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Properties

Label 2300.4.a.l
Level $2300$
Weight $4$
Character orbit 2300.a
Self dual yes
Analytic conductor $135.704$
Analytic rank $0$
Dimension $16$
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] = [2300,4,Mod(1,2300)] 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("2300.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2300.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,12,0,0,0,40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(135.704393013\)
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: \( x^{16} - 4 x^{15} - 259 x^{14} + 890 x^{13} + 26158 x^{12} - 73156 x^{11} - 1317747 x^{10} + \cdots + 2184881904 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{9}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 460)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + ( - \beta_{3} + 2) q^{7} + (\beta_{2} - \beta_1 + 7) q^{9} + (\beta_{6} - \beta_1) q^{11} + ( - \beta_{7} - \beta_1 + 7) q^{13} + (\beta_{10} + \beta_{6} - 3 \beta_1 - 1) q^{17}+ \cdots + (8 \beta_{15} - 3 \beta_{14} + \cdots - 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 12 q^{3} + 40 q^{7} + 110 q^{9} - 4 q^{11} + 104 q^{13} - 30 q^{17} - 72 q^{19} - 16 q^{21} - 368 q^{23} + 456 q^{27} + 38 q^{29} + 326 q^{31} + 590 q^{33} + 524 q^{37} + 564 q^{39} - 280 q^{41}+ \cdots - 976 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 259 x^{14} + 890 x^{13} + 26158 x^{12} - 73156 x^{11} - 1317747 x^{10} + \cdots + 2184881904 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 33 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 12\!\cdots\!09 \nu^{15} + \cdots + 53\!\cdots\!68 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 20\!\cdots\!12 \nu^{15} + \cdots - 19\!\cdots\!16 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 38\!\cdots\!37 \nu^{15} + \cdots - 10\!\cdots\!36 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 59\!\cdots\!33 \nu^{15} + \cdots + 13\!\cdots\!04 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 60\!\cdots\!51 \nu^{15} + \cdots + 24\!\cdots\!28 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 29\!\cdots\!21 \nu^{15} + \cdots - 25\!\cdots\!68 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 31\!\cdots\!86 \nu^{15} + \cdots + 21\!\cdots\!72 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 35\!\cdots\!51 \nu^{15} + \cdots + 27\!\cdots\!32 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 29\!\cdots\!52 \nu^{15} + \cdots - 50\!\cdots\!36 ) / 61\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15\!\cdots\!98 \nu^{15} + \cdots - 32\!\cdots\!36 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 27\!\cdots\!58 \nu^{15} + \cdots - 13\!\cdots\!64 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 50\!\cdots\!92 \nu^{15} + \cdots - 55\!\cdots\!84 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 16\!\cdots\!44 \nu^{15} + \cdots - 20\!\cdots\!88 ) / 15\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 33 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} + \beta_{14} - \beta_{12} - \beta_{9} - \beta_{8} - \beta_{5} + 2\beta_{4} + \beta_{2} + 62\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 6 \beta_{15} + 4 \beta_{14} + 3 \beta_{13} - 3 \beta_{12} - 4 \beta_{11} - 2 \beta_{10} - \beta_{9} + \cdots + 1964 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 111 \beta_{15} + 112 \beta_{14} + 4 \beta_{13} - 112 \beta_{12} - 8 \beta_{11} - 32 \beta_{10} + \cdots + 1437 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 727 \beta_{15} + 557 \beta_{14} + 390 \beta_{13} - 491 \beta_{12} - 354 \beta_{11} - 354 \beta_{10} + \cdots + 138493 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 10860 \beta_{15} + 10501 \beta_{14} + 668 \beta_{13} - 10144 \beta_{12} - 677 \beta_{11} + \cdots + 125620 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 71511 \beta_{15} + 57941 \beta_{14} + 39245 \beta_{13} - 57773 \beta_{12} - 25175 \beta_{11} + \cdots + 10431156 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1018717 \beta_{15} + 940707 \beta_{14} + 75016 \beta_{13} - 869137 \beta_{12} - 32748 \beta_{11} + \cdots + 11111248 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 6610448 \beta_{15} + 5489010 \beta_{14} + 3616325 \beta_{13} - 5980053 \beta_{12} - 1670038 \beta_{11} + \cdots + 809954108 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 93259415 \beta_{15} + 82859466 \beta_{14} + 7243978 \beta_{13} - 73342386 \beta_{12} + \cdots + 1007751613 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 596730875 \beta_{15} + 501104279 \beta_{14} + 319297412 \beta_{13} - 580400429 \beta_{12} + \cdots + 63945247107 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 8398600532 \beta_{15} + 7238308533 \beta_{14} + 649294776 \beta_{13} - 6175521816 \beta_{12} + \cdots + 92580855366 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 53306349963 \beta_{15} + 44989492595 \beta_{14} + 27509328647 \beta_{13} - 54275822179 \beta_{12} + \cdots + 5102456495748 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 747668104873 \beta_{15} + 629253342637 \beta_{14} + 55883147430 \beta_{13} - 521215556683 \beta_{12} + \cdots + 8522419693442 \) 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
9.27761
8.81841
7.51398
5.94851
4.67762
2.97415
1.70998
1.29215
−0.277145
−2.51185
−2.93269
−4.50846
−5.05177
−5.52653
−8.40839
−8.99558
0 −8.27761 0 0 0 19.6800 0 41.5188 0
1.2 0 −7.81841 0 0 0 −6.53013 0 34.1276 0
1.3 0 −6.51398 0 0 0 2.24515 0 15.4319 0
1.4 0 −4.94851 0 0 0 15.6910 0 −2.51221 0
1.5 0 −3.67762 0 0 0 −21.0303 0 −13.4751 0
1.6 0 −1.97415 0 0 0 −13.0256 0 −23.1027 0
1.7 0 −0.709976 0 0 0 33.8714 0 −26.4959 0
1.8 0 −0.292148 0 0 0 −7.10909 0 −26.9146 0
1.9 0 1.27714 0 0 0 16.7357 0 −25.3689 0
1.10 0 3.51185 0 0 0 −1.79731 0 −14.6669 0
1.11 0 3.93269 0 0 0 −29.6587 0 −11.5340 0
1.12 0 5.50846 0 0 0 16.7469 0 3.34309 0
1.13 0 6.05177 0 0 0 −20.6013 0 9.62397 0
1.14 0 6.52653 0 0 0 34.5907 0 15.5956 0
1.15 0 9.40839 0 0 0 −22.7593 0 61.5178 0
1.16 0 9.99558 0 0 0 22.9508 0 72.9116 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( -1 \)
\(23\) \( +1 \)

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 2300.4.a.l 16
5.b even 2 1 2300.4.a.k 16
5.c odd 4 2 460.4.c.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.4.c.a ✓ 32 5.c odd 4 2
2300.4.a.k 16 5.b even 2 1
2300.4.a.l 16 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 12 T_{3}^{15} - 199 T_{3}^{14} + 2596 T_{3}^{13} + 14159 T_{3}^{12} - 214792 T_{3}^{11} + \cdots + 1133764000 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2300))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 1133764000 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 16\!\cdots\!52 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots - 22\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 23\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( (T + 23)^{16} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 67\!\cdots\!60 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 45\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots - 20\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 82\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 42\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots - 71\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots - 48\!\cdots\!60 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots - 41\!\cdots\!40 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots - 26\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 57\!\cdots\!20 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots - 28\!\cdots\!00 \) Copy content Toggle raw display
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