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Properties

Label 125.8.a.c
Level $125$
Weight $8$
Character orbit 125.a
Self dual yes
Analytic conductor $39.048$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [125,8,Mod(1,125)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(125, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("125.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 125 = 5^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 125.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: \(39.0481281855\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 3 x^{13} - 1290 x^{12} + 2493 x^{11} + 635719 x^{10} - 554520 x^{9} - 149769280 x^{8} + \cdots + 5997063924716 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2}\cdot 5^{17} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{6} + 7) q^{3} + ( - \beta_{5} + \beta_{3} + 2 \beta_1 + 60) q^{4} + (\beta_{12} - 2 \beta_{6} + \beta_{5} + \cdots + 20) q^{6}+ \cdots + (\beta_{12} - \beta_{11} + \beta_{10} + \cdots + 714) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{6} + 7) q^{3} + ( - \beta_{5} + \beta_{3} + 2 \beta_1 + 60) q^{4} + (\beta_{12} - 2 \beta_{6} + \beta_{5} + \cdots + 20) q^{6}+ \cdots + (420 \beta_{13} - 5755 \beta_{12} + \cdots - 1897755) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 3 q^{2} + 101 q^{3} + 857 q^{4} + 383 q^{6} + 2658 q^{7} + 3540 q^{8} + 10313 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 3 q^{2} + 101 q^{3} + 857 q^{4} + 383 q^{6} + 2658 q^{7} + 3540 q^{8} + 10313 q^{9} - 6702 q^{11} + 27328 q^{12} + 22131 q^{13} - 22236 q^{14} + 78049 q^{16} + 32108 q^{17} + 56471 q^{18} - 48545 q^{19} + 51833 q^{21} + 105396 q^{22} + 190226 q^{23} - 35875 q^{24} + 104748 q^{26} + 340160 q^{27} + 410184 q^{28} + 261180 q^{29} + 126563 q^{31} - 239817 q^{32} - 203343 q^{33} + 26139 q^{34} + 773804 q^{36} + 936648 q^{37} + 36400 q^{38} - 596159 q^{39} + 394968 q^{41} + 3233471 q^{42} + 2816526 q^{43} - 3422376 q^{44} - 2014817 q^{46} + 3379863 q^{47} + 10288111 q^{48} - 292178 q^{49} - 418277 q^{51} + 10738323 q^{52} + 4340581 q^{53} + 3495205 q^{54} - 1620645 q^{56} + 7664705 q^{57} + 13085145 q^{58} + 4294755 q^{59} + 2805083 q^{61} + 14457826 q^{62} + 14276221 q^{63} + 7814892 q^{64} - 7095319 q^{66} + 13030083 q^{67} + 16016999 q^{68} - 4136184 q^{69} + 7158138 q^{71} + 25289925 q^{72} + 7441161 q^{73} - 17066391 q^{74} - 1948615 q^{76} + 4280806 q^{77} + 12105662 q^{78} + 1378010 q^{79} - 10692906 q^{81} + 9206886 q^{82} + 20668776 q^{83} + 33483044 q^{84} + 15580383 q^{86} + 22995735 q^{87} + 16245255 q^{88} + 12050490 q^{89} + 5605653 q^{91} + 1736863 q^{92} + 5519142 q^{93} - 21198721 q^{94} - 27110212 q^{96} + 10284933 q^{97} - 29538631 q^{98} - 28303359 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 3 x^{13} - 1290 x^{12} + 2493 x^{11} + 635719 x^{10} - 554520 x^{9} - 149769280 x^{8} + \cdots + 5997063924716 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 10\!\cdots\!83 \nu^{13} + \cdots - 76\!\cdots\!92 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10\!\cdots\!83 \nu^{13} + \cdots + 31\!\cdots\!92 ) / 88\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 16\!\cdots\!87 \nu^{13} + \cdots - 22\!\cdots\!88 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14\!\cdots\!19 \nu^{13} + \cdots - 33\!\cdots\!44 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12\!\cdots\!67 \nu^{13} + \cdots - 81\!\cdots\!08 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 41\!\cdots\!62 \nu^{13} + \cdots + 47\!\cdots\!88 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 82\!\cdots\!59 \nu^{13} + \cdots - 49\!\cdots\!16 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 26\!\cdots\!53 \nu^{13} + \cdots - 91\!\cdots\!72 ) / 57\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 44\!\cdots\!66 \nu^{13} + \cdots - 11\!\cdots\!56 ) / 57\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 25\!\cdots\!62 \nu^{13} + \cdots - 25\!\cdots\!12 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 16\!\cdots\!37 \nu^{13} + \cdots + 13\!\cdots\!48 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 27\!\cdots\!86 \nu^{13} + \cdots + 18\!\cdots\!64 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 32\!\cdots\!22 \nu^{13} + \cdots + 85\!\cdots\!48 ) / 57\!\cdots\!00 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( 2\beta_{2} + 5\beta _1 + 1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + 5\beta_{3} + 8\beta _1 + 921 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5 \beta_{12} - 5 \beta_{11} - \beta_{10} - \beta_{6} + 19 \beta_{5} + 5 \beta_{4} + 3 \beta_{3} + \cdots + 1494 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11 \beta_{13} + 15 \beta_{12} + 24 \beta_{11} - 51 \beta_{10} - 10 \beta_{9} - 52 \beta_{8} + \cdots + 286612 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{13} + 605 \beta_{12} - 464 \beta_{11} - 175 \beta_{10} + 16 \beta_{9} - 62 \beta_{8} + \cdots + 160608 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 7461 \beta_{13} + 12360 \beta_{12} + 12459 \beta_{11} - 37586 \beta_{10} - 5310 \beta_{9} + \cdots + 102689933 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 52807 \beta_{13} + 1440750 \beta_{12} - 937632 \beta_{11} - 562304 \beta_{10} + 68140 \beta_{9} + \cdots + 362813939 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3829283 \beta_{13} + 7483340 \beta_{12} + 4893742 \beta_{11} - 21052288 \beta_{10} - 2234920 \beta_{9} + \cdots + 39183772209 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 37102873 \beta_{13} + 632739205 \beta_{12} - 370927603 \beta_{11} - 311839745 \beta_{10} + \cdots + 166356935296 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 361299598 \beta_{13} + 793869875 \beta_{12} + 342290182 \beta_{11} - 2129348983 \beta_{10} + \cdots + 3097439818692 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 21914305695 \beta_{13} + 268942680625 \beta_{12} - 147100369270 \beta_{11} - 160331366755 \beta_{10} + \cdots + 78740487824346 ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 827626176500 \beta_{13} + 1969268971000 \beta_{12} + 542526517025 \beta_{11} - 5125082022650 \beta_{10} + \cdots + 62\!\cdots\!01 ) / 5 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 11802953560440 \beta_{13} + 112909503650830 \beta_{12} - 58842031545670 \beta_{11} + \cdots + 37\!\cdots\!29 ) / 5 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−19.8887
−19.1310
−10.9083
−9.12377
−13.3910
−9.22957
2.30933
4.37783
0.200777
6.49090
14.6312
21.2205
16.8288
18.6130
−22.1248 54.1032 361.507 0 −1197.02 916.766 −5166.30 740.161 0
1.2 −16.8950 10.1315 157.439 0 −171.170 −455.932 −497.378 −2084.35 0
1.3 −13.1443 2.09361 44.7736 0 −27.5191 −334.821 1093.96 −2182.62 0
1.4 −11.3598 −69.0570 1.04590 0 784.476 890.180 1442.18 2581.87 0
1.5 −11.1549 −61.7817 −3.56778 0 689.170 1693.32 1467.63 1629.98 0
1.6 −6.99350 81.4767 −79.0909 0 −569.808 −568.007 1448.29 4451.46 0
1.7 0.0732645 78.1852 −127.995 0 5.72820 1233.53 −18.7553 3925.92 0
1.8 2.14176 6.90445 −123.413 0 14.7877 −1271.56 −538.467 −2139.33 0
1.9 2.43684 −37.3012 −122.062 0 −90.8972 −845.440 −609.362 −795.620 0
1.10 8.72697 26.3332 −51.8400 0 229.809 827.868 −1569.46 −1493.56 0
1.11 12.3951 −58.6428 25.6388 0 −726.885 533.666 −1268.78 1251.98 0
1.12 18.9844 44.7396 232.408 0 849.355 300.382 1982.11 −185.368 0
1.13 19.0649 −54.0578 235.471 0 −1030.61 −888.169 2048.92 735.248 0
1.14 20.8491 77.8731 306.685 0 1623.58 626.212 3725.42 3877.22 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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 125.8.a.c yes 14
5.b even 2 1 125.8.a.b ✓ 14
5.c odd 4 2 125.8.b.c 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
125.8.a.b ✓ 14 5.b even 2 1
125.8.a.c yes 14 1.a even 1 1 trivial
125.8.b.c 28 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 3 T_{2}^{13} - 1320 T_{2}^{12} + 2533 T_{2}^{11} + 659029 T_{2}^{10} - 444240 T_{2}^{9} + \cdots + 1359046823936 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(125))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots + 1359046823936 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots - 23\!\cdots\!36 \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 11\!\cdots\!81 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 10\!\cdots\!49 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 21\!\cdots\!99 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 44\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots - 19\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 68\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 12\!\cdots\!89 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots - 75\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 26\!\cdots\!91 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 19\!\cdots\!81 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 22\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 16\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 42\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 55\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 35\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 33\!\cdots\!24 \) Copy content Toggle raw display
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