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Properties

Label 9025.2.a.cp.1.14
Level $9025$
Weight $2$
Character 9025.1
Self dual yes
Analytic conductor $72.065$
Analytic rank $1$
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: \(1\)
Dimension: \(21\)
Twist minimal: no (minimal twist has level 475)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.14
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 \(-0.436719\pi\)
0.197496 + 0.980304i \(0.436719\pi\)
\(3\) −0.670247 −0.386967 −0.193484 − 0.981104i \(-0.561979\pi\)
−0.193484 + 0.981104i \(0.561979\pi\)
\(4\) −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 \(-0.300572\pi\)
0.586330 + 0.810072i \(0.300572\pi\)
\(14\) 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 \(-0.709279\pi\)
−0.611117 + 0.791540i \(0.709279\pi\)
\(30\) 0 0
\(31\) 6.07501 1.09110 0.545552 − 0.838077i \(-0.316320\pi\)
0.545552 + 0.838077i \(0.316320\pi\)
\(32\) 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 \(-0.302880\pi\)
0.580443 + 0.814301i \(0.302880\pi\)
\(38\) 0 0
\(39\) −2.83386 −0.453781
\(40\) 0 0
\(41\) −0.883936 −0.138048 −0.0690238 − 0.997615i \(-0.521988\pi\)
−0.0690238 + 0.997615i \(0.521988\pi\)
\(42\) −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 \(-0.212326\pi\)
0.785655 + 0.618665i \(0.212326\pi\)
\(54\) 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 \(-0.474351\pi\)
0.0804913 + 0.996755i \(0.474351\pi\)
\(60\) 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 \(-0.438426\pi\)
0.192235 + 0.981349i \(0.438426\pi\)
\(68\) 9.35780 1.13480
\(69\) −3.76154 −0.452837
\(70\) 0 0
\(71\) −7.40790 −0.879156 −0.439578 − 0.898204i \(-0.644872\pi\)
−0.439578 + 0.898204i \(0.644872\pi\)
\(72\) 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 \(-0.730281\pi\)
−0.661974 + 0.749527i \(0.730281\pi\)
\(80\) 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 \(-0.536229\pi\)
−0.113571 + 0.993530i \(0.536229\pi\)
\(90\) 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 \(-0.578570\pi\)
−0.244337 + 0.969690i \(0.578570\pi\)
\(98\) −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.cp.1.14 21
5.4 even 2 9025.2.a.cr.1.8 21
19.2 odd 18 475.2.l.e.251.5 yes 42
19.10 odd 18 475.2.l.e.176.5 yes 42
19.18 odd 2 9025.2.a.cs.1.8 21
95.2 even 36 475.2.u.d.99.6 84
95.29 odd 18 475.2.l.d.176.3 ✓ 42
95.48 even 36 475.2.u.d.24.6 84
95.59 odd 18 475.2.l.d.251.3 yes 42
95.67 even 36 475.2.u.d.24.9 84
95.78 even 36 475.2.u.d.99.9 84
95.94 odd 2 9025.2.a.cq.1.14 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.l.d.176.3 ✓ 42 95.29 odd 18
475.2.l.d.251.3 yes 42 95.59 odd 18
475.2.l.e.176.5 yes 42 19.10 odd 18
475.2.l.e.251.5 yes 42 19.2 odd 18
475.2.u.d.24.6 84 95.48 even 36
475.2.u.d.24.9 84 95.67 even 36
475.2.u.d.99.6 84 95.2 even 36
475.2.u.d.99.9 84 95.78 even 36
9025.2.a.cp.1.14 21 1.1 even 1 trivial
9025.2.a.cq.1.14 21 95.94 odd 2
9025.2.a.cr.1.8 21 5.4 even 2
9025.2.a.cs.1.8 21 19.18 odd 2