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Properties

Label 21.7.b.a
Level $21$
Weight $7$
Character orbit 21.b
Analytic conductor $4.831$
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] = [21,7,Mod(8,21)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(21, 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("21.8");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 21.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: \(4.83113575602\)
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: \( x^{12} + 642x^{10} + 155265x^{8} + 17813036x^{6} + 1003321428x^{4} + 26369892864x^{2} + 256461520896 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{8}\cdot 7^{6} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} + 4) q^{3} + (\beta_{2} - 43) q^{4} + ( - \beta_{10} + \beta_1) q^{5} + (\beta_{9} - \beta_{6} + \beta_{5} + \cdots + 29) q^{6}+ \cdots + (\beta_{11} + 2 \beta_{6} + \beta_{5} + \cdots - 56) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} + 4) q^{3} + (\beta_{2} - 43) q^{4} + ( - \beta_{10} + \beta_1) q^{5} + (\beta_{9} - \beta_{6} + \beta_{5} + \cdots + 29) q^{6}+ \cdots + ( - 32 \beta_{11} + 3090 \beta_{10} + \cdots + 436198) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 52 q^{3} - 516 q^{4} + 350 q^{6} - 644 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 52 q^{3} - 516 q^{4} + 350 q^{6} - 644 q^{9} - 1092 q^{10} - 1726 q^{12} + 384 q^{13} - 632 q^{15} + 5892 q^{16} - 40 q^{18} + 11304 q^{19} - 5488 q^{21} + 15312 q^{22} + 12138 q^{24} - 77292 q^{25} + 114796 q^{27} + 41160 q^{28} - 101908 q^{30} - 9360 q^{31} - 65744 q^{33} - 169008 q^{34} - 195652 q^{36} + 212016 q^{37} + 159544 q^{39} + 196644 q^{40} - 37730 q^{42} + 28080 q^{43} - 4760 q^{45} - 418512 q^{46} + 865742 q^{48} + 201684 q^{49} - 371880 q^{51} - 138300 q^{52} - 254170 q^{54} - 732144 q^{55} - 440624 q^{57} + 1164240 q^{58} + 677660 q^{60} + 926736 q^{61} + 2744 q^{63} - 1380108 q^{64} - 1414588 q^{66} - 223104 q^{67} + 153048 q^{69} - 382788 q^{70} - 540192 q^{72} + 600984 q^{73} + 594716 q^{75} + 604596 q^{76} + 1866140 q^{78} - 276864 q^{79} + 617596 q^{81} + 1138200 q^{82} + 1398754 q^{84} - 3002472 q^{85} - 3372824 q^{87} - 1599048 q^{88} - 788032 q^{90} - 1243032 q^{91} + 408168 q^{93} + 8059296 q^{94} - 1141658 q^{96} + 1621416 q^{97} + 5211904 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 642x^{10} + 155265x^{8} + 17813036x^{6} + 1003321428x^{4} + 26369892864x^{2} + 256461520896 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 107 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 8717 \nu^{11} + 39992 \nu^{10} - 4877786 \nu^{9} + 23410928 \nu^{8} + \cdots + 231704485963776 ) / 838396403712 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8717 \nu^{11} + 90904 \nu^{10} + 4877786 \nu^{9} + 14227504 \nu^{8} + 988016909 \nu^{7} + \cdots - 19\!\cdots\!48 ) / 838396403712 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 8717 \nu^{11} - 289192 \nu^{10} - 4877786 \nu^{9} - 163888720 \nu^{8} + \cdots - 18\!\cdots\!36 ) / 419198201856 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 43585 \nu^{11} + 269368 \nu^{10} - 24388930 \nu^{9} + 107895664 \nu^{8} + \cdots - 525309830019072 ) / 838396403712 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8717 \nu^{11} - 1078816 \nu^{10} - 4877786 \nu^{9} - 618140608 \nu^{8} + \cdots - 55\!\cdots\!64 ) / 419198201856 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 53833 \nu^{11} - 39992 \nu^{10} - 32188498 \nu^{9} - 23410928 \nu^{8} + \cdots - 231704485963776 ) / 838396403712 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 66143 \nu^{11} + 129224 \nu^{10} - 38044286 \nu^{9} + 70245008 \nu^{8} + \cdots + 929885722859520 ) / 838396403712 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 15539 \nu^{11} - 8910438 \nu^{9} - 1831850931 \nu^{7} - 165238859332 \nu^{5} + \cdots - 80801728729344 \nu ) / 139732733952 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 149263 \nu^{11} - 1288024 \nu^{10} + 104296798 \nu^{9} - 735207472 \nu^{8} + \cdots - 69\!\cdots\!48 ) / 838396403712 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 107 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{9} + \beta_{8} + \beta_{6} - \beta_{4} + 3\beta_{3} - 160\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{7} + 15\beta_{6} + 8\beta_{5} - 19\beta_{4} - 104\beta_{3} - 251\beta_{2} + 20\beta _1 + 16904 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8 \beta_{11} - 60 \beta_{10} + 738 \beta_{9} - 251 \beta_{8} - 4 \beta_{7} - 371 \beta_{6} - 6 \beta_{5} + \cdots - 375 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1413 \beta_{7} - 5805 \beta_{6} - 3966 \beta_{5} + 7285 \beta_{4} + 41416 \beta_{3} + 56383 \beta_{2} + \cdots - 3153838 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2960 \beta_{11} + 29472 \beta_{10} - 209182 \beta_{9} + 56383 \beta_{8} + 1480 \beta_{7} + \cdots + 99287 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 453357 \beta_{7} + 1670049 \beta_{6} + 1298200 \beta_{5} - 2105949 \beta_{4} - 12145880 \beta_{3} + \cdots + 636751766 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 871800 \beta_{11} - 9572868 \beta_{10} + 53984434 \beta_{9} - 12641423 \beta_{8} - 435900 \beta_{7} + \cdots - 24021011 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 124917849 \beta_{7} - 432817833 \beta_{6} - 361948690 \beta_{5} + 550835465 \beta_{4} + \cdots - 134853558850 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 236035264 \beta_{11} + 2643975480 \beta_{10} - 13331938422 \beta_{9} + 2865198511 \beta_{8} + \cdots + 5651285103 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
− 15.2632i
− 12.1723i
− 11.9090i
− 7.64160i
− 5.92187i
− 5.05795i
5.05795i
5.92187i
7.64160i
11.9090i
12.1723i
15.2632i
− 15.2632i 26.9946 + 0.538759i −168.965 − 118.492i 8.22319 − 412.025i −129.642 1602.11i 728.419 + 29.0872i −1808.56
8.2 − 12.1723i −18.4482 − 19.7146i −84.1649 − 4.50929i −239.971 + 224.557i 129.642 245.453i −48.3270 + 727.396i −54.8885
8.3 − 11.9090i −13.4494 + 23.4118i −77.8243 201.466i 278.811 + 160.169i −129.642 164.633i −367.226 − 629.751i 2399.25
8.4 − 7.64160i −4.69404 + 26.5888i 5.60598 − 237.150i 203.181 + 35.8700i 129.642 − 531.901i −684.932 − 249.618i −1812.21
8.5 − 5.92187i 25.5592 + 8.70200i 28.9315 144.537i 51.5321 − 151.359i 129.642 − 550.328i 577.550 + 444.833i 855.929
8.6 − 5.05795i 10.0378 − 25.0648i 38.4172 − 24.8176i −126.776 − 50.7707i −129.642 − 518.021i −527.485 − 503.190i −125.526
8.7 5.05795i 10.0378 + 25.0648i 38.4172 24.8176i −126.776 + 50.7707i −129.642 518.021i −527.485 + 503.190i −125.526
8.8 5.92187i 25.5592 − 8.70200i 28.9315 − 144.537i 51.5321 + 151.359i 129.642 550.328i 577.550 − 444.833i 855.929
8.9 7.64160i −4.69404 − 26.5888i 5.60598 237.150i 203.181 − 35.8700i 129.642 531.901i −684.932 + 249.618i −1812.21
8.10 11.9090i −13.4494 − 23.4118i −77.8243 − 201.466i 278.811 − 160.169i −129.642 − 164.633i −367.226 + 629.751i 2399.25
8.11 12.1723i −18.4482 + 19.7146i −84.1649 4.50929i −239.971 − 224.557i 129.642 − 245.453i −48.3270 − 727.396i −54.8885
8.12 15.2632i 26.9946 − 0.538759i −168.965 118.492i 8.22319 + 412.025i −129.642 − 1602.11i 728.419 − 29.0872i −1808.56
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.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.b.a ✓ 12
3.b odd 2 1 inner 21.7.b.a ✓ 12
4.b odd 2 1 336.7.d.a 12
7.b odd 2 1 147.7.b.b 12
12.b even 2 1 336.7.d.a 12
21.c even 2 1 147.7.b.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.b.a ✓ 12 1.a even 1 1 trivial
21.7.b.a ✓ 12 3.b odd 2 1 inner
147.7.b.b 12 7.b odd 2 1
147.7.b.b 12 21.c even 2 1
336.7.d.a 12 4.b odd 2 1
336.7.d.a 12 12.b even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 256461520896 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{2} - 16807)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots - 25\!\cdots\!28)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 16\!\cdots\!72)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 47\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 57\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 72\!\cdots\!68)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 14\!\cdots\!68)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 98\!\cdots\!52)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 26\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 54\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 17\!\cdots\!56)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 77\!\cdots\!64)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 59\!\cdots\!48)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 26\!\cdots\!24)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 37\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 98\!\cdots\!08)^{2} \) Copy content Toggle raw display
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