Properties

Label 1617.4.a.v
Level $1617$
Weight $4$
Character orbit 1617.a
Self dual yes
Analytic conductor $95.406$
Analytic rank $1$
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] = [1617,4,Mod(1,1617)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1617, 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("1617.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1617.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: \(95.4060884793\)
Analytic rank: \(1\)
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: \( x^{8} - 4x^{7} - 39x^{6} + 130x^{5} + 495x^{4} - 1290x^{3} - 2045x^{2} + 3952x + 1488 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 231)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + \beta_1 + 3) q^{4} + ( - \beta_{3} + \beta_1 - 3) q^{5} - 3 \beta_1 q^{6} + ( - \beta_{6} + \beta_{5} + \beta_{3} + \cdots - 8) q^{8}+ \cdots + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + \beta_1 + 3) q^{4} + ( - \beta_{3} + \beta_1 - 3) q^{5} - 3 \beta_1 q^{6} + ( - \beta_{6} + \beta_{5} + \beta_{3} + \cdots - 8) q^{8}+ \cdots + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} + 24 q^{3} + 30 q^{4} - 20 q^{5} - 12 q^{6} - 78 q^{8} + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} + 24 q^{3} + 30 q^{4} - 20 q^{5} - 12 q^{6} - 78 q^{8} + 72 q^{9} - 94 q^{10} + 88 q^{11} + 90 q^{12} + 94 q^{13} - 60 q^{15} - 10 q^{16} - 144 q^{17} - 36 q^{18} + 8 q^{19} + 144 q^{20} - 44 q^{22} - 286 q^{23} - 234 q^{24} - 206 q^{25} - 132 q^{26} + 216 q^{27} + 20 q^{29} - 282 q^{30} - 192 q^{31} - 948 q^{32} + 264 q^{33} + 486 q^{34} + 270 q^{36} - 666 q^{37} - 200 q^{38} + 282 q^{39} - 486 q^{40} - 60 q^{41} - 246 q^{43} + 330 q^{44} - 180 q^{45} + 454 q^{46} - 922 q^{47} - 30 q^{48} + 1486 q^{50} - 432 q^{51} + 1714 q^{52} - 1064 q^{53} - 108 q^{54} - 220 q^{55} + 24 q^{57} - 1832 q^{58} - 50 q^{59} + 432 q^{60} - 1174 q^{61} + 694 q^{62} + 1602 q^{64} - 1840 q^{65} - 132 q^{66} - 766 q^{67} - 2806 q^{68} - 858 q^{69} - 786 q^{71} - 702 q^{72} + 398 q^{73} - 1504 q^{74} - 618 q^{75} + 1518 q^{76} - 396 q^{78} - 1886 q^{79} + 504 q^{80} + 648 q^{81} - 2738 q^{82} + 782 q^{83} - 1642 q^{85} - 356 q^{86} + 60 q^{87} - 858 q^{88} - 2380 q^{89} - 846 q^{90} - 2256 q^{92} - 576 q^{93} + 2588 q^{94} + 58 q^{95} - 2844 q^{96} + 252 q^{97} + 792 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 39x^{6} + 130x^{5} + 495x^{4} - 1290x^{3} - 2045x^{2} + 3952x + 1488 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{7} + 3\nu^{6} - 390\nu^{5} + 8\nu^{4} + 5585\nu^{3} - 365\nu^{2} - 18928\nu + 688 ) / 608 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{7} + 63\nu^{6} - 134\nu^{5} - 1352\nu^{4} + 4957\nu^{3} + 7839\nu^{2} - 24936\nu - 11088 ) / 608 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 43\nu^{6} - 118\nu^{5} - 1304\nu^{4} + 2937\nu^{3} + 11995\nu^{2} - 13104\nu - 25200 ) / 608 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} + 23\nu^{6} - 254\nu^{5} - 648\nu^{4} + 4565\nu^{3} + 4903\nu^{2} - 20576\nu - 4656 ) / 304 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -33\nu^{7} + 51\nu^{6} + 1426\nu^{5} - 776\nu^{4} - 19207\nu^{3} - 3469\nu^{2} + 70840\nu + 26896 ) / 608 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} - \beta_{3} + 3\beta_{2} + 18\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + 3\beta_{6} - 3\beta_{5} - \beta_{4} + \beta_{3} + 30\beta_{2} + 42\beta _1 + 188 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{7} + 34\beta_{6} - 28\beta_{5} - 8\beta_{4} - 20\beta_{3} + 118\beta_{2} + 399\beta _1 + 358 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 46\beta_{7} + 140\beta_{6} - 102\beta_{5} - 62\beta_{4} + 38\beta_{3} + 839\beta_{2} + 1397\beta _1 + 3937 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 202\beta_{7} + 1033\beta_{6} - 715\beta_{5} - 418\beta_{4} - 247\beta_{3} + 3839\beta_{2} + 9978\beta _1 + 12136 \) 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
5.44233
4.30750
2.47499
2.16654
−0.332658
−2.48441
−3.64724
−3.92705
−5.44233 3.00000 21.6189 2.47884 −16.3270 0 −74.1188 9.00000 −13.4907
1.2 −4.30750 3.00000 10.5546 9.64153 −12.9225 0 −11.0037 9.00000 −41.5309
1.3 −2.47499 3.00000 −1.87441 −8.80061 −7.42498 0 24.4391 9.00000 21.7814
1.4 −2.16654 3.00000 −3.30612 2.12451 −6.49961 0 24.4951 9.00000 −4.60282
1.5 0.332658 3.00000 −7.88934 −14.4186 0.997975 0 −5.28572 9.00000 −4.79647
1.6 2.48441 3.00000 −1.82769 4.95852 7.45324 0 −24.4160 9.00000 12.3190
1.7 3.64724 3.00000 5.30235 3.24850 10.9417 0 −9.83896 9.00000 11.8481
1.8 3.92705 3.00000 7.42171 −19.2327 11.7811 0 −2.27098 9.00000 −75.5277
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(7\) \( -1 \)
\(11\) \( -1 \)

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 1617.4.a.v 8
7.b odd 2 1 1617.4.a.u 8
7.d odd 6 2 231.4.i.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.4.i.a 16 7.d odd 6 2
1617.4.a.u 8 7.b odd 2 1
1617.4.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1617))\):

\( T_{2}^{8} + 4T_{2}^{7} - 39T_{2}^{6} - 130T_{2}^{5} + 495T_{2}^{4} + 1290T_{2}^{3} - 2045T_{2}^{2} - 3952T_{2} + 1488 \) Copy content Toggle raw display
\( T_{5}^{8} + 20 T_{5}^{7} - 197 T_{5}^{6} - 3368 T_{5}^{5} + 21980 T_{5}^{4} + 102244 T_{5}^{3} + \cdots - 1996016 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 4 T^{7} + \cdots + 1488 \) Copy content Toggle raw display
$3$ \( (T - 3)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 20 T^{7} + \cdots - 1996016 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T - 11)^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 1416035880512 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 1234387794689 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 56218350671676 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots - 28\!\cdots\!37 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots - 87\!\cdots\!68 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 23\!\cdots\!72 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots - 99\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 83\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 53\!\cdots\!43 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 62\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots - 81\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots - 24\!\cdots\!12 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 35\!\cdots\!40 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 26\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 31\!\cdots\!12 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 18\!\cdots\!92 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 51\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots - 13\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots - 15\!\cdots\!31 \) Copy content Toggle raw display
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