LMFDB - Embedded newform 6534.2.a.ca.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6534,2,Mod(1,6534)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6534, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6534.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6534 = 2 \cdot 3^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6534.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:	$52.1742526807$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 594)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.16228$ of defining polynomial
Character	$\chi$	$=$	6534.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.16228 q^{5} -2.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +2.16228 q^{5} -2.00000 q^{7} +1.00000 q^{8} +2.16228 q^{10} -4.16228 q^{13} -2.00000 q^{14} +1.00000 q^{16} +7.32456 q^{17} -6.32456 q^{19} +2.16228 q^{20} -2.16228 q^{23} -0.324555 q^{25} -4.16228 q^{26} -2.00000 q^{28} -6.00000 q^{29} -8.32456 q^{31} +1.00000 q^{32} +7.32456 q^{34} -4.32456 q^{35} -2.32456 q^{37} -6.32456 q^{38} +2.16228 q^{40} -3.00000 q^{41} -12.3246 q^{43} -2.16228 q^{46} +6.00000 q^{47} -3.00000 q^{49} -0.324555 q^{50} -4.16228 q^{52} -3.83772 q^{53} -2.00000 q^{56} -6.00000 q^{58} +6.00000 q^{59} -5.83772 q^{61} -8.32456 q^{62} +1.00000 q^{64} -9.00000 q^{65} +3.32456 q^{67} +7.32456 q^{68} -4.32456 q^{70} +6.00000 q^{71} +8.32456 q^{73} -2.32456 q^{74} -6.32456 q^{76} +4.48683 q^{79} +2.16228 q^{80} -3.00000 q^{82} -7.32456 q^{83} +15.8377 q^{85} -12.3246 q^{86} +4.32456 q^{89} +8.32456 q^{91} -2.16228 q^{92} +6.00000 q^{94} -13.6754 q^{95} -2.67544 q^{97} -3.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 4 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{7} + 2 q^{8} - 2 q^{10} - 2 q^{13} - 4 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{20} + 2 q^{23} + 12 q^{25} - 2 q^{26} - 4 q^{28} - 12 q^{29} - 4 q^{31} + 2 q^{32} + 2 q^{34} + 4 q^{35} + 8 q^{37} - 2 q^{40} - 6 q^{41} - 12 q^{43} + 2 q^{46} + 12 q^{47} - 6 q^{49} + 12 q^{50} - 2 q^{52} - 14 q^{53} - 4 q^{56} - 12 q^{58} + 12 q^{59} - 18 q^{61} - 4 q^{62} + 2 q^{64} - 18 q^{65} - 6 q^{67} + 2 q^{68} + 4 q^{70} + 12 q^{71} + 4 q^{73} + 8 q^{74} - 10 q^{79} - 2 q^{80} - 6 q^{82} - 2 q^{83} + 38 q^{85} - 12 q^{86} - 4 q^{89} + 4 q^{91} + 2 q^{92} + 12 q^{94} - 40 q^{95} - 18 q^{97} - 6 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 4 * q^7 + 2 * q^8 - 2 * q^10 - 2 * q^13 - 4 * q^14 + 2 * q^16 + 2 * q^17 - 2 * q^20 + 2 * q^23 + 12 * q^25 - 2 * q^26 - 4 * q^28 - 12 * q^29 - 4 * q^31 + 2 * q^32 + 2 * q^34 + 4 * q^35 + 8 * q^37 - 2 * q^40 - 6 * q^41 - 12 * q^43 + 2 * q^46 + 12 * q^47 - 6 * q^49 + 12 * q^50 - 2 * q^52 - 14 * q^53 - 4 * q^56 - 12 * q^58 + 12 * q^59 - 18 * q^61 - 4 * q^62 + 2 * q^64 - 18 * q^65 - 6 * q^67 + 2 * q^68 + 4 * q^70 + 12 * q^71 + 4 * q^73 + 8 * q^74 - 10 * q^79 - 2 * q^80 - 6 * q^82 - 2 * q^83 + 38 * q^85 - 12 * q^86 - 4 * q^89 + 4 * q^91 + 2 * q^92 + 12 * q^94 - 40 * q^95 - 18 * q^97 - 6 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.16228	0.967000	0.483500	−	0.875344i	$-0.339365\pi$
$5$	2.16228	0.967000	0.483500	+	0.875344i	$0.339365\pi$
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	2.16228	0.683772
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−4.16228	−1.15441	−0.577204	−	0.816600i	$-0.695856\pi$
$13$	−4.16228	−1.15441	−0.577204	+	0.816600i	$0.695856\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	7.32456	1.77647	0.888233	−	0.459394i	$-0.151933\pi$
$17$	7.32456	1.77647	0.888233	+	0.459394i	$0.151933\pi$
$18$	0	0
$19$	−6.32456	−1.45095	−0.725476	−	0.688247i	$-0.758380\pi$
$19$	−6.32456	−1.45095	−0.725476	+	0.688247i	$0.758380\pi$
$20$	2.16228	0.483500
$21$	0	0
$22$	0	0
$23$	−2.16228	−0.450866	−0.225433	−	0.974259i	$-0.572380\pi$
$23$	−2.16228	−0.450866	−0.225433	+	0.974259i	$0.572380\pi$
$24$	0	0
$25$	−0.324555	−0.0649111
$26$	−4.16228	−0.816290
$27$	0	0
$28$	−2.00000	−0.377964
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−8.32456	−1.49513	−0.747567	−	0.664186i	$-0.768778\pi$
$31$	−8.32456	−1.49513	−0.747567	+	0.664186i	$0.768778\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	7.32456	1.25615
$35$	−4.32456	−0.730983
$36$	0	0
$37$	−2.32456	−0.382155	−0.191077	−	0.981575i	$-0.561198\pi$
$37$	−2.32456	−0.382155	−0.191077	+	0.981575i	$0.561198\pi$
$38$	−6.32456	−1.02598
$39$	0	0
$40$	2.16228	0.341886
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	−12.3246	−1.87948	−0.939739	−	0.341894i	$-0.888932\pi$
$43$	−12.3246	−1.87948	−0.939739	+	0.341894i	$0.888932\pi$
$44$	0	0
$45$	0	0
$46$	−2.16228	−0.318810
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	−0.324555	−0.0458991
$51$	0	0
$52$	−4.16228	−0.577204
$53$	−3.83772	−0.527152	−0.263576	−	0.964639i	$-0.584902\pi$
$53$	−3.83772	−0.527152	−0.263576	+	0.964639i	$0.584902\pi$
$54$	0	0
$55$	0	0
$56$	−2.00000	−0.267261
$57$	0	0
$58$	−6.00000	−0.787839
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	−5.83772	−0.747444	−0.373722	−	0.927541i	$-0.621919\pi$
$61$	−5.83772	−0.747444	−0.373722	+	0.927541i	$0.621919\pi$
$62$	−8.32456	−1.05722
$63$	0	0
$64$	1.00000	0.125000
$65$	−9.00000	−1.11631
$66$	0	0
$67$	3.32456	0.406159	0.203080	−	0.979162i	$-0.434905\pi$
$67$	3.32456	0.406159	0.203080	+	0.979162i	$0.434905\pi$
$68$	7.32456	0.888233
$69$	0	0
$70$	−4.32456	−0.516883
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	8.32456	0.974316	0.487158	−	0.873314i	$-0.338034\pi$
$73$	8.32456	0.974316	0.487158	+	0.873314i	$0.338034\pi$
$74$	−2.32456	−0.270224
$75$	0	0
$76$	−6.32456	−0.725476
$77$	0	0
$78$	0	0
$79$	4.48683	0.504808	0.252404	−	0.967622i	$-0.418779\pi$
$79$	4.48683	0.504808	0.252404	+	0.967622i	$0.418779\pi$
$80$	2.16228	0.241750
$81$	0	0
$82$	−3.00000	−0.331295
$83$	−7.32456	−0.803974	−0.401987	−	0.915645i	$-0.631680\pi$
$83$	−7.32456	−0.803974	−0.401987	+	0.915645i	$0.631680\pi$
$84$	0	0
$85$	15.8377	1.71784
$86$	−12.3246	−1.32899
$87$	0	0
$88$	0	0
$89$	4.32456	0.458402	0.229201	−	0.973379i	$-0.426389\pi$
$89$	4.32456	0.458402	0.229201	+	0.973379i	$0.426389\pi$
$90$	0	0
$91$	8.32456	0.872651
$92$	−2.16228	−0.225433
$93$	0	0
$94$	6.00000	0.618853
$95$	−13.6754	−1.40307
$96$	0	0
$97$	−2.67544	−0.271650	−0.135825	−	0.990733i	$-0.543369\pi$
$97$	−2.67544	−0.271650	−0.135825	+	0.990733i	$0.543369\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	−0.324555	−0.0324555
$101$	16.3246	1.62435	0.812177	−	0.583411i	$-0.198282\pi$
$101$	16.3246	1.62435	0.812177	+	0.583411i	$0.198282\pi$
$102$	0	0
$103$	−14.3246	−1.41144	−0.705720	−	0.708491i	$-0.749376\pi$
$103$	−14.3246	−1.41144	−0.705720	+	0.708491i	$0.749376\pi$
$104$	−4.16228	−0.408145
$105$	0	0
$106$	−3.83772	−0.372753
$107$	11.6491	1.12616	0.563081	−	0.826402i	$-0.309616\pi$
$107$	11.6491	1.12616	0.563081	+	0.826402i	$0.309616\pi$
$108$	0	0
$109$	2.32456	0.222652	0.111326	−	0.993784i	$-0.464490\pi$
$109$	2.32456	0.222652	0.111326	+	0.993784i	$0.464490\pi$
$110$	0	0
$111$	0	0
$112$	−2.00000	−0.188982
$113$	−16.3246	−1.53568	−0.767842	−	0.640639i	$-0.778670\pi$
$113$	−16.3246	−1.53568	−0.767842	+	0.640639i	$0.778670\pi$
$114$	0	0
$115$	−4.67544	−0.435987
$116$	−6.00000	−0.557086
$117$	0	0
$118$	6.00000	0.552345
$119$	−14.6491	−1.34288
$120$	0	0
$121$	0	0
$122$	−5.83772	−0.528523
$123$	0	0
$124$	−8.32456	−0.747567
$125$	−11.5132	−1.02977
$126$	0	0
$127$	−4.16228	−0.369342	−0.184671	−	0.982800i	$-0.559122\pi$
$127$	−4.16228	−0.369342	−0.184671	+	0.982800i	$0.559122\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−9.00000	−0.789352
$131$	−19.3246	−1.68839	−0.844197	−	0.536033i	$-0.819922\pi$
$131$	−19.3246	−1.68839	−0.844197	+	0.536033i	$0.819922\pi$
$132$	0	0
$133$	12.6491	1.09682
$134$	3.32456	0.287198
$135$	0	0
$136$	7.32456	0.628075
$137$	14.6491	1.25156	0.625779	−	0.780000i	$-0.284781\pi$
$137$	14.6491	1.25156	0.625779	+	0.780000i	$0.284781\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	−4.32456	−0.365492
$141$	0	0
$142$	6.00000	0.503509
$143$	0	0
$144$	0	0
$145$	−12.9737	−1.07740
$146$	8.32456	0.688945
$147$	0	0
$148$	−2.32456	−0.191077
$149$	4.32456	0.354281	0.177141	−	0.984186i	$-0.443315\pi$
$149$	4.32456	0.354281	0.177141	+	0.984186i	$0.443315\pi$
$150$	0	0
$151$	6.16228	0.501479	0.250740	−	0.968055i	$-0.419326\pi$
$151$	6.16228	0.501479	0.250740	+	0.968055i	$0.419326\pi$
$152$	−6.32456	−0.512989
$153$	0	0
$154$	0	0
$155$	−18.0000	−1.44579
$156$	0	0
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	4.48683	0.356953
$159$	0	0
$160$	2.16228	0.170943
$161$	4.32456	0.340823
$162$	0	0
$163$	−21.6491	−1.69569	−0.847845	−	0.530245i	$-0.822100\pi$
$163$	−21.6491	−1.69569	−0.847845	+	0.530245i	$0.822100\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	−7.32456	−0.568496
$167$	−12.9737	−1.00393	−0.501966	−	0.864887i	$-0.667390\pi$
$167$	−12.9737	−1.00393	−0.501966	+	0.864887i	$0.667390\pi$
$168$	0	0
$169$	4.32456	0.332658
$170$	15.8377	1.21470
$171$	0	0
$172$	−12.3246	−0.939739
$173$	16.3246	1.24113	0.620566	−	0.784154i	$-0.286903\pi$
$173$	16.3246	1.24113	0.620566	+	0.784154i	$0.286903\pi$
$174$	0	0
$175$	0.649111	0.0490682
$176$	0	0
$177$	0	0
$178$	4.32456	0.324139
$179$	−4.32456	−0.323232	−0.161616	−	0.986854i	$-0.551671\pi$
$179$	−4.32456	−0.323232	−0.161616	+	0.986854i	$0.551671\pi$
$180$	0	0
$181$	−12.6491	−0.940201	−0.470100	−	0.882613i	$-0.655782\pi$
$181$	−12.6491	−0.940201	−0.470100	+	0.882613i	$0.655782\pi$
$182$	8.32456	0.617057
$183$	0	0
$184$	−2.16228	−0.159405
$185$	−5.02633	−0.369543
$186$	0	0
$187$	0	0
$188$	6.00000	0.437595
$189$	0	0
$190$	−13.6754	−0.992121
$191$	0.486833	0.0352260	0.0176130	−	0.999845i	$-0.494393\pi$
$191$	0.486833	0.0352260	0.0176130	+	0.999845i	$0.494393\pi$
$192$	0	0
$193$	−0.324555	−0.0233620	−0.0116810	−	0.999932i	$-0.503718\pi$
$193$	−0.324555	−0.0233620	−0.0116810	+	0.999932i	$0.503718\pi$
$194$	−2.67544	−0.192086
$195$	0	0
$196$	−3.00000	−0.214286
$197$	20.6491	1.47119	0.735594	−	0.677423i	$-0.236903\pi$
$197$	20.6491	1.47119	0.735594	+	0.677423i	$0.236903\pi$
$198$	0	0
$199$	−10.9737	−0.777903	−0.388951	−	0.921258i	$-0.627163\pi$
$199$	−10.9737	−0.777903	−0.388951	+	0.921258i	$0.627163\pi$
$200$	−0.324555	−0.0229495
$201$	0	0
$202$	16.3246	1.14859
$203$	12.0000	0.842235
$204$	0	0
$205$	−6.48683	−0.453060
$206$	−14.3246	−0.998039
$207$	0	0
$208$	−4.16228	−0.288602
$209$	0	0
$210$	0	0
$211$	−14.0000	−0.963800	−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000	−0.963800	−0.481900	+	0.876226i	$0.660053\pi$
$212$	−3.83772	−0.263576
$213$	0	0
$214$	11.6491	0.796317
$215$	−26.6491	−1.81745
$216$	0	0
$217$	16.6491	1.13022
$218$	2.32456	0.157439
$219$	0	0
$220$	0	0
$221$	−30.4868	−2.05077
$222$	0	0
$223$	3.67544	0.246126	0.123063	−	0.992399i	$-0.460728\pi$
$223$	3.67544	0.246126	0.123063	+	0.992399i	$0.460728\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	−16.3246	−1.08589
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	26.9737	1.78247	0.891235	−	0.453542i	$-0.149840\pi$
$229$	26.9737	1.78247	0.891235	+	0.453542i	$0.149840\pi$
$230$	−4.67544	−0.308290
$231$	0	0
$232$	−6.00000	−0.393919
$233$	23.6491	1.54930	0.774652	−	0.632387i	$-0.217925\pi$
$233$	23.6491	1.54930	0.774652	+	0.632387i	$0.217925\pi$
$234$	0	0
$235$	12.9737	0.846309
$236$	6.00000	0.390567
$237$	0	0
$238$	−14.6491	−0.949561
$239$	−13.6754	−0.884591	−0.442295	−	0.896869i	$-0.645836\pi$
$239$	−13.6754	−0.884591	−0.442295	+	0.896869i	$0.645836\pi$
$240$	0	0
$241$	−18.3246	−1.18039	−0.590194	−	0.807261i	$-0.700949\pi$
$241$	−18.3246	−1.18039	−0.590194	+	0.807261i	$0.700949\pi$
$242$	0	0
$243$	0	0
$244$	−5.83772	−0.373722
$245$	−6.48683	−0.414429
$246$	0	0
$247$	26.3246	1.67499
$248$	−8.32456	−0.528610
$249$	0	0
$250$	−11.5132	−0.728157
$251$	−22.3246	−1.40911	−0.704557	−	0.709648i	$-0.748854\pi$
$251$	−22.3246	−1.40911	−0.704557	+	0.709648i	$0.748854\pi$
$252$	0	0
$253$	0	0
$254$	−4.16228	−0.261165
$255$	0	0
$256$	1.00000	0.0625000
$257$	−23.2982	−1.45330	−0.726652	−	0.687006i	$-0.758925\pi$
$257$	−23.2982	−1.45330	−0.726652	+	0.687006i	$0.758925\pi$
$258$	0	0
$259$	4.64911	0.288882
$260$	−9.00000	−0.558156
$261$	0	0
$262$	−19.3246	−1.19388
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	0	0
$265$	−8.29822	−0.509756
$266$	12.6491	0.775567
$267$	0	0
$268$	3.32456	0.203080
$269$	32.1623	1.96097	0.980484	−	0.196597i	$-0.0629891\pi$
$269$	32.1623	1.96097	0.980484	+	0.196597i	$0.0629891\pi$
$270$	0	0
$271$	3.51317	0.213410	0.106705	−	0.994291i	$-0.465970\pi$
$271$	3.51317	0.213410	0.106705	+	0.994291i	$0.465970\pi$
$272$	7.32456	0.444116
$273$	0	0
$274$	14.6491	0.884985
$275$	0	0
$276$	0	0
$277$	29.4605	1.77011	0.885055	−	0.465487i	$-0.154121\pi$
$277$	29.4605	1.77011	0.885055	+	0.465487i	$0.154121\pi$
$278$	16.0000	0.959616
$279$	0	0
$280$	−4.32456	−0.258442
$281$	7.32456	0.436946	0.218473	−	0.975843i	$-0.429892\pi$
$281$	7.32456	0.436946	0.218473	+	0.975843i	$0.429892\pi$
$282$	0	0
$283$	12.6491	0.751912	0.375956	−	0.926638i	$-0.377314\pi$
$283$	12.6491	0.751912	0.375956	+	0.926638i	$0.377314\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	0	0
$287$	6.00000	0.354169
$288$	0	0
$289$	36.6491	2.15583
$290$	−12.9737	−0.761840
$291$	0	0
$292$	8.32456	0.487158
$293$	−29.2982	−1.71162	−0.855810	−	0.517290i	$-0.826941\pi$
$293$	−29.2982	−1.71162	−0.855810	+	0.517290i	$0.826941\pi$
$294$	0	0
$295$	12.9737	0.755356
$296$	−2.32456	−0.135112
$297$	0	0
$298$	4.32456	0.250515
$299$	9.00000	0.520483
$300$	0	0
$301$	24.6491	1.42075
$302$	6.16228	0.354599
$303$	0	0
$304$	−6.32456	−0.362738
$305$	−12.6228	−0.722778
$306$	0	0
$307$	−24.3246	−1.38828	−0.694138	−	0.719842i	$-0.744214\pi$
$307$	−24.3246	−1.38828	−0.694138	+	0.719842i	$0.744214\pi$
$308$	0	0
$309$	0	0
$310$	−18.0000	−1.02233
$311$	16.8114	0.953286	0.476643	−	0.879097i	$-0.341853\pi$
$311$	16.8114	0.953286	0.476643	+	0.879097i	$0.341853\pi$
$312$	0	0
$313$	−31.0000	−1.75222	−0.876112	−	0.482108i	$-0.839871\pi$
$313$	−31.0000	−1.75222	−0.876112	+	0.482108i	$0.839871\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	4.48683	0.252404
$317$	6.97367	0.391680	0.195840	−	0.980636i	$-0.437257\pi$
$317$	6.97367	0.391680	0.195840	+	0.980636i	$0.437257\pi$
$318$	0	0
$319$	0	0
$320$	2.16228	0.120875
$321$	0	0
$322$	4.32456	0.240998
$323$	−46.3246	−2.57757
$324$	0	0
$325$	1.35089	0.0749339
$326$	−21.6491	−1.19903
$327$	0	0
$328$	−3.00000	−0.165647
$329$	−12.0000	−0.661581
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	−7.32456	−0.401987
$333$	0	0
$334$	−12.9737	−0.709887
$335$	7.18861	0.392756
$336$	0	0
$337$	−9.67544	−0.527055	−0.263528	−	0.964652i	$-0.584886\pi$
$337$	−9.67544	−0.527055	−0.263528	+	0.964652i	$0.584886\pi$
$338$	4.32456	0.235225
$339$	0	0
$340$	15.8377	0.858921
$341$	0	0
$342$	0	0
$343$	20.0000	1.07990
$344$	−12.3246	−0.664495
$345$	0	0
$346$	16.3246	0.877614
$347$	29.2982	1.57281	0.786405	−	0.617711i	$-0.211940\pi$
$347$	29.2982	1.57281	0.786405	+	0.617711i	$0.211940\pi$
$348$	0	0
$349$	14.3246	0.766776	0.383388	−	0.923587i	$-0.374757\pi$
$349$	14.3246	0.766776	0.383388	+	0.923587i	$0.374757\pi$
$350$	0.649111	0.0346964
$351$	0	0
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	12.9737	0.688571
$356$	4.32456	0.229201
$357$	0	0
$358$	−4.32456	−0.228560
$359$	−18.0000	−0.950004	−0.475002	−	0.879985i	$-0.657553\pi$
$359$	−18.0000	−0.950004	−0.475002	+	0.879985i	$0.657553\pi$
$360$	0	0
$361$	21.0000	1.10526
$362$	−12.6491	−0.664822
$363$	0	0
$364$	8.32456	0.436325
$365$	18.0000	0.942163
$366$	0	0
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	−2.16228	−0.112717
$369$	0	0
$370$	−5.02633	−0.261307
$371$	7.67544	0.398489
$372$	0	0
$373$	−18.8114	−0.974017	−0.487008	−	0.873397i	$-0.661912\pi$
$373$	−18.8114	−0.974017	−0.487008	+	0.873397i	$0.661912\pi$
$374$	0	0
$375$	0	0
$376$	6.00000	0.309426
$377$	24.9737	1.28621
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	−13.6754	−0.701536
$381$	0	0
$382$	0.486833	0.0249085
$383$	−23.2982	−1.19048	−0.595242	−	0.803547i	$-0.702944\pi$
$383$	−23.2982	−1.19048	−0.595242	+	0.803547i	$0.702944\pi$
$384$	0	0
$385$	0	0
$386$	−0.324555	−0.0165194
$387$	0	0
$388$	−2.67544	−0.135825
$389$	18.9737	0.962003	0.481002	−	0.876720i	$-0.340273\pi$
$389$	18.9737	0.962003	0.481002	+	0.876720i	$0.340273\pi$
$390$	0	0
$391$	−15.8377	−0.800948
$392$	−3.00000	−0.151523
$393$	0	0
$394$	20.6491	1.04029
$395$	9.70178	0.488149
$396$	0	0
$397$	6.32456	0.317420	0.158710	−	0.987325i	$-0.449266\pi$
$397$	6.32456	0.317420	0.158710	+	0.987325i	$0.449266\pi$
$398$	−10.9737	−0.550060
$399$	0	0
$400$	−0.324555	−0.0162278
$401$	−32.6491	−1.63042	−0.815209	−	0.579166i	$-0.803378\pi$
$401$	−32.6491	−1.63042	−0.815209	+	0.579166i	$0.803378\pi$
$402$	0	0
$403$	34.6491	1.72600
$404$	16.3246	0.812177
$405$	0	0
$406$	12.0000	0.595550
$407$	0	0
$408$	0	0
$409$	12.6491	0.625458	0.312729	−	0.949842i	$-0.398757\pi$
$409$	12.6491	0.625458	0.312729	+	0.949842i	$0.398757\pi$
$410$	−6.48683	−0.320362
$411$	0	0
$412$	−14.3246	−0.705720
$413$	−12.0000	−0.590481
$414$	0	0
$415$	−15.8377	−0.777443
$416$	−4.16228	−0.204072
$417$	0	0
$418$	0	0
$419$	18.9737	0.926924	0.463462	−	0.886117i	$-0.346607\pi$
$419$	18.9737	0.926924	0.463462	+	0.886117i	$0.346607\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	−14.0000	−0.681509
$423$	0	0
$424$	−3.83772	−0.186376
$425$	−2.37722	−0.115312
$426$	0	0
$427$	11.6754	0.565014
$428$	11.6491	0.563081
$429$	0	0
$430$	−26.6491	−1.28513
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−25.0000	−1.20142	−0.600712	−	0.799466i	$-0.705116\pi$
$433$	−25.0000	−1.20142	−0.600712	+	0.799466i	$0.705116\pi$
$434$	16.6491	0.799183
$435$	0	0
$436$	2.32456	0.111326
$437$	13.6754	0.654185
$438$	0	0
$439$	−5.83772	−0.278619	−0.139310	−	0.990249i	$-0.544488\pi$
$439$	−5.83772	−0.278619	−0.139310	+	0.990249i	$0.544488\pi$
$440$	0	0
$441$	0	0
$442$	−30.4868	−1.45011
$443$	−1.67544	−0.0796028	−0.0398014	−	0.999208i	$-0.512673\pi$
$443$	−1.67544	−0.0796028	−0.0398014	+	0.999208i	$0.512673\pi$
$444$	0	0
$445$	9.35089	0.443275
$446$	3.67544	0.174037
$447$	0	0
$448$	−2.00000	−0.0944911
$449$	−28.3246	−1.33672	−0.668359	−	0.743839i	$-0.733003\pi$
$449$	−28.3246	−1.33672	−0.668359	+	0.743839i	$0.733003\pi$
$450$	0	0
$451$	0	0
$452$	−16.3246	−0.767842
$453$	0	0
$454$	−12.0000	−0.563188
$455$	18.0000	0.843853
$456$	0	0
$457$	−12.3246	−0.576518	−0.288259	−	0.957552i	$-0.593076\pi$
$457$	−12.3246	−0.576518	−0.288259	+	0.957552i	$0.593076\pi$
$458$	26.9737	1.26040
$459$	0	0
$460$	−4.67544	−0.217994
$461$	17.2982	0.805658	0.402829	−	0.915275i	$-0.368027\pi$
$461$	17.2982	0.805658	0.402829	+	0.915275i	$0.368027\pi$
$462$	0	0
$463$	−7.35089	−0.341625	−0.170812	−	0.985304i	$-0.554639\pi$
$463$	−7.35089	−0.341625	−0.170812	+	0.985304i	$0.554639\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	23.6491	1.09552
$467$	30.0000	1.38823	0.694117	−	0.719862i	$-0.255795\pi$
$467$	30.0000	1.38823	0.694117	+	0.719862i	$0.255795\pi$
$468$	0	0
$469$	−6.64911	−0.307027
$470$	12.9737	0.598431
$471$	0	0
$472$	6.00000	0.276172
$473$	0	0
$474$	0	0
$475$	2.05267	0.0941829
$476$	−14.6491	−0.671441
$477$	0	0
$478$	−13.6754	−0.625500
$479$	−6.97367	−0.318635	−0.159317	−	0.987227i	$-0.550929\pi$
$479$	−6.97367	−0.318635	−0.159317	+	0.987227i	$0.550929\pi$
$480$	0	0
$481$	9.67544	0.441162
$482$	−18.3246	−0.834661
$483$	0	0
$484$	0	0
$485$	−5.78505	−0.262686
$486$	0	0
$487$	−22.9737	−1.04104	−0.520518	−	0.853851i	$-0.674261\pi$
$487$	−22.9737	−1.04104	−0.520518	+	0.853851i	$0.674261\pi$
$488$	−5.83772	−0.264261
$489$	0	0
$490$	−6.48683	−0.293045
$491$	−8.64911	−0.390329	−0.195164	−	0.980771i	$-0.562524\pi$
$491$	−8.64911	−0.390329	−0.195164	+	0.980771i	$0.562524\pi$
$492$	0	0
$493$	−43.9473	−1.97929
$494$	26.3246	1.18440
$495$	0	0
$496$	−8.32456	−0.373784
$497$	−12.0000	−0.538274
$498$	0	0
$499$	5.00000	0.223831	0.111915	−	0.993718i	$-0.464301\pi$
$499$	5.00000	0.223831	0.111915	+	0.993718i	$0.464301\pi$
$500$	−11.5132	−0.514884
$501$	0	0
$502$	−22.3246	−0.996394
$503$	29.2982	1.30634	0.653172	−	0.757210i	$-0.273438\pi$
$503$	29.2982	1.30634	0.653172	+	0.757210i	$0.273438\pi$
$504$	0	0
$505$	35.2982	1.57075
$506$	0	0
$507$	0	0
$508$	−4.16228	−0.184671
$509$	2.16228	0.0958413	0.0479206	−	0.998851i	$-0.484741\pi$
$509$	2.16228	0.0958413	0.0479206	+	0.998851i	$0.484741\pi$
$510$	0	0
$511$	−16.6491	−0.736513
$512$	1.00000	0.0441942
$513$	0	0
$514$	−23.2982	−1.02764
$515$	−30.9737	−1.36486
$516$	0	0
$517$	0	0
$518$	4.64911	0.204270
$519$	0	0
$520$	−9.00000	−0.394676
$521$	−30.9737	−1.35698	−0.678490	−	0.734609i	$-0.737366\pi$
$521$	−30.9737	−1.35698	−0.678490	+	0.734609i	$0.737366\pi$
$522$	0	0
$523$	28.9737	1.26693	0.633465	−	0.773771i	$-0.281632\pi$
$523$	28.9737	1.26693	0.633465	+	0.773771i	$0.281632\pi$
$524$	−19.3246	−0.844197
$525$	0	0
$526$	−6.00000	−0.261612
$527$	−60.9737	−2.65605
$528$	0	0
$529$	−18.3246	−0.796720
$530$	−8.29822	−0.360452
$531$	0	0
$532$	12.6491	0.548408
$533$	12.4868	0.540865
$534$	0	0
$535$	25.1886	1.08900
$536$	3.32456	0.143599
$537$	0	0
$538$	32.1623	1.38661
$539$	0	0
$540$	0	0
$541$	43.1359	1.85456	0.927280	−	0.374370i	$-0.122141\pi$
$541$	43.1359	1.85456	0.927280	+	0.374370i	$0.122141\pi$
$542$	3.51317	0.150903
$543$	0	0
$544$	7.32456	0.314038
$545$	5.02633	0.215304
$546$	0	0
$547$	−26.9737	−1.15331	−0.576655	−	0.816988i	$-0.695642\pi$
$547$	−26.9737	−1.15331	−0.576655	+	0.816988i	$0.695642\pi$
$548$	14.6491	0.625779
$549$	0	0
$550$	0	0
$551$	37.9473	1.61661
$552$	0	0
$553$	−8.97367	−0.381599
$554$	29.4605	1.25166
$555$	0	0
$556$	16.0000	0.678551
$557$	−30.9737	−1.31240	−0.656198	−	0.754589i	$-0.727836\pi$
$557$	−30.9737	−1.31240	−0.656198	+	0.754589i	$0.727836\pi$
$558$	0	0
$559$	51.2982	2.16968
$560$	−4.32456	−0.182746
$561$	0	0
$562$	7.32456	0.308968
$563$	23.6491	0.996691	0.498346	−	0.866978i	$-0.333941\pi$
$563$	23.6491	0.996691	0.498346	+	0.866978i	$0.333941\pi$
$564$	0	0
$565$	−35.2982	−1.48501
$566$	12.6491	0.531682
$567$	0	0
$568$	6.00000	0.251754
$569$	−37.3246	−1.56473	−0.782363	−	0.622822i	$-0.785986\pi$
$569$	−37.3246	−1.56473	−0.782363	+	0.622822i	$0.785986\pi$
$570$	0	0
$571$	−12.3246	−0.515767	−0.257883	−	0.966176i	$-0.583025\pi$
$571$	−12.3246	−0.515767	−0.257883	+	0.966176i	$0.583025\pi$
$572$	0	0
$573$	0	0
$574$	6.00000	0.250435
$575$	0.701779	0.0292662
$576$	0	0
$577$	27.9473	1.16346	0.581731	−	0.813381i	$-0.302376\pi$
$577$	27.9473	1.16346	0.581731	+	0.813381i	$0.302376\pi$
$578$	36.6491	1.52440
$579$	0	0
$580$	−12.9737	−0.538702
$581$	14.6491	0.607748
$582$	0	0
$583$	0	0
$584$	8.32456	0.344473
$585$	0	0
$586$	−29.2982	−1.21030
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	0	0
$589$	52.6491	2.16937
$590$	12.9737	0.534117
$591$	0	0
$592$	−2.32456	−0.0955386
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	−31.6754	−1.29857
$596$	4.32456	0.177141
$597$	0	0
$598$	9.00000	0.368037
$599$	−12.4868	−0.510198	−0.255099	−	0.966915i	$-0.582108\pi$
$599$	−12.4868	−0.510198	−0.255099	+	0.966915i	$0.582108\pi$
$600$	0	0
$601$	−30.3246	−1.23696	−0.618482	−	0.785799i	$-0.712252\pi$
$601$	−30.3246	−1.23696	−0.618482	+	0.785799i	$0.712252\pi$
$602$	24.6491	1.00462
$603$	0	0
$604$	6.16228	0.250740
$605$	0	0
$606$	0	0
$607$	−12.8114	−0.519998	−0.259999	−	0.965609i	$-0.583722\pi$
$607$	−12.8114	−0.519998	−0.259999	+	0.965609i	$0.583722\pi$
$608$	−6.32456	−0.256495
$609$	0	0
$610$	−12.6228	−0.511081
$611$	−24.9737	−1.01033
$612$	0	0
$613$	−11.1359	−0.449777	−0.224888	−	0.974385i	$-0.572202\pi$
$613$	−11.1359	−0.449777	−0.224888	+	0.974385i	$0.572202\pi$
$614$	−24.3246	−0.981659
$615$	0	0
$616$	0	0
$617$	−28.3246	−1.14030	−0.570152	−	0.821539i	$-0.693116\pi$
$617$	−28.3246	−1.14030	−0.570152	+	0.821539i	$0.693116\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	−18.0000	−0.722897
$621$	0	0
$622$	16.8114	0.674075
$623$	−8.64911	−0.346519
$624$	0	0
$625$	−23.2719	−0.930875
$626$	−31.0000	−1.23901
$627$	0	0
$628$	−4.00000	−0.159617
$629$	−17.0263	−0.678884
$630$	0	0
$631$	−31.6228	−1.25888	−0.629441	−	0.777048i	$-0.716716\pi$
$631$	−31.6228	−1.25888	−0.629441	+	0.777048i	$0.716716\pi$
$632$	4.48683	0.178477
$633$	0	0
$634$	6.97367	0.276960
$635$	−9.00000	−0.357154
$636$	0	0
$637$	12.4868	0.494746
$638$	0	0
$639$	0	0
$640$	2.16228	0.0854715
$641$	12.9737	0.512429	0.256214	−	0.966620i	$-0.417525\pi$
$641$	12.9737	0.512429	0.256214	+	0.966620i	$0.417525\pi$
$642$	0	0
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	4.32456	0.170411
$645$	0	0
$646$	−46.3246	−1.82262
$647$	38.6491	1.51945	0.759727	−	0.650243i	$-0.225333\pi$
$647$	38.6491	1.51945	0.759727	+	0.650243i	$0.225333\pi$
$648$	0	0
$649$	0	0
$650$	1.35089	0.0529862
$651$	0	0
$652$	−21.6491	−0.847845
$653$	12.4868	0.488648	0.244324	−	0.969694i	$-0.421434\pi$
$653$	12.4868	0.488648	0.244324	+	0.969694i	$0.421434\pi$
$654$	0	0
$655$	−41.7851	−1.63268
$656$	−3.00000	−0.117130
$657$	0	0
$658$	−12.0000	−0.467809
$659$	9.00000	0.350590	0.175295	−	0.984516i	$-0.443912\pi$
$659$	9.00000	0.350590	0.175295	+	0.984516i	$0.443912\pi$
$660$	0	0
$661$	47.6228	1.85231	0.926156	−	0.377141i	$-0.123093\pi$
$661$	47.6228	1.85231	0.926156	+	0.377141i	$0.123093\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	−7.32456	−0.284248
$665$	27.3509	1.06062
$666$	0	0
$667$	12.9737	0.502342
$668$	−12.9737	−0.501966
$669$	0	0
$670$	7.18861	0.277720
$671$	0	0
$672$	0	0
$673$	−27.9473	−1.07729	−0.538645	−	0.842533i	$-0.681064\pi$
$673$	−27.9473	−1.07729	−0.538645	+	0.842533i	$0.681064\pi$
$674$	−9.67544	−0.372684
$675$	0	0
$676$	4.32456	0.166329
$677$	−22.3246	−0.858002	−0.429001	−	0.903304i	$-0.641134\pi$
$677$	−22.3246	−0.858002	−0.429001	+	0.903304i	$0.641134\pi$
$678$	0	0
$679$	5.35089	0.205348
$680$	15.8377	0.607349
$681$	0	0
$682$	0	0
$683$	24.9737	0.955591	0.477795	−	0.878471i	$-0.341436\pi$
$683$	24.9737	0.955591	0.477795	+	0.878471i	$0.341436\pi$
$684$	0	0
$685$	31.6754	1.21026
$686$	20.0000	0.763604
$687$	0	0
$688$	−12.3246	−0.469869
$689$	15.9737	0.608548
$690$	0	0
$691$	−5.32456	−0.202556	−0.101278	−	0.994858i	$-0.532293\pi$
$691$	−5.32456	−0.202556	−0.101278	+	0.994858i	$0.532293\pi$
$692$	16.3246	0.620566
$693$	0	0
$694$	29.2982	1.11215
$695$	34.5964	1.31232
$696$	0	0
$697$	−21.9737	−0.832312
$698$	14.3246	0.542192
$699$	0	0
$700$	0.649111	0.0245341
$701$	−25.9473	−0.980017	−0.490009	−	0.871718i	$-0.663006\pi$
$701$	−25.9473	−0.980017	−0.490009	+	0.871718i	$0.663006\pi$
$702$	0	0
$703$	14.7018	0.554488
$704$	0	0
$705$	0	0
$706$	−6.00000	−0.225813
$707$	−32.6491	−1.22790
$708$	0	0
$709$	35.6228	1.33784	0.668921	−	0.743334i	$-0.266757\pi$
$709$	35.6228	1.33784	0.668921	+	0.743334i	$0.266757\pi$
$710$	12.9737	0.486893
$711$	0	0
$712$	4.32456	0.162070
$713$	18.0000	0.674105
$714$	0	0
$715$	0	0
$716$	−4.32456	−0.161616
$717$	0	0
$718$	−18.0000	−0.671754
$719$	6.48683	0.241918	0.120959	−	0.992658i	$-0.461403\pi$
$719$	6.48683	0.241918	0.120959	+	0.992658i	$0.461403\pi$
$720$	0	0
$721$	28.6491	1.06695
$722$	21.0000	0.781539
$723$	0	0
$724$	−12.6491	−0.470100
$725$	1.94733	0.0723221
$726$	0	0
$727$	16.6491	0.617481	0.308741	−	0.951146i	$-0.400092\pi$
$727$	16.6491	0.617481	0.308741	+	0.951146i	$0.400092\pi$
$728$	8.32456	0.308529
$729$	0	0
$730$	18.0000	0.666210
$731$	−90.2719	−3.33883
$732$	0	0
$733$	28.4868	1.05219	0.526093	−	0.850427i	$-0.323657\pi$
$733$	28.4868	1.05219	0.526093	+	0.850427i	$0.323657\pi$
$734$	−22.0000	−0.812035
$735$	0	0
$736$	−2.16228	−0.0797026
$737$	0	0
$738$	0	0
$739$	23.9473	0.880917	0.440458	−	0.897773i	$-0.354816\pi$
$739$	23.9473	0.880917	0.440458	+	0.897773i	$0.354816\pi$
$740$	−5.02633	−0.184772
$741$	0	0
$742$	7.67544	0.281774
$743$	−10.3246	−0.378771	−0.189386	−	0.981903i	$-0.560650\pi$
$743$	−10.3246	−0.378771	−0.189386	+	0.981903i	$0.560650\pi$
$744$	0	0
$745$	9.35089	0.342590
$746$	−18.8114	−0.688734
$747$	0	0
$748$	0	0
$749$	−23.2982	−0.851298
$750$	0	0
$751$	13.0263	0.475338	0.237669	−	0.971346i	$-0.423617\pi$
$751$	13.0263	0.475338	0.237669	+	0.971346i	$0.423617\pi$
$752$	6.00000	0.218797
$753$	0	0
$754$	24.9737	0.909487
$755$	13.3246	0.484930
$756$	0	0
$757$	44.9737	1.63460	0.817298	−	0.576215i	$-0.195471\pi$
$757$	44.9737	1.63460	0.817298	+	0.576215i	$0.195471\pi$
$758$	8.00000	0.290573
$759$	0	0
$760$	−13.6754	−0.496061
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	−4.64911	−0.168309
$764$	0.486833	0.0176130
$765$	0	0
$766$	−23.2982	−0.841799
$767$	−24.9737	−0.901747
$768$	0	0
$769$	5.67544	0.204662	0.102331	−	0.994750i	$-0.467370\pi$
$769$	5.67544	0.204662	0.102331	+	0.994750i	$0.467370\pi$
$770$	0	0
$771$	0	0
$772$	−0.324555	−0.0116810
$773$	54.9737	1.97727	0.988633	−	0.150351i	$-0.0480404\pi$
$773$	54.9737	1.97727	0.988633	+	0.150351i	$0.0480404\pi$
$774$	0	0
$775$	2.70178	0.0970508
$776$	−2.67544	−0.0960429
$777$	0	0
$778$	18.9737	0.680239
$779$	18.9737	0.679802
$780$	0	0
$781$	0	0
$782$	−15.8377	−0.566356
$783$	0	0
$784$	−3.00000	−0.107143
$785$	−8.64911	−0.308700
$786$	0	0
$787$	−6.32456	−0.225446	−0.112723	−	0.993626i	$-0.535957\pi$
$787$	−6.32456	−0.225446	−0.112723	+	0.993626i	$0.535957\pi$
$788$	20.6491	0.735594
$789$	0	0
$790$	9.70178	0.345174
$791$	32.6491	1.16087
$792$	0	0
$793$	24.2982	0.862855
$794$	6.32456	0.224450
$795$	0	0
$796$	−10.9737	−0.388951
$797$	−48.4868	−1.71749	−0.858746	−	0.512402i	$-0.828756\pi$
$797$	−48.4868	−1.71749	−0.858746	+	0.512402i	$0.828756\pi$
$798$	0	0
$799$	43.9473	1.55474
$800$	−0.324555	−0.0114748
$801$	0	0
$802$	−32.6491	−1.15288
$803$	0	0
$804$	0	0
$805$	9.35089	0.329576
$806$	34.6491	1.22046
$807$	0	0
$808$	16.3246	0.574296
$809$	−21.3509	−0.750657	−0.375329	−	0.926892i	$-0.622470\pi$
$809$	−21.3509	−0.750657	−0.375329	+	0.926892i	$0.622470\pi$
$810$	0	0
$811$	−38.0000	−1.33436	−0.667180	−	0.744896i	$-0.732499\pi$
$811$	−38.0000	−1.33436	−0.667180	+	0.744896i	$0.732499\pi$
$812$	12.0000	0.421117
$813$	0	0
$814$	0	0
$815$	−46.8114	−1.63973
$816$	0	0
$817$	77.9473	2.72703
$818$	12.6491	0.442266
$819$	0	0
$820$	−6.48683	−0.226530
$821$	0.701779	0.0244922	0.0122461	−	0.999925i	$-0.496102\pi$
$821$	0.701779	0.0244922	0.0122461	+	0.999925i	$0.496102\pi$
$822$	0	0
$823$	39.9473	1.39248	0.696238	−	0.717811i	$-0.254856\pi$
$823$	39.9473	1.39248	0.696238	+	0.717811i	$0.254856\pi$
$824$	−14.3246	−0.499020
$825$	0	0
$826$	−12.0000	−0.417533
$827$	−12.6228	−0.438937	−0.219468	−	0.975620i	$-0.570432\pi$
$827$	−12.6228	−0.438937	−0.219468	+	0.975620i	$0.570432\pi$
$828$	0	0
$829$	24.3246	0.844827	0.422413	−	0.906403i	$-0.361183\pi$
$829$	24.3246	0.844827	0.422413	+	0.906403i	$0.361183\pi$
$830$	−15.8377	−0.549735
$831$	0	0
$832$	−4.16228	−0.144301
$833$	−21.9737	−0.761342
$834$	0	0
$835$	−28.0527	−0.970803
$836$	0	0
$837$	0	0
$838$	18.9737	0.655434
$839$	−42.0000	−1.45000	−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000	−1.45000	−0.725001	+	0.688748i	$0.758161\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	14.0000	0.482472
$843$	0	0
$844$	−14.0000	−0.481900
$845$	9.35089	0.321680
$846$	0	0
$847$	0	0
$848$	−3.83772	−0.131788
$849$	0	0
$850$	−2.37722	−0.0815381
$851$	5.02633	0.172301
$852$	0	0
$853$	2.32456	0.0795913	0.0397956	−	0.999208i	$-0.487329\pi$
$853$	2.32456	0.0795913	0.0397956	+	0.999208i	$0.487329\pi$
$854$	11.6754	0.399525
$855$	0	0
$856$	11.6491	0.398158
$857$	−29.6491	−1.01279	−0.506397	−	0.862300i	$-0.669023\pi$
$857$	−29.6491	−1.01279	−0.506397	+	0.862300i	$0.669023\pi$
$858$	0	0
$859$	−57.6491	−1.96696	−0.983481	−	0.181011i	$-0.942063\pi$
$859$	−57.6491	−1.96696	−0.983481	+	0.181011i	$0.942063\pi$
$860$	−26.6491	−0.908727
$861$	0	0
$862$	0	0
$863$	45.1359	1.53645	0.768223	−	0.640183i	$-0.221141\pi$
$863$	45.1359	1.53645	0.768223	+	0.640183i	$0.221141\pi$
$864$	0	0
$865$	35.2982	1.20018
$866$	−25.0000	−0.849535
$867$	0	0
$868$	16.6491	0.565108
$869$	0	0
$870$	0	0
$871$	−13.8377	−0.468873
$872$	2.32456	0.0787194
$873$	0	0
$874$	13.6754	0.462579
$875$	23.0263	0.778432
$876$	0	0
$877$	−16.1623	−0.545761	−0.272881	−	0.962048i	$-0.587976\pi$
$877$	−16.1623	−0.545761	−0.272881	+	0.962048i	$0.587976\pi$
$878$	−5.83772	−0.197014
$879$	0	0
$880$	0	0
$881$	35.2982	1.18923	0.594614	−	0.804012i	$-0.297305\pi$
$881$	35.2982	1.18923	0.594614	+	0.804012i	$0.297305\pi$
$882$	0	0
$883$	−12.6491	−0.425676	−0.212838	−	0.977087i	$-0.568271\pi$
$883$	−12.6491	−0.425676	−0.212838	+	0.977087i	$0.568271\pi$
$884$	−30.4868	−1.02538
$885$	0	0
$886$	−1.67544	−0.0562877
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	8.32456	0.279197
$890$	9.35089	0.313443
$891$	0	0
$892$	3.67544	0.123063
$893$	−37.9473	−1.26986
$894$	0	0
$895$	−9.35089	−0.312566
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	−28.3246	−0.945203
$899$	49.9473	1.66584
$900$	0	0
$901$	−28.1096	−0.936467
$902$	0	0
$903$	0	0
$904$	−16.3246	−0.542947
$905$	−27.3509	−0.909174
$906$	0	0
$907$	−22.6228	−0.751177	−0.375588	−	0.926787i	$-0.622559\pi$
$907$	−22.6228	−0.751177	−0.375588	+	0.926787i	$0.622559\pi$
$908$	−12.0000	−0.398234
$909$	0	0
$910$	18.0000	0.596694
$911$	6.48683	0.214918	0.107459	−	0.994209i	$-0.465729\pi$
$911$	6.48683	0.214918	0.107459	+	0.994209i	$0.465729\pi$
$912$	0	0
$913$	0	0
$914$	−12.3246	−0.407660
$915$	0	0
$916$	26.9737	0.891235
$917$	38.6491	1.27631
$918$	0	0
$919$	−44.4868	−1.46749	−0.733743	−	0.679428i	$-0.762228\pi$
$919$	−44.4868	−1.46749	−0.733743	+	0.679428i	$0.762228\pi$
$920$	−4.67544	−0.154145
$921$	0	0
$922$	17.2982	0.569687
$923$	−24.9737	−0.822018
$924$	0	0
$925$	0.754447	0.0248061
$926$	−7.35089	−0.241565
$927$	0	0
$928$	−6.00000	−0.196960
$929$	18.9737	0.622506	0.311253	−	0.950327i	$-0.399251\pi$
$929$	18.9737	0.622506	0.311253	+	0.950327i	$0.399251\pi$
$930$	0	0
$931$	18.9737	0.621837
$932$	23.6491	0.774652
$933$	0	0
$934$	30.0000	0.981630
$935$	0	0
$936$	0	0
$937$	−18.3246	−0.598637	−0.299319	−	0.954153i	$-0.596759\pi$
$937$	−18.3246	−0.598637	−0.299319	+	0.954153i	$0.596759\pi$
$938$	−6.64911	−0.217101
$939$	0	0
$940$	12.9737	0.423154
$941$	48.9737	1.59650	0.798248	−	0.602329i	$-0.205760\pi$
$941$	48.9737	1.59650	0.798248	+	0.602329i	$0.205760\pi$
$942$	0	0
$943$	6.48683	0.211240
$944$	6.00000	0.195283
$945$	0	0
$946$	0	0
$947$	−42.9737	−1.39646	−0.698228	−	0.715875i	$-0.746028\pi$
$947$	−42.9737	−1.39646	−0.698228	+	0.715875i	$0.746028\pi$
$948$	0	0
$949$	−34.6491	−1.12476
$950$	2.05267	0.0665973
$951$	0	0
$952$	−14.6491	−0.474780
$953$	39.0000	1.26333	0.631667	−	0.775240i	$-0.282371\pi$
$953$	39.0000	1.26333	0.631667	+	0.775240i	$0.282371\pi$
$954$	0	0
$955$	1.05267	0.0340635
$956$	−13.6754	−0.442295
$957$	0	0
$958$	−6.97367	−0.225309
$959$	−29.2982	−0.946089
$960$	0	0
$961$	38.2982	1.23543
$962$	9.67544	0.311949
$963$	0	0
$964$	−18.3246	−0.590194
$965$	−0.701779	−0.0225911
$966$	0	0
$967$	6.16228	0.198165	0.0990827	−	0.995079i	$-0.468409\pi$
$967$	6.16228	0.198165	0.0990827	+	0.995079i	$0.468409\pi$
$968$	0	0
$969$	0	0
$970$	−5.78505	−0.185747
$971$	−18.9737	−0.608894	−0.304447	−	0.952529i	$-0.598472\pi$
$971$	−18.9737	−0.608894	−0.304447	+	0.952529i	$0.598472\pi$
$972$	0	0
$973$	−32.0000	−1.02587
$974$	−22.9737	−0.736124
$975$	0	0
$976$	−5.83772	−0.186861
$977$	20.6491	0.660624	0.330312	−	0.943872i	$-0.392846\pi$
$977$	20.6491	0.660624	0.330312	+	0.943872i	$0.392846\pi$
$978$	0	0
$979$	0	0
$980$	−6.48683	−0.207214
$981$	0	0
$982$	−8.64911	−0.276004
$983$	−31.4605	−1.00343	−0.501717	−	0.865032i	$-0.667298\pi$
$983$	−31.4605	−1.00343	−0.501717	+	0.865032i	$0.667298\pi$
$984$	0	0
$985$	44.6491	1.42264
$986$	−43.9473	−1.39957
$987$	0	0
$988$	26.3246	0.837496
$989$	26.6491	0.847392
$990$	0	0
$991$	15.9473	0.506584	0.253292	−	0.967390i	$-0.418487\pi$
$991$	15.9473	0.506584	0.253292	+	0.967390i	$0.418487\pi$
$992$	−8.32456	−0.264305
$993$	0	0
$994$	−12.0000	−0.380617
$995$	−23.7281	−0.752232
$996$	0	0
$997$	−18.8114	−0.595763	−0.297881	−	0.954603i	$-0.596280\pi$
$997$	−18.8114	−0.595763	−0.297881	+	0.954603i	$0.596280\pi$
$998$	5.00000	0.158272
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6534.2.a.ca.1.2		2
3.2	odd	2		6534.2.a.bu.1.1		2
11.10	odd	2		594.2.a.i.1.2	✓	2
33.32	even	2		594.2.a.j.1.1	yes	2
44.43	even	2		4752.2.a.u.1.2		2
99.32	even	6		1782.2.e.y.1189.2		4
99.43	odd	6		1782.2.e.bb.595.1		4
99.65	even	6		1782.2.e.y.595.2		4
99.76	odd	6		1782.2.e.bb.1189.1		4
132.131	odd	2		4752.2.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
594.2.a.i.1.2	✓	2	11.10	odd	2
594.2.a.j.1.1	yes	2	33.32	even	2
1782.2.e.y.595.2		4	99.65	even	6
1782.2.e.y.1189.2		4	99.32	even	6
1782.2.e.bb.595.1		4	99.43	odd	6
1782.2.e.bb.1189.1		4	99.76	odd	6
4752.2.a.u.1.2		2	44.43	even	2
4752.2.a.bd.1.1		2	132.131	odd	2
6534.2.a.bu.1.1		2	3.2	odd	2
6534.2.a.ca.1.2		2	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:	\( N \)	\(=\)	\( 6534 = 2 \cdot 3^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6534.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.16228 q^{5} -2.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.16228 q^{5} -2.00000 q^{7} +1.00000 q^{8} +2.16228 q^{10} -4.16228 q^{13} -2.00000 q^{14} +1.00000 q^{16} +7.32456 q^{17} -6.32456 q^{19} +2.16228 q^{20} -2.16228 q^{23} -0.324555 q^{25} -4.16228 q^{26} -2.00000 q^{28} -6.00000 q^{29} -8.32456 q^{31} +1.00000 q^{32} +7.32456 q^{34} -4.32456 q^{35} -2.32456 q^{37} -6.32456 q^{38} +2.16228 q^{40} -3.00000 q^{41} -12.3246 q^{43} -2.16228 q^{46} +6.00000 q^{47} -3.00000 q^{49} -0.324555 q^{50} -4.16228 q^{52} -3.83772 q^{53} -2.00000 q^{56} -6.00000 q^{58} +6.00000 q^{59} -5.83772 q^{61} -8.32456 q^{62} +1.00000 q^{64} -9.00000 q^{65} +3.32456 q^{67} +7.32456 q^{68} -4.32456 q^{70} +6.00000 q^{71} +8.32456 q^{73} -2.32456 q^{74} -6.32456 q^{76} +4.48683 q^{79} +2.16228 q^{80} -3.00000 q^{82} -7.32456 q^{83} +15.8377 q^{85} -12.3246 q^{86} +4.32456 q^{89} +8.32456 q^{91} -2.16228 q^{92} +6.00000 q^{94} -13.6754 q^{95} -2.67544 q^{97} -3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 4 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{7} + 2 q^{8} - 2 q^{10} - 2 q^{13} - 4 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{20} + 2 q^{23} + 12 q^{25} - 2 q^{26} - 4 q^{28} - 12 q^{29} - 4 q^{31} + 2 q^{32} + 2 q^{34} + 4 q^{35} + 8 q^{37} - 2 q^{40} - 6 q^{41} - 12 q^{43} + 2 q^{46} + 12 q^{47} - 6 q^{49} + 12 q^{50} - 2 q^{52} - 14 q^{53} - 4 q^{56} - 12 q^{58} + 12 q^{59} - 18 q^{61} - 4 q^{62} + 2 q^{64} - 18 q^{65} - 6 q^{67} + 2 q^{68} + 4 q^{70} + 12 q^{71} + 4 q^{73} + 8 q^{74} - 10 q^{79} - 2 q^{80} - 6 q^{82} - 2 q^{83} + 38 q^{85} - 12 q^{86} - 4 q^{89} + 4 q^{91} + 2 q^{92} + 12 q^{94} - 40 q^{95} - 18 q^{97} - 6 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 4 * q^7 + 2 * q^8 - 2 * q^10 - 2 * q^13 - 4 * q^14 + 2 * q^16 + 2 * q^17 - 2 * q^20 + 2 * q^23 + 12 * q^25 - 2 * q^26 - 4 * q^28 - 12 * q^29 - 4 * q^31 + 2 * q^32 + 2 * q^34 + 4 * q^35 + 8 * q^37 - 2 * q^40 - 6 * q^41 - 12 * q^43 + 2 * q^46 + 12 * q^47 - 6 * q^49 + 12 * q^50 - 2 * q^52 - 14 * q^53 - 4 * q^56 - 12 * q^58 + 12 * q^59 - 18 * q^61 - 4 * q^62 + 2 * q^64 - 18 * q^65 - 6 * q^67 + 2 * q^68 + 4 * q^70 + 12 * q^71 + 4 * q^73 + 8 * q^74 - 10 * q^79 - 2 * q^80 - 6 * q^82 - 2 * q^83 + 38 * q^85 - 12 * q^86 - 4 * q^89 + 4 * q^91 + 2 * q^92 + 12 * q^94 - 40 * q^95 - 18 * q^97 - 6 * q^98

Introduction

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