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Properties

Label 1449.4.a.o
Level $1449$
Weight $4$
Character orbit 1449.a
Self dual yes
Analytic conductor $85.494$
Analytic rank $0$
Dimension $12$
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] = [1449,4,Mod(1,1449)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1449, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1449.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1449.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-4,0,72,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(85.4937675983\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 76 x^{10} + 311 x^{9} + 1979 x^{8} - 8466 x^{7} - 19942 x^{6} + 95647 x^{5} + \cdots - 70656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 161)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 6) q^{4} + ( - \beta_{7} - \beta_1 - 1) q^{5} + 7 q^{7} + ( - \beta_{9} - \beta_{7} - \beta_{6} + \cdots + 5) q^{8} + (\beta_{11} - \beta_{9} - \beta_{7} + \cdots + 9) q^{10}+ \cdots - 49 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{2} + 72 q^{4} - 16 q^{5} + 84 q^{7} + 21 q^{8} + 106 q^{10} - 50 q^{11} + 21 q^{13} - 28 q^{14} + 516 q^{16} - 26 q^{17} + 292 q^{19} + 12 q^{20} + 206 q^{22} - 276 q^{23} + 408 q^{25} - 409 q^{26}+ \cdots - 196 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 76 x^{10} + 311 x^{9} + 1979 x^{8} - 8466 x^{7} - 19942 x^{6} + 95647 x^{5} + \cdots - 70656 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1057087 \nu^{11} - 736072 \nu^{10} - 81046848 \nu^{9} + 59112285 \nu^{8} + 2160058053 \nu^{7} + \cdots + 27581677952 ) / 466611584 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1117105 \nu^{11} - 226867 \nu^{10} - 85952171 \nu^{9} + 21399060 \nu^{8} + 2305747495 \nu^{7} + \cdots + 19319250176 ) / 233305792 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1519271 \nu^{11} - 498841 \nu^{10} - 117032797 \nu^{9} + 43441900 \nu^{8} + 3145054901 \nu^{7} + \cdots + 27799261696 ) / 233305792 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1605177 \nu^{11} - 326483 \nu^{10} - 123035627 \nu^{9} + 30155740 \nu^{8} + 3282153271 \nu^{7} + \cdots + 31578353216 ) / 233305792 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 99065 \nu^{11} - 21862 \nu^{10} - 7609238 \nu^{9} + 2030749 \nu^{8} + 203556131 \nu^{7} + \cdots + 1798578304 ) / 6964352 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 7097167 \nu^{11} - 2425124 \nu^{10} - 546101432 \nu^{9} + 210879061 \nu^{8} + \cdots + 160329330944 ) / 466611584 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5914582 \nu^{11} + 1189665 \nu^{10} + 454454473 \nu^{9} - 112071589 \nu^{8} + \cdots - 101406339072 ) / 233305792 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7229129 \nu^{11} - 1769452 \nu^{10} - 555592220 \nu^{9} + 160203215 \nu^{8} + \cdots + 135239697088 ) / 233305792 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 22127067 \nu^{11} - 4919124 \nu^{10} - 1700971564 \nu^{9} + 457650357 \nu^{8} + \cdots + 427632581376 ) / 466611584 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 23\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + 2\beta_{9} - 2\beta_{7} + 2\beta_{6} - 2\beta_{5} + 6\beta_{4} + \beta_{3} + 31\beta_{2} - 2\beta _1 + 315 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{11} - 6 \beta_{10} + 36 \beta_{9} + 40 \beta_{7} + 44 \beta_{6} + 44 \beta_{5} + 14 \beta_{4} + \cdots - 185 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 27 \beta_{11} - 10 \beta_{10} + 75 \beta_{9} + 16 \beta_{8} - 65 \beta_{7} + 83 \beta_{6} - 109 \beta_{5} + \cdots + 8130 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 58 \beta_{11} - 354 \beta_{10} + 1026 \beta_{9} + 48 \beta_{8} + 1242 \beta_{7} + 1570 \beta_{6} + \cdots - 6210 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 476 \beta_{11} - 748 \beta_{10} + 2172 \beta_{9} + 1184 \beta_{8} - 1532 \beta_{7} + 2692 \beta_{6} + \cdots + 224802 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2552 \beta_{11} - 15264 \beta_{10} + 26977 \beta_{9} + 3408 \beta_{8} + 36409 \beta_{7} + \cdots - 202781 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3329 \beta_{11} - 36960 \beta_{10} + 56722 \beta_{9} + 57632 \beta_{8} - 29538 \beta_{7} + \cdots + 6459459 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 100129 \beta_{11} - 579662 \beta_{10} + 683620 \beta_{9} + 162544 \beta_{8} + 1056368 \beta_{7} + \cdots - 6540689 \) 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
5.53660
4.95487
3.76781
3.76511
2.11879
1.40034
0.285899
−0.718333
−3.06154
−3.54240
−4.89764
−5.60949
−5.53660 0 22.6539 −3.61751 0 7.00000 −81.1328 0 20.0287
1.2 −4.95487 0 16.5508 2.30377 0 7.00000 −42.3679 0 −11.4149
1.3 −3.76781 0 6.19640 −21.4417 0 7.00000 6.79562 0 80.7883
1.4 −3.76511 0 6.17602 −1.25334 0 7.00000 6.86747 0 4.71897
1.5 −2.11879 0 −3.51074 13.8523 0 7.00000 24.3888 0 −29.3501
1.6 −1.40034 0 −6.03905 −13.7756 0 7.00000 19.6594 0 19.2904
1.7 −0.285899 0 −7.91826 −1.18183 0 7.00000 4.55102 0 0.337884
1.8 0.718333 0 −7.48400 17.9779 0 7.00000 −11.1227 0 12.9141
1.9 3.06154 0 1.37302 −21.1978 0 7.00000 −20.2888 0 −64.8978
1.10 3.54240 0 4.54860 −6.95559 0 7.00000 −12.2262 0 −24.6395
1.11 4.89764 0 15.9869 14.0191 0 7.00000 39.1171 0 68.6607
1.12 5.60949 0 23.4664 5.27021 0 7.00000 86.7589 0 29.5632
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(7\) \( -1 \)
\(23\) \( +1 \)

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 1449.4.a.o 12
3.b odd 2 1 161.4.a.d ✓ 12
21.c even 2 1 1127.4.a.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.4.a.d ✓ 12 3.b odd 2 1
1127.4.a.h 12 21.c even 2 1
1449.4.a.o 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 4 T_{2}^{11} - 76 T_{2}^{10} - 311 T_{2}^{9} + 1979 T_{2}^{8} + 8466 T_{2}^{7} + \cdots - 70656 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1449))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 4 T^{11} + \cdots - 70656 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots - 9891773184 \) Copy content Toggle raw display
$7$ \( (T - 7)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots - 23\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 82\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots - 14\!\cdots\!72 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 63\!\cdots\!52 \) Copy content Toggle raw display
$23$ \( (T + 23)^{12} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 52\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots - 33\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots - 28\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 32\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 75\!\cdots\!08 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 36\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 39\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 41\!\cdots\!32 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 60\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 10\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 32\!\cdots\!88 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 13\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 26\!\cdots\!72 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 76\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots - 81\!\cdots\!12 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 22\!\cdots\!52 \) Copy content Toggle raw display
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