LMFDB - Embedded newform 8304.2.a.y.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8304,2,Mod(1,8304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8304, 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("8304.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8304 = 2^{4} \cdot 3 \cdot 173 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8304.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:	$66.3077738385$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.245992.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 6x^{3} - x^{2} + 7x + 1 $ x^5 - 6x^3 - x^2 + 7x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4152)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.142427$ of defining polynomial
Character	$\chi$	$=$	8304.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.433180 q^{5} -2.61702 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.433180 q^{5} -2.61702 q^{7} +1.00000 q^{9} +0.162713 q^{11} -3.32627 q^{13} +0.433180 q^{15} -3.61702 q^{17} +1.63316 q^{19} +2.61702 q^{21} +0.476051 q^{23} -4.81236 q^{25} -1.00000 q^{27} -5.22236 q^{29} +8.68461 q^{31} -0.162713 q^{33} +1.13364 q^{35} -1.37562 q^{37} +3.32627 q^{39} -7.14038 q^{41} +4.97093 q^{43} -0.433180 q^{45} -11.0998 q^{47} -0.151215 q^{49} +3.61702 q^{51} +4.21002 q^{53} -0.0704839 q^{55} -1.63316 q^{57} +7.10921 q^{59} -9.87745 q^{61} -2.61702 q^{63} +1.44087 q^{65} -14.4160 q^{67} -0.476051 q^{69} +5.73037 q^{71} -1.24633 q^{73} +4.81236 q^{75} -0.425823 q^{77} +14.7379 q^{79} +1.00000 q^{81} -14.0076 q^{83} +1.56682 q^{85} +5.22236 q^{87} +2.98470 q^{89} +8.70490 q^{91} -8.68461 q^{93} -0.707454 q^{95} +3.19638 q^{97} +0.162713 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} - 2 q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 - 2 * q^5 + 4 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} - 2 q^{5} + 4 q^{7} + 5 q^{9} + 12 q^{11} + q^{13} + 2 q^{15} - q^{17} + 2 q^{19} - 4 q^{21} - 4 q^{23} - 3 q^{25} - 5 q^{27} + 4 q^{29} + 24 q^{31} - 12 q^{33} + 6 q^{35} - 12 q^{37} - q^{39} + 2 q^{41} + 14 q^{43} - 2 q^{45} + 6 q^{47} + q^{49} + q^{51} + 18 q^{53} + 5 q^{55} - 2 q^{57} + 23 q^{59} - 18 q^{61} + 4 q^{63} - 18 q^{65} + 10 q^{67} + 4 q^{69} - 18 q^{73} + 3 q^{75} + 10 q^{77} + 40 q^{79} + 5 q^{81} + 8 q^{83} + 8 q^{85} - 4 q^{87} - 36 q^{89} + 36 q^{91} - 24 q^{93} - 30 q^{95} - 22 q^{97} + 12 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 - 2 * q^5 + 4 * q^7 + 5 * q^9 + 12 * q^11 + q^13 + 2 * q^15 - q^17 + 2 * q^19 - 4 * q^21 - 4 * q^23 - 3 * q^25 - 5 * q^27 + 4 * q^29 + 24 * q^31 - 12 * q^33 + 6 * q^35 - 12 * q^37 - q^39 + 2 * q^41 + 14 * q^43 - 2 * q^45 + 6 * q^47 + q^49 + q^51 + 18 * q^53 + 5 * q^55 - 2 * q^57 + 23 * q^59 - 18 * q^61 + 4 * q^63 - 18 * q^65 + 10 * q^67 + 4 * q^69 - 18 * q^73 + 3 * q^75 + 10 * q^77 + 40 * q^79 + 5 * q^81 + 8 * q^83 + 8 * q^85 - 4 * q^87 - 36 * q^89 + 36 * q^91 - 24 * q^93 - 30 * q^95 - 22 * q^97 + 12 * 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
$5$	−0.433180	−0.193724	−0.0968620	−	0.995298i	$-0.530881\pi$
$5$	−0.433180	−0.193724	−0.0968620	+	0.995298i	$0.530881\pi$
$6$	0	0
$7$	−2.61702	−0.989140	−0.494570	−	0.869138i	$-0.664674\pi$
$7$	−2.61702	−0.989140	−0.494570	+	0.869138i	$0.664674\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.162713	0.0490598	0.0245299	−	0.999699i	$-0.492191\pi$
$11$	0.162713	0.0490598	0.0245299	+	0.999699i	$0.492191\pi$
$12$	0	0
$13$	−3.32627	−0.922540	−0.461270	−	0.887260i	$-0.652606\pi$
$13$	−3.32627	−0.922540	−0.461270	+	0.887260i	$0.652606\pi$
$14$	0	0
$15$	0.433180	0.111847
$16$	0	0
$17$	−3.61702	−0.877256	−0.438628	−	0.898669i	$-0.644535\pi$
$17$	−3.61702	−0.877256	−0.438628	+	0.898669i	$0.644535\pi$
$18$	0	0
$19$	1.63316	0.374673	0.187337	−	0.982296i	$-0.440014\pi$
$19$	1.63316	0.374673	0.187337	+	0.982296i	$0.440014\pi$
$20$	0	0
$21$	2.61702	0.571080
$22$	0	0
$23$	0.476051	0.0992634	0.0496317	−	0.998768i	$-0.484195\pi$
$23$	0.476051	0.0992634	0.0496317	+	0.998768i	$0.484195\pi$
$24$	0	0
$25$	−4.81236	−0.962471
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.22236	−0.969768	−0.484884	−	0.874578i	$-0.661138\pi$
$29$	−5.22236	−0.969768	−0.484884	+	0.874578i	$0.661138\pi$
$30$	0	0
$31$	8.68461	1.55980	0.779901	−	0.625903i	$-0.215269\pi$
$31$	8.68461	1.55980	0.779901	+	0.625903i	$0.215269\pi$
$32$	0	0
$33$	−0.162713	−0.0283247
$34$	0	0
$35$	1.13364	0.191620
$36$	0	0
$37$	−1.37562	−0.226151	−0.113076	−	0.993586i	$-0.536070\pi$
$37$	−1.37562	−0.226151	−0.113076	+	0.993586i	$0.536070\pi$
$38$	0	0
$39$	3.32627	0.532629
$40$	0	0
$41$	−7.14038	−1.11514	−0.557570	−	0.830130i	$-0.688266\pi$
$41$	−7.14038	−1.11514	−0.557570	+	0.830130i	$0.688266\pi$
$42$	0	0
$43$	4.97093	0.758059	0.379030	−	0.925385i	$-0.376258\pi$
$43$	4.97093	0.758059	0.379030	+	0.925385i	$0.376258\pi$
$44$	0	0
$45$	−0.433180	−0.0645747
$46$	0	0
$47$	−11.0998	−1.61907	−0.809537	−	0.587069i	$-0.800282\pi$
$47$	−11.0998	−1.61907	−0.809537	+	0.587069i	$0.800282\pi$
$48$	0	0
$49$	−0.151215	−0.0216021
$50$	0	0
$51$	3.61702	0.506484
$52$	0	0
$53$	4.21002	0.578291	0.289146	−	0.957285i	$-0.406629\pi$
$53$	4.21002	0.578291	0.289146	+	0.957285i	$0.406629\pi$
$54$	0	0
$55$	−0.0704839	−0.00950405
$56$	0	0
$57$	−1.63316	−0.216318
$58$	0	0
$59$	7.10921	0.925541	0.462770	−	0.886478i	$-0.346855\pi$
$59$	7.10921	0.925541	0.462770	+	0.886478i	$0.346855\pi$
$60$	0	0
$61$	−9.87745	−1.26468	−0.632339	−	0.774692i	$-0.717905\pi$
$61$	−9.87745	−1.26468	−0.632339	+	0.774692i	$0.717905\pi$
$62$	0	0
$63$	−2.61702	−0.329713
$64$	0	0
$65$	1.44087	0.178718
$66$	0	0
$67$	−14.4160	−1.76120	−0.880599	−	0.473861i	$-0.842860\pi$
$67$	−14.4160	−1.76120	−0.880599	+	0.473861i	$0.842860\pi$
$68$	0	0
$69$	−0.476051	−0.0573097
$70$	0	0
$71$	5.73037	0.680070	0.340035	−	0.940413i	$-0.389561\pi$
$71$	5.73037	0.680070	0.340035	+	0.940413i	$0.389561\pi$
$72$	0	0
$73$	−1.24633	−0.145872	−0.0729361	−	0.997337i	$-0.523237\pi$
$73$	−1.24633	−0.145872	−0.0729361	+	0.997337i	$0.523237\pi$
$74$	0	0
$75$	4.81236	0.555683
$76$	0	0
$77$	−0.425823	−0.0485270
$78$	0	0
$79$	14.7379	1.65814	0.829072	−	0.559142i	$-0.188869\pi$
$79$	14.7379	1.65814	0.829072	+	0.559142i	$0.188869\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.0076	−1.53753	−0.768766	−	0.639530i	$-0.779129\pi$
$83$	−14.0076	−1.53753	−0.768766	+	0.639530i	$0.779129\pi$
$84$	0	0
$85$	1.56682	0.169945
$86$	0	0
$87$	5.22236	0.559896
$88$	0	0
$89$	2.98470	0.316377	0.158189	−	0.987409i	$-0.449435\pi$
$89$	2.98470	0.316377	0.158189	+	0.987409i	$0.449435\pi$
$90$	0	0
$91$	8.70490	0.912521
$92$	0	0
$93$	−8.68461	−0.900552
$94$	0	0
$95$	−0.707454	−0.0725832
$96$	0	0
$97$	3.19638	0.324544	0.162272	−	0.986746i	$-0.448118\pi$
$97$	3.19638	0.324544	0.162272	+	0.986746i	$0.448118\pi$
$98$	0	0
$99$	0.162713	0.0163533
$100$	0	0
$101$	7.87050	0.783144	0.391572	−	0.920147i	$-0.371931\pi$
$101$	7.87050	0.783144	0.391572	+	0.920147i	$0.371931\pi$
$102$	0	0
$103$	−13.7871	−1.35849	−0.679243	−	0.733913i	$-0.737692\pi$
$103$	−13.7871	−1.35849	−0.679243	+	0.733913i	$0.737692\pi$
$104$	0	0
$105$	−1.13364	−0.110632
$106$	0	0
$107$	−3.21166	−0.310483	−0.155241	−	0.987877i	$-0.549616\pi$
$107$	−3.21166	−0.310483	−0.155241	+	0.987877i	$0.549616\pi$
$108$	0	0
$109$	−5.39925	−0.517155	−0.258577	−	0.965991i	$-0.583254\pi$
$109$	−5.39925	−0.517155	−0.258577	+	0.965991i	$0.583254\pi$
$110$	0	0
$111$	1.37562	0.130568
$112$	0	0
$113$	3.34915	0.315061	0.157531	−	0.987514i	$-0.449647\pi$
$113$	3.34915	0.315061	0.157531	+	0.987514i	$0.449647\pi$
$114$	0	0
$115$	−0.206216	−0.0192297
$116$	0	0
$117$	−3.32627	−0.307513
$118$	0	0
$119$	9.46580	0.867729
$120$	0	0
$121$	−10.9735	−0.997593
$122$	0	0
$123$	7.14038	0.643826
$124$	0	0
$125$	4.25052	0.380178
$126$	0	0
$127$	19.0320	1.68882	0.844409	−	0.535700i	$-0.179952\pi$
$127$	19.0320	1.68882	0.844409	+	0.535700i	$0.179952\pi$
$128$	0	0
$129$	−4.97093	−0.437666
$130$	0	0
$131$	9.15226	0.799637	0.399818	−	0.916594i	$-0.369073\pi$
$131$	9.15226	0.799637	0.399818	+	0.916594i	$0.369073\pi$
$132$	0	0
$133$	−4.27402	−0.370604
$134$	0	0
$135$	0.433180	0.0372822
$136$	0	0
$137$	6.90861	0.590242	0.295121	−	0.955460i	$-0.404640\pi$
$137$	6.90861	0.590242	0.295121	+	0.955460i	$0.404640\pi$
$138$	0	0
$139$	11.8584	1.00582	0.502909	−	0.864340i	$-0.332263\pi$
$139$	11.8584	1.00582	0.502909	+	0.864340i	$0.332263\pi$
$140$	0	0
$141$	11.0998	0.934772
$142$	0	0
$143$	−0.541226	−0.0452596
$144$	0	0
$145$	2.26222	0.187867
$146$	0	0
$147$	0.151215	0.0124720
$148$	0	0
$149$	5.95353	0.487732	0.243866	−	0.969809i	$-0.421584\pi$
$149$	5.95353	0.487732	0.243866	+	0.969809i	$0.421584\pi$
$150$	0	0
$151$	−19.8358	−1.61421	−0.807105	−	0.590408i	$-0.798967\pi$
$151$	−19.8358	−1.61421	−0.807105	+	0.590408i	$0.798967\pi$
$152$	0	0
$153$	−3.61702	−0.292419
$154$	0	0
$155$	−3.76200	−0.302171
$156$	0	0
$157$	10.2449	0.817633	0.408817	−	0.912617i	$-0.365942\pi$
$157$	10.2449	0.817633	0.408817	+	0.912617i	$0.365942\pi$
$158$	0	0
$159$	−4.21002	−0.333876
$160$	0	0
$161$	−1.24583	−0.0981854
$162$	0	0
$163$	−14.6658	−1.14871	−0.574355	−	0.818606i	$-0.694747\pi$
$163$	−14.6658	−1.14871	−0.574355	+	0.818606i	$0.694747\pi$
$164$	0	0
$165$	0.0704839	0.00548717
$166$	0	0
$167$	−10.7676	−0.833225	−0.416613	−	0.909084i	$-0.636783\pi$
$167$	−10.7676	−0.833225	−0.416613	+	0.909084i	$0.636783\pi$
$168$	0	0
$169$	−1.93596	−0.148920
$170$	0	0
$171$	1.63316	0.124891
$172$	0	0
$173$	1.00000	0.0760286
$173$	1.00000	0.0760286
$174$	0	0
$175$	12.5940	0.952019
$176$	0	0
$177$	−7.10921	−0.534361
$178$	0	0
$179$	14.6356	1.09391	0.546956	−	0.837161i	$-0.315786\pi$
$179$	14.6356	1.09391	0.546956	+	0.837161i	$0.315786\pi$
$180$	0	0
$181$	12.0837	0.898172	0.449086	−	0.893489i	$-0.351750\pi$
$181$	12.0837	0.898172	0.449086	+	0.893489i	$0.351750\pi$
$182$	0	0
$183$	9.87745	0.730162
$184$	0	0
$185$	0.595893	0.0438109
$186$	0	0
$187$	−0.588535	−0.0430380
$188$	0	0
$189$	2.61702	0.190360
$190$	0	0
$191$	0.612008	0.0442833	0.0221417	−	0.999755i	$-0.492952\pi$
$191$	0.612008	0.0442833	0.0221417	+	0.999755i	$0.492952\pi$
$192$	0	0
$193$	−14.8275	−1.06730	−0.533652	−	0.845704i	$-0.679181\pi$
$193$	−14.8275	−1.06730	−0.533652	+	0.845704i	$0.679181\pi$
$194$	0	0
$195$	−1.44087	−0.103183
$196$	0	0
$197$	12.4623	0.887902	0.443951	−	0.896051i	$-0.353576\pi$
$197$	12.4623	0.887902	0.443951	+	0.896051i	$0.353576\pi$
$198$	0	0
$199$	16.9760	1.20340	0.601698	−	0.798723i	$-0.294491\pi$
$199$	16.9760	1.20340	0.601698	+	0.798723i	$0.294491\pi$
$200$	0	0
$201$	14.4160	1.01683
$202$	0	0
$203$	13.6670	0.959236
$204$	0	0
$205$	3.09307	0.216029
$206$	0	0
$207$	0.476051	0.0330878
$208$	0	0
$209$	0.265737	0.0183814
$210$	0	0
$211$	−2.71803	−0.187117	−0.0935586	−	0.995614i	$-0.529824\pi$
$211$	−2.71803	−0.187117	−0.0935586	+	0.995614i	$0.529824\pi$
$212$	0	0
$213$	−5.73037	−0.392639
$214$	0	0
$215$	−2.15331	−0.146854
$216$	0	0
$217$	−22.7278	−1.54286
$218$	0	0
$219$	1.24633	0.0842194
$220$	0	0
$221$	12.0312	0.809304
$222$	0	0
$223$	1.50132	0.100536	0.0502678	−	0.998736i	$-0.483993\pi$
$223$	1.50132	0.100536	0.0502678	+	0.998736i	$0.483993\pi$
$224$	0	0
$225$	−4.81236	−0.320824
$226$	0	0
$227$	12.3024	0.816541	0.408270	−	0.912861i	$-0.366132\pi$
$227$	12.3024	0.816541	0.408270	+	0.912861i	$0.366132\pi$
$228$	0	0
$229$	−0.510224	−0.0337165	−0.0168583	−	0.999858i	$-0.505366\pi$
$229$	−0.510224	−0.0337165	−0.0168583	+	0.999858i	$0.505366\pi$
$230$	0	0
$231$	0.425823	0.0280171
$232$	0	0
$233$	−17.2638	−1.13099	−0.565494	−	0.824753i	$-0.691314\pi$
$233$	−17.2638	−1.13099	−0.565494	+	0.824753i	$0.691314\pi$
$234$	0	0
$235$	4.80821	0.313653
$236$	0	0
$237$	−14.7379	−0.957330
$238$	0	0
$239$	−18.5728	−1.20138	−0.600689	−	0.799483i	$-0.705107\pi$
$239$	−18.5728	−1.20138	−0.600689	+	0.799483i	$0.705107\pi$
$240$	0	0
$241$	7.79701	0.502249	0.251125	−	0.967955i	$-0.419200\pi$
$241$	7.79701	0.502249	0.251125	+	0.967955i	$0.419200\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0.0655031	0.00418484
$246$	0	0
$247$	−5.43233	−0.345651
$248$	0	0
$249$	14.0076	0.887694
$250$	0	0
$251$	−10.0511	−0.634418	−0.317209	−	0.948356i	$-0.602746\pi$
$251$	−10.0511	−0.634418	−0.317209	+	0.948356i	$0.602746\pi$
$252$	0	0
$253$	0.0774595	0.00486984
$254$	0	0
$255$	−1.56682	−0.0981181
$256$	0	0
$257$	0.675528	0.0421383	0.0210691	−	0.999778i	$-0.493293\pi$
$257$	0.675528	0.0421383	0.0210691	+	0.999778i	$0.493293\pi$
$258$	0	0
$259$	3.60003	0.223695
$260$	0	0
$261$	−5.22236	−0.323256
$262$	0	0
$263$	16.5739	1.02199	0.510995	−	0.859584i	$-0.329277\pi$
$263$	16.5739	1.02199	0.510995	+	0.859584i	$0.329277\pi$
$264$	0	0
$265$	−1.82370	−0.112029
$266$	0	0
$267$	−2.98470	−0.182660
$268$	0	0
$269$	21.7862	1.32833	0.664163	−	0.747588i	$-0.268788\pi$
$269$	21.7862	1.32833	0.664163	+	0.747588i	$0.268788\pi$
$270$	0	0
$271$	27.1533	1.64944	0.824721	−	0.565539i	$-0.191332\pi$
$271$	27.1533	1.64944	0.824721	+	0.565539i	$0.191332\pi$
$272$	0	0
$273$	−8.70490	−0.526844
$274$	0	0
$275$	−0.783032	−0.0472186
$276$	0	0
$277$	−3.90974	−0.234913	−0.117457	−	0.993078i	$-0.537474\pi$
$277$	−3.90974	−0.234913	−0.117457	+	0.993078i	$0.537474\pi$
$278$	0	0
$279$	8.68461	0.519934
$280$	0	0
$281$	−26.7943	−1.59842	−0.799208	−	0.601054i	$-0.794748\pi$
$281$	−26.7943	−1.59842	−0.799208	+	0.601054i	$0.794748\pi$
$282$	0	0
$283$	30.6991	1.82487	0.912437	−	0.409217i	$-0.134198\pi$
$283$	30.6991	1.82487	0.912437	+	0.409217i	$0.134198\pi$
$284$	0	0
$285$	0.707454	0.0419059
$286$	0	0
$287$	18.6865	1.10303
$288$	0	0
$289$	−3.91718	−0.230422
$290$	0	0
$291$	−3.19638	−0.187375
$292$	0	0
$293$	−30.5778	−1.78637	−0.893187	−	0.449686i	$-0.851536\pi$
$293$	−30.5778	−1.78637	−0.893187	+	0.449686i	$0.851536\pi$
$294$	0	0
$295$	−3.07957	−0.179299
$296$	0	0
$297$	−0.162713	−0.00944156
$298$	0	0
$299$	−1.58347	−0.0915745
$300$	0	0
$301$	−13.0090	−0.749827
$302$	0	0
$303$	−7.87050	−0.452148
$304$	0	0
$305$	4.27871	0.244998
$306$	0	0
$307$	27.0111	1.54160	0.770802	−	0.637075i	$-0.219856\pi$
$307$	27.0111	1.54160	0.770802	+	0.637075i	$0.219856\pi$
$308$	0	0
$309$	13.7871	0.784322
$310$	0	0
$311$	24.5397	1.39152	0.695759	−	0.718275i	$-0.255068\pi$
$311$	24.5397	1.39152	0.695759	+	0.718275i	$0.255068\pi$
$312$	0	0
$313$	32.5389	1.83921	0.919604	−	0.392846i	$-0.128509\pi$
$313$	32.5389	1.83921	0.919604	+	0.392846i	$0.128509\pi$
$314$	0	0
$315$	1.13364	0.0638734
$316$	0	0
$317$	31.9465	1.79429	0.897147	−	0.441731i	$-0.145635\pi$
$317$	31.9465	1.79429	0.897147	+	0.441731i	$0.145635\pi$
$318$	0	0
$319$	−0.849745	−0.0475766
$320$	0	0
$321$	3.21166	0.179257
$322$	0	0
$323$	−5.90718	−0.328684
$324$	0	0
$325$	16.0072	0.887918
$326$	0	0
$327$	5.39925	0.298579
$328$	0	0
$329$	29.0484	1.60149
$330$	0	0
$331$	−14.0862	−0.774246	−0.387123	−	0.922028i	$-0.626531\pi$
$331$	−14.0862	−0.774246	−0.387123	+	0.922028i	$0.626531\pi$
$332$	0	0
$333$	−1.37562	−0.0753837
$334$	0	0
$335$	6.24474	0.341186
$336$	0	0
$337$	−13.2237	−0.720342	−0.360171	−	0.932886i	$-0.617282\pi$
$337$	−13.2237	−0.720342	−0.360171	+	0.932886i	$0.617282\pi$
$338$	0	0
$339$	−3.34915	−0.181901
$340$	0	0
$341$	1.41310	0.0765236
$342$	0	0
$343$	18.7149	1.01051
$344$	0	0
$345$	0.206216	0.0111023
$346$	0	0
$347$	−28.2637	−1.51727	−0.758636	−	0.651514i	$-0.774134\pi$
$347$	−28.2637	−1.51727	−0.758636	+	0.651514i	$0.774134\pi$
$348$	0	0
$349$	24.5308	1.31310	0.656552	−	0.754281i	$-0.272014\pi$
$349$	24.5308	1.31310	0.656552	+	0.754281i	$0.272014\pi$
$350$	0	0
$351$	3.32627	0.177543
$352$	0	0
$353$	−27.2040	−1.44792	−0.723961	−	0.689841i	$-0.757680\pi$
$353$	−27.2040	−1.44792	−0.723961	+	0.689841i	$0.757680\pi$
$354$	0	0
$355$	−2.48228	−0.131746
$356$	0	0
$357$	−9.46580	−0.500983
$358$	0	0
$359$	−22.3748	−1.18090	−0.590448	−	0.807076i	$-0.701049\pi$
$359$	−22.3748	−1.18090	−0.590448	+	0.807076i	$0.701049\pi$
$360$	0	0
$361$	−16.3328	−0.859620
$362$	0	0
$363$	10.9735	0.575961
$364$	0	0
$365$	0.539886	0.0282589
$366$	0	0
$367$	6.46325	0.337379	0.168689	−	0.985669i	$-0.446047\pi$
$367$	6.46325	0.337379	0.168689	+	0.985669i	$0.446047\pi$
$368$	0	0
$369$	−7.14038	−0.371713
$370$	0	0
$371$	−11.0177	−0.572011
$372$	0	0
$373$	23.5508	1.21941	0.609706	−	0.792627i	$-0.291287\pi$
$373$	23.5508	1.21941	0.609706	+	0.792627i	$0.291287\pi$
$374$	0	0
$375$	−4.25052	−0.219496
$376$	0	0
$377$	17.3710	0.894650
$378$	0	0
$379$	11.9868	0.615719	0.307860	−	0.951432i	$-0.400387\pi$
$379$	11.9868	0.615719	0.307860	+	0.951432i	$0.400387\pi$
$380$	0	0
$381$	−19.0320	−0.975039
$382$	0	0
$383$	−4.97314	−0.254116	−0.127058	−	0.991895i	$-0.540553\pi$
$383$	−4.97314	−0.254116	−0.127058	+	0.991895i	$0.540553\pi$
$384$	0	0
$385$	0.184458	0.00940084
$386$	0	0
$387$	4.97093	0.252686
$388$	0	0
$389$	−17.3537	−0.879870	−0.439935	−	0.898030i	$-0.644999\pi$
$389$	−17.3537	−0.879870	−0.439935	+	0.898030i	$0.644999\pi$
$390$	0	0
$391$	−1.72188	−0.0870794
$392$	0	0
$393$	−9.15226	−0.461671
$394$	0	0
$395$	−6.38417	−0.321222
$396$	0	0
$397$	24.8380	1.24658	0.623292	−	0.781989i	$-0.285795\pi$
$397$	24.8380	1.24658	0.623292	+	0.781989i	$0.285795\pi$
$398$	0	0
$399$	4.27402	0.213969
$400$	0	0
$401$	2.23499	0.111610	0.0558050	−	0.998442i	$-0.482227\pi$
$401$	2.23499	0.111610	0.0558050	+	0.998442i	$0.482227\pi$
$402$	0	0
$403$	−28.8873	−1.43898
$404$	0	0
$405$	−0.433180	−0.0215249
$406$	0	0
$407$	−0.223832	−0.0110949
$408$	0	0
$409$	13.1303	0.649250	0.324625	−	0.945843i	$-0.394762\pi$
$409$	13.1303	0.649250	0.324625	+	0.945843i	$0.394762\pi$
$410$	0	0
$411$	−6.90861	−0.340777
$412$	0	0
$413$	−18.6049	−0.915489
$414$	0	0
$415$	6.06780	0.297857
$416$	0	0
$417$	−11.8584	−0.580709
$418$	0	0
$419$	0.449295	0.0219495	0.0109748	−	0.999940i	$-0.496507\pi$
$419$	0.449295	0.0219495	0.0109748	+	0.999940i	$0.496507\pi$
$420$	0	0
$421$	31.3361	1.52723	0.763614	−	0.645673i	$-0.223423\pi$
$421$	31.3361	1.52723	0.763614	+	0.645673i	$0.223423\pi$
$422$	0	0
$423$	−11.0998	−0.539691
$424$	0	0
$425$	17.4064	0.844333
$426$	0	0
$427$	25.8495	1.25094
$428$	0	0
$429$	0.541226	0.0261306
$430$	0	0
$431$	6.91408	0.333040	0.166520	−	0.986038i	$-0.446747\pi$
$431$	6.91408	0.333040	0.166520	+	0.986038i	$0.446747\pi$
$432$	0	0
$433$	−15.3997	−0.740064	−0.370032	−	0.929019i	$-0.620653\pi$
$433$	−15.3997	−0.740064	−0.370032	+	0.929019i	$0.620653\pi$
$434$	0	0
$435$	−2.26222	−0.108465
$436$	0	0
$437$	0.777468	0.0371913
$438$	0	0
$439$	28.1147	1.34184	0.670920	−	0.741530i	$-0.265900\pi$
$439$	28.1147	1.34184	0.670920	+	0.741530i	$0.265900\pi$
$440$	0	0
$441$	−0.151215	−0.00720069
$442$	0	0
$443$	−14.3981	−0.684074	−0.342037	−	0.939686i	$-0.611117\pi$
$443$	−14.3981	−0.684074	−0.342037	+	0.939686i	$0.611117\pi$
$444$	0	0
$445$	−1.29291	−0.0612898
$446$	0	0
$447$	−5.95353	−0.281592
$448$	0	0
$449$	24.9913	1.17941	0.589706	−	0.807618i	$-0.299244\pi$
$449$	24.9913	1.17941	0.589706	+	0.807618i	$0.299244\pi$
$450$	0	0
$451$	−1.16183	−0.0547085
$452$	0	0
$453$	19.8358	0.931965
$454$	0	0
$455$	−3.77079	−0.176777
$456$	0	0
$457$	26.2162	1.22634	0.613172	−	0.789950i	$-0.289893\pi$
$457$	26.2162	1.22634	0.613172	+	0.789950i	$0.289893\pi$
$458$	0	0
$459$	3.61702	0.168828
$460$	0	0
$461$	−11.1136	−0.517611	−0.258805	−	0.965929i	$-0.583329\pi$
$461$	−11.1136	−0.517611	−0.258805	+	0.965929i	$0.583329\pi$
$462$	0	0
$463$	34.8002	1.61730	0.808651	−	0.588289i	$-0.200198\pi$
$463$	34.8002	1.61730	0.808651	+	0.588289i	$0.200198\pi$
$464$	0	0
$465$	3.76200	0.174459
$466$	0	0
$467$	36.5078	1.68938	0.844690	−	0.535255i	$-0.179785\pi$
$467$	36.5078	1.68938	0.844690	+	0.535255i	$0.179785\pi$
$468$	0	0
$469$	37.7270	1.74207
$470$	0	0
$471$	−10.2449	−0.472061
$472$	0	0
$473$	0.808834	0.0371902
$474$	0	0
$475$	−7.85936	−0.360612
$476$	0	0
$477$	4.21002	0.192764
$478$	0	0
$479$	−4.59229	−0.209827	−0.104914	−	0.994481i	$-0.533457\pi$
$479$	−4.59229	−0.209827	−0.104914	+	0.994481i	$0.533457\pi$
$480$	0	0
$481$	4.57569	0.208634
$482$	0	0
$483$	1.24583	0.0566874
$484$	0	0
$485$	−1.38461	−0.0628719
$486$	0	0
$487$	−22.7338	−1.03016	−0.515082	−	0.857141i	$-0.672239\pi$
$487$	−22.7338	−1.03016	−0.515082	+	0.857141i	$0.672239\pi$
$488$	0	0
$489$	14.6658	0.663208
$490$	0	0
$491$	−0.160122	−0.00722618	−0.00361309	−	0.999993i	$-0.501150\pi$
$491$	−0.160122	−0.00722618	−0.00361309	+	0.999993i	$0.501150\pi$
$492$	0	0
$493$	18.8894	0.850735
$494$	0	0
$495$	−0.0704839	−0.00316802
$496$	0	0
$497$	−14.9965	−0.672685
$498$	0	0
$499$	4.52788	0.202696	0.101348	−	0.994851i	$-0.467684\pi$
$499$	4.52788	0.202696	0.101348	+	0.994851i	$0.467684\pi$
$500$	0	0
$501$	10.7676	0.481063
$502$	0	0
$503$	−27.1855	−1.21214	−0.606072	−	0.795410i	$-0.707256\pi$
$503$	−27.1855	−1.21214	−0.606072	+	0.795410i	$0.707256\pi$
$504$	0	0
$505$	−3.40934	−0.151714
$506$	0	0
$507$	1.93596	0.0859788
$508$	0	0
$509$	−31.9041	−1.41412	−0.707062	−	0.707151i	$-0.749980\pi$
$509$	−31.9041	−1.41412	−0.707062	+	0.707151i	$0.749980\pi$
$510$	0	0
$511$	3.26168	0.144288
$512$	0	0
$513$	−1.63316	−0.0721059
$514$	0	0
$515$	5.97231	0.263171
$516$	0	0
$517$	−1.80608	−0.0794314
$518$	0	0
$519$	−1.00000	−0.0438951
$520$	0	0
$521$	36.5098	1.59952	0.799761	−	0.600319i	$-0.204960\pi$
$521$	36.5098	1.59952	0.799761	+	0.600319i	$0.204960\pi$
$522$	0	0
$523$	−8.38019	−0.366440	−0.183220	−	0.983072i	$-0.558652\pi$
$523$	−8.38019	−0.366440	−0.183220	+	0.983072i	$0.558652\pi$
$524$	0	0
$525$	−12.5940	−0.549648
$526$	0	0
$527$	−31.4124	−1.36835
$528$	0	0
$529$	−22.7734	−0.990147
$530$	0	0
$531$	7.10921	0.308514
$532$	0	0
$533$	23.7508	1.02876
$534$	0	0
$535$	1.39123	0.0601480
$536$	0	0
$537$	−14.6356	−0.631571
$538$	0	0
$539$	−0.0246046	−0.00105979
$540$	0	0
$541$	14.1349	0.607705	0.303853	−	0.952719i	$-0.401727\pi$
$541$	14.1349	0.607705	0.303853	+	0.952719i	$0.401727\pi$
$542$	0	0
$543$	−12.0837	−0.518560
$544$	0	0
$545$	2.33885	0.100185
$546$	0	0
$547$	30.6292	1.30961	0.654805	−	0.755798i	$-0.272751\pi$
$547$	30.6292	1.30961	0.654805	+	0.755798i	$0.272751\pi$
$548$	0	0
$549$	−9.87745	−0.421559
$550$	0	0
$551$	−8.52897	−0.363346
$552$	0	0
$553$	−38.5694	−1.64014
$554$	0	0
$555$	−0.595893	−0.0252942
$556$	0	0
$557$	10.0698	0.426673	0.213337	−	0.976979i	$-0.431567\pi$
$557$	10.0698	0.426673	0.213337	+	0.976979i	$0.431567\pi$
$558$	0	0
$559$	−16.5346	−0.699340
$560$	0	0
$561$	0.588535	0.0248480
$562$	0	0
$563$	−15.6261	−0.658563	−0.329282	−	0.944232i	$-0.606807\pi$
$563$	−15.6261	−0.658563	−0.329282	+	0.944232i	$0.606807\pi$
$564$	0	0
$565$	−1.45078	−0.0610349
$566$	0	0
$567$	−2.61702	−0.109904
$568$	0	0
$569$	35.4104	1.48448	0.742241	−	0.670133i	$-0.233763\pi$
$569$	35.4104	1.48448	0.742241	+	0.670133i	$0.233763\pi$
$570$	0	0
$571$	26.2654	1.09917	0.549587	−	0.835437i	$-0.314785\pi$
$571$	26.2654	1.09917	0.549587	+	0.835437i	$0.314785\pi$
$572$	0	0
$573$	−0.612008	−0.0255670
$574$	0	0
$575$	−2.29092	−0.0955381
$576$	0	0
$577$	−18.0055	−0.749579	−0.374789	−	0.927110i	$-0.622285\pi$
$577$	−18.0055	−0.749579	−0.374789	+	0.927110i	$0.622285\pi$
$578$	0	0
$579$	14.8275	0.616209
$580$	0	0
$581$	36.6581	1.52083
$582$	0	0
$583$	0.685025	0.0283708
$584$	0	0
$585$	1.44087	0.0595727
$586$	0	0
$587$	43.9199	1.81277	0.906384	−	0.422454i	$-0.138831\pi$
$587$	43.9199	1.81277	0.906384	+	0.422454i	$0.138831\pi$
$588$	0	0
$589$	14.1834	0.584416
$590$	0	0
$591$	−12.4623	−0.512630
$592$	0	0
$593$	10.2397	0.420493	0.210246	−	0.977648i	$-0.432573\pi$
$593$	10.2397	0.420493	0.210246	+	0.977648i	$0.432573\pi$
$594$	0	0
$595$	−4.10040	−0.168100
$596$	0	0
$597$	−16.9760	−0.694781
$598$	0	0
$599$	−1.06437	−0.0434888	−0.0217444	−	0.999764i	$-0.506922\pi$
$599$	−1.06437	−0.0434888	−0.0217444	+	0.999764i	$0.506922\pi$
$600$	0	0
$601$	−14.7468	−0.601536	−0.300768	−	0.953697i	$-0.597243\pi$
$601$	−14.7468	−0.601536	−0.300768	+	0.953697i	$0.597243\pi$
$602$	0	0
$603$	−14.4160	−0.587066
$604$	0	0
$605$	4.75351	0.193258
$606$	0	0
$607$	−0.481877	−0.0195588	−0.00977938	−	0.999952i	$-0.503113\pi$
$607$	−0.481877	−0.0195588	−0.00977938	+	0.999952i	$0.503113\pi$
$608$	0	0
$609$	−13.6670	−0.553815
$610$	0	0
$611$	36.9209	1.49366
$612$	0	0
$613$	−29.7504	−1.20161	−0.600803	−	0.799397i	$-0.705152\pi$
$613$	−29.7504	−1.20161	−0.600803	+	0.799397i	$0.705152\pi$
$614$	0	0
$615$	−3.09307	−0.124725
$616$	0	0
$617$	0.195723	0.00787950	0.00393975	−	0.999992i	$-0.498746\pi$
$617$	0.195723	0.00787950	0.00393975	+	0.999992i	$0.498746\pi$
$618$	0	0
$619$	−36.7645	−1.47769	−0.738845	−	0.673875i	$-0.764628\pi$
$619$	−36.7645	−1.47769	−0.738845	+	0.673875i	$0.764628\pi$
$620$	0	0
$621$	−0.476051	−0.0191032
$622$	0	0
$623$	−7.81100	−0.312941
$624$	0	0
$625$	22.2205	0.888821
$626$	0	0
$627$	−0.265737	−0.0106125
$628$	0	0
$629$	4.97566	0.198392
$630$	0	0
$631$	8.82037	0.351134	0.175567	−	0.984468i	$-0.443824\pi$
$631$	8.82037	0.351134	0.175567	+	0.984468i	$0.443824\pi$
$632$	0	0
$633$	2.71803	0.108032
$634$	0	0
$635$	−8.24428	−0.327164
$636$	0	0
$637$	0.502980	0.0199288
$638$	0	0
$639$	5.73037	0.226690
$640$	0	0
$641$	−15.8937	−0.627764	−0.313882	−	0.949462i	$-0.601630\pi$
$641$	−15.8937	−0.627764	−0.313882	+	0.949462i	$0.601630\pi$
$642$	0	0
$643$	20.5897	0.811979	0.405990	−	0.913878i	$-0.366927\pi$
$643$	20.5897	0.811979	0.405990	+	0.913878i	$0.366927\pi$
$644$	0	0
$645$	2.15331	0.0847863
$646$	0	0
$647$	−7.48295	−0.294185	−0.147093	−	0.989123i	$-0.546992\pi$
$647$	−7.48295	−0.294185	−0.147093	+	0.989123i	$0.546992\pi$
$648$	0	0
$649$	1.15676	0.0454068
$650$	0	0
$651$	22.7278	0.890772
$652$	0	0
$653$	−36.3302	−1.42171	−0.710855	−	0.703339i	$-0.751692\pi$
$653$	−36.3302	−1.42171	−0.710855	+	0.703339i	$0.751692\pi$
$654$	0	0
$655$	−3.96458	−0.154909
$656$	0	0
$657$	−1.24633	−0.0486241
$658$	0	0
$659$	3.52682	0.137385	0.0686926	−	0.997638i	$-0.478117\pi$
$659$	3.52682	0.137385	0.0686926	+	0.997638i	$0.478117\pi$
$660$	0	0
$661$	15.5343	0.604215	0.302107	−	0.953274i	$-0.402310\pi$
$661$	15.5343	0.604215	0.302107	+	0.953274i	$0.402310\pi$
$662$	0	0
$663$	−12.0312	−0.467252
$664$	0	0
$665$	1.85142	0.0717950
$666$	0	0
$667$	−2.48611	−0.0962625
$668$	0	0
$669$	−1.50132	−0.0580443
$670$	0	0
$671$	−1.60719	−0.0620448
$672$	0	0
$673$	15.7983	0.608979	0.304489	−	0.952516i	$-0.401514\pi$
$673$	15.7983	0.608979	0.304489	+	0.952516i	$0.401514\pi$
$674$	0	0
$675$	4.81236	0.185228
$676$	0	0
$677$	21.1236	0.811848	0.405924	−	0.913907i	$-0.366950\pi$
$677$	21.1236	0.811848	0.405924	+	0.913907i	$0.366950\pi$
$678$	0	0
$679$	−8.36500	−0.321019
$680$	0	0
$681$	−12.3024	−0.471430
$682$	0	0
$683$	34.1481	1.30664	0.653321	−	0.757081i	$-0.273375\pi$
$683$	34.1481	1.30664	0.653321	+	0.757081i	$0.273375\pi$
$684$	0	0
$685$	−2.99267	−0.114344
$686$	0	0
$687$	0.510224	0.0194663
$688$	0	0
$689$	−14.0037	−0.533497
$690$	0	0
$691$	20.5700	0.782518	0.391259	−	0.920281i	$-0.372040\pi$
$691$	20.5700	0.782518	0.391259	+	0.920281i	$0.372040\pi$
$692$	0	0
$693$	−0.425823	−0.0161757
$694$	0	0
$695$	−5.13683	−0.194851
$696$	0	0
$697$	25.8269	0.978263
$698$	0	0
$699$	17.2638	0.652976
$700$	0	0
$701$	5.27163	0.199107	0.0995533	−	0.995032i	$-0.468259\pi$
$701$	5.27163	0.199107	0.0995533	+	0.995032i	$0.468259\pi$
$702$	0	0
$703$	−2.24662	−0.0847328
$704$	0	0
$705$	−4.80821	−0.181088
$706$	0	0
$707$	−20.5972	−0.774639
$708$	0	0
$709$	3.56370	0.133838	0.0669189	−	0.997758i	$-0.478683\pi$
$709$	3.56370	0.133838	0.0669189	+	0.997758i	$0.478683\pi$
$710$	0	0
$711$	14.7379	0.552715
$712$	0	0
$713$	4.13431	0.154831
$714$	0	0
$715$	0.234448	0.00876787
$716$	0	0
$717$	18.5728	0.693616
$718$	0	0
$719$	−26.6631	−0.994368	−0.497184	−	0.867645i	$-0.665633\pi$
$719$	−26.6631	−0.994368	−0.497184	+	0.867645i	$0.665633\pi$
$720$	0	0
$721$	36.0812	1.34373
$722$	0	0
$723$	−7.79701	−0.289974
$724$	0	0
$725$	25.1319	0.933374
$726$	0	0
$727$	34.5605	1.28178	0.640889	−	0.767633i	$-0.278566\pi$
$727$	34.5605	1.28178	0.640889	+	0.767633i	$0.278566\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−17.9799	−0.665012
$732$	0	0
$733$	3.91839	0.144729	0.0723645	−	0.997378i	$-0.476946\pi$
$733$	3.91839	0.144729	0.0723645	+	0.997378i	$0.476946\pi$
$734$	0	0
$735$	−0.0655031	−0.00241612
$736$	0	0
$737$	−2.34567	−0.0864040
$738$	0	0
$739$	−4.24145	−0.156024	−0.0780122	−	0.996952i	$-0.524857\pi$
$739$	−4.24145	−0.156024	−0.0780122	+	0.996952i	$0.524857\pi$
$740$	0	0
$741$	5.43233	0.199562
$742$	0	0
$743$	4.06662	0.149190	0.0745950	−	0.997214i	$-0.476234\pi$
$743$	4.06662	0.149190	0.0745950	+	0.997214i	$0.476234\pi$
$744$	0	0
$745$	−2.57895	−0.0944854
$746$	0	0
$747$	−14.0076	−0.512510
$748$	0	0
$749$	8.40497	0.307111
$750$	0	0
$751$	−46.3208	−1.69027	−0.845134	−	0.534554i	$-0.820480\pi$
$751$	−46.3208	−1.69027	−0.845134	+	0.534554i	$0.820480\pi$
$752$	0	0
$753$	10.0511	0.366281
$754$	0	0
$755$	8.59245	0.312711
$756$	0	0
$757$	19.5918	0.712075	0.356037	−	0.934472i	$-0.384128\pi$
$757$	19.5918	0.712075	0.356037	+	0.934472i	$0.384128\pi$
$758$	0	0
$759$	−0.0774595	−0.00281160
$760$	0	0
$761$	−53.5877	−1.94255	−0.971277	−	0.237952i	$-0.923524\pi$
$761$	−53.5877	−1.94255	−0.971277	+	0.237952i	$0.923524\pi$
$762$	0	0
$763$	14.1299	0.511538
$764$	0	0
$765$	1.56682	0.0566485
$766$	0	0
$767$	−23.6471	−0.853849
$768$	0	0
$769$	8.51704	0.307132	0.153566	−	0.988138i	$-0.450924\pi$
$769$	8.51704	0.307132	0.153566	+	0.988138i	$0.450924\pi$
$770$	0	0
$771$	−0.675528	−0.0243285
$772$	0	0
$773$	1.36755	0.0491872	0.0245936	−	0.999698i	$-0.492171\pi$
$773$	1.36755	0.0491872	0.0245936	+	0.999698i	$0.492171\pi$
$774$	0	0
$775$	−41.7934	−1.50126
$776$	0	0
$777$	−3.60003	−0.129150
$778$	0	0
$779$	−11.6614	−0.417813
$780$	0	0
$781$	0.932405	0.0333641
$782$	0	0
$783$	5.22236	0.186632
$784$	0	0
$785$	−4.43789	−0.158395
$786$	0	0
$787$	−26.8682	−0.957747	−0.478873	−	0.877884i	$-0.658955\pi$
$787$	−26.8682	−0.957747	−0.478873	+	0.877884i	$0.658955\pi$
$788$	0	0
$789$	−16.5739	−0.590046
$790$	0	0
$791$	−8.76478	−0.311640
$792$	0	0
$793$	32.8550	1.16672
$794$	0	0
$795$	1.82370	0.0646799
$796$	0	0
$797$	−0.540767	−0.0191550	−0.00957748	−	0.999954i	$-0.503049\pi$
$797$	−0.540767	−0.0191550	−0.00957748	+	0.999954i	$0.503049\pi$
$798$	0	0
$799$	40.1482	1.42034
$800$	0	0
$801$	2.98470	0.105459
$802$	0	0
$803$	−0.202794	−0.00715646
$804$	0	0
$805$	0.539670	0.0190209
$806$	0	0
$807$	−21.7862	−0.766910
$808$	0	0
$809$	34.9816	1.22989	0.614944	−	0.788571i	$-0.289179\pi$
$809$	34.9816	1.22989	0.614944	+	0.788571i	$0.289179\pi$
$810$	0	0
$811$	47.9020	1.68207	0.841034	−	0.540983i	$-0.181948\pi$
$811$	47.9020	1.68207	0.841034	+	0.540983i	$0.181948\pi$
$812$	0	0
$813$	−27.1533	−0.952306
$814$	0	0
$815$	6.35291	0.222533
$816$	0	0
$817$	8.11834	0.284025
$818$	0	0
$819$	8.70490	0.304174
$820$	0	0
$821$	12.6084	0.440036	0.220018	−	0.975496i	$-0.429388\pi$
$821$	12.6084	0.440036	0.220018	+	0.975496i	$0.429388\pi$
$822$	0	0
$823$	20.8277	0.726008	0.363004	−	0.931788i	$-0.381751\pi$
$823$	20.8277	0.726008	0.363004	+	0.931788i	$0.381751\pi$
$824$	0	0
$825$	0.783032	0.0272617
$826$	0	0
$827$	−47.6331	−1.65637	−0.828183	−	0.560457i	$-0.810625\pi$
$827$	−47.6331	−1.65637	−0.828183	+	0.560457i	$0.810625\pi$
$828$	0	0
$829$	−25.8592	−0.898127	−0.449063	−	0.893500i	$-0.648242\pi$
$829$	−25.8592	−0.898127	−0.449063	+	0.893500i	$0.648242\pi$
$830$	0	0
$831$	3.90974	0.135627
$832$	0	0
$833$	0.546946	0.0189506
$834$	0	0
$835$	4.66433	0.161416
$836$	0	0
$837$	−8.68461	−0.300184
$838$	0	0
$839$	−48.7794	−1.68405	−0.842025	−	0.539438i	$-0.818637\pi$
$839$	−48.7794	−1.68405	−0.842025	+	0.539438i	$0.818637\pi$
$840$	0	0
$841$	−1.72695	−0.0595501
$842$	0	0
$843$	26.7943	0.922846
$844$	0	0
$845$	0.838618	0.0288493
$846$	0	0
$847$	28.7179	0.986759
$848$	0	0
$849$	−30.6991	−1.05359
$850$	0	0
$851$	−0.654867	−0.0224485
$852$	0	0
$853$	35.3527	1.21045	0.605227	−	0.796053i	$-0.293082\pi$
$853$	35.3527	1.21045	0.605227	+	0.796053i	$0.293082\pi$
$854$	0	0
$855$	−0.707454	−0.0241944
$856$	0	0
$857$	−49.7250	−1.69858	−0.849288	−	0.527930i	$-0.822968\pi$
$857$	−49.7250	−1.69858	−0.849288	+	0.527930i	$0.822968\pi$
$858$	0	0
$859$	39.7226	1.35532	0.677659	−	0.735376i	$-0.262995\pi$
$859$	39.7226	1.35532	0.677659	+	0.735376i	$0.262995\pi$
$860$	0	0
$861$	−18.6865	−0.636834
$862$	0	0
$863$	18.1836	0.618976	0.309488	−	0.950903i	$-0.399842\pi$
$863$	18.1836	0.618976	0.309488	+	0.950903i	$0.399842\pi$
$864$	0	0
$865$	−0.433180	−0.0147286
$866$	0	0
$867$	3.91718	0.133034
$868$	0	0
$869$	2.39805	0.0813482
$870$	0	0
$871$	47.9516	1.62478
$872$	0	0
$873$	3.19638	0.108181
$874$	0	0
$875$	−11.1237	−0.376049
$876$	0	0
$877$	−15.7787	−0.532810	−0.266405	−	0.963861i	$-0.585836\pi$
$877$	−15.7787	−0.532810	−0.266405	+	0.963861i	$0.585836\pi$
$878$	0	0
$879$	30.5778	1.03136
$880$	0	0
$881$	23.2006	0.781649	0.390824	−	0.920465i	$-0.372190\pi$
$881$	23.2006	0.781649	0.390824	+	0.920465i	$0.372190\pi$
$882$	0	0
$883$	−21.1463	−0.711629	−0.355814	−	0.934557i	$-0.615796\pi$
$883$	−21.1463	−0.711629	−0.355814	+	0.934557i	$0.615796\pi$
$884$	0	0
$885$	3.07957	0.103519
$886$	0	0
$887$	−11.0855	−0.372215	−0.186107	−	0.982529i	$-0.559587\pi$
$887$	−11.0855	−0.372215	−0.186107	+	0.982529i	$0.559587\pi$
$888$	0	0
$889$	−49.8071	−1.67048
$890$	0	0
$891$	0.162713	0.00545109
$892$	0	0
$893$	−18.1278	−0.606624
$894$	0	0
$895$	−6.33983	−0.211917
$896$	0	0
$897$	1.58347	0.0528705
$898$	0	0
$899$	−45.3542	−1.51265
$900$	0	0
$901$	−15.2277	−0.507309
$902$	0	0
$903$	13.0090	0.432913
$904$	0	0
$905$	−5.23440	−0.173997
$906$	0	0
$907$	−6.57698	−0.218385	−0.109193	−	0.994021i	$-0.534827\pi$
$907$	−6.57698	−0.218385	−0.109193	+	0.994021i	$0.534827\pi$
$908$	0	0
$909$	7.87050	0.261048
$910$	0	0
$911$	−33.0661	−1.09553	−0.547764	−	0.836633i	$-0.684521\pi$
$911$	−33.0661	−1.09553	−0.547764	+	0.836633i	$0.684521\pi$
$912$	0	0
$913$	−2.27921	−0.0754309
$914$	0	0
$915$	−4.27871	−0.141450
$916$	0	0
$917$	−23.9516	−0.790953
$918$	0	0
$919$	−9.99042	−0.329554	−0.164777	−	0.986331i	$-0.552690\pi$
$919$	−9.99042	−0.329554	−0.164777	+	0.986331i	$0.552690\pi$
$920$	0	0
$921$	−27.0111	−0.890046
$922$	0	0
$923$	−19.0607	−0.627392
$924$	0	0
$925$	6.61999	0.217664
$926$	0	0
$927$	−13.7871	−0.452829
$928$	0	0
$929$	−27.7533	−0.910556	−0.455278	−	0.890349i	$-0.650460\pi$
$929$	−27.7533	−0.910556	−0.455278	+	0.890349i	$0.650460\pi$
$930$	0	0
$931$	−0.246958	−0.00809373
$932$	0	0
$933$	−24.5397	−0.803394
$934$	0	0
$935$	0.254942	0.00833749
$936$	0	0
$937$	−16.0840	−0.525441	−0.262721	−	0.964872i	$-0.584620\pi$
$937$	−16.0840	−0.525441	−0.262721	+	0.964872i	$0.584620\pi$
$938$	0	0
$939$	−32.5389	−1.06187
$940$	0	0
$941$	11.9053	0.388103	0.194051	−	0.980991i	$-0.437837\pi$
$941$	11.9053	0.388103	0.194051	+	0.980991i	$0.437837\pi$
$942$	0	0
$943$	−3.39918	−0.110693
$944$	0	0
$945$	−1.13364	−0.0368773
$946$	0	0
$947$	−56.6650	−1.84136	−0.920682	−	0.390313i	$-0.872367\pi$
$947$	−56.6650	−1.84136	−0.920682	+	0.390313i	$0.872367\pi$
$948$	0	0
$949$	4.14563	0.134573
$950$	0	0
$951$	−31.9465	−1.03594
$952$	0	0
$953$	17.1988	0.557125	0.278562	−	0.960418i	$-0.410142\pi$
$953$	17.1988	0.557125	0.278562	+	0.960418i	$0.410142\pi$
$954$	0	0
$955$	−0.265110	−0.00857875
$956$	0	0
$957$	0.849745	0.0274684
$958$	0	0
$959$	−18.0800	−0.583832
$960$	0	0
$961$	44.4225	1.43298
$962$	0	0
$963$	−3.21166	−0.103494
$964$	0	0
$965$	6.42296	0.206762
$966$	0	0
$967$	55.3643	1.78040	0.890198	−	0.455574i	$-0.150566\pi$
$967$	55.3643	1.78040	0.890198	+	0.455574i	$0.150566\pi$
$968$	0	0
$969$	5.90718	0.189766
$970$	0	0
$971$	30.6626	0.984010	0.492005	−	0.870592i	$-0.336264\pi$
$971$	30.6626	0.984010	0.492005	+	0.870592i	$0.336264\pi$
$972$	0	0
$973$	−31.0337	−0.994894
$974$	0	0
$975$	−16.0072	−0.512640
$976$	0	0
$977$	24.4115	0.780992	0.390496	−	0.920605i	$-0.372303\pi$
$977$	24.4115	0.780992	0.390496	+	0.920605i	$0.372303\pi$
$978$	0	0
$979$	0.485648	0.0155214
$980$	0	0
$981$	−5.39925	−0.172385
$982$	0	0
$983$	−58.3212	−1.86016	−0.930079	−	0.367359i	$-0.880262\pi$
$983$	−58.3212	−1.86016	−0.930079	+	0.367359i	$0.880262\pi$
$984$	0	0
$985$	−5.39842	−0.172008
$986$	0	0
$987$	−29.0484	−0.924621
$988$	0	0
$989$	2.36641	0.0752475
$990$	0	0
$991$	38.8286	1.23343	0.616715	−	0.787187i	$-0.288463\pi$
$991$	38.8286	1.23343	0.616715	+	0.787187i	$0.288463\pi$
$992$	0	0
$993$	14.0862	0.447011
$994$	0	0
$995$	−7.35366	−0.233127
$996$	0	0
$997$	44.5660	1.41142	0.705710	−	0.708501i	$-0.250628\pi$
$997$	44.5660	1.41142	0.705710	+	0.708501i	$0.250628\pi$
$998$	0	0
$999$	1.37562	0.0435228

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8304.2.a.y.1.3		5
4.3	odd	2		4152.2.a.j.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4152.2.a.j.1.3	✓	5	4.3	odd	2
8304.2.a.y.1.3		5	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:	\( N \)	\(=\)	\( 8304 = 2^{4} \cdot 3 \cdot 173 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8304.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.433180 q^{5} -2.61702 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.433180 q^{5} -2.61702 q^{7} +1.00000 q^{9} +0.162713 q^{11} -3.32627 q^{13} +0.433180 q^{15} -3.61702 q^{17} +1.63316 q^{19} +2.61702 q^{21} +0.476051 q^{23} -4.81236 q^{25} -1.00000 q^{27} -5.22236 q^{29} +8.68461 q^{31} -0.162713 q^{33} +1.13364 q^{35} -1.37562 q^{37} +3.32627 q^{39} -7.14038 q^{41} +4.97093 q^{43} -0.433180 q^{45} -11.0998 q^{47} -0.151215 q^{49} +3.61702 q^{51} +4.21002 q^{53} -0.0704839 q^{55} -1.63316 q^{57} +7.10921 q^{59} -9.87745 q^{61} -2.61702 q^{63} +1.44087 q^{65} -14.4160 q^{67} -0.476051 q^{69} +5.73037 q^{71} -1.24633 q^{73} +4.81236 q^{75} -0.425823 q^{77} +14.7379 q^{79} +1.00000 q^{81} -14.0076 q^{83} +1.56682 q^{85} +5.22236 q^{87} +2.98470 q^{89} +8.70490 q^{91} -8.68461 q^{93} -0.707454 q^{95} +3.19638 q^{97} +0.162713 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} - 2 q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 - 2 * q^5 + 4 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} - 2 q^{5} + 4 q^{7} + 5 q^{9} + 12 q^{11} + q^{13} + 2 q^{15} - q^{17} + 2 q^{19} - 4 q^{21} - 4 q^{23} - 3 q^{25} - 5 q^{27} + 4 q^{29} + 24 q^{31} - 12 q^{33} + 6 q^{35} - 12 q^{37} - q^{39} + 2 q^{41} + 14 q^{43} - 2 q^{45} + 6 q^{47} + q^{49} + q^{51} + 18 q^{53} + 5 q^{55} - 2 q^{57} + 23 q^{59} - 18 q^{61} + 4 q^{63} - 18 q^{65} + 10 q^{67} + 4 q^{69} - 18 q^{73} + 3 q^{75} + 10 q^{77} + 40 q^{79} + 5 q^{81} + 8 q^{83} + 8 q^{85} - 4 q^{87} - 36 q^{89} + 36 q^{91} - 24 q^{93} - 30 q^{95} - 22 q^{97} + 12 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 - 2 * q^5 + 4 * q^7 + 5 * q^9 + 12 * q^11 + q^13 + 2 * q^15 - q^17 + 2 * q^19 - 4 * q^21 - 4 * q^23 - 3 * q^25 - 5 * q^27 + 4 * q^29 + 24 * q^31 - 12 * q^33 + 6 * q^35 - 12 * q^37 - q^39 + 2 * q^41 + 14 * q^43 - 2 * q^45 + 6 * q^47 + q^49 + q^51 + 18 * q^53 + 5 * q^55 - 2 * q^57 + 23 * q^59 - 18 * q^61 + 4 * q^63 - 18 * q^65 + 10 * q^67 + 4 * q^69 - 18 * q^73 + 3 * q^75 + 10 * q^77 + 40 * q^79 + 5 * q^81 + 8 * q^83 + 8 * q^85 - 4 * q^87 - 36 * q^89 + 36 * q^91 - 24 * q^93 - 30 * q^95 - 22 * q^97 + 12 * q^99

Introduction

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