LMFDB - Embedded newform 2070.2.a.x.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2070,2,Mod(1,2070)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2070.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2070, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2070.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,0,2,2,0,1,2,0,2,-3,0,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$16.5290332184$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.79129$ of defining polynomial
Character	$\chi$	$=$	2070.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.79129 q^{7} +1.00000 q^{8} +1.00000 q^{10} -3.79129 q^{11} +1.20871 q^{13} +2.79129 q^{14} +1.00000 q^{16} +3.79129 q^{17} +1.20871 q^{19} +1.00000 q^{20} -3.79129 q^{22} -1.00000 q^{23} +1.00000 q^{25} +1.20871 q^{26} +2.79129 q^{28} +1.58258 q^{29} +10.3739 q^{31} +1.00000 q^{32} +3.79129 q^{34} +2.79129 q^{35} -4.00000 q^{37} +1.20871 q^{38} +1.00000 q^{40} +2.20871 q^{41} -7.16515 q^{43} -3.79129 q^{44} -1.00000 q^{46} +13.5826 q^{47} +0.791288 q^{49} +1.00000 q^{50} +1.20871 q^{52} -6.00000 q^{53} -3.79129 q^{55} +2.79129 q^{56} +1.58258 q^{58} +4.41742 q^{59} -3.37386 q^{61} +10.3739 q^{62} +1.00000 q^{64} +1.20871 q^{65} -7.16515 q^{67} +3.79129 q^{68} +2.79129 q^{70} +5.37386 q^{71} -14.7477 q^{73} -4.00000 q^{74} +1.20871 q^{76} -10.5826 q^{77} +8.00000 q^{79} +1.00000 q^{80} +2.20871 q^{82} +6.00000 q^{83} +3.79129 q^{85} -7.16515 q^{86} -3.79129 q^{88} +3.16515 q^{89} +3.37386 q^{91} -1.00000 q^{92} +13.5826 q^{94} +1.20871 q^{95} +14.9564 q^{97} +0.791288 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + q^{7} + 2 q^{8} + 2 q^{10} - 3 q^{11} + 7 q^{13} + q^{14} + 2 q^{16} + 3 q^{17} + 7 q^{19} + 2 q^{20} - 3 q^{22} - 2 q^{23} + 2 q^{25} + 7 q^{26} + q^{28} - 6 q^{29}+ \cdots - 3 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + q^7 + 2 * q^8 + 2 * q^10 - 3 * q^11 + 7 * q^13 + q^14 + 2 * q^16 + 3 * q^17 + 7 * q^19 + 2 * q^20 - 3 * q^22 - 2 * q^23 + 2 * q^25 + 7 * q^26 + q^28 - 6 * q^29 + 7 * q^31 + 2 * q^32 + 3 * q^34 + q^35 - 8 * q^37 + 7 * q^38 + 2 * q^40 + 9 * q^41 + 4 * q^43 - 3 * q^44 - 2 * q^46 + 18 * q^47 - 3 * q^49 + 2 * q^50 + 7 * q^52 - 12 * q^53 - 3 * q^55 + q^56 - 6 * q^58 + 18 * q^59 + 7 * q^61 + 7 * q^62 + 2 * q^64 + 7 * q^65 + 4 * q^67 + 3 * q^68 + q^70 - 3 * q^71 - 2 * q^73 - 8 * q^74 + 7 * q^76 - 12 * q^77 + 16 * q^79 + 2 * q^80 + 9 * q^82 + 12 * q^83 + 3 * q^85 + 4 * q^86 - 3 * q^88 - 12 * q^89 - 7 * q^91 - 2 * q^92 + 18 * q^94 + 7 * q^95 + 7 * q^97 - 3 * q^98

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.79129	1.05501	0.527504	−	0.849553i	$-0.323128\pi$
$7$	2.79129	1.05501	0.527504	+	0.849553i	$0.323128\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−3.79129	−1.14312	−0.571558	−	0.820562i	$-0.693661\pi$
$11$	−3.79129	−1.14312	−0.571558	+	0.820562i	$0.693661\pi$
$12$	0	0
$13$	1.20871	0.335236	0.167618	−	0.985852i	$-0.446392\pi$
$13$	1.20871	0.335236	0.167618	+	0.985852i	$0.446392\pi$
$14$	2.79129	0.746003
$15$	0	0
$16$	1.00000	0.250000
$17$	3.79129	0.919522	0.459761	−	0.888043i	$-0.347935\pi$
$17$	3.79129	0.919522	0.459761	+	0.888043i	$0.347935\pi$
$18$	0	0
$19$	1.20871	0.277298	0.138649	−	0.990342i	$-0.455724\pi$
$19$	1.20871	0.277298	0.138649	+	0.990342i	$0.455724\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−3.79129	−0.808305
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	1.20871	0.237048
$27$	0	0
$28$	2.79129	0.527504
$29$	1.58258	0.293877	0.146938	−	0.989146i	$-0.453058\pi$
$29$	1.58258	0.293877	0.146938	+	0.989146i	$0.453058\pi$
$30$	0	0
$31$	10.3739	1.86320	0.931600	−	0.363484i	$-0.118413\pi$
$31$	10.3739	1.86320	0.931600	+	0.363484i	$0.118413\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.79129	0.650201
$35$	2.79129	0.471814
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	1.20871	0.196079
$39$	0	0
$40$	1.00000	0.158114
$41$	2.20871	0.344943	0.172471	−	0.985015i	$-0.444825\pi$
$41$	2.20871	0.344943	0.172471	+	0.985015i	$0.444825\pi$
$42$	0	0
$43$	−7.16515	−1.09268	−0.546338	−	0.837565i	$-0.683978\pi$
$43$	−7.16515	−1.09268	−0.546338	+	0.837565i	$0.683978\pi$
$44$	−3.79129	−0.571558
$45$	0	0
$46$	−1.00000	−0.147442
$47$	13.5826	1.98122	0.990611	−	0.136710i	$-0.0436528\pi$
$47$	13.5826	1.98122	0.990611	+	0.136710i	$0.0436528\pi$
$48$	0	0
$49$	0.791288	0.113041
$50$	1.00000	0.141421
$51$	0	0
$52$	1.20871	0.167618
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	−3.79129	−0.511217
$56$	2.79129	0.373002
$57$	0	0
$58$	1.58258	0.207802
$59$	4.41742	0.575100	0.287550	−	0.957766i	$-0.407159\pi$
$59$	4.41742	0.575100	0.287550	+	0.957766i	$0.407159\pi$
$60$	0	0
$61$	−3.37386	−0.431979	−0.215989	−	0.976396i	$-0.569298\pi$
$61$	−3.37386	−0.431979	−0.215989	+	0.976396i	$0.569298\pi$
$62$	10.3739	1.31748
$63$	0	0
$64$	1.00000	0.125000
$65$	1.20871	0.149922
$66$	0	0
$67$	−7.16515	−0.875363	−0.437681	−	0.899130i	$-0.644200\pi$
$67$	−7.16515	−0.875363	−0.437681	+	0.899130i	$0.644200\pi$
$68$	3.79129	0.459761
$69$	0	0
$70$	2.79129	0.333623
$71$	5.37386	0.637760	0.318880	−	0.947795i	$-0.396693\pi$
$71$	5.37386	0.637760	0.318880	+	0.947795i	$0.396693\pi$
$72$	0	0
$73$	−14.7477	−1.72609	−0.863045	−	0.505126i	$-0.831446\pi$
$73$	−14.7477	−1.72609	−0.863045	+	0.505126i	$0.831446\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	1.20871	0.138649
$77$	−10.5826	−1.20600
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	2.20871	0.243911
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	3.79129	0.411223
$86$	−7.16515	−0.772638
$87$	0	0
$88$	−3.79129	−0.404153
$89$	3.16515	0.335505	0.167753	−	0.985829i	$-0.446349\pi$
$89$	3.16515	0.335505	0.167753	+	0.985829i	$0.446349\pi$
$90$	0	0
$91$	3.37386	0.353677
$92$	−1.00000	−0.104257
$93$	0	0
$94$	13.5826	1.40094
$95$	1.20871	0.124011
$96$	0	0
$97$	14.9564	1.51860	0.759298	−	0.650743i	$-0.225542\pi$
$97$	14.9564	1.51860	0.759298	+	0.650743i	$0.225542\pi$
$98$	0.791288	0.0799321
$99$	0	0
$100$	1.00000	0.100000
$101$	−13.5826	−1.35152	−0.675758	−	0.737123i	$-0.736184\pi$
$101$	−13.5826	−1.35152	−0.675758	+	0.737123i	$0.736184\pi$
$102$	0	0
$103$	7.37386	0.726568	0.363284	−	0.931678i	$-0.381655\pi$
$103$	7.37386	0.726568	0.363284	+	0.931678i	$0.381655\pi$
$104$	1.20871	0.118524
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−13.5826	−1.31308	−0.656539	−	0.754292i	$-0.727980\pi$
$107$	−13.5826	−1.31308	−0.656539	+	0.754292i	$0.727980\pi$
$108$	0	0
$109$	10.3739	0.993636	0.496818	−	0.867855i	$-0.334502\pi$
$109$	10.3739	0.993636	0.496818	+	0.867855i	$0.334502\pi$
$110$	−3.79129	−0.361485
$111$	0	0
$112$	2.79129	0.263752
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	1.58258	0.146938
$117$	0	0
$118$	4.41742	0.406657
$119$	10.5826	0.970103
$120$	0	0
$121$	3.37386	0.306715
$122$	−3.37386	−0.305455
$123$	0	0
$124$	10.3739	0.931600
$125$	1.00000	0.0894427
$126$	0	0
$127$	−14.7477	−1.30865	−0.654325	−	0.756214i	$-0.727047\pi$
$127$	−14.7477	−1.30865	−0.654325	+	0.756214i	$0.727047\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	1.20871	0.106011
$131$	−9.16515	−0.800763	−0.400381	−	0.916349i	$-0.631122\pi$
$131$	−9.16515	−0.800763	−0.400381	+	0.916349i	$0.631122\pi$
$132$	0	0
$133$	3.37386	0.292551
$134$	−7.16515	−0.618975
$135$	0	0
$136$	3.79129	0.325100
$137$	0.791288	0.0676043	0.0338021	−	0.999429i	$-0.489238\pi$
$137$	0.791288	0.0676043	0.0338021	+	0.999429i	$0.489238\pi$
$138$	0	0
$139$	−14.7477	−1.25089	−0.625443	−	0.780270i	$-0.715082\pi$
$139$	−14.7477	−1.25089	−0.625443	+	0.780270i	$0.715082\pi$
$140$	2.79129	0.235907
$141$	0	0
$142$	5.37386	0.450965
$143$	−4.58258	−0.383214
$144$	0	0
$145$	1.58258	0.131426
$146$	−14.7477	−1.22053
$147$	0	0
$148$	−4.00000	−0.328798
$149$	−12.7913	−1.04790	−0.523952	−	0.851748i	$-0.675543\pi$
$149$	−12.7913	−1.04790	−0.523952	+	0.851748i	$0.675543\pi$
$150$	0	0
$151$	−6.20871	−0.505258	−0.252629	−	0.967563i	$-0.581295\pi$
$151$	−6.20871	−0.505258	−0.252629	+	0.967563i	$0.581295\pi$
$152$	1.20871	0.0980395
$153$	0	0
$154$	−10.5826	−0.852768
$155$	10.3739	0.833249
$156$	0	0
$157$	12.7477	1.01738	0.508690	−	0.860950i	$-0.330130\pi$
$157$	12.7477	1.01738	0.508690	+	0.860950i	$0.330130\pi$
$158$	8.00000	0.636446
$159$	0	0
$160$	1.00000	0.0790569
$161$	−2.79129	−0.219984
$162$	0	0
$163$	22.3739	1.75246	0.876228	−	0.481897i	$-0.160052\pi$
$163$	22.3739	1.75246	0.876228	+	0.481897i	$0.160052\pi$
$164$	2.20871	0.172471
$165$	0	0
$166$	6.00000	0.465690
$167$	18.3303	1.41844	0.709221	−	0.704987i	$-0.249047\pi$
$167$	18.3303	1.41844	0.709221	+	0.704987i	$0.249047\pi$
$168$	0	0
$169$	−11.5390	−0.887617
$170$	3.79129	0.290779
$171$	0	0
$172$	−7.16515	−0.546338
$173$	14.2087	1.08027	0.540134	−	0.841579i	$-0.318373\pi$
$173$	14.2087	1.08027	0.540134	+	0.841579i	$0.318373\pi$
$174$	0	0
$175$	2.79129	0.211002
$176$	−3.79129	−0.285779
$177$	0	0
$178$	3.16515	0.237238
$179$	16.7477	1.25178	0.625892	−	0.779910i	$-0.284735\pi$
$179$	16.7477	1.25178	0.625892	+	0.779910i	$0.284735\pi$
$180$	0	0
$181$	13.5390	1.00635	0.503174	−	0.864185i	$-0.332166\pi$
$181$	13.5390	1.00635	0.503174	+	0.864185i	$0.332166\pi$
$182$	3.37386	0.250087
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	−4.00000	−0.294086
$186$	0	0
$187$	−14.3739	−1.05112
$188$	13.5826	0.990611
$189$	0	0
$190$	1.20871	0.0876892
$191$	−16.4174	−1.18792	−0.593962	−	0.804493i	$-0.702437\pi$
$191$	−16.4174	−1.18792	−0.593962	+	0.804493i	$0.702437\pi$
$192$	0	0
$193$	6.74773	0.485712	0.242856	−	0.970062i	$-0.421916\pi$
$193$	6.74773	0.485712	0.242856	+	0.970062i	$0.421916\pi$
$194$	14.9564	1.07381
$195$	0	0
$196$	0.791288	0.0565206
$197$	−20.5390	−1.46334	−0.731672	−	0.681657i	$-0.761260\pi$
$197$	−20.5390	−1.46334	−0.731672	+	0.681657i	$0.761260\pi$
$198$	0	0
$199$	20.3303	1.44118	0.720588	−	0.693363i	$-0.243872\pi$
$199$	20.3303	1.44118	0.720588	+	0.693363i	$0.243872\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−13.5826	−0.955667
$203$	4.41742	0.310042
$204$	0	0
$205$	2.20871	0.154263
$206$	7.37386	0.513761
$207$	0	0
$208$	1.20871	0.0838091
$209$	−4.58258	−0.316983
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	−13.5826	−0.928486
$215$	−7.16515	−0.488659
$216$	0	0
$217$	28.9564	1.96569
$218$	10.3739	0.702607
$219$	0	0
$220$	−3.79129	−0.255609
$221$	4.58258	0.308257
$222$	0	0
$223$	11.1652	0.747674	0.373837	−	0.927494i	$-0.378042\pi$
$223$	11.1652	0.747674	0.373837	+	0.927494i	$0.378042\pi$
$224$	2.79129	0.186501
$225$	0	0
$226$	−6.00000	−0.399114
$227$	−4.74773	−0.315118	−0.157559	−	0.987510i	$-0.550362\pi$
$227$	−4.74773	−0.315118	−0.157559	+	0.987510i	$0.550362\pi$
$228$	0	0
$229$	−16.3303	−1.07914	−0.539568	−	0.841942i	$-0.681413\pi$
$229$	−16.3303	−1.07914	−0.539568	+	0.841942i	$0.681413\pi$
$230$	−1.00000	−0.0659380
$231$	0	0
$232$	1.58258	0.103901
$233$	−7.58258	−0.496751	−0.248376	−	0.968664i	$-0.579897\pi$
$233$	−7.58258	−0.496751	−0.248376	+	0.968664i	$0.579897\pi$
$234$	0	0
$235$	13.5826	0.886030
$236$	4.41742	0.287550
$237$	0	0
$238$	10.5826	0.685966
$239$	−3.16515	−0.204737	−0.102368	−	0.994747i	$-0.532642\pi$
$239$	−3.16515	−0.204737	−0.102368	+	0.994747i	$0.532642\pi$
$240$	0	0
$241$	−28.0000	−1.80364	−0.901819	−	0.432113i	$-0.857768\pi$
$241$	−28.0000	−1.80364	−0.901819	+	0.432113i	$0.857768\pi$
$242$	3.37386	0.216880
$243$	0	0
$244$	−3.37386	−0.215989
$245$	0.791288	0.0505535
$246$	0	0
$247$	1.46099	0.0929603
$248$	10.3739	0.658741
$249$	0	0
$250$	1.00000	0.0632456
$251$	−30.7913	−1.94353	−0.971764	−	0.235953i	$-0.924179\pi$
$251$	−30.7913	−1.94353	−0.971764	+	0.235953i	$0.924179\pi$
$252$	0	0
$253$	3.79129	0.238356
$254$	−14.7477	−0.925355
$255$	0	0
$256$	1.00000	0.0625000
$257$	−22.7477	−1.41896	−0.709482	−	0.704723i	$-0.751071\pi$
$257$	−22.7477	−1.41896	−0.709482	+	0.704723i	$0.751071\pi$
$258$	0	0
$259$	−11.1652	−0.693769
$260$	1.20871	0.0749611
$261$	0	0
$262$	−9.16515	−0.566225
$263$	−15.7913	−0.973733	−0.486866	−	0.873477i	$-0.661860\pi$
$263$	−15.7913	−0.973733	−0.486866	+	0.873477i	$0.661860\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	3.37386	0.206865
$267$	0	0
$268$	−7.16515	−0.437681
$269$	−16.7477	−1.02113	−0.510563	−	0.859840i	$-0.670563\pi$
$269$	−16.7477	−1.02113	−0.510563	+	0.859840i	$0.670563\pi$
$270$	0	0
$271$	−23.1216	−1.40454	−0.702268	−	0.711912i	$-0.747829\pi$
$271$	−23.1216	−1.40454	−0.702268	+	0.711912i	$0.747829\pi$
$272$	3.79129	0.229881
$273$	0	0
$274$	0.791288	0.0478034
$275$	−3.79129	−0.228623
$276$	0	0
$277$	−1.16515	−0.0700072	−0.0350036	−	0.999387i	$-0.511144\pi$
$277$	−1.16515	−0.0700072	−0.0350036	+	0.999387i	$0.511144\pi$
$278$	−14.7477	−0.884510
$279$	0	0
$280$	2.79129	0.166811
$281$	16.7477	0.999086	0.499543	−	0.866289i	$-0.333501\pi$
$281$	16.7477	0.999086	0.499543	+	0.866289i	$0.333501\pi$
$282$	0	0
$283$	−28.3303	−1.68406	−0.842031	−	0.539429i	$-0.818640\pi$
$283$	−28.3303	−1.68406	−0.842031	+	0.539429i	$0.818640\pi$
$284$	5.37386	0.318880
$285$	0	0
$286$	−4.58258	−0.270973
$287$	6.16515	0.363917
$288$	0	0
$289$	−2.62614	−0.154479
$290$	1.58258	0.0929320
$291$	0	0
$292$	−14.7477	−0.863045
$293$	27.4955	1.60630	0.803151	−	0.595776i	$-0.203155\pi$
$293$	27.4955	1.60630	0.803151	+	0.595776i	$0.203155\pi$
$294$	0	0
$295$	4.41742	0.257192
$296$	−4.00000	−0.232495
$297$	0	0
$298$	−12.7913	−0.740979
$299$	−1.20871	−0.0699016
$300$	0	0
$301$	−20.0000	−1.15278
$302$	−6.20871	−0.357271
$303$	0	0
$304$	1.20871	0.0693244
$305$	−3.37386	−0.193187
$306$	0	0
$307$	16.5390	0.943931	0.471966	−	0.881617i	$-0.343545\pi$
$307$	16.5390	0.943931	0.471966	+	0.881617i	$0.343545\pi$
$308$	−10.5826	−0.602998
$309$	0	0
$310$	10.3739	0.589196
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	−18.3739	−1.03855	−0.519276	−	0.854607i	$-0.673798\pi$
$313$	−18.3739	−1.03855	−0.519276	+	0.854607i	$0.673798\pi$
$314$	12.7477	0.719396
$315$	0	0
$316$	8.00000	0.450035
$317$	−5.20871	−0.292550	−0.146275	−	0.989244i	$-0.546729\pi$
$317$	−5.20871	−0.292550	−0.146275	+	0.989244i	$0.546729\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	1.00000	0.0559017
$321$	0	0
$322$	−2.79129	−0.155552
$323$	4.58258	0.254981
$324$	0	0
$325$	1.20871	0.0670473
$326$	22.3739	1.23917
$327$	0	0
$328$	2.20871	0.121956
$329$	37.9129	2.09020
$330$	0	0
$331$	6.74773	0.370889	0.185444	−	0.982655i	$-0.440628\pi$
$331$	6.74773	0.370889	0.185444	+	0.982655i	$0.440628\pi$
$332$	6.00000	0.329293
$333$	0	0
$334$	18.3303	1.00299
$335$	−7.16515	−0.391474
$336$	0	0
$337$	−16.7913	−0.914680	−0.457340	−	0.889292i	$-0.651198\pi$
$337$	−16.7913	−0.914680	−0.457340	+	0.889292i	$0.651198\pi$
$338$	−11.5390	−0.627640
$339$	0	0
$340$	3.79129	0.205611
$341$	−39.3303	−2.12986
$342$	0	0
$343$	−17.3303	−0.935748
$344$	−7.16515	−0.386319
$345$	0	0
$346$	14.2087	0.763865
$347$	−9.79129	−0.525624	−0.262812	−	0.964847i	$-0.584650\pi$
$347$	−9.79129	−0.525624	−0.262812	+	0.964847i	$0.584650\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	2.79129	0.149201
$351$	0	0
$352$	−3.79129	−0.202076
$353$	15.1652	0.807160	0.403580	−	0.914944i	$-0.367766\pi$
$353$	15.1652	0.807160	0.403580	+	0.914944i	$0.367766\pi$
$354$	0	0
$355$	5.37386	0.285215
$356$	3.16515	0.167753
$357$	0	0
$358$	16.7477	0.885145
$359$	9.16515	0.483718	0.241859	−	0.970311i	$-0.422243\pi$
$359$	9.16515	0.483718	0.241859	+	0.970311i	$0.422243\pi$
$360$	0	0
$361$	−17.5390	−0.923106
$362$	13.5390	0.711595
$363$	0	0
$364$	3.37386	0.176838
$365$	−14.7477	−0.771931
$366$	0	0
$367$	−0.834849	−0.0435787	−0.0217894	−	0.999763i	$-0.506936\pi$
$367$	−0.834849	−0.0435787	−0.0217894	+	0.999763i	$0.506936\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	−4.00000	−0.207950
$371$	−16.7477	−0.869499
$372$	0	0
$373$	−14.7477	−0.763608	−0.381804	−	0.924243i	$-0.624697\pi$
$373$	−14.7477	−0.763608	−0.381804	+	0.924243i	$0.624697\pi$
$374$	−14.3739	−0.743255
$375$	0	0
$376$	13.5826	0.700468
$377$	1.91288	0.0985183
$378$	0	0
$379$	7.37386	0.378770	0.189385	−	0.981903i	$-0.439351\pi$
$379$	7.37386	0.378770	0.189385	+	0.981903i	$0.439351\pi$
$380$	1.20871	0.0620056
$381$	0	0
$382$	−16.4174	−0.839989
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	−10.5826	−0.539338
$386$	6.74773	0.343450
$387$	0	0
$388$	14.9564	0.759298
$389$	−29.7042	−1.50606	−0.753031	−	0.657986i	$-0.771409\pi$
$389$	−29.7042	−1.50606	−0.753031	+	0.657986i	$0.771409\pi$
$390$	0	0
$391$	−3.79129	−0.191734
$392$	0.791288	0.0399661
$393$	0	0
$394$	−20.5390	−1.03474
$395$	8.00000	0.402524
$396$	0	0
$397$	16.5390	0.830069	0.415035	−	0.909806i	$-0.363769\pi$
$397$	16.5390	0.830069	0.415035	+	0.909806i	$0.363769\pi$
$398$	20.3303	1.01907
$399$	0	0
$400$	1.00000	0.0500000
$401$	−22.7477	−1.13597	−0.567984	−	0.823040i	$-0.692276\pi$
$401$	−22.7477	−1.13597	−0.567984	+	0.823040i	$0.692276\pi$
$402$	0	0
$403$	12.5390	0.624613
$404$	−13.5826	−0.675758
$405$	0	0
$406$	4.41742	0.219233
$407$	15.1652	0.751709
$408$	0	0
$409$	−22.7913	−1.12696	−0.563478	−	0.826131i	$-0.690537\pi$
$409$	−22.7913	−1.12696	−0.563478	+	0.826131i	$0.690537\pi$
$410$	2.20871	0.109081
$411$	0	0
$412$	7.37386	0.363284
$413$	12.3303	0.606735
$414$	0	0
$415$	6.00000	0.294528
$416$	1.20871	0.0592620
$417$	0	0
$418$	−4.58258	−0.224141
$419$	−39.1652	−1.91334	−0.956671	−	0.291170i	$-0.905956\pi$
$419$	−39.1652	−1.91334	−0.956671	+	0.291170i	$0.905956\pi$
$420$	0	0
$421$	−23.1216	−1.12688	−0.563439	−	0.826158i	$-0.690522\pi$
$421$	−23.1216	−1.12688	−0.563439	+	0.826158i	$0.690522\pi$
$422$	−10.0000	−0.486792
$423$	0	0
$424$	−6.00000	−0.291386
$425$	3.79129	0.183904
$426$	0	0
$427$	−9.41742	−0.455741
$428$	−13.5826	−0.656539
$429$	0	0
$430$	−7.16515	−0.345534
$431$	19.9129	0.959170	0.479585	−	0.877496i	$-0.340787\pi$
$431$	19.9129	0.959170	0.479585	+	0.877496i	$0.340787\pi$
$432$	0	0
$433$	1.53901	0.0739603	0.0369802	−	0.999316i	$-0.488226\pi$
$433$	1.53901	0.0739603	0.0369802	+	0.999316i	$0.488226\pi$
$434$	28.9564	1.38995
$435$	0	0
$436$	10.3739	0.496818
$437$	−1.20871	−0.0578205
$438$	0	0
$439$	25.5390	1.21891	0.609455	−	0.792820i	$-0.291388\pi$
$439$	25.5390	1.21891	0.609455	+	0.792820i	$0.291388\pi$
$440$	−3.79129	−0.180743
$441$	0	0
$442$	4.58258	0.217971
$443$	35.2087	1.67282	0.836408	−	0.548107i	$-0.184651\pi$
$443$	35.2087	1.67282	0.836408	+	0.548107i	$0.184651\pi$
$444$	0	0
$445$	3.16515	0.150043
$446$	11.1652	0.528685
$447$	0	0
$448$	2.79129	0.131876
$449$	−25.1216	−1.18556	−0.592781	−	0.805364i	$-0.701970\pi$
$449$	−25.1216	−1.18556	−0.592781	+	0.805364i	$0.701970\pi$
$450$	0	0
$451$	−8.37386	−0.394310
$452$	−6.00000	−0.282216
$453$	0	0
$454$	−4.74773	−0.222822
$455$	3.37386	0.158169
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	−16.3303	−0.763065
$459$	0	0
$460$	−1.00000	−0.0466252
$461$	1.25227	0.0583242	0.0291621	−	0.999575i	$-0.490716\pi$
$461$	1.25227	0.0583242	0.0291621	+	0.999575i	$0.490716\pi$
$462$	0	0
$463$	−10.0000	−0.464739	−0.232370	−	0.972628i	$-0.574648\pi$
$463$	−10.0000	−0.464739	−0.232370	+	0.972628i	$0.574648\pi$
$464$	1.58258	0.0734692
$465$	0	0
$466$	−7.58258	−0.351256
$467$	−25.9129	−1.19911	−0.599553	−	0.800335i	$-0.704655\pi$
$467$	−25.9129	−1.19911	−0.599553	+	0.800335i	$0.704655\pi$
$468$	0	0
$469$	−20.0000	−0.923514
$470$	13.5826	0.626517
$471$	0	0
$472$	4.41742	0.203328
$473$	27.1652	1.24905
$474$	0	0
$475$	1.20871	0.0554595
$476$	10.5826	0.485052
$477$	0	0
$478$	−3.16515	−0.144771
$479$	39.4955	1.80459	0.902297	−	0.431116i	$-0.141880\pi$
$479$	39.4955	1.80459	0.902297	+	0.431116i	$0.141880\pi$
$480$	0	0
$481$	−4.83485	−0.220450
$482$	−28.0000	−1.27537
$483$	0	0
$484$	3.37386	0.153357
$485$	14.9564	0.679137
$486$	0	0
$487$	15.5826	0.706114	0.353057	−	0.935602i	$-0.385142\pi$
$487$	15.5826	0.706114	0.353057	+	0.935602i	$0.385142\pi$
$488$	−3.37386	−0.152728
$489$	0	0
$490$	0.791288	0.0357467
$491$	16.7477	0.755814	0.377907	−	0.925843i	$-0.376644\pi$
$491$	16.7477	0.755814	0.377907	+	0.925843i	$0.376644\pi$
$492$	0	0
$493$	6.00000	0.270226
$494$	1.46099	0.0657328
$495$	0	0
$496$	10.3739	0.465800
$497$	15.0000	0.672842
$498$	0	0
$499$	23.1652	1.03701	0.518507	−	0.855073i	$-0.326488\pi$
$499$	23.1652	1.03701	0.518507	+	0.855073i	$0.326488\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−30.7913	−1.37428
$503$	−18.7913	−0.837862	−0.418931	−	0.908018i	$-0.637595\pi$
$503$	−18.7913	−0.837862	−0.418931	+	0.908018i	$0.637595\pi$
$504$	0	0
$505$	−13.5826	−0.604417
$506$	3.79129	0.168543
$507$	0	0
$508$	−14.7477	−0.654325
$509$	−7.25227	−0.321451	−0.160726	−	0.986999i	$-0.551383\pi$
$509$	−7.25227	−0.321451	−0.160726	+	0.986999i	$0.551383\pi$
$510$	0	0
$511$	−41.1652	−1.82104
$512$	1.00000	0.0441942
$513$	0	0
$514$	−22.7477	−1.00336
$515$	7.37386	0.324931
$516$	0	0
$517$	−51.4955	−2.26477
$518$	−11.1652	−0.490569
$519$	0	0
$520$	1.20871	0.0530055
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	−1.16515	−0.0509485	−0.0254743	−	0.999675i	$-0.508110\pi$
$523$	−1.16515	−0.0509485	−0.0254743	+	0.999675i	$0.508110\pi$
$524$	−9.16515	−0.400381
$525$	0	0
$526$	−15.7913	−0.688533
$527$	39.3303	1.71325
$528$	0	0
$529$	1.00000	0.0434783
$530$	−6.00000	−0.260623
$531$	0	0
$532$	3.37386	0.146276
$533$	2.66970	0.115637
$534$	0	0
$535$	−13.5826	−0.587226
$536$	−7.16515	−0.309487
$537$	0	0
$538$	−16.7477	−0.722046
$539$	−3.00000	−0.129219
$540$	0	0
$541$	38.3303	1.64795	0.823974	−	0.566627i	$-0.191752\pi$
$541$	38.3303	1.64795	0.823974	+	0.566627i	$0.191752\pi$
$542$	−23.1216	−0.993157
$543$	0	0
$544$	3.79129	0.162550
$545$	10.3739	0.444367
$546$	0	0
$547$	15.1216	0.646553	0.323276	−	0.946305i	$-0.395216\pi$
$547$	15.1216	0.646553	0.323276	+	0.946305i	$0.395216\pi$
$548$	0.791288	0.0338021
$549$	0	0
$550$	−3.79129	−0.161661
$551$	1.91288	0.0814914
$552$	0	0
$553$	22.3303	0.949581
$554$	−1.16515	−0.0495025
$555$	0	0
$556$	−14.7477	−0.625443
$557$	−30.3303	−1.28514	−0.642568	−	0.766229i	$-0.722131\pi$
$557$	−30.3303	−1.28514	−0.642568	+	0.766229i	$0.722131\pi$
$558$	0	0
$559$	−8.66061	−0.366305
$560$	2.79129	0.117953
$561$	0	0
$562$	16.7477	0.706460
$563$	−3.16515	−0.133395	−0.0666976	−	0.997773i	$-0.521246\pi$
$563$	−3.16515	−0.133395	−0.0666976	+	0.997773i	$0.521246\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	−28.3303	−1.19081
$567$	0	0
$568$	5.37386	0.225482
$569$	−15.4955	−0.649603	−0.324802	−	0.945782i	$-0.605298\pi$
$569$	−15.4955	−0.649603	−0.324802	+	0.945782i	$0.605298\pi$
$570$	0	0
$571$	30.1216	1.26055	0.630275	−	0.776372i	$-0.282942\pi$
$571$	30.1216	1.26055	0.630275	+	0.776372i	$0.282942\pi$
$572$	−4.58258	−0.191607
$573$	0	0
$574$	6.16515	0.257328
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	22.8348	0.950627	0.475314	−	0.879816i	$-0.342335\pi$
$577$	22.8348	0.950627	0.475314	+	0.879816i	$0.342335\pi$
$578$	−2.62614	−0.109233
$579$	0	0
$580$	1.58258	0.0657129
$581$	16.7477	0.694813
$582$	0	0
$583$	22.7477	0.942115
$584$	−14.7477	−0.610265
$585$	0	0
$586$	27.4955	1.13583
$587$	26.2087	1.08175	0.540875	−	0.841103i	$-0.318093\pi$
$587$	26.2087	1.08175	0.540875	+	0.841103i	$0.318093\pi$
$588$	0	0
$589$	12.5390	0.516661
$590$	4.41742	0.181862
$591$	0	0
$592$	−4.00000	−0.164399
$593$	−13.9129	−0.571333	−0.285667	−	0.958329i	$-0.592215\pi$
$593$	−13.9129	−0.571333	−0.285667	+	0.958329i	$0.592215\pi$
$594$	0	0
$595$	10.5826	0.433843
$596$	−12.7913	−0.523952
$597$	0	0
$598$	−1.20871	−0.0494279
$599$	40.1216	1.63932	0.819662	−	0.572848i	$-0.194161\pi$
$599$	40.1216	1.63932	0.819662	+	0.572848i	$0.194161\pi$
$600$	0	0
$601$	−22.7913	−0.929676	−0.464838	−	0.885396i	$-0.653887\pi$
$601$	−22.7913	−0.929676	−0.464838	+	0.885396i	$0.653887\pi$
$602$	−20.0000	−0.815139
$603$	0	0
$604$	−6.20871	−0.252629
$605$	3.37386	0.137167
$606$	0	0
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	1.20871	0.0490198
$609$	0	0
$610$	−3.37386	−0.136604
$611$	16.4174	0.664178
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	16.5390	0.667460
$615$	0	0
$616$	−10.5826	−0.426384
$617$	−44.8693	−1.80637	−0.903185	−	0.429251i	$-0.858778\pi$
$617$	−44.8693	−1.80637	−0.903185	+	0.429251i	$0.858778\pi$
$618$	0	0
$619$	2.79129	0.112191	0.0560957	−	0.998425i	$-0.482135\pi$
$619$	2.79129	0.112191	0.0560957	+	0.998425i	$0.482135\pi$
$620$	10.3739	0.416624
$621$	0	0
$622$	−12.0000	−0.481156
$623$	8.83485	0.353961
$624$	0	0
$625$	1.00000	0.0400000
$626$	−18.3739	−0.734367
$627$	0	0
$628$	12.7477	0.508690
$629$	−15.1652	−0.604674
$630$	0	0
$631$	−17.9129	−0.713100	−0.356550	−	0.934276i	$-0.616047\pi$
$631$	−17.9129	−0.713100	−0.356550	+	0.934276i	$0.616047\pi$
$632$	8.00000	0.318223
$633$	0	0
$634$	−5.20871	−0.206864
$635$	−14.7477	−0.585246
$636$	0	0
$637$	0.956439	0.0378955
$638$	−6.00000	−0.237542
$639$	0	0
$640$	1.00000	0.0395285
$641$	3.16515	0.125016	0.0625080	−	0.998044i	$-0.480090\pi$
$641$	3.16515	0.125016	0.0625080	+	0.998044i	$0.480090\pi$
$642$	0	0
$643$	−20.7477	−0.818210	−0.409105	−	0.912487i	$-0.634159\pi$
$643$	−20.7477	−0.818210	−0.409105	+	0.912487i	$0.634159\pi$
$644$	−2.79129	−0.109992
$645$	0	0
$646$	4.58258	0.180299
$647$	−2.83485	−0.111449	−0.0557247	−	0.998446i	$-0.517747\pi$
$647$	−2.83485	−0.111449	−0.0557247	+	0.998446i	$0.517747\pi$
$648$	0	0
$649$	−16.7477	−0.657406
$650$	1.20871	0.0474096
$651$	0	0
$652$	22.3739	0.876228
$653$	−35.5390	−1.39075	−0.695375	−	0.718647i	$-0.744762\pi$
$653$	−35.5390	−1.39075	−0.695375	+	0.718647i	$0.744762\pi$
$654$	0	0
$655$	−9.16515	−0.358112
$656$	2.20871	0.0862357
$657$	0	0
$658$	37.9129	1.47800
$659$	27.1652	1.05820	0.529102	−	0.848558i	$-0.322529\pi$
$659$	27.1652	1.05820	0.529102	+	0.848558i	$0.322529\pi$
$660$	0	0
$661$	−39.3739	−1.53147	−0.765733	−	0.643159i	$-0.777624\pi$
$661$	−39.3739	−1.53147	−0.765733	+	0.643159i	$0.777624\pi$
$662$	6.74773	0.262258
$663$	0	0
$664$	6.00000	0.232845
$665$	3.37386	0.130833
$666$	0	0
$667$	−1.58258	−0.0612776
$668$	18.3303	0.709221
$669$	0	0
$670$	−7.16515	−0.276814
$671$	12.7913	0.493802
$672$	0	0
$673$	38.0000	1.46479	0.732396	−	0.680879i	$-0.238402\pi$
$673$	38.0000	1.46479	0.732396	+	0.680879i	$0.238402\pi$
$674$	−16.7913	−0.646776
$675$	0	0
$676$	−11.5390	−0.443808
$677$	30.6606	1.17838	0.589191	−	0.807993i	$-0.299446\pi$
$677$	30.6606	1.17838	0.589191	+	0.807993i	$0.299446\pi$
$678$	0	0
$679$	41.7477	1.60213
$680$	3.79129	0.145389
$681$	0	0
$682$	−39.3303	−1.50604
$683$	−2.37386	−0.0908334	−0.0454167	−	0.998968i	$-0.514462\pi$
$683$	−2.37386	−0.0908334	−0.0454167	+	0.998968i	$0.514462\pi$
$684$	0	0
$685$	0.791288	0.0302336
$686$	−17.3303	−0.661674
$687$	0	0
$688$	−7.16515	−0.273169
$689$	−7.25227	−0.276290
$690$	0	0
$691$	15.2523	0.580224	0.290112	−	0.956993i	$-0.406307\pi$
$691$	15.2523	0.580224	0.290112	+	0.956993i	$0.406307\pi$
$692$	14.2087	0.540134
$693$	0	0
$694$	−9.79129	−0.371672
$695$	−14.7477	−0.559413
$696$	0	0
$697$	8.37386	0.317183
$698$	26.0000	0.984115
$699$	0	0
$700$	2.79129	0.105501
$701$	−9.62614	−0.363574	−0.181787	−	0.983338i	$-0.558188\pi$
$701$	−9.62614	−0.363574	−0.181787	+	0.983338i	$0.558188\pi$
$702$	0	0
$703$	−4.83485	−0.182350
$704$	−3.79129	−0.142890
$705$	0	0
$706$	15.1652	0.570748
$707$	−37.9129	−1.42586
$708$	0	0
$709$	34.5390	1.29714	0.648570	−	0.761155i	$-0.275367\pi$
$709$	34.5390	1.29714	0.648570	+	0.761155i	$0.275367\pi$
$710$	5.37386	0.201678
$711$	0	0
$712$	3.16515	0.118619
$713$	−10.3739	−0.388504
$714$	0	0
$715$	−4.58258	−0.171379
$716$	16.7477	0.625892
$717$	0	0
$718$	9.16515	0.342040
$719$	29.5390	1.10162	0.550810	−	0.834631i	$-0.314319\pi$
$719$	29.5390	1.10162	0.550810	+	0.834631i	$0.314319\pi$
$720$	0	0
$721$	20.5826	0.766535
$722$	−17.5390	−0.652735
$723$	0	0
$724$	13.5390	0.503174
$725$	1.58258	0.0587754
$726$	0	0
$727$	−2.12159	−0.0786854	−0.0393427	−	0.999226i	$-0.512526\pi$
$727$	−2.12159	−0.0786854	−0.0393427	+	0.999226i	$0.512526\pi$
$728$	3.37386	0.125044
$729$	0	0
$730$	−14.7477	−0.545838
$731$	−27.1652	−1.00474
$732$	0	0
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	−0.834849	−0.0308148
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	27.1652	1.00064
$738$	0	0
$739$	8.00000	0.294285	0.147142	−	0.989115i	$-0.452992\pi$
$739$	8.00000	0.294285	0.147142	+	0.989115i	$0.452992\pi$
$740$	−4.00000	−0.147043
$741$	0	0
$742$	−16.7477	−0.614828
$743$	−9.95644	−0.365266	−0.182633	−	0.983181i	$-0.558462\pi$
$743$	−9.95644	−0.365266	−0.182633	+	0.983181i	$0.558462\pi$
$744$	0	0
$745$	−12.7913	−0.468637
$746$	−14.7477	−0.539953
$747$	0	0
$748$	−14.3739	−0.525561
$749$	−37.9129	−1.38531
$750$	0	0
$751$	18.7477	0.684114	0.342057	−	0.939679i	$-0.388876\pi$
$751$	18.7477	0.684114	0.342057	+	0.939679i	$0.388876\pi$
$752$	13.5826	0.495306
$753$	0	0
$754$	1.91288	0.0696629
$755$	−6.20871	−0.225958
$756$	0	0
$757$	26.3303	0.956991	0.478496	−	0.878090i	$-0.341182\pi$
$757$	26.3303	0.956991	0.478496	+	0.878090i	$0.341182\pi$
$758$	7.37386	0.267831
$759$	0	0
$760$	1.20871	0.0438446
$761$	33.9564	1.23092	0.615460	−	0.788168i	$-0.288970\pi$
$761$	33.9564	1.23092	0.615460	+	0.788168i	$0.288970\pi$
$762$	0	0
$763$	28.9564	1.04829
$764$	−16.4174	−0.593962
$765$	0	0
$766$	24.0000	0.867155
$767$	5.33939	0.192794
$768$	0	0
$769$	−3.66970	−0.132333	−0.0661663	−	0.997809i	$-0.521077\pi$
$769$	−3.66970	−0.132333	−0.0661663	+	0.997809i	$0.521077\pi$
$770$	−10.5826	−0.381370
$771$	0	0
$772$	6.74773	0.242856
$773$	21.4955	0.773138	0.386569	−	0.922261i	$-0.373660\pi$
$773$	21.4955	0.773138	0.386569	+	0.922261i	$0.373660\pi$
$774$	0	0
$775$	10.3739	0.372640
$776$	14.9564	0.536905
$777$	0	0
$778$	−29.7042	−1.06495
$779$	2.66970	0.0956518
$780$	0	0
$781$	−20.3739	−0.729034
$782$	−3.79129	−0.135576
$783$	0	0
$784$	0.791288	0.0282603
$785$	12.7477	0.454986
$786$	0	0
$787$	−8.41742	−0.300049	−0.150024	−	0.988682i	$-0.547935\pi$
$787$	−8.41742	−0.300049	−0.150024	+	0.988682i	$0.547935\pi$
$788$	−20.5390	−0.731672
$789$	0	0
$790$	8.00000	0.284627
$791$	−16.7477	−0.595481
$792$	0	0
$793$	−4.07803	−0.144815
$794$	16.5390	0.586948
$795$	0	0
$796$	20.3303	0.720588
$797$	49.9129	1.76800	0.884002	−	0.467482i	$-0.154839\pi$
$797$	49.9129	1.76800	0.884002	+	0.467482i	$0.154839\pi$
$798$	0	0
$799$	51.4955	1.82178
$800$	1.00000	0.0353553
$801$	0	0
$802$	−22.7477	−0.803250
$803$	55.9129	1.97312
$804$	0	0
$805$	−2.79129	−0.0983800
$806$	12.5390	0.441668
$807$	0	0
$808$	−13.5826	−0.477833
$809$	−11.0436	−0.388271	−0.194135	−	0.980975i	$-0.562190\pi$
$809$	−11.0436	−0.388271	−0.194135	+	0.980975i	$0.562190\pi$
$810$	0	0
$811$	−47.9129	−1.68245	−0.841224	−	0.540686i	$-0.818165\pi$
$811$	−47.9129	−1.68245	−0.841224	+	0.540686i	$0.818165\pi$
$812$	4.41742	0.155021
$813$	0	0
$814$	15.1652	0.531538
$815$	22.3739	0.783722
$816$	0	0
$817$	−8.66061	−0.302996
$818$	−22.7913	−0.796879
$819$	0	0
$820$	2.20871	0.0771316
$821$	−2.83485	−0.0989369	−0.0494684	−	0.998776i	$-0.515753\pi$
$821$	−2.83485	−0.0989369	−0.0494684	+	0.998776i	$0.515753\pi$
$822$	0	0
$823$	41.1652	1.43493	0.717463	−	0.696596i	$-0.245303\pi$
$823$	41.1652	1.43493	0.717463	+	0.696596i	$0.245303\pi$
$824$	7.37386	0.256881
$825$	0	0
$826$	12.3303	0.429026
$827$	−41.0780	−1.42842	−0.714212	−	0.699930i	$-0.753215\pi$
$827$	−41.0780	−1.42842	−0.714212	+	0.699930i	$0.753215\pi$
$828$	0	0
$829$	−31.4955	−1.09388	−0.546941	−	0.837171i	$-0.684208\pi$
$829$	−31.4955	−1.09388	−0.546941	+	0.837171i	$0.684208\pi$
$830$	6.00000	0.208263
$831$	0	0
$832$	1.20871	0.0419046
$833$	3.00000	0.103944
$834$	0	0
$835$	18.3303	0.634346
$836$	−4.58258	−0.158492
$837$	0	0
$838$	−39.1652	−1.35294
$839$	22.4174	0.773935	0.386968	−	0.922093i	$-0.373522\pi$
$839$	22.4174	0.773935	0.386968	+	0.922093i	$0.373522\pi$
$840$	0	0
$841$	−26.4955	−0.913636
$842$	−23.1216	−0.796823
$843$	0	0
$844$	−10.0000	−0.344214
$845$	−11.5390	−0.396954
$846$	0	0
$847$	9.41742	0.323587
$848$	−6.00000	−0.206041
$849$	0	0
$850$	3.79129	0.130040
$851$	4.00000	0.137118
$852$	0	0
$853$	8.46099	0.289699	0.144849	−	0.989454i	$-0.453730\pi$
$853$	8.46099	0.289699	0.144849	+	0.989454i	$0.453730\pi$
$854$	−9.41742	−0.322258
$855$	0	0
$856$	−13.5826	−0.464243
$857$	−9.16515	−0.313076	−0.156538	−	0.987672i	$-0.550033\pi$
$857$	−9.16515	−0.313076	−0.156538	+	0.987672i	$0.550033\pi$
$858$	0	0
$859$	0.747727	0.0255121	0.0127561	−	0.999919i	$-0.495940\pi$
$859$	0.747727	0.0255121	0.0127561	+	0.999919i	$0.495940\pi$
$860$	−7.16515	−0.244330
$861$	0	0
$862$	19.9129	0.678235
$863$	31.5826	1.07508	0.537542	−	0.843237i	$-0.319353\pi$
$863$	31.5826	1.07508	0.537542	+	0.843237i	$0.319353\pi$
$864$	0	0
$865$	14.2087	0.483111
$866$	1.53901	0.0522979
$867$	0	0
$868$	28.9564	0.982846
$869$	−30.3303	−1.02889
$870$	0	0
$871$	−8.66061	−0.293453
$872$	10.3739	0.351303
$873$	0	0
$874$	−1.20871	−0.0408853
$875$	2.79129	0.0943628
$876$	0	0
$877$	7.70417	0.260151	0.130076	−	0.991504i	$-0.458478\pi$
$877$	7.70417	0.260151	0.130076	+	0.991504i	$0.458478\pi$
$878$	25.5390	0.861900
$879$	0	0
$880$	−3.79129	−0.127804
$881$	6.33030	0.213273	0.106637	−	0.994298i	$-0.465992\pi$
$881$	6.33030	0.213273	0.106637	+	0.994298i	$0.465992\pi$
$882$	0	0
$883$	−12.0436	−0.405298	−0.202649	−	0.979251i	$-0.564955\pi$
$883$	−12.0436	−0.405298	−0.202649	+	0.979251i	$0.564955\pi$
$884$	4.58258	0.154129
$885$	0	0
$886$	35.2087	1.18286
$887$	3.16515	0.106275	0.0531377	−	0.998587i	$-0.483078\pi$
$887$	3.16515	0.106275	0.0531377	+	0.998587i	$0.483078\pi$
$888$	0	0
$889$	−41.1652	−1.38063
$890$	3.16515	0.106096
$891$	0	0
$892$	11.1652	0.373837
$893$	16.4174	0.549388
$894$	0	0
$895$	16.7477	0.559815
$896$	2.79129	0.0932504
$897$	0	0
$898$	−25.1216	−0.838318
$899$	16.4174	0.547552
$900$	0	0
$901$	−22.7477	−0.757837
$902$	−8.37386	−0.278819
$903$	0	0
$904$	−6.00000	−0.199557
$905$	13.5390	0.450052
$906$	0	0
$907$	−20.7477	−0.688917	−0.344458	−	0.938802i	$-0.611937\pi$
$907$	−20.7477	−0.688917	−0.344458	+	0.938802i	$0.611937\pi$
$908$	−4.74773	−0.157559
$909$	0	0
$910$	3.37386	0.111842
$911$	−13.5826	−0.450011	−0.225005	−	0.974358i	$-0.572240\pi$
$911$	−13.5826	−0.450011	−0.225005	+	0.974358i	$0.572240\pi$
$912$	0	0
$913$	−22.7477	−0.752840
$914$	−10.0000	−0.330771
$915$	0	0
$916$	−16.3303	−0.539568
$917$	−25.5826	−0.844811
$918$	0	0
$919$	−36.8348	−1.21507	−0.607535	−	0.794293i	$-0.707841\pi$
$919$	−36.8348	−1.21507	−0.607535	+	0.794293i	$0.707841\pi$
$920$	−1.00000	−0.0329690
$921$	0	0
$922$	1.25227	0.0412414
$923$	6.49545	0.213800
$924$	0	0
$925$	−4.00000	−0.131519
$926$	−10.0000	−0.328620
$927$	0	0
$928$	1.58258	0.0519506
$929$	39.4955	1.29580	0.647902	−	0.761724i	$-0.275647\pi$
$929$	39.4955	1.29580	0.647902	+	0.761724i	$0.275647\pi$
$930$	0	0
$931$	0.956439	0.0313460
$932$	−7.58258	−0.248376
$933$	0	0
$934$	−25.9129	−0.847895
$935$	−14.3739	−0.470076
$936$	0	0
$937$	58.3739	1.90699	0.953495	−	0.301407i	$-0.0974564\pi$
$937$	58.3739	1.90699	0.953495	+	0.301407i	$0.0974564\pi$
$938$	−20.0000	−0.653023
$939$	0	0
$940$	13.5826	0.443015
$941$	−54.9564	−1.79153	−0.895764	−	0.444529i	$-0.853371\pi$
$941$	−54.9564	−1.79153	−0.895764	+	0.444529i	$0.853371\pi$
$942$	0	0
$943$	−2.20871	−0.0719256
$944$	4.41742	0.143775
$945$	0	0
$946$	27.1652	0.883215
$947$	29.5390	0.959889	0.479945	−	0.877299i	$-0.340657\pi$
$947$	29.5390	0.959889	0.479945	+	0.877299i	$0.340657\pi$
$948$	0	0
$949$	−17.8258	−0.578649
$950$	1.20871	0.0392158
$951$	0	0
$952$	10.5826	0.342983
$953$	26.5390	0.859683	0.429842	−	0.902904i	$-0.358569\pi$
$953$	26.5390	0.859683	0.429842	+	0.902904i	$0.358569\pi$
$954$	0	0
$955$	−16.4174	−0.531255
$956$	−3.16515	−0.102368
$957$	0	0
$958$	39.4955	1.27604
$959$	2.20871	0.0713230
$960$	0	0
$961$	76.6170	2.47152
$962$	−4.83485	−0.155882
$963$	0	0
$964$	−28.0000	−0.901819
$965$	6.74773	0.217217
$966$	0	0
$967$	−5.25227	−0.168902	−0.0844509	−	0.996428i	$-0.526914\pi$
$967$	−5.25227	−0.168902	−0.0844509	+	0.996428i	$0.526914\pi$
$968$	3.37386	0.108440
$969$	0	0
$970$	14.9564	0.480222
$971$	6.95644	0.223243	0.111621	−	0.993751i	$-0.464396\pi$
$971$	6.95644	0.223243	0.111621	+	0.993751i	$0.464396\pi$
$972$	0	0
$973$	−41.1652	−1.31969
$974$	15.5826	0.499298
$975$	0	0
$976$	−3.37386	−0.107995
$977$	−7.12159	−0.227840	−0.113920	−	0.993490i	$-0.536341\pi$
$977$	−7.12159	−0.227840	−0.113920	+	0.993490i	$0.536341\pi$
$978$	0	0
$979$	−12.0000	−0.383522
$980$	0.791288	0.0252768
$981$	0	0
$982$	16.7477	0.534441
$983$	0.626136	0.0199707	0.00998533	−	0.999950i	$-0.496822\pi$
$983$	0.626136	0.0199707	0.00998533	+	0.999950i	$0.496822\pi$
$984$	0	0
$985$	−20.5390	−0.654427
$986$	6.00000	0.191079
$987$	0	0
$988$	1.46099	0.0464801
$989$	7.16515	0.227839
$990$	0	0
$991$	−37.7913	−1.20048	−0.600240	−	0.799820i	$-0.704928\pi$
$991$	−37.7913	−1.20048	−0.600240	+	0.799820i	$0.704928\pi$
$992$	10.3739	0.329370
$993$	0	0
$994$	15.0000	0.475771
$995$	20.3303	0.644514
$996$	0	0
$997$	11.4955	0.364065	0.182032	−	0.983293i	$-0.441732\pi$
$997$	11.4955	0.364065	0.182032	+	0.983293i	$0.441732\pi$
$998$	23.1652	0.733280
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.2.a.x.1.2		2
3.2	odd	2		230.2.a.a.1.2	✓	2
12.11	even	2		1840.2.a.n.1.1		2
15.2	even	4		1150.2.b.g.599.1		4
15.8	even	4		1150.2.b.g.599.4		4
15.14	odd	2		1150.2.a.o.1.1		2
24.5	odd	2		7360.2.a.bq.1.1		2
24.11	even	2		7360.2.a.bk.1.2		2
60.59	even	2		9200.2.a.bs.1.2		2
69.68	even	2		5290.2.a.e.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.a.a.1.2	✓	2	3.2	odd	2
1150.2.a.o.1.1		2	15.14	odd	2
1150.2.b.g.599.1		4	15.2	even	4
1150.2.b.g.599.4		4	15.8	even	4
1840.2.a.n.1.1		2	12.11	even	2
2070.2.a.x.1.2		2	1.1	even	1	trivial
5290.2.a.e.1.2		2	69.68	even	2
7360.2.a.bk.1.2		2	24.11	even	2
7360.2.a.bq.1.1		2	24.5	odd	2
9200.2.a.bs.1.2		2	60.59	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2070.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.79129 q^{7} +1.00000 q^{8} +1.00000 q^{10} -3.79129 q^{11} +1.20871 q^{13} +2.79129 q^{14} +1.00000 q^{16} +3.79129 q^{17} +1.20871 q^{19} +1.00000 q^{20} -3.79129 q^{22} -1.00000 q^{23} +1.00000 q^{25} +1.20871 q^{26} +2.79129 q^{28} +1.58258 q^{29} +10.3739 q^{31} +1.00000 q^{32} +3.79129 q^{34} +2.79129 q^{35} -4.00000 q^{37} +1.20871 q^{38} +1.00000 q^{40} +2.20871 q^{41} -7.16515 q^{43} -3.79129 q^{44} -1.00000 q^{46} +13.5826 q^{47} +0.791288 q^{49} +1.00000 q^{50} +1.20871 q^{52} -6.00000 q^{53} -3.79129 q^{55} +2.79129 q^{56} +1.58258 q^{58} +4.41742 q^{59} -3.37386 q^{61} +10.3739 q^{62} +1.00000 q^{64} +1.20871 q^{65} -7.16515 q^{67} +3.79129 q^{68} +2.79129 q^{70} +5.37386 q^{71} -14.7477 q^{73} -4.00000 q^{74} +1.20871 q^{76} -10.5826 q^{77} +8.00000 q^{79} +1.00000 q^{80} +2.20871 q^{82} +6.00000 q^{83} +3.79129 q^{85} -7.16515 q^{86} -3.79129 q^{88} +3.16515 q^{89} +3.37386 q^{91} -1.00000 q^{92} +13.5826 q^{94} +1.20871 q^{95} +14.9564 q^{97} +0.791288 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + q^{7} + 2 q^{8} + 2 q^{10} - 3 q^{11} + 7 q^{13} + q^{14} + 2 q^{16} + 3 q^{17} + 7 q^{19} + 2 q^{20} - 3 q^{22} - 2 q^{23} + 2 q^{25} + 7 q^{26} + q^{28} - 6 q^{29}+ \cdots - 3 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + q^7 + 2 * q^8 + 2 * q^10 - 3 * q^11 + 7 * q^13 + q^14 + 2 * q^16 + 3 * q^17 + 7 * q^19 + 2 * q^20 - 3 * q^22 - 2 * q^23 + 2 * q^25 + 7 * q^26 + q^28 - 6 * q^29 + 7 * q^31 + 2 * q^32 + 3 * q^34 + q^35 - 8 * q^37 + 7 * q^38 + 2 * q^40 + 9 * q^41 + 4 * q^43 - 3 * q^44 - 2 * q^46 + 18 * q^47 - 3 * q^49 + 2 * q^50 + 7 * q^52 - 12 * q^53 - 3 * q^55 + q^56 - 6 * q^58 + 18 * q^59 + 7 * q^61 + 7 * q^62 + 2 * q^64 + 7 * q^65 + 4 * q^67 + 3 * q^68 + q^70 - 3 * q^71 - 2 * q^73 - 8 * q^74 + 7 * q^76 - 12 * q^77 + 16 * q^79 + 2 * q^80 + 9 * q^82 + 12 * q^83 + 3 * q^85 + 4 * q^86 - 3 * q^88 - 12 * q^89 - 7 * q^91 - 2 * q^92 + 18 * q^94 + 7 * q^95 + 7 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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