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Properties

Label 875.2.bb.b.82.13
Level $875$
Weight $2$
Character 875.82
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 = [288,2] 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 82.13
Character \(\chi\) \(=\) 875.82
Dual form 875.2.bb.b.843.13

$q$-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+(0.782618 + 0.300419i) q^{2} +(-2.02474 + 0.106112i) q^{3} +(-0.964050 - 0.868035i) q^{4} +(-1.61648 - 0.525226i) q^{6} +(-0.522392 - 2.59367i) q^{7} +(-1.25487 - 2.46282i) q^{8} +(1.10477 - 0.116116i) q^{9} +(-0.278246 + 2.64733i) q^{11} +(2.04406 + 1.65525i) q^{12} +(-0.829534 - 0.131385i) q^{13} +(0.370353 - 2.18679i) q^{14} +(0.0289953 + 0.275872i) q^{16} +(-0.809314 + 1.24623i) q^{17} +(0.899494 + 0.241019i) q^{18} +(1.58911 + 1.76489i) q^{19} +(1.33293 + 5.19608i) q^{21} +(-1.01307 + 1.98826i) q^{22} +(1.23425 - 3.21533i) q^{23} +(2.80212 + 4.85342i) q^{24} +(-0.609738 - 0.352032i) q^{26} +(3.78314 - 0.599190i) q^{27} +(-1.74778 + 2.95388i) q^{28} +(-4.32645 + 1.40575i) q^{29} +(2.23423 + 10.5112i) q^{31} +(-1.49098 + 5.56441i) q^{32} +(0.282462 - 5.38970i) q^{33} +(-1.00778 + 0.732192i) q^{34} +(-1.16584 - 0.847034i) q^{36} +(-0.0447857 + 0.0553058i) q^{37} +(0.713462 + 1.85863i) q^{38} +(1.69354 + 0.177998i) q^{39} +(3.62586 + 4.99057i) q^{41} +(-0.517826 + 4.46698i) q^{42} +(-2.31853 + 2.31853i) q^{43} +(2.56622 - 2.31063i) q^{44} +(1.93189 - 2.14559i) q^{46} +(-8.38452 + 5.44497i) q^{47} +(-0.0879815 - 0.555493i) q^{48} +(-6.45421 + 2.70982i) q^{49} +(1.50641 - 2.60919i) q^{51} +(0.685666 + 0.846727i) q^{52} +(0.280376 + 5.34990i) q^{53} +(3.14076 + 0.667589i) q^{54} +(-5.73219 + 4.54126i) q^{56} +(-3.40482 - 3.40482i) q^{57} +(-3.80827 - 0.199583i) q^{58} +(12.4469 + 5.54170i) q^{59} +(-3.76609 - 8.45877i) q^{61} +(-1.40922 + 8.89747i) q^{62} +(-0.878286 - 2.80474i) q^{63} +(-2.51243 + 3.45806i) q^{64} +(1.84023 - 4.13322i) q^{66} +(-11.5149 - 7.47789i) q^{67} +(1.86199 - 0.498920i) q^{68} +(-2.15786 + 6.64120i) q^{69} +(-2.20429 - 6.78410i) q^{71} +(-1.67231 - 2.57513i) q^{72} +(-8.83233 + 7.15228i) q^{73} +(-0.0516650 + 0.0298288i) q^{74} -3.08085i q^{76} +(7.01165 - 0.661268i) q^{77} +(1.27192 + 0.648075i) q^{78} +(0.511075 - 2.40442i) q^{79} +(-10.8560 + 2.30752i) q^{81} +(1.33840 + 4.99499i) q^{82} +(8.10334 - 4.12886i) q^{83} +(3.22537 - 6.16631i) q^{84} +(-2.51105 + 1.11799i) q^{86} +(8.61078 - 3.30537i) q^{87} +(6.86905 - 2.63678i) q^{88} +(-0.184924 + 0.0823337i) q^{89} +(0.0925721 + 2.22017i) q^{91} +(-3.98090 + 2.02837i) q^{92} +(-5.63911 - 21.0454i) q^{93} +(-8.19765 + 1.74246i) q^{94} +(2.42840 - 11.4247i) q^{96} +(1.22921 + 0.626312i) q^{97} +(-5.86527 + 0.181786i) q^{98} +2.95699i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 288 q + 2 q^{2} + 6 q^{3} + 10 q^{4} - 10 q^{7} + 64 q^{8} + 10 q^{9} - 6 q^{11} - 6 q^{12} + 20 q^{14} - 30 q^{16} - 12 q^{17} - 14 q^{18} + 30 q^{19} - 12 q^{21} - 8 q^{22} + 30 q^{23} - 48 q^{26} - 58 q^{28}+ \cdots + 62 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(626\)
\(\chi(n)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.782618 + 0.300419i 0.553395 + 0.212428i 0.618963 − 0.785420i \(-0.287553\pi\)
−0.0655689 + 0.997848i \(0.520886\pi\)
\(3\) −2.02474 + 0.106112i −1.16899 + 0.0612640i −0.626929 − 0.779077i \(-0.715688\pi\)
−0.542058 + 0.840341i \(0.682355\pi\)
\(4\) −0.964050 − 0.868035i −0.482025 − 0.434017i
\(5\) 0 0
\(6\) −1.61648 − 0.525226i −0.659925 − 0.214423i
\(7\) −0.522392 − 2.59367i −0.197446 − 0.980314i
\(8\) −1.25487 − 2.46282i −0.443663 − 0.870737i
\(9\) 1.10477 − 0.116116i 0.368256 − 0.0387052i
\(10\) 0 0
\(11\) −0.278246 + 2.64733i −0.0838943 + 0.798201i 0.868982 + 0.494843i \(0.164774\pi\)
−0.952877 + 0.303358i \(0.901892\pi\)
\(12\) 2.04406 + 1.65525i 0.590071 + 0.477830i
\(13\) −0.829534 − 0.131385i −0.230071 − 0.0364397i 0.0403341 − 0.999186i \(-0.487158\pi\)
−0.270406 + 0.962747i \(0.587158\pi\)
\(14\) 0.370353 − 2.18679i 0.0989811 − 0.584443i
\(15\) 0 0
\(16\) 0.0289953 + 0.275872i 0.00724882 + 0.0689679i
\(17\) −0.809314 + 1.24623i −0.196288 + 0.302256i −0.923085 − 0.384595i \(-0.874341\pi\)
0.726798 + 0.686851i \(0.241008\pi\)
\(18\) 0.899494 + 0.241019i 0.212013 + 0.0568086i
\(19\) 1.58911 + 1.76489i 0.364567 + 0.404893i 0.897322 − 0.441377i \(-0.145510\pi\)
−0.532754 + 0.846270i \(0.678843\pi\)
\(20\) 0 0
\(21\) 1.33293 + 5.19608i 0.290869 + 1.13388i
\(22\) −1.01307 + 1.98826i −0.215987 + 0.423898i
\(23\) 1.23425 − 3.21533i 0.257359 − 0.670443i −0.742641 − 0.669690i \(-0.766427\pi\)
1.00000 0.000753198i \(-0.000239751\pi\)
\(24\) 2.80212 + 4.85342i 0.571981 + 0.990699i
\(25\) 0 0
\(26\) −0.609738 − 0.352032i −0.119579 − 0.0690392i
\(27\) 3.78314 − 0.599190i 0.728065 − 0.115314i
\(28\) −1.74778 + 2.95388i −0.330299 + 0.558231i
\(29\) −4.32645 + 1.40575i −0.803401 + 0.261041i −0.681800 − 0.731538i \(-0.738803\pi\)
−0.121601 + 0.992579i \(0.538803\pi\)
\(30\) 0 0
\(31\) 2.23423 + 10.5112i 0.401279 + 1.88787i 0.457076 + 0.889428i \(0.348897\pi\)
−0.0557970 + 0.998442i \(0.517770\pi\)
\(32\) −1.49098 + 5.56441i −0.263570 + 0.983658i
\(33\) 0.282462 − 5.38970i 0.0491703 − 0.938226i
\(34\) −1.00778 + 0.732192i −0.172832 + 0.125570i
\(35\) 0 0
\(36\) −1.16584 − 0.847034i −0.194307 − 0.141172i
\(37\) −0.0447857 + 0.0553058i −0.00736273 + 0.00909221i −0.780814 − 0.624764i \(-0.785195\pi\)
0.773451 + 0.633856i \(0.218529\pi\)
\(38\) 0.713462 + 1.85863i 0.115739 + 0.301510i
\(39\) 1.69354 + 0.177998i 0.271183 + 0.0285025i
\(40\) 0 0
\(41\) 3.62586 + 4.99057i 0.566264 + 0.779396i 0.992106 − 0.125402i \(-0.0400220\pi\)
−0.425842 + 0.904798i \(0.640022\pi\)
\(42\) −0.517826 + 4.46698i −0.0799023 + 0.689271i
\(43\) −2.31853 + 2.31853i −0.353572 + 0.353572i −0.861437 − 0.507865i \(-0.830435\pi\)
0.507865 + 0.861437i \(0.330435\pi\)
\(44\) 2.56622 − 2.31063i 0.386872 − 0.348341i
\(45\) 0 0
\(46\) 1.93189 − 2.14559i 0.284842 − 0.316349i
\(47\) −8.38452 + 5.44497i −1.22301 + 0.794231i −0.984239 − 0.176846i \(-0.943411\pi\)
−0.238769 + 0.971076i \(0.576744\pi\)
\(48\) −0.0879815 − 0.555493i −0.0126990 − 0.0801785i
\(49\) −6.45421 + 2.70982i −0.922031 + 0.387117i
\(50\) 0 0
\(51\) 1.50641 − 2.60919i 0.210940 − 0.365359i
\(52\) 0.685666 + 0.846727i 0.0950847 + 0.117420i
\(53\) 0.280376 + 5.34990i 0.0385126 + 0.734865i 0.947833 + 0.318768i \(0.103269\pi\)
−0.909320 + 0.416097i \(0.863398\pi\)
\(54\) 3.14076 + 0.667589i 0.427403 + 0.0908473i
\(55\) 0 0
\(56\) −5.73219 + 4.54126i −0.765996 + 0.606852i
\(57\) −3.40482 − 3.40482i −0.450980 − 0.450980i
\(58\) −3.80827 − 0.199583i −0.500050 − 0.0262065i
\(59\) 12.4469 + 5.54170i 1.62044 + 0.721468i 0.998120 − 0.0612916i \(-0.0195220\pi\)
0.622324 + 0.782760i \(0.286189\pi\)
\(60\) 0 0
\(61\) −3.76609 − 8.45877i −0.482198 − 1.08303i −0.976846 − 0.213942i \(-0.931370\pi\)
0.494648 − 0.869093i \(-0.335297\pi\)
\(62\) −1.40922 + 8.89747i −0.178971 + 1.12998i
\(63\) −0.878286 − 2.80474i −0.110654 − 0.353364i
\(64\) −2.51243 + 3.45806i −0.314054 + 0.432258i
\(65\) 0 0
\(66\) 1.84023 − 4.13322i 0.226516 − 0.508764i
\(67\) −11.5149 − 7.47789i −1.40677 − 0.913570i −1.00000 4.11590e-5i \(-0.999987\pi\)
−0.406774 − 0.913529i \(-0.633346\pi\)
\(68\) 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 −0.992457 − 0.122593i \(-0.960879\pi\)
0.730856 − 0.682531i \(-0.239121\pi\)
\(72\) −1.67231 − 2.57513i −0.197083 − 0.303482i
\(73\) −8.83233 + 7.15228i −1.03375 + 0.837111i −0.986826 − 0.161788i \(-0.948274\pi\)
−0.0469205 + 0.998899i \(0.514941\pi\)
\(74\) −0.0516650 + 0.0298288i −0.00600594 + 0.00346753i
\(75\) 0 0
\(76\) − 3.08085i − 0.353397i
\(77\) 7.01165 − 0.661268i 0.799052 − 0.0753584i
\(78\) 1.27192 + 0.648075i 0.144016 + 0.0733801i
\(79\) 0.511075 − 2.40442i 0.0575005 − 0.270518i −0.940000 − 0.341173i \(-0.889176\pi\)
0.997501 + 0.0706551i \(0.0225090\pi\)
\(80\) 0 0
\(81\) −10.8560 + 2.30752i −1.20623 + 0.256391i
\(82\) 1.33840 + 4.99499i 0.147802 + 0.551604i
\(83\) 8.10334 − 4.12886i 0.889457 − 0.453201i 0.0513335 − 0.998682i \(-0.483653\pi\)
0.838123 + 0.545481i \(0.183653\pi\)
\(84\) 3.22537 − 6.16631i 0.351916 − 0.672800i
\(85\) 0 0
\(86\) −2.51105 + 1.11799i −0.270773 + 0.120556i
\(87\) 8.61078 − 3.30537i 0.923173 − 0.354373i
\(88\) 6.86905 − 2.63678i 0.732243 − 0.281082i
\(89\) −0.184924 + 0.0823337i −0.0196020 + 0.00872735i −0.416514 − 0.909129i \(-0.636748\pi\)
0.396912 + 0.917857i \(0.370082\pi\)
\(90\) 0 0
\(91\) 0.0925721 + 2.22017i 0.00970420 + 0.232737i
\(92\) −3.98090 + 2.02837i −0.415038 + 0.211472i
\(93\) −5.63911 − 21.0454i −0.584749 − 2.18231i
\(94\) −8.19765 + 1.74246i −0.845523 + 0.179721i
\(95\) 0 0
\(96\) 2.42840 − 11.4247i 0.247848 − 1.16603i
\(97\) 1.22921 + 0.626312i 0.124807 + 0.0635923i 0.515279 − 0.857022i \(-0.327688\pi\)
−0.390472 + 0.920615i \(0.627688\pi\)
\(98\) −5.86527 + 0.181786i −0.592481 + 0.0183632i
\(99\) 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.b.82.13 288
5.2 odd 4 875.2.bb.c.418.6 288
5.3 odd 4 175.2.x.a.138.13 yes 288
5.4 even 2 875.2.bb.a.82.6 288
7.3 odd 6 inner 875.2.bb.b.332.6 288
25.2 odd 20 875.2.bb.a.593.13 288
25.11 even 5 175.2.x.a.152.13 yes 288
25.14 even 10 875.2.bb.c.782.6 288
25.23 odd 20 inner 875.2.bb.b.593.6 288
35.3 even 12 175.2.x.a.38.13 ✓ 288
35.17 even 12 875.2.bb.c.668.6 288
35.24 odd 6 875.2.bb.a.332.13 288
175.52 even 60 875.2.bb.a.843.6 288
175.73 even 60 inner 875.2.bb.b.843.13 288
175.136 odd 30 175.2.x.a.52.13 yes 288
175.164 odd 30 875.2.bb.c.157.6 288
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.x.a.38.13 ✓ 288 35.3 even 12
175.2.x.a.52.13 yes 288 175.136 odd 30
175.2.x.a.138.13 yes 288 5.3 odd 4
175.2.x.a.152.13 yes 288 25.11 even 5
875.2.bb.a.82.6 288 5.4 even 2
875.2.bb.a.332.13 288 35.24 odd 6
875.2.bb.a.593.13 288 25.2 odd 20
875.2.bb.a.843.6 288 175.52 even 60
875.2.bb.b.82.13 288 1.1 even 1 trivial
875.2.bb.b.332.6 288 7.3 odd 6 inner
875.2.bb.b.593.6 288 25.23 odd 20 inner
875.2.bb.b.843.13 288 175.73 even 60 inner
875.2.bb.c.157.6 288 175.164 odd 30
875.2.bb.c.418.6 288 5.2 odd 4
875.2.bb.c.668.6 288 35.17 even 12
875.2.bb.c.782.6 288 25.14 even 10