LMFDB - Embedded newform 8046.2.a.n.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 31 x^{10} + 82 x^{9} + 334 x^{8} - 684 x^{7} - 1561 x^{6} + 1551 x^{5} + 3573 x^{4} + \cdots - 3 $ x^12 - 3x^11 - 31x^10 + 82x^9 + 334x^8 - 684x^7 - 1561x^6 + 1551x^5 + 3573x^4 + 345x^3 - 1607x^2 - 594*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.539392$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +0.539392 q^{5} +0.739124 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +0.539392 q^{5} +0.739124 q^{7} +1.00000 q^{8} +0.539392 q^{10} -0.996296 q^{11} -1.00958 q^{13} +0.739124 q^{14} +1.00000 q^{16} +2.13949 q^{17} -4.61478 q^{19} +0.539392 q^{20} -0.996296 q^{22} -2.91642 q^{23} -4.70906 q^{25} -1.00958 q^{26} +0.739124 q^{28} +8.59260 q^{29} -10.1248 q^{31} +1.00000 q^{32} +2.13949 q^{34} +0.398678 q^{35} -0.483652 q^{37} -4.61478 q^{38} +0.539392 q^{40} -9.71579 q^{41} -5.50325 q^{43} -0.996296 q^{44} -2.91642 q^{46} -11.6002 q^{47} -6.45370 q^{49} -4.70906 q^{50} -1.00958 q^{52} -6.43001 q^{53} -0.537395 q^{55} +0.739124 q^{56} +8.59260 q^{58} -1.94920 q^{59} +3.68667 q^{61} -10.1248 q^{62} +1.00000 q^{64} -0.544557 q^{65} +8.13749 q^{67} +2.13949 q^{68} +0.398678 q^{70} -0.798603 q^{71} +8.61849 q^{73} -0.483652 q^{74} -4.61478 q^{76} -0.736387 q^{77} +5.01938 q^{79} +0.539392 q^{80} -9.71579 q^{82} -6.37875 q^{83} +1.15403 q^{85} -5.50325 q^{86} -0.996296 q^{88} +1.68698 q^{89} -0.746202 q^{91} -2.91642 q^{92} -11.6002 q^{94} -2.48918 q^{95} +10.1785 q^{97} -6.45370 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 12 q^{4} - 3 q^{5} - 6 q^{7} + 12 q^{8}+O(q^{10}) $ 12 * q + 12 * q^2 + 12 * q^4 - 3 * q^5 - 6 * q^7 + 12 * q^8	$ 12 q + 12 q^{2} + 12 q^{4} - 3 q^{5} - 6 q^{7} + 12 q^{8} - 3 q^{10} - 14 q^{11} - 3 q^{13} - 6 q^{14} + 12 q^{16} - 8 q^{17} - 4 q^{19} - 3 q^{20} - 14 q^{22} - 13 q^{23} + 11 q^{25} - 3 q^{26} - 6 q^{28} - 23 q^{29} - 14 q^{31} + 12 q^{32} - 8 q^{34} - 32 q^{35} - 19 q^{37} - 4 q^{38} - 3 q^{40} - 30 q^{41} - 15 q^{43} - 14 q^{44} - 13 q^{46} + q^{47} + 14 q^{49} + 11 q^{50} - 3 q^{52} - 16 q^{53} - 7 q^{55} - 6 q^{56} - 23 q^{58} - 26 q^{59} - 16 q^{61} - 14 q^{62} + 12 q^{64} - 8 q^{65} - 39 q^{67} - 8 q^{68} - 32 q^{70} - 15 q^{71} - 2 q^{73} - 19 q^{74} - 4 q^{76} - 34 q^{77} - 13 q^{79} - 3 q^{80} - 30 q^{82} - 6 q^{83} - 11 q^{85} - 15 q^{86} - 14 q^{88} - 18 q^{89} - 35 q^{91} - 13 q^{92} + q^{94} - 51 q^{95} + 19 q^{97} + 14 q^{98}+O(q^{100}) $ 12 * q + 12 * q^2 + 12 * q^4 - 3 * q^5 - 6 * q^7 + 12 * q^8 - 3 * q^10 - 14 * q^11 - 3 * q^13 - 6 * q^14 + 12 * q^16 - 8 * q^17 - 4 * q^19 - 3 * q^20 - 14 * q^22 - 13 * q^23 + 11 * q^25 - 3 * q^26 - 6 * q^28 - 23 * q^29 - 14 * q^31 + 12 * q^32 - 8 * q^34 - 32 * q^35 - 19 * q^37 - 4 * q^38 - 3 * q^40 - 30 * q^41 - 15 * q^43 - 14 * q^44 - 13 * q^46 + q^47 + 14 * q^49 + 11 * q^50 - 3 * q^52 - 16 * q^53 - 7 * q^55 - 6 * q^56 - 23 * q^58 - 26 * q^59 - 16 * q^61 - 14 * q^62 + 12 * q^64 - 8 * q^65 - 39 * q^67 - 8 * q^68 - 32 * q^70 - 15 * q^71 - 2 * q^73 - 19 * q^74 - 4 * q^76 - 34 * q^77 - 13 * q^79 - 3 * q^80 - 30 * q^82 - 6 * q^83 - 11 * q^85 - 15 * q^86 - 14 * q^88 - 18 * q^89 - 35 * q^91 - 13 * q^92 + q^94 - 51 * q^95 + 19 * 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
$5$	0.539392	0.241224	0.120612	−	0.992700i	$-0.461514\pi$
$5$	0.539392	0.241224	0.120612	+	0.992700i	$0.461514\pi$
$6$	0	0
$7$	0.739124	0.279363	0.139681	−	0.990197i	$-0.455392\pi$
$7$	0.739124	0.279363	0.139681	+	0.990197i	$0.455392\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0.539392	0.170571
$11$	−0.996296	−0.300395	−0.150197	−	0.988656i	$-0.547991\pi$
$11$	−0.996296	−0.300395	−0.150197	+	0.988656i	$0.547991\pi$
$12$	0	0
$13$	−1.00958	−0.280006	−0.140003	−	0.990151i	$-0.544711\pi$
$13$	−1.00958	−0.280006	−0.140003	+	0.990151i	$0.544711\pi$
$14$	0.739124	0.197539
$15$	0	0
$16$	1.00000	0.250000
$17$	2.13949	0.518903	0.259452	−	0.965756i	$-0.416458\pi$
$17$	2.13949	0.518903	0.259452	+	0.965756i	$0.416458\pi$
$18$	0	0
$19$	−4.61478	−1.05870	−0.529352	−	0.848403i	$-0.677565\pi$
$19$	−4.61478	−1.05870	−0.529352	+	0.848403i	$0.677565\pi$
$20$	0.539392	0.120612
$21$	0	0
$22$	−0.996296	−0.212411
$23$	−2.91642	−0.608116	−0.304058	−	0.952653i	$-0.598342\pi$
$23$	−2.91642	−0.608116	−0.304058	+	0.952653i	$0.598342\pi$
$24$	0	0
$25$	−4.70906	−0.941811
$26$	−1.00958	−0.197994
$27$	0	0
$28$	0.739124	0.139681
$29$	8.59260	1.59561	0.797803	−	0.602918i	$-0.205995\pi$
$29$	8.59260	1.59561	0.797803	+	0.602918i	$0.205995\pi$
$30$	0	0
$31$	−10.1248	−1.81847	−0.909237	−	0.416278i	$-0.863334\pi$
$31$	−10.1248	−1.81847	−0.909237	+	0.416278i	$0.863334\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.13949	0.366920
$35$	0.398678	0.0673889
$36$	0	0
$37$	−0.483652	−0.0795119	−0.0397559	−	0.999209i	$-0.512658\pi$
$37$	−0.483652	−0.0795119	−0.0397559	+	0.999209i	$0.512658\pi$
$38$	−4.61478	−0.748616
$39$	0	0
$40$	0.539392	0.0852854
$41$	−9.71579	−1.51735	−0.758676	−	0.651468i	$-0.774153\pi$
$41$	−9.71579	−1.51735	−0.758676	+	0.651468i	$0.774153\pi$
$42$	0	0
$43$	−5.50325	−0.839238	−0.419619	−	0.907700i	$-0.637836\pi$
$43$	−5.50325	−0.839238	−0.419619	+	0.907700i	$0.637836\pi$
$44$	−0.996296	−0.150197
$45$	0	0
$46$	−2.91642	−0.430003
$47$	−11.6002	−1.69206	−0.846031	−	0.533133i	$-0.821014\pi$
$47$	−11.6002	−1.69206	−0.846031	+	0.533133i	$0.821014\pi$
$48$	0	0
$49$	−6.45370	−0.921956
$50$	−4.70906	−0.665961
$51$	0	0
$52$	−1.00958	−0.140003
$53$	−6.43001	−0.883229	−0.441615	−	0.897205i	$-0.645594\pi$
$53$	−6.43001	−0.883229	−0.441615	+	0.897205i	$0.645594\pi$
$54$	0	0
$55$	−0.537395	−0.0724623
$56$	0.739124	0.0987696
$57$	0	0
$58$	8.59260	1.12826
$59$	−1.94920	−0.253764	−0.126882	−	0.991918i	$-0.540497\pi$
$59$	−1.94920	−0.253764	−0.126882	+	0.991918i	$0.540497\pi$
$60$	0	0
$61$	3.68667	0.472030	0.236015	−	0.971749i	$-0.424159\pi$
$61$	3.68667	0.472030	0.236015	+	0.971749i	$0.424159\pi$
$62$	−10.1248	−1.28586
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.544557	−0.0675440
$66$	0	0
$67$	8.13749	0.994153	0.497077	−	0.867707i	$-0.334407\pi$
$67$	8.13749	0.994153	0.497077	+	0.867707i	$0.334407\pi$
$68$	2.13949	0.259452
$69$	0	0
$70$	0.398678	0.0476511
$71$	−0.798603	−0.0947768	−0.0473884	−	0.998877i	$-0.515090\pi$
$71$	−0.798603	−0.0947768	−0.0473884	+	0.998877i	$0.515090\pi$
$72$	0	0
$73$	8.61849	1.00872	0.504359	−	0.863494i	$-0.331729\pi$
$73$	8.61849	1.00872	0.504359	+	0.863494i	$0.331729\pi$
$74$	−0.483652	−0.0562234
$75$	0	0
$76$	−4.61478	−0.529352
$77$	−0.736387	−0.0839191
$78$	0	0
$79$	5.01938	0.564724	0.282362	−	0.959308i	$-0.408882\pi$
$79$	5.01938	0.564724	0.282362	+	0.959308i	$0.408882\pi$
$80$	0.539392	0.0603059
$81$	0	0
$82$	−9.71579	−1.07293
$83$	−6.37875	−0.700158	−0.350079	−	0.936720i	$-0.613845\pi$
$83$	−6.37875	−0.700158	−0.350079	+	0.936720i	$0.613845\pi$
$84$	0	0
$85$	1.15403	0.125172
$86$	−5.50325	−0.593431
$87$	0	0
$88$	−0.996296	−0.106206
$89$	1.68698	0.178819	0.0894095	−	0.995995i	$-0.471502\pi$
$89$	1.68698	0.178819	0.0894095	+	0.995995i	$0.471502\pi$
$90$	0	0
$91$	−0.746202	−0.0782232
$92$	−2.91642	−0.304058
$93$	0	0
$94$	−11.6002	−1.19647
$95$	−2.48918	−0.255384
$96$	0	0
$97$	10.1785	1.03347	0.516735	−	0.856145i	$-0.327147\pi$
$97$	10.1785	1.03347	0.516735	+	0.856145i	$0.327147\pi$
$98$	−6.45370	−0.651922
$99$	0	0
$100$	−4.70906	−0.470906
$101$	9.06625	0.902125	0.451063	−	0.892492i	$-0.351045\pi$
$101$	9.06625	0.902125	0.451063	+	0.892492i	$0.351045\pi$
$102$	0	0
$103$	9.11239	0.897870	0.448935	−	0.893564i	$-0.351803\pi$
$103$	9.11239	0.897870	0.448935	+	0.893564i	$0.351803\pi$
$104$	−1.00958	−0.0989970
$105$	0	0
$106$	−6.43001	−0.624537
$107$	−3.19978	−0.309335	−0.154667	−	0.987967i	$-0.549431\pi$
$107$	−3.19978	−0.309335	−0.154667	+	0.987967i	$0.549431\pi$
$108$	0	0
$109$	13.4465	1.28794	0.643971	−	0.765050i	$-0.277286\pi$
$109$	13.4465	1.28794	0.643971	+	0.765050i	$0.277286\pi$
$110$	−0.537395	−0.0512386
$111$	0	0
$112$	0.739124	0.0698407
$113$	−0.0270826	−0.00254772	−0.00127386	−	0.999999i	$-0.500405\pi$
$113$	−0.0270826	−0.00254772	−0.00127386	+	0.999999i	$0.500405\pi$
$114$	0	0
$115$	−1.57310	−0.146692
$116$	8.59260	0.797803
$117$	0	0
$118$	−1.94920	−0.179438
$119$	1.58135	0.144962
$120$	0	0
$121$	−10.0074	−0.909763
$122$	3.68667	0.333776
$123$	0	0
$124$	−10.1248	−0.909237
$125$	−5.23699	−0.468411
$126$	0	0
$127$	−15.3499	−1.36209	−0.681043	−	0.732243i	$-0.738473\pi$
$127$	−15.3499	−1.36209	−0.681043	+	0.732243i	$0.738473\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−0.544557	−0.0477609
$131$	−5.38655	−0.470625	−0.235313	−	0.971920i	$-0.575611\pi$
$131$	−5.38655	−0.470625	−0.235313	+	0.971920i	$0.575611\pi$
$132$	0	0
$133$	−3.41090	−0.295762
$134$	8.13749	0.702972
$135$	0	0
$136$	2.13949	0.183460
$137$	−13.2411	−1.13127	−0.565633	−	0.824657i	$-0.691368\pi$
$137$	−13.2411	−1.13127	−0.565633	+	0.824657i	$0.691368\pi$
$138$	0	0
$139$	7.61593	0.645975	0.322987	−	0.946403i	$-0.395313\pi$
$139$	7.61593	0.645975	0.322987	+	0.946403i	$0.395313\pi$
$140$	0.398678	0.0336944
$141$	0	0
$142$	−0.798603	−0.0670173
$143$	1.00584	0.0841123
$144$	0	0
$145$	4.63479	0.384898
$146$	8.61849	0.713271
$147$	0	0
$148$	−0.483652	−0.0397559
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−3.81684	−0.310610	−0.155305	−	0.987867i	$-0.549636\pi$
$151$	−3.81684	−0.310610	−0.155305	+	0.987867i	$0.549636\pi$
$152$	−4.61478	−0.374308
$153$	0	0
$154$	−0.736387	−0.0593397
$155$	−5.46126	−0.438659
$156$	0	0
$157$	−10.7714	−0.859650	−0.429825	−	0.902912i	$-0.641425\pi$
$157$	−10.7714	−0.859650	−0.429825	+	0.902912i	$0.641425\pi$
$158$	5.01938	0.399320
$159$	0	0
$160$	0.539392	0.0426427
$161$	−2.15560	−0.169885
$162$	0	0
$163$	−6.86588	−0.537777	−0.268889	−	0.963171i	$-0.586656\pi$
$163$	−6.86588	−0.537777	−0.268889	+	0.963171i	$0.586656\pi$
$164$	−9.71579	−0.758676
$165$	0	0
$166$	−6.37875	−0.495087
$167$	−3.89722	−0.301576	−0.150788	−	0.988566i	$-0.548181\pi$
$167$	−3.89722	−0.301576	−0.150788	+	0.988566i	$0.548181\pi$
$168$	0	0
$169$	−11.9808	−0.921597
$170$	1.15403	0.0885098
$171$	0	0
$172$	−5.50325	−0.419619
$173$	3.85757	0.293286	0.146643	−	0.989189i	$-0.453153\pi$
$173$	3.85757	0.293286	0.146643	+	0.989189i	$0.453153\pi$
$174$	0	0
$175$	−3.48058	−0.263107
$176$	−0.996296	−0.0750987
$177$	0	0
$178$	1.68698	0.126444
$179$	−13.0076	−0.972231	−0.486115	−	0.873895i	$-0.661587\pi$
$179$	−13.0076	−0.972231	−0.486115	+	0.873895i	$0.661587\pi$
$180$	0	0
$181$	−4.88562	−0.363146	−0.181573	−	0.983378i	$-0.558119\pi$
$181$	−4.88562	−0.363146	−0.181573	+	0.983378i	$0.558119\pi$
$182$	−0.746202	−0.0553122
$183$	0	0
$184$	−2.91642	−0.215002
$185$	−0.260878	−0.0191801
$186$	0	0
$187$	−2.13157	−0.155876
$188$	−11.6002	−0.846031
$189$	0	0
$190$	−2.48918	−0.180584
$191$	−24.9561	−1.80576	−0.902882	−	0.429889i	$-0.858553\pi$
$191$	−24.9561	−1.80576	−0.902882	+	0.429889i	$0.858553\pi$
$192$	0	0
$193$	16.2571	1.17021	0.585107	−	0.810956i	$-0.301053\pi$
$193$	16.2571	1.17021	0.585107	+	0.810956i	$0.301053\pi$
$194$	10.1785	0.730774
$195$	0	0
$196$	−6.45370	−0.460978
$197$	5.84834	0.416677	0.208339	−	0.978057i	$-0.433194\pi$
$197$	5.84834	0.416677	0.208339	+	0.978057i	$0.433194\pi$
$198$	0	0
$199$	0.839240	0.0594921	0.0297461	−	0.999557i	$-0.490530\pi$
$199$	0.839240	0.0594921	0.0297461	+	0.999557i	$0.490530\pi$
$200$	−4.70906	−0.332981
$201$	0	0
$202$	9.06625	0.637899
$203$	6.35100	0.445753
$204$	0	0
$205$	−5.24063	−0.366021
$206$	9.11239	0.634890
$207$	0	0
$208$	−1.00958	−0.0700015
$209$	4.59769	0.318029
$210$	0	0
$211$	−7.93985	−0.546602	−0.273301	−	0.961929i	$-0.588116\pi$
$211$	−7.93985	−0.546602	−0.273301	+	0.961929i	$0.588116\pi$
$212$	−6.43001	−0.441615
$213$	0	0
$214$	−3.19978	−0.218733
$215$	−2.96841	−0.202444
$216$	0	0
$217$	−7.48351	−0.508014
$218$	13.4465	0.910712
$219$	0	0
$220$	−0.537395	−0.0362311
$221$	−2.15998	−0.145296
$222$	0	0
$223$	13.3254	0.892332	0.446166	−	0.894950i	$-0.352789\pi$
$223$	13.3254	0.892332	0.446166	+	0.894950i	$0.352789\pi$
$224$	0.739124	0.0493848
$225$	0	0
$226$	−0.0270826	−0.00180151
$227$	−3.70206	−0.245715	−0.122857	−	0.992424i	$-0.539206\pi$
$227$	−3.70206	−0.245715	−0.122857	+	0.992424i	$0.539206\pi$
$228$	0	0
$229$	13.5088	0.892685	0.446343	−	0.894862i	$-0.352726\pi$
$229$	13.5088	0.892685	0.446343	+	0.894862i	$0.352726\pi$
$230$	−1.57310	−0.103727
$231$	0	0
$232$	8.59260	0.564132
$233$	−11.7237	−0.768048	−0.384024	−	0.923323i	$-0.625462\pi$
$233$	−11.7237	−0.768048	−0.384024	+	0.923323i	$0.625462\pi$
$234$	0	0
$235$	−6.25706	−0.408166
$236$	−1.94920	−0.126882
$237$	0	0
$238$	1.58135	0.102504
$239$	−9.01299	−0.583002	−0.291501	−	0.956570i	$-0.594155\pi$
$239$	−9.01299	−0.583002	−0.291501	+	0.956570i	$0.594155\pi$
$240$	0	0
$241$	19.2343	1.23899	0.619496	−	0.785000i	$-0.287337\pi$
$241$	19.2343	1.23899	0.619496	+	0.785000i	$0.287337\pi$
$242$	−10.0074	−0.643300
$243$	0	0
$244$	3.68667	0.236015
$245$	−3.48107	−0.222398
$246$	0	0
$247$	4.65897	0.296443
$248$	−10.1248	−0.642928
$249$	0	0
$250$	−5.23699	−0.331216
$251$	23.7302	1.49784	0.748918	−	0.662663i	$-0.230574\pi$
$251$	23.7302	1.49784	0.748918	+	0.662663i	$0.230574\pi$
$252$	0	0
$253$	2.90562	0.182675
$254$	−15.3499	−0.963141
$255$	0	0
$256$	1.00000	0.0625000
$257$	21.3888	1.33420	0.667100	−	0.744969i	$-0.267536\pi$
$257$	21.3888	1.33420	0.667100	+	0.744969i	$0.267536\pi$
$258$	0	0
$259$	−0.357479	−0.0222127
$260$	−0.544557	−0.0337720
$261$	0	0
$262$	−5.38655	−0.332782
$263$	10.5270	0.649125	0.324563	−	0.945864i	$-0.394783\pi$
$263$	10.5270	0.649125	0.324563	+	0.945864i	$0.394783\pi$
$264$	0	0
$265$	−3.46830	−0.213056
$266$	−3.41090	−0.209135
$267$	0	0
$268$	8.13749	0.497077
$269$	1.79113	0.109207	0.0546035	−	0.998508i	$-0.482611\pi$
$269$	1.79113	0.109207	0.0546035	+	0.998508i	$0.482611\pi$
$270$	0	0
$271$	8.17611	0.496664	0.248332	−	0.968675i	$-0.420118\pi$
$271$	8.17611	0.496664	0.248332	+	0.968675i	$0.420118\pi$
$272$	2.13949	0.129726
$273$	0	0
$274$	−13.2411	−0.799925
$275$	4.69161	0.282915
$276$	0	0
$277$	10.9172	0.655952	0.327976	−	0.944686i	$-0.393633\pi$
$277$	10.9172	0.655952	0.327976	+	0.944686i	$0.393633\pi$
$278$	7.61593	0.456773
$279$	0	0
$280$	0.398678	0.0238256
$281$	−10.6997	−0.638289	−0.319145	−	0.947706i	$-0.603396\pi$
$281$	−10.6997	−0.638289	−0.319145	+	0.947706i	$0.603396\pi$
$282$	0	0
$283$	33.0320	1.96355	0.981776	−	0.190043i	$-0.0608629\pi$
$283$	33.0320	1.96355	0.981776	+	0.190043i	$0.0608629\pi$
$284$	−0.798603	−0.0473884
$285$	0	0
$286$	1.00584	0.0594764
$287$	−7.18118	−0.423892
$288$	0	0
$289$	−12.4226	−0.730739
$290$	4.63479	0.272164
$291$	0	0
$292$	8.61849	0.504359
$293$	−13.6940	−0.800013	−0.400006	−	0.916512i	$-0.630992\pi$
$293$	−13.6940	−0.800013	−0.400006	+	0.916512i	$0.630992\pi$
$294$	0	0
$295$	−1.05138	−0.0612140
$296$	−0.483652	−0.0281117
$297$	0	0
$298$	1.00000	0.0579284
$299$	2.94435	0.170276
$300$	0	0
$301$	−4.06759	−0.234452
$302$	−3.81684	−0.219634
$303$	0	0
$304$	−4.61478	−0.264676
$305$	1.98856	0.113865
$306$	0	0
$307$	−25.7587	−1.47013	−0.735063	−	0.677999i	$-0.762847\pi$
$307$	−25.7587	−1.47013	−0.735063	+	0.677999i	$0.762847\pi$
$308$	−0.736387	−0.0419595
$309$	0	0
$310$	−5.46126	−0.310179
$311$	−7.74808	−0.439353	−0.219677	−	0.975573i	$-0.570500\pi$
$311$	−7.74808	−0.439353	−0.219677	+	0.975573i	$0.570500\pi$
$312$	0	0
$313$	−6.77274	−0.382818	−0.191409	−	0.981510i	$-0.561306\pi$
$313$	−6.77274	−0.382818	−0.191409	+	0.981510i	$0.561306\pi$
$314$	−10.7714	−0.607865
$315$	0	0
$316$	5.01938	0.282362
$317$	25.4402	1.42886	0.714432	−	0.699705i	$-0.246685\pi$
$317$	25.4402	1.42886	0.714432	+	0.699705i	$0.246685\pi$
$318$	0	0
$319$	−8.56078	−0.479312
$320$	0.539392	0.0301530
$321$	0	0
$322$	−2.15560	−0.120127
$323$	−9.87329	−0.549365
$324$	0	0
$325$	4.75415	0.263713
$326$	−6.86588	−0.380266
$327$	0	0
$328$	−9.71579	−0.536465
$329$	−8.57399	−0.472699
$330$	0	0
$331$	20.7971	1.14311	0.571555	−	0.820564i	$-0.306340\pi$
$331$	20.7971	1.14311	0.571555	+	0.820564i	$0.306340\pi$
$332$	−6.37875	−0.350079
$333$	0	0
$334$	−3.89722	−0.213246
$335$	4.38930	0.239813
$336$	0	0
$337$	−17.6532	−0.961628	−0.480814	−	0.876823i	$-0.659659\pi$
$337$	−17.6532	−0.961628	−0.480814	+	0.876823i	$0.659659\pi$
$338$	−11.9808	−0.651667
$339$	0	0
$340$	1.15403	0.0625859
$341$	10.0873	0.546260
$342$	0	0
$343$	−9.94395	−0.536923
$344$	−5.50325	−0.296715
$345$	0	0
$346$	3.85757	0.207385
$347$	−24.6442	−1.32297	−0.661484	−	0.749959i	$-0.730073\pi$
$347$	−24.6442	−1.32297	−0.661484	+	0.749959i	$0.730073\pi$
$348$	0	0
$349$	37.1661	1.98945	0.994727	−	0.102561i	$-0.0327038\pi$
$349$	37.1661	1.98945	0.994727	+	0.102561i	$0.0327038\pi$
$350$	−3.48058	−0.186045
$351$	0	0
$352$	−0.996296	−0.0531028
$353$	−20.2331	−1.07690	−0.538451	−	0.842657i	$-0.680990\pi$
$353$	−20.2331	−1.07690	−0.538451	+	0.842657i	$0.680990\pi$
$354$	0	0
$355$	−0.430761	−0.0228624
$356$	1.68698	0.0894095
$357$	0	0
$358$	−13.0076	−0.687471
$359$	24.9326	1.31589	0.657946	−	0.753065i	$-0.271426\pi$
$359$	24.9326	1.31589	0.657946	+	0.753065i	$0.271426\pi$
$360$	0	0
$361$	2.29619	0.120852
$362$	−4.88562	−0.256783
$363$	0	0
$364$	−0.746202	−0.0391116
$365$	4.64875	0.243327
$366$	0	0
$367$	−13.2668	−0.692522	−0.346261	−	0.938138i	$-0.612549\pi$
$367$	−13.2668	−0.692522	−0.346261	+	0.938138i	$0.612549\pi$
$368$	−2.91642	−0.152029
$369$	0	0
$370$	−0.260878	−0.0135624
$371$	−4.75257	−0.246741
$372$	0	0
$373$	5.55470	0.287611	0.143806	−	0.989606i	$-0.454066\pi$
$373$	5.55470	0.287611	0.143806	+	0.989606i	$0.454066\pi$
$374$	−2.13157	−0.110221
$375$	0	0
$376$	−11.6002	−0.598234
$377$	−8.67488	−0.446779
$378$	0	0
$379$	−18.9561	−0.973709	−0.486854	−	0.873483i	$-0.661856\pi$
$379$	−18.9561	−0.973709	−0.486854	+	0.873483i	$0.661856\pi$
$380$	−2.48918	−0.127692
$381$	0	0
$382$	−24.9561	−1.27687
$383$	−25.8100	−1.31883	−0.659415	−	0.751779i	$-0.729196\pi$
$383$	−25.8100	−1.31883	−0.659415	+	0.751779i	$0.729196\pi$
$384$	0	0
$385$	−0.397201	−0.0202433
$386$	16.2571	0.827466
$387$	0	0
$388$	10.1785	0.516735
$389$	−30.8291	−1.56310	−0.781549	−	0.623844i	$-0.785570\pi$
$389$	−30.8291	−1.56310	−0.781549	+	0.623844i	$0.785570\pi$
$390$	0	0
$391$	−6.23967	−0.315554
$392$	−6.45370	−0.325961
$393$	0	0
$394$	5.84834	0.294635
$395$	2.70741	0.136225
$396$	0	0
$397$	−31.3598	−1.57390	−0.786952	−	0.617014i	$-0.788342\pi$
$397$	−31.3598	−1.57390	−0.786952	+	0.617014i	$0.788342\pi$
$398$	0.839240	0.0420673
$399$	0	0
$400$	−4.70906	−0.235453
$401$	−12.9298	−0.645685	−0.322842	−	0.946453i	$-0.604638\pi$
$401$	−12.9298	−0.645685	−0.322842	+	0.946453i	$0.604638\pi$
$402$	0	0
$403$	10.2218	0.509184
$404$	9.06625	0.451063
$405$	0	0
$406$	6.35100	0.315195
$407$	0.481861	0.0238849
$408$	0	0
$409$	−7.08612	−0.350386	−0.175193	−	0.984534i	$-0.556055\pi$
$409$	−7.08612	−0.350386	−0.175193	+	0.984534i	$0.556055\pi$
$410$	−5.24063	−0.258816
$411$	0	0
$412$	9.11239	0.448935
$413$	−1.44070	−0.0708923
$414$	0	0
$415$	−3.44065	−0.168895
$416$	−1.00958	−0.0494985
$417$	0	0
$418$	4.59769	0.224880
$419$	−10.3277	−0.504542	−0.252271	−	0.967657i	$-0.581177\pi$
$419$	−10.3277	−0.504542	−0.252271	+	0.967657i	$0.581177\pi$
$420$	0	0
$421$	−17.8422	−0.869575	−0.434788	−	0.900533i	$-0.643177\pi$
$421$	−17.8422	−0.869575	−0.434788	+	0.900533i	$0.643177\pi$
$422$	−7.93985	−0.386506
$423$	0	0
$424$	−6.43001	−0.312269
$425$	−10.0750	−0.488709
$426$	0	0
$427$	2.72491	0.131868
$428$	−3.19978	−0.154667
$429$	0	0
$430$	−2.96841	−0.143150
$431$	−18.1849	−0.875934	−0.437967	−	0.898991i	$-0.644301\pi$
$431$	−18.1849	−0.875934	−0.437967	+	0.898991i	$0.644301\pi$
$432$	0	0
$433$	14.4191	0.692938	0.346469	−	0.938062i	$-0.387381\pi$
$433$	14.4191	0.692938	0.346469	+	0.938062i	$0.387381\pi$
$434$	−7.48351	−0.359220
$435$	0	0
$436$	13.4465	0.643971
$437$	13.4586	0.643815
$438$	0	0
$439$	34.1052	1.62775	0.813876	−	0.581038i	$-0.197353\pi$
$439$	34.1052	1.62775	0.813876	+	0.581038i	$0.197353\pi$
$440$	−0.537395	−0.0256193
$441$	0	0
$442$	−2.15998	−0.102740
$443$	19.1664	0.910622	0.455311	−	0.890333i	$-0.349528\pi$
$443$	19.1664	0.910622	0.455311	+	0.890333i	$0.349528\pi$
$444$	0	0
$445$	0.909942	0.0431354
$446$	13.3254	0.630974
$447$	0	0
$448$	0.739124	0.0349203
$449$	11.0461	0.521297	0.260648	−	0.965434i	$-0.416064\pi$
$449$	11.0461	0.521297	0.260648	+	0.965434i	$0.416064\pi$
$450$	0	0
$451$	9.67981	0.455804
$452$	−0.0270826	−0.00127386
$453$	0	0
$454$	−3.70206	−0.173746
$455$	−0.402496	−0.0188693
$456$	0	0
$457$	10.9710	0.513202	0.256601	−	0.966517i	$-0.417397\pi$
$457$	10.9710	0.513202	0.256601	+	0.966517i	$0.417397\pi$
$458$	13.5088	0.631224
$459$	0	0
$460$	−1.57310	−0.0733460
$461$	17.3967	0.810247	0.405123	−	0.914262i	$-0.367229\pi$
$461$	17.3967	0.810247	0.405123	+	0.914262i	$0.367229\pi$
$462$	0	0
$463$	3.53858	0.164452	0.0822258	−	0.996614i	$-0.473797\pi$
$463$	3.53858	0.164452	0.0822258	+	0.996614i	$0.473797\pi$
$464$	8.59260	0.398902
$465$	0	0
$466$	−11.7237	−0.543092
$467$	27.3710	1.26658	0.633290	−	0.773914i	$-0.281704\pi$
$467$	27.3710	1.26658	0.633290	+	0.773914i	$0.281704\pi$
$468$	0	0
$469$	6.01462	0.277729
$470$	−6.25706	−0.288617
$471$	0	0
$472$	−1.94920	−0.0897192
$473$	5.48287	0.252103
$474$	0	0
$475$	21.7313	0.997098
$476$	1.58135	0.0724811
$477$	0	0
$478$	−9.01299	−0.412245
$479$	5.10713	0.233351	0.116675	−	0.993170i	$-0.462776\pi$
$479$	5.10713	0.233351	0.116675	+	0.993170i	$0.462776\pi$
$480$	0	0
$481$	0.488283	0.0222638
$482$	19.2343	0.876100
$483$	0	0
$484$	−10.0074	−0.454882
$485$	5.49021	0.249297
$486$	0	0
$487$	6.60830	0.299450	0.149725	−	0.988728i	$-0.452161\pi$
$487$	6.60830	0.299450	0.149725	+	0.988728i	$0.452161\pi$
$488$	3.68667	0.166888
$489$	0	0
$490$	−3.48107	−0.157259
$491$	−17.4201	−0.786157	−0.393079	−	0.919505i	$-0.628590\pi$
$491$	−17.4201	−0.786157	−0.393079	+	0.919505i	$0.628590\pi$
$492$	0	0
$493$	18.3838	0.827966
$494$	4.65897	0.209617
$495$	0	0
$496$	−10.1248	−0.454619
$497$	−0.590267	−0.0264771
$498$	0	0
$499$	10.2470	0.458717	0.229358	−	0.973342i	$-0.426337\pi$
$499$	10.2470	0.458717	0.229358	+	0.973342i	$0.426337\pi$
$500$	−5.23699	−0.234205
$501$	0	0
$502$	23.7302	1.05913
$503$	13.5410	0.603763	0.301882	−	0.953345i	$-0.402385\pi$
$503$	13.5410	0.603763	0.301882	+	0.953345i	$0.402385\pi$
$504$	0	0
$505$	4.89027	0.217614
$506$	2.90562	0.129171
$507$	0	0
$508$	−15.3499	−0.681043
$509$	−40.0033	−1.77311	−0.886557	−	0.462619i	$-0.846910\pi$
$509$	−40.0033	−1.77311	−0.886557	+	0.462619i	$0.846910\pi$
$510$	0	0
$511$	6.37014	0.281798
$512$	1.00000	0.0441942
$513$	0	0
$514$	21.3888	0.943421
$515$	4.91515	0.216588
$516$	0	0
$517$	11.5572	0.508287
$518$	−0.357479	−0.0157067
$519$	0	0
$520$	−0.544557	−0.0238804
$521$	23.6758	1.03726	0.518628	−	0.855000i	$-0.326443\pi$
$521$	23.6758	1.03726	0.518628	+	0.855000i	$0.326443\pi$
$522$	0	0
$523$	−24.3609	−1.06523	−0.532613	−	0.846359i	$-0.678790\pi$
$523$	−24.3609	−1.06523	−0.532613	+	0.846359i	$0.678790\pi$
$524$	−5.38655	−0.235313
$525$	0	0
$526$	10.5270	0.459001
$527$	−21.6620	−0.943613
$528$	0	0
$529$	−14.4945	−0.630195
$530$	−3.46830	−0.150653
$531$	0	0
$532$	−3.41090	−0.147881
$533$	9.80883	0.424867
$534$	0	0
$535$	−1.72594	−0.0746189
$536$	8.13749	0.351486
$537$	0	0
$538$	1.79113	0.0772210
$539$	6.42979	0.276951
$540$	0	0
$541$	4.98322	0.214245	0.107123	−	0.994246i	$-0.465836\pi$
$541$	4.98322	0.214245	0.107123	+	0.994246i	$0.465836\pi$
$542$	8.17611	0.351194
$543$	0	0
$544$	2.13949	0.0917300
$545$	7.25295	0.310682
$546$	0	0
$547$	−24.5618	−1.05019	−0.525093	−	0.851045i	$-0.675969\pi$
$547$	−24.5618	−1.05019	−0.525093	+	0.851045i	$0.675969\pi$
$548$	−13.2411	−0.565633
$549$	0	0
$550$	4.69161	0.200051
$551$	−39.6530	−1.68927
$552$	0	0
$553$	3.70994	0.157763
$554$	10.9172	0.463828
$555$	0	0
$556$	7.61593	0.322987
$557$	−28.3600	−1.20165	−0.600825	−	0.799380i	$-0.705161\pi$
$557$	−28.3600	−1.20165	−0.600825	+	0.799380i	$0.705161\pi$
$558$	0	0
$559$	5.55595	0.234992
$560$	0.398678	0.0168472
$561$	0	0
$562$	−10.6997	−0.451339
$563$	−9.62910	−0.405818	−0.202909	−	0.979198i	$-0.565040\pi$
$563$	−9.62910	−0.405818	−0.202909	+	0.979198i	$0.565040\pi$
$564$	0	0
$565$	−0.0146081	−0.000614569 0
$566$	33.0320	1.38844
$567$	0	0
$568$	−0.798603	−0.0335087
$569$	−32.7841	−1.37438	−0.687190	−	0.726478i	$-0.741156\pi$
$569$	−32.7841	−1.37438	−0.687190	+	0.726478i	$0.741156\pi$
$570$	0	0
$571$	23.0993	0.966677	0.483338	−	0.875434i	$-0.339424\pi$
$571$	23.0993	0.966677	0.483338	+	0.875434i	$0.339424\pi$
$572$	1.00584	0.0420561
$573$	0	0
$574$	−7.18118	−0.299737
$575$	13.7336	0.572731
$576$	0	0
$577$	−22.6421	−0.942603	−0.471302	−	0.881972i	$-0.656216\pi$
$577$	−22.6421	−0.942603	−0.471302	+	0.881972i	$0.656216\pi$
$578$	−12.4226	−0.516711
$579$	0	0
$580$	4.63479	0.192449
$581$	−4.71469	−0.195598
$582$	0	0
$583$	6.40619	0.265317
$584$	8.61849	0.356636
$585$	0	0
$586$	−13.6940	−0.565695
$587$	−13.9302	−0.574960	−0.287480	−	0.957787i	$-0.592817\pi$
$587$	−13.9302	−0.574960	−0.287480	+	0.957787i	$0.592817\pi$
$588$	0	0
$589$	46.7239	1.92522
$590$	−1.05138	−0.0432848
$591$	0	0
$592$	−0.483652	−0.0198780
$593$	−20.7554	−0.852321	−0.426160	−	0.904648i	$-0.640134\pi$
$593$	−20.7554	−0.852321	−0.426160	+	0.904648i	$0.640134\pi$
$594$	0	0
$595$	0.852969	0.0349683
$596$	1.00000	0.0409616
$597$	0	0
$598$	2.94435	0.120403
$599$	−28.3064	−1.15657	−0.578283	−	0.815836i	$-0.696277\pi$
$599$	−28.3064	−1.15657	−0.578283	+	0.815836i	$0.696277\pi$
$600$	0	0
$601$	47.3778	1.93258	0.966291	−	0.257451i	$-0.0828827\pi$
$601$	47.3778	1.93258	0.966291	+	0.257451i	$0.0828827\pi$
$602$	−4.06759	−0.165782
$603$	0	0
$604$	−3.81684	−0.155305
$605$	−5.39791	−0.219456
$606$	0	0
$607$	11.9324	0.484321	0.242161	−	0.970236i	$-0.422144\pi$
$607$	11.9324	0.484321	0.242161	+	0.970236i	$0.422144\pi$
$608$	−4.61478	−0.187154
$609$	0	0
$610$	1.98856	0.0805146
$611$	11.7113	0.473787
$612$	0	0
$613$	−23.3292	−0.942259	−0.471129	−	0.882064i	$-0.656154\pi$
$613$	−23.3292	−0.942259	−0.471129	+	0.882064i	$0.656154\pi$
$614$	−25.7587	−1.03954
$615$	0	0
$616$	−0.736387	−0.0296699
$617$	−31.4323	−1.26542	−0.632709	−	0.774390i	$-0.718057\pi$
$617$	−31.4323	−1.26542	−0.632709	+	0.774390i	$0.718057\pi$
$618$	0	0
$619$	−16.2850	−0.654551	−0.327275	−	0.944929i	$-0.606130\pi$
$619$	−16.2850	−0.654551	−0.327275	+	0.944929i	$0.606130\pi$
$620$	−5.46126	−0.219330
$621$	0	0
$622$	−7.74808	−0.310670
$623$	1.24688	0.0499554
$624$	0	0
$625$	20.7205	0.828819
$626$	−6.77274	−0.270693
$627$	0	0
$628$	−10.7714	−0.429825
$629$	−1.03477	−0.0412590
$630$	0	0
$631$	47.5657	1.89356	0.946780	−	0.321881i	$-0.104315\pi$
$631$	47.5657	1.89356	0.946780	+	0.321881i	$0.104315\pi$
$632$	5.01938	0.199660
$633$	0	0
$634$	25.4402	1.01036
$635$	−8.27964	−0.328567
$636$	0	0
$637$	6.51549	0.258153
$638$	−8.56078	−0.338924
$639$	0	0
$640$	0.539392	0.0213214
$641$	−17.1266	−0.676460	−0.338230	−	0.941064i	$-0.609828\pi$
$641$	−17.1266	−0.676460	−0.338230	+	0.941064i	$0.609828\pi$
$642$	0	0
$643$	3.93044	0.155001	0.0775006	−	0.996992i	$-0.475306\pi$
$643$	3.93044	0.155001	0.0775006	+	0.996992i	$0.475306\pi$
$644$	−2.15560	−0.0849425
$645$	0	0
$646$	−9.87329	−0.388459
$647$	18.9388	0.744561	0.372280	−	0.928120i	$-0.378576\pi$
$647$	18.9388	0.744561	0.372280	+	0.928120i	$0.378576\pi$
$648$	0	0
$649$	1.94198	0.0762294
$650$	4.75415	0.186473
$651$	0	0
$652$	−6.86588	−0.268889
$653$	37.2808	1.45891	0.729455	−	0.684029i	$-0.239774\pi$
$653$	37.2808	1.45891	0.729455	+	0.684029i	$0.239774\pi$
$654$	0	0
$655$	−2.90547	−0.113526
$656$	−9.71579	−0.379338
$657$	0	0
$658$	−8.57399	−0.334249
$659$	32.4816	1.26530	0.632651	−	0.774437i	$-0.281967\pi$
$659$	32.4816	1.26530	0.632651	+	0.774437i	$0.281967\pi$
$660$	0	0
$661$	−37.3784	−1.45385	−0.726925	−	0.686717i	$-0.759051\pi$
$661$	−37.3784	−1.45385	−0.726925	+	0.686717i	$0.759051\pi$
$662$	20.7971	0.808301
$663$	0	0
$664$	−6.37875	−0.247543
$665$	−1.83981	−0.0713448
$666$	0	0
$667$	−25.0597	−0.970314
$668$	−3.89722	−0.150788
$669$	0	0
$670$	4.38930	0.169574
$671$	−3.67302	−0.141795
$672$	0	0
$673$	1.90378	0.0733852	0.0366926	−	0.999327i	$-0.488318\pi$
$673$	1.90378	0.0733852	0.0366926	+	0.999327i	$0.488318\pi$
$674$	−17.6532	−0.679974
$675$	0	0
$676$	−11.9808	−0.460798
$677$	13.8313	0.531580	0.265790	−	0.964031i	$-0.414367\pi$
$677$	13.8313	0.531580	0.265790	+	0.964031i	$0.414367\pi$
$678$	0	0
$679$	7.52318	0.288713
$680$	1.15403	0.0442549
$681$	0	0
$682$	10.0873	0.386264
$683$	22.9283	0.877328	0.438664	−	0.898651i	$-0.355452\pi$
$683$	22.9283	0.877328	0.438664	+	0.898651i	$0.355452\pi$
$684$	0	0
$685$	−7.14216	−0.272888
$686$	−9.94395	−0.379662
$687$	0	0
$688$	−5.50325	−0.209810
$689$	6.49158	0.247309
$690$	0	0
$691$	28.3516	1.07855	0.539273	−	0.842131i	$-0.318699\pi$
$691$	28.3516	1.07855	0.539273	+	0.842131i	$0.318699\pi$
$692$	3.85757	0.146643
$693$	0	0
$694$	−24.6442	−0.935479
$695$	4.10797	0.155824
$696$	0	0
$697$	−20.7869	−0.787359
$698$	37.1661	1.40676
$699$	0	0
$700$	−3.48058	−0.131553
$701$	−25.9937	−0.981767	−0.490884	−	0.871225i	$-0.663326\pi$
$701$	−25.9937	−0.981767	−0.490884	+	0.871225i	$0.663326\pi$
$702$	0	0
$703$	2.23195	0.0841795
$704$	−0.996296	−0.0375493
$705$	0	0
$706$	−20.2331	−0.761485
$707$	6.70108	0.252020
$708$	0	0
$709$	−40.8007	−1.53230	−0.766152	−	0.642659i	$-0.777831\pi$
$709$	−40.8007	−1.53230	−0.766152	+	0.642659i	$0.777831\pi$
$710$	−0.430761	−0.0161662
$711$	0	0
$712$	1.68698	0.0632221
$713$	29.5283	1.10584
$714$	0	0
$715$	0.542541	0.0202899
$716$	−13.0076	−0.486115
$717$	0	0
$718$	24.9326	0.930476
$719$	30.4190	1.13444	0.567218	−	0.823568i	$-0.308020\pi$
$719$	30.4190	1.13444	0.567218	+	0.823568i	$0.308020\pi$
$720$	0	0
$721$	6.73519	0.250832
$722$	2.29619	0.0854555
$723$	0	0
$724$	−4.88562	−0.181573
$725$	−40.4630	−1.50276
$726$	0	0
$727$	−6.53801	−0.242481	−0.121241	−	0.992623i	$-0.538687\pi$
$727$	−6.53801	−0.242481	−0.121241	+	0.992623i	$0.538687\pi$
$728$	−0.746202	−0.0276561
$729$	0	0
$730$	4.64875	0.172058
$731$	−11.7742	−0.435484
$732$	0	0
$733$	7.89240	0.291512	0.145756	−	0.989321i	$-0.453438\pi$
$733$	7.89240	0.291512	0.145756	+	0.989321i	$0.453438\pi$
$734$	−13.2668	−0.489687
$735$	0	0
$736$	−2.91642	−0.107501
$737$	−8.10736	−0.298638
$738$	0	0
$739$	13.5289	0.497668	0.248834	−	0.968546i	$-0.419953\pi$
$739$	13.5289	0.497668	0.248834	+	0.968546i	$0.419953\pi$
$740$	−0.260878	−0.00959007
$741$	0	0
$742$	−4.75257	−0.174472
$743$	−16.8256	−0.617272	−0.308636	−	0.951180i	$-0.599873\pi$
$743$	−16.8256	−0.617272	−0.308636	+	0.951180i	$0.599873\pi$
$744$	0	0
$745$	0.539392	0.0197618
$746$	5.55470	0.203372
$747$	0	0
$748$	−2.13157	−0.0779379
$749$	−2.36504	−0.0864166
$750$	0	0
$751$	−29.7052	−1.08396	−0.541978	−	0.840392i	$-0.682325\pi$
$751$	−29.7052	−1.08396	−0.541978	+	0.840392i	$0.682325\pi$
$752$	−11.6002	−0.423016
$753$	0	0
$754$	−8.67488	−0.315921
$755$	−2.05877	−0.0749264
$756$	0	0
$757$	12.0245	0.437039	0.218519	−	0.975833i	$-0.429877\pi$
$757$	12.0245	0.437039	0.218519	+	0.975833i	$0.429877\pi$
$758$	−18.9561	−0.688516
$759$	0	0
$760$	−2.48918	−0.0902920
$761$	−6.26920	−0.227258	−0.113629	−	0.993523i	$-0.536248\pi$
$761$	−6.26920	−0.227258	−0.113629	+	0.993523i	$0.536248\pi$
$762$	0	0
$763$	9.93864	0.359803
$764$	−24.9561	−0.902882
$765$	0	0
$766$	−25.8100	−0.932554
$767$	1.96787	0.0710555
$768$	0	0
$769$	35.8248	1.29187	0.645937	−	0.763391i	$-0.276467\pi$
$769$	35.8248	1.29187	0.645937	+	0.763391i	$0.276467\pi$
$770$	−0.397201	−0.0143141
$771$	0	0
$772$	16.2571	0.585107
$773$	−20.4307	−0.734843	−0.367421	−	0.930055i	$-0.619759\pi$
$773$	−20.4307	−0.734843	−0.367421	+	0.930055i	$0.619759\pi$
$774$	0	0
$775$	47.6784	1.71266
$776$	10.1785	0.365387
$777$	0	0
$778$	−30.8291	−1.10528
$779$	44.8362	1.60643
$780$	0	0
$781$	0.795646	0.0284704
$782$	−6.23967	−0.223130
$783$	0	0
$784$	−6.45370	−0.230489
$785$	−5.81001	−0.207368
$786$	0	0
$787$	−19.4628	−0.693775	−0.346887	−	0.937907i	$-0.612761\pi$
$787$	−19.4628	−0.693775	−0.346887	+	0.937907i	$0.612761\pi$
$788$	5.84834	0.208339
$789$	0	0
$790$	2.70741	0.0963255
$791$	−0.0200174	−0.000711737 0
$792$	0	0
$793$	−3.72197	−0.132171
$794$	−31.3598	−1.11292
$795$	0	0
$796$	0.839240	0.0297461
$797$	−33.3154	−1.18009	−0.590046	−	0.807370i	$-0.700890\pi$
$797$	−33.3154	−1.18009	−0.590046	+	0.807370i	$0.700890\pi$
$798$	0	0
$799$	−24.8185	−0.878017
$800$	−4.70906	−0.166490
$801$	0	0
$802$	−12.9298	−0.456568
$803$	−8.58657	−0.303013
$804$	0	0
$805$	−1.16271	−0.0409803
$806$	10.2218	0.360047
$807$	0	0
$808$	9.06625	0.318949
$809$	10.1111	0.355486	0.177743	−	0.984077i	$-0.443120\pi$
$809$	10.1111	0.355486	0.177743	+	0.984077i	$0.443120\pi$
$810$	0	0
$811$	−25.9473	−0.911134	−0.455567	−	0.890202i	$-0.650563\pi$
$811$	−25.9473	−0.911134	−0.455567	+	0.890202i	$0.650563\pi$
$812$	6.35100	0.222876
$813$	0	0
$814$	0.481861	0.0168892
$815$	−3.70340	−0.129725
$816$	0	0
$817$	25.3963	0.888504
$818$	−7.08612	−0.247760
$819$	0	0
$820$	−5.24063	−0.183011
$821$	50.8048	1.77310	0.886550	−	0.462632i	$-0.153095\pi$
$821$	50.8048	1.77310	0.886550	+	0.462632i	$0.153095\pi$
$822$	0	0
$823$	−30.3539	−1.05807	−0.529035	−	0.848600i	$-0.677446\pi$
$823$	−30.3539	−1.05807	−0.529035	+	0.848600i	$0.677446\pi$
$824$	9.11239	0.317445
$825$	0	0
$826$	−1.44070	−0.0501284
$827$	21.2942	0.740470	0.370235	−	0.928938i	$-0.379277\pi$
$827$	21.2942	0.740470	0.370235	+	0.928938i	$0.379277\pi$
$828$	0	0
$829$	15.7265	0.546203	0.273102	−	0.961985i	$-0.411950\pi$
$829$	15.7265	0.546203	0.273102	+	0.961985i	$0.411950\pi$
$830$	−3.44065	−0.119427
$831$	0	0
$832$	−1.00958	−0.0350007
$833$	−13.8076	−0.478406
$834$	0	0
$835$	−2.10213	−0.0727472
$836$	4.59769	0.159014
$837$	0	0
$838$	−10.3277	−0.356765
$839$	34.1108	1.17764	0.588818	−	0.808266i	$-0.299594\pi$
$839$	34.1108	1.17764	0.588818	+	0.808266i	$0.299594\pi$
$840$	0	0
$841$	44.8328	1.54596
$842$	−17.8422	−0.614882
$843$	0	0
$844$	−7.93985	−0.273301
$845$	−6.46233	−0.222311
$846$	0	0
$847$	−7.39671	−0.254154
$848$	−6.43001	−0.220807
$849$	0	0
$850$	−10.0750	−0.345569
$851$	1.41053	0.0483525
$852$	0	0
$853$	32.5738	1.11531	0.557653	−	0.830074i	$-0.311702\pi$
$853$	32.5738	1.11531	0.557653	+	0.830074i	$0.311702\pi$
$854$	2.72491	0.0932445
$855$	0	0
$856$	−3.19978	−0.109366
$857$	48.5874	1.65971	0.829857	−	0.557976i	$-0.188422\pi$
$857$	48.5874	1.65971	0.829857	+	0.557976i	$0.188422\pi$
$858$	0	0
$859$	−49.0510	−1.67360	−0.836799	−	0.547510i	$-0.815576\pi$
$859$	−49.0510	−1.67360	−0.836799	+	0.547510i	$0.815576\pi$
$860$	−2.96841	−0.101222
$861$	0	0
$862$	−18.1849	−0.619379
$863$	13.0663	0.444781	0.222391	−	0.974958i	$-0.428614\pi$
$863$	13.0663	0.444781	0.222391	+	0.974958i	$0.428614\pi$
$864$	0	0
$865$	2.08075	0.0707475
$866$	14.4191	0.489981
$867$	0	0
$868$	−7.48351	−0.254007
$869$	−5.00079	−0.169640
$870$	0	0
$871$	−8.21542	−0.278369
$872$	13.4465	0.455356
$873$	0	0
$874$	13.4586	0.455246
$875$	−3.87079	−0.130856
$876$	0	0
$877$	52.3057	1.76624	0.883118	−	0.469150i	$-0.155440\pi$
$877$	52.3057	1.76624	0.883118	+	0.469150i	$0.155440\pi$
$878$	34.1052	1.15100
$879$	0	0
$880$	−0.537395	−0.0181156
$881$	−32.4569	−1.09350	−0.546750	−	0.837296i	$-0.684135\pi$
$881$	−32.4569	−1.09350	−0.546750	+	0.837296i	$0.684135\pi$
$882$	0	0
$883$	5.56233	0.187187	0.0935937	−	0.995610i	$-0.470165\pi$
$883$	5.56233	0.187187	0.0935937	+	0.995610i	$0.470165\pi$
$884$	−2.15998	−0.0726480
$885$	0	0
$886$	19.1664	0.643907
$887$	47.7942	1.60477	0.802386	−	0.596806i	$-0.203564\pi$
$887$	47.7942	1.60477	0.802386	+	0.596806i	$0.203564\pi$
$888$	0	0
$889$	−11.3455	−0.380516
$890$	0.909942	0.0305013
$891$	0	0
$892$	13.3254	0.446166
$893$	53.5324	1.79139
$894$	0	0
$895$	−7.01618	−0.234525
$896$	0.739124	0.0246924
$897$	0	0
$898$	11.0461	0.368613
$899$	−86.9987	−2.90157
$900$	0	0
$901$	−13.7570	−0.458311
$902$	9.67981	0.322302
$903$	0	0
$904$	−0.0270826	−0.000900754 0
$905$	−2.63527	−0.0875993
$906$	0	0
$907$	20.5607	0.682708	0.341354	−	0.939935i	$-0.389114\pi$
$907$	20.5607	0.682708	0.341354	+	0.939935i	$0.389114\pi$
$908$	−3.70206	−0.122857
$909$	0	0
$910$	−0.402496	−0.0133426
$911$	−21.2139	−0.702848	−0.351424	−	0.936216i	$-0.614302\pi$
$911$	−21.2139	−0.702848	−0.351424	+	0.936216i	$0.614302\pi$
$912$	0	0
$913$	6.35512	0.210324
$914$	10.9710	0.362888
$915$	0	0
$916$	13.5088	0.446343
$917$	−3.98133	−0.131475
$918$	0	0
$919$	4.00896	0.132244	0.0661218	−	0.997812i	$-0.478937\pi$
$919$	4.00896	0.132244	0.0661218	+	0.997812i	$0.478937\pi$
$920$	−1.57310	−0.0518635
$921$	0	0
$922$	17.3967	0.572931
$923$	0.806251	0.0265381
$924$	0	0
$925$	2.27754	0.0748852
$926$	3.53858	0.116285
$927$	0	0
$928$	8.59260	0.282066
$929$	−15.5771	−0.511070	−0.255535	−	0.966800i	$-0.582252\pi$
$929$	−15.5771	−0.511070	−0.255535	+	0.966800i	$0.582252\pi$
$930$	0	0
$931$	29.7824	0.976078
$932$	−11.7237	−0.384024
$933$	0	0
$934$	27.3710	0.895608
$935$	−1.14975	−0.0376009
$936$	0	0
$937$	−24.4636	−0.799191	−0.399596	−	0.916691i	$-0.630849\pi$
$937$	−24.4636	−0.799191	−0.399596	+	0.916691i	$0.630849\pi$
$938$	6.01462	0.196384
$939$	0	0
$940$	−6.25706	−0.204083
$941$	3.22350	0.105083	0.0525416	−	0.998619i	$-0.483268\pi$
$941$	3.22350	0.105083	0.0525416	+	0.998619i	$0.483268\pi$
$942$	0	0
$943$	28.3354	0.922726
$944$	−1.94920	−0.0634411
$945$	0	0
$946$	5.48287	0.178263
$947$	−3.72328	−0.120990	−0.0604951	−	0.998168i	$-0.519268\pi$
$947$	−3.72328	−0.120990	−0.0604951	+	0.998168i	$0.519268\pi$
$948$	0	0
$949$	−8.70102	−0.282447
$950$	21.7313	0.705055
$951$	0	0
$952$	1.58135	0.0512519
$953$	−12.9216	−0.418571	−0.209285	−	0.977855i	$-0.567114\pi$
$953$	−12.9216	−0.418571	−0.209285	+	0.977855i	$0.567114\pi$
$954$	0	0
$955$	−13.4612	−0.435593
$956$	−9.01299	−0.291501
$957$	0	0
$958$	5.10713	0.165004
$959$	−9.78683	−0.316033
$960$	0	0
$961$	71.5123	2.30685
$962$	0.488283	0.0157429
$963$	0	0
$964$	19.2343	0.619496
$965$	8.76897	0.282283
$966$	0	0
$967$	−35.2762	−1.13441	−0.567203	−	0.823578i	$-0.691975\pi$
$967$	−35.2762	−1.13441	−0.567203	+	0.823578i	$0.691975\pi$
$968$	−10.0074	−0.321650
$969$	0	0
$970$	5.49021	0.176280
$971$	−50.0322	−1.60561	−0.802805	−	0.596242i	$-0.796660\pi$
$971$	−50.0322	−1.60561	−0.802805	+	0.596242i	$0.796660\pi$
$972$	0	0
$973$	5.62912	0.180461
$974$	6.60830	0.211743
$975$	0	0
$976$	3.68667	0.118008
$977$	0.876869	0.0280535	0.0140268	−	0.999902i	$-0.495535\pi$
$977$	0.876869	0.0280535	0.0140268	+	0.999902i	$0.495535\pi$
$978$	0	0
$979$	−1.68073	−0.0537163
$980$	−3.48107	−0.111199
$981$	0	0
$982$	−17.4201	−0.555897
$983$	−40.1427	−1.28035	−0.640176	−	0.768228i	$-0.721139\pi$
$983$	−40.1427	−1.28035	−0.640176	+	0.768228i	$0.721139\pi$
$984$	0	0
$985$	3.15455	0.100512
$986$	18.3838	0.585460
$987$	0	0
$988$	4.65897	0.148222
$989$	16.0498	0.510354
$990$	0	0
$991$	52.3631	1.66337	0.831685	−	0.555248i	$-0.187376\pi$
$991$	52.3631	1.66337	0.831685	+	0.555248i	$0.187376\pi$
$992$	−10.1248	−0.321464
$993$	0	0
$994$	−0.590267	−0.0187221
$995$	0.452679	0.0143509
$996$	0	0
$997$	6.42979	0.203633	0.101817	−	0.994803i	$-0.467534\pi$
$997$	6.42979	0.203633	0.101817	+	0.994803i	$0.467534\pi$
$998$	10.2470	0.324362
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.n.1.7	yes	12
3.2	odd	2		8046.2.a.k.1.6	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.k.1.6	✓	12	3.2	odd	2
8046.2.a.n.1.7	yes	12	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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.539392 q^{5} +0.739124 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.539392 q^{5} +0.739124 q^{7} +1.00000 q^{8} +0.539392 q^{10} -0.996296 q^{11} -1.00958 q^{13} +0.739124 q^{14} +1.00000 q^{16} +2.13949 q^{17} -4.61478 q^{19} +0.539392 q^{20} -0.996296 q^{22} -2.91642 q^{23} -4.70906 q^{25} -1.00958 q^{26} +0.739124 q^{28} +8.59260 q^{29} -10.1248 q^{31} +1.00000 q^{32} +2.13949 q^{34} +0.398678 q^{35} -0.483652 q^{37} -4.61478 q^{38} +0.539392 q^{40} -9.71579 q^{41} -5.50325 q^{43} -0.996296 q^{44} -2.91642 q^{46} -11.6002 q^{47} -6.45370 q^{49} -4.70906 q^{50} -1.00958 q^{52} -6.43001 q^{53} -0.537395 q^{55} +0.739124 q^{56} +8.59260 q^{58} -1.94920 q^{59} +3.68667 q^{61} -10.1248 q^{62} +1.00000 q^{64} -0.544557 q^{65} +8.13749 q^{67} +2.13949 q^{68} +0.398678 q^{70} -0.798603 q^{71} +8.61849 q^{73} -0.483652 q^{74} -4.61478 q^{76} -0.736387 q^{77} +5.01938 q^{79} +0.539392 q^{80} -9.71579 q^{82} -6.37875 q^{83} +1.15403 q^{85} -5.50325 q^{86} -0.996296 q^{88} +1.68698 q^{89} -0.746202 q^{91} -2.91642 q^{92} -11.6002 q^{94} -2.48918 q^{95} +10.1785 q^{97} -6.45370 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} - 3 q^{5} - 6 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 + 12 * q^4 - 3 * q^5 - 6 * q^7 + 12 * q^8	\( 12 q + 12 q^{2} + 12 q^{4} - 3 q^{5} - 6 q^{7} + 12 q^{8} - 3 q^{10} - 14 q^{11} - 3 q^{13} - 6 q^{14} + 12 q^{16} - 8 q^{17} - 4 q^{19} - 3 q^{20} - 14 q^{22} - 13 q^{23} + 11 q^{25} - 3 q^{26} - 6 q^{28} - 23 q^{29} - 14 q^{31} + 12 q^{32} - 8 q^{34} - 32 q^{35} - 19 q^{37} - 4 q^{38} - 3 q^{40} - 30 q^{41} - 15 q^{43} - 14 q^{44} - 13 q^{46} + q^{47} + 14 q^{49} + 11 q^{50} - 3 q^{52} - 16 q^{53} - 7 q^{55} - 6 q^{56} - 23 q^{58} - 26 q^{59} - 16 q^{61} - 14 q^{62} + 12 q^{64} - 8 q^{65} - 39 q^{67} - 8 q^{68} - 32 q^{70} - 15 q^{71} - 2 q^{73} - 19 q^{74} - 4 q^{76} - 34 q^{77} - 13 q^{79} - 3 q^{80} - 30 q^{82} - 6 q^{83} - 11 q^{85} - 15 q^{86} - 14 q^{88} - 18 q^{89} - 35 q^{91} - 13 q^{92} + q^{94} - 51 q^{95} + 19 q^{97} + 14 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 - 3 * q^5 - 6 * q^7 + 12 * q^8 - 3 * q^10 - 14 * q^11 - 3 * q^13 - 6 * q^14 + 12 * q^16 - 8 * q^17 - 4 * q^19 - 3 * q^20 - 14 * q^22 - 13 * q^23 + 11 * q^25 - 3 * q^26 - 6 * q^28 - 23 * q^29 - 14 * q^31 + 12 * q^32 - 8 * q^34 - 32 * q^35 - 19 * q^37 - 4 * q^38 - 3 * q^40 - 30 * q^41 - 15 * q^43 - 14 * q^44 - 13 * q^46 + q^47 + 14 * q^49 + 11 * q^50 - 3 * q^52 - 16 * q^53 - 7 * q^55 - 6 * q^56 - 23 * q^58 - 26 * q^59 - 16 * q^61 - 14 * q^62 + 12 * q^64 - 8 * q^65 - 39 * q^67 - 8 * q^68 - 32 * q^70 - 15 * q^71 - 2 * q^73 - 19 * q^74 - 4 * q^76 - 34 * q^77 - 13 * q^79 - 3 * q^80 - 30 * q^82 - 6 * q^83 - 11 * q^85 - 15 * q^86 - 14 * q^88 - 18 * q^89 - 35 * q^91 - 13 * q^92 + q^94 - 51 * q^95 + 19 * q^97 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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