LMFDB - Embedded newform 8450.2.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8450,2,Mod(1,8450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8450, 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("8450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8450 = 2 \cdot 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8450.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:	$67.4735897080$
Analytic rank:	$1$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 650)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.79129$ of defining polynomial
Character	$\chi$	$=$	8450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +2.79129 q^{3} +1.00000 q^{4} -2.79129 q^{6} -1.79129 q^{7} -1.00000 q^{8} +4.79129 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +2.79129 q^{3} +1.00000 q^{4} -2.79129 q^{6} -1.79129 q^{7} -1.00000 q^{8} +4.79129 q^{9} +3.79129 q^{11} +2.79129 q^{12} +1.79129 q^{14} +1.00000 q^{16} +0.791288 q^{17} -4.79129 q^{18} -5.00000 q^{19} -5.00000 q^{21} -3.79129 q^{22} -4.58258 q^{23} -2.79129 q^{24} +5.00000 q^{27} -1.79129 q^{28} -3.79129 q^{29} -9.58258 q^{31} -1.00000 q^{32} +10.5826 q^{33} -0.791288 q^{34} +4.79129 q^{36} -6.37386 q^{37} +5.00000 q^{38} -12.1652 q^{41} +5.00000 q^{42} +10.9564 q^{43} +3.79129 q^{44} +4.58258 q^{46} +0.791288 q^{47} +2.79129 q^{48} -3.79129 q^{49} +2.20871 q^{51} -4.58258 q^{53} -5.00000 q^{54} +1.79129 q^{56} -13.9564 q^{57} +3.79129 q^{58} +4.58258 q^{59} -8.58258 q^{61} +9.58258 q^{62} -8.58258 q^{63} +1.00000 q^{64} -10.5826 q^{66} +11.0000 q^{67} +0.791288 q^{68} -12.7913 q^{69} -7.58258 q^{71} -4.79129 q^{72} -7.16515 q^{73} +6.37386 q^{74} -5.00000 q^{76} -6.79129 q^{77} +14.9564 q^{79} -0.417424 q^{81} +12.1652 q^{82} -6.16515 q^{83} -5.00000 q^{84} -10.9564 q^{86} -10.5826 q^{87} -3.79129 q^{88} -0.626136 q^{89} -4.58258 q^{92} -26.7477 q^{93} -0.791288 q^{94} -2.79129 q^{96} -9.20871 q^{97} +3.79129 q^{98} +18.1652 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} + q^{7} - 2 q^{8} + 5 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + q^3 + 2 * q^4 - q^6 + q^7 - 2 * q^8 + 5 * q^9	$ 2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} + q^{7} - 2 q^{8} + 5 q^{9} + 3 q^{11} + q^{12} - q^{14} + 2 q^{16} - 3 q^{17} - 5 q^{18} - 10 q^{19} - 10 q^{21} - 3 q^{22} - q^{24} + 10 q^{27} + q^{28} - 3 q^{29} - 10 q^{31} - 2 q^{32} + 12 q^{33} + 3 q^{34} + 5 q^{36} + q^{37} + 10 q^{38} - 6 q^{41} + 10 q^{42} - q^{43} + 3 q^{44} - 3 q^{47} + q^{48} - 3 q^{49} + 9 q^{51} - 10 q^{54} - q^{56} - 5 q^{57} + 3 q^{58} - 8 q^{61} + 10 q^{62} - 8 q^{63} + 2 q^{64} - 12 q^{66} + 22 q^{67} - 3 q^{68} - 21 q^{69} - 6 q^{71} - 5 q^{72} + 4 q^{73} - q^{74} - 10 q^{76} - 9 q^{77} + 7 q^{79} - 10 q^{81} + 6 q^{82} + 6 q^{83} - 10 q^{84} + q^{86} - 12 q^{87} - 3 q^{88} - 15 q^{89} - 26 q^{93} + 3 q^{94} - q^{96} - 23 q^{97} + 3 q^{98} + 18 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + q^3 + 2 * q^4 - q^6 + q^7 - 2 * q^8 + 5 * q^9 + 3 * q^11 + q^12 - q^14 + 2 * q^16 - 3 * q^17 - 5 * q^18 - 10 * q^19 - 10 * q^21 - 3 * q^22 - q^24 + 10 * q^27 + q^28 - 3 * q^29 - 10 * q^31 - 2 * q^32 + 12 * q^33 + 3 * q^34 + 5 * q^36 + q^37 + 10 * q^38 - 6 * q^41 + 10 * q^42 - q^43 + 3 * q^44 - 3 * q^47 + q^48 - 3 * q^49 + 9 * q^51 - 10 * q^54 - q^56 - 5 * q^57 + 3 * q^58 - 8 * q^61 + 10 * q^62 - 8 * q^63 + 2 * q^64 - 12 * q^66 + 22 * q^67 - 3 * q^68 - 21 * q^69 - 6 * q^71 - 5 * q^72 + 4 * q^73 - q^74 - 10 * q^76 - 9 * q^77 + 7 * q^79 - 10 * q^81 + 6 * q^82 + 6 * q^83 - 10 * q^84 + q^86 - 12 * q^87 - 3 * q^88 - 15 * q^89 - 26 * q^93 + 3 * q^94 - q^96 - 23 * q^97 + 3 * q^98 + 18 * q^99

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
$3$	2.79129	1.61155	0.805775	−	0.592221i	$-0.201749\pi$
$3$	2.79129	1.61155	0.805775	+	0.592221i	$0.201749\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−2.79129	−1.13954
$7$	−1.79129	−0.677043	−0.338522	−	0.940959i	$-0.609927\pi$
$7$	−1.79129	−0.677043	−0.338522	+	0.940959i	$0.609927\pi$
$8$	−1.00000	−0.353553
$9$	4.79129	1.59710
$10$	0	0
$11$	3.79129	1.14312	0.571558	−	0.820562i	$-0.306339\pi$
$11$	3.79129	1.14312	0.571558	+	0.820562i	$0.306339\pi$
$12$	2.79129	0.805775
$13$	0	0
$13$	0	0
$14$	1.79129	0.478742
$15$	0	0
$16$	1.00000	0.250000
$17$	0.791288	0.191915	0.0959577	−	0.995385i	$-0.469409\pi$
$17$	0.791288	0.191915	0.0959577	+	0.995385i	$0.469409\pi$
$18$	−4.79129	−1.12932
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	−5.00000	−1.09109
$22$	−3.79129	−0.808305
$23$	−4.58258	−0.955533	−0.477767	−	0.878487i	$-0.658554\pi$
$23$	−4.58258	−0.955533	−0.477767	+	0.878487i	$0.658554\pi$
$24$	−2.79129	−0.569769
$25$	0	0
$26$	0	0
$27$	5.00000	0.962250
$28$	−1.79129	−0.338522
$29$	−3.79129	−0.704024	−0.352012	−	0.935995i	$-0.614502\pi$
$29$	−3.79129	−0.704024	−0.352012	+	0.935995i	$0.614502\pi$
$30$	0	0
$31$	−9.58258	−1.72108	−0.860541	−	0.509382i	$-0.829874\pi$
$31$	−9.58258	−1.72108	−0.860541	+	0.509382i	$0.829874\pi$
$32$	−1.00000	−0.176777
$33$	10.5826	1.84219
$34$	−0.791288	−0.135705
$35$	0	0
$36$	4.79129	0.798548
$37$	−6.37386	−1.04786	−0.523928	−	0.851762i	$-0.675534\pi$
$37$	−6.37386	−1.04786	−0.523928	+	0.851762i	$0.675534\pi$
$38$	5.00000	0.811107
$39$	0	0
$40$	0	0
$41$	−12.1652	−1.89988	−0.949939	−	0.312436i	$-0.898855\pi$
$41$	−12.1652	−1.89988	−0.949939	+	0.312436i	$0.898855\pi$
$42$	5.00000	0.771517
$43$	10.9564	1.67084	0.835421	−	0.549611i	$-0.185224\pi$
$43$	10.9564	1.67084	0.835421	+	0.549611i	$0.185224\pi$
$44$	3.79129	0.571558
$45$	0	0
$46$	4.58258	0.675664
$47$	0.791288	0.115421	0.0577106	−	0.998333i	$-0.481620\pi$
$47$	0.791288	0.115421	0.0577106	+	0.998333i	$0.481620\pi$
$48$	2.79129	0.402888
$49$	−3.79129	−0.541613
$50$	0	0
$51$	2.20871	0.309282
$52$	0	0
$53$	−4.58258	−0.629465	−0.314733	−	0.949180i	$-0.601915\pi$
$53$	−4.58258	−0.629465	−0.314733	+	0.949180i	$0.601915\pi$
$54$	−5.00000	−0.680414
$55$	0	0
$56$	1.79129	0.239371
$57$	−13.9564	−1.84858
$58$	3.79129	0.497820
$59$	4.58258	0.596601	0.298300	−	0.954472i	$-0.403580\pi$
$59$	4.58258	0.596601	0.298300	+	0.954472i	$0.403580\pi$
$60$	0	0
$61$	−8.58258	−1.09889	−0.549443	−	0.835531i	$-0.685160\pi$
$61$	−8.58258	−1.09889	−0.549443	+	0.835531i	$0.685160\pi$
$62$	9.58258	1.21699
$63$	−8.58258	−1.08130
$64$	1.00000	0.125000
$65$	0	0
$66$	−10.5826	−1.30263
$67$	11.0000	1.34386	0.671932	−	0.740613i	$-0.265465\pi$
$67$	11.0000	1.34386	0.671932	+	0.740613i	$0.265465\pi$
$68$	0.791288	0.0959577
$69$	−12.7913	−1.53989
$70$	0	0
$71$	−7.58258	−0.899886	−0.449943	−	0.893057i	$-0.648556\pi$
$71$	−7.58258	−0.899886	−0.449943	+	0.893057i	$0.648556\pi$
$72$	−4.79129	−0.564659
$73$	−7.16515	−0.838618	−0.419309	−	0.907844i	$-0.637728\pi$
$73$	−7.16515	−0.838618	−0.419309	+	0.907844i	$0.637728\pi$
$74$	6.37386	0.740947
$75$	0	0
$76$	−5.00000	−0.573539
$77$	−6.79129	−0.773939
$78$	0	0
$79$	14.9564	1.68273	0.841365	−	0.540467i	$-0.181752\pi$
$79$	14.9564	1.68273	0.841365	+	0.540467i	$0.181752\pi$
$80$	0	0
$81$	−0.417424	−0.0463805
$82$	12.1652	1.34342
$83$	−6.16515	−0.676713	−0.338357	−	0.941018i	$-0.609871\pi$
$83$	−6.16515	−0.676713	−0.338357	+	0.941018i	$0.609871\pi$
$84$	−5.00000	−0.545545
$85$	0	0
$86$	−10.9564	−1.18146
$87$	−10.5826	−1.13457
$88$	−3.79129	−0.404153
$89$	−0.626136	−0.0663703	−0.0331852	−	0.999449i	$-0.510565\pi$
$89$	−0.626136	−0.0663703	−0.0331852	+	0.999449i	$0.510565\pi$
$90$	0	0
$91$	0	0
$92$	−4.58258	−0.477767
$93$	−26.7477	−2.77361
$94$	−0.791288	−0.0816151
$95$	0	0
$96$	−2.79129	−0.284885
$97$	−9.20871	−0.935003	−0.467502	−	0.883992i	$-0.654846\pi$
$97$	−9.20871	−0.935003	−0.467502	+	0.883992i	$0.654846\pi$
$98$	3.79129	0.382978
$99$	18.1652	1.82567
$100$	0	0
$101$	−19.7477	−1.96497	−0.982486	−	0.186336i	$-0.940339\pi$
$101$	−19.7477	−1.96497	−0.982486	+	0.186336i	$0.940339\pi$
$102$	−2.20871	−0.218695
$103$	2.58258	0.254469	0.127234	−	0.991873i	$-0.459390\pi$
$103$	2.58258	0.254469	0.127234	+	0.991873i	$0.459390\pi$
$104$	0	0
$105$	0	0
$106$	4.58258	0.445099
$107$	10.4174	1.00709	0.503545	−	0.863969i	$-0.332029\pi$
$107$	10.4174	1.00709	0.503545	+	0.863969i	$0.332029\pi$
$108$	5.00000	0.481125
$109$	−1.83485	−0.175747	−0.0878733	−	0.996132i	$-0.528007\pi$
$109$	−1.83485	−0.175747	−0.0878733	+	0.996132i	$0.528007\pi$
$110$	0	0
$111$	−17.7913	−1.68867
$112$	−1.79129	−0.169261
$113$	0.626136	0.0589020	0.0294510	−	0.999566i	$-0.490624\pi$
$113$	0.626136	0.0589020	0.0294510	+	0.999566i	$0.490624\pi$
$114$	13.9564	1.30714
$115$	0	0
$116$	−3.79129	−0.352012
$117$	0	0
$118$	−4.58258	−0.421860
$119$	−1.41742	−0.129935
$120$	0	0
$121$	3.37386	0.306715
$122$	8.58258	0.777030
$123$	−33.9564	−3.06175
$124$	−9.58258	−0.860541
$125$	0	0
$126$	8.58258	0.764597
$127$	−16.3739	−1.45295	−0.726473	−	0.687195i	$-0.758842\pi$
$127$	−16.3739	−1.45295	−0.726473	+	0.687195i	$0.758842\pi$
$128$	−1.00000	−0.0883883
$129$	30.5826	2.69265
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	10.5826	0.921095
$133$	8.95644	0.776622
$134$	−11.0000	−0.950255
$135$	0	0
$136$	−0.791288	−0.0678524
$137$	15.1652	1.29565	0.647823	−	0.761791i	$-0.275680\pi$
$137$	15.1652	1.29565	0.647823	+	0.761791i	$0.275680\pi$
$138$	12.7913	1.08887
$139$	21.5826	1.83061	0.915305	−	0.402761i	$-0.131949\pi$
$139$	21.5826	1.83061	0.915305	+	0.402761i	$0.131949\pi$
$140$	0	0
$141$	2.20871	0.186007
$142$	7.58258	0.636316
$143$	0	0
$144$	4.79129	0.399274
$145$	0	0
$146$	7.16515	0.592992
$147$	−10.5826	−0.872836
$148$	−6.37386	−0.523928
$149$	3.95644	0.324124	0.162062	−	0.986781i	$-0.448186\pi$
$149$	3.95644	0.324124	0.162062	+	0.986781i	$0.448186\pi$
$150$	0	0
$151$	6.37386	0.518698	0.259349	−	0.965784i	$-0.416492\pi$
$151$	6.37386	0.518698	0.259349	+	0.965784i	$0.416492\pi$
$152$	5.00000	0.405554
$153$	3.79129	0.306507
$154$	6.79129	0.547258
$155$	0	0
$156$	0	0
$157$	−5.16515	−0.412224	−0.206112	−	0.978528i	$-0.566081\pi$
$157$	−5.16515	−0.412224	−0.206112	+	0.978528i	$0.566081\pi$
$158$	−14.9564	−1.18987
$159$	−12.7913	−1.01442
$160$	0	0
$161$	8.20871	0.646937
$162$	0.417424	0.0327960
$163$	−4.62614	−0.362347	−0.181173	−	0.983451i	$-0.557990\pi$
$163$	−4.62614	−0.362347	−0.181173	+	0.983451i	$0.557990\pi$
$164$	−12.1652	−0.949939
$165$	0	0
$166$	6.16515	0.478509
$167$	−1.41742	−0.109684	−0.0548418	−	0.998495i	$-0.517465\pi$
$167$	−1.41742	−0.109684	−0.0548418	+	0.998495i	$0.517465\pi$
$168$	5.00000	0.385758
$169$	0	0
$170$	0	0
$171$	−23.9564	−1.83199
$172$	10.9564	0.835421
$173$	19.1216	1.45379	0.726894	−	0.686750i	$-0.240963\pi$
$173$	19.1216	1.45379	0.726894	+	0.686750i	$0.240963\pi$
$174$	10.5826	0.802263
$175$	0	0
$176$	3.79129	0.285779
$177$	12.7913	0.961452
$178$	0.626136	0.0469309
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	7.37386	0.548095	0.274047	−	0.961716i	$-0.411637\pi$
$181$	7.37386	0.548095	0.274047	+	0.961716i	$0.411637\pi$
$182$	0	0
$183$	−23.9564	−1.77091
$184$	4.58258	0.337832
$185$	0	0
$186$	26.7477	1.96124
$187$	3.00000	0.219382
$188$	0.791288	0.0577106
$189$	−8.95644	−0.651485
$190$	0	0
$191$	−1.74773	−0.126461	−0.0632305	−	0.997999i	$-0.520140\pi$
$191$	−1.74773	−0.126461	−0.0632305	+	0.997999i	$0.520140\pi$
$192$	2.79129	0.201444
$193$	17.9564	1.29253	0.646266	−	0.763112i	$-0.276329\pi$
$193$	17.9564	1.29253	0.646266	+	0.763112i	$0.276329\pi$
$194$	9.20871	0.661147
$195$	0	0
$196$	−3.79129	−0.270806
$197$	−9.95644	−0.709367	−0.354683	−	0.934986i	$-0.615411\pi$
$197$	−9.95644	−0.709367	−0.354683	+	0.934986i	$0.615411\pi$
$198$	−18.1652	−1.29094
$199$	−14.1216	−1.00105	−0.500527	−	0.865721i	$-0.666860\pi$
$199$	−14.1216	−1.00105	−0.500527	+	0.865721i	$0.666860\pi$
$200$	0	0
$201$	30.7042	2.16570
$202$	19.7477	1.38945
$203$	6.79129	0.476655
$204$	2.20871	0.154641
$205$	0	0
$206$	−2.58258	−0.179937
$207$	−21.9564	−1.52608
$208$	0	0
$209$	−18.9564	−1.31124
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	−4.58258	−0.314733
$213$	−21.1652	−1.45021
$214$	−10.4174	−0.712120
$215$	0	0
$216$	−5.00000	−0.340207
$217$	17.1652	1.16525
$218$	1.83485	0.124272
$219$	−20.0000	−1.35147
$220$	0	0
$221$	0	0
$222$	17.7913	1.19407
$223$	9.74773	0.652756	0.326378	−	0.945239i	$-0.394172\pi$
$223$	9.74773	0.652756	0.326378	+	0.945239i	$0.394172\pi$
$224$	1.79129	0.119685
$225$	0	0
$226$	−0.626136	−0.0416500
$227$	16.5826	1.10062	0.550312	−	0.834959i	$-0.314509\pi$
$227$	16.5826	1.10062	0.550312	+	0.834959i	$0.314509\pi$
$228$	−13.9564	−0.924288
$229$	12.3739	0.817688	0.408844	−	0.912604i	$-0.365932\pi$
$229$	12.3739	0.817688	0.408844	+	0.912604i	$0.365932\pi$
$230$	0	0
$231$	−18.9564	−1.24724
$232$	3.79129	0.248910
$233$	15.0000	0.982683	0.491341	−	0.870967i	$-0.336507\pi$
$233$	15.0000	0.982683	0.491341	+	0.870967i	$0.336507\pi$
$234$	0	0
$235$	0	0
$236$	4.58258	0.298300
$237$	41.7477	2.71181
$238$	1.41742	0.0918780
$239$	18.1652	1.17501	0.587503	−	0.809222i	$-0.300111\pi$
$239$	18.1652	1.17501	0.587503	+	0.809222i	$0.300111\pi$
$240$	0	0
$241$	7.00000	0.450910	0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000	0.450910	0.225455	+	0.974254i	$0.427613\pi$
$242$	−3.37386	−0.216880
$243$	−16.1652	−1.03699
$244$	−8.58258	−0.549443
$245$	0	0
$246$	33.9564	2.16498
$247$	0	0
$248$	9.58258	0.608494
$249$	−17.2087	−1.09056
$250$	0	0
$251$	−20.3739	−1.28599	−0.642993	−	0.765872i	$-0.722308\pi$
$251$	−20.3739	−1.28599	−0.642993	+	0.765872i	$0.722308\pi$
$252$	−8.58258	−0.540651
$253$	−17.3739	−1.09229
$254$	16.3739	1.02739
$255$	0	0
$256$	1.00000	0.0625000
$257$	−22.9129	−1.42927	−0.714633	−	0.699500i	$-0.753406\pi$
$257$	−22.9129	−1.42927	−0.714633	+	0.699500i	$0.753406\pi$
$258$	−30.5826	−1.90399
$259$	11.4174	0.709444
$260$	0	0
$261$	−18.1652	−1.12439
$262$	15.0000	0.926703
$263$	9.95644	0.613940	0.306970	−	0.951719i	$-0.400685\pi$
$263$	9.95644	0.613940	0.306970	+	0.951719i	$0.400685\pi$
$264$	−10.5826	−0.651313
$265$	0	0
$266$	−8.95644	−0.549155
$267$	−1.74773	−0.106959
$268$	11.0000	0.671932
$269$	−15.3303	−0.934705	−0.467353	−	0.884071i	$-0.654792\pi$
$269$	−15.3303	−0.934705	−0.467353	+	0.884071i	$0.654792\pi$
$270$	0	0
$271$	−11.1652	−0.678235	−0.339117	−	0.940744i	$-0.610128\pi$
$271$	−11.1652	−0.678235	−0.339117	+	0.940744i	$0.610128\pi$
$272$	0.791288	0.0479789
$273$	0	0
$274$	−15.1652	−0.916160
$275$	0	0
$276$	−12.7913	−0.769945
$277$	7.62614	0.458210	0.229105	−	0.973402i	$-0.426420\pi$
$277$	7.62614	0.458210	0.229105	+	0.973402i	$0.426420\pi$
$278$	−21.5826	−1.29444
$279$	−45.9129	−2.74873
$280$	0	0
$281$	14.5390	0.867325	0.433662	−	0.901075i	$-0.357221\pi$
$281$	14.5390	0.867325	0.433662	+	0.901075i	$0.357221\pi$
$282$	−2.20871	−0.131527
$283$	12.5390	0.745367	0.372684	−	0.927959i	$-0.378438\pi$
$283$	12.5390	0.745367	0.372684	+	0.927959i	$0.378438\pi$
$284$	−7.58258	−0.449943
$285$	0	0
$286$	0	0
$287$	21.7913	1.28630
$288$	−4.79129	−0.282329
$289$	−16.3739	−0.963168
$290$	0	0
$291$	−25.7042	−1.50680
$292$	−7.16515	−0.419309
$293$	6.62614	0.387103	0.193552	−	0.981090i	$-0.437999\pi$
$293$	6.62614	0.387103	0.193552	+	0.981090i	$0.437999\pi$
$294$	10.5826	0.617188
$295$	0	0
$296$	6.37386	0.370473
$297$	18.9564	1.09996
$298$	−3.95644	−0.229190
$299$	0	0
$300$	0	0
$301$	−19.6261	−1.13123
$302$	−6.37386	−0.366775
$303$	−55.1216	−3.16665
$304$	−5.00000	−0.286770
$305$	0	0
$306$	−3.79129	−0.216734
$307$	−13.1652	−0.751375	−0.375687	−	0.926746i	$-0.622593\pi$
$307$	−13.1652	−0.751375	−0.375687	+	0.926746i	$0.622593\pi$
$308$	−6.79129	−0.386970
$309$	7.20871	0.410089
$310$	0	0
$311$	−17.3739	−0.985181	−0.492591	−	0.870261i	$-0.663950\pi$
$311$	−17.3739	−0.985181	−0.492591	+	0.870261i	$0.663950\pi$
$312$	0	0
$313$	4.62614	0.261485	0.130742	−	0.991416i	$-0.458264\pi$
$313$	4.62614	0.261485	0.130742	+	0.991416i	$0.458264\pi$
$314$	5.16515	0.291486
$315$	0	0
$316$	14.9564	0.841365
$317$	26.5390	1.49058	0.745290	−	0.666741i	$-0.232311\pi$
$317$	26.5390	1.49058	0.745290	+	0.666741i	$0.232311\pi$
$318$	12.7913	0.717300
$319$	−14.3739	−0.804782
$320$	0	0
$321$	29.0780	1.62298
$322$	−8.20871	−0.457454
$323$	−3.95644	−0.220142
$324$	−0.417424	−0.0231902
$325$	0	0
$326$	4.62614	0.256218
$327$	−5.12159	−0.283225
$328$	12.1652	0.671708
$329$	−1.41742	−0.0781451
$330$	0	0
$331$	−11.9564	−0.657185	−0.328593	−	0.944472i	$-0.606574\pi$
$331$	−11.9564	−0.657185	−0.328593	+	0.944472i	$0.606574\pi$
$332$	−6.16515	−0.338357
$333$	−30.5390	−1.67353
$334$	1.41742	0.0775580
$335$	0	0
$336$	−5.00000	−0.272772
$337$	−0.252273	−0.0137422	−0.00687109	−	0.999976i	$-0.502187\pi$
$337$	−0.252273	−0.0137422	−0.00687109	+	0.999976i	$0.502187\pi$
$338$	0	0
$339$	1.74773	0.0949235
$340$	0	0
$341$	−36.3303	−1.96740
$342$	23.9564	1.29542
$343$	19.3303	1.04374
$344$	−10.9564	−0.590732
$345$	0	0
$346$	−19.1216	−1.02798
$347$	−28.5826	−1.53439	−0.767197	−	0.641412i	$-0.778349\pi$
$347$	−28.5826	−1.53439	−0.767197	+	0.641412i	$0.778349\pi$
$348$	−10.5826	−0.567286
$349$	−26.9564	−1.44295	−0.721473	−	0.692443i	$-0.756534\pi$
$349$	−26.9564	−1.44295	−0.721473	+	0.692443i	$0.756534\pi$
$350$	0	0
$351$	0	0
$352$	−3.79129	−0.202076
$353$	−18.6261	−0.991369	−0.495685	−	0.868503i	$-0.665083\pi$
$353$	−18.6261	−0.991369	−0.495685	+	0.868503i	$0.665083\pi$
$354$	−12.7913	−0.679849
$355$	0	0
$356$	−0.626136	−0.0331852
$357$	−3.95644	−0.209397
$358$	−6.00000	−0.317110
$359$	33.7913	1.78344	0.891718	−	0.452591i	$-0.149500\pi$
$359$	33.7913	1.78344	0.891718	+	0.452591i	$0.149500\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	−7.37386	−0.387562
$363$	9.41742	0.494287
$364$	0	0
$365$	0	0
$366$	23.9564	1.25222
$367$	−24.7477	−1.29182	−0.645910	−	0.763413i	$-0.723522\pi$
$367$	−24.7477	−1.29182	−0.645910	+	0.763413i	$0.723522\pi$
$368$	−4.58258	−0.238883
$369$	−58.2867	−3.03429
$370$	0	0
$371$	8.20871	0.426175
$372$	−26.7477	−1.38681
$373$	13.1652	0.681666	0.340833	−	0.940124i	$-0.389291\pi$
$373$	13.1652	0.681666	0.340833	+	0.940124i	$0.389291\pi$
$374$	−3.00000	−0.155126
$375$	0	0
$376$	−0.791288	−0.0408076
$377$	0	0
$378$	8.95644	0.460670
$379$	−17.0000	−0.873231	−0.436616	−	0.899648i	$-0.643823\pi$
$379$	−17.0000	−0.873231	−0.436616	+	0.899648i	$0.643823\pi$
$380$	0	0
$381$	−45.7042	−2.34150
$382$	1.74773	0.0894215
$383$	1.74773	0.0893047	0.0446523	−	0.999003i	$-0.485782\pi$
$383$	1.74773	0.0893047	0.0446523	+	0.999003i	$0.485782\pi$
$384$	−2.79129	−0.142442
$385$	0	0
$386$	−17.9564	−0.913959
$387$	52.4955	2.66849
$388$	−9.20871	−0.467502
$389$	−6.33030	−0.320959	−0.160480	−	0.987039i	$-0.551304\pi$
$389$	−6.33030	−0.320959	−0.160480	+	0.987039i	$0.551304\pi$
$390$	0	0
$391$	−3.62614	−0.183382
$392$	3.79129	0.191489
$393$	−41.8693	−2.11203
$394$	9.95644	0.501598
$395$	0	0
$396$	18.1652	0.912833
$397$	−6.20871	−0.311606	−0.155803	−	0.987788i	$-0.549797\pi$
$397$	−6.20871	−0.311606	−0.155803	+	0.987788i	$0.549797\pi$
$398$	14.1216	0.707851
$399$	25.0000	1.25157
$400$	0	0
$401$	20.3739	1.01742	0.508711	−	0.860937i	$-0.330122\pi$
$401$	20.3739	1.01742	0.508711	+	0.860937i	$0.330122\pi$
$402$	−30.7042	−1.53138
$403$	0	0
$404$	−19.7477	−0.982486
$405$	0	0
$406$	−6.79129	−0.337046
$407$	−24.1652	−1.19782
$408$	−2.20871	−0.109348
$409$	22.4955	1.11233	0.556164	−	0.831072i	$-0.312273\pi$
$409$	22.4955	1.11233	0.556164	+	0.831072i	$0.312273\pi$
$410$	0	0
$411$	42.3303	2.08800
$412$	2.58258	0.127234
$413$	−8.20871	−0.403924
$414$	21.9564	1.07910
$415$	0	0
$416$	0	0
$417$	60.2432	2.95012
$418$	18.9564	0.927190
$419$	−3.46099	−0.169080	−0.0845401	−	0.996420i	$-0.526942\pi$
$419$	−3.46099	−0.169080	−0.0845401	+	0.996420i	$0.526942\pi$
$420$	0	0
$421$	−1.04356	−0.0508600	−0.0254300	−	0.999677i	$-0.508095\pi$
$421$	−1.04356	−0.0508600	−0.0254300	+	0.999677i	$0.508095\pi$
$422$	−5.00000	−0.243396
$423$	3.79129	0.184339
$424$	4.58258	0.222550
$425$	0	0
$426$	21.1652	1.02545
$427$	15.3739	0.743993
$428$	10.4174	0.503545
$429$	0	0
$430$	0	0
$431$	6.79129	0.327125	0.163562	−	0.986533i	$-0.447701\pi$
$431$	6.79129	0.327125	0.163562	+	0.986533i	$0.447701\pi$
$432$	5.00000	0.240563
$433$	−30.5826	−1.46970	−0.734852	−	0.678227i	$-0.762749\pi$
$433$	−30.5826	−1.46970	−0.734852	+	0.678227i	$0.762749\pi$
$434$	−17.1652	−0.823954
$435$	0	0
$436$	−1.83485	−0.0878733
$437$	22.9129	1.09607
$438$	20.0000	0.955637
$439$	−17.5826	−0.839171	−0.419585	−	0.907716i	$-0.637824\pi$
$439$	−17.5826	−0.839171	−0.419585	+	0.907716i	$0.637824\pi$
$440$	0	0
$441$	−18.1652	−0.865007
$442$	0	0
$443$	−2.04356	−0.0970925	−0.0485463	−	0.998821i	$-0.515459\pi$
$443$	−2.04356	−0.0970925	−0.0485463	+	0.998821i	$0.515459\pi$
$444$	−17.7913	−0.844337
$445$	0	0
$446$	−9.74773	−0.461568
$447$	11.0436	0.522343
$448$	−1.79129	−0.0846304
$449$	−32.3739	−1.52782	−0.763909	−	0.645325i	$-0.776722\pi$
$449$	−32.3739	−1.52782	−0.763909	+	0.645325i	$0.776722\pi$
$450$	0	0
$451$	−46.1216	−2.17178
$452$	0.626136	0.0294510
$453$	17.7913	0.835908
$454$	−16.5826	−0.778259
$455$	0	0
$456$	13.9564	0.653570
$457$	7.37386	0.344935	0.172467	−	0.985015i	$-0.444826\pi$
$457$	7.37386	0.344935	0.172467	+	0.985015i	$0.444826\pi$
$458$	−12.3739	−0.578193
$459$	3.95644	0.184671
$460$	0	0
$461$	−29.2087	−1.36039	−0.680193	−	0.733033i	$-0.738104\pi$
$461$	−29.2087	−1.36039	−0.680193	+	0.733033i	$0.738104\pi$
$462$	18.9564	0.881933
$463$	17.9564	0.834507	0.417253	−	0.908790i	$-0.362993\pi$
$463$	17.9564	0.834507	0.417253	+	0.908790i	$0.362993\pi$
$464$	−3.79129	−0.176006
$465$	0	0
$466$	−15.0000	−0.694862
$467$	8.37386	0.387496	0.193748	−	0.981051i	$-0.437936\pi$
$467$	8.37386	0.387496	0.193748	+	0.981051i	$0.437936\pi$
$468$	0	0
$469$	−19.7042	−0.909854
$470$	0	0
$471$	−14.4174	−0.664320
$472$	−4.58258	−0.210930
$473$	41.5390	1.90997
$474$	−41.7477	−1.91754
$475$	0	0
$476$	−1.41742	−0.0649675
$477$	−21.9564	−1.00532
$478$	−18.1652	−0.830855
$479$	−18.9564	−0.866142	−0.433071	−	0.901360i	$-0.642570\pi$
$479$	−18.9564	−0.866142	−0.433071	+	0.901360i	$0.642570\pi$
$480$	0	0
$481$	0	0
$482$	−7.00000	−0.318841
$483$	22.9129	1.04257
$484$	3.37386	0.153357
$485$	0	0
$486$	16.1652	0.733266
$487$	21.1216	0.957111	0.478555	−	0.878057i	$-0.341161\pi$
$487$	21.1216	0.957111	0.478555	+	0.878057i	$0.341161\pi$
$488$	8.58258	0.388515
$489$	−12.9129	−0.583941
$490$	0	0
$491$	34.9129	1.57560	0.787798	−	0.615934i	$-0.211221\pi$
$491$	34.9129	1.57560	0.787798	+	0.615934i	$0.211221\pi$
$492$	−33.9564	−1.53087
$493$	−3.00000	−0.135113
$494$	0	0
$495$	0	0
$496$	−9.58258	−0.430270
$497$	13.5826	0.609262
$498$	17.2087	0.771141
$499$	−9.74773	−0.436368	−0.218184	−	0.975908i	$-0.570013\pi$
$499$	−9.74773	−0.436368	−0.218184	+	0.975908i	$0.570013\pi$
$500$	0	0
$501$	−3.95644	−0.176761
$502$	20.3739	0.909330
$503$	43.9129	1.95798	0.978989	−	0.203912i	$-0.0653656\pi$
$503$	43.9129	1.95798	0.978989	+	0.203912i	$0.0653656\pi$
$504$	8.58258	0.382298
$505$	0	0
$506$	17.3739	0.772362
$507$	0	0
$508$	−16.3739	−0.726473
$509$	−25.9129	−1.14857	−0.574284	−	0.818656i	$-0.694720\pi$
$509$	−25.9129	−1.14857	−0.574284	+	0.818656i	$0.694720\pi$
$510$	0	0
$511$	12.8348	0.567780
$512$	−1.00000	−0.0441942
$513$	−25.0000	−1.10378
$514$	22.9129	1.01064
$515$	0	0
$516$	30.5826	1.34632
$517$	3.00000	0.131940
$518$	−11.4174	−0.501653
$519$	53.3739	2.34285
$520$	0	0
$521$	36.3303	1.59166	0.795830	−	0.605520i	$-0.207035\pi$
$521$	36.3303	1.59166	0.795830	+	0.605520i	$0.207035\pi$
$522$	18.1652	0.795067
$523$	−14.0000	−0.612177	−0.306089	−	0.952003i	$-0.599020\pi$
$523$	−14.0000	−0.612177	−0.306089	+	0.952003i	$0.599020\pi$
$524$	−15.0000	−0.655278
$525$	0	0
$526$	−9.95644	−0.434121
$527$	−7.58258	−0.330302
$528$	10.5826	0.460547
$529$	−2.00000	−0.0869565
$530$	0	0
$531$	21.9564	0.952828
$532$	8.95644	0.388311
$533$	0	0
$534$	1.74773	0.0756315
$535$	0	0
$536$	−11.0000	−0.475128
$537$	16.7477	0.722718
$538$	15.3303	0.660936
$539$	−14.3739	−0.619126
$540$	0	0
$541$	−27.9129	−1.20007	−0.600034	−	0.799974i	$-0.704846\pi$
$541$	−27.9129	−1.20007	−0.600034	+	0.799974i	$0.704846\pi$
$542$	11.1652	0.479584
$543$	20.5826	0.883283
$544$	−0.791288	−0.0339262
$545$	0	0
$546$	0	0
$547$	−41.3303	−1.76716	−0.883578	−	0.468284i	$-0.844872\pi$
$547$	−41.3303	−1.76716	−0.883578	+	0.468284i	$0.844872\pi$
$548$	15.1652	0.647823
$549$	−41.1216	−1.75503
$550$	0	0
$551$	18.9564	0.807571
$552$	12.7913	0.544433
$553$	−26.7913	−1.13928
$554$	−7.62614	−0.324003
$555$	0	0
$556$	21.5826	0.915305
$557$	19.2523	0.815745	0.407872	−	0.913039i	$-0.366271\pi$
$557$	19.2523	0.815745	0.407872	+	0.913039i	$0.366271\pi$
$558$	45.9129	1.94365
$559$	0	0
$560$	0	0
$561$	8.37386	0.353545
$562$	−14.5390	−0.613291
$563$	−14.7042	−0.619707	−0.309853	−	0.950784i	$-0.600280\pi$
$563$	−14.7042	−0.619707	−0.309853	+	0.950784i	$0.600280\pi$
$564$	2.20871	0.0930036
$565$	0	0
$566$	−12.5390	−0.527054
$567$	0.747727	0.0314016
$568$	7.58258	0.318158
$569$	6.62614	0.277782	0.138891	−	0.990308i	$-0.455646\pi$
$569$	6.62614	0.277782	0.138891	+	0.990308i	$0.455646\pi$
$570$	0	0
$571$	−6.37386	−0.266738	−0.133369	−	0.991066i	$-0.542580\pi$
$571$	−6.37386	−0.266738	−0.133369	+	0.991066i	$0.542580\pi$
$572$	0	0
$573$	−4.87841	−0.203798
$574$	−21.7913	−0.909551
$575$	0	0
$576$	4.79129	0.199637
$577$	−2.25227	−0.0937633	−0.0468817	−	0.998900i	$-0.514928\pi$
$577$	−2.25227	−0.0937633	−0.0468817	+	0.998900i	$0.514928\pi$
$578$	16.3739	0.681063
$579$	50.1216	2.08298
$580$	0	0
$581$	11.0436	0.458164
$582$	25.7042	1.06547
$583$	−17.3739	−0.719552
$584$	7.16515	0.296496
$585$	0	0
$586$	−6.62614	−0.273723
$587$	22.2867	0.919872	0.459936	−	0.887952i	$-0.347872\pi$
$587$	22.2867	0.919872	0.459936	+	0.887952i	$0.347872\pi$
$588$	−10.5826	−0.436418
$589$	47.9129	1.97422
$590$	0	0
$591$	−27.7913	−1.14318
$592$	−6.37386	−0.261964
$593$	1.58258	0.0649886	0.0324943	−	0.999472i	$-0.489655\pi$
$593$	1.58258	0.0649886	0.0324943	+	0.999472i	$0.489655\pi$
$594$	−18.9564	−0.777792
$595$	0	0
$596$	3.95644	0.162062
$597$	−39.4174	−1.61325
$598$	0	0
$599$	7.74773	0.316564	0.158282	−	0.987394i	$-0.449405\pi$
$599$	7.74773	0.316564	0.158282	+	0.987394i	$0.449405\pi$
$600$	0	0
$601$	−17.7477	−0.723945	−0.361972	−	0.932189i	$-0.617897\pi$
$601$	−17.7477	−0.723945	−0.361972	+	0.932189i	$0.617897\pi$
$602$	19.6261	0.799902
$603$	52.7042	2.14628
$604$	6.37386	0.259349
$605$	0	0
$606$	55.1216	2.23916
$607$	−5.00000	−0.202944	−0.101472	−	0.994838i	$-0.532355\pi$
$607$	−5.00000	−0.202944	−0.101472	+	0.994838i	$0.532355\pi$
$608$	5.00000	0.202777
$609$	18.9564	0.768154
$610$	0	0
$611$	0	0
$612$	3.79129	0.153254
$613$	9.58258	0.387037	0.193518	−	0.981097i	$-0.438010\pi$
$613$	9.58258	0.387037	0.193518	+	0.981097i	$0.438010\pi$
$614$	13.1652	0.531302
$615$	0	0
$616$	6.79129	0.273629
$617$	30.1652	1.21440	0.607202	−	0.794548i	$-0.292292\pi$
$617$	30.1652	1.21440	0.607202	+	0.794548i	$0.292292\pi$
$618$	−7.20871	−0.289977
$619$	−31.2087	−1.25438	−0.627192	−	0.778865i	$-0.715796\pi$
$619$	−31.2087	−1.25438	−0.627192	+	0.778865i	$0.715796\pi$
$620$	0	0
$621$	−22.9129	−0.919462
$622$	17.3739	0.696628
$623$	1.12159	0.0449356
$624$	0	0
$625$	0	0
$626$	−4.62614	−0.184898
$627$	−52.9129	−2.11314
$628$	−5.16515	−0.206112
$629$	−5.04356	−0.201100
$630$	0	0
$631$	−10.3739	−0.412977	−0.206488	−	0.978449i	$-0.566204\pi$
$631$	−10.3739	−0.412977	−0.206488	+	0.978449i	$0.566204\pi$
$632$	−14.9564	−0.594935
$633$	13.9564	0.554719
$634$	−26.5390	−1.05400
$635$	0	0
$636$	−12.7913	−0.507208
$637$	0	0
$638$	14.3739	0.569067
$639$	−36.3303	−1.43720
$640$	0	0
$641$	−3.79129	−0.149747	−0.0748734	−	0.997193i	$-0.523855\pi$
$641$	−3.79129	−0.149747	−0.0748734	+	0.997193i	$0.523855\pi$
$642$	−29.0780	−1.14762
$643$	−2.25227	−0.0888210	−0.0444105	−	0.999013i	$-0.514141\pi$
$643$	−2.25227	−0.0888210	−0.0444105	+	0.999013i	$0.514141\pi$
$644$	8.20871	0.323469
$645$	0	0
$646$	3.95644	0.155664
$647$	−45.9564	−1.80673	−0.903367	−	0.428868i	$-0.858912\pi$
$647$	−45.9564	−1.80673	−0.903367	+	0.428868i	$0.858912\pi$
$648$	0.417424	0.0163980
$649$	17.3739	0.681984
$650$	0	0
$651$	47.9129	1.87785
$652$	−4.62614	−0.181173
$653$	31.4174	1.22946	0.614729	−	0.788738i	$-0.289265\pi$
$653$	31.4174	1.22946	0.614729	+	0.788738i	$0.289265\pi$
$654$	5.12159	0.200270
$655$	0	0
$656$	−12.1652	−0.474969
$657$	−34.3303	−1.33935
$658$	1.41742	0.0552570
$659$	21.4955	0.837344	0.418672	−	0.908138i	$-0.362496\pi$
$659$	21.4955	0.837344	0.418672	+	0.908138i	$0.362496\pi$
$660$	0	0
$661$	7.95644	0.309470	0.154735	−	0.987956i	$-0.450548\pi$
$661$	7.95644	0.309470	0.154735	+	0.987956i	$0.450548\pi$
$662$	11.9564	0.464700
$663$	0	0
$664$	6.16515	0.239254
$665$	0	0
$666$	30.5390	1.18336
$667$	17.3739	0.672719
$668$	−1.41742	−0.0548418
$669$	27.2087	1.05195
$670$	0	0
$671$	−32.5390	−1.25615
$672$	5.00000	0.192879
$673$	23.9129	0.921774	0.460887	−	0.887459i	$-0.347531\pi$
$673$	23.9129	0.921774	0.460887	+	0.887459i	$0.347531\pi$
$674$	0.252273	0.00971719
$675$	0	0
$676$	0	0
$677$	4.25227	0.163428	0.0817141	−	0.996656i	$-0.473961\pi$
$677$	4.25227	0.163428	0.0817141	+	0.996656i	$0.473961\pi$
$678$	−1.74773	−0.0671211
$679$	16.4955	0.633037
$680$	0	0
$681$	46.2867	1.77371
$682$	36.3303	1.39116
$683$	−28.9129	−1.10632	−0.553160	−	0.833075i	$-0.686578\pi$
$683$	−28.9129	−1.10632	−0.553160	+	0.833075i	$0.686578\pi$
$684$	−23.9564	−0.915997
$685$	0	0
$686$	−19.3303	−0.738034
$687$	34.5390	1.31775
$688$	10.9564	0.417710
$689$	0	0
$690$	0	0
$691$	−3.25227	−0.123722	−0.0618611	−	0.998085i	$-0.519704\pi$
$691$	−3.25227	−0.123722	−0.0618611	+	0.998085i	$0.519704\pi$
$692$	19.1216	0.726894
$693$	−32.5390	−1.23605
$694$	28.5826	1.08498
$695$	0	0
$696$	10.5826	0.401131
$697$	−9.62614	−0.364616
$698$	26.9564	1.02032
$699$	41.8693	1.58364
$700$	0	0
$701$	4.87841	0.184255	0.0921275	−	0.995747i	$-0.470633\pi$
$701$	4.87841	0.184255	0.0921275	+	0.995747i	$0.470633\pi$
$702$	0	0
$703$	31.8693	1.20197
$704$	3.79129	0.142890
$705$	0	0
$706$	18.6261	0.701004
$707$	35.3739	1.33037
$708$	12.7913	0.480726
$709$	10.7913	0.405275	0.202638	−	0.979254i	$-0.435049\pi$
$709$	10.7913	0.405275	0.202638	+	0.979254i	$0.435049\pi$
$710$	0	0
$711$	71.6606	2.68748
$712$	0.626136	0.0234655
$713$	43.9129	1.64455
$714$	3.95644	0.148066
$715$	0	0
$716$	6.00000	0.224231
$717$	50.7042	1.89358
$718$	−33.7913	−1.26108
$719$	11.5390	0.430333	0.215166	−	0.976577i	$-0.430971\pi$
$719$	11.5390	0.430333	0.215166	+	0.976577i	$0.430971\pi$
$720$	0	0
$721$	−4.62614	−0.172286
$722$	−6.00000	−0.223297
$723$	19.5390	0.726664
$724$	7.37386	0.274047
$725$	0	0
$726$	−9.41742	−0.349513
$727$	−23.1652	−0.859148	−0.429574	−	0.903032i	$-0.641336\pi$
$727$	−23.1652	−0.859148	−0.429574	+	0.903032i	$0.641336\pi$
$728$	0	0
$729$	−43.8693	−1.62479
$730$	0	0
$731$	8.66970	0.320660
$732$	−23.9564	−0.885455
$733$	19.3739	0.715590	0.357795	−	0.933800i	$-0.383529\pi$
$733$	19.3739	0.715590	0.357795	+	0.933800i	$0.383529\pi$
$734$	24.7477	0.913455
$735$	0	0
$736$	4.58258	0.168916
$737$	41.7042	1.53619
$738$	58.2867	2.14556
$739$	−34.3739	−1.26446	−0.632232	−	0.774780i	$-0.717861\pi$
$739$	−34.3739	−1.26446	−0.632232	+	0.774780i	$0.717861\pi$
$740$	0	0
$741$	0	0
$742$	−8.20871	−0.301351
$743$	−42.0000	−1.54083	−0.770415	−	0.637542i	$-0.779951\pi$
$743$	−42.0000	−1.54083	−0.770415	+	0.637542i	$0.779951\pi$
$744$	26.7477	0.980619
$745$	0	0
$746$	−13.1652	−0.482010
$747$	−29.5390	−1.08078
$748$	3.00000	0.109691
$749$	−18.6606	−0.681844
$750$	0	0
$751$	2.49545	0.0910604	0.0455302	−	0.998963i	$-0.485502\pi$
$751$	2.49545	0.0910604	0.0455302	+	0.998963i	$0.485502\pi$
$752$	0.791288	0.0288553
$753$	−56.8693	−2.07243
$754$	0	0
$755$	0	0
$756$	−8.95644	−0.325743
$757$	47.7477	1.73542	0.867710	−	0.497070i	$-0.165591\pi$
$757$	47.7477	1.73542	0.867710	+	0.497070i	$0.165591\pi$
$758$	17.0000	0.617468
$759$	−48.4955	−1.76027
$760$	0	0
$761$	10.9129	0.395592	0.197796	−	0.980243i	$-0.436622\pi$
$761$	10.9129	0.395592	0.197796	+	0.980243i	$0.436622\pi$
$762$	45.7042	1.65569
$763$	3.28674	0.118988
$764$	−1.74773	−0.0632305
$765$	0	0
$766$	−1.74773	−0.0631479
$767$	0	0
$768$	2.79129	0.100722
$769$	−36.7477	−1.32516	−0.662578	−	0.748993i	$-0.730538\pi$
$769$	−36.7477	−1.32516	−0.662578	+	0.748993i	$0.730538\pi$
$770$	0	0
$771$	−63.9564	−2.30333
$772$	17.9564	0.646266
$773$	36.6606	1.31859	0.659295	−	0.751884i	$-0.270855\pi$
$773$	36.6606	1.31859	0.659295	+	0.751884i	$0.270855\pi$
$774$	−52.4955	−1.88691
$775$	0	0
$776$	9.20871	0.330574
$777$	31.8693	1.14331
$778$	6.33030	0.226952
$779$	60.8258	2.17931
$780$	0	0
$781$	−28.7477	−1.02867
$782$	3.62614	0.129670
$783$	−18.9564	−0.677448
$784$	−3.79129	−0.135403
$785$	0	0
$786$	41.8693	1.49343
$787$	43.5390	1.55200	0.775999	−	0.630734i	$-0.217246\pi$
$787$	43.5390	1.55200	0.775999	+	0.630734i	$0.217246\pi$
$788$	−9.95644	−0.354683
$789$	27.7913	0.989396
$790$	0	0
$791$	−1.12159	−0.0398792
$792$	−18.1652	−0.645471
$793$	0	0
$794$	6.20871	0.220339
$795$	0	0
$796$	−14.1216	−0.500527
$797$	−44.0780	−1.56132	−0.780662	−	0.624954i	$-0.785118\pi$
$797$	−44.0780	−1.56132	−0.780662	+	0.624954i	$0.785118\pi$
$798$	−25.0000	−0.884990
$799$	0.626136	0.0221511
$800$	0	0
$801$	−3.00000	−0.106000
$802$	−20.3739	−0.719426
$803$	−27.1652	−0.958637
$804$	30.7042	1.08285
$805$	0	0
$806$	0	0
$807$	−42.7913	−1.50632
$808$	19.7477	0.694723
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	18.7042	0.656792	0.328396	−	0.944540i	$-0.393492\pi$
$811$	18.7042	0.656792	0.328396	+	0.944540i	$0.393492\pi$
$812$	6.79129	0.238327
$813$	−31.1652	−1.09301
$814$	24.1652	0.846988
$815$	0	0
$816$	2.20871	0.0773204
$817$	−54.7822	−1.91659
$818$	−22.4955	−0.786535
$819$	0	0
$820$	0	0
$821$	−10.9129	−0.380862	−0.190431	−	0.981701i	$-0.560989\pi$
$821$	−10.9129	−0.380862	−0.190431	+	0.981701i	$0.560989\pi$
$822$	−42.3303	−1.47644
$823$	−21.2523	−0.740808	−0.370404	−	0.928871i	$-0.620781\pi$
$823$	−21.2523	−0.740808	−0.370404	+	0.928871i	$0.620781\pi$
$824$	−2.58258	−0.0899683
$825$	0	0
$826$	8.20871	0.285618
$827$	−45.3303	−1.57629	−0.788145	−	0.615490i	$-0.788958\pi$
$827$	−45.3303	−1.57629	−0.788145	+	0.615490i	$0.788958\pi$
$828$	−21.9564	−0.763039
$829$	−44.2867	−1.53814	−0.769071	−	0.639163i	$-0.779281\pi$
$829$	−44.2867	−1.53814	−0.769071	+	0.639163i	$0.779281\pi$
$830$	0	0
$831$	21.2867	0.738429
$832$	0	0
$833$	−3.00000	−0.103944
$834$	−60.2432	−2.08605
$835$	0	0
$836$	−18.9564	−0.655622
$837$	−47.9129	−1.65611
$838$	3.46099	0.119558
$839$	5.70417	0.196930	0.0984648	−	0.995141i	$-0.468607\pi$
$839$	5.70417	0.196930	0.0984648	+	0.995141i	$0.468607\pi$
$840$	0	0
$841$	−14.6261	−0.504350
$842$	1.04356	0.0359635
$843$	40.5826	1.39774
$844$	5.00000	0.172107
$845$	0	0
$846$	−3.79129	−0.130347
$847$	−6.04356	−0.207659
$848$	−4.58258	−0.157366
$849$	35.0000	1.20120
$850$	0	0
$851$	29.2087	1.00126
$852$	−21.1652	−0.725106
$853$	−32.1216	−1.09982	−0.549911	−	0.835223i	$-0.685338\pi$
$853$	−32.1216	−1.09982	−0.549911	+	0.835223i	$0.685338\pi$
$854$	−15.3739	−0.526083
$855$	0	0
$856$	−10.4174	−0.356060
$857$	24.1652	0.825466	0.412733	−	0.910852i	$-0.364574\pi$
$857$	24.1652	0.825466	0.412733	+	0.910852i	$0.364574\pi$
$858$	0	0
$859$	21.2867	0.726294	0.363147	−	0.931732i	$-0.381702\pi$
$859$	21.2867	0.726294	0.363147	+	0.931732i	$0.381702\pi$
$860$	0	0
$861$	60.8258	2.07294
$862$	−6.79129	−0.231312
$863$	−22.5826	−0.768720	−0.384360	−	0.923183i	$-0.625578\pi$
$863$	−22.5826	−0.768720	−0.384360	+	0.923183i	$0.625578\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	30.5826	1.03924
$867$	−45.7042	−1.55219
$868$	17.1652	0.582623
$869$	56.7042	1.92356
$870$	0	0
$871$	0	0
$872$	1.83485	0.0621358
$873$	−44.1216	−1.49329
$874$	−22.9129	−0.775040
$875$	0	0
$876$	−20.0000	−0.675737
$877$	18.2523	0.616335	0.308168	−	0.951332i	$-0.400284\pi$
$877$	18.2523	0.616335	0.308168	+	0.951332i	$0.400284\pi$
$878$	17.5826	0.593383
$879$	18.4955	0.623836
$880$	0	0
$881$	−5.04356	−0.169922	−0.0849609	−	0.996384i	$-0.527077\pi$
$881$	−5.04356	−0.169922	−0.0849609	+	0.996384i	$0.527077\pi$
$882$	18.1652	0.611652
$883$	19.9564	0.671588	0.335794	−	0.941936i	$-0.390995\pi$
$883$	19.9564	0.671588	0.335794	+	0.941936i	$0.390995\pi$
$884$	0	0
$885$	0	0
$886$	2.04356	0.0686548
$887$	−51.1652	−1.71796	−0.858979	−	0.512011i	$-0.828901\pi$
$887$	−51.1652	−1.71796	−0.858979	+	0.512011i	$0.828901\pi$
$888$	17.7913	0.597037
$889$	29.3303	0.983707
$890$	0	0
$891$	−1.58258	−0.0530183
$892$	9.74773	0.326378
$893$	−3.95644	−0.132397
$894$	−11.0436	−0.369352
$895$	0	0
$896$	1.79129	0.0598427
$897$	0	0
$898$	32.3739	1.08033
$899$	36.3303	1.21168
$900$	0	0
$901$	−3.62614	−0.120804
$902$	46.1216	1.53568
$903$	−54.7822	−1.82304
$904$	−0.626136	−0.0208250
$905$	0	0
$906$	−17.7913	−0.591076
$907$	−32.4955	−1.07899	−0.539497	−	0.841988i	$-0.681386\pi$
$907$	−32.4955	−1.07899	−0.539497	+	0.841988i	$0.681386\pi$
$908$	16.5826	0.550312
$909$	−94.6170	−3.13825
$910$	0	0
$911$	45.3303	1.50186	0.750930	−	0.660382i	$-0.229606\pi$
$911$	45.3303	1.50186	0.750930	+	0.660382i	$0.229606\pi$
$912$	−13.9564	−0.462144
$913$	−23.3739	−0.773562
$914$	−7.37386	−0.243906
$915$	0	0
$916$	12.3739	0.408844
$917$	26.8693	0.887303
$918$	−3.95644	−0.130582
$919$	−44.7477	−1.47609	−0.738046	−	0.674751i	$-0.764251\pi$
$919$	−44.7477	−1.47609	−0.738046	+	0.674751i	$0.764251\pi$
$920$	0	0
$921$	−36.7477	−1.21088
$922$	29.2087	0.961938
$923$	0	0
$924$	−18.9564	−0.623621
$925$	0	0
$926$	−17.9564	−0.590085
$927$	12.3739	0.406411
$928$	3.79129	0.124455
$929$	22.7477	0.746329	0.373164	−	0.927765i	$-0.378273\pi$
$929$	22.7477	0.746329	0.373164	+	0.927765i	$0.378273\pi$
$930$	0	0
$931$	18.9564	0.621272
$932$	15.0000	0.491341
$933$	−48.4955	−1.58767
$934$	−8.37386	−0.274001
$935$	0	0
$936$	0	0
$937$	−32.6261	−1.06585	−0.532925	−	0.846163i	$-0.678907\pi$
$937$	−32.6261	−1.06585	−0.532925	+	0.846163i	$0.678907\pi$
$938$	19.7042	0.643364
$939$	12.9129	0.421396
$940$	0	0
$941$	47.3739	1.54434	0.772172	−	0.635414i	$-0.219170\pi$
$941$	47.3739	1.54434	0.772172	+	0.635414i	$0.219170\pi$
$942$	14.4174	0.469745
$943$	55.7477	1.81540
$944$	4.58258	0.149150
$945$	0	0
$946$	−41.5390	−1.35055
$947$	−36.7913	−1.19556	−0.597778	−	0.801662i	$-0.703950\pi$
$947$	−36.7913	−1.19556	−0.597778	+	0.801662i	$0.703950\pi$
$948$	41.7477	1.35590
$949$	0	0
$950$	0	0
$951$	74.0780	2.40214
$952$	1.41742	0.0459390
$953$	−30.4955	−0.987845	−0.493922	−	0.869506i	$-0.664437\pi$
$953$	−30.4955	−0.987845	−0.493922	+	0.869506i	$0.664437\pi$
$954$	21.9564	0.710866
$955$	0	0
$956$	18.1652	0.587503
$957$	−40.1216	−1.29695
$958$	18.9564	0.612455
$959$	−27.1652	−0.877208
$960$	0	0
$961$	60.8258	1.96212
$962$	0	0
$963$	49.9129	1.60842
$964$	7.00000	0.225455
$965$	0	0
$966$	−22.9129	−0.737210
$967$	6.74773	0.216992	0.108496	−	0.994097i	$-0.465396\pi$
$967$	6.74773	0.216992	0.108496	+	0.994097i	$0.465396\pi$
$968$	−3.37386	−0.108440
$969$	−11.0436	−0.354770
$970$	0	0
$971$	18.4955	0.593547	0.296774	−	0.954948i	$-0.404089\pi$
$971$	18.4955	0.593547	0.296774	+	0.954948i	$0.404089\pi$
$972$	−16.1652	−0.518497
$973$	−38.6606	−1.23940
$974$	−21.1216	−0.676779
$975$	0	0
$976$	−8.58258	−0.274722
$977$	−10.5826	−0.338567	−0.169283	−	0.985567i	$-0.554145\pi$
$977$	−10.5826	−0.338567	−0.169283	+	0.985567i	$0.554145\pi$
$978$	12.9129	0.412908
$979$	−2.37386	−0.0758690
$980$	0	0
$981$	−8.79129	−0.280684
$982$	−34.9129	−1.11411
$983$	−18.4610	−0.588814	−0.294407	−	0.955680i	$-0.595122\pi$
$983$	−18.4610	−0.588814	−0.294407	+	0.955680i	$0.595122\pi$
$984$	33.9564	1.08249
$985$	0	0
$986$	3.00000	0.0955395
$987$	−3.95644	−0.125935
$988$	0	0
$989$	−50.2087	−1.59654
$990$	0	0
$991$	26.0000	0.825917	0.412959	−	0.910750i	$-0.364495\pi$
$991$	26.0000	0.825917	0.412959	+	0.910750i	$0.364495\pi$
$992$	9.58258	0.304247
$993$	−33.3739	−1.05909
$994$	−13.5826	−0.430813
$995$	0	0
$996$	−17.2087	−0.545279
$997$	5.12159	0.162202	0.0811012	−	0.996706i	$-0.474156\pi$
$997$	5.12159	0.162202	0.0811012	+	0.996706i	$0.474156\pi$
$998$	9.74773	0.308559
$999$	−31.8693	−1.00830

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.be.1.2		2
5.4	even	2		8450.2.a.bh.1.1		2
13.4	even	6		650.2.e.d.601.1	yes	4
13.10	even	6		650.2.e.d.451.1	✓	4
13.12	even	2		8450.2.a.bk.1.2		2
65.4	even	6		650.2.e.i.601.2	yes	4
65.17	odd	12		650.2.o.f.549.3		8
65.23	odd	12		650.2.o.f.399.3		8
65.43	odd	12		650.2.o.f.549.2		8
65.49	even	6		650.2.e.i.451.2	yes	4
65.62	odd	12		650.2.o.f.399.2		8
65.64	even	2		8450.2.a.bb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
650.2.e.d.451.1	✓	4	13.10	even	6
650.2.e.d.601.1	yes	4	13.4	even	6
650.2.e.i.451.2	yes	4	65.49	even	6
650.2.e.i.601.2	yes	4	65.4	even	6
650.2.o.f.399.2		8	65.62	odd	12
650.2.o.f.399.3		8	65.23	odd	12
650.2.o.f.549.2		8	65.43	odd	12
650.2.o.f.549.3		8	65.17	odd	12
8450.2.a.bb.1.1		2	65.64	even	2
8450.2.a.be.1.2		2	1.1	even	1	trivial
8450.2.a.bh.1.1		2	5.4	even	2
8450.2.a.bk.1.2		2	13.12	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8450.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +2.79129 q^{3} +1.00000 q^{4} -2.79129 q^{6} -1.79129 q^{7} -1.00000 q^{8} +4.79129 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +2.79129 q^{3} +1.00000 q^{4} -2.79129 q^{6} -1.79129 q^{7} -1.00000 q^{8} +4.79129 q^{9} +3.79129 q^{11} +2.79129 q^{12} +1.79129 q^{14} +1.00000 q^{16} +0.791288 q^{17} -4.79129 q^{18} -5.00000 q^{19} -5.00000 q^{21} -3.79129 q^{22} -4.58258 q^{23} -2.79129 q^{24} +5.00000 q^{27} -1.79129 q^{28} -3.79129 q^{29} -9.58258 q^{31} -1.00000 q^{32} +10.5826 q^{33} -0.791288 q^{34} +4.79129 q^{36} -6.37386 q^{37} +5.00000 q^{38} -12.1652 q^{41} +5.00000 q^{42} +10.9564 q^{43} +3.79129 q^{44} +4.58258 q^{46} +0.791288 q^{47} +2.79129 q^{48} -3.79129 q^{49} +2.20871 q^{51} -4.58258 q^{53} -5.00000 q^{54} +1.79129 q^{56} -13.9564 q^{57} +3.79129 q^{58} +4.58258 q^{59} -8.58258 q^{61} +9.58258 q^{62} -8.58258 q^{63} +1.00000 q^{64} -10.5826 q^{66} +11.0000 q^{67} +0.791288 q^{68} -12.7913 q^{69} -7.58258 q^{71} -4.79129 q^{72} -7.16515 q^{73} +6.37386 q^{74} -5.00000 q^{76} -6.79129 q^{77} +14.9564 q^{79} -0.417424 q^{81} +12.1652 q^{82} -6.16515 q^{83} -5.00000 q^{84} -10.9564 q^{86} -10.5826 q^{87} -3.79129 q^{88} -0.626136 q^{89} -4.58258 q^{92} -26.7477 q^{93} -0.791288 q^{94} -2.79129 q^{96} -9.20871 q^{97} +3.79129 q^{98} +18.1652 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} + q^{7} - 2 q^{8} + 5 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + q^3 + 2 * q^4 - q^6 + q^7 - 2 * q^8 + 5 * q^9	\( 2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} + q^{7} - 2 q^{8} + 5 q^{9} + 3 q^{11} + q^{12} - q^{14} + 2 q^{16} - 3 q^{17} - 5 q^{18} - 10 q^{19} - 10 q^{21} - 3 q^{22} - q^{24} + 10 q^{27} + q^{28} - 3 q^{29} - 10 q^{31} - 2 q^{32} + 12 q^{33} + 3 q^{34} + 5 q^{36} + q^{37} + 10 q^{38} - 6 q^{41} + 10 q^{42} - q^{43} + 3 q^{44} - 3 q^{47} + q^{48} - 3 q^{49} + 9 q^{51} - 10 q^{54} - q^{56} - 5 q^{57} + 3 q^{58} - 8 q^{61} + 10 q^{62} - 8 q^{63} + 2 q^{64} - 12 q^{66} + 22 q^{67} - 3 q^{68} - 21 q^{69} - 6 q^{71} - 5 q^{72} + 4 q^{73} - q^{74} - 10 q^{76} - 9 q^{77} + 7 q^{79} - 10 q^{81} + 6 q^{82} + 6 q^{83} - 10 q^{84} + q^{86} - 12 q^{87} - 3 q^{88} - 15 q^{89} - 26 q^{93} + 3 q^{94} - q^{96} - 23 q^{97} + 3 q^{98} + 18 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + q^3 + 2 * q^4 - q^6 + q^7 - 2 * q^8 + 5 * q^9 + 3 * q^11 + q^12 - q^14 + 2 * q^16 - 3 * q^17 - 5 * q^18 - 10 * q^19 - 10 * q^21 - 3 * q^22 - q^24 + 10 * q^27 + q^28 - 3 * q^29 - 10 * q^31 - 2 * q^32 + 12 * q^33 + 3 * q^34 + 5 * q^36 + q^37 + 10 * q^38 - 6 * q^41 + 10 * q^42 - q^43 + 3 * q^44 - 3 * q^47 + q^48 - 3 * q^49 + 9 * q^51 - 10 * q^54 - q^56 - 5 * q^57 + 3 * q^58 - 8 * q^61 + 10 * q^62 - 8 * q^63 + 2 * q^64 - 12 * q^66 + 22 * q^67 - 3 * q^68 - 21 * q^69 - 6 * q^71 - 5 * q^72 + 4 * q^73 - q^74 - 10 * q^76 - 9 * q^77 + 7 * q^79 - 10 * q^81 + 6 * q^82 + 6 * q^83 - 10 * q^84 + q^86 - 12 * q^87 - 3 * q^88 - 15 * q^89 - 26 * q^93 + 3 * q^94 - q^96 - 23 * q^97 + 3 * q^98 + 18 * q^99

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