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Properties

Label 245.10.a.n
Level $245$
Weight $10$
Character orbit 245.a
Self dual yes
Analytic conductor $126.184$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [245,10,Mod(1,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 245.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: \(126.183779860\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 6 x^{17} - 6671 x^{16} + 28472 x^{15} + 18323094 x^{14} - 49525664 x^{13} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{4}\cdot 5^{4}\cdot 7^{16} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 4) q^{2} + (\beta_{4} - 6) q^{3} + (\beta_{2} - 6 \beta_1 + 249) q^{4} - 625 q^{5} + ( - \beta_{8} + 3 \beta_{4} + \cdots - 288) q^{6} + ( - \beta_{6} - 14 \beta_{4} + \cdots + 3311) q^{8}+ \cdots + ( - 2758 \beta_{17} - 2931 \beta_{16} + \cdots - 184223813) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 66 q^{2} - 112 q^{3} + 4438 q^{4} - 11250 q^{5} - 5184 q^{6} + 58542 q^{8} + 117250 q^{9} - 41250 q^{10} - 50448 q^{11} + 51200 q^{12} - 265416 q^{13} + 70000 q^{15} + 275354 q^{16} - 742108 q^{17}+ \cdots - 3327698948 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 6 x^{17} - 6671 x^{16} + 28472 x^{15} + 18323094 x^{14} - 49525664 x^{13} + \cdots + 24\!\cdots\!76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 69\!\cdots\!49 \nu^{17} + \cdots - 67\!\cdots\!16 ) / 48\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 16\!\cdots\!53 \nu^{17} + \cdots - 17\!\cdots\!88 ) / 24\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 69\!\cdots\!49 \nu^{17} + \cdots - 67\!\cdots\!16 ) / 98\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 71\!\cdots\!26 \nu^{17} + \cdots - 85\!\cdots\!76 ) / 61\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 29\!\cdots\!27 \nu^{17} + \cdots + 35\!\cdots\!24 ) / 61\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 22\!\cdots\!36 \nu^{17} + \cdots + 99\!\cdots\!40 ) / 30\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 89\!\cdots\!04 \nu^{17} + \cdots - 85\!\cdots\!36 ) / 61\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 19\!\cdots\!14 \nu^{17} + \cdots + 82\!\cdots\!60 ) / 61\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 27\!\cdots\!07 \nu^{17} + \cdots - 77\!\cdots\!36 ) / 61\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 15\!\cdots\!69 \nu^{17} + \cdots + 83\!\cdots\!36 ) / 87\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 81\!\cdots\!38 \nu^{17} + \cdots + 22\!\cdots\!76 ) / 30\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 21\!\cdots\!75 \nu^{17} + \cdots - 43\!\cdots\!96 ) / 61\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 17\!\cdots\!48 \nu^{17} + \cdots - 55\!\cdots\!08 ) / 30\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 38\!\cdots\!47 \nu^{17} + \cdots + 21\!\cdots\!32 ) / 61\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 53\!\cdots\!35 \nu^{17} + \cdots - 34\!\cdots\!40 ) / 61\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 12\!\cdots\!03 \nu^{17} + \cdots - 86\!\cdots\!16 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 33\!\cdots\!58 \nu^{17} + \cdots - 11\!\cdots\!60 ) / 30\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{3} + 49\beta_1 ) / 49 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} + 2\beta_{4} + 49\beta_{2} + 96\beta _1 + 36409 ) / 49 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{10} + 3 \beta_{7} + 49 \beta_{6} - 21 \beta_{5} + 680 \beta_{4} - 3465 \beta_{3} + \cdots + 75793 ) / 49 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 126 \beta_{17} + 7 \beta_{16} + 126 \beta_{14} + 42 \beta_{13} - 77 \beta_{12} - 112 \beta_{11} + \cdots + 43486085 ) / 49 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2625 \beta_{17} - 1855 \beta_{16} - 588 \beta_{15} - 2303 \beta_{14} + 728 \beta_{13} + \cdots + 206929734 ) / 49 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 304374 \beta_{17} + 66073 \beta_{16} - 4851 \beta_{15} + 337708 \beta_{14} - 13447 \beta_{13} + \cdots + 59040018814 ) / 49 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 7351225 \beta_{17} - 4075575 \beta_{16} - 1784384 \beta_{15} - 5437775 \beta_{14} + \cdots + 444933494852 ) / 49 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 596663536 \beta_{17} + 202840827 \beta_{16} - 48871963 \beta_{15} + 711831414 \beta_{14} + \cdots + 86076348319334 ) / 49 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 15779375689 \beta_{17} - 6089101249 \beta_{16} - 4222316574 \beta_{15} - 9147811379 \beta_{14} + \cdots + 889540871191984 ) / 49 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1100509094202 \beta_{17} + 487723488609 \beta_{16} - 165930060191 \beta_{15} + 1385865900216 \beta_{14} + \cdots + 13\!\cdots\!10 ) / 49 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 31117927770797 \beta_{17} - 6712731346727 \beta_{16} - 9095268487172 \beta_{15} + \cdots + 17\!\cdots\!88 ) / 49 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 19\!\cdots\!00 \beta_{17} + \cdots + 20\!\cdots\!86 ) / 49 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 59\!\cdots\!09 \beta_{17} + \cdots + 32\!\cdots\!16 ) / 49 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 35\!\cdots\!18 \beta_{17} + \cdots + 34\!\cdots\!90 ) / 49 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 11\!\cdots\!05 \beta_{17} + \cdots + 62\!\cdots\!36 ) / 49 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 63\!\cdots\!08 \beta_{17} + \cdots + 56\!\cdots\!22 ) / 49 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 20\!\cdots\!77 \beta_{17} + \cdots + 11\!\cdots\!04 ) / 49 \) 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
42.8212
38.4700
33.9826
33.5835
26.5934
21.0159
13.8289
5.14270
2.78902
0.00962792
−11.6331
−12.2437
−22.5048
−25.9772
−30.1160
−35.2840
−34.4597
−40.0183
−37.4069 204.085 887.280 −625.000 −7634.21 0 −14038.1 21967.8 23379.3
1.2 −35.8842 −83.1852 775.679 −625.000 2985.04 0 −9461.93 −12763.2 22427.7
1.3 −31.3968 188.550 473.762 −625.000 −5919.87 0 1200.56 15868.0 19623.0
1.4 −28.1693 −195.917 281.510 −625.000 5518.85 0 6492.74 18700.5 17605.8
1.5 −21.1791 13.1571 −63.4437 −625.000 −278.657 0 12187.4 −19509.9 13237.0
1.6 −18.4301 −270.651 −172.331 −625.000 4988.12 0 12612.3 53568.7 11518.8
1.7 −11.2432 91.7473 −385.591 −625.000 −1031.53 0 10091.8 −11265.4 7026.97
1.8 −2.55692 13.8472 −505.462 −625.000 −35.4062 0 2601.56 −19491.3 1598.07
1.9 2.62520 196.431 −505.108 −625.000 515.670 0 −2670.11 18902.1 −1640.75
1.10 5.40459 −144.425 −482.790 −625.000 −780.557 0 −5376.43 1175.54 −3377.87
1.11 14.2189 −145.482 −309.823 −625.000 −2068.59 0 −11685.4 1481.93 −8886.81
1.12 17.6580 −17.2490 −200.197 −625.000 −304.582 0 −12575.9 −19385.5 −11036.2
1.13 25.0905 161.096 117.535 −625.000 4041.98 0 −9897.33 6268.80 −15681.6
1.14 31.3915 228.471 473.424 −625.000 7172.04 0 −1210.97 32516.0 −19619.7
1.15 35.5302 −250.181 750.395 −625.000 −8888.97 0 8470.23 42907.4 −22206.4
1.16 37.8698 −196.434 922.123 −625.000 −7438.93 0 15531.3 18903.5 −23668.6
1.17 39.8739 12.8650 1077.93 −625.000 512.977 0 22565.9 −19517.5 −24921.2
1.18 42.6041 81.2744 1303.11 −625.000 3462.62 0 33704.5 −13077.5 −26627.6
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
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.n ✓ 18
7.b odd 2 1 245.10.a.o yes 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.10.a.n ✓ 18 1.a even 1 1 trivial
245.10.a.o yes 18 7.b odd 2 1

Hecke kernels

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

\( T_{2}^{18} - 66 T_{2}^{17} - 4649 T_{2}^{16} + 360624 T_{2}^{15} + 7867818 T_{2}^{14} + \cdots + 85\!\cdots\!12 \) Copy content Toggle raw display
\( T_{3}^{18} + 112 T_{3}^{17} - 229500 T_{3}^{16} - 20488668 T_{3}^{15} + 21664362983 T_{3}^{14} + \cdots + 38\!\cdots\!36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + \cdots + 85\!\cdots\!12 \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 38\!\cdots\!36 \) Copy content Toggle raw display
$5$ \( (T + 625)^{18} \) Copy content Toggle raw display
$7$ \( T^{18} \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots - 58\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots - 16\!\cdots\!48 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots - 26\!\cdots\!72 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 28\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 58\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots - 57\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 51\!\cdots\!08 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 55\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots - 33\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots - 60\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 21\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 45\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 22\!\cdots\!72 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 88\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 12\!\cdots\!08 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 41\!\cdots\!88 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots - 81\!\cdots\!32 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 25\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 65\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 12\!\cdots\!12 \) Copy content Toggle raw display
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