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Properties

Label 245.10.a.i
Level $245$
Weight $10$
Character orbit 245.a
Self dual yes
Analytic conductor $126.184$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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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] = [245,10,Mod(1,245)] 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("245.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(245, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,1,268] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(126.183779860\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 3242 x^{7} - 1690 x^{6} + 3235604 x^{5} + 4945456 x^{4} - 1138644128 x^{3} + \cdots + 183792896000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{4}\cdot 5^{2}\cdot 7^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} + \beta_1 + 30) q^{3} + (\beta_{2} + 2 \beta_1 + 208) q^{4} - 625 q^{5} + (\beta_{4} - 4 \beta_{3} + \beta_{2} + \cdots + 845) q^{6} + ( - \beta_{7} + \beta_{6} + \cdots + 1506) q^{8}+ \cdots + (25328 \beta_{8} - 8002 \beta_{7} + \cdots - 254113812) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + q^{2} + 268 q^{3} + 1877 q^{4} - 5625 q^{5} + 7624 q^{6} + 13773 q^{8} + 17015 q^{9} - 625 q^{10} - 69434 q^{11} + 64966 q^{12} + 153108 q^{13} - 167500 q^{15} + 496777 q^{16} - 380338 q^{17} + 1151915 q^{18}+ \cdots - 2283620136 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 3242 x^{7} - 1690 x^{6} + 3235604 x^{5} + 4945456 x^{4} - 1138644128 x^{3} + \cdots + 183792896000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 720 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1513909 \nu^{8} + 14128713 \nu^{7} + 2570808782 \nu^{6} - 11273540310 \nu^{5} + \cdots + 23\!\cdots\!00 ) / 886831521408000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 777935 \nu^{8} + 99741627 \nu^{7} + 1004803402 \nu^{6} - 251844599250 \nu^{5} + \cdots + 22\!\cdots\!00 ) / 36951313392000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4323719 \nu^{8} - 466886961 \nu^{7} + 16719992158 \nu^{6} + 1334341889190 \nu^{5} + \cdots + 13\!\cdots\!00 ) / 110853940176000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3818203 \nu^{8} - 134025417 \nu^{7} + 10867300466 \nu^{6} + 475616221830 \nu^{5} + \cdots - 23\!\cdots\!00 ) / 73902626784000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 54168203 \nu^{8} - 7807843305 \nu^{7} + 227198217298 \nu^{6} + 22482767859030 \nu^{5} + \cdots + 48\!\cdots\!00 ) / 886831521408000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4307707 \nu^{8} + 41153067 \nu^{7} + 12468286694 \nu^{6} - 89435582730 \nu^{5} + \cdots - 74\!\cdots\!00 ) / 5834417904000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 720 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 5\beta_{3} + 4\beta_{2} + 1247\beta _1 + 1506 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{8} + 4 \beta_{7} - 23 \beta_{6} + 23 \beta_{5} + 36 \beta_{4} + 16 \beta_{3} + 1818 \beta_{2} + \cdots + 898600 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 59 \beta_{8} - 1844 \beta_{7} + 2395 \beta_{6} + 2189 \beta_{5} - 704 \beta_{4} - 22320 \beta_{3} + \cdots + 4137804 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2045 \beta_{8} + 11730 \beta_{7} - 54305 \beta_{6} + 60765 \beta_{5} + 105350 \beta_{4} + \cdots + 1349299680 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 160965 \beta_{8} - 3127706 \beta_{7} + 4505091 \beta_{6} + 4212481 \beta_{5} + 528994 \beta_{4} + \cdots + 8944534336 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 5262641 \beta_{8} + 22771954 \beta_{7} - 100133133 \beta_{6} + 127520153 \beta_{5} + \cdots + 2192642715960 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−40.7561
−26.8187
−24.3613
−12.2917
−1.60246
13.9714
18.5703
31.7536
42.5350
−40.7561 −27.9526 1149.06 −625.000 1139.24 0 −25964.0 −18901.7 25472.6
1.2 −26.8187 −117.107 207.243 −625.000 3140.66 0 8173.20 −5968.89 16761.7
1.3 −24.3613 124.658 81.4729 −625.000 −3036.83 0 10488.2 −4143.41 15225.8
1.4 −12.2917 154.861 −360.914 −625.000 −1903.51 0 10729.6 4298.95 7682.33
1.5 −1.60246 −208.582 −509.432 −625.000 334.245 0 1636.80 23823.6 1001.54
1.6 13.9714 49.7594 −316.800 −625.000 695.208 0 −11579.5 −17207.0 −8732.13
1.7 18.5703 260.186 −167.144 −625.000 4831.74 0 −12611.9 48014.0 −11606.4
1.8 31.7536 −97.8140 496.292 −625.000 −3105.95 0 −498.796 −10115.4 −19846.0
1.9 42.5350 129.992 1297.22 −625.000 5529.19 0 33399.4 −2785.18 −26584.4
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(7\) \( -1 \)

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 245.10.a.i yes 9
7.b odd 2 1 245.10.a.h ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.10.a.h ✓ 9 7.b odd 2 1
245.10.a.i yes 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(245))\):

\( T_{2}^{9} - T_{2}^{8} - 3242 T_{2}^{7} - 1690 T_{2}^{6} + 3235604 T_{2}^{5} + 4945456 T_{2}^{4} + \cdots + 183792896000 \) Copy content Toggle raw display
\( T_{3}^{9} - 268 T_{3}^{8} - 61169 T_{3}^{7} + 18160842 T_{3}^{6} + 720738315 T_{3}^{5} + \cdots - 21\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + \cdots + 183792896000 \) Copy content Toggle raw display
$3$ \( T^{9} + \cdots - 21\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T + 625)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots - 17\!\cdots\!12 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots - 68\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots - 39\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{9} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots + 43\!\cdots\!88 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots - 70\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots - 86\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots - 29\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots - 43\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots - 79\!\cdots\!72 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots - 49\!\cdots\!00 \) Copy content Toggle raw display
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