LMFDB - Embedded newform 2471.1.d.b.2470.6

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			2470.6
Root			$1.22421$ 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)

$f(q)$	$=$	$q-0.880788 q^{2} +0.101298 q^{3} -0.224212 q^{4} -0.694611 q^{5} -0.0892224 q^{6} -1.00000 q^{7} +1.07827 q^{8} -0.989739 q^{9} +O(q^{10})$	\(q-0.880788 q^{2} +0.101298 q^{3} -0.224212 q^{4} -0.694611 q^{5} -0.0892224 q^{6} -1.00000 q^{7} +1.07827 q^{8} -0.989739 q^{9} +0.611805 q^{10} +1.64153 q^{11} -0.0227123 q^{12} -1.95906 q^{13} +0.880788 q^{14} -0.0703629 q^{15} -0.725517 q^{16} +0.871750 q^{18} +0.155740 q^{20} -0.101298 q^{21} -1.44584 q^{22} +1.37793 q^{23} +0.109227 q^{24} -0.517516 q^{25} +1.72552 q^{26} -0.201557 q^{27} +0.224212 q^{28} -1.74869 q^{29} +0.0619748 q^{30} +1.90828 q^{31} -0.439245 q^{32} +0.166284 q^{33} +0.694611 q^{35} +0.221911 q^{36} -0.198450 q^{39} -0.748979 q^{40} +0.0892224 q^{42} +1.05793 q^{43} -0.368050 q^{44} +0.687483 q^{45} -1.21367 q^{46} -0.0734937 q^{48} +1.00000 q^{49} +0.455822 q^{50} +0.439245 q^{52} +0.177529 q^{54} -1.14022 q^{55} -1.07827 q^{56} +1.54023 q^{58} +1.51752 q^{59} +0.0157762 q^{60} -1.68079 q^{62} +0.989739 q^{63} +1.11240 q^{64} +1.36078 q^{65} -0.146461 q^{66} +0.139582 q^{69} -0.611805 q^{70} -1.06721 q^{72} -0.0524235 q^{75} -1.64153 q^{77} +0.174792 q^{78} +0.503952 q^{80} +0.969321 q^{81} +0.0227123 q^{84} -0.931811 q^{86} -0.177140 q^{87} +1.77001 q^{88} +1.98974 q^{89} -0.605527 q^{90} +1.95906 q^{91} -0.308949 q^{92} +0.193305 q^{93} -0.0444948 q^{96} -0.880788 q^{98} -1.62468 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.880788	−0.880788	−0.440394	−	0.897805i	$-0.645161\pi$
$2$	−0.880788	−0.880788	−0.440394	+	0.897805i	$0.645161\pi$
$3$	0.101298	0.101298	0.0506492	−	0.998717i	$-0.483871\pi$
$3$	0.101298	0.101298	0.0506492	+	0.998717i	$0.483871\pi$
$4$	−0.224212	−0.224212
$5$	−0.694611	−0.694611	−0.347305	−	0.937752i	$-0.612903\pi$
$5$	−0.694611	−0.694611	−0.347305	+	0.937752i	$0.612903\pi$
$6$	−0.0892224	−0.0892224
$7$	−1.00000	−1.00000
$7$	−1.00000	−1.00000
$8$	1.07827	1.07827
$9$	−0.989739	−0.989739
$10$	0.611805	0.611805
$11$	1.64153	1.64153	0.820763	−	0.571268i	$-0.193548\pi$
$11$	1.64153	1.64153	0.820763	+	0.571268i	$0.193548\pi$
$12$	−0.0227123	−0.0227123
$13$	−1.95906	−1.95906	−0.979530	−	0.201299i	$-0.935484\pi$
$13$	−1.95906	−1.95906	−0.979530	+	0.201299i	$0.935484\pi$
$14$	0.880788	0.880788
$15$	−0.0703629	−0.0703629
$16$	−0.725517	−0.725517
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0.871750	0.871750
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0.155740	0.155740
$21$	−0.101298	−0.101298
$22$	−1.44584	−1.44584
$23$	1.37793	1.37793	0.688967	−	0.724793i	$-0.258065\pi$
$23$	1.37793	1.37793	0.688967	+	0.724793i	$0.258065\pi$
$24$	0.109227	0.109227
$25$	−0.517516	−0.517516
$26$	1.72552	1.72552
$27$	−0.201557	−0.201557
$28$	0.224212	0.224212
$29$	−1.74869	−1.74869	−0.874347	−	0.485302i	$-0.838710\pi$
$29$	−1.74869	−1.74869	−0.874347	+	0.485302i	$0.838710\pi$
$30$	0.0619748	0.0619748
$31$	1.90828	1.90828	0.954139	−	0.299363i	$-0.0967742\pi$
$31$	1.90828	1.90828	0.954139	+	0.299363i	$0.0967742\pi$
$32$	−0.439245	−0.439245
$33$	0.166284	0.166284
$34$	0	0
$35$	0.694611	0.694611
$36$	0.221911	0.221911
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	−0.198450	−0.198450
$40$	−0.748979	−0.748979
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0.0892224	0.0892224
$43$	1.05793	1.05793	0.528964	−	0.848644i	$-0.322581\pi$
$43$	1.05793	1.05793	0.528964	+	0.848644i	$0.322581\pi$
$44$	−0.368050	−0.368050
$45$	0.687483	0.687483
$46$	−1.21367	−1.21367
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−0.0734937	−0.0734937
$49$	1.00000	1.00000
$50$	0.455822	0.455822
$51$	0	0
$52$	0.439245	0.439245
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0.177529	0.177529
$55$	−1.14022	−1.14022
$56$	−1.07827	−1.07827
$57$	0	0
$58$	1.54023	1.54023
$59$	1.51752	1.51752	0.758758	−	0.651372i	$-0.225806\pi$
$59$	1.51752	1.51752	0.758758	+	0.651372i	$0.225806\pi$
$60$	0.0157762	0.0157762
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−1.68079	−1.68079
$63$	0.989739	0.989739
$64$	1.11240	1.11240
$65$	1.36078	1.36078
$66$	−0.146461	−0.146461
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0.139582	0.139582
$70$	−0.611805	−0.611805
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−1.06721	−1.06721
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−0.0524235	−0.0524235
$76$	0	0
$77$	−1.64153	−1.64153
$78$	0.174792	0.174792
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0.503952	0.503952
$81$	0.969321	0.969321
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0.0227123	0.0227123
$85$	0	0
$86$	−0.931811	−0.931811
$87$	−0.177140	−0.177140
$88$	1.77001	1.77001
$89$	1.98974	1.98974	0.994869	−	0.101168i	$-0.0322581\pi$
$89$	1.98974	1.98974	0.994869	+	0.101168i	$0.0322581\pi$
$90$	−0.605527	−0.605527
$91$	1.95906	1.95906
$92$	−0.308949	−0.308949
$93$	0.193305	0.193305
$94$	0	0
$95$	0	0
$96$	−0.0444948	−0.0444948
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−0.880788	−0.880788
$99$	−1.62468	−1.62468
$100$	0.116033	0.116033
$101$	−0.302856	−0.302856	−0.151428	−	0.988468i	$-0.548387\pi$
$101$	−0.302856	−0.302856	−0.151428	+	0.988468i	$0.548387\pi$
$102$	0	0
$103$	1.74869	1.74869	0.874347	−	0.485302i	$-0.161290\pi$
$103$	1.74869	1.74869	0.874347	+	0.485302i	$0.161290\pi$
$104$	−2.11240	−2.11240
$105$	0.0703629	0.0703629
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0.0451915	0.0451915
$109$	1.37793	1.37793	0.688967	−	0.724793i	$-0.258065\pi$
$109$	1.37793	1.37793	0.688967	+	0.724793i	$0.258065\pi$
$110$	1.00429	1.00429
$111$	0	0
$112$	0.725517	0.725517
$113$	−0.501305	−0.501305	−0.250653	−	0.968077i	$-0.580645\pi$
$113$	−0.501305	−0.501305	−0.250653	+	0.968077i	$0.580645\pi$
$114$	0	0
$115$	−0.957127	−0.957127
$116$	0.392078	0.392078
$117$	1.93896	1.93896
$118$	−1.33661	−1.33661
$119$	0	0
$120$	−0.0758703	−0.0758703
$121$	1.69461	1.69461
$122$	0	0
$123$	0	0
$124$	−0.427859	−0.427859
$125$	1.05408	1.05408
$126$	−0.871750	−0.871750
$127$	−0.101298	−0.101298	−0.0506492	−	0.998717i	$-0.516129\pi$
$127$	−0.101298	−0.101298	−0.0506492	+	0.998717i	$0.516129\pi$
$128$	−0.540543	−0.540543
$129$	0.107166	0.107166
$130$	−1.19856	−1.19856
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	−0.0372828	−0.0372828
$133$	0	0
$134$	0	0
$135$	0.140004	0.140004
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	−0.122943	−0.122943
$139$	−1.05793	−1.05793	−0.528964	−	0.848644i	$-0.677419\pi$
$139$	−1.05793	−1.05793	−0.528964	+	0.848644i	$0.677419\pi$
$140$	−0.155740	−0.155740
$141$	0	0
$142$	0	0
$143$	−3.21585	−3.21585
$144$	0.718072	0.718072
$145$	1.21466	1.21466
$146$	0	0
$147$	0.101298	0.101298
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0.0461740	0.0461740
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	1.44584	1.44584
$155$	−1.32551	−1.32551
$156$	0.0444948	0.0444948
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0.305104	0.305104
$161$	−1.37793	−1.37793
$162$	−0.853767	−0.853767
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	−0.115503	−0.115503
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	−0.109227	−0.109227
$169$	2.83792	2.83792
$170$	0	0
$171$	0	0
$172$	−0.237200	−0.237200
$173$	−1.83792	−1.83792	−0.918958	−	0.394356i	$-0.870968\pi$
$173$	−1.83792	−1.83792	−0.918958	+	0.394356i	$0.870968\pi$
$174$	0.156023	0.156023
$175$	0.517516	0.517516
$176$	−1.19096	−1.19096
$177$	0.153722	0.153722
$178$	−1.75254	−1.75254
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−0.154142	−0.154142
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	−1.72552	−1.72552
$183$	0	0
$184$	1.48579	1.48579
$185$	0	0
$186$	−0.170261	−0.170261
$187$	0	0
$188$	0	0
$189$	0.201557	0.201557
$190$	0	0
$191$	−1.51752	−1.51752	−0.758758	−	0.651372i	$-0.774194\pi$
$191$	−1.51752	−1.51752	−0.758758	+	0.651372i	$0.774194\pi$
$192$	0.112684	0.112684
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0.137845	0.137845
$196$	−0.224212	−0.224212
$197$	1.83792	1.83792	0.918958	−	0.394356i	$-0.129032\pi$
$197$	1.83792	1.83792	0.918958	+	0.394356i	$0.129032\pi$
$198$	1.43100	1.43100
$199$	1.22421	1.22421	0.612106	−	0.790776i	$-0.290323\pi$
$199$	1.22421	1.22421	0.612106	+	0.790776i	$0.290323\pi$
$200$	−0.558023	−0.558023
$201$	0	0
$202$	0.266752	0.266752
$203$	1.74869	1.74869
$204$	0	0
$205$	0	0
$206$	−1.54023	−1.54023
$207$	−1.36379	−1.36379
$208$	1.42133	1.42133
$209$	0	0
$210$	−0.0619748	−0.0619748
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.734848	−0.734848
$216$	−0.217333	−0.217333
$217$	−1.90828	−1.90828
$218$	−1.21367	−1.21367
$219$	0	0
$220$	0.255651	0.255651
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0.439245	0.439245
$225$	0.512206	0.512206
$226$	0.441544	0.441544
$227$	0.501305	0.501305	0.250653	−	0.968077i	$-0.419355\pi$
$227$	0.501305	0.501305	0.250653	+	0.968077i	$0.419355\pi$
$228$	0	0
$229$	−1.83792	−1.83792	−0.918958	−	0.394356i	$-0.870968\pi$
$229$	−1.83792	−1.83792	−0.918958	+	0.394356i	$0.870968\pi$
$230$	0.843027	0.843027
$231$	−0.166284	−0.166284
$232$	−1.88557	−1.88557
$233$	−1.90828	−1.90828	−0.954139	−	0.299363i	$-0.903226\pi$
$233$	−1.90828	−1.90828	−0.954139	+	0.299363i	$0.903226\pi$
$234$	−1.70781	−1.70781
$235$	0	0
$236$	−0.340245	−0.340245
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0.0510495	0.0510495
$241$	0.880788	0.880788	0.440394	−	0.897805i	$-0.354839\pi$
$241$	0.880788	0.880788	0.440394	+	0.897805i	$0.354839\pi$
$242$	−1.49259	−1.49259
$243$	0.299748	0.299748
$244$	0	0
$245$	−0.694611	−0.694611
$246$	0	0
$247$	0	0
$248$	2.05764	2.05764
$249$	0	0
$250$	−0.928424	−0.928424
$251$	1.98974	1.98974	0.994869	−	0.101168i	$-0.0322581\pi$
$251$	1.98974	1.98974	0.994869	+	0.101168i	$0.0322581\pi$
$252$	−0.221911	−0.221911
$253$	2.26192	2.26192
$254$	0.0892224	0.0892224
$255$	0	0
$256$	−0.636295	−0.636295
$257$	0.501305	0.501305	0.250653	−	0.968077i	$-0.419355\pi$
$257$	0.501305	0.501305	0.250653	+	0.968077i	$0.419355\pi$
$258$	−0.0943909	−0.0943909
$259$	0	0
$260$	−0.305104	−0.305104
$261$	1.73075	1.73075
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0.179299	0.179299
$265$	0	0
$266$	0	0
$267$	0.201557	0.201557
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−0.123314	−0.123314
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0.198450	0.198450
$274$	0	0
$275$	−0.849517	−0.849517
$276$	−0.0312960	−0.0312960
$277$	1.64153	1.64153	0.820763	−	0.571268i	$-0.193548\pi$
$277$	1.64153	1.64153	0.820763	+	0.571268i	$0.193548\pi$
$278$	0.931811	0.931811
$279$	−1.88870	−1.88870
$280$	0.748979	0.748979
$281$	−1.22421	−1.22421	−0.612106	−	0.790776i	$-0.709677\pi$
$281$	−1.22421	−1.22421	−0.612106	+	0.790776i	$0.709677\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	2.83248	2.83248
$287$	0	0
$288$	0.434737	0.434737
$289$	1.00000	1.00000
$290$	−1.06986	−1.06986
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−0.0892224	−0.0892224
$295$	−1.05408	−1.05408
$296$	0	0
$297$	−0.330862	−0.330862
$298$	0	0
$299$	−2.69945	−2.69945
$300$	0.0117540	0.0117540
$301$	−1.05793	−1.05793
$302$	0	0
$303$	−0.0306788	−0.0306788
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0.368050	0.368050
$309$	0.177140	0.177140
$310$	1.16749	1.16749
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	−0.213982	−0.213982
$313$	1.74869	1.74869	0.874347	−	0.485302i	$-0.161290\pi$
$313$	1.74869	1.74869	0.874347	+	0.485302i	$0.161290\pi$
$314$	0	0
$315$	−0.687483	−0.687483
$316$	0	0
$317$	−1.22421	−1.22421	−0.612106	−	0.790776i	$-0.709677\pi$
$317$	−1.22421	−1.22421	−0.612106	+	0.790776i	$0.709677\pi$
$318$	0	0
$319$	−2.87053	−2.87053
$320$	−0.772684	−0.772684
$321$	0	0
$322$	1.21367	1.21367
$323$	0	0
$324$	−0.217333	−0.217333
$325$	1.01385	1.01385
$326$	0	0
$327$	0.139582	0.139582
$328$	0	0
$329$	0	0
$330$	0.101733	0.101733
$331$	0.302856	0.302856	0.151428	−	0.988468i	$-0.451613\pi$
$331$	0.302856	0.302856	0.151428	+	0.988468i	$0.451613\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0.0734937	0.0734937
$337$	−0.880788	−0.880788	−0.440394	−	0.897805i	$-0.645161\pi$
$337$	−0.880788	−0.880788	−0.440394	+	0.897805i	$0.645161\pi$
$338$	−2.49960	−2.49960
$339$	−0.0507814	−0.0507814
$340$	0	0
$341$	3.13249	3.13249
$342$	0	0
$343$	−1.00000	−1.00000
$344$	1.14073	1.14073
$345$	−0.0969554	−0.0969554
$346$	1.61881	1.61881
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0.0397168	0.0397168
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	−0.455822	−0.455822
$351$	0.394863	0.394863
$352$	−0.721032	−0.721032
$353$	−1.00000	−1.00000
$353$	−1.00000	−1.00000
$354$	−0.135396	−0.135396
$355$	0	0
$356$	−0.446123	−0.446123
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0.741293	0.741293
$361$	1.00000	1.00000
$362$	0	0
$363$	0.171661	0.171661
$364$	−0.439245	−0.439245
$365$	0	0
$366$	0	0
$367$	−0.302856	−0.302856	−0.151428	−	0.988468i	$-0.548387\pi$
$367$	−0.302856	−0.302856	−0.151428	+	0.988468i	$0.548387\pi$
$368$	−0.999714	−0.999714
$369$	0	0
$370$	0	0
$371$	0	0
$372$	−0.0433414	−0.0433414
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0.106777	0.106777
$376$	0	0
$377$	3.42579	3.42579
$378$	−0.177529	−0.177529
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−0.0102614	−0.0102614
$382$	1.33661	1.33661
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−0.0547561	−0.0547561
$385$	1.14022	1.14022
$386$	0	0
$387$	−1.04707	−1.04707
$388$	0	0
$389$	−1.90828	−1.90828	−0.954139	−	0.299363i	$-0.903226\pi$
$389$	−1.90828	−1.90828	−0.954139	+	0.299363i	$0.903226\pi$
$390$	−0.121412	−0.121412
$391$	0	0
$392$	1.07827	1.07827
$393$	0	0
$394$	−1.61881	−1.61881
$395$	0	0
$396$	0.364273	0.364273
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	−1.07827	−1.07827
$399$	0	0
$400$	0.375467	0.375467
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	−3.73843	−3.73843
$404$	0.0679038	0.0679038
$405$	−0.673301	−0.673301
$406$	−1.54023	−1.54023
$407$	0	0
$408$	0	0
$409$	−1.64153	−1.64153	−0.820763	−	0.571268i	$-0.806452\pi$
$409$	−1.64153	−1.64153	−0.820763	+	0.571268i	$0.806452\pi$
$410$	0	0
$411$	0	0
$412$	−0.392078	−0.392078
$413$	−1.51752	−1.51752
$414$	1.20121	1.20121
$415$	0	0
$416$	0.860507	0.860507
$417$	−0.107166	−0.107166
$418$	0	0
$419$	0.880788	0.880788	0.440394	−	0.897805i	$-0.354839\pi$
$419$	0.880788	0.880788	0.440394	+	0.897805i	$0.354839\pi$
$420$	−0.0157762	−0.0157762
$421$	−1.90828	−1.90828	−0.954139	−	0.299363i	$-0.903226\pi$
$421$	−1.90828	−1.90828	−0.954139	+	0.299363i	$0.903226\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−0.325760	−0.325760
$430$	0.647245	0.647245
$431$	1.05793	1.05793	0.528964	−	0.848644i	$-0.322581\pi$
$431$	1.05793	1.05793	0.528964	+	0.848644i	$0.322581\pi$
$432$	0.146233	0.146233
$433$	−1.83792	−1.83792	−0.918958	−	0.394356i	$-0.870968\pi$
$433$	−1.83792	−1.83792	−0.918958	+	0.394356i	$0.870968\pi$
$434$	1.68079	1.68079
$435$	0.123043	0.123043
$436$	−0.308949	−0.308949
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	−1.22947	−1.22947
$441$	−0.989739	−0.989739
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	−1.38209	−1.38209
$446$	0	0
$447$	0	0
$448$	−1.11240	−1.11240
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−0.451145	−0.451145
$451$	0	0
$452$	0.112399	0.112399
$453$	0	0
$454$	−0.441544	−0.441544
$455$	−1.36078	−1.36078
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	1.61881	1.61881
$459$	0	0
$460$	0.214599	0.214599
$461$	−1.05793	−1.05793	−0.528964	−	0.848644i	$-0.677419\pi$
$461$	−1.05793	−1.05793	−0.528964	+	0.848644i	$0.677419\pi$
$462$	0.146461	0.146461
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	1.26871	1.26871
$465$	−0.134272	−0.134272
$466$	1.68079	1.68079
$467$	−1.95906	−1.95906	−0.979530	−	0.201299i	$-0.935484\pi$
$467$	−1.95906	−1.95906	−0.979530	+	0.201299i	$0.935484\pi$
$468$	−0.434737	−0.434737
$469$	0	0
$470$	0	0
$471$	0	0
$472$	1.63629	1.63629
$473$	1.73662	1.73662
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.74869	1.74869	0.874347	−	0.485302i	$-0.161290\pi$
$479$	1.74869	1.74869	0.874347	+	0.485302i	$0.161290\pi$
$480$	0.0309065	0.0309065
$481$	0	0
$482$	−0.775788	−0.775788
$483$	−0.139582	−0.139582
$484$	−0.379952	−0.379952
$485$	0	0
$486$	−0.264014	−0.264014
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0.611805	0.611805
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	1.12852	1.12852
$496$	−1.38449	−1.38449
$497$	0	0
$498$	0	0
$499$	−0.880788	−0.880788	−0.440394	−	0.897805i	$-0.645161\pi$
$499$	−0.880788	−0.880788	−0.440394	+	0.897805i	$0.645161\pi$
$500$	−0.236338	−0.236338
$501$	0	0
$502$	−1.75254	−1.75254
$503$	0.880788	0.880788	0.440394	−	0.897805i	$-0.354839\pi$
$503$	0.880788	0.880788	0.440394	+	0.897805i	$0.354839\pi$
$504$	1.06721	1.06721
$505$	0.210367	0.210367
$506$	−1.99227	−1.99227
$507$	0.287476	0.287476
$508$	0.0227123	0.0227123
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	1.10098	1.10098
$513$	0	0
$514$	−0.441544	−0.441544
$515$	−1.21466	−1.21466
$516$	−0.0240280	−0.0240280
$517$	0	0
$518$	0	0
$519$	−0.186178	−0.186178
$520$	1.46729	1.46729
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−1.52442	−1.52442
$523$	−0.694611	−0.694611	−0.347305	−	0.937752i	$-0.612903\pi$
$523$	−0.694611	−0.694611	−0.347305	+	0.937752i	$0.612903\pi$
$524$	0	0
$525$	0.0524235	0.0524235
$526$	0	0
$527$	0	0
$528$	−0.120642	−0.120642
$529$	0.898702	0.898702
$530$	0	0
$531$	−1.50194	−1.50194
$532$	0	0
$533$	0	0
$534$	−0.177529	−0.177529
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.64153	1.64153
$540$	−0.0313905	−0.0313905
$541$	1.05793	1.05793	0.528964	−	0.848644i	$-0.322581\pi$
$541$	1.05793	1.05793	0.528964	+	0.848644i	$0.322581\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−0.957127	−0.957127
$546$	−0.174792	−0.174792
$547$	−0.501305	−0.501305	−0.250653	−	0.968077i	$-0.580645\pi$
$547$	−0.501305	−0.501305	−0.250653	+	0.968077i	$0.580645\pi$
$548$	0	0
$549$	0	0
$550$	0.748244	0.748244
$551$	0	0
$552$	0.150508	0.150508
$553$	0	0
$554$	−1.44584	−1.44584
$555$	0	0
$556$	0.237200	0.237200
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	1.66354	1.66354
$559$	−2.07254	−2.07254
$560$	−0.503952	−0.503952
$561$	0	0
$562$	1.07827	1.07827
$563$	1.90828	1.90828	0.954139	−	0.299363i	$-0.0967742\pi$
$563$	1.90828	1.90828	0.954139	+	0.299363i	$0.0967742\pi$
$564$	0	0
$565$	0.348212	0.348212
$566$	0	0
$567$	−0.969321	−0.969321
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	1.37793	1.37793	0.688967	−	0.724793i	$-0.258065\pi$
$571$	1.37793	1.37793	0.688967	+	0.724793i	$0.258065\pi$
$572$	0.721032	0.721032
$573$	−0.153722	−0.153722
$574$	0	0
$575$	−0.713103	−0.713103
$576$	−1.10098	−1.10098
$577$	−1.37793	−1.37793	−0.688967	−	0.724793i	$-0.741935\pi$
$577$	−1.37793	−1.37793	−0.688967	+	0.724793i	$0.741935\pi$
$578$	−0.880788	−0.880788
$579$	0	0
$580$	−0.272341	−0.272341
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−1.34682	−1.34682
$586$	0	0
$587$	1.22421	1.22421	0.612106	−	0.790776i	$-0.290323\pi$
$587$	1.22421	1.22421	0.612106	+	0.790776i	$0.290323\pi$
$588$	−0.0227123	−0.0227123
$589$	0	0
$590$	0.928424	0.928424
$591$	0.186178	0.186178
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0.291419	0.291419
$595$	0	0
$596$	0	0
$597$	0.124011	0.124011
$598$	2.37765	2.37765
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−0.0565268	−0.0565268
$601$	0.501305	0.501305	0.250653	−	0.968077i	$-0.419355\pi$
$601$	0.501305	0.501305	0.250653	+	0.968077i	$0.419355\pi$
$602$	0.931811	0.931811
$603$	0	0
$604$	0	0
$605$	−1.17709	−1.17709
$606$	0.0270215	0.0270215
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0.177140	0.177140
$610$	0	0
$611$	0	0
$612$	0	0
$613$	1.83792	1.83792	0.918958	−	0.394356i	$-0.129032\pi$
$613$	1.83792	1.83792	0.918958	+	0.394356i	$0.129032\pi$
$614$	0	0
$615$	0	0
$616$	−1.77001	−1.77001
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	−0.156023	−0.156023
$619$	1.22421	1.22421	0.612106	−	0.790776i	$-0.290323\pi$
$619$	1.22421	1.22421	0.612106	+	0.790776i	$0.290323\pi$
$620$	0.297195	0.297195
$621$	−0.277733	−0.277733
$622$	0	0
$623$	−1.98974	−1.98974
$624$	0.143978	0.143978
$625$	−0.214661	−0.214661
$626$	−1.54023	−1.54023
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0.605527	0.605527
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	1.07827	1.07827
$635$	0.0703629	0.0703629
$636$	0	0
$637$	−1.95906	−1.95906
$638$	2.52833	2.52833
$639$	0	0
$640$	0.375467	0.375467
$641$	0.694611	0.694611	0.347305	−	0.937752i	$-0.387097\pi$
$641$	0.694611	0.694611	0.347305	+	0.937752i	$0.387097\pi$
$642$	0	0
$643$	1.98974	1.98974	0.994869	−	0.101168i	$-0.0322581\pi$
$643$	1.98974	1.98974	0.994869	+	0.101168i	$0.0322581\pi$
$644$	0.308949	0.308949
$645$	−0.0744389	−0.0744389
$646$	0	0
$647$	1.98974	1.98974	0.994869	−	0.101168i	$-0.0322581\pi$
$647$	1.98974	1.98974	0.994869	+	0.101168i	$0.0322581\pi$
$648$	1.04519	1.04519
$649$	2.49104	2.49104
$650$	−0.892983	−0.892983
$651$	−0.193305	−0.193305
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	−0.122943	−0.122943
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.83792	1.83792	0.918958	−	0.394356i	$-0.129032\pi$
$659$	1.83792	1.83792	0.918958	+	0.394356i	$0.129032\pi$
$660$	0.0258971	0.0258971
$661$	−0.302856	−0.302856	−0.151428	−	0.988468i	$-0.548387\pi$
$661$	−0.302856	−0.302856	−0.151428	+	0.988468i	$0.548387\pi$
$662$	−0.266752	−0.266752
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.40958	−2.40958
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0.0444948	0.0444948
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0.775788	0.775788
$675$	0.104309	0.104309
$676$	−0.636295	−0.636295
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0.0447276	0.0447276
$679$	0	0
$680$	0	0
$681$	0.0507814	0.0507814
$682$	−2.75906	−2.75906
$683$	−1.90828	−1.90828	−0.954139	−	0.299363i	$-0.903226\pi$
$683$	−1.90828	−1.90828	−0.954139	+	0.299363i	$0.903226\pi$
$684$	0	0
$685$	0	0
$686$	0.880788	0.880788
$687$	−0.186178	−0.186178
$688$	−0.767545	−0.767545
$689$	0	0
$690$	0.0853972	0.0853972
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0.412083	0.412083
$693$	1.62468	1.62468
$694$	0	0
$695$	0.734848	0.734848
$696$	−0.191005	−0.191005
$697$	0	0
$698$	0	0
$699$	−0.193305	−0.193305
$700$	−0.116033	−0.116033
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	−0.347790	−0.347790
$703$	0	0
$704$	1.82603	1.82603
$705$	0	0
$706$	0.880788	0.880788
$707$	0.302856	0.302856
$708$	−0.0344663	−0.0344663
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	2.14548	2.14548
$713$	2.62948	2.62948
$714$	0	0
$715$	2.23376	2.23376
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.64153	−1.64153	−0.820763	−	0.571268i	$-0.806452\pi$
$719$	−1.64153	−1.64153	−0.820763	+	0.571268i	$0.806452\pi$
$720$	−0.498781	−0.498781
$721$	−1.74869	−1.74869
$722$	−0.880788	−0.880788
$723$	0.0892224	0.0892224
$724$	0	0
$725$	0.904977	0.904977
$726$	−0.151197	−0.151197
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	2.11240	2.11240
$729$	−0.938957	−0.938957
$730$	0	0
$731$	0	0
$732$	0	0
$733$	1.90828	1.90828	0.954139	−	0.299363i	$-0.0967742\pi$
$733$	1.90828	1.90828	0.954139	+	0.299363i	$0.0967742\pi$
$734$	0.266752	0.266752
$735$	−0.0703629	−0.0703629
$736$	−0.605250	−0.605250
$737$	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			$\pi$
$744$	0.208436	0.208436
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	−0.0940478	−0.0940478
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0.201557	0.201557
$754$	−3.01740	−3.01740
$755$	0	0
$756$	−0.0451915	−0.0451915
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0.229128	0.229128
$760$	0	0
$761$	−1.95906	−1.95906	−0.979530	−	0.201299i	$-0.935484\pi$
$761$	−1.95906	−1.95906	−0.979530	+	0.201299i	$0.935484\pi$
$762$	0.00903808	0.00903808
$763$	−1.37793	−1.37793
$764$	0.340245	0.340245
$765$	0	0
$766$	0	0
$767$	−2.97291	−2.97291
$768$	−0.0644556	−0.0644556
$769$	1.90828	1.90828	0.954139	−	0.299363i	$-0.0967742\pi$
$769$	1.90828	1.90828	0.954139	+	0.299363i	$0.0967742\pi$
$770$	−1.00429	−1.00429
$771$	0.0507814	0.0507814
$772$	0	0
$773$	−1.05793	−1.05793	−0.528964	−	0.848644i	$-0.677419\pi$
$773$	−1.05793	−1.05793	−0.528964	+	0.848644i	$0.677419\pi$
$774$	0.922249	0.922249
$775$	−0.987565	−0.987565
$776$	0	0
$777$	0	0
$778$	1.68079	1.68079
$779$	0	0
$780$	−0.0309065	−0.0309065
$781$	0	0
$782$	0	0
$783$	0.352462	0.352462
$784$	−0.725517	−0.725517
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−0.412083	−0.412083
$789$	0	0
$790$	0	0
$791$	0.501305	0.501305
$792$	−1.75185	−1.75185
$793$	0	0
$794$	0	0
$795$	0	0
$796$	−0.274483	−0.274483
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	0.227316	0.227316
$801$	−1.96932	−1.96932
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0.957127	0.957127
$806$	3.29277	3.29277
$807$	0	0
$808$	−0.326561	−0.326561
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0.593035	0.593035
$811$	1.98974	1.98974	0.994869	−	0.101168i	$-0.0322581\pi$
$811$	1.98974	1.98974	0.994869	+	0.101168i	$0.0322581\pi$
$812$	−0.392078	−0.392078
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	1.44584	1.44584
$819$	−1.93896	−1.93896
$820$	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			$\pi$
$824$	1.88557	1.88557
$825$	−0.0860546	−0.0860546
$826$	1.33661	1.33661
$827$	1.95906	1.95906	0.979530	−	0.201299i	$-0.0645161\pi$
$827$	1.95906	1.95906	0.979530	+	0.201299i	$0.0645161\pi$
$828$	0.305779	0.305779
$829$	0.101298	0.101298	0.0506492	−	0.998717i	$-0.483871\pi$
$829$	0.101298	0.101298	0.0506492	+	0.998717i	$0.483871\pi$
$830$	0	0
$831$	0.166284	0.166284
$832$	−2.17926	−2.17926
$833$	0	0
$834$	0.0943909	0.0943909
$835$	0	0
$836$	0	0
$837$	−0.384627	−0.384627
$838$	−0.775788	−0.775788
$839$	−0.302856	−0.302856	−0.151428	−	0.988468i	$-0.548387\pi$
$839$	−0.302856	−0.302856	−0.151428	+	0.988468i	$0.548387\pi$
$840$	0.0758703	0.0758703
$841$	2.05793	2.05793
$842$	1.68079	1.68079
$843$	−0.124011	−0.124011
$844$	0	0
$845$	−1.97125	−1.97125
$846$	0	0
$847$	−1.69461	−1.69461
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−1.37793	−1.37793	−0.688967	−	0.724793i	$-0.741935\pi$
$853$	−1.37793	−1.37793	−0.688967	+	0.724793i	$0.741935\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0.101298	0.101298	0.0506492	−	0.998717i	$-0.483871\pi$
$857$	0.101298	0.101298	0.0506492	+	0.998717i	$0.483871\pi$
$858$	0.286926	0.286926
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0.164762	0.164762
$861$	0	0
$862$	−0.931811	−0.931811
$863$	1.05793	1.05793	0.528964	−	0.848644i	$-0.322581\pi$
$863$	1.05793	1.05793	0.528964	+	0.848644i	$0.322581\pi$
$864$	0.0885329	0.0885329
$865$	1.27664	1.27664
$866$	1.61881	1.61881
$867$	0.101298	0.101298
$868$	0.427859	0.427859
$869$	0	0
$870$	−0.108375	−0.108375
$871$	0	0
$872$	1.48579	1.48579
$873$	0	0
$874$	0	0
$875$	−1.05408	−1.05408
$876$	0	0
$877$	0.694611	0.694611	0.347305	−	0.937752i	$-0.387097\pi$
$877$	0.694611	0.694611	0.347305	+	0.937752i	$0.387097\pi$
$878$	0	0
$879$	0	0
$880$	0.827250	0.827250
$881$	−0.302856	−0.302856	−0.151428	−	0.988468i	$-0.548387\pi$
$881$	−0.302856	−0.302856	−0.151428	+	0.988468i	$0.548387\pi$
$882$	0.871750	0.871750
$883$	−1.98974	−1.98974	−0.994869	−	0.101168i	$-0.967742\pi$
$883$	−1.98974	−1.98974	−0.994869	+	0.101168i	$0.967742\pi$
$884$	0	0
$885$	−0.106777	−0.106777
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0.101298	0.101298
$890$	1.21733	1.21733
$891$	1.59117	1.59117
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0.540543	0.540543
$897$	−0.273450	−0.273450
$898$	0	0
$899$	−3.33699	−3.33699
$900$	−0.114843	−0.114843
$901$	0	0
$902$	0	0
$903$	−0.107166	−0.107166
$904$	−0.540543	−0.540543
$905$	0	0
$906$	0	0
$907$	1.83792	1.83792	0.918958	−	0.394356i	$-0.129032\pi$
$907$	1.83792	1.83792	0.918958	+	0.394356i	$0.129032\pi$
$908$	−0.112399	−0.112399
$909$	0.299748	0.299748
$910$	1.19856	1.19856
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0.412083	0.412083
$917$	0	0
$918$	0	0
$919$	1.64153	1.64153	0.820763	−	0.571268i	$-0.193548\pi$
$919$	1.64153	1.64153	0.820763	+	0.571268i	$0.193548\pi$
$920$	−1.03204	−1.03204
$921$	0	0
$922$	0.931811	0.931811
$923$	0	0
$924$	0.0372828	0.0372828
$925$	0	0
$926$	0	0
$927$	−1.73075	−1.73075
$928$	0.768104	0.768104
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0.118265	0.118265
$931$	0	0
$932$	0.427859	0.427859
$933$	0	0
$934$	1.72552	1.72552
$935$	0	0
$936$	2.09072	2.09072
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0.177140	0.177140
$940$	0	0
$941$	0.501305	0.501305	0.250653	−	0.968077i	$-0.419355\pi$
$941$	0.501305	0.501305	0.250653	+	0.968077i	$0.419355\pi$
$942$	0	0
$943$	0	0
$944$	−1.10098	−1.10098
$945$	−0.140004	−0.140004
$946$	−1.52959	−1.52959
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−0.124011	−0.124011
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	1.05408	1.05408
$956$	0	0
$957$	−0.290780	−0.290780
$958$	−1.54023	−1.54023
$959$	0	0
$960$	−0.0782716	−0.0782716
$961$	2.64153	2.64153
$962$	0	0
$963$	0	0
$964$	−0.197483	−0.197483
$965$	0	0
$966$	0.122943	0.122943
$967$	−1.51752	−1.51752	−0.758758	−	0.651372i	$-0.774194\pi$
$967$	−1.51752	−1.51752	−0.758758	+	0.651372i	$0.774194\pi$
$968$	1.82725	1.82725
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−0.0672071	−0.0672071
$973$	1.05793	1.05793
$974$	0	0
$975$	0.102701	0.102701
$976$	0	0
$977$	1.95906	1.95906	0.979530	−	0.201299i	$-0.0645161\pi$
$977$	1.95906	1.95906	0.979530	+	0.201299i	$0.0645161\pi$
$978$	0	0
$979$	3.26621	3.26621
$980$	0.155740	0.155740
$981$	−1.36379	−1.36379
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	−1.27664	−1.27664
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.45775	1.45775
$990$	−0.993989	−0.993989
$991$	1.95906	1.95906	0.979530	−	0.201299i	$-0.0645161\pi$
$991$	1.95906	1.95906	0.979530	+	0.201299i	$0.0645161\pi$
$992$	−0.838201	−0.838201
$993$	0.0306788	0.0306788
$994$	0	0
$995$	−0.850350	−0.850350
$996$	0	0
$997$	0.880788	0.880788	0.440394	−	0.897805i	$-0.354839\pi$
$997$	0.880788	0.880788	0.440394	+	0.897805i	$0.354839\pi$
$998$	0.775788	0.775788
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2471.1.d.b.2470.6	yes	15
7.6	odd	2		2471.1.d.a.2470.6	✓	15
353.352	even	2		2471.1.d.a.2470.6	✓	15
2471.2470	odd	2	CM	2471.1.d.b.2470.6	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2471.1.d.a.2470.6	✓	15	7.6	odd	2
2471.1.d.a.2470.6	✓	15	353.352	even	2
2471.1.d.b.2470.6	yes	15	1.1	even	1	trivial
2471.1.d.b.2470.6	yes	15	2471.2470	odd	2	CM

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)

\(f(q)\)	\(=\)	\(q-0.880788 q^{2} +0.101298 q^{3} -0.224212 q^{4} -0.694611 q^{5} -0.0892224 q^{6} -1.00000 q^{7} +1.07827 q^{8} -0.989739 q^{9} +O(q^{10})\)	\(q-0.880788 q^{2} +0.101298 q^{3} -0.224212 q^{4} -0.694611 q^{5} -0.0892224 q^{6} -1.00000 q^{7} +1.07827 q^{8} -0.989739 q^{9} +0.611805 q^{10} +1.64153 q^{11} -0.0227123 q^{12} -1.95906 q^{13} +0.880788 q^{14} -0.0703629 q^{15} -0.725517 q^{16} +0.871750 q^{18} +0.155740 q^{20} -0.101298 q^{21} -1.44584 q^{22} +1.37793 q^{23} +0.109227 q^{24} -0.517516 q^{25} +1.72552 q^{26} -0.201557 q^{27} +0.224212 q^{28} -1.74869 q^{29} +0.0619748 q^{30} +1.90828 q^{31} -0.439245 q^{32} +0.166284 q^{33} +0.694611 q^{35} +0.221911 q^{36} -0.198450 q^{39} -0.748979 q^{40} +0.0892224 q^{42} +1.05793 q^{43} -0.368050 q^{44} +0.687483 q^{45} -1.21367 q^{46} -0.0734937 q^{48} +1.00000 q^{49} +0.455822 q^{50} +0.439245 q^{52} +0.177529 q^{54} -1.14022 q^{55} -1.07827 q^{56} +1.54023 q^{58} +1.51752 q^{59} +0.0157762 q^{60} -1.68079 q^{62} +0.989739 q^{63} +1.11240 q^{64} +1.36078 q^{65} -0.146461 q^{66} +0.139582 q^{69} -0.611805 q^{70} -1.06721 q^{72} -0.0524235 q^{75} -1.64153 q^{77} +0.174792 q^{78} +0.503952 q^{80} +0.969321 q^{81} +0.0227123 q^{84} -0.931811 q^{86} -0.177140 q^{87} +1.77001 q^{88} +1.98974 q^{89} -0.605527 q^{90} +1.95906 q^{91} -0.308949 q^{92} +0.193305 q^{93} -0.0444948 q^{96} -0.880788 q^{98} -1.62468 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{2} + q^{3} + 14 q^{4} + q^{5} + 2 q^{6} - 15 q^{7} - 2 q^{8} + 14 q^{9}+O(q^{10}) \) 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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