	odd	2
9702.2.a.ea.1.4		4	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:	\( N \)	\(=\)	\( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9702.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.79793 q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.79793 q^{5} -1.00000 q^{8} -3.79793 q^{10} -1.00000 q^{11} -0.361009 q^{13} +1.00000 q^{16} -4.11582 q^{17} -4.15894 q^{19} +3.79793 q^{20} +1.00000 q^{22} -0.542661 q^{23} +9.42429 q^{25} +0.361009 q^{26} -0.767438 q^{29} +8.80801 q^{31} -1.00000 q^{32} +4.11582 q^{34} +2.28577 q^{37} +4.15894 q^{38} -3.79793 q^{40} +9.13690 q^{41} -10.1995 q^{43} -1.00000 q^{44} +0.542661 q^{46} -2.98737 q^{47} -9.42429 q^{50} -0.361009 q^{52} +12.4564 q^{53} -3.79793 q^{55} +0.767438 q^{58} +2.25689 q^{59} -12.2748 q^{61} -8.80801 q^{62} +1.00000 q^{64} -1.37109 q^{65} +0.603650 q^{67} -4.11582 q^{68} +6.87385 q^{71} -9.45479 q^{73} -2.28577 q^{74} -4.15894 q^{76} +0.0321169 q^{79} +3.79793 q^{80} -9.13690 q^{82} +10.8690 q^{83} -15.6316 q^{85} +10.1995 q^{86} +1.00000 q^{88} -1.29585 q^{89} -0.542661 q^{92} +2.98737 q^{94} -15.7954 q^{95} +18.2628 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} + 4 q^{5} - 4 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^5 - 4 * q^8	\( 4 q - 4 q^{2} + 4 q^{4} + 4 q^{5} - 4 q^{8} - 4 q^{10} - 4 q^{11} - 8 q^{13} + 4 q^{16} + 4 q^{17} - 12 q^{19} + 4 q^{20} + 4 q^{22} + 8 q^{23} + 4 q^{25} + 8 q^{26} + 8 q^{29} - 4 q^{31} - 4 q^{32} - 4 q^{34} + 8 q^{37} + 12 q^{38} - 4 q^{40} + 12 q^{41} - 8 q^{43} - 4 q^{44} - 8 q^{46} + 4 q^{47} - 4 q^{50} - 8 q^{52} + 8 q^{53} - 4 q^{55} - 8 q^{58} - 24 q^{61} + 4 q^{62} + 4 q^{64} + 16 q^{65} - 8 q^{67} + 4 q^{68} - 8 q^{71} - 4 q^{73} - 8 q^{74} - 12 q^{76} - 8 q^{79} + 4 q^{80} - 12 q^{82} + 4 q^{83} - 8 q^{85} + 8 q^{86} + 4 q^{88} + 24 q^{89} + 8 q^{92} - 4 q^{94} - 8 q^{95}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^5 - 4 * q^8 - 4 * q^10 - 4 * q^11 - 8 * q^13 + 4 * q^16 + 4 * q^17 - 12 * q^19 + 4 * q^20 + 4 * q^22 + 8 * q^23 + 4 * q^25 + 8 * q^26 + 8 * q^29 - 4 * q^31 - 4 * q^32 - 4 * q^34 + 8 * q^37 + 12 * q^38 - 4 * q^40 + 12 * q^41 - 8 * q^43 - 4 * q^44 - 8 * q^46 + 4 * q^47 - 4 * q^50 - 8 * q^52 + 8 * q^53 - 4 * q^55 - 8 * q^58 - 24 * q^61 + 4 * q^62 + 4 * q^64 + 16 * q^65 - 8 * q^67 + 4 * q^68 - 8 * q^71 - 4 * q^73 - 8 * q^74 - 12 * q^76 - 8 * q^79 + 4 * q^80 - 12 * q^82 + 4 * q^83 - 8 * q^85 + 8 * q^86 + 4 * q^88 + 24 * q^89 + 8 * q^92 - 4 * q^94 - 8 * q^95

Introduction

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