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Properties

Label 5625.2.a.z.1.4
Level $5625$
Weight $2$
Character 5625.1
Self dual yes
Analytic conductor $44.916$
Analytic rank $1$
Dimension $8$
Inner twists $2$

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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] = [5625,2,Mod(1,5625)] 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("5625.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5625, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5625 = 3^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5625.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,14,0,0,-10,0,0,0,0,0,-10,0,0,22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(16)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(44.9158511370\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 15x^{6} + 70x^{4} - 105x^{2} + 45 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.856773\) of defining polynomial
Character \(\chi\) \(=\) 5625.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.856773 q^{2} -1.26594 q^{4} +0.647907 q^{7} +2.79817 q^{8} +5.91382 q^{11} +2.88397 q^{13} -0.555109 q^{14} +0.134488 q^{16} -4.20027 q^{17} -1.64791 q^{19} -5.06680 q^{22} -6.42752 q^{23} -2.47091 q^{26} -0.820212 q^{28} -2.25888 q^{29} -5.28440 q^{31} -5.71156 q^{32} +3.59868 q^{34} -1.71560 q^{37} +1.41188 q^{38} +0.176662 q^{41} +7.95077 q^{43} -7.48654 q^{44} +5.50692 q^{46} -1.05903 q^{47} -6.58022 q^{49} -3.65094 q^{52} +4.48612 q^{53} +1.81295 q^{56} +1.93534 q^{58} -9.74542 q^{59} -11.2844 q^{61} +4.52753 q^{62} +4.62453 q^{64} -12.6171 q^{67} +5.31730 q^{68} +6.09048 q^{71} -7.08312 q^{73} +1.46988 q^{74} +2.08615 q^{76} +3.83160 q^{77} +1.58111 q^{79} -0.151359 q^{82} -11.5102 q^{83} -6.81200 q^{86} +16.5479 q^{88} +15.5918 q^{89} +1.86855 q^{91} +8.13685 q^{92} +0.907347 q^{94} -7.55034 q^{97} +5.63775 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 14 q^{4} - 10 q^{7} - 10 q^{13} + 22 q^{16} + 2 q^{19} - 10 q^{22} - 70 q^{28} - 6 q^{31} - 50 q^{34} - 50 q^{37} + 14 q^{49} - 80 q^{52} + 30 q^{58} - 54 q^{61} + 36 q^{64} - 10 q^{67} - 30 q^{73}+ \cdots - 40 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.856773 −0.605830 −0.302915 − 0.953018i \(-0.597960\pi\)
−0.302915 + 0.953018i \(0.597960\pi\)
\(3\) 0 0
\(4\) −1.26594 −0.632970
\(5\) 0 0
\(6\) 0 0
\(7\) 0.647907 0.244886 0.122443 − 0.992476i \(-0.460927\pi\)
0.122443 + 0.992476i \(0.460927\pi\)
\(8\) 2.79817 0.989302
\(9\) 0 0
\(10\) 0 0
\(11\) 5.91382 1.78308 0.891542 − 0.452939i \(-0.149624\pi\)
0.891542 + 0.452939i \(0.149624\pi\)
\(12\) 0 0
\(13\) 2.88397 0.799871 0.399935 − 0.916543i \(-0.369033\pi\)
0.399935 + 0.916543i \(0.369033\pi\)
\(14\) −0.555109 −0.148359
\(15\) 0 0
\(16\) 0.134488 0.0336219
\(17\) −4.20027 −1.01872 −0.509358 − 0.860555i \(-0.670117\pi\)
−0.509358 + 0.860555i \(0.670117\pi\)
\(18\) 0 0
\(19\) −1.64791 −0.378056 −0.189028 − 0.981972i \(-0.560534\pi\)
−0.189028 + 0.981972i \(0.560534\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −5.06680 −1.08024
\(23\) −6.42752 −1.34023 −0.670115 − 0.742257i \(-0.733755\pi\)
−0.670115 + 0.742257i \(0.733755\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.47091 −0.484585
\(27\) 0 0
\(28\) −0.820212 −0.155005
\(29\) −2.25888 −0.419463 −0.209732 − 0.977759i \(-0.567259\pi\)
−0.209732 + 0.977759i \(0.567259\pi\)
\(30\) 0 0
\(31\) −5.28440 −0.949107 −0.474553 − 0.880227i \(-0.657390\pi\)
−0.474553 + 0.880227i \(0.657390\pi\)
\(32\) −5.71156 −1.00967
\(33\) 0 0
\(34\) 3.59868 0.617168
\(35\) 0 0
\(36\) 0 0
\(37\) −1.71560 −0.282042 −0.141021 − 0.990007i \(-0.545039\pi\)
−0.141021 + 0.990007i \(0.545039\pi\)
\(38\) 1.41188 0.229037
\(39\) 0 0
\(40\) 0 0
\(41\) 0.176662 0.0275900 0.0137950 − 0.999905i \(-0.495609\pi\)
0.0137950 + 0.999905i \(0.495609\pi\)
\(42\) 0 0
\(43\) 7.95077 1.21248 0.606241 − 0.795281i \(-0.292677\pi\)
0.606241 + 0.795281i \(0.292677\pi\)
\(44\) −7.48654 −1.12864
\(45\) 0 0
\(46\) 5.50692 0.811951
\(47\) −1.05903 −0.154475 −0.0772376 − 0.997013i \(-0.524610\pi\)
−0.0772376 + 0.997013i \(0.524610\pi\)
\(48\) 0 0
\(49\) −6.58022 −0.940031
\(50\) 0 0
\(51\) 0 0
\(52\) −3.65094 −0.506294
\(53\) 4.48612 0.616216 0.308108 − 0.951351i \(-0.400304\pi\)
0.308108 + 0.951351i \(0.400304\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.81295 0.242266
\(57\) 0 0
\(58\) 1.93534 0.254123
\(59\) −9.74542 −1.26875 −0.634373 − 0.773027i \(-0.718742\pi\)
−0.634373 + 0.773027i \(0.718742\pi\)
\(60\) 0 0
\(61\) −11.2844 −1.44482 −0.722410 − 0.691465i \(-0.756966\pi\)
−0.722410 + 0.691465i \(0.756966\pi\)
\(62\) 4.52753 0.574997
\(63\) 0 0
\(64\) 4.62453 0.578067
\(65\) 0 0
\(66\) 0 0
\(67\) −12.6171 −1.54143 −0.770715 − 0.637181i \(-0.780101\pi\)
−0.770715 + 0.637181i \(0.780101\pi\)
\(68\) 5.31730 0.644817
\(69\) 0 0
\(70\) 0 0
\(71\) 6.09048 0.722807 0.361404 − 0.932409i \(-0.382298\pi\)
0.361404 + 0.932409i \(0.382298\pi\)
\(72\) 0 0
\(73\) −7.08312 −0.829016 −0.414508 − 0.910046i \(-0.636046\pi\)
−0.414508 + 0.910046i \(0.636046\pi\)
\(74\) 1.46988 0.170870
\(75\) 0 0
\(76\) 2.08615 0.239298
\(77\) 3.83160 0.436652
\(78\) 0 0
\(79\) 1.58111 0.177889 0.0889443 − 0.996037i \(-0.471651\pi\)
0.0889443 + 0.996037i \(0.471651\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −0.151359 −0.0167149
\(83\) −11.5102 −1.26340 −0.631702 − 0.775211i \(-0.717643\pi\)
−0.631702 + 0.775211i \(0.717643\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −6.81200 −0.734557
\(87\) 0 0
\(88\) 16.5479 1.76401
\(89\) 15.5918 1.65272 0.826362 − 0.563140i \(-0.190407\pi\)
0.826362 + 0.563140i \(0.190407\pi\)
\(90\) 0 0
\(91\) 1.86855 0.195877
\(92\) 8.13685 0.848326
\(93\) 0 0
\(94\) 0.907347 0.0935857
\(95\) 0 0
\(96\) 0 0
\(97\) −7.55034 −0.766621 −0.383311 − 0.923619i \(-0.625216\pi\)
−0.383311 + 0.923619i \(0.625216\pi\)
\(98\) 5.63775 0.569499
\(99\) 0 0
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 5625.2.a.z.1.4 ✓ 8
3.2 odd 2 inner 5625.2.a.z.1.5 yes 8
5.4 even 2 5625.2.a.bb.1.5 yes 8
15.14 odd 2 5625.2.a.bb.1.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5625.2.a.z.1.4 ✓ 8 1.1 even 1 trivial
5625.2.a.z.1.5 yes 8 3.2 odd 2 inner
5625.2.a.bb.1.4 yes 8 15.14 odd 2
5625.2.a.bb.1.5 yes 8 5.4 even 2