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Properties

Label 42.7.b.a
Level $42$
Weight $7$
Character orbit 42.b
Analytic conductor $9.662$
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] = [42,7,Mod(29,42)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(42, 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("42.29");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 42.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: \(9.66227151203\)
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: \( x^{12} + 532x^{10} + 74137x^{8} + 4103612x^{6} + 100648268x^{4} + 983303384x^{2} + 1835779716 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{9}\cdot 7^{6} \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + (\beta_1 - 7) q^{3} - 32 q^{4} + ( - \beta_{6} - 7 \beta_{3} + \beta_1) q^{5} + (\beta_{7} + 7 \beta_{3} + 13) q^{6} + ( - \beta_{2} + \beta_1) q^{7} + 32 \beta_{3} q^{8} + (\beta_{11} - 2 \beta_{9} + 3 \beta_{8} + \cdots - 86) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + (\beta_1 - 7) q^{3} - 32 q^{4} + ( - \beta_{6} - 7 \beta_{3} + \beta_1) q^{5} + (\beta_{7} + 7 \beta_{3} + 13) q^{6} + ( - \beta_{2} + \beta_1) q^{7} + 32 \beta_{3} q^{8} + (\beta_{11} - 2 \beta_{9} + 3 \beta_{8} + \cdots - 86) q^{9}+ \cdots + (2751 \beta_{11} + 197 \beta_{10} + \cdots - 325867) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 84 q^{3} - 384 q^{4} + 160 q^{6} - 1012 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 84 q^{3} - 384 q^{4} + 160 q^{6} - 1012 q^{9} - 2496 q^{10} + 2688 q^{12} + 1440 q^{13} - 3736 q^{15} + 12288 q^{16} + 6144 q^{18} - 11400 q^{19} + 10976 q^{21} + 5472 q^{22} - 5120 q^{24} + 6132 q^{25} - 72828 q^{27} + 25376 q^{30} - 65328 q^{31} + 62656 q^{33} + 148416 q^{34} + 32384 q^{36} - 182160 q^{37} + 5336 q^{39} + 79872 q^{40} - 209328 q^{43} - 247688 q^{45} + 61728 q^{46} - 86016 q^{48} + 201684 q^{49} + 484168 q^{51} - 46080 q^{52} + 221152 q^{54} + 907728 q^{55} - 448032 q^{57} - 465792 q^{58} + 119552 q^{60} - 146160 q^{61} - 57624 q^{63} - 393216 q^{64} + 62272 q^{66} - 657600 q^{67} - 920984 q^{69} + 98784 q^{70} - 196608 q^{72} - 199176 q^{73} + 1584884 q^{75} + 364800 q^{76} - 1459136 q^{78} + 99456 q^{79} + 271484 q^{81} - 598464 q^{82} - 351232 q^{84} + 1772952 q^{85} + 3538040 q^{87} - 175104 q^{88} - 374720 q^{90} + 2033304 q^{91} + 23912 q^{93} - 1358016 q^{94} + 163840 q^{96} - 1694136 q^{97} - 3919552 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 532x^{10} + 74137x^{8} + 4103612x^{6} + 100648268x^{4} + 983303384x^{2} + 1835779716 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 529666689486 \nu^{11} + 39695648290178 \nu^{10} - 304634313057903 \nu^{9} + \cdots - 40\!\cdots\!68 ) / 93\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 529666689486 \nu^{11} - 938303053717394 \nu^{10} - 304634313057903 \nu^{9} + \cdots - 49\!\cdots\!96 ) / 93\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 102452836 \nu^{11} - 51784409123 \nu^{9} - 6212220217890 \nu^{7} + \cdots + 10\!\cdots\!46 \nu ) / 19\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 678195546829 \nu^{11} - 9075932281366 \nu^{10} + 347381055217967 \nu^{9} + \cdots - 77\!\cdots\!64 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 53474655498722 \nu^{11} + 568945557355794 \nu^{10} + \cdots + 46\!\cdots\!36 ) / 93\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 95571191794796 \nu^{11} + 39695648290178 \nu^{10} + \cdots - 40\!\cdots\!68 ) / 93\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8639093790499 \nu^{11} - 62584190450120 \nu^{10} + \cdots - 47\!\cdots\!50 ) / 78\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 9039885902116 \nu^{11} + 14736506883192 \nu^{10} + \cdots + 18\!\cdots\!48 ) / 78\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 82609455718340 \nu^{11} - 9027744718796 \nu^{10} + \cdots - 12\!\cdots\!24 ) / 46\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 180067256772169 \nu^{11} - 259574186731100 \nu^{10} + \cdots - 47\!\cdots\!60 ) / 93\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 241132495874995 \nu^{11} - 190637575868394 \nu^{10} + \cdots - 18\!\cdots\!16 ) / 93\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 69 \beta_{11} + 23 \beta_{10} - 46 \beta_{9} - 69 \beta_{8} + 173 \beta_{7} - 23 \beta_{5} + \cdots + 23 ) / 3528 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 63 \beta_{11} - 707 \beta_{10} + 504 \beta_{9} - 259 \beta_{8} - 721 \beta_{7} + 77 \beta_{5} + \cdots - 155687 ) / 1764 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 12273 \beta_{11} - 1711 \beta_{10} + 7678 \beta_{9} + 16529 \beta_{8} - 19779 \beta_{7} + 8316 \beta_{6} + \cdots - 5281 ) / 1764 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3675 \beta_{11} + 48069 \beta_{10} - 33908 \beta_{9} + 13377 \beta_{8} + 46060 \beta_{7} + \cdots + 6561835 ) / 294 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 4215261 \beta_{11} + 574467 \beta_{10} - 2887706 \beta_{9} - 5954033 \beta_{8} + 6277175 \beta_{7} + \cdots + 1820397 ) / 1764 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 4387572 \beta_{11} - 52348142 \beta_{10} + 36059604 \beta_{9} - 13000204 \beta_{8} + \cdots - 6507582998 ) / 882 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 105448278 \beta_{11} - 14771908 \beta_{10} + 74437280 \beta_{9} + 150341742 \beta_{8} + \cdots - 45338185 ) / 126 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1609270425 \beta_{11} + 18576782483 \beta_{10} - 12700840698 \beta_{9} + 4464499543 \beta_{8} + \cdots + 2262940394351 ) / 882 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 259775855601 \beta_{11} + 36747900757 \beta_{10} - 184875244276 \beta_{9} - 371155298363 \beta_{8} + \cdots + 111513977422 ) / 882 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 191126251995 \beta_{11} - 2186106582401 \beta_{10} + 1491655201414 \beta_{9} + \cdots - 265033587359527 ) / 294 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 91526824956468 \beta_{11} - 12982438381446 \beta_{10} + 65273523442460 \beta_{9} + \cdots - 39272193287511 ) / 882 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(31\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
1.54992i
18.7737i
− 6.08251i
4.78309i
− 5.38801i
− 9.39357i
− 1.54992i
− 18.7737i
6.08251i
− 4.78309i
5.38801i
9.39357i
− 5.65685i −26.9791 + 1.06089i −32.0000 − 222.080i 6.00130 + 152.617i −129.642 181.019i 726.749 − 57.2438i −1256.27
29.2 − 5.65685i −23.6675 + 12.9942i −32.0000 54.1650i 73.5063 + 133.884i 129.642 181.019i 391.302 − 615.080i 306.403
29.3 − 5.65685i −17.0635 − 20.9245i −32.0000 111.171i −118.367 + 96.5260i −129.642 181.019i −146.671 + 714.093i 628.877
29.4 − 5.65685i 1.87668 + 26.9347i −32.0000 − 33.0748i 152.366 − 10.6161i −129.642 181.019i −721.956 + 101.095i −187.099
29.5 − 5.65685i 6.11719 − 26.2979i −32.0000 − 156.019i −148.763 − 34.6041i 129.642 181.019i −654.160 − 321.739i −882.576
29.6 − 5.65685i 17.7163 + 20.3748i −32.0000 25.2202i 115.257 − 100.219i 129.642 181.019i −101.264 + 721.933i 142.667
29.7 5.65685i −26.9791 − 1.06089i −32.0000 222.080i 6.00130 − 152.617i −129.642 − 181.019i 726.749 + 57.2438i −1256.27
29.8 5.65685i −23.6675 − 12.9942i −32.0000 − 54.1650i 73.5063 − 133.884i 129.642 − 181.019i 391.302 + 615.080i 306.403
29.9 5.65685i −17.0635 + 20.9245i −32.0000 − 111.171i −118.367 − 96.5260i −129.642 − 181.019i −146.671 − 714.093i 628.877
29.10 5.65685i 1.87668 − 26.9347i −32.0000 33.0748i 152.366 + 10.6161i −129.642 − 181.019i −721.956 − 101.095i −187.099
29.11 5.65685i 6.11719 + 26.2979i −32.0000 156.019i −148.763 + 34.6041i 129.642 − 181.019i −654.160 + 321.739i −882.576
29.12 5.65685i 17.7163 − 20.3748i −32.0000 − 25.2202i 115.257 + 100.219i 129.642 − 181.019i −101.264 − 721.933i 142.667
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.12
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 42.7.b.a ✓ 12
3.b odd 2 1 inner 42.7.b.a ✓ 12
4.b odd 2 1 336.7.d.c 12
7.b odd 2 1 294.7.b.c 12
12.b even 2 1 336.7.d.c 12
21.c even 2 1 294.7.b.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.7.b.a ✓ 12 1.a even 1 1 trivial
42.7.b.a ✓ 12 3.b odd 2 1 inner
294.7.b.c 12 7.b odd 2 1
294.7.b.c 12 21.c even 2 1
336.7.d.c 12 4.b odd 2 1
336.7.d.c 12 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(42, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 32)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{2} - 16807)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 77\!\cdots\!04)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 14\!\cdots\!96)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 48\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 46\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 29\!\cdots\!64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots - 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 27\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 88\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 98\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 29\!\cdots\!88)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 10\!\cdots\!48)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 28\!\cdots\!24)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 47\!\cdots\!92)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 25\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 30\!\cdots\!04)^{2} \) Copy content Toggle raw display
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