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Properties

Label 304.4.i.h
Level $304$
Weight $4$
Character orbit 304.i
Analytic conductor $17.937$
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] = [304,4,Mod(49,304)] 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("304.49"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(304, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 304.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(17.9365806417\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
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: \( x^{16} + 160 x^{14} - 16 x^{13} + 17994 x^{12} - 1968 x^{11} + 980960 x^{10} - 136456 x^{9} + \cdots + 94780858225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{6} - \beta_{2} - 1) q^{5} + (\beta_{11} + \beta_{9} + \beta_{2} + 1) q^{7} + (\beta_{10} - \beta_{7} + \cdots + 13 \beta_{2}) q^{9} + ( - \beta_{12} + \beta_{6} + \beta_{5} + \cdots - 3) q^{11}+ \cdots + (7 \beta_{15} - 5 \beta_{14} + \cdots + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 5 q^{5} + 8 q^{7} - 104 q^{9} - 46 q^{11} + 21 q^{13} + 45 q^{15} - 75 q^{17} - 241 q^{19} + 80 q^{21} + 251 q^{23} - 275 q^{25} - 48 q^{27} - 69 q^{29} - 124 q^{31} + 245 q^{33} + 240 q^{35} - 224 q^{37}+ \cdots + 682 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 160 x^{14} - 16 x^{13} + 17994 x^{12} - 1968 x^{11} + 980960 x^{10} - 136456 x^{9} + \cdots + 94780858225 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 76\!\cdots\!56 \nu^{15} + \cdots + 50\!\cdots\!20 ) / 25\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 46\!\cdots\!31 \nu^{15} + \cdots - 40\!\cdots\!30 ) / 16\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 29\!\cdots\!44 \nu^{15} + \cdots - 17\!\cdots\!25 ) / 82\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14\!\cdots\!28 \nu^{15} + \cdots + 11\!\cdots\!00 ) / 41\!\cdots\!35 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 80\!\cdots\!06 \nu^{15} + \cdots + 15\!\cdots\!25 ) / 16\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16\!\cdots\!31 \nu^{15} + \cdots - 10\!\cdots\!65 ) / 24\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 19\!\cdots\!82 \nu^{15} + \cdots + 20\!\cdots\!05 ) / 24\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11\!\cdots\!28 \nu^{15} + \cdots - 97\!\cdots\!25 ) / 82\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 23\!\cdots\!02 \nu^{15} + \cdots - 80\!\cdots\!75 ) / 14\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 15\!\cdots\!54 \nu^{15} + \cdots + 17\!\cdots\!50 ) / 82\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 37\!\cdots\!36 \nu^{15} + \cdots + 22\!\cdots\!15 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 13\!\cdots\!02 \nu^{15} + \cdots - 12\!\cdots\!00 ) / 41\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 14\!\cdots\!96 \nu^{15} + \cdots + 26\!\cdots\!05 ) / 24\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 67\!\cdots\!61 \nu^{15} + \cdots - 36\!\cdots\!20 ) / 41\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - \beta_{7} - \beta_{3} + 40\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4 \beta_{14} - 2 \beta_{12} + 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{8} - \beta_{7} + \beta_{6} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{14} + 2 \beta_{13} - 2 \beta_{11} - 88 \beta_{10} - 8 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} + \cdots - 2577 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 62 \beta_{15} - 246 \beta_{14} + 228 \beta_{12} + 78 \beta_{11} - 78 \beta_{10} + 246 \beta_{9} + \cdots + 246 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 258 \beta_{15} + 452 \beta_{14} - 258 \beta_{13} + 260 \beta_{12} + 1048 \beta_{11} + 226 \beta_{10} + \cdots + 197014 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 24488 \beta_{14} + 8446 \beta_{13} - 24488 \beta_{11} - 19049 \beta_{10} - 11642 \beta_{9} + \cdots - 70541 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 26564 \beta_{15} - 21540 \beta_{14} - 10728 \beta_{12} - 85260 \beta_{11} + 645260 \beta_{10} + \cdots + 21540 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 880476 \beta_{15} + 4567000 \beta_{14} - 880476 \beta_{13} - 1995368 \beta_{12} + 823856 \beta_{11} + \cdots + 5748804 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2035876 \beta_{14} + 2548868 \beta_{13} - 2035876 \beta_{11} - 58952561 \beta_{10} + \cdots - 1411514312 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 84034588 \beta_{15} - 207586958 \beta_{14} + 179517098 \beta_{12} + 144789180 \beta_{11} + \cdots + 207586958 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 238259654 \beta_{15} + 390018508 \beta_{14} - 238259654 \beta_{13} - 34685780 \beta_{12} + \cdots + 123634650187 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 18666878050 \beta_{14} + 7734456026 \beta_{13} - 18666878050 \beta_{11} - 17053728632 \beta_{10} + \cdots - 91118425480 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 22050378502 \beta_{15} - 18838784966 \beta_{14} + 6048133844 \beta_{12} - 64918882530 \beta_{11} + \cdots + 18838784966 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 700429606106 \beta_{15} + 3340865737832 \beta_{14} - 700429606106 \beta_{13} - 1431950517550 \beta_{12} + \cdots + 7595456545257 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
4.69226 − 8.12723i
3.32866 − 5.76540i
2.39028 − 4.14008i
0.975263 − 1.68921i
−0.833875 + 1.44431i
−2.60830 + 4.51770i
−3.20276 + 5.54735i
−4.74152 + 8.21255i
4.69226 + 8.12723i
3.32866 + 5.76540i
2.39028 + 4.14008i
0.975263 + 1.68921i
−0.833875 − 1.44431i
−2.60830 − 4.51770i
−3.20276 − 5.54735i
−4.74152 − 8.21255i
0 −4.69226 + 8.12723i 0 8.47088 − 14.6720i 0 7.17469 0 −30.5345 − 52.8874i 0
49.2 0 −3.32866 + 5.76540i 0 −8.70072 + 15.0701i 0 −23.4805 0 −8.65991 − 14.9994i 0
49.3 0 −2.39028 + 4.14008i 0 −3.88147 + 6.72290i 0 24.6839 0 2.07317 + 3.59083i 0
49.4 0 −0.975263 + 1.68921i 0 4.98299 − 8.63079i 0 −13.4762 0 11.5977 + 20.0878i 0
49.5 0 0.833875 − 1.44431i 0 1.92544 − 3.33497i 0 11.8842 0 12.1093 + 20.9739i 0
49.6 0 2.60830 − 4.51770i 0 −1.99502 + 3.45548i 0 −32.7070 0 −0.106416 − 0.184318i 0
49.7 0 3.20276 − 5.54735i 0 −9.34053 + 16.1783i 0 16.3159 0 −7.01538 − 12.1510i 0
49.8 0 4.74152 − 8.21255i 0 6.03843 − 10.4589i 0 13.6051 0 −31.4640 − 54.4972i 0
273.1 0 −4.69226 − 8.12723i 0 8.47088 + 14.6720i 0 7.17469 0 −30.5345 + 52.8874i 0
273.2 0 −3.32866 − 5.76540i 0 −8.70072 − 15.0701i 0 −23.4805 0 −8.65991 + 14.9994i 0
273.3 0 −2.39028 − 4.14008i 0 −3.88147 − 6.72290i 0 24.6839 0 2.07317 − 3.59083i 0
273.4 0 −0.975263 − 1.68921i 0 4.98299 + 8.63079i 0 −13.4762 0 11.5977 − 20.0878i 0
273.5 0 0.833875 + 1.44431i 0 1.92544 + 3.33497i 0 11.8842 0 12.1093 − 20.9739i 0
273.6 0 2.60830 + 4.51770i 0 −1.99502 − 3.45548i 0 −32.7070 0 −0.106416 + 0.184318i 0
273.7 0 3.20276 + 5.54735i 0 −9.34053 − 16.1783i 0 16.3159 0 −7.01538 + 12.1510i 0
273.8 0 4.74152 + 8.21255i 0 6.03843 + 10.4589i 0 13.6051 0 −31.4640 + 54.4972i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.4.i.h 16
4.b odd 2 1 152.4.i.b ✓ 16
19.c even 3 1 inner 304.4.i.h 16
76.g odd 6 1 152.4.i.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.4.i.b ✓ 16 4.b odd 2 1
152.4.i.b ✓ 16 76.g odd 6 1
304.4.i.h 16 1.a even 1 1 trivial
304.4.i.h 16 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 160 T_{3}^{14} + 16 T_{3}^{13} + 17994 T_{3}^{12} + 1968 T_{3}^{11} + 980960 T_{3}^{10} + \cdots + 94780858225 \) acting on \(S_{4}^{\mathrm{new}}(304, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 94780858225 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{8} - 4 T^{7} + \cdots - 4835205120)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 23 T^{7} + \cdots + 281975709376)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 48\!\cdots\!21 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 42\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 22\!\cdots\!80)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 18\!\cdots\!40)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 50\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 32\!\cdots\!49 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 11\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 23\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 16\!\cdots\!48)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 58\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 97\!\cdots\!69 \) Copy content Toggle raw display
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