LMFDB - Embedded newform 1775.1.c.a.1774.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1775,1,Mod(1774,1775)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1775, 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("1775.1774");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1775 = 5^{2} \cdot 71 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1775.c (of order $2$, degree $1$, not 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:	no
Analytic conductor:	$0.885840397424$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 71)
Projective image:	$D_{7}$
Projective field:	Galois closure of 7.1.357911.1
Artin image:	$C_4\times D_7$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{28} - \cdots)$

Embedding invariants

Embedding label			1774.4
Root			$-0.445042i$ of defining polynomial
Character	$\chi$	$=$	1775.1774
Dual form			1775.1.c.a.1774.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.445042i q^{2} +1.24698i q^{3} +0.801938 q^{4} -0.554958 q^{6} +0.801938i q^{8} -0.554958 q^{9} +O(q^{10})$	$q+0.445042i q^{2} +1.24698i q^{3} +0.801938 q^{4} -0.554958 q^{6} +0.801938i q^{8} -0.554958 q^{9} +1.00000i q^{12} +0.445042 q^{16} -0.246980i q^{18} +0.445042 q^{19} -1.00000 q^{24} +0.554958i q^{27} -1.24698 q^{29} +1.00000i q^{32} -0.445042 q^{36} +0.445042i q^{37} +0.198062i q^{38} -1.80194i q^{43} +0.554958i q^{48} -1.00000 q^{49} -0.246980 q^{54} +0.554958i q^{57} -0.554958i q^{58} +1.00000 q^{71} -0.445042i q^{72} -1.80194i q^{73} -0.198062 q^{74} +0.356896 q^{76} +1.80194 q^{79} -1.24698 q^{81} -0.445042i q^{83} +0.801938 q^{86} -1.55496i q^{87} -1.24698 q^{89} -1.24698 q^{96} -0.445042i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) $ 6 * q - 4 * q^4 - 4 * q^6 - 4 * q^9	$ 6 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 2 q^{16} + 2 q^{19} - 6 q^{24} + 2 q^{29} - 2 q^{36} - 6 q^{49} + 8 q^{54} + 6 q^{71} - 10 q^{74} - 6 q^{76} + 2 q^{79} + 2 q^{81} - 4 q^{86} + 2 q^{89} + 2 q^{96}+O(q^{100}) $ 6 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 + 2 * q^16 + 2 * q^19 - 6 * q^24 + 2 * q^29 - 2 * q^36 - 6 * q^49 + 8 * q^54 + 6 * q^71 - 10 * q^74 - 6 * q^76 + 2 * q^79 + 2 * q^81 - 4 * q^86 + 2 * q^89 + 2 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1775\mathbb{Z}\right)^\times$.

$n$	$427$	$1001$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.445042i	0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$2$	0.445042i	0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$3$	1.24698i	1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$3$	1.24698i	1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$4$	0.801938	0.801938
$5$	0	0
$5$	0	0
$6$	−0.554958	−0.554958
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0.801938i	0.801938i
$9$	−0.554958	−0.554958
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	1.00000i	1.00000i
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0.445042	0.445042
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	− 0.246980i	− 0.246980i
$19$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$19$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−1.00000	−1.00000
$25$	0	0
$26$	0	0
$27$	0.554958i	0.554958i
$28$	0	0
$29$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$29$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	1.00000i	1.00000i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−0.445042	−0.445042
$37$	0.445042i	0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$37$	0.445042i	0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$38$	0.198062i	0.198062i
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	− 1.80194i	− 1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$43$	− 1.80194i	− 1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0.554958i	0.554958i
$49$	−1.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	−0.246980	−0.246980
$55$	0	0
$56$	0	0
$57$	0.554958i	0.554958i
$58$	− 0.554958i	− 0.554958i
$59$	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			$\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.00000	1.00000
$71$	1.00000	1.00000
$72$	− 0.445042i	− 0.445042i
$73$	− 1.80194i	− 1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$73$	− 1.80194i	− 1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$74$	−0.198062	−0.198062
$75$	0	0
$76$	0.356896	0.356896
$77$	0	0
$78$	0	0
$79$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$79$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$80$	0	0
$81$	−1.24698	−1.24698
$82$	0	0
$83$	− 0.445042i	− 0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$83$	− 0.445042i	− 0.445042i	0.974928	−	0.222521i	$-0.0714286\pi$
$84$	0	0
$85$	0	0
$86$	0.801938	0.801938
$87$	− 1.55496i	− 1.55496i
$88$	0	0
$89$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$89$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	−1.24698	−1.24698
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	− 0.445042i	− 0.445042i
$99$	0	0
$100$	0	0
$101$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$101$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$102$	0	0
$103$	1.24698i	1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$103$	1.24698i	1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$107$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$108$	0.445042i	0.445042i
$109$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$109$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$110$	0	0
$111$	−0.554958	−0.554958
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	−0.246980	−0.246980
$115$	0	0
$116$	−1.00000	−1.00000
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	1.00000i	1.00000i
$129$	2.24698	2.24698
$130$	0	0
$131$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$131$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0.445042i	0.445042i
$143$	0	0
$144$	−0.246980	−0.246980
$145$	0	0
$146$	0.801938	0.801938
$147$	− 1.24698i	− 1.24698i
$148$	0.356896i	0.356896i
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$151$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$152$	0.356896i	0.356896i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 1.24698i	− 1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$157$	− 1.24698i	− 1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$158$	0.801938i	0.801938i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	− 0.554958i	− 0.554958i
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0.198062	0.198062
$167$	1.80194i	1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$167$	1.80194i	1.80194i	−0.433884	+	0.900969i	$0.642857\pi$
$168$	0	0
$169$	−1.00000	−1.00000
$170$	0	0
$171$	−0.246980	−0.246980
$172$	− 1.44504i	− 1.44504i
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0.692021	0.692021
$175$	0	0
$176$	0	0
$177$	0	0
$178$	− 0.554958i	− 0.554958i
$179$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$179$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$191$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$192$	0	0
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	−0.801938	−0.801938
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$199$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$200$	0	0
$201$	0	0
$202$	− 0.198062i	− 0.198062i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−0.554958	−0.554958
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	1.24698i	1.24698i
$214$	0.890084	0.890084
$215$	0	0
$216$	−0.445042	−0.445042
$217$	0	0
$218$	− 0.554958i	− 0.554958i
$219$	2.24698	2.24698
$220$	0	0
$221$	0	0
$222$	− 0.246980i	− 0.246980i
$223$	− 0.445042i	− 0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$223$	− 0.445042i	− 0.445042i	0.974928	−	0.222521i	$-0.0714286\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0.445042i	0.445042i
$229$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$229$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$230$	0	0
$231$	0	0
$232$	− 1.00000i	− 1.00000i
$233$	− 0.445042i	− 0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$233$	− 0.445042i	− 0.445042i	0.974928	−	0.222521i	$-0.0714286\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.24698i	2.24698i
$238$	0	0
$239$	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			$\pi$
$242$	0.445042i	0.445042i
$243$	− 1.00000i	− 1.00000i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0.554958	0.554958
$250$	0	0
$251$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$251$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−0.445042	−0.445042
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	1.00000i	1.00000i
$259$	0	0
$260$	0	0
$261$	0.692021	0.692021
$262$	− 0.801938i	− 0.801938i
$263$	− 1.80194i	− 1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$263$	− 1.80194i	− 1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 1.55496i	− 1.55496i
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$271$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0.445042i	0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$277$	0.445042i	0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0.801938	0.801938
$285$	0	0
$286$	0	0
$287$	0	0
$288$	− 0.554958i	− 0.554958i
$289$	−1.00000	−1.00000
$290$	0	0
$291$	0	0
$292$	− 1.44504i	− 1.44504i
$293$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$293$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$294$	0.554958	0.554958
$295$	0	0
$296$	−0.356896	−0.356896
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0.554958i	0.554958i
$303$	− 0.554958i	− 0.554958i
$304$	0.198062	0.198062
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	−1.55496	−1.55496
$310$	0	0
$311$	−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
$313$	− 0.445042i	− 0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$313$	− 0.445042i	− 0.445042i	0.974928	−	0.222521i	$-0.0714286\pi$
$314$	0.554958	0.554958
$315$	0	0
$316$	1.44504	1.44504
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	2.49396	2.49396
$322$	0	0
$323$	0	0
$324$	−1.00000	−1.00000
$325$	0	0
$326$	0	0
$327$	− 1.55496i	− 1.55496i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	− 0.356896i	− 0.356896i
$333$	− 0.246980i	− 0.246980i
$334$	−0.801938	−0.801938
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	− 0.445042i	− 0.445042i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	− 0.109916i	− 0.109916i
$343$	0	0
$344$	1.44504	1.44504
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	− 1.24698i	− 1.24698i
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	−1.00000	−1.00000
$357$	0	0
$358$	0.801938i	0.801938i
$359$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$359$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$360$	0	0
$361$	−0.801938	−0.801938
$362$	0	0
$363$	1.24698i	1.24698i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.80194i	1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$367$	1.80194i	1.80194i	−0.433884	+	0.900969i	$0.642857\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 1.80194i	− 1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$373$	− 1.80194i	− 1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$379$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$380$	0	0
$381$	0	0
$382$	0.554958i	0.554958i
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	−1.24698	−1.24698
$385$	0	0
$386$	0	0
$387$	1.00000i	1.00000i
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	− 0.801938i	− 0.801938i
$393$	− 2.24698i	− 2.24698i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0.198062i	0.198062i
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	−0.356896	−0.356896
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$409$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$410$	0	0
$411$	0	0
$412$	1.00000i	1.00000i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$419$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	−0.554958	−0.554958
$427$	0	0
$428$	− 1.60388i	− 1.60388i
$429$	0	0
$430$	0	0
$431$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$431$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$432$	0.246980i	0.246980i
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	−1.00000	−1.00000
$437$	0	0
$438$	1.00000i	1.00000i
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0.554958	0.554958
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	−0.445042	−0.445042
$445$	0	0
$446$	0.198062	0.198062
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	1.55496i	1.55496i
$454$	0	0
$455$	0	0
$456$	−0.445042	−0.445042
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0.801938i	0.801938i
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	− 0.445042i	− 0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$463$	− 0.445042i	− 0.445042i	0.974928	−	0.222521i	$-0.0714286\pi$
$464$	−0.554958	−0.554958
$465$	0	0
$466$	0.198062	0.198062
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	1.55496	1.55496
$472$	0	0
$473$	0	0
$474$	−1.00000	−1.00000
$475$	0	0
$476$	0	0
$477$	0	0
$478$	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
$484$	0.801938	0.801938
$485$	0	0
$486$	0.445042	0.445042
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0.246980i	0.246980i
$499$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$499$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$500$	0	0
$501$	−2.24698	−2.24698
$502$	− 0.198062i	− 0.198062i
$503$	1.24698i	1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$503$	1.24698i	1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 1.24698i	− 1.24698i
$508$	0	0
$509$	−2.00000	−2.00000	−1.00000			$\pi$
$509$	−2.00000	−2.00000	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	0.801938i	0.801938i
$513$	0.246980i	0.246980i
$514$	0	0
$515$	0	0
$516$	1.80194	1.80194
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$521$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$522$	0.307979i	0.307979i
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	−1.44504	−1.44504
$525$	0	0
$526$	0.801938	0.801938
$527$	0	0
$528$	0	0
$529$	−1.00000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0.692021	0.692021
$535$	0	0
$536$	0	0
$537$	2.24698i	2.24698i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0.554958i	0.554958i
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0.445042i	0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$547$	0.445042i	0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.554958	−0.554958
$552$	0	0
$553$	0	0
$554$	−0.198062	−0.198062
$555$	0	0
$556$	0	0
$557$	− 1.24698i	− 1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$557$	− 1.24698i	− 1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0.801938i	0.801938i
$569$	1.80194	1.80194	0.900969	−	0.433884i	$-0.142857\pi$
$569$	1.80194	1.80194	0.900969	+	0.433884i	$0.142857\pi$
$570$	0	0
$571$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$571$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$572$	0	0
$573$	1.55496i	1.55496i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.80194i	1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$577$	1.80194i	1.80194i	−0.433884	+	0.900969i	$0.642857\pi$
$578$	− 0.445042i	− 0.445042i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	1.44504	1.44504
$585$	0	0
$586$	−0.890084	−0.890084
$587$	− 1.24698i	− 1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$587$	− 1.24698i	− 1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$588$	− 1.00000i	− 1.00000i
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0.198062i	0.198062i
$593$	1.24698i	1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$593$	1.24698i	1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.554958i	0.554958i
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	1.00000	1.00000
$605$	0	0
$606$	0.246980	0.246980
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0.445042i	0.445042i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	1.24698i	1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$613$	1.24698i	1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.80194i	1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$617$	1.80194i	1.80194i	−0.433884	+	0.900969i	$0.642857\pi$
$618$	− 0.692021i	− 0.692021i
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	− 0.198062i	− 0.198062i
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0.198062	0.198062
$627$	0	0
$628$	− 1.00000i	− 1.00000i
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	1.44504i	1.44504i
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−0.554958	−0.554958
$640$	0	0
$641$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$641$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$642$	1.10992i	1.10992i
$643$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$643$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$647$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$648$	− 1.00000i	− 1.00000i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0.692021	0.692021
$655$	0	0
$656$	0	0
$657$	1.00000i	1.00000i
$658$	0	0
$659$	−1.24698	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$659$	−1.24698	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0.356896	0.356896
$665$	0	0
$666$	0.109916	0.109916
$667$	0	0
$668$	1.44504i	1.44504i
$669$	0.554958	0.554958
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	0	0
$676$	−0.801938	−0.801938
$677$	− 1.24698i	− 1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$677$	− 1.24698i	− 1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	−0.198062	−0.198062
$685$	0	0
$686$	0	0
$687$	2.24698i	2.24698i
$688$	− 0.801938i	− 0.801938i
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	1.24698	1.24698
$697$	0	0
$698$	0	0
$699$	0.554958	0.554958
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0.198062i	0.198062i
$704$	0	0
$705$	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
$711$	−1.00000	−1.00000
$712$	− 1.00000i	− 1.00000i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	1.44504	1.44504
$717$	0	0
$718$	0.801938i	0.801938i
$719$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$719$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$720$	0	0
$721$	0	0
$722$	− 0.356896i	− 0.356896i
$723$	0	0
$724$	0	0
$725$	0	0
$726$	−0.554958	−0.554958
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	−0.801938	−0.801938
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−2.00000	−2.00000	−1.00000			$\pi$
$739$	−2.00000	−2.00000	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0.801938	0.801938
$747$	0.246980i	0.246980i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	− 0.554958i	− 0.554958i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	− 0.554958i	− 0.554958i
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	1.00000	1.00000
$765$	0	0
$766$	0	0
$767$	0	0
$768$	− 0.554958i	− 0.554958i
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	−0.445042	−0.445042
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 0.692021i	− 0.692021i
$784$	−0.445042	−0.445042
$785$	0	0
$786$	1.00000	1.00000
$787$	− 1.24698i	− 1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$787$	− 1.24698i	− 1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$788$	0	0
$789$	2.24698	2.24698
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0.356896	0.356896
$797$	0.445042i	0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$797$	0.445042i	0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0.692021	0.692021
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	− 0.356896i	− 0.356896i
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$811$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$812$	0	0
$813$	1.55496i	1.55496i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 0.801938i	− 0.801938i
$818$	− 0.554958i	− 0.554958i
$819$	0	0
$820$	0	0
$821$	−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	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	−1.00000	−1.00000
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$829$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$830$	0	0
$831$	−0.554958	−0.554958
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	− 0.554958i	− 0.554958i
$839$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$839$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$840$	0	0
$841$	0.554958	0.554958
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	1.00000i	1.00000i
$853$	1.24698i	1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$853$	1.24698i	1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$854$	0	0
$855$	0	0
$856$	1.60388	1.60388
$857$	− 1.24698i	− 1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$857$	− 1.24698i	− 1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	− 0.198062i	− 0.198062i
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	−0.554958	−0.554958
$865$	0	0
$866$	0	0
$867$	− 1.24698i	− 1.24698i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	− 1.00000i	− 1.00000i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	1.80194	1.80194
$877$	1.80194i	1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$877$	1.80194i	1.80194i	−0.433884	+	0.900969i	$0.642857\pi$
$878$	0	0
$879$	−2.49396	−2.49396
$880$	0	0
$881$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$881$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$882$	0.246980i	0.246980i
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	− 0.445042i	− 0.445042i
$889$	0	0
$890$	0	0
$891$	0	0
$892$	− 0.356896i	− 0.356896i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	−0.692021	−0.692021
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	0.246980	0.246980
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0.246980i	0.246980i
$913$	0	0
$914$	0	0
$915$	0	0
$916$	1.44504	1.44504
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0.198062	0.198062
$927$	− 0.692021i	− 0.692021i
$928$	− 1.24698i	− 1.24698i
$929$	0.445042	0.445042	0.222521	−	0.974928i	$-0.428571\pi$
$929$	0.445042	0.445042	0.222521	+	0.974928i	$0.428571\pi$
$930$	0	0
$931$	−0.445042	−0.445042
$932$	− 0.356896i	− 0.356896i
$933$	− 0.554958i	− 0.554958i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	0.554958	0.554958
$940$	0	0
$941$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$941$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$942$	0.692021i	0.692021i
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.80194i	1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$947$	1.80194i	1.80194i	−0.433884	+	0.900969i	$0.642857\pi$
$948$	1.80194i	1.80194i
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1.80194i	− 1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$953$	− 1.80194i	− 1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.00000	1.00000
$962$	0	0
$963$	1.10992i	1.10992i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	0.801938i	0.801938i
$969$	0	0
$970$	0	0
$971$	2.00000	2.00000	1.00000			$0$
$971$	2.00000	2.00000	1.00000			$0$
$972$	− 0.801938i	− 0.801938i
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1.24698i	− 1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$977$	− 1.24698i	− 1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0.692021	0.692021
$982$	0	0
$983$	− 0.445042i	− 0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$983$	− 0.445042i	− 0.445042i	0.974928	−	0.222521i	$-0.0714286\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0.445042	0.445042
$997$	1.80194i	1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$997$	1.80194i	1.80194i	−0.433884	+	0.900969i	$0.642857\pi$
$998$	− 0.554958i	− 0.554958i
$999$	−0.246980	−0.246980

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1775.1.c.a.1774.4		6
5.2	odd	4		71.1.b.a.70.2	✓	3
5.3	odd	4		1775.1.d.b.851.2		3
5.4	even	2	inner	1775.1.c.a.1774.3		6
15.2	even	4		639.1.d.a.496.2		3
20.7	even	4		1136.1.h.a.993.1		3
35.2	odd	12		3479.1.g.e.851.2		6
35.12	even	12		3479.1.g.d.851.2		6
35.17	even	12		3479.1.g.d.1206.2		6
35.27	even	4		3479.1.d.e.638.2		3
35.32	odd	12		3479.1.g.e.1206.2		6
71.70	odd	2	CM	1775.1.c.a.1774.4		6
355.212	even	4		71.1.b.a.70.2	✓	3
355.283	even	4		1775.1.d.b.851.2		3
355.354	odd	2	inner	1775.1.c.a.1774.3		6
1065.212	odd	4		639.1.d.a.496.2		3
1420.567	odd	4		1136.1.h.a.993.1		3
2485.212	even	12		3479.1.g.e.851.2		6
2485.922	odd	12		3479.1.g.d.851.2		6
2485.1277	odd	12		3479.1.g.d.1206.2		6
2485.1987	odd	4		3479.1.d.e.638.2		3
2485.2342	even	12		3479.1.g.e.1206.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
71.1.b.a.70.2	✓	3	5.2	odd	4
71.1.b.a.70.2	✓	3	355.212	even	4
639.1.d.a.496.2		3	15.2	even	4
639.1.d.a.496.2		3	1065.212	odd	4
1136.1.h.a.993.1		3	20.7	even	4
1136.1.h.a.993.1		3	1420.567	odd	4
1775.1.c.a.1774.3		6	5.4	even	2	inner
1775.1.c.a.1774.3		6	355.354	odd	2	inner
1775.1.c.a.1774.4		6	1.1	even	1	trivial
1775.1.c.a.1774.4		6	71.70	odd	2	CM
1775.1.d.b.851.2		3	5.3	odd	4
1775.1.d.b.851.2		3	355.283	even	4
3479.1.d.e.638.2		3	35.27	even	4
3479.1.d.e.638.2		3	2485.1987	odd	4
3479.1.g.d.851.2		6	35.12	even	12
3479.1.g.d.851.2		6	2485.922	odd	12
3479.1.g.d.1206.2		6	35.17	even	12
3479.1.g.d.1206.2		6	2485.1277	odd	12
3479.1.g.e.851.2		6	35.2	odd	12
3479.1.g.e.851.2		6	2485.212	even	12
3479.1.g.e.1206.2		6	35.32	odd	12
3479.1.g.e.1206.2		6	2485.2342	even	12

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:	\( N \)	\(=\)	\( 1775 = 5^{2} \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1775.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.445042i q^{2} +1.24698i q^{3} +0.801938 q^{4} -0.554958 q^{6} +0.801938i q^{8} -0.554958 q^{9} +O(q^{10})\)	\(q+0.445042i q^{2} +1.24698i q^{3} +0.801938 q^{4} -0.554958 q^{6} +0.801938i q^{8} -0.554958 q^{9} +1.00000i q^{12} +0.445042 q^{16} -0.246980i q^{18} +0.445042 q^{19} -1.00000 q^{24} +0.554958i q^{27} -1.24698 q^{29} +1.00000i q^{32} -0.445042 q^{36} +0.445042i q^{37} +0.198062i q^{38} -1.80194i q^{43} +0.554958i q^{48} -1.00000 q^{49} -0.246980 q^{54} +0.554958i q^{57} -0.554958i q^{58} +1.00000 q^{71} -0.445042i q^{72} -1.80194i q^{73} -0.198062 q^{74} +0.356896 q^{76} +1.80194 q^{79} -1.24698 q^{81} -0.445042i q^{83} +0.801938 q^{86} -1.55496i q^{87} -1.24698 q^{89} -1.24698 q^{96} -0.445042i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) \) 6 * q - 4 * q^4 - 4 * q^6 - 4 * q^9	\( 6 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 2 q^{16} + 2 q^{19} - 6 q^{24} + 2 q^{29} - 2 q^{36} - 6 q^{49} + 8 q^{54} + 6 q^{71} - 10 q^{74} - 6 q^{76} + 2 q^{79} + 2 q^{81} - 4 q^{86} + 2 q^{89} + 2 q^{96}+O(q^{100}) \) 6 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 + 2 * q^16 + 2 * q^19 - 6 * q^24 + 2 * q^29 - 2 * q^36 - 6 * q^49 + 8 * q^54 + 6 * q^71 - 10 * q^74 - 6 * q^76 + 2 * q^79 + 2 * q^81 - 4 * q^86 + 2 * q^89 + 2 * q^96

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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