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Properties

Label 465.2.n.d
Level $465$
Weight $2$
Character orbit 465.n
Analytic conductor $3.713$
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] = [465,2,Mod(16,465)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("465.16"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.n (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
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: \( x^{16} - 6 x^{15} + 18 x^{14} - 7 x^{13} + 168 x^{12} - 290 x^{11} + 2849 x^{10} - 4031 x^{9} + \cdots + 259081 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{13} q^{2} - \beta_{5} q^{3} + ( - \beta_{8} - \beta_{4} - \beta_{2}) q^{4} - q^{5} + \beta_{8} q^{6} + ( - \beta_{14} + \beta_{11} - \beta_{10} + \cdots + 3) q^{7} + ( - \beta_{13} + \beta_{11} + \beta_{8} + \cdots - 2) q^{8}+ \cdots + ( - \beta_{15} + \beta_{14} - \beta_{13} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} - 4 q^{3} - 4 q^{4} - 16 q^{5} + 4 q^{6} + 7 q^{7} - 8 q^{8} - 4 q^{9} - 4 q^{10} + 4 q^{11} + 6 q^{12} + 8 q^{13} - q^{14} + 4 q^{15} + 8 q^{16} + 4 q^{17} + 4 q^{18} + 17 q^{19} + 4 q^{20}+ \cdots + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 18 x^{14} - 7 x^{13} + 168 x^{12} - 290 x^{11} + 2849 x^{10} - 4031 x^{9} + \cdots + 259081 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 20\!\cdots\!52 \nu^{15} + \cdots - 17\!\cdots\!19 ) / 41\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 25\!\cdots\!79 \nu^{15} + \cdots - 68\!\cdots\!07 ) / 41\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 28\!\cdots\!09 \nu^{15} + \cdots - 13\!\cdots\!05 ) / 41\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 61\!\cdots\!69 \nu^{15} + \cdots - 36\!\cdots\!98 ) / 41\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 73\!\cdots\!90 \nu^{15} + \cdots + 14\!\cdots\!70 ) / 41\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15\!\cdots\!07 \nu^{15} + \cdots - 44\!\cdots\!18 ) / 82\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 19\!\cdots\!00 \nu^{15} + \cdots - 78\!\cdots\!89 ) / 82\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 22\!\cdots\!27 \nu^{15} + \cdots + 23\!\cdots\!31 ) / 82\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 35\!\cdots\!01 \nu^{15} + \cdots - 81\!\cdots\!40 ) / 82\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 23\!\cdots\!08 \nu^{15} + \cdots - 42\!\cdots\!96 ) / 41\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 47\!\cdots\!38 \nu^{15} + \cdots - 31\!\cdots\!21 ) / 82\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 26\!\cdots\!96 \nu^{15} + \cdots - 51\!\cdots\!29 ) / 41\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 73\!\cdots\!60 \nu^{15} + \cdots + 36\!\cdots\!34 ) / 41\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 15\!\cdots\!41 \nu^{15} + \cdots + 57\!\cdots\!96 ) / 82\!\cdots\!38 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 2\beta_{8} - \beta_{6} + 2\beta_{4} - 5\beta_{3} + 10\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{15} + 5 \beta_{14} - 4 \beta_{13} + 4 \beta_{12} - 9 \beta_{11} + 2 \beta_{10} + 16 \beta_{9} + \cdots - 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 14 \beta_{15} + 14 \beta_{14} - 68 \beta_{13} + 33 \beta_{12} - 99 \beta_{11} + 4 \beta_{10} + \cdots - 174 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 126 \beta_{15} + 68 \beta_{14} - 318 \beta_{13} + 255 \beta_{12} - 342 \beta_{11} + 250 \beta_{9} + \cdots - 749 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 318 \beta_{15} - 1211 \beta_{13} + 620 \beta_{12} - 1181 \beta_{11} - 169 \beta_{10} + 318 \beta_{9} + \cdots - 2663 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1757 \beta_{14} + 1757 \beta_{11} - 1757 \beta_{10} - 3102 \beta_{9} - 5292 \beta_{8} + 1211 \beta_{7} + \cdots + 568 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 8260 \beta_{15} - 13104 \beta_{14} + 27351 \beta_{13} - 14875 \beta_{12} + 38735 \beta_{11} + \cdots + 70049 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 70049 \beta_{15} - 70049 \beta_{14} + 203884 \beta_{13} - 123657 \beta_{12} + 251920 \beta_{11} + \cdots + 504898 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 327638 \beta_{15} - 203884 \beta_{14} + 1024135 \beta_{13} - 572725 \beta_{12} + 1153380 \beta_{11} + \cdots + 2429088 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1024135 \beta_{15} + 3039279 \beta_{13} - 1782212 \beta_{12} + 2745708 \beta_{11} + 640390 \beta_{10} + \cdots + 6722337 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4947940 \beta_{14} - 4947940 \beta_{11} + 4947940 \beta_{10} + 8628765 \beta_{9} + 16537550 \beta_{8} + \cdots - 4018508 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 24524769 \beta_{15} + 39739938 \beta_{14} - 73581116 \beta_{13} + 42633297 \beta_{12} + \cdots - 192884783 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 192884783 \beta_{15} + 192884783 \beta_{14} - 587638044 \beta_{13} + 335342524 \beta_{12} + \cdots - 1443780001 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 951274224 \beta_{15} + 587638044 \beta_{14} - 2868971934 \beta_{13} + 1653785588 \beta_{12} + \cdots - 6766310177 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{11}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
−2.65422 + 1.92841i
3.96324 − 2.87946i
−0.895887 + 0.650900i
2.20490 − 1.60196i
0.507945 − 1.56330i
−0.316962 + 0.975510i
−0.681323 + 2.09690i
0.872306 − 2.68468i
0.507945 + 1.56330i
−0.316962 − 0.975510i
−0.681323 − 2.09690i
0.872306 + 2.68468i
−2.65422 − 1.92841i
3.96324 + 2.87946i
−0.895887 − 0.650900i
2.20490 + 1.60196i
−0.596764 + 0.433574i −0.809017 − 0.587785i −0.449894 + 1.38463i −1.00000 0.737640 0.254538 − 0.783387i −0.787747 − 2.42443i 0.309017 + 0.951057i 0.596764 − 0.433574i
16.2 −0.596764 + 0.433574i −0.809017 − 0.587785i −0.449894 + 1.38463i −1.00000 0.737640 0.664389 − 2.04478i −0.787747 − 2.42443i 0.309017 + 0.951057i 0.596764 − 0.433574i
16.3 1.09676 − 0.796845i −0.809017 − 0.587785i −0.0501062 + 0.154211i −1.00000 −1.35567 −0.726146 + 2.23485i 0.905781 + 2.78771i 0.309017 + 0.951057i −1.09676 + 0.796845i
16.4 1.09676 − 0.796845i −0.809017 − 0.587785i −0.0501062 + 0.154211i −1.00000 −1.35567 0.998202 − 3.07215i 0.905781 + 2.78771i 0.309017 + 0.951057i −1.09676 + 0.796845i
256.1 −0.147481 + 0.453901i 0.309017 + 0.951057i 1.43376 + 1.04169i −1.00000 −0.477260 −1.48348 − 1.07781i −1.45650 + 1.05821i −0.809017 + 0.587785i 0.147481 − 0.453901i
256.2 −0.147481 + 0.453901i 0.309017 + 0.951057i 1.43376 + 1.04169i −1.00000 −0.477260 3.67861 + 2.67266i −1.45650 + 1.05821i −0.809017 + 0.587785i 0.147481 − 0.453901i
256.3 0.647481 − 1.99274i 0.309017 + 0.951057i −1.93376 − 1.40496i −1.00000 2.09529 −3.54727 − 2.57724i −0.661536 + 0.480634i −0.809017 + 0.587785i −0.647481 + 1.99274i
256.4 0.647481 − 1.99274i 0.309017 + 0.951057i −1.93376 − 1.40496i −1.00000 2.09529 3.66116 + 2.65999i −0.661536 + 0.480634i −0.809017 + 0.587785i −0.647481 + 1.99274i
376.1 −0.147481 − 0.453901i 0.309017 − 0.951057i 1.43376 − 1.04169i −1.00000 −0.477260 −1.48348 + 1.07781i −1.45650 − 1.05821i −0.809017 − 0.587785i 0.147481 + 0.453901i
376.2 −0.147481 − 0.453901i 0.309017 − 0.951057i 1.43376 − 1.04169i −1.00000 −0.477260 3.67861 − 2.67266i −1.45650 − 1.05821i −0.809017 − 0.587785i 0.147481 + 0.453901i
376.3 0.647481 + 1.99274i 0.309017 − 0.951057i −1.93376 + 1.40496i −1.00000 2.09529 −3.54727 + 2.57724i −0.661536 − 0.480634i −0.809017 − 0.587785i −0.647481 − 1.99274i
376.4 0.647481 + 1.99274i 0.309017 − 0.951057i −1.93376 + 1.40496i −1.00000 2.09529 3.66116 − 2.65999i −0.661536 − 0.480634i −0.809017 − 0.587785i −0.647481 − 1.99274i
436.1 −0.596764 − 0.433574i −0.809017 + 0.587785i −0.449894 − 1.38463i −1.00000 0.737640 0.254538 + 0.783387i −0.787747 + 2.42443i 0.309017 − 0.951057i 0.596764 + 0.433574i
436.2 −0.596764 − 0.433574i −0.809017 + 0.587785i −0.449894 − 1.38463i −1.00000 0.737640 0.664389 + 2.04478i −0.787747 + 2.42443i 0.309017 − 0.951057i 0.596764 + 0.433574i
436.3 1.09676 + 0.796845i −0.809017 + 0.587785i −0.0501062 − 0.154211i −1.00000 −1.35567 −0.726146 − 2.23485i 0.905781 − 2.78771i 0.309017 − 0.951057i −1.09676 − 0.796845i
436.4 1.09676 + 0.796845i −0.809017 + 0.587785i −0.0501062 − 0.154211i −1.00000 −1.35567 0.998202 + 3.07215i 0.905781 − 2.78771i 0.309017 − 0.951057i −1.09676 − 0.796845i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 465.2.n.d ✓ 16
31.d even 5 1 inner 465.2.n.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.n.d ✓ 16 1.a even 1 1 trivial
465.2.n.d ✓ 16 31.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 2T_{2}^{7} + 5T_{2}^{6} - 2T_{2}^{5} - T_{2}^{4} + 2T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(465, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 2 T^{7} + 5 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$5$ \( (T + 1)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 7 T^{15} + \cdots + 4946176 \) Copy content Toggle raw display
$11$ \( T^{16} - 4 T^{15} + \cdots + 5683456 \) Copy content Toggle raw display
$13$ \( T^{16} - 8 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 1173473536 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 10836601801 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 115971361 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 1111725619456 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 852891037441 \) Copy content Toggle raw display
$37$ \( (T^{8} - 22 T^{7} + \cdots + 2243824)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} - 14 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 118060960000 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 1297152256 \) Copy content Toggle raw display
$53$ \( T^{16} + 240 T^{14} + \cdots + 29691601 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 7920036833536 \) Copy content Toggle raw display
$61$ \( (T^{8} - 15 T^{7} + \cdots - 8136599)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 15 T^{7} + \cdots + 39056)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 9417631641856 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 2054475022336 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 20131203741961 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 5383880261761 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 13956380958976 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 278969886976 \) Copy content Toggle raw display
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