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Properties

Label 625.6.a.d
Level $625$
Weight $6$
Character orbit 625.a
Self dual yes
Analytic conductor $100.240$
Analytic rank $1$
Dimension $22$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [625,6,Mod(1,625)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(625, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("625.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 625.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(100.239887383\)
Analytic rank: \(1\)
Dimension: \(22\)
Twist minimal: no (minimal twist has level 25)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 9 q^{2} + 2 q^{3} + 289 q^{4} - 111 q^{6} - 101 q^{7} + 540 q^{8} + 1166 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 9 q^{2} + 2 q^{3} + 289 q^{4} - 111 q^{6} - 101 q^{7} + 540 q^{8} + 1166 q^{9} - 1046 q^{11} - 1271 q^{12} - 1453 q^{13} - 2222 q^{14} + 1317 q^{16} + 2174 q^{17} + 4272 q^{18} - 3765 q^{19} - 4811 q^{21} - 12 q^{22} - 1488 q^{23} - 9810 q^{24} - 8941 q^{26} - 13570 q^{27} - 5587 q^{28} - 9035 q^{29} - 10426 q^{31} + 15339 q^{32} + 6874 q^{33} - 16027 q^{34} - 7098 q^{36} + 24944 q^{37} - 47645 q^{38} - 28703 q^{39} - 36606 q^{41} - 34782 q^{42} + 30597 q^{43} - 47892 q^{44} - 30881 q^{46} - 81481 q^{47} - 20803 q^{48} - 12241 q^{49} - 60566 q^{51} - 79836 q^{52} + 75802 q^{53} - 65085 q^{54} - 83335 q^{56} + 146075 q^{57} - 30030 q^{58} - 87095 q^{59} - 71476 q^{61} + 131553 q^{62} - 111758 q^{63} - 30736 q^{64} - 64222 q^{66} - 85706 q^{67} + 61943 q^{68} - 93448 q^{69} - 125571 q^{71} + 318570 q^{72} + 157982 q^{73} - 89682 q^{74} - 134640 q^{76} - 236842 q^{77} + 145204 q^{78} - 173645 q^{79} - 156538 q^{81} - 176667 q^{82} - 166408 q^{83} - 213437 q^{84} - 182891 q^{86} - 175085 q^{87} - 612480 q^{88} - 280940 q^{89} - 182911 q^{91} - 534131 q^{92} - 190216 q^{93} - 376702 q^{94} - 484051 q^{96} + 636469 q^{97} + 819128 q^{98} - 340388 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −10.1039 −3.91763 70.0891 0 39.5834 54.0512 −384.849 −227.652 0
1.2 −9.04635 −2.46017 49.8365 0 22.2555 88.6608 −161.355 −236.948 0
1.3 −8.76311 16.3027 44.7921 0 −142.862 −72.9673 −112.098 22.7766 0
1.4 −7.99068 17.6131 31.8510 0 −140.741 40.0850 1.19042 67.2208 0
1.5 −6.87658 −25.8543 15.2873 0 177.789 −5.99482 114.926 425.443 0
1.6 −6.27899 −11.9788 7.42574 0 75.2146 −78.1631 154.302 −99.5089 0
1.7 −3.29869 −25.1493 −21.1186 0 82.9597 70.1855 175.222 389.487 0
1.8 −2.82379 25.6096 −24.0262 0 −72.3161 −43.3673 158.206 412.849 0
1.9 −2.55720 −10.5385 −25.4607 0 26.9491 −36.6142 146.939 −131.940 0
1.10 −1.87633 13.9964 −28.4794 0 −26.2619 −222.130 113.479 −47.1016 0
1.11 −1.44858 21.3070 −29.9016 0 −30.8649 250.563 89.6695 210.987 0
1.12 0.257959 6.09028 −31.9335 0 1.57104 127.575 −16.4922 −205.908 0
1.13 3.08370 −0.735439 −22.4908 0 −2.26787 57.2936 −168.033 −242.459 0
1.14 4.19638 −16.0327 −14.3904 0 −67.2792 −192.901 −194.672 14.0464 0
1.15 4.25718 −16.4202 −13.8765 0 −69.9036 −22.2636 −195.304 26.6221 0
1.16 5.55593 21.9790 −1.13164 0 122.114 −153.453 −184.077 240.076 0
1.17 7.43988 5.83849 23.3518 0 43.4377 204.573 −64.3418 −208.912 0
1.18 7.56817 −25.8619 25.2772 0 −195.728 130.915 −50.8794 425.840 0
1.19 7.65919 24.3690 26.6632 0 186.647 −209.197 −40.8754 350.846 0
1.20 8.77272 6.93578 44.9606 0 60.8456 102.277 113.699 −194.895 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.22
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 625.6.a.d 22
5.b even 2 1 625.6.a.c 22
25.d even 5 2 25.6.d.a ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.6.d.a ✓ 44 25.d even 5 2
625.6.a.c 22 5.b even 2 1
625.6.a.d 22 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{22} - 9 T_{2}^{21} - 456 T_{2}^{20} + 3975 T_{2}^{19} + 88240 T_{2}^{18} - 739185 T_{2}^{17} + \cdots - 604657485234176 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(625))\). Copy content Toggle raw display