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Properties

Label 25.7.c.d
Level $25$
Weight $7$
Character orbit 25.c
Analytic conductor $5.751$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [25,7,Mod(7,25)] 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("25.7"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(25, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 25.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0] 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: no
Analytic conductor: \(5.75135209050\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.333061916000256.23
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 90x^{6} - 12x^{5} + 3011x^{4} + 528x^{3} + 41202x^{2} + 17580x + 243850 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{2}\cdot 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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)
 

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.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_{2}) q^{2} + (3 \beta_{5} + 4 \beta_{4}) q^{3} + ( - \beta_{6} + 22 \beta_1) q^{4} + ( - 3 \beta_{7} + 306) q^{6} + (28 \beta_{3} - 24 \beta_{2}) q^{7} + ( - 43 \beta_{5} + 5 \beta_{4}) q^{8}+ \cdots + ( - 384 \beta_{6} + 392574 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2436 q^{6} + 7896 q^{11} + 28 q^{16} - 27264 q^{21} - 192744 q^{26} + 108656 q^{31} + 391608 q^{36} + 392136 q^{41} + 16776 q^{46} - 1182264 q^{51} - 1156080 q^{56} + 392896 q^{61} + 1332 q^{66} + 815376 q^{71}+ \cdots + 3595356 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 90x^{6} - 12x^{5} + 3011x^{4} + 528x^{3} + 41202x^{2} + 17580x + 243850 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 150 \nu^{7} - 462 \nu^{6} + 10174 \nu^{5} - 53319 \nu^{4} + 232394 \nu^{3} - 1499697 \nu^{2} + \cdots - 13362675 ) / 5194175 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 282286 \nu^{7} - 15339522 \nu^{6} + 35907577 \nu^{5} - 1002858672 \nu^{4} + 1947856606 \nu^{3} + \cdots - 99219003000 ) / 8378204275 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 121302 \nu^{7} - 349419 \nu^{6} - 14676598 \nu^{5} - 37861257 \nu^{4} - 662937606 \nu^{3} + \cdots - 13779356625 ) / 1675640855 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 963177 \nu^{7} - 1768660 \nu^{6} + 45747936 \nu^{5} - 88514646 \nu^{4} + 488036934 \nu^{3} + \cdots + 98741527500 ) / 8378204275 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 679482 \nu^{7} + 81620 \nu^{6} - 44765601 \nu^{5} + 16276001 \nu^{4} - 929477859 \nu^{3} + \cdots - 1500016825 ) / 1675640855 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3755130 \nu^{7} - 720363 \nu^{6} + 351434266 \nu^{5} - 120109521 \nu^{4} + 12942876266 \nu^{3} + \cdots + 34258391175 ) / 8378204275 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 144 \nu^{7} + 534 \nu^{6} + 12924 \nu^{5} + 34101 \nu^{4} + 351360 \nu^{3} + 846702 \nu^{2} + \cdots + 6566615 ) / 56455 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + 3\beta_{3} - 8\beta_1 ) / 15 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 18\beta_{5} + 30\beta_{4} + 45\beta _1 - 338 ) / 15 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{7} - 23\beta_{6} - 9\beta_{5} - 225\beta_{3} - 45\beta_{2} + 514\beta _1 + 72 ) / 15 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -15\beta_{7} + 6\beta_{6} - 364\beta_{5} - 460\beta_{4} + 36\beta_{3} + 60\beta_{2} - 2028\beta _1 + 2605 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 690 \beta_{7} + 506 \beta_{6} + 2250 \beta_{5} + 450 \beta_{4} + 9588 \beta_{3} + 3375 \beta_{2} + \cdots - 15420 ) / 15 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1406 \beta_{7} - 2025 \beta_{6} + 47088 \beta_{5} + 48780 \beta_{4} - 16380 \beta_{3} - 20700 \beta_{2} + \cdots - 148528 ) / 15 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 34524 \beta_{7} - 6418 \beta_{6} - 201834 \beta_{5} - 70875 \beta_{4} - 337470 \beta_{3} + \cdots + 1211742 ) / 15 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{1}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−1.22474 − 6.44174i
1.22474 + 5.44174i
−1.22474 + 2.99225i
1.22474 − 3.99225i
−1.22474 + 6.44174i
1.22474 − 5.44174i
−1.22474 − 2.99225i
1.22474 + 3.99225i
−8.83897 − 8.83897i −29.2322 + 29.2322i 92.2549i 0 516.765 −106.298 − 106.298i 249.744 − 249.744i − 980.039i 0
7.2 −2.71525 − 2.71525i −16.9847 + 16.9847i − 49.2549i 0 92.2354 383.600 + 383.600i −307.515 + 307.515i 152.039i 0
7.3 2.71525 + 2.71525i 16.9847 − 16.9847i − 49.2549i 0 92.2354 −383.600 − 383.600i 307.515 − 307.515i 152.039i 0
7.4 8.83897 + 8.83897i 29.2322 − 29.2322i 92.2549i 0 516.765 106.298 + 106.298i −249.744 + 249.744i − 980.039i 0
18.1 −8.83897 + 8.83897i −29.2322 − 29.2322i − 92.2549i 0 516.765 −106.298 + 106.298i 249.744 + 249.744i 980.039i 0
18.2 −2.71525 + 2.71525i −16.9847 − 16.9847i 49.2549i 0 92.2354 383.600 − 383.600i −307.515 − 307.515i − 152.039i 0
18.3 2.71525 − 2.71525i 16.9847 + 16.9847i 49.2549i 0 92.2354 −383.600 + 383.600i 307.515 + 307.515i − 152.039i 0
18.4 8.83897 − 8.83897i 29.2322 + 29.2322i − 92.2549i 0 516.765 106.298 − 106.298i −249.744 − 249.744i 980.039i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.7.c.d ✓ 8
3.b odd 2 1 225.7.g.f 8
5.b even 2 1 inner 25.7.c.d ✓ 8
5.c odd 4 2 inner 25.7.c.d ✓ 8
15.d odd 2 1 225.7.g.f 8
15.e even 4 2 225.7.g.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.7.c.d ✓ 8 1.a even 1 1 trivial
25.7.c.d ✓ 8 5.b even 2 1 inner
25.7.c.d ✓ 8 5.c odd 4 2 inner
225.7.g.f 8 3.b odd 2 1
225.7.g.f 8 15.d odd 2 1
225.7.g.f 8 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 24633T_{2}^{4} + 5308416 \) acting on \(S_{7}^{\mathrm{new}}(25, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 24633 T^{4} + 5308416 \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 972292630401 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 44\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( (T^{2} - 1974 T - 1028331)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 24\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 685239255555625)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 86\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 220068324090000)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 27164 T + 176460724)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 56\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{2} - 98034 T + 2274506289)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 78\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 71\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 98224 T - 31430261456)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 14\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( (T^{2} - 203844 T - 28860905916)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 20\!\cdots\!21 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 12\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 17\!\cdots\!25)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 38\!\cdots\!16 \) Copy content Toggle raw display
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