LMFDB - Embedded newform 2205.2.a.n.1.1

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			1.1
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)

$f(q)$	$=$	$q-2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} -4.41421 q^{8} +O(q^{10})$	$q-2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} -4.41421 q^{8} +2.41421 q^{10} +0.828427 q^{11} -4.82843 q^{13} +3.00000 q^{16} -4.82843 q^{17} +2.82843 q^{19} -3.82843 q^{20} -2.00000 q^{22} -0.414214 q^{23} +1.00000 q^{25} +11.6569 q^{26} +1.00000 q^{29} -6.00000 q^{31} +1.58579 q^{32} +11.6569 q^{34} -6.82843 q^{38} +4.41421 q^{40} +7.82843 q^{41} +3.58579 q^{43} +3.17157 q^{44} +1.00000 q^{46} -2.00000 q^{47} -2.41421 q^{50} -18.4853 q^{52} +1.17157 q^{53} -0.828427 q^{55} -2.41421 q^{58} -4.48528 q^{59} +5.48528 q^{61} +14.4853 q^{62} -9.82843 q^{64} +4.82843 q^{65} +9.58579 q^{67} -18.4853 q^{68} -4.48528 q^{71} -0.828427 q^{73} +10.8284 q^{76} +14.8284 q^{79} -3.00000 q^{80} -18.8995 q^{82} -13.7279 q^{83} +4.82843 q^{85} -8.65685 q^{86} -3.65685 q^{88} +8.65685 q^{89} -1.58579 q^{92} +4.82843 q^{94} -2.82843 q^{95} +11.6569 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.41421	−1.70711	−0.853553	−	0.521005i	$-0.825557\pi$
$2$	−2.41421	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−4.41421	−1.56066
$9$	0	0
$10$	2.41421	0.763441
$11$	0.828427	0.249780	0.124890	−	0.992171i	$-0.460142\pi$
$11$	0.828427	0.249780	0.124890	+	0.992171i	$0.460142\pi$
$12$	0	0
$13$	−4.82843	−1.33916	−0.669582	−	0.742738i	$-0.733527\pi$
$13$	−4.82843	−1.33916	−0.669582	+	0.742738i	$0.733527\pi$
$14$	0	0
$15$	0	0
$16$	3.00000	0.750000
$17$	−4.82843	−1.17107	−0.585533	−	0.810649i	$-0.699115\pi$
$17$	−4.82843	−1.17107	−0.585533	+	0.810649i	$0.699115\pi$
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	−3.82843	−0.856062
$21$	0	0
$22$	−2.00000	−0.426401
$23$	−0.414214	−0.0863695	−0.0431847	−	0.999067i	$-0.513750\pi$
$23$	−0.414214	−0.0863695	−0.0431847	+	0.999067i	$0.513750\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	11.6569	2.28610
$27$	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	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	1.58579	0.280330
$33$	0	0
$34$	11.6569	1.99913
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	−6.82843	−1.10772
$39$	0	0
$40$	4.41421	0.697948
$41$	7.82843	1.22259	0.611297	−	0.791401i	$-0.290648\pi$
$41$	7.82843	1.22259	0.611297	+	0.791401i	$0.290648\pi$
$42$	0	0
$43$	3.58579	0.546827	0.273414	−	0.961897i	$-0.411847\pi$
$43$	3.58579	0.546827	0.273414	+	0.961897i	$0.411847\pi$
$44$	3.17157	0.478133
$45$	0	0
$46$	1.00000	0.147442
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	0	0
$50$	−2.41421	−0.341421
$51$	0	0
$52$	−18.4853	−2.56345
$53$	1.17157	0.160928	0.0804640	−	0.996758i	$-0.474360\pi$
$53$	1.17157	0.160928	0.0804640	+	0.996758i	$0.474360\pi$
$54$	0	0
$55$	−0.828427	−0.111705
$56$	0	0
$57$	0	0
$58$	−2.41421	−0.317002
$59$	−4.48528	−0.583934	−0.291967	−	0.956428i	$-0.594310\pi$
$59$	−4.48528	−0.583934	−0.291967	+	0.956428i	$0.594310\pi$
$60$	0	0
$61$	5.48528	0.702318	0.351159	−	0.936316i	$-0.385788\pi$
$61$	5.48528	0.702318	0.351159	+	0.936316i	$0.385788\pi$
$62$	14.4853	1.83963
$63$	0	0
$64$	−9.82843	−1.22855
$65$	4.82843	0.598893
$66$	0	0
$67$	9.58579	1.17109	0.585545	−	0.810640i	$-0.300881\pi$
$67$	9.58579	1.17109	0.585545	+	0.810640i	$0.300881\pi$
$68$	−18.4853	−2.24167
$69$	0	0
$70$	0	0
$71$	−4.48528	−0.532305	−0.266152	−	0.963931i	$-0.585752\pi$
$71$	−4.48528	−0.532305	−0.266152	+	0.963931i	$0.585752\pi$
$72$	0	0
$73$	−0.828427	−0.0969601	−0.0484800	−	0.998824i	$-0.515438\pi$
$73$	−0.828427	−0.0969601	−0.0484800	+	0.998824i	$0.515438\pi$
$74$	0	0
$75$	0	0
$76$	10.8284	1.24211
$77$	0	0
$78$	0	0
$79$	14.8284	1.66833	0.834164	−	0.551516i	$-0.185951\pi$
$79$	14.8284	1.66833	0.834164	+	0.551516i	$0.185951\pi$
$80$	−3.00000	−0.335410
$81$	0	0
$82$	−18.8995	−2.08710
$83$	−13.7279	−1.50684	−0.753418	−	0.657542i	$-0.771596\pi$
$83$	−13.7279	−1.50684	−0.753418	+	0.657542i	$0.771596\pi$
$84$	0	0
$85$	4.82843	0.523716
$86$	−8.65685	−0.933493
$87$	0	0
$88$	−3.65685	−0.389822
$89$	8.65685	0.917625	0.458812	−	0.888533i	$-0.348275\pi$
$89$	8.65685	0.917625	0.458812	+	0.888533i	$0.348275\pi$
$90$	0	0
$91$	0	0
$92$	−1.58579	−0.165330
$93$	0	0
$94$	4.82843	0.498014
$95$	−2.82843	−0.290191
$96$	0	0
$97$	11.6569	1.18357	0.591787	−	0.806094i	$-0.298423\pi$
$97$	11.6569	1.18357	0.591787	+	0.806094i	$0.298423\pi$
$98$	0	0
$99$	0	0
$100$	3.82843	0.382843
$101$	10.3137	1.02625	0.513126	−	0.858313i	$-0.328487\pi$
$101$	10.3137	1.02625	0.513126	+	0.858313i	$0.328487\pi$
$102$	0	0
$103$	−2.41421	−0.237880	−0.118940	−	0.992901i	$-0.537950\pi$
$103$	−2.41421	−0.237880	−0.118940	+	0.992901i	$0.537950\pi$
$104$	21.3137	2.08998
$105$	0	0
$106$	−2.82843	−0.274721
$107$	−11.2426	−1.08687	−0.543434	−	0.839452i	$-0.682876\pi$
$107$	−11.2426	−1.08687	−0.543434	+	0.839452i	$0.682876\pi$
$108$	0	0
$109$	−13.4853	−1.29166	−0.645828	−	0.763483i	$-0.723488\pi$
$109$	−13.4853	−1.29166	−0.645828	+	0.763483i	$0.723488\pi$
$110$	2.00000	0.190693
$111$	0	0
$112$	0	0
$113$	4.48528	0.421940	0.210970	−	0.977493i	$-0.432338\pi$
$113$	4.48528	0.421940	0.210970	+	0.977493i	$0.432338\pi$
$114$	0	0
$115$	0.414214	0.0386256
$116$	3.82843	0.355461
$117$	0	0
$118$	10.8284	0.996838
$119$	0	0
$120$	0	0
$121$	−10.3137	−0.937610
$122$	−13.2426	−1.19893
$123$	0	0
$124$	−22.9706	−2.06282
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−9.31371	−0.826458	−0.413229	−	0.910627i	$-0.635599\pi$
$127$	−9.31371	−0.826458	−0.413229	+	0.910627i	$0.635599\pi$
$128$	20.5563	1.81694
$129$	0	0
$130$	−11.6569	−1.02237
$131$	19.3137	1.68745	0.843723	−	0.536778i	$-0.180359\pi$
$131$	19.3137	1.68745	0.843723	+	0.536778i	$0.180359\pi$
$132$	0	0
$133$	0	0
$134$	−23.1421	−1.99918
$135$	0	0
$136$	21.3137	1.82764
$137$	9.65685	0.825041	0.412520	−	0.910948i	$-0.364649\pi$
$137$	9.65685	0.825041	0.412520	+	0.910948i	$0.364649\pi$
$138$	0	0
$139$	−16.1421	−1.36916	−0.684579	−	0.728939i	$-0.740014\pi$
$139$	−16.1421	−1.36916	−0.684579	+	0.728939i	$0.740014\pi$
$140$	0	0
$141$	0	0
$142$	10.8284	0.908701
$143$	−4.00000	−0.334497
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	2.00000	0.165521
$147$	0	0
$148$	0	0
$149$	2.17157	0.177902	0.0889511	−	0.996036i	$-0.471649\pi$
$149$	2.17157	0.177902	0.0889511	+	0.996036i	$0.471649\pi$
$150$	0	0
$151$	11.6569	0.948621	0.474311	−	0.880358i	$-0.342697\pi$
$151$	11.6569	0.948621	0.474311	+	0.880358i	$0.342697\pi$
$152$	−12.4853	−1.01269
$153$	0	0
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	17.3137	1.38178	0.690892	−	0.722958i	$-0.257218\pi$
$157$	17.3137	1.38178	0.690892	+	0.722958i	$0.257218\pi$
$158$	−35.7990	−2.84801
$159$	0	0
$160$	−1.58579	−0.125367
$161$	0	0
$162$	0	0
$163$	12.3431	0.966790	0.483395	−	0.875402i	$-0.339404\pi$
$163$	12.3431	0.966790	0.483395	+	0.875402i	$0.339404\pi$
$164$	29.9706	2.34031
$165$	0	0
$166$	33.1421	2.57233
$167$	−22.4142	−1.73446	−0.867232	−	0.497904i	$-0.834103\pi$
$167$	−22.4142	−1.73446	−0.867232	+	0.497904i	$0.834103\pi$
$168$	0	0
$169$	10.3137	0.793362
$170$	−11.6569	−0.894040
$171$	0	0
$172$	13.7279	1.04674
$173$	−3.31371	−0.251937	−0.125968	−	0.992034i	$-0.540204\pi$
$173$	−3.31371	−0.251937	−0.125968	+	0.992034i	$0.540204\pi$
$174$	0	0
$175$	0	0
$176$	2.48528	0.187335
$177$	0	0
$178$	−20.8995	−1.56648
$179$	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
$181$	2.65685	0.197482	0.0987412	−	0.995113i	$-0.468518\pi$
$181$	2.65685	0.197482	0.0987412	+	0.995113i	$0.468518\pi$
$182$	0	0
$183$	0	0
$184$	1.82843	0.134793
$185$	0	0
$186$	0	0
$187$	−4.00000	−0.292509
$188$	−7.65685	−0.558433
$189$	0	0
$190$	6.82843	0.495386
$191$	12.8284	0.928232	0.464116	−	0.885774i	$-0.346372\pi$
$191$	12.8284	0.928232	0.464116	+	0.885774i	$0.346372\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	−28.1421	−2.02049
$195$	0	0
$196$	0	0
$197$	12.3431	0.879413	0.439706	−	0.898142i	$-0.355083\pi$
$197$	12.3431	0.879413	0.439706	+	0.898142i	$0.355083\pi$
$198$	0	0
$199$	−9.65685	−0.684556	−0.342278	−	0.939599i	$-0.611199\pi$
$199$	−9.65685	−0.684556	−0.342278	+	0.939599i	$0.611199\pi$
$200$	−4.41421	−0.312132
$201$	0	0
$202$	−24.8995	−1.75192
$203$	0	0
$204$	0	0
$205$	−7.82843	−0.546761
$206$	5.82843	0.406086
$207$	0	0
$208$	−14.4853	−1.00437
$209$	2.34315	0.162079
$210$	0	0
$211$	20.4853	1.41026	0.705132	−	0.709076i	$-0.250888\pi$
$211$	20.4853	1.41026	0.705132	+	0.709076i	$0.250888\pi$
$212$	4.48528	0.308050
$213$	0	0
$214$	27.1421	1.85540
$215$	−3.58579	−0.244549
$216$	0	0
$217$	0	0
$218$	32.5563	2.20499
$219$	0	0
$220$	−3.17157	−0.213827
$221$	23.3137	1.56825
$222$	0	0
$223$	0.343146	0.0229787	0.0114894	−	0.999934i	$-0.496343\pi$
$223$	0.343146	0.0229787	0.0114894	+	0.999934i	$0.496343\pi$
$224$	0	0
$225$	0	0
$226$	−10.8284	−0.720296
$227$	6.97056	0.462652	0.231326	−	0.972876i	$-0.425694\pi$
$227$	6.97056	0.462652	0.231326	+	0.972876i	$0.425694\pi$
$228$	0	0
$229$	−11.6569	−0.770307	−0.385153	−	0.922853i	$-0.625851\pi$
$229$	−11.6569	−0.770307	−0.385153	+	0.922853i	$0.625851\pi$
$230$	−1.00000	−0.0659380
$231$	0	0
$232$	−4.41421	−0.289807
$233$	16.8284	1.10247	0.551233	−	0.834351i	$-0.314157\pi$
$233$	16.8284	1.10247	0.551233	+	0.834351i	$0.314157\pi$
$234$	0	0
$235$	2.00000	0.130466
$236$	−17.1716	−1.11777
$237$	0	0
$238$	0	0
$239$	21.3137	1.37867	0.689335	−	0.724443i	$-0.257903\pi$
$239$	21.3137	1.37867	0.689335	+	0.724443i	$0.257903\pi$
$240$	0	0
$241$	27.6569	1.78153	0.890767	−	0.454460i	$-0.150168\pi$
$241$	27.6569	1.78153	0.890767	+	0.454460i	$0.150168\pi$
$242$	24.8995	1.60060
$243$	0	0
$244$	21.0000	1.34439
$245$	0	0
$246$	0	0
$247$	−13.6569	−0.868965
$248$	26.4853	1.68182
$249$	0	0
$250$	2.41421	0.152688
$251$	9.31371	0.587876	0.293938	−	0.955824i	$-0.405034\pi$
$251$	9.31371	0.587876	0.293938	+	0.955824i	$0.405034\pi$
$252$	0	0
$253$	−0.343146	−0.0215734
$254$	22.4853	1.41085
$255$	0	0
$256$	−29.9706	−1.87316
$257$	−6.34315	−0.395675	−0.197837	−	0.980235i	$-0.563392\pi$
$257$	−6.34315	−0.395675	−0.197837	+	0.980235i	$0.563392\pi$
$258$	0	0
$259$	0	0
$260$	18.4853	1.14641
$261$	0	0
$262$	−46.6274	−2.88065
$263$	29.0416	1.79078	0.895392	−	0.445279i	$-0.146896\pi$
$263$	29.0416	1.79078	0.895392	+	0.445279i	$0.146896\pi$
$264$	0	0
$265$	−1.17157	−0.0719691
$266$	0	0
$267$	0	0
$268$	36.6985	2.24172
$269$	−20.4558	−1.24721	−0.623607	−	0.781738i	$-0.714334\pi$
$269$	−20.4558	−1.24721	−0.623607	+	0.781738i	$0.714334\pi$
$270$	0	0
$271$	16.4853	1.00141	0.500705	−	0.865618i	$-0.333074\pi$
$271$	16.4853	1.00141	0.500705	+	0.865618i	$0.333074\pi$
$272$	−14.4853	−0.878299
$273$	0	0
$274$	−23.3137	−1.40843
$275$	0.828427	0.0499560
$276$	0	0
$277$	16.1421	0.969887	0.484943	−	0.874546i	$-0.338840\pi$
$277$	16.1421	0.969887	0.484943	+	0.874546i	$0.338840\pi$
$278$	38.9706	2.33730
$279$	0	0
$280$	0	0
$281$	30.2843	1.80661	0.903304	−	0.429001i	$-0.141134\pi$
$281$	30.2843	1.80661	0.903304	+	0.429001i	$0.141134\pi$
$282$	0	0
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	−17.1716	−1.01895
$285$	0	0
$286$	9.65685	0.571022
$287$	0	0
$288$	0	0
$289$	6.31371	0.371395
$290$	2.41421	0.141768
$291$	0	0
$292$	−3.17157	−0.185602
$293$	16.0000	0.934730	0.467365	−	0.884064i	$-0.345203\pi$
$293$	16.0000	0.934730	0.467365	+	0.884064i	$0.345203\pi$
$294$	0	0
$295$	4.48528	0.261143
$296$	0	0
$297$	0	0
$298$	−5.24264	−0.303698
$299$	2.00000	0.115663
$300$	0	0
$301$	0	0
$302$	−28.1421	−1.61940
$303$	0	0
$304$	8.48528	0.486664
$305$	−5.48528	−0.314086
$306$	0	0
$307$	−4.75736	−0.271517	−0.135758	−	0.990742i	$-0.543347\pi$
$307$	−4.75736	−0.271517	−0.135758	+	0.990742i	$0.543347\pi$
$308$	0	0
$309$	0	0
$310$	−14.4853	−0.822709
$311$	13.1716	0.746891	0.373446	−	0.927652i	$-0.378176\pi$
$311$	13.1716	0.746891	0.373446	+	0.927652i	$0.378176\pi$
$312$	0	0
$313$	−6.34315	−0.358536	−0.179268	−	0.983800i	$-0.557373\pi$
$313$	−6.34315	−0.358536	−0.179268	+	0.983800i	$0.557373\pi$
$314$	−41.7990	−2.35885
$315$	0	0
$316$	56.7696	3.19354
$317$	13.7990	0.775028	0.387514	−	0.921864i	$-0.373334\pi$
$317$	13.7990	0.775028	0.387514	+	0.921864i	$0.373334\pi$
$318$	0	0
$319$	0.828427	0.0463830
$320$	9.82843	0.549426
$321$	0	0
$322$	0	0
$323$	−13.6569	−0.759888
$324$	0	0
$325$	−4.82843	−0.267833
$326$	−29.7990	−1.65041
$327$	0	0
$328$	−34.5563	−1.90806
$329$	0	0
$330$	0	0
$331$	22.9706	1.26258	0.631288	−	0.775548i	$-0.282527\pi$
$331$	22.9706	1.26258	0.631288	+	0.775548i	$0.282527\pi$
$332$	−52.5563	−2.88440
$333$	0	0
$334$	54.1127	2.96092
$335$	−9.58579	−0.523727
$336$	0	0
$337$	9.17157	0.499607	0.249804	−	0.968296i	$-0.419634\pi$
$337$	9.17157	0.499607	0.249804	+	0.968296i	$0.419634\pi$
$338$	−24.8995	−1.35435
$339$	0	0
$340$	18.4853	1.00251
$341$	−4.97056	−0.269171
$342$	0	0
$343$	0	0
$344$	−15.8284	−0.853412
$345$	0	0
$346$	8.00000	0.430083
$347$	−7.92893	−0.425647	−0.212824	−	0.977091i	$-0.568266\pi$
$347$	−7.92893	−0.425647	−0.212824	+	0.977091i	$0.568266\pi$
$348$	0	0
$349$	−15.3431	−0.821300	−0.410650	−	0.911793i	$-0.634698\pi$
$349$	−15.3431	−0.821300	−0.410650	+	0.911793i	$0.634698\pi$
$350$	0	0
$351$	0	0
$352$	1.31371	0.0700209
$353$	26.8284	1.42793	0.713967	−	0.700180i	$-0.246897\pi$
$353$	26.8284	1.42793	0.713967	+	0.700180i	$0.246897\pi$
$354$	0	0
$355$	4.48528	0.238054
$356$	33.1421	1.75653
$357$	0	0
$358$	−24.1421	−1.27595
$359$	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
$362$	−6.41421	−0.337124
$363$	0	0
$364$	0	0
$365$	0.828427	0.0433619
$366$	0	0
$367$	−2.75736	−0.143933	−0.0719665	−	0.997407i	$-0.522927\pi$
$367$	−2.75736	−0.143933	−0.0719665	+	0.997407i	$0.522927\pi$
$368$	−1.24264	−0.0647771
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−20.9706	−1.08581	−0.542907	−	0.839793i	$-0.682677\pi$
$373$	−20.9706	−1.08581	−0.542907	+	0.839793i	$0.682677\pi$
$374$	9.65685	0.499344
$375$	0	0
$376$	8.82843	0.455291
$377$	−4.82843	−0.248677
$378$	0	0
$379$	26.8284	1.37808	0.689042	−	0.724722i	$-0.258032\pi$
$379$	26.8284	1.37808	0.689042	+	0.724722i	$0.258032\pi$
$380$	−10.8284	−0.555487
$381$	0	0
$382$	−30.9706	−1.58459
$383$	2.89949	0.148157	0.0740786	−	0.997252i	$-0.476398\pi$
$383$	2.89949	0.148157	0.0740786	+	0.997252i	$0.476398\pi$
$384$	0	0
$385$	0	0
$386$	4.82843	0.245760
$387$	0	0
$388$	44.6274	2.26561
$389$	−23.6569	−1.19945	−0.599725	−	0.800206i	$-0.704723\pi$
$389$	−23.6569	−1.19945	−0.599725	+	0.800206i	$0.704723\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	0	0
$393$	0	0
$394$	−29.7990	−1.50125
$395$	−14.8284	−0.746099
$396$	0	0
$397$	−16.6274	−0.834506	−0.417253	−	0.908790i	$-0.637007\pi$
$397$	−16.6274	−0.834506	−0.417253	+	0.908790i	$0.637007\pi$
$398$	23.3137	1.16861
$399$	0	0
$400$	3.00000	0.150000
$401$	−30.3137	−1.51379	−0.756897	−	0.653534i	$-0.773286\pi$
$401$	−30.3137	−1.51379	−0.756897	+	0.653534i	$0.773286\pi$
$402$	0	0
$403$	28.9706	1.44313
$404$	39.4853	1.96447
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−14.7990	−0.731763	−0.365881	−	0.930661i	$-0.619232\pi$
$409$	−14.7990	−0.731763	−0.365881	+	0.930661i	$0.619232\pi$
$410$	18.8995	0.933380
$411$	0	0
$412$	−9.24264	−0.455352
$413$	0	0
$414$	0	0
$415$	13.7279	0.673877
$416$	−7.65685	−0.375408
$417$	0	0
$418$	−5.65685	−0.276686
$419$	0.686292	0.0335275	0.0167638	−	0.999859i	$-0.494664\pi$
$419$	0.686292	0.0335275	0.0167638	+	0.999859i	$0.494664\pi$
$420$	0	0
$421$	13.4853	0.657232	0.328616	−	0.944464i	$-0.393418\pi$
$421$	13.4853	0.657232	0.328616	+	0.944464i	$0.393418\pi$
$422$	−49.4558	−2.40747
$423$	0	0
$424$	−5.17157	−0.251154
$425$	−4.82843	−0.234213
$426$	0	0
$427$	0	0
$428$	−43.0416	−2.08050
$429$	0	0
$430$	8.65685	0.417471
$431$	−17.7990	−0.857347	−0.428674	−	0.903459i	$-0.641019\pi$
$431$	−17.7990	−0.857347	−0.428674	+	0.903459i	$0.641019\pi$
$432$	0	0
$433$	7.79899	0.374796	0.187398	−	0.982284i	$-0.439995\pi$
$433$	7.79899	0.374796	0.187398	+	0.982284i	$0.439995\pi$
$434$	0	0
$435$	0	0
$436$	−51.6274	−2.47250
$437$	−1.17157	−0.0560439
$438$	0	0
$439$	−33.9411	−1.61992	−0.809961	−	0.586484i	$-0.800512\pi$
$439$	−33.9411	−1.61992	−0.809961	+	0.586484i	$0.800512\pi$
$440$	3.65685	0.174334
$441$	0	0
$442$	−56.2843	−2.67717
$443$	−30.2132	−1.43547	−0.717736	−	0.696315i	$-0.754822\pi$
$443$	−30.2132	−1.43547	−0.717736	+	0.696315i	$0.754822\pi$
$444$	0	0
$445$	−8.65685	−0.410374
$446$	−0.828427	−0.0392272
$447$	0	0
$448$	0	0
$449$	−3.82843	−0.180675	−0.0903373	−	0.995911i	$-0.528795\pi$
$449$	−3.82843	−0.180675	−0.0903373	+	0.995911i	$0.528795\pi$
$450$	0	0
$451$	6.48528	0.305380
$452$	17.1716	0.807683
$453$	0	0
$454$	−16.8284	−0.789797
$455$	0	0
$456$	0	0
$457$	24.2843	1.13597	0.567985	−	0.823039i	$-0.307723\pi$
$457$	24.2843	1.13597	0.567985	+	0.823039i	$0.307723\pi$
$458$	28.1421	1.31500
$459$	0	0
$460$	1.58579	0.0739377
$461$	−41.3137	−1.92417	−0.962086	−	0.272748i	$-0.912068\pi$
$461$	−41.3137	−1.92417	−0.962086	+	0.272748i	$0.912068\pi$
$462$	0	0
$463$	−37.0416	−1.72147	−0.860735	−	0.509053i	$-0.829996\pi$
$463$	−37.0416	−1.72147	−0.860735	+	0.509053i	$0.829996\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	−40.6274	−1.88203
$467$	3.10051	0.143474	0.0717371	−	0.997424i	$-0.477146\pi$
$467$	3.10051	0.143474	0.0717371	+	0.997424i	$0.477146\pi$
$468$	0	0
$469$	0	0
$470$	−4.82843	−0.222719
$471$	0	0
$472$	19.7990	0.911322
$473$	2.97056	0.136587
$474$	0	0
$475$	2.82843	0.129777
$476$	0	0
$477$	0	0
$478$	−51.4558	−2.35354
$479$	35.6569	1.62920	0.814602	−	0.580021i	$-0.196956\pi$
$479$	35.6569	1.62920	0.814602	+	0.580021i	$0.196956\pi$
$480$	0	0
$481$	0	0
$482$	−66.7696	−3.04127
$483$	0	0
$484$	−39.4853	−1.79479
$485$	−11.6569	−0.529310
$486$	0	0
$487$	4.34315	0.196807	0.0984034	−	0.995147i	$-0.468626\pi$
$487$	4.34315	0.196807	0.0984034	+	0.995147i	$0.468626\pi$
$488$	−24.2132	−1.09608
$489$	0	0
$490$	0	0
$491$	−9.31371	−0.420322	−0.210161	−	0.977667i	$-0.567399\pi$
$491$	−9.31371	−0.420322	−0.210161	+	0.977667i	$0.567399\pi$
$492$	0	0
$493$	−4.82843	−0.217461
$494$	32.9706	1.48342
$495$	0	0
$496$	−18.0000	−0.808224
$497$	0	0
$498$	0	0
$499$	−0.828427	−0.0370855	−0.0185427	−	0.999828i	$-0.505903\pi$
$499$	−0.828427	−0.0370855	−0.0185427	+	0.999828i	$0.505903\pi$
$500$	−3.82843	−0.171212
$501$	0	0
$502$	−22.4853	−1.00357
$503$	15.8701	0.707611	0.353805	−	0.935319i	$-0.384887\pi$
$503$	15.8701	0.707611	0.353805	+	0.935319i	$0.384887\pi$
$504$	0	0
$505$	−10.3137	−0.458954
$506$	0.828427	0.0368281
$507$	0	0
$508$	−35.6569	−1.58202
$509$	−13.3431	−0.591425	−0.295712	−	0.955277i	$-0.595557\pi$
$509$	−13.3431	−0.591425	−0.295712	+	0.955277i	$0.595557\pi$
$510$	0	0
$511$	0	0
$512$	31.2426	1.38074
$513$	0	0
$514$	15.3137	0.675459
$515$	2.41421	0.106383
$516$	0	0
$517$	−1.65685	−0.0728684
$518$	0	0
$519$	0	0
$520$	−21.3137	−0.934668
$521$	−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
$523$	35.6569	1.55917	0.779583	−	0.626299i	$-0.215431\pi$
$523$	35.6569	1.55917	0.779583	+	0.626299i	$0.215431\pi$
$524$	73.9411	3.23013
$525$	0	0
$526$	−70.1127	−3.05706
$527$	28.9706	1.26198
$528$	0	0
$529$	−22.8284	−0.992540
$530$	2.82843	0.122859
$531$	0	0
$532$	0	0
$533$	−37.7990	−1.63726
$534$	0	0
$535$	11.2426	0.486062
$536$	−42.3137	−1.82767
$537$	0	0
$538$	49.3848	2.12913
$539$	0	0
$540$	0	0
$541$	7.34315	0.315706	0.157853	−	0.987463i	$-0.449543\pi$
$541$	7.34315	0.315706	0.157853	+	0.987463i	$0.449543\pi$
$542$	−39.7990	−1.70951
$543$	0	0
$544$	−7.65685	−0.328285
$545$	13.4853	0.577646
$546$	0	0
$547$	24.8995	1.06463	0.532313	−	0.846548i	$-0.321323\pi$
$547$	24.8995	1.06463	0.532313	+	0.846548i	$0.321323\pi$
$548$	36.9706	1.57930
$549$	0	0
$550$	−2.00000	−0.0852803
$551$	2.82843	0.120495
$552$	0	0
$553$	0	0
$554$	−38.9706	−1.65570
$555$	0	0
$556$	−61.7990	−2.62086
$557$	−22.2843	−0.944215	−0.472107	−	0.881541i	$-0.656507\pi$
$557$	−22.2843	−0.944215	−0.472107	+	0.881541i	$0.656507\pi$
$558$	0	0
$559$	−17.3137	−0.732292
$560$	0	0
$561$	0	0
$562$	−73.1127	−3.08407
$563$	−41.7279	−1.75862	−0.879311	−	0.476248i	$-0.841997\pi$
$563$	−41.7279	−1.75862	−0.879311	+	0.476248i	$0.841997\pi$
$564$	0	0
$565$	−4.48528	−0.188697
$566$	−33.7990	−1.42068
$567$	0	0
$568$	19.7990	0.830747
$569$	−7.65685	−0.320992	−0.160496	−	0.987036i	$-0.551309\pi$
$569$	−7.65685	−0.320992	−0.160496	+	0.987036i	$0.551309\pi$
$570$	0	0
$571$	9.17157	0.383818	0.191909	−	0.981413i	$-0.438532\pi$
$571$	9.17157	0.383818	0.191909	+	0.981413i	$0.438532\pi$
$572$	−15.3137	−0.640298
$573$	0	0
$574$	0	0
$575$	−0.414214	−0.0172739
$576$	0	0
$577$	−43.9411	−1.82929	−0.914646	−	0.404255i	$-0.867531\pi$
$577$	−43.9411	−1.82929	−0.914646	+	0.404255i	$0.867531\pi$
$578$	−15.2426	−0.634010
$579$	0	0
$580$	−3.82843	−0.158967
$581$	0	0
$582$	0	0
$583$	0.970563	0.0401966
$584$	3.65685	0.151322
$585$	0	0
$586$	−38.6274	−1.59568
$587$	34.2843	1.41506	0.707532	−	0.706682i	$-0.249809\pi$
$587$	34.2843	1.41506	0.707532	+	0.706682i	$0.249809\pi$
$588$	0	0
$589$	−16.9706	−0.699260
$590$	−10.8284	−0.445799
$591$	0	0
$592$	0	0
$593$	−4.20101	−0.172515	−0.0862574	−	0.996273i	$-0.527491\pi$
$593$	−4.20101	−0.172515	−0.0862574	+	0.996273i	$0.527491\pi$
$594$	0	0
$595$	0	0
$596$	8.31371	0.340543
$597$	0	0
$598$	−4.82843	−0.197449
$599$	−6.34315	−0.259174	−0.129587	−	0.991568i	$-0.541365\pi$
$599$	−6.34315	−0.259174	−0.129587	+	0.991568i	$0.541365\pi$
$600$	0	0
$601$	19.6569	0.801820	0.400910	−	0.916117i	$-0.368694\pi$
$601$	19.6569	0.801820	0.400910	+	0.916117i	$0.368694\pi$
$602$	0	0
$603$	0	0
$604$	44.6274	1.81586
$605$	10.3137	0.419312
$606$	0	0
$607$	38.2132	1.55103	0.775513	−	0.631332i	$-0.217491\pi$
$607$	38.2132	1.55103	0.775513	+	0.631332i	$0.217491\pi$
$608$	4.48528	0.181902
$609$	0	0
$610$	13.2426	0.536179
$611$	9.65685	0.390675
$612$	0	0
$613$	35.4558	1.43205	0.716024	−	0.698076i	$-0.245960\pi$
$613$	35.4558	1.43205	0.716024	+	0.698076i	$0.245960\pi$
$614$	11.4853	0.463508
$615$	0	0
$616$	0	0
$617$	−11.3137	−0.455473	−0.227736	−	0.973723i	$-0.573132\pi$
$617$	−11.3137	−0.455473	−0.227736	+	0.973723i	$0.573132\pi$
$618$	0	0
$619$	25.5147	1.02552	0.512762	−	0.858531i	$-0.328622\pi$
$619$	25.5147	1.02552	0.512762	+	0.858531i	$0.328622\pi$
$620$	22.9706	0.922520
$621$	0	0
$622$	−31.7990	−1.27502
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	15.3137	0.612059
$627$	0	0
$628$	66.2843	2.64503
$629$	0	0
$630$	0	0
$631$	−20.1421	−0.801846	−0.400923	−	0.916112i	$-0.631310\pi$
$631$	−20.1421	−0.801846	−0.400923	+	0.916112i	$0.631310\pi$
$632$	−65.4558	−2.60369
$633$	0	0
$634$	−33.3137	−1.32306
$635$	9.31371	0.369603
$636$	0	0
$637$	0	0
$638$	−2.00000	−0.0791808
$639$	0	0
$640$	−20.5563	−0.812561
$641$	−31.4853	−1.24359	−0.621797	−	0.783179i	$-0.713597\pi$
$641$	−31.4853	−1.24359	−0.621797	+	0.783179i	$0.713597\pi$
$642$	0	0
$643$	26.2843	1.03655	0.518275	−	0.855214i	$-0.326574\pi$
$643$	26.2843	1.03655	0.518275	+	0.855214i	$0.326574\pi$
$644$	0	0
$645$	0	0
$646$	32.9706	1.29721
$647$	31.0416	1.22037	0.610186	−	0.792258i	$-0.291095\pi$
$647$	31.0416	1.22037	0.610186	+	0.792258i	$0.291095\pi$
$648$	0	0
$649$	−3.71573	−0.145855
$650$	11.6569	0.457219
$651$	0	0
$652$	47.2548	1.85064
$653$	19.1716	0.750242	0.375121	−	0.926976i	$-0.377601\pi$
$653$	19.1716	0.750242	0.375121	+	0.926976i	$0.377601\pi$
$654$	0	0
$655$	−19.3137	−0.754649
$656$	23.4853	0.916946
$657$	0	0
$658$	0	0
$659$	−21.1716	−0.824727	−0.412364	−	0.911019i	$-0.635297\pi$
$659$	−21.1716	−0.824727	−0.412364	+	0.911019i	$0.635297\pi$
$660$	0	0
$661$	31.8284	1.23798	0.618991	−	0.785398i	$-0.287542\pi$
$661$	31.8284	1.23798	0.618991	+	0.785398i	$0.287542\pi$
$662$	−55.4558	−2.15535
$663$	0	0
$664$	60.5980	2.35166
$665$	0	0
$666$	0	0
$667$	−0.414214	−0.0160384
$668$	−85.8112	−3.32013
$669$	0	0
$670$	23.1421	0.894059
$671$	4.54416	0.175425
$672$	0	0
$673$	29.6569	1.14319	0.571594	−	0.820537i	$-0.306325\pi$
$673$	29.6569	1.14319	0.571594	+	0.820537i	$0.306325\pi$
$674$	−22.1421	−0.852883
$675$	0	0
$676$	39.4853	1.51866
$677$	28.1421	1.08159	0.540795	−	0.841154i	$-0.318123\pi$
$677$	28.1421	1.08159	0.540795	+	0.841154i	$0.318123\pi$
$678$	0	0
$679$	0	0
$680$	−21.3137	−0.817343
$681$	0	0
$682$	12.0000	0.459504
$683$	34.7574	1.32995	0.664977	−	0.746864i	$-0.268441\pi$
$683$	34.7574	1.32995	0.664977	+	0.746864i	$0.268441\pi$
$684$	0	0
$685$	−9.65685	−0.368969
$686$	0	0
$687$	0	0
$688$	10.7574	0.410120
$689$	−5.65685	−0.215509
$690$	0	0
$691$	0.828427	0.0315149	0.0157574	−	0.999876i	$-0.494984\pi$
$691$	0.828427	0.0315149	0.0157574	+	0.999876i	$0.494984\pi$
$692$	−12.6863	−0.482260
$693$	0	0
$694$	19.1421	0.726626
$695$	16.1421	0.612306
$696$	0	0
$697$	−37.7990	−1.43174
$698$	37.0416	1.40205
$699$	0	0
$700$	0	0
$701$	3.20101	0.120900	0.0604502	−	0.998171i	$-0.480746\pi$
$701$	3.20101	0.120900	0.0604502	+	0.998171i	$0.480746\pi$
$702$	0	0
$703$	0	0
$704$	−8.14214	−0.306868
$705$	0	0
$706$	−64.7696	−2.43763
$707$	0	0
$708$	0	0
$709$	15.6863	0.589111	0.294556	−	0.955634i	$-0.404828\pi$
$709$	15.6863	0.589111	0.294556	+	0.955634i	$0.404828\pi$
$710$	−10.8284	−0.406384
$711$	0	0
$712$	−38.2132	−1.43210
$713$	2.48528	0.0930745
$714$	0	0
$715$	4.00000	0.149592
$716$	38.2843	1.43075
$717$	0	0
$718$	−24.1421	−0.900976
$719$	−21.1127	−0.787371	−0.393685	−	0.919245i	$-0.628800\pi$
$719$	−21.1127	−0.787371	−0.393685	+	0.919245i	$0.628800\pi$
$720$	0	0
$721$	0	0
$722$	26.5563	0.988325
$723$	0	0
$724$	10.1716	0.378024
$725$	1.00000	0.0371391
$726$	0	0
$727$	−37.5858	−1.39398	−0.696990	−	0.717081i	$-0.745478\pi$
$727$	−37.5858	−1.39398	−0.696990	+	0.717081i	$0.745478\pi$
$728$	0	0
$729$	0	0
$730$	−2.00000	−0.0740233
$731$	−17.3137	−0.640371
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	6.65685	0.245709
$735$	0	0
$736$	−0.656854	−0.0242120
$737$	7.94113	0.292515
$738$	0	0
$739$	−21.1127	−0.776643	−0.388322	−	0.921524i	$-0.626945\pi$
$739$	−21.1127	−0.776643	−0.388322	+	0.921524i	$0.626945\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.0711	0.589590	0.294795	−	0.955560i	$-0.404749\pi$
$743$	16.0711	0.589590	0.294795	+	0.955560i	$0.404749\pi$
$744$	0	0
$745$	−2.17157	−0.0795603
$746$	50.6274	1.85360
$747$	0	0
$748$	−15.3137	−0.559925
$749$	0	0
$750$	0	0
$751$	−30.3431	−1.10724	−0.553619	−	0.832770i	$-0.686753\pi$
$751$	−30.3431	−1.10724	−0.553619	+	0.832770i	$0.686753\pi$
$752$	−6.00000	−0.218797
$753$	0	0
$754$	11.6569	0.424518
$755$	−11.6569	−0.424236
$756$	0	0
$757$	−31.4558	−1.14328	−0.571641	−	0.820504i	$-0.693693\pi$
$757$	−31.4558	−1.14328	−0.571641	+	0.820504i	$0.693693\pi$
$758$	−64.7696	−2.35254
$759$	0	0
$760$	12.4853	0.452889
$761$	−9.31371	−0.337622	−0.168811	−	0.985648i	$-0.553993\pi$
$761$	−9.31371	−0.337622	−0.168811	+	0.985648i	$0.553993\pi$
$762$	0	0
$763$	0	0
$764$	49.1127	1.77684
$765$	0	0
$766$	−7.00000	−0.252920
$767$	21.6569	0.781984
$768$	0	0
$769$	−0.627417	−0.0226252	−0.0113126	−	0.999936i	$-0.503601\pi$
$769$	−0.627417	−0.0226252	−0.0113126	+	0.999936i	$0.503601\pi$
$770$	0	0
$771$	0	0
$772$	−7.65685	−0.275576
$773$	37.1127	1.33485	0.667425	−	0.744677i	$-0.267396\pi$
$773$	37.1127	1.33485	0.667425	+	0.744677i	$0.267396\pi$
$774$	0	0
$775$	−6.00000	−0.215526
$776$	−51.4558	−1.84716
$777$	0	0
$778$	57.1127	2.04759
$779$	22.1421	0.793324
$780$	0	0
$781$	−3.71573	−0.132959
$782$	−4.82843	−0.172664
$783$	0	0
$784$	0	0
$785$	−17.3137	−0.617953
$786$	0	0
$787$	2.55635	0.0911240	0.0455620	−	0.998962i	$-0.485492\pi$
$787$	2.55635	0.0911240	0.0455620	+	0.998962i	$0.485492\pi$
$788$	47.2548	1.68338
$789$	0	0
$790$	35.7990	1.27367
$791$	0	0
$792$	0	0
$793$	−26.4853	−0.940520
$794$	40.1421	1.42459
$795$	0	0
$796$	−36.9706	−1.31039
$797$	8.00000	0.283375	0.141687	−	0.989911i	$-0.454747\pi$
$797$	8.00000	0.283375	0.141687	+	0.989911i	$0.454747\pi$
$798$	0	0
$799$	9.65685	0.341635
$800$	1.58579	0.0560660
$801$	0	0
$802$	73.1838	2.58421
$803$	−0.686292	−0.0242187
$804$	0	0
$805$	0	0
$806$	−69.9411	−2.46357
$807$	0	0
$808$	−45.5269	−1.60163
$809$	−35.6274	−1.25259	−0.626297	−	0.779585i	$-0.715430\pi$
$809$	−35.6274	−1.25259	−0.626297	+	0.779585i	$0.715430\pi$
$810$	0	0
$811$	−20.6274	−0.724327	−0.362163	−	0.932115i	$-0.617962\pi$
$811$	−20.6274	−0.724327	−0.362163	+	0.932115i	$0.617962\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−12.3431	−0.432362
$816$	0	0
$817$	10.1421	0.354828
$818$	35.7279	1.24920
$819$	0	0
$820$	−29.9706	−1.04662
$821$	−47.9411	−1.67316	−0.836578	−	0.547847i	$-0.815448\pi$
$821$	−47.9411	−1.67316	−0.836578	+	0.547847i	$0.815448\pi$
$822$	0	0
$823$	2.07107	0.0721929	0.0360964	−	0.999348i	$-0.488508\pi$
$823$	2.07107	0.0721929	0.0360964	+	0.999348i	$0.488508\pi$
$824$	10.6569	0.371249
$825$	0	0
$826$	0	0
$827$	26.2132	0.911522	0.455761	−	0.890102i	$-0.349367\pi$
$827$	26.2132	0.911522	0.455761	+	0.890102i	$0.349367\pi$
$828$	0	0
$829$	29.3137	1.01811	0.509054	−	0.860735i	$-0.329995\pi$
$829$	29.3137	1.01811	0.509054	+	0.860735i	$0.329995\pi$
$830$	−33.1421	−1.15038
$831$	0	0
$832$	47.4558	1.64524
$833$	0	0
$834$	0	0
$835$	22.4142	0.775676
$836$	8.97056	0.310253
$837$	0	0
$838$	−1.65685	−0.0572351
$839$	−15.1716	−0.523781	−0.261890	−	0.965098i	$-0.584346\pi$
$839$	−15.1716	−0.523781	−0.261890	+	0.965098i	$0.584346\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	−32.5563	−1.12197
$843$	0	0
$844$	78.4264	2.69955
$845$	−10.3137	−0.354802
$846$	0	0
$847$	0	0
$848$	3.51472	0.120696
$849$	0	0
$850$	11.6569	0.399827
$851$	0	0
$852$	0	0
$853$	2.54416	0.0871102	0.0435551	−	0.999051i	$-0.486132\pi$
$853$	2.54416	0.0871102	0.0435551	+	0.999051i	$0.486132\pi$
$854$	0	0
$855$	0	0
$856$	49.6274	1.69623
$857$	−34.2843	−1.17113	−0.585564	−	0.810626i	$-0.699127\pi$
$857$	−34.2843	−1.17113	−0.585564	+	0.810626i	$0.699127\pi$
$858$	0	0
$859$	1.37258	0.0468319	0.0234160	−	0.999726i	$-0.492546\pi$
$859$	1.37258	0.0468319	0.0234160	+	0.999726i	$0.492546\pi$
$860$	−13.7279	−0.468118
$861$	0	0
$862$	42.9706	1.46358
$863$	−14.5563	−0.495504	−0.247752	−	0.968823i	$-0.579692\pi$
$863$	−14.5563	−0.495504	−0.247752	+	0.968823i	$0.579692\pi$
$864$	0	0
$865$	3.31371	0.112669
$866$	−18.8284	−0.639816
$867$	0	0
$868$	0	0
$869$	12.2843	0.416715
$870$	0	0
$871$	−46.2843	−1.56828
$872$	59.5269	2.01584
$873$	0	0
$874$	2.82843	0.0956730
$875$	0	0
$876$	0	0
$877$	25.1716	0.849984	0.424992	−	0.905197i	$-0.360277\pi$
$877$	25.1716	0.849984	0.424992	+	0.905197i	$0.360277\pi$
$878$	81.9411	2.76538
$879$	0	0
$880$	−2.48528	−0.0837788
$881$	−1.82843	−0.0616013	−0.0308006	−	0.999526i	$-0.509806\pi$
$881$	−1.82843	−0.0616013	−0.0308006	+	0.999526i	$0.509806\pi$
$882$	0	0
$883$	18.2843	0.615315	0.307657	−	0.951497i	$-0.400455\pi$
$883$	18.2843	0.615315	0.307657	+	0.951497i	$0.400455\pi$
$884$	89.2548	3.00196
$885$	0	0
$886$	72.9411	2.45051
$887$	−29.9289	−1.00492	−0.502458	−	0.864602i	$-0.667571\pi$
$887$	−29.9289	−1.00492	−0.502458	+	0.864602i	$0.667571\pi$
$888$	0	0
$889$	0	0
$890$	20.8995	0.700553
$891$	0	0
$892$	1.31371	0.0439862
$893$	−5.65685	−0.189299
$894$	0	0
$895$	−10.0000	−0.334263
$896$	0	0
$897$	0	0
$898$	9.24264	0.308431
$899$	−6.00000	−0.200111
$900$	0	0
$901$	−5.65685	−0.188457
$902$	−15.6569	−0.521316
$903$	0	0
$904$	−19.7990	−0.658505
$905$	−2.65685	−0.0883168
$906$	0	0
$907$	−14.2132	−0.471942	−0.235971	−	0.971760i	$-0.575827\pi$
$907$	−14.2132	−0.471942	−0.235971	+	0.971760i	$0.575827\pi$
$908$	26.6863	0.885616
$909$	0	0
$910$	0	0
$911$	10.2010	0.337975	0.168987	−	0.985618i	$-0.445950\pi$
$911$	10.2010	0.337975	0.168987	+	0.985618i	$0.445950\pi$
$912$	0	0
$913$	−11.3726	−0.376378
$914$	−58.6274	−1.93922
$915$	0	0
$916$	−44.6274	−1.47453
$917$	0	0
$918$	0	0
$919$	−43.1127	−1.42216	−0.711078	−	0.703113i	$-0.751793\pi$
$919$	−43.1127	−1.42216	−0.711078	+	0.703113i	$0.751793\pi$
$920$	−1.82843	−0.0602815
$921$	0	0
$922$	99.7401	3.28477
$923$	21.6569	0.712844
$924$	0	0
$925$	0	0
$926$	89.4264	2.93873
$927$	0	0
$928$	1.58579	0.0520560
$929$	−5.48528	−0.179966	−0.0899831	−	0.995943i	$-0.528681\pi$
$929$	−5.48528	−0.179966	−0.0899831	+	0.995943i	$0.528681\pi$
$930$	0	0
$931$	0	0
$932$	64.4264	2.11036
$933$	0	0
$934$	−7.48528	−0.244926
$935$	4.00000	0.130814
$936$	0	0
$937$	34.6274	1.13123	0.565614	−	0.824670i	$-0.308639\pi$
$937$	34.6274	1.13123	0.565614	+	0.824670i	$0.308639\pi$
$938$	0	0
$939$	0	0
$940$	7.65685	0.249739
$941$	46.2843	1.50882	0.754412	−	0.656401i	$-0.227922\pi$
$941$	46.2843	1.50882	0.754412	+	0.656401i	$0.227922\pi$
$942$	0	0
$943$	−3.24264	−0.105595
$944$	−13.4558	−0.437950
$945$	0	0
$946$	−7.17157	−0.233168
$947$	−33.1838	−1.07833	−0.539164	−	0.842201i	$-0.681260\pi$
$947$	−33.1838	−1.07833	−0.539164	+	0.842201i	$0.681260\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	−6.82843	−0.221543
$951$	0	0
$952$	0	0
$953$	13.6569	0.442389	0.221194	−	0.975230i	$-0.429004\pi$
$953$	13.6569	0.442389	0.221194	+	0.975230i	$0.429004\pi$
$954$	0	0
$955$	−12.8284	−0.415118
$956$	81.5980	2.63907
$957$	0	0
$958$	−86.0833	−2.78122
$959$	0	0
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	0	0
$964$	105.882	3.41024
$965$	2.00000	0.0643823
$966$	0	0
$967$	37.5269	1.20678	0.603392	−	0.797445i	$-0.293815\pi$
$967$	37.5269	1.20678	0.603392	+	0.797445i	$0.293815\pi$
$968$	45.5269	1.46329
$969$	0	0
$970$	28.1421	0.903590
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	0	0
$973$	0	0
$974$	−10.4853	−0.335970
$975$	0	0
$976$	16.4558	0.526739
$977$	1.31371	0.0420293	0.0210146	−	0.999779i	$-0.493310\pi$
$977$	1.31371	0.0420293	0.0210146	+	0.999779i	$0.493310\pi$
$978$	0	0
$979$	7.17157	0.229204
$980$	0	0
$981$	0	0
$982$	22.4853	0.717534
$983$	−28.2132	−0.899861	−0.449931	−	0.893063i	$-0.648551\pi$
$983$	−28.2132	−0.899861	−0.449931	+	0.893063i	$0.648551\pi$
$984$	0	0
$985$	−12.3431	−0.393285
$986$	11.6569	0.371230
$987$	0	0
$988$	−52.2843	−1.66338
$989$	−1.48528	−0.0472292
$990$	0	0
$991$	−4.34315	−0.137965	−0.0689823	−	0.997618i	$-0.521975\pi$
$991$	−4.34315	−0.137965	−0.0689823	+	0.997618i	$0.521975\pi$
$992$	−9.51472	−0.302093
$993$	0	0
$994$	0	0
$995$	9.65685	0.306143
$996$	0	0
$997$	33.4558	1.05956	0.529779	−	0.848136i	$-0.322275\pi$
$997$	33.4558	1.05956	0.529779	+	0.848136i	$0.322275\pi$
$998$	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.n.1.1		2
3.2	odd	2		245.2.a.h.1.2		2
7.2	even	3		315.2.j.e.46.2		4
7.4	even	3		315.2.j.e.226.2		4
7.6	odd	2		2205.2.a.q.1.1		2
12.11	even	2		3920.2.a.bq.1.2		2
15.2	even	4		1225.2.b.g.99.4		4
15.8	even	4		1225.2.b.g.99.1		4
15.14	odd	2		1225.2.a.k.1.1		2
21.2	odd	6		35.2.e.a.11.1	✓	4
21.5	even	6		245.2.e.e.116.1		4
21.11	odd	6		35.2.e.a.16.1	yes	4
21.17	even	6		245.2.e.e.226.1		4
21.20	even	2		245.2.a.g.1.2		2
84.11	even	6		560.2.q.k.401.1		4
84.23	even	6		560.2.q.k.81.1		4
84.83	odd	2		3920.2.a.bv.1.1		2
105.2	even	12		175.2.k.a.74.4		8
105.23	even	12		175.2.k.a.74.1		8
105.32	even	12		175.2.k.a.149.1		8
105.44	odd	6		175.2.e.c.151.2		4
105.53	even	12		175.2.k.a.149.4		8
105.62	odd	4		1225.2.b.h.99.4		4
105.74	odd	6		175.2.e.c.51.2		4
105.83	odd	4		1225.2.b.h.99.1		4
105.104	even	2		1225.2.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.2.e.a.11.1	✓	4	21.2	odd	6
35.2.e.a.16.1	yes	4	21.11	odd	6
175.2.e.c.51.2		4	105.74	odd	6
175.2.e.c.151.2		4	105.44	odd	6
175.2.k.a.74.1		8	105.23	even	12
175.2.k.a.74.4		8	105.2	even	12
175.2.k.a.149.1		8	105.32	even	12
175.2.k.a.149.4		8	105.53	even	12
245.2.a.g.1.2		2	21.20	even	2
245.2.a.h.1.2		2	3.2	odd	2
245.2.e.e.116.1		4	21.5	even	6
245.2.e.e.226.1		4	21.17	even	6
315.2.j.e.46.2		4	7.2	even	3
315.2.j.e.226.2		4	7.4	even	3
560.2.q.k.81.1		4	84.23	even	6
560.2.q.k.401.1		4	84.11	even	6
1225.2.a.k.1.1		2	15.14	odd	2
1225.2.a.m.1.1		2	105.104	even	2
1225.2.b.g.99.1		4	15.8	even	4
1225.2.b.g.99.4		4	15.2	even	4
1225.2.b.h.99.1		4	105.83	odd	4
1225.2.b.h.99.4		4	105.62	odd	4
2205.2.a.n.1.1		2	1.1	even	1	trivial
2205.2.a.q.1.1		2	7.6	odd	2
3920.2.a.bq.1.2		2	12.11	even	2
3920.2.a.bv.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)

\(f(q)\)	\(=\)	\(q-2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} -4.41421 q^{8} +O(q^{10})\)	\(q-2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} -4.41421 q^{8} +2.41421 q^{10} +0.828427 q^{11} -4.82843 q^{13} +3.00000 q^{16} -4.82843 q^{17} +2.82843 q^{19} -3.82843 q^{20} -2.00000 q^{22} -0.414214 q^{23} +1.00000 q^{25} +11.6569 q^{26} +1.00000 q^{29} -6.00000 q^{31} +1.58579 q^{32} +11.6569 q^{34} -6.82843 q^{38} +4.41421 q^{40} +7.82843 q^{41} +3.58579 q^{43} +3.17157 q^{44} +1.00000 q^{46} -2.00000 q^{47} -2.41421 q^{50} -18.4853 q^{52} +1.17157 q^{53} -0.828427 q^{55} -2.41421 q^{58} -4.48528 q^{59} +5.48528 q^{61} +14.4853 q^{62} -9.82843 q^{64} +4.82843 q^{65} +9.58579 q^{67} -18.4853 q^{68} -4.48528 q^{71} -0.828427 q^{73} +10.8284 q^{76} +14.8284 q^{79} -3.00000 q^{80} -18.8995 q^{82} -13.7279 q^{83} +4.82843 q^{85} -8.65685 q^{86} -3.65685 q^{88} +8.65685 q^{89} -1.58579 q^{92} +4.82843 q^{94} -2.82843 q^{95} +11.6569 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 6 q^{8}+O(q^{10}) \) 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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