LMFDB - Embedded newform 175.3.d.e.76.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [175,3,Mod(76,175)] 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("175.76"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 175 = 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	175.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0] 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:	no
Analytic conductor:	$4.76840462631$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			76.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	175.76
Dual form			175.3.d.e.76.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -4.00000 q^{4} -7.00000i q^{7} +8.00000 q^{9} -13.0000 q^{11} -4.00000i q^{12} -19.0000i q^{13} +16.0000 q^{16} -29.0000i q^{17} +7.00000 q^{21} +17.0000i q^{27} +28.0000i q^{28} -23.0000 q^{29} -13.0000i q^{33} -32.0000 q^{36} +19.0000 q^{39} +52.0000 q^{44} +31.0000i q^{47} +16.0000i q^{48} -49.0000 q^{49} +29.0000 q^{51} +76.0000i q^{52} -56.0000i q^{63} -64.0000 q^{64} +116.000i q^{68} +2.00000 q^{71} -34.0000i q^{73} +91.0000i q^{77} +157.000 q^{79} +55.0000 q^{81} +86.0000i q^{83} -28.0000 q^{84} -23.0000i q^{87} -133.000 q^{91} -149.000i q^{97} -104.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} + 16 q^{9} - 26 q^{11} + 32 q^{16} + 14 q^{21} - 46 q^{29} - 64 q^{36} + 38 q^{39} + 104 q^{44} - 98 q^{49} + 58 q^{51} - 128 q^{64} + 4 q^{71} + 314 q^{79} + 110 q^{81} - 56 q^{84} - 266 q^{91}+ \cdots - 208 q^{99}+O(q^{100}) $ 2 * q - 8 * q^4 + 16 * q^9 - 26 * q^11 + 32 * q^16 + 14 * q^21 - 46 * q^29 - 64 * q^36 + 38 * q^39 + 104 * q^44 - 98 * q^49 + 58 * q^51 - 128 * q^64 + 4 * q^71 + 314 * q^79 + 110 * q^81 - 56 * q^84 - 266 * q^91 - 208 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$1$

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	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	1.00000i	0.333333i	0.986013	+	0.166667i	$0.0533004\pi$
$3$	1.00000i	0.333333i	−0.986013	+	0.166667i	$0.946700\pi$
$4$	−4.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 7.00000i	− 1.00000i
$7$	− 7.00000i	− 1.00000i
$8$	0	0
$9$	8.00000	0.888889
$10$	0	0
$11$	−13.0000	−1.18182	−0.590909	−	0.806738i	$-0.701231\pi$
$11$	−13.0000	−1.18182	−0.590909	+	0.806738i	$0.701231\pi$
$12$	− 4.00000i	− 0.333333i
$13$	− 19.0000i	− 1.46154i	−0.682625	−	0.730769i	$-0.739162\pi$
$13$	− 19.0000i	− 1.46154i	0.682625	−	0.730769i	$-0.260838\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	1.00000
$17$	− 29.0000i	− 1.70588i	−0.522007	−	0.852941i	$-0.674817\pi$
$17$	− 29.0000i	− 1.70588i	0.522007	−	0.852941i	$-0.325183\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	7.00000	0.333333
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	17.0000i	0.629630i
$28$	28.0000i	1.00000i
$29$	−23.0000	−0.793103	−0.396552	−	0.918012i	$-0.629793\pi$
$29$	−23.0000	−0.793103	−0.396552	+	0.918012i	$0.629793\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	− 13.0000i	− 0.393939i
$34$	0	0
$35$	0	0
$36$	−32.0000	−0.888889
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	19.0000	0.487179
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	52.0000	1.18182
$45$	0	0
$46$	0	0
$47$	31.0000i	0.659574i	0.944055	+	0.329787i	$0.106977\pi$
$47$	31.0000i	0.659574i	−0.944055	+	0.329787i	$0.893023\pi$
$48$	16.0000i	0.333333i
$49$	−49.0000	−1.00000
$50$	0	0
$51$	29.0000	0.568627
$52$	76.0000i	1.46154i
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	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			$\pi$
$62$	0	0
$63$	− 56.0000i	− 0.888889i
$64$	−64.0000	−1.00000
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	116.000i	1.70588i
$69$	0	0
$70$	0	0
$71$	2.00000	0.0281690	0.0140845	−	0.999901i	$-0.495517\pi$
$71$	2.00000	0.0281690	0.0140845	+	0.999901i	$0.495517\pi$
$72$	0	0
$73$	− 34.0000i	− 0.465753i	−0.972506	−	0.232877i	$-0.925186\pi$
$73$	− 34.0000i	− 0.465753i	0.972506	−	0.232877i	$-0.0748139\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	91.0000i	1.18182i
$78$	0	0
$79$	157.000	1.98734	0.993671	−	0.112331i	$-0.0358316\pi$
$79$	157.000	1.98734	0.993671	+	0.112331i	$0.0358316\pi$
$80$	0	0
$81$	55.0000	0.679012
$82$	0	0
$83$	86.0000i	1.03614i	0.855337	+	0.518072i	$0.173350\pi$
$83$	86.0000i	1.03614i	−0.855337	+	0.518072i	$0.826650\pi$
$84$	−28.0000	−0.333333
$85$	0	0
$86$	0	0
$87$	− 23.0000i	− 0.264368i
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	−133.000	−1.46154
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 149.000i	− 1.53608i	−0.640400	−	0.768041i	$-0.721232\pi$
$97$	− 149.000i	− 1.53608i	0.640400	−	0.768041i	$-0.278768\pi$
$98$	0	0
$99$	−104.000	−1.05051
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	− 199.000i	− 1.93204i	−0.258470	−	0.966019i	$-0.583218\pi$
$103$	− 199.000i	− 1.93204i	0.258470	−	0.966019i	$-0.416782\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	− 68.0000i	− 0.629630i
$109$	97.0000	0.889908	0.444954	−	0.895553i	$-0.353220\pi$
$109$	97.0000	0.889908	0.444954	+	0.895553i	$0.353220\pi$
$110$	0	0
$111$	0	0
$112$	− 112.000i	− 1.00000i
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	92.0000	0.793103
$117$	− 152.000i	− 1.29915i
$118$	0	0
$119$	−203.000	−1.70588
$120$	0	0
$121$	48.0000	0.396694
$122$	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	$0.5\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	52.0000i	0.393939i
$133$	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
$141$	−31.0000	−0.219858
$142$	0	0
$143$	247.000i	1.72727i
$144$	128.000	0.888889
$145$	0	0
$146$	0	0
$147$	− 49.0000i	− 0.333333i
$148$	0	0
$149$	262.000	1.75839	0.879195	−	0.476463i	$-0.158081\pi$
$149$	262.000	1.75839	0.879195	+	0.476463i	$0.158081\pi$
$150$	0	0
$151$	−13.0000	−0.0860927	−0.0430464	−	0.999073i	$-0.513706\pi$
$151$	−13.0000	−0.0860927	−0.0430464	+	0.999073i	$0.513706\pi$
$152$	0	0
$153$	− 232.000i	− 1.51634i
$154$	0	0
$155$	0	0
$156$	−76.0000	−0.487179
$157$	− 134.000i	− 0.853503i	−0.904369	−	0.426752i	$-0.859658\pi$
$157$	− 134.000i	− 0.853503i	0.904369	−	0.426752i	$-0.140342\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	271.000i	1.62275i	0.584523	+	0.811377i	$0.301282\pi$
$167$	271.000i	1.62275i	−0.584523	+	0.811377i	$0.698718\pi$
$168$	0	0
$169$	−192.000	−1.13609
$170$	0	0
$171$	0	0
$172$	0	0
$173$	221.000i	1.27746i	0.769432	+	0.638728i	$0.220539\pi$
$173$	221.000i	1.27746i	−0.769432	+	0.638728i	$0.779461\pi$
$174$	0	0
$175$	0	0
$176$	−208.000	−1.18182
$177$	0	0
$178$	0	0
$179$	−218.000	−1.21788	−0.608939	−	0.793217i	$-0.708404\pi$
$179$	−218.000	−1.21788	−0.608939	+	0.793217i	$0.708404\pi$
$180$	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
$187$	377.000i	2.01604i
$188$	− 124.000i	− 0.659574i
$189$	119.000	0.629630
$190$	0	0
$191$	347.000	1.81675	0.908377	−	0.418152i	$-0.137322\pi$
$191$	347.000	1.81675	0.908377	+	0.418152i	$0.137322\pi$
$192$	− 64.0000i	− 0.333333i
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	196.000	1.00000
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	161.000i	0.793103i
$204$	−116.000	−0.568627
$205$	0	0
$206$	0	0
$207$	0	0
$208$	− 304.000i	− 1.46154i
$209$	0	0
$210$	0	0
$211$	107.000	0.507109	0.253555	−	0.967321i	$-0.418400\pi$
$211$	107.000	0.507109	0.253555	+	0.967321i	$0.418400\pi$
$212$	0	0
$213$	2.00000i	0.00938967i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	34.0000	0.155251
$220$	0	0
$221$	−551.000	−2.49321
$222$	0	0
$223$	401.000i	1.79821i	0.437737	+	0.899103i	$0.355780\pi$
$223$	401.000i	1.79821i	−0.437737	+	0.899103i	$0.644220\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	391.000i	1.72247i	0.508209	+	0.861233i	$0.330308\pi$
$227$	391.000i	1.72247i	−0.508209	+	0.861233i	$0.669692\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	−91.0000	−0.393939
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	157.000i	0.662447i
$238$	0	0
$239$	397.000	1.66109	0.830544	−	0.556953i	$-0.188030\pi$
$239$	397.000	1.66109	0.830544	+	0.556953i	$0.188030\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	208.000i	0.855967i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−86.0000	−0.345382
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	224.000i	0.888889i
$253$	0	0
$254$	0	0
$255$	0	0
$256$	256.000	1.00000
$257$	− 494.000i	− 1.92218i	−0.276237	−	0.961089i	$-0.589088\pi$
$257$	− 494.000i	− 1.92218i	0.276237	−	0.961089i	$-0.410912\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−184.000	−0.704981
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	− 464.000i	− 1.70588i
$273$	− 133.000i	− 0.487179i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	527.000	1.87544	0.937722	−	0.347385i	$-0.112930\pi$
$281$	527.000	1.87544	0.937722	+	0.347385i	$0.112930\pi$
$282$	0	0
$283$	− 559.000i	− 1.97527i	−0.156787	−	0.987633i	$-0.550113\pi$
$283$	− 559.000i	− 1.97527i	0.156787	−	0.987633i	$-0.449887\pi$
$284$	−8.00000	−0.0281690
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−552.000	−1.91003
$290$	0	0
$291$	149.000	0.512027
$292$	136.000i	0.465753i
$293$	− 19.0000i	− 0.0648464i	−0.999474	−	0.0324232i	$-0.989678\pi$
$293$	− 19.0000i	− 0.0648464i	0.999474	−	0.0324232i	$-0.0103224\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 221.000i	− 0.744108i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 569.000i	− 1.85342i	−0.375777	−	0.926710i	$-0.622624\pi$
$307$	− 569.000i	− 1.85342i	0.375777	−	0.926710i	$-0.377376\pi$
$308$	− 364.000i	− 1.18182i
$309$	199.000	0.644013
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	221.000i	0.706070i	0.935610	+	0.353035i	$0.114850\pi$
$313$	221.000i	0.706070i	−0.935610	+	0.353035i	$0.885150\pi$
$314$	0	0
$315$	0	0
$316$	−628.000	−1.98734
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	299.000	0.937304
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−220.000	−0.679012
$325$	0	0
$326$	0	0
$327$	97.0000i	0.296636i
$328$	0	0
$329$	217.000	0.659574
$330$	0	0
$331$	−598.000	−1.80665	−0.903323	−	0.428960i	$-0.858880\pi$
$331$	−598.000	−1.80665	−0.903323	+	0.428960i	$0.858880\pi$
$332$	− 344.000i	− 1.03614i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	112.000	0.333333
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	343.000i	1.00000i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	92.0000i	0.264368i
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	323.000	0.920228
$352$	0	0
$353$	− 139.000i	− 0.393768i	−0.980427	−	0.196884i	$-0.936918\pi$
$353$	− 139.000i	− 0.393768i	0.980427	−	0.196884i	$-0.0630822\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 203.000i	− 0.568627i
$358$	0	0
$359$	−578.000	−1.61003	−0.805014	−	0.593256i	$-0.797842\pi$
$359$	−578.000	−1.61003	−0.805014	+	0.593256i	$0.797842\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	0	0
$363$	48.0000i	0.132231i
$364$	532.000	1.46154
$365$	0	0
$366$	0	0
$367$	391.000i	1.06540i	0.846306	+	0.532698i	$0.178822\pi$
$367$	391.000i	1.06540i	−0.846306	+	0.532698i	$0.821178\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	437.000i	1.15915i
$378$	0	0
$379$	502.000	1.32454	0.662269	−	0.749266i	$-0.269594\pi$
$379$	502.000	1.32454	0.662269	+	0.749266i	$0.269594\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 514.000i	− 1.34204i	−0.741441	−	0.671018i	$-0.765857\pi$
$383$	− 514.000i	− 1.34204i	0.741441	−	0.671018i	$-0.234143\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	596.000i	1.53608i
$389$	−743.000	−1.91003	−0.955013	−	0.296564i	$-0.904159\pi$
$389$	−743.000	−1.91003	−0.955013	+	0.296564i	$0.904159\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	416.000	1.05051
$397$	− 389.000i	− 0.979849i	−0.871765	−	0.489924i	$-0.837024\pi$
$397$	− 389.000i	− 0.979849i	0.871765	−	0.489924i	$-0.162976\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−73.0000	−0.182045	−0.0910224	−	0.995849i	$-0.529014\pi$
$401$	−73.0000	−0.182045	−0.0910224	+	0.995849i	$0.529014\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	796.000i	1.93204i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	527.000	1.25178	0.625891	−	0.779911i	$-0.284736\pi$
$421$	527.000	1.25178	0.625891	+	0.779911i	$0.284736\pi$
$422$	0	0
$423$	248.000i	0.586288i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−247.000	−0.575758
$430$	0	0
$431$	−853.000	−1.97912	−0.989559	−	0.144127i	$-0.953963\pi$
$431$	−853.000	−1.97912	−0.989559	+	0.144127i	$0.953963\pi$
$432$	272.000i	0.629630i
$433$	− 754.000i	− 1.74134i	−0.491868	−	0.870670i	$-0.663686\pi$
$433$	− 754.000i	− 1.74134i	0.491868	−	0.870670i	$-0.336314\pi$
$434$	0	0
$435$	0	0
$436$	−388.000	−0.889908
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−392.000	−0.888889
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	262.000i	0.586130i
$448$	448.000i	1.00000i
$449$	817.000	1.81960	0.909800	−	0.415048i	$-0.136235\pi$
$449$	817.000	1.81960	0.909800	+	0.415048i	$0.136235\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 13.0000i	− 0.0286976i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	493.000	1.07407
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	−368.000	−0.793103
$465$	0	0
$466$	0	0
$467$	871.000i	1.86510i	0.361046	+	0.932548i	$0.382420\pi$
$467$	871.000i	1.86510i	−0.361046	+	0.932548i	$0.617580\pi$
$468$	608.000i	1.29915i
$469$	0	0
$470$	0	0
$471$	134.000	0.284501
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	812.000	1.70588
$477$	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
$484$	−192.000	−0.396694
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	107.000	0.217923	0.108961	−	0.994046i	$-0.465248\pi$
$491$	107.000	0.217923	0.108961	+	0.994046i	$0.465248\pi$
$492$	0	0
$493$	667.000i	1.35294i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 14.0000i	− 0.0281690i
$498$	0	0
$499$	−683.000	−1.36874	−0.684369	−	0.729136i	$-0.739922\pi$
$499$	−683.000	−1.36874	−0.684369	+	0.729136i	$0.739922\pi$
$500$	0	0
$501$	−271.000	−0.540918
$502$	0	0
$503$	− 439.000i	− 0.872763i	−0.899762	−	0.436382i	$-0.856260\pi$
$503$	− 439.000i	− 0.872763i	0.899762	−	0.436382i	$-0.143740\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 192.000i	− 0.378698i
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	−238.000	−0.465753
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 403.000i	− 0.779497i
$518$	0	0
$519$	−221.000	−0.425819
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	326.000i	0.623327i	0.950193	+	0.311663i	$0.100886\pi$
$523$	326.000i	0.623327i	−0.950193	+	0.311663i	$0.899114\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	− 208.000i	− 0.393939i
$529$	−529.000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 218.000i	− 0.405959i
$538$	0	0
$539$	637.000	1.18182
$540$	0	0
$541$	767.000	1.41774	0.708872	−	0.705337i	$-0.249204\pi$
$541$	767.000	1.41774	0.708872	+	0.705337i	$0.249204\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 1099.00i	− 1.98734i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−377.000	−0.672014
$562$	0	0
$563$	− 874.000i	− 1.55240i	−0.630488	−	0.776199i	$-0.717145\pi$
$563$	− 874.000i	− 1.55240i	0.630488	−	0.776199i	$-0.282855\pi$
$564$	124.000	0.219858
$565$	0	0
$566$	0	0
$567$	− 385.000i	− 0.679012i
$568$	0	0
$569$	1102.00	1.93673	0.968366	−	0.249536i	$-0.0802781\pi$
$569$	1102.00	1.93673	0.968366	+	0.249536i	$0.0802781\pi$
$570$	0	0
$571$	−118.000	−0.206655	−0.103327	−	0.994647i	$-0.532949\pi$
$571$	−118.000	−0.206655	−0.103327	+	0.994647i	$0.532949\pi$
$572$	− 988.000i	− 1.72727i
$573$	347.000i	0.605585i
$574$	0	0
$575$	0	0
$576$	−512.000	−0.888889
$577$	− 29.0000i	− 0.0502600i	−0.999684	−	0.0251300i	$-0.992000\pi$
$577$	− 29.0000i	− 0.0502600i	0.999684	−	0.0251300i	$-0.00799996\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	602.000	1.03614
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1094.00i	− 1.86371i	−0.362826	−	0.931857i	$-0.618188\pi$
$587$	− 1094.00i	− 1.86371i	0.362826	−	0.931857i	$-0.381812\pi$
$588$	196.000i	0.333333i
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 619.000i	− 1.04384i	−0.852993	−	0.521922i	$-0.825215\pi$
$593$	− 619.000i	− 1.04384i	0.852993	−	0.521922i	$-0.174785\pi$
$594$	0	0
$595$	0	0
$596$	−1048.00	−1.75839
$597$	0	0
$598$	0	0
$599$	−323.000	−0.539232	−0.269616	−	0.962968i	$-0.586897\pi$
$599$	−323.000	−0.539232	−0.269616	+	0.962968i	$0.586897\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	52.0000	0.0860927
$605$	0	0
$606$	0	0
$607$	− 809.000i	− 1.33278i	−0.745601	−	0.666392i	$-0.767838\pi$
$607$	− 809.000i	− 1.33278i	0.745601	−	0.666392i	$-0.232162\pi$
$608$	0	0
$609$	−161.000	−0.264368
$610$	0	0
$611$	589.000	0.963993
$612$	928.000i	1.51634i
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	304.000	0.487179
$625$	0	0
$626$	0	0
$627$	0	0
$628$	536.000i	0.853503i
$629$	0	0
$630$	0	0
$631$	947.000	1.50079	0.750396	−	0.660988i	$-0.229863\pi$
$631$	947.000	1.50079	0.750396	+	0.660988i	$0.229863\pi$
$632$	0	0
$633$	107.000i	0.169036i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	931.000i	1.46154i
$638$	0	0
$639$	16.0000	0.0250391
$640$	0	0
$641$	−958.000	−1.49454	−0.747270	−	0.664521i	$-0.768636\pi$
$641$	−958.000	−1.49454	−0.747270	+	0.664521i	$0.768636\pi$
$642$	0	0
$643$	1241.00i	1.93002i	0.262221	+	0.965008i	$0.415545\pi$
$643$	1241.00i	1.93002i	−0.262221	+	0.965008i	$0.584455\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 974.000i	− 1.50541i	−0.658358	−	0.752705i	$-0.728749\pi$
$647$	− 974.000i	− 1.50541i	0.658358	−	0.752705i	$-0.271251\pi$
$648$	0	0
$649$	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	0
$655$	0	0
$656$	0	0
$657$	− 272.000i	− 0.414003i
$658$	0	0
$659$	−1283.00	−1.94689	−0.973445	−	0.228923i	$-0.926480\pi$
$659$	−1283.00	−1.94689	−0.973445	+	0.228923i	$0.926480\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	− 551.000i	− 0.831071i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	− 1084.00i	− 1.62275i
$669$	−401.000	−0.599402
$670$	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
$676$	768.000	1.13609
$677$	1291.00i	1.90694i	0.301484	+	0.953471i	$0.402518\pi$
$677$	1291.00i	1.90694i	−0.301484	+	0.953471i	$0.597482\pi$
$678$	0	0
$679$	−1043.00	−1.53608
$680$	0	0
$681$	−391.000	−0.574156
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	− 884.000i	− 1.27746i
$693$	728.000i	1.05051i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−313.000	−0.446505	−0.223252	−	0.974761i	$-0.571667\pi$
$701$	−313.000	−0.446505	−0.223252	+	0.974761i	$0.571667\pi$
$702$	0	0
$703$	0	0
$704$	832.000	1.18182
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1417.00	1.99859	0.999295	−	0.0375492i	$-0.0119551\pi$
$709$	1417.00	1.99859	0.999295	+	0.0375492i	$0.0119551\pi$
$710$	0	0
$711$	1256.00	1.76653
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	872.000	1.21788
$717$	397.000i	0.553696i
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	−1393.00	−1.93204
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1426.00i	1.96149i	0.195304	+	0.980743i	$0.437431\pi$
$727$	1426.00i	1.96149i	−0.195304	+	0.980743i	$0.562569\pi$
$728$	0	0
$729$	287.000	0.393690
$730$	0	0
$731$	0	0
$732$	0	0
$733$	1061.00i	1.44748i	0.690075	+	0.723738i	$0.257578\pi$
$733$	1061.00i	1.44748i	−0.690075	+	0.723738i	$0.742422\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	1357.00	1.83627	0.918133	−	0.396273i	$-0.129697\pi$
$739$	1357.00	1.83627	0.918133	+	0.396273i	$0.129697\pi$
$740$	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	0
$747$	688.000i	0.921017i
$748$	− 1508.00i	− 2.01604i
$749$	0	0
$750$	0	0
$751$	−1333.00	−1.77497	−0.887483	−	0.460840i	$-0.847548\pi$
$751$	−1333.00	−1.77497	−0.887483	+	0.460840i	$0.847548\pi$
$752$	496.000i	0.659574i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	−476.000	−0.629630
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	− 679.000i	− 0.889908i
$764$	−1388.00	−1.81675
$765$	0	0
$766$	0	0
$767$	0	0
$768$	256.000i	0.333333i
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	494.000	0.640726
$772$	0	0
$773$	1541.00i	1.99353i	0.0803607	+	0.996766i	$0.474393\pi$
$773$	1541.00i	1.99353i	−0.0803607	+	0.996766i	$0.525607\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−26.0000	−0.0332907
$782$	0	0
$783$	− 391.000i	− 0.499361i
$784$	−784.000	−1.00000
$785$	0	0
$786$	0	0
$787$	− 449.000i	− 0.570521i	−0.958450	−	0.285260i	$-0.907920\pi$
$787$	− 449.000i	− 0.570521i	0.958450	−	0.285260i	$-0.0920801\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1531.00i	1.92095i	0.278360	+	0.960477i	$0.410209\pi$
$797$	1531.00i	1.92095i	−0.278360	+	0.960477i	$0.589791\pi$
$798$	0	0
$799$	899.000	1.12516
$800$	0	0
$801$	0	0
$802$	0	0
$803$	442.000i	0.550436i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	97.0000	0.119901	0.0599506	−	0.998201i	$-0.480906\pi$
$809$	97.0000	0.119901	0.0599506	+	0.998201i	$0.480906\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	− 644.000i	− 0.793103i
$813$	0	0
$814$	0	0
$815$	0	0
$816$	464.000	0.568627
$817$	0	0
$818$	0	0
$819$	−1064.00	−1.29915
$820$	0	0
$821$	1607.00	1.95737	0.978685	−	0.205369i	$-0.0658396\pi$
$821$	1607.00	1.95737	0.978685	+	0.205369i	$0.0658396\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	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	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	1216.00i	1.46154i
$833$	1421.00i	1.70588i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	−312.000	−0.370987
$842$	0	0
$843$	527.000i	0.625148i
$844$	−428.000	−0.507109
$845$	0	0
$846$	0	0
$847$	− 336.000i	− 0.396694i
$848$	0	0
$849$	559.000	0.658422
$850$	0	0
$851$	0	0
$852$	− 8.00000i	− 0.00938967i
$853$	86.0000i	0.100821i	0.998729	+	0.0504103i	$0.0160529\pi$
$853$	86.0000i	0.100821i	−0.998729	+	0.0504103i	$0.983947\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	706.000i	0.823804i	0.911228	+	0.411902i	$0.135135\pi$
$857$	706.000i	0.823804i	−0.911228	+	0.411902i	$0.864865\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 552.000i	− 0.636678i
$868$	0	0
$869$	−2041.00	−2.34868
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 1192.00i	− 1.36541i
$874$	0	0
$875$	0	0
$876$	−136.000	−0.155251
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	19.0000	0.0216155
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	2204.00	2.49321
$885$	0	0
$886$	0	0
$887$	− 494.000i	− 0.556933i	−0.960446	−	0.278467i	$-0.910174\pi$
$887$	− 494.000i	− 0.556933i	0.960446	−	0.278467i	$-0.0898262\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−715.000	−0.802469
$892$	− 1604.00i	− 1.79821i
$893$	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	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	− 1564.00i	− 1.72247i
$909$	0	0
$910$	0	0
$911$	−1678.00	−1.84193	−0.920966	−	0.389643i	$-0.872598\pi$
$911$	−1678.00	−1.84193	−0.920966	+	0.389643i	$0.872598\pi$
$912$	0	0
$913$	− 1118.00i	− 1.22453i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	997.000	1.08487	0.542437	−	0.840096i	$-0.317502\pi$
$919$	997.000	1.08487	0.542437	+	0.840096i	$0.317502\pi$
$920$	0	0
$921$	569.000	0.617807
$922$	0	0
$923$	− 38.0000i	− 0.0411701i
$924$	364.000	0.393939
$925$	0	0
$926$	0	0
$927$	− 1592.00i	− 1.71737i
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1829.00i	− 1.95197i	−0.217828	−	0.975987i	$-0.569897\pi$
$937$	− 1829.00i	− 1.95197i	0.217828	−	0.975987i	$-0.430103\pi$
$938$	0	0
$939$	−221.000	−0.235357
$940$	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	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	− 628.000i	− 0.662447i
$949$	−646.000	−0.680717
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	−1588.00	−1.66109
$957$	299.000i	0.312435i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	0	0
$964$	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	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	− 832.000i	− 0.855967i
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	776.000	0.791030
$982$	0	0
$983$	1121.00i	1.14039i	0.821511	+	0.570193i	$0.193132\pi$
$983$	1121.00i	1.14039i	−0.821511	+	0.570193i	$0.806868\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	217.000i	0.219858i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	722.000	0.728557	0.364279	−	0.931290i	$-0.381316\pi$
$991$	722.000	0.728557	0.364279	+	0.931290i	$0.381316\pi$
$992$	0	0
$993$	− 598.000i	− 0.602216i
$994$	0	0
$995$	0	0
$996$	344.000	0.345382
$997$	1651.00i	1.65597i	0.560752	+	0.827984i	$0.310512\pi$
$997$	1651.00i	1.65597i	−0.560752	+	0.827984i	$0.689488\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.3.d.e.76.2		2
5.2	odd	4		35.3.c.b.34.1	yes	1
5.3	odd	4		35.3.c.a.34.1	✓	1
5.4	even	2	inner	175.3.d.e.76.1		2
7.6	odd	2	inner	175.3.d.e.76.1		2
15.2	even	4		315.3.e.b.244.1		1
15.8	even	4		315.3.e.a.244.1		1
20.3	even	4		560.3.p.b.209.1		1
20.7	even	4		560.3.p.a.209.1		1
35.2	odd	12		245.3.i.a.129.1		2
35.3	even	12		245.3.i.a.19.1		2
35.12	even	12		245.3.i.b.129.1		2
35.13	even	4		35.3.c.b.34.1	yes	1
35.17	even	12		245.3.i.b.19.1		2
35.18	odd	12		245.3.i.b.19.1		2
35.23	odd	12		245.3.i.b.129.1		2
35.27	even	4		35.3.c.a.34.1	✓	1
35.32	odd	12		245.3.i.a.19.1		2
35.33	even	12		245.3.i.a.129.1		2
35.34	odd	2	CM	175.3.d.e.76.2		2
105.62	odd	4		315.3.e.a.244.1		1
105.83	odd	4		315.3.e.b.244.1		1
140.27	odd	4		560.3.p.b.209.1		1
140.83	odd	4		560.3.p.a.209.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.3.c.a.34.1	✓	1	5.3	odd	4
35.3.c.a.34.1	✓	1	35.27	even	4
35.3.c.b.34.1	yes	1	5.2	odd	4
35.3.c.b.34.1	yes	1	35.13	even	4
175.3.d.e.76.1		2	5.4	even	2	inner
175.3.d.e.76.1		2	7.6	odd	2	inner
175.3.d.e.76.2		2	1.1	even	1	trivial
175.3.d.e.76.2		2	35.34	odd	2	CM
245.3.i.a.19.1		2	35.3	even	12
245.3.i.a.19.1		2	35.32	odd	12
245.3.i.a.129.1		2	35.2	odd	12
245.3.i.a.129.1		2	35.33	even	12
245.3.i.b.19.1		2	35.17	even	12
245.3.i.b.19.1		2	35.18	odd	12
245.3.i.b.129.1		2	35.12	even	12
245.3.i.b.129.1		2	35.23	odd	12
315.3.e.a.244.1		1	15.8	even	4
315.3.e.a.244.1		1	105.62	odd	4
315.3.e.b.244.1		1	15.2	even	4
315.3.e.b.244.1		1	105.83	odd	4
560.3.p.a.209.1		1	20.7	even	4
560.3.p.a.209.1		1	140.83	odd	4
560.3.p.b.209.1		1	20.3	even	4
560.3.p.b.209.1		1	140.27	odd	4

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:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	175.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -4.00000 q^{4} -7.00000i q^{7} +8.00000 q^{9} -13.0000 q^{11} -4.00000i q^{12} -19.0000i q^{13} +16.0000 q^{16} -29.0000i q^{17} +7.00000 q^{21} +17.0000i q^{27} +28.0000i q^{28} -23.0000 q^{29} -13.0000i q^{33} -32.0000 q^{36} +19.0000 q^{39} +52.0000 q^{44} +31.0000i q^{47} +16.0000i q^{48} -49.0000 q^{49} +29.0000 q^{51} +76.0000i q^{52} -56.0000i q^{63} -64.0000 q^{64} +116.000i q^{68} +2.00000 q^{71} -34.0000i q^{73} +91.0000i q^{77} +157.000 q^{79} +55.0000 q^{81} +86.0000i q^{83} -28.0000 q^{84} -23.0000i q^{87} -133.000 q^{91} -149.000i q^{97} -104.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} + 16 q^{9} - 26 q^{11} + 32 q^{16} + 14 q^{21} - 46 q^{29} - 64 q^{36} + 38 q^{39} + 104 q^{44} - 98 q^{49} + 58 q^{51} - 128 q^{64} + 4 q^{71} + 314 q^{79} + 110 q^{81} - 56 q^{84} - 266 q^{91}+ \cdots - 208 q^{99}+O(q^{100}) \) 2 * q - 8 * q^4 + 16 * q^9 - 26 * q^11 + 32 * q^16 + 14 * q^21 - 46 * q^29 - 64 * q^36 + 38 * q^39 + 104 * q^44 - 98 * q^49 + 58 * q^51 - 128 * q^64 + 4 * q^71 + 314 * q^79 + 110 * q^81 - 56 * q^84 - 266 * q^91 - 208 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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