Properties

Label 363.4.a.s
Level $363$
Weight $4$
Character orbit 363.a
Self dual yes
Analytic conductor $21.418$
Analytic rank $0$
Dimension $4$
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] = [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: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 28x^{2} - 2x + 72 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 3 q^{3} + (\beta_{3} + \beta_1 + 6) q^{4} + (\beta_{3} - \beta_{2} + 3) q^{5} + 3 \beta_1 q^{6} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 6) q^{7} + (\beta_{3} + 2 \beta_{2} + 7 \beta_1 + 18) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 3 q^{3} + (\beta_{3} + \beta_1 + 6) q^{4} + (\beta_{3} - \beta_{2} + 3) q^{5} + 3 \beta_1 q^{6} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 6) q^{7} + (\beta_{3} + 2 \beta_{2} + 7 \beta_1 + 18) q^{8} + 9 q^{9} + ( - 3 \beta_{3} + 2 \beta_{2} + 9 \beta_1 - 2) q^{10} + (3 \beta_{3} + 3 \beta_1 + 18) q^{12} + (\beta_{3} - 3 \beta_{2} - 2 \beta_1 - 9) q^{13} + ( - \beta_{3} - 2 \beta_{2} - 14 \beta_1 + 18) q^{14} + (3 \beta_{3} - 3 \beta_{2} + 9) q^{15} + (5 \beta_{3} + 2 \beta_{2} + 29 \beta_1 + 66) q^{16} + (7 \beta_{3} - \beta_{2} - 10 \beta_1 - 21) q^{17} + 9 \beta_1 q^{18} + (3 \beta_{3} - 7 \beta_{2} - 12 \beta_1 + 8) q^{19} + (7 \beta_{3} + 2 \beta_{2} - 13 \beta_1 + 102) q^{20} + ( - 3 \beta_{3} - 3 \beta_{2} + 6 \beta_1 - 18) q^{21} + ( - 2 \beta_{3} - 4 \beta_{2} + 38 \beta_1 - 18) q^{23} + (3 \beta_{3} + 6 \beta_{2} + 21 \beta_1 + 54) q^{24} + ( - \beta_{3} + 5 \beta_{2} - 40 \beta_1 + 96) q^{25} + ( - 11 \beta_{3} + 2 \beta_{2} - 9 \beta_1 - 42) q^{26} + 27 q^{27} + ( - 12 \beta_{3} + 6 \beta_{2} - 24 \beta_1 - 164) q^{28} + ( - 7 \beta_{3} + 7 \beta_{2} - 4 \beta_1 - 39) q^{29} + ( - 9 \beta_{3} + 6 \beta_{2} + 27 \beta_1 - 6) q^{30} + ( - 13 \beta_{3} - 3 \beta_{2} - 16 \beta_1 + 108) q^{31} + (27 \beta_{3} - 6 \beta_{2} + 83 \beta_1 + 294) q^{32} + ( - 13 \beta_{3} + 14 \beta_{2} + 23 \beta_1 - 118) q^{34} + ( - 24 \beta_{3} + 6 \beta_{2} + 22 \beta_1 + 18) q^{35} + (9 \beta_{3} + 9 \beta_1 + 54) q^{36} + ( - 4 \beta_{3} + 14 \beta_{2} - 34 \beta_1 + 145) q^{37} + ( - 33 \beta_{3} + 6 \beta_{2} + 6 \beta_1 - 198) q^{38} + (3 \beta_{3} - 9 \beta_{2} - 6 \beta_1 - 27) q^{39} + (17 \beta_{3} - 2 \beta_{2} + 77 \beta_1 - 126) q^{40} + (5 \beta_{3} + \beta_{2} + 42 \beta_1 - 117) q^{41} + ( - 3 \beta_{3} - 6 \beta_{2} - 42 \beta_1 + 54) q^{42} + ( - 16 \beta_{3} + 6 \beta_{2} - 22 \beta_1 + 146) q^{43} + (9 \beta_{3} - 9 \beta_{2} + 27) q^{45} + (26 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 500) q^{46} + ( - 14 \beta_{3} + 2 \beta_{2} + 32 \beta_1 + 132) q^{47} + (15 \beta_{3} + 6 \beta_{2} + 87 \beta_1 + 198) q^{48} + ( - 7 \beta_{3} - 17 \beta_{2} + 20 \beta_1 + 57) q^{49} + ( - 25 \beta_{3} - 2 \beta_{2} + 58 \beta_1 - 534) q^{50} + (21 \beta_{3} - 3 \beta_{2} - 30 \beta_1 - 63) q^{51} + ( - 11 \beta_{3} + 2 \beta_{2} - 119 \beta_1 - 86) q^{52} + ( - 5 \beta_{3} - 7 \beta_{2} - 56 \beta_1 + 231) q^{53} + 27 \beta_1 q^{54} + (2 \beta_{3} - 8 \beta_{2} - 160 \beta_1 - 492) q^{56} + (9 \beta_{3} - 21 \beta_{2} - 36 \beta_1 + 24) q^{57} + (17 \beta_{3} - 14 \beta_{2} - 85 \beta_1 - 42) q^{58} + ( - 14 \beta_{3} - 28 \beta_{2} + 2 \beta_1 - 54) q^{59} + (21 \beta_{3} + 6 \beta_{2} - 39 \beta_1 + 306) q^{60} + (19 \beta_{3} + \beta_{2} - 56 \beta_1 - 58) q^{61} + ( - 25 \beta_{3} - 26 \beta_{2} - 18 \beta_1 - 294) q^{62} + ( - 9 \beta_{3} - 9 \beta_{2} + 18 \beta_1 - 54) q^{63} + (25 \beta_{3} + 38 \beta_{2} + 349 \beta_1 + 706) q^{64} + ( - 23 \beta_{3} + 17 \beta_{2} - 94 \beta_1 + 441) q^{65} + (23 \beta_{3} - 5 \beta_{2} + 62 \beta_1 + 230) q^{67} + (9 \beta_{3} - 18 \beta_{2} - 91 \beta_1 + 522) q^{68} + ( - 6 \beta_{3} - 12 \beta_{2} + 114 \beta_1 - 54) q^{69} + (40 \beta_{3} - 48 \beta_{2} - 140 \beta_1 + 248) q^{70} + (64 \beta_{3} + 2 \beta_{2} - 110 \beta_1 + 18) q^{71} + (9 \beta_{3} + 18 \beta_{2} + 63 \beta_1 + 162) q^{72} + ( - 3 \beta_{3} + 19 \beta_{2} - 108 \beta_1 - 282) q^{73} + (8 \beta_{3} - 8 \beta_{2} + 107 \beta_1 - 408) q^{74} + ( - 3 \beta_{3} + 15 \beta_{2} - 120 \beta_1 + 288) q^{75} + ( - 10 \beta_{2} - 348 \beta_1 - 76) q^{76} + ( - 33 \beta_{3} + 6 \beta_{2} - 27 \beta_1 - 126) q^{78} + ( - 31 \beta_{3} + 43 \beta_{2} - 4 \beta_1 - 68) q^{79} + (15 \beta_{3} + 18 \beta_{2} + 187 \beta_1 + 318) q^{80} + 81 q^{81} + (45 \beta_{3} + 10 \beta_{2} - 33 \beta_1 + 614) q^{82} + (54 \beta_{3} + 6 \beta_{2} + 48 \beta_1 - 384) q^{83} + ( - 36 \beta_{3} + 18 \beta_{2} - 72 \beta_1 - 492) q^{84} + (29 \beta_{3} + 45 \beta_{2} - 262 \beta_1 + 685) q^{85} + ( - 4 \beta_{3} - 32 \beta_{2} + 8 \beta_1 - 336) q^{86} + ( - 21 \beta_{3} + 21 \beta_{2} - 12 \beta_1 - 117) q^{87} + (5 \beta_{3} + 49 \beta_{2} + 98 \beta_1 + 357) q^{89} + ( - 27 \beta_{3} + 18 \beta_{2} + 81 \beta_1 - 18) q^{90} + ( - 42 \beta_{3} + 12 \beta_{2} + 102 \beta_1 + 418) q^{91} + (84 \beta_{2} + 392 \beta_1 + 168) q^{92} + ( - 39 \beta_{3} - 9 \beta_{2} - 48 \beta_1 + 324) q^{93} + (38 \beta_{3} - 28 \beta_{2} + 56 \beta_1 + 404) q^{94} + ( - 300 \beta_1 + 1188) q^{95} + (81 \beta_{3} - 18 \beta_{2} + 249 \beta_1 + 882) q^{96} + (18 \beta_{3} - 78 \beta_{2} - 60 \beta_1 + 239) q^{97} + ( - 31 \beta_{3} - 14 \beta_{2} - 13 \beta_1 + 150) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + 12 q^{3} + 25 q^{4} + 14 q^{5} + 3 q^{6} - 20 q^{7} + 75 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + 12 q^{3} + 25 q^{4} + 14 q^{5} + 3 q^{6} - 20 q^{7} + 75 q^{8} + 36 q^{9} - 3 q^{10} + 75 q^{12} - 32 q^{13} + 62 q^{14} + 42 q^{15} + 289 q^{16} - 92 q^{17} + 9 q^{18} + 34 q^{19} + 391 q^{20} - 60 q^{21} - 26 q^{23} + 225 q^{24} + 334 q^{25} - 181 q^{26} + 108 q^{27} - 692 q^{28} - 174 q^{29} - 9 q^{30} + 422 q^{31} + 1271 q^{32} - 477 q^{34} + 82 q^{35} + 225 q^{36} + 518 q^{37} - 798 q^{38} - 96 q^{39} - 423 q^{40} - 428 q^{41} + 186 q^{42} + 550 q^{43} + 126 q^{45} + 2004 q^{46} + 556 q^{47} + 867 q^{48} + 282 q^{49} - 2074 q^{50} - 276 q^{51} - 467 q^{52} + 882 q^{53} + 27 q^{54} - 2112 q^{56} + 102 q^{57} - 225 q^{58} - 158 q^{59} + 1173 q^{60} - 290 q^{61} - 1142 q^{62} - 180 q^{63} + 3097 q^{64} + 1636 q^{65} + 992 q^{67} + 2033 q^{68} - 78 q^{69} + 948 q^{70} - 42 q^{71} + 675 q^{72} - 1274 q^{73} - 1509 q^{74} + 1002 q^{75} - 632 q^{76} - 543 q^{78} - 362 q^{79} + 1423 q^{80} + 324 q^{81} + 2403 q^{82} - 1500 q^{83} - 2076 q^{84} + 2388 q^{85} - 1272 q^{86} - 522 q^{87} + 1428 q^{89} - 27 q^{90} + 1750 q^{91} + 896 q^{92} + 1266 q^{93} + 1728 q^{94} + 4452 q^{95} + 3813 q^{96} + 1052 q^{97} + 615 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 28x^{2} - 2x + 72 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu^{2} - 22\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 14 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 23\beta _1 + 18 \) 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
−4.40829
−1.81838
1.59490
5.63177
−4.40829 3.00000 11.4330 18.8997 −13.2249 −18.5994 −15.1336 9.00000 −83.3152
1.2 −1.81838 3.00000 −4.69349 −19.2178 −5.45514 −14.1043 23.0816 9.00000 34.9453
1.3 1.59490 3.00000 −5.45630 8.73607 4.78470 29.0283 −21.4614 9.00000 13.9331
1.4 5.63177 3.00000 23.7168 5.58204 16.8953 −16.3245 88.5134 9.00000 31.4368
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 363.4.a.s yes 4
3.b odd 2 1 1089.4.a.ba 4
11.b odd 2 1 363.4.a.q 4
33.d even 2 1 1089.4.a.bf 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.4.a.q 4 11.b odd 2 1
363.4.a.s yes 4 1.a even 1 1 trivial
1089.4.a.ba 4 3.b odd 2 1
1089.4.a.bf 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(363))\):

\( T_{2}^{4} - T_{2}^{3} - 28T_{2}^{2} - 2T_{2} + 72 \) Copy content Toggle raw display
\( T_{5}^{4} - 14T_{5}^{3} - 319T_{5}^{2} + 5216T_{5} - 17712 \) Copy content Toggle raw display
\( T_{7}^{4} + 20T_{7}^{3} - 627T_{7}^{2} - 18830T_{7} - 124312 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} - 28 T^{2} - 2 T + 72 \) Copy content Toggle raw display
$3$ \( (T - 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 14 T^{3} - 319 T^{2} + \cdots - 17712 \) Copy content Toggle raw display
$7$ \( T^{4} + 20 T^{3} - 627 T^{2} + \cdots - 124312 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 32 T^{3} - 2505 T^{2} + \cdots + 1724492 \) Copy content Toggle raw display
$17$ \( T^{4} + 92 T^{3} - 9637 T^{2} + \cdots - 1870416 \) Copy content Toggle raw display
$19$ \( T^{4} - 34 T^{3} - 19647 T^{2} + \cdots + 87435072 \) Copy content Toggle raw display
$23$ \( T^{4} + 26 T^{3} + \cdots + 427537152 \) Copy content Toggle raw display
$29$ \( T^{4} + 174 T^{3} + \cdots - 111819852 \) Copy content Toggle raw display
$31$ \( T^{4} - 422 T^{3} + \cdots - 755977024 \) Copy content Toggle raw display
$37$ \( T^{4} - 518 T^{3} + \cdots - 336093777 \) Copy content Toggle raw display
$41$ \( T^{4} + 428 T^{3} + \cdots - 1476203796 \) Copy content Toggle raw display
$43$ \( T^{4} - 550 T^{3} + \cdots - 474515712 \) Copy content Toggle raw display
$47$ \( T^{4} - 556 T^{3} + \cdots - 1899747648 \) Copy content Toggle raw display
$53$ \( T^{4} - 882 T^{3} + \cdots - 9562089852 \) Copy content Toggle raw display
$59$ \( T^{4} + 158 T^{3} + \cdots - 2988982656 \) Copy content Toggle raw display
$61$ \( T^{4} + 290 T^{3} + \cdots - 2291291676 \) Copy content Toggle raw display
$67$ \( T^{4} - 992 T^{3} + \cdots - 3253123496 \) Copy content Toggle raw display
$71$ \( T^{4} + 42 T^{3} + \cdots + 71278424064 \) Copy content Toggle raw display
$73$ \( T^{4} + 1274 T^{3} + \cdots + 5540854172 \) Copy content Toggle raw display
$79$ \( T^{4} + 362 T^{3} + \cdots - 13109080736 \) Copy content Toggle raw display
$83$ \( T^{4} + 1500 T^{3} + \cdots - 58659282432 \) Copy content Toggle raw display
$89$ \( T^{4} - 1428 T^{3} + \cdots + 172456334064 \) Copy content Toggle raw display
$97$ \( T^{4} - 1052 T^{3} + \cdots + 840759943813 \) Copy content Toggle raw display
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