Properties

Label 363.6.a.r
Level $363$
Weight $6$
Character orbit 363.a
Self dual yes
Analytic conductor $58.219$
Analytic rank $1$
Dimension $10$
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] = [363,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(58.2193265921\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3 x^{9} - 228 x^{8} + 523 x^{7} + 17396 x^{6} - 31445 x^{5} - 508100 x^{4} + 960757 x^{3} + 4870759 x^{2} - 11540360 x + 5059564 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 11^{4} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} - 9 q^{3} + ( - \beta_{3} + \beta_{2} - \beta_1 + 15) q^{4} + (\beta_{8} - \beta_{2} - \beta_1 - 3) q^{5} + ( - 9 \beta_1 + 9) q^{6} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + 4 \beta_1 + 7) q^{7} + ( - \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} + 15 \beta_1 - 48) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} - 9 q^{3} + ( - \beta_{3} + \beta_{2} - \beta_1 + 15) q^{4} + (\beta_{8} - \beta_{2} - \beta_1 - 3) q^{5} + ( - 9 \beta_1 + 9) q^{6} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + 4 \beta_1 + 7) q^{7} + ( - \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} + 15 \beta_1 - 48) q^{8} + 81 q^{9} + (\beta_{9} - \beta_{8} + \beta_{5} - 2 \beta_{4} + 11 \beta_{3} + \beta_{2} - 25 \beta_1 - 8) q^{10} + (9 \beta_{3} - 9 \beta_{2} + 9 \beta_1 - 135) q^{12} + (\beta_{9} - \beta_{8} + 4 \beta_{6} + 3 \beta_{4} - 12 \beta_{3} + 2 \beta_{2} + 17 \beta_1 - 112) q^{13} + ( - 4 \beta_{9} + 4 \beta_{8} - 7 \beta_{7} + 6 \beta_{6} + 8 \beta_{5} + \cdots + 144) q^{14}+ \cdots + ( - 367 \beta_{9} + 274 \beta_{8} - 309 \beta_{7} + 113 \beta_{6} + \cdots - 117918) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 7 q^{2} - 90 q^{3} + 149 q^{4} - 33 q^{5} + 63 q^{6} + 78 q^{7} - 438 q^{8} + 810 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 7 q^{2} - 90 q^{3} + 149 q^{4} - 33 q^{5} + 63 q^{6} + 78 q^{7} - 438 q^{8} + 810 q^{9} - 212 q^{10} - 1341 q^{12} - 1016 q^{13} + 1566 q^{14} + 297 q^{15} + 2361 q^{16} - 1669 q^{17} - 567 q^{18} + 2929 q^{19} - 10189 q^{20} - 702 q^{21} + 4070 q^{23} + 3942 q^{24} + 2425 q^{25} + 8481 q^{26} - 7290 q^{27} + 3272 q^{28} - 11940 q^{29} + 1908 q^{30} - 16085 q^{31} - 2313 q^{32} + 8270 q^{34} + 6987 q^{35} + 12069 q^{36} + 16136 q^{37} + 10721 q^{38} + 9144 q^{39} + 9332 q^{40} - 16278 q^{41} - 14094 q^{42} - 10844 q^{43} - 2673 q^{45} - 25995 q^{46} - 22411 q^{47} - 21249 q^{48} + 75150 q^{49} + 738 q^{50} + 15021 q^{51} - 8677 q^{52} - 27511 q^{53} + 5103 q^{54} + 84447 q^{56} - 26361 q^{57} + 16853 q^{58} - 39641 q^{59} + 91701 q^{60} - 3509 q^{61} - 227845 q^{62} + 6318 q^{63} - 22980 q^{64} - 67097 q^{65} + 10089 q^{67} - 273621 q^{68} - 36630 q^{69} - 38919 q^{70} + 60681 q^{71} - 35478 q^{72} - 133740 q^{73} - 317933 q^{74} - 21825 q^{75} + 23434 q^{76} - 76329 q^{78} - 12386 q^{79} - 289014 q^{80} + 65610 q^{81} + 385033 q^{82} - 187242 q^{83} - 29448 q^{84} - 191504 q^{85} - 43793 q^{86} + 107460 q^{87} + 102746 q^{89} - 17172 q^{90} - 435248 q^{91} + 30867 q^{92} + 144765 q^{93} - 404734 q^{94} - 648147 q^{95} + 20817 q^{96} - 120631 q^{97} - 1148087 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3 x^{9} - 228 x^{8} + 523 x^{7} + 17396 x^{6} - 31445 x^{5} - 508100 x^{4} + 960757 x^{3} + 4870759 x^{2} - 11540360 x + 5059564 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10753 \nu^{9} - 296252 \nu^{8} - 1630248 \nu^{7} + 57573155 \nu^{6} + 73965785 \nu^{5} - 3358085302 \nu^{4} - 1435937310 \nu^{3} + \cdots - 247021531028 ) / 3973778688 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10753 \nu^{9} - 296252 \nu^{8} - 1630248 \nu^{7} + 57573155 \nu^{6} + 73965785 \nu^{5} - 3358085302 \nu^{4} - 1435937310 \nu^{3} + \cdots - 64227711380 ) / 3973778688 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 519533 \nu^{9} - 14486692 \nu^{8} - 212554608 \nu^{7} + 3386645791 \nu^{6} + 29011252981 \nu^{5} - 231974286038 \nu^{4} + \cdots - 9403256470084 ) / 115239581952 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 48247 \nu^{9} - 4026100 \nu^{8} + 18089376 \nu^{7} + 853512643 \nu^{6} - 1526550335 \nu^{5} - 54131712926 \nu^{4} + \cdots - 749819187028 ) / 10476325632 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 454029 \nu^{9} + 7069700 \nu^{8} - 152938312 \nu^{7} - 1444123929 \nu^{6} + 15982771477 \nu^{5} + 88696694130 \nu^{4} + \cdots - 3502807289892 ) / 38413193984 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1679081 \nu^{9} + 25969108 \nu^{8} + 188783208 \nu^{7} - 4628495563 \nu^{6} + 6141870071 \nu^{5} + 236898347246 \nu^{4} + \cdots - 24594764837372 ) / 57619790976 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2254471 \nu^{9} + 653136 \nu^{8} + 491336420 \nu^{7} + 263165775 \nu^{6} - 33881768431 \nu^{5} - 37321971170 \nu^{4} + \cdots + 2117426070700 ) / 28809895488 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 18645735 \nu^{9} - 4735412 \nu^{8} + 4158442448 \nu^{7} + 3646561091 \nu^{6} - 294057013535 \nu^{5} - 336750092382 \nu^{4} + \cdots + 58402093950476 ) / 115239581952 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + \beta _1 + 46 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2} + 79\beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{9} - 6 \beta_{8} - 2 \beta_{7} + 4 \beta_{6} + 4 \beta_{5} + \beta_{4} - 141 \beta_{3} + 113 \beta_{2} + 189 \beta _1 + 3545 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 17 \beta_{9} - 50 \beta_{8} + 22 \beta_{7} - 168 \beta_{6} - 124 \beta_{5} + 131 \beta_{4} - 186 \beta_{3} + 298 \beta_{2} + 7311 \beta _1 + 6463 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 391 \beta_{9} - 886 \beta_{8} - 234 \beta_{7} + 293 \beta_{6} + 709 \beta_{5} + 236 \beta_{4} - 17026 \beta_{3} + 11797 \beta_{2} + 25281 \beta _1 + 321948 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3378 \beta_{9} - 10236 \beta_{8} + 4192 \beta_{7} - 22566 \beta_{6} - 12442 \beta_{5} + 14408 \beta_{4} - 34284 \beta_{3} + 38786 \beta_{2} + 723351 \beta _1 + 914046 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 45416 \beta_{9} - 115856 \beta_{8} - 19212 \beta_{7} - 2084 \beta_{6} + 92880 \beta_{5} + 39144 \beta_{4} - 1964949 \beta_{3} + 1225193 \beta_{2} + 3066769 \beta _1 + 31352122 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 489844 \beta_{9} - 1529824 \beta_{8} + 583196 \beta_{7} - 2776141 \beta_{6} - 1154921 \beta_{5} + 1543973 \beta_{4} - 5524037 \beta_{3} + 4811154 \beta_{2} + 73875371 \beta _1 + 113088743 \) 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
−9.84718
−7.59133
−5.81452
−5.33226
0.605686
1.70765
4.35046
4.81631
9.48352
10.6217
−10.8472 −9.00000 85.6613 −38.7942 97.6246 −219.048 −582.073 81.0000 420.808
1.2 −8.59133 −9.00000 41.8109 −68.5404 77.3220 242.743 −84.2889 81.0000 588.853
1.3 −6.81452 −9.00000 14.4377 77.3762 61.3307 192.167 119.679 81.0000 −527.281
1.4 −6.33226 −9.00000 8.09756 28.0024 56.9904 −150.231 151.357 81.0000 −177.319
1.5 −0.394314 −9.00000 −31.8445 21.3907 3.54883 −169.303 25.1748 81.0000 −8.43465
1.6 0.707653 −9.00000 −31.4992 −68.0022 −6.36887 −121.575 −44.9354 81.0000 −48.1219
1.7 3.35046 −9.00000 −20.7744 106.493 −30.1541 40.5248 −176.818 81.0000 356.801
1.8 3.81631 −9.00000 −17.4358 −4.66892 −34.3468 159.925 −188.662 81.0000 −17.8180
1.9 8.48352 −9.00000 39.9701 −26.7516 −76.3517 72.2591 67.6142 81.0000 −226.948
1.10 9.62167 −9.00000 60.5765 −59.5053 −86.5950 30.5377 274.954 81.0000 −572.540
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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 363.6.a.r 10
3.b odd 2 1 1089.6.a.bk 10
11.b odd 2 1 363.6.a.t 10
11.c even 5 2 33.6.e.b 20
33.d even 2 1 1089.6.a.bi 10
33.h odd 10 2 99.6.f.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.e.b 20 11.c even 5 2
99.6.f.b 20 33.h odd 10 2
363.6.a.r 10 1.a even 1 1 trivial
363.6.a.t 10 11.b odd 2 1
1089.6.a.bi 10 33.d even 2 1
1089.6.a.bk 10 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 7 T_{2}^{9} - 210 T_{2}^{8} - 1289 T_{2}^{7} + 14631 T_{2}^{6} + 71020 T_{2}^{5} - 402208 T_{2}^{4} - 1032768 T_{2}^{3} + 4655456 T_{2}^{2} - 1000000 T_{2} - 1171136 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(363))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 7 T^{9} - 210 T^{8} + \cdots - 1171136 \) Copy content Toggle raw display
$3$ \( (T + 9)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 33 T^{9} + \cdots + 66\!\cdots\!81 \) Copy content Toggle raw display
$7$ \( T^{10} - 78 T^{9} + \cdots + 45\!\cdots\!81 \) Copy content Toggle raw display
$11$ \( T^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + 1016 T^{9} + \cdots - 38\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{10} + 1669 T^{9} + \cdots - 31\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{10} - 2929 T^{9} + \cdots - 14\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{10} - 4070 T^{9} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{10} + 11940 T^{9} + \cdots + 36\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{10} + 16085 T^{9} + \cdots + 95\!\cdots\!01 \) Copy content Toggle raw display
$37$ \( T^{10} - 16136 T^{9} + \cdots + 38\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( T^{10} + 16278 T^{9} + \cdots - 73\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{10} + 10844 T^{9} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} + 22411 T^{9} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{10} + 27511 T^{9} + \cdots - 40\!\cdots\!69 \) Copy content Toggle raw display
$59$ \( T^{10} + 39641 T^{9} + \cdots + 13\!\cdots\!05 \) Copy content Toggle raw display
$61$ \( T^{10} + 3509 T^{9} + \cdots - 35\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{10} - 10089 T^{9} + \cdots + 33\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{10} - 60681 T^{9} + \cdots - 17\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{10} + 133740 T^{9} + \cdots + 27\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{10} + 12386 T^{9} + \cdots + 23\!\cdots\!45 \) Copy content Toggle raw display
$83$ \( T^{10} + 187242 T^{9} + \cdots + 27\!\cdots\!19 \) Copy content Toggle raw display
$89$ \( T^{10} - 102746 T^{9} + \cdots - 11\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{10} + 120631 T^{9} + \cdots - 69\!\cdots\!75 \) Copy content Toggle raw display
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