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 1.1
Root \(-886.582\) 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)
 
\(f(q)\) \(=\) \(q-2501.16 q^{2} +181393. q^{3} +4.15867e6 q^{4} -4.53695e8 q^{6} -9.51907e8 q^{7} -5.15621e9 q^{8} +2.24432e10 q^{9} +6.26620e10 q^{11} +7.54356e11 q^{12} -3.65290e11 q^{13} +2.38088e12 q^{14} +4.17515e12 q^{16} +4.06125e12 q^{17} -5.61342e13 q^{18} -3.33870e12 q^{19} -1.72670e14 q^{21} -1.56728e14 q^{22} -1.41899e14 q^{23} -9.35302e14 q^{24} +9.13651e14 q^{26} +2.17361e15 q^{27} -3.95867e15 q^{28} -2.13837e15 q^{29} +2.75237e15 q^{31} +3.70604e14 q^{32} +1.13665e16 q^{33} -1.01579e16 q^{34} +9.33340e16 q^{36} -3.43997e16 q^{37} +8.35063e15 q^{38} -6.62613e16 q^{39} -1.12163e17 q^{41} +4.31875e17 q^{42} -4.13618e16 q^{43} +2.60591e17 q^{44} +3.54913e17 q^{46} +5.44036e17 q^{47} +7.57345e17 q^{48} +3.47581e17 q^{49} +7.36684e17 q^{51} -1.51912e18 q^{52} -1.34608e18 q^{53} -5.43656e18 q^{54} +4.90823e18 q^{56} -6.05617e17 q^{57} +5.34841e18 q^{58} +7.47699e18 q^{59} -1.07752e19 q^{61} -6.88413e18 q^{62} -2.13639e19 q^{63} -9.68287e18 q^{64} -2.84294e19 q^{66} +1.74771e18 q^{67} +1.68894e19 q^{68} -2.57396e19 q^{69} -2.17207e19 q^{71} -1.15722e20 q^{72} +1.65237e19 q^{73} +8.60394e19 q^{74} -1.38845e19 q^{76} -5.96484e19 q^{77} +1.65730e20 q^{78} +7.06800e19 q^{79} +1.59515e20 q^{81} +2.80539e20 q^{82} -1.53121e19 q^{83} -7.18077e20 q^{84} +1.03453e20 q^{86} -3.87886e20 q^{87} -3.23098e20 q^{88} -1.94580e20 q^{89} +3.47722e20 q^{91} -5.90113e20 q^{92} +4.99262e20 q^{93} -1.36072e21 q^{94} +6.72251e19 q^{96} +7.93694e20 q^{97} -8.69358e20 q^{98} +1.40634e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2910 q^{2} - 83240 q^{3} + 9165268 q^{4} - 158524712 q^{6} - 512613800 q^{7} - 5167363080 q^{8} + 21732888532 q^{9} + 33727076448 q^{11} + 142435377680 q^{12} - 863532165080 q^{13} + 2725405637616 q^{14}+ \cdots - 15\!\cdots\!16 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2501.16 −1.72714 −0.863570 − 0.504230i \(-0.831777\pi\)
−0.863570 + 0.504230i \(0.831777\pi\)
\(3\) 181393. 1.77357 0.886785 − 0.462182i \(-0.152933\pi\)
0.886785 + 0.462182i \(0.152933\pi\)
\(4\) 4.15867e6 1.98301
\(5\) 0 0
\(6\) −4.53695e8 −3.06320
\(7\) −9.51907e8 −1.27369 −0.636847 − 0.770990i \(-0.719762\pi\)
−0.636847 + 0.770990i \(0.719762\pi\)
\(8\) −5.15621e9 −1.69780
\(9\) 2.24432e10 2.14555
\(10\) 0 0
\(11\) 6.26620e10 0.728418 0.364209 − 0.931317i \(-0.381339\pi\)
0.364209 + 0.931317i \(0.381339\pi\)
\(12\) 7.54356e11 3.51701
\(13\) −3.65290e11 −0.734908 −0.367454 − 0.930042i \(-0.619770\pi\)
−0.367454 + 0.930042i \(0.619770\pi\)
\(14\) 2.38088e12 2.19985
\(15\) 0 0
\(16\) 4.17515e12 0.949320
\(17\) 4.06125e12 0.488592 0.244296 − 0.969701i \(-0.421443\pi\)
0.244296 + 0.969701i \(0.421443\pi\)
\(18\) −5.61342e13 −3.70566
\(19\) −3.33870e12 −0.124929 −0.0624646 − 0.998047i \(-0.519896\pi\)
−0.0624646 + 0.998047i \(0.519896\pi\)
\(20\) 0 0
\(21\) −1.72670e14 −2.25899
\(22\) −1.56728e14 −1.25808
\(23\) −1.41899e14 −0.714230 −0.357115 − 0.934060i \(-0.616240\pi\)
−0.357115 + 0.934060i \(0.616240\pi\)
\(24\) −9.35302e14 −3.01116
\(25\) 0 0
\(26\) 9.13651e14 1.26929
\(27\) 2.17361e15 2.03171
\(28\) −3.95867e15 −2.52575
\(29\) −2.13837e15 −0.943851 −0.471925 − 0.881638i \(-0.656441\pi\)
−0.471925 + 0.881638i \(0.656441\pi\)
\(30\) 0 0
\(31\) 2.75237e15 0.603128 0.301564 − 0.953446i \(-0.402491\pi\)
0.301564 + 0.953446i \(0.402491\pi\)
\(32\) 3.70604e14 0.0581883
\(33\) 1.13665e16 1.29190
\(34\) −1.01579e16 −0.843866
\(35\) 0 0
\(36\) 9.33340e16 4.25465
\(37\) −3.43997e16 −1.17608 −0.588040 − 0.808832i \(-0.700100\pi\)
−0.588040 + 0.808832i \(0.700100\pi\)
\(38\) 8.35063e15 0.215770
\(39\) −6.62613e16 −1.30341
\(40\) 0 0
\(41\) −1.12163e17 −1.30503 −0.652514 − 0.757777i \(-0.726286\pi\)
−0.652514 + 0.757777i \(0.726286\pi\)
\(42\) 4.31875e17 3.90158
\(43\) −4.13618e16 −0.291864 −0.145932 − 0.989295i \(-0.546618\pi\)
−0.145932 + 0.989295i \(0.546618\pi\)
\(44\) 2.60591e17 1.44446
\(45\) 0 0
\(46\) 3.54913e17 1.23357
\(47\) 5.44036e17 1.50869 0.754345 − 0.656479i \(-0.227955\pi\)
0.754345 + 0.656479i \(0.227955\pi\)
\(48\) 7.57345e17 1.68369
\(49\) 3.47581e17 0.622296
\(50\) 0 0
\(51\) 7.36684e17 0.866552
\(52\) −1.51912e18 −1.45733
\(53\) −1.34608e18 −1.05724 −0.528620 − 0.848859i \(-0.677290\pi\)
−0.528620 + 0.848859i \(0.677290\pi\)
\(54\) −5.43656e18 −3.50905
\(55\) 0 0
\(56\) 4.90823e18 2.16247
\(57\) −6.05617e17 −0.221571
\(58\) 5.34841e18 1.63016
\(59\) 7.47699e18 1.90450 0.952249 − 0.305323i \(-0.0987643\pi\)
0.952249 + 0.305323i \(0.0987643\pi\)
\(60\) 0 0
\(61\) −1.07752e19 −1.93402 −0.967010 − 0.254738i \(-0.918011\pi\)
−0.967010 + 0.254738i \(0.918011\pi\)
\(62\) −6.88413e18 −1.04169
\(63\) −2.13639e19 −2.73277
\(64\) −9.68287e18 −1.04982
\(65\) 0 0
\(66\) −2.84294e19 −2.23129
\(67\) 1.74771e18 0.117135 0.0585673 − 0.998283i \(-0.481347\pi\)
0.0585673 + 0.998283i \(0.481347\pi\)
\(68\) 1.68894e19 0.968883
\(69\) −2.57396e19 −1.26674
\(70\) 0 0
\(71\) −2.17207e19 −0.791885 −0.395942 − 0.918275i \(-0.629582\pi\)
−0.395942 + 0.918275i \(0.629582\pi\)
\(72\) −1.15722e20 −3.64271
\(73\) 1.65237e19 0.450004 0.225002 − 0.974358i \(-0.427761\pi\)
0.225002 + 0.974358i \(0.427761\pi\)
\(74\) 8.60394e19 2.03125
\(75\) 0 0
\(76\) −1.38845e19 −0.247736
\(77\) −5.96484e19 −0.927782
\(78\) 1.65730e20 2.25117
\(79\) 7.06800e19 0.839870 0.419935 − 0.907554i \(-0.362053\pi\)
0.419935 + 0.907554i \(0.362053\pi\)
\(80\) 0 0
\(81\) 1.59515e20 1.45784
\(82\) 2.80539e20 2.25397
\(83\) −1.53121e19 −0.108322 −0.0541608 − 0.998532i \(-0.517248\pi\)
−0.0541608 + 0.998532i \(0.517248\pi\)
\(84\) −7.18077e20 −4.47959
\(85\) 0 0
\(86\) 1.03453e20 0.504091
\(87\) −3.87886e20 −1.67399
\(88\) −3.23098e20 −1.23671
\(89\) −1.94580e20 −0.661461 −0.330730 − 0.943725i \(-0.607295\pi\)
−0.330730 + 0.943725i \(0.607295\pi\)
\(90\) 0 0
\(91\) 3.47722e20 0.936048
\(92\) −5.90113e20 −1.41632
\(93\) 4.99262e20 1.06969
\(94\) −1.36072e21 −2.60572
\(95\) 0 0
\(96\) 6.72251e19 0.103201
\(97\) 7.93694e20 1.09282 0.546412 − 0.837517i \(-0.315993\pi\)
0.546412 + 0.837517i \(0.315993\pi\)
\(98\) −8.69358e20 −1.07479
\(99\) 1.40634e21 1.56286
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.1.1 4
5.2 odd 4 25.22.b.c.24.1 8
5.3 odd 4 25.22.b.c.24.8 8
5.4 even 2 5.22.a.b.1.4 ✓ 4
15.14 odd 2 45.22.a.f.1.1 4
20.19 odd 2 80.22.a.g.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.22.a.b.1.4 ✓ 4 5.4 even 2
25.22.a.c.1.1 4 1.1 even 1 trivial
25.22.b.c.24.1 8 5.2 odd 4
25.22.b.c.24.8 8 5.3 odd 4
45.22.a.f.1.1 4 15.14 odd 2
80.22.a.g.1.4 4 20.19 odd 2