LMFDB - Embedded newform 8004.2.a.g.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
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} + 80 x^{9} + 347 x^{8} - 697 x^{7} - 1714 x^{6} + 2146 x^{5} + 3304 x^{4} + \cdots + 16 $ x^12 - 3x^11 - 31x^10 + 80x^9 + 347x^8 - 697x^7 - 1714x^6 + 2146x^5 + 3304x^4 - 1156x^3 - 616x^2 + 136*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.0891245$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +0.0891245 q^{5} +2.52085 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.0891245 q^{5} +2.52085 q^{7} +1.00000 q^{9} -1.55149 q^{11} +6.31044 q^{13} -0.0891245 q^{15} -6.02488 q^{17} +4.11930 q^{19} -2.52085 q^{21} +1.00000 q^{23} -4.99206 q^{25} -1.00000 q^{27} -1.00000 q^{29} -6.28492 q^{31} +1.55149 q^{33} +0.224669 q^{35} -3.14158 q^{37} -6.31044 q^{39} -3.88280 q^{41} +1.91638 q^{43} +0.0891245 q^{45} -12.5261 q^{47} -0.645334 q^{49} +6.02488 q^{51} -3.50409 q^{53} -0.138276 q^{55} -4.11930 q^{57} -9.16270 q^{59} +10.3666 q^{61} +2.52085 q^{63} +0.562415 q^{65} -1.85364 q^{67} -1.00000 q^{69} +2.06730 q^{71} -0.852008 q^{73} +4.99206 q^{75} -3.91107 q^{77} -14.6715 q^{79} +1.00000 q^{81} +1.87772 q^{83} -0.536965 q^{85} +1.00000 q^{87} -11.8205 q^{89} +15.9077 q^{91} +6.28492 q^{93} +0.367131 q^{95} -5.06804 q^{97} -1.55149 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9	$ 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9} - 5 q^{11} - 6 q^{13} + 3 q^{15} - 7 q^{17} - 3 q^{19} - 4 q^{21} + 12 q^{23} + 11 q^{25} - 12 q^{27} - 12 q^{29} + 2 q^{31} + 5 q^{33} - 9 q^{35} - 20 q^{37} + 6 q^{39} - 3 q^{41} + 5 q^{43} - 3 q^{45} - 2 q^{49} + 7 q^{51} - 3 q^{53} + 19 q^{55} + 3 q^{57} - 20 q^{59} - 17 q^{61} + 4 q^{63} - 4 q^{65} - 9 q^{67} - 12 q^{69} + 7 q^{71} - 9 q^{73} - 11 q^{75} - 34 q^{77} + 14 q^{79} + 12 q^{81} + 5 q^{83} - 12 q^{85} + 12 q^{87} - 22 q^{89} - 3 q^{91} - 2 q^{93} - 27 q^{95} + 17 q^{97} - 5 q^{99}+O(q^{100}) $ 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9 - 5 * q^11 - 6 * q^13 + 3 * q^15 - 7 * q^17 - 3 * q^19 - 4 * q^21 + 12 * q^23 + 11 * q^25 - 12 * q^27 - 12 * q^29 + 2 * q^31 + 5 * q^33 - 9 * q^35 - 20 * q^37 + 6 * q^39 - 3 * q^41 + 5 * q^43 - 3 * q^45 - 2 * q^49 + 7 * q^51 - 3 * q^53 + 19 * q^55 + 3 * q^57 - 20 * q^59 - 17 * q^61 + 4 * q^63 - 4 * q^65 - 9 * q^67 - 12 * q^69 + 7 * q^71 - 9 * q^73 - 11 * q^75 - 34 * q^77 + 14 * q^79 + 12 * q^81 + 5 * q^83 - 12 * q^85 + 12 * q^87 - 22 * q^89 - 3 * q^91 - 2 * q^93 - 27 * q^95 + 17 * q^97 - 5 * q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0.0891245	0.0398577	0.0199289	−	0.999801i	$-0.493656\pi$
$5$	0.0891245	0.0398577	0.0199289	+	0.999801i	$0.493656\pi$
$6$	0	0
$7$	2.52085	0.952790	0.476395	−	0.879231i	$-0.341943\pi$
$7$	2.52085	0.952790	0.476395	+	0.879231i	$0.341943\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.55149	−0.467792	−0.233896	−	0.972262i	$-0.575148\pi$
$11$	−1.55149	−0.467792	−0.233896	+	0.972262i	$0.575148\pi$
$12$	0	0
$13$	6.31044	1.75020	0.875101	−	0.483941i	$-0.160795\pi$
$13$	6.31044	1.75020	0.875101	+	0.483941i	$0.160795\pi$
$14$	0	0
$15$	−0.0891245	−0.0230119
$16$	0	0
$17$	−6.02488	−1.46125	−0.730624	−	0.682780i	$-0.760771\pi$
$17$	−6.02488	−1.46125	−0.730624	+	0.682780i	$0.760771\pi$
$18$	0	0
$19$	4.11930	0.945033	0.472516	−	0.881322i	$-0.343346\pi$
$19$	4.11930	0.945033	0.472516	+	0.881322i	$0.343346\pi$
$20$	0	0
$21$	−2.52085	−0.550094
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.99206	−0.998411
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−6.28492	−1.12881	−0.564403	−	0.825500i	$-0.690893\pi$
$31$	−6.28492	−1.12881	−0.564403	+	0.825500i	$0.690893\pi$
$32$	0	0
$33$	1.55149	0.270080
$34$	0	0
$35$	0.224669	0.0379760
$36$	0	0
$37$	−3.14158	−0.516472	−0.258236	−	0.966082i	$-0.583141\pi$
$37$	−3.14158	−0.516472	−0.258236	+	0.966082i	$0.583141\pi$
$38$	0	0
$39$	−6.31044	−1.01048
$40$	0	0
$41$	−3.88280	−0.606392	−0.303196	−	0.952928i	$-0.598054\pi$
$41$	−3.88280	−0.606392	−0.303196	+	0.952928i	$0.598054\pi$
$42$	0	0
$43$	1.91638	0.292245	0.146122	−	0.989267i	$-0.453321\pi$
$43$	1.91638	0.292245	0.146122	+	0.989267i	$0.453321\pi$
$44$	0	0
$45$	0.0891245	0.0132859
$46$	0	0
$47$	−12.5261	−1.82712	−0.913562	−	0.406700i	$-0.866680\pi$
$47$	−12.5261	−1.82712	−0.913562	+	0.406700i	$0.866680\pi$
$48$	0	0
$49$	−0.645334	−0.0921905
$50$	0	0
$51$	6.02488	0.843652
$52$	0	0
$53$	−3.50409	−0.481324	−0.240662	−	0.970609i	$-0.577365\pi$
$53$	−3.50409	−0.481324	−0.240662	+	0.970609i	$0.577365\pi$
$54$	0	0
$55$	−0.138276	−0.0186451
$56$	0	0
$57$	−4.11930	−0.545615
$58$	0	0
$59$	−9.16270	−1.19288	−0.596441	−	0.802657i	$-0.703419\pi$
$59$	−9.16270	−1.19288	−0.596441	+	0.802657i	$0.703419\pi$
$60$	0	0
$61$	10.3666	1.32731	0.663656	−	0.748038i	$-0.269004\pi$
$61$	10.3666	1.32731	0.663656	+	0.748038i	$0.269004\pi$
$62$	0	0
$63$	2.52085	0.317597
$64$	0	0
$65$	0.562415	0.0697590
$66$	0	0
$67$	−1.85364	−0.226458	−0.113229	−	0.993569i	$-0.536119\pi$
$67$	−1.85364	−0.226458	−0.113229	+	0.993569i	$0.536119\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	2.06730	0.245343	0.122672	−	0.992447i	$-0.460854\pi$
$71$	2.06730	0.245343	0.122672	+	0.992447i	$0.460854\pi$
$72$	0	0
$73$	−0.852008	−0.0997200	−0.0498600	−	0.998756i	$-0.515878\pi$
$73$	−0.852008	−0.0997200	−0.0498600	+	0.998756i	$0.515878\pi$
$74$	0	0
$75$	4.99206	0.576433
$76$	0	0
$77$	−3.91107	−0.445708
$78$	0	0
$79$	−14.6715	−1.65068	−0.825339	−	0.564638i	$-0.809016\pi$
$79$	−14.6715	−1.65068	−0.825339	+	0.564638i	$0.809016\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.87772	0.206106	0.103053	−	0.994676i	$-0.467139\pi$
$83$	1.87772	0.206106	0.103053	+	0.994676i	$0.467139\pi$
$84$	0	0
$85$	−0.536965	−0.0582420
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	−11.8205	−1.25297	−0.626484	−	0.779434i	$-0.715507\pi$
$89$	−11.8205	−1.25297	−0.626484	+	0.779434i	$0.715507\pi$
$90$	0	0
$91$	15.9077	1.66757
$92$	0	0
$93$	6.28492	0.651716
$94$	0	0
$95$	0.367131	0.0376668
$96$	0	0
$97$	−5.06804	−0.514581	−0.257291	−	0.966334i	$-0.582830\pi$
$97$	−5.06804	−0.514581	−0.257291	+	0.966334i	$0.582830\pi$
$98$	0	0
$99$	−1.55149	−0.155931
$100$	0	0
$101$	2.45871	0.244651	0.122325	−	0.992490i	$-0.460965\pi$
$101$	2.45871	0.244651	0.122325	+	0.992490i	$0.460965\pi$
$102$	0	0
$103$	17.4216	1.71660	0.858299	−	0.513150i	$-0.171522\pi$
$103$	17.4216	1.71660	0.858299	+	0.513150i	$0.171522\pi$
$104$	0	0
$105$	−0.224669	−0.0219255
$106$	0	0
$107$	−7.09609	−0.686005	−0.343003	−	0.939334i	$-0.611444\pi$
$107$	−7.09609	−0.686005	−0.343003	+	0.939334i	$0.611444\pi$
$108$	0	0
$109$	2.52032	0.241403	0.120702	−	0.992689i	$-0.461486\pi$
$109$	2.52032	0.241403	0.120702	+	0.992689i	$0.461486\pi$
$110$	0	0
$111$	3.14158	0.298185
$112$	0	0
$113$	9.51675	0.895260	0.447630	−	0.894219i	$-0.352268\pi$
$113$	9.51675	0.895260	0.447630	+	0.894219i	$0.352268\pi$
$114$	0	0
$115$	0.0891245	0.00831091
$116$	0	0
$117$	6.31044	0.583400
$118$	0	0
$119$	−15.1878	−1.39226
$120$	0	0
$121$	−8.59288	−0.781171
$122$	0	0
$123$	3.88280	0.350101
$124$	0	0
$125$	−0.890537	−0.0796521
$126$	0	0
$127$	−6.81952	−0.605135	−0.302567	−	0.953128i	$-0.597844\pi$
$127$	−6.81952	−0.605135	−0.302567	+	0.953128i	$0.597844\pi$
$128$	0	0
$129$	−1.91638	−0.168728
$130$	0	0
$131$	8.83290	0.771734	0.385867	−	0.922554i	$-0.373902\pi$
$131$	8.83290	0.771734	0.385867	+	0.922554i	$0.373902\pi$
$132$	0	0
$133$	10.3841	0.900418
$134$	0	0
$135$	−0.0891245	−0.00767062
$136$	0	0
$137$	−12.8582	−1.09855	−0.549273	−	0.835643i	$-0.685095\pi$
$137$	−12.8582	−1.09855	−0.549273	+	0.835643i	$0.685095\pi$
$138$	0	0
$139$	7.85086	0.665901	0.332951	−	0.942944i	$-0.391956\pi$
$139$	7.85086	0.665901	0.332951	+	0.942944i	$0.391956\pi$
$140$	0	0
$141$	12.5261	1.05489
$142$	0	0
$143$	−9.79059	−0.818730
$144$	0	0
$145$	−0.0891245	−0.00740139
$146$	0	0
$147$	0.645334	0.0532262
$148$	0	0
$149$	−6.39581	−0.523965	−0.261983	−	0.965073i	$-0.584376\pi$
$149$	−6.39581	−0.523965	−0.261983	+	0.965073i	$0.584376\pi$
$150$	0	0
$151$	14.8499	1.20847	0.604234	−	0.796807i	$-0.293479\pi$
$151$	14.8499	1.20847	0.604234	+	0.796807i	$0.293479\pi$
$152$	0	0
$153$	−6.02488	−0.487083
$154$	0	0
$155$	−0.560141	−0.0449916
$156$	0	0
$157$	−17.9017	−1.42871	−0.714357	−	0.699781i	$-0.753281\pi$
$157$	−17.9017	−1.42871	−0.714357	+	0.699781i	$0.753281\pi$
$158$	0	0
$159$	3.50409	0.277893
$160$	0	0
$161$	2.52085	0.198671
$162$	0	0
$163$	−4.91880	−0.385270	−0.192635	−	0.981270i	$-0.561703\pi$
$163$	−4.91880	−0.385270	−0.192635	+	0.981270i	$0.561703\pi$
$164$	0	0
$165$	0.138276	0.0107648
$166$	0	0
$167$	−3.61590	−0.279807	−0.139903	−	0.990165i	$-0.544679\pi$
$167$	−3.61590	−0.279807	−0.139903	+	0.990165i	$0.544679\pi$
$168$	0	0
$169$	26.8217	2.06320
$170$	0	0
$171$	4.11930	0.315011
$172$	0	0
$173$	4.94379	0.375869	0.187935	−	0.982182i	$-0.439821\pi$
$173$	4.94379	0.375869	0.187935	+	0.982182i	$0.439821\pi$
$174$	0	0
$175$	−12.5842	−0.951277
$176$	0	0
$177$	9.16270	0.688710
$178$	0	0
$179$	4.60179	0.343954	0.171977	−	0.985101i	$-0.444985\pi$
$179$	4.60179	0.343954	0.171977	+	0.985101i	$0.444985\pi$
$180$	0	0
$181$	0.0656052	0.00487640	0.00243820	−	0.999997i	$-0.499224\pi$
$181$	0.0656052	0.00487640	0.00243820	+	0.999997i	$0.499224\pi$
$182$	0	0
$183$	−10.3666	−0.766324
$184$	0	0
$185$	−0.279992	−0.0205854
$186$	0	0
$187$	9.34754	0.683560
$188$	0	0
$189$	−2.52085	−0.183365
$190$	0	0
$191$	−1.40793	−0.101874	−0.0509372	−	0.998702i	$-0.516221\pi$
$191$	−1.40793	−0.101874	−0.0509372	+	0.998702i	$0.516221\pi$
$192$	0	0
$193$	12.5383	0.902530	0.451265	−	0.892390i	$-0.350973\pi$
$193$	12.5383	0.902530	0.451265	+	0.892390i	$0.350973\pi$
$194$	0	0
$195$	−0.562415	−0.0402754
$196$	0	0
$197$	4.95060	0.352715	0.176358	−	0.984326i	$-0.443568\pi$
$197$	4.95060	0.352715	0.176358	+	0.984326i	$0.443568\pi$
$198$	0	0
$199$	−5.40788	−0.383354	−0.191677	−	0.981458i	$-0.561393\pi$
$199$	−5.40788	−0.383354	−0.191677	+	0.981458i	$0.561393\pi$
$200$	0	0
$201$	1.85364	0.130745
$202$	0	0
$203$	−2.52085	−0.176929
$204$	0	0
$205$	−0.346053	−0.0241694
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−6.39106	−0.442079
$210$	0	0
$211$	−7.75222	−0.533685	−0.266843	−	0.963740i	$-0.585980\pi$
$211$	−7.75222	−0.533685	−0.266843	+	0.963740i	$0.585980\pi$
$212$	0	0
$213$	−2.06730	−0.141649
$214$	0	0
$215$	0.170796	0.0116482
$216$	0	0
$217$	−15.8433	−1.07551
$218$	0	0
$219$	0.852008	0.0575734
$220$	0	0
$221$	−38.0196	−2.55748
$222$	0	0
$223$	19.2273	1.28755	0.643777	−	0.765214i	$-0.277367\pi$
$223$	19.2273	1.28755	0.643777	+	0.765214i	$0.277367\pi$
$224$	0	0
$225$	−4.99206	−0.332804
$226$	0	0
$227$	−24.0362	−1.59534	−0.797668	−	0.603096i	$-0.793933\pi$
$227$	−24.0362	−1.59534	−0.797668	+	0.603096i	$0.793933\pi$
$228$	0	0
$229$	−1.05473	−0.0696985	−0.0348492	−	0.999393i	$-0.511095\pi$
$229$	−1.05473	−0.0696985	−0.0348492	+	0.999393i	$0.511095\pi$
$230$	0	0
$231$	3.91107	0.257329
$232$	0	0
$233$	−5.04352	−0.330412	−0.165206	−	0.986259i	$-0.552829\pi$
$233$	−5.04352	−0.330412	−0.165206	+	0.986259i	$0.552829\pi$
$234$	0	0
$235$	−1.11639	−0.0728250
$236$	0	0
$237$	14.6715	0.953019
$238$	0	0
$239$	8.65130	0.559606	0.279803	−	0.960057i	$-0.409731\pi$
$239$	8.65130	0.559606	0.279803	+	0.960057i	$0.409731\pi$
$240$	0	0
$241$	−25.9981	−1.67469	−0.837344	−	0.546677i	$-0.815893\pi$
$241$	−25.9981	−1.67469	−0.837344	+	0.546677i	$0.815893\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−0.0575151	−0.00367450
$246$	0	0
$247$	25.9946	1.65400
$248$	0	0
$249$	−1.87772	−0.118995
$250$	0	0
$251$	5.75346	0.363155	0.181577	−	0.983377i	$-0.441880\pi$
$251$	5.75346	0.363155	0.181577	+	0.983377i	$0.441880\pi$
$252$	0	0
$253$	−1.55149	−0.0975414
$254$	0	0
$255$	0.536965	0.0336260
$256$	0	0
$257$	4.56911	0.285013	0.142507	−	0.989794i	$-0.454484\pi$
$257$	4.56911	0.285013	0.142507	+	0.989794i	$0.454484\pi$
$258$	0	0
$259$	−7.91944	−0.492090
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	12.3698	0.762752	0.381376	−	0.924420i	$-0.375450\pi$
$263$	12.3698	0.762752	0.381376	+	0.924420i	$0.375450\pi$
$264$	0	0
$265$	−0.312301	−0.0191845
$266$	0	0
$267$	11.8205	0.723401
$268$	0	0
$269$	−1.92180	−0.117174	−0.0585870	−	0.998282i	$-0.518660\pi$
$269$	−1.92180	−0.117174	−0.0585870	+	0.998282i	$0.518660\pi$
$270$	0	0
$271$	−9.32469	−0.566435	−0.283217	−	0.959056i	$-0.591402\pi$
$271$	−9.32469	−0.566435	−0.283217	+	0.959056i	$0.591402\pi$
$272$	0	0
$273$	−15.9077	−0.962775
$274$	0	0
$275$	7.74513	0.467049
$276$	0	0
$277$	−19.6085	−1.17816	−0.589079	−	0.808075i	$-0.700509\pi$
$277$	−19.6085	−1.17816	−0.589079	+	0.808075i	$0.700509\pi$
$278$	0	0
$279$	−6.28492	−0.376268
$280$	0	0
$281$	12.2808	0.732610	0.366305	−	0.930495i	$-0.380623\pi$
$281$	12.2808	0.732610	0.366305	+	0.930495i	$0.380623\pi$
$282$	0	0
$283$	12.0243	0.714770	0.357385	−	0.933957i	$-0.383668\pi$
$283$	12.0243	0.714770	0.357385	+	0.933957i	$0.383668\pi$
$284$	0	0
$285$	−0.367131	−0.0217470
$286$	0	0
$287$	−9.78795	−0.577764
$288$	0	0
$289$	19.2992	1.13525
$290$	0	0
$291$	5.06804	0.297094
$292$	0	0
$293$	15.8746	0.927406	0.463703	−	0.885991i	$-0.346520\pi$
$293$	15.8746	0.927406	0.463703	+	0.885991i	$0.346520\pi$
$294$	0	0
$295$	−0.816621	−0.0475455
$296$	0	0
$297$	1.55149	0.0900266
$298$	0	0
$299$	6.31044	0.364942
$300$	0	0
$301$	4.83089	0.278448
$302$	0	0
$303$	−2.45871	−0.141249
$304$	0	0
$305$	0.923922	0.0529036
$306$	0	0
$307$	6.32225	0.360830	0.180415	−	0.983591i	$-0.442256\pi$
$307$	6.32225	0.360830	0.180415	+	0.983591i	$0.442256\pi$
$308$	0	0
$309$	−17.4216	−0.991078
$310$	0	0
$311$	0.581953	0.0329995	0.0164998	−	0.999864i	$-0.494748\pi$
$311$	0.581953	0.0329995	0.0164998	+	0.999864i	$0.494748\pi$
$312$	0	0
$313$	−32.7136	−1.84908	−0.924542	−	0.381081i	$-0.875552\pi$
$313$	−32.7136	−1.84908	−0.924542	+	0.381081i	$0.875552\pi$
$314$	0	0
$315$	0.224669	0.0126587
$316$	0	0
$317$	−34.5562	−1.94087	−0.970434	−	0.241366i	$-0.922405\pi$
$317$	−34.5562	−1.94087	−0.970434	+	0.241366i	$0.922405\pi$
$318$	0	0
$319$	1.55149	0.0868668
$320$	0	0
$321$	7.09609	0.396065
$322$	0	0
$323$	−24.8183	−1.38093
$324$	0	0
$325$	−31.5021	−1.74742
$326$	0	0
$327$	−2.52032	−0.139374
$328$	0	0
$329$	−31.5765	−1.74087
$330$	0	0
$331$	27.7031	1.52270	0.761351	−	0.648340i	$-0.224536\pi$
$331$	27.7031	1.52270	0.761351	+	0.648340i	$0.224536\pi$
$332$	0	0
$333$	−3.14158	−0.172157
$334$	0	0
$335$	−0.165205	−0.00902609
$336$	0	0
$337$	15.6495	0.852482	0.426241	−	0.904610i	$-0.359838\pi$
$337$	15.6495	0.852482	0.426241	+	0.904610i	$0.359838\pi$
$338$	0	0
$339$	−9.51675	−0.516879
$340$	0	0
$341$	9.75100	0.528046
$342$	0	0
$343$	−19.2727	−1.04063
$344$	0	0
$345$	−0.0891245	−0.00479830
$346$	0	0
$347$	21.9827	1.18010	0.590048	−	0.807368i	$-0.299109\pi$
$347$	21.9827	1.18010	0.590048	+	0.807368i	$0.299109\pi$
$348$	0	0
$349$	−7.13201	−0.381768	−0.190884	−	0.981613i	$-0.561135\pi$
$349$	−7.13201	−0.381768	−0.190884	+	0.981613i	$0.561135\pi$
$350$	0	0
$351$	−6.31044	−0.336826
$352$	0	0
$353$	27.1766	1.44647	0.723233	−	0.690604i	$-0.242655\pi$
$353$	27.1766	1.44647	0.723233	+	0.690604i	$0.242655\pi$
$354$	0	0
$355$	0.184247	0.00977882
$356$	0	0
$357$	15.1878	0.803823
$358$	0	0
$359$	15.8778	0.837996	0.418998	−	0.907987i	$-0.362381\pi$
$359$	15.8778	0.837996	0.418998	+	0.907987i	$0.362381\pi$
$360$	0	0
$361$	−2.03135	−0.106913
$362$	0	0
$363$	8.59288	0.451009
$364$	0	0
$365$	−0.0759348	−0.00397461
$366$	0	0
$367$	−8.03489	−0.419418	−0.209709	−	0.977764i	$-0.567252\pi$
$367$	−8.03489	−0.419418	−0.209709	+	0.977764i	$0.567252\pi$
$368$	0	0
$369$	−3.88280	−0.202131
$370$	0	0
$371$	−8.83328	−0.458601
$372$	0	0
$373$	−29.2738	−1.51574	−0.757869	−	0.652406i	$-0.773760\pi$
$373$	−29.2738	−1.51574	−0.757869	+	0.652406i	$0.773760\pi$
$374$	0	0
$375$	0.890537	0.0459872
$376$	0	0
$377$	−6.31044	−0.325004
$378$	0	0
$379$	−18.0054	−0.924877	−0.462439	−	0.886651i	$-0.653025\pi$
$379$	−18.0054	−0.924877	−0.462439	+	0.886651i	$0.653025\pi$
$380$	0	0
$381$	6.81952	0.349375
$382$	0	0
$383$	8.65577	0.442289	0.221145	−	0.975241i	$-0.429021\pi$
$383$	8.65577	0.442289	0.221145	+	0.975241i	$0.429021\pi$
$384$	0	0
$385$	−0.348572	−0.0177649
$386$	0	0
$387$	1.91638	0.0974149
$388$	0	0
$389$	−13.1237	−0.665399	−0.332699	−	0.943033i	$-0.607959\pi$
$389$	−13.1237	−0.665399	−0.332699	+	0.943033i	$0.607959\pi$
$390$	0	0
$391$	−6.02488	−0.304691
$392$	0	0
$393$	−8.83290	−0.445561
$394$	0	0
$395$	−1.30759	−0.0657922
$396$	0	0
$397$	−25.4689	−1.27825	−0.639123	−	0.769104i	$-0.720703\pi$
$397$	−25.4689	−1.27825	−0.639123	+	0.769104i	$0.720703\pi$
$398$	0	0
$399$	−10.3841	−0.519857
$400$	0	0
$401$	−12.4628	−0.622363	−0.311181	−	0.950351i	$-0.600725\pi$
$401$	−12.4628	−0.622363	−0.311181	+	0.950351i	$0.600725\pi$
$402$	0	0
$403$	−39.6606	−1.97564
$404$	0	0
$405$	0.0891245	0.00442863
$406$	0	0
$407$	4.87413	0.241602
$408$	0	0
$409$	−16.1131	−0.796741	−0.398371	−	0.917225i	$-0.630424\pi$
$409$	−16.1131	−0.796741	−0.398371	+	0.917225i	$0.630424\pi$
$410$	0	0
$411$	12.8582	0.634246
$412$	0	0
$413$	−23.0978	−1.13657
$414$	0	0
$415$	0.167351	0.00821492
$416$	0	0
$417$	−7.85086	−0.384458
$418$	0	0
$419$	−14.2081	−0.694109	−0.347055	−	0.937845i	$-0.612818\pi$
$419$	−14.2081	−0.694109	−0.347055	+	0.937845i	$0.612818\pi$
$420$	0	0
$421$	14.8848	0.725439	0.362720	−	0.931898i	$-0.381848\pi$
$421$	14.8848	0.725439	0.362720	+	0.931898i	$0.381848\pi$
$422$	0	0
$423$	−12.5261	−0.609041
$424$	0	0
$425$	30.0765	1.45893
$426$	0	0
$427$	26.1327	1.26465
$428$	0	0
$429$	9.79059	0.472694
$430$	0	0
$431$	16.6692	0.802925	0.401462	−	0.915876i	$-0.368502\pi$
$431$	16.6692	0.802925	0.401462	+	0.915876i	$0.368502\pi$
$432$	0	0
$433$	−9.60301	−0.461491	−0.230746	−	0.973014i	$-0.574117\pi$
$433$	−9.60301	−0.461491	−0.230746	+	0.973014i	$0.574117\pi$
$434$	0	0
$435$	0.0891245	0.00427319
$436$	0	0
$437$	4.11930	0.197053
$438$	0	0
$439$	−25.0898	−1.19747	−0.598736	−	0.800947i	$-0.704330\pi$
$439$	−25.0898	−1.19747	−0.598736	+	0.800947i	$0.704330\pi$
$440$	0	0
$441$	−0.645334	−0.0307302
$442$	0	0
$443$	−26.7780	−1.27226	−0.636132	−	0.771581i	$-0.719466\pi$
$443$	−26.7780	−1.27226	−0.636132	+	0.771581i	$0.719466\pi$
$444$	0	0
$445$	−1.05349	−0.0499404
$446$	0	0
$447$	6.39581	0.302511
$448$	0	0
$449$	21.0330	0.992610	0.496305	−	0.868148i	$-0.334690\pi$
$449$	21.0330	0.992610	0.496305	+	0.868148i	$0.334690\pi$
$450$	0	0
$451$	6.02413	0.283665
$452$	0	0
$453$	−14.8499	−0.697709
$454$	0	0
$455$	1.41776	0.0664657
$456$	0	0
$457$	−6.28056	−0.293792	−0.146896	−	0.989152i	$-0.546928\pi$
$457$	−6.28056	−0.293792	−0.146896	+	0.989152i	$0.546928\pi$
$458$	0	0
$459$	6.02488	0.281217
$460$	0	0
$461$	−17.1246	−0.797570	−0.398785	−	0.917044i	$-0.630568\pi$
$461$	−17.1246	−0.797570	−0.398785	+	0.917044i	$0.630568\pi$
$462$	0	0
$463$	−17.9716	−0.835210	−0.417605	−	0.908629i	$-0.637130\pi$
$463$	−17.9716	−0.835210	−0.417605	+	0.908629i	$0.637130\pi$
$464$	0	0
$465$	0.560141	0.0259759
$466$	0	0
$467$	28.0902	1.29986	0.649931	−	0.759993i	$-0.274798\pi$
$467$	28.0902	1.29986	0.649931	+	0.759993i	$0.274798\pi$
$468$	0	0
$469$	−4.67273	−0.215767
$470$	0	0
$471$	17.9017	0.824869
$472$	0	0
$473$	−2.97324	−0.136710
$474$	0	0
$475$	−20.5638	−0.943532
$476$	0	0
$477$	−3.50409	−0.160441
$478$	0	0
$479$	26.5967	1.21523	0.607617	−	0.794230i	$-0.292125\pi$
$479$	26.5967	1.21523	0.607617	+	0.794230i	$0.292125\pi$
$480$	0	0
$481$	−19.8247	−0.903931
$482$	0	0
$483$	−2.52085	−0.114702
$484$	0	0
$485$	−0.451687	−0.0205100
$486$	0	0
$487$	−10.5066	−0.476101	−0.238050	−	0.971253i	$-0.576508\pi$
$487$	−10.5066	−0.476101	−0.238050	+	0.971253i	$0.576508\pi$
$488$	0	0
$489$	4.91880	0.222436
$490$	0	0
$491$	31.5247	1.42269	0.711344	−	0.702844i	$-0.248087\pi$
$491$	31.5247	1.42269	0.711344	+	0.702844i	$0.248087\pi$
$492$	0	0
$493$	6.02488	0.271347
$494$	0	0
$495$	−0.138276	−0.00621504
$496$	0	0
$497$	5.21134	0.233761
$498$	0	0
$499$	−6.46966	−0.289622	−0.144811	−	0.989459i	$-0.546257\pi$
$499$	−6.46966	−0.289622	−0.144811	+	0.989459i	$0.546257\pi$
$500$	0	0
$501$	3.61590	0.161546
$502$	0	0
$503$	−39.1789	−1.74690	−0.873451	−	0.486912i	$-0.838123\pi$
$503$	−39.1789	−1.74690	−0.873451	+	0.486912i	$0.838123\pi$
$504$	0	0
$505$	0.219131	0.00975121
$506$	0	0
$507$	−26.8217	−1.19119
$508$	0	0
$509$	−42.2035	−1.87064	−0.935318	−	0.353808i	$-0.884887\pi$
$509$	−42.2035	−1.87064	−0.935318	+	0.353808i	$0.884887\pi$
$510$	0	0
$511$	−2.14778	−0.0950122
$512$	0	0
$513$	−4.11930	−0.181872
$514$	0	0
$515$	1.55269	0.0684197
$516$	0	0
$517$	19.4342	0.854714
$518$	0	0
$519$	−4.94379	−0.217008
$520$	0	0
$521$	34.0529	1.49188	0.745942	−	0.666011i	$-0.232000\pi$
$521$	34.0529	1.49188	0.745942	+	0.666011i	$0.232000\pi$
$522$	0	0
$523$	9.94115	0.434696	0.217348	−	0.976094i	$-0.430259\pi$
$523$	9.94115	0.434696	0.217348	+	0.976094i	$0.430259\pi$
$524$	0	0
$525$	12.5842	0.549220
$526$	0	0
$527$	37.8659	1.64946
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−9.16270	−0.397627
$532$	0	0
$533$	−24.5022	−1.06131
$534$	0	0
$535$	−0.632436	−0.0273426
$536$	0	0
$537$	−4.60179	−0.198582
$538$	0	0
$539$	1.00123	0.0431260
$540$	0	0
$541$	34.1745	1.46928	0.734638	−	0.678459i	$-0.237352\pi$
$541$	34.1745	1.46928	0.734638	+	0.678459i	$0.237352\pi$
$542$	0	0
$543$	−0.0656052	−0.00281539
$544$	0	0
$545$	0.224623	0.00962177
$546$	0	0
$547$	38.3745	1.64078	0.820388	−	0.571808i	$-0.193758\pi$
$547$	38.3745	1.64078	0.820388	+	0.571808i	$0.193758\pi$
$548$	0	0
$549$	10.3666	0.442438
$550$	0	0
$551$	−4.11930	−0.175488
$552$	0	0
$553$	−36.9847	−1.57275
$554$	0	0
$555$	0.279992	0.0118850
$556$	0	0
$557$	−30.0871	−1.27483	−0.637415	−	0.770521i	$-0.719996\pi$
$557$	−30.0871	−1.27483	−0.637415	+	0.770521i	$0.719996\pi$
$558$	0	0
$559$	12.0932	0.511487
$560$	0	0
$561$	−9.34754	−0.394654
$562$	0	0
$563$	−27.9924	−1.17974	−0.589870	−	0.807498i	$-0.700821\pi$
$563$	−27.9924	−1.17974	−0.589870	+	0.807498i	$0.700821\pi$
$564$	0	0
$565$	0.848176	0.0356830
$566$	0	0
$567$	2.52085	0.105866
$568$	0	0
$569$	−38.7969	−1.62645	−0.813225	−	0.581950i	$-0.802290\pi$
$569$	−38.7969	−1.62645	−0.813225	+	0.581950i	$0.802290\pi$
$570$	0	0
$571$	1.50405	0.0629424	0.0314712	−	0.999505i	$-0.489981\pi$
$571$	1.50405	0.0629424	0.0314712	+	0.999505i	$0.489981\pi$
$572$	0	0
$573$	1.40793	0.0588172
$574$	0	0
$575$	−4.99206	−0.208183
$576$	0	0
$577$	29.7062	1.23669	0.618343	−	0.785908i	$-0.287804\pi$
$577$	29.7062	1.23669	0.618343	+	0.785908i	$0.287804\pi$
$578$	0	0
$579$	−12.5383	−0.521076
$580$	0	0
$581$	4.73344	0.196376
$582$	0	0
$583$	5.43657	0.225160
$584$	0	0
$585$	0.562415	0.0232530
$586$	0	0
$587$	19.9946	0.825264	0.412632	−	0.910898i	$-0.364609\pi$
$587$	19.9946	0.825264	0.412632	+	0.910898i	$0.364609\pi$
$588$	0	0
$589$	−25.8895	−1.06676
$590$	0	0
$591$	−4.95060	−0.203640
$592$	0	0
$593$	23.6830	0.972543	0.486272	−	0.873808i	$-0.338357\pi$
$593$	23.6830	0.972543	0.486272	+	0.873808i	$0.338357\pi$
$594$	0	0
$595$	−1.35361	−0.0554924
$596$	0	0
$597$	5.40788	0.221330
$598$	0	0
$599$	−30.3527	−1.24018	−0.620088	−	0.784532i	$-0.712903\pi$
$599$	−30.3527	−1.24018	−0.620088	+	0.784532i	$0.712903\pi$
$600$	0	0
$601$	−38.9978	−1.59075	−0.795377	−	0.606114i	$-0.792727\pi$
$601$	−38.9978	−1.59075	−0.795377	+	0.606114i	$0.792727\pi$
$602$	0	0
$603$	−1.85364	−0.0754859
$604$	0	0
$605$	−0.765836	−0.0311357
$606$	0	0
$607$	−44.0987	−1.78991	−0.894956	−	0.446154i	$-0.852793\pi$
$607$	−44.0987	−1.78991	−0.894956	+	0.446154i	$0.852793\pi$
$608$	0	0
$609$	2.52085	0.102150
$610$	0	0
$611$	−79.0454	−3.19783
$612$	0	0
$613$	−29.1904	−1.17899	−0.589494	−	0.807773i	$-0.700673\pi$
$613$	−29.1904	−1.17899	−0.589494	+	0.807773i	$0.700673\pi$
$614$	0	0
$615$	0.346053	0.0139542
$616$	0	0
$617$	18.9272	0.761980	0.380990	−	0.924579i	$-0.375583\pi$
$617$	18.9272	0.761980	0.380990	+	0.924579i	$0.375583\pi$
$618$	0	0
$619$	−2.81697	−0.113223	−0.0566117	−	0.998396i	$-0.518030\pi$
$619$	−2.81697	−0.113223	−0.0566117	+	0.998396i	$0.518030\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−29.7976	−1.19382
$624$	0	0
$625$	24.8809	0.995237
$626$	0	0
$627$	6.39106	0.255234
$628$	0	0
$629$	18.9276	0.754694
$630$	0	0
$631$	13.5049	0.537622	0.268811	−	0.963193i	$-0.413369\pi$
$631$	13.5049	0.537622	0.268811	+	0.963193i	$0.413369\pi$
$632$	0	0
$633$	7.75222	0.308123
$634$	0	0
$635$	−0.607787	−0.0241193
$636$	0	0
$637$	−4.07234	−0.161352
$638$	0	0
$639$	2.06730	0.0817811
$640$	0	0
$641$	−33.2190	−1.31207	−0.656037	−	0.754729i	$-0.727768\pi$
$641$	−33.2190	−1.31207	−0.656037	+	0.754729i	$0.727768\pi$
$642$	0	0
$643$	46.8166	1.84627	0.923133	−	0.384481i	$-0.125619\pi$
$643$	46.8166	1.84627	0.923133	+	0.384481i	$0.125619\pi$
$644$	0	0
$645$	−0.170796	−0.00672510
$646$	0	0
$647$	−31.6817	−1.24554	−0.622769	−	0.782406i	$-0.713992\pi$
$647$	−31.6817	−1.24554	−0.622769	+	0.782406i	$0.713992\pi$
$648$	0	0
$649$	14.2158	0.558020
$650$	0	0
$651$	15.8433	0.620949
$652$	0	0
$653$	8.43461	0.330072	0.165036	−	0.986288i	$-0.447226\pi$
$653$	8.43461	0.330072	0.165036	+	0.986288i	$0.447226\pi$
$654$	0	0
$655$	0.787228	0.0307595
$656$	0	0
$657$	−0.852008	−0.0332400
$658$	0	0
$659$	23.9452	0.932771	0.466385	−	0.884582i	$-0.345556\pi$
$659$	23.9452	0.932771	0.466385	+	0.884582i	$0.345556\pi$
$660$	0	0
$661$	−16.1122	−0.626690	−0.313345	−	0.949639i	$-0.601450\pi$
$661$	−16.1122	−0.626690	−0.313345	+	0.949639i	$0.601450\pi$
$662$	0	0
$663$	38.0196	1.47656
$664$	0	0
$665$	0.925481	0.0358886
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	−19.2273	−0.743369
$670$	0	0
$671$	−16.0837	−0.620906
$672$	0	0
$673$	26.6979	1.02913	0.514564	−	0.857452i	$-0.327954\pi$
$673$	26.6979	1.02913	0.514564	+	0.857452i	$0.327954\pi$
$674$	0	0
$675$	4.99206	0.192144
$676$	0	0
$677$	32.2930	1.24112	0.620560	−	0.784159i	$-0.286906\pi$
$677$	32.2930	1.24112	0.620560	+	0.784159i	$0.286906\pi$
$678$	0	0
$679$	−12.7757	−0.490288
$680$	0	0
$681$	24.0362	0.921068
$682$	0	0
$683$	−22.2531	−0.851490	−0.425745	−	0.904843i	$-0.639988\pi$
$683$	−22.2531	−0.851490	−0.425745	+	0.904843i	$0.639988\pi$
$684$	0	0
$685$	−1.14598	−0.0437855
$686$	0	0
$687$	1.05473	0.0402404
$688$	0	0
$689$	−22.1124	−0.842414
$690$	0	0
$691$	11.3411	0.431435	0.215718	−	0.976456i	$-0.430791\pi$
$691$	11.3411	0.431435	0.215718	+	0.976456i	$0.430791\pi$
$692$	0	0
$693$	−3.91107	−0.148569
$694$	0	0
$695$	0.699704	0.0265413
$696$	0	0
$697$	23.3934	0.886089
$698$	0	0
$699$	5.04352	0.190763
$700$	0	0
$701$	−10.9018	−0.411754	−0.205877	−	0.978578i	$-0.566005\pi$
$701$	−10.9018	−0.411754	−0.205877	+	0.978578i	$0.566005\pi$
$702$	0	0
$703$	−12.9411	−0.488083
$704$	0	0
$705$	1.11639	0.0420455
$706$	0	0
$707$	6.19802	0.233101
$708$	0	0
$709$	11.6697	0.438266	0.219133	−	0.975695i	$-0.429677\pi$
$709$	11.6697	0.438266	0.219133	+	0.975695i	$0.429677\pi$
$710$	0	0
$711$	−14.6715	−0.550226
$712$	0	0
$713$	−6.28492	−0.235372
$714$	0	0
$715$	−0.872582	−0.0326327
$716$	0	0
$717$	−8.65130	−0.323089
$718$	0	0
$719$	−9.96159	−0.371505	−0.185752	−	0.982597i	$-0.559472\pi$
$719$	−9.96159	−0.371505	−0.185752	+	0.982597i	$0.559472\pi$
$720$	0	0
$721$	43.9171	1.63556
$722$	0	0
$723$	25.9981	0.966881
$724$	0	0
$725$	4.99206	0.185400
$726$	0	0
$727$	0.970079	0.0359782	0.0179891	−	0.999838i	$-0.494274\pi$
$727$	0.970079	0.0359782	0.0179891	+	0.999838i	$0.494274\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−11.5459	−0.427042
$732$	0	0
$733$	−35.5445	−1.31287	−0.656433	−	0.754385i	$-0.727935\pi$
$733$	−35.5445	−1.31287	−0.656433	+	0.754385i	$0.727935\pi$
$734$	0	0
$735$	0.0575151	0.00212148
$736$	0	0
$737$	2.87590	0.105935
$738$	0	0
$739$	−26.2939	−0.967236	−0.483618	−	0.875279i	$-0.660678\pi$
$739$	−26.2939	−0.967236	−0.483618	+	0.875279i	$0.660678\pi$
$740$	0	0
$741$	−25.9946	−0.954936
$742$	0	0
$743$	−10.2289	−0.375262	−0.187631	−	0.982240i	$-0.560081\pi$
$743$	−10.2289	−0.375262	−0.187631	+	0.982240i	$0.560081\pi$
$744$	0	0
$745$	−0.570024	−0.0208840
$746$	0	0
$747$	1.87772	0.0687021
$748$	0	0
$749$	−17.8882	−0.653619
$750$	0	0
$751$	−9.74607	−0.355639	−0.177820	−	0.984063i	$-0.556904\pi$
$751$	−9.74607	−0.355639	−0.177820	+	0.984063i	$0.556904\pi$
$752$	0	0
$753$	−5.75346	−0.209668
$754$	0	0
$755$	1.32349	0.0481668
$756$	0	0
$757$	−1.26270	−0.0458936	−0.0229468	−	0.999737i	$-0.507305\pi$
$757$	−1.26270	−0.0458936	−0.0229468	+	0.999737i	$0.507305\pi$
$758$	0	0
$759$	1.55149	0.0563155
$760$	0	0
$761$	−15.3340	−0.555858	−0.277929	−	0.960602i	$-0.589648\pi$
$761$	−15.3340	−0.555858	−0.277929	+	0.960602i	$0.589648\pi$
$762$	0	0
$763$	6.35334	0.230007
$764$	0	0
$765$	−0.536965	−0.0194140
$766$	0	0
$767$	−57.8207	−2.08778
$768$	0	0
$769$	49.0640	1.76929	0.884647	−	0.466261i	$-0.154399\pi$
$769$	49.0640	1.76929	0.884647	+	0.466261i	$0.154399\pi$
$770$	0	0
$771$	−4.56911	−0.164553
$772$	0	0
$773$	−35.8777	−1.29043	−0.645216	−	0.764000i	$-0.723233\pi$
$773$	−35.8777	−1.29043	−0.645216	+	0.764000i	$0.723233\pi$
$774$	0	0
$775$	31.3747	1.12701
$776$	0	0
$777$	7.91944	0.284108
$778$	0	0
$779$	−15.9944	−0.573060
$780$	0	0
$781$	−3.20739	−0.114770
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−1.59548	−0.0569453
$786$	0	0
$787$	30.6960	1.09419	0.547097	−	0.837069i	$-0.315733\pi$
$787$	30.6960	1.09419	0.547097	+	0.837069i	$0.315733\pi$
$788$	0	0
$789$	−12.3698	−0.440375
$790$	0	0
$791$	23.9903	0.852995
$792$	0	0
$793$	65.4181	2.32306
$794$	0	0
$795$	0.312301	0.0110762
$796$	0	0
$797$	−0.541641	−0.0191859	−0.00959296	−	0.999954i	$-0.503054\pi$
$797$	−0.541641	−0.0191859	−0.00959296	+	0.999954i	$0.503054\pi$
$798$	0	0
$799$	75.4684	2.66988
$800$	0	0
$801$	−11.8205	−0.417656
$802$	0	0
$803$	1.32188	0.0466482
$804$	0	0
$805$	0.224669	0.00791855
$806$	0	0
$807$	1.92180	0.0676505
$808$	0	0
$809$	−49.2737	−1.73237	−0.866186	−	0.499722i	$-0.833436\pi$
$809$	−49.2737	−1.73237	−0.866186	+	0.499722i	$0.833436\pi$
$810$	0	0
$811$	21.8789	0.768274	0.384137	−	0.923276i	$-0.374499\pi$
$811$	21.8789	0.768274	0.384137	+	0.923276i	$0.374499\pi$
$812$	0	0
$813$	9.32469	0.327031
$814$	0	0
$815$	−0.438386	−0.0153560
$816$	0	0
$817$	7.89414	0.276181
$818$	0	0
$819$	15.9077	0.555858
$820$	0	0
$821$	−40.0618	−1.39817	−0.699083	−	0.715040i	$-0.746408\pi$
$821$	−40.0618	−1.39817	−0.699083	+	0.715040i	$0.746408\pi$
$822$	0	0
$823$	34.0723	1.18769	0.593844	−	0.804580i	$-0.297610\pi$
$823$	34.0723	1.18769	0.593844	+	0.804580i	$0.297610\pi$
$824$	0	0
$825$	−7.74513	−0.269651
$826$	0	0
$827$	−4.55925	−0.158541	−0.0792704	−	0.996853i	$-0.525259\pi$
$827$	−4.55925	−0.158541	−0.0792704	+	0.996853i	$0.525259\pi$
$828$	0	0
$829$	−28.3451	−0.984465	−0.492233	−	0.870464i	$-0.663819\pi$
$829$	−28.3451	−0.984465	−0.492233	+	0.870464i	$0.663819\pi$
$830$	0	0
$831$	19.6085	0.680210
$832$	0	0
$833$	3.88806	0.134713
$834$	0	0
$835$	−0.322265	−0.0111525
$836$	0	0
$837$	6.28492	0.217239
$838$	0	0
$839$	−42.2492	−1.45860	−0.729302	−	0.684192i	$-0.760155\pi$
$839$	−42.2492	−1.45860	−0.729302	+	0.684192i	$0.760155\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−12.2808	−0.422972
$844$	0	0
$845$	2.39047	0.0822346
$846$	0	0
$847$	−21.6613	−0.744292
$848$	0	0
$849$	−12.0243	−0.412673
$850$	0	0
$851$	−3.14158	−0.107692
$852$	0	0
$853$	−29.4475	−1.00826	−0.504131	−	0.863627i	$-0.668187\pi$
$853$	−29.4475	−1.00826	−0.504131	+	0.863627i	$0.668187\pi$
$854$	0	0
$855$	0.367131	0.0125556
$856$	0	0
$857$	12.9477	0.442284	0.221142	−	0.975242i	$-0.429022\pi$
$857$	12.9477	0.442284	0.221142	+	0.975242i	$0.429022\pi$
$858$	0	0
$859$	9.37239	0.319782	0.159891	−	0.987135i	$-0.448886\pi$
$859$	9.37239	0.319782	0.159891	+	0.987135i	$0.448886\pi$
$860$	0	0
$861$	9.78795	0.333572
$862$	0	0
$863$	19.2209	0.654286	0.327143	−	0.944975i	$-0.393914\pi$
$863$	19.2209	0.654286	0.327143	+	0.944975i	$0.393914\pi$
$864$	0	0
$865$	0.440613	0.0149813
$866$	0	0
$867$	−19.2992	−0.655434
$868$	0	0
$869$	22.7628	0.772174
$870$	0	0
$871$	−11.6973	−0.396347
$872$	0	0
$873$	−5.06804	−0.171527
$874$	0	0
$875$	−2.24491	−0.0758917
$876$	0	0
$877$	−42.0093	−1.41855	−0.709277	−	0.704930i	$-0.750978\pi$
$877$	−42.0093	−1.41855	−0.709277	+	0.704930i	$0.750978\pi$
$878$	0	0
$879$	−15.8746	−0.535438
$880$	0	0
$881$	24.8647	0.837714	0.418857	−	0.908052i	$-0.362431\pi$
$881$	24.8647	0.837714	0.418857	+	0.908052i	$0.362431\pi$
$882$	0	0
$883$	−3.82093	−0.128585	−0.0642923	−	0.997931i	$-0.520479\pi$
$883$	−3.82093	−0.128585	−0.0642923	+	0.997931i	$0.520479\pi$
$884$	0	0
$885$	0.816621	0.0274504
$886$	0	0
$887$	28.7748	0.966163	0.483081	−	0.875576i	$-0.339518\pi$
$887$	28.7748	0.966163	0.483081	+	0.875576i	$0.339518\pi$
$888$	0	0
$889$	−17.1910	−0.576567
$890$	0	0
$891$	−1.55149	−0.0519769
$892$	0	0
$893$	−51.5989	−1.72669
$894$	0	0
$895$	0.410132	0.0137092
$896$	0	0
$897$	−6.31044	−0.210699
$898$	0	0
$899$	6.28492	0.209614
$900$	0	0
$901$	21.1117	0.703334
$902$	0	0
$903$	−4.83089	−0.160762
$904$	0	0
$905$	0.00584703	0.000194362 0
$906$	0	0
$907$	−54.9930	−1.82601	−0.913006	−	0.407947i	$-0.866245\pi$
$907$	−54.9930	−1.82601	−0.913006	+	0.407947i	$0.866245\pi$
$908$	0	0
$909$	2.45871	0.0815502
$910$	0	0
$911$	25.6668	0.850378	0.425189	−	0.905105i	$-0.360208\pi$
$911$	25.6668	0.850378	0.425189	+	0.905105i	$0.360208\pi$
$912$	0	0
$913$	−2.91326	−0.0964148
$914$	0	0
$915$	−0.923922	−0.0305439
$916$	0	0
$917$	22.2664	0.735300
$918$	0	0
$919$	29.0602	0.958607	0.479303	−	0.877649i	$-0.340889\pi$
$919$	29.0602	0.958607	0.479303	+	0.877649i	$0.340889\pi$
$920$	0	0
$921$	−6.32225	−0.208325
$922$	0	0
$923$	13.0456	0.429400
$924$	0	0
$925$	15.6829	0.515652
$926$	0	0
$927$	17.4216	0.572199
$928$	0	0
$929$	56.6942	1.86008	0.930039	−	0.367462i	$-0.119773\pi$
$929$	56.6942	1.86008	0.930039	+	0.367462i	$0.119773\pi$
$930$	0	0
$931$	−2.65832	−0.0871231
$932$	0	0
$933$	−0.581953	−0.0190523
$934$	0	0
$935$	0.833095	0.0272451
$936$	0	0
$937$	−24.8864	−0.813003	−0.406502	−	0.913650i	$-0.633251\pi$
$937$	−24.8864	−0.813003	−0.406502	+	0.913650i	$0.633251\pi$
$938$	0	0
$939$	32.7136	1.06757
$940$	0	0
$941$	−23.5747	−0.768513	−0.384257	−	0.923226i	$-0.625542\pi$
$941$	−23.5747	−0.768513	−0.384257	+	0.923226i	$0.625542\pi$
$942$	0	0
$943$	−3.88280	−0.126441
$944$	0	0
$945$	−0.224669	−0.00730849
$946$	0	0
$947$	−34.4993	−1.12108	−0.560539	−	0.828128i	$-0.689406\pi$
$947$	−34.4993	−1.12108	−0.560539	+	0.828128i	$0.689406\pi$
$948$	0	0
$949$	−5.37655	−0.174530
$950$	0	0
$951$	34.5562	1.12056
$952$	0	0
$953$	20.7822	0.673201	0.336601	−	0.941647i	$-0.390723\pi$
$953$	20.7822	0.673201	0.336601	+	0.941647i	$0.390723\pi$
$954$	0	0
$955$	−0.125481	−0.00406048
$956$	0	0
$957$	−1.55149	−0.0501526
$958$	0	0
$959$	−32.4134	−1.04668
$960$	0	0
$961$	8.50026	0.274202
$962$	0	0
$963$	−7.09609	−0.228668
$964$	0	0
$965$	1.11747	0.0359728
$966$	0	0
$967$	6.46309	0.207839	0.103920	−	0.994586i	$-0.466862\pi$
$967$	6.46309	0.207839	0.103920	+	0.994586i	$0.466862\pi$
$968$	0	0
$969$	24.8183	0.797279
$970$	0	0
$971$	35.0870	1.12600	0.562998	−	0.826459i	$-0.309648\pi$
$971$	35.0870	1.12600	0.562998	+	0.826459i	$0.309648\pi$
$972$	0	0
$973$	19.7908	0.634464
$974$	0	0
$975$	31.5021	1.00887
$976$	0	0
$977$	20.3200	0.650094	0.325047	−	0.945698i	$-0.394620\pi$
$977$	20.3200	0.650094	0.325047	+	0.945698i	$0.394620\pi$
$978$	0	0
$979$	18.3394	0.586128
$980$	0	0
$981$	2.52032	0.0804677
$982$	0	0
$983$	−12.7194	−0.405687	−0.202843	−	0.979211i	$-0.565018\pi$
$983$	−12.7194	−0.405687	−0.202843	+	0.979211i	$0.565018\pi$
$984$	0	0
$985$	0.441220	0.0140584
$986$	0	0
$987$	31.5765	1.00509
$988$	0	0
$989$	1.91638	0.0609373
$990$	0	0
$991$	−8.66921	−0.275387	−0.137693	−	0.990475i	$-0.543969\pi$
$991$	−8.66921	−0.275387	−0.137693	+	0.990475i	$0.543969\pi$
$992$	0	0
$993$	−27.7031	−0.879132
$994$	0	0
$995$	−0.481974	−0.0152796
$996$	0	0
$997$	−33.1550	−1.05003	−0.525014	−	0.851094i	$-0.675940\pi$
$997$	−33.1550	−1.05003	−0.525014	+	0.851094i	$0.675940\pi$
$998$	0	0
$999$	3.14158	0.0993952

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.g.1.7	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.g.1.7	✓	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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.0891245 q^{5} +2.52085 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.0891245 q^{5} +2.52085 q^{7} +1.00000 q^{9} -1.55149 q^{11} +6.31044 q^{13} -0.0891245 q^{15} -6.02488 q^{17} +4.11930 q^{19} -2.52085 q^{21} +1.00000 q^{23} -4.99206 q^{25} -1.00000 q^{27} -1.00000 q^{29} -6.28492 q^{31} +1.55149 q^{33} +0.224669 q^{35} -3.14158 q^{37} -6.31044 q^{39} -3.88280 q^{41} +1.91638 q^{43} +0.0891245 q^{45} -12.5261 q^{47} -0.645334 q^{49} +6.02488 q^{51} -3.50409 q^{53} -0.138276 q^{55} -4.11930 q^{57} -9.16270 q^{59} +10.3666 q^{61} +2.52085 q^{63} +0.562415 q^{65} -1.85364 q^{67} -1.00000 q^{69} +2.06730 q^{71} -0.852008 q^{73} +4.99206 q^{75} -3.91107 q^{77} -14.6715 q^{79} +1.00000 q^{81} +1.87772 q^{83} -0.536965 q^{85} +1.00000 q^{87} -11.8205 q^{89} +15.9077 q^{91} +6.28492 q^{93} +0.367131 q^{95} -5.06804 q^{97} -1.55149 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9	\( 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9} - 5 q^{11} - 6 q^{13} + 3 q^{15} - 7 q^{17} - 3 q^{19} - 4 q^{21} + 12 q^{23} + 11 q^{25} - 12 q^{27} - 12 q^{29} + 2 q^{31} + 5 q^{33} - 9 q^{35} - 20 q^{37} + 6 q^{39} - 3 q^{41} + 5 q^{43} - 3 q^{45} - 2 q^{49} + 7 q^{51} - 3 q^{53} + 19 q^{55} + 3 q^{57} - 20 q^{59} - 17 q^{61} + 4 q^{63} - 4 q^{65} - 9 q^{67} - 12 q^{69} + 7 q^{71} - 9 q^{73} - 11 q^{75} - 34 q^{77} + 14 q^{79} + 12 q^{81} + 5 q^{83} - 12 q^{85} + 12 q^{87} - 22 q^{89} - 3 q^{91} - 2 q^{93} - 27 q^{95} + 17 q^{97} - 5 q^{99}+O(q^{100}) \) 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9 - 5 * q^11 - 6 * q^13 + 3 * q^15 - 7 * q^17 - 3 * q^19 - 4 * q^21 + 12 * q^23 + 11 * q^25 - 12 * q^27 - 12 * q^29 + 2 * q^31 + 5 * q^33 - 9 * q^35 - 20 * q^37 + 6 * q^39 - 3 * q^41 + 5 * q^43 - 3 * q^45 - 2 * q^49 + 7 * q^51 - 3 * q^53 + 19 * q^55 + 3 * q^57 - 20 * q^59 - 17 * q^61 + 4 * q^63 - 4 * q^65 - 9 * q^67 - 12 * q^69 + 7 * q^71 - 9 * q^73 - 11 * q^75 - 34 * q^77 + 14 * q^79 + 12 * q^81 + 5 * q^83 - 12 * q^85 + 12 * q^87 - 22 * q^89 - 3 * q^91 - 2 * q^93 - 27 * q^95 + 17 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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