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Properties

Label 21.7.d.a
Level $21$
Weight $7$
Character orbit 21.d
Analytic conductor $4.831$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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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] = [21,7,Mod(13,21)] 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("21.13"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(21, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 21.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.83113575602\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 212x^{6} - 1123x^{5} + 44168x^{4} - 138697x^{3} + 660109x^{2} + 680340x + 1040400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{7}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

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

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} - \beta_{4} q^{3} + ( - \beta_{2} + \beta_1 + 44) q^{4} + ( - \beta_{6} - 2 \beta_{4}) q^{5} + ( - \beta_{4} - \beta_{3}) q^{6} + (\beta_{7} + 4 \beta_{4} + \cdots - 14 \beta_1) q^{7}+ \cdots + ( - 729 \beta_{7} + 729 \beta_{5} + \cdots + 208251) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 10 q^{2} + 346 q^{4} + 28 q^{7} - 1462 q^{8} - 1944 q^{9} - 6848 q^{11} + 12082 q^{14} - 4536 q^{15} + 28466 q^{16} - 2430 q^{18} + 6804 q^{21} - 7764 q^{22} - 24320 q^{23} - 77056 q^{25} - 30142 q^{28}+ \cdots + 1664064 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 212x^{6} - 1123x^{5} + 44168x^{4} - 138697x^{3} + 660109x^{2} + 680340x + 1040400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 9082495 \nu^{7} + 20968969 \nu^{6} + 1852872025 \nu^{5} - 4069000805 \nu^{4} + \cdots + 6424259322060 ) / 6112279547031 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 57298949 \nu^{7} - 9710008 \nu^{6} + 11689257155 \nu^{5} - 25670200711 \nu^{4} + \cdots + 663837261700497 ) / 6112279547031 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 27247485 \nu^{7} + 62906907 \nu^{6} + 5558616075 \nu^{5} - 12207002415 \nu^{4} + \cdots + 19272777966180 ) / 2037426515677 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6298293453 \nu^{7} + 15562438353 \nu^{6} - 1313849863656 \nu^{5} + \cdots - 10\!\cdots\!10 ) / 346362507665090 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15106603126 \nu^{7} + 163811250526 \nu^{6} - 3388053356392 \nu^{5} + \cdots + 31\!\cdots\!65 ) / 222661612070415 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 236085965056 \nu^{7} - 195554084551 \nu^{6} + 52131292245397 \nu^{5} + \cdots + 84\!\cdots\!70 ) / 15\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 76111842031 \nu^{7} + 12621498841 \nu^{6} - 15833411217367 \nu^{5} + \cdots - 53\!\cdots\!05 ) / 222661612070415 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} - 9\beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -6\beta_{7} - 3\beta_{6} - 6\beta_{5} + 106\beta_{4} - 9\beta_{3} + 9\beta_{2} - 27\beta _1 - 963 ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{5} - 2\beta_{2} + 197\beta _1 + 392 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1266 \beta_{7} + 669 \beta_{6} + 1248 \beta_{5} - 20974 \beta_{4} + 2379 \beta_{3} + 1881 \beta_{2} + \cdots - 190485 ) / 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 6552 \beta_{7} + 6264 \beta_{6} - 2736 \beta_{5} + 128220 \beta_{4} - 42061 \beta_{3} + \cdots - 1196244 ) / 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 244\beta_{7} - 244\beta_{5} - 43533\beta_{2} + 319627\beta _1 + 4485223 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1163448 \beta_{7} - 959112 \beta_{6} + 1933866 \beta_{5} - 36227340 \beta_{4} + 9695467 \beta_{3} + \cdots - 335495412 ) / 18 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
−7.66250 − 13.2718i
−7.66250 + 13.2718i
−0.564972 − 0.978561i
−0.564972 + 0.978561i
2.28617 + 3.95977i
2.28617 − 3.95977i
6.44130 + 11.1567i
6.44130 − 11.1567i
−14.3250 − 15.5885i 141.206 − 175.511i 223.305i −202.362 + 276.945i −1105.97 −243.000 2514.19i
13.2 −14.3250 15.5885i 141.206 175.511i − 223.305i −202.362 − 276.945i −1105.97 −243.000 − 2514.19i
13.3 −0.129945 − 15.5885i −63.9831 126.447i 2.02564i −213.284 + 268.624i 16.6307 −243.000 − 16.4311i
13.4 −0.129945 15.5885i −63.9831 − 126.447i − 2.02564i −213.284 − 268.624i 16.6307 −243.000 16.4311i
13.5 5.57235 − 15.5885i −32.9490 − 205.672i − 86.8643i 342.970 + 4.53729i −540.233 −243.000 − 1146.08i
13.6 5.57235 15.5885i −32.9490 205.672i 86.8643i 342.970 − 4.53729i −540.233 −243.000 1146.08i
13.7 13.8826 − 15.5885i 128.727 109.244i − 216.408i 86.6767 − 331.868i 898.572 −243.000 1516.59i
13.8 13.8826 15.5885i 128.727 − 109.244i 216.408i 86.6767 + 331.868i 898.572 −243.000 − 1516.59i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.7.d.a ✓ 8
3.b odd 2 1 63.7.d.f 8
4.b odd 2 1 336.7.f.a 8
7.b odd 2 1 inner 21.7.d.a ✓ 8
7.c even 3 1 147.7.f.b 8
7.c even 3 1 147.7.f.c 8
7.d odd 6 1 147.7.f.b 8
7.d odd 6 1 147.7.f.c 8
21.c even 2 1 63.7.d.f 8
28.d even 2 1 336.7.f.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.d.a ✓ 8 1.a even 1 1 trivial
21.7.d.a ✓ 8 7.b odd 2 1 inner
63.7.d.f 8 3.b odd 2 1
63.7.d.f 8 21.c even 2 1
147.7.f.b 8 7.c even 3 1
147.7.f.b 8 7.d odd 6 1
147.7.f.c 8 7.c even 3 1
147.7.f.c 8 7.d odd 6 1
336.7.f.a 8 4.b odd 2 1
336.7.f.a 8 28.d even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 5 T^{3} + \cdots + 144)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 243)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots - 3045338564760)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 32\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 94\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 14\!\cdots\!32)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 18\!\cdots\!24)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 55\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 14\!\cdots\!72)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 20\!\cdots\!96)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 30\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 66\!\cdots\!40)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 76\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 64\!\cdots\!72)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 36\!\cdots\!08)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 36\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 16\!\cdots\!52)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 36\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 28\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 39\!\cdots\!44 \) Copy content Toggle raw display
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