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Properties

Label 875.2.bb.c.782.6
Level $875$
Weight $2$
Character 875.782
Analytic conductor $6.987$
Analytic rank $0$
Dimension $288$
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] = [875,2,Mod(82,875)] 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("875.82"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(875, base_ring=CyclotomicField(60)) chi = DirichletCharacter(H, H._module([27, 50])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 875 = 5^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 875.bb (of order \(60\), degree \(16\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [288,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.98691017686\)
Analytic rank: \(0\)
Dimension: \(288\)
Relative dimension: \(18\) over \(\Q(\zeta_{60})\)
Twist minimal: no (minimal twist has level 175)
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label 782.6
Character \(\chi\) \(=\) 875.782
Dual form 875.2.bb.c.668.6

$q$-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+(-0.527558 + 0.651479i) q^{2} +(-1.70042 - 1.10427i) q^{3} +(0.269715 + 1.26891i) q^{4} +(1.61648 - 0.525226i) q^{6} +(0.522392 + 2.59367i) q^{7} +(-2.46282 - 1.25487i) q^{8} +(0.451824 + 1.01481i) q^{9} +(2.43178 + 1.08270i) q^{11} +(0.942586 - 2.45552i) q^{12} +(0.131385 + 0.829534i) q^{13} +(-1.96531 - 1.02798i) q^{14} +(-0.253410 + 0.112825i) q^{16} +(0.0777694 + 1.48393i) q^{17} +(-0.899494 - 0.241019i) q^{18} +(-2.32299 - 0.493768i) q^{19} +(1.97582 - 4.98719i) q^{21} +(-1.98826 + 1.01307i) q^{22} +(2.67656 + 2.16743i) q^{23} +(2.80212 + 4.85342i) q^{24} +(-0.609738 - 0.352032i) q^{26} +(-0.599190 + 3.78314i) q^{27} +(-3.15023 + 1.36242i) q^{28} +(4.32645 + 1.40575i) q^{29} +(-7.98586 - 7.19050i) q^{31} +(1.49098 - 5.56441i) q^{32} +(-2.93947 - 4.52638i) q^{33} +(-1.00778 - 0.732192i) q^{34} +(-1.16584 + 0.847034i) q^{36} +(0.0664385 + 0.0255033i) q^{37} +(1.54719 - 1.25289i) q^{38} +(0.692618 - 1.55565i) q^{39} +(-3.62586 + 4.99057i) q^{41} +(2.20670 + 3.91824i) q^{42} +(2.31853 - 2.31853i) q^{43} +(-0.717958 + 3.37773i) q^{44} +(-2.82408 + 0.600276i) q^{46} +(-9.98369 - 0.523223i) q^{47} +(0.555493 + 0.0879815i) q^{48} +(-6.45421 + 2.70982i) q^{49} +(1.50641 - 2.60919i) q^{51} +(-1.01717 + 0.390454i) q^{52} +(-2.91776 + 4.49296i) q^{53} +(-2.14853 - 2.38618i) q^{54} +(1.96815 - 7.04325i) q^{56} +(3.40482 + 3.40482i) q^{57} +(-3.19827 + 2.07698i) q^{58} +(-1.42418 + 13.5502i) q^{59} +(-9.20856 + 0.967858i) q^{61} +(8.89747 - 1.40922i) q^{62} +(-2.39606 + 1.70201i) q^{63} +(2.51243 + 3.45806i) q^{64} +(4.49959 + 0.472925i) q^{66} +(-13.7112 + 0.718572i) q^{67} +(-1.86199 + 0.498920i) q^{68} +(-2.15786 - 6.64120i) q^{69} +(-2.20429 + 6.78410i) q^{71} +(0.160697 - 3.06628i) q^{72} +(-4.07288 - 10.6102i) q^{73} +(-0.0516650 + 0.0298288i) q^{74} -3.08085i q^{76} +(-1.53782 + 6.87282i) q^{77} +(0.648075 + 1.27192i) q^{78} +(-1.82675 + 1.64481i) q^{79} +(7.42639 - 8.24784i) q^{81} +(-1.33840 - 4.99499i) q^{82} +(4.12886 - 8.10334i) q^{83} +(6.86120 + 1.16201i) q^{84} +(0.287316 + 2.73363i) q^{86} +(-5.80447 - 7.16792i) q^{87} +(-4.63038 - 5.71805i) q^{88} +(0.0211592 + 0.201316i) q^{89} +(-2.08290 + 0.774112i) q^{91} +(-2.02837 + 3.98090i) q^{92} +(5.63911 + 21.0454i) q^{93} +(5.60784 - 6.22814i) q^{94} +(-8.67990 + 7.81542i) q^{96} +(0.626312 + 1.22921i) q^{97} +(1.63958 - 5.63437i) q^{98} +2.95699i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 288 q + 8 q^{2} + 24 q^{3} - 10 q^{4} + 10 q^{7} + 36 q^{8} - 10 q^{9} - 6 q^{11} + 36 q^{12} - 20 q^{14} - 30 q^{16} + 42 q^{17} + 14 q^{18} - 30 q^{19} - 12 q^{21} - 32 q^{22} + 40 q^{23} - 48 q^{26}+ \cdots - 222 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/875\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(626\)
\(\chi(n)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.527558 + 0.651479i −0.373040 + 0.460666i −0.928748 − 0.370712i \(-0.879114\pi\)
0.555708 + 0.831377i \(0.312447\pi\)
\(3\) −1.70042 − 1.10427i −0.981740 − 0.637550i −0.0492660 − 0.998786i \(-0.515688\pi\)
−0.932474 + 0.361236i \(0.882355\pi\)
\(4\) 0.269715 + 1.26891i 0.134857 + 0.634455i
\(5\) 0 0
\(6\) 1.61648 − 0.525226i 0.659925 − 0.214423i
\(7\) 0.522392 + 2.59367i 0.197446 + 0.980314i
\(8\) −2.46282 − 1.25487i −0.870737 − 0.443663i
\(9\) 0.451824 + 1.01481i 0.150608 + 0.338271i
\(10\) 0 0
\(11\) 2.43178 + 1.08270i 0.733209 + 0.326446i 0.739154 − 0.673537i \(-0.235226\pi\)
−0.00594471 + 0.999982i \(0.501892\pi\)
\(12\) 0.942586 − 2.45552i 0.272101 − 0.708848i
\(13\) 0.131385 + 0.829534i 0.0364397 + 0.230071i 0.999186 − 0.0403341i \(-0.0128422\pi\)
−0.962747 + 0.270406i \(0.912842\pi\)
\(14\) −1.96531 − 1.02798i −0.525252 − 0.274740i
\(15\) 0 0
\(16\) −0.253410 + 0.112825i −0.0633524 + 0.0282063i
\(17\) 0.0777694 + 1.48393i 0.0188618 + 0.359905i 0.991713 + 0.128473i \(0.0410077\pi\)
−0.972851 + 0.231432i \(0.925659\pi\)
\(18\) −0.899494 − 0.241019i −0.212013 − 0.0568086i
\(19\) −2.32299 − 0.493768i −0.532931 − 0.113278i −0.0664175 − 0.997792i \(-0.521157\pi\)
−0.466514 + 0.884514i \(0.654490\pi\)
\(20\) 0 0
\(21\) 1.97582 − 4.98719i 0.431159 − 1.08830i
\(22\) −1.98826 + 1.01307i −0.423898 + 0.215987i
\(23\) 2.67656 + 2.16743i 0.558101 + 0.451941i 0.866402 − 0.499348i \(-0.166427\pi\)
−0.308301 + 0.951289i \(0.599760\pi\)
\(24\) 2.80212 + 4.85342i 0.571981 + 0.990699i
\(25\) 0 0
\(26\) −0.609738 − 0.352032i −0.119579 − 0.0690392i
\(27\) −0.599190 + 3.78314i −0.115314 + 0.728065i
\(28\) −3.15023 + 1.36242i −0.595338 + 0.257473i
\(29\) 4.32645 + 1.40575i 0.803401 + 0.261041i 0.681800 − 0.731538i \(-0.261197\pi\)
0.121601 + 0.992579i \(0.461197\pi\)
\(30\) 0 0
\(31\) −7.98586 − 7.19050i −1.43430 − 1.29145i −0.892575 − 0.450899i \(-0.851103\pi\)
−0.541729 − 0.840553i \(-0.682230\pi\)
\(32\) 1.49098 − 5.56441i 0.263570 − 0.983658i
\(33\) −2.93947 − 4.52638i −0.511696 − 0.787942i
\(34\) −1.00778 − 0.732192i −0.172832 − 0.125570i
\(35\) 0 0
\(36\) −1.16584 + 0.847034i −0.194307 + 0.141172i
\(37\) 0.0664385 + 0.0255033i 0.0109224 + 0.00419272i 0.363823 − 0.931468i \(-0.381471\pi\)
−0.352901 + 0.935661i \(0.614805\pi\)
\(38\) 1.54719 − 1.25289i 0.250988 − 0.203246i
\(39\) 0.692618 − 1.55565i 0.110908 − 0.249103i
\(40\) 0 0
\(41\) −3.62586 + 4.99057i −0.566264 + 0.779396i −0.992106 − 0.125402i \(-0.959978\pi\)
0.425842 + 0.904798i \(0.359978\pi\)
\(42\) 2.20670 + 3.91824i 0.340501 + 0.604597i
\(43\) 2.31853 − 2.31853i 0.353572 − 0.353572i −0.507865 − 0.861437i \(-0.669565\pi\)
0.861437 + 0.507865i \(0.169565\pi\)
\(44\) −0.717958 + 3.37773i −0.108236 + 0.509212i
\(45\) 0 0
\(46\) −2.82408 + 0.600276i −0.416388 + 0.0885059i
\(47\) −9.98369 − 0.523223i −1.45627 − 0.0763199i −0.692319 − 0.721592i \(-0.743411\pi\)
−0.763953 + 0.645272i \(0.776744\pi\)
\(48\) 0.555493 + 0.0879815i 0.0801785 + 0.0126990i
\(49\) −6.45421 + 2.70982i −0.922031 + 0.387117i
\(50\) 0 0
\(51\) 1.50641 − 2.60919i 0.210940 − 0.365359i
\(52\) −1.01717 + 0.390454i −0.141056 + 0.0541462i
\(53\) −2.91776 + 4.49296i −0.400785 + 0.617155i −0.980231 − 0.197855i \(-0.936602\pi\)
0.579446 + 0.815011i \(0.303269\pi\)
\(54\) −2.14853 − 2.38618i −0.292378 − 0.324718i
\(55\) 0 0
\(56\) 1.96815 − 7.04325i 0.263005 − 0.941195i
\(57\) 3.40482 + 3.40482i 0.450980 + 0.450980i
\(58\) −3.19827 + 2.07698i −0.419953 + 0.272721i
\(59\) −1.42418 + 13.5502i −0.185412 + 1.76408i 0.366728 + 0.930328i \(0.380478\pi\)
−0.552140 + 0.833751i \(0.686189\pi\)
\(60\) 0 0
\(61\) −9.20856 + 0.967858i −1.17903 + 0.123922i −0.673702 − 0.739003i \(-0.735297\pi\)
−0.505333 + 0.862925i \(0.668630\pi\)
\(62\) 8.89747 − 1.40922i 1.12998 − 0.178971i
\(63\) −2.39606 + 1.70201i −0.301875 + 0.214433i
\(64\) 2.51243 + 3.45806i 0.314054 + 0.432258i
\(65\) 0 0
\(66\) 4.49959 + 0.472925i 0.553861 + 0.0582131i
\(67\) −13.7112 + 0.718572i −1.67509 + 0.0877876i −0.866046 − 0.499964i \(-0.833346\pi\)
−0.809041 + 0.587752i \(0.800013\pi\)
\(68\) −1.86199 + 0.498920i −0.225800 + 0.0605029i
\(69\) −2.15786 − 6.64120i −0.259775 − 0.799506i
\(70\) 0 0
\(71\) −2.20429 + 6.78410i −0.261601 + 0.805124i 0.730856 + 0.682531i \(0.239121\pi\)
−0.992457 + 0.122593i \(0.960879\pi\)
\(72\) 0.160697 − 3.06628i 0.0189383 − 0.361364i
\(73\) −4.07288 − 10.6102i −0.476695 − 1.24183i −0.935510 − 0.353301i \(-0.885059\pi\)
0.458815 − 0.888532i \(-0.348274\pi\)
\(74\) −0.0516650 + 0.0298288i −0.00600594 + 0.00346753i
\(75\) 0 0
\(76\) − 3.08085i − 0.353397i
\(77\) −1.53782 + 6.87282i −0.175250 + 0.783230i
\(78\) 0.648075 + 1.27192i 0.0733801 + 0.144016i
\(79\) −1.82675 + 1.64481i −0.205526 + 0.185056i −0.765465 − 0.643477i \(-0.777491\pi\)
0.559940 + 0.828533i \(0.310824\pi\)
\(80\) 0 0
\(81\) 7.42639 − 8.24784i 0.825154 − 0.916427i
\(82\) −1.33840 − 4.99499i −0.147802 − 0.551604i
\(83\) 4.12886 − 8.10334i 0.453201 − 0.889457i −0.545481 − 0.838123i \(-0.683653\pi\)
0.998682 − 0.0513335i \(-0.0163471\pi\)
\(84\) 6.86120 + 1.16201i 0.748619 + 0.126786i
\(85\) 0 0
\(86\) 0.287316 + 2.73363i 0.0309821 + 0.294775i
\(87\) −5.80447 − 7.16792i −0.622305 − 0.768482i
\(88\) −4.63038 − 5.71805i −0.493600 − 0.609546i
\(89\) 0.0211592 + 0.201316i 0.00224287 + 0.0213395i 0.995586 − 0.0938527i \(-0.0299183\pi\)
−0.993343 + 0.115192i \(0.963252\pi\)
\(90\) 0 0
\(91\) −2.08290 + 0.774112i −0.218347 + 0.0811490i
\(92\) −2.02837 + 3.98090i −0.211472 + 0.415038i
\(93\) 5.63911 + 21.0454i 0.584749 + 2.18231i
\(94\) 5.60784 − 6.22814i 0.578405 − 0.642384i
\(95\) 0 0
\(96\) −8.67990 + 7.81542i −0.885889 + 0.797658i
\(97\) 0.626312 + 1.22921i 0.0635923 + 0.124807i 0.920615 − 0.390472i \(-0.127688\pi\)
−0.857022 + 0.515279i \(0.827688\pi\)
\(98\) 1.63958 − 5.63437i 0.165622 − 0.569158i
\(99\) 2.95699i 0.297189i
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 875.2.bb.c.782.6 288
5.2 odd 4 875.2.bb.b.593.6 288
5.3 odd 4 875.2.bb.a.593.13 288
5.4 even 2 175.2.x.a.152.13 yes 288
7.3 odd 6 inner 875.2.bb.c.157.6 288
25.9 even 10 875.2.bb.b.82.13 288
25.12 odd 20 175.2.x.a.138.13 yes 288
25.13 odd 20 inner 875.2.bb.c.418.6 288
25.16 even 5 875.2.bb.a.82.6 288
35.3 even 12 875.2.bb.a.843.6 288
35.17 even 12 875.2.bb.b.843.13 288
35.24 odd 6 175.2.x.a.52.13 yes 288
175.38 even 60 inner 875.2.bb.c.668.6 288
175.59 odd 30 875.2.bb.b.332.6 288
175.66 odd 30 875.2.bb.a.332.13 288
175.87 even 60 175.2.x.a.38.13 ✓ 288
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.x.a.38.13 ✓ 288 175.87 even 60
175.2.x.a.52.13 yes 288 35.24 odd 6
175.2.x.a.138.13 yes 288 25.12 odd 20
175.2.x.a.152.13 yes 288 5.4 even 2
875.2.bb.a.82.6 288 25.16 even 5
875.2.bb.a.332.13 288 175.66 odd 30
875.2.bb.a.593.13 288 5.3 odd 4
875.2.bb.a.843.6 288 35.3 even 12
875.2.bb.b.82.13 288 25.9 even 10
875.2.bb.b.332.6 288 175.59 odd 30
875.2.bb.b.593.6 288 5.2 odd 4
875.2.bb.b.843.13 288 35.17 even 12
875.2.bb.c.157.6 288 7.3 odd 6 inner
875.2.bb.c.418.6 288 25.13 odd 20 inner
875.2.bb.c.668.6 288 175.38 even 60 inner
875.2.bb.c.782.6 288 1.1 even 1 trivial