LMFDB - Embedded newform 3584.2.a.o.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3584,2,Mod(1,3584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3584, 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("3584.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3584 = 2^{9} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3584.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:	$28.6183840844$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 19x^{8} + 44x^{7} + 86x^{6} - 236x^{5} - 58x^{4} + 368x^{3} - 194x^{2} - 12x + 18 $ x^10 - 2x^9 - 19x^8 + 44x^7 + 86x^6 - 236x^5 - 58x^4 + 368x^3 - 194x^2 - 12*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.79086$ of defining polynomial
Character	$\chi$	$=$	3584.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.460386 q^{3} +1.55819 q^{5} -1.00000 q^{7} -2.78804 q^{9} +O(q^{10})$	$q+0.460386 q^{3} +1.55819 q^{5} -1.00000 q^{7} -2.78804 q^{9} -5.69782 q^{11} +0.801102 q^{13} +0.717367 q^{15} +4.85469 q^{17} +4.04590 q^{19} -0.460386 q^{21} +5.34057 q^{23} -2.57206 q^{25} -2.66474 q^{27} +8.47347 q^{29} -10.9126 q^{31} -2.62320 q^{33} -1.55819 q^{35} -7.26476 q^{37} +0.368817 q^{39} +11.6105 q^{41} +6.90654 q^{43} -4.34429 q^{45} +5.07068 q^{47} +1.00000 q^{49} +2.23503 q^{51} +12.2227 q^{53} -8.87826 q^{55} +1.86268 q^{57} -1.61096 q^{59} -7.21504 q^{61} +2.78804 q^{63} +1.24827 q^{65} +9.44702 q^{67} +2.45872 q^{69} +1.51816 q^{71} +4.21599 q^{73} -1.18414 q^{75} +5.69782 q^{77} -2.48184 q^{79} +7.13732 q^{81} +6.11724 q^{83} +7.56451 q^{85} +3.90107 q^{87} +10.0579 q^{89} -0.801102 q^{91} -5.02402 q^{93} +6.30426 q^{95} +12.0236 q^{97} +15.8858 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{7} + 22 q^{9}+O(q^{10}) $ 10 * q - 10 * q^7 + 22 * q^9	$ 10 q - 10 q^{7} + 22 q^{9} - 16 q^{15} + 16 q^{17} - 8 q^{23} + 30 q^{25} + 8 q^{31} + 12 q^{33} + 20 q^{41} + 24 q^{47} + 10 q^{49} + 32 q^{55} + 28 q^{57} - 22 q^{63} + 32 q^{65} + 48 q^{73} - 40 q^{79} + 62 q^{81} + 8 q^{87} + 16 q^{89} + 32 q^{95} + 32 q^{97}+O(q^{100}) $ 10 * q - 10 * q^7 + 22 * q^9 - 16 * q^15 + 16 * q^17 - 8 * q^23 + 30 * q^25 + 8 * q^31 + 12 * q^33 + 20 * q^41 + 24 * q^47 + 10 * q^49 + 32 * q^55 + 28 * q^57 - 22 * q^63 + 32 * q^65 + 48 * q^73 - 40 * q^79 + 62 * q^81 + 8 * q^87 + 16 * q^89 + 32 * q^95 + 32 * 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
$3$	0.460386	0.265804	0.132902	−	0.991129i	$-0.457570\pi$
$3$	0.460386	0.265804	0.132902	+	0.991129i	$0.457570\pi$
$4$	0	0
$5$	1.55819	0.696842	0.348421	−	0.937338i	$-0.386718\pi$
$5$	1.55819	0.696842	0.348421	+	0.937338i	$0.386718\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.78804	−0.929348
$10$	0	0
$11$	−5.69782	−1.71796	−0.858979	−	0.512011i	$-0.828901\pi$
$11$	−5.69782	−1.71796	−0.858979	+	0.512011i	$0.828901\pi$
$12$	0	0
$13$	0.801102	0.222186	0.111093	−	0.993810i	$-0.464565\pi$
$13$	0.801102	0.222186	0.111093	+	0.993810i	$0.464565\pi$
$14$	0	0
$15$	0.717367	0.185223
$16$	0	0
$17$	4.85469	1.17744	0.588718	−	0.808339i	$-0.299633\pi$
$17$	4.85469	1.17744	0.588718	+	0.808339i	$0.299633\pi$
$18$	0	0
$19$	4.04590	0.928192	0.464096	−	0.885785i	$-0.346379\pi$
$19$	4.04590	0.928192	0.464096	+	0.885785i	$0.346379\pi$
$20$	0	0
$21$	−0.460386	−0.100465
$22$	0	0
$23$	5.34057	1.11359	0.556793	−	0.830652i	$-0.312032\pi$
$23$	5.34057	1.11359	0.556793	+	0.830652i	$0.312032\pi$
$24$	0	0
$25$	−2.57206	−0.514412
$26$	0	0
$27$	−2.66474	−0.512829
$28$	0	0
$29$	8.47347	1.57348	0.786742	−	0.617282i	$-0.211766\pi$
$29$	8.47347	1.57348	0.786742	+	0.617282i	$0.211766\pi$
$30$	0	0
$31$	−10.9126	−1.95997	−0.979983	−	0.199083i	$-0.936204\pi$
$31$	−10.9126	−1.95997	−0.979983	+	0.199083i	$0.936204\pi$
$32$	0	0
$33$	−2.62320	−0.456640
$34$	0	0
$35$	−1.55819	−0.263381
$36$	0	0
$37$	−7.26476	−1.19432	−0.597159	−	0.802123i	$-0.703704\pi$
$37$	−7.26476	−1.19432	−0.597159	+	0.802123i	$0.703704\pi$
$38$	0	0
$39$	0.368817	0.0590579
$40$	0	0
$41$	11.6105	1.81325	0.906624	−	0.421939i	$-0.138650\pi$
$41$	11.6105	1.81325	0.906624	+	0.421939i	$0.138650\pi$
$42$	0	0
$43$	6.90654	1.05324	0.526618	−	0.850102i	$-0.323460\pi$
$43$	6.90654	1.05324	0.526618	+	0.850102i	$0.323460\pi$
$44$	0	0
$45$	−4.34429	−0.647608
$46$	0	0
$47$	5.07068	0.739634	0.369817	−	0.929105i	$-0.379420\pi$
$47$	5.07068	0.739634	0.369817	+	0.929105i	$0.379420\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.23503	0.312967
$52$	0	0
$53$	12.2227	1.67891	0.839457	−	0.543427i	$-0.182873\pi$
$53$	12.2227	1.67891	0.839457	+	0.543427i	$0.182873\pi$
$54$	0	0
$55$	−8.87826	−1.19714
$56$	0	0
$57$	1.86268	0.246717
$58$	0	0
$59$	−1.61096	−0.209729	−0.104864	−	0.994487i	$-0.533441\pi$
$59$	−1.61096	−0.209729	−0.104864	+	0.994487i	$0.533441\pi$
$60$	0	0
$61$	−7.21504	−0.923791	−0.461896	−	0.886934i	$-0.652831\pi$
$61$	−7.21504	−0.923791	−0.461896	+	0.886934i	$0.652831\pi$
$62$	0	0
$63$	2.78804	0.351261
$64$	0	0
$65$	1.24827	0.154828
$66$	0	0
$67$	9.44702	1.15414	0.577069	−	0.816696i	$-0.304197\pi$
$67$	9.44702	1.15414	0.577069	+	0.816696i	$0.304197\pi$
$68$	0	0
$69$	2.45872	0.295996
$70$	0	0
$71$	1.51816	0.180172	0.0900859	−	0.995934i	$-0.471286\pi$
$71$	1.51816	0.180172	0.0900859	+	0.995934i	$0.471286\pi$
$72$	0	0
$73$	4.21599	0.493444	0.246722	−	0.969086i	$-0.420647\pi$
$73$	4.21599	0.493444	0.246722	+	0.969086i	$0.420647\pi$
$74$	0	0
$75$	−1.18414	−0.136733
$76$	0	0
$77$	5.69782	0.649327
$78$	0	0
$79$	−2.48184	−0.279229	−0.139615	−	0.990206i	$-0.544586\pi$
$79$	−2.48184	−0.279229	−0.139615	+	0.990206i	$0.544586\pi$
$80$	0	0
$81$	7.13732	0.793036
$82$	0	0
$83$	6.11724	0.671454	0.335727	−	0.941959i	$-0.391018\pi$
$83$	6.11724	0.671454	0.335727	+	0.941959i	$0.391018\pi$
$84$	0	0
$85$	7.56451	0.820486
$86$	0	0
$87$	3.90107	0.418239
$88$	0	0
$89$	10.0579	1.06614	0.533069	−	0.846072i	$-0.321038\pi$
$89$	10.0579	1.06614	0.533069	+	0.846072i	$0.321038\pi$
$90$	0	0
$91$	−0.801102	−0.0839783
$92$	0	0
$93$	−5.02402	−0.520967
$94$	0	0
$95$	6.30426	0.646803
$96$	0	0
$97$	12.0236	1.22081	0.610404	−	0.792090i	$-0.291007\pi$
$97$	12.0236	1.22081	0.610404	+	0.792090i	$0.291007\pi$
$98$	0	0
$99$	15.8858	1.59658
$100$	0	0
$101$	−18.7744	−1.86812	−0.934060	−	0.357117i	$-0.883760\pi$
$101$	−18.7744	−1.86812	−0.934060	+	0.357117i	$0.883760\pi$
$102$	0	0
$103$	16.2148	1.59769	0.798846	−	0.601536i	$-0.205444\pi$
$103$	16.2148	1.59769	0.798846	+	0.601536i	$0.205444\pi$
$104$	0	0
$105$	−0.717367	−0.0700079
$106$	0	0
$107$	−9.75247	−0.942807	−0.471403	−	0.881918i	$-0.656252\pi$
$107$	−9.75247	−0.942807	−0.471403	+	0.881918i	$0.656252\pi$
$108$	0	0
$109$	7.59214	0.727195	0.363597	−	0.931556i	$-0.381548\pi$
$109$	7.59214	0.727195	0.363597	+	0.931556i	$0.381548\pi$
$110$	0	0
$111$	−3.34460	−0.317455
$112$	0	0
$113$	13.2814	1.24941	0.624706	−	0.780860i	$-0.285219\pi$
$113$	13.2814	1.24941	0.624706	+	0.780860i	$0.285219\pi$
$114$	0	0
$115$	8.32159	0.775992
$116$	0	0
$117$	−2.23351	−0.206488
$118$	0	0
$119$	−4.85469	−0.445029
$120$	0	0
$121$	21.4651	1.95138
$122$	0	0
$123$	5.34530	0.481969
$124$	0	0
$125$	−11.7987	−1.05531
$126$	0	0
$127$	12.9167	1.14617	0.573084	−	0.819497i	$-0.305747\pi$
$127$	12.9167	1.14617	0.573084	+	0.819497i	$0.305747\pi$
$128$	0	0
$129$	3.17968	0.279955
$130$	0	0
$131$	−13.3527	−1.16663	−0.583315	−	0.812246i	$-0.698245\pi$
$131$	−13.3527	−1.16663	−0.583315	+	0.812246i	$0.698245\pi$
$132$	0	0
$133$	−4.04590	−0.350824
$134$	0	0
$135$	−4.15215	−0.357361
$136$	0	0
$137$	13.3601	1.14143	0.570715	−	0.821148i	$-0.306666\pi$
$137$	13.3601	1.14143	0.570715	+	0.821148i	$0.306666\pi$
$138$	0	0
$139$	−9.68524	−0.821492	−0.410746	−	0.911750i	$-0.634732\pi$
$139$	−9.68524	−0.821492	−0.410746	+	0.911750i	$0.634732\pi$
$140$	0	0
$141$	2.33447	0.196598
$142$	0	0
$143$	−4.56454	−0.381706
$144$	0	0
$145$	13.2032	1.09647
$146$	0	0
$147$	0.460386	0.0379720
$148$	0	0
$149$	0.299751	0.0245565	0.0122783	−	0.999925i	$-0.496092\pi$
$149$	0.299751	0.0245565	0.0122783	+	0.999925i	$0.496092\pi$
$150$	0	0
$151$	−13.8803	−1.12957	−0.564783	−	0.825239i	$-0.691040\pi$
$151$	−13.8803	−1.12957	−0.564783	+	0.825239i	$0.691040\pi$
$152$	0	0
$153$	−13.5351	−1.09425
$154$	0	0
$155$	−17.0039	−1.36579
$156$	0	0
$157$	−9.48629	−0.757088	−0.378544	−	0.925583i	$-0.623575\pi$
$157$	−9.48629	−0.757088	−0.378544	+	0.925583i	$0.623575\pi$
$158$	0	0
$159$	5.62715	0.446262
$160$	0	0
$161$	−5.34057	−0.420896
$162$	0	0
$163$	1.66068	0.130074	0.0650371	−	0.997883i	$-0.479283\pi$
$163$	1.66068	0.130074	0.0650371	+	0.997883i	$0.479283\pi$
$164$	0	0
$165$	−4.08743	−0.318206
$166$	0	0
$167$	−18.8959	−1.46221	−0.731105	−	0.682265i	$-0.760995\pi$
$167$	−18.8959	−1.46221	−0.731105	+	0.682265i	$0.760995\pi$
$168$	0	0
$169$	−12.3582	−0.950633
$170$	0	0
$171$	−11.2801	−0.862614
$172$	0	0
$173$	−4.61641	−0.350979	−0.175490	−	0.984481i	$-0.556151\pi$
$173$	−4.61641	−0.350979	−0.175490	+	0.984481i	$0.556151\pi$
$174$	0	0
$175$	2.57206	0.194429
$176$	0	0
$177$	−0.741663	−0.0557468
$178$	0	0
$179$	−3.79017	−0.283290	−0.141645	−	0.989918i	$-0.545239\pi$
$179$	−3.79017	−0.283290	−0.141645	+	0.989918i	$0.545239\pi$
$180$	0	0
$181$	18.6282	1.38462	0.692311	−	0.721599i	$-0.256593\pi$
$181$	18.6282	1.38462	0.692311	+	0.721599i	$0.256593\pi$
$182$	0	0
$183$	−3.32171	−0.245548
$184$	0	0
$185$	−11.3198	−0.832251
$186$	0	0
$187$	−27.6612	−2.02278
$188$	0	0
$189$	2.66474	0.193831
$190$	0	0
$191$	−4.89496	−0.354187	−0.177093	−	0.984194i	$-0.556669\pi$
$191$	−4.89496	−0.354187	−0.177093	+	0.984194i	$0.556669\pi$
$192$	0	0
$193$	10.7517	0.773927	0.386963	−	0.922095i	$-0.373524\pi$
$193$	10.7517	0.773927	0.386963	+	0.922095i	$0.373524\pi$
$194$	0	0
$195$	0.574685	0.0411540
$196$	0	0
$197$	−4.28829	−0.305528	−0.152764	−	0.988263i	$-0.548817\pi$
$197$	−4.28829	−0.305528	−0.152764	+	0.988263i	$0.548817\pi$
$198$	0	0
$199$	10.0177	0.710134	0.355067	−	0.934841i	$-0.384458\pi$
$199$	10.0177	0.710134	0.355067	+	0.934841i	$0.384458\pi$
$200$	0	0
$201$	4.34928	0.306775
$202$	0	0
$203$	−8.47347	−0.594721
$204$	0	0
$205$	18.0912	1.26355
$206$	0	0
$207$	−14.8897	−1.03491
$208$	0	0
$209$	−23.0528	−1.59459
$210$	0	0
$211$	−1.51264	−0.104134	−0.0520671	−	0.998644i	$-0.516581\pi$
$211$	−1.51264	−0.104134	−0.0520671	+	0.998644i	$0.516581\pi$
$212$	0	0
$213$	0.698938	0.0478904
$214$	0	0
$215$	10.7617	0.733939
$216$	0	0
$217$	10.9126	0.740797
$218$	0	0
$219$	1.94098	0.131159
$220$	0	0
$221$	3.88910	0.261609
$222$	0	0
$223$	3.86743	0.258983	0.129491	−	0.991581i	$-0.458666\pi$
$223$	3.86743	0.258983	0.129491	+	0.991581i	$0.458666\pi$
$224$	0	0
$225$	7.17101	0.478068
$226$	0	0
$227$	16.0174	1.06311	0.531557	−	0.847023i	$-0.321607\pi$
$227$	16.0174	1.06311	0.531557	+	0.847023i	$0.321607\pi$
$228$	0	0
$229$	1.04044	0.0687544	0.0343772	−	0.999409i	$-0.489055\pi$
$229$	1.04044	0.0687544	0.0343772	+	0.999409i	$0.489055\pi$
$230$	0	0
$231$	2.62320	0.172594
$232$	0	0
$233$	−1.65072	−0.108142	−0.0540711	−	0.998537i	$-0.517220\pi$
$233$	−1.65072	−0.108142	−0.0540711	+	0.998537i	$0.517220\pi$
$234$	0	0
$235$	7.90105	0.515408
$236$	0	0
$237$	−1.14261	−0.0742204
$238$	0	0
$239$	18.5102	1.19732	0.598662	−	0.801002i	$-0.295699\pi$
$239$	18.5102	1.19732	0.598662	+	0.801002i	$0.295699\pi$
$240$	0	0
$241$	4.31491	0.277948	0.138974	−	0.990296i	$-0.455620\pi$
$241$	4.31491	0.277948	0.138974	+	0.990296i	$0.455620\pi$
$242$	0	0
$243$	11.2801	0.723621
$244$	0	0
$245$	1.55819	0.0995488
$246$	0	0
$247$	3.24118	0.206231
$248$	0	0
$249$	2.81629	0.178475
$250$	0	0
$251$	−7.95879	−0.502354	−0.251177	−	0.967941i	$-0.580818\pi$
$251$	−7.95879	−0.502354	−0.251177	+	0.967941i	$0.580818\pi$
$252$	0	0
$253$	−30.4296	−1.91309
$254$	0	0
$255$	3.48260	0.218089
$256$	0	0
$257$	5.44352	0.339558	0.169779	−	0.985482i	$-0.445695\pi$
$257$	5.44352	0.339558	0.169779	+	0.985482i	$0.445695\pi$
$258$	0	0
$259$	7.26476	0.451410
$260$	0	0
$261$	−23.6244	−1.46231
$262$	0	0
$263$	−6.91382	−0.426324	−0.213162	−	0.977017i	$-0.568376\pi$
$263$	−6.91382	−0.426324	−0.213162	+	0.977017i	$0.568376\pi$
$264$	0	0
$265$	19.0452	1.16994
$266$	0	0
$267$	4.63054	0.283384
$268$	0	0
$269$	4.61641	0.281468	0.140734	−	0.990047i	$-0.455054\pi$
$269$	4.61641	0.281468	0.140734	+	0.990047i	$0.455054\pi$
$270$	0	0
$271$	4.13257	0.251035	0.125518	−	0.992091i	$-0.459941\pi$
$271$	4.13257	0.251035	0.125518	+	0.992091i	$0.459941\pi$
$272$	0	0
$273$	−0.368817	−0.0223218
$274$	0	0
$275$	14.6551	0.883737
$276$	0	0
$277$	−9.74830	−0.585719	−0.292859	−	0.956156i	$-0.594607\pi$
$277$	−9.74830	−0.585719	−0.292859	+	0.956156i	$0.594607\pi$
$278$	0	0
$279$	30.4249	1.82149
$280$	0	0
$281$	16.5478	0.987161	0.493581	−	0.869700i	$-0.335688\pi$
$281$	16.5478	0.987161	0.493581	+	0.869700i	$0.335688\pi$
$282$	0	0
$283$	21.4101	1.27270	0.636350	−	0.771401i	$-0.280444\pi$
$283$	21.4101	1.27270	0.636350	+	0.771401i	$0.280444\pi$
$284$	0	0
$285$	2.90239	0.171923
$286$	0	0
$287$	−11.6105	−0.685344
$288$	0	0
$289$	6.56803	0.386355
$290$	0	0
$291$	5.53549	0.324496
$292$	0	0
$293$	−9.09788	−0.531504	−0.265752	−	0.964041i	$-0.585620\pi$
$293$	−9.09788	−0.531504	−0.265752	+	0.964041i	$0.585620\pi$
$294$	0	0
$295$	−2.51017	−0.146148
$296$	0	0
$297$	15.1832	0.881018
$298$	0	0
$299$	4.27834	0.247423
$300$	0	0
$301$	−6.90654	−0.398086
$302$	0	0
$303$	−8.64346	−0.496554
$304$	0	0
$305$	−11.2424	−0.643736
$306$	0	0
$307$	−18.0068	−1.02770	−0.513852	−	0.857879i	$-0.671782\pi$
$307$	−18.0068	−1.02770	−0.513852	+	0.857879i	$0.671782\pi$
$308$	0	0
$309$	7.46507	0.424673
$310$	0	0
$311$	−34.6644	−1.96564	−0.982820	−	0.184567i	$-0.940912\pi$
$311$	−34.6644	−1.96564	−0.982820	+	0.184567i	$0.940912\pi$
$312$	0	0
$313$	10.9134	0.616859	0.308430	−	0.951247i	$-0.400197\pi$
$313$	10.9134	0.616859	0.308430	+	0.951247i	$0.400197\pi$
$314$	0	0
$315$	4.34429	0.244773
$316$	0	0
$317$	24.2997	1.36481	0.682405	−	0.730974i	$-0.260934\pi$
$317$	24.2997	1.36481	0.682405	+	0.730974i	$0.260934\pi$
$318$	0	0
$319$	−48.2803	−2.70318
$320$	0	0
$321$	−4.48990	−0.250602
$322$	0	0
$323$	19.6416	1.09289
$324$	0	0
$325$	−2.06048	−0.114295
$326$	0	0
$327$	3.49532	0.193291
$328$	0	0
$329$	−5.07068	−0.279555
$330$	0	0
$331$	−14.9808	−0.823420	−0.411710	−	0.911315i	$-0.635068\pi$
$331$	−14.9808	−0.823420	−0.411710	+	0.911315i	$0.635068\pi$
$332$	0	0
$333$	20.2545	1.10994
$334$	0	0
$335$	14.7202	0.804251
$336$	0	0
$337$	16.3178	0.888885	0.444442	−	0.895807i	$-0.353402\pi$
$337$	16.3178	0.888885	0.444442	+	0.895807i	$0.353402\pi$
$338$	0	0
$339$	6.11460	0.332099
$340$	0	0
$341$	62.1782	3.36714
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	3.83115	0.206262
$346$	0	0
$347$	−20.4158	−1.09598	−0.547990	−	0.836485i	$-0.684607\pi$
$347$	−20.4158	−1.09598	−0.547990	+	0.836485i	$0.684607\pi$
$348$	0	0
$349$	0.958684	0.0513172	0.0256586	−	0.999671i	$-0.491832\pi$
$349$	0.958684	0.0513172	0.0256586	+	0.999671i	$0.491832\pi$
$350$	0	0
$351$	−2.13473	−0.113943
$352$	0	0
$353$	9.08835	0.483724	0.241862	−	0.970311i	$-0.422242\pi$
$353$	9.08835	0.483724	0.241862	+	0.970311i	$0.422242\pi$
$354$	0	0
$355$	2.36557	0.125551
$356$	0	0
$357$	−2.23503	−0.118291
$358$	0	0
$359$	−14.6180	−0.771507	−0.385754	−	0.922602i	$-0.626059\pi$
$359$	−14.6180	−0.771507	−0.385754	+	0.922602i	$0.626059\pi$
$360$	0	0
$361$	−2.63072	−0.138459
$362$	0	0
$363$	9.88226	0.518684
$364$	0	0
$365$	6.56929	0.343852
$366$	0	0
$367$	9.43473	0.492489	0.246245	−	0.969208i	$-0.420803\pi$
$367$	9.43473	0.492489	0.246245	+	0.969208i	$0.420803\pi$
$368$	0	0
$369$	−32.3705	−1.68514
$370$	0	0
$371$	−12.2227	−0.634570
$372$	0	0
$373$	9.04020	0.468084	0.234042	−	0.972227i	$-0.424805\pi$
$373$	9.04020	0.468084	0.234042	+	0.972227i	$0.424805\pi$
$374$	0	0
$375$	−5.43195	−0.280505
$376$	0	0
$377$	6.78812	0.349606
$378$	0	0
$379$	25.9672	1.33384	0.666921	−	0.745128i	$-0.267612\pi$
$379$	25.9672	1.33384	0.666921	+	0.745128i	$0.267612\pi$
$380$	0	0
$381$	5.94665	0.304656
$382$	0	0
$383$	6.05912	0.309607	0.154803	−	0.987945i	$-0.450526\pi$
$383$	6.05912	0.309607	0.154803	+	0.987945i	$0.450526\pi$
$384$	0	0
$385$	8.87826	0.452478
$386$	0	0
$387$	−19.2557	−0.978824
$388$	0	0
$389$	−33.4354	−1.69524	−0.847620	−	0.530604i	$-0.821965\pi$
$389$	−33.4354	−1.69524	−0.847620	+	0.530604i	$0.821965\pi$
$390$	0	0
$391$	25.9268	1.31117
$392$	0	0
$393$	−6.14740	−0.310095
$394$	0	0
$395$	−3.86717	−0.194579
$396$	0	0
$397$	−9.85496	−0.494606	−0.247303	−	0.968938i	$-0.579544\pi$
$397$	−9.85496	−0.494606	−0.247303	+	0.968938i	$0.579544\pi$
$398$	0	0
$399$	−1.86268	−0.0932504
$400$	0	0
$401$	7.22278	0.360688	0.180344	−	0.983604i	$-0.442279\pi$
$401$	7.22278	0.360688	0.180344	+	0.983604i	$0.442279\pi$
$402$	0	0
$403$	−8.74213	−0.435476
$404$	0	0
$405$	11.1213	0.552620
$406$	0	0
$407$	41.3933	2.05179
$408$	0	0
$409$	12.4416	0.615197	0.307598	−	0.951516i	$-0.400475\pi$
$409$	12.4416	0.615197	0.307598	+	0.951516i	$0.400475\pi$
$410$	0	0
$411$	6.15081	0.303397
$412$	0	0
$413$	1.61096	0.0792700
$414$	0	0
$415$	9.53179	0.467897
$416$	0	0
$417$	−4.45896	−0.218356
$418$	0	0
$419$	−5.27840	−0.257867	−0.128933	−	0.991653i	$-0.541155\pi$
$419$	−5.27840	−0.257867	−0.128933	+	0.991653i	$0.541155\pi$
$420$	0	0
$421$	−9.74830	−0.475103	−0.237552	−	0.971375i	$-0.576345\pi$
$421$	−9.74830	−0.475103	−0.237552	+	0.971375i	$0.576345\pi$
$422$	0	0
$423$	−14.1373	−0.687378
$424$	0	0
$425$	−12.4866	−0.605687
$426$	0	0
$427$	7.21504	0.349160
$428$	0	0
$429$	−2.10145	−0.101459
$430$	0	0
$431$	0.170962	0.00823495	0.00411747	−	0.999992i	$-0.498689\pi$
$431$	0.170962	0.00823495	0.00411747	+	0.999992i	$0.498689\pi$
$432$	0	0
$433$	21.8097	1.04811	0.524055	−	0.851685i	$-0.324419\pi$
$433$	21.8097	1.04811	0.524055	+	0.851685i	$0.324419\pi$
$434$	0	0
$435$	6.07859	0.291446
$436$	0	0
$437$	21.6074	1.03362
$438$	0	0
$439$	10.0646	0.480355	0.240178	−	0.970729i	$-0.422794\pi$
$439$	10.0646	0.480355	0.240178	+	0.970729i	$0.422794\pi$
$440$	0	0
$441$	−2.78804	−0.132764
$442$	0	0
$443$	1.26719	0.0602059	0.0301030	−	0.999547i	$-0.490416\pi$
$443$	1.26719	0.0602059	0.0301030	+	0.999547i	$0.490416\pi$
$444$	0	0
$445$	15.6721	0.742930
$446$	0	0
$447$	0.138001	0.00652723
$448$	0	0
$449$	−11.9899	−0.565840	−0.282920	−	0.959144i	$-0.591303\pi$
$449$	−11.9899	−0.565840	−0.282920	+	0.959144i	$0.591303\pi$
$450$	0	0
$451$	−66.1543	−3.11508
$452$	0	0
$453$	−6.39032	−0.300244
$454$	0	0
$455$	−1.24827	−0.0585196
$456$	0	0
$457$	−10.2451	−0.479247	−0.239624	−	0.970866i	$-0.577024\pi$
$457$	−10.2451	−0.479247	−0.239624	+	0.970866i	$0.577024\pi$
$458$	0	0
$459$	−12.9365	−0.603823
$460$	0	0
$461$	−23.6741	−1.10261	−0.551307	−	0.834302i	$-0.685871\pi$
$461$	−23.6741	−1.10261	−0.551307	+	0.834302i	$0.685871\pi$
$462$	0	0
$463$	−13.2464	−0.615612	−0.307806	−	0.951449i	$-0.599595\pi$
$463$	−13.2464	−0.615612	−0.307806	+	0.951449i	$0.599595\pi$
$464$	0	0
$465$	−7.82836	−0.363032
$466$	0	0
$467$	2.95855	0.136905	0.0684526	−	0.997654i	$-0.478194\pi$
$467$	2.95855	0.136905	0.0684526	+	0.997654i	$0.478194\pi$
$468$	0	0
$469$	−9.44702	−0.436223
$470$	0	0
$471$	−4.36736	−0.201237
$472$	0	0
$473$	−39.3522	−1.80942
$474$	0	0
$475$	−10.4063	−0.477473
$476$	0	0
$477$	−34.0774	−1.56029
$478$	0	0
$479$	28.7626	1.31420	0.657099	−	0.753804i	$-0.271783\pi$
$479$	28.7626	1.31420	0.657099	+	0.753804i	$0.271783\pi$
$480$	0	0
$481$	−5.81981	−0.265361
$482$	0	0
$483$	−2.45872	−0.111876
$484$	0	0
$485$	18.7349	0.850710
$486$	0	0
$487$	10.6260	0.481512	0.240756	−	0.970586i	$-0.422605\pi$
$487$	10.6260	0.481512	0.240756	+	0.970586i	$0.422605\pi$
$488$	0	0
$489$	0.764553	0.0345743
$490$	0	0
$491$	14.7765	0.666854	0.333427	−	0.942776i	$-0.391795\pi$
$491$	14.7765	0.666854	0.333427	+	0.942776i	$0.391795\pi$
$492$	0	0
$493$	41.1361	1.85268
$494$	0	0
$495$	24.7530	1.11256
$496$	0	0
$497$	−1.51816	−0.0680986
$498$	0	0
$499$	−20.7038	−0.926829	−0.463414	−	0.886142i	$-0.653376\pi$
$499$	−20.7038	−0.926829	−0.463414	+	0.886142i	$0.653376\pi$
$500$	0	0
$501$	−8.69943	−0.388662
$502$	0	0
$503$	−2.28256	−0.101774	−0.0508871	−	0.998704i	$-0.516205\pi$
$503$	−2.28256	−0.101774	−0.0508871	+	0.998704i	$0.516205\pi$
$504$	0	0
$505$	−29.2539	−1.30178
$506$	0	0
$507$	−5.68956	−0.252682
$508$	0	0
$509$	−34.2009	−1.51593	−0.757963	−	0.652297i	$-0.773806\pi$
$509$	−34.2009	−1.51593	−0.757963	+	0.652297i	$0.773806\pi$
$510$	0	0
$511$	−4.21599	−0.186504
$512$	0	0
$513$	−10.7813	−0.476004
$514$	0	0
$515$	25.2657	1.11334
$516$	0	0
$517$	−28.8918	−1.27066
$518$	0	0
$519$	−2.12533	−0.0932918
$520$	0	0
$521$	−35.5003	−1.55529	−0.777647	−	0.628701i	$-0.783587\pi$
$521$	−35.5003	−1.55529	−0.777647	+	0.628701i	$0.783587\pi$
$522$	0	0
$523$	17.1836	0.751388	0.375694	−	0.926744i	$-0.377404\pi$
$523$	17.1836	0.751388	0.375694	+	0.926744i	$0.377404\pi$
$524$	0	0
$525$	1.18414	0.0516802
$526$	0	0
$527$	−52.9774	−2.30773
$528$	0	0
$529$	5.52165	0.240072
$530$	0	0
$531$	4.49142	0.194911
$532$	0	0
$533$	9.30116	0.402878
$534$	0	0
$535$	−15.1962	−0.656987
$536$	0	0
$537$	−1.74494	−0.0752998
$538$	0	0
$539$	−5.69782	−0.245422
$540$	0	0
$541$	2.89210	0.124341	0.0621705	−	0.998066i	$-0.480198\pi$
$541$	2.89210	0.124341	0.0621705	+	0.998066i	$0.480198\pi$
$542$	0	0
$543$	8.57616	0.368038
$544$	0	0
$545$	11.8300	0.506740
$546$	0	0
$547$	−31.8601	−1.36224	−0.681119	−	0.732172i	$-0.738507\pi$
$547$	−31.8601	−1.36224	−0.681119	+	0.732172i	$0.738507\pi$
$548$	0	0
$549$	20.1158	0.858524
$550$	0	0
$551$	34.2828	1.46050
$552$	0	0
$553$	2.48184	0.105539
$554$	0	0
$555$	−5.21150	−0.221216
$556$	0	0
$557$	12.9735	0.549704	0.274852	−	0.961487i	$-0.411371\pi$
$557$	12.9735	0.549704	0.274852	+	0.961487i	$0.411371\pi$
$558$	0	0
$559$	5.53284	0.234014
$560$	0	0
$561$	−12.7348	−0.537665
$562$	0	0
$563$	44.3192	1.86783	0.933916	−	0.357493i	$-0.116368\pi$
$563$	44.3192	1.86783	0.933916	+	0.357493i	$0.116368\pi$
$564$	0	0
$565$	20.6949	0.870643
$566$	0	0
$567$	−7.13732	−0.299739
$568$	0	0
$569$	−34.3003	−1.43794	−0.718972	−	0.695039i	$-0.755387\pi$
$569$	−34.3003	−1.43794	−0.718972	+	0.695039i	$0.755387\pi$
$570$	0	0
$571$	31.9170	1.33569	0.667843	−	0.744302i	$-0.267218\pi$
$571$	31.9170	1.33569	0.667843	+	0.744302i	$0.267218\pi$
$572$	0	0
$573$	−2.25357	−0.0941443
$574$	0	0
$575$	−13.7363	−0.572841
$576$	0	0
$577$	−9.11936	−0.379644	−0.189822	−	0.981819i	$-0.560791\pi$
$577$	−9.11936	−0.379644	−0.189822	+	0.981819i	$0.560791\pi$
$578$	0	0
$579$	4.94995	0.205713
$580$	0	0
$581$	−6.11724	−0.253786
$582$	0	0
$583$	−69.6426	−2.88430
$584$	0	0
$585$	−3.48022	−0.143889
$586$	0	0
$587$	19.1574	0.790711	0.395355	−	0.918528i	$-0.370621\pi$
$587$	19.1574	0.790711	0.395355	+	0.918528i	$0.370621\pi$
$588$	0	0
$589$	−44.1514	−1.81922
$590$	0	0
$591$	−1.97427	−0.0812107
$592$	0	0
$593$	−10.4877	−0.430680	−0.215340	−	0.976539i	$-0.569086\pi$
$593$	−10.4877	−0.430680	−0.215340	+	0.976539i	$0.569086\pi$
$594$	0	0
$595$	−7.56451	−0.310115
$596$	0	0
$597$	4.61200	0.188757
$598$	0	0
$599$	13.2087	0.539692	0.269846	−	0.962903i	$-0.413027\pi$
$599$	13.2087	0.539692	0.269846	+	0.962903i	$0.413027\pi$
$600$	0	0
$601$	−0.130554	−0.00532540	−0.00266270	−	0.999996i	$-0.500848\pi$
$601$	−0.130554	−0.00532540	−0.00266270	+	0.999996i	$0.500848\pi$
$602$	0	0
$603$	−26.3387	−1.07260
$604$	0	0
$605$	33.4467	1.35980
$606$	0	0
$607$	42.3130	1.71743	0.858716	−	0.512452i	$-0.171263\pi$
$607$	42.3130	1.71743	0.858716	+	0.512452i	$0.171263\pi$
$608$	0	0
$609$	−3.90107	−0.158079
$610$	0	0
$611$	4.06213	0.164336
$612$	0	0
$613$	−12.8036	−0.517134	−0.258567	−	0.965993i	$-0.583250\pi$
$613$	−12.8036	−0.517134	−0.258567	+	0.965993i	$0.583250\pi$
$614$	0	0
$615$	8.32896	0.335856
$616$	0	0
$617$	9.74659	0.392383	0.196191	−	0.980566i	$-0.437143\pi$
$617$	9.74659	0.392383	0.196191	+	0.980566i	$0.437143\pi$
$618$	0	0
$619$	−2.51608	−0.101130	−0.0505649	−	0.998721i	$-0.516102\pi$
$619$	−2.51608	−0.101130	−0.0505649	+	0.998721i	$0.516102\pi$
$620$	0	0
$621$	−14.2312	−0.571079
$622$	0	0
$623$	−10.0579	−0.402963
$624$	0	0
$625$	−5.52422	−0.220969
$626$	0	0
$627$	−10.6132	−0.423850
$628$	0	0
$629$	−35.2682	−1.40623
$630$	0	0
$631$	−3.25431	−0.129552	−0.0647760	−	0.997900i	$-0.520633\pi$
$631$	−3.25431	−0.129552	−0.0647760	+	0.997900i	$0.520633\pi$
$632$	0	0
$633$	−0.696398	−0.0276793
$634$	0	0
$635$	20.1265	0.798697
$636$	0	0
$637$	0.801102	0.0317408
$638$	0	0
$639$	−4.23268	−0.167442
$640$	0	0
$641$	−4.61844	−0.182417	−0.0912087	−	0.995832i	$-0.529073\pi$
$641$	−4.61844	−0.182417	−0.0912087	+	0.995832i	$0.529073\pi$
$642$	0	0
$643$	36.0982	1.42358	0.711788	−	0.702394i	$-0.247886\pi$
$643$	36.0982	1.42358	0.711788	+	0.702394i	$0.247886\pi$
$644$	0	0
$645$	4.95452	0.195084
$646$	0	0
$647$	−19.8076	−0.778716	−0.389358	−	0.921086i	$-0.627303\pi$
$647$	−19.8076	−0.778716	−0.389358	+	0.921086i	$0.627303\pi$
$648$	0	0
$649$	9.17895	0.360305
$650$	0	0
$651$	5.02402	0.196907
$652$	0	0
$653$	3.94237	0.154277	0.0771385	−	0.997020i	$-0.475422\pi$
$653$	3.94237	0.154277	0.0771385	+	0.997020i	$0.475422\pi$
$654$	0	0
$655$	−20.8060	−0.812956
$656$	0	0
$657$	−11.7544	−0.458581
$658$	0	0
$659$	−22.9519	−0.894079	−0.447040	−	0.894514i	$-0.647522\pi$
$659$	−22.9519	−0.894079	−0.447040	+	0.894514i	$0.647522\pi$
$660$	0	0
$661$	−30.9033	−1.20200	−0.601000	−	0.799249i	$-0.705231\pi$
$661$	−30.9033	−1.20200	−0.601000	+	0.799249i	$0.705231\pi$
$662$	0	0
$663$	1.79049	0.0695369
$664$	0	0
$665$	−6.30426	−0.244469
$666$	0	0
$667$	45.2531	1.75221
$668$	0	0
$669$	1.78051	0.0688387
$670$	0	0
$671$	41.1100	1.58703
$672$	0	0
$673$	−32.2803	−1.24432	−0.622158	−	0.782892i	$-0.713744\pi$
$673$	−32.2803	−1.24432	−0.622158	+	0.782892i	$0.713744\pi$
$674$	0	0
$675$	6.85386	0.263805
$676$	0	0
$677$	34.7676	1.33623	0.668114	−	0.744059i	$-0.267102\pi$
$677$	34.7676	1.33623	0.668114	+	0.744059i	$0.267102\pi$
$678$	0	0
$679$	−12.0236	−0.461422
$680$	0	0
$681$	7.37420	0.282580
$682$	0	0
$683$	7.00766	0.268141	0.134070	−	0.990972i	$-0.457195\pi$
$683$	7.00766	0.268141	0.134070	+	0.990972i	$0.457195\pi$
$684$	0	0
$685$	20.8175	0.795396
$686$	0	0
$687$	0.479006	0.0182752
$688$	0	0
$689$	9.79161	0.373031
$690$	0	0
$691$	−31.0950	−1.18291	−0.591454	−	0.806339i	$-0.701446\pi$
$691$	−31.0950	−1.18291	−0.591454	+	0.806339i	$0.701446\pi$
$692$	0	0
$693$	−15.8858	−0.603451
$694$	0	0
$695$	−15.0914	−0.572450
$696$	0	0
$697$	56.3652	2.13498
$698$	0	0
$699$	−0.759969	−0.0287447
$700$	0	0
$701$	25.0206	0.945016	0.472508	−	0.881327i	$-0.343349\pi$
$701$	25.0206	0.945016	0.472508	+	0.881327i	$0.343349\pi$
$702$	0	0
$703$	−29.3925	−1.10856
$704$	0	0
$705$	3.63754	0.136998
$706$	0	0
$707$	18.7744	0.706083
$708$	0	0
$709$	−17.3014	−0.649768	−0.324884	−	0.945754i	$-0.605325\pi$
$709$	−17.3014	−0.649768	−0.324884	+	0.945754i	$0.605325\pi$
$710$	0	0
$711$	6.91949	0.259501
$712$	0	0
$713$	−58.2796	−2.18259
$714$	0	0
$715$	−7.11239	−0.265988
$716$	0	0
$717$	8.52183	0.318254
$718$	0	0
$719$	20.8103	0.776095	0.388047	−	0.921639i	$-0.373150\pi$
$719$	20.8103	0.776095	0.388047	+	0.921639i	$0.373150\pi$
$720$	0	0
$721$	−16.2148	−0.603871
$722$	0	0
$723$	1.98653	0.0738797
$724$	0	0
$725$	−21.7943	−0.809419
$726$	0	0
$727$	14.2308	0.527790	0.263895	−	0.964551i	$-0.414993\pi$
$727$	14.2308	0.527790	0.263895	+	0.964551i	$0.414993\pi$
$728$	0	0
$729$	−16.2187	−0.600694
$730$	0	0
$731$	33.5291	1.24012
$732$	0	0
$733$	−46.8888	−1.73188	−0.865939	−	0.500149i	$-0.833279\pi$
$733$	−46.8888	−1.73188	−0.865939	+	0.500149i	$0.833279\pi$
$734$	0	0
$735$	0.717367	0.0264605
$736$	0	0
$737$	−53.8274	−1.98276
$738$	0	0
$739$	−33.4989	−1.23228	−0.616138	−	0.787638i	$-0.711304\pi$
$739$	−33.4989	−1.23228	−0.616138	+	0.787638i	$0.711304\pi$
$740$	0	0
$741$	1.49219	0.0548171
$742$	0	0
$743$	46.6341	1.71084	0.855420	−	0.517936i	$-0.173299\pi$
$743$	46.6341	1.71084	0.855420	+	0.517936i	$0.173299\pi$
$744$	0	0
$745$	0.467067	0.0171120
$746$	0	0
$747$	−17.0551	−0.624015
$748$	0	0
$749$	9.75247	0.356347
$750$	0	0
$751$	38.1941	1.39372	0.696861	−	0.717207i	$-0.254580\pi$
$751$	38.1941	1.39372	0.696861	+	0.717207i	$0.254580\pi$
$752$	0	0
$753$	−3.66412	−0.133528
$754$	0	0
$755$	−21.6281	−0.787129
$756$	0	0
$757$	44.4810	1.61669	0.808345	−	0.588709i	$-0.200363\pi$
$757$	44.4810	1.61669	0.808345	+	0.588709i	$0.200363\pi$
$758$	0	0
$759$	−14.0094	−0.508508
$760$	0	0
$761$	−39.8684	−1.44523	−0.722614	−	0.691251i	$-0.757060\pi$
$761$	−39.8684	−1.44523	−0.722614	+	0.691251i	$0.757060\pi$
$762$	0	0
$763$	−7.59214	−0.274854
$764$	0	0
$765$	−21.0902	−0.762517
$766$	0	0
$767$	−1.29054	−0.0465988
$768$	0	0
$769$	−38.0582	−1.37241	−0.686206	−	0.727407i	$-0.740725\pi$
$769$	−38.0582	−1.37241	−0.686206	+	0.727407i	$0.740725\pi$
$770$	0	0
$771$	2.50612	0.0902559
$772$	0	0
$773$	−38.5593	−1.38688	−0.693441	−	0.720514i	$-0.743906\pi$
$773$	−38.5593	−1.38688	−0.693441	+	0.720514i	$0.743906\pi$
$774$	0	0
$775$	28.0679	1.00823
$776$	0	0
$777$	3.34460	0.119987
$778$	0	0
$779$	46.9747	1.68304
$780$	0	0
$781$	−8.65017	−0.309528
$782$	0	0
$783$	−22.5796	−0.806928
$784$	0	0
$785$	−14.7814	−0.527571
$786$	0	0
$787$	27.8098	0.991313	0.495656	−	0.868519i	$-0.334928\pi$
$787$	27.8098	0.991313	0.495656	+	0.868519i	$0.334928\pi$
$788$	0	0
$789$	−3.18303	−0.113319
$790$	0	0
$791$	−13.2814	−0.472234
$792$	0	0
$793$	−5.77998	−0.205253
$794$	0	0
$795$	8.76815	0.310974
$796$	0	0
$797$	−24.1683	−0.856084	−0.428042	−	0.903759i	$-0.640796\pi$
$797$	−24.1683	−0.856084	−0.428042	+	0.903759i	$0.640796\pi$
$798$	0	0
$799$	24.6166	0.870872
$800$	0	0
$801$	−28.0420	−0.990814
$802$	0	0
$803$	−24.0219	−0.847715
$804$	0	0
$805$	−8.32159	−0.293298
$806$	0	0
$807$	2.12533	0.0748153
$808$	0	0
$809$	−10.4025	−0.365731	−0.182865	−	0.983138i	$-0.558537\pi$
$809$	−10.4025	−0.365731	−0.182865	+	0.983138i	$0.558537\pi$
$810$	0	0
$811$	23.0878	0.810722	0.405361	−	0.914157i	$-0.367146\pi$
$811$	23.0878	0.810722	0.405361	+	0.914157i	$0.367146\pi$
$812$	0	0
$813$	1.90258	0.0667263
$814$	0	0
$815$	2.58764	0.0906411
$816$	0	0
$817$	27.9431	0.977606
$818$	0	0
$819$	2.23351	0.0780451
$820$	0	0
$821$	−31.4927	−1.09910	−0.549551	−	0.835460i	$-0.685201\pi$
$821$	−31.4927	−1.09910	−0.549551	+	0.835460i	$0.685201\pi$
$822$	0	0
$823$	−15.8088	−0.551059	−0.275530	−	0.961293i	$-0.588853\pi$
$823$	−15.8088	−0.551059	−0.275530	+	0.961293i	$0.588853\pi$
$824$	0	0
$825$	6.74702	0.234901
$826$	0	0
$827$	18.2711	0.635348	0.317674	−	0.948200i	$-0.397098\pi$
$827$	18.2711	0.635348	0.317674	+	0.948200i	$0.397098\pi$
$828$	0	0
$829$	−4.26253	−0.148044	−0.0740219	−	0.997257i	$-0.523583\pi$
$829$	−4.26253	−0.148044	−0.0740219	+	0.997257i	$0.523583\pi$
$830$	0	0
$831$	−4.48798	−0.155686
$832$	0	0
$833$	4.85469	0.168205
$834$	0	0
$835$	−29.4434	−1.01893
$836$	0	0
$837$	29.0793	1.00513
$838$	0	0
$839$	−35.7742	−1.23506	−0.617531	−	0.786547i	$-0.711867\pi$
$839$	−35.7742	−1.23506	−0.617531	+	0.786547i	$0.711867\pi$
$840$	0	0
$841$	42.7998	1.47585
$842$	0	0
$843$	7.61840	0.262392
$844$	0	0
$845$	−19.2564	−0.662441
$846$	0	0
$847$	−21.4651	−0.737551
$848$	0	0
$849$	9.85693	0.338289
$850$	0	0
$851$	−38.7979	−1.32998
$852$	0	0
$853$	19.7771	0.677154	0.338577	−	0.940939i	$-0.390054\pi$
$853$	19.7771	0.677154	0.338577	+	0.940939i	$0.390054\pi$
$854$	0	0
$855$	−17.5765	−0.601105
$856$	0	0
$857$	−18.4409	−0.629927	−0.314964	−	0.949104i	$-0.601992\pi$
$857$	−18.4409	−0.629927	−0.314964	+	0.949104i	$0.601992\pi$
$858$	0	0
$859$	36.5239	1.24618	0.623089	−	0.782151i	$-0.285877\pi$
$859$	36.5239	1.24618	0.623089	+	0.782151i	$0.285877\pi$
$860$	0	0
$861$	−5.34530	−0.182167
$862$	0	0
$863$	−40.5455	−1.38018	−0.690092	−	0.723722i	$-0.742430\pi$
$863$	−40.5455	−1.38018	−0.690092	+	0.723722i	$0.742430\pi$
$864$	0	0
$865$	−7.19322	−0.244577
$866$	0	0
$867$	3.02383	0.102695
$868$	0	0
$869$	14.1411	0.479704
$870$	0	0
$871$	7.56803	0.256433
$872$	0	0
$873$	−33.5222	−1.13456
$874$	0	0
$875$	11.7987	0.398868
$876$	0	0
$877$	43.4834	1.46833	0.734165	−	0.678971i	$-0.237574\pi$
$877$	43.4834	1.46833	0.734165	+	0.678971i	$0.237574\pi$
$878$	0	0
$879$	−4.18854	−0.141276
$880$	0	0
$881$	10.9941	0.370401	0.185201	−	0.982701i	$-0.440706\pi$
$881$	10.9941	0.370401	0.185201	+	0.982701i	$0.440706\pi$
$882$	0	0
$883$	−37.3203	−1.25593	−0.627964	−	0.778242i	$-0.716111\pi$
$883$	−37.3203	−1.25593	−0.627964	+	0.778242i	$0.716111\pi$
$884$	0	0
$885$	−1.15565	−0.0388467
$886$	0	0
$887$	−5.10169	−0.171298	−0.0856490	−	0.996325i	$-0.527296\pi$
$887$	−5.10169	−0.171298	−0.0856490	+	0.996325i	$0.527296\pi$
$888$	0	0
$889$	−12.9167	−0.433211
$890$	0	0
$891$	−40.6672	−1.36240
$892$	0	0
$893$	20.5154	0.686523
$894$	0	0
$895$	−5.90578	−0.197408
$896$	0	0
$897$	1.96969	0.0657660
$898$	0	0
$899$	−92.4678	−3.08398
$900$	0	0
$901$	59.3373	1.97681
$902$	0	0
$903$	−3.17968	−0.105813
$904$	0	0
$905$	29.0262	0.964862
$906$	0	0
$907$	54.0312	1.79408	0.897038	−	0.441954i	$-0.145715\pi$
$907$	54.0312	1.79408	0.897038	+	0.441954i	$0.145715\pi$
$908$	0	0
$909$	52.3438	1.73613
$910$	0	0
$911$	1.23276	0.0408431	0.0204216	−	0.999791i	$-0.493499\pi$
$911$	1.23276	0.0408431	0.0204216	+	0.999791i	$0.493499\pi$
$912$	0	0
$913$	−34.8549	−1.15353
$914$	0	0
$915$	−5.17583	−0.171108
$916$	0	0
$917$	13.3527	0.440945
$918$	0	0
$919$	25.2708	0.833606	0.416803	−	0.908997i	$-0.363150\pi$
$919$	25.2708	0.833606	0.416803	+	0.908997i	$0.363150\pi$
$920$	0	0
$921$	−8.29010	−0.273168
$922$	0	0
$923$	1.21620	0.0400316
$924$	0	0
$925$	18.6854	0.614372
$926$	0	0
$927$	−45.2076	−1.48481
$928$	0	0
$929$	50.3081	1.65055	0.825277	−	0.564728i	$-0.191019\pi$
$929$	50.3081	1.65055	0.825277	+	0.564728i	$0.191019\pi$
$930$	0	0
$931$	4.04590	0.132599
$932$	0	0
$933$	−15.9590	−0.522475
$934$	0	0
$935$	−43.1012	−1.40956
$936$	0	0
$937$	−33.6694	−1.09993	−0.549965	−	0.835188i	$-0.685359\pi$
$937$	−33.6694	−1.09993	−0.549965	+	0.835188i	$0.685359\pi$
$938$	0	0
$939$	5.02436	0.163964
$940$	0	0
$941$	−34.2749	−1.11733	−0.558666	−	0.829393i	$-0.688686\pi$
$941$	−34.2749	−1.11733	−0.558666	+	0.829393i	$0.688686\pi$
$942$	0	0
$943$	62.0064	2.01921
$944$	0	0
$945$	4.15215	0.135070
$946$	0	0
$947$	53.1716	1.72784	0.863922	−	0.503626i	$-0.168001\pi$
$947$	53.1716	1.72784	0.863922	+	0.503626i	$0.168001\pi$
$948$	0	0
$949$	3.37744	0.109636
$950$	0	0
$951$	11.1873	0.362772
$952$	0	0
$953$	−32.3378	−1.04752	−0.523761	−	0.851865i	$-0.675472\pi$
$953$	−32.3378	−1.04752	−0.523761	+	0.851865i	$0.675472\pi$
$954$	0	0
$955$	−7.62725	−0.246812
$956$	0	0
$957$	−22.2276	−0.718517
$958$	0	0
$959$	−13.3601	−0.431420
$960$	0	0
$961$	88.0854	2.84146
$962$	0	0
$963$	27.1903	0.876196
$964$	0	0
$965$	16.7532	0.539304
$966$	0	0
$967$	−23.2841	−0.748765	−0.374383	−	0.927274i	$-0.622145\pi$
$967$	−23.2841	−0.748765	−0.374383	+	0.927274i	$0.622145\pi$
$968$	0	0
$969$	9.04272	0.290494
$970$	0	0
$971$	−20.3114	−0.651824	−0.325912	−	0.945400i	$-0.605671\pi$
$971$	−20.3114	−0.651824	−0.325912	+	0.945400i	$0.605671\pi$
$972$	0	0
$973$	9.68524	0.310495
$974$	0	0
$975$	−0.948618	−0.0303801
$976$	0	0
$977$	−2.54568	−0.0814434	−0.0407217	−	0.999171i	$-0.512966\pi$
$977$	−2.54568	−0.0814434	−0.0407217	+	0.999171i	$0.512966\pi$
$978$	0	0
$979$	−57.3083	−1.83158
$980$	0	0
$981$	−21.1672	−0.675817
$982$	0	0
$983$	37.0045	1.18026	0.590130	−	0.807309i	$-0.299077\pi$
$983$	37.0045	1.18026	0.590130	+	0.807309i	$0.299077\pi$
$984$	0	0
$985$	−6.68195	−0.212905
$986$	0	0
$987$	−2.33447	−0.0743070
$988$	0	0
$989$	36.8848	1.17287
$990$	0	0
$991$	−20.5737	−0.653545	−0.326772	−	0.945103i	$-0.605961\pi$
$991$	−20.5737	−0.653545	−0.326772	+	0.945103i	$0.605961\pi$
$992$	0	0
$993$	−6.89697	−0.218869
$994$	0	0
$995$	15.6094	0.494851
$996$	0	0
$997$	−36.2391	−1.14770	−0.573851	−	0.818960i	$-0.694551\pi$
$997$	−36.2391	−1.14770	−0.573851	+	0.818960i	$0.694551\pi$
$998$	0	0
$999$	19.3587	0.612481

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3584.2.a.o.1.6	yes	10
4.3	odd	2		3584.2.a.p.1.5	yes	10
8.3	odd	2		3584.2.a.p.1.6	yes	10
8.5	even	2	inner	3584.2.a.o.1.5	✓	10
16.3	odd	4		3584.2.b.g.1793.5		10
16.5	even	4		3584.2.b.h.1793.5		10
16.11	odd	4		3584.2.b.g.1793.6		10
16.13	even	4		3584.2.b.h.1793.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.2.a.o.1.5	✓	10	8.5	even	2	inner
3584.2.a.o.1.6	yes	10	1.1	even	1	trivial
3584.2.a.p.1.5	yes	10	4.3	odd	2
3584.2.a.p.1.6	yes	10	8.3	odd	2
3584.2.b.g.1793.5		10	16.3	odd	4
3584.2.b.g.1793.6		10	16.11	odd	4
3584.2.b.h.1793.5		10	16.5	even	4
3584.2.b.h.1793.6		10	16.13	even	4

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:	\( N \)	\(=\)	\( 3584 = 2^{9} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3584.a (trivial)

\(f(q)\)	\(=\)	\(q+0.460386 q^{3} +1.55819 q^{5} -1.00000 q^{7} -2.78804 q^{9} +O(q^{10})\)	\(q+0.460386 q^{3} +1.55819 q^{5} -1.00000 q^{7} -2.78804 q^{9} -5.69782 q^{11} +0.801102 q^{13} +0.717367 q^{15} +4.85469 q^{17} +4.04590 q^{19} -0.460386 q^{21} +5.34057 q^{23} -2.57206 q^{25} -2.66474 q^{27} +8.47347 q^{29} -10.9126 q^{31} -2.62320 q^{33} -1.55819 q^{35} -7.26476 q^{37} +0.368817 q^{39} +11.6105 q^{41} +6.90654 q^{43} -4.34429 q^{45} +5.07068 q^{47} +1.00000 q^{49} +2.23503 q^{51} +12.2227 q^{53} -8.87826 q^{55} +1.86268 q^{57} -1.61096 q^{59} -7.21504 q^{61} +2.78804 q^{63} +1.24827 q^{65} +9.44702 q^{67} +2.45872 q^{69} +1.51816 q^{71} +4.21599 q^{73} -1.18414 q^{75} +5.69782 q^{77} -2.48184 q^{79} +7.13732 q^{81} +6.11724 q^{83} +7.56451 q^{85} +3.90107 q^{87} +10.0579 q^{89} -0.801102 q^{91} -5.02402 q^{93} +6.30426 q^{95} +12.0236 q^{97} +15.8858 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{7} + 22 q^{9}+O(q^{10}) \) 10 * q - 10 * q^7 + 22 * q^9	\( 10 q - 10 q^{7} + 22 q^{9} - 16 q^{15} + 16 q^{17} - 8 q^{23} + 30 q^{25} + 8 q^{31} + 12 q^{33} + 20 q^{41} + 24 q^{47} + 10 q^{49} + 32 q^{55} + 28 q^{57} - 22 q^{63} + 32 q^{65} + 48 q^{73} - 40 q^{79} + 62 q^{81} + 8 q^{87} + 16 q^{89} + 32 q^{95} + 32 q^{97}+O(q^{100}) \) 10 * q - 10 * q^7 + 22 * q^9 - 16 * q^15 + 16 * q^17 - 8 * q^23 + 30 * q^25 + 8 * q^31 + 12 * q^33 + 20 * q^41 + 24 * q^47 + 10 * q^49 + 32 * q^55 + 28 * q^57 - 22 * q^63 + 32 * q^65 + 48 * q^73 - 40 * q^79 + 62 * q^81 + 8 * q^87 + 16 * q^89 + 32 * q^95 + 32 * q^97

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