Properties

Label 363.4.a.t
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: 4.4.5225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} + x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1 + 1) q^{2} + 3 q^{3} + (3 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 2) q^{4} + (3 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 4) q^{5} + (3 \beta_{2} + 3 \beta_1 + 3) q^{6} + (6 \beta_{3} + 4 \beta_{2} - 9 \beta_1 + 4) q^{7} + (8 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 7) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1 + 1) q^{2} + 3 q^{3} + (3 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 2) q^{4} + (3 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 4) q^{5} + (3 \beta_{2} + 3 \beta_1 + 3) q^{6} + (6 \beta_{3} + 4 \beta_{2} - 9 \beta_1 + 4) q^{7} + (8 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 7) q^{8} + 9 q^{9} + (11 \beta_{2} + 6 \beta_1 + 3) q^{10} + (9 \beta_{3} + 6 \beta_{2} + 9 \beta_1 - 6) q^{12} + (15 \beta_{3} - 25 \beta_{2} + 18 \beta_1 + 21) q^{13} + ( - 14 \beta_{3} + 13 \beta_{2} - 2 \beta_1 - 22) q^{14} + (9 \beta_{3} + 12 \beta_{2} - 6 \beta_1 + 12) q^{15} + ( - 18 \beta_{3} + 17 \beta_{2} + 3 \beta_1 + 41) q^{16} + (11 \beta_{2} - 25 \beta_1 + 26) q^{17} + (9 \beta_{2} + 9 \beta_1 + 9) q^{18} + ( - 15 \beta_{3} + 34 \beta_{2} + 27 \beta_1 + 61) q^{19} + ( - \beta_{3} - 23 \beta_{2} + 31 \beta_1 + 6) q^{20} + (18 \beta_{3} + 12 \beta_{2} - 27 \beta_1 + 12) q^{21} + ( - 39 \beta_{3} - 51 \beta_{2} + 18 \beta_1 - 21) q^{23} + (24 \beta_{3} + 6 \beta_{2} + 6 \beta_1 + 21) q^{24} + ( - 3 \beta_{3} - 34 \beta_{2} - 62) q^{25} + (11 \beta_{3} + 84 \beta_{2} + 87 \beta_1 + 83) q^{26} + 27 q^{27} + ( - 39 \beta_{3} - 98 \beta_{2} + 18 \beta_1 - 63) q^{28} + ( - 26 \beta_{3} + 12 \beta_{2} + 3 \beta_1 + 181) q^{29} + (33 \beta_{2} + 18 \beta_1 + 9) q^{30} + (27 \beta_{3} - 20 \beta_{2} + 12 \beta_1 - 69) q^{31} + ( - 41 \beta_{3} - 26 \beta_{2} - 5 \beta_1 - 4) q^{32} + ( - 39 \beta_{3} + \beta_{2} - 24 \beta_1 - 63) q^{34} + ( - 8 \beta_{3} - 119 \beta_{2} - 29 \beta_1 + 119) q^{35} + (27 \beta_{3} + 18 \beta_{2} + 27 \beta_1 - 18) q^{36} + (21 \beta_{3} + 173 \beta_{2} - 96 \beta_1 + 161) q^{37} + (88 \beta_{3} + 43 \beta_{2} + 85 \beta_1 + 188) q^{38} + (45 \beta_{3} - 75 \beta_{2} + 54 \beta_1 + 63) q^{39} + (39 \beta_{3} - 54 \beta_{2} + 18 \beta_1 + 82) q^{40} + ( - 97 \beta_{3} + 99 \beta_{2} - 114 \beta_1 + 101) q^{41} + ( - 42 \beta_{3} + 39 \beta_{2} - 6 \beta_1 - 66) q^{42} + ( - 135 \beta_{3} + 21 \beta_{2} - 66 \beta_1 + 175) q^{43} + (27 \beta_{3} + 36 \beta_{2} - 18 \beta_1 + 36) q^{45} + ( - 15 \beta_{3} - 120 \beta_{2} - 63 \beta_1 - 39) q^{46} + ( - 20 \beta_{3} + 177 \beta_{2} - 87 \beta_1 - 38) q^{47} + ( - 54 \beta_{3} + 51 \beta_{2} + 9 \beta_1 + 123) q^{48} + (45 \beta_{3} - 299 \beta_{2} - 51 \beta_1 + 13) q^{49} + ( - 34 \beta_{3} - 71 \beta_{2} - 68 \beta_1 - 99) q^{50} + (33 \beta_{2} - 75 \beta_1 + 78) q^{51} + (138 \beta_{3} + 403 \beta_{2} + 135 \beta_1 + 358) q^{52} + ( - 15 \beta_{3} - 172 \beta_{2} + 38 \beta_1 - 205) q^{53} + (27 \beta_{2} + 27 \beta_1 + 27) q^{54} + (50 \beta_{3} - 266 \beta_{2} - 89 \beta_1 + 48) q^{56} + ( - 45 \beta_{3} + 102 \beta_{2} + 81 \beta_1 + 183) q^{57} + (18 \beta_{3} + 106 \beta_{2} + 135 \beta_1 + 179) q^{58} + (68 \beta_{3} + 303 \beta_{2} - 6 \beta_1 - 130) q^{59} + ( - 3 \beta_{3} - 69 \beta_{2} + 93 \beta_1 + 18) q^{60} + ( - 3 \beta_{3} - 238 \beta_{2} - 21 \beta_1 - 9) q^{61} + (4 \beta_{3} + 24 \beta_{2} + 9 \beta_1 - 14) q^{62} + (54 \beta_{3} + 36 \beta_{2} - 81 \beta_1 + 36) q^{63} + (108 \beta_{3} - 268 \beta_{2} - 120 \beta_1 - 419) q^{64} + (269 \beta_{3} + 30 \beta_{2} + 3 \beta_1 - 1) q^{65} + (3 \beta_{3} - 186 \beta_{2} - 33 \beta_1 - 151) q^{67} + ( - 47 \beta_{3} - 292 \beta_{2} + 11 \beta_1 - 405) q^{68} + ( - 117 \beta_{3} - 153 \beta_{2} + 54 \beta_1 - 63) q^{69} + ( - 177 \beta_{3} + 66 \beta_{2} + 45 \beta_1 - 124) q^{70} + ( - 369 \beta_{3} + 190 \beta_{2} - 161 \beta_1 + 379) q^{71} + (72 \beta_{3} + 18 \beta_{2} + 18 \beta_1 + 63) q^{72} + (18 \beta_{3} + 250 \beta_{2} + 57 \beta_1 + 679) q^{73} + ( - 19 \beta_{3} + 128 \beta_{2} + 11 \beta_1 - 29) q^{74} + ( - 9 \beta_{3} - 102 \beta_{2} - 186) q^{75} + (333 \beta_{3} + 265 \beta_{2} + 318 \beta_1 + 171) q^{76} + (33 \beta_{3} + 252 \beta_{2} + 261 \beta_1 + 249) q^{78} + (45 \beta_{3} - 307 \beta_{2} - 99 \beta_1 + 360) q^{79} + ( - 10 \beta_{3} + 401 \beta_{2} - 52 \beta_1 + 91) q^{80} + 81 q^{81} + ( - 129 \beta_{3} - 304 \beta_{2} - 321 \beta_1 - 353) q^{82} + (27 \beta_{3} + 491 \beta_{2} + 167 \beta_1 + 350) q^{83} + ( - 117 \beta_{3} - 294 \beta_{2} + 54 \beta_1 - 189) q^{84} + ( - 27 \beta_{3} - 71 \beta_{2} - 144 \beta_1 + 273) q^{85} + ( - 111 \beta_{3} - 296 \beta_{2} - 227 \beta_1 - 203) q^{86} + ( - 78 \beta_{3} + 36 \beta_{2} + 9 \beta_1 + 543) q^{87} + (345 \beta_{3} - 495 \beta_{2} + 282 \beta_1 - 267) q^{89} + (99 \beta_{2} + 54 \beta_1 + 27) q^{90} + (501 \beta_{3} - 339 \beta_{2} - 396 \beta_1 - 421) q^{91} + (66 \beta_{3} + 261 \beta_{2} - 339 \beta_1 - 258) q^{92} + (81 \beta_{3} - 60 \beta_{2} + 36 \beta_1 - 207) q^{93} + (3 \beta_{3} - 185 \beta_{2} - 252 \beta_1 - 229) q^{94} + (22 \beta_{3} + 613 \beta_{2} + 85 \beta_1 + 140) q^{95} + ( - 123 \beta_{3} - 78 \beta_{2} - 15 \beta_1 - 12) q^{96} + ( - 177 \beta_{3} + 148 \beta_{2} - 6 \beta_1 + 522) q^{97} + ( - 401 \beta_{3} + 97 \beta_{2} + \beta_1 - 445) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} + 12 q^{3} - 3 q^{4} + 12 q^{5} + 9 q^{6} + 11 q^{7} + 42 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{2} + 12 q^{3} - 3 q^{4} + 12 q^{5} + 9 q^{6} + 11 q^{7} + 42 q^{8} + 36 q^{9} - 4 q^{10} - 9 q^{12} + 182 q^{13} - 144 q^{14} + 36 q^{15} + 97 q^{16} + 57 q^{17} + 27 q^{18} + 173 q^{19} + 99 q^{20} + 33 q^{21} - 42 q^{23} + 126 q^{24} - 186 q^{25} + 273 q^{26} + 108 q^{27} - 116 q^{28} + 651 q^{29} - 12 q^{30} - 170 q^{31} - 51 q^{32} - 356 q^{34} + 669 q^{35} - 27 q^{36} + 244 q^{37} + 927 q^{38} + 546 q^{39} + 532 q^{40} - 102 q^{41} - 432 q^{42} + 322 q^{43} + 108 q^{45} - 9 q^{46} - 633 q^{47} + 291 q^{48} + 689 q^{49} - 390 q^{50} + 171 q^{51} + 1037 q^{52} - 468 q^{53} + 81 q^{54} + 735 q^{56} + 519 q^{57} + 675 q^{58} - 996 q^{59} + 297 q^{60} + 413 q^{61} - 87 q^{62} + 99 q^{63} - 1044 q^{64} + 477 q^{65} - 259 q^{67} - 1119 q^{68} - 126 q^{69} - 937 q^{70} + 237 q^{71} + 378 q^{72} + 2309 q^{73} - 399 q^{74} - 558 q^{75} + 1138 q^{76} + 819 q^{78} + 2045 q^{79} - 510 q^{80} + 324 q^{81} - 1383 q^{82} + 639 q^{83} - 348 q^{84} + 1036 q^{85} - 669 q^{86} + 1953 q^{87} + 894 q^{89} - 36 q^{90} - 400 q^{91} - 1761 q^{92} - 510 q^{93} - 792 q^{94} - 537 q^{95} - 153 q^{96} + 1432 q^{97} - 2775 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 8x^{2} + x + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 2\nu^{2} - 4\nu + 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 4\nu^{2} + 2\nu - 11 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 4\beta_{2} + 6\beta _1 + 5 \) 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
−1.88301
−1.48718
1.26498
3.10522
−2.50105 3.00000 −1.74477 10.4342 −7.50314 32.7556 24.3721 9.00000 −26.0964
1.2 0.130849 3.00000 −7.98288 6.68911 0.392548 14.3420 −2.09135 9.00000 0.875265
1.3 0.646943 3.00000 −7.58146 −11.1424 1.94083 −26.1376 −10.0803 9.00000 −7.20851
1.4 4.72325 3.00000 14.3091 6.01909 14.1698 −9.96005 29.7996 9.00000 28.4297
\(n\): 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.t 4
3.b odd 2 1 1089.4.a.z 4
11.b odd 2 1 363.4.a.p 4
11.d odd 10 2 33.4.e.b 8
33.d even 2 1 1089.4.a.bg 4
33.f even 10 2 99.4.f.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.e.b 8 11.d odd 10 2
99.4.f.b 8 33.f even 10 2
363.4.a.p 4 11.b odd 2 1
363.4.a.t 4 1.a even 1 1 trivial
1089.4.a.z 4 3.b odd 2 1
1089.4.a.bg 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(363))\):

\( T_{2}^{4} - 3T_{2}^{3} - 10T_{2}^{2} + 9T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{4} - 12T_{5}^{3} - 85T_{5}^{2} + 1506T_{5} - 4681 \) Copy content Toggle raw display
\( T_{7}^{4} - 11T_{7}^{3} - 970T_{7}^{2} + 4697T_{7} + 122299 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 3 T^{3} - 10 T^{2} + 9 T - 1 \) Copy content Toggle raw display
$3$ \( (T - 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 12 T^{3} - 85 T^{2} + \cdots - 4681 \) Copy content Toggle raw display
$7$ \( T^{4} - 11 T^{3} - 970 T^{2} + \cdots + 122299 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 182 T^{3} + \cdots - 12384284 \) Copy content Toggle raw display
$17$ \( T^{4} - 57 T^{3} - 3631 T^{2} + \cdots + 4061936 \) Copy content Toggle raw display
$19$ \( T^{4} - 173 T^{3} + \cdots - 48930764 \) Copy content Toggle raw display
$23$ \( T^{4} + 42 T^{3} - 20241 T^{2} + \cdots - 46471644 \) Copy content Toggle raw display
$29$ \( T^{4} - 651 T^{3} + \cdots + 515311796 \) Copy content Toggle raw display
$31$ \( T^{4} + 170 T^{3} + 3521 T^{2} + \cdots + 1691239 \) Copy content Toggle raw display
$37$ \( T^{4} - 244 T^{3} + \cdots - 141225104 \) Copy content Toggle raw display
$41$ \( T^{4} + 102 T^{3} + \cdots - 108182044 \) Copy content Toggle raw display
$43$ \( T^{4} - 322 T^{3} + \cdots + 5520039844 \) Copy content Toggle raw display
$47$ \( T^{4} + 633 T^{3} + \cdots - 4530163504 \) Copy content Toggle raw display
$53$ \( T^{4} + 468 T^{3} + \cdots - 525556039 \) Copy content Toggle raw display
$59$ \( T^{4} + 996 T^{3} + \cdots + 192157031 \) Copy content Toggle raw display
$61$ \( T^{4} - 413 T^{3} + \cdots + 4008371596 \) Copy content Toggle raw display
$67$ \( T^{4} + 259 T^{3} + \cdots + 1798706704 \) Copy content Toggle raw display
$71$ \( T^{4} - 237 T^{3} + \cdots + 357347490524 \) Copy content Toggle raw display
$73$ \( T^{4} - 2309 T^{3} + \cdots + 52618219084 \) Copy content Toggle raw display
$79$ \( T^{4} - 2045 T^{3} + \cdots - 10584399491 \) Copy content Toggle raw display
$83$ \( T^{4} - 639 T^{3} + \cdots + 90842593811 \) Copy content Toggle raw display
$89$ \( T^{4} - 894 T^{3} + \cdots - 245710544796 \) Copy content Toggle raw display
$97$ \( T^{4} - 1432 T^{3} + \cdots - 31330298009 \) Copy content Toggle raw display
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