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Properties

Label 2415.4.a.f
Level $2415$
Weight $4$
Character orbit 2415.a
Self dual yes
Analytic conductor $142.490$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2415,4,Mod(1,2415)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2415, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2415.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2415.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(142.489612664\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 52 x^{10} + 192 x^{9} + 1000 x^{8} - 3270 x^{7} - 8684 x^{6} + 23388 x^{5} + \cdots - 8000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + \beta_1 + 2) q^{4} + 5 q^{5} - 3 \beta_1 q^{6} + 7 q^{7} + ( - \beta_{3} - \beta_{2} - 4) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + \beta_1 + 2) q^{4} + 5 q^{5} - 3 \beta_1 q^{6} + 7 q^{7} + ( - \beta_{3} - \beta_{2} - 4) q^{8} + 9 q^{9} - 5 \beta_1 q^{10} + ( - \beta_{11} + \beta_{8} + \beta_{6} + \cdots - 9) q^{11}+ \cdots + ( - 9 \beta_{11} + 9 \beta_{8} + \cdots - 81) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{2} + 36 q^{3} + 24 q^{4} + 60 q^{5} - 12 q^{6} + 84 q^{7} - 48 q^{8} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{2} + 36 q^{3} + 24 q^{4} + 60 q^{5} - 12 q^{6} + 84 q^{7} - 48 q^{8} + 108 q^{9} - 20 q^{10} - 101 q^{11} + 72 q^{12} - 151 q^{13} - 28 q^{14} + 180 q^{15} - 192 q^{16} - 213 q^{17} - 36 q^{18} - 271 q^{19} + 120 q^{20} + 252 q^{21} - 164 q^{22} - 276 q^{23} - 144 q^{24} + 300 q^{25} - 50 q^{26} + 324 q^{27} + 168 q^{28} - 330 q^{29} - 60 q^{30} - 330 q^{31} + 2 q^{32} - 303 q^{33} - 358 q^{34} + 420 q^{35} + 216 q^{36} - 365 q^{37} - 354 q^{38} - 453 q^{39} - 240 q^{40} - 557 q^{41} - 84 q^{42} - 1231 q^{43} - 664 q^{44} + 540 q^{45} + 92 q^{46} - 806 q^{47} - 576 q^{48} + 588 q^{49} - 100 q^{50} - 639 q^{51} - 120 q^{52} - 678 q^{53} - 108 q^{54} - 505 q^{55} - 336 q^{56} - 813 q^{57} - 1002 q^{58} - 393 q^{59} + 360 q^{60} - 2239 q^{61} + 58 q^{62} + 756 q^{63} - 2088 q^{64} - 755 q^{65} - 492 q^{66} - 1231 q^{67} - 1996 q^{68} - 828 q^{69} - 140 q^{70} - 1836 q^{71} - 432 q^{72} - 1965 q^{73} - 1188 q^{74} + 900 q^{75} - 640 q^{76} - 707 q^{77} - 150 q^{78} - 1724 q^{79} - 960 q^{80} + 972 q^{81} - 682 q^{82} - 2033 q^{83} + 504 q^{84} - 1065 q^{85} - 426 q^{86} - 990 q^{87} - 1370 q^{88} - 1674 q^{89} - 180 q^{90} - 1057 q^{91} - 552 q^{92} - 990 q^{93} - 1948 q^{94} - 1355 q^{95} + 6 q^{96} - 2058 q^{97} - 196 q^{98} - 909 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 52 x^{10} + 192 x^{9} + 1000 x^{8} - 3270 x^{7} - 8684 x^{6} + 23388 x^{5} + \cdots - 8000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 15\nu + 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} + \nu^{10} - 67 \nu^{9} - 83 \nu^{8} + 1605 \nu^{7} + 2335 \nu^{6} - 15989 \nu^{5} + \cdots - 6096 \nu ) / 1280 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{11} - \nu^{10} + 87 \nu^{9} + 3 \nu^{8} - 2705 \nu^{7} + 1345 \nu^{6} + 37689 \nu^{5} + \cdots - 180800 ) / 1280 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2 \nu^{11} + 13 \nu^{10} + 94 \nu^{9} - 639 \nu^{8} - 1610 \nu^{7} + 11085 \nu^{6} + 12808 \nu^{5} + \cdots - 55680 ) / 640 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} - 6 \nu^{10} - 43 \nu^{9} + 282 \nu^{8} + 597 \nu^{7} - 4612 \nu^{6} - 2515 \nu^{5} + \cdots + 3584 ) / 128 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{11} + 4 \nu^{10} + 51 \nu^{9} - 188 \nu^{8} - 949 \nu^{7} + 3082 \nu^{6} + 7799 \nu^{5} + \cdots - 17152 ) / 128 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7 \nu^{11} - 38 \nu^{10} - 299 \nu^{9} + 1754 \nu^{8} + 3965 \nu^{7} - 27880 \nu^{6} - 11923 \nu^{5} + \cdots - 16800 ) / 640 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7 \nu^{11} - 23 \nu^{10} - 379 \nu^{9} + 1069 \nu^{8} + 7725 \nu^{7} - 17485 \nu^{6} - 72493 \nu^{5} + \cdots + 170080 ) / 640 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7 \nu^{11} + 23 \nu^{10} + 389 \nu^{9} - 1109 \nu^{8} - 8115 \nu^{7} + 18845 \nu^{6} + \cdots - 200320 ) / 640 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 16\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 2\beta_{3} + 21\beta_{2} + 25\beta _1 + 158 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{11} + 2\beta_{9} + 6\beta_{8} - 2\beta_{5} + 2\beta_{4} + 28\beta_{3} + 36\beta_{2} + 291\beta _1 + 140 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 4 \beta_{11} + 4 \beta_{10} + 34 \beta_{9} + 42 \beta_{8} - 36 \beta_{7} - 36 \beta_{6} - 22 \beta_{5} + \cdots + 2820 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 76 \beta_{11} + 16 \beta_{10} + 92 \beta_{9} + 252 \beta_{8} - 30 \beta_{7} - 30 \beta_{6} + \cdots + 3922 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 224 \beta_{11} + 168 \beta_{10} + 937 \beta_{9} + 1361 \beta_{8} - 983 \beta_{7} - 967 \beta_{6} + \cdots + 53364 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2170 \beta_{11} + 816 \beta_{10} + 3072 \beta_{9} + 7756 \beta_{8} - 1648 \beta_{7} - 1584 \beta_{6} + \cdots + 100766 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 8056 \beta_{11} + 5284 \beta_{10} + 24156 \beta_{9} + 39044 \beta_{8} - 24784 \beta_{7} + \cdots + 1048802 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 56584 \beta_{11} + 28312 \beta_{10} + 89784 \beta_{9} + 212392 \beta_{8} - 60428 \beta_{7} + \cdots + 2475944 \) 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.

comment: embeddings in the coefficient field
 
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
4.83461
4.34570
4.04936
2.52410
2.34312
0.358604
0.290091
−1.66838
−1.72738
−3.17690
−4.01021
−4.16272
−4.83461 3.00000 15.3735 5.00000 −14.5038 7.00000 −35.6479 9.00000 −24.1731
1.2 −4.34570 3.00000 10.8851 5.00000 −13.0371 7.00000 −12.5377 9.00000 −21.7285
1.3 −4.04936 3.00000 8.39735 5.00000 −12.1481 7.00000 −1.60900 9.00000 −20.2468
1.4 −2.52410 3.00000 −1.62890 5.00000 −7.57231 7.00000 24.3043 9.00000 −12.6205
1.5 −2.34312 3.00000 −2.50979 5.00000 −7.02936 7.00000 24.6257 9.00000 −11.7156
1.6 −0.358604 3.00000 −7.87140 5.00000 −1.07581 7.00000 5.69154 9.00000 −1.79302
1.7 −0.290091 3.00000 −7.91585 5.00000 −0.870273 7.00000 4.61704 9.00000 −1.45045
1.8 1.66838 3.00000 −5.21651 5.00000 5.00514 7.00000 −22.0502 9.00000 8.34190
1.9 1.72738 3.00000 −5.01614 5.00000 5.18215 7.00000 −22.4839 9.00000 8.63692
1.10 3.17690 3.00000 2.09268 5.00000 9.53070 7.00000 −18.7669 9.00000 15.8845
1.11 4.01021 3.00000 8.08179 5.00000 12.0306 7.00000 0.328002 9.00000 20.0511
1.12 4.16272 3.00000 9.32821 5.00000 12.4881 7.00000 5.52896 9.00000 20.8136
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(-1\)
\(23\) \(1\)

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 2415.4.a.f ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2415.4.a.f ✓ 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2415))\):

\( T_{2}^{12} + 4 T_{2}^{11} - 52 T_{2}^{10} - 192 T_{2}^{9} + 1000 T_{2}^{8} + 3270 T_{2}^{7} + \cdots - 8000 \) Copy content Toggle raw display
\( T_{11}^{12} + 101 T_{11}^{11} - 2461 T_{11}^{10} - 481708 T_{11}^{9} - 3202204 T_{11}^{8} + \cdots - 64\!\cdots\!44 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 4 T^{11} + \cdots - 8000 \) Copy content Toggle raw display
$3$ \( (T - 3)^{12} \) Copy content Toggle raw display
$5$ \( (T - 5)^{12} \) Copy content Toggle raw display
$7$ \( (T - 7)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots - 64\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 37\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 54\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 22\!\cdots\!68 \) Copy content Toggle raw display
$23$ \( (T + 23)^{12} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 26\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots - 40\!\cdots\!88 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots - 30\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots - 11\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 90\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 95\!\cdots\!08 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 48\!\cdots\!60 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 65\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 18\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 43\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 33\!\cdots\!40 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots - 46\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 68\!\cdots\!00 \) Copy content Toggle raw display
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