LMFDB - Embedded newform 215.1.d.a.214.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [215,1,Mod(214,215)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("215.214"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(215, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 215 = 5 \cdot 43 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	215.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [3,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$0.107298977716$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{7}$
Projective field:	Galois closure of 7.1.9938375.1
Artin image:	$D_7$
Artin field:	Galois closure of 7.1.9938375.1

Embedding invariants

Embedding label			214.3
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	215.214

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.24698 q^{2} -0.445042 q^{3} +0.554958 q^{4} +1.00000 q^{5} -0.554958 q^{6} -1.80194 q^{7} -0.554958 q^{8} -0.801938 q^{9} +1.24698 q^{10} +1.24698 q^{11} -0.246980 q^{12} -2.24698 q^{14} -0.445042 q^{15} -1.24698 q^{16} -1.00000 q^{18} +0.554958 q^{20} +0.801938 q^{21} +1.55496 q^{22} +0.246980 q^{24} +1.00000 q^{25} +0.801938 q^{27} -1.00000 q^{28} -0.554958 q^{30} -1.80194 q^{31} -1.00000 q^{32} -0.554958 q^{33} -1.80194 q^{35} -0.445042 q^{36} +1.24698 q^{37} -0.554958 q^{40} -0.445042 q^{41} +1.00000 q^{42} +1.00000 q^{43} +0.692021 q^{44} -0.801938 q^{45} +0.554958 q^{48} +2.24698 q^{49} +1.24698 q^{50} +1.00000 q^{54} +1.24698 q^{55} +1.00000 q^{56} -0.445042 q^{59} -0.246980 q^{60} -2.24698 q^{62} +1.44504 q^{63} -0.692021 q^{66} -2.24698 q^{70} +0.445042 q^{72} -0.445042 q^{73} +1.55496 q^{74} -0.445042 q^{75} -2.24698 q^{77} +1.24698 q^{79} -1.24698 q^{80} +0.445042 q^{81} -0.554958 q^{82} +0.445042 q^{84} +1.24698 q^{86} -0.692021 q^{88} -1.00000 q^{90} +0.801938 q^{93} +0.445042 q^{96} +2.80194 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} - q^{3} + 2 q^{4} + 3 q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} - q^{10} - q^{11} + 4 q^{12} - 2 q^{14} - q^{15} + q^{16} - 3 q^{18} + 2 q^{20} - 2 q^{21} + 5 q^{22} - 4 q^{24} + 3 q^{25}+ \cdots - 3 q^{99}+O(q^{100}) $ 3 * q - q^2 - q^3 + 2 * q^4 + 3 * q^5 - 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9 - q^10 - q^11 + 4 * q^12 - 2 * q^14 - q^15 + q^16 - 3 * q^18 + 2 * q^20 - 2 * q^21 + 5 * q^22 - 4 * q^24 + 3 * q^25 - 2 * q^27 - 3 * q^28 - 2 * q^30 - q^31 - 3 * q^32 - 2 * q^33 - q^35 - q^36 - q^37 - 2 * q^40 - q^41 + 3 * q^42 + 3 * q^43 - 3 * q^44 + 2 * q^45 + 2 * q^48 + 2 * q^49 - q^50 + 3 * q^54 - q^55 + 3 * q^56 - q^59 + 4 * q^60 - 2 * q^62 + 4 * q^63 + 3 * q^66 - 2 * q^70 + q^72 - q^73 + 5 * q^74 - q^75 - 2 * q^77 - q^79 + q^80 + q^81 - 2 * q^82 + q^84 - q^86 + 3 * q^88 - 3 * q^90 - 2 * q^93 + q^96 + 4 * q^98 - 3 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/215\mathbb{Z}\right)^\times$.

$n$	$46$	$87$
$\chi(n)$	$-1$	$-1$

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 $

$2$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$2$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$3$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$3$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$4$	0.554958	0.554958
$5$	1.00000	1.00000
$5$	1.00000	1.00000
$6$	−0.554958	−0.554958
$7$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$7$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$8$	−0.554958	−0.554958
$9$	−0.801938	−0.801938
$10$	1.24698	1.24698
$11$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$11$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$12$	−0.246980	−0.246980
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−2.24698	−2.24698
$15$	−0.445042	−0.445042
$16$	−1.24698	−1.24698
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−1.00000	−1.00000
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0.554958	0.554958
$21$	0.801938	0.801938
$22$	1.55496	1.55496
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0.246980	0.246980
$25$	1.00000	1.00000
$26$	0	0
$27$	0.801938	0.801938
$28$	−1.00000	−1.00000
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	−0.554958	−0.554958
$31$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$31$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$32$	−1.00000	−1.00000
$33$	−0.554958	−0.554958
$34$	0	0
$35$	−1.80194	−1.80194
$36$	−0.445042	−0.445042
$37$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$37$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$38$	0	0
$39$	0	0
$40$	−0.554958	−0.554958
$41$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$41$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$42$	1.00000	1.00000
$43$	1.00000	1.00000
$43$	1.00000	1.00000
$44$	0.692021	0.692021
$45$	−0.801938	−0.801938
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0.554958	0.554958
$49$	2.24698	2.24698
$50$	1.24698	1.24698
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	1.00000	1.00000
$55$	1.24698	1.24698
$56$	1.00000	1.00000
$57$	0	0
$58$	0	0
$59$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$59$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$60$	−0.246980	−0.246980
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−2.24698	−2.24698
$63$	1.44504	1.44504
$64$	0	0
$65$	0	0
$66$	−0.692021	−0.692021
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	−2.24698	−2.24698
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0.445042	0.445042
$73$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$73$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$74$	1.55496	1.55496
$75$	−0.445042	−0.445042
$76$	0	0
$77$	−2.24698	−2.24698
$78$	0	0
$79$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$79$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$80$	−1.24698	−1.24698
$81$	0.445042	0.445042
$82$	−0.554958	−0.554958
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0.445042	0.445042
$85$	0	0
$86$	1.24698	1.24698
$87$	0	0
$88$	−0.692021	−0.692021
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	−1.00000	−1.00000
$91$	0	0
$92$	0	0
$93$	0.801938	0.801938
$94$	0	0
$95$	0	0
$96$	0.445042	0.445042
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	2.80194	2.80194
$99$	−1.00000	−1.00000
$100$	0.554958	0.554958
$101$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$101$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0.801938	0.801938
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0.445042	0.445042
$109$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$109$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$110$	1.55496	1.55496
$111$	−0.554958	−0.554958
$112$	2.24698	2.24698
$113$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$113$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−0.554958	−0.554958
$119$	0	0
$120$	0.246980	0.246980
$121$	0.554958	0.554958
$122$	0	0
$123$	0.198062	0.198062
$124$	−1.00000	−1.00000
$125$	1.00000	1.00000
$126$	1.80194	1.80194
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	1.00000	1.00000
$129$	−0.445042	−0.445042
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	−0.307979	−0.307979
$133$	0	0
$134$	0	0
$135$	0.801938	0.801938
$136$	0	0
$137$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$137$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$138$	0	0
$139$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$139$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$140$	−1.00000	−1.00000
$141$	0	0
$142$	0	0
$143$	0	0
$144$	1.00000	1.00000
$145$	0	0
$146$	−0.554958	−0.554958
$147$	−1.00000	−1.00000
$148$	0.692021	0.692021
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−0.554958	−0.554958
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	−2.80194	−2.80194
$155$	−1.80194	−1.80194
$156$	0	0
$157$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$157$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$158$	1.55496	1.55496
$159$	0	0
$160$	−1.00000	−1.00000
$161$	0	0
$162$	0.554958	0.554958
$163$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$163$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$164$	−0.246980	−0.246980
$165$	−0.554958	−0.554958
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	−0.445042	−0.445042
$169$	1.00000	1.00000
$170$	0	0
$171$	0	0
$172$	0.554958	0.554958
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	−1.80194	−1.80194
$176$	−1.55496	−1.55496
$177$	0.198062	0.198062
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−0.445042	−0.445042
$181$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$181$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.24698	1.24698
$186$	1.00000	1.00000
$187$	0	0
$188$	0	0
$189$	−1.44504	−1.44504
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	1.24698	1.24698
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−1.24698	−1.24698
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−0.554958	−0.554958
$201$	0	0
$202$	−2.24698	−2.24698
$203$	0	0
$204$	0	0
$205$	−0.445042	−0.445042
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	1.00000	1.00000
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.00000	1.00000
$216$	−0.445042	−0.445042
$217$	3.24698	3.24698
$218$	−2.24698	−2.24698
$219$	0.198062	0.198062
$220$	0.692021	0.692021
$221$	0	0
$222$	−0.692021	−0.692021
$223$	2.00000	2.00000	1.00000			$0$
$223$	2.00000	2.00000	1.00000			$0$
$224$	1.80194	1.80194
$225$	−0.801938	−0.801938
$226$	−2.24698	−2.24698
$227$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$227$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$228$	0	0
$229$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$229$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$230$	0	0
$231$	1.00000	1.00000
$232$	0	0
$233$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$233$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$234$	0	0
$235$	0	0
$236$	−0.246980	−0.246980
$237$	−0.554958	−0.554958
$238$	0	0
$239$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$239$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$240$	0.554958	0.554958
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0.692021	0.692021
$243$	−1.00000	−1.00000
$244$	0	0
$245$	2.24698	2.24698
$246$	0.246980	0.246980
$247$	0	0
$248$	1.00000	1.00000
$249$	0	0
$250$	1.24698	1.24698
$251$	2.00000	2.00000	1.00000			$0$
$251$	2.00000	2.00000	1.00000			$0$
$252$	0.801938	0.801938
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.24698	1.24698
$257$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$257$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$258$	−0.554958	−0.554958
$259$	−2.24698	−2.24698
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$263$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$264$	0.307979	0.307979
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$269$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$270$	1.00000	1.00000
$271$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$271$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$272$	0	0
$273$	0	0
$274$	−0.554958	−0.554958
$275$	1.24698	1.24698
$276$	0	0
$277$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$277$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$278$	−0.554958	−0.554958
$279$	1.44504	1.44504
$280$	1.00000	1.00000
$281$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$281$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.801938	0.801938
$288$	0.801938	0.801938
$289$	1.00000	1.00000
$290$	0	0
$291$	0	0
$292$	−0.246980	−0.246980
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−1.24698	−1.24698
$295$	−0.445042	−0.445042
$296$	−0.692021	−0.692021
$297$	1.00000	1.00000
$298$	0	0
$299$	0	0
$300$	−0.246980	−0.246980
$301$	−1.80194	−1.80194
$302$	0	0
$303$	0.801938	0.801938
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	−1.24698	−1.24698
$309$	0	0
$310$	−2.24698	−2.24698
$311$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$311$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$312$	0	0
$313$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$313$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$314$	−0.554958	−0.554958
$315$	1.44504	1.44504
$316$	0.692021	0.692021
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0.246980	0.246980
$325$	0	0
$326$	1.55496	1.55496
$327$	0.801938	0.801938
$328$	0.246980	0.246980
$329$	0	0
$330$	−0.692021	−0.692021
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	−1.00000	−1.00000
$334$	0	0
$335$	0	0
$336$	−1.00000	−1.00000
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	1.24698	1.24698
$339$	0.801938	0.801938
$340$	0	0
$341$	−2.24698	−2.24698
$342$	0	0
$343$	−2.24698	−2.24698
$344$	−0.554958	−0.554958
$345$	0	0
$346$	0	0
$347$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$347$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	−2.24698	−2.24698
$351$	0	0
$352$	−1.24698	−1.24698
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0.246980	0.246980
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.00000	2.00000	1.00000			$0$
$359$	2.00000	2.00000	1.00000			$0$
$360$	0.445042	0.445042
$361$	1.00000	1.00000
$362$	−0.554958	−0.554958
$363$	−0.246980	−0.246980
$364$	0	0
$365$	−0.445042	−0.445042
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0.356896	0.356896
$370$	1.55496	1.55496
$371$	0	0
$372$	0.445042	0.445042
$373$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$373$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$374$	0	0
$375$	−0.445042	−0.445042
$376$	0	0
$377$	0	0
$378$	−1.80194	−1.80194
$379$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$379$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$383$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$384$	−0.445042	−0.445042
$385$	−2.24698	−2.24698
$386$	0	0
$387$	−0.801938	−0.801938
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−1.24698	−1.24698
$393$	0	0
$394$	0	0
$395$	1.24698	1.24698
$396$	−0.554958	−0.554958
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	−1.24698	−1.24698
$401$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$401$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$402$	0	0
$403$	0	0
$404$	−1.00000	−1.00000
$405$	0.445042	0.445042
$406$	0	0
$407$	1.55496	1.55496
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	−0.554958	−0.554958
$411$	0.198062	0.198062
$412$	0	0
$413$	0.801938	0.801938
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0.198062	0.198062
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0.445042	0.445042
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	1.24698	1.24698
$431$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$431$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$432$	−1.00000	−1.00000
$433$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$433$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$434$	4.04892	4.04892
$435$	0	0
$436$	−1.00000	−1.00000
$437$	0	0
$438$	0.246980	0.246980
$439$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$439$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$440$	−0.692021	−0.692021
$441$	−1.80194	−1.80194
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	−0.307979	−0.307979
$445$	0	0
$446$	2.49396	2.49396
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−1.00000	−1.00000
$451$	−0.554958	−0.554958
$452$	−1.00000	−1.00000
$453$	0	0
$454$	−2.24698	−2.24698
$455$	0	0
$456$	0	0
$457$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$457$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$458$	1.55496	1.55496
$459$	0	0
$460$	0	0
$461$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$461$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$462$	1.24698	1.24698
$463$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$463$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$464$	0	0
$465$	0.801938	0.801938
$466$	−0.554958	−0.554958
$467$	2.00000	2.00000	1.00000			$0$
$467$	2.00000	2.00000	1.00000			$0$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0.198062	0.198062
$472$	0.246980	0.246980
$473$	1.24698	1.24698
$474$	−0.692021	−0.692021
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−2.24698	−2.24698
$479$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$479$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$480$	0.445042	0.445042
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0.307979	0.307979
$485$	0	0
$486$	−1.24698	−1.24698
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	−0.554958	−0.554958
$490$	2.80194	2.80194
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0.109916	0.109916
$493$	0	0
$494$	0	0
$495$	−1.00000	−1.00000
$496$	2.24698	2.24698
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0.554958	0.554958
$501$	0	0
$502$	2.49396	2.49396
$503$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$503$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$504$	−0.801938	−0.801938
$505$	−1.80194	−1.80194
$506$	0	0
$507$	−0.445042	−0.445042
$508$	0	0
$509$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$509$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$510$	0	0
$511$	0.801938	0.801938
$512$	0.554958	0.554958
$513$	0	0
$514$	1.55496	1.55496
$515$	0	0
$516$	−0.246980	−0.246980
$517$	0	0
$518$	−2.80194	−2.80194
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$523$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$524$	0	0
$525$	0.801938	0.801938
$526$	1.55496	1.55496
$527$	0	0
$528$	0.692021	0.692021
$529$	1.00000	1.00000
$530$	0	0
$531$	0.356896	0.356896
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	1.55496	1.55496
$539$	2.80194	2.80194
$540$	0.445042	0.445042
$541$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$541$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$542$	−0.554958	−0.554958
$543$	0.198062	0.198062
$544$	0	0
$545$	−1.80194	−1.80194
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	−0.246980	−0.246980
$549$	0	0
$550$	1.55496	1.55496
$551$	0	0
$552$	0	0
$553$	−2.24698	−2.24698
$554$	−2.24698	−2.24698
$555$	−0.554958	−0.554958
$556$	−0.246980	−0.246980
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	1.80194	1.80194
$559$	0	0
$560$	2.24698	2.24698
$561$	0	0
$562$	1.55496	1.55496
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	−1.80194	−1.80194
$566$	0	0
$567$	−0.801938	−0.801938
$568$	0	0
$569$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$569$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	1.00000	1.00000
$575$	0	0
$576$	0	0
$577$	2.00000	2.00000	1.00000			$0$
$577$	2.00000	2.00000	1.00000			$0$
$578$	1.24698	1.24698
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0.246980	0.246980
$585$	0	0
$586$	0	0
$587$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$587$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$588$	−0.554958	−0.554958
$589$	0	0
$590$	−0.554958	−0.554958
$591$	0	0
$592$	−1.55496	−1.55496
$593$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$593$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$594$	1.24698	1.24698
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$599$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$600$	0.246980	0.246980
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	−2.24698	−2.24698
$603$	0	0
$604$	0	0
$605$	0.554958	0.554958
$606$	1.00000	1.00000
$607$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$607$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0.198062	0.198062
$616$	1.24698	1.24698
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$619$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$620$	−1.00000	−1.00000
$621$	0	0
$622$	−0.554958	−0.554958
$623$	0	0
$624$	0	0
$625$	1.00000	1.00000
$626$	−2.24698	−2.24698
$627$	0	0
$628$	−0.246980	−0.246980
$629$	0	0
$630$	1.80194	1.80194
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	−0.692021	−0.692021
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	1.00000	1.00000
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	−0.445042	−0.445042
$646$	0	0
$647$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$647$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$648$	−0.246980	−0.246980
$649$	−0.554958	−0.554958
$650$	0	0
$651$	−1.44504	−1.44504
$652$	0.692021	0.692021
$653$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$653$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$654$	1.00000	1.00000
$655$	0	0
$656$	0.554958	0.554958
$657$	0.356896	0.356896
$658$	0	0
$659$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$659$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$660$	−0.307979	−0.307979
$661$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$661$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−1.24698	−1.24698
$667$	0	0
$668$	0	0
$669$	−0.890084	−0.890084
$670$	0	0
$671$	0	0
$672$	−0.801938	−0.801938
$673$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$673$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$674$	0	0
$675$	0.801938	0.801938
$676$	0.554958	0.554958
$677$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$677$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$678$	1.00000	1.00000
$679$	0	0
$680$	0	0
$681$	0.801938	0.801938
$682$	−2.80194	−2.80194
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	−0.445042	−0.445042
$686$	−2.80194	−2.80194
$687$	−0.554958	−0.554958
$688$	−1.24698	−1.24698
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	1.80194	1.80194
$694$	1.55496	1.55496
$695$	−0.445042	−0.445042
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0.198062	0.198062
$700$	−1.00000	−1.00000
$701$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$701$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.24698	3.24698
$708$	0.109916	0.109916
$709$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$709$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$710$	0	0
$711$	−1.00000	−1.00000
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.801938	0.801938
$718$	2.49396	2.49396
$719$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$719$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$720$	1.00000	1.00000
$721$	0	0
$722$	1.24698	1.24698
$723$	0	0
$724$	−0.246980	−0.246980
$725$	0	0
$726$	−0.307979	−0.307979
$727$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$727$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$728$	0	0
$729$	0	0
$730$	−0.554958	−0.554958
$731$	0	0
$732$	0	0
$733$	2.00000	2.00000	1.00000			$0$
$733$	2.00000	2.00000	1.00000			$0$
$734$	0	0
$735$	−1.00000	−1.00000
$736$	0	0
$737$	0	0
$738$	0.445042	0.445042
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0.692021	0.692021
$741$	0	0
$742$	0	0
$743$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$743$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$744$	−0.445042	−0.445042
$745$	0	0
$746$	−2.24698	−2.24698
$747$	0	0
$748$	0	0
$749$	0	0
$750$	−0.554958	−0.554958
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	−0.890084	−0.890084
$754$	0	0
$755$	0	0
$756$	−0.801938	−0.801938
$757$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$757$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$758$	1.55496	1.55496
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	3.24698	3.24698
$764$	0	0
$765$	0	0
$766$	−0.554958	−0.554958
$767$	0	0
$768$	−0.554958	−0.554958
$769$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$769$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$770$	−2.80194	−2.80194
$771$	−0.554958	−0.554958
$772$	0	0
$773$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$773$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$774$	−1.00000	−1.00000
$775$	−1.80194	−1.80194
$776$	0	0
$777$	1.00000	1.00000
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−2.80194	−2.80194
$785$	−0.445042	−0.445042
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	−0.554958	−0.554958
$790$	1.55496	1.55496
$791$	3.24698	3.24698
$792$	0.554958	0.554958
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−1.00000	−1.00000
$801$	0	0
$802$	−2.24698	−2.24698
$803$	−0.554958	−0.554958
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−0.554958	−0.554958
$808$	1.00000	1.00000
$809$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$809$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$810$	0.554958	0.554958
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0.198062	0.198062
$814$	1.93900	1.93900
$815$	1.24698	1.24698
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	−0.246980	−0.246980
$821$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$821$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$822$	0.246980	0.246980
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	−0.554958	−0.554958
$826$	1.00000	1.00000
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0.801938	0.801938
$832$	0	0
$833$	0	0
$834$	0.246980	0.246980
$835$	0	0
$836$	0	0
$837$	−1.44504	−1.44504
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	−0.445042	−0.445042
$841$	1.00000	1.00000
$842$	0	0
$843$	−0.554958	−0.554958
$844$	0	0
$845$	1.00000	1.00000
$846$	0	0
$847$	−1.00000	−1.00000
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0.554958	0.554958
$861$	−0.356896	−0.356896
$862$	−0.554958	−0.554958
$863$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$863$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$864$	−0.801938	−0.801938
$865$	0	0
$866$	1.55496	1.55496
$867$	−0.445042	−0.445042
$868$	1.80194	1.80194
$869$	1.55496	1.55496
$870$	0	0
$871$	0	0
$872$	1.00000	1.00000
$873$	0	0
$874$	0	0
$875$	−1.80194	−1.80194
$876$	0.109916	0.109916
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	1.55496	1.55496
$879$	0	0
$880$	−1.55496	−1.55496
$881$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$881$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$882$	−2.24698	−2.24698
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0.198062	0.198062
$886$	0	0
$887$	2.00000	2.00000	1.00000			$0$
$887$	2.00000	2.00000	1.00000			$0$
$888$	0.307979	0.307979
$889$	0	0
$890$	0	0
$891$	0.554958	0.554958
$892$	1.10992	1.10992
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−1.80194	−1.80194
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−0.445042	−0.445042
$901$	0	0
$902$	−0.692021	−0.692021
$903$	0.801938	0.801938
$904$	1.00000	1.00000
$905$	−0.445042	−0.445042
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	−1.00000	−1.00000
$909$	1.44504	1.44504
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	−0.554958	−0.554958
$915$	0	0
$916$	0.692021	0.692021
$917$	0	0
$918$	0	0
$919$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$919$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$920$	0	0
$921$	0	0
$922$	1.55496	1.55496
$923$	0	0
$924$	0.554958	0.554958
$925$	1.24698	1.24698
$926$	−0.554958	−0.554958
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	1.00000	1.00000
$931$	0	0
$932$	−0.246980	−0.246980
$933$	0.198062	0.198062
$934$	2.49396	2.49396
$935$	0	0
$936$	0	0
$937$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$937$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$938$	0	0
$939$	0.801938	0.801938
$940$	0	0
$941$	2.00000	2.00000	1.00000			$0$
$941$	2.00000	2.00000	1.00000			$0$
$942$	0.246980	0.246980
$943$	0	0
$944$	0.554958	0.554958
$945$	−1.44504	−1.44504
$946$	1.55496	1.55496
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	−0.307979	−0.307979
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$953$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$954$	0	0
$955$	0	0
$956$	−1.00000	−1.00000
$957$	0	0
$958$	−2.24698	−2.24698
$959$	0.801938	0.801938
$960$	0	0
$961$	2.24698	2.24698
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−0.307979	−0.307979
$969$	0	0
$970$	0	0
$971$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$971$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$972$	−0.554958	−0.554958
$973$	0.801938	0.801938
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	−0.692021	−0.692021
$979$	0	0
$980$	1.24698	1.24698
$981$	1.44504	1.44504
$982$	0	0
$983$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$983$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$984$	−0.109916	−0.109916
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	−1.24698	−1.24698
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	1.80194	1.80194
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$997$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$998$	0	0
$999$	1.00000	1.00000

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	215.1.d.a.214.3	✓	3
3.2	odd	2		1935.1.h.b.1504.1		3
4.3	odd	2		3440.1.p.b.3009.2		3
5.2	odd	4		1075.1.c.c.601.5		6
5.3	odd	4		1075.1.c.c.601.2		6
5.4	even	2		215.1.d.b.214.1	yes	3
15.14	odd	2		1935.1.h.a.1504.3		3
20.19	odd	2		3440.1.p.a.3009.2		3
43.42	odd	2		215.1.d.b.214.1	yes	3
129.128	even	2		1935.1.h.a.1504.3		3
172.171	even	2		3440.1.p.a.3009.2		3
215.42	even	4		1075.1.c.c.601.2		6
215.128	even	4		1075.1.c.c.601.5		6
215.214	odd	2	CM	215.1.d.a.214.3	✓	3
645.644	even	2		1935.1.h.b.1504.1		3
860.859	even	2		3440.1.p.b.3009.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
215.1.d.a.214.3	✓	3	1.1	even	1	trivial
215.1.d.a.214.3	✓	3	215.214	odd	2	CM
215.1.d.b.214.1	yes	3	5.4	even	2
215.1.d.b.214.1	yes	3	43.42	odd	2
1075.1.c.c.601.2		6	5.3	odd	4
1075.1.c.c.601.2		6	215.42	even	4
1075.1.c.c.601.5		6	5.2	odd	4
1075.1.c.c.601.5		6	215.128	even	4
1935.1.h.a.1504.3		3	15.14	odd	2
1935.1.h.a.1504.3		3	129.128	even	2
1935.1.h.b.1504.1		3	3.2	odd	2
1935.1.h.b.1504.1		3	645.644	even	2
3440.1.p.a.3009.2		3	20.19	odd	2
3440.1.p.a.3009.2		3	172.171	even	2
3440.1.p.b.3009.2		3	4.3	odd	2
3440.1.p.b.3009.2		3	860.859	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 215 = 5 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	215.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.24698 q^{2} -0.445042 q^{3} +0.554958 q^{4} +1.00000 q^{5} -0.554958 q^{6} -1.80194 q^{7} -0.554958 q^{8} -0.801938 q^{9} +1.24698 q^{10} +1.24698 q^{11} -0.246980 q^{12} -2.24698 q^{14} -0.445042 q^{15} -1.24698 q^{16} -1.00000 q^{18} +0.554958 q^{20} +0.801938 q^{21} +1.55496 q^{22} +0.246980 q^{24} +1.00000 q^{25} +0.801938 q^{27} -1.00000 q^{28} -0.554958 q^{30} -1.80194 q^{31} -1.00000 q^{32} -0.554958 q^{33} -1.80194 q^{35} -0.445042 q^{36} +1.24698 q^{37} -0.554958 q^{40} -0.445042 q^{41} +1.00000 q^{42} +1.00000 q^{43} +0.692021 q^{44} -0.801938 q^{45} +0.554958 q^{48} +2.24698 q^{49} +1.24698 q^{50} +1.00000 q^{54} +1.24698 q^{55} +1.00000 q^{56} -0.445042 q^{59} -0.246980 q^{60} -2.24698 q^{62} +1.44504 q^{63} -0.692021 q^{66} -2.24698 q^{70} +0.445042 q^{72} -0.445042 q^{73} +1.55496 q^{74} -0.445042 q^{75} -2.24698 q^{77} +1.24698 q^{79} -1.24698 q^{80} +0.445042 q^{81} -0.554958 q^{82} +0.445042 q^{84} +1.24698 q^{86} -0.692021 q^{88} -1.00000 q^{90} +0.801938 q^{93} +0.445042 q^{96} +2.80194 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} - q^{3} + 2 q^{4} + 3 q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} - q^{10} - q^{11} + 4 q^{12} - 2 q^{14} - q^{15} + q^{16} - 3 q^{18} + 2 q^{20} - 2 q^{21} + 5 q^{22} - 4 q^{24} + 3 q^{25}+ \cdots - 3 q^{99}+O(q^{100}) \) 3 * q - q^2 - q^3 + 2 * q^4 + 3 * q^5 - 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9 - q^10 - q^11 + 4 * q^12 - 2 * q^14 - q^15 + q^16 - 3 * q^18 + 2 * q^20 - 2 * q^21 + 5 * q^22 - 4 * q^24 + 3 * q^25 - 2 * q^27 - 3 * q^28 - 2 * q^30 - q^31 - 3 * q^32 - 2 * q^33 - q^35 - q^36 - q^37 - 2 * q^40 - q^41 + 3 * q^42 + 3 * q^43 - 3 * q^44 + 2 * q^45 + 2 * q^48 + 2 * q^49 - q^50 + 3 * q^54 - q^55 + 3 * q^56 - q^59 + 4 * q^60 - 2 * q^62 + 4 * q^63 + 3 * q^66 - 2 * q^70 + q^72 - q^73 + 5 * q^74 - q^75 - 2 * q^77 - q^79 + q^80 + q^81 - 2 * q^82 + q^84 - q^86 + 3 * q^88 - 3 * q^90 - 2 * q^93 + q^96 + 4 * q^98 - 3 * q^99

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