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Properties

Label 1840.4.a.v
Level $1840$
Weight $4$
Character orbit 1840.a
Self dual yes
Analytic conductor $108.564$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1840,4,Mod(1,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1840.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

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: yes
Analytic conductor: \(108.563514411\)
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: \( x^{8} - 2x^{7} - 49x^{6} + 31x^{5} + 750x^{4} + 249x^{3} - 2892x^{2} - 620x + 2400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 115)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + 5 q^{5} + (\beta_{6} + \beta_{3} - 1) q^{7} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \cdots + 20) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + 5 q^{5} + (\beta_{6} + \beta_{3} - 1) q^{7} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \cdots + 20) q^{9}+ \cdots + ( - 5 \beta_{7} + 45 \beta_{6} + \cdots - 162) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 40 q^{5} - 11 q^{7} + 158 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 40 q^{5} - 11 q^{7} + 158 q^{9} - 41 q^{11} + 28 q^{13} + 71 q^{17} - 177 q^{19} + 292 q^{21} - 184 q^{23} + 200 q^{25} + 495 q^{27} + 225 q^{29} + 36 q^{31} - 53 q^{33} - 55 q^{35} - 348 q^{37} + 1077 q^{39} + 620 q^{41} + 390 q^{43} + 790 q^{45} - 123 q^{47} + 881 q^{49} - 957 q^{51} + 1406 q^{53} - 205 q^{55} - 1142 q^{57} + 676 q^{59} + 1447 q^{61} + 58 q^{63} + 140 q^{65} + 1582 q^{67} - 1396 q^{71} + 17 q^{73} + 488 q^{77} - 708 q^{79} + 4316 q^{81} - 1486 q^{83} + 355 q^{85} + 803 q^{87} + 1360 q^{89} - 2693 q^{91} + 3833 q^{93} - 885 q^{95} - 855 q^{97} - 1319 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 49x^{6} + 31x^{5} + 750x^{4} + 249x^{3} - 2892x^{2} - 620x + 2400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -11\nu^{7} - 68\nu^{6} + 819\nu^{5} + 3349\nu^{4} - 15140\nu^{3} - 42139\nu^{2} + 52362\nu + 38800 ) / 1840 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -79\nu^{7} + 348\nu^{6} + 2871\nu^{5} - 8399\nu^{4} - 33460\nu^{3} + 36929\nu^{2} + 71618\nu - 30800 ) / 3680 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{7} + 36\nu^{6} + 113\nu^{5} - 945\nu^{4} - 860\nu^{3} + 6631\nu^{2} + 3586\nu - 7488 ) / 184 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -143\nu^{7} + 956\nu^{6} + 3287\nu^{5} - 22703\nu^{4} - 23860\nu^{3} + 129313\nu^{2} + 21986\nu - 161680 ) / 3680 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 29\nu^{7} - 172\nu^{6} - 821\nu^{5} + 4469\nu^{4} + 7012\nu^{3} - 28211\nu^{2} - 6686\nu + 32096 ) / 368 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} - 16\nu^{6} - 87\nu^{5} + 363\nu^{4} + 800\nu^{3} - 1553\nu^{2} - 866\nu + 480 ) / 40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 2\beta _1 + 51 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} - 2\beta_{6} + 4\beta_{5} - 6\beta_{4} + 25\beta _1 + 107 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13\beta_{7} - 9\beta_{6} + 20\beta_{5} - 19\beta_{4} - 2\beta_{3} + 8\beta_{2} + 45\beta _1 + 594 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 107\beta_{7} - 51\beta_{6} + 205\beta_{5} - 231\beta_{4} + 111\beta_{3} - 11\beta_{2} + 705\beta _1 + 4636 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 202\beta_{7} - 74\beta_{6} + 381\beta_{5} - 340\beta_{4} + 167\beta_{3} + 28\beta_{2} + 833\beta _1 + 8573 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4304 \beta_{7} - 864 \beta_{6} + 8684 \beta_{5} - 8272 \beta_{4} + 6748 \beta_{3} - 1140 \beta_{2} + \cdots + 171103 ) / 4 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.16661
4.85207
−3.57019
−3.81977
1.76920
6.03341
−3.05234
0.954235
0 −8.59146 0 5.00000 0 −9.86011 0 46.8131 0
1.2 0 −8.45088 0 5.00000 0 −16.9190 0 44.4174 0
1.3 0 −4.96142 0 5.00000 0 18.4259 0 −2.38431 0
1.4 0 −1.39521 0 5.00000 0 −13.5587 0 −25.0534 0
1.5 0 0.0181662 0 5.00000 0 −10.0918 0 −26.9997 0
1.6 0 4.21852 0 5.00000 0 19.9088 0 −9.20409 0
1.7 0 8.94300 0 5.00000 0 −32.7645 0 52.9772 0
1.8 0 10.2193 0 5.00000 0 33.8594 0 77.4338 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(23\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1840.4.a.v 8
4.b odd 2 1 115.4.a.f ✓ 8
12.b even 2 1 1035.4.a.r 8
20.d odd 2 1 575.4.a.n 8
20.e even 4 2 575.4.b.k 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.a.f ✓ 8 4.b odd 2 1
575.4.a.n 8 20.d odd 2 1
575.4.b.k 16 20.e even 4 2
1035.4.a.r 8 12.b even 2 1
1840.4.a.v 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1840))\):

\( T_{3}^{8} - 187T_{3}^{6} - 165T_{3}^{5} + 10290T_{3}^{4} + 15441T_{3}^{3} - 137140T_{3}^{2} - 191280T_{3} + 3520 \) Copy content Toggle raw display
\( T_{7}^{8} + 11 T_{7}^{7} - 1752 T_{7}^{6} - 22539 T_{7}^{5} + 783793 T_{7}^{4} + 12717810 T_{7}^{3} + \cdots - 9289584128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 187 T^{6} + \cdots + 3520 \) Copy content Toggle raw display
$5$ \( (T - 5)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots - 9289584128 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots - 11345758080 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots - 5251923133176 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 52834929586560 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots - 54772847726720 \) Copy content Toggle raw display
$23$ \( (T + 23)^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots - 15\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots - 18\!\cdots\!22 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots - 30\!\cdots\!80 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots - 62\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots - 94\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots - 27\!\cdots\!80 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 30\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots - 59\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 11\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 84\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 13\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
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