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Properties

Label 7.15.b.b
Level $7$
Weight $15$
Character orbit 7.b
Analytic conductor $8.703$
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] = [7,15,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, 15, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7.6");
 
S:= CuspForms(chi, 15);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 15 \)
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: \(8.70302777063\)
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: \( x^{8} + 310048x^{6} + 25253565716x^{4} + 269454557317610x^{2} + 8231577821610000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{25}\cdot 3^{4}\cdot 5^{2}\cdot 7^{5} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 22) q^{2} - \beta_{3} q^{3} + (18 \beta_{2} + \beta_1 + 10352) q^{4} + ( - \beta_{5} + 5 \beta_{3}) q^{5} + (\beta_{6} - 42 \beta_{3}) q^{6} + (\beta_{7} + 3 \beta_{5} + \cdots + 184457) q^{7}+ \cdots + (55 \beta_{4} - 6174 \beta_{2} + \cdots - 3441495) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 22) q^{2} - \beta_{3} q^{3} + (18 \beta_{2} + \beta_1 + 10352) q^{4} + ( - \beta_{5} + 5 \beta_{3}) q^{5} + (\beta_{6} - 42 \beta_{3}) q^{6} + (\beta_{7} + 3 \beta_{5} + \cdots + 184457) q^{7}+ \cdots + ( - 273393692 \beta_{4} + \cdots - 43897994781366) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 176 q^{2} + 82816 q^{4} + 1475656 q^{7} + 2816512 q^{8} - 27531960 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 176 q^{2} + 82816 q^{4} + 1475656 q^{7} + 2816512 q^{8} - 27531960 q^{9} - 50103856 q^{11} - 327713680 q^{14} + 358604160 q^{15} + 843603968 q^{16} - 1899978576 q^{18} - 3461974656 q^{21} + 18666471648 q^{22} + 922763024 q^{23} - 11827622200 q^{25} - 29234622592 q^{28} - 88461515056 q^{29} + 28450320000 q^{30} + 104004747264 q^{32} + 191911251840 q^{35} - 100162908288 q^{36} - 105597579184 q^{37} + 498015493248 q^{39} - 800893726080 q^{42} - 603785575216 q^{43} + 603119438592 q^{44} + 401041794912 q^{46} - 2745819180664 q^{49} + 3952648414640 q^{50} - 2130176489472 q^{51} + 404919387728 q^{53} - 3586370375168 q^{56} + 3524551966080 q^{57} - 10095456944928 q^{58} + 34054684231680 q^{60} - 8564148201336 q^{63} - 12027763490816 q^{64} - 20955121299840 q^{65} + 31164769903696 q^{67} - 34600029179520 q^{70} + 25369793024144 q^{71} - 123614737393152 q^{72} + 118993722892768 q^{74} - 47452474665904 q^{77} + 121635975377280 q^{78} - 36814047747056 q^{79} + 174098072793096 q^{81} - 315335206803456 q^{84} + 58215702758400 q^{85} - 201018954326048 q^{86} + 342477310264320 q^{88} - 98358431773056 q^{91} + 133790712616704 q^{92} - 363290481600000 q^{93} + 396115416984960 q^{95} - 142322390748496 q^{98} - 351183958250928 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 310048x^{6} + 25253565716x^{4} + 269454557317610x^{2} + 8231577821610000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 98288152 \nu^{6} - 484993392071240 \nu^{4} + \cdots - 61\!\cdots\!20 ) / 25\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 833050156 \nu^{6} - 203137980332916 \nu^{4} + \cdots - 47\!\cdots\!30 ) / 76\!\cdots\!35 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 210377059177 \nu^{7} + \cdots + 56\!\cdots\!70 \nu ) / 73\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 317287331344 \nu^{6} + \cdots - 39\!\cdots\!60 ) / 25\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2106847799678 \nu^{7} + \cdots + 53\!\cdots\!80 \nu ) / 36\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8631613842073 \nu^{7} + \cdots + 23\!\cdots\!30 \nu ) / 36\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10784005053872 \nu^{7} + 23003331522440 \nu^{6} + \cdots + 28\!\cdots\!00 ) / 36\!\cdots\!25 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - 2\beta_{5} - 42\beta_{3} ) / 1680 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 236\beta_{4} - 265518\beta_{2} - 11699\beta _1 - 260440320 ) / 3360 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 22230\beta_{7} - 315258\beta_{6} + 288826\beta_{5} + 11115\beta_{4} + 17805596\beta_{3} ) / 3360 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -18749741\beta_{4} + 21258875808\beta_{2} + 466166894\beta _1 + 9580752483120 ) / 840 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3881869890 \beta_{7} + 98345662024 \beta_{6} - 37656000428 \beta_{5} + \cdots - 6917904588288 \beta_{3} ) / 6720 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1076260706014\beta_{4} - 1442069265878982\beta_{2} - 21074151428701\beta _1 - 427425095126419080 ) / 240 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 20859547596090 \beta_{7} + \cdots + 32\!\cdots\!08 \beta_{3} ) / 1680 \) Copy content Toggle raw display

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

Character values

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

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

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
352.493i
− 352.493i
5.53507i
− 5.53507i
− 111.708i
111.708i
− 416.278i
416.278i
−205.381 − 1784.29i 25797.5 121785.i 366460.i 334708. − 752458.i −1.93336e6 1.59928e6 − 2.50124e7i
6.2 −205.381 1784.29i 25797.5 − 121785.i − 366460.i 334708. + 752458.i −1.93336e6 1.59928e6 2.50124e7i
6.3 −40.3423 − 4206.48i −14756.5 − 59167.8i 169699.i 632945. + 526881.i 1.25628e6 −1.29115e7 2.38697e6i
6.4 −40.3423 4206.48i −14756.5 59167.8i − 169699.i 632945. − 526881.i 1.25628e6 −1.29115e7 − 2.38697e6i
6.5 105.832 − 68.7293i −5183.52 − 89854.3i − 7273.78i 140951. − 811391.i −2.28254e6 4.77825e6 − 9.50949e6i
6.6 105.832 68.7293i −5183.52 89854.3i 7273.78i 140951. + 811391.i −2.28254e6 4.77825e6 9.50949e6i
6.7 227.891 − 3466.26i 35550.5 62622.6i − 789930.i −370776. − 735356.i 4.36788e6 −7.23196e6 1.42711e7i
6.8 227.891 3466.26i 35550.5 − 62622.6i 789930.i −370776. + 735356.i 4.36788e6 −7.23196e6 − 1.42711e7i
\(n\): 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 7.15.b.b ✓ 8
3.b odd 2 1 63.15.d.c 8
4.b odd 2 1 112.15.c.b 8
7.b odd 2 1 inner 7.15.b.b ✓ 8
7.c even 3 2 49.15.d.b 16
7.d odd 6 2 49.15.d.b 16
21.c even 2 1 63.15.d.c 8
28.d even 2 1 112.15.c.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.15.b.b ✓ 8 1.a even 1 1 trivial
7.15.b.b ✓ 8 7.b odd 2 1 inner
49.15.d.b 16 7.c even 3 2
49.15.d.b 16 7.d odd 6 2
63.15.d.c 8 3.b odd 2 1
63.15.d.c 8 21.c even 2 1
112.15.c.b 8 4.b odd 2 1
112.15.c.b 8 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 88T_{2}^{3} - 49600T_{2}^{2} + 3161344T_{2} + 199833600 \) acting on \(S_{15}^{\mathrm{new}}(7, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 88 T^{3} + \cdots + 199833600)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 21\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots - 60\!\cdots\!96)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots - 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 10\!\cdots\!36)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 89\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 20\!\cdots\!16)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 38\!\cdots\!44)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
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