LMFDB - Embedded newform 8512.2.a.bn.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8512,2,Mod(1,8512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8512, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8512.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8512 = 2^{6} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8512.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:	$67.9686622005$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.733.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 8 $ x^3 - x^2 - 7*x + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 532)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.69639$ of defining polynomial
Character	$\chi$	$=$	8512.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69639 q^{3} -1.42586 q^{5} -1.00000 q^{7} +4.27053 q^{9} +O(q^{10})$	$q-2.69639 q^{3} -1.42586 q^{5} -1.00000 q^{7} +4.27053 q^{9} +3.27053 q^{11} -3.42586 q^{13} +3.84468 q^{15} +5.27053 q^{17} +1.00000 q^{19} +2.69639 q^{21} +0.574141 q^{23} -2.96693 q^{25} -3.42586 q^{27} +0.122252 q^{29} +2.12225 q^{31} -8.81864 q^{33} +1.42586 q^{35} +3.96693 q^{37} +9.23746 q^{39} -4.66332 q^{41} +11.9339 q^{43} -6.08918 q^{45} +3.11521 q^{47} +1.00000 q^{49} -14.2114 q^{51} +6.08918 q^{53} -4.66332 q^{55} -2.69639 q^{57} +1.72242 q^{59} -12.2114 q^{61} -4.27053 q^{63} +4.88479 q^{65} +5.27053 q^{67} -1.54811 q^{69} -6.81864 q^{71} -11.5481 q^{73} +8.00000 q^{75} -3.27053 q^{77} +11.0821 q^{79} -3.57414 q^{81} -5.23746 q^{83} -7.51504 q^{85} -0.329638 q^{87} -1.45893 q^{89} +3.42586 q^{91} -5.72242 q^{93} -1.42586 q^{95} +17.6563 q^{97} +13.9669 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} - 2 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10}) $ 3 * q + q^3 - 2 * q^5 - 3 * q^7 + 6 * q^9	$ 3 q + q^{3} - 2 q^{5} - 3 q^{7} + 6 q^{9} + 3 q^{11} - 8 q^{13} + 7 q^{15} + 9 q^{17} + 3 q^{19} - q^{21} + 4 q^{23} + 7 q^{25} - 8 q^{27} - 11 q^{29} - 5 q^{31} - 6 q^{33} + 2 q^{35} - 4 q^{37} + 5 q^{39} + 11 q^{41} + 4 q^{43} + 9 q^{45} - 2 q^{47} + 3 q^{49} - 4 q^{51} - 9 q^{53} + 11 q^{55} + q^{57} + 12 q^{59} + 2 q^{61} - 6 q^{63} + 26 q^{65} + 9 q^{67} + 9 q^{69} - 21 q^{73} + 24 q^{75} - 3 q^{77} + 6 q^{79} - 13 q^{81} + 7 q^{83} + 7 q^{85} - 26 q^{87} - 18 q^{89} + 8 q^{91} - 24 q^{93} - 2 q^{95} + 28 q^{97} + 26 q^{99}+O(q^{100}) $ 3 * q + q^3 - 2 * q^5 - 3 * q^7 + 6 * q^9 + 3 * q^11 - 8 * q^13 + 7 * q^15 + 9 * q^17 + 3 * q^19 - q^21 + 4 * q^23 + 7 * q^25 - 8 * q^27 - 11 * q^29 - 5 * q^31 - 6 * q^33 + 2 * q^35 - 4 * q^37 + 5 * q^39 + 11 * q^41 + 4 * q^43 + 9 * q^45 - 2 * q^47 + 3 * q^49 - 4 * q^51 - 9 * q^53 + 11 * q^55 + q^57 + 12 * q^59 + 2 * q^61 - 6 * q^63 + 26 * q^65 + 9 * q^67 + 9 * q^69 - 21 * q^73 + 24 * q^75 - 3 * q^77 + 6 * q^79 - 13 * q^81 + 7 * q^83 + 7 * q^85 - 26 * q^87 - 18 * q^89 + 8 * q^91 - 24 * q^93 - 2 * q^95 + 28 * q^97 + 26 * 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 $

$2$	0	0
$2$	0	0
$3$	−2.69639	−1.55676	−0.778382	−	0.627791i	$-0.783959\pi$
$3$	−2.69639	−1.55676	−0.778382	+	0.627791i	$0.783959\pi$
$4$	0	0
$5$	−1.42586	−0.637663	−0.318832	−	0.947811i	$-0.603290\pi$
$5$	−1.42586	−0.637663	−0.318832	+	0.947811i	$0.603290\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	4.27053	1.42351
$10$	0	0
$11$	3.27053	0.986103	0.493052	−	0.870000i	$-0.335881\pi$
$11$	3.27053	0.986103	0.493052	+	0.870000i	$0.335881\pi$
$12$	0	0
$13$	−3.42586	−0.950162	−0.475081	−	0.879942i	$-0.657581\pi$
$13$	−3.42586	−0.950162	−0.475081	+	0.879942i	$0.657581\pi$
$14$	0	0
$15$	3.84468	0.992691
$16$	0	0
$17$	5.27053	1.27829	0.639146	−	0.769085i	$-0.279288\pi$
$17$	5.27053	1.27829	0.639146	+	0.769085i	$0.279288\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.69639	0.588401
$22$	0	0
$23$	0.574141	0.119717	0.0598584	−	0.998207i	$-0.480935\pi$
$23$	0.574141	0.119717	0.0598584	+	0.998207i	$0.480935\pi$
$24$	0	0
$25$	−2.96693	−0.593385
$26$	0	0
$27$	−3.42586	−0.659307
$28$	0	0
$29$	0.122252	0.0227015	0.0113508	−	0.999936i	$-0.496387\pi$
$29$	0.122252	0.0227015	0.0113508	+	0.999936i	$0.496387\pi$
$30$	0	0
$31$	2.12225	0.381168	0.190584	−	0.981671i	$-0.438962\pi$
$31$	2.12225	0.381168	0.190584	+	0.981671i	$0.438962\pi$
$32$	0	0
$33$	−8.81864	−1.53513
$34$	0	0
$35$	1.42586	0.241014
$36$	0	0
$37$	3.96693	0.652159	0.326079	−	0.945342i	$-0.394272\pi$
$37$	3.96693	0.652159	0.326079	+	0.945342i	$0.394272\pi$
$38$	0	0
$39$	9.23746	1.47918
$40$	0	0
$41$	−4.66332	−0.728288	−0.364144	−	0.931343i	$-0.618638\pi$
$41$	−4.66332	−0.728288	−0.364144	+	0.931343i	$0.618638\pi$
$42$	0	0
$43$	11.9339	1.81990	0.909948	−	0.414723i	$-0.136121\pi$
$43$	11.9339	1.81990	0.909948	+	0.414723i	$0.136121\pi$
$44$	0	0
$45$	−6.08918	−0.907721
$46$	0	0
$47$	3.11521	0.454400	0.227200	−	0.973848i	$-0.427043\pi$
$47$	3.11521	0.454400	0.227200	+	0.973848i	$0.427043\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−14.2114	−1.99000
$52$	0	0
$53$	6.08918	0.836413	0.418206	−	0.908352i	$-0.362659\pi$
$53$	6.08918	0.836413	0.418206	+	0.908352i	$0.362659\pi$
$54$	0	0
$55$	−4.66332	−0.628802
$56$	0	0
$57$	−2.69639	−0.357146
$58$	0	0
$59$	1.72242	0.224240	0.112120	−	0.993695i	$-0.464236\pi$
$59$	1.72242	0.224240	0.112120	+	0.993695i	$0.464236\pi$
$60$	0	0
$61$	−12.2114	−1.56351	−0.781757	−	0.623584i	$-0.785676\pi$
$61$	−12.2114	−1.56351	−0.781757	+	0.623584i	$0.785676\pi$
$62$	0	0
$63$	−4.27053	−0.538037
$64$	0	0
$65$	4.88479	0.605884
$66$	0	0
$67$	5.27053	0.643898	0.321949	−	0.946757i	$-0.395662\pi$
$67$	5.27053	0.643898	0.321949	+	0.946757i	$0.395662\pi$
$68$	0	0
$69$	−1.54811	−0.186371
$70$	0	0
$71$	−6.81864	−0.809224	−0.404612	−	0.914488i	$-0.632593\pi$
$71$	−6.81864	−0.809224	−0.404612	+	0.914488i	$0.632593\pi$
$72$	0	0
$73$	−11.5481	−1.35160	−0.675802	−	0.737083i	$-0.736203\pi$
$73$	−11.5481	−1.35160	−0.675802	+	0.737083i	$0.736203\pi$
$74$	0	0
$75$	8.00000	0.923760
$76$	0	0
$77$	−3.27053	−0.372712
$78$	0	0
$79$	11.0821	1.24684	0.623419	−	0.781888i	$-0.285743\pi$
$79$	11.0821	1.24684	0.623419	+	0.781888i	$0.285743\pi$
$80$	0	0
$81$	−3.57414	−0.397127
$82$	0	0
$83$	−5.23746	−0.574886	−0.287443	−	0.957798i	$-0.592805\pi$
$83$	−5.23746	−0.574886	−0.287443	+	0.957798i	$0.592805\pi$
$84$	0	0
$85$	−7.51504	−0.815120
$86$	0	0
$87$	−0.329638	−0.0353409
$88$	0	0
$89$	−1.45893	−0.154646	−0.0773232	−	0.997006i	$-0.524637\pi$
$89$	−1.45893	−0.154646	−0.0773232	+	0.997006i	$0.524637\pi$
$90$	0	0
$91$	3.42586	0.359128
$92$	0	0
$93$	−5.72242	−0.593388
$94$	0	0
$95$	−1.42586	−0.146290
$96$	0	0
$97$	17.6563	1.79272	0.896362	−	0.443324i	$-0.146201\pi$
$97$	17.6563	1.79272	0.896362	+	0.443324i	$0.146201\pi$
$98$	0	0
$99$	13.9669	1.40373
$100$	0	0
$101$	−17.0491	−1.69645	−0.848223	−	0.529640i	$-0.822327\pi$
$101$	−17.0491	−1.69645	−0.848223	+	0.529640i	$0.822327\pi$
$102$	0	0
$103$	−11.3597	−1.11931	−0.559653	−	0.828727i	$-0.689066\pi$
$103$	−11.3597	−1.11931	−0.559653	+	0.828727i	$0.689066\pi$
$104$	0	0
$105$	−3.84468	−0.375202
$106$	0	0
$107$	−10.8186	−1.04588	−0.522939	−	0.852370i	$-0.675164\pi$
$107$	−10.8186	−1.04588	−0.522939	+	0.852370i	$0.675164\pi$
$108$	0	0
$109$	12.2776	1.17598	0.587989	−	0.808869i	$-0.299920\pi$
$109$	12.2776	1.17598	0.587989	+	0.808869i	$0.299920\pi$
$110$	0	0
$111$	−10.6964	−1.01526
$112$	0	0
$113$	−0.696393	−0.0655111	−0.0327556	−	0.999463i	$-0.510428\pi$
$113$	−0.696393	−0.0655111	−0.0327556	+	0.999463i	$0.510428\pi$
$114$	0	0
$115$	−0.818644	−0.0763390
$116$	0	0
$117$	−14.6302	−1.35257
$118$	0	0
$119$	−5.27053	−0.483149
$120$	0	0
$121$	−0.303607	−0.0276007
$122$	0	0
$123$	12.5741	1.13377
$124$	0	0
$125$	11.3597	1.01604
$126$	0	0
$127$	−5.68935	−0.504848	−0.252424	−	0.967617i	$-0.581228\pi$
$127$	−5.68935	−0.504848	−0.252424	+	0.967617i	$0.581228\pi$
$128$	0	0
$129$	−32.1784	−2.83315
$130$	0	0
$131$	15.2044	1.32841	0.664207	−	0.747549i	$-0.268769\pi$
$131$	15.2044	1.32841	0.664207	+	0.747549i	$0.268769\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	4.88479	0.420416
$136$	0	0
$137$	3.70344	0.316406	0.158203	−	0.987407i	$-0.449430\pi$
$137$	3.70344	0.316406	0.158203	+	0.987407i	$0.449430\pi$
$138$	0	0
$139$	11.6704	0.989867	0.494934	−	0.868931i	$-0.335192\pi$
$139$	11.6704	0.989867	0.494934	+	0.868931i	$0.335192\pi$
$140$	0	0
$141$	−8.39983	−0.707393
$142$	0	0
$143$	−11.2044	−0.936958
$144$	0	0
$145$	−0.174313	−0.0144759
$146$	0	0
$147$	−2.69639	−0.222395
$148$	0	0
$149$	−18.8186	−1.54168	−0.770842	−	0.637027i	$-0.780164\pi$
$149$	−18.8186	−1.54168	−0.770842	+	0.637027i	$0.780164\pi$
$150$	0	0
$151$	−4.43290	−0.360744	−0.180372	−	0.983598i	$-0.557730\pi$
$151$	−4.43290	−0.360744	−0.180372	+	0.983598i	$0.557730\pi$
$152$	0	0
$153$	22.5080	1.81966
$154$	0	0
$155$	−3.02603	−0.243057
$156$	0	0
$157$	9.82569	0.784175	0.392088	−	0.919928i	$-0.371753\pi$
$157$	9.82569	0.784175	0.392088	+	0.919928i	$0.371753\pi$
$158$	0	0
$159$	−16.4188	−1.30210
$160$	0	0
$161$	−0.574141	−0.0452487
$162$	0	0
$163$	10.9409	0.856957	0.428479	−	0.903552i	$-0.359050\pi$
$163$	10.9409	0.856957	0.428479	+	0.903552i	$0.359050\pi$
$164$	0	0
$165$	12.5741	0.978896
$166$	0	0
$167$	−12.7525	−0.986818	−0.493409	−	0.869797i	$-0.664249\pi$
$167$	−12.7525	−0.986818	−0.493409	+	0.869797i	$0.664249\pi$
$168$	0	0
$169$	−1.26349	−0.0971917
$170$	0	0
$171$	4.27053	0.326576
$172$	0	0
$173$	−22.5411	−1.71377	−0.856883	−	0.515511i	$-0.827602\pi$
$173$	−22.5411	−1.71377	−0.856883	+	0.515511i	$0.827602\pi$
$174$	0	0
$175$	2.96693	0.224279
$176$	0	0
$177$	−4.64433	−0.349089
$178$	0	0
$179$	−6.41882	−0.479765	−0.239882	−	0.970802i	$-0.577109\pi$
$179$	−6.41882	−0.479765	−0.239882	+	0.970802i	$0.577109\pi$
$180$	0	0
$181$	−9.81160	−0.729291	−0.364645	−	0.931146i	$-0.618810\pi$
$181$	−9.81160	−0.729291	−0.364645	+	0.931146i	$0.618810\pi$
$182$	0	0
$183$	32.9268	2.43402
$184$	0	0
$185$	−5.65628	−0.415858
$186$	0	0
$187$	17.2375	1.26053
$188$	0	0
$189$	3.42586	0.249195
$190$	0	0
$191$	15.2515	1.10356	0.551782	−	0.833989i	$-0.313948\pi$
$191$	15.2515	1.10356	0.551782	+	0.833989i	$0.313948\pi$
$192$	0	0
$193$	−15.7455	−1.13338	−0.566691	−	0.823930i	$-0.691777\pi$
$193$	−15.7455	−1.13338	−0.566691	+	0.823930i	$0.691777\pi$
$194$	0	0
$195$	−13.1713	−0.943217
$196$	0	0
$197$	−24.8086	−1.76754	−0.883770	−	0.467922i	$-0.845003\pi$
$197$	−24.8086	−1.76754	−0.883770	+	0.467922i	$0.845003\pi$
$198$	0	0
$199$	−2.58823	−0.183474	−0.0917372	−	0.995783i	$-0.529242\pi$
$199$	−2.58823	−0.183474	−0.0917372	+	0.995783i	$0.529242\pi$
$200$	0	0
$201$	−14.2114	−1.00240
$202$	0	0
$203$	−0.122252	−0.00858038
$204$	0	0
$205$	6.64924	0.464403
$206$	0	0
$207$	2.45189	0.170418
$208$	0	0
$209$	3.27053	0.226228
$210$	0	0
$211$	−21.7785	−1.49930	−0.749648	−	0.661837i	$-0.769777\pi$
$211$	−21.7785	−1.49930	−0.749648	+	0.661837i	$0.769777\pi$
$212$	0	0
$213$	18.3857	1.25977
$214$	0	0
$215$	−17.0160	−1.16048
$216$	0	0
$217$	−2.12225	−0.144068
$218$	0	0
$219$	31.1382	2.10413
$220$	0	0
$221$	−18.0561	−1.21459
$222$	0	0
$223$	−9.90078	−0.663005	−0.331503	−	0.943454i	$-0.607556\pi$
$223$	−9.90078	−0.663005	−0.331503	+	0.943454i	$0.607556\pi$
$224$	0	0
$225$	−12.6704	−0.844691
$226$	0	0
$227$	−2.71538	−0.180226	−0.0901131	−	0.995932i	$-0.528723\pi$
$227$	−2.71538	−0.180226	−0.0901131	+	0.995932i	$0.528723\pi$
$228$	0	0
$229$	8.29656	0.548252	0.274126	−	0.961694i	$-0.411611\pi$
$229$	8.29656	0.548252	0.274126	+	0.961694i	$0.411611\pi$
$230$	0	0
$231$	8.81864	0.580224
$232$	0	0
$233$	14.7295	0.964959	0.482480	−	0.875907i	$-0.339736\pi$
$233$	14.7295	0.964959	0.482480	+	0.875907i	$0.339736\pi$
$234$	0	0
$235$	−4.44185	−0.289754
$236$	0	0
$237$	−29.8818	−1.94103
$238$	0	0
$239$	28.9970	1.87566	0.937830	−	0.347095i	$-0.112832\pi$
$239$	28.9970	1.87566	0.937830	+	0.347095i	$0.112832\pi$
$240$	0	0
$241$	−4.31065	−0.277673	−0.138837	−	0.990315i	$-0.544336\pi$
$241$	−4.31065	−0.277673	−0.138837	+	0.990315i	$0.544336\pi$
$242$	0	0
$243$	19.9149	1.27754
$244$	0	0
$245$	−1.42586	−0.0910948
$246$	0	0
$247$	−3.42586	−0.217982
$248$	0	0
$249$	14.1223	0.894961
$250$	0	0
$251$	10.4329	0.658519	0.329259	−	0.944239i	$-0.393201\pi$
$251$	10.4329	0.658519	0.329259	+	0.944239i	$0.393201\pi$
$252$	0	0
$253$	1.87775	0.118053
$254$	0	0
$255$	20.2635	1.26895
$256$	0	0
$257$	23.7124	1.47914	0.739569	−	0.673081i	$-0.235029\pi$
$257$	23.7124	1.47914	0.739569	+	0.673081i	$0.235029\pi$
$258$	0	0
$259$	−3.96693	−0.246493
$260$	0	0
$261$	0.522079	0.0323159
$262$	0	0
$263$	30.9409	1.90790	0.953949	−	0.299970i	$-0.0969766\pi$
$263$	30.9409	1.90790	0.953949	+	0.299970i	$0.0969766\pi$
$264$	0	0
$265$	−8.68231	−0.533350
$266$	0	0
$267$	3.93385	0.240748
$268$	0	0
$269$	9.17131	0.559185	0.279592	−	0.960119i	$-0.409801\pi$
$269$	9.17131	0.559185	0.279592	+	0.960119i	$0.409801\pi$
$270$	0	0
$271$	20.0420	1.21747	0.608733	−	0.793375i	$-0.291678\pi$
$271$	20.0420	1.21747	0.608733	+	0.793375i	$0.291678\pi$
$272$	0	0
$273$	−9.23746	−0.559077
$274$	0	0
$275$	−9.70344	−0.585139
$276$	0	0
$277$	7.45893	0.448164	0.224082	−	0.974570i	$-0.428062\pi$
$277$	7.45893	0.448164	0.224082	+	0.974570i	$0.428062\pi$
$278$	0	0
$279$	9.06315	0.542596
$280$	0	0
$281$	12.2114	0.728473	0.364236	−	0.931307i	$-0.381330\pi$
$281$	12.2114	0.728473	0.364236	+	0.931307i	$0.381330\pi$
$282$	0	0
$283$	8.02303	0.476920	0.238460	−	0.971152i	$-0.423357\pi$
$283$	8.02303	0.476920	0.238460	+	0.971152i	$0.423357\pi$
$284$	0	0
$285$	3.84468	0.227739
$286$	0	0
$287$	4.66332	0.275267
$288$	0	0
$289$	10.7785	0.634031
$290$	0	0
$291$	−47.6083	−2.79085
$292$	0	0
$293$	−14.0331	−0.819821	−0.409910	−	0.912126i	$-0.634440\pi$
$293$	−14.0331	−0.819821	−0.409910	+	0.912126i	$0.634440\pi$
$294$	0	0
$295$	−2.45593	−0.142990
$296$	0	0
$297$	−11.2044	−0.650145
$298$	0	0
$299$	−1.96693	−0.113750
$300$	0	0
$301$	−11.9339	−0.687856
$302$	0	0
$303$	45.9710	2.64096
$304$	0	0
$305$	17.4118	0.996995
$306$	0	0
$307$	−16.0702	−0.917174	−0.458587	−	0.888649i	$-0.651644\pi$
$307$	−16.0702	−0.917174	−0.458587	+	0.888649i	$0.651644\pi$
$308$	0	0
$309$	30.6302	1.74249
$310$	0	0
$311$	−1.56710	−0.0888620	−0.0444310	−	0.999012i	$-0.514147\pi$
$311$	−1.56710	−0.0888620	−0.0444310	+	0.999012i	$0.514147\pi$
$312$	0	0
$313$	29.6232	1.67440	0.837201	−	0.546895i	$-0.184190\pi$
$313$	29.6232	1.67440	0.837201	+	0.546895i	$0.184190\pi$
$314$	0	0
$315$	6.08918	0.343086
$316$	0	0
$317$	22.4890	1.26311	0.631554	−	0.775332i	$-0.282417\pi$
$317$	22.4890	1.26311	0.631554	+	0.775332i	$0.282417\pi$
$318$	0	0
$319$	0.399828	0.0223861
$320$	0	0
$321$	29.1713	1.62818
$322$	0	0
$323$	5.27053	0.293260
$324$	0	0
$325$	10.1643	0.563812
$326$	0	0
$327$	−33.1052	−1.83072
$328$	0	0
$329$	−3.11521	−0.171747
$330$	0	0
$331$	−28.3857	−1.56022	−0.780111	−	0.625641i	$-0.784837\pi$
$331$	−28.3857	−1.56022	−0.780111	+	0.625641i	$0.784837\pi$
$332$	0	0
$333$	16.9409	0.928355
$334$	0	0
$335$	−7.51504	−0.410590
$336$	0	0
$337$	29.6272	1.61390	0.806950	−	0.590620i	$-0.201117\pi$
$337$	29.6272	1.61390	0.806950	+	0.590620i	$0.201117\pi$
$338$	0	0
$339$	1.87775	0.101985
$340$	0	0
$341$	6.94090	0.375871
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	2.20739	0.118842
$346$	0	0
$347$	32.8607	1.76405	0.882026	−	0.471200i	$-0.156179\pi$
$347$	32.8607	1.76405	0.882026	+	0.471200i	$0.156179\pi$
$348$	0	0
$349$	32.5641	1.74312	0.871558	−	0.490292i	$-0.163110\pi$
$349$	32.5641	1.74312	0.871558	+	0.490292i	$0.163110\pi$
$350$	0	0
$351$	11.7365	0.626448
$352$	0	0
$353$	−30.6633	−1.63204	−0.816022	−	0.578021i	$-0.803825\pi$
$353$	−30.6633	−1.63204	−0.816022	+	0.578021i	$0.803825\pi$
$354$	0	0
$355$	9.72242	0.516013
$356$	0	0
$357$	14.2114	0.752149
$358$	0	0
$359$	−6.67740	−0.352420	−0.176210	−	0.984353i	$-0.556384\pi$
$359$	−6.67740	−0.352420	−0.176210	+	0.984353i	$0.556384\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.818644	0.0429677
$364$	0	0
$365$	16.4660	0.861868
$366$	0	0
$367$	5.19544	0.271200	0.135600	−	0.990764i	$-0.456704\pi$
$367$	5.19544	0.271200	0.135600	+	0.990764i	$0.456704\pi$
$368$	0	0
$369$	−19.9149	−1.03673
$370$	0	0
$371$	−6.08918	−0.316134
$372$	0	0
$373$	1.31769	0.0682275	0.0341137	−	0.999418i	$-0.489139\pi$
$373$	1.31769	0.0682275	0.0341137	+	0.999418i	$0.489139\pi$
$374$	0	0
$375$	−30.6302	−1.58174
$376$	0	0
$377$	−0.418817	−0.0215701
$378$	0	0
$379$	15.6563	0.804209	0.402104	−	0.915594i	$-0.368279\pi$
$379$	15.6563	0.804209	0.402104	+	0.915594i	$0.368279\pi$
$380$	0	0
$381$	15.3407	0.785929
$382$	0	0
$383$	29.0631	1.48506	0.742529	−	0.669814i	$-0.233626\pi$
$383$	29.0631	1.48506	0.742529	+	0.669814i	$0.233626\pi$
$384$	0	0
$385$	4.66332	0.237665
$386$	0	0
$387$	50.9639	2.59064
$388$	0	0
$389$	−10.7295	−0.544006	−0.272003	−	0.962296i	$-0.587686\pi$
$389$	−10.7295	−0.544006	−0.272003	+	0.962296i	$0.587686\pi$
$390$	0	0
$391$	3.02603	0.153033
$392$	0	0
$393$	−40.9970	−2.06803
$394$	0	0
$395$	−15.8016	−0.795063
$396$	0	0
$397$	−10.7997	−0.542019	−0.271010	−	0.962577i	$-0.587358\pi$
$397$	−10.7997	−0.542019	−0.271010	+	0.962577i	$0.587358\pi$
$398$	0	0
$399$	2.69639	0.134988
$400$	0	0
$401$	−6.66332	−0.332750	−0.166375	−	0.986063i	$-0.553206\pi$
$401$	−6.66332	−0.332750	−0.166375	+	0.986063i	$0.553206\pi$
$402$	0	0
$403$	−7.27053	−0.362171
$404$	0	0
$405$	5.09622	0.253233
$406$	0	0
$407$	12.9740	0.643096
$408$	0	0
$409$	−25.5481	−1.26327	−0.631636	−	0.775265i	$-0.717616\pi$
$409$	−25.5481	−1.26327	−0.631636	+	0.775265i	$0.717616\pi$
$410$	0	0
$411$	−9.98592	−0.492569
$412$	0	0
$413$	−1.72242	−0.0847549
$414$	0	0
$415$	7.46788	0.366584
$416$	0	0
$417$	−31.4679	−1.54099
$418$	0	0
$419$	−33.2415	−1.62395	−0.811977	−	0.583690i	$-0.801608\pi$
$419$	−33.2415	−1.62395	−0.811977	+	0.583690i	$0.801608\pi$
$420$	0	0
$421$	−2.95094	−0.143820	−0.0719099	−	0.997411i	$-0.522909\pi$
$421$	−2.95094	−0.143820	−0.0719099	+	0.997411i	$0.522909\pi$
$422$	0	0
$423$	13.3036	0.646844
$424$	0	0
$425$	−15.6373	−0.758520
$426$	0	0
$427$	12.2114	0.590953
$428$	0	0
$429$	30.2114	1.45862
$430$	0	0
$431$	4.29656	0.206958	0.103479	−	0.994632i	$-0.467003\pi$
$431$	4.29656	0.206958	0.103479	+	0.994632i	$0.467003\pi$
$432$	0	0
$433$	5.96693	0.286752	0.143376	−	0.989668i	$-0.454204\pi$
$433$	5.96693	0.286752	0.143376	+	0.989668i	$0.454204\pi$
$434$	0	0
$435$	0.470017	0.0225356
$436$	0	0
$437$	0.574141	0.0274649
$438$	0	0
$439$	37.5711	1.79317	0.896586	−	0.442869i	$-0.146039\pi$
$439$	37.5711	1.79317	0.896586	+	0.442869i	$0.146039\pi$
$440$	0	0
$441$	4.27053	0.203359
$442$	0	0
$443$	15.2515	0.724623	0.362311	−	0.932057i	$-0.381988\pi$
$443$	15.2515	0.724623	0.362311	+	0.932057i	$0.381988\pi$
$444$	0	0
$445$	2.08023	0.0986124
$446$	0	0
$447$	50.7425	2.40004
$448$	0	0
$449$	−35.7926	−1.68916	−0.844579	−	0.535431i	$-0.820149\pi$
$449$	−35.7926	−1.68916	−0.844579	+	0.535431i	$0.820149\pi$
$450$	0	0
$451$	−15.2515	−0.718167
$452$	0	0
$453$	11.9528	0.561594
$454$	0	0
$455$	−4.88479	−0.229003
$456$	0	0
$457$	18.6443	0.872145	0.436073	−	0.899912i	$-0.356369\pi$
$457$	18.6443	0.872145	0.436073	+	0.899912i	$0.356369\pi$
$458$	0	0
$459$	−18.0561	−0.842787
$460$	0	0
$461$	11.8306	0.551006	0.275503	−	0.961300i	$-0.411156\pi$
$461$	11.8306	0.551006	0.275503	+	0.961300i	$0.411156\pi$
$462$	0	0
$463$	0.752498	0.0349715	0.0174858	−	0.999847i	$-0.494434\pi$
$463$	0.752498	0.0349715	0.0174858	+	0.999847i	$0.494434\pi$
$464$	0	0
$465$	8.15937	0.378382
$466$	0	0
$467$	13.8588	0.641307	0.320653	−	0.947197i	$-0.396098\pi$
$467$	13.8588	0.641307	0.320653	+	0.947197i	$0.396098\pi$
$468$	0	0
$469$	−5.27053	−0.243371
$470$	0	0
$471$	−26.4939	−1.22077
$472$	0	0
$473$	39.0301	1.79460
$474$	0	0
$475$	−2.96693	−0.136132
$476$	0	0
$477$	26.0040	1.19064
$478$	0	0
$479$	−7.58118	−0.346393	−0.173197	−	0.984887i	$-0.555410\pi$
$479$	−7.58118	−0.346393	−0.173197	+	0.984887i	$0.555410\pi$
$480$	0	0
$481$	−13.5901	−0.619657
$482$	0	0
$483$	1.54811	0.0704415
$484$	0	0
$485$	−25.1754	−1.14315
$486$	0	0
$487$	−10.6864	−0.484245	−0.242122	−	0.970246i	$-0.577844\pi$
$487$	−10.6864	−0.484245	−0.242122	+	0.970246i	$0.577844\pi$
$488$	0	0
$489$	−29.5010	−1.33408
$490$	0	0
$491$	14.1643	0.639225	0.319612	−	0.947548i	$-0.396447\pi$
$491$	14.1643	0.639225	0.319612	+	0.947548i	$0.396447\pi$
$492$	0	0
$493$	0.644331	0.0290192
$494$	0	0
$495$	−19.9149	−0.895107
$496$	0	0
$497$	6.81864	0.305858
$498$	0	0
$499$	23.9108	1.07040	0.535198	−	0.844727i	$-0.320237\pi$
$499$	23.9108	1.07040	0.535198	+	0.844727i	$0.320237\pi$
$500$	0	0
$501$	34.3857	1.53624
$502$	0	0
$503$	−37.4388	−1.66932	−0.834658	−	0.550769i	$-0.814335\pi$
$503$	−37.4388	−1.66932	−0.834658	+	0.550769i	$0.814335\pi$
$504$	0	0
$505$	24.3096	1.08176
$506$	0	0
$507$	3.40687	0.151304
$508$	0	0
$509$	−31.3126	−1.38790	−0.693952	−	0.720021i	$-0.744132\pi$
$509$	−31.3126	−1.38790	−0.693952	+	0.720021i	$0.744132\pi$
$510$	0	0
$511$	11.5481	0.510858
$512$	0	0
$513$	−3.42586	−0.151255
$514$	0	0
$515$	16.1973	0.713740
$516$	0	0
$517$	10.1884	0.448085
$518$	0	0
$519$	60.7796	2.66793
$520$	0	0
$521$	21.8346	0.956593	0.478296	−	0.878199i	$-0.341254\pi$
$521$	21.8346	0.956593	0.478296	+	0.878199i	$0.341254\pi$
$522$	0	0
$523$	−9.37380	−0.409888	−0.204944	−	0.978774i	$-0.565701\pi$
$523$	−9.37380	−0.409888	−0.204944	+	0.978774i	$0.565701\pi$
$524$	0	0
$525$	−8.00000	−0.349149
$526$	0	0
$527$	11.1854	0.487244
$528$	0	0
$529$	−22.6704	−0.985668
$530$	0	0
$531$	7.35567	0.319209
$532$	0	0
$533$	15.9759	0.691992
$534$	0	0
$535$	15.4259	0.666918
$536$	0	0
$537$	17.3077	0.746880
$538$	0	0
$539$	3.27053	0.140872
$540$	0	0
$541$	30.4608	1.30961	0.654807	−	0.755796i	$-0.272750\pi$
$541$	30.4608	1.30961	0.654807	+	0.755796i	$0.272750\pi$
$542$	0	0
$543$	26.4559	1.13533
$544$	0	0
$545$	−17.5061	−0.749878
$546$	0	0
$547$	−34.2675	−1.46517	−0.732587	−	0.680673i	$-0.761687\pi$
$547$	−34.2675	−1.46517	−0.732587	+	0.680673i	$0.761687\pi$
$548$	0	0
$549$	−52.1493	−2.22568
$550$	0	0
$551$	0.122252	0.00520809
$552$	0	0
$553$	−11.0821	−0.471260
$554$	0	0
$555$	15.2515	0.647392
$556$	0	0
$557$	1.44889	0.0613915	0.0306957	−	0.999529i	$-0.490228\pi$
$557$	1.44889	0.0613915	0.0306957	+	0.999529i	$0.490228\pi$
$558$	0	0
$559$	−40.8837	−1.72920
$560$	0	0
$561$	−46.4790	−1.96234
$562$	0	0
$563$	10.8707	0.458146	0.229073	−	0.973409i	$-0.426431\pi$
$563$	10.8707	0.458146	0.229073	+	0.973409i	$0.426431\pi$
$564$	0	0
$565$	0.992958	0.0417740
$566$	0	0
$567$	3.57414	0.150100
$568$	0	0
$569$	25.0772	1.05129	0.525646	−	0.850703i	$-0.323824\pi$
$569$	25.0772	1.05129	0.525646	+	0.850703i	$0.323824\pi$
$570$	0	0
$571$	11.6042	0.485621	0.242811	−	0.970074i	$-0.421931\pi$
$571$	11.6042	0.485621	0.242811	+	0.970074i	$0.421931\pi$
$572$	0	0
$573$	−41.1242	−1.71799
$574$	0	0
$575$	−1.70344	−0.0710382
$576$	0	0
$577$	26.5782	1.10646	0.553232	−	0.833027i	$-0.313394\pi$
$577$	26.5782	1.10646	0.553232	+	0.833027i	$0.313394\pi$
$578$	0	0
$579$	42.4559	1.76441
$580$	0	0
$581$	5.23746	0.217286
$582$	0	0
$583$	19.9149	0.824789
$584$	0	0
$585$	20.8607	0.862482
$586$	0	0
$587$	16.5552	0.683304	0.341652	−	0.939826i	$-0.389014\pi$
$587$	16.5552	0.683304	0.341652	+	0.939826i	$0.389014\pi$
$588$	0	0
$589$	2.12225	0.0874459
$590$	0	0
$591$	66.8937	2.75164
$592$	0	0
$593$	26.6864	1.09588	0.547939	−	0.836519i	$-0.315413\pi$
$593$	26.6864	1.09588	0.547939	+	0.836519i	$0.315413\pi$
$594$	0	0
$595$	7.51504	0.308086
$596$	0	0
$597$	6.97887	0.285626
$598$	0	0
$599$	−19.6603	−0.803299	−0.401649	−	0.915793i	$-0.631563\pi$
$599$	−19.6603	−0.803299	−0.401649	+	0.915793i	$0.631563\pi$
$600$	0	0
$601$	−23.1854	−0.945752	−0.472876	−	0.881129i	$-0.656784\pi$
$601$	−23.1854	−0.945752	−0.472876	+	0.881129i	$0.656784\pi$
$602$	0	0
$603$	22.5080	0.916596
$604$	0	0
$605$	0.432901	0.0175999
$606$	0	0
$607$	0.441848	0.0179341	0.00896704	−	0.999960i	$-0.497146\pi$
$607$	0.441848	0.0179341	0.00896704	+	0.999960i	$0.497146\pi$
$608$	0	0
$609$	0.329638	0.0133576
$610$	0	0
$611$	−10.6723	−0.431754
$612$	0	0
$613$	28.3478	1.14496	0.572478	−	0.819920i	$-0.305982\pi$
$613$	28.3478	1.14496	0.572478	+	0.819920i	$0.305982\pi$
$614$	0	0
$615$	−17.9289	−0.722965
$616$	0	0
$617$	31.7455	1.27802	0.639012	−	0.769197i	$-0.279343\pi$
$617$	31.7455	1.27802	0.639012	+	0.769197i	$0.279343\pi$
$618$	0	0
$619$	−22.2535	−0.894442	−0.447221	−	0.894424i	$-0.647586\pi$
$619$	−22.2535	−0.894442	−0.447221	+	0.894424i	$0.647586\pi$
$620$	0	0
$621$	−1.96693	−0.0789301
$622$	0	0
$623$	1.45893	0.0584509
$624$	0	0
$625$	−1.36271	−0.0545085
$626$	0	0
$627$	−8.81864	−0.352183
$628$	0	0
$629$	20.9078	0.833649
$630$	0	0
$631$	−19.4719	−0.775165	−0.387583	−	0.921835i	$-0.626690\pi$
$631$	−19.4719	−0.775165	−0.387583	+	0.921835i	$0.626690\pi$
$632$	0	0
$633$	58.7235	2.33405
$634$	0	0
$635$	8.11221	0.321923
$636$	0	0
$637$	−3.42586	−0.135737
$638$	0	0
$639$	−29.1193	−1.15194
$640$	0	0
$641$	40.2345	1.58917	0.794583	−	0.607156i	$-0.207690\pi$
$641$	40.2345	1.58917	0.794583	+	0.607156i	$0.207690\pi$
$642$	0	0
$643$	17.1011	0.674403	0.337201	−	0.941433i	$-0.390520\pi$
$643$	17.1011	0.674403	0.337201	+	0.941433i	$0.390520\pi$
$644$	0	0
$645$	45.8818	1.80659
$646$	0	0
$647$	−5.37380	−0.211266	−0.105633	−	0.994405i	$-0.533687\pi$
$647$	−5.37380	−0.211266	−0.105633	+	0.994405i	$0.533687\pi$
$648$	0	0
$649$	5.63325	0.221124
$650$	0	0
$651$	5.72242	0.224279
$652$	0	0
$653$	11.6183	0.454659	0.227330	−	0.973818i	$-0.427001\pi$
$653$	11.6183	0.454659	0.227330	+	0.973818i	$0.427001\pi$
$654$	0	0
$655$	−21.6793	−0.847081
$656$	0	0
$657$	−49.3166	−1.92402
$658$	0	0
$659$	41.8718	1.63109	0.815546	−	0.578692i	$-0.196437\pi$
$659$	41.8718	1.63109	0.815546	+	0.578692i	$0.196437\pi$
$660$	0	0
$661$	−6.71942	−0.261355	−0.130678	−	0.991425i	$-0.541715\pi$
$661$	−6.71942	−0.261355	−0.130678	+	0.991425i	$0.541715\pi$
$662$	0	0
$663$	48.6864	1.89082
$664$	0	0
$665$	1.42586	0.0552924
$666$	0	0
$667$	0.0701896	0.00271775
$668$	0	0
$669$	26.6964	1.03214
$670$	0	0
$671$	−39.9379	−1.54179
$672$	0	0
$673$	43.5280	1.67788	0.838941	−	0.544222i	$-0.183175\pi$
$673$	43.5280	1.67788	0.838941	+	0.544222i	$0.183175\pi$
$674$	0	0
$675$	10.1643	0.391223
$676$	0	0
$677$	−18.1033	−0.695765	−0.347882	−	0.937538i	$-0.613099\pi$
$677$	−18.1033	−0.695765	−0.347882	+	0.937538i	$0.613099\pi$
$678$	0	0
$679$	−17.6563	−0.677586
$680$	0	0
$681$	7.32173	0.280569
$682$	0	0
$683$	−27.3456	−1.04635	−0.523176	−	0.852225i	$-0.675253\pi$
$683$	−27.3456	−1.04635	−0.523176	+	0.852225i	$0.675253\pi$
$684$	0	0
$685$	−5.28058	−0.201760
$686$	0	0
$687$	−22.3708	−0.853499
$688$	0	0
$689$	−20.8607	−0.794728
$690$	0	0
$691$	30.8316	1.17289	0.586445	−	0.809989i	$-0.300527\pi$
$691$	30.8316	1.17289	0.586445	+	0.809989i	$0.300527\pi$
$692$	0	0
$693$	−13.9669	−0.530560
$694$	0	0
$695$	−16.6403	−0.631202
$696$	0	0
$697$	−24.5782	−0.930965
$698$	0	0
$699$	−39.7164	−1.50221
$700$	0	0
$701$	8.23042	0.310859	0.155429	−	0.987847i	$-0.450324\pi$
$701$	8.23042	0.310859	0.155429	+	0.987847i	$0.450324\pi$
$702$	0	0
$703$	3.96693	0.149615
$704$	0	0
$705$	11.9770	0.451079
$706$	0	0
$707$	17.0491	0.641196
$708$	0	0
$709$	15.6183	0.586558	0.293279	−	0.956027i	$-0.405254\pi$
$709$	15.6183	0.586558	0.293279	+	0.956027i	$0.405254\pi$
$710$	0	0
$711$	47.3266	1.77489
$712$	0	0
$713$	1.21847	0.0456321
$714$	0	0
$715$	15.9759	0.597464
$716$	0	0
$717$	−78.1873	−2.91996
$718$	0	0
$719$	16.3567	0.610002	0.305001	−	0.952352i	$-0.401343\pi$
$719$	16.3567	0.610002	0.305001	+	0.952352i	$0.401343\pi$
$720$	0	0
$721$	11.3597	0.423058
$722$	0	0
$723$	11.6232	0.432272
$724$	0	0
$725$	−0.362711	−0.0134708
$726$	0	0
$727$	−17.6183	−0.653427	−0.326713	−	0.945123i	$-0.605941\pi$
$727$	−17.6183	−0.653427	−0.326713	+	0.945123i	$0.605941\pi$
$728$	0	0
$729$	−42.9759	−1.59170
$730$	0	0
$731$	62.8978	2.32636
$732$	0	0
$733$	42.8837	1.58395	0.791973	−	0.610556i	$-0.209054\pi$
$733$	42.8837	1.58395	0.791973	+	0.610556i	$0.209054\pi$
$734$	0	0
$735$	3.84468	0.141813
$736$	0	0
$737$	17.2375	0.634950
$738$	0	0
$739$	−19.4259	−0.714592	−0.357296	−	0.933991i	$-0.616301\pi$
$739$	−19.4259	−0.714592	−0.357296	+	0.933991i	$0.616301\pi$
$740$	0	0
$741$	9.23746	0.339347
$742$	0	0
$743$	51.7305	1.89781	0.948904	−	0.315564i	$-0.102194\pi$
$743$	51.7305	1.89781	0.948904	+	0.315564i	$0.102194\pi$
$744$	0	0
$745$	26.8327	0.983075
$746$	0	0
$747$	−22.3668	−0.818357
$748$	0	0
$749$	10.8186	0.395305
$750$	0	0
$751$	23.9428	0.873685	0.436843	−	0.899538i	$-0.356097\pi$
$751$	23.9428	0.873685	0.436843	+	0.899538i	$0.356097\pi$
$752$	0	0
$753$	−28.1312	−1.02516
$754$	0	0
$755$	6.32069	0.230033
$756$	0	0
$757$	2.48901	0.0904645	0.0452322	−	0.998976i	$-0.485597\pi$
$757$	2.48901	0.0904645	0.0452322	+	0.998976i	$0.485597\pi$
$758$	0	0
$759$	−5.06315	−0.183781
$760$	0	0
$761$	−4.14528	−0.150266	−0.0751332	−	0.997174i	$-0.523938\pi$
$761$	−4.14528	−0.150266	−0.0751332	+	0.997174i	$0.523938\pi$
$762$	0	0
$763$	−12.2776	−0.444478
$764$	0	0
$765$	−32.0932	−1.16033
$766$	0	0
$767$	−5.90078	−0.213065
$768$	0	0
$769$	−11.0442	−0.398263	−0.199131	−	0.979973i	$-0.563812\pi$
$769$	−11.0442	−0.398263	−0.199131	+	0.979973i	$0.563812\pi$
$770$	0	0
$771$	−63.9379	−2.30267
$772$	0	0
$773$	42.3426	1.52296	0.761479	−	0.648189i	$-0.224473\pi$
$773$	42.3426	1.52296	0.761479	+	0.648189i	$0.224473\pi$
$774$	0	0
$775$	−6.29656	−0.226179
$776$	0	0
$777$	10.6964	0.383731
$778$	0	0
$779$	−4.66332	−0.167081
$780$	0	0
$781$	−22.3006	−0.797979
$782$	0	0
$783$	−0.418817	−0.0149673
$784$	0	0
$785$	−14.0100	−0.500040
$786$	0	0
$787$	−49.5711	−1.76702	−0.883510	−	0.468412i	$-0.844826\pi$
$787$	−49.5711	−1.76702	−0.883510	+	0.468412i	$0.844826\pi$
$788$	0	0
$789$	−83.4288	−2.97014
$790$	0	0
$791$	0.696393	0.0247609
$792$	0	0
$793$	41.8346	1.48559
$794$	0	0
$795$	23.4109	0.830300
$796$	0	0
$797$	−38.8276	−1.37534	−0.687672	−	0.726022i	$-0.741367\pi$
$797$	−38.8276	−1.37534	−0.687672	+	0.726022i	$0.741367\pi$
$798$	0	0
$799$	16.4188	0.580856
$800$	0	0
$801$	−6.23042	−0.220141
$802$	0	0
$803$	−37.7685	−1.33282
$804$	0	0
$805$	0.818644	0.0288534
$806$	0	0
$807$	−24.7295	−0.870518
$808$	0	0
$809$	13.4920	0.474354	0.237177	−	0.971466i	$-0.423778\pi$
$809$	13.4920	0.474354	0.237177	+	0.971466i	$0.423778\pi$
$810$	0	0
$811$	−24.5552	−0.862248	−0.431124	−	0.902293i	$-0.641883\pi$
$811$	−24.5552	−0.862248	−0.431124	+	0.902293i	$0.641883\pi$
$812$	0	0
$813$	−54.0412	−1.89531
$814$	0	0
$815$	−15.6002	−0.546450
$816$	0	0
$817$	11.9339	0.417513
$818$	0	0
$819$	14.6302	0.511222
$820$	0	0
$821$	37.9339	1.32390	0.661950	−	0.749548i	$-0.269729\pi$
$821$	37.9339	1.32390	0.661950	+	0.749548i	$0.269729\pi$
$822$	0	0
$823$	−10.0141	−0.349069	−0.174535	−	0.984651i	$-0.555842\pi$
$823$	−10.0141	−0.349069	−0.174535	+	0.984651i	$0.555842\pi$
$824$	0	0
$825$	26.1643	0.910923
$826$	0	0
$827$	35.7685	1.24379	0.621896	−	0.783100i	$-0.286363\pi$
$827$	35.7685	1.24379	0.621896	+	0.783100i	$0.286363\pi$
$828$	0	0
$829$	−25.6753	−0.891739	−0.445869	−	0.895098i	$-0.647105\pi$
$829$	−25.6753	−0.891739	−0.445869	+	0.895098i	$0.647105\pi$
$830$	0	0
$831$	−20.1122	−0.697685
$832$	0	0
$833$	5.27053	0.182613
$834$	0	0
$835$	18.1833	0.629258
$836$	0	0
$837$	−7.27053	−0.251306
$838$	0	0
$839$	−42.7856	−1.47712	−0.738561	−	0.674187i	$-0.764494\pi$
$839$	−42.7856	−1.47712	−0.738561	+	0.674187i	$0.764494\pi$
$840$	0	0
$841$	−28.9851	−0.999485
$842$	0	0
$843$	−32.9268	−1.13406
$844$	0	0
$845$	1.80156	0.0619756
$846$	0	0
$847$	0.303607	0.0104321
$848$	0	0
$849$	−21.6332	−0.742451
$850$	0	0
$851$	2.27758	0.0780743
$852$	0	0
$853$	−17.8447	−0.610990	−0.305495	−	0.952194i	$-0.598822\pi$
$853$	−17.8447	−0.610990	−0.305495	+	0.952194i	$0.598822\pi$
$854$	0	0
$855$	−6.08918	−0.208246
$856$	0	0
$857$	37.8116	1.29162	0.645810	−	0.763498i	$-0.276520\pi$
$857$	37.8116	1.29162	0.645810	+	0.763498i	$0.276520\pi$
$858$	0	0
$859$	−22.8558	−0.779828	−0.389914	−	0.920851i	$-0.627495\pi$
$859$	−22.8558	−0.779828	−0.389914	+	0.920851i	$0.627495\pi$
$860$	0	0
$861$	−12.5741	−0.428526
$862$	0	0
$863$	13.3177	0.453339	0.226670	−	0.973972i	$-0.427216\pi$
$863$	13.3177	0.453339	0.226670	+	0.973972i	$0.427216\pi$
$864$	0	0
$865$	32.1404	1.09281
$866$	0	0
$867$	−29.0631	−0.987036
$868$	0	0
$869$	36.2445	1.22951
$870$	0	0
$871$	−18.0561	−0.611808
$872$	0	0
$873$	75.4017	2.55196
$874$	0	0
$875$	−11.3597	−0.384028
$876$	0	0
$877$	33.5571	1.13314	0.566571	−	0.824013i	$-0.308270\pi$
$877$	33.5571	1.13314	0.566571	+	0.824013i	$0.308270\pi$
$878$	0	0
$879$	37.8387	1.27627
$880$	0	0
$881$	−52.5641	−1.77093	−0.885465	−	0.464707i	$-0.846160\pi$
$881$	−52.5641	−1.77093	−0.885465	+	0.464707i	$0.846160\pi$
$882$	0	0
$883$	5.68445	0.191297	0.0956484	−	0.995415i	$-0.469508\pi$
$883$	5.68445	0.191297	0.0956484	+	0.995415i	$0.469508\pi$
$884$	0	0
$885$	6.62216	0.222601
$886$	0	0
$887$	26.3898	0.886082	0.443041	−	0.896501i	$-0.353900\pi$
$887$	26.3898	0.886082	0.443041	+	0.896501i	$0.353900\pi$
$888$	0	0
$889$	5.68935	0.190815
$890$	0	0
$891$	−11.6894	−0.391608
$892$	0	0
$893$	3.11521	0.104247
$894$	0	0
$895$	9.15233	0.305929
$896$	0	0
$897$	5.30361	0.177082
$898$	0	0
$899$	0.259449	0.00865309
$900$	0	0
$901$	32.0932	1.06918
$902$	0	0
$903$	32.1784	1.07083
$904$	0	0
$905$	13.9900	0.465042
$906$	0	0
$907$	−0.211430	−0.00702041	−0.00351021	−	0.999994i	$-0.501117\pi$
$907$	−0.211430	−0.00702041	−0.00351021	+	0.999994i	$0.501117\pi$
$908$	0	0
$909$	−72.8086	−2.41491
$910$	0	0
$911$	−21.3787	−0.708308	−0.354154	−	0.935187i	$-0.615231\pi$
$911$	−21.3787	−0.708308	−0.354154	+	0.935187i	$0.615231\pi$
$912$	0	0
$913$	−17.1293	−0.566897
$914$	0	0
$915$	−46.9490	−1.55209
$916$	0	0
$917$	−15.2044	−0.502093
$918$	0	0
$919$	−16.8186	−0.554796	−0.277398	−	0.960755i	$-0.589472\pi$
$919$	−16.8186	−0.554796	−0.277398	+	0.960755i	$0.589472\pi$
$920$	0	0
$921$	43.3315	1.42782
$922$	0	0
$923$	23.3597	0.768894
$924$	0	0
$925$	−11.7696	−0.386981
$926$	0	0
$927$	−48.5120	−1.59334
$928$	0	0
$929$	−35.3497	−1.15979	−0.579893	−	0.814693i	$-0.696905\pi$
$929$	−35.3497	−1.15979	−0.579893	+	0.814693i	$0.696905\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	4.22551	0.138337
$934$	0	0
$935$	−24.5782	−0.803793
$936$	0	0
$937$	4.78153	0.156206	0.0781029	−	0.996945i	$-0.475114\pi$
$937$	4.78153	0.156206	0.0781029	+	0.996945i	$0.475114\pi$
$938$	0	0
$939$	−79.8758	−2.60665
$940$	0	0
$941$	13.6373	0.444563	0.222281	−	0.974983i	$-0.428650\pi$
$941$	13.6373	0.444563	0.222281	+	0.974983i	$0.428650\pi$
$942$	0	0
$943$	−2.67740	−0.0871883
$944$	0	0
$945$	−4.88479	−0.158902
$946$	0	0
$947$	59.3638	1.92906	0.964531	−	0.263968i	$-0.0850313\pi$
$947$	59.3638	1.92906	0.964531	+	0.263968i	$0.0850313\pi$
$948$	0	0
$949$	39.5622	1.28424
$950$	0	0
$951$	−60.6392	−1.96636
$952$	0	0
$953$	−52.4790	−1.69996	−0.849980	−	0.526815i	$-0.823386\pi$
$953$	−52.4790	−1.69996	−0.849980	+	0.526815i	$0.823386\pi$
$954$	0	0
$955$	−21.7465	−0.703702
$956$	0	0
$957$	−1.07809	−0.0348498
$958$	0	0
$959$	−3.70344	−0.119590
$960$	0	0
$961$	−26.4960	−0.854711
$962$	0	0
$963$	−46.2014	−1.48882
$964$	0	0
$965$	22.4508	0.722717
$966$	0	0
$967$	21.9900	0.707149	0.353575	−	0.935406i	$-0.384966\pi$
$967$	21.9900	0.707149	0.353575	+	0.935406i	$0.384966\pi$
$968$	0	0
$969$	−14.2114	−0.456537
$970$	0	0
$971$	40.9309	1.31353	0.656767	−	0.754094i	$-0.271924\pi$
$971$	40.9309	1.31353	0.656767	+	0.754094i	$0.271924\pi$
$972$	0	0
$973$	−11.6704	−0.374135
$974$	0	0
$975$	−27.4069	−0.877722
$976$	0	0
$977$	8.56924	0.274154	0.137077	−	0.990560i	$-0.456229\pi$
$977$	8.56924	0.274154	0.137077	+	0.990560i	$0.456229\pi$
$978$	0	0
$979$	−4.77149	−0.152497
$980$	0	0
$981$	52.4318	1.67402
$982$	0	0
$983$	35.2654	1.12479	0.562396	−	0.826868i	$-0.309880\pi$
$983$	35.2654	1.12479	0.562396	+	0.826868i	$0.309880\pi$
$984$	0	0
$985$	35.3736	1.12710
$986$	0	0
$987$	8.39983	0.267369
$988$	0	0
$989$	6.85172	0.217872
$990$	0	0
$991$	1.78367	0.0566600	0.0283300	−	0.999599i	$-0.490981\pi$
$991$	1.78367	0.0566600	0.0283300	+	0.999599i	$0.490981\pi$
$992$	0	0
$993$	76.5391	2.42890
$994$	0	0
$995$	3.69044	0.116995
$996$	0	0
$997$	37.4299	1.18542	0.592708	−	0.805417i	$-0.298059\pi$
$997$	37.4299	1.18542	0.592708	+	0.805417i	$0.298059\pi$
$998$	0	0
$999$	−13.5901	−0.429973

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.bn.1.1		3
4.3	odd	2		8512.2.a.bl.1.3		3
8.3	odd	2		2128.2.a.r.1.1		3
8.5	even	2		532.2.a.e.1.3	✓	3
24.5	odd	2		4788.2.a.o.1.2		3
56.13	odd	2		3724.2.a.i.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
532.2.a.e.1.3	✓	3	8.5	even	2
2128.2.a.r.1.1		3	8.3	odd	2
3724.2.a.i.1.1		3	56.13	odd	2
4788.2.a.o.1.2		3	24.5	odd	2
8512.2.a.bl.1.3		3	4.3	odd	2
8512.2.a.bn.1.1		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8512.a (trivial)

\(f(q)\)	\(=\)	\(q-2.69639 q^{3} -1.42586 q^{5} -1.00000 q^{7} +4.27053 q^{9} +O(q^{10})\)	\(q-2.69639 q^{3} -1.42586 q^{5} -1.00000 q^{7} +4.27053 q^{9} +3.27053 q^{11} -3.42586 q^{13} +3.84468 q^{15} +5.27053 q^{17} +1.00000 q^{19} +2.69639 q^{21} +0.574141 q^{23} -2.96693 q^{25} -3.42586 q^{27} +0.122252 q^{29} +2.12225 q^{31} -8.81864 q^{33} +1.42586 q^{35} +3.96693 q^{37} +9.23746 q^{39} -4.66332 q^{41} +11.9339 q^{43} -6.08918 q^{45} +3.11521 q^{47} +1.00000 q^{49} -14.2114 q^{51} +6.08918 q^{53} -4.66332 q^{55} -2.69639 q^{57} +1.72242 q^{59} -12.2114 q^{61} -4.27053 q^{63} +4.88479 q^{65} +5.27053 q^{67} -1.54811 q^{69} -6.81864 q^{71} -11.5481 q^{73} +8.00000 q^{75} -3.27053 q^{77} +11.0821 q^{79} -3.57414 q^{81} -5.23746 q^{83} -7.51504 q^{85} -0.329638 q^{87} -1.45893 q^{89} +3.42586 q^{91} -5.72242 q^{93} -1.42586 q^{95} +17.6563 q^{97} +13.9669 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} - 2 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10}) \) 3 * q + q^3 - 2 * q^5 - 3 * q^7 + 6 * q^9	\( 3 q + q^{3} - 2 q^{5} - 3 q^{7} + 6 q^{9} + 3 q^{11} - 8 q^{13} + 7 q^{15} + 9 q^{17} + 3 q^{19} - q^{21} + 4 q^{23} + 7 q^{25} - 8 q^{27} - 11 q^{29} - 5 q^{31} - 6 q^{33} + 2 q^{35} - 4 q^{37} + 5 q^{39} + 11 q^{41} + 4 q^{43} + 9 q^{45} - 2 q^{47} + 3 q^{49} - 4 q^{51} - 9 q^{53} + 11 q^{55} + q^{57} + 12 q^{59} + 2 q^{61} - 6 q^{63} + 26 q^{65} + 9 q^{67} + 9 q^{69} - 21 q^{73} + 24 q^{75} - 3 q^{77} + 6 q^{79} - 13 q^{81} + 7 q^{83} + 7 q^{85} - 26 q^{87} - 18 q^{89} + 8 q^{91} - 24 q^{93} - 2 q^{95} + 28 q^{97} + 26 q^{99}+O(q^{100}) \) 3 * q + q^3 - 2 * q^5 - 3 * q^7 + 6 * q^9 + 3 * q^11 - 8 * q^13 + 7 * q^15 + 9 * q^17 + 3 * q^19 - q^21 + 4 * q^23 + 7 * q^25 - 8 * q^27 - 11 * q^29 - 5 * q^31 - 6 * q^33 + 2 * q^35 - 4 * q^37 + 5 * q^39 + 11 * q^41 + 4 * q^43 + 9 * q^45 - 2 * q^47 + 3 * q^49 - 4 * q^51 - 9 * q^53 + 11 * q^55 + q^57 + 12 * q^59 + 2 * q^61 - 6 * q^63 + 26 * q^65 + 9 * q^67 + 9 * q^69 - 21 * q^73 + 24 * q^75 - 3 * q^77 + 6 * q^79 - 13 * q^81 + 7 * q^83 + 7 * q^85 - 26 * q^87 - 18 * q^89 + 8 * q^91 - 24 * q^93 - 2 * q^95 + 28 * q^97 + 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more