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Properties

Label 9025.2.a.cs.1.8
Level $9025$
Weight $2$
Character 9025.1
Self dual yes
Analytic conductor $72.065$
Analytic rank $0$
Dimension $21$
CM no
Inner twists $1$

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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] = [9025,2,Mod(1,9025)] 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("9025.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9025, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [21,6,9,24,0,0,0,18,24,0,0,24,12,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(72.0649878242\)
Analytic rank: \(0\)
Dimension: \(21\)
Twist minimal: no (minimal twist has level 475)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Character \(\chi\) \(=\) 9025.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.558603 q^{2} +0.670247 q^{3} -1.68796 q^{4} -0.374402 q^{6} +2.55490 q^{7} +2.06011 q^{8} -2.55077 q^{9} -4.90513 q^{11} -1.13135 q^{12} -4.22808 q^{13} -1.42718 q^{14} +2.22514 q^{16} -5.54384 q^{17} +1.42487 q^{18} +1.71241 q^{21} +2.74002 q^{22} +5.61218 q^{23} +1.38078 q^{24} +2.36182 q^{26} -3.72039 q^{27} -4.31258 q^{28} +6.58193 q^{29} -6.07501 q^{31} -5.36319 q^{32} -3.28765 q^{33} +3.09681 q^{34} +4.30560 q^{36} -7.06139 q^{37} -2.83386 q^{39} +0.883936 q^{41} -0.956560 q^{42} +3.46323 q^{43} +8.27968 q^{44} -3.13498 q^{46} -1.37641 q^{47} +1.49139 q^{48} -0.472478 q^{49} -3.71574 q^{51} +7.13685 q^{52} -11.4393 q^{53} +2.07822 q^{54} +5.26337 q^{56} -3.67669 q^{58} -1.23653 q^{59} -10.3556 q^{61} +3.39352 q^{62} -6.51696 q^{63} -1.45439 q^{64} +1.83649 q^{66} -3.14703 q^{67} +9.35780 q^{68} +3.76154 q^{69} +7.40790 q^{71} -5.25486 q^{72} +10.5700 q^{73} +3.94451 q^{74} -12.5321 q^{77} +1.58300 q^{78} +11.7675 q^{79} +5.15873 q^{81} -0.493769 q^{82} -2.30665 q^{83} -2.89049 q^{84} -1.93457 q^{86} +4.41152 q^{87} -10.1051 q^{88} +2.14285 q^{89} -10.8023 q^{91} -9.47314 q^{92} -4.07175 q^{93} +0.768865 q^{94} -3.59466 q^{96} +4.81289 q^{97} +0.263928 q^{98} +12.5119 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21 q + 6 q^{2} + 9 q^{3} + 24 q^{4} + 18 q^{8} + 24 q^{9} + 24 q^{12} + 12 q^{13} + 6 q^{14} + 30 q^{16} + 9 q^{17} + 18 q^{18} + 12 q^{22} + 6 q^{23} + 18 q^{24} + 18 q^{26} + 30 q^{27} + 15 q^{28} + 3 q^{29}+ \cdots + 84 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.558603 −0.394992 −0.197496 − 0.980304i \(-0.563281\pi\)
−0.197496 + 0.980304i \(0.563281\pi\)
\(3\) 0.670247 0.386967 0.193484 − 0.981104i \(-0.438021\pi\)
0.193484 + 0.981104i \(0.438021\pi\)
\(4\) −1.68796 −0.843981
\(5\) 0 0
\(6\) −0.374402 −0.152849
\(7\) 2.55490 0.965662 0.482831 − 0.875714i \(-0.339609\pi\)
0.482831 + 0.875714i \(0.339609\pi\)
\(8\) 2.06011 0.728358
\(9\) −2.55077 −0.850256
\(10\) 0 0
\(11\) −4.90513 −1.47895 −0.739477 − 0.673182i \(-0.764927\pi\)
−0.739477 + 0.673182i \(0.764927\pi\)
\(12\) −1.13135 −0.326593
\(13\) −4.22808 −1.17266 −0.586330 − 0.810072i \(-0.699428\pi\)
−0.586330 + 0.810072i \(0.699428\pi\)
\(14\) −1.42718 −0.381429
\(15\) 0 0
\(16\) 2.22514 0.556286
\(17\) −5.54384 −1.34458 −0.672289 − 0.740288i \(-0.734689\pi\)
−0.672289 + 0.740288i \(0.734689\pi\)
\(18\) 1.42487 0.335845
\(19\) 0 0
\(20\) 0 0
\(21\) 1.71241 0.373679
\(22\) 2.74002 0.584175
\(23\) 5.61218 1.17022 0.585110 − 0.810954i \(-0.301051\pi\)
0.585110 + 0.810954i \(0.301051\pi\)
\(24\) 1.38078 0.281851
\(25\) 0 0
\(26\) 2.36182 0.463191
\(27\) −3.72039 −0.715988
\(28\) −4.31258 −0.815001
\(29\) 6.58193 1.22223 0.611117 − 0.791540i \(-0.290721\pi\)
0.611117 + 0.791540i \(0.290721\pi\)
\(30\) 0 0
\(31\) −6.07501 −1.09110 −0.545552 − 0.838077i \(-0.683680\pi\)
−0.545552 + 0.838077i \(0.683680\pi\)
\(32\) −5.36319 −0.948086
\(33\) −3.28765 −0.572306
\(34\) 3.09681 0.531098
\(35\) 0 0
\(36\) 4.30560 0.717601
\(37\) −7.06139 −1.16089 −0.580443 − 0.814301i \(-0.697120\pi\)
−0.580443 + 0.814301i \(0.697120\pi\)
\(38\) 0 0
\(39\) −2.83386 −0.453781
\(40\) 0 0
\(41\) 0.883936 0.138048 0.0690238 − 0.997615i \(-0.478012\pi\)
0.0690238 + 0.997615i \(0.478012\pi\)
\(42\) −0.956560 −0.147600
\(43\) 3.46323 0.528138 0.264069 − 0.964504i \(-0.414935\pi\)
0.264069 + 0.964504i \(0.414935\pi\)
\(44\) 8.27968 1.24821
\(45\) 0 0
\(46\) −3.13498 −0.462227
\(47\) −1.37641 −0.200770 −0.100385 − 0.994949i \(-0.532007\pi\)
−0.100385 + 0.994949i \(0.532007\pi\)
\(48\) 1.49139 0.215264
\(49\) −0.472478 −0.0674968
\(50\) 0 0
\(51\) −3.71574 −0.520308
\(52\) 7.13685 0.989703
\(53\) −11.4393 −1.57131 −0.785655 − 0.618665i \(-0.787674\pi\)
−0.785655 + 0.618665i \(0.787674\pi\)
\(54\) 2.07822 0.282810
\(55\) 0 0
\(56\) 5.26337 0.703348
\(57\) 0 0
\(58\) −3.67669 −0.482773
\(59\) −1.23653 −0.160983 −0.0804913 − 0.996755i \(-0.525649\pi\)
−0.0804913 + 0.996755i \(0.525649\pi\)
\(60\) 0 0
\(61\) −10.3556 −1.32590 −0.662949 − 0.748665i \(-0.730695\pi\)
−0.662949 + 0.748665i \(0.730695\pi\)
\(62\) 3.39352 0.430977
\(63\) −6.51696 −0.821060
\(64\) −1.45439 −0.181799
\(65\) 0 0
\(66\) 1.83649 0.226056
\(67\) −3.14703 −0.384470 −0.192235 − 0.981349i \(-0.561574\pi\)
−0.192235 + 0.981349i \(0.561574\pi\)
\(68\) 9.35780 1.13480
\(69\) 3.76154 0.452837
\(70\) 0 0
\(71\) 7.40790 0.879156 0.439578 − 0.898204i \(-0.355128\pi\)
0.439578 + 0.898204i \(0.355128\pi\)
\(72\) −5.25486 −0.619291
\(73\) 10.5700 1.23713 0.618565 − 0.785734i \(-0.287714\pi\)
0.618565 + 0.785734i \(0.287714\pi\)
\(74\) 3.94451 0.458540
\(75\) 0 0
\(76\) 0 0
\(77\) −12.5321 −1.42817
\(78\) 1.58300 0.179240
\(79\) 11.7675 1.32395 0.661974 − 0.749527i \(-0.269719\pi\)
0.661974 + 0.749527i \(0.269719\pi\)
\(80\) 0 0
\(81\) 5.15873 0.573192
\(82\) −0.493769 −0.0545277
\(83\) −2.30665 −0.253187 −0.126594 − 0.991955i \(-0.540404\pi\)
−0.126594 + 0.991955i \(0.540404\pi\)
\(84\) −2.89049 −0.315378
\(85\) 0 0
\(86\) −1.93457 −0.208610
\(87\) 4.41152 0.472964
\(88\) −10.1051 −1.07721
\(89\) 2.14285 0.227141 0.113571 − 0.993530i \(-0.463771\pi\)
0.113571 + 0.993530i \(0.463771\pi\)
\(90\) 0 0
\(91\) −10.8023 −1.13239
\(92\) −9.47314 −0.987644
\(93\) −4.07175 −0.422221
\(94\) 0.768865 0.0793024
\(95\) 0 0
\(96\) −3.59466 −0.366878
\(97\) 4.81289 0.488675 0.244337 − 0.969690i \(-0.421430\pi\)
0.244337 + 0.969690i \(0.421430\pi\)
\(98\) 0.263928 0.0266607
\(99\) 12.5119 1.25749
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 9025.2.a.cs.1.8 21
5.4 even 2 9025.2.a.cq.1.14 21
19.9 even 9 475.2.l.e.176.5 yes 42
19.17 even 9 475.2.l.e.251.5 yes 42
19.18 odd 2 9025.2.a.cp.1.14 21
95.9 even 18 475.2.l.d.176.3 ✓ 42
95.17 odd 36 475.2.u.d.99.6 84
95.28 odd 36 475.2.u.d.24.6 84
95.47 odd 36 475.2.u.d.24.9 84
95.74 even 18 475.2.l.d.251.3 yes 42
95.93 odd 36 475.2.u.d.99.9 84
95.94 odd 2 9025.2.a.cr.1.8 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.l.d.176.3 ✓ 42 95.9 even 18
475.2.l.d.251.3 yes 42 95.74 even 18
475.2.l.e.176.5 yes 42 19.9 even 9
475.2.l.e.251.5 yes 42 19.17 even 9
475.2.u.d.24.6 84 95.28 odd 36
475.2.u.d.24.9 84 95.47 odd 36
475.2.u.d.99.6 84 95.17 odd 36
475.2.u.d.99.9 84 95.93 odd 36
9025.2.a.cp.1.14 21 19.18 odd 2
9025.2.a.cq.1.14 21 5.4 even 2
9025.2.a.cr.1.8 21 95.94 odd 2
9025.2.a.cs.1.8 21 1.1 even 1 trivial