LMFDB - Embedded newform 175.5.c.a.174.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [175,5,Mod(174,175)] 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("175.174"); 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([1, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 175 = 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	175.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == 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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 7)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			174.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	175.174
Dual form			175.5.c.a.174.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +15.0000 q^{4} -49.0000i q^{7} -31.0000i q^{8} -81.0000 q^{9} -206.000 q^{11} -49.0000 q^{14} +209.000 q^{16} +81.0000i q^{18} +206.000i q^{22} -734.000i q^{23} -735.000i q^{28} -1234.00 q^{29} -705.000i q^{32} -1215.00 q^{36} +1294.00i q^{37} -334.000i q^{43} -3090.00 q^{44} -734.000 q^{46} -2401.00 q^{49} -5582.00i q^{53} -1519.00 q^{56} +1234.00i q^{58} +3969.00i q^{63} +2639.00 q^{64} -4946.00i q^{67} +2914.00 q^{71} +2511.00i q^{72} +1294.00 q^{74} +10094.0i q^{77} +3646.00 q^{79} +6561.00 q^{81} -334.000 q^{86} +6386.00i q^{88} -11010.0i q^{92} +2401.00i q^{98} +16686.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 30 q^{4} - 162 q^{9} - 412 q^{11} - 98 q^{14} + 418 q^{16} - 2468 q^{29} - 2430 q^{36} - 6180 q^{44} - 1468 q^{46} - 4802 q^{49} - 3038 q^{56} + 5278 q^{64} + 5828 q^{71} + 2588 q^{74} + 7292 q^{79}+ \cdots + 33372 q^{99}+O(q^{100}) $ 2 * q + 30 * q^4 - 162 * q^9 - 412 * q^11 - 98 * q^14 + 418 * q^16 - 2468 * q^29 - 2430 * q^36 - 6180 * q^44 - 1468 * q^46 - 4802 * q^49 - 3038 * q^56 + 5278 * q^64 + 5828 * q^71 + 2588 * q^74 + 7292 * q^79 + 13122 * q^81 - 668 * q^86 + 33372 * 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 $

$2$	− 1.00000i	− 0.250000i	−0.992157	−	0.125000i	$-0.960107\pi$
$2$	− 1.00000i	− 0.250000i	0.992157	−	0.125000i	$-0.0398931\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	15.0000	0.937500
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 49.0000i	− 1.00000i
$7$	− 49.0000i	− 1.00000i
$8$	− 31.0000i	− 0.484375i
$9$	−81.0000	−1.00000
$10$	0	0
$11$	−206.000	−1.70248	−0.851240	−	0.524777i	$-0.824149\pi$
$11$	−206.000	−1.70248	−0.851240	+	0.524777i	$0.824149\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−49.0000	−0.250000
$15$	0	0
$16$	209.000	0.816406
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	81.0000i	0.250000i
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	206.000i	0.425620i
$23$	− 734.000i	− 1.38752i	−0.720205	−	0.693762i	$-0.755952\pi$
$23$	− 734.000i	− 1.38752i	0.720205	−	0.693762i	$-0.244048\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	− 735.000i	− 0.937500i
$29$	−1234.00	−1.46730	−0.733650	−	0.679527i	$-0.762185\pi$
$29$	−1234.00	−1.46730	−0.733650	+	0.679527i	$0.762185\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	− 705.000i	− 0.688477i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−1215.00	−0.937500
$37$	1294.00i	0.945215i	0.881273	+	0.472608i	$0.156687\pi$
$37$	1294.00i	0.945215i	−0.881273	+	0.472608i	$0.843313\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	− 334.000i	− 0.180638i	−0.995913	−	0.0903191i	$-0.971211\pi$
$43$	− 334.000i	− 0.180638i	0.995913	−	0.0903191i	$-0.0287887\pi$
$44$	−3090.00	−1.59607
$45$	0	0
$46$	−734.000	−0.346881
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−2401.00	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 5582.00i	− 1.98718i	−0.113026	−	0.993592i	$-0.536054\pi$
$53$	− 5582.00i	− 1.98718i	0.113026	−	0.993592i	$-0.463946\pi$
$54$	0	0
$55$	0	0
$56$	−1519.00	−0.484375
$57$	0	0
$58$	1234.00i	0.366825i
$59$	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
$63$	3969.00i	1.00000i
$64$	2639.00	0.644287
$65$	0	0
$66$	0	0
$67$	− 4946.00i	− 1.10180i	−0.834570	−	0.550902i	$-0.814284\pi$
$67$	− 4946.00i	− 1.10180i	0.834570	−	0.550902i	$-0.185716\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2914.00	0.578060	0.289030	−	0.957320i	$-0.406667\pi$
$71$	2914.00	0.578060	0.289030	+	0.957320i	$0.406667\pi$
$72$	2511.00i	0.484375i
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	1294.00	0.236304
$75$	0	0
$76$	0	0
$77$	10094.0i	1.70248i
$78$	0	0
$79$	3646.00	0.584201	0.292101	−	0.956388i	$-0.405646\pi$
$79$	3646.00	0.584201	0.292101	+	0.956388i	$0.405646\pi$
$80$	0	0
$81$	6561.00	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	−334.000	−0.0451595
$87$	0	0
$88$	6386.00i	0.824638i
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	− 11010.0i	− 1.30080i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	2401.00i	0.250000i
$99$	16686.0	1.70248
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	−5582.00	−0.496796
$107$	− 11698.0i	− 1.02175i	−0.859655	−	0.510874i	$-0.829322\pi$
$107$	− 11698.0i	− 1.02175i	0.859655	−	0.510874i	$-0.170678\pi$
$108$	0	0
$109$	12526.0	1.05429	0.527144	−	0.849776i	$-0.323263\pi$
$109$	12526.0	1.05429	0.527144	+	0.849776i	$0.323263\pi$
$110$	0	0
$111$	0	0
$112$	− 10241.0i	− 0.816406i
$113$	23746.0i	1.85966i	0.367989	+	0.929830i	$0.380046\pi$
$113$	23746.0i	1.85966i	−0.367989	+	0.929830i	$0.619954\pi$
$114$	0	0
$115$	0	0
$116$	−18510.0	−1.37559
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	27795.0	1.89844
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	3969.00	0.250000
$127$	32254.0i	1.99975i	0.0157475	+	0.999876i	$0.494987\pi$
$127$	32254.0i	1.99975i	−0.0157475	+	0.999876i	$0.505013\pi$
$128$	− 13919.0i	− 0.849548i
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	−4946.00	−0.275451
$135$	0	0
$136$	0	0
$137$	7262.00i	0.386915i	0.981109	+	0.193457i	$0.0619701\pi$
$137$	7262.00i	0.386915i	−0.981109	+	0.193457i	$0.938030\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	− 2914.00i	− 0.144515i
$143$	0	0
$144$	−16929.0	−0.816406
$145$	0	0
$146$	0	0
$147$	0	0
$148$	19410.0i	0.886140i
$149$	9806.00	0.441692	0.220846	−	0.975309i	$-0.429118\pi$
$149$	9806.00	0.441692	0.220846	+	0.975309i	$0.429118\pi$
$150$	0	0
$151$	29474.0	1.29266	0.646331	−	0.763057i	$-0.276302\pi$
$151$	29474.0	1.29266	0.646331	+	0.763057i	$0.276302\pi$
$152$	0	0
$153$	0	0
$154$	10094.0	0.425620
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	− 3646.00i	− 0.146050i
$159$	0	0
$160$	0	0
$161$	−35966.0	−1.38752
$162$	− 6561.00i	− 0.250000i
$163$	− 47662.0i	− 1.79390i	−0.442137	−	0.896948i	$-0.645779\pi$
$163$	− 47662.0i	− 1.79390i	0.442137	−	0.896948i	$-0.354221\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−28561.0	−1.00000
$170$	0	0
$171$	0	0
$172$	− 5010.00i	− 0.169348i
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	−43054.0	−1.38991
$177$	0	0
$178$	0	0
$179$	−52882.0	−1.65045	−0.825224	−	0.564806i	$-0.808951\pi$
$179$	−52882.0	−1.65045	−0.825224	+	0.564806i	$0.808951\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	−22754.0	−0.672082
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	28162.0	0.771963	0.385982	−	0.922506i	$-0.373863\pi$
$191$	28162.0	0.771963	0.385982	+	0.922506i	$0.373863\pi$
$192$	0	0
$193$	− 70654.0i	− 1.89680i	−0.317073	−	0.948401i	$-0.602700\pi$
$193$	− 70654.0i	− 1.89680i	0.317073	−	0.948401i	$-0.397300\pi$
$194$	0	0
$195$	0	0
$196$	−36015.0	−0.937500
$197$	− 1906.00i	− 0.0491123i	−0.999698	−	0.0245562i	$-0.992183\pi$
$197$	− 1906.00i	− 0.0491123i	0.999698	−	0.0245562i	$-0.00781725\pi$
$198$	− 16686.0i	− 0.425620i
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	60466.0i	1.46730i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	59454.0i	1.38752i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−11758.0	−0.264100	−0.132050	−	0.991243i	$-0.542156\pi$
$211$	−11758.0	−0.264100	−0.132050	+	0.991243i	$0.542156\pi$
$212$	− 83730.0i	− 1.86299i
$213$	0	0
$214$	−11698.0	−0.255437
$215$	0	0
$216$	0	0
$217$	0	0
$218$	− 12526.0i	− 0.263572i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	−34545.0	−0.688477
$225$	0	0
$226$	23746.0	0.464915
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	38254.0i	0.710724i
$233$	− 108254.i	− 1.99403i	−0.0771956	−	0.997016i	$-0.524597\pi$
$233$	− 108254.i	− 1.99403i	0.0771956	−	0.997016i	$-0.475403\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	64958.0	1.13720	0.568600	−	0.822614i	$-0.307485\pi$
$239$	64958.0	1.13720	0.568600	+	0.822614i	$0.307485\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	− 27795.0i	− 0.474609i
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	59535.0i	0.937500i
$253$	151204.i	2.36223i
$254$	32254.0	0.499938
$255$	0	0
$256$	28305.0	0.431900
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	63406.0	0.945215
$260$	0	0
$261$	99954.0	1.46730
$262$	0	0
$263$	109666.i	1.58548i	0.609561	+	0.792740i	$0.291346\pi$
$263$	109666.i	1.58548i	−0.609561	+	0.792740i	$0.708654\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	− 74190.0i	− 1.03294i
$269$	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			$\pi$
$272$	0	0
$273$	0	0
$274$	7262.00	0.0967286
$275$	0	0
$276$	0	0
$277$	− 52658.0i	− 0.686285i	−0.939283	−	0.343143i	$-0.888509\pi$
$277$	− 52658.0i	− 0.686285i	0.939283	−	0.343143i	$-0.111491\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−144926.	−1.83541	−0.917706	−	0.397260i	$-0.869961\pi$
$281$	−144926.	−1.83541	−0.917706	+	0.397260i	$0.869961\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	43710.0	0.541931
$285$	0	0
$286$	0	0
$287$	0	0
$288$	57105.0i	0.688477i
$289$	−83521.0	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	40114.0	0.457839
$297$	0	0
$298$	− 9806.00i	− 0.110423i
$299$	0	0
$300$	0	0
$301$	−16366.0	−0.180638
$302$	− 29474.0i	− 0.323166i
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	151410.i	1.59607i
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	54690.0	0.547689
$317$	− 71506.0i	− 0.711580i	−0.934566	−	0.355790i	$-0.884212\pi$
$317$	− 71506.0i	− 0.711580i	0.934566	−	0.355790i	$-0.115788\pi$
$318$	0	0
$319$	254204.	2.49805
$320$	0	0
$321$	0	0
$322$	35966.0i	0.346881i
$323$	0	0
$324$	98415.0	0.937500
$325$	0	0
$326$	−47662.0	−0.448474
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	182834.	1.66879	0.834394	−	0.551169i	$-0.185818\pi$
$331$	182834.	1.66879	0.834394	+	0.551169i	$0.185818\pi$
$332$	0	0
$333$	− 104814.i	− 0.945215i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	176062.i	1.55026i	0.631799	+	0.775132i	$0.282317\pi$
$337$	176062.i	1.55026i	−0.631799	+	0.775132i	$0.717683\pi$
$338$	28561.0i	0.250000i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	117649.i	1.00000i
$344$	−10354.0	−0.0874966
$345$	0	0
$346$	0	0
$347$	− 218866.i	− 1.81769i	−0.417136	−	0.908844i	$-0.636966\pi$
$347$	− 218866.i	− 1.81769i	0.417136	−	0.908844i	$-0.363034\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	145230.i	1.17212i
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	52882.0i	0.412612i
$359$	−169954.	−1.31869	−0.659345	−	0.751841i	$-0.729166\pi$
$359$	−169954.	−1.31869	−0.659345	+	0.751841i	$0.729166\pi$
$360$	0	0
$361$	130321.	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	− 153406.i	− 1.13278i
$369$	0	0
$370$	0	0
$371$	−273518.	−1.98718
$372$	0	0
$373$	− 209614.i	− 1.50662i	−0.657668	−	0.753308i	$-0.728457\pi$
$373$	− 209614.i	− 1.50662i	0.657668	−	0.753308i	$-0.271543\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−89714.0	−0.624571	−0.312285	−	0.949988i	$-0.601095\pi$
$379$	−89714.0	−0.624571	−0.312285	+	0.949988i	$0.601095\pi$
$380$	0	0
$381$	0	0
$382$	− 28162.0i	− 0.192991i
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	−70654.0	−0.474201
$387$	27054.0i	0.180638i
$388$	0	0
$389$	−140914.	−0.931226	−0.465613	−	0.884989i	$-0.654166\pi$
$389$	−140914.	−0.931226	−0.465613	+	0.884989i	$0.654166\pi$
$390$	0	0
$391$	0	0
$392$	74431.0i	0.484375i
$393$	0	0
$394$	−1906.00	−0.0122781
$395$	0	0
$396$	250290.	1.59607
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−196286.	−1.22068	−0.610338	−	0.792141i	$-0.708966\pi$
$401$	−196286.	−1.22068	−0.610338	+	0.792141i	$0.708966\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	60466.0	0.366825
$407$	− 266564.i	− 1.60921i
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	59454.0	0.346881
$415$	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
$421$	−326926.	−1.84453	−0.922264	−	0.386561i	$-0.873663\pi$
$421$	−326926.	−1.84453	−0.922264	+	0.386561i	$0.873663\pi$
$422$	11758.0i	0.0660250i
$423$	0	0
$424$	−173042.	−0.962542
$425$	0	0
$426$	0	0
$427$	0	0
$428$	− 175470.i	− 0.957889i
$429$	0	0
$430$	0	0
$431$	−345278.	−1.85872	−0.929361	−	0.369173i	$-0.879641\pi$
$431$	−345278.	−1.85872	−0.929361	+	0.369173i	$0.879641\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	187890.	0.988395
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	194481.	1.00000
$442$	0	0
$443$	− 156302.i	− 0.796447i	−0.917288	−	0.398224i	$-0.869627\pi$
$443$	− 156302.i	− 0.796447i	0.917288	−	0.398224i	$-0.130373\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	− 129311.i	− 0.644287i
$449$	−396034.	−1.96444	−0.982222	−	0.187721i	$-0.939890\pi$
$449$	−396034.	−1.96444	−0.982222	+	0.187721i	$0.939890\pi$
$450$	0	0
$451$	0	0
$452$	356190.i	1.74343i
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 353186.i	− 1.69111i	−0.533891	−	0.845553i	$-0.679271\pi$
$457$	− 353186.i	− 1.69111i	0.533891	−	0.845553i	$-0.320729\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	25538.0i	0.119131i	0.998224	+	0.0595655i	$0.0189715\pi$
$463$	25538.0i	0.119131i	−0.998224	+	0.0595655i	$0.981028\pi$
$464$	−257906.	−1.19791
$465$	0	0
$466$	−108254.	−0.498508
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	−242354.	−1.10180
$470$	0	0
$471$	0	0
$472$	0	0
$473$	68804.0i	0.307533i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	452142.i	1.98718i
$478$	− 64958.0i	− 0.284300i
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	416925.	1.77978
$485$	0	0
$486$	0	0
$487$	315934.i	1.33210i	0.745905	+	0.666052i	$0.232017\pi$
$487$	315934.i	1.33210i	−0.745905	+	0.666052i	$0.767983\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	427954.	1.77515	0.887573	−	0.460667i	$-0.152390\pi$
$491$	427954.	1.77515	0.887573	+	0.460667i	$0.152390\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 142786.i	− 0.578060i
$498$	0	0
$499$	409198.	1.64336	0.821679	−	0.569950i	$-0.193037\pi$
$499$	409198.	1.64336	0.821679	+	0.569950i	$0.193037\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	123039.	0.484375
$505$	0	0
$506$	151204.	0.590558
$507$	0	0
$508$	483810.i	1.87477i
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	− 251009.i	− 0.957523i
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	− 63406.0i	− 0.236304i
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	− 99954.0i	− 0.366825i
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	109666.	0.396370
$527$	0	0
$528$	0	0
$529$	−258915.	−0.925222
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−153326.	−0.533687
$537$	0	0
$538$	0	0
$539$	494606.	1.70248
$540$	0	0
$541$	−579886.	−1.98129	−0.990645	−	0.136463i	$-0.956426\pi$
$541$	−579886.	−1.98129	−0.990645	+	0.136463i	$0.956426\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 110546.i	− 0.369461i	−0.982789	−	0.184730i	$-0.940859\pi$
$547$	− 110546.i	− 0.369461i	0.982789	−	0.184730i	$-0.0591412\pi$
$548$	108930.i	0.362732i
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 178654.i	− 0.584201i
$554$	−52658.0	−0.171571
$555$	0	0
$556$	0	0
$557$	− 383506.i	− 1.23612i	−0.786130	−	0.618062i	$-0.787918\pi$
$557$	− 383506.i	− 1.23612i	0.786130	−	0.618062i	$-0.212082\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	144926.i	0.458853i
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 321489.i	− 1.00000i
$568$	− 90334.0i	− 0.279998i
$569$	219806.	0.678914	0.339457	−	0.940622i	$-0.389757\pi$
$569$	219806.	0.678914	0.339457	+	0.940622i	$0.389757\pi$
$570$	0	0
$571$	615794.	1.88870	0.944351	−	0.328941i	$-0.106692\pi$
$571$	615794.	1.88870	0.944351	+	0.328941i	$0.106692\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−213759.	−0.644287
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	83521.0i	0.250000i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.14989e6i	3.38314i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	270446.i	0.771680i
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	147090.	0.414086
$597$	0	0
$598$	0	0
$599$	687326.	1.91562	0.957809	−	0.287404i	$-0.0927922\pi$
$599$	687326.	1.91562	0.957809	+	0.287404i	$0.0927922\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	16366.0i	0.0451595i
$603$	400626.i	1.10180i
$604$	442110.	1.21187
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 704014.i	− 1.87353i	−0.349961	−	0.936764i	$-0.613805\pi$
$613$	− 704014.i	− 1.87353i	0.349961	−	0.936764i	$-0.386195\pi$
$614$	0	0
$615$	0	0
$616$	312914.	0.824638
$617$	450014.i	1.18210i	0.806633	+	0.591052i	$0.201287\pi$
$617$	450014.i	1.18210i	−0.806633	+	0.591052i	$0.798713\pi$
$618$	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
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	215714.	0.541776	0.270888	−	0.962611i	$-0.412683\pi$
$631$	215714.	0.541776	0.270888	+	0.962611i	$0.412683\pi$
$632$	− 113026.i	− 0.282972i
$633$	0	0
$634$	−71506.0	−0.177895
$635$	0	0
$636$	0	0
$637$	0	0
$638$	− 254204.i	− 0.624512i
$639$	−236034.	−0.578060
$640$	0	0
$641$	−126206.	−0.307159	−0.153580	−	0.988136i	$-0.549080\pi$
$641$	−126206.	−0.307159	−0.153580	+	0.988136i	$0.549080\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	−539490.	−1.30080
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	− 203391.i	− 0.484375i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	− 714930.i	− 1.68178i
$653$	572818.i	1.34335i	0.740844	+	0.671677i	$0.234426\pi$
$653$	572818.i	1.34335i	−0.740844	+	0.671677i	$0.765574\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	486638.	1.12056	0.560280	−	0.828303i	$-0.310693\pi$
$659$	486638.	1.12056	0.560280	+	0.828303i	$0.310693\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	− 182834.i	− 0.417197i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−104814.	−0.236304
$667$	905756.i	2.03591i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 706942.i	− 1.56082i	−0.625266	−	0.780411i	$-0.715010\pi$
$673$	− 706942.i	− 1.56082i	0.625266	−	0.780411i	$-0.284990\pi$
$674$	176062.	0.387566
$675$	0	0
$676$	−428415.	−0.937500
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	857266.i	1.83770i	0.394609	+	0.918849i	$0.370880\pi$
$683$	857266.i	1.83770i	−0.394609	+	0.918849i	$0.629120\pi$
$684$	0	0
$685$	0	0
$686$	117649.	0.250000
$687$	0	0
$688$	− 69806.0i	− 0.147474i
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	− 817614.i	− 1.70248i
$694$	−218866.	−0.454422
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	971602.	1.97721	0.988604	−	0.150539i	$-0.0481010\pi$
$701$	971602.	1.97721	0.988604	+	0.150539i	$0.0481010\pi$
$702$	0	0
$703$	0	0
$704$	−543634.	−1.09689
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−904562.	−1.79948	−0.899738	−	0.436431i	$-0.856242\pi$
$709$	−904562.	−1.79948	−0.899738	+	0.436431i	$0.856242\pi$
$710$	0	0
$711$	−295326.	−0.584201
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−793230.	−1.54729
$717$	0	0
$718$	169954.i	0.329672i
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	− 130321.i	− 0.250000i
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−531441.	−1.00000
$730$	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	0
$735$	0	0
$736$	−517470.	−0.955277
$737$	1.01888e6i	1.87580i
$738$	0	0
$739$	−410834.	−0.752277	−0.376138	−	0.926564i	$-0.622748\pi$
$739$	−410834.	−0.752277	−0.376138	+	0.926564i	$0.622748\pi$
$740$	0	0
$741$	0	0
$742$	273518.i	0.496796i
$743$	− 1.09110e6i	− 1.97646i	−0.152980	−	0.988229i	$-0.548887\pi$
$743$	− 1.09110e6i	− 1.97646i	0.152980	−	0.988229i	$-0.451113\pi$
$744$	0	0
$745$	0	0
$746$	−209614.	−0.376654
$747$	0	0
$748$	0	0
$749$	−573202.	−1.02175
$750$	0	0
$751$	−484798.	−0.859569	−0.429785	−	0.902931i	$-0.641411\pi$
$751$	−484798.	−0.859569	−0.429785	+	0.902931i	$0.641411\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 819506.i	− 1.43008i	−0.699083	−	0.715040i	$-0.746408\pi$
$757$	− 819506.i	− 1.43008i	0.699083	−	0.715040i	$-0.253592\pi$
$758$	89714.0i	0.156143i
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	− 613774.i	− 1.05429i
$764$	422430.	0.723716
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	− 1.05981e6i	− 1.77825i
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	27054.0	0.0451595
$775$	0	0
$776$	0	0
$777$	0	0
$778$	140914.i	0.232806i
$779$	0	0
$780$	0	0
$781$	−600284.	−0.984135
$782$	0	0
$783$	0	0
$784$	−501809.	−0.816406
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	− 28590.0i	− 0.0460428i
$789$	0	0
$790$	0	0
$791$	1.16355e6	1.85966
$792$	− 517266.i	− 0.824638i
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	196286.i	0.305169i
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.27869e6	1.95374	0.976870	−	0.213833i	$-0.0685949\pi$
$809$	1.27869e6	1.95374	0.976870	+	0.213833i	$0.0685949\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	906990.i	1.37559i
$813$	0	0
$814$	−266564.	−0.402302
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−7118.00	−0.0105602	−0.00528009	−	0.999986i	$-0.501681\pi$
$821$	−7118.00	−0.0105602	−0.00528009	+	0.999986i	$0.501681\pi$
$822$	0	0
$823$	− 967774.i	− 1.42881i	−0.699733	−	0.714405i	$-0.746698\pi$
$823$	− 967774.i	− 1.42881i	0.699733	−	0.714405i	$-0.253302\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.28833e6i	1.88372i	0.335999	+	0.941862i	$0.390926\pi$
$827$	1.28833e6i	1.88372i	−0.335999	+	0.941862i	$0.609074\pi$
$828$	891810.i	1.30080i
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	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
$841$	815475.	1.15297
$842$	326926.i	0.461132i
$843$	0	0
$844$	−176370.	−0.247594
$845$	0	0
$846$	0	0
$847$	− 1.36196e6i	− 1.89844i
$848$	− 1.16664e6i	− 1.62235i
$849$	0	0
$850$	0	0
$851$	949796.	1.31151
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	−362638.	−0.494909
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	345278.i	0.464680i
$863$	1.27271e6i	1.70886i	0.519566	+	0.854430i	$0.326093\pi$
$863$	1.27271e6i	1.70886i	−0.519566	+	0.854430i	$0.673907\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−751076.	−0.994591
$870$	0	0
$871$	0	0
$872$	− 388306.i	− 0.510671i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	981742.i	1.27643i	0.769857	+	0.638217i	$0.220328\pi$
$877$	981742.i	1.27643i	−0.769857	+	0.638217i	$0.779672\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	− 194481.i	− 0.250000i
$883$	1.07151e6i	1.37427i	0.726528	+	0.687137i	$0.241133\pi$
$883$	1.07151e6i	1.37427i	−0.726528	+	0.687137i	$0.758867\pi$
$884$	0	0
$885$	0	0
$886$	−156302.	−0.199112
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	1.58045e6	1.99975
$890$	0	0
$891$	−1.35157e6	−1.70248
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−682031.	−0.849548
$897$	0	0
$898$	396034.i	0.491111i
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	736126.	0.900773
$905$	0	0
$906$	0	0
$907$	− 1.54450e6i	− 1.87747i	−0.344641	−	0.938735i	$-0.611999\pi$
$907$	− 1.54450e6i	− 1.87747i	0.344641	−	0.938735i	$-0.388001\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	792514.	0.954927	0.477464	−	0.878652i	$-0.341556\pi$
$911$	792514.	0.954927	0.477464	+	0.878652i	$0.341556\pi$
$912$	0	0
$913$	0	0
$914$	−353186.	−0.422777
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1.47197e6	1.74288	0.871439	−	0.490505i	$-0.163188\pi$
$919$	1.47197e6	1.74288	0.871439	+	0.490505i	$0.163188\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	25538.0	0.0297828
$927$	0	0
$928$	869970.i	1.01020i
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	− 1.62381e6i	− 1.86940i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	242354.i	0.275451i
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	68804.0	0.0768832
$947$	− 438418.i	− 0.488864i	−0.969666	−	0.244432i	$-0.921398\pi$
$947$	− 438418.i	− 0.488864i	0.969666	−	0.244432i	$-0.0786016\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	309346.i	0.340611i	0.985391	+	0.170306i	$0.0544755\pi$
$953$	309346.i	0.340611i	−0.985391	+	0.170306i	$0.945525\pi$
$954$	452142.	0.496796
$955$	0	0
$956$	974370.	1.06612
$957$	0	0
$958$	0	0
$959$	355838.	0.386915
$960$	0	0
$961$	923521.	1.00000
$962$	0	0
$963$	947538.i	1.02175i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.75862e6i	1.88070i	0.340208	+	0.940350i	$0.389502\pi$
$967$	1.75862e6i	1.88070i	−0.340208	+	0.940350i	$0.610498\pi$
$968$	− 861645.i	− 0.919555i
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	315934.	0.333026
$975$	0	0
$976$	0	0
$977$	1.88281e6i	1.97251i	0.165243	+	0.986253i	$0.447159\pi$
$977$	1.88281e6i	1.97251i	−0.165243	+	0.986253i	$0.552841\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−1.01461e6	−1.05429
$982$	− 427954.i	− 0.443787i
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−245156.	−0.250640
$990$	0	0
$991$	12674.0	0.0129052	0.00645262	−	0.999979i	$-0.497946\pi$
$991$	12674.0	0.0129052	0.00645262	+	0.999979i	$0.497946\pi$
$992$	0	0
$993$	0	0
$994$	−142786.	−0.144515
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	− 409198.i	− 0.410840i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.5.c.a.174.1		2
5.2	odd	4		7.5.b.a.6.1	✓	1
5.3	odd	4		175.5.d.a.76.1		1
5.4	even	2	inner	175.5.c.a.174.2		2
7.6	odd	2	CM	175.5.c.a.174.1		2
15.2	even	4		63.5.d.a.55.1		1
20.7	even	4		112.5.c.a.97.1		1
35.2	odd	12		49.5.d.a.31.1		2
35.12	even	12		49.5.d.a.31.1		2
35.13	even	4		175.5.d.a.76.1		1
35.17	even	12		49.5.d.a.19.1		2
35.27	even	4		7.5.b.a.6.1	✓	1
35.32	odd	12		49.5.d.a.19.1		2
35.34	odd	2	inner	175.5.c.a.174.2		2
40.27	even	4		448.5.c.a.321.1		1
40.37	odd	4		448.5.c.b.321.1		1
60.47	odd	4		1008.5.f.a.433.1		1
105.62	odd	4		63.5.d.a.55.1		1
140.27	odd	4		112.5.c.a.97.1		1
280.27	odd	4		448.5.c.a.321.1		1
280.237	even	4		448.5.c.b.321.1		1
420.167	even	4		1008.5.f.a.433.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.5.b.a.6.1	✓	1	5.2	odd	4
7.5.b.a.6.1	✓	1	35.27	even	4
49.5.d.a.19.1		2	35.17	even	12
49.5.d.a.19.1		2	35.32	odd	12
49.5.d.a.31.1		2	35.2	odd	12
49.5.d.a.31.1		2	35.12	even	12
63.5.d.a.55.1		1	15.2	even	4
63.5.d.a.55.1		1	105.62	odd	4
112.5.c.a.97.1		1	20.7	even	4
112.5.c.a.97.1		1	140.27	odd	4
175.5.c.a.174.1		2	1.1	even	1	trivial
175.5.c.a.174.1		2	7.6	odd	2	CM
175.5.c.a.174.2		2	5.4	even	2	inner
175.5.c.a.174.2		2	35.34	odd	2	inner
175.5.d.a.76.1		1	5.3	odd	4
175.5.d.a.76.1		1	35.13	even	4
448.5.c.a.321.1		1	40.27	even	4
448.5.c.a.321.1		1	280.27	odd	4
448.5.c.b.321.1		1	40.37	odd	4
448.5.c.b.321.1		1	280.237	even	4
1008.5.f.a.433.1		1	60.47	odd	4
1008.5.f.a.433.1		1	420.167	even	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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +15.0000 q^{4} -49.0000i q^{7} -31.0000i q^{8} -81.0000 q^{9} -206.000 q^{11} -49.0000 q^{14} +209.000 q^{16} +81.0000i q^{18} +206.000i q^{22} -734.000i q^{23} -735.000i q^{28} -1234.00 q^{29} -705.000i q^{32} -1215.00 q^{36} +1294.00i q^{37} -334.000i q^{43} -3090.00 q^{44} -734.000 q^{46} -2401.00 q^{49} -5582.00i q^{53} -1519.00 q^{56} +1234.00i q^{58} +3969.00i q^{63} +2639.00 q^{64} -4946.00i q^{67} +2914.00 q^{71} +2511.00i q^{72} +1294.00 q^{74} +10094.0i q^{77} +3646.00 q^{79} +6561.00 q^{81} -334.000 q^{86} +6386.00i q^{88} -11010.0i q^{92} +2401.00i q^{98} +16686.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 30 q^{4} - 162 q^{9} - 412 q^{11} - 98 q^{14} + 418 q^{16} - 2468 q^{29} - 2430 q^{36} - 6180 q^{44} - 1468 q^{46} - 4802 q^{49} - 3038 q^{56} + 5278 q^{64} + 5828 q^{71} + 2588 q^{74} + 7292 q^{79}+ \cdots + 33372 q^{99}+O(q^{100}) \) 2 * q + 30 * q^4 - 162 * q^9 - 412 * q^11 - 98 * q^14 + 418 * q^16 - 2468 * q^29 - 2430 * q^36 - 6180 * q^44 - 1468 * q^46 - 4802 * q^49 - 3038 * q^56 + 5278 * q^64 + 5828 * q^71 + 2588 * q^74 + 7292 * q^79 + 13122 * q^81 - 668 * q^86 + 33372 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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