LMFDB - Embedded newform 9000.2.a.r.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9000,2,Mod(1,9000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9000, 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("9000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9000 = 2^{3} \cdot 3^{2} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9000.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:	$71.8653618192$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 11 $ x^4 - 7*x^2 + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1000)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.54336$ of defining polynomial
Character	$\chi$	$=$	9000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.11525 q^{7} +O(q^{10})$	$q+2.11525 q^{7} +4.99442 q^{11} -6.32279 q^{13} +3.75836 q^{17} +1.90770 q^{19} +9.35131 q^{23} -7.22705 q^{29} +0.994424 q^{31} +6.37984 q^{37} -4.18247 q^{41} -1.45664 q^{43} +2.78501 q^{47} -2.52573 q^{49} +1.52786 q^{53} -1.47771 q^{59} +8.79148 q^{61} +12.0811 q^{67} +12.3798 q^{71} -9.09362 q^{73} +10.5644 q^{77} -8.91328 q^{79} +12.1964 q^{83} -3.08328 q^{89} -13.3743 q^{91} -10.1382 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{7}+O(q^{10}) $ 4 * q - 6 * q^7	$ 4 q - 6 q^{7} - 4 q^{13} + 4 q^{17} + 14 q^{23} - 10 q^{29} - 16 q^{31} - 2 q^{41} - 12 q^{43} + 16 q^{47} + 2 q^{49} + 24 q^{53} - 8 q^{59} + 6 q^{61} + 16 q^{67} + 24 q^{71} - 4 q^{73} + 32 q^{77} - 48 q^{79} + 2 q^{83} - 10 q^{89} - 8 q^{91} - 4 q^{97}+O(q^{100}) $ 4 * q - 6 * q^7 - 4 * q^13 + 4 * q^17 + 14 * q^23 - 10 * q^29 - 16 * q^31 - 2 * q^41 - 12 * q^43 + 16 * q^47 + 2 * q^49 + 24 * q^53 - 8 * q^59 + 6 * q^61 + 16 * q^67 + 24 * q^71 - 4 * q^73 + 32 * q^77 - 48 * q^79 + 2 * q^83 - 10 * q^89 - 8 * q^91 - 4 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.11525	0.799488	0.399744	−	0.916627i	$-0.369099\pi$
$7$	2.11525	0.799488	0.399744	+	0.916627i	$0.369099\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.99442	1.50588	0.752938	−	0.658092i	$-0.228636\pi$
$11$	4.99442	1.50588	0.752938	+	0.658092i	$0.228636\pi$
$12$	0	0
$13$	−6.32279	−1.75363	−0.876813	−	0.480831i	$-0.840335\pi$
$13$	−6.32279	−1.75363	−0.876813	+	0.480831i	$0.840335\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.75836	0.911535	0.455768	−	0.890099i	$-0.349365\pi$
$17$	3.75836	0.911535	0.455768	+	0.890099i	$0.349365\pi$
$18$	0	0
$19$	1.90770	0.437656	0.218828	−	0.975763i	$-0.429777\pi$
$19$	1.90770	0.437656	0.218828	+	0.975763i	$0.429777\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	9.35131	1.94988	0.974942	−	0.222460i	$-0.0714086\pi$
$23$	9.35131	1.94988	0.974942	+	0.222460i	$0.0714086\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.22705	−1.34203	−0.671014	−	0.741444i	$-0.734141\pi$
$29$	−7.22705	−1.34203	−0.671014	+	0.741444i	$0.734141\pi$
$30$	0	0
$31$	0.994424	0.178604	0.0893019	−	0.996005i	$-0.471536\pi$
$31$	0.994424	0.178604	0.0893019	+	0.996005i	$0.471536\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.37984	1.04884	0.524419	−	0.851460i	$-0.324282\pi$
$37$	6.37984	1.04884	0.524419	+	0.851460i	$0.324282\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.18247	−0.653192	−0.326596	−	0.945164i	$-0.605902\pi$
$41$	−4.18247	−0.653192	−0.326596	+	0.945164i	$0.605902\pi$
$42$	0	0
$43$	−1.45664	−0.222135	−0.111068	−	0.993813i	$-0.535427\pi$
$43$	−1.45664	−0.222135	−0.111068	+	0.993813i	$0.535427\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.78501	0.406235	0.203117	−	0.979154i	$-0.434893\pi$
$47$	2.78501	0.406235	0.203117	+	0.979154i	$0.434893\pi$
$48$	0	0
$49$	−2.52573	−0.360819
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.52786	0.209868	0.104934	−	0.994479i	$-0.466537\pi$
$53$	1.52786	0.209868	0.104934	+	0.994479i	$0.466537\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.47771	−0.192382	−0.0961908	−	0.995363i	$-0.530666\pi$
$59$	−1.47771	−0.192382	−0.0961908	+	0.995363i	$0.530666\pi$
$60$	0	0
$61$	8.79148	1.12563	0.562817	−	0.826582i	$-0.309718\pi$
$61$	8.79148	1.12563	0.562817	+	0.826582i	$0.309718\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.0811	1.47595	0.737974	−	0.674830i	$-0.235783\pi$
$67$	12.0811	1.47595	0.737974	+	0.674830i	$0.235783\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.3798	1.46922	0.734608	−	0.678492i	$-0.237366\pi$
$71$	12.3798	1.46922	0.734608	+	0.678492i	$0.237366\pi$
$72$	0	0
$73$	−9.09362	−1.06433	−0.532164	−	0.846642i	$-0.678621\pi$
$73$	−9.09362	−1.06433	−0.532164	+	0.846642i	$0.678621\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.5644	1.20393
$78$	0	0
$79$	−8.91328	−1.00282	−0.501411	−	0.865209i	$-0.667186\pi$
$79$	−8.91328	−1.00282	−0.501411	+	0.865209i	$0.667186\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.1964	1.33873	0.669364	−	0.742935i	$-0.266567\pi$
$83$	12.1964	1.33873	0.669364	+	0.742935i	$0.266567\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.08328	−0.326827	−0.163413	−	0.986558i	$-0.552250\pi$
$89$	−3.08328	−0.326827	−0.163413	+	0.986558i	$0.552250\pi$
$90$	0	0
$91$	−13.3743	−1.40200
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−10.1382	−1.02938	−0.514689	−	0.857377i	$-0.672093\pi$
$97$	−10.1382	−1.02938	−0.514689	+	0.857377i	$0.672093\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11.5202	−1.14630	−0.573149	−	0.819451i	$-0.694278\pi$
$101$	−11.5202	−1.14630	−0.573149	+	0.819451i	$0.694278\pi$
$102$	0	0
$103$	6.20639	0.611534	0.305767	−	0.952106i	$-0.401087\pi$
$103$	6.20639	0.611534	0.305767	+	0.952106i	$0.401087\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.7168	1.22938	0.614690	−	0.788769i	$-0.289281\pi$
$107$	12.7168	1.22938	0.614690	+	0.788769i	$0.289281\pi$
$108$	0	0
$109$	0.381966	0.0365857	0.0182929	−	0.999833i	$-0.494177\pi$
$109$	0.381966	0.0365857	0.0182929	+	0.999833i	$0.494177\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−8.31164	−0.781893	−0.390947	−	0.920413i	$-0.627852\pi$
$113$	−8.31164	−0.781893	−0.390947	+	0.920413i	$0.627852\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	7.94985	0.728761
$120$	0	0
$121$	13.9443	1.26766
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−10.3017	−0.914130	−0.457065	−	0.889433i	$-0.651099\pi$
$127$	−10.3017	−0.914130	−0.457065	+	0.889433i	$0.651099\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−3.08672	−0.269688	−0.134844	−	0.990867i	$-0.543053\pi$
$131$	−3.08672	−0.269688	−0.134844	+	0.990867i	$0.543053\pi$
$132$	0	0
$133$	4.03526	0.349901
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−16.1053	−1.37596	−0.687982	−	0.725728i	$-0.741503\pi$
$137$	−16.1053	−1.37596	−0.687982	+	0.725728i	$0.741503\pi$
$138$	0	0
$139$	15.5477	1.31874	0.659370	−	0.751819i	$-0.270823\pi$
$139$	15.5477	1.31874	0.659370	+	0.751819i	$0.270823\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−31.5787	−2.64074
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.2636	1.25044	0.625222	−	0.780447i	$-0.285008\pi$
$149$	15.2636	1.25044	0.625222	+	0.780447i	$0.285008\pi$
$150$	0	0
$151$	−16.1313	−1.31275	−0.656373	−	0.754436i	$-0.727910\pi$
$151$	−16.1313	−1.31275	−0.656373	+	0.754436i	$0.727910\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.241644	0.0192853	0.00964264	−	0.999954i	$-0.496931\pi$
$157$	0.241644	0.0192853	0.00964264	+	0.999954i	$0.496931\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	19.7803	1.55891
$162$	0	0
$163$	−0.314742	−0.0246525	−0.0123263	−	0.999924i	$-0.503924\pi$
$163$	−0.314742	−0.0246525	−0.0123263	+	0.999924i	$0.503924\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.71813	0.597247	0.298623	−	0.954371i	$-0.403473\pi$
$167$	7.71813	0.597247	0.298623	+	0.954371i	$0.403473\pi$
$168$	0	0
$169$	26.9777	2.07521
$170$	0	0
$171$	0	0
$172$	0	0
$173$	13.7472	1.04518	0.522590	−	0.852584i	$-0.324966\pi$
$173$	13.7472	1.04518	0.522590	+	0.852584i	$0.324966\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.29311	−0.0966518	−0.0483259	−	0.998832i	$-0.515389\pi$
$179$	−1.29311	−0.0966518	−0.0483259	+	0.998832i	$0.515389\pi$
$180$	0	0
$181$	−18.4589	−1.37204	−0.686018	−	0.727585i	$-0.740643\pi$
$181$	−18.4589	−1.37204	−0.686018	+	0.727585i	$0.740643\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	18.7708	1.37266
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.90770	−0.427466	−0.213733	−	0.976892i	$-0.568562\pi$
$191$	−5.90770	−0.427466	−0.213733	+	0.976892i	$0.568562\pi$
$192$	0	0
$193$	20.8520	1.50096	0.750479	−	0.660894i	$-0.229823\pi$
$193$	20.8520	1.50096	0.750479	+	0.660894i	$0.229823\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.4735	1.38743	0.693713	−	0.720252i	$-0.255974\pi$
$197$	19.4735	1.38743	0.693713	+	0.720252i	$0.255974\pi$
$198$	0	0
$199$	−23.1989	−1.64452	−0.822262	−	0.569109i	$-0.807288\pi$
$199$	−23.1989	−1.64452	−0.822262	+	0.569109i	$0.807288\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−15.2870	−1.07294
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	9.52786	0.659056
$210$	0	0
$211$	23.7323	1.63380	0.816900	−	0.576780i	$-0.195691\pi$
$211$	23.7323	1.63380	0.816900	+	0.576780i	$0.195691\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.10345	0.142792
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−23.7633	−1.59849
$222$	0	0
$223$	−10.6314	−0.711931	−0.355966	−	0.934499i	$-0.615848\pi$
$223$	−10.6314	−0.711931	−0.355966	+	0.934499i	$0.615848\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	25.0243	1.66092	0.830459	−	0.557079i	$-0.188078\pi$
$227$	25.0243	1.66092	0.830459	+	0.557079i	$0.188078\pi$
$228$	0	0
$229$	11.8962	0.786126	0.393063	−	0.919511i	$-0.371415\pi$
$229$	11.8962	0.786126	0.393063	+	0.919511i	$0.371415\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.0347	−1.05047	−0.525235	−	0.850957i	$-0.676023\pi$
$233$	−16.0347	−1.05047	−0.525235	+	0.850957i	$0.676023\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	11.0867	0.717141	0.358570	−	0.933503i	$-0.383264\pi$
$239$	11.0867	0.717141	0.358570	+	0.933503i	$0.383264\pi$
$240$	0	0
$241$	6.56473	0.422872	0.211436	−	0.977392i	$-0.432186\pi$
$241$	6.56473	0.422872	0.211436	+	0.977392i	$0.432186\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−12.0620	−0.767486
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.22918	−0.0775849	−0.0387924	−	0.999247i	$-0.512351\pi$
$251$	−1.22918	−0.0775849	−0.0387924	+	0.999247i	$0.512351\pi$
$252$	0	0
$253$	46.7044	2.93628
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.52786	0.344819	0.172409	−	0.985025i	$-0.444845\pi$
$257$	5.52786	0.344819	0.172409	+	0.985025i	$0.444845\pi$
$258$	0	0
$259$	13.4949	0.838534
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6.91631	0.426478	0.213239	−	0.977000i	$-0.431599\pi$
$263$	6.91631	0.426478	0.213239	+	0.977000i	$0.431599\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2.94427	−0.179515	−0.0897577	−	0.995964i	$-0.528609\pi$
$269$	−2.94427	−0.179515	−0.0897577	+	0.995964i	$0.528609\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2.81408	−0.169082	−0.0845410	−	0.996420i	$-0.526942\pi$
$277$	−2.81408	−0.169082	−0.0845410	+	0.996420i	$0.526942\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.59393	0.154741	0.0773705	−	0.997002i	$-0.475348\pi$
$281$	2.59393	0.154741	0.0773705	+	0.997002i	$0.475348\pi$
$282$	0	0
$283$	17.6090	1.04675	0.523374	−	0.852103i	$-0.324673\pi$
$283$	17.6090	1.04675	0.523374	+	0.852103i	$0.324673\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.84695	−0.522219
$288$	0	0
$289$	−2.87476	−0.169103
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−17.7392	−1.03634	−0.518168	−	0.855279i	$-0.673386\pi$
$293$	−17.7392	−1.03634	−0.518168	+	0.855279i	$0.673386\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−59.1264	−3.41937
$300$	0	0
$301$	−3.08115	−0.177594
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.4770	0.712102	0.356051	−	0.934466i	$-0.384123\pi$
$307$	12.4770	0.712102	0.356051	+	0.934466i	$0.384123\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−20.2262	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$311$	−20.2262	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$312$	0	0
$313$	−12.6456	−0.714771	−0.357385	−	0.933957i	$-0.616332\pi$
$313$	−12.6456	−0.714771	−0.357385	+	0.933957i	$0.616332\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.67852	0.375103	0.187552	−	0.982255i	$-0.439945\pi$
$317$	6.67852	0.375103	0.187552	+	0.982255i	$0.439945\pi$
$318$	0	0
$319$	−36.0949	−2.02093
$320$	0	0
$321$	0	0
$322$	0	0
$323$	7.16982	0.398939
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.89097	0.324780
$330$	0	0
$331$	27.2628	1.49850	0.749250	−	0.662288i	$-0.230414\pi$
$331$	27.2628	1.49850	0.749250	+	0.662288i	$0.230414\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	29.3835	1.60062	0.800310	−	0.599587i	$-0.204668\pi$
$337$	29.3835	1.60062	0.800310	+	0.599587i	$0.204668\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.96658	0.268955
$342$	0	0
$343$	−20.1493	−1.08796
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.5007	0.832119	0.416059	−	0.909337i	$-0.363411\pi$
$347$	15.5007	0.832119	0.416059	+	0.909337i	$0.363411\pi$
$348$	0	0
$349$	5.98540	0.320391	0.160196	−	0.987085i	$-0.448787\pi$
$349$	5.98540	0.320391	0.160196	+	0.987085i	$0.448787\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	22.8841	1.21800	0.608998	−	0.793171i	$-0.291572\pi$
$353$	22.8841	1.21800	0.608998	+	0.793171i	$0.291572\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.3525	−0.599161	−0.299580	−	0.954071i	$-0.596847\pi$
$359$	−11.3525	−0.599161	−0.299580	+	0.954071i	$0.596847\pi$
$360$	0	0
$361$	−15.3607	−0.808457
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	17.8123	0.929794	0.464897	−	0.885365i	$-0.346091\pi$
$367$	17.8123	0.929794	0.464897	+	0.885365i	$0.346091\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.23181	0.167787
$372$	0	0
$373$	−13.0366	−0.675008	−0.337504	−	0.941324i	$-0.609583\pi$
$373$	−13.0366	−0.675008	−0.337504	+	0.941324i	$0.609583\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	45.6951	2.35342
$378$	0	0
$379$	−31.2800	−1.60675	−0.803373	−	0.595476i	$-0.796964\pi$
$379$	−31.2800	−1.60675	−0.803373	+	0.595476i	$0.796964\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	13.3291	0.681084	0.340542	−	0.940229i	$-0.389389\pi$
$383$	13.3291	0.681084	0.340542	+	0.940229i	$0.389389\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.66950	−0.135349	−0.0676746	−	0.997707i	$-0.521558\pi$
$389$	−2.66950	−0.135349	−0.0676746	+	0.997707i	$0.521558\pi$
$390$	0	0
$391$	35.1456	1.77739
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	24.9572	1.25257	0.626284	−	0.779595i	$-0.284575\pi$
$397$	24.9572	1.25257	0.626284	+	0.779595i	$0.284575\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	28.3039	1.41343	0.706716	−	0.707498i	$-0.250176\pi$
$401$	28.3039	1.41343	0.706716	+	0.707498i	$0.250176\pi$
$402$	0	0
$403$	−6.28754	−0.313204
$404$	0	0
$405$	0	0
$406$	0	0
$407$	31.8636	1.57942
$408$	0	0
$409$	−14.5647	−0.720180	−0.360090	−	0.932918i	$-0.617254\pi$
$409$	−14.5647	−0.720180	−0.360090	+	0.932918i	$0.617254\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3.12572	−0.153807
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−28.4610	−1.39041	−0.695205	−	0.718811i	$-0.744686\pi$
$419$	−28.4610	−1.39041	−0.695205	+	0.718811i	$0.744686\pi$
$420$	0	0
$421$	−20.5443	−1.00127	−0.500633	−	0.865660i	$-0.666899\pi$
$421$	−20.5443	−1.00127	−0.500633	+	0.865660i	$0.666899\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	18.5961	0.899931
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−28.9753	−1.39569	−0.697845	−	0.716249i	$-0.745857\pi$
$431$	−28.9753	−1.39569	−0.697845	+	0.716249i	$0.745857\pi$
$432$	0	0
$433$	−20.9002	−1.00440	−0.502199	−	0.864752i	$-0.667476\pi$
$433$	−20.9002	−1.00440	−0.502199	+	0.864752i	$0.667476\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	17.8395	0.853379
$438$	0	0
$439$	−30.2546	−1.44397	−0.721987	−	0.691907i	$-0.756771\pi$
$439$	−30.2546	−1.44397	−0.721987	+	0.691907i	$0.756771\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.09346	0.289509	0.144754	−	0.989468i	$-0.453761\pi$
$443$	6.09346	0.289509	0.144754	+	0.989468i	$0.453761\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	19.3135	0.911459	0.455730	−	0.890118i	$-0.349378\pi$
$449$	19.3135	0.911459	0.455730	+	0.890118i	$0.349378\pi$
$450$	0	0
$451$	−20.8890	−0.983626
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−13.1289	−0.614143	−0.307071	−	0.951686i	$-0.599349\pi$
$457$	−13.1289	−0.614143	−0.307071	+	0.951686i	$0.599349\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.28409	0.292679	0.146340	−	0.989234i	$-0.453251\pi$
$461$	6.28409	0.292679	0.146340	+	0.989234i	$0.453251\pi$
$462$	0	0
$463$	−0.705730	−0.0327981	−0.0163990	−	0.999866i	$-0.505220\pi$
$463$	−0.705730	−0.0327981	−0.0163990	+	0.999866i	$0.505220\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10.1190	−0.468251	−0.234126	−	0.972206i	$-0.575223\pi$
$467$	−10.1190	−0.468251	−0.234126	+	0.972206i	$0.575223\pi$
$468$	0	0
$469$	25.5546	1.18000
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−7.27507	−0.334508
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	29.0533	1.32748	0.663739	−	0.747964i	$-0.268969\pi$
$479$	29.0533	1.32748	0.663739	+	0.747964i	$0.268969\pi$
$480$	0	0
$481$	−40.3384	−1.83927
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7.59798	0.344297	0.172149	−	0.985071i	$-0.444929\pi$
$487$	7.59798	0.344297	0.172149	+	0.985071i	$0.444929\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	11.6822	0.527208	0.263604	−	0.964631i	$-0.415089\pi$
$491$	11.6822	0.527208	0.263604	+	0.964631i	$0.415089\pi$
$492$	0	0
$493$	−27.1618	−1.22331
$494$	0	0
$495$	0	0
$496$	0	0
$497$	26.1864	1.17462
$498$	0	0
$499$	4.21754	0.188803	0.0944015	−	0.995534i	$-0.469906\pi$
$499$	4.21754	0.188803	0.0944015	+	0.995534i	$0.469906\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.12640	0.362338	0.181169	−	0.983452i	$-0.442012\pi$
$503$	8.12640	0.362338	0.181169	+	0.983452i	$0.442012\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	20.8190	0.922787	0.461394	−	0.887196i	$-0.347350\pi$
$509$	20.8190	0.922787	0.461394	+	0.887196i	$0.347350\pi$
$510$	0	0
$511$	−19.2352	−0.850917
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	13.9095	0.611739
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−15.7390	−0.689539	−0.344769	−	0.938687i	$-0.612043\pi$
$521$	−15.7390	−0.689539	−0.344769	+	0.938687i	$0.612043\pi$
$522$	0	0
$523$	−39.6698	−1.73464	−0.867319	−	0.497753i	$-0.834159\pi$
$523$	−39.6698	−1.73464	−0.867319	+	0.497753i	$0.834159\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.73740	0.162804
$528$	0	0
$529$	64.4471	2.80205
$530$	0	0
$531$	0	0
$532$	0	0
$533$	26.4449	1.14546
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−12.6146	−0.543349
$540$	0	0
$541$	10.1162	0.434930	0.217465	−	0.976068i	$-0.430221\pi$
$541$	10.1162	0.434930	0.217465	+	0.976068i	$0.430221\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3.91575	0.167425	0.0837127	−	0.996490i	$-0.473322\pi$
$547$	3.91575	0.167425	0.0837127	+	0.996490i	$0.473322\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−13.7870	−0.587348
$552$	0	0
$553$	−18.8538	−0.801744
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−25.9095	−1.09782	−0.548910	−	0.835881i	$-0.684957\pi$
$557$	−25.9095	−1.09782	−0.548910	+	0.835881i	$0.684957\pi$
$558$	0	0
$559$	9.21002	0.389542
$560$	0	0
$561$	0	0
$562$	0	0
$563$	30.7156	1.29451	0.647254	−	0.762275i	$-0.275917\pi$
$563$	30.7156	1.29451	0.647254	+	0.762275i	$0.275917\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12.3448	0.517519	0.258760	−	0.965942i	$-0.416686\pi$
$569$	12.3448	0.517519	0.258760	+	0.965942i	$0.416686\pi$
$570$	0	0
$571$	33.9368	1.42021	0.710104	−	0.704096i	$-0.248648\pi$
$571$	33.9368	1.42021	0.710104	+	0.704096i	$0.248648\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	45.2207	1.88256	0.941280	−	0.337626i	$-0.109624\pi$
$577$	45.2207	1.88256	0.941280	+	0.337626i	$0.109624\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	25.7984	1.07030
$582$	0	0
$583$	7.63080	0.316035
$584$	0	0
$585$	0	0
$586$	0	0
$587$	37.6710	1.55485	0.777424	−	0.628976i	$-0.216526\pi$
$587$	37.6710	1.55485	0.777424	+	0.628976i	$0.216526\pi$
$588$	0	0
$589$	1.89706	0.0781671
$590$	0	0
$591$	0	0
$592$	0	0
$593$	6.19524	0.254408	0.127204	−	0.991877i	$-0.459400\pi$
$593$	6.19524	0.254408	0.127204	+	0.991877i	$0.459400\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.03100	0.164702	0.0823510	−	0.996603i	$-0.473757\pi$
$599$	4.03100	0.164702	0.0823510	+	0.996603i	$0.473757\pi$
$600$	0	0
$601$	42.2631	1.72395	0.861974	−	0.506953i	$-0.169228\pi$
$601$	42.2631	1.72395	0.861974	+	0.506953i	$0.169228\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−30.9920	−1.25793	−0.628963	−	0.777436i	$-0.716520\pi$
$607$	−30.9920	−1.25793	−0.628963	+	0.777436i	$0.716520\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−17.6090	−0.712384
$612$	0	0
$613$	6.63443	0.267962	0.133981	−	0.990984i	$-0.457224\pi$
$613$	6.63443	0.267962	0.133981	+	0.990984i	$0.457224\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.2862	0.856951	0.428475	−	0.903553i	$-0.359051\pi$
$617$	21.2862	0.856951	0.428475	+	0.903553i	$0.359051\pi$
$618$	0	0
$619$	17.4244	0.700346	0.350173	−	0.936685i	$-0.386123\pi$
$619$	17.4244	0.700346	0.350173	+	0.936685i	$0.386123\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6.52189	−0.261294
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	23.9777	0.956053
$630$	0	0
$631$	11.5148	0.458396	0.229198	−	0.973380i	$-0.426390\pi$
$631$	11.5148	0.458396	0.229198	+	0.973380i	$0.426390\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	15.9697	0.632742
$638$	0	0
$639$	0	0
$640$	0	0
$641$	45.2710	1.78810	0.894048	−	0.447970i	$-0.147853\pi$
$641$	45.2710	1.78810	0.894048	+	0.447970i	$0.147853\pi$
$642$	0	0
$643$	23.6449	0.932464	0.466232	−	0.884662i	$-0.345611\pi$
$643$	23.6449	0.932464	0.466232	+	0.884662i	$0.345611\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−24.6264	−0.968165	−0.484082	−	0.875022i	$-0.660846\pi$
$647$	−24.6264	−0.968165	−0.484082	+	0.875022i	$0.660846\pi$
$648$	0	0
$649$	−7.38032	−0.289703
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.93000	0.310325	0.155163	−	0.987889i	$-0.450410\pi$
$653$	7.93000	0.310325	0.155163	+	0.987889i	$0.450410\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	20.3575	0.793017	0.396508	−	0.918031i	$-0.370222\pi$
$659$	20.3575	0.793017	0.396508	+	0.918031i	$0.370222\pi$
$660$	0	0
$661$	−39.5617	−1.53877	−0.769385	−	0.638785i	$-0.779437\pi$
$661$	−39.5617	−1.53877	−0.769385	+	0.638785i	$0.779437\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−67.5824	−2.61680
$668$	0	0
$669$	0	0
$670$	0	0
$671$	43.9084	1.69506
$672$	0	0
$673$	−1.46282	−0.0563874	−0.0281937	−	0.999602i	$-0.508976\pi$
$673$	−1.46282	−0.0563874	−0.0281937	+	0.999602i	$0.508976\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−13.0148	−0.500199	−0.250099	−	0.968220i	$-0.580463\pi$
$677$	−13.0148	−0.500199	−0.250099	+	0.968220i	$0.580463\pi$
$678$	0	0
$679$	−21.4448	−0.822975
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.11337	0.0808660	0.0404330	−	0.999182i	$-0.487126\pi$
$683$	2.11337	0.0808660	0.0404330	+	0.999182i	$0.487126\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−9.66037	−0.368031
$690$	0	0
$691$	−6.92443	−0.263418	−0.131709	−	0.991288i	$-0.542046\pi$
$691$	−6.92443	−0.263418	−0.131709	+	0.991288i	$0.542046\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−15.7192	−0.595408
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1.65310	−0.0624369	−0.0312185	−	0.999513i	$-0.509939\pi$
$701$	−1.65310	−0.0624369	−0.0312185	+	0.999513i	$0.509939\pi$
$702$	0	0
$703$	12.1708	0.459031
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−24.3680	−0.916452
$708$	0	0
$709$	−1.67982	−0.0630870	−0.0315435	−	0.999502i	$-0.510042\pi$
$709$	−1.67982	−0.0630870	−0.0315435	+	0.999502i	$0.510042\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	9.29917	0.348257
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	50.2441	1.87379	0.936895	−	0.349611i	$-0.113686\pi$
$719$	50.2441	1.87379	0.936895	+	0.349611i	$0.113686\pi$
$720$	0	0
$721$	13.1280	0.488914
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	18.2324	0.676203	0.338101	−	0.941110i	$-0.390215\pi$
$727$	18.2324	0.676203	0.338101	+	0.941110i	$0.390215\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5.47456	−0.202484
$732$	0	0
$733$	51.5137	1.90270	0.951350	−	0.308112i	$-0.0996971\pi$
$733$	51.5137	1.90270	0.951350	+	0.308112i	$0.0996971\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	60.3384	2.22259
$738$	0	0
$739$	−13.3261	−0.490207	−0.245103	−	0.969497i	$-0.578822\pi$
$739$	−13.3261	−0.490207	−0.245103	+	0.969497i	$0.578822\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	23.9448	0.878448	0.439224	−	0.898378i	$-0.355253\pi$
$743$	23.9448	0.878448	0.439224	+	0.898378i	$0.355253\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	26.8992	0.982875
$750$	0	0
$751$	0.0607894	0.00221823	0.00110912	−	0.999999i	$-0.499647\pi$
$751$	0.0607894	0.00221823	0.00110912	+	0.999999i	$0.499647\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−29.3402	−1.06639	−0.533194	−	0.845993i	$-0.679008\pi$
$757$	−29.3402	−1.06639	−0.533194	+	0.845993i	$0.679008\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−33.1931	−1.20325	−0.601625	−	0.798779i	$-0.705480\pi$
$761$	−33.1931	−1.20325	−0.601625	+	0.798779i	$0.705480\pi$
$762$	0	0
$763$	0.807952	0.0292498
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.34326	0.337366
$768$	0	0
$769$	7.90557	0.285082	0.142541	−	0.989789i	$-0.454473\pi$
$769$	7.90557	0.285082	0.142541	+	0.989789i	$0.454473\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−27.7151	−0.996843	−0.498421	−	0.866935i	$-0.666087\pi$
$773$	−27.7151	−0.996843	−0.498421	+	0.866935i	$0.666087\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.97890	−0.285874
$780$	0	0
$781$	61.8302	2.21246
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	17.1468	0.611217	0.305609	−	0.952157i	$-0.401140\pi$
$787$	17.1468	0.611217	0.305609	+	0.952157i	$0.401140\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−17.5812	−0.625114
$792$	0	0
$793$	−55.5867	−1.97394
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−21.0575	−0.745896	−0.372948	−	0.927852i	$-0.621653\pi$
$797$	−21.0575	−0.745896	−0.372948	+	0.927852i	$0.621653\pi$
$798$	0	0
$799$	10.4670	0.370297
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−45.4174	−1.60274
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−38.8466	−1.36577	−0.682887	−	0.730524i	$-0.739276\pi$
$809$	−38.8466	−1.36577	−0.682887	+	0.730524i	$0.739276\pi$
$810$	0	0
$811$	31.2820	1.09846	0.549229	−	0.835672i	$-0.314921\pi$
$811$	31.2820	1.09846	0.549229	+	0.835672i	$0.314921\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2.77883	−0.0972189
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−8.50658	−0.296882	−0.148441	−	0.988921i	$-0.547425\pi$
$821$	−8.50658	−0.296882	−0.148441	+	0.988921i	$0.547425\pi$
$822$	0	0
$823$	44.6746	1.55726	0.778630	−	0.627483i	$-0.215915\pi$
$823$	44.6746	1.55726	0.778630	+	0.627483i	$0.215915\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−26.6536	−0.926836	−0.463418	−	0.886140i	$-0.653377\pi$
$827$	−26.6536	−0.926836	−0.463418	+	0.886140i	$0.653377\pi$
$828$	0	0
$829$	4.21967	0.146555	0.0732776	−	0.997312i	$-0.476654\pi$
$829$	4.21967	0.146555	0.0732776	+	0.997312i	$0.476654\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−9.49261	−0.328899
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−17.1897	−0.593453	−0.296726	−	0.954963i	$-0.595895\pi$
$839$	−17.1897	−0.593453	−0.296726	+	0.954963i	$0.595895\pi$
$840$	0	0
$841$	23.2302	0.801041
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	29.4956	1.01348
$848$	0	0
$849$	0	0
$850$	0	0
$851$	59.6599	2.04511
$852$	0	0
$853$	−28.2243	−0.966381	−0.483191	−	0.875515i	$-0.660522\pi$
$853$	−28.2243	−0.966381	−0.483191	+	0.875515i	$0.660522\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−51.2310	−1.75002	−0.875008	−	0.484108i	$-0.839144\pi$
$857$	−51.2310	−1.75002	−0.875008	+	0.484108i	$0.839144\pi$
$858$	0	0
$859$	−53.1686	−1.81409	−0.907044	−	0.421036i	$-0.861667\pi$
$859$	−53.1686	−1.81409	−0.907044	+	0.421036i	$0.861667\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−26.8953	−0.915526	−0.457763	−	0.889074i	$-0.651349\pi$
$863$	−26.8953	−0.915526	−0.457763	+	0.889074i	$0.651349\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−44.5167	−1.51012
$870$	0	0
$871$	−76.3866	−2.58826
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−15.7008	−0.530178	−0.265089	−	0.964224i	$-0.585401\pi$
$877$	−15.7008	−0.530178	−0.265089	+	0.964224i	$0.585401\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2.32804	−0.0784336	−0.0392168	−	0.999231i	$-0.512486\pi$
$881$	−2.32804	−0.0784336	−0.0392168	+	0.999231i	$0.512486\pi$
$882$	0	0
$883$	45.0863	1.51727	0.758637	−	0.651514i	$-0.225866\pi$
$883$	45.0863	1.51727	0.758637	+	0.651514i	$0.225866\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	45.9907	1.54422	0.772108	−	0.635491i	$-0.219202\pi$
$887$	45.9907	1.54422	0.772108	+	0.635491i	$0.219202\pi$
$888$	0	0
$889$	−21.7907	−0.730836
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.31296	0.177791
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7.18675	−0.239691
$900$	0	0
$901$	5.74226	0.191302
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−41.9462	−1.39280	−0.696401	−	0.717653i	$-0.745216\pi$
$907$	−41.9462	−1.39280	−0.696401	+	0.717653i	$0.745216\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	38.6141	1.27934	0.639671	−	0.768649i	$-0.279071\pi$
$911$	38.6141	1.27934	0.639671	+	0.768649i	$0.279071\pi$
$912$	0	0
$913$	60.9140	2.01596
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6.52918	−0.215613
$918$	0	0
$919$	−46.7156	−1.54100	−0.770502	−	0.637437i	$-0.779995\pi$
$919$	−46.7156	−1.54100	−0.770502	+	0.637437i	$0.779995\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−78.2751	−2.57646
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	5.49036	0.180133	0.0900665	−	0.995936i	$-0.471292\pi$
$929$	5.49036	0.180133	0.0900665	+	0.995936i	$0.471292\pi$
$930$	0	0
$931$	−4.81834	−0.157915
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−17.6902	−0.577912	−0.288956	−	0.957342i	$-0.593308\pi$
$937$	−17.6902	−0.577912	−0.288956	+	0.957342i	$0.593308\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−41.0472	−1.33810	−0.669050	−	0.743217i	$-0.733299\pi$
$941$	−41.0472	−1.33810	−0.669050	+	0.743217i	$0.733299\pi$
$942$	0	0
$943$	−39.1116	−1.27365
$944$	0	0
$945$	0	0
$946$	0	0
$947$	26.6027	0.864472	0.432236	−	0.901760i	$-0.357725\pi$
$947$	26.6027	0.864472	0.432236	+	0.901760i	$0.357725\pi$
$948$	0	0
$949$	57.4970	1.86643
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−22.1029	−0.715984	−0.357992	−	0.933725i	$-0.616539\pi$
$953$	−22.1029	−0.715984	−0.357992	+	0.933725i	$0.616539\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−34.0666	−1.10007
$960$	0	0
$961$	−30.0111	−0.968101
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1.05390	−0.0338912	−0.0169456	−	0.999856i	$-0.505394\pi$
$967$	−1.05390	−0.0338912	−0.0169456	+	0.999856i	$0.505394\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.47456	0.0473210	0.0236605	−	0.999720i	$-0.492468\pi$
$971$	1.47456	0.0473210	0.0236605	+	0.999720i	$0.492468\pi$
$972$	0	0
$973$	32.8872	1.05432
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5.53670	−0.177135	−0.0885674	−	0.996070i	$-0.528229\pi$
$977$	−5.53670	−0.177135	−0.0885674	+	0.996070i	$0.528229\pi$
$978$	0	0
$979$	−15.3992	−0.492160
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−4.18145	−0.133368	−0.0666838	−	0.997774i	$-0.521242\pi$
$983$	−4.18145	−0.133368	−0.0666838	+	0.997774i	$0.521242\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−13.6215	−0.433138
$990$	0	0
$991$	−21.5056	−0.683147	−0.341573	−	0.939855i	$-0.610960\pi$
$991$	−21.5056	−0.683147	−0.341573	+	0.939855i	$0.610960\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	26.3446	0.834341	0.417171	−	0.908828i	$-0.363022\pi$
$997$	26.3446	0.834341	0.417171	+	0.908828i	$0.363022\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9000.2.a.r.1.4		4
3.2	odd	2		1000.2.a.e.1.3	✓	4
5.4	even	2		9000.2.a.ba.1.1		4
12.11	even	2		2000.2.a.r.1.2		4
15.2	even	4		1000.2.c.d.249.4		8
15.8	even	4		1000.2.c.d.249.5		8
15.14	odd	2		1000.2.a.h.1.2	yes	4
24.5	odd	2		8000.2.a.br.1.2		4
24.11	even	2		8000.2.a.bb.1.3		4
60.23	odd	4		2000.2.c.j.1249.4		8
60.47	odd	4		2000.2.c.j.1249.5		8
60.59	even	2		2000.2.a.m.1.3		4
120.29	odd	2		8000.2.a.ba.1.3		4
120.59	even	2		8000.2.a.bq.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1000.2.a.e.1.3	✓	4	3.2	odd	2
1000.2.a.h.1.2	yes	4	15.14	odd	2
1000.2.c.d.249.4		8	15.2	even	4
1000.2.c.d.249.5		8	15.8	even	4
2000.2.a.m.1.3		4	60.59	even	2
2000.2.a.r.1.2		4	12.11	even	2
2000.2.c.j.1249.4		8	60.23	odd	4
2000.2.c.j.1249.5		8	60.47	odd	4
8000.2.a.ba.1.3		4	120.29	odd	2
8000.2.a.bb.1.3		4	24.11	even	2
8000.2.a.bq.1.2		4	120.59	even	2
8000.2.a.br.1.2		4	24.5	odd	2
9000.2.a.r.1.4		4	1.1	even	1	trivial
9000.2.a.ba.1.1		4	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9000 = 2^{3} \cdot 3^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9000.a (trivial)

\(f(q)\)	\(=\)	\(q+2.11525 q^{7} +O(q^{10})\)	\(q+2.11525 q^{7} +4.99442 q^{11} -6.32279 q^{13} +3.75836 q^{17} +1.90770 q^{19} +9.35131 q^{23} -7.22705 q^{29} +0.994424 q^{31} +6.37984 q^{37} -4.18247 q^{41} -1.45664 q^{43} +2.78501 q^{47} -2.52573 q^{49} +1.52786 q^{53} -1.47771 q^{59} +8.79148 q^{61} +12.0811 q^{67} +12.3798 q^{71} -9.09362 q^{73} +10.5644 q^{77} -8.91328 q^{79} +12.1964 q^{83} -3.08328 q^{89} -13.3743 q^{91} -10.1382 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{7}+O(q^{10}) \) 4 * q - 6 * q^7	\( 4 q - 6 q^{7} - 4 q^{13} + 4 q^{17} + 14 q^{23} - 10 q^{29} - 16 q^{31} - 2 q^{41} - 12 q^{43} + 16 q^{47} + 2 q^{49} + 24 q^{53} - 8 q^{59} + 6 q^{61} + 16 q^{67} + 24 q^{71} - 4 q^{73} + 32 q^{77} - 48 q^{79} + 2 q^{83} - 10 q^{89} - 8 q^{91} - 4 q^{97}+O(q^{100}) \) 4 * q - 6 * q^7 - 4 * q^13 + 4 * q^17 + 14 * q^23 - 10 * q^29 - 16 * q^31 - 2 * q^41 - 12 * q^43 + 16 * q^47 + 2 * q^49 + 24 * q^53 - 8 * q^59 + 6 * q^61 + 16 * q^67 + 24 * q^71 - 4 * q^73 + 32 * q^77 - 48 * q^79 + 2 * q^83 - 10 * q^89 - 8 * q^91 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more