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Properties

Label 637.2.e.m
Level $637$
Weight $2$
Character orbit 637.e
Analytic conductor $5.086$
Analytic rank $0$
Dimension $10$
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] = [637,2,Mod(79,637)]
 
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([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.79");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.e (of order \(3\), degree \(2\), not 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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 8x^{8} + 7x^{7} + 41x^{6} + 18x^{5} + 58x^{4} + 28x^{3} + 64x^{2} + 16x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 91)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_1) q^{2} + ( - \beta_{9} + \beta_{4}) q^{3} + (\beta_{9} + 2 \beta_{7} - \beta_{6} + \cdots - 2) q^{4}+ \cdots + (\beta_{8} - \beta_{7}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} + \beta_1) q^{2} + ( - \beta_{9} + \beta_{4}) q^{3} + (\beta_{9} + 2 \beta_{7} - \beta_{6} + \cdots - 2) q^{4}+ \cdots + ( - 3 \beta_{5} + 2 \beta_{4} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 4 q^{2} - 8 q^{4} + 2 q^{5} + 10 q^{6} + 18 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 4 q^{2} - 8 q^{4} + 2 q^{5} + 10 q^{6} + 18 q^{8} - 3 q^{9} - 5 q^{10} - 11 q^{11} + 5 q^{12} - 10 q^{13} - 10 q^{16} - 5 q^{17} - 9 q^{18} + 9 q^{19} - 2 q^{20} + 16 q^{22} - 10 q^{23} - 9 q^{25} + 4 q^{26} - 6 q^{29} + 13 q^{30} - 6 q^{31} - 22 q^{32} + 8 q^{33} + 44 q^{34} + 14 q^{36} - 4 q^{37} - 10 q^{38} + 28 q^{40} - 28 q^{41} + 4 q^{43} - 32 q^{45} - 3 q^{46} + q^{47} + 46 q^{48} + 18 q^{50} + 8 q^{51} + 8 q^{52} - 17 q^{53} + 23 q^{54} - 32 q^{57} + 27 q^{58} + 11 q^{59} + 29 q^{60} - 11 q^{61} + 46 q^{62} + 18 q^{64} - 2 q^{65} + 21 q^{66} - 13 q^{67} - 32 q^{68} - 36 q^{69} + 30 q^{71} + 19 q^{72} + 33 q^{74} - 20 q^{75} - 16 q^{76} - 10 q^{78} - 2 q^{79} + 55 q^{80} + 19 q^{81} + 34 q^{82} - 12 q^{83} - 44 q^{85} - 28 q^{86} - 8 q^{87} + 3 q^{88} - 4 q^{89} + 68 q^{90} + 42 q^{92} - 18 q^{93} + 20 q^{94} + 12 q^{95} - 37 q^{96} + 24 q^{97} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 8x^{8} + 7x^{7} + 41x^{6} + 18x^{5} + 58x^{4} + 28x^{3} + 64x^{2} + 16x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 364 \nu^{9} + 176 \nu^{8} - 220 \nu^{7} - 5913 \nu^{6} + 880 \nu^{5} + 6908 \nu^{4} + \cdots + 30518 ) / 118350 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 983 \nu^{9} - 7328 \nu^{8} + 9160 \nu^{7} - 87336 \nu^{6} - 36640 \nu^{5} - 287624 \nu^{4} + \cdots - 22604 ) / 118350 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1159 \nu^{9} - 9844 \nu^{8} + 12305 \nu^{7} - 109053 \nu^{6} - 49220 \nu^{5} - 386377 \nu^{4} + \cdots - 348592 ) / 118350 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 916 \nu^{9} - 3044 \nu^{8} + 3805 \nu^{7} - 3978 \nu^{6} - 15220 \nu^{5} - 119477 \nu^{4} + \cdots - 180842 ) / 59175 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2249 \nu^{9} - 9541 \nu^{8} + 26720 \nu^{7} - 34617 \nu^{6} + 31195 \nu^{5} - 152578 \nu^{4} + \cdots - 3988 ) / 78900 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 15259 \nu^{9} + 14531 \nu^{8} - 121720 \nu^{7} - 107253 \nu^{6} - 637445 \nu^{5} - 272902 \nu^{4} + \cdots - 2692 ) / 236700 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 18139 \nu^{9} - 21776 \nu^{8} + 145570 \nu^{7} + 96813 \nu^{6} + 719570 \nu^{5} + 92092 \nu^{4} + \cdots + 532 ) / 118350 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1058 \nu^{9} - 1552 \nu^{8} + 9041 \nu^{7} + 3726 \nu^{6} + 39580 \nu^{5} + 2993 \nu^{4} + 53832 \nu^{3} + \cdots - 136 ) / 4734 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 3\beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - 2\beta_{4} + 2\beta_{3} + 6\beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{9} + 2\beta_{8} - 19\beta_{7} + 9\beta_{6} - 13\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 22 \beta_{9} + 9 \beta_{8} - 45 \beta_{7} + 23 \beta_{6} - 9 \beta_{5} + 22 \beta_{4} - 23 \beta_{3} + \cdots + 45 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -23\beta_{5} + 70\beta_{4} - 78\beta_{3} - 128\beta_{2} + 154 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 206\beta_{9} - 78\beta_{8} + 431\beta_{7} - 221\beta_{6} + 407\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 628 \beta_{9} - 221 \beta_{8} + 1349 \beta_{7} - 691 \beta_{6} + 221 \beta_{5} - 628 \beta_{4} + \cdots - 1349 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 691\beta_{5} - 1878\beta_{4} + 2036\beta_{3} + 3634\beta_{2} - 3968 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(197\) \(248\)
\(\chi(n)\) \(1\) \(-\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−0.862625 + 1.49411i
−0.606661 + 1.05077i
−0.132804 + 0.230024i
0.597828 − 1.03547i
1.50426 − 2.60546i
−0.862625 − 1.49411i
−0.606661 − 1.05077i
−0.132804 − 0.230024i
0.597828 + 1.03547i
1.50426 + 2.60546i
−1.36263 + 2.36014i −0.673208 − 1.16603i −2.71349 − 4.69991i 1.09358 − 1.89414i 3.66932 0 9.33940 0.593582 − 1.02811i 2.98028 + 5.16200i
79.2 −1.10666 + 1.91679i 1.23721 + 2.14292i −1.44940 − 2.51043i −1.06140 + 1.83839i −5.47671 0 1.98932 −1.56140 + 2.70442i −2.34921 − 4.06896i
79.3 −0.632804 + 1.09605i −1.31364 − 2.27529i 0.199118 + 0.344882i −1.45130 + 2.51373i 3.32511 0 −3.03523 −1.95130 + 3.37975i −1.83678 − 3.18139i
79.4 0.0978281 − 0.169443i −0.129894 − 0.224983i 0.980859 + 1.69890i 1.96625 − 3.40565i −0.0508292 0 0.775135 1.46625 − 2.53963i −0.384710 − 0.666337i
79.5 1.00426 − 1.73943i 0.879528 + 1.52339i −1.01709 − 1.76164i 0.452861 − 0.784378i 3.53311 0 −0.0686323 −0.0471392 + 0.0816475i −0.909582 − 1.57544i
508.1 −1.36263 − 2.36014i −0.673208 + 1.16603i −2.71349 + 4.69991i 1.09358 + 1.89414i 3.66932 0 9.33940 0.593582 + 1.02811i 2.98028 − 5.16200i
508.2 −1.10666 − 1.91679i 1.23721 − 2.14292i −1.44940 + 2.51043i −1.06140 − 1.83839i −5.47671 0 1.98932 −1.56140 − 2.70442i −2.34921 + 4.06896i
508.3 −0.632804 − 1.09605i −1.31364 + 2.27529i 0.199118 − 0.344882i −1.45130 − 2.51373i 3.32511 0 −3.03523 −1.95130 − 3.37975i −1.83678 + 3.18139i
508.4 0.0978281 + 0.169443i −0.129894 + 0.224983i 0.980859 − 1.69890i 1.96625 + 3.40565i −0.0508292 0 0.775135 1.46625 + 2.53963i −0.384710 + 0.666337i
508.5 1.00426 + 1.73943i 0.879528 − 1.52339i −1.01709 + 1.76164i 0.452861 + 0.784378i 3.53311 0 −0.0686323 −0.0471392 − 0.0816475i −0.909582 + 1.57544i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.e.m 10
7.b odd 2 1 91.2.e.c ✓ 10
7.c even 3 1 637.2.a.k 5
7.c even 3 1 inner 637.2.e.m 10
7.d odd 6 1 91.2.e.c ✓ 10
7.d odd 6 1 637.2.a.l 5
21.c even 2 1 819.2.j.h 10
21.g even 6 1 819.2.j.h 10
21.g even 6 1 5733.2.a.bl 5
21.h odd 6 1 5733.2.a.bm 5
28.d even 2 1 1456.2.r.p 10
28.f even 6 1 1456.2.r.p 10
91.b odd 2 1 1183.2.e.f 10
91.r even 6 1 8281.2.a.bx 5
91.s odd 6 1 1183.2.e.f 10
91.s odd 6 1 8281.2.a.bw 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.c ✓ 10 7.b odd 2 1
91.2.e.c ✓ 10 7.d odd 6 1
637.2.a.k 5 7.c even 3 1
637.2.a.l 5 7.d odd 6 1
637.2.e.m 10 1.a even 1 1 trivial
637.2.e.m 10 7.c even 3 1 inner
819.2.j.h 10 21.c even 2 1
819.2.j.h 10 21.g even 6 1
1183.2.e.f 10 91.b odd 2 1
1183.2.e.f 10 91.s odd 6 1
1456.2.r.p 10 28.d even 2 1
1456.2.r.p 10 28.f even 6 1
5733.2.a.bl 5 21.g even 6 1
5733.2.a.bm 5 21.h odd 6 1
8281.2.a.bw 5 91.s odd 6 1
8281.2.a.bx 5 91.r even 6 1

Hecke kernels

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

\( T_{2}^{10} + 4T_{2}^{9} + 17T_{2}^{8} + 30T_{2}^{7} + 81T_{2}^{6} + 116T_{2}^{5} + 265T_{2}^{4} + 210T_{2}^{3} + 195T_{2}^{2} - 36T_{2} + 9 \) Copy content Toggle raw display
\( T_{3}^{10} + 9T_{3}^{8} + 65T_{3}^{6} + 4T_{3}^{5} + 144T_{3}^{4} + 72T_{3}^{3} + 256T_{3}^{2} + 64T_{3} + 16 \) Copy content Toggle raw display
\( T_{5}^{10} - 2 T_{5}^{9} + 19 T_{5}^{8} - 10 T_{5}^{7} + 217 T_{5}^{6} - 156 T_{5}^{5} + 1024 T_{5}^{4} + \cdots + 2304 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 4 T^{9} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{10} + 9 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{10} - 2 T^{9} + \cdots + 2304 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} + 11 T^{9} + \cdots + 1089 \) Copy content Toggle raw display
$13$ \( (T + 1)^{10} \) Copy content Toggle raw display
$17$ \( T^{10} + 5 T^{9} + \cdots + 184041 \) Copy content Toggle raw display
$19$ \( T^{10} - 9 T^{9} + \cdots + 49729 \) Copy content Toggle raw display
$23$ \( T^{10} + 10 T^{9} + \cdots + 144 \) Copy content Toggle raw display
$29$ \( (T^{5} + 3 T^{4} + \cdots - 108)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + 6 T^{9} + \cdots + 126736 \) Copy content Toggle raw display
$37$ \( T^{10} + 4 T^{9} + \cdots + 49505296 \) Copy content Toggle raw display
$41$ \( (T^{5} + 14 T^{4} + \cdots + 1584)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} - 2 T^{4} - 72 T^{3} + \cdots + 64)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} - T^{9} + \cdots + 26718561 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 398361681 \) Copy content Toggle raw display
$59$ \( T^{10} - 11 T^{9} + \cdots + 1089 \) Copy content Toggle raw display
$61$ \( T^{10} + 11 T^{9} + \cdots + 71588521 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 515244601 \) Copy content Toggle raw display
$71$ \( (T^{5} - 15 T^{4} + \cdots - 6336)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 75 T^{8} + \cdots + 506944 \) Copy content Toggle raw display
$79$ \( T^{10} + 2 T^{9} + \cdots + 1000000 \) Copy content Toggle raw display
$83$ \( (T^{5} + 6 T^{4} + \cdots + 7488)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + 4 T^{9} + \cdots + 59166864 \) Copy content Toggle raw display
$97$ \( (T^{5} - 12 T^{4} + \cdots + 2384)^{2} \) Copy content Toggle raw display
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