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Properties

Label 8464.2.a.cj
Level $8464$
Weight $2$
Character orbit 8464.a
Self dual yes
Analytic conductor $67.585$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8464,2,Mod(1,8464)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8464.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8464 = 2^{4} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8464.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [15,0,1,0,10,0,-10,0,16,0,9,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(67.5853802708\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 30 x^{13} + 26 x^{12} + 338 x^{11} - 238 x^{10} - 1773 x^{9} + 894 x^{8} + 4319 x^{7} + \cdots + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 184)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{13} + 1) q^{5} + (\beta_{8} - \beta_{3} - \beta_{2} + \cdots - 1) q^{7} + ( - \beta_{14} + \beta_{13} - \beta_{11} + \cdots + 1) q^{9} + (\beta_{14} + \beta_{12} - \beta_{11} + \cdots + 1) q^{11}+ \cdots + ( - \beta_{14} + \beta_{12} - \beta_{11} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + q^{3} + 10 q^{5} - 10 q^{7} + 16 q^{9} + 9 q^{11} - 12 q^{15} + 20 q^{17} + 3 q^{19} + 21 q^{21} + 11 q^{25} + 7 q^{27} - 4 q^{29} + 14 q^{31} + 34 q^{33} - 18 q^{35} + 26 q^{37} - 33 q^{39} - 11 q^{41}+ \cdots + 29 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - x^{14} - 30 x^{13} + 26 x^{12} + 338 x^{11} - 238 x^{10} - 1773 x^{9} + 894 x^{8} + 4319 x^{7} + \cdots + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 508389479 \nu^{14} + 568266619 \nu^{13} + 15215978119 \nu^{12} - 15009500516 \nu^{11} + \cdots - 30853284209 ) / 1528511161 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 549376471 \nu^{14} + 1190902065 \nu^{13} + 15456200893 \nu^{12} - 32986274347 \nu^{11} + \cdots + 18411647678 ) / 1528511161 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2448466278 \nu^{14} - 3204093500 \nu^{13} - 72449091968 \nu^{12} + 86033275875 \nu^{11} + \cdots + 85647402533 ) / 1528511161 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2804844019 \nu^{14} - 3313233498 \nu^{13} - 83577053951 \nu^{12} + 88141922613 \nu^{11} + \cdots + 147524248104 ) / 1528511161 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3745271983 \nu^{14} + 4992416686 \nu^{13} + 110678099550 \nu^{12} - 134210034723 \nu^{11} + \cdots - 128298257451 ) / 1528511161 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3883107043 \nu^{14} + 5110197511 \nu^{13} + 114983892721 \nu^{12} - 137484562764 \nu^{11} + \cdots - 90263918511 ) / 1528511161 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5036080442 \nu^{14} + 6528262146 \nu^{13} + 149120721092 \nu^{12} - 175114269267 \nu^{11} + \cdots - 190016460029 ) / 1528511161 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5749701249 \nu^{14} + 7040509708 \nu^{13} + 170955192010 \nu^{12} - 187934854016 \nu^{11} + \cdots - 264866788602 ) / 1528511161 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 6683252117 \nu^{14} - 8762476802 \nu^{13} - 197741661483 \nu^{12} + 235220206047 \nu^{11} + \cdots + 249483851366 ) / 1528511161 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 7786127503 \nu^{14} - 10234593781 \nu^{13} - 230379731590 \nu^{12} + 274888407046 \nu^{11} + \cdots + 265901966233 ) / 1528511161 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 12864184682 \nu^{14} + 15982608011 \nu^{13} + 382064984918 \nu^{12} - 427142706241 \nu^{11} + \cdots - 577821304499 ) / 1528511161 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 19213039351 \nu^{14} + 24473542413 \nu^{13} + 569749325367 \nu^{12} - 655621816054 \nu^{11} + \cdots - 759887684986 ) / 1528511161 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 24030078499 \nu^{14} + 32073557199 \nu^{13} + 710173732981 \nu^{12} - 862533990085 \nu^{11} + \cdots - 778889382220 ) / 1528511161 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{14} + \beta_{13} - \beta_{11} - \beta_{9} + \beta_{8} + \beta_{5} + \beta_{3} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} + \beta_{11} + 3\beta_{10} + 2\beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + 2\beta_{5} + \beta_{2} + 7\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 8 \beta_{14} + 9 \beta_{13} + \beta_{12} - 9 \beta_{11} + 3 \beta_{10} - 11 \beta_{9} + 9 \beta_{8} + \cdots + 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12 \beta_{14} - 2 \beta_{13} + 3 \beta_{12} + 6 \beta_{11} + 41 \beta_{10} + 27 \beta_{9} + 14 \beta_{8} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 71 \beta_{14} + 80 \beta_{13} + 17 \beta_{12} - 91 \beta_{11} + 36 \beta_{10} - 122 \beta_{9} + \cdots + 275 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 130 \beta_{14} - 35 \beta_{13} + 59 \beta_{12} + 11 \beta_{11} + 477 \beta_{10} + 287 \beta_{9} + \cdots + 86 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 679 \beta_{14} + 736 \beta_{13} + 212 \beta_{12} - 962 \beta_{11} + 340 \beta_{10} - 1349 \beta_{9} + \cdots + 2444 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1374 \beta_{14} - 460 \beta_{13} + 850 \beta_{12} - 340 \beta_{11} + 5285 \beta_{10} + 2840 \beta_{9} + \cdots + 656 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 6747 \beta_{14} + 6985 \beta_{13} + 2410 \beta_{12} - 10253 \beta_{11} + 3017 \beta_{10} + \cdots + 22356 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 14420 \beta_{14} - 5485 \beta_{13} + 10807 \beta_{12} - 7335 \beta_{11} + 57348 \beta_{10} + 27532 \beta_{9} + \cdots + 4439 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 68381 \beta_{14} + 67926 \beta_{13} + 26436 \beta_{12} - 109178 \beta_{11} + 26578 \beta_{10} + \cdots + 209885 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 150980 \beta_{14} - 62429 \beta_{13} + 128603 \beta_{12} - 107568 \beta_{11} + 615494 \beta_{10} + \cdots + 25031 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 700659 \beta_{14} + 672976 \beta_{13} + 285475 \beta_{12} - 1159849 \beta_{11} + 238315 \beta_{10} + \cdots + 2015839 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−3.23165
−2.75575
−2.34707
−1.99277
−0.560175
−0.407461
−0.322533
−0.273598
−0.209555
0.881524
1.43553
1.95660
2.77527
2.80169
3.24993
0 −3.23165 0 2.81977 0 −2.60337 0 7.44356 0
1.2 0 −2.75575 0 3.14527 0 −3.24881 0 4.59413 0
1.3 0 −2.34707 0 1.00622 0 −0.336699 0 2.50876 0
1.4 0 −1.99277 0 −1.23791 0 −1.08361 0 0.971119 0
1.5 0 −0.560175 0 3.75004 0 −3.37557 0 −2.68620 0
1.6 0 −0.407461 0 −2.35109 0 −3.33026 0 −2.83398 0
1.7 0 −0.322533 0 −2.48772 0 0.274819 0 −2.89597 0
1.8 0 −0.273598 0 4.06550 0 3.04802 0 −2.92514 0
1.9 0 −0.209555 0 0.132367 0 2.45042 0 −2.95609 0
1.10 0 0.881524 0 −2.46842 0 1.39730 0 −2.22292 0
1.11 0 1.43553 0 −0.827591 0 −5.07987 0 −0.939239 0
1.12 0 1.95660 0 1.53132 0 0.941394 0 0.828299 0
1.13 0 2.77527 0 3.46220 0 −3.66987 0 4.70214 0
1.14 0 2.80169 0 0.737681 0 3.85057 0 4.84949 0
1.15 0 3.24993 0 −1.27764 0 0.765530 0 7.56204 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

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

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 8464.2.a.cj 15
4.b odd 2 1 4232.2.a.z 15
23.b odd 2 1 8464.2.a.ci 15
23.d odd 22 2 368.2.m.f 30
92.b even 2 1 4232.2.a.y 15
92.h even 22 2 184.2.i.a ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.2.i.a ✓ 30 92.h even 22 2
368.2.m.f 30 23.d odd 22 2
4232.2.a.y 15 92.b even 2 1
4232.2.a.z 15 4.b odd 2 1
8464.2.a.ci 15 23.b odd 2 1
8464.2.a.cj 15 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8464))\):

\( T_{3}^{15} - T_{3}^{14} - 30 T_{3}^{13} + 26 T_{3}^{12} + 338 T_{3}^{11} - 238 T_{3}^{10} - 1773 T_{3}^{9} + \cdots + 11 \) Copy content Toggle raw display
\( T_{5}^{15} - 10 T_{5}^{14} + 7 T_{5}^{13} + 206 T_{5}^{12} - 459 T_{5}^{11} - 1594 T_{5}^{10} + \cdots - 1331 \) Copy content Toggle raw display
\( T_{7}^{15} + 10 T_{7}^{14} - 7 T_{7}^{13} - 335 T_{7}^{12} - 527 T_{7}^{11} + 3920 T_{7}^{10} + \cdots - 5147 \) Copy content Toggle raw display
\( T_{13}^{15} - 74 T_{13}^{13} + 2 T_{13}^{12} + 2065 T_{13}^{11} - 125 T_{13}^{10} - 27575 T_{13}^{9} + \cdots - 8119 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} \) Copy content Toggle raw display
$3$ \( T^{15} - T^{14} + \cdots + 11 \) Copy content Toggle raw display
$5$ \( T^{15} - 10 T^{14} + \cdots - 1331 \) Copy content Toggle raw display
$7$ \( T^{15} + 10 T^{14} + \cdots - 5147 \) Copy content Toggle raw display
$11$ \( T^{15} - 9 T^{14} + \cdots + 4007047 \) Copy content Toggle raw display
$13$ \( T^{15} - 74 T^{13} + \cdots - 8119 \) Copy content Toggle raw display
$17$ \( T^{15} - 20 T^{14} + \cdots - 58533793 \) Copy content Toggle raw display
$19$ \( T^{15} - 3 T^{14} + \cdots + 87968 \) Copy content Toggle raw display
$23$ \( T^{15} \) Copy content Toggle raw display
$29$ \( T^{15} + 4 T^{14} + \cdots + 20393 \) Copy content Toggle raw display
$31$ \( T^{15} - 14 T^{14} + \cdots + 5530437 \) Copy content Toggle raw display
$37$ \( T^{15} - 26 T^{14} + \cdots - 142231 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 952608271 \) Copy content Toggle raw display
$43$ \( T^{15} + 10 T^{14} + \cdots - 1624127 \) Copy content Toggle raw display
$47$ \( T^{15} + 12 T^{14} + \cdots + 55081984 \) Copy content Toggle raw display
$53$ \( T^{15} - 25 T^{14} + \cdots + 2035661 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 5833783759 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 1166425151 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 11262048063509 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 11900110073171 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 2111467144181 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 31798242673 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 1045117690969 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 18586214417 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 649941054247 \) Copy content Toggle raw display
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