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Properties

Label 2415.4.a.q
Level $2415$
Weight $4$
Character orbit 2415.a
Self dual yes
Analytic conductor $142.490$
Analytic rank $0$
Dimension $19$
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: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 120 x^{17} + 5938 x^{15} - 5 x^{14} - 157040 x^{13} + 1378 x^{12} + 2407387 x^{11} + \cdots + 1059840 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{12}\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + 5) q^{4} - 5 q^{5} - 3 \beta_1 q^{6} + 7 q^{7} + ( - \beta_{3} - 5 \beta_1) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + 5) q^{4} - 5 q^{5} - 3 \beta_1 q^{6} + 7 q^{7} + ( - \beta_{3} - 5 \beta_1) q^{8} + 9 q^{9} + 5 \beta_1 q^{10} + ( - \beta_{9} - \beta_1 + 2) q^{11} + (3 \beta_{2} + 15) q^{12} + (\beta_{13} + \beta_{2} - \beta_1 + 8) q^{13} - 7 \beta_1 q^{14} - 15 q^{15} + (\beta_{4} + 5 \beta_{2} + 28) q^{16} + (\beta_{7} - \beta_{3} - 4 \beta_1 + 3) q^{17} - 9 \beta_1 q^{18} + ( - \beta_{8} + \beta_{2} - 3 \beta_1 + 14) q^{19} + ( - 5 \beta_{2} - 25) q^{20} + 21 q^{21} + ( - \beta_{17} - \beta_{10} + \cdots + 12) q^{22}+ \cdots + ( - 9 \beta_{9} - 9 \beta_1 + 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q + 57 q^{3} + 88 q^{4} - 95 q^{5} + 133 q^{7} + 171 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q + 57 q^{3} + 88 q^{4} - 95 q^{5} + 133 q^{7} + 171 q^{9} + 31 q^{11} + 264 q^{12} + 139 q^{13} - 285 q^{15} + 504 q^{16} + 55 q^{17} + 257 q^{19} - 440 q^{20} + 399 q^{21} + 218 q^{22} - 437 q^{23} + 475 q^{25} + 275 q^{26} + 513 q^{27} + 616 q^{28} - 84 q^{29} + 164 q^{31} - 25 q^{32} + 93 q^{33} + 1005 q^{34} - 665 q^{35} + 792 q^{36} + 775 q^{37} + 760 q^{38} + 417 q^{39} + 321 q^{41} + 1265 q^{43} + 351 q^{44} - 855 q^{45} - 542 q^{47} + 1512 q^{48} + 931 q^{49} + 165 q^{51} + 2696 q^{52} + 644 q^{53} - 155 q^{55} + 771 q^{57} - 6 q^{58} + 789 q^{59} - 1320 q^{60} + 977 q^{61} + 674 q^{62} + 1197 q^{63} + 3392 q^{64} - 695 q^{65} + 654 q^{66} + 4025 q^{67} + 62 q^{68} - 1311 q^{69} + 402 q^{71} + 789 q^{73} - 1495 q^{74} + 1425 q^{75} + 2941 q^{76} + 217 q^{77} + 825 q^{78} + 2850 q^{79} - 2520 q^{80} + 1539 q^{81} + 5062 q^{82} - 615 q^{83} + 1848 q^{84} - 275 q^{85} + 1246 q^{86} - 252 q^{87} + 563 q^{88} - 6 q^{89} + 973 q^{91} - 2024 q^{92} + 492 q^{93} + 2754 q^{94} - 1285 q^{95} - 75 q^{96} + 4050 q^{97} + 279 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{19} - 120 x^{17} + 5938 x^{15} - 5 x^{14} - 157040 x^{13} + 1378 x^{12} + 2407387 x^{11} + \cdots + 1059840 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 21\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 29\nu^{2} + 101 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 22\!\cdots\!83 \nu^{18} + \cdots - 52\!\cdots\!16 ) / 46\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17\!\cdots\!37 \nu^{18} + \cdots + 67\!\cdots\!04 ) / 77\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 62\!\cdots\!63 \nu^{18} + \cdots + 20\!\cdots\!00 ) / 23\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 39\!\cdots\!59 \nu^{18} + \cdots - 13\!\cdots\!36 ) / 11\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 19\!\cdots\!95 \nu^{18} + \cdots - 19\!\cdots\!04 ) / 46\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 31\!\cdots\!63 \nu^{18} + \cdots - 53\!\cdots\!28 ) / 58\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 15\!\cdots\!41 \nu^{18} + \cdots - 41\!\cdots\!32 ) / 23\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 18\!\cdots\!09 \nu^{18} + \cdots - 10\!\cdots\!72 ) / 23\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 42\!\cdots\!29 \nu^{18} + \cdots + 31\!\cdots\!84 ) / 46\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 27\!\cdots\!99 \nu^{18} + \cdots + 15\!\cdots\!76 ) / 23\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 63\!\cdots\!41 \nu^{18} + \cdots + 77\!\cdots\!36 ) / 46\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 89\!\cdots\!25 \nu^{18} + \cdots + 44\!\cdots\!44 ) / 46\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 96\!\cdots\!45 \nu^{18} + \cdots - 38\!\cdots\!48 ) / 46\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 10\!\cdots\!39 \nu^{18} + \cdots + 30\!\cdots\!48 ) / 46\!\cdots\!72 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + 21\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 29\beta_{2} + 276 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{18} + \beta_{17} - \beta_{16} + \beta_{13} - \beta_{12} + \beta_{10} + 2 \beta_{9} + \cdots + 511 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{18} - 4 \beta_{17} + 3 \beta_{16} - 2 \beta_{13} - 2 \beta_{11} - 2 \beta_{10} - 5 \beta_{9} + \cdots + 6750 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 47 \beta_{18} + 44 \beta_{17} - 46 \beta_{16} + 3 \beta_{15} + 44 \beta_{13} - 40 \beta_{12} + \cdots - 357 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 54 \beta_{18} - 252 \beta_{17} + 195 \beta_{16} + 12 \beta_{15} - 6 \beta_{14} - 116 \beta_{13} + \cdots + 176294 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1686 \beta_{18} + 1418 \beta_{17} - 1576 \beta_{16} + 169 \beta_{15} + 16 \beta_{14} + 1468 \beta_{13} + \cdots - 23316 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2148 \beta_{18} - 10949 \beta_{17} + 8700 \beta_{16} + 822 \beta_{15} - 434 \beta_{14} - 4847 \beta_{13} + \cdots + 4759724 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 55480 \beta_{18} + 40820 \beta_{17} - 48651 \beta_{16} + 6623 \beta_{15} + 1616 \beta_{14} + \cdots - 1061702 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 76895 \beta_{18} - 408542 \beta_{17} + 331483 \beta_{16} + 37799 \beta_{15} - 21078 \beta_{14} + \cdots + 130846159 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1758742 \beta_{18} + 1112236 \beta_{17} - 1428461 \beta_{16} + 222689 \beta_{15} + 98122 \beta_{14} + \cdots - 41679394 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 2624699 \beta_{18} - 14077994 \beta_{17} + 11611885 \beta_{16} + 1468409 \beta_{15} - 871186 \beta_{14} + \cdots + 3636201774 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 54652525 \beta_{18} + 29327965 \beta_{17} - 40752464 \beta_{16} + 6863951 \beta_{15} + \cdots - 1509310266 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 87285036 \beta_{18} - 462792618 \beta_{17} + 386811770 \beta_{16} + 52132499 \beta_{15} + \cdots + 101784988632 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 1677430525 \beta_{18} + 755572308 \beta_{17} - 1141560747 \beta_{16} + 199428920 \beta_{15} + \cdots - 51944936203 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 2854425073 \beta_{18} - 14761125802 \beta_{17} + 12474846148 \beta_{16} + 1753658297 \beta_{15} + \cdots + 2864345007756 \) 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
5.38042
5.15150
4.87655
3.51660
3.49548
3.21835
1.77496
1.14448
0.911725
0.0339312
−0.168108
−1.56798
−2.49062
−3.05045
−3.36855
−3.54519
−4.57673
−5.31810
−5.41827
−5.38042 3.00000 20.9489 −5.00000 −16.1413 7.00000 −69.6708 9.00000 26.9021
1.2 −5.15150 3.00000 18.5379 −5.00000 −15.4545 7.00000 −54.2860 9.00000 25.7575
1.3 −4.87655 3.00000 15.7807 −5.00000 −14.6296 7.00000 −37.9431 9.00000 24.3827
1.4 −3.51660 3.00000 4.36650 −5.00000 −10.5498 7.00000 12.7776 9.00000 17.5830
1.5 −3.49548 3.00000 4.21838 −5.00000 −10.4864 7.00000 13.2186 9.00000 17.4774
1.6 −3.21835 3.00000 2.35780 −5.00000 −9.65506 7.00000 18.1586 9.00000 16.0918
1.7 −1.77496 3.00000 −4.84953 −5.00000 −5.32487 7.00000 22.8074 9.00000 8.87479
1.8 −1.14448 3.00000 −6.69017 −5.00000 −3.43344 7.00000 16.8126 9.00000 5.72240
1.9 −0.911725 3.00000 −7.16876 −5.00000 −2.73518 7.00000 13.8297 9.00000 4.55863
1.10 −0.0339312 3.00000 −7.99885 −5.00000 −0.101794 7.00000 0.542861 9.00000 0.169656
1.11 0.168108 3.00000 −7.97174 −5.00000 0.504323 7.00000 −2.68497 9.00000 −0.840539
1.12 1.56798 3.00000 −5.54144 −5.00000 4.70393 7.00000 −21.2327 9.00000 −7.83989
1.13 2.49062 3.00000 −1.79680 −5.00000 7.47187 7.00000 −24.4001 9.00000 −12.4531
1.14 3.05045 3.00000 1.30524 −5.00000 9.15135 7.00000 −20.4220 9.00000 −15.2522
1.15 3.36855 3.00000 3.34714 −5.00000 10.1057 7.00000 −15.6734 9.00000 −16.8428
1.16 3.54519 3.00000 4.56840 −5.00000 10.6356 7.00000 −12.1657 9.00000 −17.7260
1.17 4.57673 3.00000 12.9464 −5.00000 13.7302 7.00000 22.6385 9.00000 −22.8836
1.18 5.31810 3.00000 20.2822 −5.00000 15.9543 7.00000 65.3178 9.00000 −26.5905
1.19 5.41827 3.00000 21.3576 −5.00000 16.2548 7.00000 72.3752 9.00000 −27.0913
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
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.q ✓ 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2415.4.a.q ✓ 19 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2415))\):

\( T_{2}^{19} - 120 T_{2}^{17} + 5938 T_{2}^{15} + 5 T_{2}^{14} - 157040 T_{2}^{13} - 1378 T_{2}^{12} + \cdots - 1059840 \) Copy content Toggle raw display
\( T_{11}^{19} - 31 T_{11}^{18} - 17146 T_{11}^{17} + 464519 T_{11}^{16} + 119109364 T_{11}^{15} + \cdots - 22\!\cdots\!92 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{19} - 120 T^{17} + \cdots - 1059840 \) Copy content Toggle raw display
$3$ \( (T - 3)^{19} \) Copy content Toggle raw display
$5$ \( (T + 5)^{19} \) Copy content Toggle raw display
$7$ \( (T - 7)^{19} \) Copy content Toggle raw display
$11$ \( T^{19} + \cdots - 22\!\cdots\!92 \) Copy content Toggle raw display
$13$ \( T^{19} + \cdots + 88\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{19} + \cdots + 79\!\cdots\!28 \) Copy content Toggle raw display
$19$ \( T^{19} + \cdots + 81\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( (T + 23)^{19} \) Copy content Toggle raw display
$29$ \( T^{19} + \cdots + 33\!\cdots\!48 \) Copy content Toggle raw display
$31$ \( T^{19} + \cdots - 24\!\cdots\!72 \) Copy content Toggle raw display
$37$ \( T^{19} + \cdots - 37\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( T^{19} + \cdots - 52\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{19} + \cdots - 31\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{19} + \cdots - 19\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{19} + \cdots + 12\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( T^{19} + \cdots + 44\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{19} + \cdots + 25\!\cdots\!20 \) Copy content Toggle raw display
$67$ \( T^{19} + \cdots - 83\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{19} + \cdots - 31\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{19} + \cdots - 53\!\cdots\!88 \) Copy content Toggle raw display
$79$ \( T^{19} + \cdots - 61\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{19} + \cdots - 36\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{19} + \cdots - 25\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{19} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
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