LMFDB - Embedded newform 2471.1.d.a.2470.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2471,1,Mod(2470,2471)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2471, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2471.2470");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2471 = 7 \cdot 353 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2471.d (of order $2$, degree $1$, minimal)

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:	$1.23318964622$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\Q(\zeta_{62})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 14 x^{13} + 13 x^{12} + 78 x^{11} - 66 x^{10} - 220 x^{9} + 165 x^{8} + 330 x^{7} + \cdots + 1 $ x^15 - x^14 - 14x^13 + 13x^12 + 78x^11 - 66x^10 - 220x^9 + 165x^8 + 330x^7 - 210x^6 - 252x^5 + 126x^4 + 84x^3 - 28x^2 - 8*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{31}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{31} - \cdots)$

Embedding invariants

Embedding label			2470.10
Root			$1.51752$ of defining polynomial
Character	$\chi$	$=$	2471.2470

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.694611 q^{2} -1.22421 q^{3} -0.517516 q^{4} -1.98974 q^{5} -0.850350 q^{6} +1.00000 q^{7} -1.05408 q^{8} +0.498695 q^{9} +O(q^{10})$	\(q+0.694611 q^{2} -1.22421 q^{3} -0.517516 q^{4} -1.98974 q^{5} -0.850350 q^{6} +1.00000 q^{7} -1.05408 q^{8} +0.498695 q^{9} -1.38209 q^{10} -0.101298 q^{11} +0.633550 q^{12} -1.74869 q^{13} +0.694611 q^{14} +2.43586 q^{15} -0.214661 q^{16} +0.346399 q^{18} +1.02972 q^{20} -1.22421 q^{21} -0.0703629 q^{22} -0.880788 q^{23} +1.29042 q^{24} +2.95906 q^{25} -1.21466 q^{26} +0.613704 q^{27} -0.517516 q^{28} -1.90828 q^{29} +1.69198 q^{30} +1.37793 q^{31} +0.904977 q^{32} +0.124011 q^{33} -1.98974 q^{35} -0.258083 q^{36} +2.14077 q^{39} +2.09735 q^{40} -0.850350 q^{42} +1.64153 q^{43} +0.0524235 q^{44} -0.992273 q^{45} -0.611805 q^{46} +0.262790 q^{48} +1.00000 q^{49} +2.05539 q^{50} +0.904977 q^{52} +0.426285 q^{54} +0.201557 q^{55} -1.05408 q^{56} -1.32551 q^{58} +1.95906 q^{59} -1.26060 q^{60} +0.957127 q^{62} +0.498695 q^{63} +0.843267 q^{64} +3.47944 q^{65} +0.0861391 q^{66} +1.07827 q^{69} -1.38209 q^{70} -0.525666 q^{72} -3.62252 q^{75} -0.101298 q^{77} +1.48700 q^{78} +0.427119 q^{80} -1.25000 q^{81} +0.633550 q^{84} +1.14022 q^{86} +2.33614 q^{87} +0.106777 q^{88} -0.501305 q^{89} -0.689243 q^{90} -1.74869 q^{91} +0.455822 q^{92} -1.68688 q^{93} -1.10788 q^{96} +0.694611 q^{98} -0.0505170 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{2} - q^{3} + 14 q^{4} - q^{5} - 2 q^{6} + 15 q^{7} - 2 q^{8} + 14 q^{9}+O(q^{10}) $ 15 * q - q^2 - q^3 + 14 * q^4 - q^5 - 2 * q^6 + 15 * q^7 - 2 * q^8 + 14 * q^9	$ 15 q - q^{2} - q^{3} + 14 q^{4} - q^{5} - 2 q^{6} + 15 q^{7} - 2 q^{8} + 14 q^{9} - 2 q^{10} - q^{11} - 3 q^{12} - q^{13} - q^{14} - 2 q^{15} + 13 q^{16} - 3 q^{18} - 3 q^{20} - q^{21} - 2 q^{22} - q^{23} - 4 q^{24} + 14 q^{25} - 2 q^{26} - 2 q^{27} + 14 q^{28} - q^{29} - 4 q^{30} - q^{31} - 3 q^{32} - 2 q^{33} - q^{35} + 11 q^{36} - 2 q^{39} - 4 q^{40} - 2 q^{42} - q^{43} - 3 q^{44} - 3 q^{45} - 2 q^{46} - 5 q^{48} + 15 q^{49} - 3 q^{50} - 3 q^{52} - 4 q^{54} - 2 q^{55} - 2 q^{56} - 2 q^{58} - q^{59} - 6 q^{60} - 2 q^{62} + 14 q^{63} + 12 q^{64} - 2 q^{65} - 4 q^{66} - 2 q^{69} - 2 q^{70} - 6 q^{72} - 3 q^{75} - q^{77} - 4 q^{78} - 5 q^{80} + 13 q^{81} - 3 q^{84} - 2 q^{86} - 2 q^{87} - 4 q^{88} - q^{89} + 25 q^{90} - q^{91} - 3 q^{92} - 2 q^{93} - 6 q^{96} - q^{98} - 3 q^{99}+O(q^{100}) $ 15 * q - q^2 - q^3 + 14 * q^4 - q^5 - 2 * q^6 + 15 * q^7 - 2 * q^8 + 14 * q^9 - 2 * q^10 - q^11 - 3 * q^12 - q^13 - q^14 - 2 * q^15 + 13 * q^16 - 3 * q^18 - 3 * q^20 - q^21 - 2 * q^22 - q^23 - 4 * q^24 + 14 * q^25 - 2 * q^26 - 2 * q^27 + 14 * q^28 - q^29 - 4 * q^30 - q^31 - 3 * q^32 - 2 * q^33 - q^35 + 11 * q^36 - 2 * q^39 - 4 * q^40 - 2 * q^42 - q^43 - 3 * q^44 - 3 * q^45 - 2 * q^46 - 5 * q^48 + 15 * q^49 - 3 * q^50 - 3 * q^52 - 4 * q^54 - 2 * q^55 - 2 * q^56 - 2 * q^58 - q^59 - 6 * q^60 - 2 * q^62 + 14 * q^63 + 12 * q^64 - 2 * q^65 - 4 * q^66 - 2 * q^69 - 2 * q^70 - 6 * q^72 - 3 * q^75 - q^77 - 4 * q^78 - 5 * q^80 + 13 * q^81 - 3 * q^84 - 2 * q^86 - 2 * q^87 - 4 * q^88 - q^89 + 25 * q^90 - q^91 - 3 * q^92 - 2 * q^93 - 6 * q^96 - q^98 - 3 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2471\mathbb{Z}\right)^\times$.

$n$	$1060$	$1415$
$\chi(n)$	$-1$	$-1$

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 $

$2$	0.694611	0.694611	0.347305	−	0.937752i	$-0.387097\pi$
$2$	0.694611	0.694611	0.347305	+	0.937752i	$0.387097\pi$
$3$	−1.22421	−1.22421	−0.612106	−	0.790776i	$-0.709677\pi$
$3$	−1.22421	−1.22421	−0.612106	+	0.790776i	$0.709677\pi$
$4$	−0.517516	−0.517516
$5$	−1.98974	−1.98974	−0.994869	−	0.101168i	$-0.967742\pi$
$5$	−1.98974	−1.98974	−0.994869	+	0.101168i	$0.967742\pi$
$6$	−0.850350	−0.850350
$7$	1.00000	1.00000
$7$	1.00000	1.00000
$8$	−1.05408	−1.05408
$9$	0.498695	0.498695
$10$	−1.38209	−1.38209
$11$	−0.101298	−0.101298	−0.0506492	−	0.998717i	$-0.516129\pi$
$11$	−0.101298	−0.101298	−0.0506492	+	0.998717i	$0.516129\pi$
$12$	0.633550	0.633550
$13$	−1.74869	−1.74869	−0.874347	−	0.485302i	$-0.838710\pi$
$13$	−1.74869	−1.74869	−0.874347	+	0.485302i	$0.838710\pi$
$14$	0.694611	0.694611
$15$	2.43586	2.43586
$16$	−0.214661	−0.214661
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0.346399	0.346399
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	1.02972	1.02972
$21$	−1.22421	−1.22421
$22$	−0.0703629	−0.0703629
$23$	−0.880788	−0.880788	−0.440394	−	0.897805i	$-0.645161\pi$
$23$	−0.880788	−0.880788	−0.440394	+	0.897805i	$0.645161\pi$
$24$	1.29042	1.29042
$25$	2.95906	2.95906
$26$	−1.21466	−1.21466
$27$	0.613704	0.613704
$28$	−0.517516	−0.517516
$29$	−1.90828	−1.90828	−0.954139	−	0.299363i	$-0.903226\pi$
$29$	−1.90828	−1.90828	−0.954139	+	0.299363i	$0.903226\pi$
$30$	1.69198	1.69198
$31$	1.37793	1.37793	0.688967	−	0.724793i	$-0.258065\pi$
$31$	1.37793	1.37793	0.688967	+	0.724793i	$0.258065\pi$
$32$	0.904977	0.904977
$33$	0.124011	0.124011
$34$	0	0
$35$	−1.98974	−1.98974
$36$	−0.258083	−0.258083
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	2.14077	2.14077
$40$	2.09735	2.09735
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	−0.850350	−0.850350
$43$	1.64153	1.64153	0.820763	−	0.571268i	$-0.193548\pi$
$43$	1.64153	1.64153	0.820763	+	0.571268i	$0.193548\pi$
$44$	0.0524235	0.0524235
$45$	−0.992273	−0.992273
$46$	−0.611805	−0.611805
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0.262790	0.262790
$49$	1.00000	1.00000
$50$	2.05539	2.05539
$51$	0	0
$52$	0.904977	0.904977
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0.426285	0.426285
$55$	0.201557	0.201557
$56$	−1.05408	−1.05408
$57$	0	0
$58$	−1.32551	−1.32551
$59$	1.95906	1.95906	0.979530	−	0.201299i	$-0.0645161\pi$
$59$	1.95906	1.95906	0.979530	+	0.201299i	$0.0645161\pi$
$60$	−1.26060	−1.26060
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0.957127	0.957127
$63$	0.498695	0.498695
$64$	0.843267	0.843267
$65$	3.47944	3.47944
$66$	0.0861391	0.0861391
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	1.07827	1.07827
$70$	−1.38209	−1.38209
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−0.525666	−0.525666
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−3.62252	−3.62252
$76$	0	0
$77$	−0.101298	−0.101298
$78$	1.48700	1.48700
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0.427119	0.427119
$81$	−1.25000	−1.25000
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0.633550	0.633550
$85$	0	0
$86$	1.14022	1.14022
$87$	2.33614	2.33614
$88$	0.106777	0.106777
$89$	−0.501305	−0.501305	−0.250653	−	0.968077i	$-0.580645\pi$
$89$	−0.501305	−0.501305	−0.250653	+	0.968077i	$0.580645\pi$
$90$	−0.689243	−0.689243
$91$	−1.74869	−1.74869
$92$	0.455822	0.455822
$93$	−1.68688	−1.68688
$94$	0	0
$95$	0	0
$96$	−1.10788	−1.10788
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0.694611	0.694611
$99$	−0.0505170	−0.0505170
$100$	−1.53136	−1.53136
$101$	1.83792	1.83792	0.918958	−	0.394356i	$-0.129032\pi$
$101$	1.83792	1.83792	0.918958	+	0.394356i	$0.129032\pi$
$102$	0	0
$103$	−1.90828	−1.90828	−0.954139	−	0.299363i	$-0.903226\pi$
$103$	−1.90828	−1.90828	−0.954139	+	0.299363i	$0.903226\pi$
$104$	1.84327	1.84327
$105$	2.43586	2.43586
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−0.317602	−0.317602
$109$	−0.880788	−0.880788	−0.440394	−	0.897805i	$-0.645161\pi$
$109$	−0.880788	−0.880788	−0.440394	+	0.897805i	$0.645161\pi$
$110$	0.140004	0.140004
$111$	0	0
$112$	−0.214661	−0.214661
$113$	0.302856	0.302856	0.151428	−	0.988468i	$-0.451613\pi$
$113$	0.302856	0.302856	0.151428	+	0.988468i	$0.451613\pi$
$114$	0	0
$115$	1.75254	1.75254
$116$	0.987565	0.987565
$117$	−0.872064	−0.872064
$118$	1.36078	1.36078
$119$	0	0
$120$	−2.56760	−2.56760
$121$	−0.989739	−0.989739
$122$	0	0
$123$	0	0
$124$	−0.713103	−0.713103
$125$	−3.89802	−3.89802
$126$	0.346399	0.346399
$127$	−1.22421	−1.22421	−0.612106	−	0.790776i	$-0.709677\pi$
$127$	−1.22421	−1.22421	−0.612106	+	0.790776i	$0.709677\pi$
$128$	−0.319235	−0.319235
$129$	−2.00958	−2.00958
$130$	2.41686	2.41686
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	−0.0641775	−0.0641775
$133$	0	0
$134$	0	0
$135$	−1.22111	−1.22111
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0.748979	0.748979
$139$	1.64153	1.64153	0.820763	−	0.571268i	$-0.193548\pi$
$139$	1.64153	1.64153	0.820763	+	0.571268i	$0.193548\pi$
$140$	1.02972	1.02972
$141$	0	0
$142$	0	0
$143$	0.177140	0.177140
$144$	−0.107050	−0.107050
$145$	3.79698	3.79698
$146$	0	0
$147$	−1.22421	−1.22421
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−2.51624	−2.51624
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	−0.0703629	−0.0703629
$155$	−2.74173	−2.74173
$156$	−1.10788	−1.10788
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	−1.80067	−1.80067
$161$	−0.880788	−0.880788
$162$	−0.868262	−0.868262
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	−0.246749	−0.246749
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	1.29042	1.29042
$169$	2.05793	2.05793
$170$	0	0
$171$	0	0
$172$	−0.849517	−0.849517
$173$	1.05793	1.05793	0.528964	−	0.848644i	$-0.322581\pi$
$173$	1.05793	1.05793	0.528964	+	0.848644i	$0.322581\pi$
$174$	1.62271	1.62271
$175$	2.95906	2.95906
$176$	0.0217448	0.0217448
$177$	−2.39830	−2.39830
$178$	−0.348212	−0.348212
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0.513517	0.513517
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	−1.21466	−1.21466
$183$	0	0
$184$	0.928424	0.928424
$185$	0	0
$186$	−1.17173	−1.17173
$187$	0	0
$188$	0	0
$189$	0.613704	0.613704
$190$	0	0
$191$	1.95906	1.95906	0.979530	−	0.201299i	$-0.0645161\pi$
$191$	1.95906	1.95906	0.979530	+	0.201299i	$0.0645161\pi$
$192$	−1.03234	−1.03234
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	−4.25958	−4.25958
$196$	−0.517516	−0.517516
$197$	1.05793	1.05793	0.528964	−	0.848644i	$-0.322581\pi$
$197$	1.05793	1.05793	0.528964	+	0.848644i	$0.322581\pi$
$198$	−0.0350896	−0.0350896
$199$	−1.51752	−1.51752	−0.758758	−	0.651372i	$-0.774194\pi$
$199$	−1.51752	−1.51752	−0.758758	+	0.651372i	$0.774194\pi$
$200$	−3.11909	−3.11909
$201$	0	0
$202$	1.27664	1.27664
$203$	−1.90828	−1.90828
$204$	0	0
$205$	0	0
$206$	−1.32551	−1.32551
$207$	−0.439245	−0.439245
$208$	0.375376	0.375376
$209$	0	0
$210$	1.69198	1.69198
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.26621	−3.26621
$216$	−0.646894	−0.646894
$217$	1.37793	1.37793
$218$	−0.611805	−0.611805
$219$	0	0
$220$	−0.104309	−0.104309
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0.904977	0.904977
$225$	1.47567	1.47567
$226$	0.210367	0.210367
$227$	0.302856	0.302856	0.151428	−	0.988468i	$-0.451613\pi$
$227$	0.302856	0.302856	0.151428	+	0.988468i	$0.451613\pi$
$228$	0	0
$229$	1.05793	1.05793	0.528964	−	0.848644i	$-0.322581\pi$
$229$	1.05793	1.05793	0.528964	+	0.848644i	$0.322581\pi$
$230$	1.21733	1.21733
$231$	0.124011	0.124011
$232$	2.01148	2.01148
$233$	1.37793	1.37793	0.688967	−	0.724793i	$-0.258065\pi$
$233$	1.37793	1.37793	0.688967	+	0.724793i	$0.258065\pi$
$234$	−0.605745	−0.605745
$235$	0	0
$236$	−1.01385	−1.01385
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	−0.522884	−0.522884
$241$	0.694611	0.694611	0.347305	−	0.937752i	$-0.387097\pi$
$241$	0.694611	0.694611	0.347305	+	0.937752i	$0.387097\pi$
$242$	−0.687483	−0.687483
$243$	0.916559	0.916559
$244$	0	0
$245$	−1.98974	−1.98974
$246$	0	0
$247$	0	0
$248$	−1.45246	−1.45246
$249$	0	0
$250$	−2.70760	−2.70760
$251$	−0.501305	−0.501305	−0.250653	−	0.968077i	$-0.580645\pi$
$251$	−0.501305	−0.501305	−0.250653	+	0.968077i	$0.580645\pi$
$252$	−0.258083	−0.258083
$253$	0.0892224	0.0892224
$254$	−0.850350	−0.850350
$255$	0	0
$256$	−1.06501	−1.06501
$257$	0.302856	0.302856	0.151428	−	0.988468i	$-0.451613\pi$
$257$	0.302856	0.302856	0.151428	+	0.988468i	$0.451613\pi$
$258$	−1.39587	−1.39587
$259$	0	0
$260$	−1.80067	−1.80067
$261$	−0.951649	−0.951649
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−0.130717	−0.130717
$265$	0	0
$266$	0	0
$267$	0.613704	0.613704
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−0.848196	−0.848196
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	2.14077	2.14077
$274$	0	0
$275$	−0.299748	−0.299748
$276$	−0.558023	−0.558023
$277$	−0.101298	−0.101298	−0.0506492	−	0.998717i	$-0.516129\pi$
$277$	−0.101298	−0.101298	−0.0506492	+	0.998717i	$0.516129\pi$
$278$	1.14022	1.14022
$279$	0.687169	0.687169
$280$	2.09735	2.09735
$281$	−1.51752	−1.51752	−0.758758	−	0.651372i	$-0.774194\pi$
$281$	−1.51752	−1.51752	−0.758758	+	0.651372i	$0.774194\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0.123043	0.123043
$287$	0	0
$288$	0.451308	0.451308
$289$	1.00000	1.00000
$290$	2.63742	2.63742
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−0.850350	−0.850350
$295$	−3.89802	−3.89802
$296$	0	0
$297$	−0.0621672	−0.0621672
$298$	0	0
$299$	1.54023	1.54023
$300$	1.87471	1.87471
$301$	1.64153	1.64153
$302$	0	0
$303$	−2.25000	−2.25000
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0.0524235	0.0524235
$309$	2.33614	2.33614
$310$	−1.90443	−1.90443
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	−2.25655	−2.25655
$313$	−1.90828	−1.90828	−0.954139	−	0.299363i	$-0.903226\pi$
$313$	−1.90828	−1.90828	−0.954139	+	0.299363i	$0.903226\pi$
$314$	0	0
$315$	−0.992273	−0.992273
$316$	0	0
$317$	−1.51752	−1.51752	−0.758758	−	0.651372i	$-0.774194\pi$
$317$	−1.51752	−1.51752	−0.758758	+	0.651372i	$0.774194\pi$
$318$	0	0
$319$	0.193305	0.193305
$320$	−1.67788	−1.67788
$321$	0	0
$322$	−0.611805	−0.611805
$323$	0	0
$324$	0.646894	0.646894
$325$	−5.17449	−5.17449
$326$	0	0
$327$	1.07827	1.07827
$328$	0	0
$329$	0	0
$330$	−0.171394	−0.171394
$331$	1.83792	1.83792	0.918958	−	0.394356i	$-0.129032\pi$
$331$	1.83792	1.83792	0.918958	+	0.394356i	$0.129032\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0.262790	0.262790
$337$	0.694611	0.694611	0.347305	−	0.937752i	$-0.387097\pi$
$337$	0.694611	0.694611	0.347305	+	0.937752i	$0.387097\pi$
$338$	1.42946	1.42946
$339$	−0.370759	−0.370759
$340$	0	0
$341$	−0.139582	−0.139582
$342$	0	0
$343$	1.00000	1.00000
$344$	−1.73031	−1.73031
$345$	−2.14548	−2.14548
$346$	0.734848	0.734848
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	−1.20899	−1.20899
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	2.05539	2.05539
$351$	−1.07318	−1.07318
$352$	−0.0916727	−0.0916727
$353$	1.00000	1.00000
$353$	1.00000	1.00000
$354$	−1.66589	−1.66589
$355$	0	0
$356$	0.259434	0.259434
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	1.04594	1.04594
$361$	1.00000	1.00000
$362$	0	0
$363$	1.21165	1.21165
$364$	0.904977	0.904977
$365$	0	0
$366$	0	0
$367$	1.83792	1.83792	0.918958	−	0.394356i	$-0.129032\pi$
$367$	1.83792	1.83792	0.918958	+	0.394356i	$0.129032\pi$
$368$	0.189071	0.189071
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0.872989	0.872989
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	4.77200	4.77200
$376$	0	0
$377$	3.33699	3.33699
$378$	0.426285	0.426285
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	1.49869	1.49869
$382$	1.36078	1.36078
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0.390811	0.390811
$385$	0.201557	0.201557
$386$	0	0
$387$	0.818621	0.818621
$388$	0	0
$389$	1.37793	1.37793	0.688967	−	0.724793i	$-0.258065\pi$
$389$	1.37793	1.37793	0.688967	+	0.724793i	$0.258065\pi$
$390$	−2.95875	−2.95875
$391$	0	0
$392$	−1.05408	−1.05408
$393$	0	0
$394$	0.734848	0.734848
$395$	0	0
$396$	0.0261434	0.0261434
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	−1.05408	−1.05408
$399$	0	0
$400$	−0.635194	−0.635194
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	−2.40958	−2.40958
$404$	−0.951151	−0.951151
$405$	2.48717	2.48717
$406$	−1.32551	−1.32551
$407$	0	0
$408$	0	0
$409$	−0.101298	−0.101298	−0.0506492	−	0.998717i	$-0.516129\pi$
$409$	−0.101298	−0.101298	−0.0506492	+	0.998717i	$0.516129\pi$
$410$	0	0
$411$	0	0
$412$	0.987565	0.987565
$413$	1.95906	1.95906
$414$	−0.305104	−0.305104
$415$	0	0
$416$	−1.58253	−1.58253
$417$	−2.00958	−2.00958
$418$	0	0
$419$	0.694611	0.694611	0.347305	−	0.937752i	$-0.387097\pi$
$419$	0.694611	0.694611	0.347305	+	0.937752i	$0.387097\pi$
$420$	−1.26060	−1.26060
$421$	1.37793	1.37793	0.688967	−	0.724793i	$-0.258065\pi$
$421$	1.37793	1.37793	0.688967	+	0.724793i	$0.258065\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−0.216857	−0.216857
$430$	−2.26874	−2.26874
$431$	1.64153	1.64153	0.820763	−	0.571268i	$-0.193548\pi$
$431$	1.64153	1.64153	0.820763	+	0.571268i	$0.193548\pi$
$432$	−0.131738	−0.131738
$433$	1.05793	1.05793	0.528964	−	0.848644i	$-0.322581\pi$
$433$	1.05793	1.05793	0.528964	+	0.848644i	$0.322581\pi$
$434$	0.957127	0.957127
$435$	−4.64830	−4.64830
$436$	0.455822	0.455822
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	−0.212458	−0.212458
$441$	0.498695	0.498695
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0.997466	0.997466
$446$	0	0
$447$	0	0
$448$	0.843267	0.843267
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	1.02501	1.02501
$451$	0	0
$452$	−0.156733	−0.156733
$453$	0	0
$454$	0.210367	0.210367
$455$	3.47944	3.47944
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0.734848	0.734848
$459$	0	0
$460$	−0.906967	−0.906967
$461$	1.64153	1.64153	0.820763	−	0.571268i	$-0.193548\pi$
$461$	1.64153	1.64153	0.820763	+	0.571268i	$0.193548\pi$
$462$	0.0861391	0.0861391
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0.409632	0.409632
$465$	3.35646	3.35646
$466$	0.957127	0.957127
$467$	−1.74869	−1.74869	−0.874347	−	0.485302i	$-0.838710\pi$
$467$	−1.74869	−1.74869	−0.874347	+	0.485302i	$0.838710\pi$
$468$	0.451308	0.451308
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−2.06501	−2.06501
$473$	−0.166284	−0.166284
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.90828	−1.90828	−0.954139	−	0.299363i	$-0.903226\pi$
$479$	−1.90828	−1.90828	−0.954139	+	0.299363i	$0.903226\pi$
$480$	2.20440	2.20440
$481$	0	0
$482$	0.482484	0.482484
$483$	1.07827	1.07827
$484$	0.512206	0.512206
$485$	0	0
$486$	0.636652	0.636652
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	−1.38209	−1.38209
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0.100516	0.100516
$496$	−0.295788	−0.295788
$497$	0	0
$498$	0	0
$499$	0.694611	0.694611	0.347305	−	0.937752i	$-0.387097\pi$
$499$	0.694611	0.694611	0.347305	+	0.937752i	$0.387097\pi$
$500$	2.01729	2.01729
$501$	0	0
$502$	−0.348212	−0.348212
$503$	0.694611	0.694611	0.347305	−	0.937752i	$-0.387097\pi$
$503$	0.694611	0.694611	0.347305	+	0.937752i	$0.387097\pi$
$504$	−0.525666	−0.525666
$505$	−3.65697	−3.65697
$506$	0.0619748	0.0619748
$507$	−2.51934	−2.51934
$508$	0.633550	0.633550
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−0.420533	−0.420533
$513$	0	0
$514$	0.210367	0.210367
$515$	3.79698	3.79698
$516$	1.03999	1.03999
$517$	0	0
$518$	0	0
$519$	−1.29513	−1.29513
$520$	−3.66762	−3.66762
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−0.661025	−0.661025
$523$	−1.98974	−1.98974	−0.994869	−	0.101168i	$-0.967742\pi$
$523$	−1.98974	−1.98974	−0.994869	+	0.101168i	$0.967742\pi$
$524$	0	0
$525$	−3.62252	−3.62252
$526$	0	0
$527$	0	0
$528$	−0.0266202	−0.0266202
$529$	−0.224212	−0.224212
$530$	0	0
$531$	0.976973	0.976973
$532$	0	0
$533$	0	0
$534$	0.426285	0.426285
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.101298	−0.101298
$540$	0.631944	0.631944
$541$	1.64153	1.64153	0.820763	−	0.571268i	$-0.193548\pi$
$541$	1.64153	1.64153	0.820763	+	0.571268i	$0.193548\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.75254	1.75254
$546$	1.48700	1.48700
$547$	0.302856	0.302856	0.151428	−	0.988468i	$-0.451613\pi$
$547$	0.302856	0.302856	0.151428	+	0.988468i	$0.451613\pi$
$548$	0	0
$549$	0	0
$550$	−0.208208	−0.208208
$551$	0	0
$552$	−1.13659	−1.13659
$553$	0	0
$554$	−0.0703629	−0.0703629
$555$	0	0
$556$	−0.849517	−0.849517
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0.477315	0.477315
$559$	−2.87053	−2.87053
$560$	0.427119	0.427119
$561$	0	0
$562$	−1.05408	−1.05408
$563$	1.37793	1.37793	0.688967	−	0.724793i	$-0.258065\pi$
$563$	1.37793	1.37793	0.688967	+	0.724793i	$0.258065\pi$
$564$	0	0
$565$	−0.602603	−0.602603
$566$	0	0
$567$	−1.25000	−1.25000
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	−0.880788	−0.880788	−0.440394	−	0.897805i	$-0.645161\pi$
$571$	−0.880788	−0.880788	−0.440394	+	0.897805i	$0.645161\pi$
$572$	−0.0916727	−0.0916727
$573$	−2.39830	−2.39830
$574$	0	0
$575$	−2.60631	−2.60631
$576$	0.420533	0.420533
$577$	−0.880788	−0.880788	−0.440394	−	0.897805i	$-0.645161\pi$
$577$	−0.880788	−0.880788	−0.440394	+	0.897805i	$0.645161\pi$
$578$	0.694611	0.694611
$579$	0	0
$580$	−1.96500	−1.96500
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	1.73518	1.73518
$586$	0	0
$587$	−1.51752	−1.51752	−0.758758	−	0.651372i	$-0.774194\pi$
$587$	−1.51752	−1.51752	−0.758758	+	0.651372i	$0.774194\pi$
$588$	0.633550	0.633550
$589$	0	0
$590$	−2.70760	−2.70760
$591$	−1.29513	−1.29513
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	−0.0431820	−0.0431820
$595$	0	0
$596$	0	0
$597$	1.85776	1.85776
$598$	1.06986	1.06986
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	3.81843	3.81843
$601$	0.302856	0.302856	0.151428	−	0.988468i	$-0.451613\pi$
$601$	0.302856	0.302856	0.151428	+	0.988468i	$0.451613\pi$
$602$	1.14022	1.14022
$603$	0	0
$604$	0	0
$605$	1.96932	1.96932
$606$	−1.56287	−1.56287
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	2.33614	2.33614
$610$	0	0
$611$	0	0
$612$	0	0
$613$	1.05793	1.05793	0.528964	−	0.848644i	$-0.322581\pi$
$613$	1.05793	1.05793	0.528964	+	0.848644i	$0.322581\pi$
$614$	0	0
$615$	0	0
$616$	0.106777	0.106777
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	1.62271	1.62271
$619$	−1.51752	−1.51752	−0.758758	−	0.651372i	$-0.774194\pi$
$619$	−1.51752	−1.51752	−0.758758	+	0.651372i	$0.774194\pi$
$620$	1.41889	1.41889
$621$	−0.540543	−0.540543
$622$	0	0
$623$	−0.501305	−0.501305
$624$	−0.459539	−0.459539
$625$	4.79698	4.79698
$626$	−1.32551	−1.32551
$627$	0	0
$628$	0	0
$629$	0	0
$630$	−0.689243	−0.689243
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	−1.05408	−1.05408
$635$	2.43586	2.43586
$636$	0	0
$637$	−1.74869	−1.74869
$638$	0.134272	0.134272
$639$	0	0
$640$	0.635194	0.635194
$641$	−1.98974	−1.98974	−0.994869	−	0.101168i	$-0.967742\pi$
$641$	−1.98974	−1.98974	−0.994869	+	0.101168i	$0.967742\pi$
$642$	0	0
$643$	−0.501305	−0.501305	−0.250653	−	0.968077i	$-0.580645\pi$
$643$	−0.501305	−0.501305	−0.250653	+	0.968077i	$0.580645\pi$
$644$	0.455822	0.455822
$645$	3.99853	3.99853
$646$	0	0
$647$	−0.501305	−0.501305	−0.250653	−	0.968077i	$-0.580645\pi$
$647$	−0.501305	−0.501305	−0.250653	+	0.968077i	$0.580645\pi$
$648$	1.31760	1.31760
$649$	−0.198450	−0.198450
$650$	−3.59425	−3.59425
$651$	−1.68688	−1.68688
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0.748979	0.748979
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.05793	1.05793	0.528964	−	0.848644i	$-0.322581\pi$
$659$	1.05793	1.05793	0.528964	+	0.848644i	$0.322581\pi$
$660$	0.127696	0.127696
$661$	1.83792	1.83792	0.918958	−	0.394356i	$-0.129032\pi$
$661$	1.83792	1.83792	0.918958	+	0.394356i	$0.129032\pi$
$662$	1.27664	1.27664
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.68079	1.68079
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	−1.10788	−1.10788
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0.482484	0.482484
$675$	1.81599	1.81599
$676$	−1.06501	−1.06501
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	−0.257533	−0.257533
$679$	0	0
$680$	0	0
$681$	−0.370759	−0.370759
$682$	−0.0969554	−0.0969554
$683$	1.37793	1.37793	0.688967	−	0.724793i	$-0.258065\pi$
$683$	1.37793	1.37793	0.688967	+	0.724793i	$0.258065\pi$
$684$	0	0
$685$	0	0
$686$	0.694611	0.694611
$687$	−1.29513	−1.29513
$688$	−0.352371	−0.352371
$689$	0	0
$690$	−1.49027	−1.49027
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	−0.547495	−0.547495
$693$	−0.0505170	−0.0505170
$694$	0	0
$695$	−3.26621	−3.26621
$696$	−2.46248	−2.46248
$697$	0	0
$698$	0	0
$699$	−1.68688	−1.68688
$700$	−1.53136	−1.53136
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	−0.745442	−0.745442
$703$	0	0
$704$	−0.0854216	−0.0854216
$705$	0	0
$706$	0.694611	0.694611
$707$	1.83792	1.83792
$708$	1.24116	1.24116
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0.528417	0.528417
$713$	−1.21367	−1.21367
$714$	0	0
$715$	−0.352462	−0.352462
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−0.101298	−0.101298	−0.0506492	−	0.998717i	$-0.516129\pi$
$719$	−0.101298	−0.101298	−0.0506492	+	0.998717i	$0.516129\pi$
$720$	0.213002	0.213002
$721$	−1.90828	−1.90828
$722$	0.694611	0.694611
$723$	−0.850350	−0.850350
$724$	0	0
$725$	−5.64671	−5.64671
$726$	0.841625	0.841625
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	1.84327	1.84327
$729$	0.127936	0.127936
$730$	0	0
$731$	0	0
$732$	0	0
$733$	1.37793	1.37793	0.688967	−	0.724793i	$-0.258065\pi$
$733$	1.37793	1.37793	0.688967	+	0.724793i	$0.258065\pi$
$734$	1.27664	1.27664
$735$	2.43586	2.43586
$736$	−0.797093	−0.797093
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	1.77811	1.77811
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	3.31468	3.31468
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0.613704	0.613704
$754$	2.31791	2.31791
$755$	0	0
$756$	−0.317602	−0.317602
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	−0.109227	−0.109227
$760$	0	0
$761$	−1.74869	−1.74869	−0.874347	−	0.485302i	$-0.838710\pi$
$761$	−1.74869	−1.74869	−0.874347	+	0.485302i	$0.838710\pi$
$762$	1.04101	1.04101
$763$	−0.880788	−0.880788
$764$	−1.01385	−1.01385
$765$	0	0
$766$	0	0
$767$	−3.42579	−3.42579
$768$	1.30380	1.30380
$769$	1.37793	1.37793	0.688967	−	0.724793i	$-0.258065\pi$
$769$	1.37793	1.37793	0.688967	+	0.724793i	$0.258065\pi$
$770$	0.140004	0.140004
$771$	−0.370759	−0.370759
$772$	0	0
$773$	1.64153	1.64153	0.820763	−	0.571268i	$-0.193548\pi$
$773$	1.64153	1.64153	0.820763	+	0.571268i	$0.193548\pi$
$774$	0.568623	0.568623
$775$	4.07739	4.07739
$776$	0	0
$777$	0	0
$778$	0.957127	0.957127
$779$	0	0
$780$	2.20440	2.20440
$781$	0	0
$782$	0	0
$783$	−1.17112	−1.17112
$784$	−0.214661	−0.214661
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−0.547495	−0.547495
$789$	0	0
$790$	0	0
$791$	0.302856	0.302856
$792$	0.0532491	0.0532491
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0.785339	0.785339
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	2.67788	2.67788
$801$	−0.249998	−0.249998
$802$	0	0
$803$	0	0
$804$	0	0
$805$	1.75254	1.75254
$806$	−1.67372	−1.67372
$807$	0	0
$808$	−1.93732	−1.93732
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	1.72761	1.72761
$811$	−0.501305	−0.501305	−0.250653	−	0.968077i	$-0.580645\pi$
$811$	−0.501305	−0.501305	−0.250653	+	0.968077i	$0.580645\pi$
$812$	0.987565	0.987565
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	−0.0703629	−0.0703629
$819$	−0.872064	−0.872064
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	2.01148	2.01148
$825$	0.366955	0.366955
$826$	1.36078	1.36078
$827$	−1.74869	−1.74869	−0.874347	−	0.485302i	$-0.838710\pi$
$827$	−1.74869	−1.74869	−0.874347	+	0.485302i	$0.838710\pi$
$828$	0.227316	0.227316
$829$	−1.22421	−1.22421	−0.612106	−	0.790776i	$-0.709677\pi$
$829$	−1.22421	−1.22421	−0.612106	+	0.790776i	$0.709677\pi$
$830$	0	0
$831$	0.124011	0.124011
$832$	−1.47462	−1.47462
$833$	0	0
$834$	−1.39587	−1.39587
$835$	0	0
$836$	0	0
$837$	0.845643	0.845643
$838$	0.482484	0.482484
$839$	1.83792	1.83792	0.918958	−	0.394356i	$-0.129032\pi$
$839$	1.83792	1.83792	0.918958	+	0.394356i	$0.129032\pi$
$840$	−2.56760	−2.56760
$841$	2.64153	2.64153
$842$	0.957127	0.957127
$843$	1.85776	1.85776
$844$	0	0
$845$	−4.09474	−4.09474
$846$	0	0
$847$	−0.989739	−0.989739
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−0.880788	−0.880788	−0.440394	−	0.897805i	$-0.645161\pi$
$853$	−0.880788	−0.880788	−0.440394	+	0.897805i	$0.645161\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.22421	−1.22421	−0.612106	−	0.790776i	$-0.709677\pi$
$857$	−1.22421	−1.22421	−0.612106	+	0.790776i	$0.709677\pi$
$858$	−0.150631	−0.150631
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	1.69032	1.69032
$861$	0	0
$862$	1.14022	1.14022
$863$	1.64153	1.64153	0.820763	−	0.571268i	$-0.193548\pi$
$863$	1.64153	1.64153	0.820763	+	0.571268i	$0.193548\pi$
$864$	0.555388	0.555388
$865$	−2.10500	−2.10500
$866$	0.734848	0.734848
$867$	−1.22421	−1.22421
$868$	−0.713103	−0.713103
$869$	0	0
$870$	−3.22876	−3.22876
$871$	0	0
$872$	0.928424	0.928424
$873$	0	0
$874$	0	0
$875$	−3.89802	−3.89802
$876$	0	0
$877$	−1.98974	−1.98974	−0.994869	−	0.101168i	$-0.967742\pi$
$877$	−1.98974	−1.98974	−0.994869	+	0.101168i	$0.967742\pi$
$878$	0	0
$879$	0	0
$880$	−0.0432664	−0.0432664
$881$	1.83792	1.83792	0.918958	−	0.394356i	$-0.129032\pi$
$881$	1.83792	1.83792	0.918958	+	0.394356i	$0.129032\pi$
$882$	0.346399	0.346399
$883$	−0.501305	−0.501305	−0.250653	−	0.968077i	$-0.580645\pi$
$883$	−0.501305	−0.501305	−0.250653	+	0.968077i	$0.580645\pi$
$884$	0	0
$885$	4.77200	4.77200
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	−1.22421	−1.22421
$890$	0.692850	0.692850
$891$	0.126623	0.126623
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−0.319235	−0.319235
$897$	−1.88557	−1.88557
$898$	0	0
$899$	−2.62948	−2.62948
$900$	−0.763682	−0.763682
$901$	0	0
$902$	0	0
$903$	−2.00958	−2.00958
$904$	−0.319235	−0.319235
$905$	0	0
$906$	0	0
$907$	1.05793	1.05793	0.528964	−	0.848644i	$-0.322581\pi$
$907$	1.05793	1.05793	0.528964	+	0.848644i	$0.322581\pi$
$908$	−0.156733	−0.156733
$909$	0.916559	0.916559
$910$	2.41686	2.41686
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	−0.547495	−0.547495
$917$	0	0
$918$	0	0
$919$	−0.101298	−0.101298	−0.0506492	−	0.998717i	$-0.516129\pi$
$919$	−0.101298	−0.101298	−0.0506492	+	0.998717i	$0.516129\pi$
$920$	−1.84732	−1.84732
$921$	0	0
$922$	1.14022	1.14022
$923$	0	0
$924$	−0.0641775	−0.0641775
$925$	0	0
$926$	0	0
$927$	−0.951649	−0.951649
$928$	−1.72695	−1.72695
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	2.33143	2.33143
$931$	0	0
$932$	−0.713103	−0.713103
$933$	0	0
$934$	−1.21466	−1.21466
$935$	0	0
$936$	0.919228	0.919228
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	2.33614	2.33614
$940$	0	0
$941$	0.302856	0.302856	0.151428	−	0.988468i	$-0.451613\pi$
$941$	0.302856	0.302856	0.151428	+	0.988468i	$0.451613\pi$
$942$	0	0
$943$	0	0
$944$	−0.420533	−0.420533
$945$	−1.22111	−1.22111
$946$	−0.115503	−0.115503
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	1.85776	1.85776
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	−3.89802	−3.89802
$956$	0	0
$957$	−0.236647	−0.236647
$958$	−1.32551	−1.32551
$959$	0	0
$960$	2.05408	2.05408
$961$	0.898702	0.898702
$962$	0	0
$963$	0	0
$964$	−0.359472	−0.359472
$965$	0	0
$966$	0.748979	0.748979
$967$	1.95906	1.95906	0.979530	−	0.201299i	$-0.0645161\pi$
$967$	1.95906	1.95906	0.979530	+	0.201299i	$0.0645161\pi$
$968$	1.04327	1.04327
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−0.474334	−0.474334
$973$	1.64153	1.64153
$974$	0	0
$975$	6.33467	6.33467
$976$	0	0
$977$	−1.74869	−1.74869	−0.874347	−	0.485302i	$-0.838710\pi$
$977$	−1.74869	−1.74869	−0.874347	+	0.485302i	$0.838710\pi$
$978$	0	0
$979$	0.0507814	0.0507814
$980$	1.02972	1.02972
$981$	−0.439245	−0.439245
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	−2.10500	−2.10500
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.44584	−1.44584
$990$	0.0698192	0.0698192
$991$	−1.74869	−1.74869	−0.874347	−	0.485302i	$-0.838710\pi$
$991$	−1.74869	−1.74869	−0.874347	+	0.485302i	$0.838710\pi$
$992$	1.24700	1.24700
$993$	−2.25000	−2.25000
$994$	0	0
$995$	3.01946	3.01946
$996$	0	0
$997$	0.694611	0.694611	0.347305	−	0.937752i	$-0.387097\pi$
$997$	0.694611	0.694611	0.347305	+	0.937752i	$0.387097\pi$
$998$	0.482484	0.482484
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2471.1.d.a.2470.10	✓	15
7.6	odd	2		2471.1.d.b.2470.10	yes	15
353.352	even	2		2471.1.d.b.2470.10	yes	15
2471.2470	odd	2	CM	2471.1.d.a.2470.10	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2471.1.d.a.2470.10	✓	15	1.1	even	1	trivial
2471.1.d.a.2470.10	✓	15	2471.2470	odd	2	CM
2471.1.d.b.2470.10	yes	15	7.6	odd	2
2471.1.d.b.2470.10	yes	15	353.352	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:	\( N \)	\(=\)	\( 2471 = 7 \cdot 353 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2471.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.694611 q^{2} -1.22421 q^{3} -0.517516 q^{4} -1.98974 q^{5} -0.850350 q^{6} +1.00000 q^{7} -1.05408 q^{8} +0.498695 q^{9} +O(q^{10})\)	\(q+0.694611 q^{2} -1.22421 q^{3} -0.517516 q^{4} -1.98974 q^{5} -0.850350 q^{6} +1.00000 q^{7} -1.05408 q^{8} +0.498695 q^{9} -1.38209 q^{10} -0.101298 q^{11} +0.633550 q^{12} -1.74869 q^{13} +0.694611 q^{14} +2.43586 q^{15} -0.214661 q^{16} +0.346399 q^{18} +1.02972 q^{20} -1.22421 q^{21} -0.0703629 q^{22} -0.880788 q^{23} +1.29042 q^{24} +2.95906 q^{25} -1.21466 q^{26} +0.613704 q^{27} -0.517516 q^{28} -1.90828 q^{29} +1.69198 q^{30} +1.37793 q^{31} +0.904977 q^{32} +0.124011 q^{33} -1.98974 q^{35} -0.258083 q^{36} +2.14077 q^{39} +2.09735 q^{40} -0.850350 q^{42} +1.64153 q^{43} +0.0524235 q^{44} -0.992273 q^{45} -0.611805 q^{46} +0.262790 q^{48} +1.00000 q^{49} +2.05539 q^{50} +0.904977 q^{52} +0.426285 q^{54} +0.201557 q^{55} -1.05408 q^{56} -1.32551 q^{58} +1.95906 q^{59} -1.26060 q^{60} +0.957127 q^{62} +0.498695 q^{63} +0.843267 q^{64} +3.47944 q^{65} +0.0861391 q^{66} +1.07827 q^{69} -1.38209 q^{70} -0.525666 q^{72} -3.62252 q^{75} -0.101298 q^{77} +1.48700 q^{78} +0.427119 q^{80} -1.25000 q^{81} +0.633550 q^{84} +1.14022 q^{86} +2.33614 q^{87} +0.106777 q^{88} -0.501305 q^{89} -0.689243 q^{90} -1.74869 q^{91} +0.455822 q^{92} -1.68688 q^{93} -1.10788 q^{96} +0.694611 q^{98} -0.0505170 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{2} - q^{3} + 14 q^{4} - q^{5} - 2 q^{6} + 15 q^{7} - 2 q^{8} + 14 q^{9}+O(q^{10}) \) 15 * q - q^2 - q^3 + 14 * q^4 - q^5 - 2 * q^6 + 15 * q^7 - 2 * q^8 + 14 * q^9	\( 15 q - q^{2} - q^{3} + 14 q^{4} - q^{5} - 2 q^{6} + 15 q^{7} - 2 q^{8} + 14 q^{9} - 2 q^{10} - q^{11} - 3 q^{12} - q^{13} - q^{14} - 2 q^{15} + 13 q^{16} - 3 q^{18} - 3 q^{20} - q^{21} - 2 q^{22} - q^{23} - 4 q^{24} + 14 q^{25} - 2 q^{26} - 2 q^{27} + 14 q^{28} - q^{29} - 4 q^{30} - q^{31} - 3 q^{32} - 2 q^{33} - q^{35} + 11 q^{36} - 2 q^{39} - 4 q^{40} - 2 q^{42} - q^{43} - 3 q^{44} - 3 q^{45} - 2 q^{46} - 5 q^{48} + 15 q^{49} - 3 q^{50} - 3 q^{52} - 4 q^{54} - 2 q^{55} - 2 q^{56} - 2 q^{58} - q^{59} - 6 q^{60} - 2 q^{62} + 14 q^{63} + 12 q^{64} - 2 q^{65} - 4 q^{66} - 2 q^{69} - 2 q^{70} - 6 q^{72} - 3 q^{75} - q^{77} - 4 q^{78} - 5 q^{80} + 13 q^{81} - 3 q^{84} - 2 q^{86} - 2 q^{87} - 4 q^{88} - q^{89} + 25 q^{90} - q^{91} - 3 q^{92} - 2 q^{93} - 6 q^{96} - q^{98} - 3 q^{99}+O(q^{100}) \) 15 * q - q^2 - q^3 + 14 * q^4 - q^5 - 2 * q^6 + 15 * q^7 - 2 * q^8 + 14 * q^9 - 2 * q^10 - q^11 - 3 * q^12 - q^13 - q^14 - 2 * q^15 + 13 * q^16 - 3 * q^18 - 3 * q^20 - q^21 - 2 * q^22 - q^23 - 4 * q^24 + 14 * q^25 - 2 * q^26 - 2 * q^27 + 14 * q^28 - q^29 - 4 * q^30 - q^31 - 3 * q^32 - 2 * q^33 - q^35 + 11 * q^36 - 2 * q^39 - 4 * q^40 - 2 * q^42 - q^43 - 3 * q^44 - 3 * q^45 - 2 * q^46 - 5 * q^48 + 15 * q^49 - 3 * q^50 - 3 * q^52 - 4 * q^54 - 2 * q^55 - 2 * q^56 - 2 * q^58 - q^59 - 6 * q^60 - 2 * q^62 + 14 * q^63 + 12 * q^64 - 2 * q^65 - 4 * q^66 - 2 * q^69 - 2 * q^70 - 6 * q^72 - 3 * q^75 - q^77 - 4 * q^78 - 5 * q^80 + 13 * q^81 - 3 * q^84 - 2 * q^86 - 2 * q^87 - 4 * q^88 - q^89 + 25 * q^90 - q^91 - 3 * q^92 - 2 * q^93 - 6 * q^96 - q^98 - 3 * q^99

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