LMFDB - Embedded newform 2175.2.c.f.349.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2175,2,Mod(349,2175)] 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("2175.349"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2175.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-14,0,-2,0,0,-4,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$17.3674624396$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{21})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 11x^{2} + 25 $ x^4 + 11*x^2 + 25
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 435)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.1
Root			$-2.79129i$ of defining polynomial
Character	$\chi$	$=$	2175.349
Dual form			2175.2.c.f.349.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.79129i q^{2} -1.00000i q^{3} -5.79129 q^{4} -2.79129 q^{6} +1.00000i q^{7} +10.5826i q^{8} -1.00000 q^{9} +5.00000 q^{11} +5.79129i q^{12} -4.58258i q^{13} +2.79129 q^{14} +17.9564 q^{16} -3.00000i q^{17} +2.79129i q^{18} +5.58258 q^{19} +1.00000 q^{21} -13.9564i q^{22} +4.00000i q^{23} +10.5826 q^{24} -12.7913 q^{26} +1.00000i q^{27} -5.79129i q^{28} -1.00000 q^{29} +4.00000 q^{31} -28.9564i q^{32} -5.00000i q^{33} -8.37386 q^{34} +5.79129 q^{36} -4.00000i q^{37} -15.5826i q^{38} -4.58258 q^{39} +9.16515 q^{41} -2.79129i q^{42} +0.417424i q^{43} -28.9564 q^{44} +11.1652 q^{46} +1.41742i q^{47} -17.9564i q^{48} +6.00000 q^{49} -3.00000 q^{51} +26.5390i q^{52} -9.58258i q^{53} +2.79129 q^{54} -10.5826 q^{56} -5.58258i q^{57} +2.79129i q^{58} -1.58258 q^{59} -14.7477 q^{61} -11.1652i q^{62} -1.00000i q^{63} -44.9129 q^{64} -13.9564 q^{66} +14.1652i q^{67} +17.3739i q^{68} +4.00000 q^{69} -0.417424 q^{71} -10.5826i q^{72} -4.00000i q^{73} -11.1652 q^{74} -32.3303 q^{76} +5.00000i q^{77} +12.7913i q^{78} +1.58258 q^{79} +1.00000 q^{81} -25.5826i q^{82} +2.41742i q^{83} -5.79129 q^{84} +1.16515 q^{86} +1.00000i q^{87} +52.9129i q^{88} -10.5826 q^{89} +4.58258 q^{91} -23.1652i q^{92} -4.00000i q^{93} +3.95644 q^{94} -28.9564 q^{96} +2.41742i q^{97} -16.7477i q^{98} -5.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 14 q^{4} - 2 q^{6} - 4 q^{9} + 20 q^{11} + 2 q^{14} + 26 q^{16} + 4 q^{19} + 4 q^{21} + 24 q^{24} - 42 q^{26} - 4 q^{29} + 16 q^{31} - 6 q^{34} + 14 q^{36} - 70 q^{44} + 8 q^{46} + 24 q^{49} - 12 q^{51}+ \cdots - 20 q^{99}+O(q^{100}) $ 4 * q - 14 * q^4 - 2 * q^6 - 4 * q^9 + 20 * q^11 + 2 * q^14 + 26 * q^16 + 4 * q^19 + 4 * q^21 + 24 * q^24 - 42 * q^26 - 4 * q^29 + 16 * q^31 - 6 * q^34 + 14 * q^36 - 70 * q^44 + 8 * q^46 + 24 * q^49 - 12 * q^51 + 2 * q^54 - 24 * q^56 + 12 * q^59 - 4 * q^61 - 88 * q^64 - 10 * q^66 + 16 * q^69 - 20 * q^71 - 8 * q^74 - 56 * q^76 - 12 * q^79 + 4 * q^81 - 14 * q^84 - 32 * q^86 - 24 * q^89 - 30 * q^94 - 70 * q^96 - 20 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times$.

$n$	$901$	$1451$	$2002$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 2.79129i	− 1.97374i	−0.161521	−	0.986869i	$-0.551640\pi$
$2$	− 2.79129i	− 1.97374i	0.161521	−	0.986869i	$-0.448360\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−5.79129	−2.89564
$5$	0	0
$5$	0	0
$6$	−2.79129	−1.13954
$7$	1.00000i	0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	1.00000i	0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$	10.5826i	3.74151i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	5.79129i	1.67180i
$13$	− 4.58258i	− 1.27098i	−0.772110	−	0.635489i	$-0.780799\pi$
$13$	− 4.58258i	− 1.27098i	0.772110	−	0.635489i	$-0.219201\pi$
$14$	2.79129	0.746003
$15$	0	0
$16$	17.9564	4.48911
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	2.79129i	0.657913i
$19$	5.58258	1.28073	0.640365	−	0.768070i	$-0.278783\pi$
$19$	5.58258	1.28073	0.640365	+	0.768070i	$0.278783\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	− 13.9564i	− 2.97552i
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	10.5826	2.16016
$25$	0	0
$26$	−12.7913	−2.50858
$27$	1.00000i	0.192450i
$28$	− 5.79129i	− 1.09445i
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	− 28.9564i	− 5.11882i
$33$	− 5.00000i	− 0.870388i
$34$	−8.37386	−1.43611
$35$	0	0
$36$	5.79129	0.965215
$37$	− 4.00000i	− 0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$	− 4.00000i	− 0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	− 15.5826i	− 2.52783i
$39$	−4.58258	−0.733799
$40$	0	0
$41$	9.16515	1.43136	0.715678	−	0.698430i	$-0.246118\pi$
$41$	9.16515	1.43136	0.715678	+	0.698430i	$0.246118\pi$
$42$	− 2.79129i	− 0.430705i
$43$	0.417424i	0.0636566i	0.999493	+	0.0318283i	$0.0101330\pi$
$43$	0.417424i	0.0636566i	−0.999493	+	0.0318283i	$0.989867\pi$
$44$	−28.9564	−4.36535
$45$	0	0
$46$	11.1652	1.64621
$47$	1.41742i	0.206753i	0.994642	+	0.103376i	$0.0329646\pi$
$47$	1.41742i	0.206753i	−0.994642	+	0.103376i	$0.967035\pi$
$48$	− 17.9564i	− 2.59179i
$49$	6.00000	0.857143
$50$	0	0
$51$	−3.00000	−0.420084
$52$	26.5390i	3.68030i
$53$	− 9.58258i	− 1.31627i	−0.752901	−	0.658134i	$-0.771346\pi$
$53$	− 9.58258i	− 1.31627i	0.752901	−	0.658134i	$-0.228654\pi$
$54$	2.79129	0.379846
$55$	0	0
$56$	−10.5826	−1.41416
$57$	− 5.58258i	− 0.739430i
$58$	2.79129i	0.366514i
$59$	−1.58258	−0.206034	−0.103017	−	0.994680i	$-0.532850\pi$
$59$	−1.58258	−0.206034	−0.103017	+	0.994680i	$0.532850\pi$
$60$	0	0
$61$	−14.7477	−1.88825	−0.944126	−	0.329583i	$-0.893092\pi$
$61$	−14.7477	−1.88825	−0.944126	+	0.329583i	$0.893092\pi$
$62$	− 11.1652i	− 1.41798i
$63$	− 1.00000i	− 0.125988i
$64$	−44.9129	−5.61411
$65$	0	0
$66$	−13.9564	−1.71792
$67$	14.1652i	1.73055i	0.501299	+	0.865274i	$0.332856\pi$
$67$	14.1652i	1.73055i	−0.501299	+	0.865274i	$0.667144\pi$
$68$	17.3739i	2.10689i
$69$	4.00000	0.481543
$70$	0	0
$71$	−0.417424	−0.0495392	−0.0247696	−	0.999693i	$-0.507885\pi$
$71$	−0.417424	−0.0495392	−0.0247696	+	0.999693i	$0.507885\pi$
$72$	− 10.5826i	− 1.24717i
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	−11.1652	−1.29792
$75$	0	0
$76$	−32.3303	−3.70854
$77$	5.00000i	0.569803i
$78$	12.7913i	1.44833i
$79$	1.58258	0.178054	0.0890268	−	0.996029i	$-0.471624\pi$
$79$	1.58258	0.178054	0.0890268	+	0.996029i	$0.471624\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 25.5826i	− 2.82512i
$83$	2.41742i	0.265347i	0.991160	+	0.132673i	$0.0423561\pi$
$83$	2.41742i	0.265347i	−0.991160	+	0.132673i	$0.957644\pi$
$84$	−5.79129	−0.631881
$85$	0	0
$86$	1.16515	0.125642
$87$	1.00000i	0.107211i
$88$	52.9129i	5.64053i
$89$	−10.5826	−1.12175	−0.560875	−	0.827900i	$-0.689535\pi$
$89$	−10.5826	−1.12175	−0.560875	+	0.827900i	$0.689535\pi$
$90$	0	0
$91$	4.58258	0.480384
$92$	− 23.1652i	− 2.41513i
$93$	− 4.00000i	− 0.414781i
$94$	3.95644	0.408076
$95$	0	0
$96$	−28.9564	−2.95535
$97$	2.41742i	0.245452i	0.992441	+	0.122726i	$0.0391637\pi$
$97$	2.41742i	0.245452i	−0.992441	+	0.122726i	$0.960836\pi$
$98$	− 16.7477i	− 1.69178i
$99$	−5.00000	−0.502519
$100$	0	0
$101$	8.58258	0.853998	0.426999	−	0.904252i	$-0.359571\pi$
$101$	8.58258	0.853998	0.426999	+	0.904252i	$0.359571\pi$
$102$	8.37386i	0.829136i
$103$	3.16515i	0.311872i	0.987767	+	0.155936i	$0.0498393\pi$
$103$	3.16515i	0.311872i	−0.987767	+	0.155936i	$0.950161\pi$
$104$	48.4955	4.75537
$105$	0	0
$106$	−26.7477	−2.59797
$107$	− 13.1652i	− 1.27272i	−0.771391	−	0.636362i	$-0.780439\pi$
$107$	− 13.1652i	− 1.27272i	0.771391	−	0.636362i	$-0.219561\pi$
$108$	− 5.79129i	− 0.557267i
$109$	4.16515	0.398949	0.199475	−	0.979903i	$-0.436076\pi$
$109$	4.16515	0.398949	0.199475	+	0.979903i	$0.436076\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	17.9564i	1.69672i
$113$	− 4.16515i	− 0.391824i	−0.980621	−	0.195912i	$-0.937233\pi$
$113$	− 4.16515i	− 0.391824i	0.980621	−	0.195912i	$-0.0627668\pi$
$114$	−15.5826	−1.45944
$115$	0	0
$116$	5.79129	0.537708
$117$	4.58258i	0.423659i
$118$	4.41742i	0.406657i
$119$	3.00000	0.275010
$120$	0	0
$121$	14.0000	1.27273
$122$	41.1652i	3.72692i
$123$	− 9.16515i	− 0.826394i
$124$	−23.1652	−2.08029
$125$	0	0
$126$	−2.79129	−0.248668
$127$	2.00000i	0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	2.00000i	0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	67.4519i	5.96196i
$129$	0.417424	0.0367522
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	28.9564i	2.52033i
$133$	5.58258i	0.484071i
$134$	39.5390	3.41565
$135$	0	0
$136$	31.7477	2.72235
$137$	− 20.3303i	− 1.73693i	−0.495746	−	0.868467i	$-0.665105\pi$
$137$	− 20.3303i	− 1.73693i	0.495746	−	0.868467i	$-0.334895\pi$
$138$	− 11.1652i	− 0.950441i
$139$	18.5826	1.57615	0.788077	−	0.615577i	$-0.211077\pi$
$139$	18.5826	1.57615	0.788077	+	0.615577i	$0.211077\pi$
$140$	0	0
$141$	1.41742	0.119369
$142$	1.16515i	0.0977773i
$143$	− 22.9129i	− 1.91607i
$144$	−17.9564	−1.49637
$145$	0	0
$146$	−11.1652	−0.924035
$147$	− 6.00000i	− 0.494872i
$148$	23.1652i	1.90416i
$149$	10.7477	0.880488	0.440244	−	0.897878i	$-0.354892\pi$
$149$	10.7477	0.880488	0.440244	+	0.897878i	$0.354892\pi$
$150$	0	0
$151$	−11.1652	−0.908607	−0.454304	−	0.890847i	$-0.650112\pi$
$151$	−11.1652	−0.908607	−0.454304	+	0.890847i	$0.650112\pi$
$152$	59.0780i	4.79186i
$153$	3.00000i	0.242536i
$154$	13.9564	1.12464
$155$	0	0
$156$	26.5390	2.12482
$157$	− 16.7477i	− 1.33661i	−0.743886	−	0.668307i	$-0.767019\pi$
$157$	− 16.7477i	− 1.33661i	0.743886	−	0.668307i	$-0.232981\pi$
$158$	− 4.41742i	− 0.351431i
$159$	−9.58258	−0.759948
$160$	0	0
$161$	−4.00000	−0.315244
$162$	− 2.79129i	− 0.219304i
$163$	1.58258i	0.123957i	0.998077	+	0.0619784i	$0.0197410\pi$
$163$	1.58258i	0.123957i	−0.998077	+	0.0619784i	$0.980259\pi$
$164$	−53.0780	−4.14470
$165$	0	0
$166$	6.74773	0.523725
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	10.5826i	0.816463i
$169$	−8.00000	−0.615385
$170$	0	0
$171$	−5.58258	−0.426910
$172$	− 2.41742i	− 0.184327i
$173$	− 15.1652i	− 1.15299i	−0.817102	−	0.576493i	$-0.804421\pi$
$173$	− 15.1652i	− 1.15299i	0.817102	−	0.576493i	$-0.195579\pi$
$174$	2.79129	0.211607
$175$	0	0
$176$	89.7822	6.76759
$177$	1.58258i	0.118954i
$178$	29.5390i	2.21404i
$179$	22.7477	1.70024	0.850122	−	0.526585i	$-0.176528\pi$
$179$	22.7477	1.70024	0.850122	+	0.526585i	$0.176528\pi$
$180$	0	0
$181$	−2.16515	−0.160934	−0.0804672	−	0.996757i	$-0.525641\pi$
$181$	−2.16515	−0.160934	−0.0804672	+	0.996757i	$0.525641\pi$
$182$	− 12.7913i	− 0.948153i
$183$	14.7477i	1.09018i
$184$	−42.3303	−3.12063
$185$	0	0
$186$	−11.1652	−0.818669
$187$	− 15.0000i	− 1.09691i
$188$	− 8.20871i	− 0.598682i
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	44.9129i	3.24131i
$193$	− 16.3303i	− 1.17548i	−0.809050	−	0.587740i	$-0.800018\pi$
$193$	− 16.3303i	− 1.17548i	0.809050	−	0.587740i	$-0.199982\pi$
$194$	6.74773	0.484459
$195$	0	0
$196$	−34.7477	−2.48198
$197$	20.3303i	1.44847i	0.689551	+	0.724237i	$0.257808\pi$
$197$	20.3303i	1.44847i	−0.689551	+	0.724237i	$0.742192\pi$
$198$	13.9564i	0.991841i
$199$	−22.5826	−1.60084	−0.800418	−	0.599442i	$-0.795389\pi$
$199$	−22.5826	−1.60084	−0.800418	+	0.599442i	$0.795389\pi$
$200$	0	0
$201$	14.1652	0.999133
$202$	− 23.9564i	− 1.68557i
$203$	− 1.00000i	− 0.0701862i
$204$	17.3739	1.21641
$205$	0	0
$206$	8.83485	0.615553
$207$	− 4.00000i	− 0.278019i
$208$	− 82.2867i	− 5.70556i
$209$	27.9129	1.93077
$210$	0	0
$211$	−16.3303	−1.12422	−0.562112	−	0.827061i	$-0.690011\pi$
$211$	−16.3303	−1.12422	−0.562112	+	0.827061i	$0.690011\pi$
$212$	55.4955i	3.81144i
$213$	0.417424i	0.0286014i
$214$	−36.7477	−2.51202
$215$	0	0
$216$	−10.5826	−0.720053
$217$	4.00000i	0.271538i
$218$	− 11.6261i	− 0.787421i
$219$	−4.00000	−0.270295
$220$	0	0
$221$	−13.7477	−0.924772
$222$	11.1652i	0.749356i
$223$	− 7.00000i	− 0.468755i	−0.972146	−	0.234377i	$-0.924695\pi$
$223$	− 7.00000i	− 0.468755i	0.972146	−	0.234377i	$-0.0753051\pi$
$224$	28.9564	1.93473
$225$	0	0
$226$	−11.6261	−0.773359
$227$	1.58258i	0.105039i	0.998620	+	0.0525196i	$0.0167252\pi$
$227$	1.58258i	0.105039i	−0.998620	+	0.0525196i	$0.983275\pi$
$228$	32.3303i	2.14113i
$229$	−1.16515	−0.0769954	−0.0384977	−	0.999259i	$-0.512257\pi$
$229$	−1.16515	−0.0769954	−0.0384977	+	0.999259i	$0.512257\pi$
$230$	0	0
$231$	5.00000	0.328976
$232$	− 10.5826i	− 0.694780i
$233$	− 5.16515i	− 0.338380i	−0.985583	−	0.169190i	$-0.945885\pi$
$233$	− 5.16515i	− 0.338380i	0.985583	−	0.169190i	$-0.0541152\pi$
$234$	12.7913	0.836193
$235$	0	0
$236$	9.16515	0.596601
$237$	− 1.58258i	− 0.102799i
$238$	− 8.37386i	− 0.542797i
$239$	26.0000	1.68180	0.840900	−	0.541190i	$-0.182026\pi$
$239$	26.0000	1.68180	0.840900	+	0.541190i	$0.182026\pi$
$240$	0	0
$241$	−7.33030	−0.472186	−0.236093	−	0.971730i	$-0.575867\pi$
$241$	−7.33030	−0.472186	−0.236093	+	0.971730i	$0.575867\pi$
$242$	− 39.0780i	− 2.51203i
$243$	− 1.00000i	− 0.0641500i
$244$	85.4083	5.46771
$245$	0	0
$246$	−25.5826	−1.63109
$247$	− 25.5826i	− 1.62778i
$248$	42.3303i	2.68798i
$249$	2.41742	0.153198
$250$	0	0
$251$	−2.16515	−0.136663	−0.0683316	−	0.997663i	$-0.521768\pi$
$251$	−2.16515	−0.136663	−0.0683316	+	0.997663i	$0.521768\pi$
$252$	5.79129i	0.364817i
$253$	20.0000i	1.25739i
$254$	5.58258	0.350282
$255$	0	0
$256$	98.4519	6.15324
$257$	− 14.7477i	− 0.919938i	−0.887935	−	0.459969i	$-0.847861\pi$
$257$	− 14.7477i	− 0.919938i	0.887935	−	0.459969i	$-0.152139\pi$
$258$	− 1.16515i	− 0.0725392i
$259$	4.00000	0.248548
$260$	0	0
$261$	1.00000	0.0618984
$262$	41.8693i	2.58670i
$263$	6.33030i	0.390343i	0.980769	+	0.195172i	$0.0625264\pi$
$263$	6.33030i	0.390343i	−0.980769	+	0.195172i	$0.937474\pi$
$264$	52.9129	3.25656
$265$	0	0
$266$	15.5826	0.955429
$267$	10.5826i	0.647643i
$268$	− 82.0345i	− 5.01105i
$269$	−13.4174	−0.818075	−0.409037	−	0.912518i	$-0.634135\pi$
$269$	−13.4174	−0.818075	−0.409037	+	0.912518i	$0.634135\pi$
$270$	0	0
$271$	−17.1652	−1.04271	−0.521354	−	0.853340i	$-0.674573\pi$
$271$	−17.1652	−1.04271	−0.521354	+	0.853340i	$0.674573\pi$
$272$	− 53.8693i	− 3.26631i
$273$	− 4.58258i	− 0.277350i
$274$	−56.7477	−3.42826
$275$	0	0
$276$	−23.1652	−1.39438
$277$	20.9129i	1.25653i	0.777998	+	0.628267i	$0.216235\pi$
$277$	20.9129i	1.25653i	−0.777998	+	0.628267i	$0.783765\pi$
$278$	− 51.8693i	− 3.11091i
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−9.58258	−0.571649	−0.285824	−	0.958282i	$-0.592267\pi$
$281$	−9.58258	−0.571649	−0.285824	+	0.958282i	$0.592267\pi$
$282$	− 3.95644i	− 0.235603i
$283$	19.1652i	1.13925i	0.821905	+	0.569625i	$0.192912\pi$
$283$	19.1652i	1.13925i	−0.821905	+	0.569625i	$0.807088\pi$
$284$	2.41742	0.143448
$285$	0	0
$286$	−63.9564	−3.78182
$287$	9.16515i	0.541002i
$288$	28.9564i	1.70627i
$289$	8.00000	0.470588
$290$	0	0
$291$	2.41742	0.141712
$292$	23.1652i	1.35564i
$293$	− 11.8348i	− 0.691399i	−0.938345	−	0.345700i	$-0.887642\pi$
$293$	− 11.8348i	− 0.691399i	0.938345	−	0.345700i	$-0.112358\pi$
$294$	−16.7477	−0.976747
$295$	0	0
$296$	42.3303	2.46040
$297$	5.00000i	0.290129i
$298$	− 30.0000i	− 1.73785i
$299$	18.3303	1.06007
$300$	0	0
$301$	−0.417424	−0.0240599
$302$	31.1652i	1.79335i
$303$	− 8.58258i	− 0.493056i
$304$	100.243	5.74934
$305$	0	0
$306$	8.37386	0.478702
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	− 28.9564i	− 1.64995i
$309$	3.16515	0.180059
$310$	0	0
$311$	−3.00000	−0.170114	−0.0850572	−	0.996376i	$-0.527107\pi$
$311$	−3.00000	−0.170114	−0.0850572	+	0.996376i	$0.527107\pi$
$312$	− 48.4955i	− 2.74551i
$313$	− 1.41742i	− 0.0801176i	−0.999197	−	0.0400588i	$-0.987245\pi$
$313$	− 1.41742i	− 0.0801176i	0.999197	−	0.0400588i	$-0.0127545\pi$
$314$	−46.7477	−2.63813
$315$	0	0
$316$	−9.16515	−0.515580
$317$	− 25.0000i	− 1.40414i	−0.712108	−	0.702070i	$-0.752259\pi$
$317$	− 25.0000i	− 1.40414i	0.712108	−	0.702070i	$-0.247741\pi$
$318$	26.7477i	1.49994i
$319$	−5.00000	−0.279946
$320$	0	0
$321$	−13.1652	−0.734807
$322$	11.1652i	0.622210i
$323$	− 16.7477i	− 0.931868i
$324$	−5.79129	−0.321738
$325$	0	0
$326$	4.41742	0.244659
$327$	− 4.16515i	− 0.230333i
$328$	96.9909i	5.35543i
$329$	−1.41742	−0.0781451
$330$	0	0
$331$	28.3303	1.55717	0.778587	−	0.627537i	$-0.215937\pi$
$331$	28.3303	1.55717	0.778587	+	0.627537i	$0.215937\pi$
$332$	− 14.0000i	− 0.768350i
$333$	4.00000i	0.219199i
$334$	0	0
$335$	0	0
$336$	17.9564	0.979604
$337$	− 2.83485i	− 0.154424i	−0.997015	−	0.0772120i	$-0.975398\pi$
$337$	− 2.83485i	− 0.154424i	0.997015	−	0.0772120i	$-0.0246018\pi$
$338$	22.3303i	1.21461i
$339$	−4.16515	−0.226220
$340$	0	0
$341$	20.0000	1.08306
$342$	15.5826i	0.842609i
$343$	13.0000i	0.701934i
$344$	−4.41742	−0.238172
$345$	0	0
$346$	−42.3303	−2.27569
$347$	− 11.1652i	− 0.599377i	−0.954037	−	0.299688i	$-0.903117\pi$
$347$	− 11.1652i	− 0.599377i	0.954037	−	0.299688i	$-0.0968827\pi$
$348$	− 5.79129i	− 0.310446i
$349$	−32.3303	−1.73060	−0.865301	−	0.501253i	$-0.832873\pi$
$349$	−32.3303	−1.73060	−0.865301	+	0.501253i	$0.832873\pi$
$350$	0	0
$351$	4.58258	0.244600
$352$	− 144.782i	− 7.71692i
$353$	33.1652i	1.76520i	0.470122	+	0.882601i	$0.344210\pi$
$353$	33.1652i	1.76520i	−0.470122	+	0.882601i	$0.655790\pi$
$354$	4.41742	0.234783
$355$	0	0
$356$	61.2867	3.24819
$357$	− 3.00000i	− 0.158777i
$358$	− 63.4955i	− 3.35584i
$359$	8.83485	0.466285	0.233143	−	0.972443i	$-0.425099\pi$
$359$	8.83485	0.466285	0.233143	+	0.972443i	$0.425099\pi$
$360$	0	0
$361$	12.1652	0.640271
$362$	6.04356i	0.317643i
$363$	− 14.0000i	− 0.734809i
$364$	−26.5390	−1.39102
$365$	0	0
$366$	41.1652	2.15174
$367$	17.4955i	0.913255i	0.889658	+	0.456628i	$0.150943\pi$
$367$	17.4955i	0.913255i	−0.889658	+	0.456628i	$0.849057\pi$
$368$	71.8258i	3.74418i
$369$	−9.16515	−0.477119
$370$	0	0
$371$	9.58258	0.497503
$372$	23.1652i	1.20106i
$373$	8.33030i	0.431327i	0.976468	+	0.215663i	$0.0691914\pi$
$373$	8.33030i	0.431327i	−0.976468	+	0.215663i	$0.930809\pi$
$374$	−41.8693	−2.16501
$375$	0	0
$376$	−15.0000	−0.773566
$377$	4.58258i	0.236015i
$378$	2.79129i	0.143568i
$379$	26.0000	1.33553	0.667765	−	0.744372i	$-0.267251\pi$
$379$	26.0000	1.33553	0.667765	+	0.744372i	$0.267251\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	11.1652i	0.571259i
$383$	5.58258i	0.285256i	0.989776	+	0.142628i	$0.0455553\pi$
$383$	5.58258i	0.285256i	−0.989776	+	0.142628i	$0.954445\pi$
$384$	67.4519	3.44214
$385$	0	0
$386$	−45.5826	−2.32009
$387$	− 0.417424i	− 0.0212189i
$388$	− 14.0000i	− 0.710742i
$389$	−2.58258	−0.130942	−0.0654709	−	0.997854i	$-0.520855\pi$
$389$	−2.58258	−0.130942	−0.0654709	+	0.997854i	$0.520855\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	63.4955i	3.20700i
$393$	15.0000i	0.756650i
$394$	56.7477	2.85891
$395$	0	0
$396$	28.9564	1.45512
$397$	29.1652i	1.46376i	0.681435	+	0.731878i	$0.261356\pi$
$397$	29.1652i	1.46376i	−0.681435	+	0.731878i	$0.738644\pi$
$398$	63.0345i	3.15963i
$399$	5.58258	0.279478
$400$	0	0
$401$	21.5826	1.07778	0.538891	−	0.842375i	$-0.318843\pi$
$401$	21.5826	1.07778	0.538891	+	0.842375i	$0.318843\pi$
$402$	− 39.5390i	− 1.97203i
$403$	− 18.3303i	− 0.913097i
$404$	−49.7042	−2.47287
$405$	0	0
$406$	−2.79129	−0.138529
$407$	− 20.0000i	− 0.991363i
$408$	− 31.7477i	− 1.57175i
$409$	−24.7477	−1.22370	−0.611848	−	0.790975i	$-0.709574\pi$
$409$	−24.7477	−1.22370	−0.611848	+	0.790975i	$0.709574\pi$
$410$	0	0
$411$	−20.3303	−1.00282
$412$	− 18.3303i	− 0.903069i
$413$	− 1.58258i	− 0.0778735i
$414$	−11.1652	−0.548737
$415$	0	0
$416$	−132.695	−6.50591
$417$	− 18.5826i	− 0.909993i
$418$	− 77.9129i	− 3.81084i
$419$	18.8348	0.920143	0.460071	−	0.887882i	$-0.347824\pi$
$419$	18.8348	0.920143	0.460071	+	0.887882i	$0.347824\pi$
$420$	0	0
$421$	13.5826	0.661974	0.330987	−	0.943635i	$-0.392618\pi$
$421$	13.5826	0.661974	0.330987	+	0.943635i	$0.392618\pi$
$422$	45.5826i	2.21893i
$423$	− 1.41742i	− 0.0689175i
$424$	101.408	4.92482
$425$	0	0
$426$	1.16515	0.0564518
$427$	− 14.7477i	− 0.713693i
$428$	76.2432i	3.68535i
$429$	−22.9129	−1.10624
$430$	0	0
$431$	−20.8348	−1.00358	−0.501790	−	0.864990i	$-0.667325\pi$
$431$	−20.8348	−1.00358	−0.501790	+	0.864990i	$0.667325\pi$
$432$	17.9564i	0.863930i
$433$	39.0780i	1.87797i	0.343958	+	0.938985i	$0.388232\pi$
$433$	39.0780i	1.87797i	−0.343958	+	0.938985i	$0.611768\pi$
$434$	11.1652	0.535944
$435$	0	0
$436$	−24.1216	−1.15521
$437$	22.3303i	1.06820i
$438$	11.1652i	0.533492i
$439$	−28.9129	−1.37994	−0.689968	−	0.723840i	$-0.742376\pi$
$439$	−28.9129	−1.37994	−0.689968	+	0.723840i	$0.742376\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	38.3739i	1.82526i
$443$	8.58258i	0.407770i	0.978995	+	0.203885i	$0.0653569\pi$
$443$	8.58258i	0.407770i	−0.978995	+	0.203885i	$0.934643\pi$
$444$	23.1652	1.09937
$445$	0	0
$446$	−19.5390	−0.925199
$447$	− 10.7477i	− 0.508350i
$448$	− 44.9129i	− 2.12193i
$449$	−34.0780	−1.60824	−0.804121	−	0.594466i	$-0.797364\pi$
$449$	−34.0780	−1.60824	−0.804121	+	0.594466i	$0.797364\pi$
$450$	0	0
$451$	45.8258	2.15785
$452$	24.1216i	1.13458i
$453$	11.1652i	0.524585i
$454$	4.41742	0.207320
$455$	0	0
$456$	59.0780	2.76658
$457$	23.7477i	1.11087i	0.831559	+	0.555436i	$0.187449\pi$
$457$	23.7477i	1.11087i	−0.831559	+	0.555436i	$0.812551\pi$
$458$	3.25227i	0.151969i
$459$	3.00000	0.140028
$460$	0	0
$461$	9.16515	0.426864	0.213432	−	0.976958i	$-0.431536\pi$
$461$	9.16515	0.426864	0.213432	+	0.976958i	$0.431536\pi$
$462$	− 13.9564i	− 0.649312i
$463$	− 18.1652i	− 0.844206i	−0.906548	−	0.422103i	$-0.861292\pi$
$463$	− 18.1652i	− 0.844206i	0.906548	−	0.422103i	$-0.138708\pi$
$464$	−17.9564	−0.833607
$465$	0	0
$466$	−14.4174	−0.667874
$467$	− 8.00000i	− 0.370196i	−0.982720	−	0.185098i	$-0.940740\pi$
$467$	− 8.00000i	− 0.370196i	0.982720	−	0.185098i	$-0.0592602\pi$
$468$	− 26.5390i	− 1.22677i
$469$	−14.1652	−0.654086
$470$	0	0
$471$	−16.7477	−0.771695
$472$	− 16.7477i	− 0.770877i
$473$	2.08712i	0.0959659i
$474$	−4.41742	−0.202899
$475$	0	0
$476$	−17.3739	−0.796330
$477$	9.58258i	0.438756i
$478$	− 72.5735i	− 3.31943i
$479$	0.834849	0.0381452	0.0190726	−	0.999818i	$-0.493929\pi$
$479$	0.834849	0.0381452	0.0190726	+	0.999818i	$0.493929\pi$
$480$	0	0
$481$	−18.3303	−0.835790
$482$	20.4610i	0.931972i
$483$	4.00000i	0.182006i
$484$	−81.0780	−3.68536
$485$	0	0
$486$	−2.79129	−0.126615
$487$	− 34.3303i	− 1.55565i	−0.628478	−	0.777827i	$-0.716322\pi$
$487$	− 34.3303i	− 1.55565i	0.628478	−	0.777827i	$-0.283678\pi$
$488$	− 156.069i	− 7.06491i
$489$	1.58258	0.0715665
$490$	0	0
$491$	16.0000	0.722070	0.361035	−	0.932552i	$-0.382424\pi$
$491$	16.0000	0.722070	0.361035	+	0.932552i	$0.382424\pi$
$492$	53.0780i	2.39294i
$493$	3.00000i	0.135113i
$494$	−71.4083	−3.21281
$495$	0	0
$496$	71.8258	3.22507
$497$	− 0.417424i	− 0.0187240i
$498$	− 6.74773i	− 0.302373i
$499$	22.5826	1.01093	0.505467	−	0.862846i	$-0.331320\pi$
$499$	22.5826	1.01093	0.505467	+	0.862846i	$0.331320\pi$
$500$	0	0
$501$	0	0
$502$	6.04356i	0.269737i
$503$	− 22.9129i	− 1.02163i	−0.859689	−	0.510817i	$-0.829343\pi$
$503$	− 22.9129i	− 1.02163i	0.859689	−	0.510817i	$-0.170657\pi$
$504$	10.5826	0.471385
$505$	0	0
$506$	55.8258	2.48176
$507$	8.00000i	0.355292i
$508$	− 11.5826i	− 0.513894i
$509$	0.747727	0.0331424	0.0165712	−	0.999863i	$-0.494725\pi$
$509$	0.747727	0.0331424	0.0165712	+	0.999863i	$0.494725\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	− 139.904i	− 6.18293i
$513$	5.58258i	0.246477i
$514$	−41.1652	−1.81572
$515$	0	0
$516$	−2.41742	−0.106421
$517$	7.08712i	0.311691i
$518$	− 11.1652i	− 0.490569i
$519$	−15.1652	−0.665676
$520$	0	0
$521$	−21.0780	−0.923445	−0.461723	−	0.887024i	$-0.652768\pi$
$521$	−21.0780	−0.923445	−0.461723	+	0.887024i	$0.652768\pi$
$522$	− 2.79129i	− 0.122171i
$523$	− 3.33030i	− 0.145624i	−0.997346	−	0.0728120i	$-0.976803\pi$
$523$	− 3.33030i	− 0.145624i	0.997346	−	0.0728120i	$-0.0231973\pi$
$524$	86.8693	3.79490
$525$	0	0
$526$	17.6697	0.770435
$527$	− 12.0000i	− 0.522728i
$528$	− 89.7822i	− 3.90727i
$529$	7.00000	0.304348
$530$	0	0
$531$	1.58258	0.0686779
$532$	− 32.3303i	− 1.40170i
$533$	− 42.0000i	− 1.81922i
$534$	29.5390	1.27828
$535$	0	0
$536$	−149.904	−6.47486
$537$	− 22.7477i	− 0.981637i
$538$	37.4519i	1.61467i
$539$	30.0000	1.29219
$540$	0	0
$541$	23.4955	1.01015	0.505074	−	0.863076i	$-0.331465\pi$
$541$	23.4955	1.01015	0.505074	+	0.863076i	$0.331465\pi$
$542$	47.9129i	2.05803i
$543$	2.16515i	0.0929155i
$544$	−86.8693	−3.72449
$545$	0	0
$546$	−12.7913	−0.547417
$547$	− 14.1652i	− 0.605658i	−0.953045	−	0.302829i	$-0.902069\pi$
$547$	− 14.1652i	− 0.605658i	0.953045	−	0.302829i	$-0.0979311\pi$
$548$	117.739i	5.02955i
$549$	14.7477	0.629418
$550$	0	0
$551$	−5.58258	−0.237826
$552$	42.3303i	1.80170i
$553$	1.58258i	0.0672980i
$554$	58.3739	2.48007
$555$	0	0
$556$	−107.617	−4.56398
$557$	4.74773i	0.201168i	0.994929	+	0.100584i	$0.0320711\pi$
$557$	4.74773i	0.201168i	−0.994929	+	0.100584i	$0.967929\pi$
$558$	11.1652i	0.472659i
$559$	1.91288	0.0809061
$560$	0	0
$561$	−15.0000	−0.633300
$562$	26.7477i	1.12828i
$563$	5.41742i	0.228317i	0.993463	+	0.114159i	$0.0364172\pi$
$563$	5.41742i	0.228317i	−0.993463	+	0.114159i	$0.963583\pi$
$564$	−8.20871	−0.345649
$565$	0	0
$566$	53.4955	2.24858
$567$	1.00000i	0.0419961i
$568$	− 4.41742i	− 0.185351i
$569$	−19.4174	−0.814021	−0.407010	−	0.913424i	$-0.633429\pi$
$569$	−19.4174	−0.814021	−0.407010	+	0.913424i	$0.633429\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	132.695i	5.54826i
$573$	4.00000i	0.167102i
$574$	25.5826	1.06780
$575$	0	0
$576$	44.9129	1.87137
$577$	− 0.834849i	− 0.0347552i	−0.999849	−	0.0173776i	$-0.994468\pi$
$577$	− 0.834849i	− 0.0347552i	0.999849	−	0.0173776i	$-0.00553174\pi$
$578$	− 22.3303i	− 0.928818i
$579$	−16.3303	−0.678664
$580$	0	0
$581$	−2.41742	−0.100292
$582$	− 6.74773i	− 0.279702i
$583$	− 47.9129i	− 1.98435i
$584$	42.3303	1.75164
$585$	0	0
$586$	−33.0345	−1.36464
$587$	− 2.41742i	− 0.0997778i	−0.998755	−	0.0498889i	$-0.984113\pi$
$587$	− 2.41742i	− 0.0997778i	0.998755	−	0.0498889i	$-0.0158867\pi$
$588$	34.7477i	1.43297i
$589$	22.3303	0.920104
$590$	0	0
$591$	20.3303	0.836277
$592$	− 71.8258i	− 2.95202i
$593$	− 17.5826i	− 0.722030i	−0.932560	−	0.361015i	$-0.882430\pi$
$593$	− 17.5826i	− 0.722030i	0.932560	−	0.361015i	$-0.117570\pi$
$594$	13.9564	0.572640
$595$	0	0
$596$	−62.2432	−2.54958
$597$	22.5826i	0.924243i
$598$	− 51.1652i	− 2.09230i
$599$	18.1652	0.742208	0.371104	−	0.928591i	$-0.378979\pi$
$599$	18.1652	0.742208	0.371104	+	0.928591i	$0.378979\pi$
$600$	0	0
$601$	4.33030	0.176637	0.0883184	−	0.996092i	$-0.471851\pi$
$601$	4.33030	0.176637	0.0883184	+	0.996092i	$0.471851\pi$
$602$	1.16515i	0.0474880i
$603$	− 14.1652i	− 0.576850i
$604$	64.6606	2.63100
$605$	0	0
$606$	−23.9564	−0.973164
$607$	− 5.58258i	− 0.226590i	−0.993561	−	0.113295i	$-0.963860\pi$
$607$	− 5.58258i	− 0.226590i	0.993561	−	0.113295i	$-0.0361405\pi$
$608$	− 161.652i	− 6.55583i
$609$	−1.00000	−0.0405220
$610$	0	0
$611$	6.49545	0.262778
$612$	− 17.3739i	− 0.702297i
$613$	16.2523i	0.656423i	0.944604	+	0.328212i	$0.106446\pi$
$613$	16.2523i	0.656423i	−0.944604	+	0.328212i	$0.893554\pi$
$614$	0	0
$615$	0	0
$616$	−52.9129	−2.13192
$617$	− 39.4955i	− 1.59003i	−0.606592	−	0.795014i	$-0.707464\pi$
$617$	− 39.4955i	− 1.59003i	0.606592	−	0.795014i	$-0.292536\pi$
$618$	− 8.83485i	− 0.355390i
$619$	5.16515	0.207605	0.103802	−	0.994598i	$-0.466899\pi$
$619$	5.16515	0.207605	0.103802	+	0.994598i	$0.466899\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	8.37386i	0.335761i
$623$	− 10.5826i	− 0.423982i
$624$	−82.2867	−3.29411
$625$	0	0
$626$	−3.95644	−0.158131
$627$	− 27.9129i	− 1.11473i
$628$	96.9909i	3.87036i
$629$	−12.0000	−0.478471
$630$	0	0
$631$	−34.9129	−1.38986	−0.694930	−	0.719078i	$-0.744565\pi$
$631$	−34.9129	−1.38986	−0.694930	+	0.719078i	$0.744565\pi$
$632$	16.7477i	0.666189i
$633$	16.3303i	0.649071i
$634$	−69.7822	−2.77141
$635$	0	0
$636$	55.4955	2.20054
$637$	− 27.4955i	− 1.08941i
$638$	13.9564i	0.552541i
$639$	0.417424	0.0165131
$640$	0	0
$641$	2.58258	0.102006	0.0510028	−	0.998699i	$-0.483758\pi$
$641$	2.58258	0.102006	0.0510028	+	0.998699i	$0.483758\pi$
$642$	36.7477i	1.45032i
$643$	29.6606i	1.16970i	0.811141	+	0.584850i	$0.198847\pi$
$643$	29.6606i	1.16970i	−0.811141	+	0.584850i	$0.801153\pi$
$644$	23.1652	0.912835
$645$	0	0
$646$	−46.7477	−1.83926
$647$	− 2.74773i	− 0.108024i	−0.998540	−	0.0540121i	$-0.982799\pi$
$647$	− 2.74773i	− 0.108024i	0.998540	−	0.0540121i	$-0.0172010\pi$
$648$	10.5826i	0.415723i
$649$	−7.91288	−0.310608
$650$	0	0
$651$	4.00000	0.156772
$652$	− 9.16515i	− 0.358935i
$653$	− 7.83485i	− 0.306601i	−0.988180	−	0.153301i	$-0.951010\pi$
$653$	− 7.83485i	− 0.306601i	0.988180	−	0.153301i	$-0.0489903\pi$
$654$	−11.6261	−0.454618
$655$	0	0
$656$	164.573	6.42552
$657$	4.00000i	0.156055i
$658$	3.95644i	0.154238i
$659$	0.165151	0.00643338	0.00321669	−	0.999995i	$-0.498976\pi$
$659$	0.165151	0.00643338	0.00321669	+	0.999995i	$0.498976\pi$
$660$	0	0
$661$	25.0000	0.972387	0.486194	−	0.873851i	$-0.338385\pi$
$661$	25.0000	0.972387	0.486194	+	0.873851i	$0.338385\pi$
$662$	− 79.0780i	− 3.07345i
$663$	13.7477i	0.533917i
$664$	−25.5826	−0.992796
$665$	0	0
$666$	11.1652	0.432641
$667$	− 4.00000i	− 0.154881i
$668$	0	0
$669$	−7.00000	−0.270636
$670$	0	0
$671$	−73.7386	−2.84665
$672$	− 28.9564i	− 1.11702i
$673$	25.7477i	0.992502i	0.868179	+	0.496251i	$0.165290\pi$
$673$	25.7477i	0.992502i	−0.868179	+	0.496251i	$0.834710\pi$
$674$	−7.91288	−0.304793
$675$	0	0
$676$	46.3303	1.78193
$677$	− 46.8258i	− 1.79966i	−0.436241	−	0.899830i	$-0.643690\pi$
$677$	− 46.8258i	− 1.79966i	0.436241	−	0.899830i	$-0.356310\pi$
$678$	11.6261i	0.446499i
$679$	−2.41742	−0.0927722
$680$	0	0
$681$	1.58258	0.0606444
$682$	− 55.8258i	− 2.13768i
$683$	11.1652i	0.427223i	0.976919	+	0.213611i	$0.0685226\pi$
$683$	11.1652i	0.427223i	−0.976919	+	0.213611i	$0.931477\pi$
$684$	32.3303	1.23618
$685$	0	0
$686$	36.2867	1.38543
$687$	1.16515i	0.0444533i
$688$	7.49545i	0.285762i
$689$	−43.9129	−1.67295
$690$	0	0
$691$	−2.91288	−0.110811	−0.0554056	−	0.998464i	$-0.517645\pi$
$691$	−2.91288	−0.110811	−0.0554056	+	0.998464i	$0.517645\pi$
$692$	87.8258i	3.33863i
$693$	− 5.00000i	− 0.189934i
$694$	−31.1652	−1.18301
$695$	0	0
$696$	−10.5826	−0.401131
$697$	− 27.4955i	− 1.04146i
$698$	90.2432i	3.41575i
$699$	−5.16515	−0.195364
$700$	0	0
$701$	41.0780	1.55150	0.775748	−	0.631043i	$-0.217373\pi$
$701$	41.0780	1.55150	0.775748	+	0.631043i	$0.217373\pi$
$702$	− 12.7913i	− 0.482776i
$703$	− 22.3303i	− 0.842203i
$704$	−224.564	−8.46359
$705$	0	0
$706$	92.5735	3.48405
$707$	8.58258i	0.322781i
$708$	− 9.16515i	− 0.344447i
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	−1.58258	−0.0593512
$712$	− 111.991i	− 4.19704i
$713$	16.0000i	0.599205i
$714$	−8.37386	−0.313384
$715$	0	0
$716$	−131.739	−4.92330
$717$	− 26.0000i	− 0.970988i
$718$	− 24.6606i	− 0.920326i
$719$	37.9129	1.41391	0.706956	−	0.707258i	$-0.250068\pi$
$719$	37.9129	1.41391	0.706956	+	0.707258i	$0.250068\pi$
$720$	0	0
$721$	−3.16515	−0.117876
$722$	− 33.9564i	− 1.26373i
$723$	7.33030i	0.272617i
$724$	12.5390	0.466009
$725$	0	0
$726$	−39.0780	−1.45032
$727$	46.3303i	1.71830i	0.511727	+	0.859148i	$0.329006\pi$
$727$	46.3303i	1.71830i	−0.511727	+	0.859148i	$0.670994\pi$
$728$	48.4955i	1.79736i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	1.25227	0.0463170
$732$	− 85.4083i	− 3.15678i
$733$	47.5826i	1.75750i	0.477280	+	0.878751i	$0.341623\pi$
$733$	47.5826i	1.75750i	−0.477280	+	0.878751i	$0.658377\pi$
$734$	48.8348	1.80253
$735$	0	0
$736$	115.826	4.26939
$737$	70.8258i	2.60890i
$738$	25.5826i	0.941708i
$739$	−46.7477	−1.71964	−0.859821	−	0.510595i	$-0.829425\pi$
$739$	−46.7477	−1.71964	−0.859821	+	0.510595i	$0.829425\pi$
$740$	0	0
$741$	−25.5826	−0.939799
$742$	− 26.7477i	− 0.981940i
$743$	− 2.25227i	− 0.0826279i	−0.999146	−	0.0413139i	$-0.986846\pi$
$743$	− 2.25227i	− 0.0826279i	0.999146	−	0.0413139i	$-0.0131544\pi$
$744$	42.3303	1.55190
$745$	0	0
$746$	23.2523	0.851326
$747$	− 2.41742i	− 0.0884489i
$748$	86.8693i	3.17626i
$749$	13.1652	0.481044
$750$	0	0
$751$	37.4955	1.36823	0.684114	−	0.729375i	$-0.260189\pi$
$751$	37.4955	1.36823	0.684114	+	0.729375i	$0.260189\pi$
$752$	25.4519i	0.928135i
$753$	2.16515i	0.0789025i
$754$	12.7913	0.465831
$755$	0	0
$756$	5.79129	0.210627
$757$	34.3303i	1.24776i	0.781522	+	0.623878i	$0.214444\pi$
$757$	34.3303i	1.24776i	−0.781522	+	0.623878i	$0.785556\pi$
$758$	− 72.5735i	− 2.63599i
$759$	20.0000	0.725954
$760$	0	0
$761$	45.5826	1.65237	0.826184	−	0.563401i	$-0.190507\pi$
$761$	45.5826	1.65237	0.826184	+	0.563401i	$0.190507\pi$
$762$	− 5.58258i	− 0.202235i
$763$	4.16515i	0.150789i
$764$	23.1652	0.838086
$765$	0	0
$766$	15.5826	0.563021
$767$	7.25227i	0.261864i
$768$	− 98.4519i	− 3.55258i
$769$	−16.4174	−0.592027	−0.296014	−	0.955184i	$-0.595657\pi$
$769$	−16.4174	−0.592027	−0.296014	+	0.955184i	$0.595657\pi$
$770$	0	0
$771$	−14.7477	−0.531126
$772$	94.5735i	3.40377i
$773$	− 13.1652i	− 0.473518i	−0.971568	−	0.236759i	$-0.923915\pi$
$773$	− 13.1652i	− 0.473518i	0.971568	−	0.236759i	$-0.0760851\pi$
$774$	−1.16515	−0.0418805
$775$	0	0
$776$	−25.5826	−0.918361
$777$	− 4.00000i	− 0.143499i
$778$	7.20871i	0.258445i
$779$	51.1652	1.83318
$780$	0	0
$781$	−2.08712	−0.0746831
$782$	− 33.4955i	− 1.19779i
$783$	− 1.00000i	− 0.0357371i
$784$	107.739	3.84781
$785$	0	0
$786$	41.8693	1.49343
$787$	− 24.0000i	− 0.855508i	−0.903895	−	0.427754i	$-0.859305\pi$
$787$	− 24.0000i	− 0.855508i	0.903895	−	0.427754i	$-0.140695\pi$
$788$	− 117.739i	− 4.19427i
$789$	6.33030	0.225365
$790$	0	0
$791$	4.16515	0.148096
$792$	− 52.9129i	− 1.88018i
$793$	67.5826i	2.39993i
$794$	81.4083	2.88907
$795$	0	0
$796$	130.782	4.63545
$797$	− 4.33030i	− 0.153387i	−0.997055	−	0.0766936i	$-0.975564\pi$
$797$	− 4.33030i	− 0.153387i	0.997055	−	0.0766936i	$-0.0244363\pi$
$798$	− 15.5826i	− 0.551617i
$799$	4.25227	0.150435
$800$	0	0
$801$	10.5826	0.373917
$802$	− 60.2432i	− 2.12726i
$803$	− 20.0000i	− 0.705785i
$804$	−82.0345	−2.89313
$805$	0	0
$806$	−51.1652	−1.80222
$807$	13.4174i	0.472316i
$808$	90.8258i	3.19524i
$809$	20.0780	0.705906	0.352953	−	0.935641i	$-0.385178\pi$
$809$	20.0780	0.705906	0.352953	+	0.935641i	$0.385178\pi$
$810$	0	0
$811$	23.4174	0.822297	0.411148	−	0.911568i	$-0.365128\pi$
$811$	23.4174	0.822297	0.411148	+	0.911568i	$0.365128\pi$
$812$	5.79129i	0.203234i
$813$	17.1652i	0.602008i
$814$	−55.8258	−1.95669
$815$	0	0
$816$	−53.8693	−1.88580
$817$	2.33030i	0.0815270i
$818$	69.0780i	2.41526i
$819$	−4.58258	−0.160128
$820$	0	0
$821$	23.4955	0.819997	0.409999	−	0.912086i	$-0.365529\pi$
$821$	23.4955	0.819997	0.409999	+	0.912086i	$0.365529\pi$
$822$	56.7477i	1.97930i
$823$	13.0780i	0.455871i	0.973676	+	0.227936i	$0.0731976\pi$
$823$	13.0780i	0.455871i	−0.973676	+	0.227936i	$0.926802\pi$
$824$	−33.4955	−1.16687
$825$	0	0
$826$	−4.41742	−0.153702
$827$	47.8258i	1.66306i	0.555476	+	0.831532i	$0.312536\pi$
$827$	47.8258i	1.66306i	−0.555476	+	0.831532i	$0.687464\pi$
$828$	23.1652i	0.805045i
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	0	0
$831$	20.9129	0.725460
$832$	205.817i	7.13541i
$833$	− 18.0000i	− 0.623663i
$834$	−51.8693	−1.79609
$835$	0	0
$836$	−161.652	−5.59083
$837$	4.00000i	0.138260i
$838$	− 52.5735i	− 1.81612i
$839$	−6.49545	−0.224248	−0.112124	−	0.993694i	$-0.535765\pi$
$839$	−6.49545	−0.224248	−0.112124	+	0.993694i	$0.535765\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	− 37.9129i	− 1.30656i
$843$	9.58258i	0.330041i
$844$	94.5735	3.25535
$845$	0	0
$846$	−3.95644	−0.136025
$847$	14.0000i	0.481046i
$848$	− 172.069i	− 5.90887i
$849$	19.1652	0.657746
$850$	0	0
$851$	16.0000	0.548473
$852$	− 2.41742i	− 0.0828196i
$853$	6.74773i	0.231038i	0.993305	+	0.115519i	$0.0368531\pi$
$853$	6.74773i	0.231038i	−0.993305	+	0.115519i	$0.963147\pi$
$854$	−41.1652	−1.40864
$855$	0	0
$856$	139.321	4.76190
$857$	26.6606i	0.910709i	0.890310	+	0.455354i	$0.150487\pi$
$857$	26.6606i	0.910709i	−0.890310	+	0.455354i	$0.849513\pi$
$858$	63.9564i	2.18344i
$859$	38.4174	1.31079	0.655393	−	0.755288i	$-0.272503\pi$
$859$	38.4174	1.31079	0.655393	+	0.755288i	$0.272503\pi$
$860$	0	0
$861$	9.16515	0.312348
$862$	58.1561i	1.98080i
$863$	15.5826i	0.530437i	0.964188	+	0.265219i	$0.0854441\pi$
$863$	15.5826i	0.530437i	−0.964188	+	0.265219i	$0.914556\pi$
$864$	28.9564	0.985118
$865$	0	0
$866$	109.078	3.70662
$867$	− 8.00000i	− 0.271694i
$868$	− 23.1652i	− 0.786276i
$869$	7.91288	0.268426
$870$	0	0
$871$	64.9129	2.19949
$872$	44.0780i	1.49267i
$873$	− 2.41742i	− 0.0818174i
$874$	62.3303	2.10835
$875$	0	0
$876$	23.1652	0.782678
$877$	56.3303i	1.90214i	0.308977	+	0.951070i	$0.400013\pi$
$877$	56.3303i	1.90214i	−0.308977	+	0.951070i	$0.599987\pi$
$878$	80.7042i	2.72363i
$879$	−11.8348	−0.399180
$880$	0	0
$881$	32.0780	1.08074	0.540368	−	0.841429i	$-0.318285\pi$
$881$	32.0780	1.08074	0.540368	+	0.841429i	$0.318285\pi$
$882$	16.7477i	0.563925i
$883$	− 16.0000i	− 0.538443i	−0.963078	−	0.269221i	$-0.913234\pi$
$883$	− 16.0000i	− 0.538443i	0.963078	−	0.269221i	$-0.0867663\pi$
$884$	79.6170	2.67781
$885$	0	0
$886$	23.9564	0.804832
$887$	0.912878i	0.0306515i	0.999883	+	0.0153257i	$0.00487852\pi$
$887$	0.912878i	0.0306515i	−0.999883	+	0.0153257i	$0.995121\pi$
$888$	− 42.3303i	− 1.42051i
$889$	−2.00000	−0.0670778
$890$	0	0
$891$	5.00000	0.167506
$892$	40.5390i	1.35735i
$893$	7.91288i	0.264794i
$894$	−30.0000	−1.00335
$895$	0	0
$896$	−67.4519	−2.25341
$897$	− 18.3303i	− 0.612031i
$898$	95.1216i	3.17425i
$899$	−4.00000	−0.133407
$900$	0	0
$901$	−28.7477	−0.957726
$902$	− 127.913i	− 4.25903i
$903$	0.417424i	0.0138910i
$904$	44.0780	1.46601
$905$	0	0
$906$	31.1652	1.03539
$907$	14.4174i	0.478723i	0.970931	+	0.239361i	$0.0769381\pi$
$907$	14.4174i	0.478723i	−0.970931	+	0.239361i	$0.923062\pi$
$908$	− 9.16515i	− 0.304156i
$909$	−8.58258	−0.284666
$910$	0	0
$911$	40.8258	1.35262	0.676309	−	0.736618i	$-0.263578\pi$
$911$	40.8258	1.35262	0.676309	+	0.736618i	$0.263578\pi$
$912$	− 100.243i	− 3.31938i
$913$	12.0871i	0.400025i
$914$	66.2867	2.19257
$915$	0	0
$916$	6.74773	0.222951
$917$	− 15.0000i	− 0.495344i
$918$	− 8.37386i	− 0.276379i
$919$	26.9129	0.887774	0.443887	−	0.896083i	$-0.353599\pi$
$919$	26.9129	0.887774	0.443887	+	0.896083i	$0.353599\pi$
$920$	0	0
$921$	0	0
$922$	− 25.5826i	− 0.842517i
$923$	1.91288i	0.0629632i
$924$	−28.9564	−0.952597
$925$	0	0
$926$	−50.7042	−1.66624
$927$	− 3.16515i	− 0.103957i
$928$	28.9564i	0.950542i
$929$	−52.3303	−1.71690	−0.858451	−	0.512896i	$-0.828573\pi$
$929$	−52.3303	−1.71690	−0.858451	+	0.512896i	$0.828573\pi$
$930$	0	0
$931$	33.4955	1.09777
$932$	29.9129i	0.979829i
$933$	3.00000i	0.0982156i
$934$	−22.3303	−0.730670
$935$	0	0
$936$	−48.4955	−1.58512
$937$	− 9.41742i	− 0.307654i	−0.988098	−	0.153827i	$-0.950840\pi$
$937$	− 9.41742i	− 0.307654i	0.988098	−	0.153827i	$-0.0491598\pi$
$938$	39.5390i	1.29099i
$939$	−1.41742	−0.0462559
$940$	0	0
$941$	−25.9129	−0.844736	−0.422368	−	0.906425i	$-0.638801\pi$
$941$	−25.9129	−0.844736	−0.422368	+	0.906425i	$0.638801\pi$
$942$	46.7477i	1.52312i
$943$	36.6606i	1.19383i
$944$	−28.4174	−0.924908
$945$	0	0
$946$	5.82576	0.189412
$947$	− 9.08712i	− 0.295292i	−0.989040	−	0.147646i	$-0.952830\pi$
$947$	− 9.08712i	− 0.295292i	0.989040	−	0.147646i	$-0.0471696\pi$
$948$	9.16515i	0.297670i
$949$	−18.3303	−0.595027
$950$	0	0
$951$	−25.0000	−0.810681
$952$	31.7477i	1.02895i
$953$	28.4174i	0.920531i	0.887781	+	0.460265i	$0.152246\pi$
$953$	28.4174i	0.920531i	−0.887781	+	0.460265i	$0.847754\pi$
$954$	26.7477	0.865990
$955$	0	0
$956$	−150.573	−4.86989
$957$	5.00000i	0.161627i
$958$	− 2.33030i	− 0.0752887i
$959$	20.3303	0.656500
$960$	0	0
$961$	−15.0000	−0.483871
$962$	51.1652i	1.64963i
$963$	13.1652i	0.424241i
$964$	42.4519	1.36728
$965$	0	0
$966$	11.1652	0.359233
$967$	36.7477i	1.18173i	0.806771	+	0.590864i	$0.201213\pi$
$967$	36.7477i	1.18173i	−0.806771	+	0.590864i	$0.798787\pi$
$968$	148.156i	4.76192i
$969$	−16.7477	−0.538015
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	5.79129i	0.185756i
$973$	18.5826i	0.595730i
$974$	−95.8258	−3.07046
$975$	0	0
$976$	−264.817	−8.47657
$977$	− 57.4955i	− 1.83944i	−0.392572	−	0.919721i	$-0.628415\pi$
$977$	− 57.4955i	− 1.83944i	0.392572	−	0.919721i	$-0.371585\pi$
$978$	− 4.41742i	− 0.141254i
$979$	−52.9129	−1.69110
$980$	0	0
$981$	−4.16515	−0.132983
$982$	− 44.6606i	− 1.42518i
$983$	− 36.8348i	− 1.17485i	−0.809279	−	0.587425i	$-0.800142\pi$
$983$	− 36.8348i	− 1.17485i	0.809279	−	0.587425i	$-0.199858\pi$
$984$	96.9909	3.09196
$985$	0	0
$986$	8.37386	0.266678
$987$	1.41742i	0.0451171i
$988$	148.156i	4.71347i
$989$	−1.66970	−0.0530933
$990$	0	0
$991$	−48.0780	−1.52725	−0.763624	−	0.645661i	$-0.776582\pi$
$991$	−48.0780	−1.52725	−0.763624	+	0.645661i	$0.776582\pi$
$992$	− 115.826i	− 3.67747i
$993$	− 28.3303i	− 0.899035i
$994$	−1.16515	−0.0369564
$995$	0	0
$996$	−14.0000	−0.443607
$997$	− 19.1652i	− 0.606966i	−0.952837	−	0.303483i	$-0.901850\pi$
$997$	− 19.1652i	− 0.606966i	0.952837	−	0.303483i	$-0.0981496\pi$
$998$	− 63.0345i	− 1.99532i
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.c.f.349.1		4
5.2	odd	4		2175.2.a.r.1.2		2
5.3	odd	4		435.2.a.f.1.1	✓	2
5.4	even	2	inner	2175.2.c.f.349.4		4
15.2	even	4		6525.2.a.t.1.1		2
15.8	even	4		1305.2.a.m.1.2		2
20.3	even	4		6960.2.a.bw.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.a.f.1.1	✓	2	5.3	odd	4
1305.2.a.m.1.2		2	15.8	even	4
2175.2.a.r.1.2		2	5.2	odd	4
2175.2.c.f.349.1		4	1.1	even	1	trivial
2175.2.c.f.349.4		4	5.4	even	2	inner
6525.2.a.t.1.1		2	15.2	even	4
6960.2.a.bw.1.2		2	20.3	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:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2175.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.79129i q^{2} -1.00000i q^{3} -5.79129 q^{4} -2.79129 q^{6} +1.00000i q^{7} +10.5826i q^{8} -1.00000 q^{9} +5.00000 q^{11} +5.79129i q^{12} -4.58258i q^{13} +2.79129 q^{14} +17.9564 q^{16} -3.00000i q^{17} +2.79129i q^{18} +5.58258 q^{19} +1.00000 q^{21} -13.9564i q^{22} +4.00000i q^{23} +10.5826 q^{24} -12.7913 q^{26} +1.00000i q^{27} -5.79129i q^{28} -1.00000 q^{29} +4.00000 q^{31} -28.9564i q^{32} -5.00000i q^{33} -8.37386 q^{34} +5.79129 q^{36} -4.00000i q^{37} -15.5826i q^{38} -4.58258 q^{39} +9.16515 q^{41} -2.79129i q^{42} +0.417424i q^{43} -28.9564 q^{44} +11.1652 q^{46} +1.41742i q^{47} -17.9564i q^{48} +6.00000 q^{49} -3.00000 q^{51} +26.5390i q^{52} -9.58258i q^{53} +2.79129 q^{54} -10.5826 q^{56} -5.58258i q^{57} +2.79129i q^{58} -1.58258 q^{59} -14.7477 q^{61} -11.1652i q^{62} -1.00000i q^{63} -44.9129 q^{64} -13.9564 q^{66} +14.1652i q^{67} +17.3739i q^{68} +4.00000 q^{69} -0.417424 q^{71} -10.5826i q^{72} -4.00000i q^{73} -11.1652 q^{74} -32.3303 q^{76} +5.00000i q^{77} +12.7913i q^{78} +1.58258 q^{79} +1.00000 q^{81} -25.5826i q^{82} +2.41742i q^{83} -5.79129 q^{84} +1.16515 q^{86} +1.00000i q^{87} +52.9129i q^{88} -10.5826 q^{89} +4.58258 q^{91} -23.1652i q^{92} -4.00000i q^{93} +3.95644 q^{94} -28.9564 q^{96} +2.41742i q^{97} -16.7477i q^{98} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 14 q^{4} - 2 q^{6} - 4 q^{9} + 20 q^{11} + 2 q^{14} + 26 q^{16} + 4 q^{19} + 4 q^{21} + 24 q^{24} - 42 q^{26} - 4 q^{29} + 16 q^{31} - 6 q^{34} + 14 q^{36} - 70 q^{44} + 8 q^{46} + 24 q^{49} - 12 q^{51}+ \cdots - 20 q^{99}+O(q^{100}) \) 4 * q - 14 * q^4 - 2 * q^6 - 4 * q^9 + 20 * q^11 + 2 * q^14 + 26 * q^16 + 4 * q^19 + 4 * q^21 + 24 * q^24 - 42 * q^26 - 4 * q^29 + 16 * q^31 - 6 * q^34 + 14 * q^36 - 70 * q^44 + 8 * q^46 + 24 * q^49 - 12 * q^51 + 2 * q^54 - 24 * q^56 + 12 * q^59 - 4 * q^61 - 88 * q^64 - 10 * q^66 + 16 * q^69 - 20 * q^71 - 8 * q^74 - 56 * q^76 - 12 * q^79 + 4 * q^81 - 14 * q^84 - 32 * q^86 - 24 * q^89 - 30 * q^94 - 70 * q^96 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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