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Properties

Label 615.2.r.a.32.16
Level $615$
Weight $2$
Character 615.32
Analytic conductor $4.911$
Analytic rank $0$
Dimension $160$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [615,2,Mod(32,615)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("615.32"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(615, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 615 = 3 \cdot 5 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 615.r (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.91079972431\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(80\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 32.16
Character \(\chi\) \(=\) 615.32
Dual form 615.2.r.a.173.16

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.29733 - 1.29733i) q^{2} +(1.71357 + 0.252374i) q^{3} +1.36614i q^{4} +(-1.49068 - 1.66670i) q^{5} +(-1.89565 - 2.55048i) q^{6} +3.24596 q^{7} +(-0.822328 + 0.822328i) q^{8} +(2.87261 + 0.864918i) q^{9} +(-0.228361 + 4.09616i) q^{10} +(-0.219091 - 0.219091i) q^{11} +(-0.344778 + 2.34097i) q^{12} -2.61160i q^{13} +(-4.21108 - 4.21108i) q^{14} +(-2.13374 - 3.23221i) q^{15} +4.86594 q^{16} +4.06277 q^{17} +(-2.60465 - 4.84882i) q^{18} +(1.48229 - 1.48229i) q^{19} +(2.27694 - 2.03647i) q^{20} +(5.56216 + 0.819194i) q^{21} +0.568468i q^{22} +(-4.72777 - 4.72777i) q^{23} +(-1.61665 + 1.20158i) q^{24} +(-0.555772 + 4.96902i) q^{25} +(-3.38812 + 3.38812i) q^{26} +(4.70413 + 2.20707i) q^{27} +4.43443i q^{28} +(3.90816 - 3.90816i) q^{29} +(-1.42508 + 6.96141i) q^{30} -5.17858 q^{31} +(-4.66808 - 4.66808i) q^{32} +(-0.320134 - 0.430720i) q^{33} +(-5.27076 - 5.27076i) q^{34} +(-4.83867 - 5.41003i) q^{35} +(-1.18160 + 3.92439i) q^{36} +(-1.54679 - 1.54679i) q^{37} -3.84606 q^{38} +(0.659101 - 4.47516i) q^{39} +(2.59640 + 0.144749i) q^{40} +(-5.93734 + 2.39750i) q^{41} +(-6.15320 - 8.27873i) q^{42} +(0.285878 + 0.285878i) q^{43} +(0.299309 - 0.299309i) q^{44} +(-2.84058 - 6.07710i) q^{45} +12.2670i q^{46} +13.3231 q^{47} +(8.33811 + 1.22804i) q^{48} +3.53623 q^{49} +(7.16748 - 5.72544i) q^{50} +(6.96183 + 1.02534i) q^{51} +3.56781 q^{52} +3.04210 q^{53} +(-3.23952 - 8.96612i) q^{54} +(-0.0385652 + 0.691753i) q^{55} +(-2.66924 + 2.66924i) q^{56} +(2.91410 - 2.16592i) q^{57} -10.1404 q^{58} -7.10552 q^{59} +(4.41564 - 2.91498i) q^{60} -9.98094i q^{61} +(6.71834 + 6.71834i) q^{62} +(9.32438 + 2.80749i) q^{63} +2.38022i q^{64} +(-4.35276 + 3.89306i) q^{65} +(-0.143466 + 0.974108i) q^{66} +5.53467i q^{67} +5.55031i q^{68} +(-6.90818 - 9.29451i) q^{69} +(-0.741249 + 13.2960i) q^{70} +(-2.07445 - 2.07445i) q^{71} +(-3.07348 + 1.65099i) q^{72} +(-5.15886 - 5.15886i) q^{73} +4.01339i q^{74} +(-2.20640 + 8.37447i) q^{75} +(2.02502 + 2.02502i) q^{76} +(-0.711161 - 0.711161i) q^{77} +(-6.66083 + 4.95069i) q^{78} +(-2.44707 - 2.44707i) q^{79} +(-7.25354 - 8.11006i) q^{80} +(7.50383 + 4.96915i) q^{81} +(10.8130 + 4.59235i) q^{82} +(4.65868 + 4.65868i) q^{83} +(-1.11913 + 7.59868i) q^{84} +(-6.05628 - 6.77142i) q^{85} -0.741757i q^{86} +(7.68321 - 5.71058i) q^{87} +0.360330 q^{88} +(-11.9607 + 11.9607i) q^{89} +(-4.19884 + 11.5692i) q^{90} -8.47715i q^{91} +(6.45879 - 6.45879i) q^{92} +(-8.87384 - 1.30694i) q^{93} +(-17.2845 - 17.2845i) q^{94} +(-4.68016 - 0.260919i) q^{95} +(-6.82097 - 9.17717i) q^{96} -2.87473i q^{97} +(-4.58766 - 4.58766i) q^{98} +(-0.439869 - 0.818861i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 160 q - 8 q^{6} - 8 q^{7} - 24 q^{10} - 14 q^{15} - 152 q^{16} + 4 q^{18} + 8 q^{24} - 8 q^{25} + 10 q^{30} - 16 q^{31} - 12 q^{33} - 16 q^{34} - 8 q^{37} + 24 q^{39} + 24 q^{40} + 12 q^{42} + 32 q^{43}+ \cdots + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/615\mathbb{Z}\right)^\times\).

\(n\) \(206\) \(211\) \(247\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{1}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.29733 − 1.29733i −0.917352 − 0.917352i 0.0794841 − 0.996836i \(-0.474673\pi\)
−0.996836 + 0.0794841i \(0.974673\pi\)
\(3\) 1.71357 + 0.252374i 0.989328 + 0.145708i
\(4\) 1.36614i 0.683069i
\(5\) −1.49068 − 1.66670i −0.666650 − 0.745370i
\(6\) −1.89565 − 2.55048i −0.773896 − 1.04123i
\(7\) 3.24596 1.22686 0.613428 − 0.789751i \(-0.289790\pi\)
0.613428 + 0.789751i \(0.289790\pi\)
\(8\) −0.822328 + 0.822328i −0.290737 + 0.290737i
\(9\) 2.87261 + 0.864918i 0.957538 + 0.288306i
\(10\) −0.228361 + 4.09616i −0.0722140 + 1.29532i
\(11\) −0.219091 − 0.219091i −0.0660585 − 0.0660585i 0.673306 − 0.739364i \(-0.264874\pi\)
−0.739364 + 0.673306i \(0.764874\pi\)
\(12\) −0.344778 + 2.34097i −0.0995287 + 0.675779i
\(13\) − 2.61160i − 0.724329i −0.932114 − 0.362164i \(-0.882038\pi\)
0.932114 − 0.362164i \(-0.117962\pi\)
\(14\) −4.21108 − 4.21108i −1.12546 − 1.12546i
\(15\) −2.13374 − 3.23221i −0.550929 − 0.834552i
\(16\) 4.86594 1.21649
\(17\) 4.06277 0.985367 0.492683 − 0.870209i \(-0.336016\pi\)
0.492683 + 0.870209i \(0.336016\pi\)
\(18\) −2.60465 − 4.84882i −0.613922 − 1.14288i
\(19\) 1.48229 − 1.48229i 0.340062 − 0.340062i −0.516329 − 0.856390i \(-0.672702\pi\)
0.856390 + 0.516329i \(0.172702\pi\)
\(20\) 2.27694 − 2.03647i 0.509140 − 0.455369i
\(21\) 5.56216 + 0.819194i 1.21376 + 0.178763i
\(22\) 0.568468i 0.121198i
\(23\) −4.72777 − 4.72777i −0.985808 − 0.985808i 0.0140923 − 0.999901i \(-0.495514\pi\)
−0.999901 + 0.0140923i \(0.995514\pi\)
\(24\) −1.61665 + 1.20158i −0.329997 + 0.245271i
\(25\) −0.555772 + 4.96902i −0.111154 + 0.993803i
\(26\) −3.38812 + 3.38812i −0.664465 + 0.664465i
\(27\) 4.70413 + 2.20707i 0.905311 + 0.424750i
\(28\) 4.43443i 0.838028i
\(29\) 3.90816 − 3.90816i 0.725728 − 0.725728i −0.244038 − 0.969766i \(-0.578472\pi\)
0.969766 + 0.244038i \(0.0784721\pi\)
\(30\) −1.42508 + 6.96141i −0.260182 + 1.27097i
\(31\) −5.17858 −0.930100 −0.465050 − 0.885284i \(-0.653964\pi\)
−0.465050 + 0.885284i \(0.653964\pi\)
\(32\) −4.66808 − 4.66808i −0.825209 − 0.825209i
\(33\) −0.320134 − 0.430720i −0.0557283 − 0.0749788i
\(34\) −5.27076 − 5.27076i −0.903928 − 0.903928i
\(35\) −4.83867 − 5.41003i −0.817884 − 0.914462i
\(36\) −1.18160 + 3.92439i −0.196933 + 0.654065i
\(37\) −1.54679 − 1.54679i −0.254290 − 0.254290i 0.568437 − 0.822727i \(-0.307548\pi\)
−0.822727 + 0.568437i \(0.807548\pi\)
\(38\) −3.84606 −0.623913
\(39\) 0.659101 − 4.47516i 0.105541 − 0.716599i
\(40\) 2.59640 + 0.144749i 0.410527 + 0.0228868i
\(41\) −5.93734 + 2.39750i −0.927257 + 0.374426i
\(42\) −6.15320 − 8.27873i −0.949459 − 1.27744i
\(43\) 0.285878 + 0.285878i 0.0435960 + 0.0435960i 0.728569 − 0.684973i \(-0.240186\pi\)
−0.684973 + 0.728569i \(0.740186\pi\)
\(44\) 0.299309 − 0.299309i 0.0451226 − 0.0451226i
\(45\) −2.84058 − 6.07710i −0.423449 − 0.905920i
\(46\) 12.2670i 1.80867i
\(47\) 13.3231 1.94338 0.971688 − 0.236268i \(-0.0759243\pi\)
0.971688 + 0.236268i \(0.0759243\pi\)
\(48\) 8.33811 + 1.22804i 1.20350 + 0.177252i
\(49\) 3.53623 0.505175
\(50\) 7.16748 − 5.72544i 1.01364 − 0.809700i
\(51\) 6.96183 + 1.02534i 0.974851 + 0.143576i
\(52\) 3.56781 0.494767
\(53\) 3.04210 0.417865 0.208932 − 0.977930i \(-0.433001\pi\)
0.208932 + 0.977930i \(0.433001\pi\)
\(54\) −3.23952 − 8.96612i −0.440843 − 1.22013i
\(55\) −0.0385652 + 0.691753i −0.00520013 + 0.0932760i
\(56\) −2.66924 + 2.66924i −0.356692 + 0.356692i
\(57\) 2.91410 − 2.16592i 0.385982 − 0.286883i
\(58\) −10.1404 −1.33150
\(59\) −7.10552 −0.925060 −0.462530 − 0.886604i \(-0.653058\pi\)
−0.462530 + 0.886604i \(0.653058\pi\)
\(60\) 4.41564 − 2.91498i 0.570057 − 0.376323i
\(61\) − 9.98094i − 1.27793i −0.769237 − 0.638964i \(-0.779363\pi\)
0.769237 − 0.638964i \(-0.220637\pi\)
\(62\) 6.71834 + 6.71834i 0.853229 + 0.853229i
\(63\) 9.32438 + 2.80749i 1.17476 + 0.353710i
\(64\) 2.38022i 0.297528i
\(65\) −4.35276 + 3.89306i −0.539893 + 0.482874i
\(66\) −0.143466 + 0.974108i −0.0176595 + 0.119904i
\(67\) 5.53467i 0.676167i 0.941116 + 0.338084i \(0.109779\pi\)
−0.941116 + 0.338084i \(0.890221\pi\)
\(68\) 5.55031i 0.673074i
\(69\) −6.90818 − 9.29451i −0.831647 − 1.11893i
\(70\) −0.741249 + 13.2960i −0.0885962 + 1.58917i
\(71\) −2.07445 − 2.07445i −0.246192 − 0.246192i 0.573214 − 0.819406i \(-0.305697\pi\)
−0.819406 + 0.573214i \(0.805697\pi\)
\(72\) −3.07348 + 1.65099i −0.362213 + 0.194571i
\(73\) −5.15886 − 5.15886i −0.603799 − 0.603799i 0.337520 − 0.941318i \(-0.390412\pi\)
−0.941318 + 0.337520i \(0.890412\pi\)
\(74\) 4.01339i 0.466548i
\(75\) −2.20640 + 8.37447i −0.254773 + 0.967001i
\(76\) 2.02502 + 2.02502i 0.232286 + 0.232286i
\(77\) −0.711161 − 0.711161i −0.0810443 − 0.0810443i
\(78\) −6.66083 + 4.95069i −0.754191 + 0.560555i
\(79\) −2.44707 − 2.44707i −0.275317 − 0.275317i 0.555919 − 0.831236i \(-0.312366\pi\)
−0.831236 + 0.555919i \(0.812366\pi\)
\(80\) −7.25354 − 8.11006i −0.810971 − 0.906732i
\(81\) 7.50383 + 4.96915i 0.833759 + 0.552128i
\(82\) 10.8130 + 4.59235i 1.19410 + 0.507140i
\(83\) 4.65868 + 4.65868i 0.511357 + 0.511357i 0.914942 − 0.403585i \(-0.132236\pi\)
−0.403585 + 0.914942i \(0.632236\pi\)
\(84\) −1.11913 + 7.59868i −0.122107 + 0.829084i
\(85\) −6.05628 − 6.77142i −0.656895 − 0.734463i
\(86\) − 0.741757i − 0.0799857i
\(87\) 7.68321 − 5.71058i 0.823727 − 0.612238i
\(88\) 0.360330 0.0384113
\(89\) −11.9607 + 11.9607i −1.26783 + 1.26783i −0.320625 + 0.947206i \(0.603893\pi\)
−0.947206 + 0.320625i \(0.896107\pi\)
\(90\) −4.19884 + 11.5692i −0.442596 + 1.21950i
\(91\) − 8.47715i − 0.888647i
\(92\) 6.45879 − 6.45879i 0.673376 − 0.673376i
\(93\) −8.87384 − 1.30694i −0.920174 − 0.135523i
\(94\) −17.2845 − 17.2845i −1.78276 − 1.78276i
\(95\) −4.68016 − 0.260919i −0.480174 − 0.0267697i
\(96\) −6.82097 − 9.17717i −0.696162 − 0.936641i
\(97\) − 2.87473i − 0.291884i −0.989293 − 0.145942i \(-0.953379\pi\)
0.989293 − 0.145942i \(-0.0466213\pi\)
\(98\) −4.58766 − 4.58766i −0.463423 − 0.463423i
\(99\) −0.439869 − 0.818861i −0.0442085 − 0.0822986i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 615.2.r.a.32.16 yes 160
3.2 odd 2 inner 615.2.r.a.32.65 yes 160
5.3 odd 4 615.2.k.a.278.16 ✓ 160
15.8 even 4 615.2.k.a.278.65 yes 160
41.9 even 4 615.2.k.a.542.65 yes 160
123.50 odd 4 615.2.k.a.542.16 yes 160
205.173 odd 4 inner 615.2.r.a.173.65 yes 160
615.173 even 4 inner 615.2.r.a.173.16 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
615.2.k.a.278.16 ✓ 160 5.3 odd 4
615.2.k.a.278.65 yes 160 15.8 even 4
615.2.k.a.542.16 yes 160 123.50 odd 4
615.2.k.a.542.65 yes 160 41.9 even 4
615.2.r.a.32.16 yes 160 1.1 even 1 trivial
615.2.r.a.32.65 yes 160 3.2 odd 2 inner
615.2.r.a.173.16 yes 160 615.173 even 4 inner
615.2.r.a.173.65 yes 160 205.173 odd 4 inner