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Properties

Label 1521.4.a.bn
Level $1521$
Weight $4$
Character orbit 1521.a
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1521,4,Mod(1,1521)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1521, 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("1521.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.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: \(89.7419051187\)
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: \( x^{18} - 100 x^{16} + 4066 x^{14} - 86197 x^{12} + 1016064 x^{10} - 6594119 x^{8} + 22251817 x^{6} + \cdots - 6889792 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 7\cdot 13^{4} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 3) q^{4} + ( - \beta_{14} + \beta_1) q^{5} + (\beta_{7} + \beta_{2} + 5) q^{7} + (\beta_{3} + 3 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 3) q^{4} + ( - \beta_{14} + \beta_1) q^{5} + (\beta_{7} + \beta_{2} + 5) q^{7} + (\beta_{3} + 3 \beta_1) q^{8} + (\beta_{11} - \beta_{4} + \beta_{2} + 8) q^{10} + (\beta_{17} - \beta_{16} + \cdots - \beta_1) q^{11}+ \cdots + ( - 32 \beta_{17} + 8 \beta_{16} + \cdots + 322 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 56 q^{4} + 94 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 56 q^{4} + 94 q^{7} + 156 q^{10} + 88 q^{16} + 448 q^{19} - 256 q^{22} - 16 q^{25} + 1300 q^{28} + 818 q^{31} + 900 q^{34} + 524 q^{37} + 800 q^{40} + 752 q^{43} + 1650 q^{46} + 2948 q^{49} - 464 q^{55} + 1626 q^{58} - 1784 q^{61} - 4274 q^{64} + 2872 q^{67} + 5772 q^{70} + 3078 q^{73} + 5726 q^{76} + 3326 q^{79} - 1038 q^{82} + 2340 q^{85} + 780 q^{88} + 1220 q^{94} + 4630 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 100 x^{16} + 4066 x^{14} - 86197 x^{12} + 1016064 x^{10} - 6594119 x^{8} + 22251817 x^{6} + \cdots - 6889792 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 19\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 181489 \nu^{16} - 21373147 \nu^{14} + 997040927 \nu^{12} - 23574535710 \nu^{10} + \cdots - 391044378816 ) / 49938423808 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1433467 \nu^{16} + 138372809 \nu^{14} - 5408740661 \nu^{12} + 109302912522 \nu^{10} + \cdots - 6066047938496 ) / 99876847616 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 116153 \nu^{17} + 10652275 \nu^{15} - 378988503 \nu^{13} + 6387634062 \nu^{11} + \cdots + 4651421005504 \nu ) / 26889920512 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2408429 \nu^{16} + 236768687 \nu^{14} - 9418555363 \nu^{12} + 193723590182 \nu^{10} + \cdots - 12024332034624 ) / 99876847616 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2578733 \nu^{16} + 251135855 \nu^{14} - 9869708963 \nu^{12} + 199518257062 \nu^{10} + \cdots - 7630928212544 ) / 99876847616 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 487481 \nu^{17} - 49197939 \nu^{15} + 2004907351 \nu^{13} - 42210338830 \nu^{11} + \cdots + 1081481943360 \nu ) / 99876847616 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 224043 \nu^{16} - 22133081 \nu^{14} + 883985541 \nu^{12} - 18217576618 \nu^{10} + \cdots + 1369985559488 ) / 7682834432 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1597405 \nu^{16} + 157633343 \nu^{14} - 6278374259 \nu^{12} + 128809709446 \nu^{10} + \cdots - 9201566989888 ) / 49938423808 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 2408429 \nu^{16} - 236768687 \nu^{14} + 9418555363 \nu^{12} - 193723590182 \nu^{10} + \cdots + 14970699039296 ) / 49938423808 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 5256429 \nu^{17} + 523191087 \nu^{15} - 21167805411 \nu^{13} + 445951059110 \nu^{11} + \cdots - 49189596316224 \nu ) / 349568966656 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 350141 \nu^{17} - 34535167 \nu^{15} + 1375479251 \nu^{13} - 28222338150 \nu^{11} + \cdots + 1371275587136 \nu ) / 13444960256 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 13068997 \nu^{17} - 1287411959 \nu^{15} + 51228961675 \nu^{13} - 1050840575606 \nu^{11} + \cdots + 71133236166720 \nu ) / 349568966656 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 27213579 \nu^{17} - 2663050105 \nu^{15} + 105137553317 \nu^{13} - 2135981850218 \nu^{11} + \cdots + 81381209244608 \nu ) / 699137933312 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 37394823 \nu^{17} - 3687858637 \nu^{15} + 147131501289 \nu^{13} - 3033885246066 \nu^{11} + \cdots + 292688312489664 \nu ) / 699137933312 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 19\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} + 2\beta_{7} + 24\beta_{2} + 205 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{17} + 2 \beta_{16} + 5 \beta_{15} - 2 \beta_{14} + 2 \beta_{13} - \beta_{9} + \cdots + 399 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 41\beta_{12} + 8\beta_{11} - 3\beta_{10} - 9\beta_{8} + 71\beta_{7} + 11\beta_{5} + \beta_{4} + 545\beta_{2} + 4249 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 131 \beta_{17} + 83 \beta_{16} + 214 \beta_{15} - 90 \beta_{14} + 68 \beta_{13} - 36 \beta_{9} + \cdots + 8698 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1211 \beta_{12} + 402 \beta_{11} - 52 \beta_{10} - 402 \beta_{8} + 1937 \beta_{7} + 541 \beta_{5} + \cdots + 91717 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4091 \beta_{17} + 2438 \beta_{16} + 6831 \beta_{15} - 3122 \beta_{14} + 1726 \beta_{13} + \cdots + 193605 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 31810 \beta_{12} + 14332 \beta_{11} + 711 \beta_{10} - 13143 \beta_{8} + 48377 \beta_{7} + \cdots + 2025527 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 113258 \beta_{17} + 63047 \beta_{16} + 195077 \beta_{15} - 95716 \beta_{14} + 38930 \beta_{13} + \cdots + 4364864 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 793077 \beta_{12} + 445782 \beta_{11} + 80599 \beta_{10} - 380837 \beta_{8} + 1162914 \beta_{7} + \cdots + 45373093 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 2964221 \beta_{17} + 1537129 \beta_{16} + 5263192 \beta_{15} - 2727302 \beta_{14} + \cdots + 99211870 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 19268646 \beta_{12} + 12892602 \beta_{11} + 3574706 \beta_{10} - 10369896 \beta_{8} + 27448921 \beta_{7} + \cdots + 1025751919 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 75245356 \beta_{17} + 36375074 \beta_{16} + 137261260 \beta_{15} - 74158776 \beta_{14} + \cdots + 2267254771 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 461812574 \beta_{12} + 356346272 \beta_{11} + 124225144 \beta_{10} - 272109608 \beta_{8} + \cdots + 23332380115 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 1874831298 \beta_{17} + 847629884 \beta_{16} + 3500293526 \beta_{15} - 1954043116 \beta_{14} + \cdots + 52007923651 \beta_1 \) 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
−4.85886
−4.80856
−4.56788
−3.66578
−3.45813
−1.86163
−1.32218
−1.15064
−0.685025
0.685025
1.15064
1.32218
1.86163
3.45813
3.66578
4.56788
4.80856
4.85886
−4.85886 0 15.6085 −14.8714 0 29.6598 −36.9687 0 72.2582
1.2 −4.80856 0 15.1223 −1.59043 0 34.5272 −34.2479 0 7.64769
1.3 −4.56788 0 12.8655 1.42360 0 3.82462 −22.2252 0 −6.50284
1.4 −3.66578 0 5.43796 9.45496 0 −15.4112 9.39187 0 −34.6598
1.5 −3.45813 0 3.95865 −9.27918 0 −27.2299 13.9755 0 32.0886
1.6 −1.86163 0 −4.53433 3.46613 0 −21.2285 23.3343 0 −6.45265
1.7 −1.32218 0 −6.25184 12.9610 0 5.54391 18.8435 0 −17.1367
1.8 −1.15064 0 −6.67603 −21.2034 0 31.2668 16.8868 0 24.3975
1.9 −0.685025 0 −7.53074 −9.28431 0 6.04730 10.6389 0 6.35999
1.10 0.685025 0 −7.53074 9.28431 0 6.04730 −10.6389 0 6.35999
1.11 1.15064 0 −6.67603 21.2034 0 31.2668 −16.8868 0 24.3975
1.12 1.32218 0 −6.25184 −12.9610 0 5.54391 −18.8435 0 −17.1367
1.13 1.86163 0 −4.53433 −3.46613 0 −21.2285 −23.3343 0 −6.45265
1.14 3.45813 0 3.95865 9.27918 0 −27.2299 −13.9755 0 32.0886
1.15 3.66578 0 5.43796 −9.45496 0 −15.4112 −9.39187 0 −34.6598
1.16 4.56788 0 12.8655 −1.42360 0 3.82462 22.2252 0 −6.50284
1.17 4.80856 0 15.1223 1.59043 0 34.5272 34.2479 0 7.64769
1.18 4.85886 0 15.6085 14.8714 0 29.6598 36.9687 0 72.2582
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(13\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.4.a.bn yes 18
3.b odd 2 1 inner 1521.4.a.bn yes 18
13.b even 2 1 1521.4.a.bm ✓ 18
39.d odd 2 1 1521.4.a.bm ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1521.4.a.bm ✓ 18 13.b even 2 1
1521.4.a.bm ✓ 18 39.d odd 2 1
1521.4.a.bn yes 18 1.a even 1 1 trivial
1521.4.a.bn yes 18 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1521))\):

\( T_{2}^{18} - 100 T_{2}^{16} + 4066 T_{2}^{14} - 86197 T_{2}^{12} + 1016064 T_{2}^{10} - 6594119 T_{2}^{8} + \cdots - 6889792 \) Copy content Toggle raw display
\( T_{5}^{18} - 1117 T_{5}^{16} + 472720 T_{5}^{14} - 99614185 T_{5}^{12} + 11320990891 T_{5}^{10} + \cdots - 682535158703125 \) Copy content Toggle raw display
\( T_{7}^{9} - 47 T_{7}^{8} - 1176 T_{7}^{7} + 68263 T_{7}^{6} + 310197 T_{7}^{5} - 29038353 T_{7}^{4} + \cdots + 36574832125 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 100 T^{16} + \cdots - 6889792 \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots - 682535158703125 \) Copy content Toggle raw display
$7$ \( (T^{9} - 47 T^{8} + \cdots + 36574832125)^{2} \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots - 11\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( T^{18} \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots - 12\!\cdots\!28 \) Copy content Toggle raw display
$19$ \( (T^{9} + \cdots + 60\!\cdots\!36)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots - 42\!\cdots\!08 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots - 86\!\cdots\!25 \) Copy content Toggle raw display
$31$ \( (T^{9} + \cdots - 10\!\cdots\!51)^{2} \) Copy content Toggle raw display
$37$ \( (T^{9} + \cdots - 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots - 34\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{9} + \cdots - 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 12\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots - 16\!\cdots\!57 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{9} + \cdots + 47\!\cdots\!52)^{2} \) Copy content Toggle raw display
$67$ \( (T^{9} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{9} + \cdots + 10\!\cdots\!75)^{2} \) Copy content Toggle raw display
$79$ \( (T^{9} + \cdots - 15\!\cdots\!37)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 10\!\cdots\!77 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 83\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{9} + \cdots - 41\!\cdots\!75)^{2} \) Copy content Toggle raw display
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