LMFDB - Embedded newform 6615.2.a.br.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6615,2,Mod(1,6615)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6615, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6615.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6615 = 3^{3} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6615.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:	$52.8210409371$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.466809.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 7x^{3} - x^{2} + 11x + 3 $ x^5 - 7x^3 - x^2 + 11x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 945)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.22697$ of defining polynomial
Character	$\chi$	$=$	6615.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.22697 q^{2} +2.95940 q^{4} +1.00000 q^{5} +2.13656 q^{8} +O(q^{10})$	$q+2.22697 q^{2} +2.95940 q^{4} +1.00000 q^{5} +2.13656 q^{8} +2.22697 q^{10} -0.394523 q^{11} +2.66209 q^{13} -1.16075 q^{16} -4.66764 q^{17} +6.49048 q^{19} +2.95940 q^{20} -0.878592 q^{22} +1.34838 q^{23} +1.00000 q^{25} +5.92841 q^{26} -4.42975 q^{29} +3.60142 q^{31} -6.85807 q^{32} -10.3947 q^{34} +10.5257 q^{37} +14.4541 q^{38} +2.13656 q^{40} +4.54069 q^{41} +9.11198 q^{43} -1.16755 q^{44} +3.00280 q^{46} +8.86487 q^{47} +2.22697 q^{50} +7.87820 q^{52} -3.07955 q^{53} -0.394523 q^{55} -9.86493 q^{58} +4.14983 q^{59} +14.1244 q^{61} +8.02025 q^{62} -12.9512 q^{64} +2.66209 q^{65} -1.79185 q^{67} -13.8134 q^{68} -13.7902 q^{71} -2.60410 q^{73} +23.4404 q^{74} +19.2079 q^{76} -8.21605 q^{79} -1.16075 q^{80} +10.1120 q^{82} +2.47133 q^{83} -4.66764 q^{85} +20.2921 q^{86} -0.842921 q^{88} +4.88232 q^{89} +3.99040 q^{92} +19.7418 q^{94} +6.49048 q^{95} +14.9814 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{4} + 5 q^{5} + 3 q^{8}+O(q^{10}) $ 5 * q + 4 * q^4 + 5 * q^5 + 3 * q^8	$ 5 q + 4 q^{4} + 5 q^{5} + 3 q^{8} + 5 q^{11} + 6 q^{13} - 10 q^{16} - q^{17} + 7 q^{19} + 4 q^{20} + 4 q^{22} + 4 q^{23} + 5 q^{25} + 7 q^{26} + 12 q^{29} + 11 q^{31} - 4 q^{32} + 8 q^{34} - 3 q^{38} + 3 q^{40} + 7 q^{41} + 2 q^{43} - 10 q^{44} + 14 q^{46} + 14 q^{47} + 7 q^{52} + 2 q^{53} + 5 q^{55} - 28 q^{58} + 2 q^{59} + 25 q^{61} - 24 q^{62} - 17 q^{64} + 6 q^{65} + 6 q^{67} - 14 q^{68} - 15 q^{71} + 17 q^{73} + 29 q^{74} + 34 q^{76} - 7 q^{79} - 10 q^{80} + 7 q^{82} + 2 q^{83} - q^{85} + 7 q^{86} + 10 q^{88} + 15 q^{89} + 21 q^{92} + 21 q^{94} + 7 q^{95} + 48 q^{97}+O(q^{100}) $ 5 * q + 4 * q^4 + 5 * q^5 + 3 * q^8 + 5 * q^11 + 6 * q^13 - 10 * q^16 - q^17 + 7 * q^19 + 4 * q^20 + 4 * q^22 + 4 * q^23 + 5 * q^25 + 7 * q^26 + 12 * q^29 + 11 * q^31 - 4 * q^32 + 8 * q^34 - 3 * q^38 + 3 * q^40 + 7 * q^41 + 2 * q^43 - 10 * q^44 + 14 * q^46 + 14 * q^47 + 7 * q^52 + 2 * q^53 + 5 * q^55 - 28 * q^58 + 2 * q^59 + 25 * q^61 - 24 * q^62 - 17 * q^64 + 6 * q^65 + 6 * q^67 - 14 * q^68 - 15 * q^71 + 17 * q^73 + 29 * q^74 + 34 * q^76 - 7 * q^79 - 10 * q^80 + 7 * q^82 + 2 * q^83 - q^85 + 7 * q^86 + 10 * q^88 + 15 * q^89 + 21 * q^92 + 21 * q^94 + 7 * q^95 + 48 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.22697	1.57471	0.787353	−	0.616502i	$-0.211451\pi$
$2$	2.22697	1.57471	0.787353	+	0.616502i	$0.211451\pi$
$3$	0	0
$3$	0	0
$4$	2.95940	1.47970
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.13656	0.755387
$9$	0	0
$10$	2.22697	0.704230
$11$	−0.394523	−0.118953	−0.0594766	−	0.998230i	$-0.518943\pi$
$11$	−0.394523	−0.118953	−0.0594766	+	0.998230i	$0.518943\pi$
$12$	0	0
$13$	2.66209	0.738332	0.369166	−	0.929363i	$-0.379643\pi$
$13$	2.66209	0.738332	0.369166	+	0.929363i	$0.379643\pi$
$14$	0	0
$15$	0	0
$16$	−1.16075	−0.290187
$17$	−4.66764	−1.13207	−0.566034	−	0.824382i	$-0.691523\pi$
$17$	−4.66764	−1.13207	−0.566034	+	0.824382i	$0.691523\pi$
$18$	0	0
$19$	6.49048	1.48902	0.744509	−	0.667612i	$-0.232684\pi$
$19$	6.49048	1.48902	0.744509	+	0.667612i	$0.232684\pi$
$20$	2.95940	0.661742
$21$	0	0
$22$	−0.878592	−0.187316
$23$	1.34838	0.281157	0.140578	−	0.990070i	$-0.455104\pi$
$23$	1.34838	0.281157	0.140578	+	0.990070i	$0.455104\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	5.92841	1.16266
$27$	0	0
$28$	0	0
$29$	−4.42975	−0.822584	−0.411292	−	0.911504i	$-0.634922\pi$
$29$	−4.42975	−0.822584	−0.411292	+	0.911504i	$0.634922\pi$
$30$	0	0
$31$	3.60142	0.646833	0.323417	−	0.946257i	$-0.395168\pi$
$31$	3.60142	0.646833	0.323417	+	0.946257i	$0.395168\pi$
$32$	−6.85807	−1.21235
$33$	0	0
$34$	−10.3947	−1.78268
$35$	0	0
$36$	0	0
$37$	10.5257	1.73042	0.865208	−	0.501413i	$-0.167186\pi$
$37$	10.5257	1.73042	0.865208	+	0.501413i	$0.167186\pi$
$38$	14.4541	2.34477
$39$	0	0
$40$	2.13656	0.337819
$41$	4.54069	0.709136	0.354568	−	0.935030i	$-0.384628\pi$
$41$	4.54069	0.709136	0.354568	+	0.935030i	$0.384628\pi$
$42$	0	0
$43$	9.11198	1.38956	0.694782	−	0.719221i	$-0.255501\pi$
$43$	9.11198	1.38956	0.694782	+	0.719221i	$0.255501\pi$
$44$	−1.16755	−0.176015
$45$	0	0
$46$	3.00280	0.442739
$47$	8.86487	1.29307	0.646537	−	0.762882i	$-0.276216\pi$
$47$	8.86487	1.29307	0.646537	+	0.762882i	$0.276216\pi$
$48$	0	0
$49$	0	0
$50$	2.22697	0.314941
$51$	0	0
$52$	7.87820	1.09251
$53$	−3.07955	−0.423009	−0.211504	−	0.977377i	$-0.567836\pi$
$53$	−3.07955	−0.423009	−0.211504	+	0.977377i	$0.567836\pi$
$54$	0	0
$55$	−0.394523	−0.0531975
$56$	0	0
$57$	0	0
$58$	−9.86493	−1.29533
$59$	4.14983	0.540262	0.270131	−	0.962824i	$-0.412933\pi$
$59$	4.14983	0.540262	0.270131	+	0.962824i	$0.412933\pi$
$60$	0	0
$61$	14.1244	1.80844	0.904221	−	0.427065i	$-0.140452\pi$
$61$	14.1244	1.80844	0.904221	+	0.427065i	$0.140452\pi$
$62$	8.02025	1.01857
$63$	0	0
$64$	−12.9512	−1.61890
$65$	2.66209	0.330192
$66$	0	0
$67$	−1.79185	−0.218909	−0.109455	−	0.993992i	$-0.534910\pi$
$67$	−1.79185	−0.218909	−0.109455	+	0.993992i	$0.534910\pi$
$68$	−13.8134	−1.67512
$69$	0	0
$70$	0	0
$71$	−13.7902	−1.63660	−0.818298	−	0.574794i	$-0.805082\pi$
$71$	−13.7902	−1.63660	−0.818298	+	0.574794i	$0.805082\pi$
$72$	0	0
$73$	−2.60410	−0.304787	−0.152394	−	0.988320i	$-0.548698\pi$
$73$	−2.60410	−0.304787	−0.152394	+	0.988320i	$0.548698\pi$
$74$	23.4404	2.72490
$75$	0	0
$76$	19.2079	2.20330
$77$	0	0
$78$	0	0
$79$	−8.21605	−0.924378	−0.462189	−	0.886781i	$-0.652936\pi$
$79$	−8.21605	−0.924378	−0.462189	+	0.886781i	$0.652936\pi$
$80$	−1.16075	−0.129776
$81$	0	0
$82$	10.1120	1.11668
$83$	2.47133	0.271264	0.135632	−	0.990759i	$-0.456694\pi$
$83$	2.47133	0.271264	0.135632	+	0.990759i	$0.456694\pi$
$84$	0	0
$85$	−4.66764	−0.506276
$86$	20.2921	2.18815
$87$	0	0
$88$	−0.842921	−0.0898557
$89$	4.88232	0.517524	0.258762	−	0.965941i	$-0.416685\pi$
$89$	4.88232	0.517524	0.258762	+	0.965941i	$0.416685\pi$
$90$	0	0
$91$	0	0
$92$	3.99040	0.416027
$93$	0	0
$94$	19.7418	2.03621
$95$	6.49048	0.665909
$96$	0	0
$97$	14.9814	1.52113	0.760563	−	0.649264i	$-0.224923\pi$
$97$	14.9814	1.52113	0.760563	+	0.649264i	$0.224923\pi$
$98$	0	0
$99$	0	0
$100$	2.95940	0.295940
$101$	6.72112	0.668777	0.334388	−	0.942435i	$-0.391470\pi$
$101$	6.72112	0.668777	0.334388	+	0.942435i	$0.391470\pi$
$102$	0	0
$103$	−1.47589	−0.145424	−0.0727121	−	0.997353i	$-0.523165\pi$
$103$	−1.47589	−0.145424	−0.0727121	+	0.997353i	$0.523165\pi$
$104$	5.68772	0.557726
$105$	0	0
$106$	−6.85807	−0.666115
$107$	10.2203	0.988038	0.494019	−	0.869451i	$-0.335527\pi$
$107$	10.2203	0.988038	0.494019	+	0.869451i	$0.335527\pi$
$108$	0	0
$109$	−3.73672	−0.357912	−0.178956	−	0.983857i	$-0.557272\pi$
$109$	−3.73672	−0.357912	−0.178956	+	0.983857i	$0.557272\pi$
$110$	−0.878592	−0.0837704
$111$	0	0
$112$	0	0
$113$	−5.69858	−0.536077	−0.268039	−	0.963408i	$-0.586375\pi$
$113$	−5.69858	−0.536077	−0.268039	+	0.963408i	$0.586375\pi$
$114$	0	0
$115$	1.34838	0.125737
$116$	−13.1094	−1.21718
$117$	0	0
$118$	9.24156	0.850754
$119$	0	0
$120$	0	0
$121$	−10.8444	−0.985850
$122$	31.4546	2.84777
$123$	0	0
$124$	10.6580	0.957120
$125$	1.00000	0.0894427
$126$	0	0
$127$	6.45543	0.572826	0.286413	−	0.958106i	$-0.407537\pi$
$127$	6.45543	0.572826	0.286413	+	0.958106i	$0.407537\pi$
$128$	−15.1259	−1.33695
$129$	0	0
$130$	5.92841	0.519956
$131$	−9.35798	−0.817611	−0.408805	−	0.912622i	$-0.634055\pi$
$131$	−9.35798	−0.817611	−0.408805	+	0.912622i	$0.634055\pi$
$132$	0	0
$133$	0	0
$134$	−3.99040	−0.344718
$135$	0	0
$136$	−9.97268	−0.855150
$137$	−5.68491	−0.485695	−0.242847	−	0.970064i	$-0.578081\pi$
$137$	−5.68491	−0.485695	−0.242847	+	0.970064i	$0.578081\pi$
$138$	0	0
$139$	−5.50650	−0.467055	−0.233528	−	0.972350i	$-0.575027\pi$
$139$	−5.50650	−0.467055	−0.233528	+	0.972350i	$0.575027\pi$
$140$	0	0
$141$	0	0
$142$	−30.7104	−2.57716
$143$	−1.05026	−0.0878269
$144$	0	0
$145$	−4.42975	−0.367871
$146$	−5.79927	−0.479951
$147$	0	0
$148$	31.1498	2.56050
$149$	19.2249	1.57496	0.787482	−	0.616338i	$-0.211384\pi$
$149$	19.2249	1.57496	0.787482	+	0.616338i	$0.211384\pi$
$150$	0	0
$151$	−9.01306	−0.733472	−0.366736	−	0.930325i	$-0.619525\pi$
$151$	−9.01306	−0.733472	−0.366736	+	0.930325i	$0.619525\pi$
$152$	13.8673	1.12479
$153$	0	0
$154$	0	0
$155$	3.60142	0.289273
$156$	0	0
$157$	−17.7972	−1.42037	−0.710185	−	0.704016i	$-0.751389\pi$
$157$	−17.7972	−1.42037	−0.710185	+	0.704016i	$0.751389\pi$
$158$	−18.2969	−1.45562
$159$	0	0
$160$	−6.85807	−0.542178
$161$	0	0
$162$	0	0
$163$	6.94535	0.544002	0.272001	−	0.962297i	$-0.412315\pi$
$163$	6.94535	0.544002	0.272001	+	0.962297i	$0.412315\pi$
$164$	13.4377	1.04931
$165$	0	0
$166$	5.50358	0.427161
$167$	−19.9707	−1.54538	−0.772688	−	0.634786i	$-0.781088\pi$
$167$	−19.9707	−1.54538	−0.772688	+	0.634786i	$0.781088\pi$
$168$	0	0
$169$	−5.91326	−0.454866
$170$	−10.3947	−0.797237
$171$	0	0
$172$	26.9660	2.05614
$173$	−24.6188	−1.87173	−0.935865	−	0.352359i	$-0.885380\pi$
$173$	−24.6188	−1.87173	−0.935865	+	0.352359i	$0.885380\pi$
$174$	0	0
$175$	0	0
$176$	0.457943	0.0345187
$177$	0	0
$178$	10.8728	0.814949
$179$	−1.27493	−0.0952931	−0.0476466	−	0.998864i	$-0.515172\pi$
$179$	−1.27493	−0.0952931	−0.0476466	+	0.998864i	$0.515172\pi$
$180$	0	0
$181$	14.1119	1.04893	0.524465	−	0.851432i	$-0.324265\pi$
$181$	14.1119	1.04893	0.524465	+	0.851432i	$0.324265\pi$
$182$	0	0
$183$	0	0
$184$	2.88089	0.212382
$185$	10.5257	0.773865
$186$	0	0
$187$	1.84149	0.134663
$188$	26.2347	1.91336
$189$	0	0
$190$	14.4541	1.04861
$191$	22.3376	1.61629	0.808147	−	0.588980i	$-0.200471\pi$
$191$	22.3376	1.61629	0.808147	+	0.588980i	$0.200471\pi$
$192$	0	0
$193$	−7.14639	−0.514408	−0.257204	−	0.966357i	$-0.582801\pi$
$193$	−7.14639	−0.514408	−0.257204	+	0.966357i	$0.582801\pi$
$194$	33.3630	2.39533
$195$	0	0
$196$	0	0
$197$	−15.2441	−1.08610	−0.543050	−	0.839700i	$-0.682731\pi$
$197$	−15.2441	−1.08610	−0.543050	+	0.839700i	$0.682731\pi$
$198$	0	0
$199$	−4.33906	−0.307588	−0.153794	−	0.988103i	$-0.549149\pi$
$199$	−4.33906	−0.307588	−0.153794	+	0.988103i	$0.549149\pi$
$200$	2.13656	0.151077
$201$	0	0
$202$	14.9677	1.05313
$203$	0	0
$204$	0	0
$205$	4.54069	0.317135
$206$	−3.28677	−0.229000
$207$	0	0
$208$	−3.09002	−0.214255
$209$	−2.56064	−0.177124
$210$	0	0
$211$	−14.8790	−1.02431	−0.512157	−	0.858892i	$-0.671153\pi$
$211$	−14.8790	−1.02431	−0.512157	+	0.858892i	$0.671153\pi$
$212$	−9.11362	−0.625926
$213$	0	0
$214$	22.7604	1.55587
$215$	9.11198	0.621432
$216$	0	0
$217$	0	0
$218$	−8.32156	−0.563607
$219$	0	0
$220$	−1.16755	−0.0787163
$221$	−12.4257	−0.835842
$222$	0	0
$223$	−3.17358	−0.212519	−0.106259	−	0.994338i	$-0.533887\pi$
$223$	−3.17358	−0.212519	−0.106259	+	0.994338i	$0.533887\pi$
$224$	0	0
$225$	0	0
$226$	−12.6906	−0.844164
$227$	−0.0848091	−0.00562898	−0.00281449	−	0.999996i	$-0.500896\pi$
$227$	−0.0848091	−0.00562898	−0.00281449	+	0.999996i	$0.500896\pi$
$228$	0	0
$229$	23.9651	1.58366	0.791830	−	0.610742i	$-0.209129\pi$
$229$	23.9651	1.58366	0.791830	+	0.610742i	$0.209129\pi$
$230$	3.00280	0.197999
$231$	0	0
$232$	−9.46442	−0.621369
$233$	8.05531	0.527721	0.263860	−	0.964561i	$-0.415004\pi$
$233$	8.05531	0.527721	0.263860	+	0.964561i	$0.415004\pi$
$234$	0	0
$235$	8.86487	0.578281
$236$	12.2810	0.799426
$237$	0	0
$238$	0	0
$239$	26.6324	1.72271	0.861354	−	0.508005i	$-0.169617\pi$
$239$	26.6324	1.72271	0.861354	+	0.508005i	$0.169617\pi$
$240$	0	0
$241$	3.28655	0.211705	0.105853	−	0.994382i	$-0.466243\pi$
$241$	3.28655	0.211705	0.105853	+	0.994382i	$0.466243\pi$
$242$	−24.1501	−1.55242
$243$	0	0
$244$	41.7997	2.67595
$245$	0	0
$246$	0	0
$247$	17.2783	1.09939
$248$	7.69463	0.488610
$249$	0	0
$250$	2.22697	0.140846
$251$	25.0271	1.57969	0.789847	−	0.613304i	$-0.210160\pi$
$251$	25.0271	1.57969	0.789847	+	0.613304i	$0.210160\pi$
$252$	0	0
$253$	−0.531967	−0.0334445
$254$	14.3760	0.902033
$255$	0	0
$256$	−7.78242	−0.486401
$257$	−28.5550	−1.78121	−0.890606	−	0.454776i	$-0.849719\pi$
$257$	−28.5550	−1.78121	−0.890606	+	0.454776i	$0.849719\pi$
$258$	0	0
$259$	0	0
$260$	7.87820	0.488585
$261$	0	0
$262$	−20.8400	−1.28750
$263$	−7.44406	−0.459020	−0.229510	−	0.973306i	$-0.573712\pi$
$263$	−7.44406	−0.459020	−0.229510	+	0.973306i	$0.573712\pi$
$264$	0	0
$265$	−3.07955	−0.189175
$266$	0	0
$267$	0	0
$268$	−5.30280	−0.323920
$269$	−30.6521	−1.86889	−0.934446	−	0.356104i	$-0.884105\pi$
$269$	−30.6521	−1.86889	−0.934446	+	0.356104i	$0.884105\pi$
$270$	0	0
$271$	−24.7117	−1.50113	−0.750564	−	0.660798i	$-0.770218\pi$
$271$	−24.7117	−1.50113	−0.750564	+	0.660798i	$0.770218\pi$
$272$	5.41796	0.328512
$273$	0	0
$274$	−12.6601	−0.764827
$275$	−0.394523	−0.0237906
$276$	0	0
$277$	−24.0693	−1.44618	−0.723092	−	0.690752i	$-0.757280\pi$
$277$	−24.0693	−1.44618	−0.723092	+	0.690752i	$0.757280\pi$
$278$	−12.2628	−0.735475
$279$	0	0
$280$	0	0
$281$	−6.11439	−0.364754	−0.182377	−	0.983229i	$-0.558379\pi$
$281$	−6.11439	−0.364754	−0.182377	+	0.983229i	$0.558379\pi$
$282$	0	0
$283$	21.2106	1.26084	0.630419	−	0.776255i	$-0.282883\pi$
$283$	21.2106	1.26084	0.630419	+	0.776255i	$0.282883\pi$
$284$	−40.8107	−2.42167
$285$	0	0
$286$	−2.33889	−0.138302
$287$	0	0
$288$	0	0
$289$	4.78684	0.281579
$290$	−9.86493	−0.579288
$291$	0	0
$292$	−7.70659	−0.450994
$293$	31.2110	1.82336	0.911682	−	0.410897i	$-0.134784\pi$
$293$	31.2110	1.82336	0.911682	+	0.410897i	$0.134784\pi$
$294$	0	0
$295$	4.14983	0.241613
$296$	22.4888	1.30713
$297$	0	0
$298$	42.8133	2.48011
$299$	3.58951	0.207587
$300$	0	0
$301$	0	0
$302$	−20.0718	−1.15500
$303$	0	0
$304$	−7.53382	−0.432094
$305$	14.1244	0.808760
$306$	0	0
$307$	21.3253	1.21710	0.608549	−	0.793516i	$-0.291752\pi$
$307$	21.3253	1.21710	0.608549	+	0.793516i	$0.291752\pi$
$308$	0	0
$309$	0	0
$310$	8.02025	0.455520
$311$	−3.23372	−0.183367	−0.0916837	−	0.995788i	$-0.529225\pi$
$311$	−3.23372	−0.183367	−0.0916837	+	0.995788i	$0.529225\pi$
$312$	0	0
$313$	0.103150	0.00583041	0.00291520	−	0.999996i	$-0.499072\pi$
$313$	0.103150	0.00583041	0.00291520	+	0.999996i	$0.499072\pi$
$314$	−39.6338	−2.23666
$315$	0	0
$316$	−24.3146	−1.36780
$317$	2.89971	0.162864	0.0814320	−	0.996679i	$-0.474051\pi$
$317$	2.89971	0.162864	0.0814320	+	0.996679i	$0.474051\pi$
$318$	0	0
$319$	1.74764	0.0978490
$320$	−12.9512	−0.723996
$321$	0	0
$322$	0	0
$323$	−30.2952	−1.68567
$324$	0	0
$325$	2.66209	0.147666
$326$	15.4671	0.856643
$327$	0	0
$328$	9.70143	0.535672
$329$	0	0
$330$	0	0
$331$	−20.7807	−1.14221	−0.571104	−	0.820878i	$-0.693485\pi$
$331$	−20.7807	−1.14221	−0.571104	+	0.820878i	$0.693485\pi$
$332$	7.31366	0.401389
$333$	0	0
$334$	−44.4741	−2.43351
$335$	−1.79185	−0.0978991
$336$	0	0
$337$	−33.7717	−1.83966	−0.919831	−	0.392315i	$-0.871674\pi$
$337$	−33.7717	−1.83966	−0.919831	+	0.392315i	$0.871674\pi$
$338$	−13.1687	−0.716280
$339$	0	0
$340$	−13.8134	−0.749137
$341$	−1.42084	−0.0769429
$342$	0	0
$343$	0	0
$344$	19.4683	1.04966
$345$	0	0
$346$	−54.8253	−2.94742
$347$	29.3453	1.57534	0.787669	−	0.616098i	$-0.211288\pi$
$347$	29.3453	1.57534	0.787669	+	0.616098i	$0.211288\pi$
$348$	0	0
$349$	−3.65107	−0.195437	−0.0977187	−	0.995214i	$-0.531155\pi$
$349$	−3.65107	−0.195437	−0.0977187	+	0.995214i	$0.531155\pi$
$350$	0	0
$351$	0	0
$352$	2.70567	0.144213
$353$	15.9837	0.850727	0.425363	−	0.905023i	$-0.360146\pi$
$353$	15.9837	0.850727	0.425363	+	0.905023i	$0.360146\pi$
$354$	0	0
$355$	−13.7902	−0.731908
$356$	14.4487	0.765781
$357$	0	0
$358$	−2.83924	−0.150059
$359$	−25.0145	−1.32021	−0.660107	−	0.751172i	$-0.729489\pi$
$359$	−25.0145	−1.32021	−0.660107	+	0.751172i	$0.729489\pi$
$360$	0	0
$361$	23.1263	1.21718
$362$	31.4268	1.65176
$363$	0	0
$364$	0	0
$365$	−2.60410	−0.136305
$366$	0	0
$367$	3.60702	0.188285	0.0941425	−	0.995559i	$-0.469989\pi$
$367$	3.60702	0.188285	0.0941425	+	0.995559i	$0.469989\pi$
$368$	−1.56513	−0.0815881
$369$	0	0
$370$	23.4404	1.21861
$371$	0	0
$372$	0	0
$373$	34.6721	1.79525	0.897627	−	0.440755i	$-0.145289\pi$
$373$	34.6721	1.79525	0.897627	+	0.440755i	$0.145289\pi$
$374$	4.10095	0.212055
$375$	0	0
$376$	18.9403	0.976772
$377$	−11.7924	−0.607340
$378$	0	0
$379$	30.8892	1.58667	0.793336	−	0.608785i	$-0.208343\pi$
$379$	30.8892	1.58667	0.793336	+	0.608785i	$0.208343\pi$
$380$	19.2079	0.985346
$381$	0	0
$382$	49.7453	2.54519
$383$	−21.4578	−1.09644	−0.548222	−	0.836333i	$-0.684695\pi$
$383$	−21.4578	−1.09644	−0.548222	+	0.836333i	$0.684695\pi$
$384$	0	0
$385$	0	0
$386$	−15.9148	−0.810042
$387$	0	0
$388$	44.3358	2.25081
$389$	37.4159	1.89706	0.948530	−	0.316688i	$-0.102571\pi$
$389$	37.4159	1.89706	0.948530	+	0.316688i	$0.102571\pi$
$390$	0	0
$391$	−6.29375	−0.318288
$392$	0	0
$393$	0	0
$394$	−33.9483	−1.71029
$395$	−8.21605	−0.413395
$396$	0	0
$397$	0.485327	0.0243579	0.0121789	−	0.999926i	$-0.496123\pi$
$397$	0.485327	0.0243579	0.0121789	+	0.999926i	$0.496123\pi$
$398$	−9.66296	−0.484361
$399$	0	0
$400$	−1.16075	−0.0580375
$401$	−14.7986	−0.739004	−0.369502	−	0.929230i	$-0.620472\pi$
$401$	−14.7986	−0.739004	−0.369502	+	0.929230i	$0.620472\pi$
$402$	0	0
$403$	9.58731	0.477578
$404$	19.8905	0.989589
$405$	0	0
$406$	0	0
$407$	−4.15264	−0.205838
$408$	0	0
$409$	−8.26077	−0.408469	−0.204234	−	0.978922i	$-0.565470\pi$
$409$	−8.26077	−0.408469	−0.204234	+	0.978922i	$0.565470\pi$
$410$	10.1120	0.499395
$411$	0	0
$412$	−4.36776	−0.215184
$413$	0	0
$414$	0	0
$415$	2.47133	0.121313
$416$	−18.2568	−0.895115
$417$	0	0
$418$	−5.70248	−0.278918
$419$	5.11517	0.249892	0.124946	−	0.992164i	$-0.460124\pi$
$419$	5.11517	0.249892	0.124946	+	0.992164i	$0.460124\pi$
$420$	0	0
$421$	−34.3722	−1.67520	−0.837599	−	0.546285i	$-0.816041\pi$
$421$	−34.3722	−1.67520	−0.837599	+	0.546285i	$0.816041\pi$
$422$	−33.1351	−1.61299
$423$	0	0
$424$	−6.57964	−0.319535
$425$	−4.66764	−0.226414
$426$	0	0
$427$	0	0
$428$	30.2461	1.46200
$429$	0	0
$430$	20.2921	0.978572
$431$	22.4572	1.08173	0.540864	−	0.841110i	$-0.318097\pi$
$431$	22.4572	1.08173	0.540864	+	0.841110i	$0.318097\pi$
$432$	0	0
$433$	−19.1810	−0.921780	−0.460890	−	0.887457i	$-0.652470\pi$
$433$	−19.1810	−0.921780	−0.460890	+	0.887457i	$0.652470\pi$
$434$	0	0
$435$	0	0
$436$	−11.0584	−0.529603
$437$	8.75163	0.418647
$438$	0	0
$439$	−8.39337	−0.400594	−0.200297	−	0.979735i	$-0.564191\pi$
$439$	−8.39337	−0.400594	−0.200297	+	0.979735i	$0.564191\pi$
$440$	−0.842921	−0.0401847
$441$	0	0
$442$	−27.6717	−1.31621
$443$	−32.7267	−1.55489	−0.777447	−	0.628949i	$-0.783486\pi$
$443$	−32.7267	−1.55489	−0.777447	+	0.628949i	$0.783486\pi$
$444$	0	0
$445$	4.88232	0.231444
$446$	−7.06748	−0.334655
$447$	0	0
$448$	0	0
$449$	17.2817	0.815575	0.407787	−	0.913077i	$-0.366300\pi$
$449$	17.2817	0.815575	0.407787	+	0.913077i	$0.366300\pi$
$450$	0	0
$451$	−1.79141	−0.0843540
$452$	−16.8644	−0.793234
$453$	0	0
$454$	−0.188867	−0.00886399
$455$	0	0
$456$	0	0
$457$	−17.3001	−0.809267	−0.404633	−	0.914479i	$-0.632601\pi$
$457$	−17.3001	−0.809267	−0.404633	+	0.914479i	$0.632601\pi$
$458$	53.3696	2.49380
$459$	0	0
$460$	3.99040	0.186053
$461$	−17.8514	−0.831424	−0.415712	−	0.909496i	$-0.636468\pi$
$461$	−17.8514	−0.831424	−0.415712	+	0.909496i	$0.636468\pi$
$462$	0	0
$463$	2.75459	0.128017	0.0640083	−	0.997949i	$-0.479612\pi$
$463$	2.75459	0.128017	0.0640083	+	0.997949i	$0.479612\pi$
$464$	5.14183	0.238703
$465$	0	0
$466$	17.9389	0.831005
$467$	−11.5696	−0.535377	−0.267688	−	0.963506i	$-0.586260\pi$
$467$	−11.5696	−0.535377	−0.267688	+	0.963506i	$0.586260\pi$
$468$	0	0
$469$	0	0
$470$	19.7418	0.910622
$471$	0	0
$472$	8.86636	0.408107
$473$	−3.59489	−0.165293
$474$	0	0
$475$	6.49048	0.297804
$476$	0	0
$477$	0	0
$478$	59.3096	2.71276
$479$	−7.34741	−0.335712	−0.167856	−	0.985812i	$-0.553684\pi$
$479$	−7.34741	−0.335712	−0.167856	+	0.985812i	$0.553684\pi$
$480$	0	0
$481$	28.0204	1.27762
$482$	7.31905	0.333373
$483$	0	0
$484$	−32.0928	−1.45876
$485$	14.9814	0.680268
$486$	0	0
$487$	−6.68491	−0.302922	−0.151461	−	0.988463i	$-0.548398\pi$
$487$	−6.68491	−0.302922	−0.151461	+	0.988463i	$0.548398\pi$
$488$	30.1776	1.36607
$489$	0	0
$490$	0	0
$491$	−20.0129	−0.903170	−0.451585	−	0.892228i	$-0.649141\pi$
$491$	−20.0129	−0.903170	−0.451585	+	0.892228i	$0.649141\pi$
$492$	0	0
$493$	20.6765	0.931221
$494$	38.4782	1.73122
$495$	0	0
$496$	−4.18034	−0.187703
$497$	0	0
$498$	0	0
$499$	−7.63037	−0.341582	−0.170791	−	0.985307i	$-0.554632\pi$
$499$	−7.63037	−0.341582	−0.170791	+	0.985307i	$0.554632\pi$
$500$	2.95940	0.132348
$501$	0	0
$502$	55.7345	2.48755
$503$	5.88626	0.262455	0.131228	−	0.991352i	$-0.458108\pi$
$503$	5.88626	0.262455	0.131228	+	0.991352i	$0.458108\pi$
$504$	0	0
$505$	6.72112	0.299086
$506$	−1.18467	−0.0526652
$507$	0	0
$508$	19.1042	0.847611
$509$	−33.9008	−1.50263	−0.751314	−	0.659945i	$-0.770580\pi$
$509$	−33.9008	−1.50263	−0.751314	+	0.659945i	$0.770580\pi$
$510$	0	0
$511$	0	0
$512$	12.9205	0.571012
$513$	0	0
$514$	−63.5912	−2.80489
$515$	−1.47589	−0.0650357
$516$	0	0
$517$	−3.49740	−0.153815
$518$	0	0
$519$	0	0
$520$	5.68772	0.249423
$521$	3.40566	0.149205	0.0746024	−	0.997213i	$-0.476231\pi$
$521$	3.40566	0.149205	0.0746024	+	0.997213i	$0.476231\pi$
$522$	0	0
$523$	0.0239825	0.00104868	0.000524341	−	1.00000i	$-0.499833\pi$
$523$	0.0239825	0.00104868	0.000524341		1.00000i	$0.499833\pi$
$524$	−27.6940	−1.20982
$525$	0	0
$526$	−16.5777	−0.722822
$527$	−16.8101	−0.732260
$528$	0	0
$529$	−21.1819	−0.920951
$530$	−6.85807	−0.297896
$531$	0	0
$532$	0	0
$533$	12.0877	0.523578
$534$	0	0
$535$	10.2203	0.441864
$536$	−3.82839	−0.165361
$537$	0	0
$538$	−68.2614	−2.94296
$539$	0	0
$540$	0	0
$541$	32.0086	1.37616	0.688078	−	0.725637i	$-0.258455\pi$
$541$	32.0086	1.37616	0.688078	+	0.725637i	$0.258455\pi$
$542$	−55.0322	−2.36384
$543$	0	0
$544$	32.0110	1.37246
$545$	−3.73672	−0.160063
$546$	0	0
$547$	−21.2558	−0.908832	−0.454416	−	0.890790i	$-0.650152\pi$
$547$	−21.2558	−0.908832	−0.454416	+	0.890790i	$0.650152\pi$
$548$	−16.8239	−0.718683
$549$	0	0
$550$	−0.878592	−0.0374633
$551$	−28.7512	−1.22484
$552$	0	0
$553$	0	0
$554$	−53.6016	−2.27731
$555$	0	0
$556$	−16.2959	−0.691102
$557$	−5.10173	−0.216167	−0.108084	−	0.994142i	$-0.534471\pi$
$557$	−5.10173	−0.216167	−0.108084	+	0.994142i	$0.534471\pi$
$558$	0	0
$559$	24.2569	1.02596
$560$	0	0
$561$	0	0
$562$	−13.6166	−0.574380
$563$	−36.3837	−1.53339	−0.766695	−	0.642011i	$-0.778100\pi$
$563$	−36.3837	−1.53339	−0.766695	+	0.642011i	$0.778100\pi$
$564$	0	0
$565$	−5.69858	−0.239741
$566$	47.2353	1.98545
$567$	0	0
$568$	−29.4636	−1.23626
$569$	−13.3680	−0.560417	−0.280209	−	0.959939i	$-0.590404\pi$
$569$	−13.3680	−0.560417	−0.280209	+	0.959939i	$0.590404\pi$
$570$	0	0
$571$	25.8072	1.08000	0.539999	−	0.841665i	$-0.318424\pi$
$571$	25.8072	1.08000	0.539999	+	0.841665i	$0.318424\pi$
$572$	−3.10813	−0.129958
$573$	0	0
$574$	0	0
$575$	1.34838	0.0562313
$576$	0	0
$577$	−1.18133	−0.0491794	−0.0245897	−	0.999698i	$-0.507828\pi$
$577$	−1.18133	−0.0491794	−0.0245897	+	0.999698i	$0.507828\pi$
$578$	10.6602	0.443404
$579$	0	0
$580$	−13.1094	−0.544338
$581$	0	0
$582$	0	0
$583$	1.21495	0.0503183
$584$	−5.56382	−0.230232
$585$	0	0
$586$	69.5059	2.87126
$587$	−14.0304	−0.579096	−0.289548	−	0.957164i	$-0.593505\pi$
$587$	−14.0304	−0.579096	−0.289548	+	0.957164i	$0.593505\pi$
$588$	0	0
$589$	23.3749	0.963147
$590$	9.24156	0.380469
$591$	0	0
$592$	−12.2177	−0.502145
$593$	1.96796	0.0808145	0.0404073	−	0.999183i	$-0.487134\pi$
$593$	1.96796	0.0808145	0.0404073	+	0.999183i	$0.487134\pi$
$594$	0	0
$595$	0	0
$596$	56.8941	2.33047
$597$	0	0
$598$	7.99374	0.326888
$599$	16.2471	0.663839	0.331920	−	0.943308i	$-0.392304\pi$
$599$	16.2471	0.663839	0.331920	+	0.943308i	$0.392304\pi$
$600$	0	0
$601$	4.50195	0.183638	0.0918192	−	0.995776i	$-0.470732\pi$
$601$	4.50195	0.183638	0.0918192	+	0.995776i	$0.470732\pi$
$602$	0	0
$603$	0	0
$604$	−26.6732	−1.08532
$605$	−10.8444	−0.440886
$606$	0	0
$607$	−45.5950	−1.85065	−0.925323	−	0.379180i	$-0.876206\pi$
$607$	−45.5950	−1.85065	−0.925323	+	0.379180i	$0.876206\pi$
$608$	−44.5122	−1.80521
$609$	0	0
$610$	31.4546	1.27356
$611$	23.5991	0.954718
$612$	0	0
$613$	16.6182	0.671203	0.335601	−	0.942004i	$-0.391060\pi$
$613$	16.6182	0.671203	0.335601	+	0.942004i	$0.391060\pi$
$614$	47.4908	1.91657
$615$	0	0
$616$	0	0
$617$	−18.9959	−0.764746	−0.382373	−	0.924008i	$-0.624893\pi$
$617$	−18.9959	−0.764746	−0.382373	+	0.924008i	$0.624893\pi$
$618$	0	0
$619$	11.9723	0.481208	0.240604	−	0.970623i	$-0.422655\pi$
$619$	11.9723	0.481208	0.240604	+	0.970623i	$0.422655\pi$
$620$	10.6580	0.428037
$621$	0	0
$622$	−7.20140	−0.288750
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0.229713	0.00918118
$627$	0	0
$628$	−52.6690	−2.10172
$629$	−49.1302	−1.95895
$630$	0	0
$631$	−40.0918	−1.59603	−0.798015	−	0.602638i	$-0.794116\pi$
$631$	−40.0918	−1.59603	−0.798015	+	0.602638i	$0.794116\pi$
$632$	−17.5541	−0.698264
$633$	0	0
$634$	6.45757	0.256463
$635$	6.45543	0.256176
$636$	0	0
$637$	0	0
$638$	3.89194	0.154083
$639$	0	0
$640$	−15.1259	−0.597902
$641$	−36.4785	−1.44082	−0.720408	−	0.693551i	$-0.756045\pi$
$641$	−36.4785	−1.44082	−0.720408	+	0.693551i	$0.756045\pi$
$642$	0	0
$643$	36.7751	1.45027	0.725134	−	0.688608i	$-0.241778\pi$
$643$	36.7751	1.45027	0.725134	+	0.688608i	$0.241778\pi$
$644$	0	0
$645$	0	0
$646$	−67.4666	−2.65444
$647$	−25.1900	−0.990323	−0.495161	−	0.868801i	$-0.664891\pi$
$647$	−25.1900	−0.990323	−0.495161	+	0.868801i	$0.664891\pi$
$648$	0	0
$649$	−1.63721	−0.0642659
$650$	5.92841	0.232531
$651$	0	0
$652$	20.5541	0.804960
$653$	−31.6250	−1.23758	−0.618791	−	0.785556i	$-0.712377\pi$
$653$	−31.6250	−1.23758	−0.618791	+	0.785556i	$0.712377\pi$
$654$	0	0
$655$	−9.35798	−0.365647
$656$	−5.27060	−0.205782
$657$	0	0
$658$	0	0
$659$	21.6764	0.844392	0.422196	−	0.906505i	$-0.361259\pi$
$659$	21.6764	0.844392	0.422196	+	0.906505i	$0.361259\pi$
$660$	0	0
$661$	26.3034	1.02308	0.511542	−	0.859258i	$-0.329074\pi$
$661$	26.3034	1.02308	0.511542	+	0.859258i	$0.329074\pi$
$662$	−46.2779	−1.79864
$663$	0	0
$664$	5.28014	0.204909
$665$	0	0
$666$	0	0
$667$	−5.97298	−0.231275
$668$	−59.1012	−2.28669
$669$	0	0
$670$	−3.99040	−0.154162
$671$	−5.57240	−0.215120
$672$	0	0
$673$	40.3549	1.55557	0.777784	−	0.628532i	$-0.216344\pi$
$673$	40.3549	1.55557	0.777784	+	0.628532i	$0.216344\pi$
$674$	−75.2086	−2.89693
$675$	0	0
$676$	−17.4997	−0.673065
$677$	18.2397	0.701009	0.350504	−	0.936561i	$-0.386010\pi$
$677$	18.2397	0.701009	0.350504	+	0.936561i	$0.386010\pi$
$678$	0	0
$679$	0	0
$680$	−9.97268	−0.382435
$681$	0	0
$682$	−3.16417	−0.121162
$683$	−36.2167	−1.38579	−0.692896	−	0.721037i	$-0.743666\pi$
$683$	−36.2167	−1.38579	−0.692896	+	0.721037i	$0.743666\pi$
$684$	0	0
$685$	−5.68491	−0.217209
$686$	0	0
$687$	0	0
$688$	−10.5767	−0.403234
$689$	−8.19805	−0.312321
$690$	0	0
$691$	−24.3214	−0.925231	−0.462615	−	0.886559i	$-0.653089\pi$
$691$	−24.3214	−0.925231	−0.462615	+	0.886559i	$0.653089\pi$
$692$	−72.8568	−2.76960
$693$	0	0
$694$	65.3512	2.48070
$695$	−5.50650	−0.208873
$696$	0	0
$697$	−21.1943	−0.802790
$698$	−8.13083	−0.307757
$699$	0	0
$700$	0	0
$701$	34.3460	1.29723	0.648615	−	0.761117i	$-0.275349\pi$
$701$	34.3460	1.29723	0.648615	+	0.761117i	$0.275349\pi$
$702$	0	0
$703$	68.3169	2.57662
$704$	5.10956	0.192574
$705$	0	0
$706$	35.5952	1.33964
$707$	0	0
$708$	0	0
$709$	31.3994	1.17923	0.589614	−	0.807685i	$-0.299280\pi$
$709$	31.3994	1.17923	0.589614	+	0.807685i	$0.299280\pi$
$710$	−30.7104	−1.15254
$711$	0	0
$712$	10.4313	0.390931
$713$	4.85608	0.181861
$714$	0	0
$715$	−1.05026	−0.0392774
$716$	−3.77304	−0.141005
$717$	0	0
$718$	−55.7065	−2.07895
$719$	−4.02717	−0.150188	−0.0750940	−	0.997176i	$-0.523926\pi$
$719$	−4.02717	−0.150188	−0.0750940	+	0.997176i	$0.523926\pi$
$720$	0	0
$721$	0	0
$722$	51.5017	1.91669
$723$	0	0
$724$	41.7628	1.55210
$725$	−4.42975	−0.164517
$726$	0	0
$727$	−9.17871	−0.340419	−0.170210	−	0.985408i	$-0.554445\pi$
$727$	−9.17871	−0.340419	−0.170210	+	0.985408i	$0.554445\pi$
$728$	0	0
$729$	0	0
$730$	−5.79927	−0.214640
$731$	−42.5314	−1.57308
$732$	0	0
$733$	15.4081	0.569113	0.284556	−	0.958659i	$-0.408154\pi$
$733$	15.4081	0.569113	0.284556	+	0.958659i	$0.408154\pi$
$734$	8.03273	0.296494
$735$	0	0
$736$	−9.24728	−0.340859
$737$	0.706926	0.0260399
$738$	0	0
$739$	18.6796	0.687140	0.343570	−	0.939127i	$-0.388364\pi$
$739$	18.6796	0.687140	0.343570	+	0.939127i	$0.388364\pi$
$740$	31.1498	1.14509
$741$	0	0
$742$	0	0
$743$	−38.2524	−1.40335	−0.701673	−	0.712499i	$-0.747563\pi$
$743$	−38.2524	−1.40335	−0.701673	+	0.712499i	$0.747563\pi$
$744$	0	0
$745$	19.2249	0.704345
$746$	77.2138	2.82700
$747$	0	0
$748$	5.44971	0.199261
$749$	0	0
$750$	0	0
$751$	−28.4641	−1.03867	−0.519335	−	0.854571i	$-0.673820\pi$
$751$	−28.4641	−1.03867	−0.519335	+	0.854571i	$0.673820\pi$
$752$	−10.2899	−0.375234
$753$	0	0
$754$	−26.2614	−0.956382
$755$	−9.01306	−0.328019
$756$	0	0
$757$	11.0135	0.400294	0.200147	−	0.979766i	$-0.435858\pi$
$757$	11.0135	0.400294	0.200147	+	0.979766i	$0.435858\pi$
$758$	68.7893	2.49854
$759$	0	0
$760$	13.8673	0.503019
$761$	−7.10595	−0.257591	−0.128795	−	0.991671i	$-0.541111\pi$
$761$	−7.10595	−0.257591	−0.128795	+	0.991671i	$0.541111\pi$
$762$	0	0
$763$	0	0
$764$	66.1060	2.39163
$765$	0	0
$766$	−47.7860	−1.72658
$767$	11.0472	0.398893
$768$	0	0
$769$	27.8897	1.00573	0.502863	−	0.864366i	$-0.332280\pi$
$769$	27.8897	1.00573	0.502863	+	0.864366i	$0.332280\pi$
$770$	0	0
$771$	0	0
$772$	−21.1490	−0.761170
$773$	−15.1410	−0.544584	−0.272292	−	0.962215i	$-0.587782\pi$
$773$	−15.1410	−0.544584	−0.272292	+	0.962215i	$0.587782\pi$
$774$	0	0
$775$	3.60142	0.129367
$776$	32.0085	1.14904
$777$	0	0
$778$	83.3241	2.98731
$779$	29.4712	1.05592
$780$	0	0
$781$	5.44055	0.194678
$782$	−14.0160	−0.501211
$783$	0	0
$784$	0	0
$785$	−17.7972	−0.635208
$786$	0	0
$787$	−35.3303	−1.25939	−0.629695	−	0.776842i	$-0.716820\pi$
$787$	−35.3303	−1.25939	−0.629695	+	0.776842i	$0.716820\pi$
$788$	−45.1135	−1.60710
$789$	0	0
$790$	−18.2969	−0.650975
$791$	0	0
$792$	0	0
$793$	37.6004	1.33523
$794$	1.08081	0.0383565
$795$	0	0
$796$	−12.8410	−0.455138
$797$	−3.50359	−0.124104	−0.0620518	−	0.998073i	$-0.519764\pi$
$797$	−3.50359	−0.124104	−0.0620518	+	0.998073i	$0.519764\pi$
$798$	0	0
$799$	−41.3780	−1.46385
$800$	−6.85807	−0.242469
$801$	0	0
$802$	−32.9559	−1.16371
$803$	1.02738	0.0362554
$804$	0	0
$805$	0	0
$806$	21.3507	0.752045
$807$	0	0
$808$	14.3601	0.505185
$809$	32.5749	1.14527	0.572636	−	0.819810i	$-0.305921\pi$
$809$	32.5749	1.14527	0.572636	+	0.819810i	$0.305921\pi$
$810$	0	0
$811$	0.141435	0.00496647	0.00248323	−	0.999997i	$-0.499210\pi$
$811$	0.141435	0.00496647	0.00248323	+	0.999997i	$0.499210\pi$
$812$	0	0
$813$	0	0
$814$	−9.24780	−0.324135
$815$	6.94535	0.243285
$816$	0	0
$817$	59.1411	2.06909
$818$	−18.3965	−0.643219
$819$	0	0
$820$	13.4377	0.469265
$821$	28.2754	0.986820	0.493410	−	0.869797i	$-0.335750\pi$
$821$	28.2754	0.986820	0.493410	+	0.869797i	$0.335750\pi$
$822$	0	0
$823$	5.16993	0.180213	0.0901063	−	0.995932i	$-0.471279\pi$
$823$	5.16993	0.180213	0.0901063	+	0.995932i	$0.471279\pi$
$824$	−3.15333	−0.109852
$825$	0	0
$826$	0	0
$827$	−9.78331	−0.340199	−0.170099	−	0.985427i	$-0.554409\pi$
$827$	−9.78331	−0.340199	−0.170099	+	0.985427i	$0.554409\pi$
$828$	0	0
$829$	27.6786	0.961318	0.480659	−	0.876908i	$-0.340397\pi$
$829$	27.6786	0.961318	0.480659	+	0.876908i	$0.340397\pi$
$830$	5.50358	0.191032
$831$	0	0
$832$	−34.4774	−1.19529
$833$	0	0
$834$	0	0
$835$	−19.9707	−0.691113
$836$	−7.57797	−0.262090
$837$	0	0
$838$	11.3913	0.393507
$839$	18.4377	0.636541	0.318270	−	0.948000i	$-0.396898\pi$
$839$	18.4377	0.636541	0.318270	+	0.948000i	$0.396898\pi$
$840$	0	0
$841$	−9.37731	−0.323356
$842$	−76.5459	−2.63795
$843$	0	0
$844$	−44.0330	−1.51568
$845$	−5.91326	−0.203422
$846$	0	0
$847$	0	0
$848$	3.57459	0.122752
$849$	0	0
$850$	−10.3947	−0.356535
$851$	14.1926	0.486518
$852$	0	0
$853$	−25.1736	−0.861928	−0.430964	−	0.902369i	$-0.641826\pi$
$853$	−25.1736	−0.861928	−0.430964	+	0.902369i	$0.641826\pi$
$854$	0	0
$855$	0	0
$856$	21.8363	0.746351
$857$	20.2904	0.693107	0.346553	−	0.938030i	$-0.387352\pi$
$857$	20.2904	0.693107	0.346553	+	0.938030i	$0.387352\pi$
$858$	0	0
$859$	32.6046	1.11246	0.556228	−	0.831030i	$-0.312248\pi$
$859$	32.6046	1.11246	0.556228	+	0.831030i	$0.312248\pi$
$860$	26.9660	0.919532
$861$	0	0
$862$	50.0116	1.70340
$863$	51.3186	1.74691	0.873453	−	0.486908i	$-0.161875\pi$
$863$	51.3186	1.74691	0.873453	+	0.486908i	$0.161875\pi$
$864$	0	0
$865$	−24.6188	−0.837063
$866$	−42.7155	−1.45153
$867$	0	0
$868$	0	0
$869$	3.24142	0.109958
$870$	0	0
$871$	−4.77007	−0.161628
$872$	−7.98371	−0.270362
$873$	0	0
$874$	19.4896	0.659247
$875$	0	0
$876$	0	0
$877$	−4.02848	−0.136032	−0.0680161	−	0.997684i	$-0.521667\pi$
$877$	−4.02848	−0.136032	−0.0680161	+	0.997684i	$0.521667\pi$
$878$	−18.6918	−0.630818
$879$	0	0
$880$	0.457943	0.0154372
$881$	−12.1103	−0.408005	−0.204003	−	0.978970i	$-0.565395\pi$
$881$	−12.1103	−0.408005	−0.204003	+	0.978970i	$0.565395\pi$
$882$	0	0
$883$	56.9539	1.91665	0.958326	−	0.285677i	$-0.0922184\pi$
$883$	56.9539	1.91665	0.958326	+	0.285677i	$0.0922184\pi$
$884$	−36.7726	−1.23680
$885$	0	0
$886$	−72.8815	−2.44850
$887$	−6.21062	−0.208532	−0.104266	−	0.994549i	$-0.533249\pi$
$887$	−6.21062	−0.208532	−0.104266	+	0.994549i	$0.533249\pi$
$888$	0	0
$889$	0	0
$890$	10.8728	0.364456
$891$	0	0
$892$	−9.39190	−0.314464
$893$	57.5373	1.92541
$894$	0	0
$895$	−1.27493	−0.0426164
$896$	0	0
$897$	0	0
$898$	38.4859	1.28429
$899$	−15.9534	−0.532075
$900$	0	0
$901$	14.3742	0.478875
$902$	−3.98941	−0.132833
$903$	0	0
$904$	−12.1753	−0.404946
$905$	14.1119	0.469096
$906$	0	0
$907$	−23.4658	−0.779169	−0.389585	−	0.920991i	$-0.627381\pi$
$907$	−23.4658	−0.779169	−0.389585	+	0.920991i	$0.627381\pi$
$908$	−0.250984	−0.00832920
$909$	0	0
$910$	0	0
$911$	−28.8860	−0.957036	−0.478518	−	0.878078i	$-0.658826\pi$
$911$	−28.8860	−0.957036	−0.478518	+	0.878078i	$0.658826\pi$
$912$	0	0
$913$	−0.974998	−0.0322677
$914$	−38.5269	−1.27436
$915$	0	0
$916$	70.9224	2.34334
$917$	0	0
$918$	0	0
$919$	53.4177	1.76209	0.881044	−	0.473033i	$-0.156841\pi$
$919$	53.4177	1.76209	0.881044	+	0.473033i	$0.156841\pi$
$920$	2.88089	0.0949801
$921$	0	0
$922$	−39.7546	−1.30925
$923$	−36.7108	−1.20835
$924$	0	0
$925$	10.5257	0.346083
$926$	6.13439	0.201589
$927$	0	0
$928$	30.3795	0.997257
$929$	31.9144	1.04708	0.523539	−	0.852001i	$-0.324611\pi$
$929$	31.9144	1.04708	0.523539	+	0.852001i	$0.324611\pi$
$930$	0	0
$931$	0	0
$932$	23.8389	0.780868
$933$	0	0
$934$	−25.7651	−0.843061
$935$	1.84149	0.0602232
$936$	0	0
$937$	9.39523	0.306929	0.153464	−	0.988154i	$-0.450957\pi$
$937$	9.39523	0.306929	0.153464	+	0.988154i	$0.450957\pi$
$938$	0	0
$939$	0	0
$940$	26.2347	0.855682
$941$	−48.8782	−1.59338	−0.796691	−	0.604386i	$-0.793418\pi$
$941$	−48.8782	−1.59338	−0.796691	+	0.604386i	$0.793418\pi$
$942$	0	0
$943$	6.12257	0.199378
$944$	−4.81692	−0.156777
$945$	0	0
$946$	−8.00571	−0.260288
$947$	−26.1425	−0.849517	−0.424759	−	0.905307i	$-0.639641\pi$
$947$	−26.1425	−0.849517	−0.424759	+	0.905307i	$0.639641\pi$
$948$	0	0
$949$	−6.93237	−0.225034
$950$	14.4541	0.468953
$951$	0	0
$952$	0	0
$953$	−12.2195	−0.395827	−0.197913	−	0.980219i	$-0.563417\pi$
$953$	−12.2195	−0.395827	−0.197913	+	0.980219i	$0.563417\pi$
$954$	0	0
$955$	22.3376	0.722829
$956$	78.8160	2.54909
$957$	0	0
$958$	−16.3625	−0.528647
$959$	0	0
$960$	0	0
$961$	−18.0298	−0.581607
$962$	62.4007	2.01188
$963$	0	0
$964$	9.72621	0.313260
$965$	−7.14639	−0.230050
$966$	0	0
$967$	−54.0385	−1.73776	−0.868880	−	0.495023i	$-0.835160\pi$
$967$	−54.0385	−1.73776	−0.868880	+	0.495023i	$0.835160\pi$
$968$	−23.1696	−0.744698
$969$	0	0
$970$	33.3630	1.07122
$971$	−52.7144	−1.69169	−0.845843	−	0.533433i	$-0.820902\pi$
$971$	−52.7144	−1.69169	−0.845843	+	0.533433i	$0.820902\pi$
$972$	0	0
$973$	0	0
$974$	−14.8871	−0.477014
$975$	0	0
$976$	−16.3949	−0.524787
$977$	−22.1436	−0.708436	−0.354218	−	0.935163i	$-0.615253\pi$
$977$	−22.1436	−0.708436	−0.354218	+	0.935163i	$0.615253\pi$
$978$	0	0
$979$	−1.92619	−0.0615612
$980$	0	0
$981$	0	0
$982$	−44.5682	−1.42223
$983$	−37.6256	−1.20007	−0.600036	−	0.799973i	$-0.704847\pi$
$983$	−37.6256	−1.20007	−0.600036	+	0.799973i	$0.704847\pi$
$984$	0	0
$985$	−15.2441	−0.485719
$986$	46.0459	1.46640
$987$	0	0
$988$	51.1333	1.62677
$989$	12.2864	0.390685
$990$	0	0
$991$	−13.3587	−0.424353	−0.212177	−	0.977231i	$-0.568055\pi$
$991$	−13.3587	−0.424353	−0.212177	+	0.977231i	$0.568055\pi$
$992$	−24.6988	−0.784187
$993$	0	0
$994$	0	0
$995$	−4.33906	−0.137557
$996$	0	0
$997$	61.0322	1.93291	0.966454	−	0.256840i	$-0.0826814\pi$
$997$	61.0322	1.93291	0.966454	+	0.256840i	$0.0826814\pi$
$998$	−16.9926	−0.537892
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6615.2.a.br.1.5		5
3.2	odd	2		6615.2.a.bk.1.1		5
7.2	even	3		945.2.j.e.676.1	yes	10
7.4	even	3		945.2.j.e.541.1	✓	10
7.6	odd	2		6615.2.a.bn.1.5		5
21.2	odd	6		945.2.j.g.676.5	yes	10
21.11	odd	6		945.2.j.g.541.5	yes	10
21.20	even	2		6615.2.a.bo.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
945.2.j.e.541.1	✓	10	7.4	even	3
945.2.j.e.676.1	yes	10	7.2	even	3
945.2.j.g.541.5	yes	10	21.11	odd	6
945.2.j.g.676.5	yes	10	21.2	odd	6
6615.2.a.bk.1.1		5	3.2	odd	2
6615.2.a.bn.1.5		5	7.6	odd	2
6615.2.a.bo.1.1		5	21.20	even	2
6615.2.a.br.1.5		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:	\( N \)	\(=\)	\( 6615 = 3^{3} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6615.a (trivial)

\(f(q)\)	\(=\)	\(q+2.22697 q^{2} +2.95940 q^{4} +1.00000 q^{5} +2.13656 q^{8} +O(q^{10})\)	\(q+2.22697 q^{2} +2.95940 q^{4} +1.00000 q^{5} +2.13656 q^{8} +2.22697 q^{10} -0.394523 q^{11} +2.66209 q^{13} -1.16075 q^{16} -4.66764 q^{17} +6.49048 q^{19} +2.95940 q^{20} -0.878592 q^{22} +1.34838 q^{23} +1.00000 q^{25} +5.92841 q^{26} -4.42975 q^{29} +3.60142 q^{31} -6.85807 q^{32} -10.3947 q^{34} +10.5257 q^{37} +14.4541 q^{38} +2.13656 q^{40} +4.54069 q^{41} +9.11198 q^{43} -1.16755 q^{44} +3.00280 q^{46} +8.86487 q^{47} +2.22697 q^{50} +7.87820 q^{52} -3.07955 q^{53} -0.394523 q^{55} -9.86493 q^{58} +4.14983 q^{59} +14.1244 q^{61} +8.02025 q^{62} -12.9512 q^{64} +2.66209 q^{65} -1.79185 q^{67} -13.8134 q^{68} -13.7902 q^{71} -2.60410 q^{73} +23.4404 q^{74} +19.2079 q^{76} -8.21605 q^{79} -1.16075 q^{80} +10.1120 q^{82} +2.47133 q^{83} -4.66764 q^{85} +20.2921 q^{86} -0.842921 q^{88} +4.88232 q^{89} +3.99040 q^{92} +19.7418 q^{94} +6.49048 q^{95} +14.9814 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{4} + 5 q^{5} + 3 q^{8}+O(q^{10}) \) 5 * q + 4 * q^4 + 5 * q^5 + 3 * q^8	\( 5 q + 4 q^{4} + 5 q^{5} + 3 q^{8} + 5 q^{11} + 6 q^{13} - 10 q^{16} - q^{17} + 7 q^{19} + 4 q^{20} + 4 q^{22} + 4 q^{23} + 5 q^{25} + 7 q^{26} + 12 q^{29} + 11 q^{31} - 4 q^{32} + 8 q^{34} - 3 q^{38} + 3 q^{40} + 7 q^{41} + 2 q^{43} - 10 q^{44} + 14 q^{46} + 14 q^{47} + 7 q^{52} + 2 q^{53} + 5 q^{55} - 28 q^{58} + 2 q^{59} + 25 q^{61} - 24 q^{62} - 17 q^{64} + 6 q^{65} + 6 q^{67} - 14 q^{68} - 15 q^{71} + 17 q^{73} + 29 q^{74} + 34 q^{76} - 7 q^{79} - 10 q^{80} + 7 q^{82} + 2 q^{83} - q^{85} + 7 q^{86} + 10 q^{88} + 15 q^{89} + 21 q^{92} + 21 q^{94} + 7 q^{95} + 48 q^{97}+O(q^{100}) \) 5 * q + 4 * q^4 + 5 * q^5 + 3 * q^8 + 5 * q^11 + 6 * q^13 - 10 * q^16 - q^17 + 7 * q^19 + 4 * q^20 + 4 * q^22 + 4 * q^23 + 5 * q^25 + 7 * q^26 + 12 * q^29 + 11 * q^31 - 4 * q^32 + 8 * q^34 - 3 * q^38 + 3 * q^40 + 7 * q^41 + 2 * q^43 - 10 * q^44 + 14 * q^46 + 14 * q^47 + 7 * q^52 + 2 * q^53 + 5 * q^55 - 28 * q^58 + 2 * q^59 + 25 * q^61 - 24 * q^62 - 17 * q^64 + 6 * q^65 + 6 * q^67 - 14 * q^68 - 15 * q^71 + 17 * q^73 + 29 * q^74 + 34 * q^76 - 7 * q^79 - 10 * q^80 + 7 * q^82 + 2 * q^83 - q^85 + 7 * q^86 + 10 * q^88 + 15 * q^89 + 21 * q^92 + 21 * q^94 + 7 * q^95 + 48 * q^97

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