Properties

Label 363.4.a.r
Level $363$
Weight $4$
Character orbit 363.a
Self dual yes
Analytic conductor $21.418$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [363,4,Mod(1,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, 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("363.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 363.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: \(21.4176933321\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 7x^{2} + 8x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_1) q^{2} - 3 q^{3} + 2 \beta_{2} q^{4} + ( - 4 \beta_{2} + 4) q^{5} + ( - 3 \beta_{3} + 3 \beta_1) q^{6} + ( - 4 \beta_{3} - 13 \beta_1) q^{7} + ( - 2 \beta_{3} - 2 \beta_1) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_1) q^{2} - 3 q^{3} + 2 \beta_{2} q^{4} + ( - 4 \beta_{2} + 4) q^{5} + ( - 3 \beta_{3} + 3 \beta_1) q^{6} + ( - 4 \beta_{3} - 13 \beta_1) q^{7} + ( - 2 \beta_{3} - 2 \beta_1) q^{8} + 9 q^{9} + ( - 8 \beta_{3} + 16 \beta_1) q^{10} - 6 \beta_{2} q^{12} + (24 \beta_{3} - 8 \beta_1) q^{13} + (9 \beta_{2} + 19) q^{14} + (12 \beta_{2} - 12) q^{15} + ( - 16 \beta_{2} - 4) q^{16} + (8 \beta_{3} - 40 \beta_1) q^{17} + (9 \beta_{3} - 9 \beta_1) q^{18} + (20 \beta_{3} - 39 \beta_1) q^{19} + (8 \beta_{2} - 120) q^{20} + (12 \beta_{3} + 39 \beta_1) q^{21} + (24 \beta_{2} + 34) q^{23} + (6 \beta_{3} + 6 \beta_1) q^{24} + ( - 32 \beta_{2} + 131) q^{25} + (32 \beta_{2} + 144) q^{26} - 27 q^{27} + (78 \beta_{3} + 40 \beta_1) q^{28} + (104 \beta_{3} + 22 \beta_1) q^{29} + (24 \beta_{3} - 48 \beta_1) q^{30} + ( - 28 \beta_{2} - 51) q^{31} + ( - 36 \beta_{3} + 100 \beta_1) q^{32} + (48 \beta_{2} + 160) q^{34} + ( - 172 \beta_{3} - 132 \beta_1) q^{35} + 18 \beta_{2} q^{36} + ( - 72 \beta_{2} - 43) q^{37} + (59 \beta_{2} + 217) q^{38} + ( - 72 \beta_{3} + 24 \beta_1) q^{39} + ( - 32 \beta_{3} - 48 \beta_1) q^{40} + (52 \beta_{3} - 186 \beta_1) q^{41} + ( - 27 \beta_{2} - 57) q^{42} + (40 \beta_{3} + 54 \beta_1) q^{43} + ( - 36 \beta_{2} + 36) q^{45} + (106 \beta_{3} - 154 \beta_1) q^{46} + ( - 28 \beta_{2} + 166) q^{47} + (48 \beta_{2} + 12) q^{48} + ( - 104 \beta_{2} + 244) q^{49} + (35 \beta_{3} + 29 \beta_1) q^{50} + ( - 24 \beta_{3} + 120 \beta_1) q^{51} + (48 \beta_{3} - 240 \beta_1) q^{52} + (24 \beta_{2} + 630) q^{53} + ( - 27 \beta_{3} + 27 \beta_1) q^{54} + ( - 34 \beta_{2} + 118) q^{56} + ( - 60 \beta_{3} + 117 \beta_1) q^{57} + (82 \beta_{2} + 454) q^{58} + ( - 24 \beta_{2} + 90) q^{59} + ( - 24 \beta_{2} + 360) q^{60} + ( - 208 \beta_{3} + 85 \beta_1) q^{61} + ( - 135 \beta_{3} + 191 \beta_1) q^{62} + ( - 36 \beta_{3} - 117 \beta_1) q^{63} + ( - 8 \beta_{2} - 448) q^{64} + 448 \beta_1 q^{65} + ( - 60 \beta_{2} + 657) q^{67} + (240 \beta_{3} - 80 \beta_1) q^{68} + ( - 72 \beta_{2} - 102) q^{69} + ( - 40 \beta_{2} - 464) q^{70} + (20 \beta_{2} + 128) q^{71} + ( - 18 \beta_{3} - 18 \beta_1) q^{72} + (16 \beta_{3} + 365 \beta_1) q^{73} + ( - 259 \beta_{3} + 403 \beta_1) q^{74} + (96 \beta_{2} - 393) q^{75} + (234 \beta_{3} - 200 \beta_1) q^{76} + ( - 96 \beta_{2} - 432) q^{78} + (276 \beta_{3} - 113 \beta_1) q^{79} + ( - 48 \beta_{2} + 944) q^{80} + 81 q^{81} + (238 \beta_{2} + 818) q^{82} + (68 \beta_{3} + 126 \beta_1) q^{83} + ( - 234 \beta_{3} - 120 \beta_1) q^{84} - 448 \beta_{3} q^{85} + ( - 14 \beta_{2} + 38) q^{86} + ( - 312 \beta_{3} - 66 \beta_1) q^{87} + ( - 76 \beta_{2} - 136) q^{89} + ( - 72 \beta_{3} + 144 \beta_1) q^{90} + (280 \beta_{2} - 168) q^{91} + (68 \beta_{2} + 720) q^{92} + (84 \beta_{2} + 153) q^{93} + (82 \beta_{3} - 26 \beta_1) q^{94} + ( - 388 \beta_{3} + 244 \beta_1) q^{95} + (108 \beta_{3} - 300 \beta_1) q^{96} + (208 \beta_{2} + 5) q^{97} + ( - 68 \beta_{3} + 276 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} + 16 q^{5} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{3} + 16 q^{5} + 36 q^{9} + 76 q^{14} - 48 q^{15} - 16 q^{16} - 480 q^{20} + 136 q^{23} + 524 q^{25} + 576 q^{26} - 108 q^{27} - 204 q^{31} + 640 q^{34} - 172 q^{37} + 868 q^{38} - 228 q^{42} + 144 q^{45} + 664 q^{47} + 48 q^{48} + 976 q^{49} + 2520 q^{53} + 472 q^{56} + 1816 q^{58} + 360 q^{59} + 1440 q^{60} - 1792 q^{64} + 2628 q^{67} - 408 q^{69} - 1856 q^{70} + 512 q^{71} - 1572 q^{75} - 1728 q^{78} + 3776 q^{80} + 324 q^{81} + 3272 q^{82} + 152 q^{86} - 544 q^{89} - 672 q^{91} + 2880 q^{92} + 612 q^{93} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 7x^{2} + 8x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} - 3\nu^{2} - 19\nu + 10 ) / 7 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{3} - 6\nu^{2} - 24\nu + 13 ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - 2\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2\beta_{2} - 2\beta _1 + 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11\beta_{3} + 3\beta_{2} - 15\beta _1 + 13 ) / 2 \) Copy content Toggle raw display

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
−2.35008
1.11402
−0.114017
3.35008
−3.96812 −3.00000 7.74597 −11.4919 11.9044 −13.5724 1.00803 9.00000 45.6014
1.2 −0.504017 −3.00000 −7.74597 19.4919 1.51205 31.4609 7.93624 9.00000 −9.82427
1.3 0.504017 −3.00000 −7.74597 19.4919 −1.51205 −31.4609 −7.93624 9.00000 9.82427
1.4 3.96812 −3.00000 7.74597 −11.4919 −11.9044 13.5724 −1.00803 9.00000 −45.6014
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.4.a.r 4
3.b odd 2 1 1089.4.a.bb 4
11.b odd 2 1 inner 363.4.a.r 4
33.d even 2 1 1089.4.a.bb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.4.a.r 4 1.a even 1 1 trivial
363.4.a.r 4 11.b odd 2 1 inner
1089.4.a.bb 4 3.b odd 2 1
1089.4.a.bb 4 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(363))\):

\( T_{2}^{4} - 16T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{5}^{2} - 8T_{5} - 224 \) Copy content Toggle raw display
\( T_{7}^{4} - 1174T_{7}^{2} + 182329 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 16T^{2} + 4 \) Copy content Toggle raw display
$3$ \( (T + 3)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 8 T - 224)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 1174 T^{2} + 182329 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 6144 T^{2} + \cdots + 7225344 \) Copy content Toggle raw display
$17$ \( T^{4} - 10240 T^{2} + \cdots + 20070400 \) Copy content Toggle raw display
$19$ \( T^{4} - 13126 T^{2} + \cdots + 6568969 \) Copy content Toggle raw display
$23$ \( (T^{2} - 68 T - 7484)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 111064 T^{2} + \cdots + 2769706384 \) Copy content Toggle raw display
$31$ \( (T^{2} + 102 T - 9159)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 86 T - 75911)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 234616 T^{2} + \cdots + 8148311824 \) Copy content Toggle raw display
$43$ \( T^{4} - 33496 T^{2} + \cdots + 559504 \) Copy content Toggle raw display
$47$ \( (T^{2} - 332 T + 15796)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 1260 T + 388260)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 180 T - 540)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} - 475990 T^{2} + \cdots + 37886676025 \) Copy content Toggle raw display
$67$ \( (T^{2} - 1314 T + 377649)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 256 T + 10384)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 801910 T^{2} + \cdots + 158718576025 \) Copy content Toggle raw display
$79$ \( T^{4} - 838374 T^{2} + \cdots + 117356260329 \) Copy content Toggle raw display
$83$ \( T^{4} - 141496 T^{2} + \cdots + 600642064 \) Copy content Toggle raw display
$89$ \( (T^{2} + 272 T - 68144)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 10 T - 648935)^{2} \) Copy content Toggle raw display
show more
show less