Properties

Label 10.26.b.a
Level $10$
Weight $26$
Character orbit 10.b
Analytic conductor $39.600$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [10,26,Mod(9,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 26, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.9");
 
S:= CuspForms(chi, 26);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 10.b (of order \(2\), degree \(1\), 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: \(39.5996779952\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 1406300109694 x^{10} + \cdots + 56\!\cdots\!01 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{90}\cdot 3^{8}\cdot 5^{29} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{3} + 33 \beta_{2}) q^{3} - 16777216 q^{4} + ( - \beta_{4} + 46 \beta_{3} + \cdots - 40857945) q^{5} + (\beta_{5} + \beta_{4} - 547099989) q^{6} + ( - \beta_{10} + \cdots - 1381463 \beta_{2}) q^{7}+ \cdots + ( - 5165376575508 \beta_{11} + \cdots + 34\!\cdots\!00) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 201326592 q^{4} - 490295340 q^{5} - 6565199872 q^{6} - 1082937564236 q^{9} + 1636528619520 q^{10} + 19723089228624 q^{11} + 278591122243584 q^{14} - 449884766537680 q^{15} + 33\!\cdots\!72 q^{16}+ \cdots + 41\!\cdots\!28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 1406300109694 x^{10} + \cdots + 56\!\cdots\!01 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 72\!\cdots\!37 \nu^{10} + \cdots - 21\!\cdots\!77 ) / 13\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 34\!\cdots\!27 \nu^{11} + \cdots + 41\!\cdots\!47 \nu ) / 58\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 38\!\cdots\!97 \nu^{11} + \cdots - 72\!\cdots\!89 \nu ) / 19\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 22\!\cdots\!71 \nu^{11} + \cdots - 16\!\cdots\!04 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 22\!\cdots\!71 \nu^{11} + \cdots + 26\!\cdots\!54 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 42\!\cdots\!49 \nu^{11} + \cdots - 51\!\cdots\!26 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 32\!\cdots\!79 \nu^{11} + \cdots + 17\!\cdots\!96 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 13\!\cdots\!13 \nu^{11} + \cdots + 49\!\cdots\!88 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 39\!\cdots\!92 \nu^{11} + \cdots + 36\!\cdots\!33 ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 18\!\cdots\!69 \nu^{11} + \cdots + 17\!\cdots\!44 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11\!\cdots\!27 \nu^{11} + \cdots + 27\!\cdots\!77 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 33\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{9} + 3 \beta_{8} - \beta_{7} + 69 \beta_{5} - 266 \beta_{4} + 135 \beta_{3} + \cdots - 937533406440 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 202358 \beta_{11} - 1537113 \beta_{10} + 412437 \beta_{9} + 338499 \beta_{8} - 1163373 \beta_{7} + \cdots - 2104932 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 67813081218 \beta_{11} + 33906540609 \beta_{10} + 1527164057033 \beta_{9} - 2875918650702 \beta_{8} + \cdots + 71\!\cdots\!43 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 64\!\cdots\!20 \beta_{11} + \cdots + 63\!\cdots\!40 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 48\!\cdots\!10 \beta_{11} + \cdots - 30\!\cdots\!33 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 40\!\cdots\!04 \beta_{11} + \cdots - 39\!\cdots\!68 ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 33\!\cdots\!32 \beta_{11} + \cdots + 14\!\cdots\!18 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 59\!\cdots\!57 \beta_{11} + \cdots + 56\!\cdots\!22 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 21\!\cdots\!60 \beta_{11} + \cdots - 70\!\cdots\!42 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 33\!\cdots\!15 \beta_{11} + \cdots - 31\!\cdots\!10 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
737346.i
540092.i
116846.i
58327.7i
508420.i
543485.i
543485.i
508420.i
58327.7i
116846.i
540092.i
737346.i
4096.00i 1.47469e6i −1.67772e7 −2.33061e8 4.93666e8i −6.04034e9 3.57876e9i 6.87195e10i −1.32743e12 −2.02205e12 + 9.54619e11i
9.2 4096.00i 1.08018e6i −1.67772e7 3.02592e8 + 4.54380e8i −4.42443e9 5.43159e10i 6.87195e10i −3.19509e11 1.86114e12 1.23942e12i
9.3 4096.00i 233693.i −1.67772e7 −3.44896e8 + 4.23166e8i −9.57206e8 6.39090e10i 6.87195e10i 7.92676e11 1.73329e12 + 1.41270e12i
9.4 4096.00i 116655.i −1.67772e7 5.03737e8 2.10410e8i −4.77821e8 1.34370e10i 6.87195e10i 8.33680e11 −8.61839e11 2.06331e12i
9.5 4096.00i 1.01684e6i −1.67772e7 −2.64029e8 4.77820e8i 4.16497e9 1.16326e10i 6.87195e10i −1.86674e11 −1.95715e12 + 1.08146e12i
9.6 4096.00i 1.08697e6i −1.67772e7 −2.09490e8 + 5.04120e8i 4.45223e9 4.89841e10i 6.87195e10i −3.34213e11 2.06488e12 + 8.58070e11i
9.7 4096.00i 1.08697e6i −1.67772e7 −2.09490e8 5.04120e8i 4.45223e9 4.89841e10i 6.87195e10i −3.34213e11 2.06488e12 8.58070e11i
9.8 4096.00i 1.01684e6i −1.67772e7 −2.64029e8 + 4.77820e8i 4.16497e9 1.16326e10i 6.87195e10i −1.86674e11 −1.95715e12 1.08146e12i
9.9 4096.00i 116655.i −1.67772e7 5.03737e8 + 2.10410e8i −4.77821e8 1.34370e10i 6.87195e10i 8.33680e11 −8.61839e11 + 2.06331e12i
9.10 4096.00i 233693.i −1.67772e7 −3.44896e8 4.23166e8i −9.57206e8 6.39090e10i 6.87195e10i 7.92676e11 1.73329e12 1.41270e12i
9.11 4096.00i 1.08018e6i −1.67772e7 3.02592e8 4.54380e8i −4.42443e9 5.43159e10i 6.87195e10i −3.19509e11 1.86114e12 + 1.23942e12i
9.12 4096.00i 1.47469e6i −1.67772e7 −2.33061e8 + 4.93666e8i −6.04034e9 3.57876e9i 6.87195e10i −1.32743e12 −2.02205e12 9.54619e11i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.12
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 10.26.b.a 12
5.b even 2 1 inner 10.26.b.a 12
5.c odd 4 1 50.26.a.k 6
5.c odd 4 1 50.26.a.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.26.b.a 12 1.a even 1 1 trivial
10.26.b.a 12 5.b even 2 1 inner
50.26.a.k 6 5.c odd 4 1
50.26.a.l 6 5.c odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{26}^{\mathrm{new}}(10, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 16777216)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 23\!\cdots\!96 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 70\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 90\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 67\!\cdots\!64)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 35\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 25\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 75\!\cdots\!36)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 93\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 47\!\cdots\!36)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 35\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 47\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 45\!\cdots\!36)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 64\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 14\!\cdots\!36)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 87\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 93\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
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