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Properties

Label 7.17.b.b
Level $7$
Weight $17$
Character orbit 7.b
Analytic conductor $11.363$
Analytic rank $0$
Dimension $8$
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] = [7,17,Mod(6,7)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 17, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7.6");
 
S:= CuspForms(chi, 17);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 17 \)
Character orbit: \([\chi]\) \(=\) 7.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: \(11.3627180700\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5365384x^{6} + 10449491370210x^{4} + 8743024230718881600x^{2} + 2655236149032650377194000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{17}\cdot 3^{5}\cdot 7^{6} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 68) q^{2} + \beta_1 q^{3} + (\beta_{5} + 139 \beta_{2} + 2348) q^{4} + \beta_{3} q^{5} + (\beta_{4} - 42 \beta_1) q^{6} + ( - 4 \beta_{7} - \beta_{6} + \cdots - 379309) q^{7}+ \cdots + (\beta_{7} - 272 \beta_{5} + \cdots - 5241735) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 68) q^{2} + \beta_1 q^{3} + (\beta_{5} + 139 \beta_{2} + 2348) q^{4} + \beta_{3} q^{5} + (\beta_{4} - 42 \beta_1) q^{6} + ( - 4 \beta_{7} - \beta_{6} + \cdots - 379309) q^{7}+ \cdots + ( - 1886147684 \beta_{7} + \cdots - 19\!\cdots\!74) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 544 q^{2} + 18784 q^{4} - 3034472 q^{7} - 35863808 q^{8} - 41933880 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 544 q^{2} + 18784 q^{4} - 3034472 q^{7} - 35863808 q^{8} - 41933880 q^{9} + 430398704 q^{11} - 2080234240 q^{14} + 83393280 q^{15} - 4357080832 q^{16} - 7232400864 q^{18} + 45847234944 q^{21} - 34275403968 q^{22} + 89765082416 q^{23} + 61966251080 q^{25} - 376785722656 q^{28} - 22437591664 q^{29} - 192018300480 q^{30} + 941689387008 q^{32} + 371925382080 q^{35} - 4527659399328 q^{36} + 5737866534416 q^{37} - 7975804007808 q^{39} - 13160568536640 q^{42} - 3976952110864 q^{43} + 45337613120448 q^{44} + 35817469755072 q^{46} - 27450534789496 q^{49} - 96564765668320 q^{50} + 58670380591488 q^{51} - 108679841507824 q^{53} - 15117119134208 q^{56} - 196163055495360 q^{57} + 650847682404672 q^{58} - 335782392744960 q^{60} + 223739049782808 q^{63} - 460533940742144 q^{64} + 573279455461440 q^{65} - 722120065643024 q^{67} + 12\!\cdots\!60 q^{70}+ \cdots - 15\!\cdots\!92 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5365384x^{6} + 10449491370210x^{4} + 8743024230718881600x^{2} + 2655236149032650377194000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 6\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2123049 \nu^{6} + 8379851087056 \nu^{4} + \cdots + 42\!\cdots\!00 ) / 36\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 20994056 \nu^{7} + 93059602745484 \nu^{5} + \cdots + 58\!\cdots\!50 \nu ) / 22\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4246098 \nu^{7} - 16759702174112 \nu^{5} + \cdots - 85\!\cdots\!00 \nu ) / 12\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 543271063 \nu^{6} + \cdots + 97\!\cdots\!00 ) / 11\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 688547536 \nu^{7} - 320507083115 \nu^{6} + \cdots - 97\!\cdots\!50 ) / 34\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4245503308 \nu^{6} + \cdots + 12\!\cdots\!00 ) / 22\!\cdots\!75 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - 272\beta_{5} + 19868\beta_{2} - 48288456 ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 1197 \beta_{7} - 2394 \beta_{6} + 1197 \beta_{5} - 3232 \beta_{4} - 6924 \beta_{3} + \cdots - 5882856 \beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 452576\beta_{7} + 736734173\beta_{5} - 64349813537\beta_{2} + 70995264543324 ) / 36 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3184217523 \beta_{7} + 6368435046 \beta_{6} - 3184217523 \beta_{5} + 10030142488 \beta_{4} + \cdots + 9315915656814 \beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -1690153456211\beta_{7} - 388738182475028\beta_{5} + 40356451128322157\beta_{2} - 28097176608290600964 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 16\!\cdots\!53 \beta_{7} + \cdots - 38\!\cdots\!54 \beta_1 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1
− 1246.39i
1246.39i
− 923.458i
923.458i
− 1024.53i
1024.53i
− 1381.83i
1381.83i
−408.297 − 7478.36i 101171. 25884.6i 3.05340e6i 879246. + 5.69736e6i −1.45496e7 −1.28792e7 − 1.05686e7i
6.2 −408.297 7478.36i 101171. − 25884.6i − 3.05340e6i 879246. − 5.69736e6i −1.45496e7 −1.28792e7 1.05686e7i
6.3 −174.041 − 5540.75i −35245.9 − 367486.i 964315.i −2.61707e6 − 5.13652e6i 1.75401e7 1.23468e7 6.39575e7i
6.4 −174.041 5540.75i −35245.9 367486.i − 964315.i −2.61707e6 + 5.13652e6i 1.75401e7 1.23468e7 − 6.39575e7i
6.5 40.3121 − 6147.19i −63910.9 623292.i − 247806.i 5.20719e6 − 2.47347e6i −5.21828e6 5.25876e6 2.51262e7i
6.6 40.3121 6147.19i −63910.9 − 623292.i 247806.i 5.20719e6 + 2.47347e6i −5.21828e6 5.25876e6 − 2.51262e7i
6.7 270.026 − 8290.96i 7378.00 − 234861.i − 2.23878e6i −4.98661e6 + 2.89252e6i −1.57042e7 −2.56934e7 − 6.34186e7i
6.8 270.026 8290.96i 7378.00 234861.i 2.23878e6i −4.98661e6 − 2.89252e6i −1.57042e7 −2.56934e7 6.34186e7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.8
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 7.17.b.b ✓ 8
3.b odd 2 1 63.17.d.c 8
4.b odd 2 1 112.17.c.b 8
7.b odd 2 1 inner 7.17.b.b ✓ 8
21.c even 2 1 63.17.d.c 8
28.d even 2 1 112.17.c.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.17.b.b ✓ 8 1.a even 1 1 trivial
7.17.b.b ✓ 8 7.b odd 2 1 inner
63.17.d.c 8 3.b odd 2 1
63.17.d.c 8 21.c even 2 1
112.17.c.b 8 4.b odd 2 1
112.17.c.b 8 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 272T_{2}^{3} - 98776T_{2}^{2} - 15713792T_{2} + 773514240 \) acting on \(S_{17}^{\mathrm{new}}(7, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 272 T^{3} + \cdots + 773514240)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots - 13\!\cdots\!36)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 23\!\cdots\!60)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 96\!\cdots\!56)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 34\!\cdots\!60)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 22\!\cdots\!80)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 43\!\cdots\!20)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 12\!\cdots\!16)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 87\!\cdots\!16)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
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