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 = [4,-2910] 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: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 1929606x^{2} - 743130000x + 239341586400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 5^{2}\cdot 7 \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-844.370\) 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)
 
\(f(q)\) \(=\) \(q-2416.74 q^{2} -157402. q^{3} +3.74348e6 q^{4} +3.80399e8 q^{6} +6.35419e8 q^{7} -3.97874e9 q^{8} +1.43149e10 q^{9} -1.50079e11 q^{11} -5.89229e11 q^{12} +4.38323e11 q^{13} -1.53564e12 q^{14} +1.76494e12 q^{16} -1.31267e13 q^{17} -3.45954e13 q^{18} +2.30078e13 q^{19} -1.00016e14 q^{21} +3.62703e14 q^{22} -1.02059e14 q^{23} +6.26260e14 q^{24} -1.05931e15 q^{26} -6.06711e14 q^{27} +2.37868e15 q^{28} -2.42933e15 q^{29} +4.77446e15 q^{31} +4.07862e15 q^{32} +2.36227e16 q^{33} +3.17238e16 q^{34} +5.35875e16 q^{36} +2.00142e16 q^{37} -5.56038e16 q^{38} -6.89927e16 q^{39} +5.81468e16 q^{41} +2.41712e17 q^{42} -4.75499e16 q^{43} -5.61819e17 q^{44} +2.46651e17 q^{46} -3.37481e16 q^{47} -2.77805e17 q^{48} -1.54789e17 q^{49} +2.06616e18 q^{51} +1.64085e18 q^{52} +2.87521e17 q^{53} +1.46626e18 q^{54} -2.52817e18 q^{56} -3.62146e18 q^{57} +5.87106e18 q^{58} +3.80923e18 q^{59} +2.95437e18 q^{61} -1.15386e19 q^{62} +9.09595e18 q^{63} -1.35583e19 q^{64} -5.70900e19 q^{66} +1.23532e18 q^{67} -4.91395e19 q^{68} +1.60643e19 q^{69} +1.69929e19 q^{71} -5.69553e19 q^{72} -8.57496e18 q^{73} -4.83691e19 q^{74} +8.61290e19 q^{76} -9.53632e19 q^{77} +1.66737e20 q^{78} -9.76372e19 q^{79} -5.42416e19 q^{81} -1.40526e20 q^{82} -5.22174e19 q^{83} -3.74407e20 q^{84} +1.14916e20 q^{86} +3.82380e20 q^{87} +5.97127e20 q^{88} +3.85562e20 q^{89} +2.78519e20 q^{91} -3.82057e20 q^{92} -7.51508e20 q^{93} +8.15604e19 q^{94} -6.41981e20 q^{96} +6.77498e20 q^{97} +3.74085e20 q^{98} -2.14837e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2910 q^{2} - 83240 q^{3} + 9165268 q^{4} - 158524712 q^{6} - 512613800 q^{7} - 5167363080 q^{8} + 21732888532 q^{9} + 33727076448 q^{11} + 142435377680 q^{12} - 863532165080 q^{13} + 2725405637616 q^{14}+ \cdots - 15\!\cdots\!16 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2416.74 −1.66884 −0.834420 − 0.551128i \(-0.814197\pi\)
−0.834420 + 0.551128i \(0.814197\pi\)
\(3\) −157402. −1.53899 −0.769495 − 0.638653i \(-0.779492\pi\)
−0.769495 + 0.638653i \(0.779492\pi\)
\(4\) 3.74348e6 1.78503
\(5\) 0 0
\(6\) 3.80399e8 2.56833
\(7\) 6.35419e8 0.850218 0.425109 − 0.905142i \(-0.360236\pi\)
0.425109 + 0.905142i \(0.360236\pi\)
\(8\) −3.97874e9 −1.31009
\(9\) 1.43149e10 1.36849
\(10\) 0 0
\(11\) −1.50079e11 −1.74461 −0.872304 − 0.488965i \(-0.837375\pi\)
−0.872304 + 0.488965i \(0.837375\pi\)
\(12\) −5.89229e11 −2.74714
\(13\) 4.38323e11 0.881838 0.440919 − 0.897547i \(-0.354653\pi\)
0.440919 + 0.897547i \(0.354653\pi\)
\(14\) −1.53564e12 −1.41888
\(15\) 0 0
\(16\) 1.76494e12 0.401302
\(17\) −1.31267e13 −1.57922 −0.789608 − 0.613611i \(-0.789716\pi\)
−0.789608 + 0.613611i \(0.789716\pi\)
\(18\) −3.45954e13 −2.28379
\(19\) 2.30078e13 0.860917 0.430459 − 0.902610i \(-0.358352\pi\)
0.430459 + 0.902610i \(0.358352\pi\)
\(20\) 0 0
\(21\) −1.00016e14 −1.30848
\(22\) 3.62703e14 2.91147
\(23\) −1.02059e14 −0.513701 −0.256851 − 0.966451i \(-0.582685\pi\)
−0.256851 + 0.966451i \(0.582685\pi\)
\(24\) 6.26260e14 2.01622
\(25\) 0 0
\(26\) −1.05931e15 −1.47165
\(27\) −6.06711e14 −0.567103
\(28\) 2.37868e15 1.51767
\(29\) −2.42933e15 −1.07228 −0.536139 − 0.844129i \(-0.680118\pi\)
−0.536139 + 0.844129i \(0.680118\pi\)
\(30\) 0 0
\(31\) 4.77446e15 1.04623 0.523114 − 0.852263i \(-0.324770\pi\)
0.523114 + 0.852263i \(0.324770\pi\)
\(32\) 4.07862e15 0.640381
\(33\) 2.36227e16 2.68493
\(34\) 3.17238e16 2.63546
\(35\) 0 0
\(36\) 5.35875e16 2.44280
\(37\) 2.00142e16 0.684259 0.342129 − 0.939653i \(-0.388852\pi\)
0.342129 + 0.939653i \(0.388852\pi\)
\(38\) −5.56038e16 −1.43673
\(39\) −6.89927e16 −1.35714
\(40\) 0 0
\(41\) 5.81468e16 0.676542 0.338271 − 0.941049i \(-0.390158\pi\)
0.338271 + 0.941049i \(0.390158\pi\)
\(42\) 2.41712e17 2.18364
\(43\) −4.75499e16 −0.335529 −0.167765 − 0.985827i \(-0.553655\pi\)
−0.167765 + 0.985827i \(0.553655\pi\)
\(44\) −5.61819e17 −3.11418
\(45\) 0 0
\(46\) 2.46651e17 0.857285
\(47\) −3.37481e16 −0.0935884 −0.0467942 − 0.998905i \(-0.514901\pi\)
−0.0467942 + 0.998905i \(0.514901\pi\)
\(48\) −2.77805e17 −0.617599
\(49\) −1.54789e17 −0.277129
\(50\) 0 0
\(51\) 2.06616e18 2.43040
\(52\) 1.64085e18 1.57411
\(53\) 2.87521e17 0.225826 0.112913 − 0.993605i \(-0.463982\pi\)
0.112913 + 0.993605i \(0.463982\pi\)
\(54\) 1.46626e18 0.946405
\(55\) 0 0
\(56\) −2.52817e18 −1.11386
\(57\) −3.62146e18 −1.32494
\(58\) 5.87106e18 1.78946
\(59\) 3.80923e18 0.970267 0.485133 − 0.874440i \(-0.338771\pi\)
0.485133 + 0.874440i \(0.338771\pi\)
\(60\) 0 0
\(61\) 2.95437e18 0.530276 0.265138 − 0.964210i \(-0.414582\pi\)
0.265138 + 0.964210i \(0.414582\pi\)
\(62\) −1.15386e19 −1.74599
\(63\) 9.09595e18 1.16352
\(64\) −1.35583e19 −1.47000
\(65\) 0 0
\(66\) −5.70900e19 −4.48073
\(67\) 1.23532e18 0.0827932 0.0413966 − 0.999143i \(-0.486819\pi\)
0.0413966 + 0.999143i \(0.486819\pi\)
\(68\) −4.91395e19 −2.81895
\(69\) 1.60643e19 0.790581
\(70\) 0 0
\(71\) 1.69929e19 0.619520 0.309760 − 0.950815i \(-0.399751\pi\)
0.309760 + 0.950815i \(0.399751\pi\)
\(72\) −5.69553e19 −1.79285
\(73\) −8.57496e18 −0.233530 −0.116765 − 0.993160i \(-0.537252\pi\)
−0.116765 + 0.993160i \(0.537252\pi\)
\(74\) −4.83691e19 −1.14192
\(75\) 0 0
\(76\) 8.61290e19 1.53676
\(77\) −9.53632e19 −1.48330
\(78\) 1.66737e20 2.26485
\(79\) −9.76372e19 −1.16019 −0.580097 − 0.814547i \(-0.696986\pi\)
−0.580097 + 0.814547i \(0.696986\pi\)
\(80\) 0 0
\(81\) −5.42416e19 −0.495724
\(82\) −1.40526e20 −1.12904
\(83\) −5.22174e19 −0.369399 −0.184699 − 0.982795i \(-0.559131\pi\)
−0.184699 + 0.982795i \(0.559131\pi\)
\(84\) −3.74407e20 −2.33567
\(85\) 0 0
\(86\) 1.14916e20 0.559945
\(87\) 3.82380e20 1.65023
\(88\) 5.97127e20 2.28559
\(89\) 3.85562e20 1.31069 0.655344 − 0.755331i \(-0.272524\pi\)
0.655344 + 0.755331i \(0.272524\pi\)
\(90\) 0 0
\(91\) 2.78519e20 0.749755
\(92\) −3.82057e20 −0.916972
\(93\) −7.51508e20 −1.61014
\(94\) 8.15604e19 0.156184
\(95\) 0 0
\(96\) −6.41981e20 −0.985541
\(97\) 6.77498e20 0.932835 0.466417 − 0.884565i \(-0.345544\pi\)
0.466417 + 0.884565i \(0.345544\pi\)
\(98\) 3.74085e20 0.462484
\(99\) −2.14837e21 −2.38748
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.c.1.2 4
5.2 odd 4 25.22.b.c.24.2 8
5.3 odd 4 25.22.b.c.24.7 8
5.4 even 2 5.22.a.b.1.3 ✓ 4
15.14 odd 2 45.22.a.f.1.2 4
20.19 odd 2 80.22.a.g.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.22.a.b.1.3 ✓ 4 5.4 even 2
25.22.a.c.1.2 4 1.1 even 1 trivial
25.22.b.c.24.2 8 5.2 odd 4
25.22.b.c.24.7 8 5.3 odd 4
45.22.a.f.1.2 4 15.14 odd 2
80.22.a.g.1.1 4 20.19 odd 2