LMFDB - Embedded newform 8037.2.a.m.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, 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("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 15 x^{10} + 14 x^{9} + 84 x^{8} - 76 x^{7} - 213 x^{6} + 196 x^{5} + 225 x^{4} + \cdots - 17 $ x^12 - x^11 - 15x^10 + 14x^9 + 84x^8 - 76x^7 - 213x^6 + 196x^5 + 225x^4 - 229x^3 - 49x^2 + 83x - 17
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 893)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$1.87636$ of defining polynomial
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.87636 q^{2} +1.52073 q^{4} -3.86690 q^{5} -2.13974 q^{7} -0.899279 q^{8} +O(q^{10})$	\(q+1.87636 q^{2} +1.52073 q^{4} -3.86690 q^{5} -2.13974 q^{7} -0.899279 q^{8} -7.25570 q^{10} +3.11715 q^{11} +2.25546 q^{13} -4.01493 q^{14} -4.72884 q^{16} +2.97044 q^{17} +1.00000 q^{19} -5.88052 q^{20} +5.84890 q^{22} +4.45369 q^{23} +9.95290 q^{25} +4.23206 q^{26} -3.25398 q^{28} +0.450045 q^{29} +4.13370 q^{31} -7.07445 q^{32} +5.57362 q^{34} +8.27417 q^{35} -2.09001 q^{37} +1.87636 q^{38} +3.47742 q^{40} -7.08376 q^{41} -6.87267 q^{43} +4.74035 q^{44} +8.35673 q^{46} -1.00000 q^{47} -2.42150 q^{49} +18.6752 q^{50} +3.42996 q^{52} +3.90497 q^{53} -12.0537 q^{55} +1.92423 q^{56} +0.844447 q^{58} -0.860132 q^{59} -3.35177 q^{61} +7.75632 q^{62} -3.81655 q^{64} -8.72164 q^{65} -5.34081 q^{67} +4.51724 q^{68} +15.5253 q^{70} +6.02358 q^{71} -14.1038 q^{73} -3.92161 q^{74} +1.52073 q^{76} -6.66990 q^{77} +3.83608 q^{79} +18.2859 q^{80} -13.2917 q^{82} +12.2174 q^{83} -11.4864 q^{85} -12.8956 q^{86} -2.80319 q^{88} -5.57382 q^{89} -4.82611 q^{91} +6.77287 q^{92} -1.87636 q^{94} -3.86690 q^{95} -10.9195 q^{97} -4.54361 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7}+O(q^{10}) $ 12 * q + q^2 + 7 * q^4 + 7 * q^5 - 13 * q^7	$ 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7} + q^{10} + 4 q^{11} - 17 q^{13} - 3 q^{14} - 19 q^{16} + 6 q^{17} + 12 q^{19} - 5 q^{20} - 8 q^{22} + 13 q^{23} - 7 q^{25} + 19 q^{26} - 29 q^{28} + 2 q^{29} - 14 q^{31} + 21 q^{32} - 6 q^{34} + 3 q^{35} - 2 q^{37} + q^{38} + 8 q^{40} - 8 q^{41} - 42 q^{43} - 24 q^{44} - 9 q^{46} - 12 q^{47} - 5 q^{49} + 33 q^{50} - 26 q^{52} - 3 q^{53} - 12 q^{55} - 7 q^{56} - 16 q^{58} - 8 q^{59} - 6 q^{61} + 24 q^{62} - 22 q^{64} - 22 q^{65} - 29 q^{67} + 30 q^{68} - 34 q^{70} + 7 q^{71} - 48 q^{73} - 25 q^{74} + 7 q^{76} + 18 q^{77} - 11 q^{79} - 3 q^{80} + 28 q^{82} + 57 q^{83} - 7 q^{85} - 9 q^{86} - 11 q^{88} + 2 q^{89} - 4 q^{91} + 13 q^{92} - q^{94} + 7 q^{95} - 14 q^{97} - 58 q^{98}+O(q^{100}) $ 12 * q + q^2 + 7 * q^4 + 7 * q^5 - 13 * q^7 + q^10 + 4 * q^11 - 17 * q^13 - 3 * q^14 - 19 * q^16 + 6 * q^17 + 12 * q^19 - 5 * q^20 - 8 * q^22 + 13 * q^23 - 7 * q^25 + 19 * q^26 - 29 * q^28 + 2 * q^29 - 14 * q^31 + 21 * q^32 - 6 * q^34 + 3 * q^35 - 2 * q^37 + q^38 + 8 * q^40 - 8 * q^41 - 42 * q^43 - 24 * q^44 - 9 * q^46 - 12 * q^47 - 5 * q^49 + 33 * q^50 - 26 * q^52 - 3 * q^53 - 12 * q^55 - 7 * q^56 - 16 * q^58 - 8 * q^59 - 6 * q^61 + 24 * q^62 - 22 * q^64 - 22 * q^65 - 29 * q^67 + 30 * q^68 - 34 * q^70 + 7 * q^71 - 48 * q^73 - 25 * q^74 + 7 * q^76 + 18 * q^77 - 11 * q^79 - 3 * q^80 + 28 * q^82 + 57 * q^83 - 7 * q^85 - 9 * q^86 - 11 * q^88 + 2 * q^89 - 4 * q^91 + 13 * q^92 - q^94 + 7 * q^95 - 14 * q^97 - 58 * 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 $

$2$	1.87636	1.32679	0.663394	−	0.748270i	$-0.269115\pi$
$2$	1.87636	1.32679	0.663394	+	0.748270i	$0.269115\pi$
$3$	0	0
$3$	0	0
$4$	1.52073	0.760366
$5$	−3.86690	−1.72933	−0.864665	−	0.502350i	$-0.832469\pi$
$5$	−3.86690	−1.72933	−0.864665	+	0.502350i	$0.832469\pi$
$6$	0	0
$7$	−2.13974	−0.808747	−0.404373	−	0.914594i	$-0.632510\pi$
$7$	−2.13974	−0.808747	−0.404373	+	0.914594i	$0.632510\pi$
$8$	−0.899279	−0.317943
$9$	0	0
$10$	−7.25570	−2.29445
$11$	3.11715	0.939856	0.469928	−	0.882705i	$-0.344280\pi$
$11$	3.11715	0.939856	0.469928	+	0.882705i	$0.344280\pi$
$12$	0	0
$13$	2.25546	0.625553	0.312776	−	0.949827i	$-0.398741\pi$
$13$	2.25546	0.625553	0.312776	+	0.949827i	$0.398741\pi$
$14$	−4.01493	−1.07304
$15$	0	0
$16$	−4.72884	−1.18221
$17$	2.97044	0.720437	0.360219	−	0.932868i	$-0.382702\pi$
$17$	2.97044	0.720437	0.360219	+	0.932868i	$0.382702\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−5.88052	−1.31492
$21$	0	0
$22$	5.84890	1.24699
$23$	4.45369	0.928659	0.464329	−	0.885663i	$-0.346295\pi$
$23$	4.45369	0.928659	0.464329	+	0.885663i	$0.346295\pi$
$24$	0	0
$25$	9.95290	1.99058
$26$	4.23206	0.829976
$27$	0	0
$28$	−3.25398	−0.614944
$29$	0.450045	0.0835712	0.0417856	−	0.999127i	$-0.486695\pi$
$29$	0.450045	0.0835712	0.0417856	+	0.999127i	$0.486695\pi$
$30$	0	0
$31$	4.13370	0.742435	0.371217	−	0.928546i	$-0.378940\pi$
$31$	4.13370	0.742435	0.371217	+	0.928546i	$0.378940\pi$
$32$	−7.07445	−1.25060
$33$	0	0
$34$	5.57362	0.955868
$35$	8.27417	1.39859
$36$	0	0
$37$	−2.09001	−0.343595	−0.171798	−	0.985132i	$-0.554958\pi$
$37$	−2.09001	−0.343595	−0.171798	+	0.985132i	$0.554958\pi$
$38$	1.87636	0.304386
$39$	0	0
$40$	3.47742	0.549829
$41$	−7.08376	−1.10630	−0.553149	−	0.833083i	$-0.686574\pi$
$41$	−7.08376	−1.10630	−0.553149	+	0.833083i	$0.686574\pi$
$42$	0	0
$43$	−6.87267	−1.04807	−0.524036	−	0.851696i	$-0.675574\pi$
$43$	−6.87267	−1.04807	−0.524036	+	0.851696i	$0.675574\pi$
$44$	4.74035	0.714635
$45$	0	0
$46$	8.35673	1.23213
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−2.42150	−0.345928
$50$	18.6752	2.64108
$51$	0	0
$52$	3.42996	0.475649
$53$	3.90497	0.536389	0.268194	−	0.963365i	$-0.413573\pi$
$53$	3.90497	0.536389	0.268194	+	0.963365i	$0.413573\pi$
$54$	0	0
$55$	−12.0537	−1.62532
$56$	1.92423	0.257136
$57$	0	0
$58$	0.844447	0.110881
$59$	−0.860132	−0.111980	−0.0559898	−	0.998431i	$-0.517831\pi$
$59$	−0.860132	−0.111980	−0.0559898	+	0.998431i	$0.517831\pi$
$60$	0	0
$61$	−3.35177	−0.429150	−0.214575	−	0.976708i	$-0.568837\pi$
$61$	−3.35177	−0.429150	−0.214575	+	0.976708i	$0.568837\pi$
$62$	7.75632	0.985054
$63$	0	0
$64$	−3.81655	−0.477069
$65$	−8.72164	−1.08179
$66$	0	0
$67$	−5.34081	−0.652484	−0.326242	−	0.945286i	$-0.605782\pi$
$67$	−5.34081	−0.652484	−0.326242	+	0.945286i	$0.605782\pi$
$68$	4.51724	0.547796
$69$	0	0
$70$	15.5253	1.85563
$71$	6.02358	0.714868	0.357434	−	0.933938i	$-0.383652\pi$
$71$	6.02358	0.714868	0.357434	+	0.933938i	$0.383652\pi$
$72$	0	0
$73$	−14.1038	−1.65073	−0.825363	−	0.564602i	$-0.809030\pi$
$73$	−14.1038	−1.65073	−0.825363	+	0.564602i	$0.809030\pi$
$74$	−3.92161	−0.455878
$75$	0	0
$76$	1.52073	0.174440
$77$	−6.66990	−0.760105
$78$	0	0
$79$	3.83608	0.431592	0.215796	−	0.976438i	$-0.430765\pi$
$79$	3.83608	0.431592	0.215796	+	0.976438i	$0.430765\pi$
$80$	18.2859	2.04443
$81$	0	0
$82$	−13.2917	−1.46782
$83$	12.2174	1.34104	0.670518	−	0.741893i	$-0.266072\pi$
$83$	12.2174	1.34104	0.670518	+	0.741893i	$0.266072\pi$
$84$	0	0
$85$	−11.4864	−1.24587
$86$	−12.8956	−1.39057
$87$	0	0
$88$	−2.80319	−0.298821
$89$	−5.57382	−0.590824	−0.295412	−	0.955370i	$-0.595457\pi$
$89$	−5.57382	−0.590824	−0.295412	+	0.955370i	$0.595457\pi$
$90$	0	0
$91$	−4.82611	−0.505914
$92$	6.77287	0.706121
$93$	0	0
$94$	−1.87636	−0.193532
$95$	−3.86690	−0.396735
$96$	0	0
$97$	−10.9195	−1.10871	−0.554356	−	0.832280i	$-0.687035\pi$
$97$	−10.9195	−1.10871	−0.554356	+	0.832280i	$0.687035\pi$
$98$	−4.54361	−0.458974
$99$	0	0
$100$	15.1357	1.51357
$101$	−5.08136	−0.505614	−0.252807	−	0.967517i	$-0.581354\pi$
$101$	−5.08136	−0.505614	−0.252807	+	0.967517i	$0.581354\pi$
$102$	0	0
$103$	−7.39532	−0.728682	−0.364341	−	0.931266i	$-0.618706\pi$
$103$	−7.39532	−0.728682	−0.364341	+	0.931266i	$0.618706\pi$
$104$	−2.02829	−0.198890
$105$	0	0
$106$	7.32714	0.711674
$107$	1.17005	0.113113	0.0565564	−	0.998399i	$-0.481988\pi$
$107$	1.17005	0.113113	0.0565564	+	0.998399i	$0.481988\pi$
$108$	0	0
$109$	1.31408	0.125866	0.0629331	−	0.998018i	$-0.479955\pi$
$109$	1.31408	0.125866	0.0629331	+	0.998018i	$0.479955\pi$
$110$	−22.6171	−2.15645
$111$	0	0
$112$	10.1185	0.956108
$113$	15.4981	1.45794	0.728970	−	0.684546i	$-0.239999\pi$
$113$	15.4981	1.45794	0.728970	+	0.684546i	$0.239999\pi$
$114$	0	0
$115$	−17.2220	−1.60596
$116$	0.684398	0.0635447
$117$	0	0
$118$	−1.61392	−0.148573
$119$	−6.35598	−0.582652
$120$	0	0
$121$	−1.28338	−0.116671
$122$	−6.28913	−0.569391
$123$	0	0
$124$	6.28625	0.564522
$125$	−19.1523	−1.71304
$126$	0	0
$127$	−19.5860	−1.73798	−0.868990	−	0.494830i	$-0.835230\pi$
$127$	−19.5860	−1.73798	−0.868990	+	0.494830i	$0.835230\pi$
$128$	6.98767	0.617629
$129$	0	0
$130$	−16.3650	−1.43530
$131$	−15.0107	−1.31149	−0.655744	−	0.754984i	$-0.727645\pi$
$131$	−15.0107	−1.31149	−0.655744	+	0.754984i	$0.727645\pi$
$132$	0	0
$133$	−2.13974	−0.185539
$134$	−10.0213	−0.865708
$135$	0	0
$136$	−2.67126	−0.229058
$137$	6.06142	0.517862	0.258931	−	0.965896i	$-0.416630\pi$
$137$	6.06142	0.517862	0.258931	+	0.965896i	$0.416630\pi$
$138$	0	0
$139$	15.5166	1.31610	0.658051	−	0.752973i	$-0.271381\pi$
$139$	15.5166	1.31610	0.658051	+	0.752973i	$0.271381\pi$
$140$	12.5828	1.06344
$141$	0	0
$142$	11.3024	0.948478
$143$	7.03061	0.587929
$144$	0	0
$145$	−1.74028	−0.144522
$146$	−26.4638	−2.19016
$147$	0	0
$148$	−3.17834	−0.261258
$149$	−6.93344	−0.568010	−0.284005	−	0.958823i	$-0.591663\pi$
$149$	−6.93344	−0.568010	−0.284005	+	0.958823i	$0.591663\pi$
$150$	0	0
$151$	−15.9549	−1.29839	−0.649196	−	0.760621i	$-0.724894\pi$
$151$	−15.9549	−1.29839	−0.649196	+	0.760621i	$0.724894\pi$
$152$	−0.899279	−0.0729412
$153$	0	0
$154$	−12.5151	−1.00850
$155$	−15.9846	−1.28391
$156$	0	0
$157$	13.2749	1.05945	0.529727	−	0.848168i	$-0.322294\pi$
$157$	13.2749	1.05945	0.529727	+	0.848168i	$0.322294\pi$
$158$	7.19787	0.572632
$159$	0	0
$160$	27.3562	2.16270
$161$	−9.52976	−0.751050
$162$	0	0
$163$	3.90534	0.305890	0.152945	−	0.988235i	$-0.451124\pi$
$163$	3.90534	0.305890	0.152945	+	0.988235i	$0.451124\pi$
$164$	−10.7725	−0.841191
$165$	0	0
$166$	22.9243	1.77927
$167$	−15.5798	−1.20560	−0.602799	−	0.797893i	$-0.705948\pi$
$167$	−15.5798	−1.20560	−0.602799	+	0.797893i	$0.705948\pi$
$168$	0	0
$169$	−7.91289	−0.608684
$170$	−21.5526	−1.65301
$171$	0	0
$172$	−10.4515	−0.796918
$173$	−22.2687	−1.69306	−0.846528	−	0.532345i	$-0.821311\pi$
$173$	−22.2687	−1.69306	−0.846528	+	0.532345i	$0.821311\pi$
$174$	0	0
$175$	−21.2966	−1.60987
$176$	−14.7405	−1.11111
$177$	0	0
$178$	−10.4585	−0.783898
$179$	14.1370	1.05665	0.528326	−	0.849042i	$-0.322820\pi$
$179$	14.1370	1.05665	0.528326	+	0.849042i	$0.322820\pi$
$180$	0	0
$181$	−11.2738	−0.837975	−0.418988	−	0.907992i	$-0.637615\pi$
$181$	−11.2738	−0.837975	−0.418988	+	0.907992i	$0.637615\pi$
$182$	−9.05553	−0.671240
$183$	0	0
$184$	−4.00511	−0.295261
$185$	8.08185	0.594189
$186$	0	0
$187$	9.25930	0.677107
$188$	−1.52073	−0.110911
$189$	0	0
$190$	−7.25570	−0.526384
$191$	4.08423	0.295525	0.147762	−	0.989023i	$-0.452793\pi$
$191$	4.08423	0.295525	0.147762	+	0.989023i	$0.452793\pi$
$192$	0	0
$193$	−22.9007	−1.64843	−0.824214	−	0.566279i	$-0.808382\pi$
$193$	−22.9007	−1.64843	−0.824214	+	0.566279i	$0.808382\pi$
$194$	−20.4890	−1.47102
$195$	0	0
$196$	−3.68245	−0.263032
$197$	19.4939	1.38888	0.694442	−	0.719549i	$-0.255651\pi$
$197$	19.4939	1.38888	0.694442	+	0.719549i	$0.255651\pi$
$198$	0	0
$199$	−20.6517	−1.46396	−0.731979	−	0.681327i	$-0.761403\pi$
$199$	−20.6517	−1.46396	−0.731979	+	0.681327i	$0.761403\pi$
$200$	−8.95043	−0.632891
$201$	0	0
$202$	−9.53447	−0.670843
$203$	−0.962980	−0.0675880
$204$	0	0
$205$	27.3922	1.91315
$206$	−13.8763	−0.966807
$207$	0	0
$208$	−10.6657	−0.739534
$209$	3.11715	0.215618
$210$	0	0
$211$	−14.1440	−0.973712	−0.486856	−	0.873482i	$-0.661856\pi$
$211$	−14.1440	−0.973712	−0.486856	+	0.873482i	$0.661856\pi$
$212$	5.93841	0.407852
$213$	0	0
$214$	2.19543	0.150077
$215$	26.5759	1.81246
$216$	0	0
$217$	−8.84506	−0.600442
$218$	2.46569	0.166998
$219$	0	0
$220$	−18.3304	−1.23584
$221$	6.69972	0.450672
$222$	0	0
$223$	5.91810	0.396305	0.198153	−	0.980171i	$-0.436506\pi$
$223$	5.91810	0.396305	0.198153	+	0.980171i	$0.436506\pi$
$224$	15.1375	1.01142
$225$	0	0
$226$	29.0801	1.93438
$227$	9.01629	0.598432	0.299216	−	0.954185i	$-0.403275\pi$
$227$	9.01629	0.598432	0.299216	+	0.954185i	$0.403275\pi$
$228$	0	0
$229$	−19.8774	−1.31353	−0.656767	−	0.754093i	$-0.728077\pi$
$229$	−19.8774	−1.31353	−0.656767	+	0.754093i	$0.728077\pi$
$230$	−32.3146	−2.13076
$231$	0	0
$232$	−0.404716	−0.0265709
$233$	−14.6922	−0.962520	−0.481260	−	0.876578i	$-0.659821\pi$
$233$	−14.6922	−0.962520	−0.481260	+	0.876578i	$0.659821\pi$
$234$	0	0
$235$	3.86690	0.252249
$236$	−1.30803	−0.0851456
$237$	0	0
$238$	−11.9261	−0.773055
$239$	26.2291	1.69662	0.848311	−	0.529499i	$-0.177620\pi$
$239$	26.2291	1.69662	0.848311	+	0.529499i	$0.177620\pi$
$240$	0	0
$241$	1.75539	0.113075	0.0565374	−	0.998400i	$-0.481994\pi$
$241$	1.75539	0.113075	0.0565374	+	0.998400i	$0.481994\pi$
$242$	−2.40809	−0.154798
$243$	0	0
$244$	−5.09714	−0.326311
$245$	9.36369	0.598224
$246$	0	0
$247$	2.25546	0.143512
$248$	−3.71735	−0.236052
$249$	0	0
$250$	−35.9367	−2.27284
$251$	−30.6806	−1.93654	−0.968272	−	0.249897i	$-0.919603\pi$
$251$	−30.6806	−1.93654	−0.968272	+	0.249897i	$0.919603\pi$
$252$	0	0
$253$	13.8828	0.872805
$254$	−36.7505	−2.30593
$255$	0	0
$256$	20.7445	1.29653
$257$	−7.82345	−0.488014	−0.244007	−	0.969774i	$-0.578462\pi$
$257$	−7.82345	−0.488014	−0.244007	+	0.969774i	$0.578462\pi$
$258$	0	0
$259$	4.47208	0.277882
$260$	−13.2633	−0.822554
$261$	0	0
$262$	−28.1654	−1.74007
$263$	−11.9466	−0.736656	−0.368328	−	0.929696i	$-0.620070\pi$
$263$	−11.9466	−0.736656	−0.368328	+	0.929696i	$0.620070\pi$
$264$	0	0
$265$	−15.1001	−0.927593
$266$	−4.01493	−0.246171
$267$	0	0
$268$	−8.12195	−0.496127
$269$	23.9381	1.45953	0.729766	−	0.683696i	$-0.239629\pi$
$269$	23.9381	1.45953	0.729766	+	0.683696i	$0.239629\pi$
$270$	0	0
$271$	32.3793	1.96690	0.983450	−	0.181181i	$-0.0579921\pi$
$271$	32.3793	1.96690	0.983450	+	0.181181i	$0.0579921\pi$
$272$	−14.0467	−0.851708
$273$	0	0
$274$	11.3734	0.687093
$275$	31.0247	1.87086
$276$	0	0
$277$	−20.0984	−1.20760	−0.603798	−	0.797138i	$-0.706347\pi$
$277$	−20.0984	−1.20760	−0.603798	+	0.797138i	$0.706347\pi$
$278$	29.1148	1.74619
$279$	0	0
$280$	−7.44079	−0.444672
$281$	−15.6719	−0.934909	−0.467455	−	0.884017i	$-0.654829\pi$
$281$	−15.6719	−0.934909	−0.467455	+	0.884017i	$0.654829\pi$
$282$	0	0
$283$	−32.0086	−1.90272	−0.951358	−	0.308087i	$-0.900311\pi$
$283$	−32.0086	−1.90272	−0.951358	+	0.308087i	$0.900311\pi$
$284$	9.16025	0.543561
$285$	0	0
$286$	13.1920	0.780058
$287$	15.1574	0.894715
$288$	0	0
$289$	−8.17649	−0.480970
$290$	−3.26539	−0.191750
$291$	0	0
$292$	−21.4481	−1.25516
$293$	−6.21552	−0.363114	−0.181557	−	0.983380i	$-0.558114\pi$
$293$	−6.21552	−0.363114	−0.181557	+	0.983380i	$0.558114\pi$
$294$	0	0
$295$	3.32604	0.193650
$296$	1.87950	0.109244
$297$	0	0
$298$	−13.0096	−0.753628
$299$	10.0451	0.580925
$300$	0	0
$301$	14.7057	0.847625
$302$	−29.9372	−1.72269
$303$	0	0
$304$	−4.72884	−0.271217
$305$	12.9609	0.742141
$306$	0	0
$307$	17.5887	1.00384	0.501921	−	0.864913i	$-0.332627\pi$
$307$	17.5887	1.00384	0.501921	+	0.864913i	$0.332627\pi$
$308$	−10.1431	−0.577958
$309$	0	0
$310$	−29.9929	−1.70348
$311$	24.5549	1.39238	0.696191	−	0.717857i	$-0.254877\pi$
$311$	24.5549	1.39238	0.696191	+	0.717857i	$0.254877\pi$
$312$	0	0
$313$	2.46100	0.139104	0.0695519	−	0.997578i	$-0.477843\pi$
$313$	2.46100	0.139104	0.0695519	+	0.997578i	$0.477843\pi$
$314$	24.9085	1.40567
$315$	0	0
$316$	5.83365	0.328168
$317$	−6.44461	−0.361965	−0.180983	−	0.983486i	$-0.557928\pi$
$317$	−6.44461	−0.361965	−0.180983	+	0.983486i	$0.557928\pi$
$318$	0	0
$319$	1.40286	0.0785449
$320$	14.7582	0.825009
$321$	0	0
$322$	−17.8813	−0.996484
$323$	2.97044	0.165280
$324$	0	0
$325$	22.4484	1.24521
$326$	7.32784	0.405851
$327$	0	0
$328$	6.37028	0.351740
$329$	2.13974	0.117968
$330$	0	0
$331$	14.4313	0.793218	0.396609	−	0.917988i	$-0.370187\pi$
$331$	14.4313	0.793218	0.396609	+	0.917988i	$0.370187\pi$
$332$	18.5794	1.01968
$333$	0	0
$334$	−29.2333	−1.59957
$335$	20.6524	1.12836
$336$	0	0
$337$	25.9954	1.41606	0.708031	−	0.706181i	$-0.249584\pi$
$337$	25.9954	1.41606	0.708031	+	0.706181i	$0.249584\pi$
$338$	−14.8474	−0.807594
$339$	0	0
$340$	−17.4677	−0.947320
$341$	12.8854	0.697782
$342$	0	0
$343$	20.1596	1.08852
$344$	6.18045	0.333227
$345$	0	0
$346$	−41.7841	−2.24633
$347$	−2.84174	−0.152552	−0.0762762	−	0.997087i	$-0.524303\pi$
$347$	−2.84174	−0.152552	−0.0762762	+	0.997087i	$0.524303\pi$
$348$	0	0
$349$	6.79002	0.363462	0.181731	−	0.983348i	$-0.441830\pi$
$349$	6.79002	0.363462	0.181731	+	0.983348i	$0.441830\pi$
$350$	−39.9602	−2.13596
$351$	0	0
$352$	−22.0521	−1.17538
$353$	−0.793877	−0.0422538	−0.0211269	−	0.999777i	$-0.506725\pi$
$353$	−0.793877	−0.0422538	−0.0211269	+	0.999777i	$0.506725\pi$
$354$	0	0
$355$	−23.2926	−1.23624
$356$	−8.47629	−0.449243
$357$	0	0
$358$	26.5262	1.40195
$359$	2.57199	0.135744	0.0678722	−	0.997694i	$-0.478379\pi$
$359$	2.57199	0.135744	0.0678722	+	0.997694i	$0.478379\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−21.1537	−1.11182
$363$	0	0
$364$	−7.33922	−0.384680
$365$	54.5380	2.85465
$366$	0	0
$367$	13.3540	0.697073	0.348536	−	0.937295i	$-0.386679\pi$
$367$	13.3540	0.697073	0.348536	+	0.937295i	$0.386679\pi$
$368$	−21.0608	−1.09787
$369$	0	0
$370$	15.1645	0.788363
$371$	−8.35563	−0.433803
$372$	0	0
$373$	−4.97885	−0.257795	−0.128897	−	0.991658i	$-0.541144\pi$
$373$	−4.97885	−0.257795	−0.128897	+	0.991658i	$0.541144\pi$
$374$	17.3738	0.898378
$375$	0	0
$376$	0.899279	0.0463768
$377$	1.01506	0.0522782
$378$	0	0
$379$	−7.41913	−0.381095	−0.190547	−	0.981678i	$-0.561026\pi$
$379$	−7.41913	−0.381095	−0.190547	+	0.981678i	$0.561026\pi$
$380$	−5.88052	−0.301664
$381$	0	0
$382$	7.66349	0.392098
$383$	−10.1713	−0.519730	−0.259865	−	0.965645i	$-0.583678\pi$
$383$	−10.1713	−0.519730	−0.259865	+	0.965645i	$0.583678\pi$
$384$	0	0
$385$	25.7918	1.31447
$386$	−42.9700	−2.18711
$387$	0	0
$388$	−16.6057	−0.843026
$389$	6.77005	0.343255	0.171628	−	0.985162i	$-0.445097\pi$
$389$	6.77005	0.343255	0.171628	+	0.985162i	$0.445097\pi$
$390$	0	0
$391$	13.2294	0.669041
$392$	2.17760	0.109986
$393$	0	0
$394$	36.5776	1.84275
$395$	−14.8337	−0.746365
$396$	0	0
$397$	−14.8986	−0.747741	−0.373870	−	0.927481i	$-0.621969\pi$
$397$	−14.8986	−0.747741	−0.373870	+	0.927481i	$0.621969\pi$
$398$	−38.7500	−1.94236
$399$	0	0
$400$	−47.0656	−2.35328
$401$	−36.9136	−1.84338	−0.921689	−	0.387930i	$-0.873190\pi$
$401$	−36.9136	−1.84338	−0.921689	+	0.387930i	$0.873190\pi$
$402$	0	0
$403$	9.32341	0.464432
$404$	−7.72739	−0.384452
$405$	0	0
$406$	−1.80690	−0.0896749
$407$	−6.51487	−0.322930
$408$	0	0
$409$	13.2635	0.655838	0.327919	−	0.944706i	$-0.393653\pi$
$409$	13.2635	0.655838	0.327919	+	0.944706i	$0.393653\pi$
$410$	51.3976	2.53835
$411$	0	0
$412$	−11.2463	−0.554065
$413$	1.84046	0.0905632
$414$	0	0
$415$	−47.2435	−2.31909
$416$	−15.9562	−0.782315
$417$	0	0
$418$	5.84890	0.286079
$419$	−6.51752	−0.318402	−0.159201	−	0.987246i	$-0.550892\pi$
$419$	−6.51752	−0.318402	−0.159201	+	0.987246i	$0.550892\pi$
$420$	0	0
$421$	15.2937	0.745369	0.372685	−	0.927958i	$-0.378437\pi$
$421$	15.2937	0.745369	0.372685	+	0.927958i	$0.378437\pi$
$422$	−26.5392	−1.29191
$423$	0	0
$424$	−3.51166	−0.170541
$425$	29.5645	1.43409
$426$	0	0
$427$	7.17192	0.347074
$428$	1.77933	0.0860072
$429$	0	0
$430$	49.8660	2.40475
$431$	−20.1813	−0.972098	−0.486049	−	0.873931i	$-0.661562\pi$
$431$	−20.1813	−0.972098	−0.486049	+	0.873931i	$0.661562\pi$
$432$	0	0
$433$	14.6928	0.706091	0.353046	−	0.935606i	$-0.385146\pi$
$433$	14.6928	0.706091	0.353046	+	0.935606i	$0.385146\pi$
$434$	−16.5965	−0.796659
$435$	0	0
$436$	1.99837	0.0957044
$437$	4.45369	0.213049
$438$	0	0
$439$	20.3320	0.970391	0.485196	−	0.874406i	$-0.338748\pi$
$439$	20.3320	0.970391	0.485196	+	0.874406i	$0.338748\pi$
$440$	10.8396	0.516759
$441$	0	0
$442$	12.5711	0.597946
$443$	−27.0619	−1.28575	−0.642875	−	0.765971i	$-0.722258\pi$
$443$	−27.0619	−1.28575	−0.642875	+	0.765971i	$0.722258\pi$
$444$	0	0
$445$	21.5534	1.02173
$446$	11.1045	0.525813
$447$	0	0
$448$	8.16644	0.385828
$449$	28.0326	1.32294	0.661470	−	0.749972i	$-0.269933\pi$
$449$	28.0326	1.32294	0.661470	+	0.749972i	$0.269933\pi$
$450$	0	0
$451$	−22.0811	−1.03976
$452$	23.5685	1.10857
$453$	0	0
$454$	16.9178	0.793993
$455$	18.6621	0.874892
$456$	0	0
$457$	−18.2724	−0.854745	−0.427372	−	0.904076i	$-0.640561\pi$
$457$	−18.2724	−0.854745	−0.427372	+	0.904076i	$0.640561\pi$
$458$	−37.2972	−1.74278
$459$	0	0
$460$	−26.1900	−1.22112
$461$	30.5335	1.42209	0.711043	−	0.703149i	$-0.248223\pi$
$461$	30.5335	1.42209	0.711043	+	0.703149i	$0.248223\pi$
$462$	0	0
$463$	−37.4170	−1.73891	−0.869457	−	0.494009i	$-0.835531\pi$
$463$	−37.4170	−1.73891	−0.869457	+	0.494009i	$0.835531\pi$
$464$	−2.12819	−0.0987987
$465$	0	0
$466$	−27.5679	−1.27706
$467$	−17.2411	−0.797822	−0.398911	−	0.916990i	$-0.630612\pi$
$467$	−17.2411	−0.797822	−0.398911	+	0.916990i	$0.630612\pi$
$468$	0	0
$469$	11.4280	0.527695
$470$	7.25570	0.334680
$471$	0	0
$472$	0.773499	0.0356032
$473$	−21.4231	−0.985036
$474$	0	0
$475$	9.95290	0.456670
$476$	−9.66574	−0.443029
$477$	0	0
$478$	49.2153	2.25106
$479$	−29.0114	−1.32557	−0.662783	−	0.748812i	$-0.730625\pi$
$479$	−29.0114	−1.32557	−0.662783	+	0.748812i	$0.730625\pi$
$480$	0	0
$481$	−4.71394	−0.214937
$482$	3.29375	0.150026
$483$	0	0
$484$	−1.95168	−0.0887129
$485$	42.2247	1.91733
$486$	0	0
$487$	40.6074	1.84009	0.920047	−	0.391807i	$-0.128150\pi$
$487$	40.6074	1.84009	0.920047	+	0.391807i	$0.128150\pi$
$488$	3.01418	0.136445
$489$	0	0
$490$	17.5697	0.793716
$491$	−22.2052	−1.00211	−0.501054	−	0.865416i	$-0.667054\pi$
$491$	−22.2052	−1.00211	−0.501054	+	0.865416i	$0.667054\pi$
$492$	0	0
$493$	1.33683	0.0602078
$494$	4.23206	0.190410
$495$	0	0
$496$	−19.5476	−0.877714
$497$	−12.8889	−0.578147
$498$	0	0
$499$	−39.8424	−1.78359	−0.891796	−	0.452437i	$-0.850555\pi$
$499$	−39.8424	−1.78359	−0.891796	+	0.452437i	$0.850555\pi$
$500$	−29.1256	−1.30254
$501$	0	0
$502$	−57.5680	−2.56938
$503$	5.76600	0.257093	0.128547	−	0.991703i	$-0.458969\pi$
$503$	5.76600	0.257093	0.128547	+	0.991703i	$0.458969\pi$
$504$	0	0
$505$	19.6491	0.874373
$506$	26.0492	1.15803
$507$	0	0
$508$	−29.7851	−1.32150
$509$	−44.1049	−1.95492	−0.977458	−	0.211131i	$-0.932285\pi$
$509$	−44.1049	−1.95492	−0.977458	+	0.211131i	$0.932285\pi$
$510$	0	0
$511$	30.1785	1.33502
$512$	24.9488	1.10259
$513$	0	0
$514$	−14.6796	−0.647491
$515$	28.5969	1.26013
$516$	0	0
$517$	−3.11715	−0.137092
$518$	8.39124	0.368690
$519$	0	0
$520$	7.84319	0.343947
$521$	24.1903	1.05980	0.529899	−	0.848061i	$-0.322230\pi$
$521$	24.1903	1.05980	0.529899	+	0.848061i	$0.322230\pi$
$522$	0	0
$523$	−43.3238	−1.89442	−0.947209	−	0.320616i	$-0.896110\pi$
$523$	−43.3238	−1.89442	−0.947209	+	0.320616i	$0.896110\pi$
$524$	−22.8272	−0.997211
$525$	0	0
$526$	−22.4161	−0.977387
$527$	12.2789	0.534878
$528$	0	0
$529$	−3.16464	−0.137593
$530$	−28.3333	−1.23072
$531$	0	0
$532$	−3.25398	−0.141078
$533$	−15.9772	−0.692047
$534$	0	0
$535$	−4.52446	−0.195609
$536$	4.80288	0.207453
$537$	0	0
$538$	44.9166	1.93649
$539$	−7.54817	−0.325123
$540$	0	0
$541$	−0.876548	−0.0376857	−0.0188429	−	0.999822i	$-0.505998\pi$
$541$	−0.876548	−0.0376857	−0.0188429	+	0.999822i	$0.505998\pi$
$542$	60.7552	2.60966
$543$	0	0
$544$	−21.0142	−0.900978
$545$	−5.08142	−0.217664
$546$	0	0
$547$	17.4216	0.744896	0.372448	−	0.928053i	$-0.378519\pi$
$547$	17.4216	0.744896	0.372448	+	0.928053i	$0.378519\pi$
$548$	9.21779	0.393765
$549$	0	0
$550$	58.2135	2.48223
$551$	0.450045	0.0191726
$552$	0	0
$553$	−8.20822	−0.349049
$554$	−37.7119	−1.60222
$555$	0	0
$556$	23.5966	1.00072
$557$	36.4447	1.54421	0.772106	−	0.635494i	$-0.219203\pi$
$557$	36.4447	1.54421	0.772106	+	0.635494i	$0.219203\pi$
$558$	0	0
$559$	−15.5010	−0.655624
$560$	−39.1272	−1.65343
$561$	0	0
$562$	−29.4062	−1.24043
$563$	−37.3548	−1.57432	−0.787159	−	0.616750i	$-0.788449\pi$
$563$	−37.3548	−1.57432	−0.787159	+	0.616750i	$0.788449\pi$
$564$	0	0
$565$	−59.9296	−2.52126
$566$	−60.0598	−2.52450
$567$	0	0
$568$	−5.41688	−0.227287
$569$	−10.9433	−0.458769	−0.229384	−	0.973336i	$-0.573671\pi$
$569$	−10.9433	−0.458769	−0.229384	+	0.973336i	$0.573671\pi$
$570$	0	0
$571$	29.7541	1.24517	0.622584	−	0.782553i	$-0.286083\pi$
$571$	29.7541	1.24517	0.622584	+	0.782553i	$0.286083\pi$
$572$	10.6917	0.447042
$573$	0	0
$574$	28.4408	1.18710
$575$	44.3271	1.84857
$576$	0	0
$577$	41.6706	1.73477	0.867384	−	0.497639i	$-0.165800\pi$
$577$	41.6706	1.73477	0.867384	+	0.497639i	$0.165800\pi$
$578$	−15.3420	−0.638145
$579$	0	0
$580$	−2.64650	−0.109890
$581$	−26.1422	−1.08456
$582$	0	0
$583$	12.1724	0.504128
$584$	12.6833	0.524837
$585$	0	0
$586$	−11.6626	−0.481776
$587$	40.3468	1.66529	0.832645	−	0.553808i	$-0.186826\pi$
$587$	40.3468	1.66529	0.832645	+	0.553808i	$0.186826\pi$
$588$	0	0
$589$	4.13370	0.170326
$590$	6.24086	0.256932
$591$	0	0
$592$	9.88331	0.406202
$593$	14.8031	0.607889	0.303945	−	0.952690i	$-0.401696\pi$
$593$	14.8031	0.607889	0.303945	+	0.952690i	$0.401696\pi$
$594$	0	0
$595$	24.5779	1.00760
$596$	−10.5439	−0.431895
$597$	0	0
$598$	18.8483	0.770764
$599$	36.1019	1.47508	0.737542	−	0.675301i	$-0.235986\pi$
$599$	36.1019	1.47508	0.737542	+	0.675301i	$0.235986\pi$
$600$	0	0
$601$	24.3569	0.993539	0.496769	−	0.867883i	$-0.334520\pi$
$601$	24.3569	0.993539	0.496769	+	0.867883i	$0.334520\pi$
$602$	27.5933	1.12462
$603$	0	0
$604$	−24.2631	−0.987253
$605$	4.96271	0.201763
$606$	0	0
$607$	−8.51330	−0.345544	−0.172772	−	0.984962i	$-0.555272\pi$
$607$	−8.51330	−0.345544	−0.172772	+	0.984962i	$0.555272\pi$
$608$	−7.07445	−0.286907
$609$	0	0
$610$	24.3194	0.984664
$611$	−2.25546	−0.0912463
$612$	0	0
$613$	−32.9272	−1.32992	−0.664958	−	0.746881i	$-0.731550\pi$
$613$	−32.9272	−1.32992	−0.664958	+	0.746881i	$0.731550\pi$
$614$	33.0028	1.33189
$615$	0	0
$616$	5.99810	0.241670
$617$	−20.3772	−0.820355	−0.410177	−	0.912006i	$-0.634533\pi$
$617$	−20.3772	−0.820355	−0.410177	+	0.912006i	$0.634533\pi$
$618$	0	0
$619$	−36.4255	−1.46406	−0.732031	−	0.681271i	$-0.761428\pi$
$619$	−36.4255	−1.46406	−0.732031	+	0.681271i	$0.761428\pi$
$620$	−24.3083	−0.976245
$621$	0	0
$622$	46.0739	1.84739
$623$	11.9265	0.477827
$624$	0	0
$625$	24.2957	0.971826
$626$	4.61772	0.184561
$627$	0	0
$628$	20.1876	0.805573
$629$	−6.20825	−0.247539
$630$	0	0
$631$	−24.1628	−0.961905	−0.480952	−	0.876747i	$-0.659709\pi$
$631$	−24.1628	−0.961905	−0.480952	+	0.876747i	$0.659709\pi$
$632$	−3.44970	−0.137222
$633$	0	0
$634$	−12.0924	−0.480251
$635$	75.7372	3.00554
$636$	0	0
$637$	−5.46160	−0.216396
$638$	2.63227	0.104212
$639$	0	0
$640$	−27.0206	−1.06808
$641$	−13.9247	−0.549991	−0.274996	−	0.961445i	$-0.588676\pi$
$641$	−13.9247	−0.549991	−0.274996	+	0.961445i	$0.588676\pi$
$642$	0	0
$643$	20.6261	0.813415	0.406708	−	0.913558i	$-0.366677\pi$
$643$	20.6261	0.813415	0.406708	+	0.913558i	$0.366677\pi$
$644$	−14.4922	−0.571073
$645$	0	0
$646$	5.57362	0.219291
$647$	−34.2331	−1.34584	−0.672921	−	0.739715i	$-0.734960\pi$
$647$	−34.2331	−1.34584	−0.672921	+	0.739715i	$0.734960\pi$
$648$	0	0
$649$	−2.68116	−0.105245
$650$	42.1213	1.65213
$651$	0	0
$652$	5.93898	0.232589
$653$	−7.38862	−0.289139	−0.144569	−	0.989495i	$-0.546180\pi$
$653$	−7.38862	−0.289139	−0.144569	+	0.989495i	$0.546180\pi$
$654$	0	0
$655$	58.0447	2.26799
$656$	33.4979	1.30788
$657$	0	0
$658$	4.01493	0.156518
$659$	−11.3598	−0.442513	−0.221257	−	0.975216i	$-0.571016\pi$
$659$	−11.3598	−0.442513	−0.221257	+	0.975216i	$0.571016\pi$
$660$	0	0
$661$	7.73149	0.300720	0.150360	−	0.988631i	$-0.451957\pi$
$661$	7.73149	0.300720	0.150360	+	0.988631i	$0.451957\pi$
$662$	27.0784	1.05243
$663$	0	0
$664$	−10.9869	−0.426374
$665$	8.27417	0.320858
$666$	0	0
$667$	2.00436	0.0776091
$668$	−23.6926	−0.916696
$669$	0	0
$670$	38.7513	1.49709
$671$	−10.4480	−0.403339
$672$	0	0
$673$	48.3229	1.86271	0.931356	−	0.364111i	$-0.118627\pi$
$673$	48.3229	1.86271	0.931356	+	0.364111i	$0.118627\pi$
$674$	48.7768	1.87881
$675$	0	0
$676$	−12.0334	−0.462823
$677$	11.4785	0.441156	0.220578	−	0.975369i	$-0.429206\pi$
$677$	11.4785	0.441156	0.220578	+	0.975369i	$0.429206\pi$
$678$	0	0
$679$	23.3650	0.896667
$680$	10.3295	0.396117
$681$	0	0
$682$	24.1776	0.925808
$683$	33.4711	1.28074	0.640368	−	0.768068i	$-0.278782\pi$
$683$	33.4711	1.28074	0.640368	+	0.768068i	$0.278782\pi$
$684$	0	0
$685$	−23.4389	−0.895553
$686$	37.8267	1.44423
$687$	0	0
$688$	32.4997	1.23904
$689$	8.80751	0.335540
$690$	0	0
$691$	16.3070	0.620346	0.310173	−	0.950680i	$-0.399613\pi$
$691$	16.3070	0.620346	0.310173	+	0.950680i	$0.399613\pi$
$692$	−33.8647	−1.28734
$693$	0	0
$694$	−5.33213	−0.202405
$695$	−60.0012	−2.27597
$696$	0	0
$697$	−21.0419	−0.797018
$698$	12.7405	0.482237
$699$	0	0
$700$	−32.3865	−1.22409
$701$	−17.8447	−0.673983	−0.336992	−	0.941508i	$-0.609409\pi$
$701$	−17.8447	−0.673983	−0.336992	+	0.941508i	$0.609409\pi$
$702$	0	0
$703$	−2.09001	−0.0788262
$704$	−11.8968	−0.448376
$705$	0	0
$706$	−1.48960	−0.0560618
$707$	10.8728	0.408914
$708$	0	0
$709$	−22.9967	−0.863659	−0.431830	−	0.901955i	$-0.642132\pi$
$709$	−22.9967	−0.863659	−0.431830	+	0.901955i	$0.642132\pi$
$710$	−43.7053	−1.64023
$711$	0	0
$712$	5.01242	0.187849
$713$	18.4102	0.689469
$714$	0	0
$715$	−27.1867	−1.01672
$716$	21.4986	0.803442
$717$	0	0
$718$	4.82598	0.180104
$719$	−25.3552	−0.945590	−0.472795	−	0.881172i	$-0.656755\pi$
$719$	−25.3552	−0.945590	−0.472795	+	0.881172i	$0.656755\pi$
$720$	0	0
$721$	15.8241	0.589320
$722$	1.87636	0.0698309
$723$	0	0
$724$	−17.1444	−0.637168
$725$	4.47925	0.166355
$726$	0	0
$727$	28.7939	1.06791	0.533954	−	0.845514i	$-0.320706\pi$
$727$	28.7939	1.06791	0.533954	+	0.845514i	$0.320706\pi$
$728$	4.34002	0.160852
$729$	0	0
$730$	102.333	3.78751
$731$	−20.4148	−0.755070
$732$	0	0
$733$	−19.8822	−0.734367	−0.367183	−	0.930149i	$-0.619678\pi$
$733$	−19.8822	−0.734367	−0.367183	+	0.930149i	$0.619678\pi$
$734$	25.0569	0.924868
$735$	0	0
$736$	−31.5074	−1.16138
$737$	−16.6481	−0.613241
$738$	0	0
$739$	25.7721	0.948041	0.474021	−	0.880514i	$-0.342802\pi$
$739$	25.7721	0.948041	0.474021	+	0.880514i	$0.342802\pi$
$740$	12.2903	0.451802
$741$	0	0
$742$	−15.6782	−0.575564
$743$	3.95001	0.144912	0.0724559	−	0.997372i	$-0.476916\pi$
$743$	3.95001	0.144912	0.0724559	+	0.997372i	$0.476916\pi$
$744$	0	0
$745$	26.8109	0.982276
$746$	−9.34212	−0.342039
$747$	0	0
$748$	14.0809	0.514849
$749$	−2.50360	−0.0914797
$750$	0	0
$751$	41.8612	1.52754	0.763769	−	0.645490i	$-0.223347\pi$
$751$	41.8612	1.52754	0.763769	+	0.645490i	$0.223347\pi$
$752$	4.72884	0.172443
$753$	0	0
$754$	1.90462	0.0693621
$755$	61.6960	2.24535
$756$	0	0
$757$	−22.9714	−0.834911	−0.417455	−	0.908697i	$-0.637078\pi$
$757$	−22.9714	−0.834911	−0.417455	+	0.908697i	$0.637078\pi$
$758$	−13.9210	−0.505632
$759$	0	0
$760$	3.47742	0.126139
$761$	−0.596642	−0.0216283	−0.0108141	−	0.999942i	$-0.503442\pi$
$761$	−0.596642	−0.0216283	−0.0108141	+	0.999942i	$0.503442\pi$
$762$	0	0
$763$	−2.81180	−0.101794
$764$	6.21102	0.224707
$765$	0	0
$766$	−19.0851	−0.689571
$767$	−1.94000	−0.0700492
$768$	0	0
$769$	−35.2647	−1.27168	−0.635838	−	0.771823i	$-0.719345\pi$
$769$	−35.2647	−1.27168	−0.635838	+	0.771823i	$0.719345\pi$
$770$	48.3948	1.74403
$771$	0	0
$772$	−34.8258	−1.25341
$773$	−0.398665	−0.0143390	−0.00716949	−	0.999974i	$-0.502282\pi$
$773$	−0.398665	−0.0143390	−0.00716949	+	0.999974i	$0.502282\pi$
$774$	0	0
$775$	41.1423	1.47788
$776$	9.81971	0.352507
$777$	0	0
$778$	12.7031	0.455427
$779$	−7.08376	−0.253802
$780$	0	0
$781$	18.7764	0.671872
$782$	24.8232	0.887675
$783$	0	0
$784$	11.4509	0.408960
$785$	−51.3327	−1.83214
$786$	0	0
$787$	13.1774	0.469725	0.234863	−	0.972029i	$-0.424536\pi$
$787$	13.1774	0.469725	0.234863	+	0.972029i	$0.424536\pi$
$788$	29.6450	1.05606
$789$	0	0
$790$	−27.8334	−0.990268
$791$	−33.1620	−1.17910
$792$	0	0
$793$	−7.55979	−0.268456
$794$	−27.9552	−0.992093
$795$	0	0
$796$	−31.4057	−1.11314
$797$	−28.9908	−1.02691	−0.513453	−	0.858118i	$-0.671634\pi$
$797$	−28.9908	−1.02691	−0.513453	+	0.858118i	$0.671634\pi$
$798$	0	0
$799$	−2.97044	−0.105087
$800$	−70.4113	−2.48941
$801$	0	0
$802$	−69.2633	−2.44577
$803$	−43.9637	−1.55144
$804$	0	0
$805$	36.8506	1.29881
$806$	17.4941	0.616203
$807$	0	0
$808$	4.56956	0.160757
$809$	−20.0485	−0.704866	−0.352433	−	0.935837i	$-0.614646\pi$
$809$	−20.0485	−0.704866	−0.352433	+	0.935837i	$0.614646\pi$
$810$	0	0
$811$	4.12297	0.144777	0.0723885	−	0.997377i	$-0.476938\pi$
$811$	4.12297	0.144777	0.0723885	+	0.997377i	$0.476938\pi$
$812$	−1.46444	−0.0513916
$813$	0	0
$814$	−12.2242	−0.428460
$815$	−15.1016	−0.528985
$816$	0	0
$817$	−6.87267	−0.240444
$818$	24.8871	0.870158
$819$	0	0
$820$	41.6562	1.45470
$821$	52.1805	1.82111	0.910556	−	0.413386i	$-0.135654\pi$
$821$	52.1805	1.82111	0.910556	+	0.413386i	$0.135654\pi$
$822$	0	0
$823$	10.0619	0.350736	0.175368	−	0.984503i	$-0.443888\pi$
$823$	10.0619	0.350736	0.175368	+	0.984503i	$0.443888\pi$
$824$	6.65046	0.231680
$825$	0	0
$826$	3.45337	0.120158
$827$	−15.2334	−0.529716	−0.264858	−	0.964287i	$-0.585325\pi$
$827$	−15.2334	−0.529716	−0.264858	+	0.964287i	$0.585325\pi$
$828$	0	0
$829$	−16.9078	−0.587233	−0.293616	−	0.955923i	$-0.594859\pi$
$829$	−16.9078	−0.587233	−0.293616	+	0.955923i	$0.594859\pi$
$830$	−88.6460	−3.07695
$831$	0	0
$832$	−8.60809	−0.298432
$833$	−7.19292	−0.249220
$834$	0	0
$835$	60.2453	2.08488
$836$	4.74035	0.163948
$837$	0	0
$838$	−12.2292	−0.422451
$839$	31.0527	1.07206	0.536029	−	0.844199i	$-0.319924\pi$
$839$	31.0527	1.07206	0.536029	+	0.844199i	$0.319924\pi$
$840$	0	0
$841$	−28.7975	−0.993016
$842$	28.6965	0.988947
$843$	0	0
$844$	−21.5092	−0.740378
$845$	30.5983	1.05261
$846$	0	0
$847$	2.74611	0.0943575
$848$	−18.4660	−0.634124
$849$	0	0
$850$	55.4736	1.90273
$851$	−9.30825	−0.319083
$852$	0	0
$853$	−14.7731	−0.505820	−0.252910	−	0.967490i	$-0.581388\pi$
$853$	−14.7731	−0.505820	−0.252910	+	0.967490i	$0.581388\pi$
$854$	13.4571	0.460493
$855$	0	0
$856$	−1.05220	−0.0359635
$857$	−15.1514	−0.517561	−0.258781	−	0.965936i	$-0.583321\pi$
$857$	−15.1514	−0.517561	−0.258781	+	0.965936i	$0.583321\pi$
$858$	0	0
$859$	−45.1787	−1.54148	−0.770738	−	0.637152i	$-0.780112\pi$
$859$	−45.1787	−1.54148	−0.770738	+	0.637152i	$0.780112\pi$
$860$	40.4148	1.37813
$861$	0	0
$862$	−37.8674	−1.28977
$863$	27.9395	0.951073	0.475537	−	0.879696i	$-0.342254\pi$
$863$	27.9395	0.951073	0.475537	+	0.879696i	$0.342254\pi$
$864$	0	0
$865$	86.1107	2.92785
$866$	27.5690	0.936833
$867$	0	0
$868$	−13.4510	−0.456556
$869$	11.9576	0.405635
$870$	0	0
$871$	−12.0460	−0.408163
$872$	−1.18173	−0.0400183
$873$	0	0
$874$	8.35673	0.282671
$875$	40.9811	1.38541
$876$	0	0
$877$	−11.1495	−0.376490	−0.188245	−	0.982122i	$-0.560280\pi$
$877$	−11.1495	−0.376490	−0.188245	+	0.982122i	$0.560280\pi$
$878$	38.1501	1.28750
$879$	0	0
$880$	57.0000	1.92147
$881$	−24.7154	−0.832684	−0.416342	−	0.909208i	$-0.636688\pi$
$881$	−24.7154	−0.832684	−0.416342	+	0.909208i	$0.636688\pi$
$882$	0	0
$883$	13.8276	0.465335	0.232668	−	0.972556i	$-0.425255\pi$
$883$	13.8276	0.465335	0.232668	+	0.972556i	$0.425255\pi$
$884$	10.1885	0.342676
$885$	0	0
$886$	−50.7779	−1.70592
$887$	−45.1358	−1.51551	−0.757755	−	0.652539i	$-0.773704\pi$
$887$	−45.1358	−1.51551	−0.757755	+	0.652539i	$0.773704\pi$
$888$	0	0
$889$	41.9091	1.40559
$890$	40.4420	1.35562
$891$	0	0
$892$	8.99985	0.301337
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	−54.6665	−1.82730
$896$	−14.9518	−0.499505
$897$	0	0
$898$	52.5993	1.75526
$899$	1.86035	0.0620462
$900$	0	0
$901$	11.5995	0.386435
$902$	−41.4322	−1.37954
$903$	0	0
$904$	−13.9371	−0.463542
$905$	43.5946	1.44913
$906$	0	0
$907$	−16.5203	−0.548546	−0.274273	−	0.961652i	$-0.588437\pi$
$907$	−16.5203	−0.548546	−0.274273	+	0.961652i	$0.588437\pi$
$908$	13.7114	0.455028
$909$	0	0
$910$	35.0168	1.16080
$911$	38.1578	1.26423	0.632113	−	0.774876i	$-0.282188\pi$
$911$	38.1578	1.26423	0.632113	+	0.774876i	$0.282188\pi$
$912$	0	0
$913$	38.0835	1.26038
$914$	−34.2856	−1.13407
$915$	0	0
$916$	−30.2282	−0.998767
$917$	32.1190	1.06066
$918$	0	0
$919$	26.1621	0.863007	0.431503	−	0.902111i	$-0.357983\pi$
$919$	26.1621	0.863007	0.431503	+	0.902111i	$0.357983\pi$
$920$	15.4874	0.510603
$921$	0	0
$922$	57.2918	1.88681
$923$	13.5860	0.447187
$924$	0	0
$925$	−20.8016	−0.683954
$926$	−70.2077	−2.30717
$927$	0	0
$928$	−3.18382	−0.104514
$929$	−32.1481	−1.05474	−0.527372	−	0.849634i	$-0.676823\pi$
$929$	−32.1481	−1.05474	−0.527372	+	0.849634i	$0.676823\pi$
$930$	0	0
$931$	−2.42150	−0.0793614
$932$	−22.3430	−0.731868
$933$	0	0
$934$	−32.3505	−1.05854
$935$	−35.8048	−1.17094
$936$	0	0
$937$	17.8430	0.582905	0.291453	−	0.956585i	$-0.405861\pi$
$937$	17.8430	0.582905	0.291453	+	0.956585i	$0.405861\pi$
$938$	21.4430	0.700139
$939$	0	0
$940$	5.88052	0.191801
$941$	−12.4387	−0.405491	−0.202746	−	0.979231i	$-0.564986\pi$
$941$	−12.4387	−0.405491	−0.202746	+	0.979231i	$0.564986\pi$
$942$	0	0
$943$	−31.5489	−1.02737
$944$	4.06743	0.132383
$945$	0	0
$946$	−40.1975	−1.30693
$947$	12.0275	0.390842	0.195421	−	0.980719i	$-0.437393\pi$
$947$	12.0275	0.390842	0.195421	+	0.980719i	$0.437393\pi$
$948$	0	0
$949$	−31.8106	−1.03262
$950$	18.6752	0.605904
$951$	0	0
$952$	5.71580	0.185250
$953$	−36.9519	−1.19699	−0.598495	−	0.801127i	$-0.704234\pi$
$953$	−36.9519	−1.19699	−0.598495	+	0.801127i	$0.704234\pi$
$954$	0	0
$955$	−15.7933	−0.511059
$956$	39.8875	1.29005
$957$	0	0
$958$	−54.4359	−1.75874
$959$	−12.9699	−0.418819
$960$	0	0
$961$	−13.9125	−0.448790
$962$	−8.84505	−0.285176
$963$	0	0
$964$	2.66948	0.0859783
$965$	88.5546	2.85067
$966$	0	0
$967$	−7.76390	−0.249670	−0.124835	−	0.992178i	$-0.539840\pi$
$967$	−7.76390	−0.249670	−0.124835	+	0.992178i	$0.539840\pi$
$968$	1.15412	0.0370948
$969$	0	0
$970$	79.2289	2.54389
$971$	2.83840	0.0910886	0.0455443	−	0.998962i	$-0.485498\pi$
$971$	2.83840	0.0910886	0.0455443	+	0.998962i	$0.485498\pi$
$972$	0	0
$973$	−33.2016	−1.06439
$974$	76.1941	2.44142
$975$	0	0
$976$	15.8500	0.507345
$977$	54.5811	1.74620	0.873102	−	0.487538i	$-0.162105\pi$
$977$	54.5811	1.74620	0.873102	+	0.487538i	$0.162105\pi$
$978$	0	0
$979$	−17.3744	−0.555289
$980$	14.2397	0.454869
$981$	0	0
$982$	−41.6651	−1.32959
$983$	−0.607233	−0.0193677	−0.00968386	−	0.999953i	$-0.503083\pi$
$983$	−0.607233	−0.0193677	−0.00968386	+	0.999953i	$0.503083\pi$
$984$	0	0
$985$	−75.3809	−2.40184
$986$	2.50838	0.0798830
$987$	0	0
$988$	3.42996	0.109121
$989$	−30.6087	−0.973301
$990$	0	0
$991$	15.3716	0.488296	0.244148	−	0.969738i	$-0.421492\pi$
$991$	15.3716	0.488296	0.244148	+	0.969738i	$0.421492\pi$
$992$	−29.2437	−0.928488
$993$	0	0
$994$	−24.1843	−0.767078
$995$	79.8579	2.53166
$996$	0	0
$997$	−40.9010	−1.29535	−0.647673	−	0.761918i	$-0.724258\pi$
$997$	−40.9010	−1.29535	−0.647673	+	0.761918i	$0.724258\pi$
$998$	−74.7588	−2.36645
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.m.1.10		12
3.2	odd	2		893.2.a.a.1.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.a.1.3	✓	12	3.2	odd	2
8037.2.a.m.1.10		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q+1.87636 q^{2} +1.52073 q^{4} -3.86690 q^{5} -2.13974 q^{7} -0.899279 q^{8} +O(q^{10})\)	\(q+1.87636 q^{2} +1.52073 q^{4} -3.86690 q^{5} -2.13974 q^{7} -0.899279 q^{8} -7.25570 q^{10} +3.11715 q^{11} +2.25546 q^{13} -4.01493 q^{14} -4.72884 q^{16} +2.97044 q^{17} +1.00000 q^{19} -5.88052 q^{20} +5.84890 q^{22} +4.45369 q^{23} +9.95290 q^{25} +4.23206 q^{26} -3.25398 q^{28} +0.450045 q^{29} +4.13370 q^{31} -7.07445 q^{32} +5.57362 q^{34} +8.27417 q^{35} -2.09001 q^{37} +1.87636 q^{38} +3.47742 q^{40} -7.08376 q^{41} -6.87267 q^{43} +4.74035 q^{44} +8.35673 q^{46} -1.00000 q^{47} -2.42150 q^{49} +18.6752 q^{50} +3.42996 q^{52} +3.90497 q^{53} -12.0537 q^{55} +1.92423 q^{56} +0.844447 q^{58} -0.860132 q^{59} -3.35177 q^{61} +7.75632 q^{62} -3.81655 q^{64} -8.72164 q^{65} -5.34081 q^{67} +4.51724 q^{68} +15.5253 q^{70} +6.02358 q^{71} -14.1038 q^{73} -3.92161 q^{74} +1.52073 q^{76} -6.66990 q^{77} +3.83608 q^{79} +18.2859 q^{80} -13.2917 q^{82} +12.2174 q^{83} -11.4864 q^{85} -12.8956 q^{86} -2.80319 q^{88} -5.57382 q^{89} -4.82611 q^{91} +6.77287 q^{92} -1.87636 q^{94} -3.86690 q^{95} -10.9195 q^{97} -4.54361 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7}+O(q^{10}) \) 12 * q + q^2 + 7 * q^4 + 7 * q^5 - 13 * q^7	\( 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7} + q^{10} + 4 q^{11} - 17 q^{13} - 3 q^{14} - 19 q^{16} + 6 q^{17} + 12 q^{19} - 5 q^{20} - 8 q^{22} + 13 q^{23} - 7 q^{25} + 19 q^{26} - 29 q^{28} + 2 q^{29} - 14 q^{31} + 21 q^{32} - 6 q^{34} + 3 q^{35} - 2 q^{37} + q^{38} + 8 q^{40} - 8 q^{41} - 42 q^{43} - 24 q^{44} - 9 q^{46} - 12 q^{47} - 5 q^{49} + 33 q^{50} - 26 q^{52} - 3 q^{53} - 12 q^{55} - 7 q^{56} - 16 q^{58} - 8 q^{59} - 6 q^{61} + 24 q^{62} - 22 q^{64} - 22 q^{65} - 29 q^{67} + 30 q^{68} - 34 q^{70} + 7 q^{71} - 48 q^{73} - 25 q^{74} + 7 q^{76} + 18 q^{77} - 11 q^{79} - 3 q^{80} + 28 q^{82} + 57 q^{83} - 7 q^{85} - 9 q^{86} - 11 q^{88} + 2 q^{89} - 4 q^{91} + 13 q^{92} - q^{94} + 7 q^{95} - 14 q^{97} - 58 q^{98}+O(q^{100}) \) 12 * q + q^2 + 7 * q^4 + 7 * q^5 - 13 * q^7 + q^10 + 4 * q^11 - 17 * q^13 - 3 * q^14 - 19 * q^16 + 6 * q^17 + 12 * q^19 - 5 * q^20 - 8 * q^22 + 13 * q^23 - 7 * q^25 + 19 * q^26 - 29 * q^28 + 2 * q^29 - 14 * q^31 + 21 * q^32 - 6 * q^34 + 3 * q^35 - 2 * q^37 + q^38 + 8 * q^40 - 8 * q^41 - 42 * q^43 - 24 * q^44 - 9 * q^46 - 12 * q^47 - 5 * q^49 + 33 * q^50 - 26 * q^52 - 3 * q^53 - 12 * q^55 - 7 * q^56 - 16 * q^58 - 8 * q^59 - 6 * q^61 + 24 * q^62 - 22 * q^64 - 22 * q^65 - 29 * q^67 + 30 * q^68 - 34 * q^70 + 7 * q^71 - 48 * q^73 - 25 * q^74 + 7 * q^76 + 18 * q^77 - 11 * q^79 - 3 * q^80 + 28 * q^82 + 57 * q^83 - 7 * q^85 - 9 * q^86 - 11 * q^88 + 2 * q^89 - 4 * q^91 + 13 * q^92 - q^94 + 7 * q^95 - 14 * q^97 - 58 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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