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Properties

Label 1452.3.e.l
Level $1452$
Weight $3$
Character orbit 1452.e
Analytic conductor $39.564$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1452,3,Mod(485,1452)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1452, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1452.485"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1452.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,-4,0,14,0,0,0,-2,0,-8,0,0,0,-2,0,1,0,0,0,-70,0, -54,0,0,0,-50] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(39.5641343851\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7 x^{14} - 18 x^{13} + 41 x^{12} + 216 x^{11} + 199 x^{10} - 1278 x^{9} - 468 x^{8} + \cdots + 43046721 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 132)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{12} q^{3} + \beta_{10} q^{5} + \beta_{5} q^{7} + ( - \beta_{11} + 1) q^{9} + (\beta_{14} - \beta_{11} + \cdots - \beta_1) q^{13} + ( - \beta_{13} + \beta_{9} - \beta_{8} + \cdots - 1) q^{15}+ \cdots + (2 \beta_{14} + 17 \beta_{12} + \cdots - 35) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{7} + 14 q^{9} - 2 q^{13} - 8 q^{15} - 2 q^{19} + q^{21} - 70 q^{25} - 54 q^{27} - 50 q^{31} + 30 q^{37} + 68 q^{39} - 50 q^{43} + 17 q^{45} + 232 q^{49} + 218 q^{51} + 205 q^{57} - 80 q^{61}+ \cdots - 416 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 7 x^{14} - 18 x^{13} + 41 x^{12} + 216 x^{11} + 199 x^{10} - 1278 x^{9} - 468 x^{8} + \cdots + 43046721 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 91 \nu^{15} - 8100 \nu^{14} - 15136 \nu^{13} + 2574 \nu^{12} + 218219 \nu^{11} + \cdots + 43850259792 ) / 50508152640 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 91 \nu^{15} - 8100 \nu^{14} - 15136 \nu^{13} + 2574 \nu^{12} + 218219 \nu^{11} + \cdots + 43850259792 ) / 50508152640 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 464 \nu^{15} + 273 \nu^{14} - 27548 \nu^{13} - 53760 \nu^{12} + 26746 \nu^{11} + \cdots + 5030089065 ) / 16836050880 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 371 \nu^{15} + 2448 \nu^{14} + 13360 \nu^{13} - 227934 \nu^{12} - 383309 \nu^{11} + \cdots - 52383076488 ) / 12627038160 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1021 \nu^{15} - 17604 \nu^{14} + 170848 \nu^{13} + 299070 \nu^{12} - 11405 \nu^{11} + \cdots + 15841193328 ) / 50508152640 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1004 \nu^{15} + 39483 \nu^{14} - 62756 \nu^{13} + 362376 \nu^{12} + 481642 \nu^{11} + \cdots + 3955515363 ) / 50508152640 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 541 \nu^{15} + 3438 \nu^{14} + 7432 \nu^{13} - 114930 \nu^{12} - 299849 \nu^{11} + \cdots - 72461980350 ) / 8418025440 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6490 \nu^{15} - 29889 \nu^{14} + 79612 \nu^{13} - 297252 \nu^{12} + 2014060 \nu^{11} + \cdots + 359253584559 ) / 50508152640 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 6202 \nu^{15} - 64215 \nu^{14} - 5996 \nu^{13} + 315468 \nu^{12} + 394528 \nu^{11} + \cdots + 94133612889 ) / 50508152640 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 734 \nu^{15} + 1845 \nu^{14} - 5948 \nu^{13} - 12276 \nu^{12} - 63056 \nu^{11} + \cdots - 2616284043 ) / 4591650240 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{14} - 7 \nu^{12} - 18 \nu^{11} + 41 \nu^{10} + 216 \nu^{9} + 199 \nu^{8} - 1278 \nu^{7} + \cdots - 3188646 ) / 531441 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - \nu^{15} + 7 \nu^{13} + 18 \nu^{12} - 41 \nu^{11} - 216 \nu^{10} - 199 \nu^{9} + \cdots + 3720087 \nu ) / 4782969 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 6761 \nu^{15} + 29853 \nu^{14} - 48380 \nu^{13} - 51462 \nu^{12} - 771101 \nu^{11} + \cdots - 142910330751 ) / 25254076320 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 6220 \nu^{15} + 4833 \nu^{14} - 26044 \nu^{13} - 134856 \nu^{12} + 281750 \nu^{11} + \cdots - 15530300343 ) / 16836050880 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 3173 \nu^{15} - 14376 \nu^{14} + 29224 \nu^{13} + 71706 \nu^{12} - 148331 \nu^{11} + \cdots + 25566563628 ) / 8418025440 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{14} - \beta_{13} + 3\beta_{12} - \beta_{8} + \beta_{3} + \beta_{2} - \beta _1 + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{15} + 3 \beta_{13} + 3 \beta_{12} - 6 \beta_{10} - 3 \beta_{9} + 3 \beta_{8} + 6 \beta_{7} + \cdots + 12 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3 \beta_{15} - 8 \beta_{14} - 5 \beta_{13} + 3 \beta_{11} + 12 \beta_{10} - 9 \beta_{9} + \beta_{8} + \cdots - 51 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9 \beta_{15} - 27 \beta_{13} + 27 \beta_{12} + 9 \beta_{11} - 54 \beta_{10} - 63 \beta_{9} - 45 \beta_{8} + \cdots - 126 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 63 \beta_{15} - 64 \beta_{14} - 139 \beta_{13} - 105 \beta_{12} + 117 \beta_{11} + 90 \beta_{10} + \cdots - 501 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3 \beta_{15} - 216 \beta_{14} - 168 \beta_{13} - 366 \beta_{12} - 513 \beta_{11} - 438 \beta_{10} + \cdots + 948 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 426 \beta_{15} + 958 \beta_{14} + 340 \beta_{13} + 3096 \beta_{12} + 291 \beta_{11} - 726 \beta_{10} + \cdots - 6171 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 378 \beta_{15} - 1134 \beta_{14} - 450 \beta_{13} - 5967 \beta_{12} + 576 \beta_{11} - 5328 \beta_{10} + \cdots + 14760 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 774 \beta_{15} - 5638 \beta_{14} - 6136 \beta_{13} - 25188 \beta_{12} + 7578 \beta_{11} - 5814 \beta_{10} + \cdots + 19353 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 14010 \beta_{15} + 27270 \beta_{14} + 4710 \beta_{13} - 22794 \beta_{12} + 2700 \beta_{11} + \cdots - 218328 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 35058 \beta_{15} - 36380 \beta_{14} + 32419 \beta_{13} - 34173 \beta_{12} - 5550 \beta_{11} + \cdots + 119253 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 86769 \beta_{15} - 162486 \beta_{14} + 116577 \beta_{13} - 366471 \beta_{12} + 182016 \beta_{11} + \cdots - 112428 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 425547 \beta_{15} - 193390 \beta_{14} - 120277 \beta_{13} - 429756 \beta_{12} + 565353 \beta_{11} + \cdots + 1370523 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1566867 \beta_{15} + 2305746 \beta_{14} + 731163 \beta_{13} - 5442603 \beta_{12} + 172125 \beta_{11} + \cdots + 10434 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1452\mathbb{Z}\right)^\times\).

\(n\) \(485\) \(727\) \(1333\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
485.1
2.93080 − 0.640644i
2.93080 + 0.640644i
2.66900 − 1.36984i
2.66900 + 1.36984i
2.14870 − 2.09358i
2.14870 + 2.09358i
0.454866 − 2.96532i
0.454866 + 2.96532i
−1.08635 − 2.79640i
−1.08635 + 2.79640i
−1.61472 − 2.52838i
−1.61472 + 2.52838i
−2.61331 − 1.47330i
−2.61331 + 1.47330i
−2.88898 − 0.808570i
−2.88898 + 0.808570i
0 −2.93080 − 0.640644i 0 6.71318i 0 −3.02703 0 8.17915 + 3.75519i 0
485.2 0 −2.93080 + 0.640644i 0 − 6.71318i 0 −3.02703 0 8.17915 − 3.75519i 0
485.3 0 −2.66900 − 1.36984i 0 − 8.10965i 0 10.6867 0 5.24710 + 7.31218i 0
485.4 0 −2.66900 + 1.36984i 0 8.10965i 0 10.6867 0 5.24710 − 7.31218i 0
485.5 0 −2.14870 − 2.09358i 0 4.10414i 0 −9.08076 0 0.233808 + 8.99696i 0
485.6 0 −2.14870 + 2.09358i 0 − 4.10414i 0 −9.08076 0 0.233808 − 8.99696i 0
485.7 0 −0.454866 − 2.96532i 0 0.778792i 0 6.58832 0 −8.58619 + 2.69764i 0
485.8 0 −0.454866 + 2.96532i 0 − 0.778792i 0 6.58832 0 −8.58619 − 2.69764i 0
485.9 0 1.08635 − 2.79640i 0 − 3.92851i 0 −4.86735 0 −6.63968 − 6.07575i 0
485.10 0 1.08635 + 2.79640i 0 3.92851i 0 −4.86735 0 −6.63968 + 6.07575i 0
485.11 0 1.61472 − 2.52838i 0 − 3.70187i 0 −13.1901 0 −3.78538 − 8.16522i 0
485.12 0 1.61472 + 2.52838i 0 3.70187i 0 −13.1901 0 −3.78538 + 8.16522i 0
485.13 0 2.61331 − 1.47330i 0 4.01278i 0 4.49752 0 4.65877 − 7.70038i 0
485.14 0 2.61331 + 1.47330i 0 − 4.01278i 0 4.49752 0 4.65877 + 7.70038i 0
485.15 0 2.88898 − 0.808570i 0 7.84073i 0 6.39278 0 7.69243 − 4.67189i 0
485.16 0 2.88898 + 0.808570i 0 − 7.84073i 0 6.39278 0 7.69243 + 4.67189i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 485.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1452.3.e.l 16
3.b odd 2 1 inner 1452.3.e.l 16
11.b odd 2 1 1452.3.e.m 16
11.c even 5 2 132.3.m.a ✓ 32
33.d even 2 1 1452.3.e.m 16
33.h odd 10 2 132.3.m.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
132.3.m.a ✓ 32 11.c even 5 2
132.3.m.a ✓ 32 33.h odd 10 2
1452.3.e.l 16 1.a even 1 1 trivial
1452.3.e.l 16 3.b odd 2 1 inner
1452.3.e.m 16 11.b odd 2 1
1452.3.e.m 16 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1452, [\chi])\):

\( T_{5}^{16} + 235 T_{5}^{14} + 22060 T_{5}^{12} + 1065987 T_{5}^{10} + 28676340 T_{5}^{8} + \cdots + 6339393280 \) Copy content Toggle raw display
\( T_{7}^{8} + 2T_{7}^{7} - 252T_{7}^{6} - 64T_{7}^{5} + 18956T_{7}^{4} - 14800T_{7}^{3} - 474045T_{7}^{2} + 348722T_{7} + 3572404 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 7 T^{14} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 6339393280 \) Copy content Toggle raw display
$7$ \( (T^{8} + 2 T^{7} + \cdots + 3572404)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( (T^{8} + T^{7} + \cdots - 3087920)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} + T^{7} + \cdots + 493196)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 55\!\cdots\!80 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 12\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( (T^{8} + 25 T^{7} + \cdots + 26256466876)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 15 T^{7} + \cdots - 17666503056)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 89\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( (T^{8} + 25 T^{7} + \cdots - 135843584976)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 47\!\cdots\!80 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 41\!\cdots\!55 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 6695512726320)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 22 T^{7} + \cdots - 14444700016)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 93\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 128619188548884)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 41732306692956)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 13\!\cdots\!55 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 14\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 140591577086469)^{2} \) Copy content Toggle raw display
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