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Properties

Label 51.4.d.a
Level $51$
Weight $4$
Character orbit 51.d
Analytic conductor $3.009$
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] = [51,4,Mod(16,51)]
 
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([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("51.16");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 51 = 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 51.d (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: \(3.00909741029\)
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: \( x^{8} + 50x^{6} + 685x^{4} + 1728x^{2} + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{2} - \beta_{4} q^{3} + (\beta_{3} - \beta_{2} + 5) q^{4} + (\beta_{6} + \beta_{4} + 2 \beta_1) q^{5} + 3 \beta_1 q^{6} + (2 \beta_{6} - \beta_{5} - \beta_1) q^{7} + ( - \beta_{7} - \beta_{3} + 7 \beta_{2} - 10) q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{2} - \beta_{4} q^{3} + (\beta_{3} - \beta_{2} + 5) q^{4} + (\beta_{6} + \beta_{4} + 2 \beta_1) q^{5} + 3 \beta_1 q^{6} + (2 \beta_{6} - \beta_{5} - \beta_1) q^{7} + ( - \beta_{7} - \beta_{3} + 7 \beta_{2} - 10) q^{8} - 9 q^{9} + ( - \beta_{6} - \beta_{5} + \cdots - 6 \beta_1) q^{10}+ \cdots + ( - 9 \beta_{6} + 27 \beta_{5} + \cdots + 45 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} + 36 q^{4} - 48 q^{8} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} + 36 q^{4} - 48 q^{8} - 72 q^{9} - 236 q^{13} + 72 q^{15} + 372 q^{16} - 160 q^{17} + 36 q^{18} + 76 q^{19} - 84 q^{21} + 380 q^{25} + 796 q^{26} - 504 q^{30} - 1208 q^{32} + 60 q^{33} + 100 q^{34} - 288 q^{35} - 324 q^{36} - 116 q^{38} + 588 q^{42} + 124 q^{43} - 472 q^{47} + 64 q^{49} + 1120 q^{50} - 84 q^{51} - 1944 q^{52} + 1944 q^{53} + 316 q^{55} + 1224 q^{59} + 1332 q^{60} - 1228 q^{64} + 1596 q^{66} + 920 q^{67} - 2480 q^{68} - 1464 q^{69} - 1872 q^{70} + 432 q^{72} + 456 q^{76} - 4360 q^{77} + 648 q^{81} - 336 q^{83} - 1500 q^{84} + 836 q^{85} + 6364 q^{86} - 900 q^{87} - 344 q^{89} - 1092 q^{93} + 1432 q^{94} + 3756 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 50x^{6} + 685x^{4} + 1728x^{2} + 324 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 35\nu^{4} + 262\nu^{2} + 348 ) / 204 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 35\nu^{4} + 58\nu^{2} - 2304 ) / 204 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{7} - 191\nu^{5} - 2425\nu^{3} - 4554\nu ) / 612 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\nu^{7} + 659\nu^{5} + 8302\nu^{3} + 3708\nu ) / 1836 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\nu^{7} + 659\nu^{5} + 9220\nu^{3} + 23904\nu ) / 918 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7\nu^{6} + 347\nu^{4} + 4384\nu^{2} + 4680 ) / 204 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} - 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - 2\beta_{5} - 22\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} + 25\beta_{3} - 39\beta_{2} + 303 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -11\beta_{6} + 70\beta_{5} + 52\beta_{4} + 532\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -70\beta_{7} - 613\beta_{3} + 1307\beta_{2} - 7547 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -81\beta_{6} - 2130\beta_{5} - 2636\beta_{4} - 13204\beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
5.18933i
− 5.18933i
1.72284i
− 1.72284i
− 0.451338i
0.451338i
− 4.46083i
4.46083i
−5.18933 − 3.00000i 18.9291 11.3455i 15.5680i − 21.0285i −56.7149 −9.00000 − 58.8758i
16.2 −5.18933 3.00000i 18.9291 − 11.3455i − 15.5680i 21.0285i −56.7149 −9.00000 58.8758i
16.3 −1.72284 − 3.00000i −5.03184 10.2055i 5.16851i 20.3112i 22.4517 −9.00000 − 17.5823i
16.4 −1.72284 3.00000i −5.03184 − 10.2055i − 5.16851i − 20.3112i 22.4517 −9.00000 17.5823i
16.5 0.451338 − 3.00000i −7.79629 − 8.74515i − 1.35401i − 20.7311i −7.12947 −9.00000 − 3.94702i
16.6 0.451338 3.00000i −7.79629 8.74515i 1.35401i 20.7311i −7.12947 −9.00000 3.94702i
16.7 4.46083 − 3.00000i 11.8990 − 0.805870i − 13.3825i 7.44837i 17.3927 −9.00000 − 3.59485i
16.8 4.46083 3.00000i 11.8990 0.805870i 13.3825i − 7.44837i 17.3927 −9.00000 3.59485i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 51.4.d.a ✓ 8
3.b odd 2 1 153.4.d.d 8
4.b odd 2 1 816.4.c.f 8
17.b even 2 1 inner 51.4.d.a ✓ 8
17.c even 4 1 867.4.a.n 4
17.c even 4 1 867.4.a.o 4
51.c odd 2 1 153.4.d.d 8
68.d odd 2 1 816.4.c.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.4.d.a ✓ 8 1.a even 1 1 trivial
51.4.d.a ✓ 8 17.b even 2 1 inner
153.4.d.d 8 3.b odd 2 1
153.4.d.d 8 51.c odd 2 1
816.4.c.f 8 4.b odd 2 1
816.4.c.f 8 68.d odd 2 1
867.4.a.n 4 17.c even 4 1
867.4.a.o 4 17.c even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(51, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{3} - 23 T^{2} + \cdots + 18)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 310 T^{6} + \cdots + 665856 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 4349666304 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 5341855744 \) Copy content Toggle raw display
$13$ \( (T^{4} + 118 T^{3} + \cdots - 1094828)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 582622237229761 \) Copy content Toggle raw display
$19$ \( (T^{4} - 38 T^{3} + \cdots + 3560224)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 36\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 34\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 30\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( (T^{4} - 62 T^{3} + \cdots - 884788304)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 236 T^{3} + \cdots - 49049856)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 972 T^{3} + \cdots - 23669953488)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 612 T^{3} + \cdots + 2682851328)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( (T^{4} - 460 T^{3} + \cdots - 16007052288)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 26\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( (T^{4} + 168 T^{3} + \cdots - 20547582336)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 172 T^{3} + \cdots + 138200373936)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 22\!\cdots\!44 \) Copy content Toggle raw display
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