LMFDB - Embedded newform 175.5.d.c.76.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [175,5,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, 5); 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, 5, names="a")

Level:	$ N $	$=$	$ 175 = 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
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:	$18.0897435397$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			76.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	175.76
Dual form			175.5.d.c.76.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-17.0000i q^{3} -16.0000 q^{4} -49.0000i q^{7} -208.000 q^{9} -73.0000 q^{11} +272.000i q^{12} +23.0000i q^{13} +256.000 q^{16} -263.000i q^{17} -833.000 q^{21} +2159.00i q^{27} +784.000i q^{28} +1153.00 q^{29} +1241.00i q^{33} +3328.00 q^{36} +391.000 q^{39} +1168.00 q^{44} +3457.00i q^{47} -4352.00i q^{48} -2401.00 q^{49} -4471.00 q^{51} -368.000i q^{52} +10192.0i q^{63} -4096.00 q^{64} +4208.00i q^{68} -10078.0 q^{71} -9502.00i q^{73} +3577.00i q^{77} -12167.0 q^{79} +19855.0 q^{81} -6382.00i q^{83} +13328.0 q^{84} -19601.0i q^{87} +1127.00 q^{91} -3383.00i q^{97} +15184.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4} - 416 q^{9} - 146 q^{11} + 512 q^{16} - 1666 q^{21} + 2306 q^{29} + 6656 q^{36} + 782 q^{39} + 2336 q^{44} - 4802 q^{49} - 8942 q^{51} - 8192 q^{64} - 20156 q^{71} - 24334 q^{79} + 39710 q^{81}+ \cdots + 30368 q^{99}+O(q^{100}) $ 2 * q - 32 * q^4 - 416 * q^9 - 146 * q^11 + 512 * q^16 - 1666 * q^21 + 2306 * q^29 + 6656 * q^36 + 782 * q^39 + 2336 * q^44 - 4802 * q^49 - 8942 * q^51 - 8192 * q^64 - 20156 * q^71 - 24334 * q^79 + 39710 * q^81 + 26656 * q^84 + 2254 * q^91 + 30368 * 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	$0.5\pi$
$3$	− 17.0000i	− 1.88889i	−0.328671	−	0.944444i	$-0.606601\pi$
$3$	− 17.0000i	− 1.88889i	0.328671	−	0.944444i	$-0.393399\pi$
$4$	−16.0000	−1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 49.0000i	− 1.00000i
$7$	− 49.0000i	− 1.00000i
$8$	0	0
$9$	−208.000	−2.56790
$10$	0	0
$11$	−73.0000	−0.603306	−0.301653	−	0.953418i	$-0.597538\pi$
$11$	−73.0000	−0.603306	−0.301653	+	0.953418i	$0.597538\pi$
$12$	272.000i	1.88889i
$13$	23.0000i	0.136095i	0.997682	+	0.0680473i	$0.0216769\pi$
$13$	23.0000i	0.136095i	−0.997682	+	0.0680473i	$0.978323\pi$
$14$	0	0
$15$	0	0
$16$	256.000	1.00000
$17$	− 263.000i	− 0.910035i	−0.890483	−	0.455017i	$-0.849633\pi$
$17$	− 263.000i	− 0.910035i	0.890483	−	0.455017i	$-0.150367\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	−833.000	−1.88889
$22$	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
$27$	2159.00i	2.96159i
$28$	784.000i	1.00000i
$29$	1153.00	1.37099	0.685493	−	0.728079i	$-0.259587\pi$
$29$	1153.00	1.37099	0.685493	+	0.728079i	$0.259587\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	1241.00i	1.13958i
$34$	0	0
$35$	0	0
$36$	3328.00	2.56790
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	391.000	0.257068
$40$	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	$0.5\pi$
$44$	1168.00	0.603306
$45$	0	0
$46$	0	0
$47$	3457.00i	1.56496i	0.622675	+	0.782481i	$0.286046\pi$
$47$	3457.00i	1.56496i	−0.622675	+	0.782481i	$0.713954\pi$
$48$	− 4352.00i	− 1.88889i
$49$	−2401.00	−1.00000
$50$	0	0
$51$	−4471.00	−1.71895
$52$	− 368.000i	− 0.136095i
$53$	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
$63$	10192.0i	2.56790i
$64$	−4096.00	−1.00000
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	4208.00i	0.910035i
$69$	0	0
$70$	0	0
$71$	−10078.0	−1.99921	−0.999603	−	0.0281662i	$-0.991033\pi$
$71$	−10078.0	−1.99921	−0.999603	+	0.0281662i	$0.991033\pi$
$72$	0	0
$73$	− 9502.00i	− 1.78307i	−0.452948	−	0.891537i	$-0.649628\pi$
$73$	− 9502.00i	− 1.78307i	0.452948	−	0.891537i	$-0.350372\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3577.00i	0.603306i
$78$	0	0
$79$	−12167.0	−1.94953	−0.974764	−	0.223239i	$-0.928337\pi$
$79$	−12167.0	−1.94953	−0.974764	+	0.223239i	$0.928337\pi$
$80$	0	0
$81$	19855.0	3.02622
$82$	0	0
$83$	− 6382.00i	− 0.926404i	−0.886253	−	0.463202i	$-0.846700\pi$
$83$	− 6382.00i	− 0.926404i	0.886253	−	0.463202i	$-0.153300\pi$
$84$	13328.0	1.88889
$85$	0	0
$86$	0	0
$87$	− 19601.0i	− 2.58964i
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	1127.00	0.136095
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 3383.00i	− 0.359549i	−0.983708	−	0.179775i	$-0.942463\pi$
$97$	− 3383.00i	− 0.359549i	0.983708	−	0.179775i	$-0.0575369\pi$
$98$	0	0
$99$	15184.0	1.54923
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	18383.0i	1.73277i	0.499373	+	0.866387i	$0.333564\pi$
$103$	18383.0i	1.73277i	−0.499373	+	0.866387i	$0.666436\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	− 34544.0i	− 2.96159i
$109$	14353.0	1.20806	0.604032	−	0.796960i	$-0.293560\pi$
$109$	14353.0	1.20806	0.604032	+	0.796960i	$0.293560\pi$
$110$	0	0
$111$	0	0
$112$	− 12544.0i	− 1.00000i
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	−18448.0	−1.37099
$117$	− 4784.00i	− 0.349478i
$118$	0	0
$119$	−12887.0	−0.910035
$120$	0	0
$121$	−9312.00	−0.636022
$122$	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			$\pi$
$132$	− 19856.0i	− 1.13958i
$133$	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
$141$	58769.0	2.95604
$142$	0	0
$143$	− 1679.00i	− 0.0821067i
$144$	−53248.0	−2.56790
$145$	0	0
$146$	0	0
$147$	40817.0i	1.88889i
$148$	0	0
$149$	−24242.0	−1.09193	−0.545966	−	0.837807i	$-0.683837\pi$
$149$	−24242.0	−1.09193	−0.545966	+	0.837807i	$0.683837\pi$
$150$	0	0
$151$	−45433.0	−1.99259	−0.996294	−	0.0860129i	$-0.972587\pi$
$151$	−45433.0	−1.99259	−0.996294	+	0.0860129i	$0.972587\pi$
$152$	0	0
$153$	54704.0i	2.33688i
$154$	0	0
$155$	0	0
$156$	−6256.00	−0.257068
$157$	31342.0i	1.27153i	0.771882	+	0.635766i	$0.219316\pi$
$157$	31342.0i	1.27153i	−0.771882	+	0.635766i	$0.780684\pi$
$158$	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
$167$	− 17663.0i	− 0.633332i	−0.948537	−	0.316666i	$-0.897437\pi$
$167$	− 17663.0i	− 0.633332i	0.948537	−	0.316666i	$-0.102563\pi$
$168$	0	0
$169$	28032.0	0.981478
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 11017.0i	− 0.368105i	−0.982916	−	0.184052i	$-0.941078\pi$
$173$	− 11017.0i	− 0.368105i	0.982916	−	0.184052i	$-0.0589216\pi$
$174$	0	0
$175$	0	0
$176$	−18688.0	−0.603306
$177$	0	0
$178$	0	0
$179$	16558.0	0.516775	0.258388	−	0.966041i	$-0.416809\pi$
$179$	16558.0	0.516775	0.258388	+	0.966041i	$0.416809\pi$
$180$	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
$187$	19199.0i	0.549029i
$188$	− 55312.0i	− 1.56496i
$189$	105791.	2.96159
$190$	0	0
$191$	47447.0	1.30059	0.650297	−	0.759680i	$-0.274644\pi$
$191$	47447.0	1.30059	0.650297	+	0.759680i	$0.274644\pi$
$192$	69632.0i	1.88889i
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	38416.0	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
$203$	− 56497.0i	− 1.37099i
$204$	71536.0	1.71895
$205$	0	0
$206$	0	0
$207$	0	0
$208$	5888.00i	0.136095i
$209$	0	0
$210$	0	0
$211$	−77593.0	−1.74284	−0.871420	−	0.490537i	$-0.836801\pi$
$211$	−77593.0	−1.74284	−0.871420	+	0.490537i	$0.836801\pi$
$212$	0	0
$213$	171326.i	3.77628i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−161534.	−3.36803
$220$	0	0
$221$	6049.00	0.123851
$222$	0	0
$223$	61343.0i	1.23355i	0.787141	+	0.616773i	$0.211560\pi$
$223$	61343.0i	1.23355i	−0.787141	+	0.616773i	$0.788440\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 49823.0i	− 0.966892i	−0.875374	−	0.483446i	$-0.839385\pi$
$227$	− 49823.0i	− 0.966892i	0.875374	−	0.483446i	$-0.160615\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	60809.0	1.13958
$232$	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
$237$	206839.i	3.68244i
$238$	0	0
$239$	−43367.0	−0.759213	−0.379606	−	0.925148i	$-0.623941\pi$
$239$	−43367.0	−0.759213	−0.379606	+	0.925148i	$0.623941\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	− 162656.i	− 2.75459i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−108494.	−1.74988
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	− 163072.i	− 2.56790i
$253$	0	0
$254$	0	0
$255$	0	0
$256$	65536.0	1.00000
$257$	− 111938.i	− 1.69477i	−0.530977	−	0.847386i	$-0.678175\pi$
$257$	− 111938.i	− 1.69477i	0.530977	−	0.847386i	$-0.321825\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−239824.	−3.52056
$262$	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			$\pi$
$272$	− 67328.0i	− 0.910035i
$273$	− 19159.0i	− 0.257068i
$274$	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
$281$	119807.	1.51729	0.758647	−	0.651502i	$-0.225861\pi$
$281$	119807.	1.51729	0.758647	+	0.651502i	$0.225861\pi$
$282$	0	0
$283$	152303.i	1.90167i	0.309695	+	0.950836i	$0.399773\pi$
$283$	152303.i	1.90167i	−0.309695	+	0.950836i	$0.600227\pi$
$284$	161248.	1.99921
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	14352.0	0.171837
$290$	0	0
$291$	−57511.0	−0.679149
$292$	152032.i	1.78307i
$293$	− 171337.i	− 1.99579i	−0.0648123	−	0.997897i	$-0.520645\pi$
$293$	− 171337.i	− 1.99579i	0.0648123	−	0.997897i	$-0.479355\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 157607.i	− 1.78675i
$298$	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
$307$	− 135263.i	− 1.43517i	−0.696473	−	0.717583i	$-0.745248\pi$
$307$	− 135263.i	− 1.43517i	0.696473	−	0.717583i	$-0.254752\pi$
$308$	− 57232.0i	− 0.603306i
$309$	312511.	3.27302
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	− 147097.i	− 1.50146i	−0.660606	−	0.750732i	$-0.729701\pi$
$313$	− 147097.i	− 1.50146i	0.660606	−	0.750732i	$-0.270299\pi$
$314$	0	0
$315$	0	0
$316$	194672.	1.94953
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	−84169.0	−0.827124
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−317680.	−3.02622
$325$	0	0
$326$	0	0
$327$	− 244001.i	− 2.28190i
$328$	0	0
$329$	169393.	1.56496
$330$	0	0
$331$	138482.	1.26397	0.631986	−	0.774980i	$-0.282240\pi$
$331$	138482.	1.26397	0.631986	+	0.774980i	$0.282240\pi$
$332$	102112.i	0.926404i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	−213248.	−1.88889
$337$	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
$343$	117649.i	1.00000i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	313616.i	2.58964i
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	−49657.0	−0.403057
$352$	0	0
$353$	− 229897.i	− 1.84495i	−0.386060	−	0.922473i	$-0.626164\pi$
$353$	− 229897.i	− 1.84495i	0.386060	−	0.922473i	$-0.373836\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	219079.i	1.71895i
$358$	0	0
$359$	−76322.0	−0.592190	−0.296095	−	0.955159i	$-0.595684\pi$
$359$	−76322.0	−0.592190	−0.296095	+	0.955159i	$0.595684\pi$
$360$	0	0
$361$	130321.	1.00000
$362$	0	0
$363$	158304.i	1.20138i
$364$	−18032.0	−0.136095
$365$	0	0
$366$	0	0
$367$	116497.i	0.864933i	0.901650	+	0.432467i	$0.142357\pi$
$367$	116497.i	0.864933i	−0.901650	+	0.432467i	$0.857643\pi$
$368$	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
$377$	26519.0i	0.186584i
$378$	0	0
$379$	35278.0	0.245598	0.122799	−	0.992432i	$-0.460813\pi$
$379$	35278.0	0.245598	0.122799	+	0.992432i	$0.460813\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 29182.0i	− 0.198938i	−0.995041	−	0.0994689i	$-0.968286\pi$
$383$	− 29182.0i	− 0.198938i	0.995041	−	0.0994689i	$-0.0317144\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	54128.0i	0.359549i
$389$	−249407.	−1.64820	−0.824099	−	0.566446i	$-0.808318\pi$
$389$	−249407.	−1.64820	−0.824099	+	0.566446i	$0.808318\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	−242944.	−1.54923
$397$	163897.i	1.03990i	0.854198	+	0.519948i	$0.174049\pi$
$397$	163897.i	1.03990i	−0.854198	+	0.519948i	$0.825951\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−316273.	−1.96686	−0.983430	−	0.181289i	$-0.941973\pi$
$401$	−316273.	−1.96686	−0.983430	+	0.181289i	$0.941973\pi$
$402$	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
$412$	− 294128.i	− 1.73277i
$413$	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
$421$	−76753.0	−0.433043	−0.216522	−	0.976278i	$-0.569471\pi$
$421$	−76753.0	−0.433043	−0.216522	+	0.976278i	$0.569471\pi$
$422$	0	0
$423$	− 719056.i	− 4.01867i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−28543.0	−0.155090
$430$	0	0
$431$	356087.	1.91691	0.958455	−	0.285245i	$-0.0920749\pi$
$431$	356087.	1.91691	0.958455	+	0.285245i	$0.0920749\pi$
$432$	552704.i	2.96159i
$433$	193538.i	1.03226i	0.856509	+	0.516132i	$0.172628\pi$
$433$	193538.i	1.03226i	−0.856509	+	0.516132i	$0.827372\pi$
$434$	0	0
$435$	0	0
$436$	−229648.	−1.20806
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	499408.	2.56790
$442$	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
$447$	412114.i	2.06254i
$448$	200704.i	1.00000i
$449$	−264287.	−1.31094	−0.655470	−	0.755221i	$-0.727530\pi$
$449$	−264287.	−1.31094	−0.655470	+	0.755221i	$0.727530\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	772361.i	3.76378i
$454$	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
$459$	567817.	2.69515
$460$	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	$0.5\pi$
$464$	295168.	1.37099
$465$	0	0
$466$	0	0
$467$	− 322463.i	− 1.47858i	−0.673385	−	0.739292i	$-0.735160\pi$
$467$	− 322463.i	− 1.47858i	0.673385	−	0.739292i	$-0.264840\pi$
$468$	76544.0i	0.349478i
$469$	0	0
$470$	0	0
$471$	532814.	2.40178
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	206192.	0.910035
$477$	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
$484$	148992.	0.636022
$485$	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
$491$	−470713.	−1.95251	−0.976255	−	0.216625i	$-0.930495\pi$
$491$	−470713.	−1.95251	−0.976255	+	0.216625i	$0.930495\pi$
$492$	0	0
$493$	− 303239.i	− 1.24765i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	493822.i	1.99921i
$498$	0	0
$499$	31513.0	0.126558	0.0632789	−	0.997996i	$-0.479844\pi$
$499$	31513.0	0.126558	0.0632789	+	0.997996i	$0.479844\pi$
$500$	0	0
$501$	−300271.	−1.19629
$502$	0	0
$503$	− 313297.i	− 1.23828i	−0.785279	−	0.619142i	$-0.787481\pi$
$503$	− 313297.i	− 1.23828i	0.785279	−	0.619142i	$-0.212519\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 476544.i	− 1.85390i
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	−465598.	−1.78307
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 252361.i	− 0.944150i
$518$	0	0
$519$	−187289.	−0.695309
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	− 440782.i	− 1.61146i	−0.592281	−	0.805732i	$-0.701772\pi$
$523$	− 440782.i	− 1.61146i	0.592281	−	0.805732i	$-0.298228\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	317696.i	1.13958i
$529$	−279841.	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 281486.i	− 0.976131i
$538$	0	0
$539$	175273.	0.603306
$540$	0	0
$541$	2927.00	0.0100006	0.00500032	−	0.999987i	$-0.498408\pi$
$541$	2927.00	0.0100006	0.00500032	+	0.999987i	$0.498408\pi$
$542$	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
$553$	596183.i	1.94953i
$554$	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
$561$	326383.	1.03706
$562$	0	0
$563$	129938.i	0.409939i	0.978768	+	0.204970i	$0.0657095\pi$
$563$	129938.i	0.409939i	−0.978768	+	0.204970i	$0.934290\pi$
$564$	−940304.	−2.95604
$565$	0	0
$566$	0	0
$567$	− 972895.i	− 3.02622i
$568$	0	0
$569$	−566882.	−1.75093	−0.875464	−	0.483284i	$-0.839444\pi$
$569$	−566882.	−1.75093	−0.875464	+	0.483284i	$0.839444\pi$
$570$	0	0
$571$	−638158.	−1.95729	−0.978647	−	0.205549i	$-0.934102\pi$
$571$	−638158.	−1.95729	−0.978647	+	0.205549i	$0.934102\pi$
$572$	26864.0i	0.0821067i
$573$	− 806599.i	− 2.45668i
$574$	0	0
$575$	0	0
$576$	851968.	2.56790
$577$	665017.i	1.99747i	0.0502441	+	0.998737i	$0.484000\pi$
$577$	665017.i	1.99747i	−0.0502441	+	0.998737i	$0.516000\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−312718.	−0.926404
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 507698.i	− 1.47343i	−0.676204	−	0.736715i	$-0.736376\pi$
$587$	− 507698.i	− 1.47343i	0.676204	−	0.736715i	$-0.263624\pi$
$588$	− 653072.i	− 1.88889i
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 320137.i	− 0.910388i	−0.890392	−	0.455194i	$-0.849570\pi$
$593$	− 320137.i	− 0.910388i	0.890392	−	0.455194i	$-0.150430\pi$
$594$	0	0
$595$	0	0
$596$	387872.	1.09193
$597$	0	0
$598$	0	0
$599$	613273.	1.70923	0.854614	−	0.519263i	$-0.173794\pi$
$599$	613273.	1.70923	0.854614	+	0.519263i	$0.173794\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	726928.	1.99259
$605$	0	0
$606$	0	0
$607$	82417.0i	0.223686i	0.993726	+	0.111843i	$0.0356754\pi$
$607$	82417.0i	0.223686i	−0.993726	+	0.111843i	$0.964325\pi$
$608$	0	0
$609$	−960449.	−2.58964
$610$	0	0
$611$	−79511.0	−0.212983
$612$	− 875264.i	− 2.33688i
$613$	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
$624$	100096.	0.257068
$625$	0	0
$626$	0	0
$627$	0	0
$628$	− 501472.i	− 1.27153i
$629$	0	0
$630$	0	0
$631$	100487.	0.252378	0.126189	−	0.992006i	$-0.459725\pi$
$631$	100487.	0.252378	0.126189	+	0.992006i	$0.459725\pi$
$632$	0	0
$633$	1.31908e6i	3.29203i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 55223.0i	− 0.136095i
$638$	0	0
$639$	2.09622e6	5.13376
$640$	0	0
$641$	96002.0	0.233649	0.116825	−	0.993153i	$-0.462728\pi$
$641$	96002.0	0.233649	0.116825	+	0.993153i	$0.462728\pi$
$642$	0	0
$643$	713183.i	1.72496i	0.506091	+	0.862480i	$0.331090\pi$
$643$	713183.i	1.72496i	−0.506091	+	0.862480i	$0.668910\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 111458.i	− 0.266258i	−0.991099	−	0.133129i	$-0.957498\pi$
$647$	− 111458.i	− 0.266258i	0.991099	−	0.133129i	$-0.0425025\pi$
$648$	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
$657$	1.97642e6i	4.57876i
$658$	0	0
$659$	−777527.	−1.79038	−0.895189	−	0.445687i	$-0.852959\pi$
$659$	−777527.	−1.79038	−0.895189	+	0.445687i	$0.852959\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	− 102833.i	− 0.233941i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	282608.i	0.633332i
$669$	1.04283e6	2.33003
$670$	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
$676$	−448512.	−0.981478
$677$	− 750023.i	− 1.63643i	−0.574913	−	0.818215i	$-0.694964\pi$
$677$	− 750023.i	− 1.63643i	0.574913	−	0.818215i	$-0.305036\pi$
$678$	0	0
$679$	−165767.	−0.359549
$680$	0	0
$681$	−846991.	−1.82635
$682$	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			$\pi$
$692$	176272.i	0.368105i
$693$	− 744016.i	− 1.54923i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−884833.	−1.80063	−0.900317	−	0.435235i	$-0.856665\pi$
$701$	−884833.	−1.80063	−0.900317	+	0.435235i	$0.856665\pi$
$702$	0	0
$703$	0	0
$704$	299008.	0.603306
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1.00253e6	−1.99436	−0.997180	−	0.0750454i	$-0.976090\pi$
$709$	−1.00253e6	−1.99436	−0.997180	+	0.0750454i	$0.976090\pi$
$710$	0	0
$711$	2.53074e6	5.00619
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−264928.	−0.516775
$717$	737239.i	1.43407i
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	900767.	1.73277
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 976418.i	− 1.84743i	−0.383086	−	0.923713i	$-0.625139\pi$
$727$	− 976418.i	− 1.84743i	0.383086	−	0.923713i	$-0.374861\pi$
$728$	0	0
$729$	−1.15690e6	−2.17691
$730$	0	0
$731$	0	0
$732$	0	0
$733$	51143.0i	0.0951871i	0.998867	+	0.0475936i	$0.0151552\pi$
$733$	51143.0i	0.0951871i	−0.998867	+	0.0475936i	$0.984845\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−749207.	−1.37187	−0.685935	−	0.727663i	$-0.740607\pi$
$739$	−749207.	−1.37187	−0.685935	+	0.727663i	$0.740607\pi$
$740$	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
$747$	1.32746e6i	2.37892i
$748$	− 307184.i	− 0.549029i
$749$	0	0
$750$	0	0
$751$	648887.	1.15051	0.575253	−	0.817975i	$-0.304903\pi$
$751$	648887.	1.15051	0.575253	+	0.817975i	$0.304903\pi$
$752$	884992.i	1.56496i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	−1.69266e6	−2.96159
$757$	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
$763$	− 703297.i	− 1.20806i
$764$	−759152.	−1.30059
$765$	0	0
$766$	0	0
$767$	0	0
$768$	− 1.11411e6i	− 1.88889i
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	−1.90295e6	−3.20124
$772$	0	0
$773$	1.17962e6i	1.97417i	0.160202	+	0.987084i	$0.448786\pi$
$773$	1.17962e6i	1.97417i	−0.160202	+	0.987084i	$0.551214\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	735694.	1.20613
$782$	0	0
$783$	2.48933e6i	4.06030i
$784$	−614656.	−1.00000
$785$	0	0
$786$	0	0
$787$	1.03714e6i	1.67451i	0.546816	+	0.837253i	$0.315840\pi$
$787$	1.03714e6i	1.67451i	−0.546816	+	0.837253i	$0.684160\pi$
$788$	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
$797$	− 1.07354e6i	− 1.69006i	−0.534717	−	0.845031i	$-0.679582\pi$
$797$	− 1.07354e6i	− 1.69006i	0.534717	−	0.845031i	$-0.320418\pi$
$798$	0	0
$799$	909191.	1.42417
$800$	0	0
$801$	0	0
$802$	0	0
$803$	693646.i	1.07574i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.29955e6	1.98562	0.992812	−	0.119685i	$-0.0381886\pi$
$809$	1.29955e6	1.98562	0.992812	+	0.119685i	$0.0381886\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	903952.i	1.37099i
$813$	0	0
$814$	0	0
$815$	0	0
$816$	−1.14458e6	−1.71895
$817$	0	0
$818$	0	0
$819$	−234416.	−0.349478
$820$	0	0
$821$	1.23437e6	1.83129	0.915647	−	0.401984i	$-0.131679\pi$
$821$	1.23437e6	1.83129	0.915647	+	0.401984i	$0.131679\pi$
$822$	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
$832$	− 94208.0i	− 0.136095i
$833$	631463.i	0.910035i
$834$	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
$841$	622128.	0.879605
$842$	0	0
$843$	− 2.03672e6i	− 2.86600i
$844$	1.24149e6	1.74284
$845$	0	0
$846$	0	0
$847$	456288.i	0.636022i
$848$	0	0
$849$	2.58915e6	3.59205
$850$	0	0
$851$	0	0
$852$	− 2.74122e6i	− 3.77628i
$853$	− 1.44782e6i	− 1.98984i	−0.100692	−	0.994918i	$-0.532106\pi$
$853$	− 1.44782e6i	− 1.98984i	0.100692	−	0.994918i	$-0.467894\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	970462.i	1.32135i	0.750673	+	0.660674i	$0.229729\pi$
$857$	970462.i	1.32135i	−0.750673	+	0.660674i	$0.770271\pi$
$858$	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
$867$	− 243984.i	− 0.324581i
$868$	0	0
$869$	888191.	1.17616
$870$	0	0
$871$	0	0
$872$	0	0
$873$	703664.i	0.923287i
$874$	0	0
$875$	0	0
$876$	2.58454e6	3.36803
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	−2.91273e6	−3.76983
$880$	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	$0.5\pi$
$884$	−96784.0	−0.123851
$885$	0	0
$886$	0	0
$887$	1.32950e6i	1.68983i	0.534904	+	0.844913i	$0.320348\pi$
$887$	1.32950e6i	1.68983i	−0.534904	+	0.844913i	$0.679652\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.44942e6	−1.82573
$892$	− 981488.i	− 1.23355i
$893$	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	$0.5\pi$
$908$	797168.i	0.966892i
$909$	0	0
$910$	0	0
$911$	1.15584e6	1.39271	0.696357	−	0.717696i	$-0.254803\pi$
$911$	1.15584e6	1.39271	0.696357	+	0.717696i	$0.254803\pi$
$912$	0	0
$913$	465886.i	0.558905i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	695113.	0.823047	0.411523	−	0.911399i	$-0.364997\pi$
$919$	695113.	0.823047	0.411523	+	0.911399i	$0.364997\pi$
$920$	0	0
$921$	−2.29947e6	−2.71087
$922$	0	0
$923$	− 231794.i	− 0.272081i
$924$	−972944.	−1.13958
$925$	0	0
$926$	0	0
$927$	− 3.82366e6i	− 4.44959i
$928$	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
$937$	− 1.58930e6i	− 1.81020i	−0.425195	−	0.905102i	$-0.639794\pi$
$937$	− 1.58930e6i	− 1.81020i	0.425195	−	0.905102i	$-0.360206\pi$
$938$	0	0
$939$	−2.50065e6	−2.83610
$940$	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	$0.5\pi$
$948$	− 3.30942e6i	− 3.68244i
$949$	218546.	0.242667
$950$	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
$956$	693872.	0.759213
$957$	1.43087e6i	1.56235i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	923521.	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			$\pi$
$972$	2.60250e6i	2.75459i
$973$	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
$981$	−2.98542e6	−3.10219
$982$	0	0
$983$	− 675937.i	− 0.699518i	−0.936840	−	0.349759i	$-0.886263\pi$
$983$	− 675937.i	− 0.699518i	0.936840	−	0.349759i	$-0.113737\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 2.87968e6i	− 2.95604i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−1.44288e6	−1.46920	−0.734602	−	0.678498i	$-0.762631\pi$
$991$	−1.44288e6	−1.46920	−0.734602	+	0.678498i	$0.762631\pi$
$992$	0	0
$993$	− 2.35419e6i	− 2.38750i
$994$	0	0
$995$	0	0
$996$	1.73590e6	1.74988
$997$	− 737783.i	− 0.742230i	−0.928587	−	0.371115i	$-0.878976\pi$
$997$	− 737783.i	− 0.742230i	0.928587	−	0.371115i	$-0.121024\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.5.d.c.76.1		2
5.2	odd	4		35.5.c.a.34.1	✓	1
5.3	odd	4		35.5.c.b.34.1	yes	1
5.4	even	2	inner	175.5.d.c.76.2		2
7.6	odd	2	inner	175.5.d.c.76.2		2
15.2	even	4		315.5.e.a.244.1		1
15.8	even	4		315.5.e.b.244.1		1
20.3	even	4		560.5.p.a.209.1		1
20.7	even	4		560.5.p.b.209.1		1
35.13	even	4		35.5.c.a.34.1	✓	1
35.27	even	4		35.5.c.b.34.1	yes	1
35.34	odd	2	CM	175.5.d.c.76.1		2
105.62	odd	4		315.5.e.b.244.1		1
105.83	odd	4		315.5.e.a.244.1		1
140.27	odd	4		560.5.p.a.209.1		1
140.83	odd	4		560.5.p.b.209.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.5.c.a.34.1	✓	1	5.2	odd	4
35.5.c.a.34.1	✓	1	35.13	even	4
35.5.c.b.34.1	yes	1	5.3	odd	4
35.5.c.b.34.1	yes	1	35.27	even	4
175.5.d.c.76.1		2	1.1	even	1	trivial
175.5.d.c.76.1		2	35.34	odd	2	CM
175.5.d.c.76.2		2	5.4	even	2	inner
175.5.d.c.76.2		2	7.6	odd	2	inner
315.5.e.a.244.1		1	15.2	even	4
315.5.e.a.244.1		1	105.83	odd	4
315.5.e.b.244.1		1	15.8	even	4
315.5.e.b.244.1		1	105.62	odd	4
560.5.p.a.209.1		1	20.3	even	4
560.5.p.a.209.1		1	140.27	odd	4
560.5.p.b.209.1		1	20.7	even	4
560.5.p.b.209.1		1	140.83	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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	175.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-17.0000i q^{3} -16.0000 q^{4} -49.0000i q^{7} -208.000 q^{9} -73.0000 q^{11} +272.000i q^{12} +23.0000i q^{13} +256.000 q^{16} -263.000i q^{17} -833.000 q^{21} +2159.00i q^{27} +784.000i q^{28} +1153.00 q^{29} +1241.00i q^{33} +3328.00 q^{36} +391.000 q^{39} +1168.00 q^{44} +3457.00i q^{47} -4352.00i q^{48} -2401.00 q^{49} -4471.00 q^{51} -368.000i q^{52} +10192.0i q^{63} -4096.00 q^{64} +4208.00i q^{68} -10078.0 q^{71} -9502.00i q^{73} +3577.00i q^{77} -12167.0 q^{79} +19855.0 q^{81} -6382.00i q^{83} +13328.0 q^{84} -19601.0i q^{87} +1127.00 q^{91} -3383.00i q^{97} +15184.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4} - 416 q^{9} - 146 q^{11} + 512 q^{16} - 1666 q^{21} + 2306 q^{29} + 6656 q^{36} + 782 q^{39} + 2336 q^{44} - 4802 q^{49} - 8942 q^{51} - 8192 q^{64} - 20156 q^{71} - 24334 q^{79} + 39710 q^{81}+ \cdots + 30368 q^{99}+O(q^{100}) \) 2 * q - 32 * q^4 - 416 * q^9 - 146 * q^11 + 512 * q^16 - 1666 * q^21 + 2306 * q^29 + 6656 * q^36 + 782 * q^39 + 2336 * q^44 - 4802 * q^49 - 8942 * q^51 - 8192 * q^64 - 20156 * q^71 - 24334 * q^79 + 39710 * q^81 + 26656 * q^84 + 2254 * q^91 + 30368 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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