Properties

Label 1305.4.a.i
Level $1305$
Weight $4$
Character orbit 1305.a
Self dual yes
Analytic conductor $76.997$
Analytic rank $0$
Dimension $6$
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] = [1305,4,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1305.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1305.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: \(76.9974925575\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 31x^{4} + 9x^{3} + 230x^{2} + 32x - 128 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 435)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{4} + \beta_{2} + \beta_1 + 2) q^{4} - 5 q^{5} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots + 3) q^{7}+ \cdots + (2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \cdots + 8) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{4} + \beta_{2} + \beta_1 + 2) q^{4} - 5 q^{5} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots + 3) q^{7}+ \cdots + (16 \beta_{5} + 60 \beta_{4} + \cdots + 332) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} + 15 q^{4} - 30 q^{5} + 23 q^{7} + 51 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} + 15 q^{4} - 30 q^{5} + 23 q^{7} + 51 q^{8} - 5 q^{10} + 111 q^{11} - 83 q^{13} + 102 q^{14} - 37 q^{16} + 35 q^{17} - 76 q^{19} - 75 q^{20} + 66 q^{22} - 166 q^{23} + 150 q^{25} + 282 q^{26} + 368 q^{28} + 174 q^{29} - 164 q^{31} + 259 q^{32} + 512 q^{34} - 115 q^{35} - 538 q^{37} - 6 q^{38} - 255 q^{40} + 1250 q^{41} - 80 q^{43} + 1004 q^{44} - 1190 q^{46} + 491 q^{47} - 969 q^{49} + 25 q^{50} - 336 q^{52} + 356 q^{53} - 555 q^{55} + 1260 q^{56} + 29 q^{58} + 646 q^{59} - 1476 q^{61} + 4 q^{62} - 845 q^{64} + 415 q^{65} + 87 q^{67} + 558 q^{68} - 510 q^{70} + 1312 q^{71} - 1142 q^{73} + 2270 q^{74} - 1594 q^{76} + 2561 q^{77} - 142 q^{79} + 185 q^{80} - 1544 q^{82} - 346 q^{83} - 175 q^{85} + 2056 q^{86} + 1400 q^{88} + 2827 q^{89} + 1379 q^{91} + 1434 q^{92} + 978 q^{94} + 380 q^{95} - 4 q^{97} + 1903 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 31x^{4} + 9x^{3} + 230x^{2} + 32x - 128 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 2\nu^{3} - 17\nu^{2} + 18\nu + 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 2\nu^{2} - 15\nu + 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{4} + 2\nu^{3} + 25\nu^{2} - 26\nu - 96 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 25\nu^{3} + 26\nu^{2} + 128\nu - 24 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{2} + \beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{4} + 2\beta_{3} + 2\beta_{2} + 17\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 21\beta_{4} + 4\beta_{3} + 29\beta_{2} + 33\beta _1 + 170 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{5} + 66\beta_{4} + 58\beta_{3} + 82\beta_{2} + 337\beta _1 + 304 \) Copy content Toggle raw display

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
−3.96002
−3.16884
−0.886494
0.691666
3.33404
4.98965
−3.96002 0 7.68179 −5.00000 0 2.19133 1.26012 0 19.8001
1.2 −3.16884 0 2.04152 −5.00000 0 −6.36645 18.8814 0 15.8442
1.3 −0.886494 0 −7.21413 −5.00000 0 17.9759 13.4872 0 4.43247
1.4 0.691666 0 −7.52160 −5.00000 0 −14.5720 −10.7358 0 −3.45833
1.5 3.33404 0 3.11579 −5.00000 0 1.26363 −16.2841 0 −16.6702
1.6 4.98965 0 16.8966 −5.00000 0 22.5076 44.3911 0 −24.9483
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( +1 \)
\(29\) \( -1 \)

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 1305.4.a.i 6
3.b odd 2 1 435.4.a.g 6
15.d odd 2 1 2175.4.a.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.4.a.g 6 3.b odd 2 1
1305.4.a.i 6 1.a even 1 1 trivial
2175.4.a.l 6 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - T_{2}^{5} - 31T_{2}^{4} + 9T_{2}^{3} + 230T_{2}^{2} + 32T_{2} - 128 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1305))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{5} + \cdots - 128 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T + 5)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 23 T^{5} + \cdots + 103936 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 4447644416 \) Copy content Toggle raw display
$13$ \( T^{6} + 83 T^{5} + \cdots - 755576576 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 35167804928 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 70167660032 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots - 17557826816 \) Copy content Toggle raw display
$29$ \( (T - 29)^{6} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 679337811968 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots - 11015322431488 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots - 2844920728576 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 1852550152192 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots - 36309886959616 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 31\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 82\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots - 99\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 136347629123072 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 859390658797568 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 17\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots - 34\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots - 45\!\cdots\!08 \) Copy content Toggle raw display
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