Properties

Label 25.22.a.d.1.4
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 1.4
Root \(619.148\) 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)
 
\(f(q)\) \(=\) \(q-724.148 q^{2} -45482.7 q^{3} -1.57276e6 q^{4} +3.29362e7 q^{6} +8.64838e8 q^{7} +2.65756e9 q^{8} -8.39168e9 q^{9} -1.29721e11 q^{11} +7.15334e10 q^{12} -4.09683e11 q^{13} -6.26271e11 q^{14} +1.37385e12 q^{16} +4.21753e12 q^{17} +6.07682e12 q^{18} +3.90456e13 q^{19} -3.93351e13 q^{21} +9.39372e13 q^{22} +7.15736e13 q^{23} -1.20873e14 q^{24} +2.96671e14 q^{26} +8.57441e14 q^{27} -1.36018e15 q^{28} +2.25439e15 q^{29} +3.23330e15 q^{31} -6.56818e15 q^{32} +5.90006e15 q^{33} -3.05412e15 q^{34} +1.31981e16 q^{36} -5.89801e15 q^{37} -2.82748e16 q^{38} +1.86335e16 q^{39} -4.48382e16 q^{41} +2.84845e16 q^{42} +2.23819e17 q^{43} +2.04020e17 q^{44} -5.18299e16 q^{46} -4.48679e17 q^{47} -6.24865e16 q^{48} +1.89398e17 q^{49} -1.91825e17 q^{51} +6.44334e17 q^{52} -4.10000e17 q^{53} -6.20914e17 q^{54} +2.29836e18 q^{56} -1.77590e18 q^{57} -1.63251e18 q^{58} +2.11360e18 q^{59} -5.18356e18 q^{61} -2.34139e18 q^{62} -7.25744e18 q^{63} +1.87516e18 q^{64} -4.27252e18 q^{66} +2.32860e19 q^{67} -6.63318e18 q^{68} -3.25536e18 q^{69} -2.64444e19 q^{71} -2.23014e19 q^{72} -5.17924e19 q^{73} +4.27103e18 q^{74} -6.14094e19 q^{76} -1.12188e20 q^{77} -1.34934e19 q^{78} +1.00279e20 q^{79} +4.87812e19 q^{81} +3.24695e19 q^{82} -2.11124e20 q^{83} +6.18648e19 q^{84} -1.62078e20 q^{86} -1.02536e20 q^{87} -3.44741e20 q^{88} +4.42627e20 q^{89} -3.54309e20 q^{91} -1.12568e20 q^{92} -1.47059e20 q^{93} +3.24910e20 q^{94} +2.98738e20 q^{96} -3.86433e20 q^{97} -1.37152e20 q^{98} +1.08858e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 737 q^{2} - 50381 q^{3} + 7749909 q^{4} - 30442931 q^{6} - 817782442 q^{7} + 3231986745 q^{8} + 1856104186 q^{9} - 17026930591 q^{11} - 286884094477 q^{12} + 325366982844 q^{13} + 13403645898 q^{14}+ \cdots + 26\!\cdots\!82 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −724.148 −0.500049 −0.250024 − 0.968240i \(-0.580439\pi\)
−0.250024 + 0.968240i \(0.580439\pi\)
\(3\) −45482.7 −0.444706 −0.222353 − 0.974966i \(-0.571374\pi\)
−0.222353 + 0.974966i \(0.571374\pi\)
\(4\) −1.57276e6 −0.749951
\(5\) 0 0
\(6\) 3.29362e7 0.222375
\(7\) 8.64838e8 1.15719 0.578596 − 0.815615i \(-0.303601\pi\)
0.578596 + 0.815615i \(0.303601\pi\)
\(8\) 2.65756e9 0.875061
\(9\) −8.39168e9 −0.802237
\(10\) 0 0
\(11\) −1.29721e11 −1.50795 −0.753975 − 0.656903i \(-0.771866\pi\)
−0.753975 + 0.656903i \(0.771866\pi\)
\(12\) 7.15334e10 0.333508
\(13\) −4.09683e11 −0.824220 −0.412110 − 0.911134i \(-0.635208\pi\)
−0.412110 + 0.911134i \(0.635208\pi\)
\(14\) −6.26271e11 −0.578652
\(15\) 0 0
\(16\) 1.37385e12 0.312378
\(17\) 4.21753e12 0.507394 0.253697 − 0.967284i \(-0.418353\pi\)
0.253697 + 0.967284i \(0.418353\pi\)
\(18\) 6.07682e12 0.401158
\(19\) 3.90456e13 1.46103 0.730515 − 0.682897i \(-0.239280\pi\)
0.730515 + 0.682897i \(0.239280\pi\)
\(20\) 0 0
\(21\) −3.93351e13 −0.514610
\(22\) 9.39372e13 0.754049
\(23\) 7.15736e13 0.360255 0.180128 − 0.983643i \(-0.442349\pi\)
0.180128 + 0.983643i \(0.442349\pi\)
\(24\) −1.20873e14 −0.389145
\(25\) 0 0
\(26\) 2.96671e14 0.412150
\(27\) 8.57441e14 0.801465
\(28\) −1.36018e15 −0.867837
\(29\) 2.25439e15 0.995060 0.497530 − 0.867447i \(-0.334240\pi\)
0.497530 + 0.867447i \(0.334240\pi\)
\(30\) 0 0
\(31\) 3.23330e15 0.708514 0.354257 − 0.935148i \(-0.384734\pi\)
0.354257 + 0.935148i \(0.384734\pi\)
\(32\) −6.56818e15 −1.03127
\(33\) 5.90006e15 0.670594
\(34\) −3.05412e15 −0.253722
\(35\) 0 0
\(36\) 1.31981e16 0.601638
\(37\) −5.89801e15 −0.201645 −0.100822 − 0.994904i \(-0.532147\pi\)
−0.100822 + 0.994904i \(0.532147\pi\)
\(38\) −2.82748e16 −0.730586
\(39\) 1.86335e16 0.366535
\(40\) 0 0
\(41\) −4.48382e16 −0.521696 −0.260848 − 0.965380i \(-0.584002\pi\)
−0.260848 + 0.965380i \(0.584002\pi\)
\(42\) 2.84845e16 0.257330
\(43\) 2.23819e17 1.57935 0.789675 − 0.613526i \(-0.210249\pi\)
0.789675 + 0.613526i \(0.210249\pi\)
\(44\) 2.04020e17 1.13089
\(45\) 0 0
\(46\) −5.18299e16 −0.180145
\(47\) −4.48679e17 −1.24425 −0.622126 − 0.782917i \(-0.713731\pi\)
−0.622126 + 0.782917i \(0.713731\pi\)
\(48\) −6.24865e16 −0.138916
\(49\) 1.89398e17 0.339092
\(50\) 0 0
\(51\) −1.91825e17 −0.225641
\(52\) 6.44334e17 0.618124
\(53\) −4.10000e17 −0.322023 −0.161012 − 0.986953i \(-0.551476\pi\)
−0.161012 + 0.986953i \(0.551476\pi\)
\(54\) −6.20914e17 −0.400772
\(55\) 0 0
\(56\) 2.29836e18 1.01261
\(57\) −1.77590e18 −0.649728
\(58\) −1.63251e18 −0.497579
\(59\) 2.11360e18 0.538365 0.269183 − 0.963089i \(-0.413247\pi\)
0.269183 + 0.963089i \(0.413247\pi\)
\(60\) 0 0
\(61\) −5.18356e18 −0.930389 −0.465194 − 0.885208i \(-0.654016\pi\)
−0.465194 + 0.885208i \(0.654016\pi\)
\(62\) −2.34139e18 −0.354292
\(63\) −7.25744e18 −0.928341
\(64\) 1.87516e18 0.203305
\(65\) 0 0
\(66\) −4.27252e18 −0.335330
\(67\) 2.32860e19 1.56066 0.780331 − 0.625367i \(-0.215051\pi\)
0.780331 + 0.625367i \(0.215051\pi\)
\(68\) −6.63318e18 −0.380520
\(69\) −3.25536e18 −0.160208
\(70\) 0 0
\(71\) −2.64444e19 −0.964098 −0.482049 − 0.876144i \(-0.660107\pi\)
−0.482049 + 0.876144i \(0.660107\pi\)
\(72\) −2.23014e19 −0.702006
\(73\) −5.17924e19 −1.41051 −0.705255 − 0.708953i \(-0.749168\pi\)
−0.705255 + 0.708953i \(0.749168\pi\)
\(74\) 4.27103e18 0.100832
\(75\) 0 0
\(76\) −6.14094e19 −1.09570
\(77\) −1.12188e20 −1.74499
\(78\) −1.34934e19 −0.183286
\(79\) 1.00279e20 1.19158 0.595790 − 0.803140i \(-0.296839\pi\)
0.595790 + 0.803140i \(0.296839\pi\)
\(80\) 0 0
\(81\) 4.87812e19 0.445821
\(82\) 3.24695e19 0.260874
\(83\) −2.11124e20 −1.49355 −0.746773 − 0.665079i \(-0.768398\pi\)
−0.746773 + 0.665079i \(0.768398\pi\)
\(84\) 6.18648e19 0.385932
\(85\) 0 0
\(86\) −1.62078e20 −0.789752
\(87\) −1.02536e20 −0.442509
\(88\) −3.44741e20 −1.31955
\(89\) 4.42627e20 1.50468 0.752338 − 0.658777i \(-0.228926\pi\)
0.752338 + 0.658777i \(0.228926\pi\)
\(90\) 0 0
\(91\) −3.54309e20 −0.953780
\(92\) −1.12568e20 −0.270174
\(93\) −1.47059e20 −0.315080
\(94\) 3.24910e20 0.622187
\(95\) 0 0
\(96\) 2.98738e20 0.458610
\(97\) −3.86433e20 −0.532073 −0.266037 − 0.963963i \(-0.585714\pi\)
−0.266037 + 0.963963i \(0.585714\pi\)
\(98\) −1.37152e20 −0.169562
\(99\) 1.08858e21 1.20973
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.1.4 ✓ 7
5.2 odd 4 25.22.b.d.24.7 14
5.3 odd 4 25.22.b.d.24.8 14
5.4 even 2 25.22.a.e.1.4 yes 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.22.a.d.1.4 ✓ 7 1.1 even 1 trivial
25.22.a.e.1.4 yes 7 5.4 even 2
25.22.b.d.24.7 14 5.2 odd 4
25.22.b.d.24.8 14 5.3 odd 4