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Properties

Label 2415.4.a.p
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: \( x^{19} - 120 x^{17} + 5966 x^{15} - 69 x^{14} - 159452 x^{13} + 5302 x^{12} + 2487727 x^{11} + \cdots - 14364672 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{4} \)
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 \( q + \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + 5) q^{4} - 5 q^{5} + 3 \beta_1 q^{6} - 7 q^{7} + (\beta_{3} + 5 \beta_1) q^{8} + 9 q^{9} - 5 \beta_1 q^{10} + (\beta_{8} - 2) q^{11} + (3 \beta_{2} + 15) q^{12} + ( - \beta_{12} + \beta_{2} + \beta_1 + 3) q^{13} - 7 \beta_1 q^{14} - 15 q^{15} + ( - \beta_{12} + \beta_{11} + \cdots + 23) q^{16}+ \cdots + (9 \beta_{8} - 18) q^{99}+O(q^{100}) \) 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 \( 19 q + 57 q^{3} + 88 q^{4} - 95 q^{5} - 133 q^{7} + 171 q^{9} - 37 q^{11} + 264 q^{12} + 47 q^{13} - 285 q^{15} + 392 q^{16} + 125 q^{17} + 11 q^{19} - 440 q^{20} - 399 q^{21} + 32 q^{22} + 437 q^{23} + 475 q^{25} + 287 q^{26} + 513 q^{27} - 616 q^{28} + 16 q^{29} + 266 q^{31} + 345 q^{32} - 111 q^{33} - 167 q^{34} + 665 q^{35} + 792 q^{36} + 199 q^{37} - 816 q^{38} + 141 q^{39} + 341 q^{41} + 169 q^{43} - 279 q^{44} - 855 q^{45} + 438 q^{47} + 1176 q^{48} + 931 q^{49} + 375 q^{51} + 1806 q^{52} - 32 q^{53} + 185 q^{55} + 33 q^{57} + 566 q^{58} + 209 q^{59} - 1320 q^{60} + 75 q^{61} + 1162 q^{62} - 1197 q^{63} + 2184 q^{64} - 235 q^{65} + 96 q^{66} + 373 q^{67} + 2212 q^{68} + 1311 q^{69} + 944 q^{71} + 1673 q^{73} + 353 q^{74} + 1425 q^{75} + 2707 q^{76} + 259 q^{77} + 861 q^{78} - 844 q^{79} - 1960 q^{80} + 1539 q^{81} + 4888 q^{82} - 487 q^{83} - 1848 q^{84} - 625 q^{85} - 2180 q^{86} + 48 q^{87} + 3319 q^{88} + 252 q^{89} - 329 q^{91} + 2024 q^{92} + 798 q^{93} + 2834 q^{94} - 55 q^{95} + 1035 q^{96} + 4224 q^{97} - 333 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{19} - 120 x^{17} + 5966 x^{15} - 69 x^{14} - 159452 x^{13} + 5302 x^{12} + 2487727 x^{11} + \cdots - 14364672 \) : 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
\(\beta_{4}\)\(=\) \( ( 78\!\cdots\!63 \nu^{18} + \cdots + 18\!\cdots\!20 ) / 16\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 80\!\cdots\!09 \nu^{18} + \cdots - 47\!\cdots\!56 ) / 33\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 83\!\cdots\!63 \nu^{18} + \cdots + 42\!\cdots\!36 ) / 33\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19\!\cdots\!91 \nu^{18} + \cdots + 11\!\cdots\!20 ) / 27\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 34\!\cdots\!91 \nu^{18} + \cdots - 72\!\cdots\!08 ) / 33\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\!\cdots\!23 \nu^{18} + \cdots - 15\!\cdots\!52 ) / 83\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 20\!\cdots\!05 \nu^{18} + \cdots + 13\!\cdots\!64 ) / 16\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13\!\cdots\!69 \nu^{18} + \cdots - 44\!\cdots\!84 ) / 55\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13\!\cdots\!69 \nu^{18} + \cdots - 63\!\cdots\!56 ) / 55\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 29\!\cdots\!43 \nu^{18} + \cdots - 38\!\cdots\!16 ) / 11\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 50\!\cdots\!67 \nu^{18} + \cdots - 42\!\cdots\!40 ) / 16\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 38\!\cdots\!27 \nu^{18} + \cdots - 29\!\cdots\!12 ) / 11\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 29\!\cdots\!81 \nu^{18} + \cdots + 62\!\cdots\!92 ) / 83\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 15\!\cdots\!95 \nu^{18} + \cdots + 35\!\cdots\!96 ) / 41\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 74\!\cdots\!07 \nu^{18} + \cdots + 14\!\cdots\!96 ) / 16\!\cdots\!36 \) 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
\(\nu^{4}\)\(=\) \( -\beta_{12} + \beta_{11} + 29\beta_{2} + \beta _1 + 271 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{18} + 3 \beta_{16} + \beta_{14} + \beta_{12} - 2 \beta_{11} + \beta_{8} - \beta_{7} - \beta_{6} + \cdots + 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{18} - 2 \beta_{17} + 3 \beta_{15} + 4 \beta_{13} - 38 \beta_{12} + 41 \beta_{11} + \beta_{10} + \cdots + 6483 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 89 \beta_{18} + 12 \beta_{17} + 143 \beta_{16} - 11 \beta_{15} + 58 \beta_{14} + 11 \beta_{13} + \cdots + 1188 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 55 \beta_{18} - 107 \beta_{17} - 20 \beta_{16} + 200 \beta_{15} - 32 \beta_{14} + 238 \beta_{13} + \cdots + 165706 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3005 \beta_{18} + 743 \beta_{17} + 5177 \beta_{16} - 744 \beta_{15} + 2354 \beta_{14} + 675 \beta_{13} + \cdots + 47949 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2109 \beta_{18} - 4126 \beta_{17} - 1475 \beta_{16} + 9141 \beta_{15} - 2267 \beta_{14} + \cdots + 4401307 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 93084 \beta_{18} + 31380 \beta_{17} + 170299 \beta_{16} - 34130 \beta_{15} + 83414 \beta_{14} + \cdots + 1646040 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 71366 \beta_{18} - 141361 \beta_{17} - 71983 \beta_{16} + 355351 \beta_{15} - 108132 \beta_{14} + \cdots + 119848960 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2792884 \beta_{18} + 1132686 \beta_{17} + 5358423 \beta_{16} - 1330472 \beta_{15} + 2763164 \beta_{14} + \cdots + 51556046 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 2296682 \beta_{18} - 4595977 \beta_{17} - 2945893 \beta_{16} + 12654867 \beta_{15} + \cdots + 3320176318 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 82773510 \beta_{18} + 37686936 \beta_{17} + 164597564 \beta_{16} - 47529166 \beta_{15} + \cdots + 1517883784 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 72368299 \beta_{18} - 145655223 \beta_{17} - 109682231 \beta_{16} + 427361178 \beta_{15} + \cdots + 93122126058 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 2441745053 \beta_{18} + 1196170430 \beta_{17} + 4984353454 \beta_{16} - 1610247367 \beta_{15} + \cdots + 42597803678 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 2259872173 \beta_{18} - 4554048892 \beta_{17} - 3857535369 \beta_{16} + 13942490575 \beta_{15} + \cdots + 2635635499649 \) 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
−5.45454
−4.86147
−4.50856
−4.37011
−3.40905
−2.77958
−2.43591
−1.12890
−0.875248
−0.0889040
0.759586
1.69278
2.36666
2.60585
3.49480
3.81384
4.50165
5.31326
5.36384
−5.45454 3.00000 21.7520 −5.00000 −16.3636 −7.00000 −75.0107 9.00000 27.2727
1.2 −4.86147 3.00000 15.6339 −5.00000 −14.5844 −7.00000 −37.1122 9.00000 24.3074
1.3 −4.50856 3.00000 12.3271 −5.00000 −13.5257 −7.00000 −19.5089 9.00000 22.5428
1.4 −4.37011 3.00000 11.0978 −5.00000 −13.1103 −7.00000 −13.5379 9.00000 21.8505
1.5 −3.40905 3.00000 3.62164 −5.00000 −10.2272 −7.00000 14.9261 9.00000 17.0453
1.6 −2.77958 3.00000 −0.273928 −5.00000 −8.33874 −7.00000 22.9981 9.00000 13.8979
1.7 −2.43591 3.00000 −2.06633 −5.00000 −7.30773 −7.00000 24.5207 9.00000 12.1796
1.8 −1.12890 3.00000 −6.72558 −5.00000 −3.38670 −7.00000 16.6237 9.00000 5.64450
1.9 −0.875248 3.00000 −7.23394 −5.00000 −2.62574 −7.00000 13.3335 9.00000 4.37624
1.10 −0.0889040 3.00000 −7.99210 −5.00000 −0.266712 −7.00000 1.42176 9.00000 0.444520
1.11 0.759586 3.00000 −7.42303 −5.00000 2.27876 −7.00000 −11.7151 9.00000 −3.79793
1.12 1.69278 3.00000 −5.13448 −5.00000 5.07835 −7.00000 −22.2338 9.00000 −8.46392
1.13 2.36666 3.00000 −2.39890 −5.00000 7.09999 −7.00000 −24.6107 9.00000 −11.8333
1.14 2.60585 3.00000 −1.20955 −5.00000 7.81755 −7.00000 −23.9987 9.00000 −13.0292
1.15 3.49480 3.00000 4.21360 −5.00000 10.4844 −7.00000 −13.2327 9.00000 −17.4740
1.16 3.81384 3.00000 6.54539 −5.00000 11.4415 −7.00000 −5.54766 9.00000 −19.0692
1.17 4.50165 3.00000 12.2648 −5.00000 13.5049 −7.00000 19.1987 9.00000 −22.5082
1.18 5.31326 3.00000 20.2308 −5.00000 15.9398 −7.00000 64.9853 9.00000 −26.5663
1.19 5.36384 3.00000 20.7708 −5.00000 16.0915 −7.00000 68.5005 9.00000 −26.8192
\(n\): 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.p ✓ 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2415.4.a.p ✓ 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2415))\):

\( T_{2}^{19} - 120 T_{2}^{17} + 5966 T_{2}^{15} - 69 T_{2}^{14} - 159452 T_{2}^{13} + 5302 T_{2}^{12} + \cdots - 14364672 \) Copy content Toggle raw display
\( T_{11}^{19} + 37 T_{11}^{18} - 12414 T_{11}^{17} - 310717 T_{11}^{16} + 64702068 T_{11}^{15} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{19} - 120 T^{17} + \cdots - 14364672 \) 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
$11$ \( T^{19} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{19} + \cdots - 38\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{19} + \cdots - 22\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{19} + \cdots - 74\!\cdots\!32 \) Copy content Toggle raw display
$23$ \( (T - 23)^{19} \) Copy content Toggle raw display
$29$ \( T^{19} + \cdots + 66\!\cdots\!20 \) Copy content Toggle raw display
$31$ \( T^{19} + \cdots + 13\!\cdots\!48 \) Copy content Toggle raw display
$37$ \( T^{19} + \cdots - 82\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{19} + \cdots + 34\!\cdots\!20 \) Copy content Toggle raw display
$43$ \( T^{19} + \cdots - 13\!\cdots\!48 \) Copy content Toggle raw display
$47$ \( T^{19} + \cdots - 29\!\cdots\!52 \) Copy content Toggle raw display
$53$ \( T^{19} + \cdots + 10\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{19} + \cdots - 20\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{19} + \cdots - 87\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{19} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{19} + \cdots + 75\!\cdots\!60 \) Copy content Toggle raw display
$73$ \( T^{19} + \cdots + 75\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{19} + \cdots + 34\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( T^{19} + \cdots + 54\!\cdots\!20 \) Copy content Toggle raw display
$89$ \( T^{19} + \cdots - 73\!\cdots\!48 \) Copy content Toggle raw display
$97$ \( T^{19} + \cdots + 67\!\cdots\!04 \) Copy content Toggle raw display
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