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Properties

Label 2400.3.g.d
Level $2400$
Weight $3$
Character orbit 2400.g
Analytic conductor $65.395$
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] = [2400,3,Mod(751,2400)] 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("2400.751"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2400.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,48,0,32,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,96] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(43)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(65.3952634465\)
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: \( x^{16} - 2 x^{15} + 4 x^{14} - 12 x^{13} + 22 x^{12} - 64 x^{11} + 144 x^{10} - 272 x^{9} + 656 x^{8} + \cdots + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{28}\cdot 3 \)
Twist minimal: no (minimal twist has level 600)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{2} q^{7} + 3 q^{9} + ( - \beta_{3} + 2) q^{11} + ( - \beta_{6} + \beta_{2}) q^{13} + (\beta_{10} - 2 \beta_1) q^{17} + (\beta_{15} - \beta_{10} - \beta_{3} + \cdots + 2) q^{19}+ \cdots + ( - 3 \beta_{3} + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 48 q^{9} + 32 q^{11} + 32 q^{19} + 96 q^{43} - 128 q^{49} - 96 q^{51} + 48 q^{57} - 128 q^{59} + 32 q^{67} + 160 q^{73} + 144 q^{81} + 480 q^{91} - 112 q^{97} + 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 4 x^{14} - 12 x^{13} + 22 x^{12} - 64 x^{11} + 144 x^{10} - 272 x^{9} + 656 x^{8} + \cdots + 65536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 17 \nu^{15} - 29 \nu^{14} - 42 \nu^{13} + 40 \nu^{12} - 82 \nu^{11} + 710 \nu^{10} + \cdots + 806912 ) / 897024 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 9 \nu^{15} + 9 \nu^{14} - 298 \nu^{13} + 408 \nu^{12} - 1018 \nu^{11} + 2010 \nu^{10} + \cdots + 544768 ) / 299008 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 20 \nu^{15} - 47 \nu^{14} - 204 \nu^{13} - 56 \nu^{12} + 1604 \nu^{11} - 994 \nu^{10} + \cdots - 950272 ) / 448512 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 23 \nu^{15} + 96 \nu^{14} + 82 \nu^{13} - 904 \nu^{12} + 3206 \nu^{11} - 2796 \nu^{10} + \cdots + 827392 ) / 448512 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{15} - 291 \nu^{14} + 194 \nu^{13} - 2096 \nu^{12} + 406 \nu^{11} - 11262 \nu^{10} + \cdots - 8544256 ) / 897024 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5 \nu^{15} - 297 \nu^{14} - 94 \nu^{13} + 1720 \nu^{12} + 1474 \nu^{11} + 7254 \nu^{10} + \cdots - 335872 ) / 897024 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 69 \nu^{15} - 150 \nu^{14} + 100 \nu^{13} - 84 \nu^{12} - 2062 \nu^{11} + 1248 \nu^{10} + \cdots + 1286144 ) / 598016 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 65 \nu^{15} + 194 \nu^{14} + 724 \nu^{13} - 36 \nu^{12} + 906 \nu^{11} - 7136 \nu^{10} + \cdots - 4882432 ) / 598016 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 23 \nu^{15} + 722 \nu^{14} - 948 \nu^{13} + 6284 \nu^{12} - 4166 \nu^{11} + 17152 \nu^{10} + \cdots + 7192576 ) / 1794048 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 119 \nu^{15} + 162 \nu^{14} - 732 \nu^{13} + 2324 \nu^{12} - 3786 \nu^{11} + 4824 \nu^{10} + \cdots + 2359296 ) / 598016 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 65 \nu^{15} + 65 \nu^{14} - 238 \nu^{13} + 416 \nu^{12} - 474 \nu^{11} + 5562 \nu^{10} + \cdots + 479232 ) / 299008 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 35 \nu^{15} - 958 \nu^{14} + 3204 \nu^{13} - 340 \nu^{12} + 10114 \nu^{11} - 14576 \nu^{10} + \cdots - 7307264 ) / 1794048 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 175 \nu^{15} + 394 \nu^{14} - 652 \nu^{13} + 2580 \nu^{12} - 6330 \nu^{11} + 9640 \nu^{10} + \cdots + 393216 ) / 598016 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 5 \nu^{15} + 506 \nu^{14} - 532 \nu^{13} + 1428 \nu^{12} - 5826 \nu^{11} + 10904 \nu^{10} + \cdots + 262144 ) / 598016 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 911 \nu^{15} + 958 \nu^{14} - 3204 \nu^{13} + 3844 \nu^{12} - 8362 \nu^{11} + 32096 \nu^{10} + \cdots + 10895360 ) / 1794048 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - \beta_{13} + \beta_{9} - \beta_{7} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 2\beta _1 + 2 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} + \beta_{2} - 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{14} + \beta_{13} + 2 \beta_{12} + \beta_{10} - \beta_{9} - 2 \beta_{8} - 2 \beta_{7} - \beta_{6} + \cdots + 10 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{15} + \beta_{14} - \beta_{12} - 2 \beta_{11} + \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{6} + \cdots - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7 \beta_{15} + 2 \beta_{14} + 3 \beta_{13} - 8 \beta_{11} - 6 \beta_{10} - 3 \beta_{9} - 11 \beta_{7} + \cdots + 66 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4 \beta_{15} + 3 \beta_{14} - 4 \beta_{13} + 3 \beta_{12} - 3 \beta_{11} + 8 \beta_{10} - 3 \beta_{9} + \cdots - 28 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 4 \beta_{15} + 23 \beta_{14} - 15 \beta_{13} - 4 \beta_{12} - 6 \beta_{11} + 13 \beta_{10} + \cdots + 18 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4 \beta_{15} - 8 \beta_{14} + 12 \beta_{13} - 3 \beta_{12} + 2 \beta_{11} - 5 \beta_{10} - 3 \beta_{9} + \cdots - 62 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 31 \beta_{15} - 30 \beta_{14} + 7 \beta_{13} - 20 \beta_{11} + 34 \beta_{10} - 39 \beta_{9} - 52 \beta_{8} + \cdots - 382 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 23 \beta_{15} + 51 \beta_{14} - 7 \beta_{13} + 15 \beta_{12} + 37 \beta_{11} - 48 \beta_{10} + \cdots - 118 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 56 \beta_{15} - 79 \beta_{14} + 25 \beta_{13} - 58 \beta_{12} + 44 \beta_{11} - 27 \beta_{10} + \cdots - 1086 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 131 \beta_{15} - 43 \beta_{14} + 102 \beta_{13} - 15 \beta_{12} - 12 \beta_{11} + 142 \beta_{10} + \cdots + 778 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 73 \beta_{15} - 266 \beta_{14} + 319 \beta_{13} + 184 \beta_{12} + 304 \beta_{11} - 290 \beta_{10} + \cdots + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 158 \beta_{15} + 55 \beta_{14} - 122 \beta_{13} - 433 \beta_{12} + 85 \beta_{11} - 320 \beta_{10} + \cdots - 296 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1178 \beta_{15} + 943 \beta_{14} - 509 \beta_{13} - 128 \beta_{12} - 2 \beta_{11} - 431 \beta_{10} + \cdots + 9150 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(1951\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
751.1
−0.935441 − 1.76775i
0.479713 + 1.94162i
−1.74272 + 0.981294i
1.83242 − 0.801397i
1.83242 + 0.801397i
−1.74272 − 0.981294i
0.479713 − 1.94162i
−0.935441 + 1.76775i
1.97080 − 0.340537i
−1.53917 − 1.27709i
−0.169347 + 1.99282i
1.10374 − 1.66786i
1.10374 + 1.66786i
−0.169347 − 1.99282i
−1.53917 + 1.27709i
1.97080 + 0.340537i
0 −1.73205 0 0 0 − 10.9881i 0 3.00000 0
751.2 0 −1.73205 0 0 0 − 8.37149i 0 3.00000 0
751.3 0 −1.73205 0 0 0 − 6.06690i 0 3.00000 0
751.4 0 −1.73205 0 0 0 − 0.610378i 0 3.00000 0
751.5 0 −1.73205 0 0 0 0.610378i 0 3.00000 0
751.6 0 −1.73205 0 0 0 6.06690i 0 3.00000 0
751.7 0 −1.73205 0 0 0 8.37149i 0 3.00000 0
751.8 0 −1.73205 0 0 0 10.9881i 0 3.00000 0
751.9 0 1.73205 0 0 0 − 11.8458i 0 3.00000 0
751.10 0 1.73205 0 0 0 − 8.70188i 0 3.00000 0
751.11 0 1.73205 0 0 0 − 2.71814i 0 3.00000 0
751.12 0 1.73205 0 0 0 − 2.13685i 0 3.00000 0
751.13 0 1.73205 0 0 0 2.13685i 0 3.00000 0
751.14 0 1.73205 0 0 0 2.71814i 0 3.00000 0
751.15 0 1.73205 0 0 0 8.70188i 0 3.00000 0
751.16 0 1.73205 0 0 0 11.8458i 0 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 751.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2400.3.g.d 16
4.b odd 2 1 600.3.g.b ✓ 16
5.b even 2 1 2400.3.g.c 16
5.c odd 4 2 2400.3.p.c 32
8.b even 2 1 600.3.g.b ✓ 16
8.d odd 2 1 inner 2400.3.g.d 16
20.d odd 2 1 600.3.g.c yes 16
20.e even 4 2 600.3.p.c 32
40.e odd 2 1 2400.3.g.c 16
40.f even 2 1 600.3.g.c yes 16
40.i odd 4 2 600.3.p.c 32
40.k even 4 2 2400.3.p.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.3.g.b ✓ 16 4.b odd 2 1
600.3.g.b ✓ 16 8.b even 2 1
600.3.g.c yes 16 20.d odd 2 1
600.3.g.c yes 16 40.f even 2 1
600.3.p.c 32 20.e even 4 2
600.3.p.c 32 40.i odd 4 2
2400.3.g.c 16 5.b even 2 1
2400.3.g.c 16 40.e odd 2 1
2400.3.g.d 16 1.a even 1 1 trivial
2400.3.g.d 16 8.d odd 2 1 inner
2400.3.p.c 32 5.c odd 4 2
2400.3.p.c 32 40.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2400, [\chi])\):

\( T_{7}^{16} + 456 T_{7}^{14} + 80796 T_{7}^{12} + 7020664 T_{7}^{10} + 309600966 T_{7}^{8} + \cdots + 41593807233 \) Copy content Toggle raw display
\( T_{17}^{8} - 1040 T_{17}^{6} + 3248 T_{17}^{5} + 147888 T_{17}^{4} + 183424 T_{17}^{3} - 3135808 T_{17}^{2} + \cdots - 3681728 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 41593807233 \) Copy content Toggle raw display
$11$ \( (T^{8} - 16 T^{7} + \cdots + 4975168)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 14328667961793 \) Copy content Toggle raw display
$17$ \( (T^{8} - 1040 T^{6} + \cdots - 3681728)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 16 T^{7} + \cdots + 67183681)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 96\!\cdots\!88 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 31\!\cdots\!32 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 36\!\cdots\!93 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 91\!\cdots\!72 \) Copy content Toggle raw display
$41$ \( (T^{8} - 6080 T^{6} + \cdots + 82365625408)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 48 T^{7} + \cdots - 256825457759)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 20\!\cdots\!12 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 51\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots - 4317848463104)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 37\!\cdots\!93 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots - 8301889426559)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 33\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 5531610691328)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 26\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 298413965580032)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 11\!\cdots\!08)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 37741159803743)^{2} \) Copy content Toggle raw display
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