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Properties

Label 20.7.d.d
Level $20$
Weight $7$
Character orbit 20.d
Analytic conductor $4.601$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [20,7,Mod(19,20)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(20, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("20.19");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 20.d (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: \(4.60108167240\)
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: \( x^{12} + 16x^{10} - 10x^{8} - 1775x^{6} - 1000x^{4} + 160000x^{2} + 1000000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{2}\cdot 5^{2}\cdot 7^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{3} + ( - \beta_{8} + \beta_{4} + 6) q^{4} + (\beta_{6} - 3 \beta_{4} - \beta_{3} + \cdots + 37) q^{5}+ \cdots + ( - 3 \beta_{11} + 6 \beta_{8} + \cdots + 162) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{3} + ( - \beta_{8} + \beta_{4} + 6) q^{4} + (\beta_{6} - 3 \beta_{4} - \beta_{3} + \cdots + 37) q^{5}+ \cdots + ( - 1611 \beta_{11} - 357 \beta_{10} + \cdots + 3024) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 64 q^{4} + 460 q^{5} - 672 q^{6} + 1956 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 64 q^{4} + 460 q^{5} - 672 q^{6} + 1956 q^{9} + 2000 q^{10} + 512 q^{14} - 20928 q^{16} - 13760 q^{20} + 51216 q^{21} - 20928 q^{24} - 55700 q^{25} + 14496 q^{26} - 16072 q^{29} + 83520 q^{30} + 257216 q^{34} - 144960 q^{36} - 36800 q^{40} - 192136 q^{41} - 165120 q^{44} - 57180 q^{45} - 49472 q^{46} + 145796 q^{49} - 36000 q^{50} - 118656 q^{54} + 1078208 q^{56} - 902400 q^{60} + 215384 q^{61} - 6656 q^{64} + 710400 q^{65} - 1403520 q^{66} + 1015824 q^{69} + 530080 q^{70} - 1020384 q^{74} + 2515200 q^{76} - 127040 q^{80} - 2327652 q^{81} + 424704 q^{84} - 44800 q^{85} - 5268832 q^{86} - 4346152 q^{89} + 3582000 q^{90} + 4292992 q^{94} + 7673088 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 16x^{10} - 10x^{8} - 1775x^{6} - 1000x^{4} + 160000x^{2} + 1000000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 9\nu^{11} + 64\nu^{9} - 970\nu^{7} - 8775\nu^{5} + 89000\nu^{3} + 770000\nu ) / 50000 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{11} + 33\nu^{9} + 260\nu^{7} - 2900\nu^{5} - 28375\nu^{3} + 230000\nu ) / 12500 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{11} - 28\nu^{9} + 350\nu^{7} + 3125\nu^{5} - 24500\nu^{3} - 290000\nu ) / 6250 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{10} - 28\nu^{8} + 350\nu^{6} + 3125\nu^{4} - 34500\nu^{2} - 327500 ) / 2500 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9\nu^{10} - 16\nu^{8} - 1050\nu^{6} - 775\nu^{4} + 87000\nu^{2} - 20000 ) / 1250 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 61 \nu^{11} - 30 \nu^{10} - 576 \nu^{9} - 280 \nu^{8} + 5810 \nu^{7} + 3500 \nu^{6} + \cdots - 3250000 ) / 50000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -103\nu^{11} + 2512\nu^{9} + 5590\nu^{7} - 226775\nu^{5} - 957000\nu^{3} + 26490000\nu ) / 50000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{10} - 56\nu^{8} + 170\nu^{6} + 5975\nu^{4} - 5000\nu^{2} - 586250 ) / 1250 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 121 \nu^{11} + 880 \nu^{10} + 1016 \nu^{9} - 1120 \nu^{8} - 12330 \nu^{7} - 76000 \nu^{6} + \cdots + 9550000 ) / 50000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 27 \nu^{11} - 3 \nu^{10} - 232 \nu^{9} - 28 \nu^{8} + 2670 \nu^{7} + 350 \nu^{6} + 25125 \nu^{5} + \cdots - 325000 ) / 5000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 61\nu^{10} + 436\nu^{8} - 6050\nu^{6} - 49675\nu^{4} + 547500\nu^{2} + 4712500 ) / 2500 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 5\beta_{7} - 42\beta_{6} + 21\beta_{4} + 62\beta_{3} - 100\beta_{2} - 151\beta _1 + 21 ) / 1680 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 33 \beta_{11} + 8 \beta_{10} + 16 \beta_{9} + 130 \beta_{8} - 8 \beta_{6} - 76 \beta_{5} + \cdots - 9346 ) / 3360 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -35\beta_{10} + 10\beta_{7} + 119\beta_{6} - 42\beta_{4} + 341\beta_{3} - 60\beta_{2} + 727\beta _1 - 42 ) / 560 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 333 \beta_{11} - 208 \beta_{10} - 416 \beta_{9} + 1870 \beta_{8} + 208 \beta_{6} - 124 \beta_{5} + \cdots + 153746 ) / 3360 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1470 \beta_{10} + 295 \beta_{7} + 168 \beta_{6} + 651 \beta_{4} - 1298 \beta_{3} - 8420 \beta_{2} + \cdots + 651 ) / 1680 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 507 \beta_{11} + 243 \beta_{10} + 486 \beta_{9} + 1980 \beta_{8} - 243 \beta_{6} - 2396 \beta_{5} + \cdots + 69834 ) / 560 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 9030 \beta_{10} - 2795 \beta_{7} - 10668 \beta_{6} + 9849 \beta_{4} + 133048 \beta_{3} + \cdots + 9849 ) / 1680 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4869 \beta_{11} - 2744 \beta_{10} - 5488 \beta_{9} + 19910 \beta_{8} + 2744 \beta_{6} - 5732 \beta_{5} + \cdots - 2702822 ) / 480 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 53865 \beta_{10} - 1360 \beta_{7} + 149681 \beta_{6} - 47908 \beta_{4} - 199641 \beta_{3} + \cdots - 47908 ) / 560 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 763167 \beta_{11} + 40708 \beta_{10} + 81416 \beta_{9} + 538130 \beta_{8} - 40708 \beta_{6} + \cdots + 126299254 ) / 3360 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 219030 \beta_{10} - 709045 \beta_{7} - 4116168 \beta_{6} + 2167599 \beta_{4} + 1912298 \beta_{3} + \cdots + 2167599 ) / 1680 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.826519 + 3.05235i
−0.826519 − 3.05235i
−0.954985 − 3.01463i
−0.954985 + 3.01463i
3.06674 − 0.771446i
3.06674 + 0.771446i
−3.06674 − 0.771446i
−3.06674 + 0.771446i
0.954985 − 3.01463i
0.954985 + 3.01463i
0.826519 + 3.05235i
0.826519 − 3.05235i
−7.42116 − 2.98770i 6.08848 46.1473 + 44.3444i 26.7958 − 122.094i −45.1836 − 18.1905i −422.345 −209.979 − 466.961i −691.930 −563.636 + 826.023i
19.2 −7.42116 + 2.98770i 6.08848 46.1473 − 44.3444i 26.7958 + 122.094i −45.1836 + 18.1905i −422.345 −209.979 + 466.961i −691.930 −563.636 − 826.023i
19.3 −5.94356 − 5.35482i 40.3729 6.65182 + 63.6534i −32.9271 + 120.585i −239.959 − 216.189i 450.705 301.317 − 413.947i 900.969 841.416 − 540.387i
19.4 −5.94356 + 5.35482i 40.3729 6.65182 − 63.6534i −32.9271 − 120.585i −239.959 + 216.189i 450.705 301.317 + 413.947i 900.969 841.416 + 540.387i
19.5 −3.68788 − 7.09927i −31.7642 −36.7991 + 52.3624i 121.131 + 30.8579i 117.142 + 225.502i 88.8045 507.445 + 68.1408i 279.961 −227.649 − 973.743i
19.6 −3.68788 + 7.09927i −31.7642 −36.7991 − 52.3624i 121.131 − 30.8579i 117.142 − 225.502i 88.8045 507.445 − 68.1408i 279.961 −227.649 + 973.743i
19.7 3.68788 − 7.09927i 31.7642 −36.7991 − 52.3624i 121.131 + 30.8579i 117.142 − 225.502i −88.8045 −507.445 + 68.1408i 279.961 665.785 − 746.143i
19.8 3.68788 + 7.09927i 31.7642 −36.7991 + 52.3624i 121.131 − 30.8579i 117.142 + 225.502i −88.8045 −507.445 − 68.1408i 279.961 665.785 + 746.143i
19.9 5.94356 − 5.35482i −40.3729 6.65182 − 63.6534i −32.9271 + 120.585i −239.959 + 216.189i −450.705 −301.317 − 413.947i 900.969 450.008 + 893.024i
19.10 5.94356 + 5.35482i −40.3729 6.65182 + 63.6534i −32.9271 − 120.585i −239.959 − 216.189i −450.705 −301.317 + 413.947i 900.969 450.008 − 893.024i
19.11 7.42116 − 2.98770i −6.08848 46.1473 − 44.3444i 26.7958 − 122.094i −45.1836 + 18.1905i 422.345 209.979 − 466.961i −691.930 −165.925 − 986.138i
19.12 7.42116 + 2.98770i −6.08848 46.1473 + 44.3444i 26.7958 + 122.094i −45.1836 − 18.1905i 422.345 209.979 + 466.961i −691.930 −165.925 + 986.138i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.7.d.d ✓ 12
3.b odd 2 1 180.7.f.e 12
4.b odd 2 1 inner 20.7.d.d ✓ 12
5.b even 2 1 inner 20.7.d.d ✓ 12
5.c odd 4 2 100.7.b.f 12
8.b even 2 1 320.7.h.f 12
8.d odd 2 1 320.7.h.f 12
12.b even 2 1 180.7.f.e 12
15.d odd 2 1 180.7.f.e 12
20.d odd 2 1 inner 20.7.d.d ✓ 12
20.e even 4 2 100.7.b.f 12
40.e odd 2 1 320.7.h.f 12
40.f even 2 1 320.7.h.f 12
60.h even 2 1 180.7.f.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.7.d.d ✓ 12 1.a even 1 1 trivial
20.7.d.d ✓ 12 4.b odd 2 1 inner
20.7.d.d ✓ 12 5.b even 2 1 inner
20.7.d.d ✓ 12 20.d odd 2 1 inner
100.7.b.f 12 5.c odd 4 2
100.7.b.f 12 20.e even 4 2
180.7.f.e 12 3.b odd 2 1
180.7.f.e 12 12.b even 2 1
180.7.f.e 12 15.d odd 2 1
180.7.f.e 12 60.h even 2 1
320.7.h.f 12 8.b even 2 1
320.7.h.f 12 8.d odd 2 1
320.7.h.f 12 40.e odd 2 1
320.7.h.f 12 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 2676T_{3}^{4} + 1742400T_{3}^{2} - 60963840 \) acting on \(S_{7}^{\mathrm{new}}(20, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 68719476736 \) Copy content Toggle raw display
$3$ \( (T^{6} - 2676 T^{4} + \cdots - 60963840)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + \cdots + 3814697265625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots - 285751234944000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 93\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots + 67\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 56\!\cdots\!40)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + \cdots - 9262562663592)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + \cdots - 3327456655368)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 29\!\cdots\!40)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 21\!\cdots\!88)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 18\!\cdots\!40)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 29\!\cdots\!40)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots - 46\!\cdots\!72)^{4} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 59\!\cdots\!00)^{2} \) Copy content Toggle raw display
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