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Properties

Label 637.2.c.g
Level $637$
Weight $2$
Character orbit 637.c
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(246,637)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.246");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.c (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: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 10x^{14} + 121x^{12} + 296x^{10} + 3468x^{8} - 1748x^{6} + 40192x^{4} - 65056x^{2} + 228484 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} - \beta_{11} q^{3} + (\beta_{2} - 1) q^{4} - \beta_{6} q^{5} - \beta_{4} q^{6} + ( - \beta_{14} + \beta_{13} - \beta_{10}) q^{8} + ( - \beta_{5} - \beta_{3} + \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{2} - \beta_{11} q^{3} + (\beta_{2} - 1) q^{4} - \beta_{6} q^{5} - \beta_{4} q^{6} + ( - \beta_{14} + \beta_{13} - \beta_{10}) q^{8} + ( - \beta_{5} - \beta_{3} + \beta_{2} + 1) q^{9} + (\beta_{11} + 2 \beta_{9} + \cdots - \beta_{7}) q^{10}+ \cdots + ( - \beta_{15} - 2 \beta_{14} + \cdots + \beta_{10}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 20 q^{4} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 20 q^{4} + 16 q^{9} + 28 q^{16} - 8 q^{22} - 36 q^{23} - 44 q^{25} + 36 q^{29} + 52 q^{36} + 32 q^{39} - 36 q^{43} - 72 q^{51} + 12 q^{53} - 164 q^{64} - 24 q^{65} + 96 q^{74} + 24 q^{78} + 36 q^{79} + 16 q^{81} + 136 q^{88} + 24 q^{92} - 84 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 10x^{14} + 121x^{12} + 296x^{10} + 3468x^{8} - 1748x^{6} + 40192x^{4} - 65056x^{2} + 228484 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 127996212 \nu^{14} + 979990851 \nu^{12} + 13620194602 \nu^{10} + 6825369227 \nu^{8} + \cdots - 24635344078600 ) / 29065582596052 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 127996212 \nu^{14} - 979990851 \nu^{12} - 13620194602 \nu^{10} - 6825369227 \nu^{8} + \cdots + 39168135376626 ) / 14532791298026 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 306986712 \nu^{14} - 5785250672 \nu^{12} - 51065796556 \nu^{10} - 245977410085 \nu^{8} + \cdots - 47980987815069 ) / 7266395649013 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1919925366 \nu^{14} - 6971345277 \nu^{12} - 76432745837 \nu^{10} + \cdots + 137905300117434 ) / 29065582596052 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 774992715 \nu^{14} - 9797848728 \nu^{12} - 101725454643 \nu^{10} - 319450784079 \nu^{8} + \cdots - 8985354172108 ) / 7266395649013 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7536716227 \nu^{14} + 51263835016 \nu^{12} + 571971494681 \nu^{10} - 973429379572 \nu^{8} + \cdots - 200469162311696 ) / 58131165192104 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14293266565 \nu^{15} + 622324322223 \nu^{14} - 4349411499250 \nu^{13} + \cdots - 43\!\cdots\!64 ) / 13\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 14293266565 \nu^{15} + 622324322223 \nu^{14} + 4349411499250 \nu^{13} + \cdots - 43\!\cdots\!64 ) / 13\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 35037469784 \nu^{15} - 319783603172 \nu^{13} - 4005316030475 \nu^{11} + \cdots + 67\!\cdots\!78 \nu ) / 69\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 35037469784 \nu^{15} + 319783603172 \nu^{13} + 4005316030475 \nu^{11} + \cdots + 148357429566350 \nu ) / 69\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 213313563659 \nu^{15} - 4532646032618 \nu^{13} - 47876731013081 \nu^{11} + \cdots - 13\!\cdots\!20 \nu ) / 13\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 144596254252 \nu^{15} + 1364700202471 \nu^{13} + 16771353642497 \nu^{11} + \cdots - 14\!\cdots\!34 \nu ) / 69\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 374090467397 \nu^{15} - 3298265569313 \nu^{13} - 31101270941936 \nu^{11} + \cdots + 98\!\cdots\!66 \nu ) / 69\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 527579471881 \nu^{15} - 4834097351350 \nu^{13} - 49372803625627 \nu^{11} + \cdots + 50\!\cdots\!72 \nu ) / 69\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 674575131380 \nu^{15} + 7249765957837 \nu^{13} + 73681722610361 \nu^{11} + \cdots + 20\!\cdots\!62 \nu ) / 69\!\cdots\!28 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{10} + \beta_{9} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} + \beta_{13} - 3\beta_{12} + \beta_{10} - 7\beta_{9} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{8} - 4\beta_{7} + \beta_{5} - 8\beta_{4} - \beta_{3} + 4\beta_{2} - 16\beta _1 - 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} - 10 \beta_{14} + 11 \beta_{13} + 20 \beta_{12} + 20 \beta_{11} - 48 \beta_{10} + \cdots - 5 \beta_{7} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta_{8} + 14\beta_{7} + 12\beta_{6} + 20\beta_{5} + 40\beta_{4} - 19\beta_{3} - 120\beta_{2} + 24\beta _1 + 183 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 31 \beta_{15} + 199 \beta_{14} - 167 \beta_{13} + 56 \beta_{12} - 14 \beta_{11} + 474 \beta_{10} + \cdots + 7 \beta_{7} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 216 \beta_{8} + 216 \beta_{7} + 48 \beta_{6} - 200 \beta_{5} + 496 \beta_{4} + 243 \beta_{3} + \cdots - 1109 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 195 \beta_{15} - 843 \beta_{14} + 707 \beta_{13} - 2304 \beta_{12} - 1926 \beta_{11} + \cdots + 507 \beta_{7} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3518 \beta_{8} - 3518 \beta_{7} - 1404 \beta_{6} - 128 \beta_{5} - 8220 \beta_{4} - 83 \beta_{3} + \cdots - 1573 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1321 \beta_{15} - 8971 \beta_{14} + 7659 \beta_{13} + 20394 \beta_{12} + 17754 \beta_{11} + \cdots - 5005 \beta_{7} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 17740 \beta_{8} + 17740 \beta_{7} + 7368 \beta_{6} + 16706 \beta_{5} + 40600 \beta_{4} - 20311 \beta_{3} + \cdots + 123603 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 27679 \beta_{15} + 165715 \beta_{14} - 143889 \beta_{13} - 17238 \beta_{12} - 14534 \beta_{11} + \cdots + 4797 \beta_{7} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 121854 \beta_{8} + 121854 \beta_{7} + 45764 \beta_{6} - 164802 \beta_{5} + 282420 \beta_{4} + \cdots - 1286237 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 163077 \beta_{15} - 994809 \beta_{14} + 868719 \beta_{13} - 1642422 \beta_{12} + \cdots + 376459 \beta_{7} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(197\) \(248\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
246.1
1.41421 − 2.73420i
−1.41421 − 2.73420i
−1.41421 − 1.65552i
1.41421 − 1.65552i
−1.41421 − 1.52340i
1.41421 − 1.52340i
1.41421 − 0.680196i
−1.41421 − 0.680196i
1.41421 + 0.680196i
−1.41421 + 0.680196i
−1.41421 + 1.52340i
1.41421 + 1.52340i
−1.41421 + 1.65552i
1.41421 + 1.65552i
1.41421 + 2.73420i
−1.41421 + 2.73420i
− 2.73420i −1.15595 −5.47586 1.87727i 3.16060i 0 9.50370i −1.66378 5.13283
246.2 − 2.73420i 1.15595 −5.47586 − 1.87727i − 3.16060i 0 9.50370i −1.66378 −5.13283
246.3 − 1.65552i −0.494977 −0.740740 2.87389i 0.819443i 0 − 2.08473i −2.75500 4.75778
246.4 − 1.65552i 0.494977 −0.740740 − 2.87389i − 0.819443i 0 − 2.08473i −2.75500 −4.75778
246.5 − 1.52340i −2.99510 −0.320733 − 2.94606i 4.56272i 0 − 2.55819i 5.97063 −4.48801
246.6 − 1.52340i 2.99510 −0.320733 2.94606i − 4.56272i 0 − 2.55819i 5.97063 4.48801
246.7 − 0.680196i −2.33413 1.53733 3.24613i 1.58766i 0 − 2.40608i 2.44815 2.20800
246.8 − 0.680196i 2.33413 1.53733 − 3.24613i − 1.58766i 0 − 2.40608i 2.44815 −2.20800
246.9 0.680196i −2.33413 1.53733 − 3.24613i − 1.58766i 0 2.40608i 2.44815 2.20800
246.10 0.680196i 2.33413 1.53733 3.24613i 1.58766i 0 2.40608i 2.44815 −2.20800
246.11 1.52340i −2.99510 −0.320733 2.94606i − 4.56272i 0 2.55819i 5.97063 −4.48801
246.12 1.52340i 2.99510 −0.320733 − 2.94606i 4.56272i 0 2.55819i 5.97063 4.48801
246.13 1.65552i −0.494977 −0.740740 − 2.87389i − 0.819443i 0 2.08473i −2.75500 4.75778
246.14 1.65552i 0.494977 −0.740740 2.87389i 0.819443i 0 2.08473i −2.75500 −4.75778
246.15 2.73420i −1.15595 −5.47586 − 1.87727i − 3.16060i 0 − 9.50370i −1.66378 5.13283
246.16 2.73420i 1.15595 −5.47586 1.87727i 3.16060i 0 − 9.50370i −1.66378 −5.13283
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 246.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.c.g ✓ 16
7.b odd 2 1 inner 637.2.c.g ✓ 16
7.c even 3 2 637.2.r.g 32
7.d odd 6 2 637.2.r.g 32
13.b even 2 1 inner 637.2.c.g ✓ 16
13.d odd 4 2 8281.2.a.cs 16
91.b odd 2 1 inner 637.2.c.g ✓ 16
91.i even 4 2 8281.2.a.cs 16
91.r even 6 2 637.2.r.g 32
91.s odd 6 2 637.2.r.g 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.c.g ✓ 16 1.a even 1 1 trivial
637.2.c.g ✓ 16 7.b odd 2 1 inner
637.2.c.g ✓ 16 13.b even 2 1 inner
637.2.c.g ✓ 16 91.b odd 2 1 inner
637.2.r.g 32 7.c even 3 2
637.2.r.g 32 7.d odd 6 2
637.2.r.g 32 91.r even 6 2
637.2.r.g 32 91.s odd 6 2
8281.2.a.cs 16 13.d odd 4 2
8281.2.a.cs 16 91.i even 4 2

Hecke kernels

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

\( T_{2}^{8} + 13T_{2}^{6} + 50T_{2}^{4} + 68T_{2}^{2} + 22 \) Copy content Toggle raw display
\( T_{3}^{8} - 16T_{3}^{6} + 72T_{3}^{4} - 82T_{3}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 13 T^{6} + \cdots + 22)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} - 16 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + 31 T^{6} + \cdots + 2662)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 46 T^{6} + \cdots + 5632)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( (T^{8} - 78 T^{6} + \cdots + 39204)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 75 T^{6} + \cdots + 7128)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 9 T^{3} + \cdots - 792)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 9 T^{3} + \cdots - 144)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 165 T^{6} + \cdots + 1824768)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 90 T^{6} + \cdots + 7128)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 154 T^{6} + \cdots + 59488)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 9 T^{3} - 21 T^{2} + \cdots + 72)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + 115 T^{6} + \cdots + 1408)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 3 T^{3} + \cdots + 2502)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} + 190 T^{6} + \cdots + 1580128)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 168 T^{6} + \cdots + 219024)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 348 T^{6} + \cdots + 456192)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 124 T^{6} + \cdots + 68992)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 189 T^{6} + \cdots + 514998)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 9 T^{3} + 11 T^{2} + \cdots - 44)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 157 T^{6} + \cdots + 25432)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 385 T^{6} + \cdots + 1888678)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 537 T^{6} + \cdots + 28741878)^{2} \) Copy content Toggle raw display
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