LMFDB - Embedded newform 8730.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8730,2,Mod(1,8730)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8730, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8730.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8730 = 2 \cdot 3^{2} \cdot 5 \cdot 97 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8730.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:	$69.7094009646$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2910)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	8730.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{8} -1.00000 q^{10} +4.82843 q^{11} -3.41421 q^{13} +1.00000 q^{16} +0.585786 q^{17} -2.58579 q^{19} -1.00000 q^{20} +4.82843 q^{22} -4.24264 q^{23} +1.00000 q^{25} -3.41421 q^{26} -10.4853 q^{29} -3.17157 q^{31} +1.00000 q^{32} +0.585786 q^{34} -4.58579 q^{37} -2.58579 q^{38} -1.00000 q^{40} +6.24264 q^{41} +5.65685 q^{43} +4.82843 q^{44} -4.24264 q^{46} +4.48528 q^{47} -7.00000 q^{49} +1.00000 q^{50} -3.41421 q^{52} +9.65685 q^{53} -4.82843 q^{55} -10.4853 q^{58} +1.17157 q^{59} -8.82843 q^{61} -3.17157 q^{62} +1.00000 q^{64} +3.41421 q^{65} -12.7279 q^{67} +0.585786 q^{68} -0.242641 q^{71} -4.00000 q^{73} -4.58579 q^{74} -2.58579 q^{76} -0.828427 q^{79} -1.00000 q^{80} +6.24264 q^{82} -13.6569 q^{83} -0.585786 q^{85} +5.65685 q^{86} +4.82843 q^{88} -2.00000 q^{89} -4.24264 q^{92} +4.48528 q^{94} +2.58579 q^{95} -1.00000 q^{97} -7.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{8} - 2 q^{10} + 4 q^{11} - 4 q^{13} + 2 q^{16} + 4 q^{17} - 8 q^{19} - 2 q^{20} + 4 q^{22} + 2 q^{25} - 4 q^{26} - 4 q^{29} - 12 q^{31} + 2 q^{32} + 4 q^{34} - 12 q^{37} - 8 q^{38} - 2 q^{40} + 4 q^{41} + 4 q^{44} - 8 q^{47} - 14 q^{49} + 2 q^{50} - 4 q^{52} + 8 q^{53} - 4 q^{55} - 4 q^{58} + 8 q^{59} - 12 q^{61} - 12 q^{62} + 2 q^{64} + 4 q^{65} + 4 q^{68} + 8 q^{71} - 8 q^{73} - 12 q^{74} - 8 q^{76} + 4 q^{79} - 2 q^{80} + 4 q^{82} - 16 q^{83} - 4 q^{85} + 4 q^{88} - 4 q^{89} - 8 q^{94} + 8 q^{95} - 2 q^{97} - 14 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^8 - 2 * q^10 + 4 * q^11 - 4 * q^13 + 2 * q^16 + 4 * q^17 - 8 * q^19 - 2 * q^20 + 4 * q^22 + 2 * q^25 - 4 * q^26 - 4 * q^29 - 12 * q^31 + 2 * q^32 + 4 * q^34 - 12 * q^37 - 8 * q^38 - 2 * q^40 + 4 * q^41 + 4 * q^44 - 8 * q^47 - 14 * q^49 + 2 * q^50 - 4 * q^52 + 8 * q^53 - 4 * q^55 - 4 * q^58 + 8 * q^59 - 12 * q^61 - 12 * q^62 + 2 * q^64 + 4 * q^65 + 4 * q^68 + 8 * q^71 - 8 * q^73 - 12 * q^74 - 8 * q^76 + 4 * q^79 - 2 * q^80 + 4 * q^82 - 16 * q^83 - 4 * q^85 + 4 * q^88 - 4 * q^89 - 8 * q^94 + 8 * q^95 - 2 * q^97 - 14 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	4.82843	1.45583	0.727913	−	0.685670i	$-0.240491\pi$
$11$	4.82843	1.45583	0.727913	+	0.685670i	$0.240491\pi$
$12$	0	0
$13$	−3.41421	−0.946932	−0.473466	−	0.880812i	$-0.656997\pi$
$13$	−3.41421	−0.946932	−0.473466	+	0.880812i	$0.656997\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	0.585786	0.142074	0.0710370	−	0.997474i	$-0.477369\pi$
$17$	0.585786	0.142074	0.0710370	+	0.997474i	$0.477369\pi$
$18$	0	0
$19$	−2.58579	−0.593220	−0.296610	−	0.954999i	$-0.595856\pi$
$19$	−2.58579	−0.593220	−0.296610	+	0.954999i	$0.595856\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	4.82843	1.02942
$23$	−4.24264	−0.884652	−0.442326	−	0.896854i	$-0.645847\pi$
$23$	−4.24264	−0.884652	−0.442326	+	0.896854i	$0.645847\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−3.41421	−0.669582
$27$	0	0
$28$	0	0
$29$	−10.4853	−1.94707	−0.973534	−	0.228543i	$-0.926604\pi$
$29$	−10.4853	−1.94707	−0.973534	+	0.228543i	$0.926604\pi$
$30$	0	0
$31$	−3.17157	−0.569631	−0.284816	−	0.958582i	$-0.591932\pi$
$31$	−3.17157	−0.569631	−0.284816	+	0.958582i	$0.591932\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0.585786	0.100462
$35$	0	0
$36$	0	0
$37$	−4.58579	−0.753899	−0.376949	−	0.926234i	$-0.623027\pi$
$37$	−4.58579	−0.753899	−0.376949	+	0.926234i	$0.623027\pi$
$38$	−2.58579	−0.419470
$39$	0	0
$40$	−1.00000	−0.158114
$41$	6.24264	0.974937	0.487468	−	0.873141i	$-0.337920\pi$
$41$	6.24264	0.974937	0.487468	+	0.873141i	$0.337920\pi$
$42$	0	0
$43$	5.65685	0.862662	0.431331	−	0.902194i	$-0.358044\pi$
$43$	5.65685	0.862662	0.431331	+	0.902194i	$0.358044\pi$
$44$	4.82843	0.727913
$45$	0	0
$46$	−4.24264	−0.625543
$47$	4.48528	0.654246	0.327123	−	0.944982i	$-0.393921\pi$
$47$	4.48528	0.654246	0.327123	+	0.944982i	$0.393921\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	1.00000	0.141421
$51$	0	0
$52$	−3.41421	−0.473466
$53$	9.65685	1.32647	0.663235	−	0.748411i	$-0.269183\pi$
$53$	9.65685	1.32647	0.663235	+	0.748411i	$0.269183\pi$
$54$	0	0
$55$	−4.82843	−0.651065
$56$	0	0
$57$	0	0
$58$	−10.4853	−1.37678
$59$	1.17157	0.152526	0.0762629	−	0.997088i	$-0.475701\pi$
$59$	1.17157	0.152526	0.0762629	+	0.997088i	$0.475701\pi$
$60$	0	0
$61$	−8.82843	−1.13036	−0.565182	−	0.824966i	$-0.691194\pi$
$61$	−8.82843	−1.13036	−0.565182	+	0.824966i	$0.691194\pi$
$62$	−3.17157	−0.402790
$63$	0	0
$64$	1.00000	0.125000
$65$	3.41421	0.423481
$66$	0	0
$67$	−12.7279	−1.55496	−0.777482	−	0.628906i	$-0.783503\pi$
$67$	−12.7279	−1.55496	−0.777482	+	0.628906i	$0.783503\pi$
$68$	0.585786	0.0710370
$69$	0	0
$70$	0	0
$71$	−0.242641	−0.0287962	−0.0143981	−	0.999896i	$-0.504583\pi$
$71$	−0.242641	−0.0287962	−0.0143981	+	0.999896i	$0.504583\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	−4.58579	−0.533087
$75$	0	0
$76$	−2.58579	−0.296610
$77$	0	0
$78$	0	0
$79$	−0.828427	−0.0932053	−0.0466027	−	0.998914i	$-0.514839\pi$
$79$	−0.828427	−0.0932053	−0.0466027	+	0.998914i	$0.514839\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	6.24264	0.689384
$83$	−13.6569	−1.49903	−0.749517	−	0.661985i	$-0.769714\pi$
$83$	−13.6569	−1.49903	−0.749517	+	0.661985i	$0.769714\pi$
$84$	0	0
$85$	−0.585786	−0.0635375
$86$	5.65685	0.609994
$87$	0	0
$88$	4.82843	0.514712
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	0	0
$92$	−4.24264	−0.442326
$93$	0	0
$94$	4.48528	0.462621
$95$	2.58579	0.265296
$96$	0	0
$97$	−1.00000	−0.101535
$97$	−1.00000	−0.101535
$98$	−7.00000	−0.707107
$99$	0	0
$100$	1.00000	0.100000
$101$	15.3137	1.52377	0.761885	−	0.647712i	$-0.224274\pi$
$101$	15.3137	1.52377	0.761885	+	0.647712i	$0.224274\pi$
$102$	0	0
$103$	−13.6569	−1.34565	−0.672825	−	0.739802i	$-0.734919\pi$
$103$	−13.6569	−1.34565	−0.672825	+	0.739802i	$0.734919\pi$
$104$	−3.41421	−0.334791
$105$	0	0
$106$	9.65685	0.937957
$107$	−6.82843	−0.660129	−0.330064	−	0.943958i	$-0.607070\pi$
$107$	−6.82843	−0.660129	−0.330064	+	0.943958i	$0.607070\pi$
$108$	0	0
$109$	−17.3137	−1.65835	−0.829176	−	0.558987i	$-0.811190\pi$
$109$	−17.3137	−1.65835	−0.829176	+	0.558987i	$0.811190\pi$
$110$	−4.82843	−0.460372
$111$	0	0
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	4.24264	0.395628
$116$	−10.4853	−0.973534
$117$	0	0
$118$	1.17157	0.107852
$119$	0	0
$120$	0	0
$121$	12.3137	1.11943
$122$	−8.82843	−0.799288
$123$	0	0
$124$	−3.17157	−0.284816
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	3.41421	0.299446
$131$	−5.65685	−0.494242	−0.247121	−	0.968985i	$-0.579484\pi$
$131$	−5.65685	−0.494242	−0.247121	+	0.968985i	$0.579484\pi$
$132$	0	0
$133$	0	0
$134$	−12.7279	−1.09952
$135$	0	0
$136$	0.585786	0.0502308
$137$	0.585786	0.0500471	0.0250236	−	0.999687i	$-0.492034\pi$
$137$	0.585786	0.0500471	0.0250236	+	0.999687i	$0.492034\pi$
$138$	0	0
$139$	−13.8995	−1.17894	−0.589470	−	0.807790i	$-0.700663\pi$
$139$	−13.8995	−1.17894	−0.589470	+	0.807790i	$0.700663\pi$
$140$	0	0
$141$	0	0
$142$	−0.242641	−0.0203620
$143$	−16.4853	−1.37857
$144$	0	0
$145$	10.4853	0.870755
$146$	−4.00000	−0.331042
$147$	0	0
$148$	−4.58579	−0.376949
$149$	−4.34315	−0.355804	−0.177902	−	0.984048i	$-0.556931\pi$
$149$	−4.34315	−0.355804	−0.177902	+	0.984048i	$0.556931\pi$
$150$	0	0
$151$	−11.3137	−0.920697	−0.460348	−	0.887738i	$-0.652275\pi$
$151$	−11.3137	−0.920697	−0.460348	+	0.887738i	$0.652275\pi$
$152$	−2.58579	−0.209735
$153$	0	0
$154$	0	0
$155$	3.17157	0.254747
$156$	0	0
$157$	14.7279	1.17542	0.587708	−	0.809073i	$-0.300030\pi$
$157$	14.7279	1.17542	0.587708	+	0.809073i	$0.300030\pi$
$158$	−0.828427	−0.0659061
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	0	0
$163$	11.3137	0.886158	0.443079	−	0.896483i	$-0.353886\pi$
$163$	11.3137	0.886158	0.443079	+	0.896483i	$0.353886\pi$
$164$	6.24264	0.487468
$165$	0	0
$166$	−13.6569	−1.05998
$167$	−14.3431	−1.10991	−0.554953	−	0.831882i	$-0.687264\pi$
$167$	−14.3431	−1.10991	−0.554953	+	0.831882i	$0.687264\pi$
$168$	0	0
$169$	−1.34315	−0.103319
$170$	−0.585786	−0.0449278
$171$	0	0
$172$	5.65685	0.431331
$173$	18.9706	1.44231	0.721153	−	0.692776i	$-0.243613\pi$
$173$	18.9706	1.44231	0.721153	+	0.692776i	$0.243613\pi$
$174$	0	0
$175$	0	0
$176$	4.82843	0.363956
$177$	0	0
$178$	−2.00000	−0.149906
$179$	16.4853	1.23217	0.616084	−	0.787681i	$-0.288718\pi$
$179$	16.4853	1.23217	0.616084	+	0.787681i	$0.288718\pi$
$180$	0	0
$181$	22.7279	1.68935	0.844677	−	0.535277i	$-0.179793\pi$
$181$	22.7279	1.68935	0.844677	+	0.535277i	$0.179793\pi$
$182$	0	0
$183$	0	0
$184$	−4.24264	−0.312772
$185$	4.58579	0.337154
$186$	0	0
$187$	2.82843	0.206835
$188$	4.48528	0.327123
$189$	0	0
$190$	2.58579	0.187593
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	0	0
$193$	−4.34315	−0.312626	−0.156313	−	0.987708i	$-0.549961\pi$
$193$	−4.34315	−0.312626	−0.156313	+	0.987708i	$0.549961\pi$
$194$	−1.00000	−0.0717958
$195$	0	0
$196$	−7.00000	−0.500000
$197$	10.9706	0.781620	0.390810	−	0.920471i	$-0.372195\pi$
$197$	10.9706	0.781620	0.390810	+	0.920471i	$0.372195\pi$
$198$	0	0
$199$	−13.1716	−0.933708	−0.466854	−	0.884334i	$-0.654613\pi$
$199$	−13.1716	−0.933708	−0.466854	+	0.884334i	$0.654613\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	15.3137	1.07747
$203$	0	0
$204$	0	0
$205$	−6.24264	−0.436005
$206$	−13.6569	−0.951518
$207$	0	0
$208$	−3.41421	−0.236733
$209$	−12.4853	−0.863625
$210$	0	0
$211$	27.5563	1.89706	0.948529	−	0.316691i	$-0.102572\pi$
$211$	27.5563	1.89706	0.948529	+	0.316691i	$0.102572\pi$
$212$	9.65685	0.663235
$213$	0	0
$214$	−6.82843	−0.466782
$215$	−5.65685	−0.385794
$216$	0	0
$217$	0	0
$218$	−17.3137	−1.17263
$219$	0	0
$220$	−4.82843	−0.325532
$221$	−2.00000	−0.134535
$222$	0	0
$223$	3.31371	0.221902	0.110951	−	0.993826i	$-0.464610\pi$
$223$	3.31371	0.221902	0.110951	+	0.993826i	$0.464610\pi$
$224$	0	0
$225$	0	0
$226$	6.00000	0.399114
$227$	−7.17157	−0.475994	−0.237997	−	0.971266i	$-0.576491\pi$
$227$	−7.17157	−0.475994	−0.237997	+	0.971266i	$0.576491\pi$
$228$	0	0
$229$	−24.8284	−1.64071	−0.820354	−	0.571856i	$-0.806224\pi$
$229$	−24.8284	−1.64071	−0.820354	+	0.571856i	$0.806224\pi$
$230$	4.24264	0.279751
$231$	0	0
$232$	−10.4853	−0.688392
$233$	8.38478	0.549305	0.274652	−	0.961544i	$-0.411437\pi$
$233$	8.38478	0.549305	0.274652	+	0.961544i	$0.411437\pi$
$234$	0	0
$235$	−4.48528	−0.292587
$236$	1.17157	0.0762629
$237$	0	0
$238$	0	0
$239$	−13.4142	−0.867693	−0.433847	−	0.900987i	$-0.642844\pi$
$239$	−13.4142	−0.867693	−0.433847	+	0.900987i	$0.642844\pi$
$240$	0	0
$241$	−15.3137	−0.986443	−0.493221	−	0.869904i	$-0.664181\pi$
$241$	−15.3137	−0.986443	−0.493221	+	0.869904i	$0.664181\pi$
$242$	12.3137	0.791555
$243$	0	0
$244$	−8.82843	−0.565182
$245$	7.00000	0.447214
$246$	0	0
$247$	8.82843	0.561739
$248$	−3.17157	−0.201395
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−11.3137	−0.714115	−0.357057	−	0.934082i	$-0.616220\pi$
$251$	−11.3137	−0.714115	−0.357057	+	0.934082i	$0.616220\pi$
$252$	0	0
$253$	−20.4853	−1.28790
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−22.2426	−1.38746	−0.693729	−	0.720236i	$-0.744033\pi$
$257$	−22.2426	−1.38746	−0.693729	+	0.720236i	$0.744033\pi$
$258$	0	0
$259$	0	0
$260$	3.41421	0.211741
$261$	0	0
$262$	−5.65685	−0.349482
$263$	17.4142	1.07381	0.536903	−	0.843644i	$-0.319594\pi$
$263$	17.4142	1.07381	0.536903	+	0.843644i	$0.319594\pi$
$264$	0	0
$265$	−9.65685	−0.593216
$266$	0	0
$267$	0	0
$268$	−12.7279	−0.777482
$269$	−22.9706	−1.40054	−0.700270	−	0.713878i	$-0.746937\pi$
$269$	−22.9706	−1.40054	−0.700270	+	0.713878i	$0.746937\pi$
$270$	0	0
$271$	−26.1421	−1.58802	−0.794011	−	0.607904i	$-0.792011\pi$
$271$	−26.1421	−1.58802	−0.794011	+	0.607904i	$0.792011\pi$
$272$	0.585786	0.0355185
$273$	0	0
$274$	0.585786	0.0353887
$275$	4.82843	0.291165
$276$	0	0
$277$	22.2426	1.33643	0.668215	−	0.743968i	$-0.267058\pi$
$277$	22.2426	1.33643	0.668215	+	0.743968i	$0.267058\pi$
$278$	−13.8995	−0.833636
$279$	0	0
$280$	0	0
$281$	−5.07107	−0.302515	−0.151257	−	0.988494i	$-0.548332\pi$
$281$	−5.07107	−0.302515	−0.151257	+	0.988494i	$0.548332\pi$
$282$	0	0
$283$	25.6569	1.52514	0.762571	−	0.646905i	$-0.223937\pi$
$283$	25.6569	1.52514	0.762571	+	0.646905i	$0.223937\pi$
$284$	−0.242641	−0.0143981
$285$	0	0
$286$	−16.4853	−0.974795
$287$	0	0
$288$	0	0
$289$	−16.6569	−0.979815
$290$	10.4853	0.615717
$291$	0	0
$292$	−4.00000	−0.234082
$293$	−10.9706	−0.640907	−0.320454	−	0.947264i	$-0.603835\pi$
$293$	−10.9706	−0.640907	−0.320454	+	0.947264i	$0.603835\pi$
$294$	0	0
$295$	−1.17157	−0.0682116
$296$	−4.58579	−0.266543
$297$	0	0
$298$	−4.34315	−0.251592
$299$	14.4853	0.837705
$300$	0	0
$301$	0	0
$302$	−11.3137	−0.651031
$303$	0	0
$304$	−2.58579	−0.148305
$305$	8.82843	0.505514
$306$	0	0
$307$	16.4853	0.940865	0.470432	−	0.882436i	$-0.344098\pi$
$307$	16.4853	0.940865	0.470432	+	0.882436i	$0.344098\pi$
$308$	0	0
$309$	0	0
$310$	3.17157	0.180133
$311$	−16.2426	−0.921036	−0.460518	−	0.887650i	$-0.652336\pi$
$311$	−16.2426	−0.921036	−0.460518	+	0.887650i	$0.652336\pi$
$312$	0	0
$313$	3.65685	0.206698	0.103349	−	0.994645i	$-0.467044\pi$
$313$	3.65685	0.206698	0.103349	+	0.994645i	$0.467044\pi$
$314$	14.7279	0.831145
$315$	0	0
$316$	−0.828427	−0.0466027
$317$	−9.79899	−0.550366	−0.275183	−	0.961392i	$-0.588738\pi$
$317$	−9.79899	−0.550366	−0.275183	+	0.961392i	$0.588738\pi$
$318$	0	0
$319$	−50.6274	−2.83459
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	0	0
$323$	−1.51472	−0.0842812
$324$	0	0
$325$	−3.41421	−0.189386
$326$	11.3137	0.626608
$327$	0	0
$328$	6.24264	0.344692
$329$	0	0
$330$	0	0
$331$	−24.7279	−1.35917	−0.679585	−	0.733597i	$-0.737840\pi$
$331$	−24.7279	−1.35917	−0.679585	+	0.733597i	$0.737840\pi$
$332$	−13.6569	−0.749517
$333$	0	0
$334$	−14.3431	−0.784822
$335$	12.7279	0.695401
$336$	0	0
$337$	−30.4853	−1.66064	−0.830320	−	0.557288i	$-0.811842\pi$
$337$	−30.4853	−1.66064	−0.830320	+	0.557288i	$0.811842\pi$
$338$	−1.34315	−0.0730575
$339$	0	0
$340$	−0.585786	−0.0317687
$341$	−15.3137	−0.829284
$342$	0	0
$343$	0	0
$344$	5.65685	0.304997
$345$	0	0
$346$	18.9706	1.01986
$347$	−31.1127	−1.67022	−0.835109	−	0.550085i	$-0.814595\pi$
$347$	−31.1127	−1.67022	−0.835109	+	0.550085i	$0.814595\pi$
$348$	0	0
$349$	−9.75736	−0.522299	−0.261150	−	0.965298i	$-0.584102\pi$
$349$	−9.75736	−0.522299	−0.261150	+	0.965298i	$0.584102\pi$
$350$	0	0
$351$	0	0
$352$	4.82843	0.257356
$353$	15.6569	0.833330	0.416665	−	0.909060i	$-0.363199\pi$
$353$	15.6569	0.833330	0.416665	+	0.909060i	$0.363199\pi$
$354$	0	0
$355$	0.242641	0.0128780
$356$	−2.00000	−0.106000
$357$	0	0
$358$	16.4853	0.871274
$359$	15.7574	0.831642	0.415821	−	0.909447i	$-0.363494\pi$
$359$	15.7574	0.831642	0.415821	+	0.909447i	$0.363494\pi$
$360$	0	0
$361$	−12.3137	−0.648090
$362$	22.7279	1.19455
$363$	0	0
$364$	0	0
$365$	4.00000	0.209370
$366$	0	0
$367$	−14.1421	−0.738213	−0.369107	−	0.929387i	$-0.620336\pi$
$367$	−14.1421	−0.738213	−0.369107	+	0.929387i	$0.620336\pi$
$368$	−4.24264	−0.221163
$369$	0	0
$370$	4.58579	0.238404
$371$	0	0
$372$	0	0
$373$	0.585786	0.0303309	0.0151654	−	0.999885i	$-0.495173\pi$
$373$	0.585786	0.0303309	0.0151654	+	0.999885i	$0.495173\pi$
$374$	2.82843	0.146254
$375$	0	0
$376$	4.48528	0.231311
$377$	35.7990	1.84374
$378$	0	0
$379$	−9.65685	−0.496039	−0.248020	−	0.968755i	$-0.579780\pi$
$379$	−9.65685	−0.496039	−0.248020	+	0.968755i	$0.579780\pi$
$380$	2.58579	0.132648
$381$	0	0
$382$	4.00000	0.204658
$383$	−34.8701	−1.78178	−0.890888	−	0.454222i	$-0.849917\pi$
$383$	−34.8701	−1.78178	−0.890888	+	0.454222i	$0.849917\pi$
$384$	0	0
$385$	0	0
$386$	−4.34315	−0.221060
$387$	0	0
$388$	−1.00000	−0.0507673
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	−2.48528	−0.125686
$392$	−7.00000	−0.353553
$393$	0	0
$394$	10.9706	0.552689
$395$	0.828427	0.0416827
$396$	0	0
$397$	10.9706	0.550597	0.275298	−	0.961359i	$-0.411223\pi$
$397$	10.9706	0.550597	0.275298	+	0.961359i	$0.411223\pi$
$398$	−13.1716	−0.660231
$399$	0	0
$400$	1.00000	0.0500000
$401$	−27.8995	−1.39323	−0.696617	−	0.717443i	$-0.745312\pi$
$401$	−27.8995	−1.39323	−0.696617	+	0.717443i	$0.745312\pi$
$402$	0	0
$403$	10.8284	0.539402
$404$	15.3137	0.761885
$405$	0	0
$406$	0	0
$407$	−22.1421	−1.09754
$408$	0	0
$409$	13.3137	0.658321	0.329160	−	0.944274i	$-0.393234\pi$
$409$	13.3137	0.658321	0.329160	+	0.944274i	$0.393234\pi$
$410$	−6.24264	−0.308302
$411$	0	0
$412$	−13.6569	−0.672825
$413$	0	0
$414$	0	0
$415$	13.6569	0.670389
$416$	−3.41421	−0.167396
$417$	0	0
$418$	−12.4853	−0.610675
$419$	12.9706	0.633653	0.316827	−	0.948483i	$-0.397383\pi$
$419$	12.9706	0.633653	0.316827	+	0.948483i	$0.397383\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	27.5563	1.34142
$423$	0	0
$424$	9.65685	0.468978
$425$	0.585786	0.0284148
$426$	0	0
$427$	0	0
$428$	−6.82843	−0.330064
$429$	0	0
$430$	−5.65685	−0.272798
$431$	−21.1716	−1.01980	−0.509899	−	0.860234i	$-0.670317\pi$
$431$	−21.1716	−1.01980	−0.509899	+	0.860234i	$0.670317\pi$
$432$	0	0
$433$	33.7990	1.62428	0.812138	−	0.583466i	$-0.198304\pi$
$433$	33.7990	1.62428	0.812138	+	0.583466i	$0.198304\pi$
$434$	0	0
$435$	0	0
$436$	−17.3137	−0.829176
$437$	10.9706	0.524793
$438$	0	0
$439$	18.6274	0.889038	0.444519	−	0.895769i	$-0.353375\pi$
$439$	18.6274	0.889038	0.444519	+	0.895769i	$0.353375\pi$
$440$	−4.82843	−0.230186
$441$	0	0
$442$	−2.00000	−0.0951303
$443$	33.4558	1.58954	0.794768	−	0.606914i	$-0.207593\pi$
$443$	33.4558	1.58954	0.794768	+	0.606914i	$0.207593\pi$
$444$	0	0
$445$	2.00000	0.0948091
$446$	3.31371	0.156909
$447$	0	0
$448$	0	0
$449$	38.4853	1.81623	0.908116	−	0.418718i	$-0.137520\pi$
$449$	38.4853	1.81623	0.908116	+	0.418718i	$0.137520\pi$
$450$	0	0
$451$	30.1421	1.41934
$452$	6.00000	0.282216
$453$	0	0
$454$	−7.17157	−0.336579
$455$	0	0
$456$	0	0
$457$	−30.2843	−1.41664	−0.708319	−	0.705892i	$-0.750546\pi$
$457$	−30.2843	−1.41664	−0.708319	+	0.705892i	$0.750546\pi$
$458$	−24.8284	−1.16016
$459$	0	0
$460$	4.24264	0.197814
$461$	−23.3137	−1.08583	−0.542914	−	0.839788i	$-0.682679\pi$
$461$	−23.3137	−1.08583	−0.542914	+	0.839788i	$0.682679\pi$
$462$	0	0
$463$	−17.7990	−0.827189	−0.413595	−	0.910461i	$-0.635727\pi$
$463$	−17.7990	−0.827189	−0.413595	+	0.910461i	$0.635727\pi$
$464$	−10.4853	−0.486767
$465$	0	0
$466$	8.38478	0.388417
$467$	−26.4853	−1.22559	−0.612796	−	0.790241i	$-0.709955\pi$
$467$	−26.4853	−1.22559	−0.612796	+	0.790241i	$0.709955\pi$
$468$	0	0
$469$	0	0
$470$	−4.48528	−0.206891
$471$	0	0
$472$	1.17157	0.0539260
$473$	27.3137	1.25589
$474$	0	0
$475$	−2.58579	−0.118644
$476$	0	0
$477$	0	0
$478$	−13.4142	−0.613552
$479$	0.686292	0.0313575	0.0156787	−	0.999877i	$-0.495009\pi$
$479$	0.686292	0.0313575	0.0156787	+	0.999877i	$0.495009\pi$
$480$	0	0
$481$	15.6569	0.713891
$482$	−15.3137	−0.697520
$483$	0	0
$484$	12.3137	0.559714
$485$	1.00000	0.0454077
$486$	0	0
$487$	7.17157	0.324975	0.162487	−	0.986711i	$-0.448048\pi$
$487$	7.17157	0.324975	0.162487	+	0.986711i	$0.448048\pi$
$488$	−8.82843	−0.399644
$489$	0	0
$490$	7.00000	0.316228
$491$	10.6274	0.479609	0.239804	−	0.970821i	$-0.422917\pi$
$491$	10.6274	0.479609	0.239804	+	0.970821i	$0.422917\pi$
$492$	0	0
$493$	−6.14214	−0.276628
$494$	8.82843	0.397210
$495$	0	0
$496$	−3.17157	−0.142408
$497$	0	0
$498$	0	0
$499$	−10.1005	−0.452161	−0.226080	−	0.974109i	$-0.572591\pi$
$499$	−10.1005	−0.452161	−0.226080	+	0.974109i	$0.572591\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−11.3137	−0.504956
$503$	7.79899	0.347740	0.173870	−	0.984769i	$-0.444373\pi$
$503$	7.79899	0.347740	0.173870	+	0.984769i	$0.444373\pi$
$504$	0	0
$505$	−15.3137	−0.681451
$506$	−20.4853	−0.910682
$507$	0	0
$508$	8.00000	0.354943
$509$	−15.3137	−0.678768	−0.339384	−	0.940648i	$-0.610219\pi$
$509$	−15.3137	−0.678768	−0.339384	+	0.940648i	$0.610219\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−22.2426	−0.981081
$515$	13.6569	0.601793
$516$	0	0
$517$	21.6569	0.952467
$518$	0	0
$519$	0	0
$520$	3.41421	0.149723
$521$	−12.1421	−0.531957	−0.265978	−	0.963979i	$-0.585695\pi$
$521$	−12.1421	−0.531957	−0.265978	+	0.963979i	$0.585695\pi$
$522$	0	0
$523$	−21.6985	−0.948808	−0.474404	−	0.880307i	$-0.657336\pi$
$523$	−21.6985	−0.948808	−0.474404	+	0.880307i	$0.657336\pi$
$524$	−5.65685	−0.247121
$525$	0	0
$526$	17.4142	0.759296
$527$	−1.85786	−0.0809298
$528$	0	0
$529$	−5.00000	−0.217391
$530$	−9.65685	−0.419467
$531$	0	0
$532$	0	0
$533$	−21.3137	−0.923199
$534$	0	0
$535$	6.82843	0.295219
$536$	−12.7279	−0.549762
$537$	0	0
$538$	−22.9706	−0.990331
$539$	−33.7990	−1.45583
$540$	0	0
$541$	−30.7279	−1.32110	−0.660548	−	0.750784i	$-0.729676\pi$
$541$	−30.7279	−1.32110	−0.660548	+	0.750784i	$0.729676\pi$
$542$	−26.1421	−1.12290
$543$	0	0
$544$	0.585786	0.0251154
$545$	17.3137	0.741638
$546$	0	0
$547$	31.7990	1.35963	0.679813	−	0.733385i	$-0.262061\pi$
$547$	31.7990	1.35963	0.679813	+	0.733385i	$0.262061\pi$
$548$	0.585786	0.0250236
$549$	0	0
$550$	4.82843	0.205885
$551$	27.1127	1.15504
$552$	0	0
$553$	0	0
$554$	22.2426	0.944999
$555$	0	0
$556$	−13.8995	−0.589470
$557$	−41.3137	−1.75052	−0.875259	−	0.483654i	$-0.839309\pi$
$557$	−41.3137	−1.75052	−0.875259	+	0.483654i	$0.839309\pi$
$558$	0	0
$559$	−19.3137	−0.816883
$560$	0	0
$561$	0	0
$562$	−5.07107	−0.213910
$563$	−18.1421	−0.764600	−0.382300	−	0.924038i	$-0.624868\pi$
$563$	−18.1421	−0.764600	−0.382300	+	0.924038i	$0.624868\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	25.6569	1.07844
$567$	0	0
$568$	−0.242641	−0.0101810
$569$	0.384776	0.0161307	0.00806533	−	0.999967i	$-0.497433\pi$
$569$	0.384776	0.0161307	0.00806533	+	0.999967i	$0.497433\pi$
$570$	0	0
$571$	30.6274	1.28172	0.640859	−	0.767659i	$-0.278578\pi$
$571$	30.6274	1.28172	0.640859	+	0.767659i	$0.278578\pi$
$572$	−16.4853	−0.689284
$573$	0	0
$574$	0	0
$575$	−4.24264	−0.176930
$576$	0	0
$577$	16.3431	0.680374	0.340187	−	0.940358i	$-0.389510\pi$
$577$	16.3431	0.680374	0.340187	+	0.940358i	$0.389510\pi$
$578$	−16.6569	−0.692834
$579$	0	0
$580$	10.4853	0.435378
$581$	0	0
$582$	0	0
$583$	46.6274	1.93111
$584$	−4.00000	−0.165521
$585$	0	0
$586$	−10.9706	−0.453190
$587$	33.6569	1.38917	0.694584	−	0.719412i	$-0.255589\pi$
$587$	33.6569	1.38917	0.694584	+	0.719412i	$0.255589\pi$
$588$	0	0
$589$	8.20101	0.337917
$590$	−1.17157	−0.0482329
$591$	0	0
$592$	−4.58579	−0.188475
$593$	−14.2843	−0.586585	−0.293292	−	0.956023i	$-0.594751\pi$
$593$	−14.2843	−0.586585	−0.293292	+	0.956023i	$0.594751\pi$
$594$	0	0
$595$	0	0
$596$	−4.34315	−0.177902
$597$	0	0
$598$	14.4853	0.592347
$599$	3.55635	0.145308	0.0726542	−	0.997357i	$-0.476853\pi$
$599$	3.55635	0.145308	0.0726542	+	0.997357i	$0.476853\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	0	0
$604$	−11.3137	−0.460348
$605$	−12.3137	−0.500623
$606$	0	0
$607$	24.0000	0.974130	0.487065	−	0.873366i	$-0.338067\pi$
$607$	24.0000	0.974130	0.487065	+	0.873366i	$0.338067\pi$
$608$	−2.58579	−0.104867
$609$	0	0
$610$	8.82843	0.357453
$611$	−15.3137	−0.619526
$612$	0	0
$613$	−9.51472	−0.384296	−0.192148	−	0.981366i	$-0.561545\pi$
$613$	−9.51472	−0.384296	−0.192148	+	0.981366i	$0.561545\pi$
$614$	16.4853	0.665292
$615$	0	0
$616$	0	0
$617$	37.1127	1.49410	0.747050	−	0.664767i	$-0.231469\pi$
$617$	37.1127	1.49410	0.747050	+	0.664767i	$0.231469\pi$
$618$	0	0
$619$	6.87006	0.276131	0.138065	−	0.990423i	$-0.455912\pi$
$619$	6.87006	0.276131	0.138065	+	0.990423i	$0.455912\pi$
$620$	3.17157	0.127373
$621$	0	0
$622$	−16.2426	−0.651271
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	3.65685	0.146157
$627$	0	0
$628$	14.7279	0.587708
$629$	−2.68629	−0.107109
$630$	0	0
$631$	−27.3137	−1.08734	−0.543671	−	0.839299i	$-0.682966\pi$
$631$	−27.3137	−1.08734	−0.543671	+	0.839299i	$0.682966\pi$
$632$	−0.828427	−0.0329531
$633$	0	0
$634$	−9.79899	−0.389168
$635$	−8.00000	−0.317470
$636$	0	0
$637$	23.8995	0.946932
$638$	−50.6274	−2.00436
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	46.2426	1.82647	0.913237	−	0.407428i	$-0.133574\pi$
$641$	46.2426	1.82647	0.913237	+	0.407428i	$0.133574\pi$
$642$	0	0
$643$	−47.5980	−1.87708	−0.938541	−	0.345169i	$-0.887822\pi$
$643$	−47.5980	−1.87708	−0.938541	+	0.345169i	$0.887822\pi$
$644$	0	0
$645$	0	0
$646$	−1.51472	−0.0595958
$647$	−13.1716	−0.517828	−0.258914	−	0.965900i	$-0.583365\pi$
$647$	−13.1716	−0.517828	−0.258914	+	0.965900i	$0.583365\pi$
$648$	0	0
$649$	5.65685	0.222051
$650$	−3.41421	−0.133916
$651$	0	0
$652$	11.3137	0.443079
$653$	−31.9411	−1.24995	−0.624976	−	0.780644i	$-0.714891\pi$
$653$	−31.9411	−1.24995	−0.624976	+	0.780644i	$0.714891\pi$
$654$	0	0
$655$	5.65685	0.221032
$656$	6.24264	0.243734
$657$	0	0
$658$	0	0
$659$	−0.485281	−0.0189039	−0.00945194	−	0.999955i	$-0.503009\pi$
$659$	−0.485281	−0.0189039	−0.00945194	+	0.999955i	$0.503009\pi$
$660$	0	0
$661$	29.3137	1.14017	0.570086	−	0.821585i	$-0.306910\pi$
$661$	29.3137	1.14017	0.570086	+	0.821585i	$0.306910\pi$
$662$	−24.7279	−0.961078
$663$	0	0
$664$	−13.6569	−0.529989
$665$	0	0
$666$	0	0
$667$	44.4853	1.72248
$668$	−14.3431	−0.554953
$669$	0	0
$670$	12.7279	0.491723
$671$	−42.6274	−1.64561
$672$	0	0
$673$	18.3431	0.707076	0.353538	−	0.935420i	$-0.384978\pi$
$673$	18.3431	0.707076	0.353538	+	0.935420i	$0.384978\pi$
$674$	−30.4853	−1.17425
$675$	0	0
$676$	−1.34315	−0.0516595
$677$	−14.3431	−0.551252	−0.275626	−	0.961265i	$-0.588885\pi$
$677$	−14.3431	−0.551252	−0.275626	+	0.961265i	$0.588885\pi$
$678$	0	0
$679$	0	0
$680$	−0.585786	−0.0224639
$681$	0	0
$682$	−15.3137	−0.586392
$683$	15.3137	0.585963	0.292981	−	0.956118i	$-0.405353\pi$
$683$	15.3137	0.585963	0.292981	+	0.956118i	$0.405353\pi$
$684$	0	0
$685$	−0.585786	−0.0223817
$686$	0	0
$687$	0	0
$688$	5.65685	0.215666
$689$	−32.9706	−1.25608
$690$	0	0
$691$	7.51472	0.285873	0.142937	−	0.989732i	$-0.454345\pi$
$691$	7.51472	0.285873	0.142937	+	0.989732i	$0.454345\pi$
$692$	18.9706	0.721153
$693$	0	0
$694$	−31.1127	−1.18102
$695$	13.8995	0.527238
$696$	0	0
$697$	3.65685	0.138513
$698$	−9.75736	−0.369321
$699$	0	0
$700$	0	0
$701$	26.9706	1.01866	0.509332	−	0.860570i	$-0.329893\pi$
$701$	26.9706	1.01866	0.509332	+	0.860570i	$0.329893\pi$
$702$	0	0
$703$	11.8579	0.447228
$704$	4.82843	0.181978
$705$	0	0
$706$	15.6569	0.589253
$707$	0	0
$708$	0	0
$709$	−10.7279	−0.402896	−0.201448	−	0.979499i	$-0.564565\pi$
$709$	−10.7279	−0.402896	−0.201448	+	0.979499i	$0.564565\pi$
$710$	0.242641	0.00910614
$711$	0	0
$712$	−2.00000	−0.0749532
$713$	13.4558	0.503925
$714$	0	0
$715$	16.4853	0.616515
$716$	16.4853	0.616084
$717$	0	0
$718$	15.7574	0.588059
$719$	−12.9289	−0.482168	−0.241084	−	0.970504i	$-0.577503\pi$
$719$	−12.9289	−0.482168	−0.241084	+	0.970504i	$0.577503\pi$
$720$	0	0
$721$	0	0
$722$	−12.3137	−0.458269
$723$	0	0
$724$	22.7279	0.844677
$725$	−10.4853	−0.389414
$726$	0	0
$727$	17.7990	0.660128	0.330064	−	0.943959i	$-0.392930\pi$
$727$	17.7990	0.660128	0.330064	+	0.943959i	$0.392930\pi$
$728$	0	0
$729$	0	0
$730$	4.00000	0.148047
$731$	3.31371	0.122562
$732$	0	0
$733$	44.8284	1.65578	0.827888	−	0.560894i	$-0.189542\pi$
$733$	44.8284	1.65578	0.827888	+	0.560894i	$0.189542\pi$
$734$	−14.1421	−0.521996
$735$	0	0
$736$	−4.24264	−0.156386
$737$	−61.4558	−2.26376
$738$	0	0
$739$	49.6985	1.82819	0.914094	−	0.405502i	$-0.132903\pi$
$739$	49.6985	1.82819	0.914094	+	0.405502i	$0.132903\pi$
$740$	4.58579	0.168577
$741$	0	0
$742$	0	0
$743$	9.85786	0.361650	0.180825	−	0.983515i	$-0.442123\pi$
$743$	9.85786	0.361650	0.180825	+	0.983515i	$0.442123\pi$
$744$	0	0
$745$	4.34315	0.159121
$746$	0.585786	0.0214472
$747$	0	0
$748$	2.82843	0.103418
$749$	0	0
$750$	0	0
$751$	−31.5980	−1.15303	−0.576513	−	0.817088i	$-0.695587\pi$
$751$	−31.5980	−1.15303	−0.576513	+	0.817088i	$0.695587\pi$
$752$	4.48528	0.163561
$753$	0	0
$754$	35.7990	1.30372
$755$	11.3137	0.411748
$756$	0	0
$757$	5.07107	0.184311	0.0921555	−	0.995745i	$-0.470624\pi$
$757$	5.07107	0.184311	0.0921555	+	0.995745i	$0.470624\pi$
$758$	−9.65685	−0.350753
$759$	0	0
$760$	2.58579	0.0937963
$761$	10.7279	0.388887	0.194443	−	0.980914i	$-0.437710\pi$
$761$	10.7279	0.388887	0.194443	+	0.980914i	$0.437710\pi$
$762$	0	0
$763$	0	0
$764$	4.00000	0.144715
$765$	0	0
$766$	−34.8701	−1.25991
$767$	−4.00000	−0.144432
$768$	0	0
$769$	46.7696	1.68655	0.843277	−	0.537480i	$-0.180624\pi$
$769$	46.7696	1.68655	0.843277	+	0.537480i	$0.180624\pi$
$770$	0	0
$771$	0	0
$772$	−4.34315	−0.156313
$773$	−16.0000	−0.575480	−0.287740	−	0.957709i	$-0.592904\pi$
$773$	−16.0000	−0.575480	−0.287740	+	0.957709i	$0.592904\pi$
$774$	0	0
$775$	−3.17157	−0.113926
$776$	−1.00000	−0.0358979
$777$	0	0
$778$	14.0000	0.501924
$779$	−16.1421	−0.578352
$780$	0	0
$781$	−1.17157	−0.0419222
$782$	−2.48528	−0.0888735
$783$	0	0
$784$	−7.00000	−0.250000
$785$	−14.7279	−0.525662
$786$	0	0
$787$	−11.1127	−0.396125	−0.198062	−	0.980189i	$-0.563465\pi$
$787$	−11.1127	−0.396125	−0.198062	+	0.980189i	$0.563465\pi$
$788$	10.9706	0.390810
$789$	0	0
$790$	0.828427	0.0294741
$791$	0	0
$792$	0	0
$793$	30.1421	1.07038
$794$	10.9706	0.389331
$795$	0	0
$796$	−13.1716	−0.466854
$797$	−23.6569	−0.837969	−0.418984	−	0.907993i	$-0.637614\pi$
$797$	−23.6569	−0.837969	−0.418984	+	0.907993i	$0.637614\pi$
$798$	0	0
$799$	2.62742	0.0929513
$800$	1.00000	0.0353553
$801$	0	0
$802$	−27.8995	−0.985165
$803$	−19.3137	−0.681566
$804$	0	0
$805$	0	0
$806$	10.8284	0.381415
$807$	0	0
$808$	15.3137	0.538734
$809$	31.1716	1.09593	0.547967	−	0.836500i	$-0.315402\pi$
$809$	31.1716	1.09593	0.547967	+	0.836500i	$0.315402\pi$
$810$	0	0
$811$	39.7990	1.39753	0.698766	−	0.715351i	$-0.253733\pi$
$811$	39.7990	1.39753	0.698766	+	0.715351i	$0.253733\pi$
$812$	0	0
$813$	0	0
$814$	−22.1421	−0.776081
$815$	−11.3137	−0.396302
$816$	0	0
$817$	−14.6274	−0.511749
$818$	13.3137	0.465503
$819$	0	0
$820$	−6.24264	−0.218002
$821$	20.1421	0.702965	0.351483	−	0.936194i	$-0.385678\pi$
$821$	20.1421	0.702965	0.351483	+	0.936194i	$0.385678\pi$
$822$	0	0
$823$	55.5980	1.93802	0.969012	−	0.247014i	$-0.0794494\pi$
$823$	55.5980	1.93802	0.969012	+	0.247014i	$0.0794494\pi$
$824$	−13.6569	−0.475759
$825$	0	0
$826$	0	0
$827$	30.3431	1.05513	0.527567	−	0.849513i	$-0.323104\pi$
$827$	30.3431	1.05513	0.527567	+	0.849513i	$0.323104\pi$
$828$	0	0
$829$	−11.4558	−0.397878	−0.198939	−	0.980012i	$-0.563750\pi$
$829$	−11.4558	−0.397878	−0.198939	+	0.980012i	$0.563750\pi$
$830$	13.6569	0.474036
$831$	0	0
$832$	−3.41421	−0.118367
$833$	−4.10051	−0.142074
$834$	0	0
$835$	14.3431	0.496365
$836$	−12.4853	−0.431812
$837$	0	0
$838$	12.9706	0.448061
$839$	−16.7279	−0.577512	−0.288756	−	0.957403i	$-0.593242\pi$
$839$	−16.7279	−0.577512	−0.288756	+	0.957403i	$0.593242\pi$
$840$	0	0
$841$	80.9411	2.79107
$842$	2.00000	0.0689246
$843$	0	0
$844$	27.5563	0.948529
$845$	1.34315	0.0462056
$846$	0	0
$847$	0	0
$848$	9.65685	0.331618
$849$	0	0
$850$	0.585786	0.0200923
$851$	19.4558	0.666938
$852$	0	0
$853$	14.2426	0.487659	0.243829	−	0.969818i	$-0.421596\pi$
$853$	14.2426	0.487659	0.243829	+	0.969818i	$0.421596\pi$
$854$	0	0
$855$	0	0
$856$	−6.82843	−0.233391
$857$	28.8284	0.984760	0.492380	−	0.870380i	$-0.336127\pi$
$857$	28.8284	0.984760	0.492380	+	0.870380i	$0.336127\pi$
$858$	0	0
$859$	−25.6985	−0.876821	−0.438410	−	0.898775i	$-0.644458\pi$
$859$	−25.6985	−0.876821	−0.438410	+	0.898775i	$0.644458\pi$
$860$	−5.65685	−0.192897
$861$	0	0
$862$	−21.1716	−0.721107
$863$	−2.78680	−0.0948637	−0.0474318	−	0.998874i	$-0.515104\pi$
$863$	−2.78680	−0.0948637	−0.0474318	+	0.998874i	$0.515104\pi$
$864$	0	0
$865$	−18.9706	−0.645018
$866$	33.7990	1.14854
$867$	0	0
$868$	0	0
$869$	−4.00000	−0.135691
$870$	0	0
$871$	43.4558	1.47245
$872$	−17.3137	−0.586316
$873$	0	0
$874$	10.9706	0.371085
$875$	0	0
$876$	0	0
$877$	0.627417	0.0211864	0.0105932	−	0.999944i	$-0.496628\pi$
$877$	0.627417	0.0211864	0.0105932	+	0.999944i	$0.496628\pi$
$878$	18.6274	0.628645
$879$	0	0
$880$	−4.82843	−0.162766
$881$	4.14214	0.139552	0.0697760	−	0.997563i	$-0.477772\pi$
$881$	4.14214	0.139552	0.0697760	+	0.997563i	$0.477772\pi$
$882$	0	0
$883$	−29.6985	−0.999434	−0.499717	−	0.866189i	$-0.666563\pi$
$883$	−29.6985	−0.999434	−0.499717	+	0.866189i	$0.666563\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	33.4558	1.12397
$887$	−22.5858	−0.758356	−0.379178	−	0.925324i	$-0.623793\pi$
$887$	−22.5858	−0.758356	−0.379178	+	0.925324i	$0.623793\pi$
$888$	0	0
$889$	0	0
$890$	2.00000	0.0670402
$891$	0	0
$892$	3.31371	0.110951
$893$	−11.5980	−0.388112
$894$	0	0
$895$	−16.4853	−0.551042
$896$	0	0
$897$	0	0
$898$	38.4853	1.28427
$899$	33.2548	1.10911
$900$	0	0
$901$	5.65685	0.188457
$902$	30.1421	1.00362
$903$	0	0
$904$	6.00000	0.199557
$905$	−22.7279	−0.755502
$906$	0	0
$907$	19.7574	0.656032	0.328016	−	0.944672i	$-0.393620\pi$
$907$	19.7574	0.656032	0.328016	+	0.944672i	$0.393620\pi$
$908$	−7.17157	−0.237997
$909$	0	0
$910$	0	0
$911$	−15.3553	−0.508745	−0.254373	−	0.967106i	$-0.581869\pi$
$911$	−15.3553	−0.508745	−0.254373	+	0.967106i	$0.581869\pi$
$912$	0	0
$913$	−65.9411	−2.18233
$914$	−30.2843	−1.00171
$915$	0	0
$916$	−24.8284	−0.820354
$917$	0	0
$918$	0	0
$919$	−11.5147	−0.379836	−0.189918	−	0.981800i	$-0.560822\pi$
$919$	−11.5147	−0.379836	−0.189918	+	0.981800i	$0.560822\pi$
$920$	4.24264	0.139876
$921$	0	0
$922$	−23.3137	−0.767796
$923$	0.828427	0.0272680
$924$	0	0
$925$	−4.58579	−0.150780
$926$	−17.7990	−0.584911
$927$	0	0
$928$	−10.4853	−0.344196
$929$	54.3259	1.78238	0.891188	−	0.453635i	$-0.149873\pi$
$929$	54.3259	1.78238	0.891188	+	0.453635i	$0.149873\pi$
$930$	0	0
$931$	18.1005	0.593220
$932$	8.38478	0.274652
$933$	0	0
$934$	−26.4853	−0.866625
$935$	−2.82843	−0.0924995
$936$	0	0
$937$	46.9117	1.53254	0.766269	−	0.642520i	$-0.222111\pi$
$937$	46.9117	1.53254	0.766269	+	0.642520i	$0.222111\pi$
$938$	0	0
$939$	0	0
$940$	−4.48528	−0.146294
$941$	−4.62742	−0.150849	−0.0754247	−	0.997151i	$-0.524031\pi$
$941$	−4.62742	−0.150849	−0.0754247	+	0.997151i	$0.524031\pi$
$942$	0	0
$943$	−26.4853	−0.862479
$944$	1.17157	0.0381314
$945$	0	0
$946$	27.3137	0.888045
$947$	−26.4264	−0.858743	−0.429371	−	0.903128i	$-0.641265\pi$
$947$	−26.4264	−0.858743	−0.429371	+	0.903128i	$0.641265\pi$
$948$	0	0
$949$	13.6569	0.443320
$950$	−2.58579	−0.0838940
$951$	0	0
$952$	0	0
$953$	3.89949	0.126317	0.0631585	−	0.998004i	$-0.479883\pi$
$953$	3.89949	0.126317	0.0631585	+	0.998004i	$0.479883\pi$
$954$	0	0
$955$	−4.00000	−0.129437
$956$	−13.4142	−0.433847
$957$	0	0
$958$	0.686292	0.0221731
$959$	0	0
$960$	0	0
$961$	−20.9411	−0.675520
$962$	15.6569	0.504797
$963$	0	0
$964$	−15.3137	−0.493221
$965$	4.34315	0.139811
$966$	0	0
$967$	7.85786	0.252692	0.126346	−	0.991986i	$-0.459675\pi$
$967$	7.85786	0.252692	0.126346	+	0.991986i	$0.459675\pi$
$968$	12.3137	0.395778
$969$	0	0
$970$	1.00000	0.0321081
$971$	19.1716	0.615245	0.307623	−	0.951508i	$-0.400467\pi$
$971$	19.1716	0.615245	0.307623	+	0.951508i	$0.400467\pi$
$972$	0	0
$973$	0	0
$974$	7.17157	0.229792
$975$	0	0
$976$	−8.82843	−0.282591
$977$	30.5269	0.976642	0.488321	−	0.872664i	$-0.337609\pi$
$977$	30.5269	0.976642	0.488321	+	0.872664i	$0.337609\pi$
$978$	0	0
$979$	−9.65685	−0.308634
$980$	7.00000	0.223607
$981$	0	0
$982$	10.6274	0.339135
$983$	−14.1005	−0.449736	−0.224868	−	0.974389i	$-0.572195\pi$
$983$	−14.1005	−0.449736	−0.224868	+	0.974389i	$0.572195\pi$
$984$	0	0
$985$	−10.9706	−0.349551
$986$	−6.14214	−0.195605
$987$	0	0
$988$	8.82843	0.280870
$989$	−24.0000	−0.763156
$990$	0	0
$991$	53.4558	1.69808	0.849040	−	0.528328i	$-0.177181\pi$
$991$	53.4558	1.69808	0.849040	+	0.528328i	$0.177181\pi$
$992$	−3.17157	−0.100698
$993$	0	0
$994$	0	0
$995$	13.1716	0.417567
$996$	0	0
$997$	−29.1127	−0.922008	−0.461004	−	0.887398i	$-0.652511\pi$
$997$	−29.1127	−0.922008	−0.461004	+	0.887398i	$0.652511\pi$
$998$	−10.1005	−0.319726
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8730.2.a.v.1.2		2
3.2	odd	2		2910.2.a.m.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2910.2.a.m.1.1	✓	2	3.2	odd	2
8730.2.a.v.1.2		2	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:	\( N \)	\(=\)	\( 8730 = 2 \cdot 3^{2} \cdot 5 \cdot 97 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8730.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{8} -1.00000 q^{10} +4.82843 q^{11} -3.41421 q^{13} +1.00000 q^{16} +0.585786 q^{17} -2.58579 q^{19} -1.00000 q^{20} +4.82843 q^{22} -4.24264 q^{23} +1.00000 q^{25} -3.41421 q^{26} -10.4853 q^{29} -3.17157 q^{31} +1.00000 q^{32} +0.585786 q^{34} -4.58579 q^{37} -2.58579 q^{38} -1.00000 q^{40} +6.24264 q^{41} +5.65685 q^{43} +4.82843 q^{44} -4.24264 q^{46} +4.48528 q^{47} -7.00000 q^{49} +1.00000 q^{50} -3.41421 q^{52} +9.65685 q^{53} -4.82843 q^{55} -10.4853 q^{58} +1.17157 q^{59} -8.82843 q^{61} -3.17157 q^{62} +1.00000 q^{64} +3.41421 q^{65} -12.7279 q^{67} +0.585786 q^{68} -0.242641 q^{71} -4.00000 q^{73} -4.58579 q^{74} -2.58579 q^{76} -0.828427 q^{79} -1.00000 q^{80} +6.24264 q^{82} -13.6569 q^{83} -0.585786 q^{85} +5.65685 q^{86} +4.82843 q^{88} -2.00000 q^{89} -4.24264 q^{92} +4.48528 q^{94} +2.58579 q^{95} -1.00000 q^{97} -7.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{8} - 2 q^{10} + 4 q^{11} - 4 q^{13} + 2 q^{16} + 4 q^{17} - 8 q^{19} - 2 q^{20} + 4 q^{22} + 2 q^{25} - 4 q^{26} - 4 q^{29} - 12 q^{31} + 2 q^{32} + 4 q^{34} - 12 q^{37} - 8 q^{38} - 2 q^{40} + 4 q^{41} + 4 q^{44} - 8 q^{47} - 14 q^{49} + 2 q^{50} - 4 q^{52} + 8 q^{53} - 4 q^{55} - 4 q^{58} + 8 q^{59} - 12 q^{61} - 12 q^{62} + 2 q^{64} + 4 q^{65} + 4 q^{68} + 8 q^{71} - 8 q^{73} - 12 q^{74} - 8 q^{76} + 4 q^{79} - 2 q^{80} + 4 q^{82} - 16 q^{83} - 4 q^{85} + 4 q^{88} - 4 q^{89} - 8 q^{94} + 8 q^{95} - 2 q^{97} - 14 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^8 - 2 * q^10 + 4 * q^11 - 4 * q^13 + 2 * q^16 + 4 * q^17 - 8 * q^19 - 2 * q^20 + 4 * q^22 + 2 * q^25 - 4 * q^26 - 4 * q^29 - 12 * q^31 + 2 * q^32 + 4 * q^34 - 12 * q^37 - 8 * q^38 - 2 * q^40 + 4 * q^41 + 4 * q^44 - 8 * q^47 - 14 * q^49 + 2 * q^50 - 4 * q^52 + 8 * q^53 - 4 * q^55 - 4 * q^58 + 8 * q^59 - 12 * q^61 - 12 * q^62 + 2 * q^64 + 4 * q^65 + 4 * q^68 + 8 * q^71 - 8 * q^73 - 12 * q^74 - 8 * q^76 + 4 * q^79 - 2 * q^80 + 4 * q^82 - 16 * q^83 - 4 * q^85 + 4 * q^88 - 4 * q^89 - 8 * q^94 + 8 * q^95 - 2 * q^97 - 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more