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Properties

Label 637.2.f.l
Level $637$
Weight $2$
Character orbit 637.f
Analytic conductor $5.086$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [637,2,Mod(295,637)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.295");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.f (of order \(3\), degree \(2\), 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: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 8x^{14} + 45x^{12} + 124x^{10} + 248x^{8} + 250x^{6} + 177x^{4} + 14x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{10} + \beta_{4} + 1) q^{2} + ( - \beta_{15} + \beta_{8}) q^{3} + (\beta_{12} - \beta_{11} - \beta_{10} + \cdots - 2) q^{4}+ \cdots + \beta_{10} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{10} + \beta_{4} + 1) q^{2} + ( - \beta_{15} + \beta_{8}) q^{3} + (\beta_{12} - \beta_{11} - \beta_{10} + \cdots - 2) q^{4}+ \cdots + (\beta_{6} - \beta_{4} - \beta_{2} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} - 12 q^{4} + 24 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} - 12 q^{4} + 24 q^{8} - 4 q^{9} - 4 q^{11} - 8 q^{15} - 4 q^{16} - 56 q^{18} + 28 q^{22} + 12 q^{23} - 24 q^{25} + 8 q^{29} + 28 q^{30} + 4 q^{36} - 8 q^{37} - 4 q^{39} + 32 q^{43} - 8 q^{44} - 4 q^{46} + 36 q^{50} - 88 q^{51} - 8 q^{53} - 96 q^{57} - 48 q^{58} + 128 q^{60} - 64 q^{64} + 16 q^{65} + 20 q^{67} + 8 q^{71} + 28 q^{72} + 76 q^{74} + 28 q^{78} - 8 q^{79} + 56 q^{81} + 36 q^{85} + 8 q^{86} + 28 q^{88} - 160 q^{92} + 8 q^{93} + 52 q^{95} + 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 8x^{14} + 45x^{12} + 124x^{10} + 248x^{8} + 250x^{6} + 177x^{4} + 14x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7068 \nu^{14} + 50635 \nu^{12} + 276768 \nu^{10} + 645048 \nu^{8} + 1213590 \nu^{6} + \cdots - 1483013 ) / 773722 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7068 \nu^{15} + 50635 \nu^{13} + 276768 \nu^{11} + 645048 \nu^{9} + 1213590 \nu^{7} + \cdots - 2256735 \nu ) / 773722 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14022 \nu^{14} + 106693 \nu^{12} + 549072 \nu^{10} + 1279692 \nu^{8} + 1796116 \nu^{6} + \cdots + 408609 ) / 773722 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 36499 \nu^{15} + 262518 \nu^{13} + 1429224 \nu^{11} + 3331014 \nu^{9} + 6036084 \nu^{7} + \cdots - 5036536 \nu ) / 773722 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 43453 \nu^{14} + 318576 \nu^{12} + 1701528 \nu^{10} + 3965658 \nu^{8} + 6618610 \nu^{6} + \cdots - 1597470 ) / 773722 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 50521 \nu^{15} + 369211 \nu^{13} + 1978296 \nu^{11} + 4610706 \nu^{9} + 7832200 \nu^{7} + \cdots - 3854205 \nu ) / 773722 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 64431 \nu^{15} - 508380 \nu^{13} - 2848760 \nu^{11} - 7712676 \nu^{9} - 15333840 \nu^{7} + \cdots - 860742 \nu ) / 773722 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 64431 \nu^{14} - 508380 \nu^{12} - 2848760 \nu^{10} - 7712676 \nu^{8} - 15333840 \nu^{6} + \cdots - 87020 ) / 773722 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 107927 \nu^{14} - 828304 \nu^{12} - 4592694 \nu^{10} - 12008036 \nu^{8} - 23561464 \nu^{6} + \cdots - 1305850 ) / 773722 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 114457 \nu^{14} - 952041 \nu^{12} - 5418506 \nu^{10} - 15617428 \nu^{8} - 31705946 \nu^{6} + \cdots - 1814963 ) / 773722 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 121794 \nu^{14} - 966125 \nu^{12} - 5420752 \nu^{10} - 14780304 \nu^{8} - 29454090 \nu^{6} + \cdots - 1657053 ) / 773722 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 186225 \nu^{15} + 1474505 \nu^{13} + 8269512 \nu^{11} + 22492980 \nu^{9} + 44787930 \nu^{7} + \cdots + 2517795 \nu ) / 773722 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1241 \nu^{15} + 9879 \nu^{13} + 55358 \nu^{11} + 150958 \nu^{9} + 297972 \nu^{7} + 289428 \nu^{5} + \cdots + 1691 \nu ) / 2734 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 408609 \nu^{15} + 3254850 \nu^{13} + 18280712 \nu^{11} + 50118444 \nu^{9} + 100055340 \nu^{7} + \cdots + 5638608 \nu ) / 773722 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} - 2\beta_{9} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{13} - 3\beta_{8} - \beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{12} + \beta_{11} + \beta_{10} + 6\beta_{9} - 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} + 6\beta_{13} + 11\beta_{8} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -6\beta_{6} + 7\beta_{4} + 23\beta_{2} + 28 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{7} - 7\beta_{5} + 29\beta_{3} + 44\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 103\beta_{12} - 29\beta_{11} - 37\beta_{10} - 81\beta_{9} + 29\beta_{6} - 37\beta_{4} - 103\beta_{2} - 37 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 37\beta_{15} - 8\beta_{14} - 132\beta_{13} - 184\beta_{8} + 37\beta_{5} - 132\beta_{3} - 184\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -456\beta_{12} + 132\beta_{11} + 177\beta_{10} + 331\beta_{9} - 331 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -177\beta_{15} + 45\beta_{14} + 588\beta_{13} + 787\beta_{8} - 45\beta_{7} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -588\beta_{6} + 810\beta_{4} + 2008\beta_{2} + 2207 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 222\beta_{7} - 810\beta_{5} + 2596\beta_{3} + 3405\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 8819 \beta_{12} - 2596 \beta_{11} - 3628 \beta_{10} - 6000 \beta_{9} + 2596 \beta_{6} - 3628 \beta_{4} + \cdots - 3628 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 3628\beta_{15} - 1032\beta_{14} - 11415\beta_{13} - 14819\beta_{8} + 3628\beta_{5} - 11415\beta_{3} - 14819\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(248\)
\(\chi(n)\) \(-1 + \beta_{9}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
295.1
0.756863 + 1.31093i
−0.756863 − 1.31093i
−1.04641 − 1.81243i
1.04641 + 1.81243i
−0.141226 − 0.244611i
0.141226 + 0.244611i
−0.558788 − 0.967849i
0.558788 + 0.967849i
0.756863 − 1.31093i
−0.756863 + 1.31093i
−1.04641 + 1.81243i
1.04641 − 1.81243i
−0.141226 + 0.244611i
0.141226 − 0.244611i
−0.558788 + 0.967849i
0.558788 − 0.967849i
−1.21605 + 2.10626i −0.376796 + 0.652630i −1.95755 − 3.39058i 0.341537 −0.916405 − 1.58726i 0 4.65773 1.21605 + 2.10626i −0.415326 + 0.719366i
295.2 −1.21605 + 2.10626i 0.376796 − 0.652630i −1.95755 − 3.39058i −0.341537 0.916405 + 1.58726i 0 4.65773 1.21605 + 2.10626i 0.415326 − 0.719366i
295.3 0.289905 − 0.502131i −0.946019 + 1.63855i 0.831910 + 1.44091i −1.47362 0.548512 + 0.950050i 0 2.12432 −0.289905 − 0.502131i −0.427209 + 0.739948i
295.4 0.289905 − 0.502131i 0.946019 − 1.63855i 0.831910 + 1.44091i 1.47362 −0.548512 − 0.950050i 0 2.12432 −0.289905 − 0.502131i 0.427209 − 0.739948i
295.5 0.760387 − 1.31703i −1.06311 + 1.84135i −0.156376 − 0.270851i −0.589391 1.61674 + 2.80028i 0 2.56592 −0.760387 − 1.31703i −0.448165 + 0.776245i
295.6 0.760387 − 1.31703i 1.06311 − 1.84135i −0.156376 − 0.270851i 0.589391 −1.61674 − 2.80028i 0 2.56592 −0.760387 − 1.31703i 0.448165 − 0.776245i
295.7 1.16576 − 2.01915i −1.15450 + 1.99966i −1.71798 − 2.97563i 3.37112 2.69174 + 4.66224i 0 −3.34797 −1.16576 − 2.01915i 3.92990 − 6.80679i
295.8 1.16576 − 2.01915i 1.15450 − 1.99966i −1.71798 − 2.97563i −3.37112 −2.69174 − 4.66224i 0 −3.34797 −1.16576 − 2.01915i −3.92990 + 6.80679i
393.1 −1.21605 − 2.10626i −0.376796 − 0.652630i −1.95755 + 3.39058i 0.341537 −0.916405 + 1.58726i 0 4.65773 1.21605 − 2.10626i −0.415326 − 0.719366i
393.2 −1.21605 − 2.10626i 0.376796 + 0.652630i −1.95755 + 3.39058i −0.341537 0.916405 − 1.58726i 0 4.65773 1.21605 − 2.10626i 0.415326 + 0.719366i
393.3 0.289905 + 0.502131i −0.946019 − 1.63855i 0.831910 − 1.44091i −1.47362 0.548512 − 0.950050i 0 2.12432 −0.289905 + 0.502131i −0.427209 − 0.739948i
393.4 0.289905 + 0.502131i 0.946019 + 1.63855i 0.831910 − 1.44091i 1.47362 −0.548512 + 0.950050i 0 2.12432 −0.289905 + 0.502131i 0.427209 + 0.739948i
393.5 0.760387 + 1.31703i −1.06311 − 1.84135i −0.156376 + 0.270851i −0.589391 1.61674 − 2.80028i 0 2.56592 −0.760387 + 1.31703i −0.448165 − 0.776245i
393.6 0.760387 + 1.31703i 1.06311 + 1.84135i −0.156376 + 0.270851i 0.589391 −1.61674 + 2.80028i 0 2.56592 −0.760387 + 1.31703i 0.448165 + 0.776245i
393.7 1.16576 + 2.01915i −1.15450 − 1.99966i −1.71798 + 2.97563i 3.37112 2.69174 − 4.66224i 0 −3.34797 −1.16576 + 2.01915i 3.92990 + 6.80679i
393.8 1.16576 + 2.01915i 1.15450 + 1.99966i −1.71798 + 2.97563i −3.37112 −2.69174 + 4.66224i 0 −3.34797 −1.16576 + 2.01915i −3.92990 − 6.80679i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 295.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
13.c even 3 1 inner
91.n odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.f.l ✓ 16
7.b odd 2 1 inner 637.2.f.l ✓ 16
7.c even 3 1 637.2.g.m 16
7.c even 3 1 637.2.h.m 16
7.d odd 6 1 637.2.g.m 16
7.d odd 6 1 637.2.h.m 16
13.c even 3 1 inner 637.2.f.l ✓ 16
13.c even 3 1 8281.2.a.ci 8
13.e even 6 1 8281.2.a.cl 8
91.g even 3 1 637.2.h.m 16
91.h even 3 1 637.2.g.m 16
91.m odd 6 1 637.2.h.m 16
91.n odd 6 1 inner 637.2.f.l ✓ 16
91.n odd 6 1 8281.2.a.ci 8
91.t odd 6 1 8281.2.a.cl 8
91.v odd 6 1 637.2.g.m 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.f.l ✓ 16 1.a even 1 1 trivial
637.2.f.l ✓ 16 7.b odd 2 1 inner
637.2.f.l ✓ 16 13.c even 3 1 inner
637.2.f.l ✓ 16 91.n odd 6 1 inner
637.2.g.m 16 7.c even 3 1
637.2.g.m 16 7.d odd 6 1
637.2.g.m 16 91.h even 3 1
637.2.g.m 16 91.v odd 6 1
637.2.h.m 16 7.c even 3 1
637.2.h.m 16 7.d odd 6 1
637.2.h.m 16 91.g even 3 1
637.2.h.m 16 91.m odd 6 1
8281.2.a.ci 8 13.c even 3 1
8281.2.a.ci 8 91.n odd 6 1
8281.2.a.cl 8 13.e even 6 1
8281.2.a.cl 8 91.t odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\):

\( T_{2}^{8} - 2T_{2}^{7} + 9T_{2}^{6} - 14T_{2}^{5} + 54T_{2}^{4} - 80T_{2}^{3} + 119T_{2}^{2} - 60T_{2} + 25 \) Copy content Toggle raw display
\( T_{3}^{16} + 14T_{3}^{14} + 129T_{3}^{12} + 698T_{3}^{10} + 2760T_{3}^{8} + 6668T_{3}^{6} + 11117T_{3}^{4} + 5880T_{3}^{2} + 2401 \) Copy content Toggle raw display
\( T_{5}^{8} - 14T_{5}^{6} + 31T_{5}^{4} - 12T_{5}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 2 T^{7} + 9 T^{6} + \cdots + 25)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + 14 T^{14} + \cdots + 2401 \) Copy content Toggle raw display
$5$ \( (T^{8} - 14 T^{6} + 31 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 2 T^{7} + 9 T^{6} + \cdots + 25)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} + 58 T^{14} + \cdots + 13845841 \) Copy content Toggle raw display
$19$ \( T^{16} + 94 T^{14} + \cdots + 1500625 \) Copy content Toggle raw display
$23$ \( (T^{8} - 6 T^{7} + \cdots + 10000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 4 T^{7} + \cdots + 3171961)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 70 T^{6} + \cdots + 26569)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 4 T^{7} + \cdots + 144400)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + 176 T^{14} + \cdots + 16777216 \) Copy content Toggle raw display
$43$ \( (T^{8} - 16 T^{7} + \cdots + 10272025)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 226 T^{6} + \cdots + 27889)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 2 T^{3} + \cdots + 271)^{4} \) Copy content Toggle raw display
$59$ \( T^{16} + 194 T^{14} + \cdots + 28561 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 384160000 \) Copy content Toggle raw display
$67$ \( (T^{8} - 10 T^{7} + \cdots + 80089)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 4 T^{7} + \cdots + 35473936)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 428 T^{6} + \cdots + 2226064)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 2 T^{3} + \cdots + 8164)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 350 T^{6} + \cdots + 405769)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 20\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 96254442001 \) Copy content Toggle raw display
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