Properties

Label 1617.4.a.q
Level $1617$
Weight $4$
Character orbit 1617.a
Self dual yes
Analytic conductor $95.406$
Analytic rank $0$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 38x^{5} + 73x^{4} + 383x^{3} - 256x^{2} - 676x - 224 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} - 3 q^{3} + (\beta_{2} - \beta_1 + 5) q^{4} + ( - \beta_{4} + \beta_{2} - \beta_1 + 4) q^{5} + ( - 3 \beta_1 + 3) q^{6} + (\beta_{5} - \beta_{4} - \beta_{2} + \cdots - 8) q^{8}+ \cdots + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} - 3 q^{3} + (\beta_{2} - \beta_1 + 5) q^{4} + ( - \beta_{4} + \beta_{2} - \beta_1 + 4) q^{5} + ( - 3 \beta_1 + 3) q^{6} + (\beta_{5} - \beta_{4} - \beta_{2} + \cdots - 8) q^{8}+ \cdots + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 4 q^{2} - 21 q^{3} + 30 q^{4} + 20 q^{5} + 12 q^{6} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 4 q^{2} - 21 q^{3} + 30 q^{4} + 20 q^{5} + 12 q^{6} - 39 q^{8} + 63 q^{9} - 49 q^{10} + 77 q^{11} - 90 q^{12} + 92 q^{13} - 60 q^{15} + 218 q^{16} + 170 q^{17} - 36 q^{18} + 76 q^{19} + 569 q^{20} - 44 q^{22} - 56 q^{23} + 117 q^{24} + 53 q^{25} + 109 q^{26} - 189 q^{27} - 472 q^{29} + 147 q^{30} + 290 q^{31} - 1046 q^{32} - 231 q^{33} + 344 q^{34} + 270 q^{36} - 66 q^{37} + 385 q^{38} - 276 q^{39} - 800 q^{40} - 166 q^{41} - 76 q^{43} + 330 q^{44} + 180 q^{45} - 528 q^{46} + 1082 q^{47} - 654 q^{48} - 569 q^{50} - 510 q^{51} + 1065 q^{52} - 150 q^{53} + 108 q^{54} + 220 q^{55} - 228 q^{57} + 1457 q^{58} + 1284 q^{59} - 1707 q^{60} - 764 q^{61} - 296 q^{62} + 1661 q^{64} + 2722 q^{65} + 132 q^{66} - 658 q^{67} - 360 q^{68} + 168 q^{69} - 272 q^{71} - 351 q^{72} + 1658 q^{73} + 613 q^{74} - 159 q^{75} - 1757 q^{76} - 327 q^{78} + 792 q^{79} + 5079 q^{80} + 567 q^{81} + 3208 q^{82} - 770 q^{83} - 776 q^{85} - 478 q^{86} + 1416 q^{87} - 429 q^{88} + 656 q^{89} - 441 q^{90} - 3916 q^{92} - 870 q^{93} - 849 q^{94} - 1636 q^{95} + 3138 q^{96} + 608 q^{97} + 693 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 38x^{5} + 73x^{4} + 383x^{3} - 256x^{2} - 676x - 224 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 3\nu^{5} + 36\nu^{4} - 73\nu^{3} - 311\nu^{2} + 290\nu + 240 ) / 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{6} - 11\nu^{5} - 108\nu^{4} + 291\nu^{3} + 1003\nu^{2} - 1414\nu - 1328 ) / 24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{6} - 11\nu^{5} - 108\nu^{4} + 315\nu^{3} + 955\nu^{2} - 1870\nu - 1064 ) / 24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{6} - 13\nu^{5} - 96\nu^{4} + 339\nu^{3} + 725\nu^{2} - 1598\nu - 676 ) / 12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{4} + 2\beta_{2} + 21\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + 2\beta_{5} - 6\beta_{4} - 3\beta_{3} + 33\beta_{2} + 41\beta _1 + 269 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 36\beta_{5} - 48\beta_{4} - 18\beta_{3} + 107\beta_{2} + 519\beta _1 + 584 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 36\beta_{6} + 107\beta_{5} - 287\beta_{4} - 174\beta_{3} + 1052\beta_{2} + 1479\beta _1 + 6995 \) 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
−4.54076
−3.02575
−0.792709
−0.462025
1.81666
4.28176
5.72283
−5.54076 −3.00000 22.7000 15.6321 16.6223 0 −81.4494 9.00000 −86.6138
1.2 −4.02575 −3.00000 8.20667 2.56007 12.0773 0 −0.831993 9.00000 −10.3062
1.3 −1.79271 −3.00000 −4.78619 −15.7758 5.37813 0 22.9219 9.00000 28.2814
1.4 −1.46203 −3.00000 −5.86248 13.7188 4.38608 0 20.2673 9.00000 −20.0572
1.5 0.816662 −3.00000 −7.33306 −2.99797 −2.44999 0 −12.5219 9.00000 −2.44833
1.6 3.28176 −3.00000 2.76992 −6.75357 −9.84527 0 −17.1638 9.00000 −22.1636
1.7 4.72283 −3.00000 14.3051 13.6164 −14.1685 0 29.7779 9.00000 64.3077
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.q 7
7.b odd 2 1 1617.4.a.r yes 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1617.4.a.q 7 1.a even 1 1 trivial
1617.4.a.r yes 7 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1617))\):

\( T_{2}^{7} + 4T_{2}^{6} - 35T_{2}^{5} - 127T_{2}^{4} + 270T_{2}^{3} + 927T_{2}^{2} + 52T_{2} - 740 \) Copy content Toggle raw display
\( T_{5}^{7} - 20T_{5}^{6} - 264T_{5}^{5} + 6386T_{5}^{4} + 4315T_{5}^{3} - 359910T_{5}^{2} - 133484T_{5} + 2387792 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} + 4 T^{6} + \cdots - 740 \) Copy content Toggle raw display
$3$ \( (T + 3)^{7} \) Copy content Toggle raw display
$5$ \( T^{7} - 20 T^{6} + \cdots + 2387792 \) Copy content Toggle raw display
$7$ \( T^{7} \) Copy content Toggle raw display
$11$ \( (T - 11)^{7} \) Copy content Toggle raw display
$13$ \( T^{7} + \cdots - 115171354880 \) Copy content Toggle raw display
$17$ \( T^{7} + \cdots - 370778627072 \) Copy content Toggle raw display
$19$ \( T^{7} + \cdots + 534703012144 \) Copy content Toggle raw display
$23$ \( T^{7} + \cdots - 76625986977792 \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots + 47\!\cdots\!60 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots + 37\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots - 68\!\cdots\!12 \) Copy content Toggle raw display
$41$ \( T^{7} + \cdots - 10\!\cdots\!08 \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots - 23\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots + 92\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots - 16\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots - 22\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots + 39\!\cdots\!40 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots + 15\!\cdots\!20 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots - 10\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots - 39\!\cdots\!60 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots + 16\!\cdots\!40 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots - 92\!\cdots\!68 \) Copy content Toggle raw display
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