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Properties

Label 625.2.b.c
Level $625$
Weight $2$
Character orbit 625.b
Analytic conductor $4.991$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [625,2,Mod(624,625)]
 
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([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("625.624");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(4.99065012633\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.58140625.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + (\beta_{6} + \beta_{5}) q^{3} + ( - \beta_{3} - 1) q^{4} + (\beta_{3} + 2 \beta_1) q^{6} + ( - \beta_{7} - \beta_{6} - \beta_{4}) q^{7} + ( - \beta_{7} + \beta_{6}) q^{8} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + (\beta_{6} + \beta_{5}) q^{3} + ( - \beta_{3} - 1) q^{4} + (\beta_{3} + 2 \beta_1) q^{6} + ( - \beta_{7} - \beta_{6} - \beta_{4}) q^{7} + ( - \beta_{7} + \beta_{6}) q^{8} + (\beta_{2} - 1) q^{9} + 2 q^{11} + (\beta_{7} + \beta_{6} - \beta_{4}) q^{12} + \beta_{5} q^{13} + ( - \beta_{2} + 2 \beta_1 - 2) q^{14} + ( - \beta_{3} + \beta_{2} - 1) q^{16} + ( - 2 \beta_{7} - \beta_{6} - \beta_{4}) q^{17} + (\beta_{7} - 2 \beta_{6} + \beta_{5}) q^{18} + (\beta_{3} + 2 \beta_1 - 2) q^{19} + ( - \beta_{3} + \beta_{2}) q^{21} - 2 \beta_{4} q^{22} + ( - \beta_{6} + \beta_{5}) q^{23} + (\beta_{2} + 2 \beta_1 - 4) q^{24} + (\beta_{3} - \beta_{2} + 3 \beta_1) q^{26} + (2 \beta_{7} + \beta_{6} + \cdots + 2 \beta_{4}) q^{27}+ \cdots + (2 \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{4} + 6 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{4} + 6 q^{6} - 4 q^{9} + 16 q^{11} - 12 q^{14} - 2 q^{16} - 10 q^{19} + 6 q^{21} - 20 q^{24} + 6 q^{26} - 20 q^{29} + 16 q^{31} - 2 q^{34} - 12 q^{36} - 18 q^{39} + 26 q^{41} - 12 q^{44} + 6 q^{46} + 14 q^{49} - 4 q^{51} + 30 q^{54} + 10 q^{56} - 30 q^{59} + 6 q^{61} + 44 q^{64} + 12 q^{66} - 8 q^{69} + 46 q^{71} - 12 q^{74} - 20 q^{76} - 10 q^{79} - 32 q^{81} + 18 q^{84} - 14 q^{86} - 30 q^{89} - 14 q^{91} + 68 q^{94} - 54 q^{96} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -14\nu^{7} + 62\nu^{6} + 49\nu^{5} + 28\nu^{4} - 194\nu^{3} - 180\nu^{2} + 10\nu + 2465 ) / 1355 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -45\nu^{7} + 238\nu^{6} - 249\nu^{5} + 632\nu^{4} - 701\nu^{3} - 927\nu^{2} + 1000\nu + 2600 ) / 1355 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 78\nu^{7} - 268\nu^{6} + 269\nu^{5} - 427\nu^{4} + 926\nu^{3} + 306\nu^{2} - 2185\nu - 2700 ) / 1355 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 163\nu^{7} - 296\nu^{6} + 378\nu^{5} - 1139\nu^{4} + 1407\nu^{3} + 973\nu^{2} + 1045\nu - 3245 ) / 1355 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 227\nu^{7} - 502\nu^{6} + 696\nu^{5} - 1538\nu^{4} + 2139\nu^{3} + 1099\nu^{2} + 1580\nu - 4835 ) / 1355 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 826\nu^{7} - 1490\nu^{6} + 1445\nu^{5} - 3820\nu^{4} + 4400\nu^{3} + 10078\nu^{2} + 2120\nu - 18065 ) / 1355 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1442\nu^{7} - 2592\nu^{6} + 2541\nu^{5} - 6678\nu^{4} + 6974\nu^{3} + 16914\nu^{2} + 4390\nu - 30320 ) / 1355 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} - \beta_{3} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{7} + 5\beta_{6} + \beta_{5} + \beta_{2} - \beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{7} + 4\beta_{6} + 5\beta_{5} - 9\beta_{4} + \beta_{3} + 4\beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{7} + 2\beta_{6} + 4\beta_{5} - 6\beta_{4} + 10\beta _1 - 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -8\beta_{7} + 14\beta_{6} - 9\beta_{5} + 16\beta_{4} + 2\beta_{3} + 7\beta_{2} + 31\beta _1 - 52 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8\beta_{7} - 11\beta_{6} - 15\beta_{5} + 23\beta_{4} + 23\beta_{3} + 38\beta_{2} + 12\beta _1 - 15 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/625\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
624.1
−0.357358 − 1.86824i
−0.983224 + 0.644389i
1.66637 − 0.917186i
1.17421 + 0.0566033i
1.17421 − 0.0566033i
1.66637 + 0.917186i
−0.983224 − 0.644389i
−0.357358 + 1.86824i
− 2.30927i 0.474903i −3.33275 0 1.09668 − 3.03582i 3.07768i 2.77447 0
624.2 − 2.08529i 2.19849i −2.34841 0 4.58448 0.992398i 0.726543i −1.83337 0
624.3 − 1.13370i − 2.60278i 0.714715 0 −2.95078 − 0.407162i − 3.07768i −3.77447 0
624.4 − 0.183172i 1.47195i 1.96645 0 0.269620 − 3.26086i − 0.726543i 0.833366 0
624.5 0.183172i − 1.47195i 1.96645 0 0.269620 3.26086i 0.726543i 0.833366 0
624.6 1.13370i 2.60278i 0.714715 0 −2.95078 0.407162i 3.07768i −3.77447 0
624.7 2.08529i − 2.19849i −2.34841 0 4.58448 − 0.992398i − 0.726543i −1.83337 0
624.8 2.30927i − 0.474903i −3.33275 0 1.09668 3.03582i − 3.07768i 2.77447 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 624.8
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 625.2.b.c 8
5.b even 2 1 inner 625.2.b.c 8
5.c odd 4 2 625.2.a.f 8
15.e even 4 2 5625.2.a.x 8
20.e even 4 2 10000.2.a.bj 8
25.d even 5 1 25.2.e.a ✓ 8
25.d even 5 1 125.2.e.b 8
25.d even 5 1 625.2.e.a 8
25.d even 5 1 625.2.e.i 8
25.e even 10 1 25.2.e.a ✓ 8
25.e even 10 1 125.2.e.b 8
25.e even 10 1 625.2.e.a 8
25.e even 10 1 625.2.e.i 8
25.f odd 20 4 125.2.d.b 16
25.f odd 20 4 625.2.d.o 16
75.h odd 10 1 225.2.m.a 8
75.j odd 10 1 225.2.m.a 8
100.h odd 10 1 400.2.y.c 8
100.j odd 10 1 400.2.y.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.e.a ✓ 8 25.d even 5 1
25.2.e.a ✓ 8 25.e even 10 1
125.2.d.b 16 25.f odd 20 4
125.2.e.b 8 25.d even 5 1
125.2.e.b 8 25.e even 10 1
225.2.m.a 8 75.h odd 10 1
225.2.m.a 8 75.j odd 10 1
400.2.y.c 8 100.h odd 10 1
400.2.y.c 8 100.j odd 10 1
625.2.a.f 8 5.c odd 4 2
625.2.b.c 8 1.a even 1 1 trivial
625.2.b.c 8 5.b even 2 1 inner
625.2.d.o 16 25.f odd 20 4
625.2.e.a 8 25.d even 5 1
625.2.e.a 8 25.e even 10 1
625.2.e.i 8 25.d even 5 1
625.2.e.i 8 25.e even 10 1
5625.2.a.x 8 15.e even 4 2
10000.2.a.bj 8 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(625, [\chi])\):

\( T_{2}^{8} + 11T_{2}^{6} + 36T_{2}^{4} + 31T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{8} + 14T_{3}^{6} + 61T_{3}^{4} + 84T_{3}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 11 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} + 14 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 21 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T - 2)^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 14 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{8} + 41 T^{6} + \cdots + 1936 \) Copy content Toggle raw display
$19$ \( (T^{4} + 5 T^{3} - 5 T^{2} + \cdots - 20)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 34 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( (T^{4} + 10 T^{3} + \cdots - 695)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 8 T^{3} - 41 T^{2} + \cdots - 44)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 111 T^{6} + \cdots + 116281 \) Copy content Toggle raw display
$41$ \( (T^{4} - 13 T^{3} + \cdots + 116)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 129 T^{6} + \cdots + 246016 \) Copy content Toggle raw display
$47$ \( T^{8} + 141 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$53$ \( T^{8} + 239 T^{6} + \cdots + 8755681 \) Copy content Toggle raw display
$59$ \( (T^{4} + 15 T^{3} + \cdots - 2020)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 3 T^{3} + \cdots + 341)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 176 T^{6} + \cdots + 246016 \) Copy content Toggle raw display
$71$ \( (T^{4} - 23 T^{3} + \cdots - 4924)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 79 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( (T^{4} + 5 T^{3} + \cdots + 5780)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 374 T^{6} + \cdots + 99856 \) Copy content Toggle raw display
$89$ \( (T^{4} + 15 T^{3} + \cdots + 1180)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 666 T^{6} + \cdots + 301334881 \) Copy content Toggle raw display
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