LMFDB - Embedded newform 1535.1.d.a.1534.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1535,1,Mod(1534,1535)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1535, 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("1535.1534");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1535 = 5 \cdot 307 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1535.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:	$0.766064794392$
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 19.1.47312447868976594992787109375.1

Embedding invariants

Embedding label			1534.1
Root			$-1.89163$ of defining polynomial
Character	$\chi$	$=$	1535.1534

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.97272 q^{2} -0.803391 q^{3} +2.89163 q^{4} +1.00000 q^{5} +1.58487 q^{6} -3.73167 q^{8} -0.354563 q^{9} +O(q^{10})$	\(q-1.97272 q^{2} -0.803391 q^{3} +2.89163 q^{4} +1.00000 q^{5} +1.58487 q^{6} -3.73167 q^{8} -0.354563 q^{9} -1.97272 q^{10} -1.75895 q^{11} -2.32311 q^{12} +0.490971 q^{13} -0.803391 q^{15} +4.46992 q^{16} +0.699455 q^{18} -0.165159 q^{19} +2.89163 q^{20} +3.46992 q^{22} +1.09390 q^{23} +2.99799 q^{24} +1.00000 q^{25} -0.968550 q^{26} +1.08824 q^{27} +1.58487 q^{30} -5.08623 q^{32} +1.41312 q^{33} -1.02527 q^{36} +0.325812 q^{38} -0.394442 q^{39} -3.73167 q^{40} +1.57828 q^{41} -1.35456 q^{43} -5.08623 q^{44} -0.354563 q^{45} -2.15795 q^{46} +1.57828 q^{47} -3.59109 q^{48} +1.00000 q^{49} -1.97272 q^{50} +1.41971 q^{52} -2.14680 q^{54} -1.75895 q^{55} +0.132687 q^{57} -2.32311 q^{60} +5.56381 q^{64} +0.490971 q^{65} -2.78770 q^{66} -0.165159 q^{67} -0.878826 q^{69} +1.09390 q^{71} +1.32311 q^{72} +1.09390 q^{73} -0.803391 q^{75} -0.477579 q^{76} +0.778124 q^{78} +0.490971 q^{79} +4.46992 q^{80} -0.519722 q^{81} -3.11351 q^{82} +2.67218 q^{86} +6.56381 q^{88} -1.97272 q^{89} +0.699455 q^{90} +3.16315 q^{92} -3.11351 q^{94} -0.165159 q^{95} +4.08623 q^{96} -1.97272 q^{98} +0.623658 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{2} - q^{3} + 8 q^{4} + 9 q^{5} - 2 q^{6} - 2 q^{8} + 8 q^{9}+O(q^{10}) $ 9 * q - q^2 - q^3 + 8 * q^4 + 9 * q^5 - 2 * q^6 - 2 * q^8 + 8 * q^9	$ 9 q - q^{2} - q^{3} + 8 q^{4} + 9 q^{5} - 2 q^{6} - 2 q^{8} + 8 q^{9} - q^{10} - q^{11} - 3 q^{12} - q^{13} - q^{15} + 7 q^{16} - 3 q^{18} - q^{19} + 8 q^{20} - 2 q^{22} - q^{23} - 4 q^{24} + 9 q^{25} - 2 q^{26} - 2 q^{27} - 2 q^{30} - 3 q^{32} - 2 q^{33} + 5 q^{36} - 2 q^{38} - 2 q^{39} - 2 q^{40} - q^{41} - q^{43} - 3 q^{44} + 8 q^{45} - 2 q^{46} - q^{47} - 5 q^{48} + 9 q^{49} - q^{50} - 3 q^{52} - 4 q^{54} - q^{55} - 2 q^{57} - 3 q^{60} + 6 q^{64} - q^{65} - 4 q^{66} - q^{67} - 2 q^{69} - q^{71} - 6 q^{72} - q^{73} - q^{75} - 3 q^{76} + 15 q^{78} - q^{79} + 7 q^{80} + 7 q^{81} - 2 q^{82} - 2 q^{86} + 15 q^{88} - q^{89} - 3 q^{90} - 3 q^{92} - 2 q^{94} - q^{95} - 6 q^{96} - q^{98} - 3 q^{99}+O(q^{100}) $ 9 * q - q^2 - q^3 + 8 * q^4 + 9 * q^5 - 2 * q^6 - 2 * q^8 + 8 * q^9 - q^10 - q^11 - 3 * q^12 - q^13 - q^15 + 7 * q^16 - 3 * q^18 - q^19 + 8 * q^20 - 2 * q^22 - q^23 - 4 * q^24 + 9 * q^25 - 2 * q^26 - 2 * q^27 - 2 * q^30 - 3 * q^32 - 2 * q^33 + 5 * q^36 - 2 * q^38 - 2 * q^39 - 2 * q^40 - q^41 - q^43 - 3 * q^44 + 8 * q^45 - 2 * q^46 - q^47 - 5 * q^48 + 9 * q^49 - q^50 - 3 * q^52 - 4 * q^54 - q^55 - 2 * q^57 - 3 * q^60 + 6 * q^64 - q^65 - 4 * q^66 - q^67 - 2 * q^69 - q^71 - 6 * q^72 - q^73 - q^75 - 3 * q^76 + 15 * q^78 - q^79 + 7 * q^80 + 7 * q^81 - 2 * q^82 - 2 * q^86 + 15 * q^88 - q^89 - 3 * q^90 - 3 * q^92 - 2 * q^94 - q^95 - 6 * q^96 - q^98 - 3 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1535\mathbb{Z}\right)^\times$.

$n$	$922$	$926$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$2$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$3$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$3$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$4$	2.89163	2.89163
$5$	1.00000	1.00000
$5$	1.00000	1.00000
$6$	1.58487	1.58487
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−3.73167	−3.73167
$9$	−0.354563	−0.354563
$10$	−1.97272	−1.97272
$11$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$11$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$12$	−2.32311	−2.32311
$13$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$13$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$14$	0	0
$15$	−0.803391	−0.803391
$16$	4.46992	4.46992
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0.699455	0.699455
$19$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$19$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$20$	2.89163	2.89163
$21$	0	0
$22$	3.46992	3.46992
$23$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$23$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$24$	2.99799	2.99799
$25$	1.00000	1.00000
$26$	−0.968550	−0.968550
$27$	1.08824	1.08824
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	1.58487	1.58487
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−5.08623	−5.08623
$33$	1.41312	1.41312
$34$	0	0
$35$	0	0
$36$	−1.02527	−1.02527
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0.325812	0.325812
$39$	−0.394442	−0.394442
$40$	−3.73167	−3.73167
$41$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$41$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$42$	0	0
$43$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$43$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$44$	−5.08623	−5.08623
$45$	−0.354563	−0.354563
$46$	−2.15795	−2.15795
$47$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$47$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$48$	−3.59109	−3.59109
$49$	1.00000	1.00000
$50$	−1.97272	−1.97272
$51$	0	0
$52$	1.41971	1.41971
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−2.14680	−2.14680
$55$	−1.75895	−1.75895
$56$	0	0
$57$	0.132687	0.132687
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	−2.32311	−2.32311
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	5.56381	5.56381
$65$	0.490971	0.490971
$66$	−2.78770	−2.78770
$67$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$67$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$68$	0	0
$69$	−0.878826	−0.878826
$70$	0	0
$71$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$71$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$72$	1.32311	1.32311
$73$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$73$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$74$	0	0
$75$	−0.803391	−0.803391
$76$	−0.477579	−0.477579
$77$	0	0
$78$	0.778124	0.778124
$79$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$79$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$80$	4.46992	4.46992
$81$	−0.519722	−0.519722
$82$	−3.11351	−3.11351
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	2.67218	2.67218
$87$	0	0
$88$	6.56381	6.56381
$89$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$89$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$90$	0.699455	0.699455
$91$	0	0
$92$	3.16315	3.16315
$93$	0	0
$94$	−3.11351	−3.11351
$95$	−0.165159	−0.165159
$96$	4.08623	4.08623
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−1.97272	−1.97272
$99$	0.623658	0.623658
$100$	2.89163	2.89163
$101$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$101$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−1.83214	−1.83214
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	3.14680	3.14680
$109$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$109$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$110$	3.46992	3.46992
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	−0.261755	−0.261755
$115$	1.09390	1.09390
$116$	0	0
$117$	−0.174080	−0.174080
$118$	0	0
$119$	0	0
$120$	2.99799	2.99799
$121$	2.09390	2.09390
$122$	0	0
$123$	−1.26798	−1.26798
$124$	0	0
$125$	1.00000	1.00000
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−5.88962	−5.88962
$129$	1.08824	1.08824
$130$	−0.968550	−0.968550
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	4.08623	4.08623
$133$	0	0
$134$	0.325812	0.325812
$135$	1.08824	1.08824
$136$	0	0
$137$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$137$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$138$	1.73368	1.73368
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	−1.26798	−1.26798
$142$	−2.15795	−2.15795
$143$	−0.863592	−0.863592
$144$	−1.58487	−1.58487
$145$	0	0
$146$	−2.15795	−2.15795
$147$	−0.803391	−0.803391
$148$	0	0
$149$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$149$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$150$	1.58487	1.58487
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0.616318	0.616318
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−1.14058	−1.14058
$157$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$157$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$158$	−0.968550	−0.968550
$159$	0	0
$160$	−5.08623	−5.08623
$161$	0	0
$162$	1.02527	1.02527
$163$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$163$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$164$	4.56381	4.56381
$165$	1.41312	1.41312
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−0.758948	−0.758948
$170$	0	0
$171$	0.0585592	0.0585592
$172$	−3.91690	−3.91690
$173$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$173$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$174$	0	0
$175$	0	0
$176$	−7.86235	−7.86235
$177$	0	0
$178$	3.89163	3.89163
$179$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$179$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$180$	−1.02527	−1.02527
$181$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$181$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$182$	0	0
$183$	0	0
$184$	−4.08206	−4.08206
$185$	0	0
$186$	0	0
$187$	0	0
$188$	4.56381	4.56381
$189$	0	0
$190$	0.325812	0.325812
$191$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$191$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$192$	−4.46992	−4.46992
$193$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$193$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$194$	0	0
$195$	−0.394442	−0.394442
$196$	2.89163	2.89163
$197$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$197$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$198$	−1.23030	−1.23030
$199$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$199$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$200$	−3.73167	−3.73167
$201$	0.132687	0.132687
$202$	−3.73167	−3.73167
$203$	0	0
$204$	0	0
$205$	1.57828	1.57828
$206$	0	0
$207$	−0.387855	−0.387855
$208$	2.19460	2.19460
$209$	0.290505	0.290505
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	−0.878826	−0.878826
$214$	0	0
$215$	−1.35456	−1.35456
$216$	−4.06097	−4.06097
$217$	0	0
$218$	1.58487	1.58487
$219$	−0.878826	−0.878826
$220$	−5.08623	−5.08623
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	−0.354563	−0.354563
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0.383682	0.383682
$229$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$229$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$230$	−2.15795	−2.15795
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0.343412	0.343412
$235$	1.57828	1.57828
$236$	0	0
$237$	−0.394442	−0.394442
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	−3.59109	−3.59109
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−4.13068	−4.13068
$243$	−0.670704	−0.670704
$244$	0	0
$245$	1.00000	1.00000
$246$	2.50137	2.50137
$247$	−0.0810881	−0.0810881
$248$	0	0
$249$	0	0
$250$	−1.97272	−1.97272
$251$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$251$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$252$	0	0
$253$	−1.92411	−1.92411
$254$	0	0
$255$	0	0
$256$	6.05478	6.05478
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	−2.14680	−2.14680
$259$	0	0
$260$	1.41971	1.41971
$261$	0	0
$262$	0	0
$263$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$263$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$264$	−5.27331	−5.27331
$265$	0	0
$266$	0	0
$267$	1.58487	1.58487
$268$	−0.477579	−0.477579
$269$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$269$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$270$	−2.14680	−2.14680
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	2.67218	2.67218
$275$	−1.75895	−1.75895
$276$	−2.54124	−2.54124
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	2.50137	2.50137
$283$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$283$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$284$	3.16315	3.16315
$285$	0.132687	0.132687
$286$	1.70363	1.70363
$287$	0	0
$288$	1.80339	1.80339
$289$	1.00000	1.00000
$290$	0	0
$291$	0	0
$292$	3.16315	3.16315
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	1.58487	1.58487
$295$	0	0
$296$	0	0
$297$	−1.91416	−1.91416
$298$	0.325812	0.325812
$299$	0.537071	0.537071
$300$	−2.32311	−2.32311
$301$	0	0
$302$	0	0
$303$	−1.51972	−1.51972
$304$	−0.738245	−0.738245
$305$	0	0
$306$	0	0
$307$	1.00000	1.00000
$307$	1.00000	1.00000
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$311$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$312$	1.47193	1.47193
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	−3.73167	−3.73167
$315$	0	0
$316$	1.41971	1.41971
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	5.56381	5.56381
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−1.50285	−1.50285
$325$	0.490971	0.490971
$326$	−3.11351	−3.11351
$327$	0.645437	0.645437
$328$	−5.88962	−5.88962
$329$	0	0
$330$	−2.78770	−2.78770
$331$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$331$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−0.165159	−0.165159
$336$	0	0
$337$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$337$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$338$	1.49719	1.49719
$339$	0	0
$340$	0	0
$341$	0	0
$342$	−0.115521	−0.115521
$343$	0	0
$344$	5.05478	5.05478
$345$	−0.878826	−0.878826
$346$	−3.73167	−3.73167
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$349$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$350$	0	0
$351$	0.534296	0.534296
$352$	8.94642	8.94642
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	1.09390	1.09390
$356$	−5.70439	−5.70439
$357$	0	0
$358$	3.89163	3.89163
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	1.32311	1.32311
$361$	−0.972723	−0.972723
$362$	−0.968550	−0.968550
$363$	−1.68222	−1.68222
$364$	0	0
$365$	1.09390	1.09390
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	4.88962	4.88962
$369$	−0.559600	−0.559600
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−0.803391	−0.803391
$376$	−5.88962	−5.88962
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	−0.477579	−0.477579
$381$	0	0
$382$	−3.73167	−3.73167
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	4.73167	4.73167
$385$	0	0
$386$	3.89163	3.89163
$387$	0.480278	0.480278
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0.778124	0.778124
$391$	0	0
$392$	−3.73167	−3.73167
$393$	0	0
$394$	0.325812	0.325812
$395$	0.490971	0.490971
$396$	1.80339	1.80339
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	2.67218	2.67218
$399$	0	0
$400$	4.46992	4.46992
$401$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$401$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$402$	−0.261755	−0.261755
$403$	0	0
$404$	5.46992	5.46992
$405$	−0.519722	−0.519722
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$409$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$410$	−3.11351	−3.11351
$411$	1.08824	1.08824
$412$	0	0
$413$	0	0
$414$	0.765131	0.765131
$415$	0	0
$416$	−2.49719	−2.49719
$417$	0	0
$418$	−0.573087	−0.573087
$419$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$419$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$420$	0	0
$421$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$421$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$422$	0	0
$423$	−0.559600	−0.559600
$424$	0	0
$425$	0	0
$426$	1.73368	1.73368
$427$	0	0
$428$	0	0
$429$	0.693802	0.693802
$430$	2.67218	2.67218
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	4.86436	4.86436
$433$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$433$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$434$	0	0
$435$	0	0
$436$	−2.32311	−2.32311
$437$	−0.180666	−0.180666
$438$	1.73368	1.73368
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	6.56381	6.56381
$441$	−0.354563	−0.354563
$442$	0	0
$443$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$443$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$444$	0	0
$445$	−1.97272	−1.97272
$446$	0	0
$447$	0.132687	0.132687
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0.699455	0.699455
$451$	−2.77611	−2.77611
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	−0.495144	−0.495144
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	−3.11351	−3.11351
$459$	0	0
$460$	3.16315	3.16315
$461$	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	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	−0.503376	−0.503376
$469$	0	0
$470$	−3.11351	−3.11351
$471$	−1.51972	−1.51972
$472$	0	0
$473$	2.38261	2.38261
$474$	0.778124	0.778124
$475$	−0.165159	−0.165159
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	4.08623	4.08623
$481$	0	0
$482$	0	0
$483$	0	0
$484$	6.05478	6.05478
$485$	0	0
$486$	1.32311	1.32311
$487$	2.00000	2.00000	1.00000			$0$
$487$	2.00000	2.00000	1.00000			$0$
$488$	0	0
$489$	−1.26798	−1.26798
$490$	−1.97272	−1.97272
$491$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$491$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$492$	−3.66652	−3.66652
$493$	0	0
$494$	0.159964	0.159964
$495$	0.623658	0.623658
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	2.89163	2.89163
$501$	0	0
$502$	−0.968550	−0.968550
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	1.89163	1.89163
$506$	3.79573	3.79573
$507$	0.609731	0.609731
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−6.05478	−6.05478
$513$	−0.179733	−0.179733
$514$	0	0
$515$	0	0
$516$	3.14680	3.14680
$517$	−2.77611	−2.77611
$518$	0	0
$519$	−1.51972	−1.51972
$520$	−1.83214	−1.83214
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	−3.73167	−3.73167
$527$	0	0
$528$	6.31654	6.31654
$529$	0.196609	0.196609
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0.774890	0.774890
$534$	−3.12650	−3.12650
$535$	0	0
$536$	0.616318	0.616318
$537$	1.58487	1.58487
$538$	1.58487	1.58487
$539$	−1.75895	−1.75895
$540$	3.14680	3.14680
$541$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$541$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$542$	0	0
$543$	−0.394442	−0.394442
$544$	0	0
$545$	−0.803391	−0.803391
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	−3.91690	−3.91690
$549$	0	0
$550$	3.46992	3.46992
$551$	0	0
$552$	3.27949	3.27949
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	−0.665051	−0.665051
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	−3.66652	−3.66652
$565$	0	0
$566$	3.89163	3.89163
$567$	0	0
$568$	−4.08206	−4.08206
$569$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$569$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$570$	−0.261755	−0.261755
$571$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$571$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$572$	−2.49719	−2.49719
$573$	−1.51972	−1.51972
$574$	0	0
$575$	1.09390	1.09390
$576$	−1.97272	−1.97272
$577$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$577$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$578$	−1.97272	−1.97272
$579$	1.58487	1.58487
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−4.08206	−4.08206
$585$	−0.174080	−0.174080
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	−2.32311	−2.32311
$589$	0	0
$590$	0	0
$591$	0.132687	0.132687
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	3.77611	3.77611
$595$	0	0
$596$	−0.477579	−0.477579
$597$	1.08824	1.08824
$598$	−1.05949	−1.05949
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	2.99799	2.99799
$601$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$601$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$602$	0	0
$603$	0.0585592	0.0585592
$604$	0	0
$605$	2.09390	2.09390
$606$	2.99799	2.99799
$607$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$607$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$608$	0.840036	0.840036
$609$	0	0
$610$	0	0
$611$	0.774890	0.774890
$612$	0	0
$613$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$613$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$614$	−1.97272	−1.97272
$615$	−1.26798	−1.26798
$616$	0	0
$617$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$617$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	1.19043	1.19043
$622$	−0.968550	−0.968550
$623$	0	0
$624$	−1.76312	−1.76312
$625$	1.00000	1.00000
$626$	0	0
$627$	−0.233389	−0.233389
$628$	5.46992	5.46992
$629$	0	0
$630$	0	0
$631$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$631$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$632$	−1.83214	−1.83214
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0.490971	0.490971
$638$	0	0
$639$	−0.387855	−0.387855
$640$	−5.88962	−5.88962
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$643$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$644$	0	0
$645$	1.08824	1.08824
$646$	0	0
$647$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$647$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$648$	1.93943	1.93943
$649$	0	0
$650$	−0.968550	−0.968550
$651$	0	0
$652$	4.56381	4.56381
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	−1.27327	−1.27327
$655$	0	0
$656$	7.05478	7.05478
$657$	−0.387855	−0.387855
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	4.08623	4.08623
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	2.67218	2.67218
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0.325812	0.325812
$671$	0	0
$672$	0	0
$673$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$673$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$674$	2.67218	2.67218
$675$	1.08824	1.08824
$676$	−2.19460	−2.19460
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0.169332	0.169332
$685$	−1.35456	−1.35456
$686$	0	0
$687$	−1.26798	−1.26798
$688$	−6.05478	−6.05478
$689$	0	0
$690$	1.73368	1.73368
$691$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$691$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$692$	5.46992	5.46992
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	3.46992	3.46992
$699$	0	0
$700$	0	0
$701$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$701$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$702$	−1.05402	−1.05402
$703$	0	0
$704$	−9.78645	−9.78645
$705$	−1.26798	−1.26798
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	−2.15795	−2.15795
$711$	−0.174080	−0.174080
$712$	7.36155	7.36155
$713$	0	0
$714$	0	0
$715$	−0.863592	−0.863592
$716$	−5.70439	−5.70439
$717$	0	0
$718$	0	0
$719$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$719$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$720$	−1.58487	−1.58487
$721$	0	0
$722$	1.91891	1.91891
$723$	0	0
$724$	1.41971	1.41971
$725$	0	0
$726$	3.31855	3.31855
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	1.05856	1.05856
$730$	−2.15795	−2.15795
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	−0.803391	−0.803391
$736$	−5.56381	−5.56381
$737$	0.290505	0.290505
$738$	1.10394	1.10394
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0.0651455	0.0651455
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	−0.165159	−0.165159
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	1.58487	1.58487
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	7.05478	7.05478
$753$	−0.394442	−0.394442
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$757$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$758$	0	0
$759$	1.54581	1.54581
$760$	0.616318	0.616318
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	5.46992	5.46992
$765$	0	0
$766$	0	0
$767$	0	0
$768$	−4.86436	−4.86436
$769$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$769$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$770$	0	0
$771$	0	0
$772$	−5.70439	−5.70439
$773$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$773$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$774$	−0.947456	−0.947456
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.260667	−0.260667
$780$	−1.14058	−1.14058
$781$	−1.92411	−1.92411
$782$	0	0
$783$	0	0
$784$	4.46992	4.46992
$785$	1.89163	1.89163
$786$	0	0
$787$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$787$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$788$	−0.477579	−0.477579
$789$	−1.51972	−1.51972
$790$	−0.968550	−0.968550
$791$	0	0
$792$	−2.32729	−2.32729
$793$	0	0
$794$	0	0
$795$	0	0
$796$	−3.91690	−3.91690
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−5.08623	−5.08623
$801$	0.699455	0.699455
$802$	0.325812	0.325812
$803$	−1.92411	−1.92411
$804$	0.383682	0.383682
$805$	0	0
$806$	0	0
$807$	0.645437	0.645437
$808$	−7.05896	−7.05896
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	1.02527	1.02527
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.57828	1.57828
$816$	0	0
$817$	0.223718	0.223718
$818$	−0.968550	−0.968550
$819$	0	0
$820$	4.56381	4.56381
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	−2.14680	−2.14680
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	1.41312	1.41312
$826$	0	0
$827$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$827$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$828$	−1.12154	−1.12154
$829$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$829$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$830$	0	0
$831$	0	0
$832$	2.73167	2.73167
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0.840036	0.840036
$837$	0	0
$838$	−3.11351	−3.11351
$839$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$839$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	3.89163	3.89163
$843$	0	0
$844$	0	0
$845$	−0.758948	−0.758948
$846$	1.10394	1.10394
$847$	0	0
$848$	0	0
$849$	1.58487	1.58487
$850$	0	0
$851$	0	0
$852$	−2.54124	−2.54124
$853$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$853$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$854$	0	0
$855$	0.0585592	0.0585592
$856$	0	0
$857$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$857$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$858$	−1.36868	−1.36868
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	−3.91690	−3.91690
$861$	0	0
$862$	0	0
$863$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$863$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$864$	−5.53506	−5.53506
$865$	1.89163	1.89163
$866$	3.46992	3.46992
$867$	−0.803391	−0.803391
$868$	0	0
$869$	−0.863592	−0.863592
$870$	0	0
$871$	−0.0810881	−0.0810881
$872$	2.99799	2.99799
$873$	0	0
$874$	0.356405	0.356405
$875$	0	0
$876$	−2.54124	−2.54124
$877$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$877$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$878$	0	0
$879$	0	0
$880$	−7.86235	−7.86235
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0.699455	0.699455
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	3.46992	3.46992
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	3.89163	3.89163
$891$	0.914163	0.914163
$892$	0	0
$893$	−0.260667	−0.260667
$894$	−0.261755	−0.261755
$895$	−1.97272	−1.97272
$896$	0	0
$897$	−0.431478	−0.431478
$898$	0	0
$899$	0	0
$900$	−1.02527	−1.02527
$901$	0	0
$902$	5.47650	5.47650
$903$	0	0
$904$	0	0
$905$	0.490971	0.490971
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	−0.670704	−0.670704
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0.593100	0.593100
$913$	0	0
$914$	0	0
$915$	0	0
$916$	4.56381	4.56381
$917$	0	0
$918$	0	0
$919$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$919$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$920$	−4.08206	−4.08206
$921$	−0.803391	−0.803391
$922$	0	0
$923$	0.537071	0.537071
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	−0.165159	−0.165159
$932$	0	0
$933$	−0.394442	−0.394442
$934$	0	0
$935$	0	0
$936$	0.649610	0.649610
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	4.56381	4.56381
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	2.99799	2.99799
$943$	1.72648	1.72648
$944$	0	0
$945$	0	0
$946$	−4.70022	−4.70022
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	−1.14058	−1.14058
$949$	0.537071	0.537071
$950$	0.325812	0.325812
$951$	0	0
$952$	0	0
$953$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$953$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$954$	0	0
$955$	1.89163	1.89163
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	−4.46992	−4.46992
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.97272	−1.97272
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−7.81373	−7.81373
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−1.93943	−1.93943
$973$	0	0
$974$	−3.94545	−3.94545
$975$	−0.394442	−0.394442
$976$	0	0
$977$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$977$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$978$	2.50137	2.50137
$979$	3.46992	3.46992
$980$	2.89163	2.89163
$981$	0.284853	0.284853
$982$	1.58487	1.58487
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	4.73167	4.73167
$985$	−0.165159	−0.165159
$986$	0	0
$987$	0	0
$988$	−0.234477	−0.234477
$989$	−1.48175	−1.48175
$990$	−1.23030	−1.23030
$991$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$991$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$992$	0	0
$993$	1.08824	1.08824
$994$	0	0
$995$	−1.35456	−1.35456
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1535.1.d.a.1534.1	✓	9
5.4	even	2		1535.1.d.b.1534.9	yes	9
307.306	odd	2		1535.1.d.b.1534.9	yes	9
1535.1534	odd	2	CM	1535.1.d.a.1534.1	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1535.1.d.a.1534.1	✓	9	1.1	even	1	trivial
1535.1.d.a.1534.1	✓	9	1535.1534	odd	2	CM
1535.1.d.b.1534.9	yes	9	5.4	even	2
1535.1.d.b.1534.9	yes	9	307.306	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1535 = 5 \cdot 307 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1535.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.97272 q^{2} -0.803391 q^{3} +2.89163 q^{4} +1.00000 q^{5} +1.58487 q^{6} -3.73167 q^{8} -0.354563 q^{9} +O(q^{10})\)	\(q-1.97272 q^{2} -0.803391 q^{3} +2.89163 q^{4} +1.00000 q^{5} +1.58487 q^{6} -3.73167 q^{8} -0.354563 q^{9} -1.97272 q^{10} -1.75895 q^{11} -2.32311 q^{12} +0.490971 q^{13} -0.803391 q^{15} +4.46992 q^{16} +0.699455 q^{18} -0.165159 q^{19} +2.89163 q^{20} +3.46992 q^{22} +1.09390 q^{23} +2.99799 q^{24} +1.00000 q^{25} -0.968550 q^{26} +1.08824 q^{27} +1.58487 q^{30} -5.08623 q^{32} +1.41312 q^{33} -1.02527 q^{36} +0.325812 q^{38} -0.394442 q^{39} -3.73167 q^{40} +1.57828 q^{41} -1.35456 q^{43} -5.08623 q^{44} -0.354563 q^{45} -2.15795 q^{46} +1.57828 q^{47} -3.59109 q^{48} +1.00000 q^{49} -1.97272 q^{50} +1.41971 q^{52} -2.14680 q^{54} -1.75895 q^{55} +0.132687 q^{57} -2.32311 q^{60} +5.56381 q^{64} +0.490971 q^{65} -2.78770 q^{66} -0.165159 q^{67} -0.878826 q^{69} +1.09390 q^{71} +1.32311 q^{72} +1.09390 q^{73} -0.803391 q^{75} -0.477579 q^{76} +0.778124 q^{78} +0.490971 q^{79} +4.46992 q^{80} -0.519722 q^{81} -3.11351 q^{82} +2.67218 q^{86} +6.56381 q^{88} -1.97272 q^{89} +0.699455 q^{90} +3.16315 q^{92} -3.11351 q^{94} -0.165159 q^{95} +4.08623 q^{96} -1.97272 q^{98} +0.623658 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{2} - q^{3} + 8 q^{4} + 9 q^{5} - 2 q^{6} - 2 q^{8} + 8 q^{9}+O(q^{10}) \) 9 * q - q^2 - q^3 + 8 * q^4 + 9 * q^5 - 2 * q^6 - 2 * q^8 + 8 * q^9	\( 9 q - q^{2} - q^{3} + 8 q^{4} + 9 q^{5} - 2 q^{6} - 2 q^{8} + 8 q^{9} - q^{10} - q^{11} - 3 q^{12} - q^{13} - q^{15} + 7 q^{16} - 3 q^{18} - q^{19} + 8 q^{20} - 2 q^{22} - q^{23} - 4 q^{24} + 9 q^{25} - 2 q^{26} - 2 q^{27} - 2 q^{30} - 3 q^{32} - 2 q^{33} + 5 q^{36} - 2 q^{38} - 2 q^{39} - 2 q^{40} - q^{41} - q^{43} - 3 q^{44} + 8 q^{45} - 2 q^{46} - q^{47} - 5 q^{48} + 9 q^{49} - q^{50} - 3 q^{52} - 4 q^{54} - q^{55} - 2 q^{57} - 3 q^{60} + 6 q^{64} - q^{65} - 4 q^{66} - q^{67} - 2 q^{69} - q^{71} - 6 q^{72} - q^{73} - q^{75} - 3 q^{76} + 15 q^{78} - q^{79} + 7 q^{80} + 7 q^{81} - 2 q^{82} - 2 q^{86} + 15 q^{88} - q^{89} - 3 q^{90} - 3 q^{92} - 2 q^{94} - q^{95} - 6 q^{96} - q^{98} - 3 q^{99}+O(q^{100}) \) 9 * q - q^2 - q^3 + 8 * q^4 + 9 * q^5 - 2 * q^6 - 2 * q^8 + 8 * q^9 - q^10 - q^11 - 3 * q^12 - q^13 - q^15 + 7 * q^16 - 3 * q^18 - q^19 + 8 * q^20 - 2 * q^22 - q^23 - 4 * q^24 + 9 * q^25 - 2 * q^26 - 2 * q^27 - 2 * q^30 - 3 * q^32 - 2 * q^33 + 5 * q^36 - 2 * q^38 - 2 * q^39 - 2 * q^40 - q^41 - q^43 - 3 * q^44 + 8 * q^45 - 2 * q^46 - q^47 - 5 * q^48 + 9 * q^49 - q^50 - 3 * q^52 - 4 * q^54 - q^55 - 2 * q^57 - 3 * q^60 + 6 * q^64 - q^65 - 4 * q^66 - q^67 - 2 * q^69 - q^71 - 6 * q^72 - q^73 - q^75 - 3 * q^76 + 15 * q^78 - q^79 + 7 * q^80 + 7 * q^81 - 2 * q^82 - 2 * q^86 + 15 * q^88 - q^89 - 3 * q^90 - 3 * q^92 - 2 * q^94 - q^95 - 6 * q^96 - q^98 - 3 * q^99

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