LMFDB - Embedded newform 8450.2.a.bd.1.1

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:	$\Q(\sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 3$ x^2 - x - 3
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			1.1
Root			$-1.30278$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.30278 q^{3} +1.00000 q^{4} +1.30278 q^{6} -4.30278 q^{7} -1.00000 q^{8} -1.30278 q^{9} -4.30278 q^{11} -1.30278 q^{12} +4.30278 q^{14} +1.00000 q^{16} +7.90833 q^{17} +1.30278 q^{18} -7.60555 q^{19} +5.60555 q^{21} +4.30278 q^{22} -1.60555 q^{23} +1.30278 q^{24} +5.60555 q^{27} -4.30278 q^{28} -0.302776 q^{29} -1.39445 q^{31} -1.00000 q^{32} +5.60555 q^{33} -7.90833 q^{34} -1.30278 q^{36} +7.30278 q^{37} +7.60555 q^{38} +1.00000 q^{41} -5.60555 q^{42} +9.51388 q^{43} -4.30278 q^{44} +1.60555 q^{46} -6.69722 q^{47} -1.30278 q^{48} +11.5139 q^{49} -10.3028 q^{51} -2.21110 q^{53} -5.60555 q^{54} +4.30278 q^{56} +9.90833 q^{57} +0.302776 q^{58} +4.21110 q^{59} -10.2111 q^{61} +1.39445 q^{62} +5.60555 q^{63} +1.00000 q^{64} -5.60555 q^{66} +1.60555 q^{67} +7.90833 q^{68} +2.09167 q^{69} -3.39445 q^{71} +1.30278 q^{72} +8.00000 q^{73} -7.30278 q^{74} -7.60555 q^{76} +18.5139 q^{77} +5.69722 q^{79} -3.39445 q^{81} -1.00000 q^{82} +4.81665 q^{83} +5.60555 q^{84} -9.51388 q^{86} +0.394449 q^{87} +4.30278 q^{88} +10.9083 q^{89} -1.60555 q^{92} +1.81665 q^{93} +6.69722 q^{94} +1.30278 q^{96} +15.3028 q^{97} -11.5139 q^{98} +5.60555 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} - 5 q^{7} - 2 q^{8} + q^{9} - 5 q^{11} + q^{12} + 5 q^{14} + 2 q^{16} + 5 q^{17} - q^{18} - 8 q^{19} + 4 q^{21} + 5 q^{22} + 4 q^{23} - q^{24} + 4 q^{27} - 5 q^{28}+ \cdots + 4 q^{99}+O(q^{100})$ 2 * q - 2 * q^2 + q^3 + 2 * q^4 - q^6 - 5 * q^7 - 2 * q^8 + q^9 - 5 * q^11 + q^12 + 5 * q^14 + 2 * q^16 + 5 * q^17 - q^18 - 8 * q^19 + 4 * q^21 + 5 * q^22 + 4 * q^23 - q^24 + 4 * q^27 - 5 * q^28 + 3 * q^29 - 10 * q^31 - 2 * q^32 + 4 * q^33 - 5 * q^34 + q^36 + 11 * q^37 + 8 * q^38 + 2 * q^41 - 4 * q^42 + q^43 - 5 * q^44 - 4 * q^46 - 17 * q^47 + q^48 + 5 * q^49 - 17 * q^51 + 10 * q^53 - 4 * q^54 + 5 * q^56 + 9 * q^57 - 3 * q^58 - 6 * q^59 - 6 * q^61 + 10 * q^62 + 4 * q^63 + 2 * q^64 - 4 * q^66 - 4 * q^67 + 5 * q^68 + 15 * q^69 - 14 * q^71 - q^72 + 16 * q^73 - 11 * q^74 - 8 * q^76 + 19 * q^77 + 15 * q^79 - 14 * q^81 - 2 * q^82 - 12 * q^83 + 4 * q^84 - q^86 + 8 * q^87 + 5 * q^88 + 11 * q^89 + 4 * q^92 - 18 * q^93 + 17 * q^94 - q^96 + 27 * q^97 - 5 * q^98 + 4 * q^99

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
$3$	−1.30278	−0.752158	−0.376079	−	0.926588i	$-0.622728\pi$
$3$	−1.30278	−0.752158	−0.376079	+	0.926588i	$0.622728\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.30278	0.531856
$7$	−4.30278	−1.62630	−0.813148	−	0.582057i	$-0.802248\pi$
$7$	−4.30278	−1.62630	−0.813148	+	0.582057i	$0.802248\pi$
$8$	−1.00000	−0.353553
$9$	−1.30278	−0.434259
$10$	0	0
$11$	−4.30278	−1.29734	−0.648668	−	0.761072i	$-0.724674\pi$
$11$	−4.30278	−1.29734	−0.648668	+	0.761072i	$0.724674\pi$
$12$	−1.30278	−0.376079
$13$	0	0
$13$	0	0
$14$	4.30278	1.14997
$15$	0	0
$16$	1.00000	0.250000
$17$	7.90833	1.91805	0.959026	−	0.283320i	$-0.0914358\pi$
$17$	7.90833	1.91805	0.959026	+	0.283320i	$0.0914358\pi$
$18$	1.30278	0.307067
$19$	−7.60555	−1.74483	−0.872417	−	0.488763i	$-0.837448\pi$
$19$	−7.60555	−1.74483	−0.872417	+	0.488763i	$0.837448\pi$
$20$	0	0
$21$	5.60555	1.22323
$22$	4.30278	0.917355
$23$	−1.60555	−0.334781	−0.167390	−	0.985891i	$-0.553534\pi$
$23$	−1.60555	−0.334781	−0.167390	+	0.985891i	$0.553534\pi$
$24$	1.30278	0.265928
$25$	0	0
$26$	0	0
$27$	5.60555	1.07879
$28$	−4.30278	−0.813148
$29$	−0.302776	−0.0562240	−0.0281120	−	0.999605i	$-0.508950\pi$
$29$	−0.302776	−0.0562240	−0.0281120	+	0.999605i	$0.508950\pi$
$30$	0	0
$31$	−1.39445	−0.250450	−0.125225	−	0.992128i	$-0.539965\pi$
$31$	−1.39445	−0.250450	−0.125225	+	0.992128i	$0.539965\pi$
$32$	−1.00000	−0.176777
$33$	5.60555	0.975801
$34$	−7.90833	−1.35627
$35$	0	0
$36$	−1.30278	−0.217129
$37$	7.30278	1.20057	0.600284	−	0.799787i	$-0.295054\pi$
$37$	7.30278	1.20057	0.600284	+	0.799787i	$0.295054\pi$
$38$	7.60555	1.23378
$39$	0	0
$40$	0	0
$41$	1.00000	0.156174	0.0780869	−	0.996947i	$-0.475119\pi$
$41$	1.00000	0.156174	0.0780869	+	0.996947i	$0.475119\pi$
$42$	−5.60555	−0.864955
$43$	9.51388	1.45085	0.725426	−	0.688300i	$-0.241643\pi$
$43$	9.51388	1.45085	0.725426	+	0.688300i	$0.241643\pi$
$44$	−4.30278	−0.648668
$45$	0	0
$46$	1.60555	0.236726
$47$	−6.69722	−0.976891	−0.488445	−	0.872595i	$-0.662436\pi$
$47$	−6.69722	−0.976891	−0.488445	+	0.872595i	$0.662436\pi$
$48$	−1.30278	−0.188039
$49$	11.5139	1.64484
$50$	0	0
$51$	−10.3028	−1.44268
$52$	0	0
$53$	−2.21110	−0.303718	−0.151859	−	0.988402i	$-0.548526\pi$
$53$	−2.21110	−0.303718	−0.151859	+	0.988402i	$0.548526\pi$
$54$	−5.60555	−0.762819
$55$	0	0
$56$	4.30278	0.574983
$57$	9.90833	1.31239
$58$	0.302776	0.0397564
$59$	4.21110	0.548239	0.274119	−	0.961696i	$-0.411614\pi$
$59$	4.21110	0.548239	0.274119	+	0.961696i	$0.411614\pi$
$60$	0	0
$61$	−10.2111	−1.30740	−0.653699	−	0.756755i	$-0.726784\pi$
$61$	−10.2111	−1.30740	−0.653699	+	0.756755i	$0.726784\pi$
$62$	1.39445	0.177095
$63$	5.60555	0.706233
$64$	1.00000	0.125000
$65$	0	0
$66$	−5.60555	−0.689996
$67$	1.60555	0.196149	0.0980747	−	0.995179i	$-0.468732\pi$
$67$	1.60555	0.196149	0.0980747	+	0.995179i	$0.468732\pi$
$68$	7.90833	0.959026
$69$	2.09167	0.251808
$70$	0	0
$71$	−3.39445	−0.402847	−0.201423	−	0.979504i	$-0.564557\pi$
$71$	−3.39445	−0.402847	−0.201423	+	0.979504i	$0.564557\pi$
$72$	1.30278	0.153534
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	−7.30278	−0.848930
$75$	0	0
$76$	−7.60555	−0.872417
$77$	18.5139	2.10985
$78$	0	0
$79$	5.69722	0.640988	0.320494	−	0.947251i	$-0.396151\pi$
$79$	5.69722	0.640988	0.320494	+	0.947251i	$0.396151\pi$
$80$	0	0
$81$	−3.39445	−0.377161
$82$	−1.00000	−0.110432
$83$	4.81665	0.528696	0.264348	−	0.964427i	$-0.414843\pi$
$83$	4.81665	0.528696	0.264348	+	0.964427i	$0.414843\pi$
$84$	5.60555	0.611616
$85$	0	0
$86$	−9.51388	−1.02591
$87$	0.394449	0.0422893
$88$	4.30278	0.458677
$89$	10.9083	1.15628	0.578140	−	0.815937i	$-0.303779\pi$
$89$	10.9083	1.15628	0.578140	+	0.815937i	$0.303779\pi$
$90$	0	0
$91$	0	0
$92$	−1.60555	−0.167390
$93$	1.81665	0.188378
$94$	6.69722	0.690766
$95$	0	0
$96$	1.30278	0.132964
$97$	15.3028	1.55376	0.776881	−	0.629648i	$-0.216801\pi$
$97$	15.3028	1.55376	0.776881	+	0.629648i	$0.216801\pi$
$98$	−11.5139	−1.16308
$99$	5.60555	0.563379
$100$	0	0
$101$	8.21110	0.817035	0.408518	−	0.912750i	$-0.366046\pi$
$101$	8.21110	0.817035	0.408518	+	0.912750i	$0.366046\pi$
$102$	10.3028	1.02013
$103$	−13.6056	−1.34059	−0.670297	−	0.742093i	$-0.733833\pi$
$103$	−13.6056	−1.34059	−0.670297	+	0.742093i	$0.733833\pi$
$104$	0	0
$105$	0	0
$106$	2.21110	0.214761
$107$	7.81665	0.755664	0.377832	−	0.925874i	$-0.376670\pi$
$107$	7.81665	0.755664	0.377832	+	0.925874i	$0.376670\pi$
$108$	5.60555	0.539394
$109$	4.39445	0.420912	0.210456	−	0.977603i	$-0.432505\pi$
$109$	4.39445	0.420912	0.210456	+	0.977603i	$0.432505\pi$
$110$	0	0
$111$	−9.51388	−0.903017
$112$	−4.30278	−0.406574
$113$	−7.51388	−0.706846	−0.353423	−	0.935464i	$-0.614982\pi$
$113$	−7.51388	−0.706846	−0.353423	+	0.935464i	$0.614982\pi$
$114$	−9.90833	−0.928000
$115$	0	0
$116$	−0.302776	−0.0281120
$117$	0	0
$118$	−4.21110	−0.387663
$119$	−34.0278	−3.11932
$120$	0	0
$121$	7.51388	0.683080
$122$	10.2111	0.924470
$123$	−1.30278	−0.117467
$124$	−1.39445	−0.125225
$125$	0	0
$126$	−5.60555	−0.499382
$127$	5.90833	0.524279	0.262140	−	0.965030i	$-0.415572\pi$
$127$	5.90833	0.524279	0.262140	+	0.965030i	$0.415572\pi$
$128$	−1.00000	−0.0883883
$129$	−12.3944	−1.09127
$130$	0	0
$131$	−12.8167	−1.11980	−0.559898	−	0.828561i	$-0.689160\pi$
$131$	−12.8167	−1.11980	−0.559898	+	0.828561i	$0.689160\pi$
$132$	5.60555	0.487901
$133$	32.7250	2.83762
$134$	−1.60555	−0.138699
$135$	0	0
$136$	−7.90833	−0.678133
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	−2.09167	−0.178055
$139$	11.8167	1.00228	0.501138	−	0.865368i	$-0.332915\pi$
$139$	11.8167	1.00228	0.501138	+	0.865368i	$0.332915\pi$
$140$	0	0
$141$	8.72498	0.734776
$142$	3.39445	0.284856
$143$	0	0
$144$	−1.30278	−0.108565
$145$	0	0
$146$	−8.00000	−0.662085
$147$	−15.0000	−1.23718
$148$	7.30278	0.600284
$149$	−4.69722	−0.384812	−0.192406	−	0.981315i	$-0.561629\pi$
$149$	−4.69722	−0.384812	−0.192406	+	0.981315i	$0.561629\pi$
$150$	0	0
$151$	13.1194	1.06764	0.533822	−	0.845597i	$-0.320755\pi$
$151$	13.1194	1.06764	0.533822	+	0.845597i	$0.320755\pi$
$152$	7.60555	0.616892
$153$	−10.3028	−0.832930
$154$	−18.5139	−1.49189
$155$	0	0
$156$	0	0
$157$	−4.42221	−0.352930	−0.176465	−	0.984307i	$-0.556466\pi$
$157$	−4.42221	−0.352930	−0.176465	+	0.984307i	$0.556466\pi$
$158$	−5.69722	−0.453247
$159$	2.88057	0.228444
$160$	0	0
$161$	6.90833	0.544452
$162$	3.39445	0.266693
$163$	−8.90833	−0.697754	−0.348877	−	0.937169i	$-0.613437\pi$
$163$	−8.90833	−0.697754	−0.348877	+	0.937169i	$0.613437\pi$
$164$	1.00000	0.0780869
$165$	0	0
$166$	−4.81665	−0.373845
$167$	−3.60555	−0.279006	−0.139503	−	0.990222i	$-0.544550\pi$
$167$	−3.60555	−0.279006	−0.139503	+	0.990222i	$0.544550\pi$
$168$	−5.60555	−0.432478
$169$	0	0
$170$	0	0
$171$	9.90833	0.757709
$172$	9.51388	0.725426
$173$	−24.5139	−1.86376	−0.931878	−	0.362772i	$-0.881830\pi$
$173$	−24.5139	−1.86376	−0.931878	+	0.362772i	$0.881830\pi$
$174$	−0.394449	−0.0299031
$175$	0	0
$176$	−4.30278	−0.324334
$177$	−5.48612	−0.412362
$178$	−10.9083	−0.817614
$179$	−7.21110	−0.538983	−0.269492	−	0.963003i	$-0.586856\pi$
$179$	−7.21110	−0.538983	−0.269492	+	0.963003i	$0.586856\pi$
$180$	0	0
$181$	−7.09167	−0.527120	−0.263560	−	0.964643i	$-0.584897\pi$
$181$	−7.09167	−0.527120	−0.263560	+	0.964643i	$0.584897\pi$
$182$	0	0
$183$	13.3028	0.983369
$184$	1.60555	0.118363
$185$	0	0
$186$	−1.81665	−0.133204
$187$	−34.0278	−2.48836
$188$	−6.69722	−0.488445
$189$	−24.1194	−1.75443
$190$	0	0
$191$	10.8167	0.782666	0.391333	−	0.920249i	$-0.372014\pi$
$191$	10.8167	0.782666	0.391333	+	0.920249i	$0.372014\pi$
$192$	−1.30278	−0.0940197
$193$	0.513878	0.0369898	0.0184949	−	0.999829i	$-0.494113\pi$
$193$	0.513878	0.0369898	0.0184949	+	0.999829i	$0.494113\pi$
$194$	−15.3028	−1.09868
$195$	0	0
$196$	11.5139	0.822420
$197$	18.6972	1.33212	0.666061	−	0.745897i	$-0.267979\pi$
$197$	18.6972	1.33212	0.666061	+	0.745897i	$0.267979\pi$
$198$	−5.60555	−0.398369
$199$	−6.90833	−0.489718	−0.244859	−	0.969559i	$-0.578742\pi$
$199$	−6.90833	−0.489718	−0.244859	+	0.969559i	$0.578742\pi$
$200$	0	0
$201$	−2.09167	−0.147535
$202$	−8.21110	−0.577731
$203$	1.30278	0.0914369
$204$	−10.3028	−0.721339
$205$	0	0
$206$	13.6056	0.947944
$207$	2.09167	0.145381
$208$	0	0
$209$	32.7250	2.26363
$210$	0	0
$211$	14.8167	1.02002	0.510010	−	0.860168i	$-0.329642\pi$
$211$	14.8167	1.02002	0.510010	+	0.860168i	$0.329642\pi$
$212$	−2.21110	−0.151859
$213$	4.42221	0.303005
$214$	−7.81665	−0.534335
$215$	0	0
$216$	−5.60555	−0.381409
$217$	6.00000	0.407307
$218$	−4.39445	−0.297630
$219$	−10.4222	−0.704267
$220$	0	0
$221$	0	0
$222$	9.51388	0.638530
$223$	−5.18335	−0.347103	−0.173551	−	0.984825i	$-0.555524\pi$
$223$	−5.18335	−0.347103	−0.173551	+	0.984825i	$0.555524\pi$
$224$	4.30278	0.287491
$225$	0	0
$226$	7.51388	0.499816
$227$	19.0000	1.26107	0.630537	−	0.776159i	$-0.282835\pi$
$227$	19.0000	1.26107	0.630537	+	0.776159i	$0.282835\pi$
$228$	9.90833	0.656195
$229$	−10.5139	−0.694777	−0.347388	−	0.937721i	$-0.612931\pi$
$229$	−10.5139	−0.694777	−0.347388	+	0.937721i	$0.612931\pi$
$230$	0	0
$231$	−24.1194	−1.58694
$232$	0.302776	0.0198782
$233$	−21.0000	−1.37576	−0.687878	−	0.725826i	$-0.741458\pi$
$233$	−21.0000	−1.37576	−0.687878	+	0.725826i	$0.741458\pi$
$234$	0	0
$235$	0	0
$236$	4.21110	0.274119
$237$	−7.42221	−0.482124
$238$	34.0278	2.20569
$239$	−29.4222	−1.90316	−0.951582	−	0.307395i	$-0.900543\pi$
$239$	−29.4222	−1.90316	−0.951582	+	0.307395i	$0.900543\pi$
$240$	0	0
$241$	5.00000	0.322078	0.161039	−	0.986948i	$-0.448515\pi$
$241$	5.00000	0.322078	0.161039	+	0.986948i	$0.448515\pi$
$242$	−7.51388	−0.483010
$243$	−12.3944	−0.795104
$244$	−10.2111	−0.653699
$245$	0	0
$246$	1.30278	0.0830619
$247$	0	0
$248$	1.39445	0.0885476
$249$	−6.27502	−0.397663
$250$	0	0
$251$	22.3305	1.40949	0.704745	−	0.709460i	$-0.251061\pi$
$251$	22.3305	1.40949	0.704745	+	0.709460i	$0.251061\pi$
$252$	5.60555	0.353117
$253$	6.90833	0.434323
$254$	−5.90833	−0.370721
$255$	0	0
$256$	1.00000	0.0625000
$257$	−1.60555	−0.100152	−0.0500758	−	0.998745i	$-0.515946\pi$
$257$	−1.60555	−0.100152	−0.0500758	+	0.998745i	$0.515946\pi$
$258$	12.3944	0.771645
$259$	−31.4222	−1.95248
$260$	0	0
$261$	0.394449	0.0244158
$262$	12.8167	0.791816
$263$	−13.7250	−0.846319	−0.423159	−	0.906055i	$-0.639079\pi$
$263$	−13.7250	−0.846319	−0.423159	+	0.906055i	$0.639079\pi$
$264$	−5.60555	−0.344998
$265$	0	0
$266$	−32.7250	−2.00650
$267$	−14.2111	−0.869705
$268$	1.60555	0.0980747
$269$	−7.18335	−0.437976	−0.218988	−	0.975728i	$-0.570276\pi$
$269$	−7.18335	−0.437976	−0.218988	+	0.975728i	$0.570276\pi$
$270$	0	0
$271$	−26.4222	−1.60503	−0.802517	−	0.596629i	$-0.796506\pi$
$271$	−26.4222	−1.60503	−0.802517	+	0.596629i	$0.796506\pi$
$272$	7.90833	0.479513
$273$	0	0
$274$	0	0
$275$	0	0
$276$	2.09167	0.125904
$277$	−6.88057	−0.413413	−0.206707	−	0.978403i	$-0.566275\pi$
$277$	−6.88057	−0.413413	−0.206707	+	0.978403i	$0.566275\pi$
$278$	−11.8167	−0.708716
$279$	1.81665	0.108760
$280$	0	0
$281$	9.69722	0.578488	0.289244	−	0.957255i	$-0.406596\pi$
$281$	9.69722	0.578488	0.289244	+	0.957255i	$0.406596\pi$
$282$	−8.72498	−0.519565
$283$	4.48612	0.266672	0.133336	−	0.991071i	$-0.457431\pi$
$283$	4.48612	0.266672	0.133336	+	0.991071i	$0.457431\pi$
$284$	−3.39445	−0.201423
$285$	0	0
$286$	0	0
$287$	−4.30278	−0.253985
$288$	1.30278	0.0767668
$289$	45.5416	2.67892
$290$	0	0
$291$	−19.9361	−1.16867
$292$	8.00000	0.468165
$293$	−15.5139	−0.906330	−0.453165	−	0.891427i	$-0.649705\pi$
$293$	−15.5139	−0.906330	−0.453165	+	0.891427i	$0.649705\pi$
$294$	15.0000	0.874818
$295$	0	0
$296$	−7.30278	−0.424465
$297$	−24.1194	−1.39955
$298$	4.69722	0.272103
$299$	0	0
$300$	0	0
$301$	−40.9361	−2.35952
$302$	−13.1194	−0.754938
$303$	−10.6972	−0.614539
$304$	−7.60555	−0.436208
$305$	0	0
$306$	10.3028	0.588970
$307$	−3.21110	−0.183267	−0.0916337	−	0.995793i	$-0.529209\pi$
$307$	−3.21110	−0.183267	−0.0916337	+	0.995793i	$0.529209\pi$
$308$	18.5139	1.05493
$309$	17.7250	1.00834
$310$	0	0
$311$	0.908327	0.0515065	0.0257532	−	0.999668i	$-0.491802\pi$
$311$	0.908327	0.0515065	0.0257532	+	0.999668i	$0.491802\pi$
$312$	0	0
$313$	−5.09167	−0.287798	−0.143899	−	0.989592i	$-0.545964\pi$
$313$	−5.09167	−0.287798	−0.143899	+	0.989592i	$0.545964\pi$
$314$	4.42221	0.249559
$315$	0	0
$316$	5.69722	0.320494
$317$	29.5139	1.65766	0.828832	−	0.559497i	$-0.189006\pi$
$317$	29.5139	1.65766	0.828832	+	0.559497i	$0.189006\pi$
$318$	−2.88057	−0.161534
$319$	1.30278	0.0729414
$320$	0	0
$321$	−10.1833	−0.568379
$322$	−6.90833	−0.384986
$323$	−60.1472	−3.34668
$324$	−3.39445	−0.188580
$325$	0	0
$326$	8.90833	0.493387
$327$	−5.72498	−0.316592
$328$	−1.00000	−0.0552158
$329$	28.8167	1.58871
$330$	0	0
$331$	−10.0917	−0.554689	−0.277344	−	0.960771i	$-0.589454\pi$
$331$	−10.0917	−0.554689	−0.277344	+	0.960771i	$0.589454\pi$
$332$	4.81665	0.264348
$333$	−9.51388	−0.521357
$334$	3.60555	0.197287
$335$	0	0
$336$	5.60555	0.305808
$337$	−22.0278	−1.19993	−0.599964	−	0.800027i	$-0.704819\pi$
$337$	−22.0278	−1.19993	−0.599964	+	0.800027i	$0.704819\pi$
$338$	0	0
$339$	9.78890	0.531660
$340$	0	0
$341$	6.00000	0.324918
$342$	−9.90833	−0.535781
$343$	−19.4222	−1.04870
$344$	−9.51388	−0.512954
$345$	0	0
$346$	24.5139	1.31787
$347$	12.2111	0.655526	0.327763	−	0.944760i	$-0.393705\pi$
$347$	12.2111	0.655526	0.327763	+	0.944760i	$0.393705\pi$
$348$	0.394449	0.0211447
$349$	−1.27502	−0.0682502	−0.0341251	−	0.999418i	$-0.510864\pi$
$349$	−1.27502	−0.0682502	−0.0341251	+	0.999418i	$0.510864\pi$
$350$	0	0
$351$	0	0
$352$	4.30278	0.229339
$353$	2.48612	0.132323	0.0661615	−	0.997809i	$-0.478925\pi$
$353$	2.48612	0.132323	0.0661615	+	0.997809i	$0.478925\pi$
$354$	5.48612	0.291584
$355$	0	0
$356$	10.9083	0.578140
$357$	44.3305	2.34622
$358$	7.21110	0.381119
$359$	23.9361	1.26330	0.631649	−	0.775254i	$-0.282378\pi$
$359$	23.9361	1.26330	0.631649	+	0.775254i	$0.282378\pi$
$360$	0	0
$361$	38.8444	2.04444
$362$	7.09167	0.372730
$363$	−9.78890	−0.513784
$364$	0	0
$365$	0	0
$366$	−13.3028	−0.695347
$367$	−4.18335	−0.218369	−0.109184	−	0.994022i	$-0.534824\pi$
$367$	−4.18335	−0.218369	−0.109184	+	0.994022i	$0.534824\pi$
$368$	−1.60555	−0.0836951
$369$	−1.30278	−0.0678198
$370$	0	0
$371$	9.51388	0.493936
$372$	1.81665	0.0941891
$373$	−0.788897	−0.0408476	−0.0204238	−	0.999791i	$-0.506502\pi$
$373$	−0.788897	−0.0408476	−0.0204238	+	0.999791i	$0.506502\pi$
$374$	34.0278	1.75953
$375$	0	0
$376$	6.69722	0.345383
$377$	0	0
$378$	24.1194	1.24057
$379$	−13.1833	−0.677183	−0.338592	−	0.940933i	$-0.609950\pi$
$379$	−13.1833	−0.677183	−0.338592	+	0.940933i	$0.609950\pi$
$380$	0	0
$381$	−7.69722	−0.394341
$382$	−10.8167	−0.553428
$383$	4.39445	0.224546	0.112273	−	0.993677i	$-0.464187\pi$
$383$	4.39445	0.224546	0.112273	+	0.993677i	$0.464187\pi$
$384$	1.30278	0.0664820
$385$	0	0
$386$	−0.513878	−0.0261557
$387$	−12.3944	−0.630045
$388$	15.3028	0.776881
$389$	5.21110	0.264213	0.132107	−	0.991236i	$-0.457826\pi$
$389$	5.21110	0.264213	0.132107	+	0.991236i	$0.457826\pi$
$390$	0	0
$391$	−12.6972	−0.642126
$392$	−11.5139	−0.581539
$393$	16.6972	0.842264
$394$	−18.6972	−0.941953
$395$	0	0
$396$	5.60555	0.281690
$397$	29.3305	1.47206	0.736029	−	0.676950i	$-0.236699\pi$
$397$	29.3305	1.47206	0.736029	+	0.676950i	$0.236699\pi$
$398$	6.90833	0.346283
$399$	−42.6333	−2.13433
$400$	0	0
$401$	−17.7250	−0.885143	−0.442572	−	0.896733i	$-0.645934\pi$
$401$	−17.7250	−0.885143	−0.442572	+	0.896733i	$0.645934\pi$
$402$	2.09167	0.104323
$403$	0	0
$404$	8.21110	0.408518
$405$	0	0
$406$	−1.30278	−0.0646557
$407$	−31.4222	−1.55754
$408$	10.3028	0.510063
$409$	−19.7889	−0.978498	−0.489249	−	0.872144i	$-0.662729\pi$
$409$	−19.7889	−0.978498	−0.489249	+	0.872144i	$0.662729\pi$
$410$	0	0
$411$	0	0
$412$	−13.6056	−0.670297
$413$	−18.1194	−0.891599
$414$	−2.09167	−0.102800
$415$	0	0
$416$	0	0
$417$	−15.3944	−0.753869
$418$	−32.7250	−1.60063
$419$	13.6972	0.669153	0.334577	−	0.942369i	$-0.391407\pi$
$419$	13.6972	0.669153	0.334577	+	0.942369i	$0.391407\pi$
$420$	0	0
$421$	−19.0917	−0.930471	−0.465236	−	0.885187i	$-0.654030\pi$
$421$	−19.0917	−0.930471	−0.465236	+	0.885187i	$0.654030\pi$
$422$	−14.8167	−0.721263
$423$	8.72498	0.424223
$424$	2.21110	0.107381
$425$	0	0
$426$	−4.42221	−0.214257
$427$	43.9361	2.12622
$428$	7.81665	0.377832
$429$	0	0
$430$	0	0
$431$	−9.90833	−0.477267	−0.238634	−	0.971110i	$-0.576700\pi$
$431$	−9.90833	−0.477267	−0.238634	+	0.971110i	$0.576700\pi$
$432$	5.60555	0.269697
$433$	15.1833	0.729665	0.364833	−	0.931073i	$-0.381126\pi$
$433$	15.1833	0.729665	0.364833	+	0.931073i	$0.381126\pi$
$434$	−6.00000	−0.288009
$435$	0	0
$436$	4.39445	0.210456
$437$	12.2111	0.584136
$438$	10.4222	0.497992
$439$	−25.0278	−1.19451	−0.597255	−	0.802052i	$-0.703742\pi$
$439$	−25.0278	−1.19451	−0.597255	+	0.802052i	$0.703742\pi$
$440$	0	0
$441$	−15.0000	−0.714286
$442$	0	0
$443$	33.1194	1.57355	0.786776	−	0.617239i	$-0.211749\pi$
$443$	33.1194	1.57355	0.786776	+	0.617239i	$0.211749\pi$
$444$	−9.51388	−0.451509
$445$	0	0
$446$	5.18335	0.245439
$447$	6.11943	0.289439
$448$	−4.30278	−0.203287
$449$	26.0917	1.23134	0.615671	−	0.788003i	$-0.288885\pi$
$449$	26.0917	1.23134	0.615671	+	0.788003i	$0.288885\pi$
$450$	0	0
$451$	−4.30278	−0.202610
$452$	−7.51388	−0.353423
$453$	−17.0917	−0.803037
$454$	−19.0000	−0.891714
$455$	0	0
$456$	−9.90833	−0.464000
$457$	−7.51388	−0.351484	−0.175742	−	0.984436i	$-0.556233\pi$
$457$	−7.51388	−0.351484	−0.175742	+	0.984436i	$0.556233\pi$
$458$	10.5139	0.491281
$459$	44.3305	2.06917
$460$	0	0
$461$	26.5139	1.23487	0.617437	−	0.786620i	$-0.288171\pi$
$461$	26.5139	1.23487	0.617437	+	0.786620i	$0.288171\pi$
$462$	24.1194	1.12214
$463$	28.5139	1.32515	0.662576	−	0.748995i	$-0.269463\pi$
$463$	28.5139	1.32515	0.662576	+	0.748995i	$0.269463\pi$
$464$	−0.302776	−0.0140560
$465$	0	0
$466$	21.0000	0.972806
$467$	34.5139	1.59711	0.798556	−	0.601921i	$-0.205598\pi$
$467$	34.5139	1.59711	0.798556	+	0.601921i	$0.205598\pi$
$468$	0	0
$469$	−6.90833	−0.318997
$470$	0	0
$471$	5.76114	0.265459
$472$	−4.21110	−0.193832
$473$	−40.9361	−1.88224
$474$	7.42221	0.340913
$475$	0	0
$476$	−34.0278	−1.55966
$477$	2.88057	0.131892
$478$	29.4222	1.34574
$479$	−0.724981	−0.0331252	−0.0165626	−	0.999863i	$-0.505272\pi$
$479$	−0.724981	−0.0331252	−0.0165626	+	0.999863i	$0.505272\pi$
$480$	0	0
$481$	0	0
$482$	−5.00000	−0.227744
$483$	−9.00000	−0.409514
$484$	7.51388	0.341540
$485$	0	0
$486$	12.3944	0.562224
$487$	3.30278	0.149663	0.0748315	−	0.997196i	$-0.476158\pi$
$487$	3.30278	0.149663	0.0748315	+	0.997196i	$0.476158\pi$
$488$	10.2111	0.462235
$489$	11.6056	0.524821
$490$	0	0
$491$	−8.57779	−0.387110	−0.193555	−	0.981089i	$-0.562002\pi$
$491$	−8.57779	−0.387110	−0.193555	+	0.981089i	$0.562002\pi$
$492$	−1.30278	−0.0587337
$493$	−2.39445	−0.107841
$494$	0	0
$495$	0	0
$496$	−1.39445	−0.0626126
$497$	14.6056	0.655149
$498$	6.27502	0.281190
$499$	14.6333	0.655077	0.327538	−	0.944838i	$-0.393781\pi$
$499$	14.6333	0.655077	0.327538	+	0.944838i	$0.393781\pi$
$500$	0	0
$501$	4.69722	0.209857
$502$	−22.3305	−0.996660
$503$	−17.3944	−0.775580	−0.387790	−	0.921748i	$-0.626761\pi$
$503$	−17.3944	−0.775580	−0.387790	+	0.921748i	$0.626761\pi$
$504$	−5.60555	−0.249691
$505$	0	0
$506$	−6.90833	−0.307113
$507$	0	0
$508$	5.90833	0.262140
$509$	−25.8167	−1.14430	−0.572152	−	0.820148i	$-0.693891\pi$
$509$	−25.8167	−1.14430	−0.572152	+	0.820148i	$0.693891\pi$
$510$	0	0
$511$	−34.4222	−1.52275
$512$	−1.00000	−0.0441942
$513$	−42.6333	−1.88231
$514$	1.60555	0.0708178
$515$	0	0
$516$	−12.3944	−0.545635
$517$	28.8167	1.26735
$518$	31.4222	1.38061
$519$	31.9361	1.40184
$520$	0	0
$521$	34.8444	1.52656	0.763281	−	0.646067i	$-0.223587\pi$
$521$	34.8444	1.52656	0.763281	+	0.646067i	$0.223587\pi$
$522$	−0.394449	−0.0172646
$523$	8.78890	0.384312	0.192156	−	0.981364i	$-0.438452\pi$
$523$	8.78890	0.384312	0.192156	+	0.981364i	$0.438452\pi$
$524$	−12.8167	−0.559898
$525$	0	0
$526$	13.7250	0.598438
$527$	−11.0278	−0.480377
$528$	5.60555	0.243950
$529$	−20.4222	−0.887922
$530$	0	0
$531$	−5.48612	−0.238077
$532$	32.7250	1.41881
$533$	0	0
$534$	14.2111	0.614975
$535$	0	0
$536$	−1.60555	−0.0693493
$537$	9.39445	0.405400
$538$	7.18335	0.309696
$539$	−49.5416	−2.13391
$540$	0	0
$541$	−3.81665	−0.164091	−0.0820454	−	0.996629i	$-0.526145\pi$
$541$	−3.81665	−0.164091	−0.0820454	+	0.996629i	$0.526145\pi$
$542$	26.4222	1.13493
$543$	9.23886	0.396477
$544$	−7.90833	−0.339067
$545$	0	0
$546$	0	0
$547$	−14.3944	−0.615462	−0.307731	−	0.951473i	$-0.599570\pi$
$547$	−14.3944	−0.615462	−0.307731	+	0.951473i	$0.599570\pi$
$548$	0	0
$549$	13.3028	0.567749
$550$	0	0
$551$	2.30278	0.0981015
$552$	−2.09167	−0.0890275
$553$	−24.5139	−1.04244
$554$	6.88057	0.292327
$555$	0	0
$556$	11.8167	0.501138
$557$	11.8167	0.500688	0.250344	−	0.968157i	$-0.419456\pi$
$557$	11.8167	0.500688	0.250344	+	0.968157i	$0.419456\pi$
$558$	−1.81665	−0.0769051
$559$	0	0
$560$	0	0
$561$	44.3305	1.87164
$562$	−9.69722	−0.409053
$563$	−11.7250	−0.494149	−0.247075	−	0.968996i	$-0.579469\pi$
$563$	−11.7250	−0.494149	−0.247075	+	0.968996i	$0.579469\pi$
$564$	8.72498	0.367388
$565$	0	0
$566$	−4.48612	−0.188566
$567$	14.6056	0.613375
$568$	3.39445	0.142428
$569$	−33.1472	−1.38960	−0.694801	−	0.719202i	$-0.744508\pi$
$569$	−33.1472	−1.38960	−0.694801	+	0.719202i	$0.744508\pi$
$570$	0	0
$571$	−4.51388	−0.188900	−0.0944500	−	0.995530i	$-0.530109\pi$
$571$	−4.51388	−0.188900	−0.0944500	+	0.995530i	$0.530109\pi$
$572$	0	0
$573$	−14.0917	−0.588688
$574$	4.30278	0.179594
$575$	0	0
$576$	−1.30278	−0.0542823
$577$	16.3944	0.682510	0.341255	−	0.939971i	$-0.389148\pi$
$577$	16.3944	0.682510	0.341255	+	0.939971i	$0.389148\pi$
$578$	−45.5416	−1.89428
$579$	−0.669468	−0.0278221
$580$	0	0
$581$	−20.7250	−0.859817
$582$	19.9361	0.826377
$583$	9.51388	0.394025
$584$	−8.00000	−0.331042
$585$	0	0
$586$	15.5139	0.640872
$587$	32.5139	1.34199	0.670996	−	0.741461i	$-0.265867\pi$
$587$	32.5139	1.34199	0.670996	+	0.741461i	$0.265867\pi$
$588$	−15.0000	−0.618590
$589$	10.6056	0.436994
$590$	0	0
$591$	−24.3583	−1.00197
$592$	7.30278	0.300142
$593$	−36.6611	−1.50549	−0.752745	−	0.658313i	$-0.771271\pi$
$593$	−36.6611	−1.50549	−0.752745	+	0.658313i	$0.771271\pi$
$594$	24.1194	0.989632
$595$	0	0
$596$	−4.69722	−0.192406
$597$	9.00000	0.368345
$598$	0	0
$599$	−41.6611	−1.70222	−0.851112	−	0.524983i	$-0.824072\pi$
$599$	−41.6611	−1.70222	−0.851112	+	0.524983i	$0.824072\pi$
$600$	0	0
$601$	−15.6056	−0.636564	−0.318282	−	0.947996i	$-0.603106\pi$
$601$	−15.6056	−0.636564	−0.318282	+	0.947996i	$0.603106\pi$
$602$	40.9361	1.66843
$603$	−2.09167	−0.0851795
$604$	13.1194	0.533822
$605$	0	0
$606$	10.6972	0.434545
$607$	41.0555	1.66639	0.833196	−	0.552978i	$-0.186509\pi$
$607$	41.0555	1.66639	0.833196	+	0.552978i	$0.186509\pi$
$608$	7.60555	0.308446
$609$	−1.69722	−0.0687750
$610$	0	0
$611$	0	0
$612$	−10.3028	−0.416465
$613$	−43.4500	−1.75493	−0.877464	−	0.479643i	$-0.840766\pi$
$613$	−43.4500	−1.75493	−0.877464	+	0.479643i	$0.840766\pi$
$614$	3.21110	0.129590
$615$	0	0
$616$	−18.5139	−0.745945
$617$	−49.0555	−1.97490	−0.987450	−	0.157930i	$-0.949518\pi$
$617$	−49.0555	−1.97490	−0.987450	+	0.157930i	$0.949518\pi$
$618$	−17.7250	−0.713003
$619$	−32.3305	−1.29947	−0.649737	−	0.760159i	$-0.725121\pi$
$619$	−32.3305	−1.29947	−0.649737	+	0.760159i	$0.725121\pi$
$620$	0	0
$621$	−9.00000	−0.361158
$622$	−0.908327	−0.0364206
$623$	−46.9361	−1.88045
$624$	0	0
$625$	0	0
$626$	5.09167	0.203504
$627$	−42.6333	−1.70261
$628$	−4.42221	−0.176465
$629$	57.7527	2.30275
$630$	0	0
$631$	−15.9083	−0.633300	−0.316650	−	0.948542i	$-0.602558\pi$
$631$	−15.9083	−0.633300	−0.316650	+	0.948542i	$0.602558\pi$
$632$	−5.69722	−0.226623
$633$	−19.3028	−0.767216
$634$	−29.5139	−1.17215
$635$	0	0
$636$	2.88057	0.114222
$637$	0	0
$638$	−1.30278	−0.0515774
$639$	4.42221	0.174940
$640$	0	0
$641$	−50.1749	−1.98179	−0.990896	−	0.134633i	$-0.957014\pi$
$641$	−50.1749	−1.98179	−0.990896	+	0.134633i	$0.957014\pi$
$642$	10.1833	0.401905
$643$	6.57779	0.259403	0.129701	−	0.991553i	$-0.458598\pi$
$643$	6.57779	0.259403	0.129701	+	0.991553i	$0.458598\pi$
$644$	6.90833	0.272226
$645$	0	0
$646$	60.1472	2.36646
$647$	−12.6972	−0.499179	−0.249590	−	0.968352i	$-0.580296\pi$
$647$	−12.6972	−0.499179	−0.249590	+	0.968352i	$0.580296\pi$
$648$	3.39445	0.133347
$649$	−18.1194	−0.711250
$650$	0	0
$651$	−7.81665	−0.306359
$652$	−8.90833	−0.348877
$653$	−6.21110	−0.243059	−0.121530	−	0.992588i	$-0.538780\pi$
$653$	−6.21110	−0.243059	−0.121530	+	0.992588i	$0.538780\pi$
$654$	5.72498	0.223864
$655$	0	0
$656$	1.00000	0.0390434
$657$	−10.4222	−0.406609
$658$	−28.8167	−1.12339
$659$	5.21110	0.202996	0.101498	−	0.994836i	$-0.467636\pi$
$659$	5.21110	0.202996	0.101498	+	0.994836i	$0.467636\pi$
$660$	0	0
$661$	−24.6972	−0.960611	−0.480305	−	0.877101i	$-0.659474\pi$
$661$	−24.6972	−0.960611	−0.480305	+	0.877101i	$0.659474\pi$
$662$	10.0917	0.392224
$663$	0	0
$664$	−4.81665	−0.186922
$665$	0	0
$666$	9.51388	0.368655
$667$	0.486122	0.0188227
$668$	−3.60555	−0.139503
$669$	6.75274	0.261076
$670$	0	0
$671$	43.9361	1.69613
$672$	−5.60555	−0.216239
$673$	21.0278	0.810560	0.405280	−	0.914193i	$-0.367174\pi$
$673$	21.0278	0.810560	0.405280	+	0.914193i	$0.367174\pi$
$674$	22.0278	0.848477
$675$	0	0
$676$	0	0
$677$	21.4222	0.823322	0.411661	−	0.911337i	$-0.364949\pi$
$677$	21.4222	0.823322	0.411661	+	0.911337i	$0.364949\pi$
$678$	−9.78890	−0.375940
$679$	−65.8444	−2.52688
$680$	0	0
$681$	−24.7527	−0.948527
$682$	−6.00000	−0.229752
$683$	26.6333	1.01910	0.509548	−	0.860442i	$-0.329813\pi$
$683$	26.6333	1.01910	0.509548	+	0.860442i	$0.329813\pi$
$684$	9.90833	0.378854
$685$	0	0
$686$	19.4222	0.741543
$687$	13.6972	0.522582
$688$	9.51388	0.362713
$689$	0	0
$690$	0	0
$691$	−30.6056	−1.16429	−0.582145	−	0.813085i	$-0.697786\pi$
$691$	−30.6056	−1.16429	−0.582145	+	0.813085i	$0.697786\pi$
$692$	−24.5139	−0.931878
$693$	−24.1194	−0.916221
$694$	−12.2111	−0.463527
$695$	0	0
$696$	−0.394449	−0.0149515
$697$	7.90833	0.299549
$698$	1.27502	0.0482602
$699$	27.3583	1.03479
$700$	0	0
$701$	−13.9083	−0.525310	−0.262655	−	0.964890i	$-0.584598\pi$
$701$	−13.9083	−0.525310	−0.262655	+	0.964890i	$0.584598\pi$
$702$	0	0
$703$	−55.5416	−2.09479
$704$	−4.30278	−0.162167
$705$	0	0
$706$	−2.48612	−0.0935664
$707$	−35.3305	−1.32874
$708$	−5.48612	−0.206181
$709$	−47.5416	−1.78546	−0.892732	−	0.450588i	$-0.851214\pi$
$709$	−47.5416	−1.78546	−0.892732	+	0.450588i	$0.851214\pi$
$710$	0	0
$711$	−7.42221	−0.278354
$712$	−10.9083	−0.408807
$713$	2.23886	0.0838459
$714$	−44.3305	−1.65903
$715$	0	0
$716$	−7.21110	−0.269492
$717$	38.3305	1.43148
$718$	−23.9361	−0.893287
$719$	38.9361	1.45207	0.726035	−	0.687657i	$-0.241361\pi$
$719$	38.9361	1.45207	0.726035	+	0.687657i	$0.241361\pi$
$720$	0	0
$721$	58.5416	2.18020
$722$	−38.8444	−1.44564
$723$	−6.51388	−0.242254
$724$	−7.09167	−0.263560
$725$	0	0
$726$	9.78890	0.363300
$727$	30.0555	1.11470	0.557349	−	0.830279i	$-0.311819\pi$
$727$	30.0555	1.11470	0.557349	+	0.830279i	$0.311819\pi$
$728$	0	0
$729$	26.3305	0.975205
$730$	0	0
$731$	75.2389	2.78281
$732$	13.3028	0.491685
$733$	−12.5416	−0.463236	−0.231618	−	0.972807i	$-0.574402\pi$
$733$	−12.5416	−0.463236	−0.231618	+	0.972807i	$0.574402\pi$
$734$	4.18335	0.154410
$735$	0	0
$736$	1.60555	0.0591814
$737$	−6.90833	−0.254471
$738$	1.30278	0.0479558
$739$	17.7250	0.652024	0.326012	−	0.945366i	$-0.394295\pi$
$739$	17.7250	0.652024	0.326012	+	0.945366i	$0.394295\pi$
$740$	0	0
$741$	0	0
$742$	−9.51388	−0.349265
$743$	20.7889	0.762671	0.381335	−	0.924437i	$-0.375464\pi$
$743$	20.7889	0.762671	0.381335	+	0.924437i	$0.375464\pi$
$744$	−1.81665	−0.0666018
$745$	0	0
$746$	0.788897	0.0288836
$747$	−6.27502	−0.229591
$748$	−34.0278	−1.24418
$749$	−33.6333	−1.22893
$750$	0	0
$751$	−10.6333	−0.388015	−0.194007	−	0.981000i	$-0.562149\pi$
$751$	−10.6333	−0.388015	−0.194007	+	0.981000i	$0.562149\pi$
$752$	−6.69722	−0.244223
$753$	−29.0917	−1.06016
$754$	0	0
$755$	0	0
$756$	−24.1194	−0.877215
$757$	−24.2111	−0.879967	−0.439984	−	0.898006i	$-0.645016\pi$
$757$	−24.2111	−0.879967	−0.439984	+	0.898006i	$0.645016\pi$
$758$	13.1833	0.478841
$759$	−9.00000	−0.326679
$760$	0	0
$761$	−12.3944	−0.449298	−0.224649	−	0.974440i	$-0.572124\pi$
$761$	−12.3944	−0.449298	−0.224649	+	0.974440i	$0.572124\pi$
$762$	7.69722	0.278841
$763$	−18.9083	−0.684527
$764$	10.8167	0.391333
$765$	0	0
$766$	−4.39445	−0.158778
$767$	0	0
$768$	−1.30278	−0.0470099
$769$	−43.4500	−1.56685	−0.783423	−	0.621489i	$-0.786528\pi$
$769$	−43.4500	−1.56685	−0.783423	+	0.621489i	$0.786528\pi$
$770$	0	0
$771$	2.09167	0.0753298
$772$	0.513878	0.0184949
$773$	16.8444	0.605851	0.302926	−	0.953014i	$-0.402037\pi$
$773$	16.8444	0.605851	0.302926	+	0.953014i	$0.402037\pi$
$774$	12.3944	0.445509
$775$	0	0
$776$	−15.3028	−0.549338
$777$	40.9361	1.46857
$778$	−5.21110	−0.186827
$779$	−7.60555	−0.272497
$780$	0	0
$781$	14.6056	0.522628
$782$	12.6972	0.454052
$783$	−1.69722	−0.0606539
$784$	11.5139	0.411210
$785$	0	0
$786$	−16.6972	−0.595570
$787$	14.6972	0.523899	0.261950	−	0.965082i	$-0.415635\pi$
$787$	14.6972	0.523899	0.261950	+	0.965082i	$0.415635\pi$
$788$	18.6972	0.666061
$789$	17.8806	0.636565
$790$	0	0
$791$	32.3305	1.14954
$792$	−5.60555	−0.199185
$793$	0	0
$794$	−29.3305	−1.04090
$795$	0	0
$796$	−6.90833	−0.244859
$797$	1.00000	0.0354218	0.0177109	−	0.999843i	$-0.494362\pi$
$797$	1.00000	0.0354218	0.0177109	+	0.999843i	$0.494362\pi$
$798$	42.6333	1.50920
$799$	−52.9638	−1.87373
$800$	0	0
$801$	−14.2111	−0.502125
$802$	17.7250	0.625891
$803$	−34.4222	−1.21473
$804$	−2.09167	−0.0737676
$805$	0	0
$806$	0	0
$807$	9.35829	0.329427
$808$	−8.21110	−0.288866
$809$	29.6333	1.04185	0.520926	−	0.853602i	$-0.325587\pi$
$809$	29.6333	1.04185	0.520926	+	0.853602i	$0.325587\pi$
$810$	0	0
$811$	−25.3028	−0.888501	−0.444250	−	0.895903i	$-0.646530\pi$
$811$	−25.3028	−0.888501	−0.444250	+	0.895903i	$0.646530\pi$
$812$	1.30278	0.0457185
$813$	34.4222	1.20724
$814$	31.4222	1.10135
$815$	0	0
$816$	−10.3028	−0.360669
$817$	−72.3583	−2.53150
$818$	19.7889	0.691903
$819$	0	0
$820$	0	0
$821$	10.5778	0.369168	0.184584	−	0.982817i	$-0.440906\pi$
$821$	10.5778	0.369168	0.184584	+	0.982817i	$0.440906\pi$
$822$	0	0
$823$	−33.8167	−1.17877	−0.589387	−	0.807851i	$-0.700631\pi$
$823$	−33.8167	−1.17877	−0.589387	+	0.807851i	$0.700631\pi$
$824$	13.6056	0.473972
$825$	0	0
$826$	18.1194	0.630456
$827$	−14.4500	−0.502474	−0.251237	−	0.967926i	$-0.580837\pi$
$827$	−14.4500	−0.502474	−0.251237	+	0.967926i	$0.580837\pi$
$828$	2.09167	0.0726907
$829$	28.0917	0.975664	0.487832	−	0.872937i	$-0.337788\pi$
$829$	28.0917	0.975664	0.487832	+	0.872937i	$0.337788\pi$
$830$	0	0
$831$	8.96384	0.310952
$832$	0	0
$833$	91.0555	3.15489
$834$	15.3944	0.533066
$835$	0	0
$836$	32.7250	1.13182
$837$	−7.81665	−0.270183
$838$	−13.6972	−0.473163
$839$	−33.9361	−1.17160	−0.585802	−	0.810454i	$-0.699220\pi$
$839$	−33.9361	−1.17160	−0.585802	+	0.810454i	$0.699220\pi$
$840$	0	0
$841$	−28.9083	−0.996839
$842$	19.0917	0.657943
$843$	−12.6333	−0.435114
$844$	14.8167	0.510010
$845$	0	0
$846$	−8.72498	−0.299971
$847$	−32.3305	−1.11089
$848$	−2.21110	−0.0759296
$849$	−5.84441	−0.200580
$850$	0	0
$851$	−11.7250	−0.401927
$852$	4.42221	0.151502
$853$	40.7250	1.39440	0.697198	−	0.716878i	$-0.254430\pi$
$853$	40.7250	1.39440	0.697198	+	0.716878i	$0.254430\pi$
$854$	−43.9361	−1.50346
$855$	0	0
$856$	−7.81665	−0.267168
$857$	−49.4222	−1.68823	−0.844115	−	0.536162i	$-0.819874\pi$
$857$	−49.4222	−1.68823	−0.844115	+	0.536162i	$0.819874\pi$
$858$	0	0
$859$	−10.1194	−0.345270	−0.172635	−	0.984986i	$-0.555228\pi$
$859$	−10.1194	−0.345270	−0.172635	+	0.984986i	$0.555228\pi$
$860$	0	0
$861$	5.60555	0.191037
$862$	9.90833	0.337479
$863$	−32.0278	−1.09024	−0.545119	−	0.838359i	$-0.683515\pi$
$863$	−32.0278	−1.09024	−0.545119	+	0.838359i	$0.683515\pi$
$864$	−5.60555	−0.190705
$865$	0	0
$866$	−15.1833	−0.515951
$867$	−59.3305	−2.01497
$868$	6.00000	0.203653
$869$	−24.5139	−0.831576
$870$	0	0
$871$	0	0
$872$	−4.39445	−0.148815
$873$	−19.9361	−0.674734
$874$	−12.2111	−0.413047
$875$	0	0
$876$	−10.4222	−0.352134
$877$	−53.4777	−1.80581	−0.902907	−	0.429836i	$-0.858571\pi$
$877$	−53.4777	−1.80581	−0.902907	+	0.429836i	$0.858571\pi$
$878$	25.0278	0.844646
$879$	20.2111	0.681704
$880$	0	0
$881$	−57.5694	−1.93956	−0.969781	−	0.243977i	$-0.921548\pi$
$881$	−57.5694	−1.93956	−0.969781	+	0.243977i	$0.921548\pi$
$882$	15.0000	0.505076
$883$	23.1194	0.778031	0.389015	−	0.921231i	$-0.372815\pi$
$883$	23.1194	0.778031	0.389015	+	0.921231i	$0.372815\pi$
$884$	0	0
$885$	0	0
$886$	−33.1194	−1.11267
$887$	−9.57779	−0.321591	−0.160795	−	0.986988i	$-0.551406\pi$
$887$	−9.57779	−0.321591	−0.160795	+	0.986988i	$0.551406\pi$
$888$	9.51388	0.319265
$889$	−25.4222	−0.852633
$890$	0	0
$891$	14.6056	0.489304
$892$	−5.18335	−0.173551
$893$	50.9361	1.70451
$894$	−6.11943	−0.204664
$895$	0	0
$896$	4.30278	0.143746
$897$	0	0
$898$	−26.0917	−0.870690
$899$	0.422205	0.0140813
$900$	0	0
$901$	−17.4861	−0.582547
$902$	4.30278	0.143267
$903$	53.3305	1.77473
$904$	7.51388	0.249908
$905$	0	0
$906$	17.0917	0.567833
$907$	−33.6611	−1.11770	−0.558849	−	0.829270i	$-0.688757\pi$
$907$	−33.6611	−1.11770	−0.558849	+	0.829270i	$0.688757\pi$
$908$	19.0000	0.630537
$909$	−10.6972	−0.354805
$910$	0	0
$911$	37.4222	1.23985	0.619926	−	0.784660i	$-0.287162\pi$
$911$	37.4222	1.23985	0.619926	+	0.784660i	$0.287162\pi$
$912$	9.90833	0.328097
$913$	−20.7250	−0.685897
$914$	7.51388	0.248537
$915$	0	0
$916$	−10.5139	−0.347388
$917$	55.1472	1.82112
$918$	−44.3305	−1.46313
$919$	−22.2389	−0.733592	−0.366796	−	0.930301i	$-0.619545\pi$
$919$	−22.2389	−0.733592	−0.366796	+	0.930301i	$0.619545\pi$
$920$	0	0
$921$	4.18335	0.137846
$922$	−26.5139	−0.873188
$923$	0	0
$924$	−24.1194	−0.793471
$925$	0	0
$926$	−28.5139	−0.937024
$927$	17.7250	0.582165
$928$	0.302776	0.00993910
$929$	38.6611	1.26843	0.634214	−	0.773157i	$-0.281324\pi$
$929$	38.6611	1.26843	0.634214	+	0.773157i	$0.281324\pi$
$930$	0	0
$931$	−87.5694	−2.86997
$932$	−21.0000	−0.687878
$933$	−1.18335	−0.0387410
$934$	−34.5139	−1.12933
$935$	0	0
$936$	0	0
$937$	−1.33053	−0.0434666	−0.0217333	−	0.999764i	$-0.506918\pi$
$937$	−1.33053	−0.0434666	−0.0217333	+	0.999764i	$0.506918\pi$
$938$	6.90833	0.225565
$939$	6.63331	0.216470
$940$	0	0
$941$	47.5694	1.55072	0.775359	−	0.631521i	$-0.217569\pi$
$941$	47.5694	1.55072	0.775359	+	0.631521i	$0.217569\pi$
$942$	−5.76114	−0.187708
$943$	−1.60555	−0.0522839
$944$	4.21110	0.137060
$945$	0	0
$946$	40.9361	1.33095
$947$	−11.6695	−0.379207	−0.189603	−	0.981861i	$-0.560720\pi$
$947$	−11.6695	−0.379207	−0.189603	+	0.981861i	$0.560720\pi$
$948$	−7.42221	−0.241062
$949$	0	0
$950$	0	0
$951$	−38.4500	−1.24683
$952$	34.0278	1.10285
$953$	43.4777	1.40838	0.704191	−	0.710011i	$-0.251310\pi$
$953$	43.4777	1.40838	0.704191	+	0.710011i	$0.251310\pi$
$954$	−2.88057	−0.0932619
$955$	0	0
$956$	−29.4222	−0.951582
$957$	−1.69722	−0.0548635
$958$	0.724981	0.0234231
$959$	0	0
$960$	0	0
$961$	−29.0555	−0.937275
$962$	0	0
$963$	−10.1833	−0.328154
$964$	5.00000	0.161039
$965$	0	0
$966$	9.00000	0.289570
$967$	−4.97224	−0.159897	−0.0799483	−	0.996799i	$-0.525476\pi$
$967$	−4.97224	−0.159897	−0.0799483	+	0.996799i	$0.525476\pi$
$968$	−7.51388	−0.241505
$969$	78.3583	2.51723
$970$	0	0
$971$	46.4500	1.49065	0.745325	−	0.666701i	$-0.232294\pi$
$971$	46.4500	1.49065	0.745325	+	0.666701i	$0.232294\pi$
$972$	−12.3944	−0.397552
$973$	−50.8444	−1.63000
$974$	−3.30278	−0.105828
$975$	0	0
$976$	−10.2111	−0.326849
$977$	4.81665	0.154098	0.0770492	−	0.997027i	$-0.475450\pi$
$977$	4.81665	0.154098	0.0770492	+	0.997027i	$0.475450\pi$
$978$	−11.6056	−0.371105
$979$	−46.9361	−1.50008
$980$	0	0
$981$	−5.72498	−0.182785
$982$	8.57779	0.273728
$983$	53.9638	1.72118	0.860590	−	0.509299i	$-0.170095\pi$
$983$	53.9638	1.72118	0.860590	+	0.509299i	$0.170095\pi$
$984$	1.30278	0.0415310
$985$	0	0
$986$	2.39445	0.0762548
$987$	−37.5416	−1.19496
$988$	0	0
$989$	−15.2750	−0.485717
$990$	0	0
$991$	−54.8444	−1.74219	−0.871095	−	0.491114i	$-0.836590\pi$
$991$	−54.8444	−1.74219	−0.871095	+	0.491114i	$0.836590\pi$
$992$	1.39445	0.0442738
$993$	13.1472	0.417213
$994$	−14.6056	−0.463260
$995$	0	0
$996$	−6.27502	−0.198832
$997$	11.9083	0.377140	0.188570	−	0.982060i	$-0.439615\pi$
$997$	11.9083	0.377140	0.188570	+	0.982060i	$0.439615\pi$
$998$	−14.6333	−0.463209
$999$	40.9361	1.29516

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.bd.1.1		2
5.4	even	2		8450.2.a.bi.1.2		2
13.3	even	3		650.2.e.g.451.2	yes	4
13.9	even	3		650.2.e.g.601.2	yes	4
13.12	even	2		8450.2.a.bl.1.1		2
65.3	odd	12		650.2.o.h.399.2		8
65.9	even	6		650.2.e.e.601.1	yes	4
65.22	odd	12		650.2.o.h.549.2		8
65.29	even	6		650.2.e.e.451.1	✓	4
65.42	odd	12		650.2.o.h.399.3		8
65.48	odd	12		650.2.o.h.549.3		8
65.64	even	2		8450.2.a.ba.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
650.2.e.e.451.1	✓	4	65.29	even	6
650.2.e.e.601.1	yes	4	65.9	even	6
650.2.e.g.451.2	yes	4	13.3	even	3
650.2.e.g.601.2	yes	4	13.9	even	3
650.2.o.h.399.2		8	65.3	odd	12
650.2.o.h.399.3		8	65.42	odd	12
650.2.o.h.549.2		8	65.22	odd	12
650.2.o.h.549.3		8	65.48	odd	12
8450.2.a.ba.1.2		2	65.64	even	2
8450.2.a.bd.1.1		2	1.1	even	1	trivial
8450.2.a.bi.1.2		2	5.4	even	2
8450.2.a.bl.1.1		2	13.12	even	2

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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

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L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	8450.2.a.bd.1.1

Level	$8450$
Weight	$2$
Character	8450.1
Self dual	yes
Analytic conductor	$67.474$
Analytic rank	$1$
Dimension	$2$
CM	no
Inner twists	$1$