LMFDB - Embedded newform 1007.1.d.e.1006.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1007,1,Mod(1006,1007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1007, 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("1007.1006");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1007 = 19 \cdot 53 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1007.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:	$0.502558467721$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{15})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 4x + 1 $ x^4 - x^3 - 4x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{15}$
Projective field:	Galois closure of 15.1.1050041089388771366543.1

Embedding invariants

Embedding label			1006.4
Root			$-1.95630$ of defining polynomial
Character	$\chi$	$=$	1007.1006

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.95630 q^{2} +0.209057 q^{3} +2.82709 q^{4} +0.408977 q^{6} -1.61803 q^{7} +3.57433 q^{8} -0.956295 q^{9} +O(q^{10})$	$q+1.95630 q^{2} +0.209057 q^{3} +2.82709 q^{4} +0.408977 q^{6} -1.61803 q^{7} +3.57433 q^{8} -0.956295 q^{9} -1.00000 q^{11} +0.591023 q^{12} -3.16535 q^{14} +4.16535 q^{16} +1.33826 q^{17} -1.87080 q^{18} -1.00000 q^{19} -0.338261 q^{21} -1.95630 q^{22} +0.747238 q^{24} +1.00000 q^{25} -0.408977 q^{27} -4.57433 q^{28} -1.82709 q^{31} +4.57433 q^{32} -0.209057 q^{33} +2.61803 q^{34} -2.70353 q^{36} -1.95630 q^{38} -0.618034 q^{41} -0.661739 q^{42} -0.209057 q^{43} -2.82709 q^{44} +0.618034 q^{47} +0.870796 q^{48} +1.61803 q^{49} +1.95630 q^{50} +0.279773 q^{51} -1.00000 q^{53} -0.800080 q^{54} -5.78339 q^{56} -0.209057 q^{57} -3.57433 q^{62} +1.54732 q^{63} +4.78339 q^{64} -0.408977 q^{66} +1.00000 q^{67} +3.78339 q^{68} +1.00000 q^{71} -3.41811 q^{72} +0.209057 q^{75} -2.82709 q^{76} +1.61803 q^{77} +1.61803 q^{79} +0.870796 q^{81} -1.20906 q^{82} -0.956295 q^{84} -0.408977 q^{86} -3.57433 q^{88} -0.381966 q^{93} +1.20906 q^{94} +0.956295 q^{96} +3.16535 q^{98} +0.956295 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} - q^{3} + 5 q^{4} - q^{6} - 2 q^{7} + q^{8} + 5 q^{9}+O(q^{10}) $ 4 * q - q^2 - q^3 + 5 * q^4 - q^6 - 2 * q^7 + q^8 + 5 * q^9	$ 4 q - q^{2} - q^{3} + 5 q^{4} - q^{6} - 2 q^{7} + q^{8} + 5 q^{9} - 4 q^{11} + 5 q^{12} - 2 q^{14} + 6 q^{16} + q^{17} - 10 q^{18} - 4 q^{19} + 3 q^{21} + q^{22} - 4 q^{24} + 4 q^{25} + q^{27} - 5 q^{28} - q^{31} + 5 q^{32} + q^{33} + 6 q^{34} + 5 q^{36} + q^{38} + 2 q^{41} - 7 q^{42} + q^{43} - 5 q^{44} - 2 q^{47} + 6 q^{48} + 2 q^{49} - q^{50} + q^{51} - 4 q^{53} - 4 q^{54} - 8 q^{56} + q^{57} - q^{62} + 4 q^{64} + q^{66} + 4 q^{67} + 4 q^{71} - 10 q^{72} - q^{75} - 5 q^{76} + 2 q^{77} + 2 q^{79} + 6 q^{81} - 3 q^{82} + 5 q^{84} + q^{86} - q^{88} - 6 q^{93} + 3 q^{94} - 5 q^{96} + 2 q^{98} - 5 q^{99}+O(q^{100}) $ 4 * q - q^2 - q^3 + 5 * q^4 - q^6 - 2 * q^7 + q^8 + 5 * q^9 - 4 * q^11 + 5 * q^12 - 2 * q^14 + 6 * q^16 + q^17 - 10 * q^18 - 4 * q^19 + 3 * q^21 + q^22 - 4 * q^24 + 4 * q^25 + q^27 - 5 * q^28 - q^31 + 5 * q^32 + q^33 + 6 * q^34 + 5 * q^36 + q^38 + 2 * q^41 - 7 * q^42 + q^43 - 5 * q^44 - 2 * q^47 + 6 * q^48 + 2 * q^49 - q^50 + q^51 - 4 * q^53 - 4 * q^54 - 8 * q^56 + q^57 - q^62 + 4 * q^64 + q^66 + 4 * q^67 + 4 * q^71 - 10 * q^72 - q^75 - 5 * q^76 + 2 * q^77 + 2 * q^79 + 6 * q^81 - 3 * q^82 + 5 * q^84 + q^86 - q^88 - 6 * q^93 + 3 * q^94 - 5 * q^96 + 2 * q^98 - 5 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1007\mathbb{Z}\right)^\times$.

$n$	$267$	$743$
$\chi(n)$	$-1$	$-1$

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 $

$2$	1.95630	1.95630	0.978148	−	0.207912i	$-0.0666667\pi$
$2$	1.95630	1.95630	0.978148	+	0.207912i	$0.0666667\pi$
$3$	0.209057	0.209057	0.104528	−	0.994522i	$-0.466667\pi$
$3$	0.209057	0.209057	0.104528	+	0.994522i	$0.466667\pi$
$4$	2.82709	2.82709
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0.408977	0.408977
$7$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$7$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$8$	3.57433	3.57433
$9$	−0.956295	−0.956295
$10$	0	0
$11$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$11$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$12$	0.591023	0.591023
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−3.16535	−3.16535
$15$	0	0
$16$	4.16535	4.16535
$17$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$17$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$18$	−1.87080	−1.87080
$19$	−1.00000	−1.00000
$19$	−1.00000	−1.00000
$20$	0	0
$21$	−0.338261	−0.338261
$22$	−1.95630	−1.95630
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0.747238	0.747238
$25$	1.00000	1.00000
$26$	0	0
$27$	−0.408977	−0.408977
$28$	−4.57433	−4.57433
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−1.82709	−1.82709	−0.913545	−	0.406737i	$-0.866667\pi$
$31$	−1.82709	−1.82709	−0.913545	+	0.406737i	$0.866667\pi$
$32$	4.57433	4.57433
$33$	−0.209057	−0.209057
$34$	2.61803	2.61803
$35$	0	0
$36$	−2.70353	−2.70353
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	−1.95630	−1.95630
$39$	0	0
$40$	0	0
$41$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$41$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$42$	−0.661739	−0.661739
$43$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$43$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$44$	−2.82709	−2.82709
$45$	0	0
$46$	0	0
$47$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$47$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$48$	0.870796	0.870796
$49$	1.61803	1.61803
$50$	1.95630	1.95630
$51$	0.279773	0.279773
$52$	0	0
$53$	−1.00000	−1.00000
$53$	−1.00000	−1.00000
$54$	−0.800080	−0.800080
$55$	0	0
$56$	−5.78339	−5.78339
$57$	−0.209057	−0.209057
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−3.57433	−3.57433
$63$	1.54732	1.54732
$64$	4.78339	4.78339
$65$	0	0
$66$	−0.408977	−0.408977
$67$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$67$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$68$	3.78339	3.78339
$69$	0	0
$70$	0	0
$71$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$71$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$72$	−3.41811	−3.41811
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0.209057	0.209057
$76$	−2.82709	−2.82709
$77$	1.61803	1.61803
$78$	0	0
$79$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$79$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$80$	0	0
$81$	0.870796	0.870796
$82$	−1.20906	−1.20906
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	−0.956295	−0.956295
$85$	0	0
$86$	−0.408977	−0.408977
$87$	0	0
$88$	−3.57433	−3.57433
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.381966	−0.381966
$94$	1.20906	1.20906
$95$	0	0
$96$	0.956295	0.956295
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	3.16535	3.16535
$99$	0.956295	0.956295
$100$	2.82709	2.82709
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0.547318	0.547318
$103$	−1.33826	−1.33826	−0.669131	−	0.743145i	$-0.733333\pi$
$103$	−1.33826	−1.33826	−0.669131	+	0.743145i	$0.733333\pi$
$104$	0	0
$105$	0	0
$106$	−1.95630	−1.95630
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−1.15622	−1.15622
$109$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$109$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$110$	0	0
$111$	0	0
$112$	−6.73968	−6.73968
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	−0.408977	−0.408977
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.16535	−2.16535
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−0.129204	−0.129204
$124$	−5.16535	−5.16535
$125$	0	0
$126$	3.02701	3.02701
$127$	1.95630	1.95630	0.978148	−	0.207912i	$-0.0666667\pi$
$127$	1.95630	1.95630	0.978148	+	0.207912i	$0.0666667\pi$
$128$	4.78339	4.78339
$129$	−0.0437048	−0.0437048
$130$	0	0
$131$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$131$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$132$	−0.591023	−0.591023
$133$	1.61803	1.61803
$134$	1.95630	1.95630
$135$	0	0
$136$	4.78339	4.78339
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0.129204	0.129204
$142$	1.95630	1.95630
$143$	0	0
$144$	−3.98331	−3.98331
$145$	0	0
$146$	0	0
$147$	0.338261	0.338261
$148$	0	0
$149$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$149$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$150$	0.408977	0.408977
$151$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$151$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$152$	−3.57433	−3.57433
$153$	−1.27977	−1.27977
$154$	3.16535	3.16535
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	3.16535	3.16535
$159$	−0.209057	−0.209057
$160$	0	0
$161$	0	0
$162$	1.70353	1.70353
$163$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$163$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$164$	−1.74724	−1.74724
$165$	0	0
$166$	0	0
$167$	−1.33826	−1.33826	−0.669131	−	0.743145i	$-0.733333\pi$
$167$	−1.33826	−1.33826	−0.669131	+	0.743145i	$0.733333\pi$
$168$	−1.20906	−1.20906
$169$	1.00000	1.00000
$170$	0	0
$171$	0.956295	0.956295
$172$	−0.591023	−0.591023
$173$	−1.82709	−1.82709	−0.913545	−	0.406737i	$-0.866667\pi$
$173$	−1.82709	−1.82709	−0.913545	+	0.406737i	$0.866667\pi$
$174$	0	0
$175$	−1.61803	−1.61803
$176$	−4.16535	−4.16535
$177$	0	0
$178$	0	0
$179$	−1.82709	−1.82709	−0.913545	−	0.406737i	$-0.866667\pi$
$179$	−1.82709	−1.82709	−0.913545	+	0.406737i	$0.866667\pi$
$180$	0	0
$181$	1.95630	1.95630	0.978148	−	0.207912i	$-0.0666667\pi$
$181$	1.95630	1.95630	0.978148	+	0.207912i	$0.0666667\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	−0.747238	−0.747238
$187$	−1.33826	−1.33826
$188$	1.74724	1.74724
$189$	0.661739	0.661739
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	1.00000	1.00000
$193$	−1.33826	−1.33826	−0.669131	−	0.743145i	$-0.733333\pi$
$193$	−1.33826	−1.33826	−0.669131	+	0.743145i	$0.733333\pi$
$194$	0	0
$195$	0	0
$196$	4.57433	4.57433
$197$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$197$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$198$	1.87080	1.87080
$199$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$199$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$200$	3.57433	3.57433
$201$	0.209057	0.209057
$202$	0	0
$203$	0	0
$204$	0.790943	0.790943
$205$	0	0
$206$	−2.61803	−2.61803
$207$	0	0
$208$	0	0
$209$	1.00000	1.00000
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−2.82709	−2.82709
$213$	0.209057	0.209057
$214$	0	0
$215$	0	0
$216$	−1.46182	−1.46182
$217$	2.95630	2.95630
$218$	−1.20906	−1.20906
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−7.40142	−7.40142
$225$	−0.956295	−0.956295
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	−0.591023	−0.591023
$229$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$229$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$230$	0	0
$231$	0.338261	0.338261
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0.338261	0.338261
$238$	−4.23607	−4.23607
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0.591023	0.591023
$244$	0	0
$245$	0	0
$246$	−0.252762	−0.252762
$247$	0	0
$248$	−6.53062	−6.53062
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	4.37441	4.37441
$253$	0	0
$254$	3.82709	3.82709
$255$	0	0
$256$	4.57433	4.57433
$257$	−1.33826	−1.33826	−0.669131	−	0.743145i	$-0.733333\pi$
$257$	−1.33826	−1.33826	−0.669131	+	0.743145i	$0.733333\pi$
$258$	−0.0854995	−0.0854995
$259$	0	0
$260$	0	0
$261$	0	0
$262$	3.57433	3.57433
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−0.747238	−0.747238
$265$	0	0
$266$	3.16535	3.16535
$267$	0	0
$268$	2.82709	2.82709
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$271$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$272$	5.57433	5.57433
$273$	0	0
$274$	0	0
$275$	−1.00000	−1.00000
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	1.74724	1.74724
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0.252762	0.252762
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	2.82709	2.82709
$285$	0	0
$286$	0	0
$287$	1.00000	1.00000
$288$	−4.37441	−4.37441
$289$	0.790943	0.790943
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0.661739	0.661739
$295$	0	0
$296$	0	0
$297$	0.408977	0.408977
$298$	−3.16535	−3.16535
$299$	0	0
$300$	0.591023	0.591023
$301$	0.338261	0.338261
$302$	−1.20906	−1.20906
$303$	0	0
$304$	−4.16535	−4.16535
$305$	0	0
$306$	−2.50361	−2.50361
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	4.57433	4.57433
$309$	−0.279773	−0.279773
$310$	0	0
$311$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$311$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	4.57433	4.57433
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	−0.408977	−0.408977
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.33826	−1.33826
$324$	2.46182	2.46182
$325$	0	0
$326$	2.61803	2.61803
$327$	−0.129204	−0.129204
$328$	−2.20906	−2.20906
$329$	−1.00000	−1.00000
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	−2.61803	−2.61803
$335$	0	0
$336$	−1.40898	−1.40898
$337$	0.209057	0.209057	0.104528	−	0.994522i	$-0.466667\pi$
$337$	0.209057	0.209057	0.104528	+	0.994522i	$0.466667\pi$
$338$	1.95630	1.95630
$339$	0	0
$340$	0	0
$341$	1.82709	1.82709
$342$	1.87080	1.87080
$343$	−1.00000	−1.00000
$344$	−0.747238	−0.747238
$345$	0	0
$346$	−3.57433	−3.57433
$347$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$347$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	−3.16535	−3.16535
$351$	0	0
$352$	−4.57433	−4.57433
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.452682	−0.452682
$358$	−3.57433	−3.57433
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	1.00000	1.00000
$362$	3.82709	3.82709
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$367$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$368$	0	0
$369$	0.591023	0.591023
$370$	0	0
$371$	1.61803	1.61803
$372$	−1.07985	−1.07985
$373$	−1.82709	−1.82709	−0.913545	−	0.406737i	$-0.866667\pi$
$373$	−1.82709	−1.82709	−0.913545	+	0.406737i	$0.866667\pi$
$374$	−2.61803	−2.61803
$375$	0	0
$376$	2.20906	2.20906
$377$	0	0
$378$	1.29456	1.29456
$379$	1.95630	1.95630	0.978148	−	0.207912i	$-0.0666667\pi$
$379$	1.95630	1.95630	0.978148	+	0.207912i	$0.0666667\pi$
$380$	0	0
$381$	0.408977	0.408977
$382$	0	0
$383$	−2.00000	−2.00000	−1.00000			$\pi$
$383$	−2.00000	−2.00000	−1.00000			$\pi$
$384$	1.00000	1.00000
$385$	0	0
$386$	−2.61803	−2.61803
$387$	0.199920	0.199920
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	5.78339	5.78339
$393$	0.381966	0.381966
$394$	1.20906	1.20906
$395$	0	0
$396$	2.70353	2.70353
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	−3.82709	−3.82709
$399$	0.338261	0.338261
$400$	4.16535	4.16535
$401$	0.209057	0.209057	0.104528	−	0.994522i	$-0.466667\pi$
$401$	0.209057	0.209057	0.104528	+	0.994522i	$0.466667\pi$
$402$	0.408977	0.408977
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	1.00000	1.00000
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	−3.78339	−3.78339
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	1.95630	1.95630
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	0.209057	0.209057	0.104528	−	0.994522i	$-0.466667\pi$
$421$	0.209057	0.209057	0.104528	+	0.994522i	$0.466667\pi$
$422$	0	0
$423$	−0.591023	−0.591023
$424$	−3.57433	−3.57433
$425$	1.33826	1.33826
$426$	0.408977	0.408977
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−1.70353	−1.70353
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	5.78339	5.78339
$435$	0	0
$436$	−1.74724	−1.74724
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−1.54732	−1.54732
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−0.338261	−0.338261
$448$	−7.73968	−7.73968
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−1.87080	−1.87080
$451$	0.618034	0.618034
$452$	0	0
$453$	−0.129204	−0.129204
$454$	0	0
$455$	0	0
$456$	−0.747238	−0.747238
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	−0.408977	−0.408977
$459$	−0.547318	−0.547318
$460$	0	0
$461$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$461$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$462$	0.661739	0.661739
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$467$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$468$	0	0
$469$	−1.61803	−1.61803
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0.209057	0.209057
$474$	0.661739	0.661739
$475$	−1.00000	−1.00000
$476$	−6.12165	−6.12165
$477$	0.956295	0.956295
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	1.15622	1.15622
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0.279773	0.279773
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	−0.365272	−0.365272
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−7.61048	−7.61048
$497$	−1.61803	−1.61803
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	−0.279773	−0.279773
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	5.53062	5.53062
$505$	0	0
$506$	0	0
$507$	0.209057	0.209057
$508$	5.53062	5.53062
$509$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$509$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$510$	0	0
$511$	0	0
$512$	4.16535	4.16535
$513$	0.408977	0.408977
$514$	−2.61803	−2.61803
$515$	0	0
$516$	−0.123557	−0.123557
$517$	−0.618034	−0.618034
$518$	0	0
$519$	−0.381966	−0.381966
$520$	0	0
$521$	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			$\pi$
$524$	5.16535	5.16535
$525$	−0.338261	−0.338261
$526$	0	0
$527$	−2.44512	−2.44512
$528$	−0.870796	−0.870796
$529$	1.00000	1.00000
$530$	0	0
$531$	0	0
$532$	4.57433	4.57433
$533$	0	0
$534$	0	0
$535$	0	0
$536$	3.57433	3.57433
$537$	−0.381966	−0.381966
$538$	0	0
$539$	−1.61803	−1.61803
$540$	0	0
$541$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$541$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$542$	2.61803	2.61803
$543$	0.408977	0.408977
$544$	6.12165	6.12165
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	−1.95630	−1.95630
$551$	0	0
$552$	0	0
$553$	−2.61803	−2.61803
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	3.41811	3.41811
$559$	0	0
$560$	0	0
$561$	−0.279773	−0.279773
$562$	0	0
$563$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$563$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$564$	0.365272	0.365272
$565$	0	0
$566$	0	0
$567$	−1.40898	−1.40898
$568$	3.57433	3.57433
$569$	−1.33826	−1.33826	−0.669131	−	0.743145i	$-0.733333\pi$
$569$	−1.33826	−1.33826	−0.669131	+	0.743145i	$0.733333\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	1.95630	1.95630
$575$	0	0
$576$	−4.57433	−4.57433
$577$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$577$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$578$	1.54732	1.54732
$579$	−0.279773	−0.279773
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.00000	1.00000
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$587$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$588$	0.956295	0.956295
$589$	1.82709	1.82709
$590$	0	0
$591$	0.129204	0.129204
$592$	0	0
$593$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$593$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$594$	0.800080	0.800080
$595$	0	0
$596$	−4.57433	−4.57433
$597$	−0.408977	−0.408977
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0.747238	0.747238
$601$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$601$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$602$	0.661739	0.661739
$603$	−0.956295	−0.956295
$604$	−1.74724	−1.74724
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	−4.57433	−4.57433
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−3.61803	−3.61803
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	5.78339	5.78339
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	−0.547318	−0.547318
$619$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$619$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$620$	0	0
$621$	0	0
$622$	3.57433	3.57433
$623$	0	0
$624$	0	0
$625$	1.00000	1.00000
$626$	0	0
$627$	0.209057	0.209057
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	5.78339	5.78339
$633$	0	0
$634$	0	0
$635$	0	0
$636$	−0.591023	−0.591023
$637$	0	0
$638$	0	0
$639$	−0.956295	−0.956295
$640$	0	0
$641$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$641$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$642$	0	0
$643$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$643$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$644$	0	0
$645$	0	0
$646$	−2.61803	−2.61803
$647$	1.33826	1.33826	0.669131	−	0.743145i	$-0.266667\pi$
$647$	1.33826	1.33826	0.669131	+	0.743145i	$0.266667\pi$
$648$	3.11251	3.11251
$649$	0	0
$650$	0	0
$651$	0.618034	0.618034
$652$	3.78339	3.78339
$653$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$653$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$654$	−0.252762	−0.252762
$655$	0	0
$656$	−2.57433	−2.57433
$657$	0	0
$658$	−1.95630	−1.95630
$659$	−0.618034	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$659$	−0.618034	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−3.78339	−3.78339
$669$	0	0
$670$	0	0
$671$	0	0
$672$	−1.54732	−1.54732
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0.408977	0.408977
$675$	−0.408977	−0.408977
$676$	2.82709	2.82709
$677$	0.209057	0.209057	0.104528	−	0.994522i	$-0.466667\pi$
$677$	0.209057	0.209057	0.104528	+	0.994522i	$0.466667\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	3.57433	3.57433
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	2.70353	2.70353
$685$	0	0
$686$	−1.95630	−1.95630
$687$	−0.0437048	−0.0437048
$688$	−0.870796	−0.870796
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	−5.16535	−5.16535
$693$	−1.54732	−1.54732
$694$	−3.82709	−3.82709
$695$	0	0
$696$	0	0
$697$	−0.827091	−0.827091
$698$	0	0
$699$	0	0
$700$	−4.57433	−4.57433
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	−4.78339	−4.78339
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	−1.54732	−1.54732
$712$	0	0
$713$	0	0
$714$	−0.885579	−0.885579
$715$	0	0
$716$	−5.16535	−5.16535
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	2.16535	2.16535
$722$	1.95630	1.95630
$723$	0	0
$724$	5.53062	5.53062
$725$	0	0
$726$	0	0
$727$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$727$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$728$	0	0
$729$	−0.747238	−0.747238
$730$	0	0
$731$	−0.279773	−0.279773
$732$	0	0
$733$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$733$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$734$	−1.95630	−1.95630
$735$	0	0
$736$	0	0
$737$	−1.00000	−1.00000
$738$	1.15622	1.15622
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	3.16535	3.16535
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	−1.36527	−1.36527
$745$	0	0
$746$	−3.57433	−3.57433
$747$	0	0
$748$	−3.78339	−3.78339
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	2.57433	2.57433
$753$	0	0
$754$	0	0
$755$	0	0
$756$	1.87080	1.87080
$757$	0.618034	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$757$	0.618034	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$758$	3.82709	3.82709
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0.800080	0.800080
$763$	1.00000	1.00000
$764$	0	0
$765$	0	0
$766$	−3.91259	−3.91259
$767$	0	0
$768$	0.956295	0.956295
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	−0.279773	−0.279773
$772$	−3.78339	−3.78339
$773$	−1.33826	−1.33826	−0.669131	−	0.743145i	$-0.733333\pi$
$773$	−1.33826	−1.33826	−0.669131	+	0.743145i	$0.733333\pi$
$774$	0.391103	0.391103
$775$	−1.82709	−1.82709
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.618034	0.618034
$780$	0	0
$781$	−1.00000	−1.00000
$782$	0	0
$783$	0	0
$784$	6.73968	6.73968
$785$	0	0
$786$	0.747238	0.747238
$787$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$787$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$788$	1.74724	1.74724
$789$	0	0
$790$	0	0
$791$	0	0
$792$	3.41811	3.41811
$793$	0	0
$794$	0	0
$795$	0	0
$796$	−5.53062	−5.53062
$797$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$797$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$798$	0.661739	0.661739
$799$	0.827091	0.827091
$800$	4.57433	4.57433
$801$	0	0
$802$	0.408977	0.408977
$803$	0	0
$804$	0.591023	0.591023
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0.279773	0.279773
$814$	0	0
$815$	0	0
$816$	1.16535	1.16535
$817$	0.209057	0.209057
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$823$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$824$	−4.78339	−4.78339
$825$	−0.209057	−0.209057
$826$	0	0
$827$	−1.82709	−1.82709	−0.913545	−	0.406737i	$-0.866667\pi$
$827$	−1.82709	−1.82709	−0.913545	+	0.406737i	$0.866667\pi$
$828$	0	0
$829$	1.95630	1.95630	0.978148	−	0.207912i	$-0.0666667\pi$
$829$	1.95630	1.95630	0.978148	+	0.207912i	$0.0666667\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.16535	2.16535
$834$	0	0
$835$	0	0
$836$	2.82709	2.82709
$837$	0.747238	0.747238
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	0.408977	0.408977
$843$	0	0
$844$	0	0
$845$	0	0
$846$	−1.15622	−1.15622
$847$	0	0
$848$	−4.16535	−4.16535
$849$	0	0
$850$	2.61803	2.61803
$851$	0	0
$852$	0.591023	0.591023
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	−1.95630	−1.95630	−0.978148	−	0.207912i	$-0.933333\pi$
$859$	−1.95630	−1.95630	−0.978148	+	0.207912i	$0.933333\pi$
$860$	0	0
$861$	0.209057	0.209057
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−1.87080	−1.87080
$865$	0	0
$866$	0	0
$867$	0.165352	0.165352
$868$	8.35772	8.35772
$869$	−1.61803	−1.61803
$870$	0	0
$871$	0	0
$872$	−2.20906	−2.20906
$873$	0	0
$874$	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	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−3.02701	−3.02701
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$887$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$888$	0	0
$889$	−3.16535	−3.16535
$890$	0	0
$891$	−0.870796	−0.870796
$892$	0	0
$893$	−0.618034	−0.618034
$894$	−0.661739	−0.661739
$895$	0	0
$896$	−7.73968	−7.73968
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−2.70353	−2.70353
$901$	−1.33826	−1.33826
$902$	1.20906	1.20906
$903$	0.0707158	0.0707158
$904$	0	0
$905$	0	0
$906$	−0.252762	−0.252762
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−0.870796	−0.870796
$913$	0	0
$914$	0	0
$915$	0	0
$916$	−0.591023	−0.591023
$917$	−2.95630	−2.95630
$918$	−1.07072	−1.07072
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	−3.82709	−3.82709
$923$	0	0
$924$	0.956295	0.956295
$925$	0	0
$926$	0	0
$927$	1.27977	1.27977
$928$	0	0
$929$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$929$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$930$	0	0
$931$	−1.61803	−1.61803
$932$	0	0
$933$	0.381966	0.381966
$934$	−0.408977	−0.408977
$935$	0	0
$936$	0	0
$937$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$937$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$938$	−3.16535	−3.16535
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0.408977	0.408977
$947$	−1.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$947$	−1.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$948$	0.956295	0.956295
$949$	0	0
$950$	−1.95630	−1.95630
$951$	0	0
$952$	−7.73968	−7.73968
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	1.87080	1.87080
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	2.33826	2.33826
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−0.209057	−0.209057	−0.104528	−	0.994522i	$-0.533333\pi$
$967$	−0.209057	−0.209057	−0.104528	+	0.994522i	$0.533333\pi$
$968$	0	0
$969$	−0.279773	−0.279773
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	1.67088	1.67088
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$977$	1.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$978$	0.547318	0.547318
$979$	0	0
$980$	0	0
$981$	0.591023	0.591023
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	−0.461819	−0.461819
$985$	0	0
$986$	0	0
$987$	−0.209057	−0.209057
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	−8.35772	−8.35772
$993$	0	0
$994$	−3.16535	−3.16535
$995$	0	0
$996$	0	0
$997$	1.82709	1.82709	0.913545	−	0.406737i	$-0.133333\pi$
$997$	1.82709	1.82709	0.913545	+	0.406737i	$0.133333\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1007.1.d.e.1006.4	✓	4
19.18	odd	2		1007.1.d.f.1006.1	yes	4
53.52	even	2		1007.1.d.f.1006.1	yes	4
1007.1006	odd	2	CM	1007.1.d.e.1006.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1007.1.d.e.1006.4	✓	4	1.1	even	1	trivial
1007.1.d.e.1006.4	✓	4	1007.1006	odd	2	CM
1007.1.d.f.1006.1	yes	4	19.18	odd	2
1007.1.d.f.1006.1	yes	4	53.52	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1007 = 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1007.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.95630 q^{2} +0.209057 q^{3} +2.82709 q^{4} +0.408977 q^{6} -1.61803 q^{7} +3.57433 q^{8} -0.956295 q^{9} +O(q^{10})\)	\(q+1.95630 q^{2} +0.209057 q^{3} +2.82709 q^{4} +0.408977 q^{6} -1.61803 q^{7} +3.57433 q^{8} -0.956295 q^{9} -1.00000 q^{11} +0.591023 q^{12} -3.16535 q^{14} +4.16535 q^{16} +1.33826 q^{17} -1.87080 q^{18} -1.00000 q^{19} -0.338261 q^{21} -1.95630 q^{22} +0.747238 q^{24} +1.00000 q^{25} -0.408977 q^{27} -4.57433 q^{28} -1.82709 q^{31} +4.57433 q^{32} -0.209057 q^{33} +2.61803 q^{34} -2.70353 q^{36} -1.95630 q^{38} -0.618034 q^{41} -0.661739 q^{42} -0.209057 q^{43} -2.82709 q^{44} +0.618034 q^{47} +0.870796 q^{48} +1.61803 q^{49} +1.95630 q^{50} +0.279773 q^{51} -1.00000 q^{53} -0.800080 q^{54} -5.78339 q^{56} -0.209057 q^{57} -3.57433 q^{62} +1.54732 q^{63} +4.78339 q^{64} -0.408977 q^{66} +1.00000 q^{67} +3.78339 q^{68} +1.00000 q^{71} -3.41811 q^{72} +0.209057 q^{75} -2.82709 q^{76} +1.61803 q^{77} +1.61803 q^{79} +0.870796 q^{81} -1.20906 q^{82} -0.956295 q^{84} -0.408977 q^{86} -3.57433 q^{88} -0.381966 q^{93} +1.20906 q^{94} +0.956295 q^{96} +3.16535 q^{98} +0.956295 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} - q^{3} + 5 q^{4} - q^{6} - 2 q^{7} + q^{8} + 5 q^{9}+O(q^{10}) \) 4 * q - q^2 - q^3 + 5 * q^4 - q^6 - 2 * q^7 + q^8 + 5 * q^9	\( 4 q - q^{2} - q^{3} + 5 q^{4} - q^{6} - 2 q^{7} + q^{8} + 5 q^{9} - 4 q^{11} + 5 q^{12} - 2 q^{14} + 6 q^{16} + q^{17} - 10 q^{18} - 4 q^{19} + 3 q^{21} + q^{22} - 4 q^{24} + 4 q^{25} + q^{27} - 5 q^{28} - q^{31} + 5 q^{32} + q^{33} + 6 q^{34} + 5 q^{36} + q^{38} + 2 q^{41} - 7 q^{42} + q^{43} - 5 q^{44} - 2 q^{47} + 6 q^{48} + 2 q^{49} - q^{50} + q^{51} - 4 q^{53} - 4 q^{54} - 8 q^{56} + q^{57} - q^{62} + 4 q^{64} + q^{66} + 4 q^{67} + 4 q^{71} - 10 q^{72} - q^{75} - 5 q^{76} + 2 q^{77} + 2 q^{79} + 6 q^{81} - 3 q^{82} + 5 q^{84} + q^{86} - q^{88} - 6 q^{93} + 3 q^{94} - 5 q^{96} + 2 q^{98} - 5 q^{99}+O(q^{100}) \) 4 * q - q^2 - q^3 + 5 * q^4 - q^6 - 2 * q^7 + q^8 + 5 * q^9 - 4 * q^11 + 5 * q^12 - 2 * q^14 + 6 * q^16 + q^17 - 10 * q^18 - 4 * q^19 + 3 * q^21 + q^22 - 4 * q^24 + 4 * q^25 + q^27 - 5 * q^28 - q^31 + 5 * q^32 + q^33 + 6 * q^34 + 5 * q^36 + q^38 + 2 * q^41 - 7 * q^42 + q^43 - 5 * q^44 - 2 * q^47 + 6 * q^48 + 2 * q^49 - q^50 + q^51 - 4 * q^53 - 4 * q^54 - 8 * q^56 + q^57 - q^62 + 4 * q^64 + q^66 + 4 * q^67 + 4 * q^71 - 10 * q^72 - q^75 - 5 * q^76 + 2 * q^77 + 2 * q^79 + 6 * q^81 - 3 * q^82 + 5 * q^84 + q^86 - q^88 - 6 * q^93 + 3 * q^94 - 5 * q^96 + 2 * q^98 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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