Properties

Label 25.22.a.d.1.7
Level $25$
Weight $22$
Character 25.1
Self dual yes
Analytic conductor $69.869$
Analytic rank $1$
Dimension $7$
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] = [25,22,Mod(1,25)] 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("25.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(25, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,-737] 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: yes
Analytic conductor: \(69.8693360718\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 2 x^{6} - 11176187 x^{5} + 2419003474 x^{4} + 32443001467723 x^{3} + \cdots + 99\!\cdots\!78 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{13}\cdot 3^{3}\cdot 5^{10}\cdot 7^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-2832.50\) of defining polynomial
Character \(\chi\) \(=\) 25.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2727.50 q^{2} -50838.6 q^{3} +5.34213e6 q^{4} -1.38663e8 q^{6} -3.19642e8 q^{7} +8.85070e9 q^{8} -7.87579e9 q^{9} -1.64577e11 q^{11} -2.71586e11 q^{12} +5.81608e11 q^{13} -8.71824e11 q^{14} +1.29371e13 q^{16} -3.89820e12 q^{17} -2.14813e13 q^{18} -3.65997e13 q^{19} +1.62501e13 q^{21} -4.48884e14 q^{22} -2.08441e14 q^{23} -4.49957e14 q^{24} +1.58634e15 q^{26} +9.32184e14 q^{27} -1.70757e15 q^{28} +8.16783e14 q^{29} -5.45592e15 q^{31} +1.67246e16 q^{32} +8.36685e15 q^{33} -1.06324e16 q^{34} -4.20735e16 q^{36} +3.79454e16 q^{37} -9.98259e16 q^{38} -2.95681e16 q^{39} +9.62478e16 q^{41} +4.43223e16 q^{42} -6.14764e16 q^{43} -8.79191e17 q^{44} -5.68523e17 q^{46} -4.95585e17 q^{47} -6.57702e17 q^{48} -4.56375e17 q^{49} +1.98179e17 q^{51} +3.10703e18 q^{52} -9.65750e17 q^{53} +2.54254e18 q^{54} -2.82905e18 q^{56} +1.86068e18 q^{57} +2.22778e18 q^{58} -3.38226e16 q^{59} +1.46967e18 q^{61} -1.48810e19 q^{62} +2.51743e18 q^{63} +1.84855e19 q^{64} +2.28206e19 q^{66} -1.95501e19 q^{67} -2.08247e19 q^{68} +1.05968e19 q^{69} +6.92876e18 q^{71} -6.97062e19 q^{72} +2.18546e19 q^{73} +1.03496e20 q^{74} -1.95520e20 q^{76} +5.26056e19 q^{77} -8.06472e19 q^{78} -6.04006e19 q^{79} +3.49926e19 q^{81} +2.62516e20 q^{82} +1.33566e19 q^{83} +8.68103e19 q^{84} -1.67677e20 q^{86} -4.15241e19 q^{87} -1.45662e21 q^{88} -8.06834e19 q^{89} -1.85906e20 q^{91} -1.11352e21 q^{92} +2.77371e20 q^{93} -1.35171e21 q^{94} -8.50257e20 q^{96} +7.05209e20 q^{97} -1.24477e21 q^{98} +1.29617e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 737 q^{2} - 50381 q^{3} + 7749909 q^{4} - 30442931 q^{6} - 817782442 q^{7} + 3231986745 q^{8} + 1856104186 q^{9} - 17026930591 q^{11} - 286884094477 q^{12} + 325366982844 q^{13} + 13403645898 q^{14}+ \cdots + 26\!\cdots\!82 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2727.50 1.88343 0.941717 − 0.336405i \(-0.109211\pi\)
0.941717 + 0.336405i \(0.109211\pi\)
\(3\) −50838.6 −0.497073 −0.248537 − 0.968623i \(-0.579950\pi\)
−0.248537 + 0.968623i \(0.579950\pi\)
\(4\) 5.34213e6 2.54733
\(5\) 0 0
\(6\) −1.38663e8 −0.936205
\(7\) −3.19642e8 −0.427695 −0.213847 − 0.976867i \(-0.568600\pi\)
−0.213847 + 0.976867i \(0.568600\pi\)
\(8\) 8.85070e9 2.91429
\(9\) −7.87579e9 −0.752918
\(10\) 0 0
\(11\) −1.64577e11 −1.91313 −0.956567 − 0.291512i \(-0.905842\pi\)
−0.956567 + 0.291512i \(0.905842\pi\)
\(12\) −2.71586e11 −1.26621
\(13\) 5.81608e11 1.17011 0.585053 − 0.810995i \(-0.301074\pi\)
0.585053 + 0.810995i \(0.301074\pi\)
\(14\) −8.71824e11 −0.805535
\(15\) 0 0
\(16\) 1.29371e13 2.94155
\(17\) −3.89820e12 −0.468976 −0.234488 − 0.972119i \(-0.575341\pi\)
−0.234488 + 0.972119i \(0.575341\pi\)
\(18\) −2.14813e13 −1.41807
\(19\) −3.65997e13 −1.36951 −0.684754 − 0.728774i \(-0.740090\pi\)
−0.684754 + 0.728774i \(0.740090\pi\)
\(20\) 0 0
\(21\) 1.62501e13 0.212596
\(22\) −4.48884e14 −3.60326
\(23\) −2.08441e14 −1.04916 −0.524578 − 0.851362i \(-0.675777\pi\)
−0.524578 + 0.851362i \(0.675777\pi\)
\(24\) −4.49957e14 −1.44862
\(25\) 0 0
\(26\) 1.58634e15 2.20382
\(27\) 9.32184e14 0.871329
\(28\) −1.70757e15 −1.08948
\(29\) 8.16783e14 0.360519 0.180259 − 0.983619i \(-0.442306\pi\)
0.180259 + 0.983619i \(0.442306\pi\)
\(30\) 0 0
\(31\) −5.45592e15 −1.19556 −0.597778 − 0.801661i \(-0.703950\pi\)
−0.597778 + 0.801661i \(0.703950\pi\)
\(32\) 1.67246e16 2.62592
\(33\) 8.36685e15 0.950968
\(34\) −1.06324e16 −0.883286
\(35\) 0 0
\(36\) −4.20735e16 −1.91793
\(37\) 3.79454e16 1.29730 0.648650 − 0.761086i \(-0.275334\pi\)
0.648650 + 0.761086i \(0.275334\pi\)
\(38\) −9.98259e16 −2.57938
\(39\) −2.95681e16 −0.581628
\(40\) 0 0
\(41\) 9.62478e16 1.11985 0.559926 − 0.828543i \(-0.310830\pi\)
0.559926 + 0.828543i \(0.310830\pi\)
\(42\) 4.43223e16 0.400410
\(43\) −6.14764e16 −0.433800 −0.216900 − 0.976194i \(-0.569595\pi\)
−0.216900 + 0.976194i \(0.569595\pi\)
\(44\) −8.79191e17 −4.87338
\(45\) 0 0
\(46\) −5.68523e17 −1.97602
\(47\) −4.95585e17 −1.37433 −0.687164 − 0.726503i \(-0.741145\pi\)
−0.687164 + 0.726503i \(0.741145\pi\)
\(48\) −6.57702e17 −1.46216
\(49\) −4.56375e17 −0.817077
\(50\) 0 0
\(51\) 1.98179e17 0.233115
\(52\) 3.10703e18 2.98064
\(53\) −9.65750e17 −0.758522 −0.379261 − 0.925290i \(-0.623822\pi\)
−0.379261 + 0.925290i \(0.623822\pi\)
\(54\) 2.54254e18 1.64109
\(55\) 0 0
\(56\) −2.82905e18 −1.24643
\(57\) 1.86068e18 0.680746
\(58\) 2.22778e18 0.679014
\(59\) −3.38226e16 −0.00861512 −0.00430756 − 0.999991i \(-0.501371\pi\)
−0.00430756 + 0.999991i \(0.501371\pi\)
\(60\) 0 0
\(61\) 1.46967e18 0.263789 0.131895 − 0.991264i \(-0.457894\pi\)
0.131895 + 0.991264i \(0.457894\pi\)
\(62\) −1.48810e19 −2.25175
\(63\) 2.51743e18 0.322019
\(64\) 1.84855e19 2.00421
\(65\) 0 0
\(66\) 2.28206e19 1.79109
\(67\) −1.95501e19 −1.31028 −0.655139 − 0.755508i \(-0.727390\pi\)
−0.655139 + 0.755508i \(0.727390\pi\)
\(68\) −2.08247e19 −1.19464
\(69\) 1.05968e19 0.521507
\(70\) 0 0
\(71\) 6.92876e18 0.252606 0.126303 − 0.991992i \(-0.459689\pi\)
0.126303 + 0.991992i \(0.459689\pi\)
\(72\) −6.97062e19 −2.19422
\(73\) 2.18546e19 0.595186 0.297593 − 0.954693i \(-0.403816\pi\)
0.297593 + 0.954693i \(0.403816\pi\)
\(74\) 1.03496e20 2.44338
\(75\) 0 0
\(76\) −1.95520e20 −3.48859
\(77\) 5.26056e19 0.818237
\(78\) −8.06472e19 −1.09546
\(79\) −6.04006e19 −0.717723 −0.358861 − 0.933391i \(-0.616835\pi\)
−0.358861 + 0.933391i \(0.616835\pi\)
\(80\) 0 0
\(81\) 3.49926e19 0.319804
\(82\) 2.62516e20 2.10917
\(83\) 1.33566e19 0.0944876 0.0472438 − 0.998883i \(-0.484956\pi\)
0.0472438 + 0.998883i \(0.484956\pi\)
\(84\) 8.68103e19 0.541550
\(85\) 0 0
\(86\) −1.67677e20 −0.817035
\(87\) −4.15241e19 −0.179204
\(88\) −1.45662e21 −5.57543
\(89\) −8.06834e19 −0.274277 −0.137138 − 0.990552i \(-0.543791\pi\)
−0.137138 + 0.990552i \(0.543791\pi\)
\(90\) 0 0
\(91\) −1.85906e20 −0.500448
\(92\) −1.11352e21 −2.67254
\(93\) 2.77371e20 0.594279
\(94\) −1.35171e21 −2.58846
\(95\) 0 0
\(96\) −8.50257e20 −1.30528
\(97\) 7.05209e20 0.970989 0.485495 − 0.874240i \(-0.338640\pi\)
0.485495 + 0.874240i \(0.338640\pi\)
\(98\) −1.24477e21 −1.53891
\(99\) 1.29617e21 1.44043
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 25.22.a.d.1.7 ✓ 7
5.2 odd 4 25.22.b.d.24.14 14
5.3 odd 4 25.22.b.d.24.1 14
5.4 even 2 25.22.a.e.1.1 yes 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.22.a.d.1.7 ✓ 7 1.1 even 1 trivial
25.22.a.e.1.1 yes 7 5.4 even 2
25.22.b.d.24.1 14 5.3 odd 4
25.22.b.d.24.14 14 5.2 odd 4