LMFDB - Embedded newform 2031.1.d.b.2030.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2031,1,Mod(2030,2031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2031, 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("2031.2030");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2031 = 3 \cdot 677 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2031.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:	$1.01360104066$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\Q(\zeta_{38})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 8x^{7} + 7x^{6} + 21x^{5} - 15x^{4} - 20x^{3} + 10x^{2} + 5x - 1 $ x^9 - x^8 - 8x^7 + 7x^6 + 21x^5 - 15x^4 - 20x^3 + 10x^2 + 5*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{19}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{19} - \cdots)$

Embedding invariants

Embedding label			2030.4
Root			$1.75895$ of defining polynomial
Character	$\chi$	$=$	2031.2030

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.490971 q^{2} -1.00000 q^{3} -0.758948 q^{4} +1.97272 q^{5} +0.490971 q^{6} +0.863592 q^{8} +1.00000 q^{9} -0.968550 q^{10} -1.09390 q^{11} +0.758948 q^{12} +1.57828 q^{13} -1.97272 q^{15} +0.334949 q^{16} +0.165159 q^{17} -0.490971 q^{18} -1.49719 q^{20} +0.537071 q^{22} +0.803391 q^{23} -0.863592 q^{24} +2.89163 q^{25} -0.774890 q^{26} -1.00000 q^{27} -1.89163 q^{29} +0.968550 q^{30} -1.02804 q^{32} +1.09390 q^{33} -0.0810881 q^{34} -0.758948 q^{36} -1.35456 q^{37} -1.57828 q^{39} +1.70363 q^{40} +0.830210 q^{44} +1.97272 q^{45} -0.394442 q^{46} +1.75895 q^{47} -0.334949 q^{48} +1.00000 q^{49} -1.41971 q^{50} -0.165159 q^{51} -1.19783 q^{52} -1.57828 q^{53} +0.490971 q^{54} -2.15795 q^{55} +0.928738 q^{58} +1.49719 q^{60} +0.169790 q^{64} +3.11351 q^{65} -0.537071 q^{66} -0.125347 q^{68} -0.803391 q^{69} +0.863592 q^{72} +0.665051 q^{74} -2.89163 q^{75} +0.774890 q^{78} +1.09390 q^{79} +0.660761 q^{80} +1.00000 q^{81} +1.35456 q^{83} +0.325812 q^{85} +1.89163 q^{87} -0.944680 q^{88} +0.803391 q^{89} -0.968550 q^{90} -0.609731 q^{92} -0.863592 q^{94} +1.02804 q^{96} -0.165159 q^{97} -0.490971 q^{98} -1.09390 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{2} - 9 q^{3} + 8 q^{4} + q^{5} - q^{6} + 2 q^{8} + 9 q^{9} - 2 q^{10} + q^{11} - 8 q^{12} - q^{13} - q^{15} + 7 q^{16} + q^{17} + q^{18} + 3 q^{20} - 2 q^{22} + q^{23} - 2 q^{24} + 8 q^{25}+ \cdots + q^{99}+O(q^{100}) $ 9 * q + q^2 - 9 * q^3 + 8 * q^4 + q^5 - q^6 + 2 * q^8 + 9 * q^9 - 2 * q^10 + q^11 - 8 * q^12 - q^13 - q^15 + 7 * q^16 + q^17 + q^18 + 3 * q^20 - 2 * q^22 + q^23 - 2 * q^24 + 8 * q^25 + 2 * q^26 - 9 * q^27 + q^29 + 2 * q^30 + 3 * q^32 - q^33 - 2 * q^34 + 8 * q^36 - q^37 + q^39 - 4 * q^40 + 3 * q^44 + q^45 - 2 * q^46 + q^47 - 7 * q^48 + 9 * q^49 + 3 * q^50 - q^51 - 3 * q^52 + q^53 - q^54 - 2 * q^55 - 2 * q^58 - 3 * q^60 + 6 * q^64 + 2 * q^65 + 2 * q^66 + 3 * q^68 - q^69 + 2 * q^72 + 2 * q^74 - 8 * q^75 - 2 * q^78 - q^79 + 5 * q^80 + 9 * q^81 + q^83 - 2 * q^85 - q^87 - 4 * q^88 + q^89 - 2 * q^90 + 3 * q^92 - 2 * q^94 - 3 * q^96 - q^97 + q^98 + q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2031\mathbb{Z}\right)^\times$.

$n$	$679$	$1355$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−0.490971	−0.490971	−0.245485	−	0.969400i	$-0.578947\pi$
$2$	−0.490971	−0.490971	−0.245485	+	0.969400i	$0.578947\pi$
$3$	−1.00000	−1.00000
$3$	−1.00000	−1.00000
$4$	−0.758948	−0.758948
$5$	1.97272	1.97272	0.986361	−	0.164595i	$-0.0526316\pi$
$5$	1.97272	1.97272	0.986361	+	0.164595i	$0.0526316\pi$
$6$	0.490971	0.490971
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0.863592	0.863592
$9$	1.00000	1.00000
$10$	−0.968550	−0.968550
$11$	−1.09390	−1.09390	−0.546948	−	0.837166i	$-0.684211\pi$
$11$	−1.09390	−1.09390	−0.546948	+	0.837166i	$0.684211\pi$
$12$	0.758948	0.758948
$13$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$13$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$14$	0	0
$15$	−1.97272	−1.97272
$16$	0.334949	0.334949
$17$	0.165159	0.165159	0.0825793	−	0.996584i	$-0.473684\pi$
$17$	0.165159	0.165159	0.0825793	+	0.996584i	$0.473684\pi$
$18$	−0.490971	−0.490971
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−1.49719	−1.49719
$21$	0	0
$22$	0.537071	0.537071
$23$	0.803391	0.803391	0.401695	−	0.915773i	$-0.368421\pi$
$23$	0.803391	0.803391	0.401695	+	0.915773i	$0.368421\pi$
$24$	−0.863592	−0.863592
$25$	2.89163	2.89163
$26$	−0.774890	−0.774890
$27$	−1.00000	−1.00000
$28$	0	0
$29$	−1.89163	−1.89163	−0.945817	−	0.324699i	$-0.894737\pi$
$29$	−1.89163	−1.89163	−0.945817	+	0.324699i	$0.894737\pi$
$30$	0.968550	0.968550
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−1.02804	−1.02804
$33$	1.09390	1.09390
$34$	−0.0810881	−0.0810881
$35$	0	0
$36$	−0.758948	−0.758948
$37$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$37$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$38$	0	0
$39$	−1.57828	−1.57828
$40$	1.70363	1.70363
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0.830210	0.830210
$45$	1.97272	1.97272
$46$	−0.394442	−0.394442
$47$	1.75895	1.75895	0.879474	−	0.475947i	$-0.157895\pi$
$47$	1.75895	1.75895	0.879474	+	0.475947i	$0.157895\pi$
$48$	−0.334949	−0.334949
$49$	1.00000	1.00000
$50$	−1.41971	−1.41971
$51$	−0.165159	−0.165159
$52$	−1.19783	−1.19783
$53$	−1.57828	−1.57828	−0.789141	−	0.614213i	$-0.789474\pi$
$53$	−1.57828	−1.57828	−0.789141	+	0.614213i	$0.789474\pi$
$54$	0.490971	0.490971
$55$	−2.15795	−2.15795
$56$	0	0
$57$	0	0
$58$	0.928738	0.928738
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	1.49719	1.49719
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	0.169790	0.169790
$65$	3.11351	3.11351
$66$	−0.537071	−0.537071
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	−0.125347	−0.125347
$69$	−0.803391	−0.803391
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0.863592	0.863592
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0.665051	0.665051
$75$	−2.89163	−2.89163
$76$	0	0
$77$	0	0
$78$	0.774890	0.774890
$79$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$79$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$80$	0.660761	0.660761
$81$	1.00000	1.00000
$82$	0	0
$83$	1.35456	1.35456	0.677282	−	0.735724i	$-0.263158\pi$
$83$	1.35456	1.35456	0.677282	+	0.735724i	$0.263158\pi$
$84$	0	0
$85$	0.325812	0.325812
$86$	0	0
$87$	1.89163	1.89163
$88$	−0.944680	−0.944680
$89$	0.803391	0.803391	0.401695	−	0.915773i	$-0.368421\pi$
$89$	0.803391	0.803391	0.401695	+	0.915773i	$0.368421\pi$
$90$	−0.968550	−0.968550
$91$	0	0
$92$	−0.609731	−0.609731
$93$	0	0
$94$	−0.863592	−0.863592
$95$	0	0
$96$	1.02804	1.02804
$97$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$97$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$98$	−0.490971	−0.490971
$99$	−1.09390	−1.09390
$100$	−2.19460	−2.19460
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0.0810881	0.0810881
$103$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$103$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$104$	1.36299	1.36299
$105$	0	0
$106$	0.774890	0.774890
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0.758948	0.758948
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	1.05949	1.05949
$111$	1.35456	1.35456
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	1.58487	1.58487
$116$	1.43565	1.43565
$117$	1.57828	1.57828
$118$	0	0
$119$	0	0
$120$	−1.70363	−1.70363
$121$	0.196609	0.196609
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.73167	3.73167
$126$	0	0
$127$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$127$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$128$	0.944680	0.944680
$129$	0	0
$130$	−1.52864	−1.52864
$131$	0.165159	0.165159	0.0825793	−	0.996584i	$-0.473684\pi$
$131$	0.165159	0.165159	0.0825793	+	0.996584i	$0.473684\pi$
$132$	−0.830210	−0.830210
$133$	0	0
$134$	0	0
$135$	−1.97272	−1.97272
$136$	0.142630	0.142630
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0.394442	0.394442
$139$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$139$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$140$	0	0
$141$	−1.75895	−1.75895
$142$	0	0
$143$	−1.72648	−1.72648
$144$	0.334949	0.334949
$145$	−3.73167	−3.73167
$146$	0	0
$147$	−1.00000	−1.00000
$148$	1.02804	1.02804
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	1.41971	1.41971
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0.165159	0.165159
$154$	0	0
$155$	0	0
$156$	1.19783	1.19783
$157$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$157$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$158$	−0.537071	−0.537071
$159$	1.57828	1.57828
$160$	−2.02804	−2.02804
$161$	0	0
$162$	−0.490971	−0.490971
$163$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$163$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$164$	0	0
$165$	2.15795	2.15795
$166$	−0.665051	−0.665051
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	1.49097	1.49097
$170$	−0.159964	−0.159964
$171$	0	0
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	−0.928738	−0.928738
$175$	0	0
$176$	−0.366399	−0.366399
$177$	0	0
$178$	−0.394442	−0.394442
$179$	1.75895	1.75895	0.879474	−	0.475947i	$-0.157895\pi$
$179$	1.75895	1.75895	0.879474	+	0.475947i	$0.157895\pi$
$180$	−1.49719	−1.49719
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0.693802	0.693802
$185$	−2.67218	−2.67218
$186$	0	0
$187$	−0.180666	−0.180666
$188$	−1.33495	−1.33495
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−0.169790	−0.169790
$193$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$193$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$194$	0.0810881	0.0810881
$195$	−3.11351	−3.11351
$196$	−0.758948	−0.758948
$197$	0.803391	0.803391	0.401695	−	0.915773i	$-0.368421\pi$
$197$	0.803391	0.803391	0.401695	+	0.915773i	$0.368421\pi$
$198$	0.537071	0.537071
$199$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$199$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$200$	2.49719	2.49719
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0.125347	0.125347
$205$	0	0
$206$	−0.241052	−0.241052
$207$	0.803391	0.803391
$208$	0.528643	0.528643
$209$	0	0
$210$	0	0
$211$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$211$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$212$	1.19783	1.19783
$213$	0	0
$214$	0	0
$215$	0	0
$216$	−0.863592	−0.863592
$217$	0	0
$218$	0	0
$219$	0	0
$220$	1.63777	1.63777
$221$	0.260667	0.260667
$222$	−0.665051	−0.665051
$223$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$223$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$224$	0	0
$225$	2.89163	2.89163
$226$	0	0
$227$	−1.57828	−1.57828	−0.789141	−	0.614213i	$-0.789474\pi$
$227$	−1.57828	−1.57828	−0.789141	+	0.614213i	$0.789474\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	−0.778124	−0.778124
$231$	0	0
$232$	−1.63360	−1.63360
$233$	−1.57828	−1.57828	−0.789141	−	0.614213i	$-0.789474\pi$
$233$	−1.57828	−1.57828	−0.789141	+	0.614213i	$0.789474\pi$
$234$	−0.774890	−0.774890
$235$	3.46992	3.46992
$236$	0	0
$237$	−1.09390	−1.09390
$238$	0	0
$239$	−1.89163	−1.89163	−0.945817	−	0.324699i	$-0.894737\pi$
$239$	−1.89163	−1.89163	−0.945817	+	0.324699i	$0.894737\pi$
$240$	−0.660761	−0.660761
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−0.0965294	−0.0965294
$243$	−1.00000	−1.00000
$244$	0	0
$245$	1.97272	1.97272
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−1.35456	−1.35456
$250$	−1.83214	−1.83214
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	−0.878826	−0.878826
$254$	0.863592	0.863592
$255$	−0.325812	−0.325812
$256$	−0.633601	−0.633601
$257$	−0.490971	−0.490971	−0.245485	−	0.969400i	$-0.578947\pi$
$257$	−0.490971	−0.490971	−0.245485	+	0.969400i	$0.578947\pi$
$258$	0	0
$259$	0	0
$260$	−2.36299	−2.36299
$261$	−1.89163	−1.89163
$262$	−0.0810881	−0.0810881
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0.944680	0.944680
$265$	−3.11351	−3.11351
$266$	0	0
$267$	−0.803391	−0.803391
$268$	0	0
$269$	−0.490971	−0.490971	−0.245485	−	0.969400i	$-0.578947\pi$
$269$	−0.490971	−0.490971	−0.245485	+	0.969400i	$0.578947\pi$
$270$	0.968550	0.968550
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0.0553197	0.0553197
$273$	0	0
$274$	0	0
$275$	−3.16315	−3.16315
$276$	0.609731	0.609731
$277$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$277$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$278$	−0.928738	−0.928738
$279$	0	0
$280$	0	0
$281$	−1.89163	−1.89163	−0.945817	−	0.324699i	$-0.894737\pi$
$281$	−1.89163	−1.89163	−0.945817	+	0.324699i	$0.894737\pi$
$282$	0.863592	0.863592
$283$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$283$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$284$	0	0
$285$	0	0
$286$	0.847650	0.847650
$287$	0	0
$288$	−1.02804	−1.02804
$289$	−0.972723	−0.972723
$290$	1.83214	1.83214
$291$	0.165159	0.165159
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0.490971	0.490971
$295$	0	0
$296$	−1.16979	−1.16979
$297$	1.09390	1.09390
$298$	0	0
$299$	1.26798	1.26798
$300$	2.19460	2.19460
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	−0.0810881	−0.0810881
$307$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$307$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$308$	0	0
$309$	−0.490971	−0.490971
$310$	0	0
$311$	1.35456	1.35456	0.677282	−	0.735724i	$-0.263158\pi$
$311$	1.35456	1.35456	0.677282	+	0.735724i	$0.263158\pi$
$312$	−1.36299	−1.36299
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0.968550	0.968550
$315$	0	0
$316$	−0.830210	−0.830210
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	−0.774890	−0.774890
$319$	2.06925	2.06925
$320$	0.334949	0.334949
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−0.758948	−0.758948
$325$	4.56381	4.56381
$326$	0.394442	0.394442
$327$	0	0
$328$	0	0
$329$	0	0
$330$	−1.05949	−1.05949
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	−1.02804	−1.02804
$333$	−1.35456	−1.35456
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$337$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$338$	−0.732023	−0.732023
$339$	0	0
$340$	−0.247274	−0.247274
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−1.58487	−1.58487
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	−1.43565	−1.43565
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	−1.57828	−1.57828
$352$	1.12457	1.12457
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	−0.609731	−0.609731
$357$	0	0
$358$	−0.863592	−0.863592
$359$	−0.490971	−0.490971	−0.245485	−	0.969400i	$-0.578947\pi$
$359$	−0.490971	−0.490971	−0.245485	+	0.969400i	$0.578947\pi$
$360$	1.70363	1.70363
$361$	1.00000	1.00000
$362$	0	0
$363$	−0.196609	−0.196609
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0.269095	0.269095
$369$	0	0
$370$	1.31196	1.31196
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0.0887020	0.0887020
$375$	−3.73167	−3.73167
$376$	1.51901	1.51901
$377$	−2.98553	−2.98553
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	1.75895	1.75895
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−0.944680	−0.944680
$385$	0	0
$386$	−0.241052	−0.241052
$387$	0	0
$388$	0.125347	0.125347
$389$	1.75895	1.75895	0.879474	−	0.475947i	$-0.157895\pi$
$389$	1.75895	1.75895	0.879474	+	0.475947i	$0.157895\pi$
$390$	1.52864	1.52864
$391$	0.132687	0.132687
$392$	0.863592	0.863592
$393$	−0.165159	−0.165159
$394$	−0.394442	−0.394442
$395$	2.15795	2.15795
$396$	0.830210	0.830210
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0.0810881	0.0810881
$399$	0	0
$400$	0.968550	0.968550
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.97272	1.97272
$406$	0	0
$407$	1.48175	1.48175
$408$	−0.142630	−0.142630
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	−0.372621	−0.372621
$413$	0	0
$414$	−0.394442	−0.394442
$415$	2.67218	2.67218
$416$	−1.62254	−1.62254
$417$	−1.89163	−1.89163
$418$	0	0
$419$	1.97272	1.97272	0.986361	−	0.164595i	$-0.0526316\pi$
$419$	1.97272	1.97272	0.986361	+	0.164595i	$0.0526316\pi$
$420$	0	0
$421$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$421$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$422$	0.665051	0.665051
$423$	1.75895	1.75895
$424$	−1.36299	−1.36299
$425$	0.477579	0.477579
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1.72648	1.72648
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−0.334949	−0.334949
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	3.73167	3.73167
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	−1.86359	−1.86359
$441$	1.00000	1.00000
$442$	−0.127980	−0.127980
$443$	−1.89163	−1.89163	−0.945817	−	0.324699i	$-0.894737\pi$
$443$	−1.89163	−1.89163	−0.945817	+	0.324699i	$0.894737\pi$
$444$	−1.02804	−1.02804
$445$	1.58487	1.58487
$446$	0.0810881	0.0810881
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−1.41971	−1.41971
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0.774890	0.774890
$455$	0	0
$456$	0	0
$457$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$457$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$458$	0	0
$459$	−0.165159	−0.165159
$460$	−1.20283	−1.20283
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	−0.633601	−0.633601
$465$	0	0
$466$	0.774890	0.774890
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	−1.19783	−1.19783
$469$	0	0
$470$	−1.70363	−1.70363
$471$	1.97272	1.97272
$472$	0	0
$473$	0	0
$474$	0.537071	0.537071
$475$	0	0
$476$	0	0
$477$	−1.57828	−1.57828
$478$	0.928738	0.928738
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	2.02804	2.02804
$481$	−2.13788	−2.13788
$482$	0	0
$483$	0	0
$484$	−0.149216	−0.149216
$485$	−0.325812	−0.325812
$486$	0.490971	0.490971
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0.803391	0.803391
$490$	−0.968550	−0.968550
$491$	−1.57828	−1.57828	−0.789141	−	0.614213i	$-0.789474\pi$
$491$	−1.57828	−1.57828	−0.789141	+	0.614213i	$0.789474\pi$
$492$	0	0
$493$	−0.312420	−0.312420
$494$	0	0
$495$	−2.15795	−2.15795
$496$	0	0
$497$	0	0
$498$	0.665051	0.665051
$499$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$499$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$500$	−2.83214	−2.83214
$501$	0	0
$502$	0	0
$503$	1.97272	1.97272	0.986361	−	0.164595i	$-0.0526316\pi$
$503$	1.97272	1.97272	0.986361	+	0.164595i	$0.0526316\pi$
$504$	0	0
$505$	0	0
$506$	0.431478	0.431478
$507$	−1.49097	−1.49097
$508$	1.33495	1.33495
$509$	0.165159	0.165159	0.0825793	−	0.996584i	$-0.473684\pi$
$509$	0.165159	0.165159	0.0825793	+	0.996584i	$0.473684\pi$
$510$	0.159964	0.159964
$511$	0	0
$512$	−0.633601	−0.633601
$513$	0	0
$514$	0.241052	0.241052
$515$	0.968550	0.968550
$516$	0	0
$517$	−1.92411	−1.92411
$518$	0	0
$519$	0	0
$520$	2.68880	2.68880
$521$	1.35456	1.35456	0.677282	−	0.735724i	$-0.263158\pi$
$521$	1.35456	1.35456	0.677282	+	0.735724i	$0.263158\pi$
$522$	0.928738	0.928738
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−0.125347	−0.125347
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0.366399	0.366399
$529$	−0.354563	−0.354563
$530$	1.52864	1.52864
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0.394442	0.394442
$535$	0	0
$536$	0	0
$537$	−1.75895	−1.75895
$538$	0.241052	0.241052
$539$	−1.09390	−1.09390
$540$	1.49719	1.49719
$541$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$541$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$542$	0	0
$543$	0	0
$544$	−0.169790	−0.169790
$545$	0	0
$546$	0	0
$547$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$547$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$548$	0	0
$549$	0	0
$550$	1.55301	1.55301
$551$	0	0
$552$	−0.693802	−0.693802
$553$	0	0
$554$	0.863592	0.863592
$555$	2.67218	2.67218
$556$	−1.43565	−1.43565
$557$	−1.09390	−1.09390	−0.546948	−	0.837166i	$-0.684211\pi$
$557$	−1.09390	−1.09390	−0.546948	+	0.837166i	$0.684211\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0.180666	0.180666
$562$	0.928738	0.928738
$563$	1.97272	1.97272	0.986361	−	0.164595i	$-0.0526316\pi$
$563$	1.97272	1.97272	0.986361	+	0.164595i	$0.0526316\pi$
$564$	1.33495	1.33495
$565$	0	0
$566$	−0.241052	−0.241052
$567$	0	0
$568$	0	0
$569$	1.75895	1.75895	0.879474	−	0.475947i	$-0.157895\pi$
$569$	1.75895	1.75895	0.879474	+	0.475947i	$0.157895\pi$
$570$	0	0
$571$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$571$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$572$	1.31030	1.31030
$573$	0	0
$574$	0	0
$575$	2.32311	2.32311
$576$	0.169790	0.169790
$577$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$577$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$578$	0.477579	0.477579
$579$	−0.490971	−0.490971
$580$	2.83214	2.83214
$581$	0	0
$582$	−0.0810881	−0.0810881
$583$	1.72648	1.72648
$584$	0	0
$585$	3.11351	3.11351
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0.758948	0.758948
$589$	0	0
$590$	0	0
$591$	−0.803391	−0.803391
$592$	−0.453709	−0.453709
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	−0.537071	−0.537071
$595$	0	0
$596$	0	0
$597$	0.165159	0.165159
$598$	−0.622540	−0.622540
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−2.49719	−2.49719
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0.387855	0.387855
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.77611	2.77611
$612$	−0.125347	−0.125347
$613$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$613$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$614$	−0.537071	−0.537071
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0.241052	0.241052
$619$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$619$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$620$	0	0
$621$	−0.803391	−0.803391
$622$	−0.665051	−0.665051
$623$	0	0
$624$	−0.528643	−0.528643
$625$	4.46992	4.46992
$626$	0	0
$627$	0	0
$628$	1.49719	1.49719
$629$	−0.223718	−0.223718
$630$	0	0
$631$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$631$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$632$	0.944680	0.944680
$633$	1.35456	1.35456
$634$	0	0
$635$	−3.46992	−3.46992
$636$	−1.19783	−1.19783
$637$	1.57828	1.57828
$638$	−1.01594	−1.01594
$639$	0	0
$640$	1.86359	1.86359
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$643$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.89163	−1.89163	−0.945817	−	0.324699i	$-0.894737\pi$
$647$	−1.89163	−1.89163	−0.945817	+	0.324699i	$0.894737\pi$
$648$	0.863592	0.863592
$649$	0	0
$650$	−2.24070	−2.24070
$651$	0	0
$652$	0.609731	0.609731
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0.325812	0.325812
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.35456	1.35456	0.677282	−	0.735724i	$-0.263158\pi$
$659$	1.35456	1.35456	0.677282	+	0.735724i	$0.263158\pi$
$660$	−1.63777	−1.63777
$661$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$661$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$662$	0	0
$663$	−0.260667	−0.260667
$664$	1.16979	1.16979
$665$	0	0
$666$	0.665051	0.665051
$667$	−1.51972	−1.51972
$668$	0	0
$669$	0.165159	0.165159
$670$	0	0
$671$	0	0
$672$	0	0
$673$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$673$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$674$	0.394442	0.394442
$675$	−2.89163	−2.89163
$676$	−1.13157	−1.13157
$677$	−1.00000	−1.00000
$677$	−1.00000	−1.00000
$678$	0	0
$679$	0	0
$680$	0.281369	0.281369
$681$	1.57828	1.57828
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2.49097	−2.49097
$690$	0.778124	0.778124
$691$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$691$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.73167	3.73167
$696$	1.63360	1.63360
$697$	0	0
$698$	0	0
$699$	1.57828	1.57828
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0.774890	0.774890
$703$	0	0
$704$	−0.185733	−0.185733
$705$	−3.46992	−3.46992
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	1.09390	1.09390
$712$	0.693802	0.693802
$713$	0	0
$714$	0	0
$715$	−3.40586	−3.40586
$716$	−1.33495	−1.33495
$717$	1.89163	1.89163
$718$	0.241052	0.241052
$719$	−0.490971	−0.490971	−0.245485	−	0.969400i	$-0.578947\pi$
$719$	−0.490971	−0.490971	−0.245485	+	0.969400i	$0.578947\pi$
$720$	0.660761	0.660761
$721$	0	0
$722$	−0.490971	−0.490971
$723$	0	0
$724$	0	0
$725$	−5.46992	−5.46992
$726$	0.0965294	0.0965294
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	1.00000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$733$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$734$	0	0
$735$	−1.97272	−1.97272
$736$	−0.825920	−0.825920
$737$	0	0
$738$	0	0
$739$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$739$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$740$	2.02804	2.02804
$741$	0	0
$742$	0	0
$743$	−1.09390	−1.09390	−0.546948	−	0.837166i	$-0.684211\pi$
$743$	−1.09390	−1.09390	−0.546948	+	0.837166i	$0.684211\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.35456	1.35456
$748$	0.137116	0.137116
$749$	0	0
$750$	1.83214	1.83214
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0.589157	0.589157
$753$	0	0
$754$	1.46581	1.46581
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0.878826	0.878826
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	−0.863592	−0.863592
$763$	0	0
$764$	0	0
$765$	0.325812	0.325812
$766$	0	0
$767$	0	0
$768$	0.633601	0.633601
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0.490971	0.490971
$772$	−0.372621	−0.372621
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	0	0
$776$	−0.142630	−0.142630
$777$	0	0
$778$	−0.863592	−0.863592
$779$	0	0
$780$	2.36299	2.36299
$781$	0	0
$782$	−0.0651455	−0.0651455
$783$	1.89163	1.89163
$784$	0.334949	0.334949
$785$	−3.89163	−3.89163
$786$	0.0810881	0.0810881
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−0.609731	−0.609731
$789$	0	0
$790$	−1.05949	−1.05949
$791$	0	0
$792$	−0.944680	−0.944680
$793$	0	0
$794$	0	0
$795$	3.11351	3.11351
$796$	0.125347	0.125347
$797$	1.35456	1.35456	0.677282	−	0.735724i	$-0.263158\pi$
$797$	1.35456	1.35456	0.677282	+	0.735724i	$0.263158\pi$
$798$	0	0
$799$	0.290505	0.290505
$800$	−2.97272	−2.97272
$801$	0.803391	0.803391
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0.490971	0.490971
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	−0.968550	−0.968550
$811$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$811$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$812$	0	0
$813$	0	0
$814$	−0.727497	−0.727497
$815$	−1.58487	−1.58487
$816$	−0.0553197	−0.0553197
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$823$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$824$	0.423999	0.423999
$825$	3.16315	3.16315
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	−0.609731	−0.609731
$829$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$829$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$830$	−1.31196	−1.31196
$831$	1.75895	1.75895
$832$	0.267977	0.267977
$833$	0.165159	0.165159
$834$	0.928738	0.928738
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−0.968550	−0.968550
$839$	−1.57828	−1.57828	−0.789141	−	0.614213i	$-0.789474\pi$
$839$	−1.57828	−1.57828	−0.789141	+	0.614213i	$0.789474\pi$
$840$	0	0
$841$	2.57828	2.57828
$842$	−0.928738	−0.928738
$843$	1.89163	1.89163
$844$	1.02804	1.02804
$845$	2.94127	2.94127
$846$	−0.863592	−0.863592
$847$	0	0
$848$	−0.528643	−0.528643
$849$	−0.490971	−0.490971
$850$	−0.234477	−0.234477
$851$	−1.08824	−1.08824
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.09390	−1.09390	−0.546948	−	0.837166i	$-0.684211\pi$
$857$	−1.09390	−1.09390	−0.546948	+	0.837166i	$0.684211\pi$
$858$	−0.847650	−0.847650
$859$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$859$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.09390	−1.09390	−0.546948	−	0.837166i	$-0.684211\pi$
$863$	−1.09390	−1.09390	−0.546948	+	0.837166i	$0.684211\pi$
$864$	1.02804	1.02804
$865$	0	0
$866$	0	0
$867$	0.972723	0.972723
$868$	0	0
$869$	−1.19661	−1.19661
$870$	−1.83214	−1.83214
$871$	0	0
$872$	0	0
$873$	−0.165159	−0.165159
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	−0.722804	−0.722804
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−0.490971	−0.490971
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	−0.197832	−0.197832
$885$	0	0
$886$	0.928738	0.928738
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	1.16979	1.16979
$889$	0	0
$890$	−0.778124	−0.778124
$891$	−1.09390	−1.09390
$892$	0.125347	0.125347
$893$	0	0
$894$	0	0
$895$	3.46992	3.46992
$896$	0	0
$897$	−1.26798	−1.26798
$898$	0	0
$899$	0	0
$900$	−2.19460	−2.19460
$901$	−0.260667	−0.260667
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	1.19783	1.19783
$909$	0	0
$910$	0	0
$911$	0.165159	0.165159	0.0825793	−	0.996584i	$-0.473684\pi$
$911$	0.165159	0.165159	0.0825793	+	0.996584i	$0.473684\pi$
$912$	0	0
$913$	−1.48175	−1.48175
$914$	0.394442	0.394442
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0.0810881	0.0810881
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	1.36868	1.36868
$921$	−1.09390	−1.09390
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−3.91690	−3.91690
$926$	0	0
$927$	0.490971	0.490971
$928$	1.94468	1.94468
$929$	1.35456	1.35456	0.677282	−	0.735724i	$-0.263158\pi$
$929$	1.35456	1.35456	0.677282	+	0.735724i	$0.263158\pi$
$930$	0	0
$931$	0	0
$932$	1.19783	1.19783
$933$	−1.35456	−1.35456
$934$	0	0
$935$	−0.356405	−0.356405
$936$	1.36299	1.36299
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	−2.63348	−2.63348
$941$	0.803391	0.803391	0.401695	−	0.915773i	$-0.368421\pi$
$941$	0.803391	0.803391	0.401695	+	0.915773i	$0.368421\pi$
$942$	−0.968550	−0.968550
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.97272	1.97272	0.986361	−	0.164595i	$-0.0526316\pi$
$947$	1.97272	1.97272	0.986361	+	0.164595i	$0.0526316\pi$
$948$	0.830210	0.830210
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0.774890	0.774890
$955$	0	0
$956$	1.43565	1.43565
$957$	−2.06925	−2.06925
$958$	0	0
$959$	0	0
$960$	−0.334949	−0.334949
$961$	1.00000	1.00000
$962$	1.04964	1.04964
$963$	0	0
$964$	0	0
$965$	0.968550	0.968550
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0.169790	0.169790
$969$	0	0
$970$	0.159964	0.159964
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0.758948	0.758948
$973$	0	0
$974$	0	0
$975$	−4.56381	−4.56381
$976$	0	0
$977$	−2.00000	−2.00000	−1.00000			$\pi$
$977$	−2.00000	−2.00000	−1.00000			$\pi$
$978$	−0.394442	−0.394442
$979$	−0.878826	−0.878826
$980$	−1.49719	−1.49719
$981$	0	0
$982$	0.774890	0.774890
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	1.58487	1.58487
$986$	0.153389	0.153389
$987$	0	0
$988$	0	0
$989$	0	0
$990$	1.05949	1.05949
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−0.325812	−0.325812
$996$	1.02804	1.02804
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0.394442	0.394442
$999$	1.35456	1.35456

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2031.1.d.b.2030.4	yes	9
3.2	odd	2		2031.1.d.a.2030.6	✓	9
677.676	even	2		2031.1.d.a.2030.6	✓	9
2031.2030	odd	2	CM	2031.1.d.b.2030.4	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2031.1.d.a.2030.6	✓	9	3.2	odd	2
2031.1.d.a.2030.6	✓	9	677.676	even	2
2031.1.d.b.2030.4	yes	9	1.1	even	1	trivial
2031.1.d.b.2030.4	yes	9	2031.2030	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:	\( N \)	\(=\)	\( 2031 = 3 \cdot 677 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2031.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.490971 q^{2} -1.00000 q^{3} -0.758948 q^{4} +1.97272 q^{5} +0.490971 q^{6} +0.863592 q^{8} +1.00000 q^{9} -0.968550 q^{10} -1.09390 q^{11} +0.758948 q^{12} +1.57828 q^{13} -1.97272 q^{15} +0.334949 q^{16} +0.165159 q^{17} -0.490971 q^{18} -1.49719 q^{20} +0.537071 q^{22} +0.803391 q^{23} -0.863592 q^{24} +2.89163 q^{25} -0.774890 q^{26} -1.00000 q^{27} -1.89163 q^{29} +0.968550 q^{30} -1.02804 q^{32} +1.09390 q^{33} -0.0810881 q^{34} -0.758948 q^{36} -1.35456 q^{37} -1.57828 q^{39} +1.70363 q^{40} +0.830210 q^{44} +1.97272 q^{45} -0.394442 q^{46} +1.75895 q^{47} -0.334949 q^{48} +1.00000 q^{49} -1.41971 q^{50} -0.165159 q^{51} -1.19783 q^{52} -1.57828 q^{53} +0.490971 q^{54} -2.15795 q^{55} +0.928738 q^{58} +1.49719 q^{60} +0.169790 q^{64} +3.11351 q^{65} -0.537071 q^{66} -0.125347 q^{68} -0.803391 q^{69} +0.863592 q^{72} +0.665051 q^{74} -2.89163 q^{75} +0.774890 q^{78} +1.09390 q^{79} +0.660761 q^{80} +1.00000 q^{81} +1.35456 q^{83} +0.325812 q^{85} +1.89163 q^{87} -0.944680 q^{88} +0.803391 q^{89} -0.968550 q^{90} -0.609731 q^{92} -0.863592 q^{94} +1.02804 q^{96} -0.165159 q^{97} -0.490971 q^{98} -1.09390 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{2} - 9 q^{3} + 8 q^{4} + q^{5} - q^{6} + 2 q^{8} + 9 q^{9} - 2 q^{10} + q^{11} - 8 q^{12} - q^{13} - q^{15} + 7 q^{16} + q^{17} + q^{18} + 3 q^{20} - 2 q^{22} + q^{23} - 2 q^{24} + 8 q^{25}+ \cdots + q^{99}+O(q^{100}) \) 9 * q + q^2 - 9 * q^3 + 8 * q^4 + q^5 - q^6 + 2 * q^8 + 9 * q^9 - 2 * q^10 + q^11 - 8 * q^12 - q^13 - q^15 + 7 * q^16 + q^17 + q^18 + 3 * q^20 - 2 * q^22 + q^23 - 2 * q^24 + 8 * q^25 + 2 * q^26 - 9 * q^27 + q^29 + 2 * q^30 + 3 * q^32 - q^33 - 2 * q^34 + 8 * q^36 - q^37 + q^39 - 4 * q^40 + 3 * q^44 + q^45 - 2 * q^46 + q^47 - 7 * q^48 + 9 * q^49 + 3 * q^50 - q^51 - 3 * q^52 + q^53 - q^54 - 2 * q^55 - 2 * q^58 - 3 * q^60 + 6 * q^64 + 2 * q^65 + 2 * q^66 + 3 * q^68 - q^69 + 2 * q^72 + 2 * q^74 - 8 * q^75 - 2 * q^78 - q^79 + 5 * q^80 + 9 * q^81 + q^83 - 2 * q^85 - q^87 - 4 * q^88 + q^89 - 2 * q^90 + 3 * q^92 - 2 * q^94 - 3 * q^96 - q^97 + q^98 + q^99

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