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Properties

Label 363.3.b.m
Level $363$
Weight $3$
Character orbit 363.b
Analytic conductor $9.891$
Analytic rank $0$
Dimension $8$
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] = [363,3,Mod(122,363)]
 
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([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.122");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 363.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(9.89103359628\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 29x^{6} + 282x^{4} + 1061x^{2} + 1331 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{5} + \beta_{4} - \beta_{2}) q^{3} + (\beta_{6} + \beta_{4} + \beta_{3} + \cdots - 4) q^{4}+ \cdots + (3 \beta_{7} + 4 \beta_{5} + \beta_{4} + \cdots + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{5} + \beta_{4} - \beta_{2}) q^{3} + (\beta_{6} + \beta_{4} + \beta_{3} + \cdots - 4) q^{4}+ \cdots + (23 \beta_{7} + 8 \beta_{6} + \cdots - 10 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 5 q^{3} - 26 q^{4} + q^{6} + 28 q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 5 q^{3} - 26 q^{4} + q^{6} + 28 q^{7} + 11 q^{9} - 6 q^{10} + 53 q^{12} + 44 q^{13} - 54 q^{15} - 14 q^{16} + q^{18} + 68 q^{19} - 6 q^{21} + 33 q^{24} + 42 q^{25} - 25 q^{27} - 118 q^{28} + 10 q^{30} + 2 q^{31} - 66 q^{34} - 7 q^{36} + 140 q^{37} + 38 q^{39} + 58 q^{40} + 174 q^{42} - 78 q^{43} - 36 q^{45} - 286 q^{46} - 285 q^{48} - 140 q^{49} + 58 q^{51} - 102 q^{52} + 523 q^{54} - 22 q^{57} - 68 q^{58} + 262 q^{60} + 22 q^{61} + 246 q^{63} - 52 q^{64} + 184 q^{67} - 176 q^{69} + 374 q^{70} + 489 q^{72} - 378 q^{73} - 33 q^{75} - 450 q^{76} - 246 q^{78} - 252 q^{79} + 11 q^{81} - 200 q^{82} + 450 q^{84} - 156 q^{85} + 66 q^{87} + 598 q^{90} - 148 q^{91} + 380 q^{93} - 460 q^{94} + 399 q^{96} - 324 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 29x^{6} + 282x^{4} + 1061x^{2} + 1331 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 30\nu^{4} + 260\nu^{2} + 541 ) / 52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{7} + 22\nu^{6} - 47\nu^{5} + 517\nu^{4} - 377\nu^{3} + 3575\nu^{2} - 1121\nu + 6897 ) / 286 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -27\nu^{7} + 11\nu^{6} - 706\nu^{5} + 330\nu^{4} - 5304\nu^{3} + 3432\nu^{2} - 9199\nu + 10527 ) / 1144 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{7} + 94\nu^{5} + 611\nu^{3} + 812\nu ) / 143 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 35\nu^{7} - 77\nu^{6} + 894\nu^{5} - 1738\nu^{4} + 6812\nu^{3} - 10868\nu^{2} + 13683\nu - 17061 ) / 1144 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -10\nu^{7} - 22\nu^{6} - 235\nu^{5} - 517\nu^{4} - 1599\nu^{3} - 3575\nu^{2} - 2745\nu - 6897 ) / 286 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} - 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} - 3\beta_{5} - 2\beta_{3} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - 15\beta_{6} + \beta_{5} - 15\beta_{4} - 16\beta_{3} + 23\beta_{2} + 85 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 33\beta_{7} + 8\beta_{6} + 36\beta_{5} - 8\beta_{4} + 41\beta_{3} + 113\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -30\beta_{7} + 190\beta_{6} - 30\beta_{5} + 190\beta_{4} + 220\beta_{3} - 378\beta_{2} - 1011 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -470\beta_{7} - 188\beta_{6} - 352\beta_{5} + 188\beta_{4} - 658\beta_{3} - 1331\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
122.1
− 3.58727i
− 3.06025i
− 2.00420i
− 1.65816i
1.65816i
2.00420i
3.06025i
3.58727i
− 3.58727i −2.74329 + 1.21424i −8.86854 2.08639i 4.35581 + 9.84093i 8.86303 17.4648i 6.05125 − 6.66201i 7.48447
122.2 − 3.06025i 0.126437 − 2.99733i −5.36516 − 6.63932i −9.17261 − 0.386930i −3.06298 4.17774i −8.96803 − 0.757950i −20.3180
122.3 − 2.00420i 2.80061 + 1.07544i −0.0168066 5.48438i 2.15539 − 5.61298i 5.59084 − 7.98311i 6.68687 + 6.02376i 10.9918
122.4 − 1.65816i 2.31624 + 1.90658i 1.25051 − 0.698503i 3.16141 − 3.84069i 2.60911 − 8.70618i 1.72990 + 8.83218i −1.15823
122.5 1.65816i 2.31624 − 1.90658i 1.25051 0.698503i 3.16141 + 3.84069i 2.60911 8.70618i 1.72990 − 8.83218i −1.15823
122.6 2.00420i 2.80061 − 1.07544i −0.0168066 − 5.48438i 2.15539 + 5.61298i 5.59084 7.98311i 6.68687 − 6.02376i 10.9918
122.7 3.06025i 0.126437 + 2.99733i −5.36516 6.63932i −9.17261 + 0.386930i −3.06298 − 4.17774i −8.96803 + 0.757950i −20.3180
122.8 3.58727i −2.74329 − 1.21424i −8.86854 − 2.08639i 4.35581 − 9.84093i 8.86303 − 17.4648i 6.05125 + 6.66201i 7.48447
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 122.8
Significant digits:
Format:

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 363.3.b.m 8
3.b odd 2 1 inner 363.3.b.m 8
11.b odd 2 1 363.3.b.l 8
11.c even 5 2 33.3.h.b ✓ 16
11.c even 5 2 363.3.h.o 16
11.d odd 10 2 363.3.h.j 16
11.d odd 10 2 363.3.h.n 16
33.d even 2 1 363.3.b.l 8
33.f even 10 2 363.3.h.j 16
33.f even 10 2 363.3.h.n 16
33.h odd 10 2 33.3.h.b ✓ 16
33.h odd 10 2 363.3.h.o 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.h.b ✓ 16 11.c even 5 2
33.3.h.b ✓ 16 33.h odd 10 2
363.3.b.l 8 11.b odd 2 1
363.3.b.l 8 33.d even 2 1
363.3.b.m 8 1.a even 1 1 trivial
363.3.b.m 8 3.b odd 2 1 inner
363.3.h.j 16 11.d odd 10 2
363.3.h.j 16 33.f even 10 2
363.3.h.n 16 11.d odd 10 2
363.3.h.n 16 33.f even 10 2
363.3.h.o 16 11.c even 5 2
363.3.h.o 16 33.h odd 10 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(363, [\chi])\):

\( T_{2}^{8} + 29T_{2}^{6} + 282T_{2}^{4} + 1061T_{2}^{2} + 1331 \) Copy content Toggle raw display
\( T_{5}^{8} + 79T_{5}^{6} + 1687T_{5}^{4} + 6576T_{5}^{2} + 2816 \) Copy content Toggle raw display
\( T_{7}^{4} - 14T_{7}^{3} + 35T_{7}^{2} + 138T_{7} - 396 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 29 T^{6} + \cdots + 1331 \) Copy content Toggle raw display
$3$ \( T^{8} - 5 T^{7} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{8} + 79 T^{6} + \cdots + 2816 \) Copy content Toggle raw display
$7$ \( (T^{4} - 14 T^{3} + \cdots - 396)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} - 22 T^{3} + \cdots + 4444)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 734 T^{6} + \cdots + 8306771 \) Copy content Toggle raw display
$19$ \( (T^{4} - 34 T^{3} + \cdots - 64999)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 25255373616 \) Copy content Toggle raw display
$29$ \( T^{8} + 184 T^{6} + \cdots + 340736 \) Copy content Toggle raw display
$31$ \( (T^{4} - T^{3} + \cdots - 97204)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 70 T^{3} + \cdots - 698036)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 3927187055891 \) Copy content Toggle raw display
$43$ \( (T^{4} + 39 T^{3} + \cdots - 684409)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 14778028911536 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 1733384590256 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 162457590843251 \) Copy content Toggle raw display
$61$ \( (T^{4} - 11 T^{3} + \cdots + 1180476)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 92 T^{3} + \cdots - 11977619)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 1657701936 \) Copy content Toggle raw display
$73$ \( (T^{4} + 189 T^{3} + \cdots + 2336356)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 126 T^{3} + \cdots - 27267284)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 46820268891 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 3248250664131 \) Copy content Toggle raw display
$97$ \( (T^{4} + 162 T^{3} + \cdots - 3010169)^{2} \) Copy content Toggle raw display
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