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Properties

Label 25.34.b.d
Level $25$
Weight $34$
Character orbit 25.b
Analytic conductor $172.457$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(172.457072203\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22q - 98676413354q^{4} - 35567955353446q^{6} - 48599275949251316q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 22q - 98676413354q^{4} - 35567955353446q^{6} - 48599275949251316q^{9} - 388975069134227046q^{11} - 19390008595334902068q^{14} + \)\(57\!\cdots\!02\)\(q^{16} + \)\(46\!\cdots\!30\)\(q^{19} + \)\(79\!\cdots\!24\)\(q^{21} + \)\(40\!\cdots\!90\)\(q^{24} + \)\(46\!\cdots\!84\)\(q^{26} - \)\(33\!\cdots\!80\)\(q^{29} - \)\(37\!\cdots\!76\)\(q^{31} + \)\(89\!\cdots\!02\)\(q^{34} + \)\(38\!\cdots\!12\)\(q^{36} + \)\(71\!\cdots\!88\)\(q^{39} + \)\(10\!\cdots\!34\)\(q^{41} + \)\(59\!\cdots\!22\)\(q^{44} + \)\(33\!\cdots\!64\)\(q^{46} + \)\(50\!\cdots\!46\)\(q^{49} + \)\(13\!\cdots\!14\)\(q^{51} + \)\(42\!\cdots\!30\)\(q^{54} + \)\(67\!\cdots\!20\)\(q^{56} + \)\(13\!\cdots\!40\)\(q^{59} - \)\(68\!\cdots\!96\)\(q^{61} - \)\(28\!\cdots\!74\)\(q^{64} - \)\(32\!\cdots\!22\)\(q^{66} - \)\(14\!\cdots\!52\)\(q^{69} - \)\(11\!\cdots\!36\)\(q^{71} - \)\(16\!\cdots\!08\)\(q^{74} - \)\(75\!\cdots\!10\)\(q^{76} - \)\(56\!\cdots\!80\)\(q^{79} - \)\(90\!\cdots\!58\)\(q^{81} + \)\(59\!\cdots\!32\)\(q^{84} + \)\(91\!\cdots\!24\)\(q^{86} + \)\(48\!\cdots\!10\)\(q^{89} - \)\(69\!\cdots\!96\)\(q^{91} - \)\(46\!\cdots\!88\)\(q^{94} - \)\(36\!\cdots\!06\)\(q^{96} - \)\(26\!\cdots\!12\)\(q^{99} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1 − 177167.i − 1.20911e8i −2.27982e10 0 −2.14214e13 − 1.11259e14i 2.51723e15i −9.06036e15 0
24.2 − 171393.i 3.66048e7i −2.07856e10 0 6.27381e12 − 4.77732e13i 2.09024e15i 4.21915e15 0
24.3 − 153994.i 7.94763e7i −1.51243e10 0 1.22389e13 4.85062e13i 1.00626e15i −7.57418e14 0
24.4 − 140551.i − 1.29136e8i −1.11647e10 0 −1.81502e13 4.00085e13i 3.61884e14i −1.11170e16 0
24.5 − 109239.i − 1.96178e7i −3.34318e9 0 −2.14302e12 − 7.24227e12i − 5.73150e14i 5.17420e15 0
24.6 − 104621.i − 2.08087e7i −2.35554e9 0 −2.17702e12 5.34780e13i − 6.52247e14i 5.12606e15 0
24.7 − 100295.i 1.18201e8i −1.46921e9 0 1.18550e13 − 1.07417e13i − 7.14175e14i −8.41244e15 0
24.8 − 61242.8i − 1.16397e8i 4.83925e9 0 −7.12847e12 1.52684e14i − 8.22441e14i −7.98915e15 0
24.9 − 41215.3i 1.10688e8i 6.89123e9 0 4.56203e12 − 7.02715e13i − 6.38061e14i −6.69270e15 0
24.10 − 34748.6i − 4.98225e7i 7.38247e9 0 −1.73126e12 − 1.47813e14i − 5.55018e14i 3.07678e15 0
24.11 − 642.905i 5.85306e7i 8.58952e9 0 3.76296e10 1.03914e14i − 1.10448e13i 2.13323e15 0
24.12 642.905i − 5.85306e7i 8.58952e9 0 3.76296e10 − 1.03914e14i 1.10448e13i 2.13323e15 0
24.13 34748.6i 4.98225e7i 7.38247e9 0 −1.73126e12 1.47813e14i 5.55018e14i 3.07678e15 0
24.14 41215.3i − 1.10688e8i 6.89123e9 0 4.56203e12 7.02715e13i 6.38061e14i −6.69270e15 0
24.15 61242.8i 1.16397e8i 4.83925e9 0 −7.12847e12 − 1.52684e14i 8.22441e14i −7.98915e15 0
24.16 100295.i − 1.18201e8i −1.46921e9 0 1.18550e13 1.07417e13i 7.14175e14i −8.41244e15 0
24.17 104621.i 2.08087e7i −2.35554e9 0 −2.17702e12 − 5.34780e13i 6.52247e14i 5.12606e15 0
24.18 109239.i 1.96178e7i −3.34318e9 0 −2.14302e12 7.24227e12i 5.73150e14i 5.17420e15 0
24.19 140551.i 1.29136e8i −1.11647e10 0 −1.81502e13 − 4.00085e13i − 3.61884e14i −1.11170e16 0
24.20 153994.i − 7.94763e7i −1.51243e10 0 1.22389e13 − 4.85062e13i − 1.00626e15i −7.57418e14 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.34.b.d 22
5.b even 2 1 inner 25.34.b.d 22
5.c odd 4 1 25.34.a.d ✓ 11
5.c odd 4 1 25.34.a.e yes 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.34.a.d ✓ 11 5.c odd 4 1
25.34.a.e yes 11 5.c odd 4 1
25.34.b.d 22 1.a even 1 1 trivial
25.34.b.d 22 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(87\!\cdots\!80\)\( T_{2}^{18} + \)\(29\!\cdots\!60\)\( T_{2}^{16} + \)\(59\!\cdots\!80\)\( T_{2}^{14} + \)\(75\!\cdots\!88\)\( T_{2}^{12} + \)\(59\!\cdots\!12\)\( T_{2}^{10} + \)\(28\!\cdots\!20\)\( T_{2}^{8} + \)\(73\!\cdots\!40\)\( T_{2}^{6} + \)\(92\!\cdots\!20\)\( T_{2}^{4} + \)\(43\!\cdots\!36\)\( T_{2}^{2} + \)\(18\!\cdots\!24\)\( \)">\(T_{2}^{22} + \cdots\) acting on \(S_{34}^{\mathrm{new}}(25, [\chi])\).