Properties

Label 1617.4.a.ba
Level $1617$
Weight $4$
Character orbit 1617.a
Self dual yes
Analytic conductor $95.406$
Analytic rank $0$
Dimension $12$
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: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 79 x^{10} + 310 x^{9} + 2225 x^{8} - 8576 x^{7} - 26761 x^{6} + 101926 x^{5} + \cdots + 275328 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3}\cdot 3\cdot 7 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 7) q^{4} + ( - \beta_{4} + \beta_1 - 2) q^{5} - 3 \beta_1 q^{6} + (\beta_{3} + 7 \beta_1 - 1) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 7) q^{4} + ( - \beta_{4} + \beta_1 - 2) q^{5} - 3 \beta_1 q^{6} + (\beta_{3} + 7 \beta_1 - 1) q^{8} + 9 q^{9} + ( - \beta_{7} + \beta_{5} - \beta_{4} + \cdots + 9) q^{10}+ \cdots + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{2} - 36 q^{3} + 78 q^{4} - 20 q^{5} - 12 q^{6} + 18 q^{8} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{2} - 36 q^{3} + 78 q^{4} - 20 q^{5} - 12 q^{6} + 18 q^{8} + 108 q^{9} + 100 q^{10} + 132 q^{11} - 234 q^{12} - 32 q^{13} + 60 q^{15} + 526 q^{16} - 100 q^{17} + 36 q^{18} - 6 q^{19} - 28 q^{20} + 44 q^{22} + 500 q^{23} - 54 q^{24} + 506 q^{25} - 38 q^{26} - 324 q^{27} + 96 q^{29} - 300 q^{30} - 226 q^{31} + 398 q^{32} - 396 q^{33} - 262 q^{34} + 702 q^{36} + 1114 q^{37} - 218 q^{38} + 96 q^{39} + 1068 q^{40} - 800 q^{41} + 604 q^{43} + 858 q^{44} - 180 q^{45} - 948 q^{46} - 428 q^{47} - 1578 q^{48} + 1010 q^{50} + 300 q^{51} - 224 q^{52} + 1028 q^{53} - 108 q^{54} - 220 q^{55} + 18 q^{57} + 1292 q^{58} - 1192 q^{59} + 84 q^{60} + 922 q^{61} - 674 q^{62} + 5414 q^{64} - 248 q^{65} - 132 q^{66} + 1424 q^{67} - 2074 q^{68} - 1500 q^{69} + 2332 q^{71} + 162 q^{72} - 284 q^{73} - 914 q^{74} - 1518 q^{75} + 1060 q^{76} + 114 q^{78} + 408 q^{79} - 2348 q^{80} + 972 q^{81} - 838 q^{82} + 1238 q^{83} + 3658 q^{85} + 1866 q^{86} - 288 q^{87} + 198 q^{88} - 700 q^{89} + 900 q^{90} + 6488 q^{92} + 678 q^{93} - 4488 q^{94} + 1762 q^{95} - 1194 q^{96} + 498 q^{97} + 1188 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 79 x^{10} + 310 x^{9} + 2225 x^{8} - 8576 x^{7} - 26761 x^{6} + 101926 x^{5} + \cdots + 275328 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 15 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 23\nu + 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1573 \nu^{11} + 3847 \nu^{10} + 130234 \nu^{9} - 285364 \nu^{8} - 3944969 \nu^{7} + \cdots + 169295232 ) / 9515520 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13999 \nu^{11} + 19189 \nu^{10} + 1161646 \nu^{9} - 1341628 \nu^{8} - 34734875 \nu^{7} + \cdots + 1189817472 ) / 9515520 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 175 \nu^{11} - 249 \nu^{10} - 14522 \nu^{9} + 17468 \nu^{8} + 434139 \nu^{7} - 424609 \nu^{6} + \cdots + 1686912 ) / 113280 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 4957 \nu^{11} + 7103 \nu^{10} + 411226 \nu^{9} - 500436 \nu^{8} - 12315681 \nu^{7} + \cdots + 465506688 ) / 3171840 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2487 \nu^{11} - 3629 \nu^{10} - 207054 \nu^{9} + 245068 \nu^{8} + 6263747 \nu^{7} + \cdots - 267421440 ) / 792960 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 31127 \nu^{11} - 45917 \nu^{10} - 2580878 \nu^{9} + 3264284 \nu^{8} + 77544355 \nu^{7} + \cdots - 2495054976 ) / 9515520 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 18329 \nu^{11} + 14939 \nu^{10} + 1525466 \nu^{9} - 904628 \nu^{8} - 45993325 \nu^{7} + \cdots + 2726477952 ) / 4757760 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 49289 \nu^{11} + 85571 \nu^{10} + 4022642 \nu^{9} - 5931812 \nu^{8} - 118366717 \nu^{7} + \cdots + 4231245696 ) / 9515520 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 23\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} + \beta_{6} + 2\beta_{5} + \beta_{4} + \beta_{3} + 32\beta_{2} + \beta _1 + 343 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{10} - 2 \beta_{9} - \beta_{8} - \beta_{7} + 2 \beta_{6} + \beta_{5} - 16 \beta_{4} + 38 \beta_{3} + \cdots - 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{10} - 6 \beta_{9} - 2 \beta_{8} - 60 \beta_{7} + 44 \beta_{6} + 96 \beta_{5} + 14 \beta_{4} + \cdots + 8927 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 60 \beta_{10} - 96 \beta_{9} - 56 \beta_{8} - 48 \beta_{7} + 116 \beta_{6} + 100 \beta_{5} + \cdots + 53 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 8 \beta_{11} - 128 \beta_{10} - 312 \beta_{9} - 176 \beta_{8} - 2539 \beta_{7} + 1563 \beta_{6} + \cdots + 245683 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 120 \beta_{11} - 2651 \beta_{10} - 3414 \beta_{9} - 2547 \beta_{8} - 1591 \beta_{7} + 4954 \beta_{6} + \cdots + 22598 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 616 \beta_{11} - 6146 \beta_{10} - 11402 \beta_{9} - 9850 \beta_{8} - 93758 \beta_{7} + 52146 \beta_{6} + \cdots + 6963187 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 9880 \beta_{11} - 103882 \beta_{10} - 108628 \beta_{9} - 105654 \beta_{8} - 45938 \beta_{7} + \cdots + 1370863 \) 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
−5.45929
−4.98543
−3.63350
−2.95517
−1.14635
−0.428288
2.09996
2.60515
2.77685
4.42348
5.06724
5.63534
−5.45929 −3.00000 21.8038 −18.9501 16.3779 0 −75.3590 9.00000 103.454
1.2 −4.98543 −3.00000 16.8545 7.45523 14.9563 0 −44.1437 9.00000 −37.1676
1.3 −3.63350 −3.00000 5.20232 9.34572 10.9005 0 10.1654 9.00000 −33.9577
1.4 −2.95517 −3.00000 0.733054 −4.94230 8.86552 0 21.4751 9.00000 14.6054
1.5 −1.14635 −3.00000 −6.68589 −17.9065 3.43904 0 16.8351 9.00000 20.5270
1.6 −0.428288 −3.00000 −7.81657 −4.03964 1.28486 0 6.77404 9.00000 1.73013
1.7 2.09996 −3.00000 −3.59017 −11.8735 −6.29988 0 −24.3389 9.00000 −24.9339
1.8 2.60515 −3.00000 −1.21321 12.9406 −7.81544 0 −24.0018 9.00000 33.7122
1.9 2.77685 −3.00000 −0.289077 15.3328 −8.33056 0 −23.0176 9.00000 42.5771
1.10 4.42348 −3.00000 11.5672 −10.8900 −13.2705 0 15.7795 9.00000 −48.1715
1.11 5.06724 −3.00000 17.6769 −13.6349 −15.2017 0 49.0353 9.00000 −69.0911
1.12 5.63534 −3.00000 23.7571 17.1624 −16.9060 0 88.7965 9.00000 96.7160
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.ba 12
7.b odd 2 1 1617.4.a.bb 12
7.d odd 6 2 231.4.i.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.4.i.d 24 7.d odd 6 2
1617.4.a.ba 12 1.a even 1 1 trivial
1617.4.a.bb 12 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1617))\):

\( T_{2}^{12} - 4 T_{2}^{11} - 79 T_{2}^{10} + 310 T_{2}^{9} + 2225 T_{2}^{8} - 8576 T_{2}^{7} + \cdots + 275328 \) Copy content Toggle raw display
\( T_{5}^{12} + 20 T_{5}^{11} - 803 T_{5}^{10} - 16668 T_{5}^{9} + 229764 T_{5}^{8} + 5157804 T_{5}^{7} + \cdots - 2833859960832 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 4 T^{11} + \cdots + 275328 \) Copy content Toggle raw display
$3$ \( (T + 3)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots - 2833859960832 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T - 11)^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots - 20\!\cdots\!88 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 12\!\cdots\!68 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots - 83\!\cdots\!28 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots - 11\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots - 81\!\cdots\!08 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots - 20\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 72\!\cdots\!60 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 76\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots - 83\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 22\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 92\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 15\!\cdots\!20 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 99\!\cdots\!08 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 16\!\cdots\!92 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 47\!\cdots\!72 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 13\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 88\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 15\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 54\!\cdots\!04 \) Copy content Toggle raw display
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