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Properties

Label 51.3.b.a
Level $51$
Weight $3$
Character orbit 51.b
Analytic conductor $1.390$
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] = [51,3,Mod(35,51)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(51, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("51.35");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 51 = 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 51.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: \(1.38964934824\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 28x^{8} + 254x^{6} + 880x^{4} + 1249x^{2} + 612 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} - 2) q^{4} + \beta_{7} q^{5} + ( - \beta_{5} + \beta_{3} + \beta_1) q^{6} + ( - \beta_{9} + \beta_{4} - \beta_{2}) q^{7} + (\beta_{8} - \beta_{6} - \beta_{4} + \cdots - 1) q^{8}+ \cdots + (\beta_{9} + \beta_{8} - \beta_{4} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} - 2) q^{4} + \beta_{7} q^{5} + ( - \beta_{5} + \beta_{3} + \beta_1) q^{6} + ( - \beta_{9} + \beta_{4} - \beta_{2}) q^{7} + (\beta_{8} - \beta_{6} - \beta_{4} + \cdots - 1) q^{8}+ \cdots + (5 \beta_{9} - 12 \beta_{8} + \cdots + 26) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{3} - 16 q^{4} - 2 q^{6} - 4 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{3} - 16 q^{4} - 2 q^{6} - 4 q^{7} + 6 q^{9} - 24 q^{10} - 46 q^{12} + 30 q^{13} + 30 q^{15} + 40 q^{16} + 80 q^{18} - 6 q^{19} - 40 q^{21} + 56 q^{22} - 30 q^{24} - 32 q^{25} - 10 q^{27} - 96 q^{28} - 20 q^{30} - 80 q^{31} - 26 q^{33} - 104 q^{37} - 24 q^{39} + 196 q^{40} + 122 q^{43} + 144 q^{45} + 324 q^{46} + 154 q^{48} - 154 q^{49} - 472 q^{52} + 70 q^{54} + 314 q^{55} - 232 q^{57} - 148 q^{58} - 244 q^{60} + 176 q^{61} - 304 q^{63} - 208 q^{64} + 320 q^{66} - 132 q^{67} + 294 q^{69} - 16 q^{70} - 480 q^{72} + 262 q^{75} + 248 q^{76} + 108 q^{78} - 92 q^{79} + 114 q^{81} - 76 q^{82} + 32 q^{84} + 34 q^{85} + 120 q^{87} + 52 q^{88} - 376 q^{90} + 156 q^{91} + 16 q^{93} + 216 q^{94} + 546 q^{96} + 464 q^{97} + 236 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 28x^{8} + 254x^{6} + 880x^{4} + 1249x^{2} + 612 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} + 25\nu^{7} + 179\nu^{5} + 331\nu^{3} + 88\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{9} + \nu^{8} - 79\nu^{7} + 29\nu^{6} - 629\nu^{5} + 271\nu^{4} - 1573\nu^{3} + 911\nu^{2} - 1100\nu + 876 ) / 48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3 \nu^{9} + 5 \nu^{8} + 79 \nu^{7} + 133 \nu^{6} + 629 \nu^{5} + 1067 \nu^{4} + 1573 \nu^{3} + \cdots + 1836 ) / 48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{9} + 5\nu^{8} + 29\nu^{7} + 133\nu^{6} + 271\nu^{5} + 1091\nu^{4} + 887\nu^{3} + 3007\nu^{2} + 708\nu + 2508 ) / 48 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{9} + 27\nu^{7} + 227\nu^{5} + 649\nu^{3} + 552\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{9} + 3\nu^{8} + 25\nu^{7} + 81\nu^{6} + 179\nu^{5} + 681\nu^{4} + 343\nu^{3} + 1935\nu^{2} + 196\nu + 1572 ) / 24 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3 \nu^{9} + 13 \nu^{8} - 79 \nu^{7} + 341 \nu^{6} - 629 \nu^{5} + 2707 \nu^{4} - 1573 \nu^{3} + \cdots + 4812 ) / 48 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - \beta_{6} - \beta_{4} - 2\beta_{3} + \beta_{2} - 9\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + \beta_{6} - 2\beta_{5} - \beta_{4} - 14\beta_{2} + 61 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{9} - 16 \beta_{8} + 4 \beta_{7} + 12 \beta_{6} - 2 \beta_{5} + 20 \beta_{4} + 34 \beta_{3} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{9} - 20\beta_{8} - 20\beta_{6} + 48\beta_{5} + 32\beta_{4} + 177\beta_{2} - 726 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 48 \beta_{9} + 225 \beta_{8} - 92 \beta_{7} - 129 \beta_{6} + 48 \beta_{5} - 321 \beta_{4} + \cdots - 225 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 108\beta_{9} + 317\beta_{8} + 317\beta_{6} - 842\beta_{5} - 633\beta_{4} - 2250\beta_{2} + 9105 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 842 \beta_{9} - 3092 \beta_{8} + 1584 \beta_{7} + 1408 \beta_{6} - 842 \beta_{5} + 4776 \beta_{4} + \cdots + 3092 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/51\mathbb{Z}\right)^\times\).

\(n\) \(35\) \(37\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1
− 3.68945i
− 2.98570i
− 1.64307i
− 1.27372i
− 1.07309i
1.07309i
1.27372i
1.64307i
2.98570i
3.68945i
− 3.68945i 2.54713 + 1.58496i −9.61201 − 5.40351i 5.84763 − 9.39751i 0.794806 20.7052i 3.97579 + 8.07423i −19.9360
35.2 − 2.98570i −1.28836 − 2.70927i −4.91441 2.35897i −8.08906 + 3.84667i 5.84735 2.73016i −5.68024 + 6.98104i 7.04319
35.3 − 1.64307i 2.62025 − 1.46092i 1.30032 4.88562i −2.40040 − 4.30525i −9.85503 − 8.70880i 4.73140 − 7.65597i 8.02741
35.4 − 1.27372i 0.116472 + 2.99774i 2.37764 1.87906i 3.81828 − 0.148352i 4.80879 − 8.12332i −8.97287 + 0.698304i 2.39340
35.5 − 1.07309i −2.99549 − 0.164435i 2.84847 − 8.87903i −0.176454 + 3.21445i −3.59593 − 7.34906i 8.94592 + 0.985125i −9.52805
35.6 1.07309i −2.99549 + 0.164435i 2.84847 8.87903i −0.176454 − 3.21445i −3.59593 7.34906i 8.94592 − 0.985125i −9.52805
35.7 1.27372i 0.116472 − 2.99774i 2.37764 − 1.87906i 3.81828 + 0.148352i 4.80879 8.12332i −8.97287 − 0.698304i 2.39340
35.8 1.64307i 2.62025 + 1.46092i 1.30032 − 4.88562i −2.40040 + 4.30525i −9.85503 8.70880i 4.73140 + 7.65597i 8.02741
35.9 2.98570i −1.28836 + 2.70927i −4.91441 − 2.35897i −8.08906 − 3.84667i 5.84735 − 2.73016i −5.68024 − 6.98104i 7.04319
35.10 3.68945i 2.54713 − 1.58496i −9.61201 5.40351i 5.84763 + 9.39751i 0.794806 − 20.7052i 3.97579 − 8.07423i −19.9360
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 51.3.b.a ✓ 10
3.b odd 2 1 inner 51.3.b.a ✓ 10
4.b odd 2 1 816.3.g.b 10
12.b even 2 1 816.3.g.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.3.b.a ✓ 10 1.a even 1 1 trivial
51.3.b.a ✓ 10 3.b odd 2 1 inner
816.3.g.b 10 4.b odd 2 1
816.3.g.b 10 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(51, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 28 T^{8} + \cdots + 612 \) Copy content Toggle raw display
$3$ \( T^{10} - 2 T^{9} + \cdots + 59049 \) Copy content Toggle raw display
$5$ \( T^{10} + 141 T^{8} + \cdots + 1079568 \) Copy content Toggle raw display
$7$ \( (T^{5} + 2 T^{4} + \cdots - 792)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 26128211472 \) Copy content Toggle raw display
$13$ \( (T^{5} - 15 T^{4} + \cdots - 114112)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 17)^{5} \) Copy content Toggle raw display
$19$ \( (T^{5} + 3 T^{4} + \cdots - 4961808)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 5828318352 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 1388276947008 \) Copy content Toggle raw display
$31$ \( (T^{5} + 40 T^{4} + \cdots + 3960944)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + 52 T^{4} + \cdots + 272940872)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 31061865961728 \) Copy content Toggle raw display
$43$ \( (T^{5} - 61 T^{4} + \cdots + 360432)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 195128082432 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 22\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 56\!\cdots\!72 \) Copy content Toggle raw display
$61$ \( (T^{5} - 88 T^{4} + \cdots - 746872)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + 66 T^{4} + \cdots - 26585888)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 13\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( (T^{5} - 13736 T^{3} + \cdots - 720824832)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + 46 T^{4} + \cdots + 142351992)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 82\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 14\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( (T^{5} - 232 T^{4} + \cdots + 11638779584)^{2} \) Copy content Toggle raw display
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