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Properties

Label 5915.2.a.bm
Level $5915$
Weight $2$
Character orbit 5915.a
Self dual yes
Analytic conductor $47.232$
Analytic rank $0$
Dimension $18$
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] = [5915,2,Mod(1,5915)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5915, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5915.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5915 = 5 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5915.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: \(47.2315127956\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 30 x^{16} + 371 x^{14} - 2439 x^{12} - 6 x^{11} + 9148 x^{10} + 106 x^{9} - 19446 x^{8} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 455)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{4} q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + \beta_{8} q^{6} - q^{7} + (\beta_{3} + \beta_1) q^{8} + ( - \beta_{6} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{4} q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + \beta_{8} q^{6} - q^{7} + (\beta_{3} + \beta_1) q^{8} + ( - \beta_{6} + 1) q^{9} - \beta_1 q^{10} + (\beta_{16} + \beta_1) q^{11} + (\beta_{17} + \beta_{16} + \cdots + \beta_1) q^{12}+ \cdots + (2 \beta_{17} + \beta_{16} + \beta_{15} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 4 q^{3} + 24 q^{4} - 18 q^{5} - 18 q^{7} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 4 q^{3} + 24 q^{4} - 18 q^{5} - 18 q^{7} + 26 q^{9} - 4 q^{11} + 8 q^{12} - 4 q^{15} + 28 q^{16} + 6 q^{17} + 26 q^{18} - 24 q^{19} - 24 q^{20} - 4 q^{21} + 34 q^{22} + 18 q^{23} + 24 q^{24} + 18 q^{25} + 10 q^{27} - 24 q^{28} + 2 q^{29} - 10 q^{31} + 6 q^{33} - 14 q^{34} + 18 q^{35} + 46 q^{36} + 10 q^{37} - 8 q^{38} - 46 q^{41} + 32 q^{43} - 8 q^{44} - 26 q^{45} + 4 q^{46} - 10 q^{47} + 90 q^{48} + 18 q^{49} - 34 q^{51} + 44 q^{53} + 16 q^{54} + 4 q^{55} + 26 q^{57} + 26 q^{58} - 46 q^{59} - 8 q^{60} + 46 q^{61} + 8 q^{62} - 26 q^{63} - 10 q^{64} + 38 q^{66} + 2 q^{67} - 58 q^{68} + 36 q^{69} + 32 q^{72} - 20 q^{73} + 50 q^{74} + 4 q^{75} - 66 q^{76} + 4 q^{77} + 32 q^{79} - 28 q^{80} + 42 q^{81} - 4 q^{82} - 14 q^{83} - 8 q^{84} - 6 q^{85} + 64 q^{86} - 8 q^{87} + 130 q^{88} - 26 q^{89} - 26 q^{90} + 4 q^{92} - 8 q^{93} + 50 q^{94} + 24 q^{95} + 26 q^{96} + 24 q^{97} - 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 30 x^{16} + 371 x^{14} - 2439 x^{12} - 6 x^{11} + 9148 x^{10} + 106 x^{9} - 19446 x^{8} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 81 \nu^{17} - 1384 \nu^{16} + 3714 \nu^{15} + 34736 \nu^{14} - 58667 \nu^{13} - 343608 \nu^{12} + \cdots - 74816 ) / 23680 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3 \nu^{17} - 3 \nu^{16} - 62 \nu^{15} + 62 \nu^{14} + 401 \nu^{13} - 441 \nu^{12} - 161 \nu^{11} + \cdots - 172 ) / 80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 412 \nu^{17} - 423 \nu^{16} + 12368 \nu^{15} + 11502 \nu^{14} - 151684 \nu^{13} - 126141 \nu^{12} + \cdots - 62172 ) / 11840 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 108 \nu^{17} - 2277 \nu^{16} + 3472 \nu^{15} + 59018 \nu^{14} - 41716 \nu^{13} - 605959 \nu^{12} + \cdots + 24812 ) / 5920 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 346 \nu^{17} + 321 \nu^{16} + 8684 \nu^{15} - 7154 \nu^{14} - 85902 \nu^{13} + 60267 \nu^{12} + \cdots + 324 ) / 5920 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 663 \nu^{17} - 2533 \nu^{16} - 22342 \nu^{15} + 69002 \nu^{14} + 311261 \nu^{13} - 755271 \nu^{12} + \cdots + 9228 ) / 11840 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 326 \nu^{17} - 1051 \nu^{16} - 11604 \nu^{15} + 29334 \nu^{14} + 168402 \nu^{13} - 328697 \nu^{12} + \cdots + 44916 ) / 5920 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1588 \nu^{17} - 3387 \nu^{16} + 41952 \nu^{15} + 90838 \nu^{14} - 436876 \nu^{13} - 967449 \nu^{12} + \cdots + 79572 ) / 11840 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 43 \nu^{17} + 12 \nu^{16} - 1302 \nu^{15} - 248 \nu^{14} + 16201 \nu^{13} + 1604 \nu^{12} + \cdots + 2448 ) / 320 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 3637 \nu^{17} - 2233 \nu^{16} + 98738 \nu^{15} + 62482 \nu^{14} - 1078759 \nu^{13} - 690371 \nu^{12} + \cdots - 83972 ) / 11840 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 7079 \nu^{17} - 3854 \nu^{16} - 214846 \nu^{15} + 110716 \nu^{14} + 2688093 \nu^{13} - 1299258 \nu^{12} + \cdots + 185384 ) / 23680 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 5109 \nu^{17} - 1574 \nu^{16} - 151706 \nu^{15} + 46956 \nu^{14} + 1854423 \nu^{13} - 581058 \nu^{12} + \cdots + 258504 ) / 11840 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 2981 \nu^{17} - 674 \nu^{16} + 86474 \nu^{15} + 15556 \nu^{14} - 1025447 \nu^{13} - 125878 \nu^{12} + \cdots - 68456 ) / 5920 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 6723 \nu^{17} + 2207 \nu^{16} - 194302 \nu^{15} - 52958 \nu^{14} + 2291201 \nu^{13} + 471429 \nu^{12} + \cdots + 253468 ) / 11840 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{17} + \beta_{16} + \beta_{11} - \beta_{8} - \beta_{7} + 2\beta_{4} + 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{17} - \beta_{16} - \beta_{14} - \beta_{13} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{4} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11 \beta_{17} + 11 \beta_{16} - \beta_{14} + \beta_{13} + 2 \beta_{12} + 9 \beta_{11} + \beta_{9} + \cdots + 84 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 14 \beta_{17} - 14 \beta_{16} + \beta_{15} - 13 \beta_{14} - 11 \beta_{13} + 13 \beta_{10} + \cdots - 11 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 96 \beta_{17} + 96 \beta_{16} - 15 \beta_{14} + 17 \beta_{13} + 34 \beta_{12} + 65 \beta_{11} + \cdots + 502 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 143 \beta_{17} - 143 \beta_{16} + 18 \beta_{15} - 127 \beta_{14} - 91 \beta_{13} - 3 \beta_{11} + \cdots - 95 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 773 \beta_{17} + 771 \beta_{16} + \beta_{15} - 160 \beta_{14} + 195 \beta_{13} + 384 \beta_{12} + \cdots + 3133 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1284 \beta_{17} - 1284 \beta_{16} + 213 \beta_{15} - 1109 \beta_{14} - 683 \beta_{13} + 2 \beta_{12} + \cdots - 764 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 5996 \beta_{17} + 5953 \beta_{16} + 20 \beta_{15} - 1482 \beta_{14} + 1892 \beta_{13} + 3662 \beta_{12} + \cdots + 20159 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 10765 \beta_{17} - 10765 \beta_{16} + 2112 \beta_{15} - 9133 \beta_{14} - 4911 \beta_{13} + \cdots - 6018 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 45584 \beta_{17} + 44995 \beta_{16} + 262 \beta_{15} - 12730 \beta_{14} + 16784 \beta_{13} + \cdots + 132693 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 86613 \beta_{17} - 86619 \beta_{16} + 19055 \beta_{15} - 72636 \beta_{14} - 34562 \beta_{13} + \cdots - 47217 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 342439 \beta_{17} + 335872 \beta_{16} + 2861 \beta_{15} - 104490 \beta_{14} + 141097 \beta_{13} + \cdots + 889133 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 678659 \beta_{17} - 678843 \beta_{16} + 162393 \beta_{15} - 564916 \beta_{14} - 240450 \beta_{13} + \cdots - 370468 \) 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
−2.70966
−2.38598
−2.31647
−2.30015
−1.66416
−1.63742
−0.694457
−0.371835
−0.343482
−0.0388534
0.633110
0.823341
1.77348
1.79616
1.86420
2.31214
2.55205
2.70800
−2.70966 1.95707 5.34224 −1.00000 −5.30299 −1.00000 −9.05633 0.830130 2.70966
1.2 −2.38598 −1.47359 3.69290 −1.00000 3.51595 −1.00000 −4.03923 −0.828545 2.38598
1.3 −2.31647 −2.40758 3.36601 −1.00000 5.57708 −1.00000 −3.16432 2.79645 2.31647
1.4 −2.30015 2.22084 3.29070 −1.00000 −5.10826 −1.00000 −2.96881 1.93212 2.30015
1.5 −1.66416 0.676913 0.769433 −1.00000 −1.12649 −1.00000 2.04786 −2.54179 1.66416
1.6 −1.63742 −1.31337 0.681150 −1.00000 2.15054 −1.00000 2.15951 −1.27506 1.63742
1.7 −0.694457 3.28723 −1.51773 −1.00000 −2.28284 −1.00000 2.44291 7.80586 0.694457
1.8 −0.371835 1.06927 −1.86174 −1.00000 −0.397592 −1.00000 1.43593 −1.85666 0.371835
1.9 −0.343482 0.877112 −1.88202 −1.00000 −0.301272 −1.00000 1.33341 −2.23068 0.343482
1.10 −0.0388534 −2.80020 −1.99849 −1.00000 0.108797 −1.00000 0.155355 4.84112 0.0388534
1.11 0.633110 0.430660 −1.59917 −1.00000 0.272655 −1.00000 −2.27867 −2.81453 −0.633110
1.12 0.823341 2.81863 −1.32211 −1.00000 2.32069 −1.00000 −2.73523 4.94467 −0.823341
1.13 1.77348 −0.485839 1.14522 −1.00000 −0.861623 −1.00000 −1.51594 −2.76396 −1.77348
1.14 1.79616 −1.87010 1.22619 −1.00000 −3.35900 −1.00000 −1.38989 0.497279 −1.79616
1.15 1.86420 −3.40797 1.47522 −1.00000 −6.35311 −1.00000 −0.978287 8.61424 −1.86420
1.16 2.31214 2.53963 3.34597 −1.00000 5.87198 −1.00000 3.11207 3.44974 −2.31214
1.17 2.55205 −1.16055 4.51298 −1.00000 −2.96178 −1.00000 6.41325 −1.65313 −2.55205
1.18 2.70800 3.04184 5.33325 −1.00000 8.23728 −1.00000 9.02642 6.25277 −2.70800
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(7\) \(1\)
\(13\) \(-1\)

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 5915.2.a.bm 18
13.b even 2 1 5915.2.a.bn 18
13.f odd 12 2 455.2.bq.b ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
455.2.bq.b ✓ 36 13.f odd 12 2
5915.2.a.bm 18 1.a even 1 1 trivial
5915.2.a.bn 18 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5915))\):

\( T_{2}^{18} - 30 T_{2}^{16} + 371 T_{2}^{14} - 2439 T_{2}^{12} - 6 T_{2}^{11} + 9148 T_{2}^{10} + \cdots + 16 \) Copy content Toggle raw display
\( T_{3}^{18} - 4 T_{3}^{17} - 32 T_{3}^{16} + 138 T_{3}^{15} + 386 T_{3}^{14} - 1872 T_{3}^{13} + \cdots + 3988 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 30 T^{16} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{18} - 4 T^{17} + \cdots + 3988 \) Copy content Toggle raw display
$5$ \( (T + 1)^{18} \) Copy content Toggle raw display
$7$ \( (T + 1)^{18} \) Copy content Toggle raw display
$11$ \( T^{18} + 4 T^{17} + \cdots + 1683600 \) Copy content Toggle raw display
$13$ \( T^{18} \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 1622225488 \) Copy content Toggle raw display
$19$ \( T^{18} + 24 T^{17} + \cdots + 11077888 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 12042059776 \) Copy content Toggle raw display
$29$ \( T^{18} - 2 T^{17} + \cdots + 17229892 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 111913292800 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots - 200787968 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 29934881701888 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots - 1311885324800 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 181104231799712 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 792585216 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 33396618018816 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 8498171392 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots - 93\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 9106844016803 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 66449378674381 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 217298646246784 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 116044368333 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 1633406266368 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 106401896492 \) Copy content Toggle raw display
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