LMFDB - Embedded newform 2205.2.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,2,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, 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("2205.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2205.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:	$17.6070136457$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	2205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} -1.58579 q^{8} +O(q^{10})$	$q+0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} -1.58579 q^{8} +0.414214 q^{10} -4.82843 q^{11} -0.828427 q^{13} +3.00000 q^{16} -0.828427 q^{17} +2.82843 q^{19} -1.82843 q^{20} -2.00000 q^{22} +2.41421 q^{23} +1.00000 q^{25} -0.343146 q^{26} +1.00000 q^{29} +6.00000 q^{31} +4.41421 q^{32} -0.343146 q^{34} +1.17157 q^{38} -1.58579 q^{40} -2.17157 q^{41} +6.41421 q^{43} +8.82843 q^{44} +1.00000 q^{46} +2.00000 q^{47} +0.414214 q^{50} +1.51472 q^{52} +6.82843 q^{53} -4.82843 q^{55} +0.414214 q^{58} -12.4853 q^{59} +11.4853 q^{61} +2.48528 q^{62} -4.17157 q^{64} -0.828427 q^{65} +12.4142 q^{67} +1.51472 q^{68} +12.4853 q^{71} -4.82843 q^{73} -5.17157 q^{76} +9.17157 q^{79} +3.00000 q^{80} -0.899495 q^{82} -11.7279 q^{83} -0.828427 q^{85} +2.65685 q^{86} +7.65685 q^{88} +2.65685 q^{89} -4.41421 q^{92} +0.828427 q^{94} +2.82843 q^{95} -0.343146 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 6 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 6 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 6 q^{8} - 2 q^{10} - 4 q^{11} + 4 q^{13} + 6 q^{16} + 4 q^{17} + 2 q^{20} - 4 q^{22} + 2 q^{23} + 2 q^{25} - 12 q^{26} + 2 q^{29} + 12 q^{31} + 6 q^{32} - 12 q^{34} + 8 q^{38} - 6 q^{40} - 10 q^{41} + 10 q^{43} + 12 q^{44} + 2 q^{46} + 4 q^{47} - 2 q^{50} + 20 q^{52} + 8 q^{53} - 4 q^{55} - 2 q^{58} - 8 q^{59} + 6 q^{61} - 12 q^{62} - 14 q^{64} + 4 q^{65} + 22 q^{67} + 20 q^{68} + 8 q^{71} - 4 q^{73} - 16 q^{76} + 24 q^{79} + 6 q^{80} + 18 q^{82} + 2 q^{83} + 4 q^{85} - 6 q^{86} + 4 q^{88} - 6 q^{89} - 6 q^{92} - 4 q^{94} - 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 6 * q^8 - 2 * q^10 - 4 * q^11 + 4 * q^13 + 6 * q^16 + 4 * q^17 + 2 * q^20 - 4 * q^22 + 2 * q^23 + 2 * q^25 - 12 * q^26 + 2 * q^29 + 12 * q^31 + 6 * q^32 - 12 * q^34 + 8 * q^38 - 6 * q^40 - 10 * q^41 + 10 * q^43 + 12 * q^44 + 2 * q^46 + 4 * q^47 - 2 * q^50 + 20 * q^52 + 8 * q^53 - 4 * q^55 - 2 * q^58 - 8 * q^59 + 6 * q^61 - 12 * q^62 - 14 * q^64 + 4 * q^65 + 22 * q^67 + 20 * q^68 + 8 * q^71 - 4 * q^73 - 16 * q^76 + 24 * q^79 + 6 * q^80 + 18 * q^82 + 2 * q^83 + 4 * q^85 - 6 * q^86 + 4 * q^88 - 6 * q^89 - 6 * q^92 - 4 * q^94 - 12 * 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 $

$2$	0.414214	0.292893	0.146447	−	0.989219i	$-0.453216\pi$
$2$	0.414214	0.292893	0.146447	+	0.989219i	$0.453216\pi$
$3$	0	0
$3$	0	0
$4$	−1.82843	−0.914214
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.58579	−0.560660
$9$	0	0
$10$	0.414214	0.130986
$11$	−4.82843	−1.45583	−0.727913	−	0.685670i	$-0.759509\pi$
$11$	−4.82843	−1.45583	−0.727913	+	0.685670i	$0.759509\pi$
$12$	0	0
$13$	−0.828427	−0.229764	−0.114882	−	0.993379i	$-0.536649\pi$
$13$	−0.828427	−0.229764	−0.114882	+	0.993379i	$0.536649\pi$
$14$	0	0
$15$	0	0
$16$	3.00000	0.750000
$17$	−0.828427	−0.200923	−0.100462	−	0.994941i	$-0.532032\pi$
$17$	−0.828427	−0.200923	−0.100462	+	0.994941i	$0.532032\pi$
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	−1.82843	−0.408849
$21$	0	0
$22$	−2.00000	−0.426401
$23$	2.41421	0.503398	0.251699	−	0.967806i	$-0.419011\pi$
$23$	2.41421	0.503398	0.251699	+	0.967806i	$0.419011\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.343146	−0.0672964
$27$	0	0
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	4.41421	0.780330
$33$	0	0
$34$	−0.343146	−0.0588490
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	1.17157	0.190054
$39$	0	0
$40$	−1.58579	−0.250735
$41$	−2.17157	−0.339143	−0.169571	−	0.985518i	$-0.554238\pi$
$41$	−2.17157	−0.339143	−0.169571	+	0.985518i	$0.554238\pi$
$42$	0	0
$43$	6.41421	0.978158	0.489079	−	0.872239i	$-0.337333\pi$
$43$	6.41421	0.978158	0.489079	+	0.872239i	$0.337333\pi$
$44$	8.82843	1.33094
$45$	0	0
$46$	1.00000	0.147442
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	0	0
$49$	0	0
$50$	0.414214	0.0585786
$51$	0	0
$52$	1.51472	0.210054
$53$	6.82843	0.937957	0.468978	−	0.883210i	$-0.344622\pi$
$53$	6.82843	0.937957	0.468978	+	0.883210i	$0.344622\pi$
$54$	0	0
$55$	−4.82843	−0.651065
$56$	0	0
$57$	0	0
$58$	0.414214	0.0543889
$59$	−12.4853	−1.62545	−0.812723	−	0.582651i	$-0.802016\pi$
$59$	−12.4853	−1.62545	−0.812723	+	0.582651i	$0.802016\pi$
$60$	0	0
$61$	11.4853	1.47054	0.735270	−	0.677775i	$-0.237055\pi$
$61$	11.4853	1.47054	0.735270	+	0.677775i	$0.237055\pi$
$62$	2.48528	0.315631
$63$	0	0
$64$	−4.17157	−0.521447
$65$	−0.828427	−0.102754
$66$	0	0
$67$	12.4142	1.51664	0.758319	−	0.651884i	$-0.226021\pi$
$67$	12.4142	1.51664	0.758319	+	0.651884i	$0.226021\pi$
$68$	1.51472	0.183687
$69$	0	0
$70$	0	0
$71$	12.4853	1.48173	0.740865	−	0.671654i	$-0.234416\pi$
$71$	12.4853	1.48173	0.740865	+	0.671654i	$0.234416\pi$
$72$	0	0
$73$	−4.82843	−0.565125	−0.282562	−	0.959249i	$-0.591184\pi$
$73$	−4.82843	−0.565125	−0.282562	+	0.959249i	$0.591184\pi$
$74$	0	0
$75$	0	0
$76$	−5.17157	−0.593220
$77$	0	0
$78$	0	0
$79$	9.17157	1.03188	0.515941	−	0.856624i	$-0.327442\pi$
$79$	9.17157	1.03188	0.515941	+	0.856624i	$0.327442\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	−0.899495	−0.0993326
$83$	−11.7279	−1.28731	−0.643653	−	0.765317i	$-0.722582\pi$
$83$	−11.7279	−1.28731	−0.643653	+	0.765317i	$0.722582\pi$
$84$	0	0
$85$	−0.828427	−0.0898555
$86$	2.65685	0.286496
$87$	0	0
$88$	7.65685	0.816223
$89$	2.65685	0.281626	0.140813	−	0.990036i	$-0.455028\pi$
$89$	2.65685	0.281626	0.140813	+	0.990036i	$0.455028\pi$
$90$	0	0
$91$	0	0
$92$	−4.41421	−0.460214
$93$	0	0
$94$	0.828427	0.0854457
$95$	2.82843	0.290191
$96$	0	0
$97$	−0.343146	−0.0348412	−0.0174206	−	0.999848i	$-0.505545\pi$
$97$	−0.343146	−0.0348412	−0.0174206	+	0.999848i	$0.505545\pi$
$98$	0	0
$99$	0	0
$100$	−1.82843	−0.182843
$101$	12.3137	1.22526	0.612630	−	0.790370i	$-0.290112\pi$
$101$	12.3137	1.22526	0.612630	+	0.790370i	$0.290112\pi$
$102$	0	0
$103$	−0.414214	−0.0408137	−0.0204068	−	0.999792i	$-0.506496\pi$
$103$	−0.414214	−0.0408137	−0.0204068	+	0.999792i	$0.506496\pi$
$104$	1.31371	0.128820
$105$	0	0
$106$	2.82843	0.274721
$107$	−2.75736	−0.266564	−0.133282	−	0.991078i	$-0.542552\pi$
$107$	−2.75736	−0.266564	−0.133282	+	0.991078i	$0.542552\pi$
$108$	0	0
$109$	3.48528	0.333829	0.166915	−	0.985971i	$-0.446620\pi$
$109$	3.48528	0.333829	0.166915	+	0.985971i	$0.446620\pi$
$110$	−2.00000	−0.190693
$111$	0	0
$112$	0	0
$113$	−12.4853	−1.17452	−0.587258	−	0.809400i	$-0.699793\pi$
$113$	−12.4853	−1.17452	−0.587258	+	0.809400i	$0.699793\pi$
$114$	0	0
$115$	2.41421	0.225127
$116$	−1.82843	−0.169765
$117$	0	0
$118$	−5.17157	−0.476082
$119$	0	0
$120$	0	0
$121$	12.3137	1.11943
$122$	4.75736	0.430711
$123$	0	0
$124$	−10.9706	−0.985186
$125$	1.00000	0.0894427
$126$	0	0
$127$	13.3137	1.18140	0.590700	−	0.806891i	$-0.298852\pi$
$127$	13.3137	1.18140	0.590700	+	0.806891i	$0.298852\pi$
$128$	−10.5563	−0.933058
$129$	0	0
$130$	−0.343146	−0.0300959
$131$	3.31371	0.289520	0.144760	−	0.989467i	$-0.453759\pi$
$131$	3.31371	0.289520	0.144760	+	0.989467i	$0.453759\pi$
$132$	0	0
$133$	0	0
$134$	5.14214	0.444213
$135$	0	0
$136$	1.31371	0.112650
$137$	−1.65685	−0.141555	−0.0707773	−	0.997492i	$-0.522548\pi$
$137$	−1.65685	−0.141555	−0.0707773	+	0.997492i	$0.522548\pi$
$138$	0	0
$139$	−12.1421	−1.02988	−0.514941	−	0.857225i	$-0.672186\pi$
$139$	−12.1421	−1.02988	−0.514941	+	0.857225i	$0.672186\pi$
$140$	0	0
$141$	0	0
$142$	5.17157	0.433989
$143$	4.00000	0.334497
$144$	0	0
$145$	1.00000	0.0830455
$146$	−2.00000	−0.165521
$147$	0	0
$148$	0	0
$149$	7.82843	0.641330	0.320665	−	0.947193i	$-0.396094\pi$
$149$	7.82843	0.641330	0.320665	+	0.947193i	$0.396094\pi$
$150$	0	0
$151$	0.343146	0.0279248	0.0139624	−	0.999903i	$-0.495555\pi$
$151$	0.343146	0.0279248	0.0139624	+	0.999903i	$0.495555\pi$
$152$	−4.48528	−0.363804
$153$	0	0
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	5.31371	0.424080	0.212040	−	0.977261i	$-0.431989\pi$
$157$	5.31371	0.424080	0.212040	+	0.977261i	$0.431989\pi$
$158$	3.79899	0.302231
$159$	0	0
$160$	4.41421	0.348974
$161$	0	0
$162$	0	0
$163$	23.6569	1.85295	0.926474	−	0.376359i	$-0.122824\pi$
$163$	23.6569	1.85295	0.926474	+	0.376359i	$0.122824\pi$
$164$	3.97056	0.310049
$165$	0	0
$166$	−4.85786	−0.377043
$167$	19.5858	1.51559	0.757797	−	0.652491i	$-0.226276\pi$
$167$	19.5858	1.51559	0.757797	+	0.652491i	$0.226276\pi$
$168$	0	0
$169$	−12.3137	−0.947208
$170$	−0.343146	−0.0263181
$171$	0	0
$172$	−11.7279	−0.894246
$173$	−19.3137	−1.46839	−0.734197	−	0.678936i	$-0.762441\pi$
$173$	−19.3137	−1.46839	−0.734197	+	0.678936i	$0.762441\pi$
$174$	0	0
$175$	0	0
$176$	−14.4853	−1.09187
$177$	0	0
$178$	1.10051	0.0824863
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	8.65685	0.643459	0.321729	−	0.946832i	$-0.395736\pi$
$181$	8.65685	0.643459	0.321729	+	0.946832i	$0.395736\pi$
$182$	0	0
$183$	0	0
$184$	−3.82843	−0.282235
$185$	0	0
$186$	0	0
$187$	4.00000	0.292509
$188$	−3.65685	−0.266704
$189$	0	0
$190$	1.17157	0.0849948
$191$	7.17157	0.518917	0.259458	−	0.965754i	$-0.416456\pi$
$191$	7.17157	0.518917	0.259458	+	0.965754i	$0.416456\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	−0.142136	−0.0102047
$195$	0	0
$196$	0	0
$197$	23.6569	1.68548	0.842741	−	0.538320i	$-0.180941\pi$
$197$	23.6569	1.68548	0.842741	+	0.538320i	$0.180941\pi$
$198$	0	0
$199$	−1.65685	−0.117451	−0.0587256	−	0.998274i	$-0.518704\pi$
$199$	−1.65685	−0.117451	−0.0587256	+	0.998274i	$0.518704\pi$
$200$	−1.58579	−0.112132
$201$	0	0
$202$	5.10051	0.358870
$203$	0	0
$204$	0	0
$205$	−2.17157	−0.151669
$206$	−0.171573	−0.0119540
$207$	0	0
$208$	−2.48528	−0.172323
$209$	−13.6569	−0.944664
$210$	0	0
$211$	3.51472	0.241963	0.120982	−	0.992655i	$-0.461396\pi$
$211$	3.51472	0.241963	0.120982	+	0.992655i	$0.461396\pi$
$212$	−12.4853	−0.857493
$213$	0	0
$214$	−1.14214	−0.0780748
$215$	6.41421	0.437446
$216$	0	0
$217$	0	0
$218$	1.44365	0.0977764
$219$	0	0
$220$	8.82843	0.595212
$221$	0.686292	0.0461650
$222$	0	0
$223$	−11.6569	−0.780601	−0.390300	−	0.920688i	$-0.627629\pi$
$223$	−11.6569	−0.780601	−0.390300	+	0.920688i	$0.627629\pi$
$224$	0	0
$225$	0	0
$226$	−5.17157	−0.344008
$227$	26.9706	1.79010	0.895050	−	0.445967i	$-0.147140\pi$
$227$	26.9706	1.79010	0.895050	+	0.445967i	$0.147140\pi$
$228$	0	0
$229$	0.343146	0.0226757	0.0113379	−	0.999936i	$-0.496391\pi$
$229$	0.343146	0.0226757	0.0113379	+	0.999936i	$0.496391\pi$
$230$	1.00000	0.0659380
$231$	0	0
$232$	−1.58579	−0.104112
$233$	11.1716	0.731874	0.365937	−	0.930640i	$-0.380749\pi$
$233$	11.1716	0.731874	0.365937	+	0.930640i	$0.380749\pi$
$234$	0	0
$235$	2.00000	0.130466
$236$	22.8284	1.48600
$237$	0	0
$238$	0	0
$239$	−1.31371	−0.0849767	−0.0424884	−	0.999097i	$-0.513529\pi$
$239$	−1.31371	−0.0849767	−0.0424884	+	0.999097i	$0.513529\pi$
$240$	0	0
$241$	−16.3431	−1.05275	−0.526377	−	0.850251i	$-0.676450\pi$
$241$	−16.3431	−1.05275	−0.526377	+	0.850251i	$0.676450\pi$
$242$	5.10051	0.327873
$243$	0	0
$244$	−21.0000	−1.34439
$245$	0	0
$246$	0	0
$247$	−2.34315	−0.149091
$248$	−9.51472	−0.604185
$249$	0	0
$250$	0.414214	0.0261972
$251$	13.3137	0.840354	0.420177	−	0.907442i	$-0.361968\pi$
$251$	13.3137	0.840354	0.420177	+	0.907442i	$0.361968\pi$
$252$	0	0
$253$	−11.6569	−0.732860
$254$	5.51472	0.346024
$255$	0	0
$256$	3.97056	0.248160
$257$	17.6569	1.10140	0.550702	−	0.834702i	$-0.314360\pi$
$257$	17.6569	1.10140	0.550702	+	0.834702i	$0.314360\pi$
$258$	0	0
$259$	0	0
$260$	1.51472	0.0939389
$261$	0	0
$262$	1.37258	0.0847985
$263$	−19.0416	−1.17416	−0.587079	−	0.809530i	$-0.699722\pi$
$263$	−19.0416	−1.17416	−0.587079	+	0.809530i	$0.699722\pi$
$264$	0	0
$265$	6.82843	0.419467
$266$	0	0
$267$	0	0
$268$	−22.6985	−1.38653
$269$	−30.4558	−1.85693	−0.928463	−	0.371425i	$-0.878869\pi$
$269$	−30.4558	−1.85693	−0.928463	+	0.371425i	$0.878869\pi$
$270$	0	0
$271$	0.485281	0.0294787	0.0147394	−	0.999891i	$-0.495308\pi$
$271$	0.485281	0.0294787	0.0147394	+	0.999891i	$0.495308\pi$
$272$	−2.48528	−0.150692
$273$	0	0
$274$	−0.686292	−0.0414604
$275$	−4.82843	−0.291165
$276$	0	0
$277$	−12.1421	−0.729550	−0.364775	−	0.931096i	$-0.618854\pi$
$277$	−12.1421	−0.729550	−0.364775	+	0.931096i	$0.618854\pi$
$278$	−5.02944	−0.301646
$279$	0	0
$280$	0	0
$281$	−26.2843	−1.56799	−0.783994	−	0.620768i	$-0.786821\pi$
$281$	−26.2843	−1.56799	−0.783994	+	0.620768i	$0.786821\pi$
$282$	0	0
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	−22.8284	−1.35462
$285$	0	0
$286$	1.65685	0.0979718
$287$	0	0
$288$	0	0
$289$	−16.3137	−0.959630
$290$	0.414214	0.0243235
$291$	0	0
$292$	8.82843	0.516645
$293$	−16.0000	−0.934730	−0.467365	−	0.884064i	$-0.654797\pi$
$293$	−16.0000	−0.934730	−0.467365	+	0.884064i	$0.654797\pi$
$294$	0	0
$295$	−12.4853	−0.726921
$296$	0	0
$297$	0	0
$298$	3.24264	0.187841
$299$	−2.00000	−0.115663
$300$	0	0
$301$	0	0
$302$	0.142136	0.00817899
$303$	0	0
$304$	8.48528	0.486664
$305$	11.4853	0.657645
$306$	0	0
$307$	13.2426	0.755797	0.377899	−	0.925847i	$-0.376647\pi$
$307$	13.2426	0.755797	0.377899	+	0.925847i	$0.376647\pi$
$308$	0	0
$309$	0	0
$310$	2.48528	0.141154
$311$	−18.8284	−1.06766	−0.533831	−	0.845591i	$-0.679248\pi$
$311$	−18.8284	−1.06766	−0.533831	+	0.845591i	$0.679248\pi$
$312$	0	0
$313$	17.6569	0.998024	0.499012	−	0.866595i	$-0.333696\pi$
$313$	17.6569	0.998024	0.499012	+	0.866595i	$0.333696\pi$
$314$	2.20101	0.124210
$315$	0	0
$316$	−16.7696	−0.943361
$317$	−25.7990	−1.44902	−0.724508	−	0.689267i	$-0.757933\pi$
$317$	−25.7990	−1.44902	−0.724508	+	0.689267i	$0.757933\pi$
$318$	0	0
$319$	−4.82843	−0.270340
$320$	−4.17157	−0.233198
$321$	0	0
$322$	0	0
$323$	−2.34315	−0.130376
$324$	0	0
$325$	−0.828427	−0.0459529
$326$	9.79899	0.542716
$327$	0	0
$328$	3.44365	0.190144
$329$	0	0
$330$	0	0
$331$	−10.9706	−0.602997	−0.301498	−	0.953467i	$-0.597487\pi$
$331$	−10.9706	−0.602997	−0.301498	+	0.953467i	$0.597487\pi$
$332$	21.4437	1.17687
$333$	0	0
$334$	8.11270	0.443907
$335$	12.4142	0.678261
$336$	0	0
$337$	14.8284	0.807756	0.403878	−	0.914813i	$-0.367662\pi$
$337$	14.8284	0.807756	0.403878	+	0.914813i	$0.367662\pi$
$338$	−5.10051	−0.277431
$339$	0	0
$340$	1.51472	0.0821472
$341$	−28.9706	−1.56884
$342$	0	0
$343$	0	0
$344$	−10.1716	−0.548414
$345$	0	0
$346$	−8.00000	−0.430083
$347$	−22.0711	−1.18484	−0.592418	−	0.805630i	$-0.701827\pi$
$347$	−22.0711	−1.18484	−0.592418	+	0.805630i	$0.701827\pi$
$348$	0	0
$349$	26.6569	1.42691	0.713454	−	0.700702i	$-0.247130\pi$
$349$	26.6569	1.42691	0.713454	+	0.700702i	$0.247130\pi$
$350$	0	0
$351$	0	0
$352$	−21.3137	−1.13602
$353$	−21.1716	−1.12685	−0.563425	−	0.826168i	$-0.690516\pi$
$353$	−21.1716	−1.12685	−0.563425	+	0.826168i	$0.690516\pi$
$354$	0	0
$355$	12.4853	0.662650
$356$	−4.85786	−0.257466
$357$	0	0
$358$	4.14214	0.218919
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	3.58579	0.188465
$363$	0	0
$364$	0	0
$365$	−4.82843	−0.252731
$366$	0	0
$367$	11.2426	0.586861	0.293431	−	0.955980i	$-0.405203\pi$
$367$	11.2426	0.586861	0.293431	+	0.955980i	$0.405203\pi$
$368$	7.24264	0.377549
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	12.9706	0.671590	0.335795	−	0.941935i	$-0.390995\pi$
$373$	12.9706	0.671590	0.335795	+	0.941935i	$0.390995\pi$
$374$	1.65685	0.0856739
$375$	0	0
$376$	−3.17157	−0.163561
$377$	−0.828427	−0.0426662
$378$	0	0
$379$	21.1716	1.08751	0.543755	−	0.839244i	$-0.317002\pi$
$379$	21.1716	1.08751	0.543755	+	0.839244i	$0.317002\pi$
$380$	−5.17157	−0.265296
$381$	0	0
$382$	2.97056	0.151987
$383$	16.8995	0.863524	0.431762	−	0.901988i	$-0.357892\pi$
$383$	16.8995	0.863524	0.431762	+	0.901988i	$0.357892\pi$
$384$	0	0
$385$	0	0
$386$	−0.828427	−0.0421658
$387$	0	0
$388$	0.627417	0.0318523
$389$	−12.3431	−0.625822	−0.312911	−	0.949782i	$-0.601304\pi$
$389$	−12.3431	−0.625822	−0.312911	+	0.949782i	$0.601304\pi$
$390$	0	0
$391$	−2.00000	−0.101144
$392$	0	0
$393$	0	0
$394$	9.79899	0.493666
$395$	9.17157	0.461472
$396$	0	0
$397$	−28.6274	−1.43677	−0.718384	−	0.695646i	$-0.755118\pi$
$397$	−28.6274	−1.43677	−0.718384	+	0.695646i	$0.755118\pi$
$398$	−0.686292	−0.0344007
$399$	0	0
$400$	3.00000	0.150000
$401$	−7.68629	−0.383835	−0.191918	−	0.981411i	$-0.561471\pi$
$401$	−7.68629	−0.383835	−0.191918	+	0.981411i	$0.561471\pi$
$402$	0	0
$403$	−4.97056	−0.247601
$404$	−22.5147	−1.12015
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−24.7990	−1.22623	−0.613116	−	0.789993i	$-0.710084\pi$
$409$	−24.7990	−1.22623	−0.613116	+	0.789993i	$0.710084\pi$
$410$	−0.899495	−0.0444229
$411$	0	0
$412$	0.757359	0.0373124
$413$	0	0
$414$	0	0
$415$	−11.7279	−0.575701
$416$	−3.65685	−0.179292
$417$	0	0
$418$	−5.65685	−0.276686
$419$	−23.3137	−1.13895	−0.569475	−	0.822009i	$-0.692853\pi$
$419$	−23.3137	−1.13895	−0.569475	+	0.822009i	$0.692853\pi$
$420$	0	0
$421$	−3.48528	−0.169862	−0.0849311	−	0.996387i	$-0.527067\pi$
$421$	−3.48528	−0.169862	−0.0849311	+	0.996387i	$0.527067\pi$
$422$	1.45584	0.0708694
$423$	0	0
$424$	−10.8284	−0.525875
$425$	−0.828427	−0.0401846
$426$	0	0
$427$	0	0
$428$	5.04163	0.243696
$429$	0	0
$430$	2.65685	0.128125
$431$	21.7990	1.05002	0.525010	−	0.851096i	$-0.324062\pi$
$431$	21.7990	1.05002	0.525010	+	0.851096i	$0.324062\pi$
$432$	0	0
$433$	31.7990	1.52816	0.764081	−	0.645120i	$-0.223193\pi$
$433$	31.7990	1.52816	0.764081	+	0.645120i	$0.223193\pi$
$434$	0	0
$435$	0	0
$436$	−6.37258	−0.305191
$437$	6.82843	0.326648
$438$	0	0
$439$	−33.9411	−1.61992	−0.809961	−	0.586484i	$-0.800512\pi$
$439$	−33.9411	−1.61992	−0.809961	+	0.586484i	$0.800512\pi$
$440$	7.65685	0.365026
$441$	0	0
$442$	0.284271	0.0135214
$443$	12.2132	0.580267	0.290133	−	0.956986i	$-0.406300\pi$
$443$	12.2132	0.580267	0.290133	+	0.956986i	$0.406300\pi$
$444$	0	0
$445$	2.65685	0.125947
$446$	−4.82843	−0.228633
$447$	0	0
$448$	0	0
$449$	1.82843	0.0862888	0.0431444	−	0.999069i	$-0.486262\pi$
$449$	1.82843	0.0862888	0.0431444	+	0.999069i	$0.486262\pi$
$450$	0	0
$451$	10.4853	0.493733
$452$	22.8284	1.07376
$453$	0	0
$454$	11.1716	0.524308
$455$	0	0
$456$	0	0
$457$	−32.2843	−1.51019	−0.755097	−	0.655613i	$-0.772410\pi$
$457$	−32.2843	−1.51019	−0.755097	+	0.655613i	$0.772410\pi$
$458$	0.142136	0.00664156
$459$	0	0
$460$	−4.41421	−0.205814
$461$	18.6863	0.870307	0.435154	−	0.900356i	$-0.356694\pi$
$461$	18.6863	0.870307	0.435154	+	0.900356i	$0.356694\pi$
$462$	0	0
$463$	11.0416	0.513148	0.256574	−	0.966525i	$-0.417406\pi$
$463$	11.0416	0.513148	0.256574	+	0.966525i	$0.417406\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	4.62742	0.214361
$467$	−22.8995	−1.05966	−0.529831	−	0.848103i	$-0.677745\pi$
$467$	−22.8995	−1.05966	−0.529831	+	0.848103i	$0.677745\pi$
$468$	0	0
$469$	0	0
$470$	0.828427	0.0382125
$471$	0	0
$472$	19.7990	0.911322
$473$	−30.9706	−1.42403
$474$	0	0
$475$	2.82843	0.129777
$476$	0	0
$477$	0	0
$478$	−0.544156	−0.0248891
$479$	−24.3431	−1.11227	−0.556133	−	0.831093i	$-0.687716\pi$
$479$	−24.3431	−1.11227	−0.556133	+	0.831093i	$0.687716\pi$
$480$	0	0
$481$	0	0
$482$	−6.76955	−0.308345
$483$	0	0
$484$	−22.5147	−1.02340
$485$	−0.343146	−0.0155814
$486$	0	0
$487$	15.6569	0.709480	0.354740	−	0.934965i	$-0.384569\pi$
$487$	15.6569	0.709480	0.354740	+	0.934965i	$0.384569\pi$
$488$	−18.2132	−0.824473
$489$	0	0
$490$	0	0
$491$	13.3137	0.600839	0.300420	−	0.953807i	$-0.402873\pi$
$491$	13.3137	0.600839	0.300420	+	0.953807i	$0.402873\pi$
$492$	0	0
$493$	−0.828427	−0.0373105
$494$	−0.970563	−0.0436677
$495$	0	0
$496$	18.0000	0.808224
$497$	0	0
$498$	0	0
$499$	4.82843	0.216150	0.108075	−	0.994143i	$-0.465531\pi$
$499$	4.82843	0.216150	0.108075	+	0.994143i	$0.465531\pi$
$500$	−1.82843	−0.0817697
$501$	0	0
$502$	5.51472	0.246134
$503$	37.8701	1.68854	0.844271	−	0.535916i	$-0.180034\pi$
$503$	37.8701	1.68854	0.844271	+	0.535916i	$0.180034\pi$
$504$	0	0
$505$	12.3137	0.547953
$506$	−4.82843	−0.214650
$507$	0	0
$508$	−24.3431	−1.08005
$509$	24.6569	1.09290	0.546448	−	0.837493i	$-0.315980\pi$
$509$	24.6569	1.09290	0.546448	+	0.837493i	$0.315980\pi$
$510$	0	0
$511$	0	0
$512$	22.7574	1.00574
$513$	0	0
$514$	7.31371	0.322594
$515$	−0.414214	−0.0182524
$516$	0	0
$517$	−9.65685	−0.424708
$518$	0	0
$519$	0	0
$520$	1.31371	0.0576099
$521$	−18.9706	−0.831115	−0.415558	−	0.909567i	$-0.636414\pi$
$521$	−18.9706	−0.831115	−0.415558	+	0.909567i	$0.636414\pi$
$522$	0	0
$523$	−24.3431	−1.06445	−0.532226	−	0.846602i	$-0.678644\pi$
$523$	−24.3431	−1.06445	−0.532226	+	0.846602i	$0.678644\pi$
$524$	−6.05887	−0.264683
$525$	0	0
$526$	−7.88730	−0.343903
$527$	−4.97056	−0.216521
$528$	0	0
$529$	−17.1716	−0.746590
$530$	2.82843	0.122859
$531$	0	0
$532$	0	0
$533$	1.79899	0.0779229
$534$	0	0
$535$	−2.75736	−0.119211
$536$	−19.6863	−0.850318
$537$	0	0
$538$	−12.6152	−0.543881
$539$	0	0
$540$	0	0
$541$	18.6569	0.802121	0.401060	−	0.916052i	$-0.368642\pi$
$541$	18.6569	0.802121	0.401060	+	0.916052i	$0.368642\pi$
$542$	0.201010	0.00863412
$543$	0	0
$544$	−3.65685	−0.156786
$545$	3.48528	0.149293
$546$	0	0
$547$	5.10051	0.218082	0.109041	−	0.994037i	$-0.465222\pi$
$547$	5.10051	0.218082	0.109041	+	0.994037i	$0.465222\pi$
$548$	3.02944	0.129411
$549$	0	0
$550$	−2.00000	−0.0852803
$551$	2.82843	0.120495
$552$	0	0
$553$	0	0
$554$	−5.02944	−0.213680
$555$	0	0
$556$	22.2010	0.941533
$557$	34.2843	1.45267	0.726336	−	0.687340i	$-0.241222\pi$
$557$	34.2843	1.45267	0.726336	+	0.687340i	$0.241222\pi$
$558$	0	0
$559$	−5.31371	−0.224746
$560$	0	0
$561$	0	0
$562$	−10.8873	−0.459253
$563$	16.2721	0.685786	0.342893	−	0.939374i	$-0.388593\pi$
$563$	16.2721	0.685786	0.342893	+	0.939374i	$0.388593\pi$
$564$	0	0
$565$	−12.4853	−0.525260
$566$	−5.79899	−0.243750
$567$	0	0
$568$	−19.7990	−0.830747
$569$	3.65685	0.153303	0.0766517	−	0.997058i	$-0.475577\pi$
$569$	3.65685	0.153303	0.0766517	+	0.997058i	$0.475577\pi$
$570$	0	0
$571$	14.8284	0.620550	0.310275	−	0.950647i	$-0.399579\pi$
$571$	14.8284	0.620550	0.310275	+	0.950647i	$0.399579\pi$
$572$	−7.31371	−0.305802
$573$	0	0
$574$	0	0
$575$	2.41421	0.100680
$576$	0	0
$577$	−23.9411	−0.996682	−0.498341	−	0.866981i	$-0.666057\pi$
$577$	−23.9411	−0.996682	−0.498341	+	0.866981i	$0.666057\pi$
$578$	−6.75736	−0.281069
$579$	0	0
$580$	−1.82843	−0.0759213
$581$	0	0
$582$	0	0
$583$	−32.9706	−1.36550
$584$	7.65685	0.316843
$585$	0	0
$586$	−6.62742	−0.273776
$587$	22.2843	0.919770	0.459885	−	0.887978i	$-0.347891\pi$
$587$	22.2843	0.919770	0.459885	+	0.887978i	$0.347891\pi$
$588$	0	0
$589$	16.9706	0.699260
$590$	−5.17157	−0.212910
$591$	0	0
$592$	0	0
$593$	43.7990	1.79861	0.899304	−	0.437323i	$-0.144073\pi$
$593$	43.7990	1.79861	0.899304	+	0.437323i	$0.144073\pi$
$594$	0	0
$595$	0	0
$596$	−14.3137	−0.586312
$597$	0	0
$598$	−0.828427	−0.0338769
$599$	−17.6569	−0.721440	−0.360720	−	0.932674i	$-0.617469\pi$
$599$	−17.6569	−0.721440	−0.360720	+	0.932674i	$0.617469\pi$
$600$	0	0
$601$	−8.34315	−0.340324	−0.170162	−	0.985416i	$-0.554429\pi$
$601$	−8.34315	−0.340324	−0.170162	+	0.985416i	$0.554429\pi$
$602$	0	0
$603$	0	0
$604$	−0.627417	−0.0255292
$605$	12.3137	0.500623
$606$	0	0
$607$	4.21320	0.171009	0.0855043	−	0.996338i	$-0.472750\pi$
$607$	4.21320	0.171009	0.0855043	+	0.996338i	$0.472750\pi$
$608$	12.4853	0.506345
$609$	0	0
$610$	4.75736	0.192620
$611$	−1.65685	−0.0670291
$612$	0	0
$613$	−15.4558	−0.624256	−0.312128	−	0.950040i	$-0.601042\pi$
$613$	−15.4558	−0.624256	−0.312128	+	0.950040i	$0.601042\pi$
$614$	5.48528	0.221368
$615$	0	0
$616$	0	0
$617$	11.3137	0.455473	0.227736	−	0.973723i	$-0.426868\pi$
$617$	11.3137	0.455473	0.227736	+	0.973723i	$0.426868\pi$
$618$	0	0
$619$	−42.4853	−1.70763	−0.853814	−	0.520578i	$-0.825716\pi$
$619$	−42.4853	−1.70763	−0.853814	+	0.520578i	$0.825716\pi$
$620$	−10.9706	−0.440588
$621$	0	0
$622$	−7.79899	−0.312711
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	7.31371	0.292315
$627$	0	0
$628$	−9.71573	−0.387700
$629$	0	0
$630$	0	0
$631$	8.14214	0.324133	0.162067	−	0.986780i	$-0.448184\pi$
$631$	8.14214	0.324133	0.162067	+	0.986780i	$0.448184\pi$
$632$	−14.5442	−0.578535
$633$	0	0
$634$	−10.6863	−0.424407
$635$	13.3137	0.528338
$636$	0	0
$637$	0	0
$638$	−2.00000	−0.0791808
$639$	0	0
$640$	−10.5563	−0.417276
$641$	−14.5147	−0.573297	−0.286648	−	0.958036i	$-0.592541\pi$
$641$	−14.5147	−0.573297	−0.286648	+	0.958036i	$0.592541\pi$
$642$	0	0
$643$	30.2843	1.19430	0.597148	−	0.802131i	$-0.296301\pi$
$643$	30.2843	1.19430	0.597148	+	0.802131i	$0.296301\pi$
$644$	0	0
$645$	0	0
$646$	−0.970563	−0.0381863
$647$	17.0416	0.669976	0.334988	−	0.942222i	$-0.391268\pi$
$647$	17.0416	0.669976	0.334988	+	0.942222i	$0.391268\pi$
$648$	0	0
$649$	60.2843	2.36636
$650$	−0.343146	−0.0134593
$651$	0	0
$652$	−43.2548	−1.69399
$653$	24.8284	0.971611	0.485806	−	0.874067i	$-0.338526\pi$
$653$	24.8284	0.971611	0.485806	+	0.874067i	$0.338526\pi$
$654$	0	0
$655$	3.31371	0.129477
$656$	−6.51472	−0.254357
$657$	0	0
$658$	0	0
$659$	−26.8284	−1.04509	−0.522544	−	0.852613i	$-0.675017\pi$
$659$	−26.8284	−1.04509	−0.522544	+	0.852613i	$0.675017\pi$
$660$	0	0
$661$	−26.1716	−1.01796	−0.508978	−	0.860779i	$-0.669977\pi$
$661$	−26.1716	−1.01796	−0.508978	+	0.860779i	$0.669977\pi$
$662$	−4.54416	−0.176614
$663$	0	0
$664$	18.5980	0.721742
$665$	0	0
$666$	0	0
$667$	2.41421	0.0934787
$668$	−35.8112	−1.38558
$669$	0	0
$670$	5.14214	0.198658
$671$	−55.4558	−2.14085
$672$	0	0
$673$	18.3431	0.707076	0.353538	−	0.935420i	$-0.384978\pi$
$673$	18.3431	0.707076	0.353538	+	0.935420i	$0.384978\pi$
$674$	6.14214	0.236586
$675$	0	0
$676$	22.5147	0.865951
$677$	0.142136	0.00546272	0.00273136	−	0.999996i	$-0.499131\pi$
$677$	0.142136	0.00546272	0.00273136	+	0.999996i	$0.499131\pi$
$678$	0	0
$679$	0	0
$680$	1.31371	0.0503784
$681$	0	0
$682$	−12.0000	−0.459504
$683$	43.2426	1.65463	0.827317	−	0.561736i	$-0.189866\pi$
$683$	43.2426	1.65463	0.827317	+	0.561736i	$0.189866\pi$
$684$	0	0
$685$	−1.65685	−0.0633051
$686$	0	0
$687$	0	0
$688$	19.2426	0.733619
$689$	−5.65685	−0.215509
$690$	0	0
$691$	4.82843	0.183682	0.0918410	−	0.995774i	$-0.470725\pi$
$691$	4.82843	0.183682	0.0918410	+	0.995774i	$0.470725\pi$
$692$	35.3137	1.34243
$693$	0	0
$694$	−9.14214	−0.347031
$695$	−12.1421	−0.460577
$696$	0	0
$697$	1.79899	0.0681416
$698$	11.0416	0.417932
$699$	0	0
$700$	0	0
$701$	42.7990	1.61650	0.808248	−	0.588843i	$-0.200416\pi$
$701$	42.7990	1.61650	0.808248	+	0.588843i	$0.200416\pi$
$702$	0	0
$703$	0	0
$704$	20.1421	0.759135
$705$	0	0
$706$	−8.76955	−0.330046
$707$	0	0
$708$	0	0
$709$	38.3137	1.43890	0.719451	−	0.694543i	$-0.244394\pi$
$709$	38.3137	1.43890	0.719451	+	0.694543i	$0.244394\pi$
$710$	5.17157	0.194086
$711$	0	0
$712$	−4.21320	−0.157896
$713$	14.4853	0.542478
$714$	0	0
$715$	4.00000	0.149592
$716$	−18.2843	−0.683315
$717$	0	0
$718$	4.14214	0.154583
$719$	−41.1127	−1.53324	−0.766622	−	0.642098i	$-0.778064\pi$
$719$	−41.1127	−1.53324	−0.766622	+	0.642098i	$0.778064\pi$
$720$	0	0
$721$	0	0
$722$	−4.55635	−0.169570
$723$	0	0
$724$	−15.8284	−0.588259
$725$	1.00000	0.0371391
$726$	0	0
$727$	40.4142	1.49888	0.749440	−	0.662072i	$-0.230323\pi$
$727$	40.4142	1.49888	0.749440	+	0.662072i	$0.230323\pi$
$728$	0	0
$729$	0	0
$730$	−2.00000	−0.0740233
$731$	−5.31371	−0.196535
$732$	0	0
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	4.65685	0.171888
$735$	0	0
$736$	10.6569	0.392817
$737$	−59.9411	−2.20796
$738$	0	0
$739$	41.1127	1.51236	0.756178	−	0.654367i	$-0.227065\pi$
$739$	41.1127	1.51236	0.756178	+	0.654367i	$0.227065\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.92893	0.0707657	0.0353828	−	0.999374i	$-0.488735\pi$
$743$	1.92893	0.0707657	0.0353828	+	0.999374i	$0.488735\pi$
$744$	0	0
$745$	7.82843	0.286811
$746$	5.37258	0.196704
$747$	0	0
$748$	−7.31371	−0.267416
$749$	0	0
$750$	0	0
$751$	−41.6569	−1.52008	−0.760040	−	0.649876i	$-0.774821\pi$
$751$	−41.6569	−1.52008	−0.760040	+	0.649876i	$0.774821\pi$
$752$	6.00000	0.218797
$753$	0	0
$754$	−0.343146	−0.0124966
$755$	0.343146	0.0124884
$756$	0	0
$757$	19.4558	0.707135	0.353567	−	0.935409i	$-0.384969\pi$
$757$	19.4558	0.707135	0.353567	+	0.935409i	$0.384969\pi$
$758$	8.76955	0.318524
$759$	0	0
$760$	−4.48528	−0.162698
$761$	−13.3137	−0.482622	−0.241311	−	0.970448i	$-0.577577\pi$
$761$	−13.3137	−0.482622	−0.241311	+	0.970448i	$0.577577\pi$
$762$	0	0
$763$	0	0
$764$	−13.1127	−0.474401
$765$	0	0
$766$	7.00000	0.252920
$767$	10.3431	0.373469
$768$	0	0
$769$	−44.6274	−1.60931	−0.804653	−	0.593745i	$-0.797649\pi$
$769$	−44.6274	−1.60931	−0.804653	+	0.593745i	$0.797649\pi$
$770$	0	0
$771$	0	0
$772$	3.65685	0.131613
$773$	25.1127	0.903241	0.451620	−	0.892210i	$-0.350846\pi$
$773$	25.1127	0.903241	0.451620	+	0.892210i	$0.350846\pi$
$774$	0	0
$775$	6.00000	0.215526
$776$	0.544156	0.0195341
$777$	0	0
$778$	−5.11270	−0.183299
$779$	−6.14214	−0.220065
$780$	0	0
$781$	−60.2843	−2.15714
$782$	−0.828427	−0.0296245
$783$	0	0
$784$	0	0
$785$	5.31371	0.189654
$786$	0	0
$787$	28.5563	1.01792	0.508962	−	0.860789i	$-0.330029\pi$
$787$	28.5563	1.01792	0.508962	+	0.860789i	$0.330029\pi$
$788$	−43.2548	−1.54089
$789$	0	0
$790$	3.79899	0.135162
$791$	0	0
$792$	0	0
$793$	−9.51472	−0.337878
$794$	−11.8579	−0.420820
$795$	0	0
$796$	3.02944	0.107376
$797$	−8.00000	−0.283375	−0.141687	−	0.989911i	$-0.545253\pi$
$797$	−8.00000	−0.283375	−0.141687	+	0.989911i	$0.545253\pi$
$798$	0	0
$799$	−1.65685	−0.0586153
$800$	4.41421	0.156066
$801$	0	0
$802$	−3.18377	−0.112423
$803$	23.3137	0.822723
$804$	0	0
$805$	0	0
$806$	−2.05887	−0.0725208
$807$	0	0
$808$	−19.5269	−0.686954
$809$	9.62742	0.338482	0.169241	−	0.985575i	$-0.445868\pi$
$809$	9.62742	0.338482	0.169241	+	0.985575i	$0.445868\pi$
$810$	0	0
$811$	−24.6274	−0.864786	−0.432393	−	0.901685i	$-0.642331\pi$
$811$	−24.6274	−0.864786	−0.432393	+	0.901685i	$0.642331\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	23.6569	0.828663
$816$	0	0
$817$	18.1421	0.634713
$818$	−10.2721	−0.359155
$819$	0	0
$820$	3.97056	0.138658
$821$	19.9411	0.695950	0.347975	−	0.937504i	$-0.386869\pi$
$821$	19.9411	0.695950	0.347975	+	0.937504i	$0.386869\pi$
$822$	0	0
$823$	−12.0711	−0.420771	−0.210385	−	0.977619i	$-0.567472\pi$
$823$	−12.0711	−0.420771	−0.210385	+	0.977619i	$0.567472\pi$
$824$	0.656854	0.0228826
$825$	0	0
$826$	0	0
$827$	−16.2132	−0.563788	−0.281894	−	0.959446i	$-0.590963\pi$
$827$	−16.2132	−0.563788	−0.281894	+	0.959446i	$0.590963\pi$
$828$	0	0
$829$	−6.68629	−0.232225	−0.116112	−	0.993236i	$-0.537043\pi$
$829$	−6.68629	−0.232225	−0.116112	+	0.993236i	$0.537043\pi$
$830$	−4.85786	−0.168619
$831$	0	0
$832$	3.45584	0.119810
$833$	0	0
$834$	0	0
$835$	19.5858	0.677794
$836$	24.9706	0.863625
$837$	0	0
$838$	−9.65685	−0.333590
$839$	20.8284	0.719077	0.359539	−	0.933130i	$-0.382934\pi$
$839$	20.8284	0.719077	0.359539	+	0.933130i	$0.382934\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	−1.44365	−0.0497515
$843$	0	0
$844$	−6.42641	−0.221206
$845$	−12.3137	−0.423604
$846$	0	0
$847$	0	0
$848$	20.4853	0.703467
$849$	0	0
$850$	−0.343146	−0.0117698
$851$	0	0
$852$	0	0
$853$	−53.4558	−1.83029	−0.915147	−	0.403121i	$-0.867925\pi$
$853$	−53.4558	−1.83029	−0.915147	+	0.403121i	$0.867925\pi$
$854$	0	0
$855$	0	0
$856$	4.37258	0.149452
$857$	−22.2843	−0.761216	−0.380608	−	0.924736i	$-0.624285\pi$
$857$	−22.2843	−0.761216	−0.380608	+	0.924736i	$0.624285\pi$
$858$	0	0
$859$	−46.6274	−1.59091	−0.795453	−	0.606015i	$-0.792767\pi$
$859$	−46.6274	−1.59091	−0.795453	+	0.606015i	$0.792767\pi$
$860$	−11.7279	−0.399919
$861$	0	0
$862$	9.02944	0.307544
$863$	16.5563	0.563585	0.281792	−	0.959475i	$-0.409071\pi$
$863$	16.5563	0.563585	0.281792	+	0.959475i	$0.409071\pi$
$864$	0	0
$865$	−19.3137	−0.656686
$866$	13.1716	0.447588
$867$	0	0
$868$	0	0
$869$	−44.2843	−1.50224
$870$	0	0
$871$	−10.2843	−0.348469
$872$	−5.52691	−0.187165
$873$	0	0
$874$	2.82843	0.0956730
$875$	0	0
$876$	0	0
$877$	30.8284	1.04100	0.520501	−	0.853861i	$-0.325745\pi$
$877$	30.8284	1.04100	0.520501	+	0.853861i	$0.325745\pi$
$878$	−14.0589	−0.474464
$879$	0	0
$880$	−14.4853	−0.488299
$881$	−3.82843	−0.128983	−0.0644915	−	0.997918i	$-0.520543\pi$
$881$	−3.82843	−0.128983	−0.0644915	+	0.997918i	$0.520543\pi$
$882$	0	0
$883$	−38.2843	−1.28837	−0.644184	−	0.764870i	$-0.722803\pi$
$883$	−38.2843	−1.28837	−0.644184	+	0.764870i	$0.722803\pi$
$884$	−1.25483	−0.0422046
$885$	0	0
$886$	5.05887	0.169956
$887$	44.0711	1.47976	0.739881	−	0.672738i	$-0.234882\pi$
$887$	44.0711	1.47976	0.739881	+	0.672738i	$0.234882\pi$
$888$	0	0
$889$	0	0
$890$	1.10051	0.0368890
$891$	0	0
$892$	21.3137	0.713636
$893$	5.65685	0.189299
$894$	0	0
$895$	10.0000	0.334263
$896$	0	0
$897$	0	0
$898$	0.757359	0.0252734
$899$	6.00000	0.200111
$900$	0	0
$901$	−5.65685	−0.188457
$902$	4.34315	0.144611
$903$	0	0
$904$	19.7990	0.658505
$905$	8.65685	0.287764
$906$	0	0
$907$	28.2132	0.936804	0.468402	−	0.883515i	$-0.344830\pi$
$907$	28.2132	0.936804	0.468402	+	0.883515i	$0.344830\pi$
$908$	−49.3137	−1.63653
$909$	0	0
$910$	0	0
$911$	49.7990	1.64991	0.824957	−	0.565195i	$-0.191199\pi$
$911$	49.7990	1.64991	0.824957	+	0.565195i	$0.191199\pi$
$912$	0	0
$913$	56.6274	1.87409
$914$	−13.3726	−0.442326
$915$	0	0
$916$	−0.627417	−0.0207304
$917$	0	0
$918$	0	0
$919$	19.1127	0.630470	0.315235	−	0.949014i	$-0.397917\pi$
$919$	19.1127	0.630470	0.315235	+	0.949014i	$0.397917\pi$
$920$	−3.82843	−0.126220
$921$	0	0
$922$	7.74012	0.254907
$923$	−10.3431	−0.340449
$924$	0	0
$925$	0	0
$926$	4.57359	0.150298
$927$	0	0
$928$	4.41421	0.144904
$929$	−11.4853	−0.376820	−0.188410	−	0.982090i	$-0.560333\pi$
$929$	−11.4853	−0.376820	−0.188410	+	0.982090i	$0.560333\pi$
$930$	0	0
$931$	0	0
$932$	−20.4264	−0.669089
$933$	0	0
$934$	−9.48528	−0.310368
$935$	4.00000	0.130814
$936$	0	0
$937$	10.6274	0.347183	0.173591	−	0.984818i	$-0.444463\pi$
$937$	10.6274	0.347183	0.173591	+	0.984818i	$0.444463\pi$
$938$	0	0
$939$	0	0
$940$	−3.65685	−0.119273
$941$	10.2843	0.335258	0.167629	−	0.985850i	$-0.446389\pi$
$941$	10.2843	0.335258	0.167629	+	0.985850i	$0.446389\pi$
$942$	0	0
$943$	−5.24264	−0.170724
$944$	−37.4558	−1.21908
$945$	0	0
$946$	−12.8284	−0.417088
$947$	43.1838	1.40328	0.701642	−	0.712530i	$-0.252451\pi$
$947$	43.1838	1.40328	0.701642	+	0.712530i	$0.252451\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	1.17157	0.0380108
$951$	0	0
$952$	0	0
$953$	2.34315	0.0759019	0.0379510	−	0.999280i	$-0.487917\pi$
$953$	2.34315	0.0759019	0.0379510	+	0.999280i	$0.487917\pi$
$954$	0	0
$955$	7.17157	0.232067
$956$	2.40202	0.0776869
$957$	0	0
$958$	−10.0833	−0.325775
$959$	0	0
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	0	0
$964$	29.8823	0.962442
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	−27.5269	−0.885206	−0.442603	−	0.896718i	$-0.645945\pi$
$967$	−27.5269	−0.885206	−0.442603	+	0.896718i	$0.645945\pi$
$968$	−19.5269	−0.627619
$969$	0	0
$970$	−0.142136	−0.00456370
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	0	0
$973$	0	0
$974$	6.48528	0.207802
$975$	0	0
$976$	34.4558	1.10290
$977$	−21.3137	−0.681886	−0.340943	−	0.940084i	$-0.610746\pi$
$977$	−21.3137	−0.681886	−0.340943	+	0.940084i	$0.610746\pi$
$978$	0	0
$979$	−12.8284	−0.409998
$980$	0	0
$981$	0	0
$982$	5.51472	0.175982
$983$	−14.2132	−0.453331	−0.226665	−	0.973973i	$-0.572782\pi$
$983$	−14.2132	−0.453331	−0.226665	+	0.973973i	$0.572782\pi$
$984$	0	0
$985$	23.6569	0.753770
$986$	−0.343146	−0.0109280
$987$	0	0
$988$	4.28427	0.136301
$989$	15.4853	0.492403
$990$	0	0
$991$	−15.6569	−0.497356	−0.248678	−	0.968586i	$-0.579996\pi$
$991$	−15.6569	−0.497356	−0.248678	+	0.968586i	$0.579996\pi$
$992$	26.4853	0.840909
$993$	0	0
$994$	0	0
$995$	−1.65685	−0.0525258
$996$	0	0
$997$	17.4558	0.552832	0.276416	−	0.961038i	$-0.410853\pi$
$997$	17.4558	0.552832	0.276416	+	0.961038i	$0.410853\pi$
$998$	2.00000	0.0633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.2.a.q.1.2		2
3.2	odd	2		245.2.a.g.1.1		2
7.3	odd	6		315.2.j.e.226.1		4
7.5	odd	6		315.2.j.e.46.1		4
7.6	odd	2		2205.2.a.n.1.2		2
12.11	even	2		3920.2.a.bv.1.2		2
15.2	even	4		1225.2.b.h.99.2		4
15.8	even	4		1225.2.b.h.99.3		4
15.14	odd	2		1225.2.a.m.1.2		2
21.2	odd	6		245.2.e.e.116.2		4
21.5	even	6		35.2.e.a.11.2	✓	4
21.11	odd	6		245.2.e.e.226.2		4
21.17	even	6		35.2.e.a.16.2	yes	4
21.20	even	2		245.2.a.h.1.1		2
84.47	odd	6		560.2.q.k.81.2		4
84.59	odd	6		560.2.q.k.401.2		4
84.83	odd	2		3920.2.a.bq.1.1		2
105.17	odd	12		175.2.k.a.149.3		8
105.38	odd	12		175.2.k.a.149.2		8
105.47	odd	12		175.2.k.a.74.2		8
105.59	even	6		175.2.e.c.51.1		4
105.62	odd	4		1225.2.b.g.99.2		4
105.68	odd	12		175.2.k.a.74.3		8
105.83	odd	4		1225.2.b.g.99.3		4
105.89	even	6		175.2.e.c.151.1		4
105.104	even	2		1225.2.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.2.e.a.11.2	✓	4	21.5	even	6
35.2.e.a.16.2	yes	4	21.17	even	6
175.2.e.c.51.1		4	105.59	even	6
175.2.e.c.151.1		4	105.89	even	6
175.2.k.a.74.2		8	105.47	odd	12
175.2.k.a.74.3		8	105.68	odd	12
175.2.k.a.149.2		8	105.38	odd	12
175.2.k.a.149.3		8	105.17	odd	12
245.2.a.g.1.1		2	3.2	odd	2
245.2.a.h.1.1		2	21.20	even	2
245.2.e.e.116.2		4	21.2	odd	6
245.2.e.e.226.2		4	21.11	odd	6
315.2.j.e.46.1		4	7.5	odd	6
315.2.j.e.226.1		4	7.3	odd	6
560.2.q.k.81.2		4	84.47	odd	6
560.2.q.k.401.2		4	84.59	odd	6
1225.2.a.k.1.2		2	105.104	even	2
1225.2.a.m.1.2		2	15.14	odd	2
1225.2.b.g.99.2		4	105.62	odd	4
1225.2.b.g.99.3		4	105.83	odd	4
1225.2.b.h.99.2		4	15.2	even	4
1225.2.b.h.99.3		4	15.8	even	4
2205.2.a.n.1.2		2	7.6	odd	2
2205.2.a.q.1.2		2	1.1	even	1	trivial
3920.2.a.bq.1.1		2	84.83	odd	2
3920.2.a.bv.1.2		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q+0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} -1.58579 q^{8} +O(q^{10})\)	\(q+0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} -1.58579 q^{8} +0.414214 q^{10} -4.82843 q^{11} -0.828427 q^{13} +3.00000 q^{16} -0.828427 q^{17} +2.82843 q^{19} -1.82843 q^{20} -2.00000 q^{22} +2.41421 q^{23} +1.00000 q^{25} -0.343146 q^{26} +1.00000 q^{29} +6.00000 q^{31} +4.41421 q^{32} -0.343146 q^{34} +1.17157 q^{38} -1.58579 q^{40} -2.17157 q^{41} +6.41421 q^{43} +8.82843 q^{44} +1.00000 q^{46} +2.00000 q^{47} +0.414214 q^{50} +1.51472 q^{52} +6.82843 q^{53} -4.82843 q^{55} +0.414214 q^{58} -12.4853 q^{59} +11.4853 q^{61} +2.48528 q^{62} -4.17157 q^{64} -0.828427 q^{65} +12.4142 q^{67} +1.51472 q^{68} +12.4853 q^{71} -4.82843 q^{73} -5.17157 q^{76} +9.17157 q^{79} +3.00000 q^{80} -0.899495 q^{82} -11.7279 q^{83} -0.828427 q^{85} +2.65685 q^{86} +7.65685 q^{88} +2.65685 q^{89} -4.41421 q^{92} +0.828427 q^{94} +2.82843 q^{95} -0.343146 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 6 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 6 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 6 q^{8} - 2 q^{10} - 4 q^{11} + 4 q^{13} + 6 q^{16} + 4 q^{17} + 2 q^{20} - 4 q^{22} + 2 q^{23} + 2 q^{25} - 12 q^{26} + 2 q^{29} + 12 q^{31} + 6 q^{32} - 12 q^{34} + 8 q^{38} - 6 q^{40} - 10 q^{41} + 10 q^{43} + 12 q^{44} + 2 q^{46} + 4 q^{47} - 2 q^{50} + 20 q^{52} + 8 q^{53} - 4 q^{55} - 2 q^{58} - 8 q^{59} + 6 q^{61} - 12 q^{62} - 14 q^{64} + 4 q^{65} + 22 q^{67} + 20 q^{68} + 8 q^{71} - 4 q^{73} - 16 q^{76} + 24 q^{79} + 6 q^{80} + 18 q^{82} + 2 q^{83} + 4 q^{85} - 6 q^{86} + 4 q^{88} - 6 q^{89} - 6 q^{92} - 4 q^{94} - 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 6 * q^8 - 2 * q^10 - 4 * q^11 + 4 * q^13 + 6 * q^16 + 4 * q^17 + 2 * q^20 - 4 * q^22 + 2 * q^23 + 2 * q^25 - 12 * q^26 + 2 * q^29 + 12 * q^31 + 6 * q^32 - 12 * q^34 + 8 * q^38 - 6 * q^40 - 10 * q^41 + 10 * q^43 + 12 * q^44 + 2 * q^46 + 4 * q^47 - 2 * q^50 + 20 * q^52 + 8 * q^53 - 4 * q^55 - 2 * q^58 - 8 * q^59 + 6 * q^61 - 12 * q^62 - 14 * q^64 + 4 * q^65 + 22 * q^67 + 20 * q^68 + 8 * q^71 - 4 * q^73 - 16 * q^76 + 24 * q^79 + 6 * q^80 + 18 * q^82 + 2 * q^83 + 4 * q^85 - 6 * q^86 + 4 * q^88 - 6 * q^89 - 6 * q^92 - 4 * q^94 - 12 * q^97

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