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Properties

Label 5625.2.a.z.1.7
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 1.7
Root \(2.37653\) 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)
 
\(f(q)\) \(=\) \(q+2.37653 q^{2} +3.64791 q^{4} -4.26594 q^{7} +3.91630 q^{8} +2.49140 q^{11} -2.02987 q^{13} -10.1381 q^{14} +2.01141 q^{16} -7.24447 q^{17} +3.26594 q^{19} +5.92090 q^{22} +6.15085 q^{23} -4.82406 q^{26} -15.5618 q^{28} -0.951631 q^{29} +2.66637 q^{31} -3.05243 q^{32} -17.2167 q^{34} -9.66637 q^{37} +7.76161 q^{38} -12.1679 q^{41} -7.95077 q^{43} +9.08840 q^{44} +14.6177 q^{46} +2.93756 q^{47} +11.1983 q^{49} -7.40479 q^{52} -12.4437 q^{53} -16.7067 q^{56} -2.26158 q^{58} +8.13677 q^{59} -3.33363 q^{61} +6.33671 q^{62} -11.2770 q^{64} +11.2352 q^{67} -26.4271 q^{68} -9.67655 q^{71} -8.24312 q^{73} -22.9724 q^{74} +11.9138 q^{76} -10.6282 q^{77} +7.65496 q^{79} -28.9175 q^{82} -10.3240 q^{83} -18.8953 q^{86} +9.75709 q^{88} -0.997628 q^{89} +8.65932 q^{91} +22.4377 q^{92} +6.98120 q^{94} +5.31428 q^{97} +26.6130 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 14 q^{4} - 10 q^{7} - 10 q^{13} + 22 q^{16} + 2 q^{19} - 10 q^{22} - 70 q^{28} - 6 q^{31} - 50 q^{34} - 50 q^{37} + 14 q^{49} - 80 q^{52} + 30 q^{58} - 54 q^{61} + 36 q^{64} - 10 q^{67} - 30 q^{73}+ \cdots - 40 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.37653 1.68046 0.840231 − 0.542228i \(-0.182419\pi\)
0.840231 + 0.542228i \(0.182419\pi\)
\(3\) 0 0
\(4\) 3.64791 1.82395
\(5\) 0 0
\(6\) 0 0
\(7\) −4.26594 −1.61237 −0.806187 − 0.591661i \(-0.798472\pi\)
−0.806187 + 0.591661i \(0.798472\pi\)
\(8\) 3.91630 1.38462
\(9\) 0 0
\(10\) 0 0
\(11\) 2.49140 0.751186 0.375593 − 0.926785i \(-0.377439\pi\)
0.375593 + 0.926785i \(0.377439\pi\)
\(12\) 0 0
\(13\) −2.02987 −0.562985 −0.281493 − 0.959563i \(-0.590830\pi\)
−0.281493 + 0.959563i \(0.590830\pi\)
\(14\) −10.1381 −2.70953
\(15\) 0 0
\(16\) 2.01141 0.502853
\(17\) −7.24447 −1.75704 −0.878521 − 0.477704i \(-0.841469\pi\)
−0.878521 + 0.477704i \(0.841469\pi\)
\(18\) 0 0
\(19\) 3.26594 0.749258 0.374629 − 0.927175i \(-0.377770\pi\)
0.374629 + 0.927175i \(0.377770\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 5.92090 1.26234
\(23\) 6.15085 1.28254 0.641270 − 0.767315i \(-0.278408\pi\)
0.641270 + 0.767315i \(0.278408\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −4.82406 −0.946076
\(27\) 0 0
\(28\) −15.5618 −2.94090
\(29\) −0.951631 −0.176713 −0.0883567 − 0.996089i \(-0.528162\pi\)
−0.0883567 + 0.996089i \(0.528162\pi\)
\(30\) 0 0
\(31\) 2.66637 0.478894 0.239447 − 0.970909i \(-0.423034\pi\)
0.239447 + 0.970909i \(0.423034\pi\)
\(32\) −3.05243 −0.539598
\(33\) 0 0
\(34\) −17.2167 −2.95264
\(35\) 0 0
\(36\) 0 0
\(37\) −9.66637 −1.58914 −0.794571 − 0.607172i \(-0.792304\pi\)
−0.794571 + 0.607172i \(0.792304\pi\)
\(38\) 7.76161 1.25910
\(39\) 0 0
\(40\) 0 0
\(41\) −12.1679 −1.90031 −0.950157 − 0.311771i \(-0.899078\pi\)
−0.950157 + 0.311771i \(0.899078\pi\)
\(42\) 0 0
\(43\) −7.95077 −1.21248 −0.606241 − 0.795281i \(-0.707323\pi\)
−0.606241 + 0.795281i \(0.707323\pi\)
\(44\) 9.08840 1.37013
\(45\) 0 0
\(46\) 14.6177 2.15526
\(47\) 2.93756 0.428487 0.214243 − 0.976780i \(-0.431271\pi\)
0.214243 + 0.976780i \(0.431271\pi\)
\(48\) 0 0
\(49\) 11.1983 1.59975
\(50\) 0 0
\(51\) 0 0
\(52\) −7.40479 −1.02686
\(53\) −12.4437 −1.70927 −0.854636 − 0.519228i \(-0.826220\pi\)
−0.854636 + 0.519228i \(0.826220\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −16.7067 −2.23253
\(57\) 0 0
\(58\) −2.26158 −0.296960
\(59\) 8.13677 1.05932 0.529659 − 0.848211i \(-0.322320\pi\)
0.529659 + 0.848211i \(0.322320\pi\)
\(60\) 0 0
\(61\) −3.33363 −0.426828 −0.213414 − 0.976962i \(-0.568458\pi\)
−0.213414 + 0.976962i \(0.568458\pi\)
\(62\) 6.33671 0.804763
\(63\) 0 0
\(64\) −11.2770 −1.40963
\(65\) 0 0
\(66\) 0 0
\(67\) 11.2352 1.37260 0.686298 − 0.727321i \(-0.259235\pi\)
0.686298 + 0.727321i \(0.259235\pi\)
\(68\) −26.4271 −3.20476
\(69\) 0 0
\(70\) 0 0
\(71\) −9.67655 −1.14839 −0.574197 − 0.818717i \(-0.694686\pi\)
−0.574197 + 0.818717i \(0.694686\pi\)
\(72\) 0 0
\(73\) −8.24312 −0.964784 −0.482392 − 0.875955i \(-0.660232\pi\)
−0.482392 + 0.875955i \(0.660232\pi\)
\(74\) −22.9724 −2.67049
\(75\) 0 0
\(76\) 11.9138 1.36661
\(77\) −10.6282 −1.21119
\(78\) 0 0
\(79\) 7.65496 0.861250 0.430625 − 0.902531i \(-0.358293\pi\)
0.430625 + 0.902531i \(0.358293\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −28.9175 −3.19341
\(83\) −10.3240 −1.13321 −0.566604 − 0.823990i \(-0.691743\pi\)
−0.566604 + 0.823990i \(0.691743\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −18.8953 −2.03753
\(87\) 0 0
\(88\) 9.75709 1.04011
\(89\) −0.997628 −0.105748 −0.0528742 − 0.998601i \(-0.516838\pi\)
−0.0528742 + 0.998601i \(0.516838\pi\)
\(90\) 0 0
\(91\) 8.65932 0.907743
\(92\) 22.4377 2.33929
\(93\) 0 0
\(94\) 6.98120 0.720055
\(95\) 0 0
\(96\) 0 0
\(97\) 5.31428 0.539583 0.269791 − 0.962919i \(-0.413045\pi\)
0.269791 + 0.962919i \(0.413045\pi\)
\(98\) 26.6130 2.68832
\(99\) 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.1.7 yes 8
3.2 odd 2 inner 5625.2.a.z.1.2 ✓ 8
5.4 even 2 5625.2.a.bb.1.2 yes 8
15.14 odd 2 5625.2.a.bb.1.7 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5625.2.a.z.1.2 ✓ 8 3.2 odd 2 inner
5625.2.a.z.1.7 yes 8 1.1 even 1 trivial
5625.2.a.bb.1.2 yes 8 5.4 even 2
5625.2.a.bb.1.7 yes 8 15.14 odd 2