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Properties

Label 2415.4.a.i
Level $2415$
Weight $4$
Character orbit 2415.a
Self dual yes
Analytic conductor $142.490$
Analytic rank $1$
Dimension $15$
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: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 3 x^{14} - 93 x^{13} + 269 x^{12} + 3403 x^{11} - 9279 x^{10} - 62727 x^{9} + 155291 x^{8} + \cdots + 6322176 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{4}\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 5) q^{4} - 5 q^{5} + 3 \beta_1 q^{6} - 7 q^{7} + ( - \beta_{3} - 4 \beta_1) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 5) q^{4} - 5 q^{5} + 3 \beta_1 q^{6} - 7 q^{7} + ( - \beta_{3} - 4 \beta_1) q^{8} + 9 q^{9} + 5 \beta_1 q^{10} + (\beta_{8} + \beta_{2} - \beta_1 + 7) q^{11} + ( - 3 \beta_{2} - 15) q^{12} + ( - \beta_{9} + \beta_{2} + 2 \beta_1 - 13) q^{13} + 7 \beta_1 q^{14} + 15 q^{15} + (\beta_{4} + 4 \beta_{2} - 3 \beta_1 + 12) q^{16} + (\beta_{6} - \beta_{4} - \beta_{2} + \cdots - 16) q^{17}+ \cdots + (9 \beta_{8} + 9 \beta_{2} - 9 \beta_1 + 63) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 3 q^{2} - 45 q^{3} + 75 q^{4} - 75 q^{5} + 9 q^{6} - 105 q^{7} - 9 q^{8} + 135 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 3 q^{2} - 45 q^{3} + 75 q^{4} - 75 q^{5} + 9 q^{6} - 105 q^{7} - 9 q^{8} + 135 q^{9} + 15 q^{10} + 97 q^{11} - 225 q^{12} - 185 q^{13} + 21 q^{14} + 225 q^{15} + 167 q^{16} - 237 q^{17} - 27 q^{18} + 121 q^{19} - 375 q^{20} + 315 q^{21} + 62 q^{22} + 345 q^{23} + 27 q^{24} + 375 q^{25} - 278 q^{26} - 405 q^{27} - 525 q^{28} + 472 q^{29} - 45 q^{30} - 122 q^{31} + 555 q^{32} - 291 q^{33} + 508 q^{34} + 525 q^{35} + 675 q^{36} - 135 q^{37} + 180 q^{38} + 555 q^{39} + 45 q^{40} + 133 q^{41} - 63 q^{42} - 79 q^{43} + 1638 q^{44} - 675 q^{45} - 69 q^{46} - 1510 q^{47} - 501 q^{48} + 735 q^{49} - 75 q^{50} + 711 q^{51} + 88 q^{52} - 774 q^{53} + 81 q^{54} - 485 q^{55} + 63 q^{56} - 363 q^{57} - 44 q^{58} - 657 q^{59} + 1125 q^{60} + 165 q^{61} - 196 q^{62} - 945 q^{63} + 571 q^{64} + 925 q^{65} - 186 q^{66} + 1253 q^{67} - 3278 q^{68} - 1035 q^{69} - 105 q^{70} + 1594 q^{71} - 81 q^{72} - 2997 q^{73} + 380 q^{74} - 1125 q^{75} + 2096 q^{76} - 679 q^{77} + 834 q^{78} + 2746 q^{79} - 835 q^{80} + 1215 q^{81} + 3258 q^{82} - 3081 q^{83} + 1575 q^{84} + 1185 q^{85} + 2146 q^{86} - 1416 q^{87} - 242 q^{88} + 720 q^{89} + 135 q^{90} + 1295 q^{91} + 1725 q^{92} + 366 q^{93} - 104 q^{94} - 605 q^{95} - 1665 q^{96} + 160 q^{97} - 147 q^{98} + 873 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 3 x^{14} - 93 x^{13} + 269 x^{12} + 3403 x^{11} - 9279 x^{10} - 62727 x^{9} + 155291 x^{8} + \cdots + 6322176 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 20\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 28\nu^{2} + 3\nu + 104 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 29308 \nu^{14} + 323509 \nu^{13} + 769046 \nu^{12} - 32254807 \nu^{11} + 58914696 \nu^{10} + \cdots - 197368815840 ) / 1760456544 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 65549 \nu^{14} + 190277 \nu^{13} + 5032939 \nu^{12} - 17523551 \nu^{11} - 136108113 \nu^{10} + \cdots - 239789850048 ) / 320083008 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 139537 \nu^{14} - 1261579 \nu^{13} - 10571615 \nu^{12} + 113311657 \nu^{11} + \cdots + 1330135043328 ) / 640166016 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1773115 \nu^{14} - 6062542 \nu^{13} - 159044363 \nu^{12} + 508276534 \nu^{11} + \cdots + 2196123847680 ) / 7041826176 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 261886 \nu^{14} - 19861 \nu^{13} - 23957894 \nu^{12} + 1127983 \nu^{11} + 852186090 \nu^{10} + \cdots - 298697587968 ) / 1005975168 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 183949 \nu^{14} - 183913 \nu^{13} - 16749893 \nu^{12} + 14380207 \nu^{11} + 591540435 \nu^{10} + \cdots - 235741056768 ) / 640166016 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 910039 \nu^{14} - 4166762 \nu^{13} - 78532161 \nu^{12} + 365445134 \nu^{11} + \cdots + 2888051461120 ) / 2347275392 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 235273 \nu^{14} + 858711 \nu^{13} + 20010739 \nu^{12} - 75967933 \nu^{11} + \cdots - 678810992896 ) / 335325056 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 458981 \nu^{14} - 219143 \nu^{13} - 41372677 \nu^{12} + 18773441 \nu^{11} + 1441826979 \nu^{10} + \cdots - 236897764992 ) / 640166016 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 6898942 \nu^{14} + 1492165 \nu^{13} + 635311742 \nu^{12} - 87639247 \nu^{11} + \cdots + 15954932297472 ) / 7041826176 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 20\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 28\beta_{2} - 3\beta _1 + 260 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} + \beta_{12} + \beta_{10} - 2\beta_{9} + \beta_{7} + \beta_{4} + 34\beta_{3} - 2\beta_{2} + 459\beta _1 - 39 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{14} + 2 \beta_{12} + 2 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} - 2 \beta_{6} + \cdots + 5965 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 4 \beta_{14} + 50 \beta_{13} + 42 \beta_{12} + 34 \beta_{10} - 112 \beta_{9} + 42 \beta_{7} + \cdots - 1934 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 108 \beta_{14} + 12 \beta_{13} + 140 \beta_{12} + 140 \beta_{11} - 324 \beta_{10} - 120 \beta_{9} + \cdots + 145968 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 240 \beta_{14} + 1845 \beta_{13} + 1297 \beta_{12} + 16 \beta_{11} + 881 \beta_{10} - 4486 \beta_{9} + \cdots - 69891 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 4170 \beta_{14} + 736 \beta_{13} + 6342 \beta_{12} + 6370 \beta_{11} - 12306 \beta_{10} + \cdots + 3706633 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 9812 \beta_{14} + 60830 \beta_{13} + 35650 \beta_{12} + 1064 \beta_{11} + 20986 \beta_{10} + \cdots - 2244238 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 141284 \beta_{14} + 29896 \beta_{13} + 239820 \beta_{12} + 241276 \beta_{11} - 408052 \beta_{10} + \cdots + 96339516 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 342360 \beta_{14} + 1898189 \beta_{13} + 923165 \beta_{12} + 42832 \beta_{11} + 485117 \beta_{10} + \cdots - 67700707 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 4481098 \beta_{14} + 1018048 \beta_{13} + 8250290 \beta_{12} + 8284874 \beta_{11} - 12663510 \beta_{10} + \cdots + 2543213077 \) 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
5.29004
4.62232
4.58447
3.79550
3.01308
2.94565
1.18774
0.679024
−1.59953
−1.78420
−1.98040
−3.59386
−3.84252
−5.03424
−5.28307
−5.29004 −3.00000 19.9845 −5.00000 15.8701 −7.00000 −63.3988 9.00000 26.4502
1.2 −4.62232 −3.00000 13.3658 −5.00000 13.8670 −7.00000 −24.8026 9.00000 23.1116
1.3 −4.58447 −3.00000 13.0173 −5.00000 13.7534 −7.00000 −23.0018 9.00000 22.9223
1.4 −3.79550 −3.00000 6.40585 −5.00000 11.3865 −7.00000 6.05061 9.00000 18.9775
1.5 −3.01308 −3.00000 1.07866 −5.00000 9.03924 −7.00000 20.8546 9.00000 15.0654
1.6 −2.94565 −3.00000 0.676865 −5.00000 8.83696 −7.00000 21.5714 9.00000 14.7283
1.7 −1.18774 −3.00000 −6.58926 −5.00000 3.56323 −7.00000 17.3283 9.00000 5.93872
1.8 −0.679024 −3.00000 −7.53893 −5.00000 2.03707 −7.00000 10.5513 9.00000 3.39512
1.9 1.59953 −3.00000 −5.44149 −5.00000 −4.79860 −7.00000 −21.5001 9.00000 −7.99767
1.10 1.78420 −3.00000 −4.81661 −5.00000 −5.35261 −7.00000 −22.8675 9.00000 −8.92102
1.11 1.98040 −3.00000 −4.07801 −5.00000 −5.94120 −7.00000 −23.9193 9.00000 −9.90200
1.12 3.59386 −3.00000 4.91584 −5.00000 −10.7816 −7.00000 −11.0840 9.00000 −17.9693
1.13 3.84252 −3.00000 6.76498 −5.00000 −11.5276 −7.00000 −4.74561 9.00000 −19.2126
1.14 5.03424 −3.00000 17.3435 −5.00000 −15.1027 −7.00000 47.0375 9.00000 −25.1712
1.15 5.28307 −3.00000 19.9109 −5.00000 −15.8492 −7.00000 62.9260 9.00000 −26.4154
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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.i ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2415.4.a.i ✓ 15 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2415))\):

\( T_{2}^{15} + 3 T_{2}^{14} - 93 T_{2}^{13} - 269 T_{2}^{12} + 3403 T_{2}^{11} + 9279 T_{2}^{10} + \cdots - 6322176 \) Copy content Toggle raw display
\( T_{11}^{15} - 97 T_{11}^{14} - 5884 T_{11}^{13} + 684129 T_{11}^{12} + 13648203 T_{11}^{11} + \cdots + 45\!\cdots\!12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + 3 T^{14} + \cdots - 6322176 \) Copy content Toggle raw display
$3$ \( (T + 3)^{15} \) Copy content Toggle raw display
$5$ \( (T + 5)^{15} \) Copy content Toggle raw display
$7$ \( (T + 7)^{15} \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots + 45\!\cdots\!12 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots + 28\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots - 10\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( (T - 23)^{15} \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots - 28\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots - 21\!\cdots\!28 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 29\!\cdots\!12 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 90\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 34\!\cdots\!28 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots - 80\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 67\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 91\!\cdots\!72 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 22\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 75\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 24\!\cdots\!12 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 46\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 45\!\cdots\!08 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 22\!\cdots\!28 \) Copy content Toggle raw display
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