LMFDB - Embedded newform 8046.2.a.k.1.6

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:	$0$
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			1.6
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	$-0.538486\pi$
$5$	−0.539392	−0.241224	−0.120612	+	0.992700i	$0.538486\pi$
$6$	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	$-0.452009\pi$
$11$	0.996296	0.300395	0.150197	+	0.988656i	$0.452009\pi$
$12$	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	$-0.583542\pi$
$17$	−2.13949	−0.518903	−0.259452	+	0.965756i	$0.583542\pi$
$18$	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	$-0.401658\pi$
$23$	2.91642	0.608116	0.304058	+	0.952653i	$0.401658\pi$
$24$	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	$-0.794005\pi$
$29$	−8.59260	−1.59561	−0.797803	+	0.602918i	$0.794005\pi$
$30$	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	$-0.225847\pi$
$41$	9.71579	1.51735	0.758676	+	0.651468i	$0.225847\pi$
$42$	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	$-0.178986\pi$
$47$	11.6002	1.69206	0.846031	+	0.533133i	$0.178986\pi$
$48$	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	$-0.354406\pi$
$53$	6.43001	0.883229	0.441615	+	0.897205i	$0.354406\pi$
$54$	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	$-0.459503\pi$
$59$	1.94920	0.253764	0.126882	+	0.991918i	$0.459503\pi$
$60$	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	$-0.484910\pi$
$71$	0.798603	0.0947768	0.0473884	+	0.998877i	$0.484910\pi$
$72$	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	$-0.386155\pi$
$83$	6.37875	0.700158	0.350079	+	0.936720i	$0.386155\pi$
$84$	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	$-0.528498\pi$
$89$	−1.68698	−0.178819	−0.0894095	+	0.995995i	$0.528498\pi$
$90$	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	$-0.648955\pi$
$101$	−9.06625	−0.902125	−0.451063	+	0.892492i	$0.648955\pi$
$102$	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	$-0.450569\pi$
$107$	3.19978	0.309335	0.154667	+	0.987967i	$0.450569\pi$
$108$	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	$-0.499595\pi$
$113$	0.0270826	0.00254772	0.00127386	+	0.999999i	$0.499595\pi$
$114$	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	$-0.424389\pi$
$131$	5.38655	0.470625	0.235313	+	0.971920i	$0.424389\pi$
$132$	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	$-0.308632\pi$
$137$	13.2411	1.13127	0.565633	+	0.824657i	$0.308632\pi$
$138$	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	$-0.451819\pi$
$167$	3.89722	0.301576	0.150788	+	0.988566i	$0.451819\pi$
$168$	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	$-0.546847\pi$
$173$	−3.85757	−0.293286	−0.146643	+	0.989189i	$0.546847\pi$
$174$	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	$-0.338413\pi$
$179$	13.0076	0.972231	0.486115	+	0.873895i	$0.338413\pi$
$180$	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	$-0.141447\pi$
$191$	24.9561	1.80576	0.902882	+	0.429889i	$0.141447\pi$
$192$	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	$-0.566806\pi$
$197$	−5.84834	−0.416677	−0.208339	+	0.978057i	$0.566806\pi$
$198$	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	$-0.460794\pi$
$227$	3.70206	0.245715	0.122857	+	0.992424i	$0.460794\pi$
$228$	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	$-0.374538\pi$
$233$	11.7237	0.768048	0.384024	+	0.923323i	$0.374538\pi$
$234$	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	$-0.405845\pi$
$239$	9.01299	0.583002	0.291501	+	0.956570i	$0.405845\pi$
$240$	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	$-0.769426\pi$
$251$	−23.7302	−1.49784	−0.748918	+	0.662663i	$0.769426\pi$
$252$	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	$-0.732464\pi$
$257$	−21.3888	−1.33420	−0.667100	+	0.744969i	$0.732464\pi$
$258$	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	$-0.605217\pi$
$263$	−10.5270	−0.649125	−0.324563	+	0.945864i	$0.605217\pi$
$264$	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	$-0.517389\pi$
$269$	−1.79113	−0.109207	−0.0546035	+	0.998508i	$0.517389\pi$
$270$	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	$-0.396604\pi$
$281$	10.6997	0.638289	0.319145	+	0.947706i	$0.396604\pi$
$282$	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	$-0.369008\pi$
$293$	13.6940	0.800013	0.400006	+	0.916512i	$0.369008\pi$
$294$	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	$-0.429500\pi$
$311$	7.74808	0.439353	0.219677	+	0.975573i	$0.429500\pi$
$312$	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	$-0.753315\pi$
$317$	−25.4402	−1.42886	−0.714432	+	0.699705i	$0.753315\pi$
$318$	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	$-0.269927\pi$
$347$	24.6442	1.32297	0.661484	+	0.749959i	$0.269927\pi$
$348$	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	$-0.319010\pi$
$353$	20.2331	1.07690	0.538451	+	0.842657i	$0.319010\pi$
$354$	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	$-0.728574\pi$
$359$	−24.9326	−1.31589	−0.657946	+	0.753065i	$0.728574\pi$
$360$	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	$-0.270804\pi$
$383$	25.8100	1.31883	0.659415	+	0.751779i	$0.270804\pi$
$384$	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	$-0.214430\pi$
$389$	30.8291	1.56310	0.781549	+	0.623844i	$0.214430\pi$
$390$	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	$-0.395362\pi$
$401$	12.9298	0.645685	0.322842	+	0.946453i	$0.395362\pi$
$402$	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	$-0.418823\pi$
$419$	10.3277	0.504542	0.252271	+	0.967657i	$0.418823\pi$
$420$	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	$-0.355699\pi$
$431$	18.1849	0.875934	0.437967	+	0.898991i	$0.355699\pi$
$432$	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	$-0.650472\pi$
$443$	−19.1664	−0.910622	−0.455311	+	0.890333i	$0.650472\pi$
$444$	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	$-0.583936\pi$
$449$	−11.0461	−0.521297	−0.260648	+	0.965434i	$0.583936\pi$
$450$	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	$-0.632771\pi$
$461$	−17.3967	−0.810247	−0.405123	+	0.914262i	$0.632771\pi$
$462$	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	$-0.718296\pi$
$467$	−27.3710	−1.26658	−0.633290	+	0.773914i	$0.718296\pi$
$468$	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	$-0.537224\pi$
$479$	−5.10713	−0.233351	−0.116675	+	0.993170i	$0.537224\pi$
$480$	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	$-0.371410\pi$
$491$	17.4201	0.786157	0.393079	+	0.919505i	$0.371410\pi$
$492$	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	$-0.597615\pi$
$503$	−13.5410	−0.603763	−0.301882	+	0.953345i	$0.597615\pi$
$504$	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	$-0.153090\pi$
$509$	40.0033	1.77311	0.886557	+	0.462619i	$0.153090\pi$
$510$	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	$-0.673557\pi$
$521$	−23.6758	−1.03726	−0.518628	+	0.855000i	$0.673557\pi$
$522$	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	$-0.294839\pi$
$557$	28.3600	1.20165	0.600825	+	0.799380i	$0.294839\pi$
$558$	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	$-0.434960\pi$
$563$	9.62910	0.405818	0.202909	+	0.979198i	$0.434960\pi$
$564$	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	$-0.258844\pi$
$569$	32.7841	1.37438	0.687190	+	0.726478i	$0.258844\pi$
$570$	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	$-0.407183\pi$
$587$	13.9302	0.574960	0.287480	+	0.957787i	$0.407183\pi$
$588$	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	$-0.359866\pi$
$593$	20.7554	0.852321	0.426160	+	0.904648i	$0.359866\pi$
$594$	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	$-0.303723\pi$
$599$	28.3064	1.15657	0.578283	+	0.815836i	$0.303723\pi$
$600$	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	$-0.281943\pi$
$617$	31.4323	1.26542	0.632709	+	0.774390i	$0.281943\pi$
$618$	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	$-0.390172\pi$
$641$	17.1266	0.676460	0.338230	+	0.941064i	$0.390172\pi$
$642$	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	$-0.621424\pi$
$647$	−18.9388	−0.744561	−0.372280	+	0.928120i	$0.621424\pi$
$648$	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	$-0.760226\pi$
$653$	−37.2808	−1.45891	−0.729455	+	0.684029i	$0.760226\pi$
$654$	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	$-0.718033\pi$
$659$	−32.4816	−1.26530	−0.632651	+	0.774437i	$0.718033\pi$
$660$	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	$-0.585633\pi$
$677$	−13.8313	−0.531580	−0.265790	+	0.964031i	$0.585633\pi$
$678$	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	$-0.644548\pi$
$683$	−22.9283	−0.877328	−0.438664	+	0.898651i	$0.644548\pi$
$684$	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	$-0.336674\pi$
$701$	25.9937	0.981767	0.490884	+	0.871225i	$0.336674\pi$
$702$	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	$-0.691980\pi$
$719$	−30.4190	−1.13444	−0.567218	+	0.823568i	$0.691980\pi$
$720$	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	$-0.400127\pi$
$743$	16.8256	0.617272	0.308636	+	0.951180i	$0.400127\pi$
$744$	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	$-0.463752\pi$
$761$	6.26920	0.227258	0.113629	+	0.993523i	$0.463752\pi$
$762$	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	$-0.380241\pi$
$773$	20.4307	0.734843	0.367421	+	0.930055i	$0.380241\pi$
$774$	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	$-0.299110\pi$
$797$	33.3154	1.18009	0.590046	+	0.807370i	$0.299110\pi$
$798$	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	$-0.556880\pi$
$809$	−10.1111	−0.355486	−0.177743	+	0.984077i	$0.556880\pi$
$810$	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	$-0.846905\pi$
$821$	−50.8048	−1.77310	−0.886550	+	0.462632i	$0.846905\pi$
$822$	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	$-0.620723\pi$
$827$	−21.2942	−0.740470	−0.370235	+	0.928938i	$0.620723\pi$
$828$	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	$-0.700406\pi$
$839$	−34.1108	−1.17764	−0.588818	+	0.808266i	$0.700406\pi$
$840$	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	$-0.811578\pi$
$857$	−48.5874	−1.65971	−0.829857	+	0.557976i	$0.811578\pi$
$858$	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	$-0.571386\pi$
$863$	−13.0663	−0.444781	−0.222391	+	0.974958i	$0.571386\pi$
$864$	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	$-0.315865\pi$
$881$	32.4569	1.09350	0.546750	+	0.837296i	$0.315865\pi$
$882$	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	$-0.796436\pi$
$887$	−47.7942	−1.60477	−0.802386	+	0.596806i	$0.796436\pi$
$888$	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	$-0.385698\pi$
$911$	21.2139	0.702848	0.351424	+	0.936216i	$0.385698\pi$
$912$	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	$-0.417748\pi$
$929$	15.5771	0.511070	0.255535	+	0.966800i	$0.417748\pi$
$930$	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	$-0.516732\pi$
$941$	−3.22350	−0.105083	−0.0525416	+	0.998619i	$0.516732\pi$
$942$	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	$-0.480732\pi$
$947$	3.72328	0.120990	0.0604951	+	0.998168i	$0.480732\pi$
$948$	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	$-0.432886\pi$
$953$	12.9216	0.418571	0.209285	+	0.977855i	$0.432886\pi$
$954$	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	$-0.203340\pi$
$971$	50.0322	1.60561	0.802805	+	0.596242i	$0.203340\pi$
$972$	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	$-0.504465\pi$
$977$	−0.876869	−0.0280535	−0.0140268	+	0.999902i	$0.504465\pi$
$978$	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	$-0.278861\pi$
$983$	40.1427	1.28035	0.640176	+	0.768228i	$0.278861\pi$
$984$	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.k.1.6	✓	12
3.2	odd	2		8046.2.a.n.1.7	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.k.1.6	✓	12	1.1	even	1	trivial
8046.2.a.n.1.7	yes	12	3.2	odd	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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
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

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Database

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