↑

Properties

Label 1503.4.a.a
Level $1503$
Weight $4$
Character orbit 1503.a
Self dual yes
Analytic conductor $88.680$
Analytic rank $1$
Dimension $15$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1503,4,Mod(1,1503)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1503, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1503.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1503 = 3^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1503.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: \(88.6798707386\)
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: \( x^{15} - 7 x^{14} - 56 x^{13} + 441 x^{12} + 1120 x^{11} - 10727 x^{10} - 9040 x^{9} + 127615 x^{8} + \cdots + 575872 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 167)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 2) q^{4} + (\beta_{11} + 2) q^{5} + (\beta_{12} - \beta_{6} + \beta_{5} + \cdots - 5) q^{7}+ \cdots + (\beta_{12} - \beta_{11} - \beta_{9} + \cdots + 4) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 2) q^{4} + (\beta_{11} + 2) q^{5} + (\beta_{12} - \beta_{6} + \beta_{5} + \cdots - 5) q^{7}+ \cdots + ( - 20 \beta_{14} + 14 \beta_{13} + \cdots - 140) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 7 q^{2} + 41 q^{4} + 26 q^{5} - 85 q^{7} + 84 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 7 q^{2} + 41 q^{4} + 26 q^{5} - 85 q^{7} + 84 q^{8} - 63 q^{10} + 39 q^{11} - 142 q^{13} + 69 q^{14} - 83 q^{16} + 154 q^{17} - 215 q^{19} + 161 q^{20} - 564 q^{22} + 150 q^{23} - 517 q^{25} + 168 q^{26} - 770 q^{28} + 461 q^{29} - 449 q^{31} + 589 q^{32} - 507 q^{34} + 212 q^{35} - 1454 q^{37} + 164 q^{38} - 744 q^{40} + 288 q^{41} - 584 q^{43} - 1278 q^{44} + 597 q^{46} - 151 q^{47} - 392 q^{49} - 1926 q^{50} + 227 q^{52} - 278 q^{53} + 382 q^{55} - 2957 q^{56} + 457 q^{58} - 250 q^{59} + 79 q^{61} - 2994 q^{62} + 472 q^{64} + 198 q^{65} - 1906 q^{67} - 2390 q^{68} + 1645 q^{70} - 936 q^{71} - 2500 q^{73} - 3251 q^{74} + 1780 q^{76} + 560 q^{77} - 764 q^{79} - 2639 q^{80} - 583 q^{82} - 2436 q^{83} - 2278 q^{85} - 4701 q^{86} - 2404 q^{88} + 1411 q^{89} - 334 q^{91} + 190 q^{92} + 505 q^{94} - 1360 q^{95} - 6311 q^{97} - 2245 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 7 x^{14} - 56 x^{13} + 441 x^{12} + 1120 x^{11} - 10727 x^{10} - 9040 x^{9} + 127615 x^{8} + \cdots + 575872 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 39283701 \nu^{14} - 4738227001 \nu^{13} + 23125327254 \nu^{12} + 266599219857 \nu^{11} + \cdots + 249185091598656 ) / 6049542168320 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 149144529 \nu^{14} + 7964983940 \nu^{13} - 34416904320 \nu^{12} - 435099249569 \nu^{11} + \cdots - 509868347208832 ) / 12099084336640 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 248454063 \nu^{14} + 538331830 \nu^{13} + 18505418944 \nu^{12} - 30135438315 \nu^{11} + \cdots - 103009451546112 ) / 12099084336640 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 277870935 \nu^{14} - 1688445820 \nu^{13} - 14934937035 \nu^{12} + 95230947064 \nu^{11} + \cdots + 26414390591808 ) / 3024771084160 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 581989964 \nu^{14} - 1072541385 \nu^{13} - 47405143536 \nu^{12} + 72195208978 \nu^{11} + \cdots - 209627279063488 ) / 6049542168320 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1166138805 \nu^{14} - 5871438452 \nu^{13} - 71553290316 \nu^{12} + 341776412377 \nu^{11} + \cdots - 76783619381376 ) / 12099084336640 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 316800129 \nu^{14} - 1048671012 \nu^{13} - 22131076505 \nu^{12} + 59751145004 \nu^{11} + \cdots - 73084826229440 ) / 3024771084160 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1757207065 \nu^{14} + 7750584908 \nu^{13} + 115535371316 \nu^{12} + \cdots + 106058520362112 ) / 12099084336640 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1815422225 \nu^{14} + 10269111036 \nu^{13} + 102161713720 \nu^{12} + \cdots + 146646404199808 ) / 12099084336640 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2156613575 \nu^{14} + 13904873928 \nu^{13} + 110387993728 \nu^{12} + \cdots - 263784016791168 ) / 12099084336640 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1087839742 \nu^{14} + 6699826979 \nu^{13} + 59599912290 \nu^{12} - 385860345462 \nu^{11} + \cdots + 84109317352640 ) / 6049542168320 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 2043061635 \nu^{14} - 7361653143 \nu^{13} - 142184352818 \nu^{12} + 436890246955 \nu^{11} + \cdots - 424464480749888 ) / 6049542168320 \) 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
\(\nu^{3}\)\(=\) \( \beta_{12} - \beta_{11} - \beta_{9} + \beta_{6} - 2\beta_{5} + 2\beta_{2} + 17\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} + 2 \beta_{12} - 2 \beta_{11} - 5 \beta_{10} - 5 \beta_{8} - 3 \beta_{7} + 2 \beta_{6} + \cdots + 163 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{14} - 7 \beta_{13} + 34 \beta_{12} - 32 \beta_{11} - 9 \beta_{10} - 30 \beta_{9} - 13 \beta_{8} + \cdots + 163 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 18 \beta_{14} + 7 \beta_{13} + 107 \beta_{12} - 89 \beta_{11} - 175 \beta_{10} - 11 \beta_{9} + \cdots + 3227 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 174 \beta_{14} - 266 \beta_{13} + 1048 \beta_{12} - 914 \beta_{11} - 514 \beta_{10} - 718 \beta_{9} + \cdots + 5416 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1108 \beta_{14} - 376 \beta_{13} + 4104 \beta_{12} - 3076 \beta_{11} - 5312 \beta_{10} - 492 \beta_{9} + \cdots + 71330 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 8176 \beta_{14} - 8096 \beta_{13} + 31569 \beta_{12} - 25369 \beta_{11} - 20464 \beta_{10} + \cdots + 168740 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 47272 \beta_{14} - 21967 \beta_{13} + 138970 \beta_{12} - 97290 \beta_{11} - 158373 \beta_{10} + \cdots + 1695739 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 311146 \beta_{14} - 235191 \beta_{13} + 939282 \beta_{12} - 698576 \beta_{11} - 708441 \beta_{10} + \cdots + 5101083 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1736858 \beta_{14} - 842225 \beta_{13} + 4428163 \beta_{12} - 2943377 \beta_{11} - 4714103 \beta_{10} + \cdots + 42497979 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 10722982 \beta_{14} - 6789778 \beta_{13} + 27722992 \beta_{12} - 19263130 \beta_{11} - 22902682 \beta_{10} + \cdots + 151739544 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 59000748 \beta_{14} - 28353056 \beta_{13} + 136470696 \beta_{12} - 86917044 \beta_{11} + \cdots + 1108534370 \) 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
−4.51597
−4.08258
−3.33004
−2.46607
−2.41423
−1.34567
−0.359816
0.849440
1.37673
2.47005
3.19469
3.25734
4.27299
4.67872
5.41441
−4.51597 0 12.3940 17.2075 0 −17.5606 −19.8431 0 −77.7084
1.2 −4.08258 0 8.66750 3.30186 0 5.57603 −2.72511 0 −13.4801
1.3 −3.33004 0 3.08915 −12.0311 0 −25.0857 16.3533 0 40.0640
1.4 −2.46607 0 −1.91852 13.3734 0 0.323120 24.4597 0 −32.9798
1.5 −2.41423 0 −2.17151 −1.07967 0 −29.3530 24.5563 0 2.60658
1.6 −1.34567 0 −6.18917 −4.56101 0 20.0773 19.0940 0 6.13762
1.7 −0.359816 0 −7.87053 12.7210 0 −19.5779 5.71047 0 −4.57721
1.8 0.849440 0 −7.27845 −11.2673 0 −21.1890 −12.9781 0 −9.57087
1.9 1.37673 0 −6.10461 −8.92350 0 13.4226 −19.4183 0 −12.2853
1.10 2.47005 0 −1.89885 11.8284 0 15.4378 −24.4507 0 29.2167
1.11 3.19469 0 2.20606 −3.23067 0 −2.30945 −18.5098 0 −10.3210
1.12 3.25734 0 2.61024 9.43027 0 22.0810 −17.5563 0 30.7176
1.13 4.27299 0 10.2584 7.64686 0 −19.0594 9.65015 0 32.6749
1.14 4.67872 0 13.8904 −2.81758 0 −10.2461 27.5596 0 −13.1827
1.15 5.41441 0 21.3159 −5.59842 0 −17.5366 72.0977 0 −30.3122
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(167\) \(1\)

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 1503.4.a.a 15
3.b odd 2 1 167.4.a.a ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
167.4.a.a ✓ 15 3.b odd 2 1
1503.4.a.a 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{15} - 7 T_{2}^{14} - 56 T_{2}^{13} + 441 T_{2}^{12} + 1120 T_{2}^{11} - 10727 T_{2}^{10} + \cdots + 575872 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1503))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} - 7 T^{14} + \cdots + 575872 \) Copy content Toggle raw display
$3$ \( T^{15} \) Copy content Toggle raw display
$5$ \( T^{15} + \cdots - 2502751002528 \) Copy content Toggle raw display
$7$ \( T^{15} + \cdots + 70\!\cdots\!41 \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots - 17\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots + 10\!\cdots\!48 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots - 74\!\cdots\!32 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots - 23\!\cdots\!08 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots - 12\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 25\!\cdots\!68 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 19\!\cdots\!88 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 21\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 39\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 48\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 52\!\cdots\!52 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 31\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 39\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 40\!\cdots\!68 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 60\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 93\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 31\!\cdots\!96 \) Copy content Toggle raw display
show more
show less