LMFDB - Embedded newform 2205.2.a.t.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 245)
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)

$f(q)$	$=$	$q+1.41421 q^{2} -1.00000 q^{5} -2.82843 q^{8} +O(q^{10})$	$q+1.41421 q^{2} -1.00000 q^{5} -2.82843 q^{8} -1.41421 q^{10} +0.171573 q^{11} +4.41421 q^{13} -4.00000 q^{16} -3.24264 q^{17} +6.00000 q^{19} +0.242641 q^{22} -7.41421 q^{23} +1.00000 q^{25} +6.24264 q^{26} +8.65685 q^{29} +10.2426 q^{31} -4.58579 q^{34} +2.24264 q^{37} +8.48528 q^{38} +2.82843 q^{40} +6.24264 q^{41} +2.00000 q^{43} -10.4853 q^{46} +7.24264 q^{47} +1.41421 q^{50} -4.24264 q^{53} -0.171573 q^{55} +12.2426 q^{58} +2.24264 q^{59} -2.82843 q^{61} +14.4853 q^{62} +8.00000 q^{64} -4.41421 q^{65} -8.24264 q^{67} +3.17157 q^{71} -8.48528 q^{73} +3.17157 q^{74} +1.48528 q^{79} +4.00000 q^{80} +8.82843 q^{82} +3.24264 q^{85} +2.82843 q^{86} -0.485281 q^{88} +8.00000 q^{89} +10.2426 q^{94} -6.00000 q^{95} +13.2426 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5}+O(q^{10}) $ 2 * q - 2 * q^5	$ 2 q - 2 q^{5} + 6 q^{11} + 6 q^{13} - 8 q^{16} + 2 q^{17} + 12 q^{19} - 8 q^{22} - 12 q^{23} + 2 q^{25} + 4 q^{26} + 6 q^{29} + 12 q^{31} - 12 q^{34} - 4 q^{37} + 4 q^{41} + 4 q^{43} - 4 q^{46} + 6 q^{47} - 6 q^{55} + 16 q^{58} - 4 q^{59} + 12 q^{62} + 16 q^{64} - 6 q^{65} - 8 q^{67} + 12 q^{71} + 12 q^{74} - 14 q^{79} + 8 q^{80} + 12 q^{82} - 2 q^{85} + 16 q^{88} + 16 q^{89} + 12 q^{94} - 12 q^{95} + 18 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 6 * q^11 + 6 * q^13 - 8 * q^16 + 2 * q^17 + 12 * q^19 - 8 * q^22 - 12 * q^23 + 2 * q^25 + 4 * q^26 + 6 * q^29 + 12 * q^31 - 12 * q^34 - 4 * q^37 + 4 * q^41 + 4 * q^43 - 4 * q^46 + 6 * q^47 - 6 * q^55 + 16 * q^58 - 4 * q^59 + 12 * q^62 + 16 * q^64 - 6 * q^65 - 8 * q^67 + 12 * q^71 + 12 * q^74 - 14 * q^79 + 8 * q^80 + 12 * q^82 - 2 * q^85 + 16 * q^88 + 16 * q^89 + 12 * q^94 - 12 * q^95 + 18 * 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 $

$2$	1.41421	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$2$	1.41421	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−2.82843	−1.00000
$9$	0	0
$10$	−1.41421	−0.447214
$11$	0.171573	0.0517312	0.0258656	−	0.999665i	$-0.491766\pi$
$11$	0.171573	0.0517312	0.0258656	+	0.999665i	$0.491766\pi$
$12$	0	0
$13$	4.41421	1.22428	0.612141	−	0.790748i	$-0.290308\pi$
$13$	4.41421	1.22428	0.612141	+	0.790748i	$0.290308\pi$
$14$	0	0
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−3.24264	−0.786456	−0.393228	−	0.919441i	$-0.628642\pi$
$17$	−3.24264	−0.786456	−0.393228	+	0.919441i	$0.628642\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	0	0
$22$	0.242641	0.0517312
$23$	−7.41421	−1.54597	−0.772985	−	0.634424i	$-0.781237\pi$
$23$	−7.41421	−1.54597	−0.772985	+	0.634424i	$0.781237\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	6.24264	1.22428
$27$	0	0
$28$	0	0
$29$	8.65685	1.60754	0.803769	−	0.594942i	$-0.202825\pi$
$29$	8.65685	1.60754	0.803769	+	0.594942i	$0.202825\pi$
$30$	0	0
$31$	10.2426	1.83963	0.919816	−	0.392349i	$-0.128338\pi$
$31$	10.2426	1.83963	0.919816	+	0.392349i	$0.128338\pi$
$32$	0	0
$33$	0	0
$34$	−4.58579	−0.786456
$35$	0	0
$36$	0	0
$37$	2.24264	0.368688	0.184344	−	0.982862i	$-0.440984\pi$
$37$	2.24264	0.368688	0.184344	+	0.982862i	$0.440984\pi$
$38$	8.48528	1.37649
$39$	0	0
$40$	2.82843	0.447214
$41$	6.24264	0.974937	0.487468	−	0.873141i	$-0.337920\pi$
$41$	6.24264	0.974937	0.487468	+	0.873141i	$0.337920\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	0	0
$46$	−10.4853	−1.54597
$47$	7.24264	1.05645	0.528224	−	0.849105i	$-0.322858\pi$
$47$	7.24264	1.05645	0.528224	+	0.849105i	$0.322858\pi$
$48$	0	0
$49$	0	0
$50$	1.41421	0.200000
$51$	0	0
$52$	0	0
$53$	−4.24264	−0.582772	−0.291386	−	0.956606i	$-0.594116\pi$
$53$	−4.24264	−0.582772	−0.291386	+	0.956606i	$0.594116\pi$
$54$	0	0
$55$	−0.171573	−0.0231349
$56$	0	0
$57$	0	0
$58$	12.2426	1.60754
$59$	2.24264	0.291967	0.145983	−	0.989287i	$-0.453365\pi$
$59$	2.24264	0.291967	0.145983	+	0.989287i	$0.453365\pi$
$60$	0	0
$61$	−2.82843	−0.362143	−0.181071	−	0.983470i	$-0.557957\pi$
$61$	−2.82843	−0.362143	−0.181071	+	0.983470i	$0.557957\pi$
$62$	14.4853	1.83963
$63$	0	0
$64$	8.00000	1.00000
$65$	−4.41421	−0.547516
$66$	0	0
$67$	−8.24264	−1.00700	−0.503499	−	0.863996i	$-0.667954\pi$
$67$	−8.24264	−1.00700	−0.503499	+	0.863996i	$0.667954\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.17157	0.376396	0.188198	−	0.982131i	$-0.439735\pi$
$71$	3.17157	0.376396	0.188198	+	0.982131i	$0.439735\pi$
$72$	0	0
$73$	−8.48528	−0.993127	−0.496564	−	0.868000i	$-0.665405\pi$
$73$	−8.48528	−0.993127	−0.496564	+	0.868000i	$0.665405\pi$
$74$	3.17157	0.368688
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.48528	0.167107	0.0835536	−	0.996503i	$-0.473373\pi$
$79$	1.48528	0.167107	0.0835536	+	0.996503i	$0.473373\pi$
$80$	4.00000	0.447214
$81$	0	0
$82$	8.82843	0.974937
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	3.24264	0.351714
$86$	2.82843	0.304997
$87$	0	0
$88$	−0.485281	−0.0517312
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	10.2426	1.05645
$95$	−6.00000	−0.615587
$96$	0	0
$97$	13.2426	1.34459	0.672293	−	0.740285i	$-0.265309\pi$
$97$	13.2426	1.34459	0.672293	+	0.740285i	$0.265309\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−2.48528	−0.247295	−0.123647	−	0.992326i	$-0.539459\pi$
$101$	−2.48528	−0.247295	−0.123647	+	0.992326i	$0.539459\pi$
$102$	0	0
$103$	19.2426	1.89603	0.948017	−	0.318220i	$-0.103085\pi$
$103$	19.2426	1.89603	0.948017	+	0.318220i	$0.103085\pi$
$104$	−12.4853	−1.22428
$105$	0	0
$106$	−6.00000	−0.582772
$107$	2.48528	0.240261	0.120131	−	0.992758i	$-0.461669\pi$
$107$	2.48528	0.240261	0.120131	+	0.992758i	$0.461669\pi$
$108$	0	0
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	−0.242641	−0.0231349
$111$	0	0
$112$	0	0
$113$	13.0711	1.22962	0.614811	−	0.788674i	$-0.289232\pi$
$113$	13.0711	1.22962	0.614811	+	0.788674i	$0.289232\pi$
$114$	0	0
$115$	7.41421	0.691379
$116$	0	0
$117$	0	0
$118$	3.17157	0.291967
$119$	0	0
$120$	0	0
$121$	−10.9706	−0.997324
$122$	−4.00000	−0.362143
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.24264	0.731416	0.365708	−	0.930730i	$-0.380827\pi$
$127$	8.24264	0.731416	0.365708	+	0.930730i	$0.380827\pi$
$128$	11.3137	1.00000
$129$	0	0
$130$	−6.24264	−0.547516
$131$	−12.2426	−1.06964	−0.534822	−	0.844965i	$-0.679621\pi$
$131$	−12.2426	−1.06964	−0.534822	+	0.844965i	$0.679621\pi$
$132$	0	0
$133$	0	0
$134$	−11.6569	−1.00700
$135$	0	0
$136$	9.17157	0.786456
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	7.75736	0.657971	0.328985	−	0.944335i	$-0.393293\pi$
$139$	7.75736	0.657971	0.328985	+	0.944335i	$0.393293\pi$
$140$	0	0
$141$	0	0
$142$	4.48528	0.376396
$143$	0.757359	0.0633336
$144$	0	0
$145$	−8.65685	−0.718913
$146$	−12.0000	−0.993127
$147$	0	0
$148$	0	0
$149$	9.17157	0.751365	0.375682	−	0.926749i	$-0.377408\pi$
$149$	9.17157	0.751365	0.375682	+	0.926749i	$0.377408\pi$
$150$	0	0
$151$	−7.48528	−0.609144	−0.304572	−	0.952489i	$-0.598513\pi$
$151$	−7.48528	−0.609144	−0.304572	+	0.952489i	$0.598513\pi$
$152$	−16.9706	−1.37649
$153$	0	0
$154$	0	0
$155$	−10.2426	−0.822709
$156$	0	0
$157$	−15.1716	−1.21082	−0.605412	−	0.795913i	$-0.706992\pi$
$157$	−15.1716	−1.21082	−0.605412	+	0.795913i	$0.706992\pi$
$158$	2.10051	0.167107
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−10.2426	−0.802266	−0.401133	−	0.916020i	$-0.631383\pi$
$163$	−10.2426	−0.802266	−0.401133	+	0.916020i	$0.631383\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−0.757359	−0.0586062	−0.0293031	−	0.999571i	$-0.509329\pi$
$167$	−0.757359	−0.0586062	−0.0293031	+	0.999571i	$0.509329\pi$
$168$	0	0
$169$	6.48528	0.498868
$170$	4.58579	0.351714
$171$	0	0
$172$	0	0
$173$	7.24264	0.550648	0.275324	−	0.961352i	$-0.411215\pi$
$173$	7.24264	0.550648	0.275324	+	0.961352i	$0.411215\pi$
$174$	0	0
$175$	0	0
$176$	−0.686292	−0.0517312
$177$	0	0
$178$	11.3137	0.847998
$179$	14.4853	1.08268	0.541340	−	0.840804i	$-0.317917\pi$
$179$	14.4853	1.08268	0.541340	+	0.840804i	$0.317917\pi$
$180$	0	0
$181$	−18.7279	−1.39204	−0.696018	−	0.718025i	$-0.745047\pi$
$181$	−18.7279	−1.39204	−0.696018	+	0.718025i	$0.745047\pi$
$182$	0	0
$183$	0	0
$184$	20.9706	1.54597
$185$	−2.24264	−0.164882
$186$	0	0
$187$	−0.556349	−0.0406843
$188$	0	0
$189$	0	0
$190$	−8.48528	−0.615587
$191$	−13.9706	−1.01087	−0.505437	−	0.862863i	$-0.668669\pi$
$191$	−13.9706	−1.01087	−0.505437	+	0.862863i	$0.668669\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	18.7279	1.34459
$195$	0	0
$196$	0	0
$197$	−13.4142	−0.955723	−0.477862	−	0.878435i	$-0.658588\pi$
$197$	−13.4142	−0.955723	−0.477862	+	0.878435i	$0.658588\pi$
$198$	0	0
$199$	−7.41421	−0.525580	−0.262790	−	0.964853i	$-0.584643\pi$
$199$	−7.41421	−0.525580	−0.262790	+	0.964853i	$0.584643\pi$
$200$	−2.82843	−0.200000
$201$	0	0
$202$	−3.51472	−0.247295
$203$	0	0
$204$	0	0
$205$	−6.24264	−0.436005
$206$	27.2132	1.89603
$207$	0	0
$208$	−17.6569	−1.22428
$209$	1.02944	0.0712077
$210$	0	0
$211$	9.00000	0.619586	0.309793	−	0.950804i	$-0.399740\pi$
$211$	9.00000	0.619586	0.309793	+	0.950804i	$0.399740\pi$
$212$	0	0
$213$	0	0
$214$	3.51472	0.240261
$215$	−2.00000	−0.136399
$216$	0	0
$217$	0	0
$218$	7.07107	0.478913
$219$	0	0
$220$	0	0
$221$	−14.3137	−0.962844
$222$	0	0
$223$	−24.2132	−1.62144	−0.810718	−	0.585437i	$-0.800923\pi$
$223$	−24.2132	−1.62144	−0.810718	+	0.585437i	$0.800923\pi$
$224$	0	0
$225$	0	0
$226$	18.4853	1.22962
$227$	−15.7279	−1.04390	−0.521949	−	0.852976i	$-0.674795\pi$
$227$	−15.7279	−1.04390	−0.521949	+	0.852976i	$0.674795\pi$
$228$	0	0
$229$	18.0416	1.19222	0.596112	−	0.802901i	$-0.296711\pi$
$229$	18.0416	1.19222	0.596112	+	0.802901i	$0.296711\pi$
$230$	10.4853	0.691379
$231$	0	0
$232$	−24.4853	−1.60754
$233$	9.17157	0.600850	0.300425	−	0.953805i	$-0.402872\pi$
$233$	9.17157	0.600850	0.300425	+	0.953805i	$0.402872\pi$
$234$	0	0
$235$	−7.24264	−0.472458
$236$	0	0
$237$	0	0
$238$	0	0
$239$	17.4853	1.13103	0.565514	−	0.824738i	$-0.308678\pi$
$239$	17.4853	1.13103	0.565514	+	0.824738i	$0.308678\pi$
$240$	0	0
$241$	−0.727922	−0.0468896	−0.0234448	−	0.999725i	$-0.507463\pi$
$241$	−0.727922	−0.0468896	−0.0234448	+	0.999725i	$0.507463\pi$
$242$	−15.5147	−0.997324
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	26.4853	1.68522
$248$	−28.9706	−1.83963
$249$	0	0
$250$	−1.41421	−0.0894427
$251$	−17.2132	−1.08649	−0.543244	−	0.839575i	$-0.682804\pi$
$251$	−17.2132	−1.08649	−0.543244	+	0.839575i	$0.682804\pi$
$252$	0	0
$253$	−1.27208	−0.0799749
$254$	11.6569	0.731416
$255$	0	0
$256$	0	0
$257$	9.51472	0.593512	0.296756	−	0.954953i	$-0.404095\pi$
$257$	9.51472	0.593512	0.296756	+	0.954953i	$0.404095\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−17.3137	−1.06964
$263$	16.6274	1.02529	0.512645	−	0.858601i	$-0.328666\pi$
$263$	16.6274	1.02529	0.512645	+	0.858601i	$0.328666\pi$
$264$	0	0
$265$	4.24264	0.260623
$266$	0	0
$267$	0	0
$268$	0	0
$269$	16.2426	0.990331	0.495166	−	0.868799i	$-0.335107\pi$
$269$	16.2426	0.990331	0.495166	+	0.868799i	$0.335107\pi$
$270$	0	0
$271$	−0.686292	−0.0416892	−0.0208446	−	0.999783i	$-0.506636\pi$
$271$	−0.686292	−0.0416892	−0.0208446	+	0.999783i	$0.506636\pi$
$272$	12.9706	0.786456
$273$	0	0
$274$	−16.9706	−1.02523
$275$	0.171573	0.0103462
$276$	0	0
$277$	15.2132	0.914073	0.457036	−	0.889448i	$-0.348911\pi$
$277$	15.2132	0.914073	0.457036	+	0.889448i	$0.348911\pi$
$278$	10.9706	0.657971
$279$	0	0
$280$	0	0
$281$	−2.31371	−0.138024	−0.0690121	−	0.997616i	$-0.521985\pi$
$281$	−2.31371	−0.138024	−0.0690121	+	0.997616i	$0.521985\pi$
$282$	0	0
$283$	24.5563	1.45972	0.729862	−	0.683595i	$-0.239584\pi$
$283$	24.5563	1.45972	0.729862	+	0.683595i	$0.239584\pi$
$284$	0	0
$285$	0	0
$286$	1.07107	0.0633336
$287$	0	0
$288$	0	0
$289$	−6.48528	−0.381487
$290$	−12.2426	−0.718913
$291$	0	0
$292$	0	0
$293$	−25.7279	−1.50304	−0.751521	−	0.659710i	$-0.770679\pi$
$293$	−25.7279	−1.50304	−0.751521	+	0.659710i	$0.770679\pi$
$294$	0	0
$295$	−2.24264	−0.130572
$296$	−6.34315	−0.368688
$297$	0	0
$298$	12.9706	0.751365
$299$	−32.7279	−1.89270
$300$	0	0
$301$	0	0
$302$	−10.5858	−0.609144
$303$	0	0
$304$	−24.0000	−1.37649
$305$	2.82843	0.161955
$306$	0	0
$307$	30.8995	1.76353	0.881764	−	0.471691i	$-0.156356\pi$
$307$	30.8995	1.76353	0.881764	+	0.471691i	$0.156356\pi$
$308$	0	0
$309$	0	0
$310$	−14.4853	−0.822709
$311$	10.0000	0.567048	0.283524	−	0.958965i	$-0.408496\pi$
$311$	10.0000	0.567048	0.283524	+	0.958965i	$0.408496\pi$
$312$	0	0
$313$	−18.2132	−1.02947	−0.514736	−	0.857349i	$-0.672110\pi$
$313$	−18.2132	−1.02947	−0.514736	+	0.857349i	$0.672110\pi$
$314$	−21.4558	−1.21082
$315$	0	0
$316$	0	0
$317$	0.343146	0.0192730	0.00963649	−	0.999954i	$-0.496933\pi$
$317$	0.343146	0.0192730	0.00963649	+	0.999954i	$0.496933\pi$
$318$	0	0
$319$	1.48528	0.0831598
$320$	−8.00000	−0.447214
$321$	0	0
$322$	0	0
$323$	−19.4558	−1.08255
$324$	0	0
$325$	4.41421	0.244857
$326$	−14.4853	−0.802266
$327$	0	0
$328$	−17.6569	−0.974937
$329$	0	0
$330$	0	0
$331$	−27.4558	−1.50911	−0.754555	−	0.656237i	$-0.772147\pi$
$331$	−27.4558	−1.50911	−0.754555	+	0.656237i	$0.772147\pi$
$332$	0	0
$333$	0	0
$334$	−1.07107	−0.0586062
$335$	8.24264	0.450344
$336$	0	0
$337$	22.2426	1.21163	0.605817	−	0.795604i	$-0.292846\pi$
$337$	22.2426	1.21163	0.605817	+	0.795604i	$0.292846\pi$
$338$	9.17157	0.498868
$339$	0	0
$340$	0	0
$341$	1.75736	0.0951663
$342$	0	0
$343$	0	0
$344$	−5.65685	−0.304997
$345$	0	0
$346$	10.2426	0.550648
$347$	−13.0711	−0.701692	−0.350846	−	0.936433i	$-0.614106\pi$
$347$	−13.0711	−0.701692	−0.350846	+	0.936433i	$0.614106\pi$
$348$	0	0
$349$	−10.9706	−0.587241	−0.293620	−	0.955922i	$-0.594860\pi$
$349$	−10.9706	−0.587241	−0.293620	+	0.955922i	$0.594860\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6.21320	0.330695	0.165348	−	0.986235i	$-0.447125\pi$
$353$	6.21320	0.330695	0.165348	+	0.986235i	$0.447125\pi$
$354$	0	0
$355$	−3.17157	−0.168330
$356$	0	0
$357$	0	0
$358$	20.4853	1.08268
$359$	11.3137	0.597115	0.298557	−	0.954392i	$-0.403495\pi$
$359$	11.3137	0.597115	0.298557	+	0.954392i	$0.403495\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−26.4853	−1.39204
$363$	0	0
$364$	0	0
$365$	8.48528	0.444140
$366$	0	0
$367$	−23.8701	−1.24601	−0.623003	−	0.782219i	$-0.714088\pi$
$367$	−23.8701	−1.24601	−0.623003	+	0.782219i	$0.714088\pi$
$368$	29.6569	1.54597
$369$	0	0
$370$	−3.17157	−0.164882
$371$	0	0
$372$	0	0
$373$	−16.4853	−0.853576	−0.426788	−	0.904352i	$-0.640355\pi$
$373$	−16.4853	−0.853576	−0.426788	+	0.904352i	$0.640355\pi$
$374$	−0.786797	−0.0406843
$375$	0	0
$376$	−20.4853	−1.05645
$377$	38.2132	1.96808
$378$	0	0
$379$	2.00000	0.102733	0.0513665	−	0.998680i	$-0.483642\pi$
$379$	2.00000	0.102733	0.0513665	+	0.998680i	$0.483642\pi$
$380$	0	0
$381$	0	0
$382$	−19.7574	−1.01087
$383$	−4.48528	−0.229187	−0.114594	−	0.993412i	$-0.536557\pi$
$383$	−4.48528	−0.229187	−0.114594	+	0.993412i	$0.536557\pi$
$384$	0	0
$385$	0	0
$386$	−22.6274	−1.15171
$387$	0	0
$388$	0	0
$389$	6.85786	0.347708	0.173854	−	0.984771i	$-0.444378\pi$
$389$	6.85786	0.347708	0.173854	+	0.984771i	$0.444378\pi$
$390$	0	0
$391$	24.0416	1.21584
$392$	0	0
$393$	0	0
$394$	−18.9706	−0.955723
$395$	−1.48528	−0.0747326
$396$	0	0
$397$	1.58579	0.0795883	0.0397942	−	0.999208i	$-0.487330\pi$
$397$	1.58579	0.0795883	0.0397942	+	0.999208i	$0.487330\pi$
$398$	−10.4853	−0.525580
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−11.8284	−0.590683	−0.295342	−	0.955392i	$-0.595434\pi$
$401$	−11.8284	−0.590683	−0.295342	+	0.955392i	$0.595434\pi$
$402$	0	0
$403$	45.2132	2.25223
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.384776	0.0190727
$408$	0	0
$409$	2.48528	0.122889	0.0614446	−	0.998110i	$-0.480429\pi$
$409$	2.48528	0.122889	0.0614446	+	0.998110i	$0.480429\pi$
$410$	−8.82843	−0.436005
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	1.45584	0.0712077
$419$	−18.7279	−0.914919	−0.457459	−	0.889230i	$-0.651241\pi$
$419$	−18.7279	−0.914919	−0.457459	+	0.889230i	$0.651241\pi$
$420$	0	0
$421$	−19.0000	−0.926003	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000	−0.926003	−0.463002	+	0.886357i	$0.653228\pi$
$422$	12.7279	0.619586
$423$	0	0
$424$	12.0000	0.582772
$425$	−3.24264	−0.157291
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	−2.82843	−0.136399
$431$	28.7990	1.38720	0.693599	−	0.720361i	$-0.256024\pi$
$431$	28.7990	1.38720	0.693599	+	0.720361i	$0.256024\pi$
$432$	0	0
$433$	10.9706	0.527212	0.263606	−	0.964630i	$-0.415088\pi$
$433$	10.9706	0.527212	0.263606	+	0.964630i	$0.415088\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−44.4853	−2.12802
$438$	0	0
$439$	−30.3848	−1.45019	−0.725093	−	0.688651i	$-0.758203\pi$
$439$	−30.3848	−1.45019	−0.725093	+	0.688651i	$0.758203\pi$
$440$	0.485281	0.0231349
$441$	0	0
$442$	−20.2426	−0.962844
$443$	−15.1716	−0.720823	−0.360412	−	0.932793i	$-0.617364\pi$
$443$	−15.1716	−0.720823	−0.360412	+	0.932793i	$0.617364\pi$
$444$	0	0
$445$	−8.00000	−0.379236
$446$	−34.2426	−1.62144
$447$	0	0
$448$	0	0
$449$	24.1716	1.14073	0.570364	−	0.821392i	$-0.306802\pi$
$449$	24.1716	1.14073	0.570364	+	0.821392i	$0.306802\pi$
$450$	0	0
$451$	1.07107	0.0504346
$452$	0	0
$453$	0	0
$454$	−22.2426	−1.04390
$455$	0	0
$456$	0	0
$457$	−11.7574	−0.549986	−0.274993	−	0.961446i	$-0.588676\pi$
$457$	−11.7574	−0.549986	−0.274993	+	0.961446i	$0.588676\pi$
$458$	25.5147	1.19222
$459$	0	0
$460$	0	0
$461$	−3.02944	−0.141095	−0.0705475	−	0.997508i	$-0.522475\pi$
$461$	−3.02944	−0.141095	−0.0705475	+	0.997508i	$0.522475\pi$
$462$	0	0
$463$	−21.4558	−0.997138	−0.498569	−	0.866850i	$-0.666141\pi$
$463$	−21.4558	−0.997138	−0.498569	+	0.866850i	$0.666141\pi$
$464$	−34.6274	−1.60754
$465$	0	0
$466$	12.9706	0.600850
$467$	5.72792	0.265057	0.132528	−	0.991179i	$-0.457690\pi$
$467$	5.72792	0.265057	0.132528	+	0.991179i	$0.457690\pi$
$468$	0	0
$469$	0	0
$470$	−10.2426	−0.472458
$471$	0	0
$472$	−6.34315	−0.291967
$473$	0.343146	0.0157779
$474$	0	0
$475$	6.00000	0.275299
$476$	0	0
$477$	0	0
$478$	24.7279	1.13103
$479$	−11.7574	−0.537207	−0.268604	−	0.963251i	$-0.586562\pi$
$479$	−11.7574	−0.537207	−0.268604	+	0.963251i	$0.586562\pi$
$480$	0	0
$481$	9.89949	0.451378
$482$	−1.02944	−0.0468896
$483$	0	0
$484$	0	0
$485$	−13.2426	−0.601317
$486$	0	0
$487$	27.6985	1.25514	0.627569	−	0.778561i	$-0.284050\pi$
$487$	27.6985	1.25514	0.627569	+	0.778561i	$0.284050\pi$
$488$	8.00000	0.362143
$489$	0	0
$490$	0	0
$491$	37.2843	1.68262	0.841308	−	0.540556i	$-0.181786\pi$
$491$	37.2843	1.68262	0.841308	+	0.540556i	$0.181786\pi$
$492$	0	0
$493$	−28.0711	−1.26426
$494$	37.4558	1.68522
$495$	0	0
$496$	−40.9706	−1.83963
$497$	0	0
$498$	0	0
$499$	3.00000	0.134298	0.0671492	−	0.997743i	$-0.478610\pi$
$499$	3.00000	0.134298	0.0671492	+	0.997743i	$0.478610\pi$
$500$	0	0
$501$	0	0
$502$	−24.3431	−1.08649
$503$	41.2426	1.83892	0.919459	−	0.393185i	$-0.128627\pi$
$503$	41.2426	1.83892	0.919459	+	0.393185i	$0.128627\pi$
$504$	0	0
$505$	2.48528	0.110594
$506$	−1.79899	−0.0799749
$507$	0	0
$508$	0	0
$509$	25.2132	1.11756	0.558778	−	0.829317i	$-0.311270\pi$
$509$	25.2132	1.11756	0.558778	+	0.829317i	$0.311270\pi$
$510$	0	0
$511$	0	0
$512$	−22.6274	−1.00000
$513$	0	0
$514$	13.4558	0.593512
$515$	−19.2426	−0.847932
$516$	0	0
$517$	1.24264	0.0546513
$518$	0	0
$519$	0	0
$520$	12.4853	0.547516
$521$	−14.9706	−0.655872	−0.327936	−	0.944700i	$-0.606353\pi$
$521$	−14.9706	−0.655872	−0.327936	+	0.944700i	$0.606353\pi$
$522$	0	0
$523$	32.4853	1.42048	0.710241	−	0.703959i	$-0.248586\pi$
$523$	32.4853	1.42048	0.710241	+	0.703959i	$0.248586\pi$
$524$	0	0
$525$	0	0
$526$	23.5147	1.02529
$527$	−33.2132	−1.44679
$528$	0	0
$529$	31.9706	1.39002
$530$	6.00000	0.260623
$531$	0	0
$532$	0	0
$533$	27.5563	1.19360
$534$	0	0
$535$	−2.48528	−0.107448
$536$	23.3137	1.00700
$537$	0	0
$538$	22.9706	0.990331
$539$	0	0
$540$	0	0
$541$	−21.9706	−0.944588	−0.472294	−	0.881441i	$-0.656574\pi$
$541$	−21.9706	−0.944588	−0.472294	+	0.881441i	$0.656574\pi$
$542$	−0.970563	−0.0416892
$543$	0	0
$544$	0	0
$545$	−5.00000	−0.214176
$546$	0	0
$547$	−24.4853	−1.04692	−0.523458	−	0.852052i	$-0.675358\pi$
$547$	−24.4853	−1.04692	−0.523458	+	0.852052i	$0.675358\pi$
$548$	0	0
$549$	0	0
$550$	0.242641	0.0103462
$551$	51.9411	2.21277
$552$	0	0
$553$	0	0
$554$	21.5147	0.914073
$555$	0	0
$556$	0	0
$557$	7.79899	0.330454	0.165227	−	0.986256i	$-0.447164\pi$
$557$	7.79899	0.330454	0.165227	+	0.986256i	$0.447164\pi$
$558$	0	0
$559$	8.82843	0.373403
$560$	0	0
$561$	0	0
$562$	−3.27208	−0.138024
$563$	31.9411	1.34616	0.673079	−	0.739571i	$-0.264971\pi$
$563$	31.9411	1.34616	0.673079	+	0.739571i	$0.264971\pi$
$564$	0	0
$565$	−13.0711	−0.549904
$566$	34.7279	1.45972
$567$	0	0
$568$	−8.97056	−0.376396
$569$	−26.1421	−1.09594	−0.547968	−	0.836500i	$-0.684598\pi$
$569$	−26.1421	−1.09594	−0.547968	+	0.836500i	$0.684598\pi$
$570$	0	0
$571$	−17.5147	−0.732968	−0.366484	−	0.930424i	$-0.619439\pi$
$571$	−17.5147	−0.732968	−0.366484	+	0.930424i	$0.619439\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7.41421	−0.309194
$576$	0	0
$577$	15.7279	0.654762	0.327381	−	0.944892i	$-0.393834\pi$
$577$	15.7279	0.654762	0.327381	+	0.944892i	$0.393834\pi$
$578$	−9.17157	−0.381487
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−0.727922	−0.0301475
$584$	24.0000	0.993127
$585$	0	0
$586$	−36.3848	−1.50304
$587$	37.4558	1.54597	0.772984	−	0.634425i	$-0.218763\pi$
$587$	37.4558	1.54597	0.772984	+	0.634425i	$0.218763\pi$
$588$	0	0
$589$	61.4558	2.53224
$590$	−3.17157	−0.130572
$591$	0	0
$592$	−8.97056	−0.368688
$593$	−19.2426	−0.790201	−0.395100	−	0.918638i	$-0.629290\pi$
$593$	−19.2426	−0.790201	−0.395100	+	0.918638i	$0.629290\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	−46.2843	−1.89270
$599$	17.8284	0.728450	0.364225	−	0.931311i	$-0.381334\pi$
$599$	17.8284	0.728450	0.364225	+	0.931311i	$0.381334\pi$
$600$	0	0
$601$	−10.9706	−0.447499	−0.223749	−	0.974647i	$-0.571830\pi$
$601$	−10.9706	−0.447499	−0.223749	+	0.974647i	$0.571830\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10.9706	0.446017
$606$	0	0
$607$	−5.10051	−0.207023	−0.103512	−	0.994628i	$-0.533008\pi$
$607$	−5.10051	−0.207023	−0.103512	+	0.994628i	$0.533008\pi$
$608$	0	0
$609$	0	0
$610$	4.00000	0.161955
$611$	31.9706	1.29339
$612$	0	0
$613$	−23.9411	−0.966973	−0.483486	−	0.875352i	$-0.660630\pi$
$613$	−23.9411	−0.966973	−0.483486	+	0.875352i	$0.660630\pi$
$614$	43.6985	1.76353
$615$	0	0
$616$	0	0
$617$	−4.58579	−0.184617	−0.0923084	−	0.995730i	$-0.529425\pi$
$617$	−4.58579	−0.184617	−0.0923084	+	0.995730i	$0.529425\pi$
$618$	0	0
$619$	16.9289	0.680431	0.340216	−	0.940347i	$-0.389500\pi$
$619$	16.9289	0.680431	0.340216	+	0.940347i	$0.389500\pi$
$620$	0	0
$621$	0	0
$622$	14.1421	0.567048
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−25.7574	−1.02947
$627$	0	0
$628$	0	0
$629$	−7.27208	−0.289957
$630$	0	0
$631$	−42.4558	−1.69014	−0.845070	−	0.534655i	$-0.820441\pi$
$631$	−42.4558	−1.69014	−0.845070	+	0.534655i	$0.820441\pi$
$632$	−4.20101	−0.167107
$633$	0	0
$634$	0.485281	0.0192730
$635$	−8.24264	−0.327099
$636$	0	0
$637$	0	0
$638$	2.10051	0.0831598
$639$	0	0
$640$	−11.3137	−0.447214
$641$	23.3137	0.920836	0.460418	−	0.887702i	$-0.347700\pi$
$641$	23.3137	0.920836	0.460418	+	0.887702i	$0.347700\pi$
$642$	0	0
$643$	2.27208	0.0896020	0.0448010	−	0.998996i	$-0.485735\pi$
$643$	2.27208	0.0896020	0.0448010	+	0.998996i	$0.485735\pi$
$644$	0	0
$645$	0	0
$646$	−27.5147	−1.08255
$647$	11.5147	0.452690	0.226345	−	0.974047i	$-0.427322\pi$
$647$	11.5147	0.452690	0.226345	+	0.974047i	$0.427322\pi$
$648$	0	0
$649$	0.384776	0.0151038
$650$	6.24264	0.244857
$651$	0	0
$652$	0	0
$653$	−34.9706	−1.36850	−0.684252	−	0.729246i	$-0.739871\pi$
$653$	−34.9706	−1.36850	−0.684252	+	0.729246i	$0.739871\pi$
$654$	0	0
$655$	12.2426	0.478360
$656$	−24.9706	−0.974937
$657$	0	0
$658$	0	0
$659$	−19.9706	−0.777943	−0.388971	−	0.921250i	$-0.627169\pi$
$659$	−19.9706	−0.777943	−0.388971	+	0.921250i	$0.627169\pi$
$660$	0	0
$661$	13.4558	0.523372	0.261686	−	0.965153i	$-0.415722\pi$
$661$	13.4558	0.523372	0.261686	+	0.965153i	$0.415722\pi$
$662$	−38.8284	−1.50911
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−64.1838	−2.48521
$668$	0	0
$669$	0	0
$670$	11.6569	0.450344
$671$	−0.485281	−0.0187341
$672$	0	0
$673$	3.51472	0.135482	0.0677412	−	0.997703i	$-0.478421\pi$
$673$	3.51472	0.135482	0.0677412	+	0.997703i	$0.478421\pi$
$674$	31.4558	1.21163
$675$	0	0
$676$	0	0
$677$	−44.2132	−1.69925	−0.849626	−	0.527386i	$-0.823172\pi$
$677$	−44.2132	−1.69925	−0.849626	+	0.527386i	$0.823172\pi$
$678$	0	0
$679$	0	0
$680$	−9.17157	−0.351714
$681$	0	0
$682$	2.48528	0.0951663
$683$	−31.7990	−1.21675	−0.608377	−	0.793648i	$-0.708179\pi$
$683$	−31.7990	−1.21675	−0.608377	+	0.793648i	$0.708179\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	0	0
$687$	0	0
$688$	−8.00000	−0.304997
$689$	−18.7279	−0.713477
$690$	0	0
$691$	14.8284	0.564100	0.282050	−	0.959400i	$-0.408986\pi$
$691$	14.8284	0.564100	0.282050	+	0.959400i	$0.408986\pi$
$692$	0	0
$693$	0	0
$694$	−18.4853	−0.701692
$695$	−7.75736	−0.294253
$696$	0	0
$697$	−20.2426	−0.766745
$698$	−15.5147	−0.587241
$699$	0	0
$700$	0	0
$701$	−46.4558	−1.75461	−0.877307	−	0.479931i	$-0.840662\pi$
$701$	−46.4558	−1.75461	−0.877307	+	0.479931i	$0.840662\pi$
$702$	0	0
$703$	13.4558	0.507497
$704$	1.37258	0.0517312
$705$	0	0
$706$	8.78680	0.330695
$707$	0	0
$708$	0	0
$709$	17.0000	0.638448	0.319224	−	0.947679i	$-0.396578\pi$
$709$	17.0000	0.638448	0.319224	+	0.947679i	$0.396578\pi$
$710$	−4.48528	−0.168330
$711$	0	0
$712$	−22.6274	−0.847998
$713$	−75.9411	−2.84402
$714$	0	0
$715$	−0.757359	−0.0283236
$716$	0	0
$717$	0	0
$718$	16.0000	0.597115
$719$	25.2132	0.940294	0.470147	−	0.882588i	$-0.344201\pi$
$719$	25.2132	0.940294	0.470147	+	0.882588i	$0.344201\pi$
$720$	0	0
$721$	0	0
$722$	24.0416	0.894737
$723$	0	0
$724$	0	0
$725$	8.65685	0.321507
$726$	0	0
$727$	−6.68629	−0.247981	−0.123990	−	0.992283i	$-0.539569\pi$
$727$	−6.68629	−0.247981	−0.123990	+	0.992283i	$0.539569\pi$
$728$	0	0
$729$	0	0
$730$	12.0000	0.444140
$731$	−6.48528	−0.239867
$732$	0	0
$733$	14.6985	0.542901	0.271450	−	0.962452i	$-0.412497\pi$
$733$	14.6985	0.542901	0.271450	+	0.962452i	$0.412497\pi$
$734$	−33.7574	−1.24601
$735$	0	0
$736$	0	0
$737$	−1.41421	−0.0520932
$738$	0	0
$739$	29.9706	1.10248	0.551242	−	0.834345i	$-0.314154\pi$
$739$	29.9706	1.10248	0.551242	+	0.834345i	$0.314154\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−11.2721	−0.413532	−0.206766	−	0.978390i	$-0.566294\pi$
$743$	−11.2721	−0.413532	−0.206766	+	0.978390i	$0.566294\pi$
$744$	0	0
$745$	−9.17157	−0.336020
$746$	−23.3137	−0.853576
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	29.4853	1.07593	0.537967	−	0.842966i	$-0.319193\pi$
$751$	29.4853	1.07593	0.537967	+	0.842966i	$0.319193\pi$
$752$	−28.9706	−1.05645
$753$	0	0
$754$	54.0416	1.96808
$755$	7.48528	0.272417
$756$	0	0
$757$	0.485281	0.0176379	0.00881893	−	0.999961i	$-0.497193\pi$
$757$	0.485281	0.0176379	0.00881893	+	0.999961i	$0.497193\pi$
$758$	2.82843	0.102733
$759$	0	0
$760$	16.9706	0.615587
$761$	12.7279	0.461387	0.230693	−	0.973026i	$-0.425901\pi$
$761$	12.7279	0.461387	0.230693	+	0.973026i	$0.425901\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−6.34315	−0.229187
$767$	9.89949	0.357450
$768$	0	0
$769$	−8.82843	−0.318361	−0.159181	−	0.987249i	$-0.550885\pi$
$769$	−8.82843	−0.318361	−0.159181	+	0.987249i	$0.550885\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	6.21320	0.223473	0.111737	−	0.993738i	$-0.464359\pi$
$773$	6.21320	0.223473	0.111737	+	0.993738i	$0.464359\pi$
$774$	0	0
$775$	10.2426	0.367927
$776$	−37.4558	−1.34459
$777$	0	0
$778$	9.69848	0.347708
$779$	37.4558	1.34199
$780$	0	0
$781$	0.544156	0.0194714
$782$	34.0000	1.21584
$783$	0	0
$784$	0	0
$785$	15.1716	0.541497
$786$	0	0
$787$	20.2721	0.722622	0.361311	−	0.932445i	$-0.382329\pi$
$787$	20.2721	0.722622	0.361311	+	0.932445i	$0.382329\pi$
$788$	0	0
$789$	0	0
$790$	−2.10051	−0.0747326
$791$	0	0
$792$	0	0
$793$	−12.4853	−0.443365
$794$	2.24264	0.0795883
$795$	0	0
$796$	0	0
$797$	21.1838	0.750367	0.375184	−	0.926950i	$-0.377580\pi$
$797$	21.1838	0.750367	0.375184	+	0.926950i	$0.377580\pi$
$798$	0	0
$799$	−23.4853	−0.830850
$800$	0	0
$801$	0	0
$802$	−16.7279	−0.590683
$803$	−1.45584	−0.0513756
$804$	0	0
$805$	0	0
$806$	63.9411	2.25223
$807$	0	0
$808$	7.02944	0.247295
$809$	48.5980	1.70861	0.854307	−	0.519769i	$-0.173982\pi$
$809$	48.5980	1.70861	0.854307	+	0.519769i	$0.173982\pi$
$810$	0	0
$811$	47.3553	1.66287	0.831435	−	0.555621i	$-0.187520\pi$
$811$	47.3553	1.66287	0.831435	+	0.555621i	$0.187520\pi$
$812$	0	0
$813$	0	0
$814$	0.544156	0.0190727
$815$	10.2426	0.358784
$816$	0	0
$817$	12.0000	0.419827
$818$	3.51472	0.122889
$819$	0	0
$820$	0	0
$821$	23.4853	0.819642	0.409821	−	0.912166i	$-0.365591\pi$
$821$	23.4853	0.819642	0.409821	+	0.912166i	$0.365591\pi$
$822$	0	0
$823$	16.7279	0.583099	0.291549	−	0.956556i	$-0.405829\pi$
$823$	16.7279	0.583099	0.291549	+	0.956556i	$0.405829\pi$
$824$	−54.4264	−1.89603
$825$	0	0
$826$	0	0
$827$	−42.0416	−1.46193	−0.730965	−	0.682415i	$-0.760930\pi$
$827$	−42.0416	−1.46193	−0.730965	+	0.682415i	$0.760930\pi$
$828$	0	0
$829$	5.95837	0.206943	0.103471	−	0.994632i	$-0.467005\pi$
$829$	5.95837	0.206943	0.103471	+	0.994632i	$0.467005\pi$
$830$	0	0
$831$	0	0
$832$	35.3137	1.22428
$833$	0	0
$834$	0	0
$835$	0.757359	0.0262095
$836$	0	0
$837$	0	0
$838$	−26.4853	−0.914919
$839$	23.2721	0.803441	0.401721	−	0.915762i	$-0.368412\pi$
$839$	23.2721	0.803441	0.401721	+	0.915762i	$0.368412\pi$
$840$	0	0
$841$	45.9411	1.58418
$842$	−26.8701	−0.926003
$843$	0	0
$844$	0	0
$845$	−6.48528	−0.223100
$846$	0	0
$847$	0	0
$848$	16.9706	0.582772
$849$	0	0
$850$	−4.58579	−0.157291
$851$	−16.6274	−0.569981
$852$	0	0
$853$	10.9706	0.375625	0.187812	−	0.982205i	$-0.439860\pi$
$853$	10.9706	0.375625	0.187812	+	0.982205i	$0.439860\pi$
$854$	0	0
$855$	0	0
$856$	−7.02944	−0.240261
$857$	22.4853	0.768083	0.384041	−	0.923316i	$-0.374532\pi$
$857$	22.4853	0.768083	0.384041	+	0.923316i	$0.374532\pi$
$858$	0	0
$859$	−24.7696	−0.845126	−0.422563	−	0.906334i	$-0.638870\pi$
$859$	−24.7696	−0.845126	−0.422563	+	0.906334i	$0.638870\pi$
$860$	0	0
$861$	0	0
$862$	40.7279	1.38720
$863$	−18.3848	−0.625825	−0.312913	−	0.949782i	$-0.601305\pi$
$863$	−18.3848	−0.625825	−0.312913	+	0.949782i	$0.601305\pi$
$864$	0	0
$865$	−7.24264	−0.246257
$866$	15.5147	0.527212
$867$	0	0
$868$	0	0
$869$	0.254834	0.00864465
$870$	0	0
$871$	−36.3848	−1.23285
$872$	−14.1421	−0.478913
$873$	0	0
$874$	−62.9117	−2.12802
$875$	0	0
$876$	0	0
$877$	13.0294	0.439973	0.219986	−	0.975503i	$-0.429399\pi$
$877$	13.0294	0.439973	0.219986	+	0.975503i	$0.429399\pi$
$878$	−42.9706	−1.45019
$879$	0	0
$880$	0.686292	0.0231349
$881$	52.9706	1.78462	0.892312	−	0.451420i	$-0.149082\pi$
$881$	52.9706	1.78462	0.892312	+	0.451420i	$0.149082\pi$
$882$	0	0
$883$	31.5147	1.06055	0.530277	−	0.847824i	$-0.322088\pi$
$883$	31.5147	1.06055	0.530277	+	0.847824i	$0.322088\pi$
$884$	0	0
$885$	0	0
$886$	−21.4558	−0.720823
$887$	−2.97056	−0.0997417	−0.0498709	−	0.998756i	$-0.515881\pi$
$887$	−2.97056	−0.0997417	−0.0498709	+	0.998756i	$0.515881\pi$
$888$	0	0
$889$	0	0
$890$	−11.3137	−0.379236
$891$	0	0
$892$	0	0
$893$	43.4558	1.45419
$894$	0	0
$895$	−14.4853	−0.484190
$896$	0	0
$897$	0	0
$898$	34.1838	1.14073
$899$	88.6690	2.95728
$900$	0	0
$901$	13.7574	0.458324
$902$	1.51472	0.0504346
$903$	0	0
$904$	−36.9706	−1.22962
$905$	18.7279	0.622537
$906$	0	0
$907$	−44.1838	−1.46710	−0.733549	−	0.679637i	$-0.762137\pi$
$907$	−44.1838	−1.46710	−0.733549	+	0.679637i	$0.762137\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−5.65685	−0.187420	−0.0937100	−	0.995600i	$-0.529873\pi$
$911$	−5.65685	−0.187420	−0.0937100	+	0.995600i	$0.529873\pi$
$912$	0	0
$913$	0	0
$914$	−16.6274	−0.549986
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	12.4558	0.410880	0.205440	−	0.978670i	$-0.434137\pi$
$919$	12.4558	0.410880	0.205440	+	0.978670i	$0.434137\pi$
$920$	−20.9706	−0.691379
$921$	0	0
$922$	−4.28427	−0.141095
$923$	14.0000	0.460816
$924$	0	0
$925$	2.24264	0.0737376
$926$	−30.3431	−0.997138
$927$	0	0
$928$	0	0
$929$	−29.2721	−0.960386	−0.480193	−	0.877163i	$-0.659433\pi$
$929$	−29.2721	−0.960386	−0.480193	+	0.877163i	$0.659433\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	8.10051	0.265057
$935$	0.556349	0.0181946
$936$	0	0
$937$	−48.5563	−1.58627	−0.793133	−	0.609048i	$-0.791552\pi$
$937$	−48.5563	−1.58627	−0.793133	+	0.609048i	$0.791552\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−28.9706	−0.944413	−0.472207	−	0.881488i	$-0.656542\pi$
$941$	−28.9706	−0.944413	−0.472207	+	0.881488i	$0.656542\pi$
$942$	0	0
$943$	−46.2843	−1.50722
$944$	−8.97056	−0.291967
$945$	0	0
$946$	0.485281	0.0157779
$947$	52.2426	1.69766	0.848829	−	0.528668i	$-0.177308\pi$
$947$	52.2426	1.69766	0.848829	+	0.528668i	$0.177308\pi$
$948$	0	0
$949$	−37.4558	−1.21587
$950$	8.48528	0.275299
$951$	0	0
$952$	0	0
$953$	−53.0122	−1.71723	−0.858617	−	0.512618i	$-0.828676\pi$
$953$	−53.0122	−1.71723	−0.858617	+	0.512618i	$0.828676\pi$
$954$	0	0
$955$	13.9706	0.452077
$956$	0	0
$957$	0	0
$958$	−16.6274	−0.537207
$959$	0	0
$960$	0	0
$961$	73.9117	2.38425
$962$	14.0000	0.451378
$963$	0	0
$964$	0	0
$965$	16.0000	0.515058
$966$	0	0
$967$	−60.4264	−1.94318	−0.971591	−	0.236666i	$-0.923945\pi$
$967$	−60.4264	−1.94318	−0.971591	+	0.236666i	$0.923945\pi$
$968$	31.0294	0.997324
$969$	0	0
$970$	−18.7279	−0.601317
$971$	17.2721	0.554287	0.277144	−	0.960828i	$-0.410612\pi$
$971$	17.2721	0.554287	0.277144	+	0.960828i	$0.410612\pi$
$972$	0	0
$973$	0	0
$974$	39.1716	1.25514
$975$	0	0
$976$	11.3137	0.362143
$977$	−42.7696	−1.36832	−0.684160	−	0.729332i	$-0.739831\pi$
$977$	−42.7696	−1.36832	−0.684160	+	0.729332i	$0.739831\pi$
$978$	0	0
$979$	1.37258	0.0438679
$980$	0	0
$981$	0	0
$982$	52.7279	1.68262
$983$	−0.213203	−0.00680013	−0.00340007	−	0.999994i	$-0.501082\pi$
$983$	−0.213203	−0.00680013	−0.00340007	+	0.999994i	$0.501082\pi$
$984$	0	0
$985$	13.4142	0.427412
$986$	−39.6985	−1.26426
$987$	0	0
$988$	0	0
$989$	−14.8284	−0.471517
$990$	0	0
$991$	19.9411	0.633451	0.316725	−	0.948517i	$-0.397417\pi$
$991$	19.9411	0.633451	0.316725	+	0.948517i	$0.397417\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	7.41421	0.235046
$996$	0	0
$997$	−21.7279	−0.688130	−0.344065	−	0.938946i	$-0.611804\pi$
$997$	−21.7279	−0.688130	−0.344065	+	0.938946i	$0.611804\pi$
$998$	4.24264	0.134298
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.2.a.t.1.2		2
3.2	odd	2		245.2.a.f.1.1	yes	2
7.6	odd	2		2205.2.a.v.1.2		2
12.11	even	2		3920.2.a.br.1.2		2
15.2	even	4		1225.2.b.j.99.1		4
15.8	even	4		1225.2.b.j.99.4		4
15.14	odd	2		1225.2.a.p.1.2		2
21.2	odd	6		245.2.e.f.116.2		4
21.5	even	6		245.2.e.g.116.2		4
21.11	odd	6		245.2.e.f.226.2		4
21.17	even	6		245.2.e.g.226.2		4
21.20	even	2		245.2.a.e.1.1	✓	2
84.83	odd	2		3920.2.a.bw.1.1		2
105.62	odd	4		1225.2.b.i.99.2		4
105.83	odd	4		1225.2.b.i.99.3		4
105.104	even	2		1225.2.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.a.e.1.1	✓	2	21.20	even	2
245.2.a.f.1.1	yes	2	3.2	odd	2
245.2.e.f.116.2		4	21.2	odd	6
245.2.e.f.226.2		4	21.11	odd	6
245.2.e.g.116.2		4	21.5	even	6
245.2.e.g.226.2		4	21.17	even	6
1225.2.a.p.1.2		2	15.14	odd	2
1225.2.a.r.1.2		2	105.104	even	2
1225.2.b.i.99.2		4	105.62	odd	4
1225.2.b.i.99.3		4	105.83	odd	4
1225.2.b.j.99.1		4	15.2	even	4
1225.2.b.j.99.4		4	15.8	even	4
2205.2.a.t.1.2		2	1.1	even	1	trivial
2205.2.a.v.1.2		2	7.6	odd	2
3920.2.a.br.1.2		2	12.11	even	2
3920.2.a.bw.1.1		2	84.83	odd	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)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} -1.00000 q^{5} -2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} -1.00000 q^{5} -2.82843 q^{8} -1.41421 q^{10} +0.171573 q^{11} +4.41421 q^{13} -4.00000 q^{16} -3.24264 q^{17} +6.00000 q^{19} +0.242641 q^{22} -7.41421 q^{23} +1.00000 q^{25} +6.24264 q^{26} +8.65685 q^{29} +10.2426 q^{31} -4.58579 q^{34} +2.24264 q^{37} +8.48528 q^{38} +2.82843 q^{40} +6.24264 q^{41} +2.00000 q^{43} -10.4853 q^{46} +7.24264 q^{47} +1.41421 q^{50} -4.24264 q^{53} -0.171573 q^{55} +12.2426 q^{58} +2.24264 q^{59} -2.82843 q^{61} +14.4853 q^{62} +8.00000 q^{64} -4.41421 q^{65} -8.24264 q^{67} +3.17157 q^{71} -8.48528 q^{73} +3.17157 q^{74} +1.48528 q^{79} +4.00000 q^{80} +8.82843 q^{82} +3.24264 q^{85} +2.82843 q^{86} -0.485281 q^{88} +8.00000 q^{89} +10.2426 q^{94} -6.00000 q^{95} +13.2426 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5}+O(q^{10}) \) 2 * q - 2 * q^5	\( 2 q - 2 q^{5} + 6 q^{11} + 6 q^{13} - 8 q^{16} + 2 q^{17} + 12 q^{19} - 8 q^{22} - 12 q^{23} + 2 q^{25} + 4 q^{26} + 6 q^{29} + 12 q^{31} - 12 q^{34} - 4 q^{37} + 4 q^{41} + 4 q^{43} - 4 q^{46} + 6 q^{47} - 6 q^{55} + 16 q^{58} - 4 q^{59} + 12 q^{62} + 16 q^{64} - 6 q^{65} - 8 q^{67} + 12 q^{71} + 12 q^{74} - 14 q^{79} + 8 q^{80} + 12 q^{82} - 2 q^{85} + 16 q^{88} + 16 q^{89} + 12 q^{94} - 12 q^{95} + 18 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 6 * q^11 + 6 * q^13 - 8 * q^16 + 2 * q^17 + 12 * q^19 - 8 * q^22 - 12 * q^23 + 2 * q^25 + 4 * q^26 + 6 * q^29 + 12 * q^31 - 12 * q^34 - 4 * q^37 + 4 * q^41 + 4 * q^43 - 4 * q^46 + 6 * q^47 - 6 * q^55 + 16 * q^58 - 4 * q^59 + 12 * q^62 + 16 * q^64 - 6 * q^65 - 8 * q^67 + 12 * q^71 + 12 * q^74 - 14 * q^79 + 8 * q^80 + 12 * q^82 - 2 * q^85 + 16 * q^88 + 16 * q^89 + 12 * q^94 - 12 * q^95 + 18 * q^97

Introduction

L-functions

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