LMFDB - Embedded newform 3726.2.a.y.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3726,2,Mod(1,3726)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3726, 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("3726.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3726 = 2 \cdot 3^{4} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3726.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:	$29.7522597931$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 34x^{6} - 4x^{5} + 370x^{4} + 104x^{3} - 1466x^{2} - 564x + 1401 $ x^8 - 34x^6 - 4x^5 + 370x^4 + 104x^3 - 1466x^2 - 564x + 1401
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$3.14503$ of defining polynomial
Character	$\chi$	$=$	3726.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +3.14503 q^{5} -4.34182 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +3.14503 q^{5} -4.34182 q^{7} -1.00000 q^{8} -3.14503 q^{10} +0.707196 q^{11} +1.81273 q^{13} +4.34182 q^{14} +1.00000 q^{16} +1.97189 q^{17} -7.10402 q^{19} +3.14503 q^{20} -0.707196 q^{22} -1.00000 q^{23} +4.89121 q^{25} -1.81273 q^{26} -4.34182 q^{28} +4.82981 q^{29} +9.88660 q^{31} -1.00000 q^{32} -1.97189 q^{34} -13.6552 q^{35} +3.37100 q^{37} +7.10402 q^{38} -3.14503 q^{40} -3.52909 q^{41} -0.625106 q^{43} +0.707196 q^{44} +1.00000 q^{46} -3.38936 q^{47} +11.8514 q^{49} -4.89121 q^{50} +1.81273 q^{52} -6.84816 q^{53} +2.22415 q^{55} +4.34182 q^{56} -4.82981 q^{58} -4.78884 q^{59} -0.494888 q^{61} -9.88660 q^{62} +1.00000 q^{64} +5.70108 q^{65} +10.4925 q^{67} +1.97189 q^{68} +13.6552 q^{70} +14.8786 q^{71} +4.29687 q^{73} -3.37100 q^{74} -7.10402 q^{76} -3.07052 q^{77} +9.50967 q^{79} +3.14503 q^{80} +3.52909 q^{82} -9.69078 q^{83} +6.20165 q^{85} +0.625106 q^{86} -0.707196 q^{88} +13.1466 q^{89} -7.87054 q^{91} -1.00000 q^{92} +3.38936 q^{94} -22.3424 q^{95} +6.20887 q^{97} -11.8514 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{4} + 6 q^{7} - 8 q^{8}+O(q^{10}) $ 8 * q - 8 * q^2 + 8 * q^4 + 6 * q^7 - 8 * q^8	$ 8 q - 8 q^{2} + 8 q^{4} + 6 q^{7} - 8 q^{8} + 6 q^{13} - 6 q^{14} + 8 q^{16} + 10 q^{17} + 12 q^{19} - 8 q^{23} + 28 q^{25} - 6 q^{26} + 6 q^{28} - 6 q^{29} + 16 q^{31} - 8 q^{32} - 10 q^{34} - 18 q^{35} + 4 q^{37} - 12 q^{38} + 4 q^{41} + 6 q^{43} + 8 q^{46} - 14 q^{47} + 28 q^{49} - 28 q^{50} + 6 q^{52} - 20 q^{53} + 16 q^{55} - 6 q^{56} + 6 q^{58} - 6 q^{59} + 12 q^{61} - 16 q^{62} + 8 q^{64} + 10 q^{65} + 10 q^{67} + 10 q^{68} + 18 q^{70} + 18 q^{71} + 44 q^{73} - 4 q^{74} + 12 q^{76} - 28 q^{77} + 22 q^{79} - 4 q^{82} - 4 q^{83} + 22 q^{85} - 6 q^{86} + 18 q^{89} + 14 q^{91} - 8 q^{92} + 14 q^{94} + 20 q^{95} + 36 q^{97} - 28 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^4 + 6 * q^7 - 8 * q^8 + 6 * q^13 - 6 * q^14 + 8 * q^16 + 10 * q^17 + 12 * q^19 - 8 * q^23 + 28 * q^25 - 6 * q^26 + 6 * q^28 - 6 * q^29 + 16 * q^31 - 8 * q^32 - 10 * q^34 - 18 * q^35 + 4 * q^37 - 12 * q^38 + 4 * q^41 + 6 * q^43 + 8 * q^46 - 14 * q^47 + 28 * q^49 - 28 * q^50 + 6 * q^52 - 20 * q^53 + 16 * q^55 - 6 * q^56 + 6 * q^58 - 6 * q^59 + 12 * q^61 - 16 * q^62 + 8 * q^64 + 10 * q^65 + 10 * q^67 + 10 * q^68 + 18 * q^70 + 18 * q^71 + 44 * q^73 - 4 * q^74 + 12 * q^76 - 28 * q^77 + 22 * q^79 - 4 * q^82 - 4 * q^83 + 22 * q^85 - 6 * q^86 + 18 * q^89 + 14 * q^91 - 8 * q^92 + 14 * q^94 + 20 * q^95 + 36 * q^97 - 28 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.14503	1.40650	0.703250	−	0.710943i	$-0.251731\pi$
$5$	3.14503	1.40650	0.703250	+	0.710943i	$0.251731\pi$
$6$	0	0
$7$	−4.34182	−1.64105	−0.820527	−	0.571608i	$-0.806320\pi$
$7$	−4.34182	−1.64105	−0.820527	+	0.571608i	$0.806320\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−3.14503	−0.994546
$11$	0.707196	0.213228	0.106614	−	0.994300i	$-0.465999\pi$
$11$	0.707196	0.213228	0.106614	+	0.994300i	$0.465999\pi$
$12$	0	0
$13$	1.81273	0.502760	0.251380	−	0.967888i	$-0.419116\pi$
$13$	1.81273	0.502760	0.251380	+	0.967888i	$0.419116\pi$
$14$	4.34182	1.16040
$15$	0	0
$16$	1.00000	0.250000
$17$	1.97189	0.478254	0.239127	−	0.970988i	$-0.423139\pi$
$17$	1.97189	0.478254	0.239127	+	0.970988i	$0.423139\pi$
$18$	0	0
$19$	−7.10402	−1.62977	−0.814887	−	0.579620i	$-0.803201\pi$
$19$	−7.10402	−1.62977	−0.814887	+	0.579620i	$0.803201\pi$
$20$	3.14503	0.703250
$21$	0	0
$22$	−0.707196	−0.150775
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	4.89121	0.978243
$26$	−1.81273	−0.355505
$27$	0	0
$28$	−4.34182	−0.820527
$29$	4.82981	0.896873	0.448436	−	0.893815i	$-0.351981\pi$
$29$	4.82981	0.896873	0.448436	+	0.893815i	$0.351981\pi$
$30$	0	0
$31$	9.88660	1.77569	0.887843	−	0.460147i	$-0.152203\pi$
$31$	9.88660	1.77569	0.887843	+	0.460147i	$0.152203\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.97189	−0.338176
$35$	−13.6552	−2.30814
$36$	0	0
$37$	3.37100	0.554189	0.277094	−	0.960843i	$-0.410629\pi$
$37$	3.37100	0.554189	0.277094	+	0.960843i	$0.410629\pi$
$38$	7.10402	1.15242
$39$	0	0
$40$	−3.14503	−0.497273
$41$	−3.52909	−0.551152	−0.275576	−	0.961279i	$-0.588869\pi$
$41$	−3.52909	−0.551152	−0.275576	+	0.961279i	$0.588869\pi$
$42$	0	0
$43$	−0.625106	−0.0953278	−0.0476639	−	0.998863i	$-0.515178\pi$
$43$	−0.625106	−0.0953278	−0.0476639	+	0.998863i	$0.515178\pi$
$44$	0.707196	0.106614
$45$	0	0
$46$	1.00000	0.147442
$47$	−3.38936	−0.494388	−0.247194	−	0.968966i	$-0.579509\pi$
$47$	−3.38936	−0.494388	−0.247194	+	0.968966i	$0.579509\pi$
$48$	0	0
$49$	11.8514	1.69306
$50$	−4.89121	−0.691722
$51$	0	0
$52$	1.81273	0.251380
$53$	−6.84816	−0.940668	−0.470334	−	0.882489i	$-0.655867\pi$
$53$	−6.84816	−0.940668	−0.470334	+	0.882489i	$0.655867\pi$
$54$	0	0
$55$	2.22415	0.299905
$56$	4.34182	0.580200
$57$	0	0
$58$	−4.82981	−0.634185
$59$	−4.78884	−0.623454	−0.311727	−	0.950172i	$-0.600907\pi$
$59$	−4.78884	−0.623454	−0.311727	+	0.950172i	$0.600907\pi$
$60$	0	0
$61$	−0.494888	−0.0633639	−0.0316819	−	0.999498i	$-0.510086\pi$
$61$	−0.494888	−0.0633639	−0.0316819	+	0.999498i	$0.510086\pi$
$62$	−9.88660	−1.25560
$63$	0	0
$64$	1.00000	0.125000
$65$	5.70108	0.707132
$66$	0	0
$67$	10.4925	1.28187	0.640934	−	0.767596i	$-0.278547\pi$
$67$	10.4925	1.28187	0.640934	+	0.767596i	$0.278547\pi$
$68$	1.97189	0.239127
$69$	0	0
$70$	13.6552	1.63210
$71$	14.8786	1.76577	0.882885	−	0.469589i	$-0.155598\pi$
$71$	14.8786	1.76577	0.882885	+	0.469589i	$0.155598\pi$
$72$	0	0
$73$	4.29687	0.502910	0.251455	−	0.967869i	$-0.419091\pi$
$73$	4.29687	0.502910	0.251455	+	0.967869i	$0.419091\pi$
$74$	−3.37100	−0.391871
$75$	0	0
$76$	−7.10402	−0.814887
$77$	−3.07052	−0.349918
$78$	0	0
$79$	9.50967	1.06992	0.534961	−	0.844877i	$-0.320326\pi$
$79$	9.50967	1.06992	0.534961	+	0.844877i	$0.320326\pi$
$80$	3.14503	0.351625
$81$	0	0
$82$	3.52909	0.389723
$83$	−9.69078	−1.06370	−0.531850	−	0.846838i	$-0.678503\pi$
$83$	−9.69078	−1.06370	−0.531850	+	0.846838i	$0.678503\pi$
$84$	0	0
$85$	6.20165	0.672664
$86$	0.625106	0.0674069
$87$	0	0
$88$	−0.707196	−0.0753874
$89$	13.1466	1.39354	0.696768	−	0.717297i	$-0.254621\pi$
$89$	13.1466	1.39354	0.696768	+	0.717297i	$0.254621\pi$
$90$	0	0
$91$	−7.87054	−0.825057
$92$	−1.00000	−0.104257
$93$	0	0
$94$	3.38936	0.349585
$95$	−22.3424	−2.29228
$96$	0	0
$97$	6.20887	0.630415	0.315208	−	0.949023i	$-0.397926\pi$
$97$	6.20887	0.630415	0.315208	+	0.949023i	$0.397926\pi$
$98$	−11.8514	−1.19717
$99$	0	0
$100$	4.89121	0.489121
$101$	17.4633	1.73766	0.868830	−	0.495110i	$-0.164872\pi$
$101$	17.4633	1.73766	0.868830	+	0.495110i	$0.164872\pi$
$102$	0	0
$103$	4.29626	0.423323	0.211661	−	0.977343i	$-0.432113\pi$
$103$	4.29626	0.423323	0.211661	+	0.977343i	$0.432113\pi$
$104$	−1.81273	−0.177753
$105$	0	0
$106$	6.84816	0.665153
$107$	3.42899	0.331493	0.165747	−	0.986168i	$-0.446997\pi$
$107$	3.42899	0.331493	0.165747	+	0.986168i	$0.446997\pi$
$108$	0	0
$109$	4.19327	0.401643	0.200821	−	0.979628i	$-0.435639\pi$
$109$	4.19327	0.401643	0.200821	+	0.979628i	$0.435639\pi$
$110$	−2.22415	−0.212065
$111$	0	0
$112$	−4.34182	−0.410263
$113$	11.5617	1.08763	0.543815	−	0.839205i	$-0.316980\pi$
$113$	11.5617	1.08763	0.543815	+	0.839205i	$0.316980\pi$
$114$	0	0
$115$	−3.14503	−0.293276
$116$	4.82981	0.448436
$117$	0	0
$118$	4.78884	0.440849
$119$	−8.56159	−0.784840
$120$	0	0
$121$	−10.4999	−0.954534
$122$	0.494888	0.0448050
$123$	0	0
$124$	9.88660	0.887843
$125$	−0.342137	−0.0306016
$126$	0	0
$127$	−9.07671	−0.805428	−0.402714	−	0.915326i	$-0.631933\pi$
$127$	−9.07671	−0.805428	−0.402714	+	0.915326i	$0.631933\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−5.70108	−0.500018
$131$	−19.8038	−1.73027	−0.865133	−	0.501542i	$-0.832766\pi$
$131$	−19.8038	−1.73027	−0.865133	+	0.501542i	$0.832766\pi$
$132$	0	0
$133$	30.8444	2.67455
$134$	−10.4925	−0.906417
$135$	0	0
$136$	−1.97189	−0.169088
$137$	6.17300	0.527395	0.263698	−	0.964605i	$-0.415058\pi$
$137$	6.17300	0.527395	0.263698	+	0.964605i	$0.415058\pi$
$138$	0	0
$139$	19.4158	1.64683	0.823415	−	0.567440i	$-0.192066\pi$
$139$	19.4158	1.64683	0.823415	+	0.567440i	$0.192066\pi$
$140$	−13.6552	−1.15407
$141$	0	0
$142$	−14.8786	−1.24859
$143$	1.28195	0.107202
$144$	0	0
$145$	15.1899	1.26145
$146$	−4.29687	−0.355611
$147$	0	0
$148$	3.37100	0.277094
$149$	3.82085	0.313016	0.156508	−	0.987677i	$-0.449976\pi$
$149$	3.82085	0.313016	0.156508	+	0.987677i	$0.449976\pi$
$150$	0	0
$151$	7.77920	0.633062	0.316531	−	0.948582i	$-0.397482\pi$
$151$	7.77920	0.633062	0.316531	+	0.948582i	$0.397482\pi$
$152$	7.10402	0.576212
$153$	0	0
$154$	3.07052	0.247430
$155$	31.0937	2.49750
$156$	0	0
$157$	4.46979	0.356728	0.178364	−	0.983965i	$-0.442920\pi$
$157$	4.46979	0.356728	0.178364	+	0.983965i	$0.442920\pi$
$158$	−9.50967	−0.756548
$159$	0	0
$160$	−3.14503	−0.248636
$161$	4.34182	0.342183
$162$	0	0
$163$	−10.0023	−0.783444	−0.391722	−	0.920084i	$-0.628120\pi$
$163$	−10.0023	−0.783444	−0.391722	+	0.920084i	$0.628120\pi$
$164$	−3.52909	−0.275576
$165$	0	0
$166$	9.69078	0.752150
$167$	1.17173	0.0906709	0.0453354	−	0.998972i	$-0.485564\pi$
$167$	1.17173	0.0906709	0.0453354	+	0.998972i	$0.485564\pi$
$168$	0	0
$169$	−9.71402	−0.747232
$170$	−6.20165	−0.475645
$171$	0	0
$172$	−0.625106	−0.0476639
$173$	−18.6599	−1.41869	−0.709344	−	0.704863i	$-0.751009\pi$
$173$	−18.6599	−1.41869	−0.709344	+	0.704863i	$0.751009\pi$
$174$	0	0
$175$	−21.2368	−1.60535
$176$	0.707196	0.0533069
$177$	0	0
$178$	−13.1466	−0.985379
$179$	−18.4875	−1.38182	−0.690911	−	0.722940i	$-0.742790\pi$
$179$	−18.4875	−1.38182	−0.690911	+	0.722940i	$0.742790\pi$
$180$	0	0
$181$	18.8670	1.40237	0.701187	−	0.712977i	$-0.252654\pi$
$181$	18.8670	1.40237	0.701187	+	0.712977i	$0.252654\pi$
$182$	7.87054	0.583403
$183$	0	0
$184$	1.00000	0.0737210
$185$	10.6019	0.779467
$186$	0	0
$187$	1.39451	0.101977
$188$	−3.38936	−0.247194
$189$	0	0
$190$	22.3424	1.62088
$191$	7.93269	0.573989	0.286995	−	0.957932i	$-0.407344\pi$
$191$	7.93269	0.573989	0.286995	+	0.957932i	$0.407344\pi$
$192$	0	0
$193$	20.1605	1.45118	0.725592	−	0.688125i	$-0.241566\pi$
$193$	20.1605	1.45118	0.725592	+	0.688125i	$0.241566\pi$
$194$	−6.20887	−0.445771
$195$	0	0
$196$	11.8514	0.846529
$197$	−16.7659	−1.19452	−0.597259	−	0.802049i	$-0.703743\pi$
$197$	−16.7659	−1.19452	−0.597259	+	0.802049i	$0.703743\pi$
$198$	0	0
$199$	27.7439	1.96671	0.983357	−	0.181683i	$-0.0581545\pi$
$199$	27.7439	1.96671	0.983357	+	0.181683i	$0.0581545\pi$
$200$	−4.89121	−0.345861
$201$	0	0
$202$	−17.4633	−1.22871
$203$	−20.9702	−1.47182
$204$	0	0
$205$	−11.0991	−0.775195
$206$	−4.29626	−0.299334
$207$	0	0
$208$	1.81273	0.125690
$209$	−5.02394	−0.347513
$210$	0	0
$211$	−14.3922	−0.990799	−0.495400	−	0.868665i	$-0.664978\pi$
$211$	−14.3922	−0.990799	−0.495400	+	0.868665i	$0.664978\pi$
$212$	−6.84816	−0.470334
$213$	0	0
$214$	−3.42899	−0.234401
$215$	−1.96598	−0.134079
$216$	0	0
$217$	−42.9258	−2.91400
$218$	−4.19327	−0.284004
$219$	0	0
$220$	2.22415	0.149952
$221$	3.57450	0.240447
$222$	0	0
$223$	21.1296	1.41494	0.707471	−	0.706743i	$-0.249836\pi$
$223$	21.1296	1.41494	0.707471	+	0.706743i	$0.249836\pi$
$224$	4.34182	0.290100
$225$	0	0
$226$	−11.5617	−0.769070
$227$	−6.86546	−0.455676	−0.227838	−	0.973699i	$-0.573166\pi$
$227$	−6.86546	−0.455676	−0.227838	+	0.973699i	$0.573166\pi$
$228$	0	0
$229$	−13.6587	−0.902590	−0.451295	−	0.892375i	$-0.649038\pi$
$229$	−13.6587	−0.902590	−0.451295	+	0.892375i	$0.649038\pi$
$230$	3.14503	0.207377
$231$	0	0
$232$	−4.82981	−0.317092
$233$	1.25786	0.0824051	0.0412026	−	0.999151i	$-0.486881\pi$
$233$	1.25786	0.0824051	0.0412026	+	0.999151i	$0.486881\pi$
$234$	0	0
$235$	−10.6596	−0.695357
$236$	−4.78884	−0.311727
$237$	0	0
$238$	8.56159	0.554966
$239$	−4.59400	−0.297161	−0.148580	−	0.988900i	$-0.547470\pi$
$239$	−4.59400	−0.297161	−0.148580	+	0.988900i	$0.547470\pi$
$240$	0	0
$241$	10.6014	0.682898	0.341449	−	0.939900i	$-0.389082\pi$
$241$	10.6014	0.682898	0.341449	+	0.939900i	$0.389082\pi$
$242$	10.4999	0.674957
$243$	0	0
$244$	−0.494888	−0.0316819
$245$	37.2730	2.38129
$246$	0	0
$247$	−12.8777	−0.819385
$248$	−9.88660	−0.627800
$249$	0	0
$250$	0.342137	0.0216386
$251$	2.07311	0.130854	0.0654268	−	0.997857i	$-0.479159\pi$
$251$	2.07311	0.130854	0.0654268	+	0.997857i	$0.479159\pi$
$252$	0	0
$253$	−0.707196	−0.0444611
$254$	9.07671	0.569524
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.1207	1.13034	0.565168	−	0.824976i	$-0.308811\pi$
$257$	18.1207	1.13034	0.565168	+	0.824976i	$0.308811\pi$
$258$	0	0
$259$	−14.6363	−0.909454
$260$	5.70108	0.353566
$261$	0	0
$262$	19.8038	1.22348
$263$	25.5877	1.57781	0.788903	−	0.614517i	$-0.210649\pi$
$263$	25.5877	1.57781	0.788903	+	0.614517i	$0.210649\pi$
$264$	0	0
$265$	−21.5377	−1.32305
$266$	−30.8444	−1.89119
$267$	0	0
$268$	10.4925	0.640934
$269$	−9.92258	−0.604990	−0.302495	−	0.953151i	$-0.597820\pi$
$269$	−9.92258	−0.604990	−0.302495	+	0.953151i	$0.597820\pi$
$270$	0	0
$271$	19.3302	1.17423	0.587113	−	0.809505i	$-0.300265\pi$
$271$	19.3302	1.17423	0.587113	+	0.809505i	$0.300265\pi$
$272$	1.97189	0.119563
$273$	0	0
$274$	−6.17300	−0.372925
$275$	3.45905	0.208589
$276$	0	0
$277$	−30.4068	−1.82697	−0.913483	−	0.406877i	$-0.866618\pi$
$277$	−30.4068	−1.82697	−0.913483	+	0.406877i	$0.866618\pi$
$278$	−19.4158	−1.16448
$279$	0	0
$280$	13.6552	0.816052
$281$	6.58304	0.392712	0.196356	−	0.980533i	$-0.437089\pi$
$281$	6.58304	0.392712	0.196356	+	0.980533i	$0.437089\pi$
$282$	0	0
$283$	16.4510	0.977908	0.488954	−	0.872309i	$-0.337379\pi$
$283$	16.4510	0.977908	0.488954	+	0.872309i	$0.337379\pi$
$284$	14.8786	0.882885
$285$	0	0
$286$	−1.28195	−0.0758036
$287$	15.3227	0.904470
$288$	0	0
$289$	−13.1116	−0.771273
$290$	−15.1899	−0.891981
$291$	0	0
$292$	4.29687	0.251455
$293$	7.13504	0.416833	0.208417	−	0.978040i	$-0.433169\pi$
$293$	7.13504	0.416833	0.208417	+	0.978040i	$0.433169\pi$
$294$	0	0
$295$	−15.0611	−0.876888
$296$	−3.37100	−0.195935
$297$	0	0
$298$	−3.82085	−0.221336
$299$	−1.81273	−0.104833
$300$	0	0
$301$	2.71410	0.156438
$302$	−7.77920	−0.447642
$303$	0	0
$304$	−7.10402	−0.407443
$305$	−1.55644	−0.0891213
$306$	0	0
$307$	−27.6948	−1.58063	−0.790313	−	0.612704i	$-0.790082\pi$
$307$	−27.6948	−1.58063	−0.790313	+	0.612704i	$0.790082\pi$
$308$	−3.07052	−0.174959
$309$	0	0
$310$	−31.0937	−1.76600
$311$	−17.4835	−0.991400	−0.495700	−	0.868494i	$-0.665088\pi$
$311$	−17.4835	−0.991400	−0.495700	+	0.868494i	$0.665088\pi$
$312$	0	0
$313$	−30.7038	−1.73548	−0.867740	−	0.497019i	$-0.834428\pi$
$313$	−30.7038	−1.73548	−0.867740	+	0.497019i	$0.834428\pi$
$314$	−4.46979	−0.252245
$315$	0	0
$316$	9.50967	0.534961
$317$	21.0549	1.18256	0.591280	−	0.806466i	$-0.298623\pi$
$317$	21.0549	1.18256	0.591280	+	0.806466i	$0.298623\pi$
$318$	0	0
$319$	3.41562	0.191238
$320$	3.14503	0.175813
$321$	0	0
$322$	−4.34182	−0.241960
$323$	−14.0083	−0.779445
$324$	0	0
$325$	8.86644	0.491821
$326$	10.0023	0.553978
$327$	0	0
$328$	3.52909	0.194862
$329$	14.7160	0.811318
$330$	0	0
$331$	−7.61610	−0.418619	−0.209309	−	0.977849i	$-0.567122\pi$
$331$	−7.61610	−0.418619	−0.209309	+	0.977849i	$0.567122\pi$
$332$	−9.69078	−0.531850
$333$	0	0
$334$	−1.17173	−0.0641140
$335$	32.9994	1.80295
$336$	0	0
$337$	27.1798	1.48058	0.740289	−	0.672289i	$-0.234689\pi$
$337$	27.1798	1.48058	0.740289	+	0.672289i	$0.234689\pi$
$338$	9.71402	0.528373
$339$	0	0
$340$	6.20165	0.336332
$341$	6.99177	0.378625
$342$	0	0
$343$	−21.0639	−1.13735
$344$	0.625106	0.0337035
$345$	0	0
$346$	18.6599	1.00316
$347$	21.5804	1.15850	0.579248	−	0.815151i	$-0.303346\pi$
$347$	21.5804	1.15850	0.579248	+	0.815151i	$0.303346\pi$
$348$	0	0
$349$	32.4685	1.73800	0.868999	−	0.494814i	$-0.164764\pi$
$349$	32.4685	1.73800	0.868999	+	0.494814i	$0.164764\pi$
$350$	21.2368	1.13515
$351$	0	0
$352$	−0.707196	−0.0376937
$353$	18.5289	0.986195	0.493098	−	0.869974i	$-0.335865\pi$
$353$	18.5289	0.986195	0.493098	+	0.869974i	$0.335865\pi$
$354$	0	0
$355$	46.7938	2.48356
$356$	13.1466	0.696768
$357$	0	0
$358$	18.4875	0.977095
$359$	−12.5444	−0.662066	−0.331033	−	0.943619i	$-0.607397\pi$
$359$	−12.5444	−0.662066	−0.331033	+	0.943619i	$0.607397\pi$
$360$	0	0
$361$	31.4671	1.65616
$362$	−18.8670	−0.991628
$363$	0	0
$364$	−7.87054	−0.412528
$365$	13.5138	0.707343
$366$	0	0
$367$	−12.8807	−0.672370	−0.336185	−	0.941796i	$-0.609137\pi$
$367$	−12.8807	−0.672370	−0.336185	+	0.941796i	$0.609137\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	−10.6019	−0.551166
$371$	29.7335	1.54369
$372$	0	0
$373$	22.6005	1.17021	0.585104	−	0.810958i	$-0.301054\pi$
$373$	22.6005	1.17021	0.585104	+	0.810958i	$0.301054\pi$
$374$	−1.39451	−0.0721086
$375$	0	0
$376$	3.38936	0.174793
$377$	8.75512	0.450912
$378$	0	0
$379$	−4.25649	−0.218641	−0.109321	−	0.994007i	$-0.534868\pi$
$379$	−4.25649	−0.218641	−0.109321	+	0.994007i	$0.534868\pi$
$380$	−22.3424	−1.14614
$381$	0	0
$382$	−7.93269	−0.405872
$383$	14.7741	0.754920	0.377460	−	0.926026i	$-0.376798\pi$
$383$	14.7741	0.754920	0.377460	+	0.926026i	$0.376798\pi$
$384$	0	0
$385$	−9.65688	−0.492160
$386$	−20.1605	−1.02614
$387$	0	0
$388$	6.20887	0.315208
$389$	−1.77165	−0.0898262	−0.0449131	−	0.998991i	$-0.514301\pi$
$389$	−1.77165	−0.0898262	−0.0449131	+	0.998991i	$0.514301\pi$
$390$	0	0
$391$	−1.97189	−0.0997228
$392$	−11.8514	−0.598586
$393$	0	0
$394$	16.7659	0.844651
$395$	29.9082	1.50484
$396$	0	0
$397$	11.1512	0.559661	0.279830	−	0.960049i	$-0.409722\pi$
$397$	11.1512	0.559661	0.279830	+	0.960049i	$0.409722\pi$
$398$	−27.7439	−1.39068
$399$	0	0
$400$	4.89121	0.244561
$401$	9.38971	0.468900	0.234450	−	0.972128i	$-0.424671\pi$
$401$	9.38971	0.468900	0.234450	+	0.972128i	$0.424671\pi$
$402$	0	0
$403$	17.9217	0.892744
$404$	17.4633	0.868830
$405$	0	0
$406$	20.9702	1.04073
$407$	2.38396	0.118168
$408$	0	0
$409$	−4.53962	−0.224470	−0.112235	−	0.993682i	$-0.535801\pi$
$409$	−4.53962	−0.224470	−0.112235	+	0.993682i	$0.535801\pi$
$410$	11.0991	0.548146
$411$	0	0
$412$	4.29626	0.211661
$413$	20.7923	1.02312
$414$	0	0
$415$	−30.4778	−1.49610
$416$	−1.81273	−0.0888763
$417$	0	0
$418$	5.02394	0.245729
$419$	35.0088	1.71029	0.855145	−	0.518389i	$-0.173468\pi$
$419$	35.0088	1.71029	0.855145	+	0.518389i	$0.173468\pi$
$420$	0	0
$421$	28.7560	1.40148	0.700740	−	0.713417i	$-0.252853\pi$
$421$	28.7560	1.40148	0.700740	+	0.713417i	$0.252853\pi$
$422$	14.3922	0.700601
$423$	0	0
$424$	6.84816	0.332576
$425$	9.64494	0.467848
$426$	0	0
$427$	2.14871	0.103984
$428$	3.42899	0.165747
$429$	0	0
$430$	1.96598	0.0948078
$431$	6.53462	0.314762	0.157381	−	0.987538i	$-0.449695\pi$
$431$	6.53462	0.314762	0.157381	+	0.987538i	$0.449695\pi$
$432$	0	0
$433$	15.1743	0.729230	0.364615	−	0.931158i	$-0.381201\pi$
$433$	15.1743	0.729230	0.364615	+	0.931158i	$0.381201\pi$
$434$	42.9258	2.06051
$435$	0	0
$436$	4.19327	0.200821
$437$	7.10402	0.339831
$438$	0	0
$439$	−30.6735	−1.46397	−0.731984	−	0.681322i	$-0.761405\pi$
$439$	−30.6735	−1.46397	−0.731984	+	0.681322i	$0.761405\pi$
$440$	−2.22415	−0.106032
$441$	0	0
$442$	−3.57450	−0.170022
$443$	−32.6186	−1.54976	−0.774878	−	0.632110i	$-0.782189\pi$
$443$	−32.6186	−1.54976	−0.774878	+	0.632110i	$0.782189\pi$
$444$	0	0
$445$	41.3464	1.96001
$446$	−21.1296	−1.00051
$447$	0	0
$448$	−4.34182	−0.205132
$449$	−13.9315	−0.657466	−0.328733	−	0.944423i	$-0.606622\pi$
$449$	−13.9315	−0.657466	−0.328733	+	0.944423i	$0.606622\pi$
$450$	0	0
$451$	−2.49576	−0.117521
$452$	11.5617	0.543815
$453$	0	0
$454$	6.86546	0.322212
$455$	−24.7531	−1.16044
$456$	0	0
$457$	−1.02615	−0.0480014	−0.0240007	−	0.999712i	$-0.507640\pi$
$457$	−1.02615	−0.0480014	−0.0240007	+	0.999712i	$0.507640\pi$
$458$	13.6587	0.638227
$459$	0	0
$460$	−3.14503	−0.146638
$461$	33.0820	1.54078	0.770390	−	0.637573i	$-0.220061\pi$
$461$	33.0820	1.54078	0.770390	+	0.637573i	$0.220061\pi$
$462$	0	0
$463$	−30.7735	−1.43017	−0.715083	−	0.699040i	$-0.753611\pi$
$463$	−30.7735	−1.43017	−0.715083	+	0.699040i	$0.753611\pi$
$464$	4.82981	0.224218
$465$	0	0
$466$	−1.25786	−0.0582692
$467$	36.4317	1.68586	0.842928	−	0.538026i	$-0.180830\pi$
$467$	36.4317	1.68586	0.842928	+	0.538026i	$0.180830\pi$
$468$	0	0
$469$	−45.5567	−2.10361
$470$	10.6596	0.491692
$471$	0	0
$472$	4.78884	0.220424
$473$	−0.442073	−0.0203265
$474$	0	0
$475$	−34.7473	−1.59431
$476$	−8.56159	−0.392420
$477$	0	0
$478$	4.59400	0.210124
$479$	−37.5424	−1.71536	−0.857678	−	0.514187i	$-0.828094\pi$
$479$	−37.5424	−1.71536	−0.857678	+	0.514187i	$0.828094\pi$
$480$	0	0
$481$	6.11070	0.278624
$482$	−10.6014	−0.482882
$483$	0	0
$484$	−10.4999	−0.477267
$485$	19.5271	0.886679
$486$	0	0
$487$	28.2332	1.27937	0.639684	−	0.768638i	$-0.279065\pi$
$487$	28.2332	1.27937	0.639684	+	0.768638i	$0.279065\pi$
$488$	0.494888	0.0224025
$489$	0	0
$490$	−37.2730	−1.68382
$491$	−7.18057	−0.324055	−0.162027	−	0.986786i	$-0.551803\pi$
$491$	−7.18057	−0.324055	−0.162027	+	0.986786i	$0.551803\pi$
$492$	0	0
$493$	9.52385	0.428933
$494$	12.8777	0.579393
$495$	0	0
$496$	9.88660	0.443921
$497$	−64.6004	−2.89772
$498$	0	0
$499$	34.8151	1.55854	0.779270	−	0.626688i	$-0.215590\pi$
$499$	34.8151	1.55854	0.779270	+	0.626688i	$0.215590\pi$
$500$	−0.342137	−0.0153008
$501$	0	0
$502$	−2.07311	−0.0925275
$503$	−26.9370	−1.20106	−0.600531	−	0.799602i	$-0.705044\pi$
$503$	−26.9370	−1.20106	−0.600531	+	0.799602i	$0.705044\pi$
$504$	0	0
$505$	54.9225	2.44402
$506$	0.707196	0.0314387
$507$	0	0
$508$	−9.07671	−0.402714
$509$	−31.2747	−1.38623	−0.693113	−	0.720829i	$-0.743761\pi$
$509$	−31.2747	−1.38623	−0.693113	+	0.720829i	$0.743761\pi$
$510$	0	0
$511$	−18.6562	−0.825303
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−18.1207	−0.799269
$515$	13.5119	0.595403
$516$	0	0
$517$	−2.39694	−0.105417
$518$	14.6363	0.643081
$519$	0	0
$520$	−5.70108	−0.250009
$521$	15.5941	0.683189	0.341595	−	0.939847i	$-0.389033\pi$
$521$	15.5941	0.683189	0.341595	+	0.939847i	$0.389033\pi$
$522$	0	0
$523$	23.5412	1.02938	0.514692	−	0.857375i	$-0.327906\pi$
$523$	23.5412	1.02938	0.514692	+	0.857375i	$0.327906\pi$
$524$	−19.8038	−0.865133
$525$	0	0
$526$	−25.5877	−1.11568
$527$	19.4953	0.849228
$528$	0	0
$529$	1.00000	0.0434783
$530$	21.5377	0.935537
$531$	0	0
$532$	30.8444	1.33727
$533$	−6.39728	−0.277097
$534$	0	0
$535$	10.7843	0.466246
$536$	−10.4925	−0.453209
$537$	0	0
$538$	9.92258	0.427793
$539$	8.38127	0.361007
$540$	0	0
$541$	−7.13258	−0.306653	−0.153327	−	0.988176i	$-0.548999\pi$
$541$	−7.13258	−0.306653	−0.153327	+	0.988176i	$0.548999\pi$
$542$	−19.3302	−0.830303
$543$	0	0
$544$	−1.97189	−0.0845441
$545$	13.1880	0.564910
$546$	0	0
$547$	−26.5871	−1.13678	−0.568391	−	0.822759i	$-0.692434\pi$
$547$	−26.5871	−1.13678	−0.568391	+	0.822759i	$0.692434\pi$
$548$	6.17300	0.263698
$549$	0	0
$550$	−3.45905	−0.147494
$551$	−34.3110	−1.46170
$552$	0	0
$553$	−41.2893	−1.75580
$554$	30.4068	1.29186
$555$	0	0
$556$	19.4158	0.823415
$557$	−17.3187	−0.733818	−0.366909	−	0.930257i	$-0.619584\pi$
$557$	−17.3187	−0.733818	−0.366909	+	0.930257i	$0.619584\pi$
$558$	0	0
$559$	−1.13315	−0.0479270
$560$	−13.6552	−0.577036
$561$	0	0
$562$	−6.58304	−0.277689
$563$	27.1401	1.14382	0.571910	−	0.820317i	$-0.306203\pi$
$563$	27.1401	1.14382	0.571910	+	0.820317i	$0.306203\pi$
$564$	0	0
$565$	36.3618	1.52975
$566$	−16.4510	−0.691486
$567$	0	0
$568$	−14.8786	−0.624294
$569$	−9.02265	−0.378249	−0.189125	−	0.981953i	$-0.560565\pi$
$569$	−9.02265	−0.378249	−0.189125	+	0.981953i	$0.560565\pi$
$570$	0	0
$571$	25.5700	1.07007	0.535035	−	0.844830i	$-0.320298\pi$
$571$	25.5700	1.07007	0.535035	+	0.844830i	$0.320298\pi$
$572$	1.28195	0.0536012
$573$	0	0
$574$	−15.3227	−0.639557
$575$	−4.89121	−0.203978
$576$	0	0
$577$	−35.6170	−1.48275	−0.741377	−	0.671089i	$-0.765827\pi$
$577$	−35.6170	−1.48275	−0.741377	+	0.671089i	$0.765827\pi$
$578$	13.1116	0.545373
$579$	0	0
$580$	15.1899	0.630726
$581$	42.0756	1.74559
$582$	0	0
$583$	−4.84300	−0.200576
$584$	−4.29687	−0.177806
$585$	0	0
$586$	−7.13504	−0.294746
$587$	−37.4857	−1.54720	−0.773600	−	0.633675i	$-0.781546\pi$
$587$	−37.4857	−1.54720	−0.773600	+	0.633675i	$0.781546\pi$
$588$	0	0
$589$	−70.2346	−2.89397
$590$	15.0611	0.620054
$591$	0	0
$592$	3.37100	0.138547
$593$	−44.4709	−1.82620	−0.913100	−	0.407735i	$-0.866319\pi$
$593$	−44.4709	−1.82620	−0.913100	+	0.407735i	$0.866319\pi$
$594$	0	0
$595$	−26.9265	−1.10388
$596$	3.82085	0.156508
$597$	0	0
$598$	1.81273	0.0741279
$599$	14.5790	0.595681	0.297840	−	0.954616i	$-0.403734\pi$
$599$	14.5790	0.595681	0.297840	+	0.954616i	$0.403734\pi$
$600$	0	0
$601$	−5.68854	−0.232040	−0.116020	−	0.993247i	$-0.537014\pi$
$601$	−5.68854	−0.232040	−0.116020	+	0.993247i	$0.537014\pi$
$602$	−2.71410	−0.110618
$603$	0	0
$604$	7.77920	0.316531
$605$	−33.0224	−1.34255
$606$	0	0
$607$	2.23855	0.0908599	0.0454299	−	0.998968i	$-0.485534\pi$
$607$	2.23855	0.0908599	0.0454299	+	0.998968i	$0.485534\pi$
$608$	7.10402	0.288106
$609$	0	0
$610$	1.55644	0.0630183
$611$	−6.14398	−0.248559
$612$	0	0
$613$	12.1293	0.489899	0.244949	−	0.969536i	$-0.421229\pi$
$613$	12.1293	0.489899	0.244949	+	0.969536i	$0.421229\pi$
$614$	27.6948	1.11767
$615$	0	0
$616$	3.07052	0.123715
$617$	−24.6521	−0.992455	−0.496227	−	0.868193i	$-0.665282\pi$
$617$	−24.6521	−0.992455	−0.496227	+	0.868193i	$0.665282\pi$
$618$	0	0
$619$	−2.75288	−0.110648	−0.0553238	−	0.998468i	$-0.517619\pi$
$619$	−2.75288	−0.110648	−0.0553238	+	0.998468i	$0.517619\pi$
$620$	31.0937	1.24875
$621$	0	0
$622$	17.4835	0.701025
$623$	−57.0801	−2.28687
$624$	0	0
$625$	−25.5321	−1.02128
$626$	30.7038	1.22717
$627$	0	0
$628$	4.46979	0.178364
$629$	6.64724	0.265043
$630$	0	0
$631$	14.8100	0.589576	0.294788	−	0.955563i	$-0.404751\pi$
$631$	14.8100	0.589576	0.294788	+	0.955563i	$0.404751\pi$
$632$	−9.50967	−0.378274
$633$	0	0
$634$	−21.0549	−0.836196
$635$	−28.5465	−1.13283
$636$	0	0
$637$	21.4834	0.851202
$638$	−3.41562	−0.135226
$639$	0	0
$640$	−3.14503	−0.124318
$641$	−24.1315	−0.953139	−0.476569	−	0.879137i	$-0.658120\pi$
$641$	−24.1315	−0.953139	−0.476569	+	0.879137i	$0.658120\pi$
$642$	0	0
$643$	−10.9263	−0.430893	−0.215447	−	0.976516i	$-0.569121\pi$
$643$	−10.9263	−0.430893	−0.215447	+	0.976516i	$0.569121\pi$
$644$	4.34182	0.171092
$645$	0	0
$646$	14.0083	0.551151
$647$	−2.77084	−0.108933	−0.0544664	−	0.998516i	$-0.517346\pi$
$647$	−2.77084	−0.108933	−0.0544664	+	0.998516i	$0.517346\pi$
$648$	0	0
$649$	−3.38665	−0.132938
$650$	−8.86644	−0.347770
$651$	0	0
$652$	−10.0023	−0.391722
$653$	−19.6137	−0.767544	−0.383772	−	0.923428i	$-0.625375\pi$
$653$	−19.6137	−0.767544	−0.383772	+	0.923428i	$0.625375\pi$
$654$	0	0
$655$	−62.2836	−2.43362
$656$	−3.52909	−0.137788
$657$	0	0
$658$	−14.7160	−0.573689
$659$	31.4247	1.22413	0.612065	−	0.790807i	$-0.290339\pi$
$659$	31.4247	1.22413	0.612065	+	0.790807i	$0.290339\pi$
$660$	0	0
$661$	−9.06295	−0.352508	−0.176254	−	0.984345i	$-0.556398\pi$
$661$	−9.06295	−0.352508	−0.176254	+	0.984345i	$0.556398\pi$
$662$	7.61610	0.296008
$663$	0	0
$664$	9.69078	0.376075
$665$	97.0065	3.76175
$666$	0	0
$667$	−4.82981	−0.187011
$668$	1.17173	0.0453354
$669$	0	0
$670$	−32.9994	−1.27488
$671$	−0.349983	−0.0135109
$672$	0	0
$673$	−49.8864	−1.92298	−0.961490	−	0.274839i	$-0.911375\pi$
$673$	−49.8864	−1.92298	−0.961490	+	0.274839i	$0.911375\pi$
$674$	−27.1798	−1.04693
$675$	0	0
$676$	−9.71402	−0.373616
$677$	−40.1482	−1.54302	−0.771511	−	0.636216i	$-0.780499\pi$
$677$	−40.1482	−1.54302	−0.771511	+	0.636216i	$0.780499\pi$
$678$	0	0
$679$	−26.9578	−1.03455
$680$	−6.20165	−0.237823
$681$	0	0
$682$	−6.99177	−0.267729
$683$	14.8136	0.566825	0.283413	−	0.958998i	$-0.408533\pi$
$683$	14.8136	0.566825	0.283413	+	0.958998i	$0.408533\pi$
$684$	0	0
$685$	19.4143	0.741781
$686$	21.0639	0.804225
$687$	0	0
$688$	−0.625106	−0.0238319
$689$	−12.4139	−0.472930
$690$	0	0
$691$	24.3066	0.924667	0.462333	−	0.886706i	$-0.347012\pi$
$691$	24.3066	0.924667	0.462333	+	0.886706i	$0.347012\pi$
$692$	−18.6599	−0.709344
$693$	0	0
$694$	−21.5804	−0.819181
$695$	61.0634	2.31627
$696$	0	0
$697$	−6.95898	−0.263590
$698$	−32.4685	−1.22895
$699$	0	0
$700$	−21.2368	−0.802675
$701$	−12.0437	−0.454885	−0.227442	−	0.973792i	$-0.573036\pi$
$701$	−12.0437	−0.454885	−0.227442	+	0.973792i	$0.573036\pi$
$702$	0	0
$703$	−23.9476	−0.903203
$704$	0.707196	0.0266535
$705$	0	0
$706$	−18.5289	−0.697345
$707$	−75.8224	−2.85159
$708$	0	0
$709$	29.4027	1.10424	0.552121	−	0.833764i	$-0.313819\pi$
$709$	29.4027	1.10424	0.552121	+	0.833764i	$0.313819\pi$
$710$	−46.7938	−1.75614
$711$	0	0
$712$	−13.1466	−0.492689
$713$	−9.88660	−0.370256
$714$	0	0
$715$	4.03179	0.150780
$716$	−18.4875	−0.690911
$717$	0	0
$718$	12.5444	0.468151
$719$	−13.7708	−0.513563	−0.256782	−	0.966469i	$-0.582662\pi$
$719$	−13.7708	−0.513563	−0.256782	+	0.966469i	$0.582662\pi$
$720$	0	0
$721$	−18.6536	−0.694695
$722$	−31.4671	−1.17108
$723$	0	0
$724$	18.8670	0.701187
$725$	23.6236	0.877359
$726$	0	0
$727$	9.27460	0.343976	0.171988	−	0.985099i	$-0.444981\pi$
$727$	9.27460	0.343976	0.171988	+	0.985099i	$0.444981\pi$
$728$	7.87054	0.291702
$729$	0	0
$730$	−13.5138	−0.500167
$731$	−1.23264	−0.0455908
$732$	0	0
$733$	−14.2081	−0.524790	−0.262395	−	0.964961i	$-0.584512\pi$
$733$	−14.2081	−0.524790	−0.262395	+	0.964961i	$0.584512\pi$
$734$	12.8807	0.475437
$735$	0	0
$736$	1.00000	0.0368605
$737$	7.42029	0.273330
$738$	0	0
$739$	42.7563	1.57282	0.786408	−	0.617707i	$-0.211938\pi$
$739$	42.7563	1.57282	0.786408	+	0.617707i	$0.211938\pi$
$740$	10.6019	0.389733
$741$	0	0
$742$	−29.7335	−1.09155
$743$	19.5812	0.718365	0.359182	−	0.933267i	$-0.383056\pi$
$743$	19.5812	0.718365	0.359182	+	0.933267i	$0.383056\pi$
$744$	0	0
$745$	12.0167	0.440257
$746$	−22.6005	−0.827462
$747$	0	0
$748$	1.39451	0.0509885
$749$	−14.8881	−0.543999
$750$	0	0
$751$	47.7990	1.74421	0.872106	−	0.489317i	$-0.162754\pi$
$751$	47.7990	1.74421	0.872106	+	0.489317i	$0.162754\pi$
$752$	−3.38936	−0.123597
$753$	0	0
$754$	−8.75512	−0.318843
$755$	24.4658	0.890402
$756$	0	0
$757$	−43.5555	−1.58305	−0.791527	−	0.611135i	$-0.790713\pi$
$757$	−43.5555	−1.58305	−0.791527	+	0.611135i	$0.790713\pi$
$758$	4.25649	0.154603
$759$	0	0
$760$	22.3424	0.810442
$761$	36.2466	1.31394	0.656970	−	0.753917i	$-0.271838\pi$
$761$	36.2466	1.31394	0.656970	+	0.753917i	$0.271838\pi$
$762$	0	0
$763$	−18.2064	−0.659117
$764$	7.93269	0.286995
$765$	0	0
$766$	−14.7741	−0.533809
$767$	−8.68087	−0.313448
$768$	0	0
$769$	29.4865	1.06331	0.531655	−	0.846961i	$-0.321570\pi$
$769$	29.4865	1.06331	0.531655	+	0.846961i	$0.321570\pi$
$770$	9.65688	0.348010
$771$	0	0
$772$	20.1605	0.725592
$773$	−18.4931	−0.665152	−0.332576	−	0.943076i	$-0.607918\pi$
$773$	−18.4931	−0.665152	−0.332576	+	0.943076i	$0.607918\pi$
$774$	0	0
$775$	48.3575	1.73705
$776$	−6.20887	−0.222885
$777$	0	0
$778$	1.77165	0.0635167
$779$	25.0707	0.898253
$780$	0	0
$781$	10.5221	0.376511
$782$	1.97189	0.0705147
$783$	0	0
$784$	11.8514	0.423265
$785$	14.0576	0.501738
$786$	0	0
$787$	−6.22756	−0.221989	−0.110994	−	0.993821i	$-0.535404\pi$
$787$	−6.22756	−0.221989	−0.110994	+	0.993821i	$0.535404\pi$
$788$	−16.7659	−0.597259
$789$	0	0
$790$	−29.9082	−1.06409
$791$	−50.1986	−1.78486
$792$	0	0
$793$	−0.897097	−0.0318568
$794$	−11.1512	−0.395740
$795$	0	0
$796$	27.7439	0.983357
$797$	−24.9414	−0.883470	−0.441735	−	0.897145i	$-0.645637\pi$
$797$	−24.9414	−0.883470	−0.441735	+	0.897145i	$0.645637\pi$
$798$	0	0
$799$	−6.68344	−0.236443
$800$	−4.89121	−0.172931
$801$	0	0
$802$	−9.38971	−0.331562
$803$	3.03873	0.107234
$804$	0	0
$805$	13.6552	0.481281
$806$	−17.9217	−0.631265
$807$	0	0
$808$	−17.4633	−0.614356
$809$	−31.9419	−1.12302	−0.561509	−	0.827471i	$-0.689779\pi$
$809$	−31.9419	−1.12302	−0.561509	+	0.827471i	$0.689779\pi$
$810$	0	0
$811$	−9.49941	−0.333569	−0.166785	−	0.985993i	$-0.553338\pi$
$811$	−9.49941	−0.333569	−0.166785	+	0.985993i	$0.553338\pi$
$812$	−20.9702	−0.735908
$813$	0	0
$814$	−2.38396	−0.0835577
$815$	−31.4577	−1.10191
$816$	0	0
$817$	4.44076	0.155363
$818$	4.53962	0.158724
$819$	0	0
$820$	−11.0991	−0.387598
$821$	44.6958	1.55990	0.779948	−	0.625845i	$-0.215246\pi$
$821$	44.6958	1.55990	0.779948	+	0.625845i	$0.215246\pi$
$822$	0	0
$823$	−22.6291	−0.788801	−0.394401	−	0.918939i	$-0.629048\pi$
$823$	−22.6291	−0.788801	−0.394401	+	0.918939i	$0.629048\pi$
$824$	−4.29626	−0.149667
$825$	0	0
$826$	−20.7923	−0.723456
$827$	−34.6396	−1.20454	−0.602269	−	0.798294i	$-0.705736\pi$
$827$	−34.6396	−1.20454	−0.602269	+	0.798294i	$0.705736\pi$
$828$	0	0
$829$	−39.3312	−1.36603	−0.683014	−	0.730405i	$-0.739331\pi$
$829$	−39.3312	−1.36603	−0.683014	+	0.730405i	$0.739331\pi$
$830$	30.4778	1.05790
$831$	0	0
$832$	1.81273	0.0628450
$833$	23.3697	0.809711
$834$	0	0
$835$	3.68511	0.127529
$836$	−5.02394	−0.173757
$837$	0	0
$838$	−35.0088	−1.20936
$839$	−51.1942	−1.76742	−0.883710	−	0.468035i	$-0.844962\pi$
$839$	−51.1942	−1.76742	−0.883710	+	0.468035i	$0.844962\pi$
$840$	0	0
$841$	−5.67296	−0.195619
$842$	−28.7560	−0.990996
$843$	0	0
$844$	−14.3922	−0.495400
$845$	−30.5509	−1.05098
$846$	0	0
$847$	45.5886	1.56644
$848$	−6.84816	−0.235167
$849$	0	0
$850$	−9.64494	−0.330819
$851$	−3.37100	−0.115556
$852$	0	0
$853$	−10.2831	−0.352087	−0.176043	−	0.984382i	$-0.556330\pi$
$853$	−10.2831	−0.352087	−0.176043	+	0.984382i	$0.556330\pi$
$854$	−2.14871	−0.0735275
$855$	0	0
$856$	−3.42899	−0.117201
$857$	2.32785	0.0795178	0.0397589	−	0.999209i	$-0.487341\pi$
$857$	2.32785	0.0795178	0.0397589	+	0.999209i	$0.487341\pi$
$858$	0	0
$859$	47.3489	1.61552	0.807762	−	0.589508i	$-0.200678\pi$
$859$	47.3489	1.61552	0.807762	+	0.589508i	$0.200678\pi$
$860$	−1.96598	−0.0670393
$861$	0	0
$862$	−6.53462	−0.222570
$863$	−44.2545	−1.50644	−0.753220	−	0.657769i	$-0.771500\pi$
$863$	−44.2545	−1.50644	−0.753220	+	0.657769i	$0.771500\pi$
$864$	0	0
$865$	−58.6860	−1.99538
$866$	−15.1743	−0.515643
$867$	0	0
$868$	−42.9258	−1.45700
$869$	6.72520	0.228137
$870$	0	0
$871$	19.0201	0.644472
$872$	−4.19327	−0.142002
$873$	0	0
$874$	−7.10402	−0.240297
$875$	1.48550	0.0502189
$876$	0	0
$877$	−11.9889	−0.404835	−0.202418	−	0.979299i	$-0.564880\pi$
$877$	−11.9889	−0.404835	−0.202418	+	0.979299i	$0.564880\pi$
$878$	30.6735	1.03518
$879$	0	0
$880$	2.22415	0.0749762
$881$	30.2259	1.01834	0.509168	−	0.860667i	$-0.329953\pi$
$881$	30.2259	1.01834	0.509168	+	0.860667i	$0.329953\pi$
$882$	0	0
$883$	−33.9452	−1.14235	−0.571173	−	0.820830i	$-0.693512\pi$
$883$	−33.9452	−1.14235	−0.571173	+	0.820830i	$0.693512\pi$
$884$	3.57450	0.120223
$885$	0	0
$886$	32.6186	1.09584
$887$	9.41969	0.316282	0.158141	−	0.987417i	$-0.449450\pi$
$887$	9.41969	0.316282	0.158141	+	0.987417i	$0.449450\pi$
$888$	0	0
$889$	39.4095	1.32175
$890$	−41.3464	−1.38594
$891$	0	0
$892$	21.1296	0.707471
$893$	24.0781	0.805741
$894$	0	0
$895$	−58.1438	−1.94353
$896$	4.34182	0.145050
$897$	0	0
$898$	13.9315	0.464899
$899$	47.7504	1.59256
$900$	0	0
$901$	−13.5038	−0.449878
$902$	2.49576	0.0830998
$903$	0	0
$904$	−11.5617	−0.384535
$905$	59.3373	1.97244
$906$	0	0
$907$	−49.0774	−1.62959	−0.814794	−	0.579751i	$-0.803150\pi$
$907$	−49.0774	−1.62959	−0.814794	+	0.579751i	$0.803150\pi$
$908$	−6.86546	−0.227838
$909$	0	0
$910$	24.7531	0.820557
$911$	−36.3736	−1.20511	−0.602555	−	0.798077i	$-0.705851\pi$
$911$	−36.3736	−1.20511	−0.602555	+	0.798077i	$0.705851\pi$
$912$	0	0
$913$	−6.85328	−0.226811
$914$	1.02615	0.0339421
$915$	0	0
$916$	−13.6587	−0.451295
$917$	85.9846	2.83946
$918$	0	0
$919$	−21.0799	−0.695363	−0.347682	−	0.937613i	$-0.613031\pi$
$919$	−21.0799	−0.695363	−0.347682	+	0.937613i	$0.613031\pi$
$920$	3.14503	0.103689
$921$	0	0
$922$	−33.0820	−1.08950
$923$	26.9709	0.887759
$924$	0	0
$925$	16.4883	0.542131
$926$	30.7735	1.01128
$927$	0	0
$928$	−4.82981	−0.158546
$929$	−49.1470	−1.61246	−0.806230	−	0.591602i	$-0.798496\pi$
$929$	−49.1470	−1.61246	−0.806230	+	0.591602i	$0.798496\pi$
$930$	0	0
$931$	−84.1926	−2.75930
$932$	1.25786	0.0412026
$933$	0	0
$934$	−36.4317	−1.19208
$935$	4.38579	0.143431
$936$	0	0
$937$	−52.2072	−1.70553	−0.852767	−	0.522292i	$-0.825077\pi$
$937$	−52.2072	−1.70553	−0.852767	+	0.522292i	$0.825077\pi$
$938$	45.5567	1.48748
$939$	0	0
$940$	−10.6596	−0.347679
$941$	−32.7482	−1.06756	−0.533780	−	0.845623i	$-0.679229\pi$
$941$	−32.7482	−1.06756	−0.533780	+	0.845623i	$0.679229\pi$
$942$	0	0
$943$	3.52909	0.114923
$944$	−4.78884	−0.155864
$945$	0	0
$946$	0.442073	0.0143730
$947$	−30.8613	−1.00286	−0.501429	−	0.865199i	$-0.667192\pi$
$947$	−30.8613	−1.00286	−0.501429	+	0.865199i	$0.667192\pi$
$948$	0	0
$949$	7.78905	0.252843
$950$	34.7473	1.12735
$951$	0	0
$952$	8.56159	0.277483
$953$	37.1868	1.20460	0.602299	−	0.798270i	$-0.294251\pi$
$953$	37.1868	1.20460	0.602299	+	0.798270i	$0.294251\pi$
$954$	0	0
$955$	24.9485	0.807316
$956$	−4.59400	−0.148580
$957$	0	0
$958$	37.5424	1.21294
$959$	−26.8021	−0.865484
$960$	0	0
$961$	66.7448	2.15306
$962$	−6.11070	−0.197017
$963$	0	0
$964$	10.6014	0.341449
$965$	63.4054	2.04109
$966$	0	0
$967$	33.3262	1.07170	0.535849	−	0.844314i	$-0.319992\pi$
$967$	33.3262	1.07170	0.535849	+	0.844314i	$0.319992\pi$
$968$	10.4999	0.337479
$969$	0	0
$970$	−19.5271	−0.626977
$971$	−1.83877	−0.0590089	−0.0295045	−	0.999565i	$-0.509393\pi$
$971$	−1.83877	−0.0590089	−0.0295045	+	0.999565i	$0.509393\pi$
$972$	0	0
$973$	−84.3000	−2.70254
$974$	−28.2332	−0.904650
$975$	0	0
$976$	−0.494888	−0.0158410
$977$	22.4620	0.718624	0.359312	−	0.933218i	$-0.383011\pi$
$977$	22.4620	0.718624	0.359312	+	0.933218i	$0.383011\pi$
$978$	0	0
$979$	9.29722	0.297141
$980$	37.2730	1.19064
$981$	0	0
$982$	7.18057	0.229141
$983$	−4.79888	−0.153061	−0.0765303	−	0.997067i	$-0.524384\pi$
$983$	−4.79888	−0.153061	−0.0765303	+	0.997067i	$0.524384\pi$
$984$	0	0
$985$	−52.7291	−1.68009
$986$	−9.52385	−0.303301
$987$	0	0
$988$	−12.8777	−0.409693
$989$	0.625106	0.0198772
$990$	0	0
$991$	14.2045	0.451220	0.225610	−	0.974218i	$-0.427562\pi$
$991$	14.2045	0.451220	0.225610	+	0.974218i	$0.427562\pi$
$992$	−9.88660	−0.313900
$993$	0	0
$994$	64.6004	2.04900
$995$	87.2555	2.76618
$996$	0	0
$997$	−21.9762	−0.695992	−0.347996	−	0.937496i	$-0.613138\pi$
$997$	−21.9762	−0.695992	−0.347996	+	0.937496i	$0.613138\pi$
$998$	−34.8151	−1.10205
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3726.2.a.y.1.7	✓	8
3.2	odd	2		3726.2.a.z.1.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3726.2.a.y.1.7	✓	8	1.1	even	1	trivial
3726.2.a.z.1.2	yes	8	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:	\( N \)	\(=\)	\( 3726 = 2 \cdot 3^{4} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3726.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.14503 q^{5} -4.34182 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.14503 q^{5} -4.34182 q^{7} -1.00000 q^{8} -3.14503 q^{10} +0.707196 q^{11} +1.81273 q^{13} +4.34182 q^{14} +1.00000 q^{16} +1.97189 q^{17} -7.10402 q^{19} +3.14503 q^{20} -0.707196 q^{22} -1.00000 q^{23} +4.89121 q^{25} -1.81273 q^{26} -4.34182 q^{28} +4.82981 q^{29} +9.88660 q^{31} -1.00000 q^{32} -1.97189 q^{34} -13.6552 q^{35} +3.37100 q^{37} +7.10402 q^{38} -3.14503 q^{40} -3.52909 q^{41} -0.625106 q^{43} +0.707196 q^{44} +1.00000 q^{46} -3.38936 q^{47} +11.8514 q^{49} -4.89121 q^{50} +1.81273 q^{52} -6.84816 q^{53} +2.22415 q^{55} +4.34182 q^{56} -4.82981 q^{58} -4.78884 q^{59} -0.494888 q^{61} -9.88660 q^{62} +1.00000 q^{64} +5.70108 q^{65} +10.4925 q^{67} +1.97189 q^{68} +13.6552 q^{70} +14.8786 q^{71} +4.29687 q^{73} -3.37100 q^{74} -7.10402 q^{76} -3.07052 q^{77} +9.50967 q^{79} +3.14503 q^{80} +3.52909 q^{82} -9.69078 q^{83} +6.20165 q^{85} +0.625106 q^{86} -0.707196 q^{88} +13.1466 q^{89} -7.87054 q^{91} -1.00000 q^{92} +3.38936 q^{94} -22.3424 q^{95} +6.20887 q^{97} -11.8514 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} + 6 q^{7} - 8 q^{8}+O(q^{10}) \) 8 * q - 8 * q^2 + 8 * q^4 + 6 * q^7 - 8 * q^8	\( 8 q - 8 q^{2} + 8 q^{4} + 6 q^{7} - 8 q^{8} + 6 q^{13} - 6 q^{14} + 8 q^{16} + 10 q^{17} + 12 q^{19} - 8 q^{23} + 28 q^{25} - 6 q^{26} + 6 q^{28} - 6 q^{29} + 16 q^{31} - 8 q^{32} - 10 q^{34} - 18 q^{35} + 4 q^{37} - 12 q^{38} + 4 q^{41} + 6 q^{43} + 8 q^{46} - 14 q^{47} + 28 q^{49} - 28 q^{50} + 6 q^{52} - 20 q^{53} + 16 q^{55} - 6 q^{56} + 6 q^{58} - 6 q^{59} + 12 q^{61} - 16 q^{62} + 8 q^{64} + 10 q^{65} + 10 q^{67} + 10 q^{68} + 18 q^{70} + 18 q^{71} + 44 q^{73} - 4 q^{74} + 12 q^{76} - 28 q^{77} + 22 q^{79} - 4 q^{82} - 4 q^{83} + 22 q^{85} - 6 q^{86} + 18 q^{89} + 14 q^{91} - 8 q^{92} + 14 q^{94} + 20 q^{95} + 36 q^{97} - 28 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^4 + 6 * q^7 - 8 * q^8 + 6 * q^13 - 6 * q^14 + 8 * q^16 + 10 * q^17 + 12 * q^19 - 8 * q^23 + 28 * q^25 - 6 * q^26 + 6 * q^28 - 6 * q^29 + 16 * q^31 - 8 * q^32 - 10 * q^34 - 18 * q^35 + 4 * q^37 - 12 * q^38 + 4 * q^41 + 6 * q^43 + 8 * q^46 - 14 * q^47 + 28 * q^49 - 28 * q^50 + 6 * q^52 - 20 * q^53 + 16 * q^55 - 6 * q^56 + 6 * q^58 - 6 * q^59 + 12 * q^61 - 16 * q^62 + 8 * q^64 + 10 * q^65 + 10 * q^67 + 10 * q^68 + 18 * q^70 + 18 * q^71 + 44 * q^73 - 4 * q^74 + 12 * q^76 - 28 * q^77 + 22 * q^79 - 4 * q^82 - 4 * q^83 + 22 * q^85 - 6 * q^86 + 18 * q^89 + 14 * q^91 - 8 * q^92 + 14 * q^94 + 20 * q^95 + 36 * q^97 - 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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