Properties

Label 25.22.a.d.1.1
Level $25$
Weight $22$
Character 25.1
Self dual yes
Analytic conductor $69.869$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.1
Root \(2328.48\) 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)
 
\(f(q)\) \(=\) \(q-2433.48 q^{2} +81248.2 q^{3} +3.82468e6 q^{4} -1.97716e8 q^{6} +9.23572e8 q^{7} -4.20390e9 q^{8} -3.85908e9 q^{9} +1.17587e11 q^{11} +3.10748e11 q^{12} +2.04518e11 q^{13} -2.24750e12 q^{14} +2.20917e12 q^{16} -1.46652e13 q^{17} +9.39101e12 q^{18} -5.51635e12 q^{19} +7.50386e13 q^{21} -2.86146e14 q^{22} -3.80086e14 q^{23} -3.41559e14 q^{24} -4.97691e14 q^{26} -1.16343e15 q^{27} +3.53236e15 q^{28} -2.37555e14 q^{29} -2.09809e15 q^{31} +3.44023e15 q^{32} +9.55373e15 q^{33} +3.56874e16 q^{34} -1.47597e16 q^{36} -1.94611e16 q^{37} +1.34239e16 q^{38} +1.66167e16 q^{39} +1.08634e17 q^{41} -1.82605e17 q^{42} +1.38626e17 q^{43} +4.49732e17 q^{44} +9.24931e17 q^{46} -6.12930e17 q^{47} +1.79491e17 q^{48} +2.94440e17 q^{49} -1.19152e18 q^{51} +7.82215e17 q^{52} +5.59966e16 q^{53} +2.83118e18 q^{54} -3.88260e18 q^{56} -4.48193e17 q^{57} +5.78085e17 q^{58} -3.50351e18 q^{59} +5.19218e18 q^{61} +5.10566e18 q^{62} -3.56414e18 q^{63} -1.30047e19 q^{64} -2.32488e19 q^{66} -1.63813e19 q^{67} -5.60895e19 q^{68} -3.08813e19 q^{69} +1.92572e18 q^{71} +1.62232e19 q^{72} -4.44035e19 q^{73} +4.73582e19 q^{74} -2.10982e19 q^{76} +1.08600e20 q^{77} -4.04365e19 q^{78} +1.36169e20 q^{79} -5.41591e19 q^{81} -2.64358e20 q^{82} +1.41806e20 q^{83} +2.86998e20 q^{84} -3.37344e20 q^{86} -1.93009e19 q^{87} -4.94324e20 q^{88} -4.36181e19 q^{89} +1.88887e20 q^{91} -1.45370e21 q^{92} -1.70466e20 q^{93} +1.49155e21 q^{94} +2.79512e20 q^{96} -4.15104e20 q^{97} -7.16514e20 q^{98} -4.53778e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 737 q^{2} - 50381 q^{3} + 7749909 q^{4} - 30442931 q^{6} - 817782442 q^{7} + 3231986745 q^{8} + 1856104186 q^{9} - 17026930591 q^{11} - 286884094477 q^{12} + 325366982844 q^{13} + 13403645898 q^{14}+ \cdots + 26\!\cdots\!82 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2433.48 −1.68040 −0.840201 − 0.542276i \(-0.817563\pi\)
−0.840201 + 0.542276i \(0.817563\pi\)
\(3\) 81248.2 0.794402 0.397201 − 0.917732i \(-0.369982\pi\)
0.397201 + 0.917732i \(0.369982\pi\)
\(4\) 3.82468e6 1.82375
\(5\) 0 0
\(6\) −1.97716e8 −1.33491
\(7\) 9.23572e8 1.23578 0.617890 − 0.786264i \(-0.287988\pi\)
0.617890 + 0.786264i \(0.287988\pi\)
\(8\) −4.20390e9 −1.38423
\(9\) −3.85908e9 −0.368925
\(10\) 0 0
\(11\) 1.17587e11 1.36690 0.683449 − 0.729998i \(-0.260479\pi\)
0.683449 + 0.729998i \(0.260479\pi\)
\(12\) 3.10748e11 1.44879
\(13\) 2.04518e11 0.411459 0.205729 − 0.978609i \(-0.434043\pi\)
0.205729 + 0.978609i \(0.434043\pi\)
\(14\) −2.24750e12 −2.07661
\(15\) 0 0
\(16\) 2.20917e12 0.502308
\(17\) −1.46652e13 −1.76430 −0.882151 − 0.470966i \(-0.843906\pi\)
−0.882151 + 0.470966i \(0.843906\pi\)
\(18\) 9.39101e12 0.619942
\(19\) −5.51635e12 −0.206414 −0.103207 − 0.994660i \(-0.532910\pi\)
−0.103207 + 0.994660i \(0.532910\pi\)
\(20\) 0 0
\(21\) 7.50386e13 0.981707
\(22\) −2.86146e14 −2.29694
\(23\) −3.80086e14 −1.91311 −0.956553 − 0.291557i \(-0.905827\pi\)
−0.956553 + 0.291557i \(0.905827\pi\)
\(24\) −3.41559e14 −1.09963
\(25\) 0 0
\(26\) −4.97691e14 −0.691416
\(27\) −1.16343e15 −1.08748
\(28\) 3.53236e15 2.25375
\(29\) −2.37555e14 −0.104854 −0.0524269 − 0.998625i \(-0.516696\pi\)
−0.0524269 + 0.998625i \(0.516696\pi\)
\(30\) 0 0
\(31\) −2.09809e15 −0.459754 −0.229877 − 0.973220i \(-0.573832\pi\)
−0.229877 + 0.973220i \(0.573832\pi\)
\(32\) 3.44023e15 0.540148
\(33\) 9.55373e15 1.08587
\(34\) 3.56874e16 2.96474
\(35\) 0 0
\(36\) −1.47597e16 −0.672826
\(37\) −1.94611e16 −0.665349 −0.332675 − 0.943042i \(-0.607951\pi\)
−0.332675 + 0.943042i \(0.607951\pi\)
\(38\) 1.34239e16 0.346858
\(39\) 1.66167e16 0.326864
\(40\) 0 0
\(41\) 1.08634e17 1.26396 0.631981 − 0.774984i \(-0.282242\pi\)
0.631981 + 0.774984i \(0.282242\pi\)
\(42\) −1.82605e17 −1.64966
\(43\) 1.38626e17 0.978196 0.489098 − 0.872229i \(-0.337326\pi\)
0.489098 + 0.872229i \(0.337326\pi\)
\(44\) 4.49732e17 2.49288
\(45\) 0 0
\(46\) 9.24931e17 3.21479
\(47\) −6.12930e17 −1.69974 −0.849871 − 0.526990i \(-0.823320\pi\)
−0.849871 + 0.526990i \(0.823320\pi\)
\(48\) 1.79491e17 0.399035
\(49\) 2.94440e17 0.527154
\(50\) 0 0
\(51\) −1.19152e18 −1.40157
\(52\) 7.82215e17 0.750397
\(53\) 5.59966e16 0.0439810 0.0219905 − 0.999758i \(-0.493000\pi\)
0.0219905 + 0.999758i \(0.493000\pi\)
\(54\) 2.83118e18 1.82740
\(55\) 0 0
\(56\) −3.88260e18 −1.71060
\(57\) −4.48193e17 −0.163976
\(58\) 5.78085e17 0.176197
\(59\) −3.50351e18 −0.892396 −0.446198 − 0.894934i \(-0.647222\pi\)
−0.446198 + 0.894934i \(0.647222\pi\)
\(60\) 0 0
\(61\) 5.19218e18 0.931937 0.465968 − 0.884801i \(-0.345706\pi\)
0.465968 + 0.884801i \(0.345706\pi\)
\(62\) 5.10566e18 0.772572
\(63\) −3.56414e18 −0.455910
\(64\) −1.30047e19 −1.40997
\(65\) 0 0
\(66\) −2.32488e19 −1.82469
\(67\) −1.63813e19 −1.09790 −0.548951 − 0.835855i \(-0.684973\pi\)
−0.548951 + 0.835855i \(0.684973\pi\)
\(68\) −5.60895e19 −3.21764
\(69\) −3.08813e19 −1.51978
\(70\) 0 0
\(71\) 1.92572e18 0.0702070 0.0351035 − 0.999384i \(-0.488824\pi\)
0.0351035 + 0.999384i \(0.488824\pi\)
\(72\) 1.62232e19 0.510676
\(73\) −4.44035e19 −1.20928 −0.604641 − 0.796498i \(-0.706683\pi\)
−0.604641 + 0.796498i \(0.706683\pi\)
\(74\) 4.73582e19 1.11805
\(75\) 0 0
\(76\) −2.10982e19 −0.376447
\(77\) 1.08600e20 1.68919
\(78\) −4.04365e19 −0.549262
\(79\) 1.36169e20 1.61806 0.809031 − 0.587767i \(-0.199993\pi\)
0.809031 + 0.587767i \(0.199993\pi\)
\(80\) 0 0
\(81\) −5.41591e19 −0.494970
\(82\) −2.64358e20 −2.12396
\(83\) 1.41806e20 1.00317 0.501584 − 0.865109i \(-0.332751\pi\)
0.501584 + 0.865109i \(0.332751\pi\)
\(84\) 2.86998e20 1.79039
\(85\) 0 0
\(86\) −3.37344e20 −1.64376
\(87\) −1.93009e19 −0.0832962
\(88\) −4.94324e20 −1.89210
\(89\) −4.36181e19 −0.148276 −0.0741381 − 0.997248i \(-0.523621\pi\)
−0.0741381 + 0.997248i \(0.523621\pi\)
\(90\) 0 0
\(91\) 1.88887e20 0.508473
\(92\) −1.45370e21 −3.48902
\(93\) −1.70466e20 −0.365230
\(94\) 1.49155e21 2.85625
\(95\) 0 0
\(96\) 2.79512e20 0.429095
\(97\) −4.15104e20 −0.571549 −0.285775 − 0.958297i \(-0.592251\pi\)
−0.285775 + 0.958297i \(0.592251\pi\)
\(98\) −7.16514e20 −0.885831
\(99\) −4.53778e20 −0.504283
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.1.1 ✓ 7
5.2 odd 4 25.22.b.d.24.2 14
5.3 odd 4 25.22.b.d.24.13 14
5.4 even 2 25.22.a.e.1.7 yes 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.22.a.d.1.1 ✓ 7 1.1 even 1 trivial
25.22.a.e.1.7 yes 7 5.4 even 2
25.22.b.d.24.2 14 5.2 odd 4
25.22.b.d.24.13 14 5.3 odd 4