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Properties

Label 20.7.b.a
Level $20$
Weight $7$
Character orbit 20.b
Analytic conductor $4.601$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [20,7,Mod(11,20)]
 
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, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("20.11");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 20.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: \(4.60108167240\)
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: \( x^{12} - x^{11} - 18 x^{10} - 30 x^{9} + 174 x^{8} + 1853 x^{7} + 10388 x^{6} - 17262 x^{5} + \cdots + 7603655 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{33}\cdot 5^{7} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{2} + ( - \beta_{6} - \beta_{2}) q^{3} + (\beta_{3} + \beta_{2} + 13) q^{4} + ( - \beta_{2} - \beta_1) q^{5} + (\beta_{10} + 2 \beta_{6} + \beta_{3} + \cdots - 56) q^{6}+ \cdots + ( - \beta_{11} + 2 \beta_{10} + \cdots - 166) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{2} + ( - \beta_{6} - \beta_{2}) q^{3} + (\beta_{3} + \beta_{2} + 13) q^{4} + ( - \beta_{2} - \beta_1) q^{5} + (\beta_{10} + 2 \beta_{6} + \beta_{3} + \cdots - 56) q^{6}+ \cdots + ( - 2632 \beta_{11} - 1164 \beta_{10} + \cdots + 7670) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 10 q^{2} + 156 q^{4} - 672 q^{6} + 440 q^{8} - 1996 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 10 q^{2} + 156 q^{4} - 672 q^{6} + 440 q^{8} - 1996 q^{9} + 750 q^{10} - 440 q^{12} - 5040 q^{13} + 6248 q^{14} + 3312 q^{16} + 6840 q^{17} + 15790 q^{18} - 3500 q^{20} - 27464 q^{21} - 26160 q^{22} + 28528 q^{24} + 37500 q^{25} - 18684 q^{26} + 19320 q^{28} - 74968 q^{29} - 13000 q^{30} - 60800 q^{32} + 112880 q^{33} + 48204 q^{34} - 128580 q^{36} + 62640 q^{37} + 74800 q^{38} + 57000 q^{40} - 16976 q^{41} - 138360 q^{42} - 222160 q^{44} + 77000 q^{45} - 144792 q^{46} + 297600 q^{48} + 72564 q^{49} - 31250 q^{50} + 548280 q^{52} + 322160 q^{53} + 150416 q^{54} - 246512 q^{56} - 1213440 q^{57} + 350700 q^{58} + 157000 q^{60} + 46464 q^{61} - 7120 q^{62} - 542784 q^{64} - 133000 q^{65} - 65200 q^{66} - 1678280 q^{68} + 41256 q^{69} - 360000 q^{70} + 2317560 q^{72} - 415080 q^{73} + 1581924 q^{74} + 208320 q^{76} + 75600 q^{77} + 473200 q^{78} + 734000 q^{80} + 2287428 q^{81} - 3169500 q^{82} - 2256224 q^{84} + 372000 q^{85} - 62512 q^{86} - 278880 q^{88} + 278392 q^{89} - 2162250 q^{90} + 4095720 q^{92} - 3646000 q^{93} + 4706568 q^{94} - 2641152 q^{96} + 2344680 q^{97} + 1050270 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - 18 x^{10} - 30 x^{9} + 174 x^{8} + 1853 x^{7} + 10388 x^{6} - 17262 x^{5} + \cdots + 7603655 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 22\!\cdots\!53 \nu^{11} + \cdots + 67\!\cdots\!55 ) / 18\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 30\!\cdots\!53 \nu^{11} + \cdots + 13\!\cdots\!45 ) / 18\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 49\!\cdots\!25 \nu^{11} + \cdots + 31\!\cdots\!15 ) / 36\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 14\!\cdots\!47 \nu^{11} + \cdots - 61\!\cdots\!00 ) / 99\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 19\!\cdots\!79 \nu^{11} + \cdots + 84\!\cdots\!10 ) / 99\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 98\!\cdots\!91 \nu^{11} + \cdots + 30\!\cdots\!05 ) / 39\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 55\!\cdots\!99 \nu^{11} + \cdots - 99\!\cdots\!85 ) / 19\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13\!\cdots\!73 \nu^{11} + \cdots - 18\!\cdots\!25 ) / 19\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 68\!\cdots\!28 \nu^{11} + \cdots - 29\!\cdots\!29 ) / 49\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 72\!\cdots\!64 \nu^{11} + \cdots + 22\!\cdots\!95 ) / 49\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 32\!\cdots\!83 \nu^{11} + \cdots + 72\!\cdots\!85 ) / 19\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 25\beta_{6} - 25\beta_{5} - 59\beta_{2} + 16\beta _1 + 50 ) / 800 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 20 \beta_{11} + 35 \beta_{10} + 39 \beta_{9} - 31 \beta_{8} - 26 \beta_{7} + 14 \beta_{6} + \cdots + 2402 ) / 800 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 130 \beta_{11} - 30 \beta_{10} + 8 \beta_{9} - 112 \beta_{8} - 82 \beta_{7} + 3 \beta_{6} + \cdots + 9644 ) / 800 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 75 \beta_{11} + 190 \beta_{10} - 16 \beta_{9} - 661 \beta_{8} - 706 \beta_{7} + 1719 \beta_{6} + \cdots + 11297 ) / 800 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 25 \beta_{11} + 35 \beta_{10} + 81 \beta_{9} - 344 \beta_{8} - 264 \beta_{7} + 261 \beta_{6} + \cdots - 34887 ) / 80 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 6110 \beta_{11} - 535 \beta_{10} - 12585 \beta_{9} - 14935 \beta_{8} - 22360 \beta_{7} + \cdots - 4568500 ) / 800 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 52365 \beta_{11} + 4835 \beta_{10} - 49969 \beta_{9} - 24294 \beta_{8} - 27494 \beta_{7} + \cdots - 9376277 ) / 800 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 493935 \beta_{11} - 237015 \beta_{10} - 582083 \beta_{9} + 252982 \beta_{8} - 150178 \beta_{7} + \cdots - 99485809 ) / 800 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 2625715 \beta_{11} + 603075 \beta_{10} - 1436079 \beta_{9} + 1908076 \beta_{8} + 931856 \beta_{7} + \cdots - 275264777 ) / 800 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 459070 \beta_{11} - 165605 \beta_{10} - 460161 \beta_{9} + 868579 \beta_{8} + 536734 \beta_{7} + \cdots - 24994968 ) / 40 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 34579650 \beta_{11} + 5947895 \beta_{10} - 19832475 \beta_{9} + 99242815 \beta_{8} + \cdots + 4234554190 ) / 800 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−2.35434 + 0.469325i
−2.35434 − 0.469325i
−4.07586 + 2.01083i
−4.07586 − 2.01083i
−0.552397 + 3.35995i
−0.552397 − 3.35995i
−0.601180 + 3.99996i
−0.601180 − 3.99996i
4.83379 + 2.37885i
4.83379 − 2.37885i
3.24999 + 1.01902i
3.24999 − 1.01902i
−7.94474 − 0.938649i − 50.7798i 62.2379 + 14.9147i −55.9017 −47.6645 + 403.433i − 91.7717i −480.464 − 176.913i −1849.59 444.125 + 52.4721i
11.2 −7.94474 + 0.938649i 50.7798i 62.2379 − 14.9147i −55.9017 −47.6645 − 403.433i 91.7717i −480.464 + 176.913i −1849.59 444.125 − 52.4721i
11.3 −6.91565 − 4.02166i 5.41240i 31.6525 + 55.6248i 55.9017 21.7668 − 37.4303i − 454.127i 4.80679 − 511.977i 699.706 −386.597 − 224.818i
11.4 −6.91565 + 4.02166i − 5.41240i 31.6525 − 55.6248i 55.9017 21.7668 + 37.4303i 454.127i 4.80679 + 511.977i 699.706 −386.597 + 224.818i
11.5 −4.34086 − 6.71989i 5.31460i −26.3138 + 58.3402i −55.9017 35.7136 − 23.0700i 502.023i 506.265 − 76.4207i 700.755 242.662 + 375.653i
11.6 −4.34086 + 6.71989i − 5.31460i −26.3138 − 58.3402i −55.9017 35.7136 + 23.0700i − 502.023i 506.265 + 76.4207i 700.755 242.662 − 375.653i
11.7 0.0337085 − 7.99993i − 38.3717i −63.9977 − 0.539332i 55.9017 −306.971 − 1.29345i 111.263i −6.47188 + 511.959i −743.391 1.88436 − 447.210i
11.8 0.0337085 + 7.99993i 38.3717i −63.9977 + 0.539332i 55.9017 −306.971 + 1.29345i − 111.263i −6.47188 − 511.959i −743.391 1.88436 + 447.210i
11.9 6.43150 − 4.75771i − 20.5795i 18.7284 − 61.1984i −55.9017 −97.9113 − 132.357i − 24.2619i −170.712 − 482.702i 305.483 −359.532 + 265.964i
11.10 6.43150 + 4.75771i 20.5795i 18.7284 + 61.1984i −55.9017 −97.9113 + 132.357i 24.2619i −170.712 + 482.702i 305.483 −359.532 − 265.964i
11.11 7.73604 − 2.03804i 28.9821i 55.6928 − 31.5327i 55.9017 59.0666 + 224.207i 435.848i 366.577 − 357.443i −110.960 432.458 − 113.930i
11.12 7.73604 + 2.03804i − 28.9821i 55.6928 + 31.5327i 55.9017 59.0666 − 224.207i − 435.848i 366.577 + 357.443i −110.960 432.458 + 113.930i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b 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.b.a ✓ 12
3.b odd 2 1 180.7.c.a 12
4.b odd 2 1 inner 20.7.b.a ✓ 12
5.b even 2 1 100.7.b.h 12
5.c odd 4 2 100.7.d.b 24
8.b even 2 1 320.7.b.d 12
8.d odd 2 1 320.7.b.d 12
12.b even 2 1 180.7.c.a 12
20.d odd 2 1 100.7.b.h 12
20.e even 4 2 100.7.d.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.7.b.a ✓ 12 1.a even 1 1 trivial
20.7.b.a ✓ 12 4.b odd 2 1 inner
100.7.b.h 12 5.b even 2 1
100.7.b.h 12 20.d odd 2 1
100.7.d.b 24 5.c odd 4 2
100.7.d.b 24 20.e even 4 2
180.7.c.a 12 3.b odd 2 1
180.7.c.a 12 12.b even 2 1
320.7.b.d 12 8.b even 2 1
320.7.b.d 12 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(20, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 68719476736 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{2} - 3125)^{6} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots - 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 19\!\cdots\!84)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 36\!\cdots\!56)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 52\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 80\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 10\!\cdots\!76)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
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