LMFDB - Embedded newform 175.7.c.a.174.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [175,7,Mod(174,175)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 7, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("175.174"); S:= CuspForms(chi, 7); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [2,0,0,-34] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$40.2594646335$
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 7)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			174.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	175.174
Dual form			175.7.c.a.174.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000i q^{2} -17.0000 q^{4} -343.000i q^{7} +423.000i q^{8} -729.000 q^{9} +1962.00 q^{11} +3087.00 q^{14} -4895.00 q^{16} -6561.00i q^{18} +17658.0i q^{22} +22734.0i q^{23} +5831.00i q^{28} +21222.0 q^{29} -16983.0i q^{32} +12393.0 q^{36} +101194. i q^{37} +126614. i q^{43} -33354.0 q^{44} -204606. q^{46} -117649. q^{49} -50346.0i q^{53} +145089. q^{56} +190998. i q^{58} +250047. i q^{63} -160433. q^{64} -53926.0i q^{67} -242478. q^{71} -308367. i q^{72} -910746. q^{74} -672966. i q^{77} -929378. q^{79} +531441. q^{81} -1.13953e6 q^{86} +829926. i q^{88} -386478. i q^{92} -1.05884e6i q^{98} -1.43030e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 34 q^{4} - 1458 q^{9} + 3924 q^{11} + 6174 q^{14} - 9790 q^{16} + 42444 q^{29} + 24786 q^{36} - 66708 q^{44} - 409212 q^{46} - 235298 q^{49} + 290178 q^{56} - 320866 q^{64} - 484956 q^{71} - 1821492 q^{74}+ \cdots - 2860596 q^{99}+O(q^{100}) $ 2 * q - 34 * q^4 - 1458 * q^9 + 3924 * q^11 + 6174 * q^14 - 9790 * q^16 + 42444 * q^29 + 24786 * q^36 - 66708 * q^44 - 409212 * q^46 - 235298 * q^49 + 290178 * q^56 - 320866 * q^64 - 484956 * q^71 - 1821492 * q^74 - 1858756 * q^79 + 1062882 * q^81 - 2279052 * q^86 - 2860596 * 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 $

$2$	9.00000i	1.12500i	0.826797	+	0.562500i	$0.190160\pi$
$2$	9.00000i	1.12500i	−0.826797	+	0.562500i	$0.809840\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	−17.0000	−0.265625
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 343.000i	− 1.00000i
$7$	− 343.000i	− 1.00000i
$8$	423.000i	0.826172i
$9$	−729.000	−1.00000
$10$	0	0
$11$	1962.00	1.47408	0.737040	−	0.675849i	$-0.236223\pi$
$11$	1962.00	1.47408	0.737040	+	0.675849i	$0.236223\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	3087.00	1.12500
$15$	0	0
$16$	−4895.00	−1.19507
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	− 6561.00i	− 1.12500i
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	17658.0i	1.65834i
$23$	22734.0i	1.86850i	0.356623	+	0.934248i	$0.383928\pi$
$23$	22734.0i	1.86850i	−0.356623	+	0.934248i	$0.616072\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	5831.00i	0.265625i
$29$	21222.0	0.870146	0.435073	−	0.900395i	$-0.356722\pi$
$29$	21222.0	0.870146	0.435073	+	0.900395i	$0.356722\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	− 16983.0i	− 0.518280i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	12393.0	0.265625
$37$	101194.i	1.99779i	0.0470096	+	0.998894i	$0.485031\pi$
$37$	101194.i	1.99779i	−0.0470096	+	0.998894i	$0.514969\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	126614.i	1.59249i	0.604975	+	0.796244i	$0.293183\pi$
$43$	126614.i	1.59249i	−0.604975	+	0.796244i	$0.706817\pi$
$44$	−33354.0	−0.391552
$45$	0	0
$46$	−204606.	−2.10206
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−117649.	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 50346.0i	− 0.338172i	−0.985601	−	0.169086i	$-0.945918\pi$
$53$	− 50346.0i	− 0.338172i	0.985601	−	0.169086i	$-0.0540815\pi$
$54$	0	0
$55$	0	0
$56$	145089.	0.826172
$57$	0	0
$58$	190998.i	0.978915i
$59$	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
$63$	250047.i	1.00000i
$64$	−160433.	−0.612003
$65$	0	0
$66$	0	0
$67$	− 53926.0i	− 0.179297i	−0.995973	−	0.0896487i	$-0.971426\pi$
$67$	− 53926.0i	− 0.179297i	0.995973	−	0.0896487i	$-0.0285744\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−242478.	−0.677481	−0.338741	−	0.940880i	$-0.610001\pi$
$71$	−242478.	−0.677481	−0.338741	+	0.940880i	$0.610001\pi$
$72$	− 308367.i	− 0.826172i
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	−910746.	−2.24751
$75$	0	0
$76$	0	0
$77$	− 672966.i	− 1.47408i
$78$	0	0
$79$	−929378.	−1.88500	−0.942499	−	0.334208i	$-0.891531\pi$
$79$	−929378.	−1.88500	−0.942499	+	0.334208i	$0.891531\pi$
$80$	0	0
$81$	531441.	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
$86$	−1.13953e6	−1.79155
$87$	0	0
$88$	829926.i	1.21784i
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	− 386478.i	− 0.496319i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	− 1.05884e6i	− 1.12500i
$99$	−1.43030e6	−1.47408
$100$	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
$106$	453114.	0.380443
$107$	46314.0i	0.0378060i	0.999821	+	0.0189030i	$0.00601737\pi$
$107$	46314.0i	0.0378060i	−0.999821	+	0.0189030i	$0.993983\pi$
$108$	0	0
$109$	2.58714e6	1.99775	0.998874	−	0.0474386i	$-0.0151058\pi$
$109$	2.58714e6	1.99775	0.998874	+	0.0474386i	$0.0151058\pi$
$110$	0	0
$111$	0	0
$112$	1.67898e6i	1.19507i
$113$	2.43689e6i	1.68889i	0.535642	+	0.844445i	$0.320069\pi$
$113$	2.43689e6i	1.68889i	−0.535642	+	0.844445i	$0.679931\pi$
$114$	0	0
$115$	0	0
$116$	−360774.	−0.231133
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2.07788e6	1.17291
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	−2.25042e6	−1.12500
$127$	− 96766.0i	− 0.0472402i	−0.999721	−	0.0236201i	$-0.992481\pi$
$127$	− 96766.0i	− 0.0472402i	0.999721	−	0.0236201i	$-0.00751921\pi$
$128$	− 2.53081e6i	− 1.20678i
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	485334.	0.201709
$135$	0	0
$136$	0	0
$137$	4.52939e6i	1.76148i	0.473598	+	0.880741i	$0.342955\pi$
$137$	4.52939e6i	1.76148i	−0.473598	+	0.880741i	$0.657045\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	− 2.18230e6i	− 0.762166i
$143$	0	0
$144$	3.56846e6	1.19507
$145$	0	0
$146$	0	0
$147$	0	0
$148$	− 1.72030e6i	− 0.530663i
$149$	5.95330e6	1.79970	0.899848	−	0.436204i	$-0.143677\pi$
$149$	5.95330e6	1.79970	0.899848	+	0.436204i	$0.143677\pi$
$150$	0	0
$151$	1.82840e6	0.531057	0.265528	−	0.964103i	$-0.414454\pi$
$151$	1.82840e6	0.531057	0.265528	+	0.964103i	$0.414454\pi$
$152$	0	0
$153$	0	0
$154$	6.05669e6	1.65834
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	− 8.36440e6i	− 2.12062i
$159$	0	0
$160$	0	0
$161$	7.79776e6	1.86850
$162$	4.78297e6i	1.12500i
$163$	5.49309e6i	1.26839i	0.773171	+	0.634197i	$0.218669\pi$
$163$	5.49309e6i	1.26839i	−0.773171	+	0.634197i	$0.781331\pi$
$164$	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
$169$	−4.82681e6	−1.00000
$170$	0	0
$171$	0	0
$172$	− 2.15244e6i	− 0.423005i
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	−9.60399e6	−1.76163
$177$	0	0
$178$	0	0
$179$	7.12762e6	1.24276	0.621378	−	0.783511i	$-0.286573\pi$
$179$	7.12762e6	1.24276	0.621378	+	0.783511i	$0.286573\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	−9.61648e6	−1.54370
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.64544e6	0.379663	0.189831	−	0.981817i	$-0.439206\pi$
$191$	2.64544e6	0.379663	0.189831	+	0.981817i	$0.439206\pi$
$192$	0	0
$193$	− 6.68999e6i	− 0.930579i	−0.885159	−	0.465290i	$-0.845950\pi$
$193$	− 6.68999e6i	− 0.930579i	0.885159	−	0.465290i	$-0.154050\pi$
$194$	0	0
$195$	0	0
$196$	2.00003e6	0.265625
$197$	− 1.04066e7i	− 1.36117i	−0.732670	−	0.680585i	$-0.761726\pi$
$197$	− 1.04066e7i	− 1.36117i	0.732670	−	0.680585i	$-0.238274\pi$
$198$	− 1.28727e7i	− 1.65834i
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 7.27915e6i	− 0.870146i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1.65731e7i	− 1.86850i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	1.56456e7	1.66550	0.832748	−	0.553652i	$-0.186766\pi$
$211$	1.56456e7	1.66550	0.832748	+	0.553652i	$0.186766\pi$
$212$	855882.i	0.0898269i
$213$	0	0
$214$	−416826.	−0.0425318
$215$	0	0
$216$	0	0
$217$	0	0
$218$	2.32843e7i	2.24747i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	−5.82517e6	−0.518280
$225$	0	0
$226$	−2.19320e7	−1.90000
$227$	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
$232$	8.97691e6i	0.718890i
$233$	2.92577e6i	0.231299i	0.993290	+	0.115649i	$0.0368949\pi$
$233$	2.92577e6i	0.231299i	−0.993290	+	0.115649i	$0.963105\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.71015e7	−1.98518	−0.992591	−	0.121505i	$-0.961228\pi$
$239$	−2.71015e7	−1.98518	−0.992591	+	0.121505i	$0.961228\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	1.87009e7i	1.31952i
$243$	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			$\pi$
$252$	− 4.25080e6i	− 0.265625i
$253$	4.46041e7i	2.75431i
$254$	870894.	0.0531452
$255$	0	0
$256$	1.25096e7	0.745628
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	3.47095e7	1.99779
$260$	0	0
$261$	−1.54708e7	−0.870146
$262$	0	0
$263$	− 2.01675e7i	− 1.10863i	−0.832308	−	0.554313i	$-0.812981\pi$
$263$	− 2.01675e7i	− 1.10863i	0.832308	−	0.554313i	$-0.187019\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	916742.i	0.0476259i
$269$	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
$274$	−4.07645e7	−1.98167
$275$	0	0
$276$	0	0
$277$	1.09282e7i	0.514175i	0.966388	+	0.257087i	$0.0827628\pi$
$277$	1.09282e7i	0.514175i	−0.966388	+	0.257087i	$0.917237\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2.55231e7	−1.15031	−0.575155	−	0.818045i	$-0.695058\pi$
$281$	−2.55231e7	−1.15031	−0.575155	+	0.818045i	$0.695058\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	4.12213e6	0.179956
$285$	0	0
$286$	0	0
$287$	0	0
$288$	1.23806e7i	0.518280i
$289$	−2.41376e7	−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
$296$	−4.28051e7	−1.65052
$297$	0	0
$298$	5.35797e7i	2.02466i
$299$	0	0
$300$	0	0
$301$	4.34286e7	1.59249
$302$	1.64556e7i	0.597439i
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	1.14404e7i	0.391552i
$309$	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
$316$	1.57994e7	0.500703
$317$	− 1.51291e7i	− 0.474937i	−0.971395	−	0.237469i	$-0.923682\pi$
$317$	− 1.51291e7i	− 0.474937i	0.971395	−	0.237469i	$-0.0763177\pi$
$318$	0	0
$319$	4.16376e7	1.28267
$320$	0	0
$321$	0	0
$322$	7.01799e7i	2.10206i
$323$	0	0
$324$	−9.03450e6	−0.265625
$325$	0	0
$326$	−4.94378e7	−1.42694
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	4.64551e7	1.28100	0.640500	−	0.767958i	$-0.278727\pi$
$331$	4.64551e7	1.28100	0.640500	+	0.767958i	$0.278727\pi$
$332$	0	0
$333$	− 7.37704e7i	− 1.99779i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 6.54566e7i	− 1.71027i	−0.518408	−	0.855133i	$-0.673475\pi$
$337$	− 6.54566e7i	− 1.71027i	0.518408	−	0.855133i	$-0.326525\pi$
$338$	− 4.34413e7i	− 1.12500i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	4.03536e7i	1.00000i
$344$	−5.35577e7	−1.31567
$345$	0	0
$346$	0	0
$347$	− 6.67538e7i	− 1.59767i	−0.601548	−	0.798836i	$-0.705449\pi$
$347$	− 6.67538e7i	− 1.59767i	0.601548	−	0.798836i	$-0.294551\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	− 3.33206e7i	− 0.763986i
$353$	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
$358$	6.41486e7i	1.39810i
$359$	2.68617e7	0.580565	0.290282	−	0.956941i	$-0.406251\pi$
$359$	2.68617e7	0.580565	0.290282	+	0.956941i	$0.406251\pi$
$360$	0	0
$361$	4.70459e7	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	$0.5\pi$
$368$	− 1.11283e8i	− 2.23298i
$369$	0	0
$370$	0	0
$371$	−1.72687e7	−0.338172
$372$	0	0
$373$	− 9.13707e7i	− 1.76068i	−0.474344	−	0.880340i	$-0.657315\pi$
$373$	− 9.13707e7i	− 1.76068i	0.474344	−	0.880340i	$-0.342685\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−3.31112e7	−0.608215	−0.304107	−	0.952638i	$-0.598358\pi$
$379$	−3.31112e7	−0.608215	−0.304107	+	0.952638i	$0.598358\pi$
$380$	0	0
$381$	0	0
$382$	2.38090e7i	0.427121i
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	6.02099e7	1.04690
$387$	− 9.23016e7i	− 1.59249i
$388$	0	0
$389$	6.93106e6	0.117747	0.0588737	−	0.998265i	$-0.481249\pi$
$389$	6.93106e6	0.117747	0.0588737	+	0.998265i	$0.481249\pi$
$390$	0	0
$391$	0	0
$392$	− 4.97655e7i	− 0.826172i
$393$	0	0
$394$	9.36598e7	1.53132
$395$	0	0
$396$	2.43151e7	0.391552
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.26409e8	−1.96040	−0.980199	−	0.198016i	$-0.936550\pi$
$401$	−1.26409e8	−1.96040	−0.980199	+	0.198016i	$0.936550\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	6.55123e7	0.978915
$407$	1.98543e8i	2.94490i
$408$	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
$414$	1.49158e8	2.10206
$415$	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
$421$	8.36917e7	1.12160	0.560798	−	0.827953i	$-0.310495\pi$
$421$	8.36917e7	1.12160	0.560798	+	0.827953i	$0.310495\pi$
$422$	1.40810e8i	1.87368i
$423$	0	0
$424$	2.12964e7	0.279388
$425$	0	0
$426$	0	0
$427$	0	0
$428$	− 787338.i	− 0.0100422i
$429$	0	0
$430$	0	0
$431$	−8.60283e7	−1.07451	−0.537254	−	0.843421i	$-0.680538\pi$
$431$	−8.60283e7	−1.07451	−0.537254	+	0.843421i	$0.680538\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	−4.39814e7	−0.530652
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	8.57661e7	1.00000
$442$	0	0
$443$	− 1.71340e8i	− 1.97082i	−0.170196	−	0.985410i	$-0.554440\pi$
$443$	− 1.71340e8i	− 1.97082i	0.170196	−	0.985410i	$-0.445560\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	5.50285e7i	0.612003i
$449$	1.73823e8	1.92030	0.960149	−	0.279490i	$-0.0901653\pi$
$449$	1.73823e8	1.92030	0.960149	+	0.279490i	$0.0901653\pi$
$450$	0	0
$451$	0	0
$452$	− 4.14272e7i	− 0.448611i
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 1.26728e8i	− 1.32777i	−0.747834	−	0.663886i	$-0.768906\pi$
$457$	− 1.26728e8i	− 1.32777i	0.747834	−	0.663886i	$-0.231094\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	1.27272e8i	1.28230i	0.767415	+	0.641151i	$0.221543\pi$
$463$	1.27272e8i	1.28230i	−0.767415	+	0.641151i	$0.778457\pi$
$464$	−1.03882e8	−1.03988
$465$	0	0
$466$	−2.63320e7	−0.260211
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	−1.84966e7	−0.179297
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.48417e8i	2.34746i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.67022e7i	0.338172i
$478$	− 2.43914e8i	− 2.23333i
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−3.53240e7	−0.311554
$485$	0	0
$486$	0	0
$487$	2.20135e8i	1.90591i	0.303113	+	0.952955i	$0.401974\pi$
$487$	2.20135e8i	1.90591i	−0.303113	+	0.952955i	$0.598026\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.78277e8	1.50609	0.753044	−	0.657970i	$-0.228585\pi$
$491$	1.78277e8	1.50609	0.753044	+	0.657970i	$0.228585\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.31700e7i	0.677481i
$498$	0	0
$499$	1.96143e8	1.57860	0.789300	−	0.614008i	$-0.210444\pi$
$499$	1.96143e8	1.57860	0.789300	+	0.614008i	$0.210444\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	−1.05770e8	−0.826172
$505$	0	0
$506$	−4.01437e8	−3.09860
$507$	0	0
$508$	1.64502e6i	0.0125482i
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	− 4.93857e7i	− 0.367952i
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	3.12386e8i	2.24751i
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	− 1.39238e8i	− 0.978915i
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	1.81508e8	1.24720
$527$	0	0
$528$	0	0
$529$	−3.68799e8	−2.49128
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	2.28107e7	0.148130
$537$	0	0
$538$	0	0
$539$	−2.30827e8	−1.47408
$540$	0	0
$541$	−6.45700e7	−0.407792	−0.203896	−	0.978993i	$-0.565360\pi$
$541$	−6.45700e7	−0.407792	−0.203896	+	0.978993i	$0.565360\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 1.58854e8i	− 0.970592i	−0.874350	−	0.485296i	$-0.838712\pi$
$547$	− 1.58854e8i	− 0.970592i	0.874350	−	0.485296i	$-0.161288\pi$
$548$	− 7.69997e7i	− 0.467894i
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	3.18777e8i	1.88500i
$554$	−9.83541e7	−0.578447
$555$	0	0
$556$	0	0
$557$	7.34035e7i	0.424767i	0.977186	+	0.212384i	$0.0681227\pi$
$557$	7.34035e7i	0.424767i	−0.977186	+	0.212384i	$0.931877\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	− 2.29708e8i	− 1.29410i
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.82284e8i	− 1.00000i
$568$	− 1.02568e8i	− 0.559716i
$569$	−3.55493e8	−1.92972	−0.964859	−	0.262767i	$-0.915365\pi$
$569$	−3.55493e8	−1.92972	−0.964859	+	0.262767i	$0.915365\pi$
$570$	0	0
$571$	−3.26262e8	−1.75250	−0.876250	−	0.481857i	$-0.839963\pi$
$571$	−3.26262e8	−1.75250	−0.876250	+	0.481857i	$0.839963\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	1.16956e8	0.612003
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	− 2.17238e8i	− 1.12500i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 9.87789e7i	− 0.498492i
$584$	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
$592$	− 4.95345e8i	− 2.38749i
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	−1.01206e8	−0.478044
$597$	0	0
$598$	0	0
$599$	−1.82026e8	−0.846941	−0.423471	−	0.905910i	$-0.639188\pi$
$599$	−1.82026e8	−0.846941	−0.423471	+	0.905910i	$0.639188\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	3.90857e8i	1.79155i
$603$	3.93121e7i	0.179297i
$604$	−3.10828e7	−0.141062
$605$	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
$613$	2.35393e8i	1.02191i	0.859608	+	0.510954i	$0.170708\pi$
$613$	2.35393e8i	1.02191i	−0.859608	+	0.510954i	$0.829292\pi$
$614$	0	0
$615$	0	0
$616$	2.84665e8	1.21784
$617$	4.63532e8i	1.97344i	0.162423	+	0.986721i	$0.448069\pi$
$617$	4.63532e8i	1.97344i	−0.162423	+	0.986721i	$0.551931\pi$
$618$	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
$631$	1.83542e8	0.730544	0.365272	−	0.930901i	$-0.380976\pi$
$631$	1.83542e8	0.730544	0.365272	+	0.930901i	$0.380976\pi$
$632$	− 3.93127e8i	− 1.55733i
$633$	0	0
$634$	1.36162e8	0.534304
$635$	0	0
$636$	0	0
$637$	0	0
$638$	3.74738e8i	1.44300i
$639$	1.76766e8	0.677481
$640$	0	0
$641$	−4.47931e8	−1.70073	−0.850367	−	0.526189i	$-0.823620\pi$
$641$	−4.47931e8	−1.70073	−0.850367	+	0.526189i	$0.823620\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	−1.32562e8	−0.496319
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	2.24800e8i	0.826172i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	− 9.33826e7i	− 0.336917i
$653$	− 1.74812e8i	− 0.627816i	−0.949453	−	0.313908i	$-0.898362\pi$
$653$	− 1.74812e8i	− 0.627816i	0.949453	−	0.313908i	$-0.101638\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	5.69128e8	1.98863	0.994314	−	0.106485i	$-0.0339598\pi$
$659$	5.69128e8	1.98863	0.994314	+	0.106485i	$0.0339598\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	4.18096e8i	1.44113i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	6.63934e8	2.24751
$667$	4.82461e8i	1.62587i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 5.17302e8i	− 1.69707i	−0.529141	−	0.848534i	$-0.677486\pi$
$673$	− 5.17302e8i	− 1.69707i	0.529141	−	0.848534i	$-0.322514\pi$
$674$	5.89109e8	1.92405
$675$	0	0
$676$	8.20558e7	0.265625
$677$	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
$683$	− 5.22860e8i	− 1.64105i	−0.571607	−	0.820527i	$-0.693680\pi$
$683$	− 5.22860e8i	− 1.64105i	0.571607	−	0.820527i	$-0.306320\pi$
$684$	0	0
$685$	0	0
$686$	−3.63182e8	−1.12500
$687$	0	0
$688$	− 6.19776e8i	− 1.90313i
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	4.90592e8i	1.47408i
$694$	6.00785e8	1.79738
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−6.71321e8	−1.94884	−0.974420	−	0.224735i	$-0.927848\pi$
$701$	−6.71321e8	−1.94884	−0.974420	+	0.224735i	$0.927848\pi$
$702$	0	0
$703$	0	0
$704$	−3.14770e8	−0.902142
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	5.55400e8	1.55836	0.779178	−	0.626802i	$-0.215637\pi$
$709$	5.55400e8	1.55836	0.779178	+	0.626802i	$0.215637\pi$
$710$	0	0
$711$	6.77517e8	1.88500
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−1.21170e8	−0.330107
$717$	0	0
$718$	2.41756e8i	0.653136i
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	4.23413e8i	1.12500i
$723$	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
$729$	−3.87420e8	−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
$736$	3.86092e8	0.968405
$737$	− 1.05803e8i	− 0.264299i
$738$	0	0
$739$	1.65862e8	0.410973	0.205486	−	0.978660i	$-0.434122\pi$
$739$	1.65862e8	0.410973	0.205486	+	0.978660i	$0.434122\pi$
$740$	0	0
$741$	0	0
$742$	− 1.55418e8i	− 0.380443i
$743$	1.87319e8i	0.456684i	0.973581	+	0.228342i	$0.0733304\pi$
$743$	1.87319e8i	0.456684i	−0.973581	+	0.228342i	$0.926670\pi$
$744$	0	0
$745$	0	0
$746$	8.22336e8	1.98076
$747$	0	0
$748$	0	0
$749$	1.58857e7	0.0378060
$750$	0	0
$751$	−8.41137e8	−1.98585	−0.992926	−	0.118736i	$-0.962116\pi$
$751$	−8.41137e8	−1.98585	−0.992926	+	0.118736i	$0.962116\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	3.45533e8i	0.796529i	0.917271	+	0.398264i	$0.130387\pi$
$757$	3.45533e8i	0.796529i	−0.917271	+	0.398264i	$0.869613\pi$
$758$	− 2.98001e8i	− 0.684242i
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	− 8.87390e8i	− 1.99775i
$764$	−4.49725e7	−0.100848
$765$	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
$772$	1.13730e8i	0.247185i
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	8.30714e8	1.79155
$775$	0	0
$776$	0	0
$777$	0	0
$778$	6.23796e7i	0.132466i
$779$	0	0
$780$	0	0
$781$	−4.75742e8	−0.998661
$782$	0	0
$783$	0	0
$784$	5.75892e8	1.19507
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	1.76913e8i	0.361561i
$789$	0	0
$790$	0	0
$791$	8.35855e8	1.68889
$792$	− 6.05016e8i	− 1.21784i
$793$	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
$802$	− 1.13768e9i	− 2.20545i
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−3.36371e8	−0.635292	−0.317646	−	0.948209i	$-0.602892\pi$
$809$	−3.36371e8	−0.635292	−0.317646	+	0.948209i	$0.602892\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	1.23745e8i	0.231133i
$813$	0	0
$814$	−1.78688e9	−3.31301
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	7.88782e8	1.42537	0.712685	−	0.701484i	$-0.247479\pi$
$821$	7.88782e8	1.42537	0.712685	+	0.701484i	$0.247479\pi$
$822$	0	0
$823$	− 1.02325e9i	− 1.83563i	−0.397014	−	0.917813i	$-0.629954\pi$
$823$	− 1.02325e9i	− 1.83563i	0.397014	−	0.917813i	$-0.370046\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 5.56178e8i	− 0.983326i	−0.870786	−	0.491663i	$-0.836389\pi$
$827$	− 5.56178e8i	− 0.983326i	0.870786	−	0.491663i	$-0.163611\pi$
$828$	2.81742e8i	0.496319i
$829$	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
$841$	−1.44450e8	−0.242845
$842$	7.53225e8i	1.26179i
$843$	0	0
$844$	−2.65975e8	−0.442398
$845$	0	0
$846$	0	0
$847$	− 7.12714e8i	− 1.17291i
$848$	2.46444e8i	0.404138i
$849$	0	0
$850$	0	0
$851$	−2.30054e9	−3.73286
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	−1.95908e7	−0.0312343
$857$	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
$862$	− 7.74255e8i	− 1.20882i
$863$	8.77431e8i	1.36515i	0.730815	+	0.682576i	$0.239140\pi$
$863$	8.77431e8i	1.36515i	−0.730815	+	0.682576i	$0.760860\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.82344e9	−2.77864
$870$	0	0
$871$	0	0
$872$	1.09436e9i	1.65048i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1.30615e9i	− 1.93640i	−0.250185	−	0.968198i	$-0.580491\pi$
$877$	− 1.30615e9i	− 1.93640i	0.250185	−	0.968198i	$-0.419509\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	7.71895e8i	1.12500i
$883$	4.73327e8i	0.687510i	0.939059	+	0.343755i	$0.111699\pi$
$883$	4.73327e8i	0.687510i	−0.939059	+	0.343755i	$0.888301\pi$
$884$	0	0
$885$	0	0
$886$	1.54206e9	2.21717
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−3.31907e7	−0.0472402
$890$	0	0
$891$	1.04269e9	1.47408
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−8.68067e8	−1.20678
$897$	0	0
$898$	1.56441e9i	2.16033i
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−1.03081e9	−1.39531
$905$	0	0
$906$	0	0
$907$	1.28922e9i	1.72785i	0.503621	+	0.863925i	$0.332001\pi$
$907$	1.28922e9i	1.72785i	−0.503621	+	0.863925i	$0.667999\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	5.85794e7	0.0774800	0.0387400	−	0.999249i	$-0.487666\pi$
$911$	5.85794e7	0.0774800	0.0387400	+	0.999249i	$0.487666\pi$
$912$	0	0
$913$	0	0
$914$	1.14055e9	1.49374
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1.07950e9	1.39084	0.695419	−	0.718604i	$-0.255219\pi$
$919$	1.07950e9	1.39084	0.695419	+	0.718604i	$0.255219\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−1.14545e9	−1.44259
$927$	0	0
$928$	− 3.60413e8i	− 0.450979i
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	− 4.97382e7i	− 0.0614387i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	− 1.66470e8i	− 0.201709i
$939$	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
$946$	−2.23575e9	−2.64089
$947$	6.84836e8i	0.806374i	0.915118	+	0.403187i	$0.132098\pi$
$947$	6.84836e8i	0.806374i	−0.915118	+	0.403187i	$0.867902\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	8.73142e8i	1.00880i	0.863469	+	0.504401i	$0.168287\pi$
$953$	8.73142e8i	1.00880i	−0.863469	+	0.504401i	$0.831713\pi$
$954$	−3.30320e8	−0.380443
$955$	0	0
$956$	4.60726e8	0.527314
$957$	0	0
$958$	0	0
$959$	1.55358e9	1.76148
$960$	0	0
$961$	8.87504e8	1.00000
$962$	0	0
$963$	− 3.37629e7i	− 0.0378060i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	8.99699e8i	0.994988i	0.867467	+	0.497494i	$0.165746\pi$
$967$	8.99699e8i	0.994988i	−0.867467	+	0.497494i	$0.834254\pi$
$968$	8.78945e8i	0.969026i
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	−1.98121e9	−2.14415
$975$	0	0
$976$	0	0
$977$	− 4.59650e8i	− 0.492882i	−0.969158	−	0.246441i	$-0.920739\pi$
$977$	− 4.59650e8i	− 0.492882i	0.969158	−	0.246441i	$-0.0792612\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−1.88603e9	−1.99775
$982$	1.60449e9i	1.69435i
$983$	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
$989$	−2.87844e9	−2.97556
$990$	0	0
$991$	1.36299e9	1.40046	0.700230	−	0.713918i	$-0.253081\pi$
$991$	1.36299e9	1.40046	0.700230	+	0.713918i	$0.253081\pi$
$992$	0	0
$993$	0	0
$994$	−7.48530e8	−0.762166
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	1.76529e9i	1.77592i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.7.c.a.174.2		2
5.2	odd	4		175.7.d.a.76.1		1
5.3	odd	4		7.7.b.a.6.1	✓	1
5.4	even	2	inner	175.7.c.a.174.1		2
7.6	odd	2	CM	175.7.c.a.174.2		2
15.8	even	4		63.7.d.a.55.1		1
20.3	even	4		112.7.c.a.97.1		1
35.3	even	12		49.7.d.a.19.1		2
35.13	even	4		7.7.b.a.6.1	✓	1
35.18	odd	12		49.7.d.a.19.1		2
35.23	odd	12		49.7.d.a.31.1		2
35.27	even	4		175.7.d.a.76.1		1
35.33	even	12		49.7.d.a.31.1		2
35.34	odd	2	inner	175.7.c.a.174.1		2
40.3	even	4		448.7.c.b.321.1		1
40.13	odd	4		448.7.c.a.321.1		1
105.83	odd	4		63.7.d.a.55.1		1
140.83	odd	4		112.7.c.a.97.1		1
280.13	even	4		448.7.c.a.321.1		1
280.83	odd	4		448.7.c.b.321.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.7.b.a.6.1	✓	1	5.3	odd	4
7.7.b.a.6.1	✓	1	35.13	even	4
49.7.d.a.19.1		2	35.3	even	12
49.7.d.a.19.1		2	35.18	odd	12
49.7.d.a.31.1		2	35.23	odd	12
49.7.d.a.31.1		2	35.33	even	12
63.7.d.a.55.1		1	15.8	even	4
63.7.d.a.55.1		1	105.83	odd	4
112.7.c.a.97.1		1	20.3	even	4
112.7.c.a.97.1		1	140.83	odd	4
175.7.c.a.174.1		2	5.4	even	2	inner
175.7.c.a.174.1		2	35.34	odd	2	inner
175.7.c.a.174.2		2	1.1	even	1	trivial
175.7.c.a.174.2		2	7.6	odd	2	CM
175.7.d.a.76.1		1	5.2	odd	4
175.7.d.a.76.1		1	35.27	even	4
448.7.c.a.321.1		1	40.13	odd	4
448.7.c.a.321.1		1	280.13	even	4
448.7.c.b.321.1		1	40.3	even	4
448.7.c.b.321.1		1	280.83	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:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	175.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+9.00000i q^{2} -17.0000 q^{4} -343.000i q^{7} +423.000i q^{8} -729.000 q^{9} +1962.00 q^{11} +3087.00 q^{14} -4895.00 q^{16} -6561.00i q^{18} +17658.0i q^{22} +22734.0i q^{23} +5831.00i q^{28} +21222.0 q^{29} -16983.0i q^{32} +12393.0 q^{36} +101194. i q^{37} +126614. i q^{43} -33354.0 q^{44} -204606. q^{46} -117649. q^{49} -50346.0i q^{53} +145089. q^{56} +190998. i q^{58} +250047. i q^{63} -160433. q^{64} -53926.0i q^{67} -242478. q^{71} -308367. i q^{72} -910746. q^{74} -672966. i q^{77} -929378. q^{79} +531441. q^{81} -1.13953e6 q^{86} +829926. i q^{88} -386478. i q^{92} -1.05884e6i q^{98} -1.43030e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 34 q^{4} - 1458 q^{9} + 3924 q^{11} + 6174 q^{14} - 9790 q^{16} + 42444 q^{29} + 24786 q^{36} - 66708 q^{44} - 409212 q^{46} - 235298 q^{49} + 290178 q^{56} - 320866 q^{64} - 484956 q^{71} - 1821492 q^{74}+ \cdots - 2860596 q^{99}+O(q^{100}) \) 2 * q - 34 * q^4 - 1458 * q^9 + 3924 * q^11 + 6174 * q^14 - 9790 * q^16 + 42444 * q^29 + 24786 * q^36 - 66708 * q^44 - 409212 * q^46 - 235298 * q^49 + 290178 * q^56 - 320866 * q^64 - 484956 * q^71 - 1821492 * q^74 - 1858756 * q^79 + 1062882 * q^81 - 2279052 * q^86 - 2860596 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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