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Properties

Label 276.2.i.a
Level $276$
Weight $2$
Character orbit 276.i
Analytic conductor $2.204$
Analytic rank $0$
Dimension $20$
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] = [276,2,Mod(13,276)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(276, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 0, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("276.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 276 = 2^{2} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 276.i (of order \(11\), degree \(10\), 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: \(2.20387109579\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{11})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 8 x^{19} + 43 x^{18} - 165 x^{17} + 538 x^{16} - 1433 x^{15} + 3444 x^{14} - 7370 x^{13} + \cdots + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{17} q^{3} + ( - \beta_{18} - \beta_{16} + \cdots + \beta_{3}) q^{5}+ \cdots - \beta_{14} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{17} q^{3} + ( - \beta_{18} - \beta_{16} + \cdots + \beta_{3}) q^{5}+ \cdots + (\beta_{17} - \beta_{15} - \beta_{12} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{3} - 4 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{3} - 4 q^{5} - 2 q^{9} - 22 q^{13} + 7 q^{15} + 7 q^{17} + 19 q^{19} + 20 q^{23} + 20 q^{25} - 2 q^{27} + 32 q^{29} - 3 q^{31} + 11 q^{33} - 26 q^{35} - 10 q^{37} - 22 q^{39} - 40 q^{41} + 8 q^{43} - 4 q^{45} - 18 q^{47} - 34 q^{49} - 26 q^{51} - 34 q^{53} - 17 q^{55} - 3 q^{57} - 32 q^{59} + 32 q^{61} + 49 q^{65} + 35 q^{67} - 2 q^{69} + 33 q^{71} - q^{73} - 2 q^{75} - 50 q^{77} + 22 q^{79} - 2 q^{81} - 14 q^{83} - 9 q^{85} - 12 q^{87} + 10 q^{89} - 72 q^{91} + 30 q^{93} - 51 q^{95} - 4 q^{97} + 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 8 x^{19} + 43 x^{18} - 165 x^{17} + 538 x^{16} - 1433 x^{15} + 3444 x^{14} - 7370 x^{13} + \cdots + 529 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 17\!\cdots\!28 \nu^{19} + \cdots - 23\!\cdots\!38 ) / 18\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 28\!\cdots\!78 \nu^{19} + \cdots - 62\!\cdots\!66 ) / 18\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 32\!\cdots\!78 \nu^{19} + \cdots + 22\!\cdots\!33 ) / 18\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 39\!\cdots\!69 \nu^{19} + \cdots - 10\!\cdots\!84 ) / 18\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 39\!\cdots\!27 \nu^{19} + \cdots - 14\!\cdots\!85 ) / 18\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 44\!\cdots\!45 \nu^{19} + \cdots - 28\!\cdots\!56 ) / 18\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 48\!\cdots\!89 \nu^{19} + \cdots - 25\!\cdots\!91 ) / 18\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11\!\cdots\!54 \nu^{19} + \cdots + 27\!\cdots\!63 ) / 18\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 19\!\cdots\!96 \nu^{19} + \cdots + 88\!\cdots\!12 ) / 18\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 24\!\cdots\!93 \nu^{19} + \cdots + 33\!\cdots\!91 ) / 18\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 27\!\cdots\!65 \nu^{19} + \cdots - 15\!\cdots\!38 ) / 18\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 32\!\cdots\!22 \nu^{19} + \cdots + 75\!\cdots\!61 ) / 18\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 39\!\cdots\!53 \nu^{19} + \cdots + 36\!\cdots\!49 ) / 18\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 42\!\cdots\!77 \nu^{19} + \cdots + 12\!\cdots\!73 ) / 18\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 43\!\cdots\!22 \nu^{19} + \cdots - 37\!\cdots\!50 ) / 18\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 47\!\cdots\!79 \nu^{19} + \cdots - 21\!\cdots\!54 ) / 18\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 55\!\cdots\!03 \nu^{19} + \cdots - 32\!\cdots\!97 ) / 20\!\cdots\!83 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 39\!\cdots\!80 \nu^{19} + \cdots + 34\!\cdots\!99 ) / 62\!\cdots\!49 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{18} + 4\beta_{17} - \beta_{16} + \beta_{15} + 2\beta_{11} - \beta_{10} - 2\beta_{9} - \beta_{8} - \beta_{6} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + 2 \beta_{18} + 2 \beta_{16} + \beta_{14} - \beta_{13} - 2 \beta_{12} - \beta_{11} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{19} - 3 \beta_{18} - 2 \beta_{17} - 3 \beta_{16} + 13 \beta_{15} - 29 \beta_{14} + 6 \beta_{13} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{19} - 11 \beta_{18} + 10 \beta_{17} - 24 \beta_{16} - 58 \beta_{14} + 41 \beta_{13} + 23 \beta_{12} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 15 \beta_{19} + 92 \beta_{18} - 54 \beta_{17} - 15 \beta_{16} - 86 \beta_{15} + 44 \beta_{14} + \cdots - 59 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{19} + 81 \beta_{18} - 180 \beta_{17} + 216 \beta_{16} - 39 \beta_{15} + 127 \beta_{14} + \cdots - 20 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 332 \beta_{19} - 1181 \beta_{18} - 190 \beta_{16} - 142 \beta_{15} - 95 \beta_{14} + 95 \beta_{12} + \cdots + 446 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1972 \beta_{19} - 2836 \beta_{18} + 1640 \beta_{17} - 3235 \beta_{16} - 203 \beta_{15} - 45 \beta_{14} + \cdots - 203 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3473 \beta_{18} + 2239 \beta_{17} + 1891 \beta_{16} + 2239 \beta_{15} + 5064 \beta_{14} - 3855 \beta_{13} + \cdots - 1891 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 14352 \beta_{19} + 14352 \beta_{18} - 6649 \beta_{17} + 21001 \beta_{16} + 8645 \beta_{15} + 7806 \beta_{14} + \cdots + 5425 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 16140 \beta_{19} - 12711 \beta_{18} + 13271 \beta_{17} - 26763 \beta_{16} + 15857 \beta_{15} + \cdots + 13271 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 7830 \beta_{19} + 49326 \beta_{18} + 32552 \beta_{17} - 66024 \beta_{16} - 57156 \beta_{15} + \cdots - 66283 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 118706 \beta_{19} + 481841 \beta_{18} - 365341 \beta_{17} + 481841 \beta_{16} - 235616 \beta_{15} + \cdots - 149143 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 246225 \beta_{19} - 234914 \beta_{18} - 521536 \beta_{17} + 699115 \beta_{16} - 43151 \beta_{15} + \cdots + 481139 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 1359003 \beta_{19} - 3875898 \beta_{18} + 2516895 \beta_{17} - 3358624 \beta_{16} - 3284227 \beta_{14} + \cdots + 815310 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 3426712 \beta_{19} + 593231 \beta_{18} + 4013236 \beta_{17} - 3426712 \beta_{16} + 84365 \beta_{15} + \cdots - 3472966 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 5253987 \beta_{19} + 14241220 \beta_{18} - 10610971 \beta_{17} + 21841851 \beta_{16} + \cdots - 10912914 \beta_1 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 1170363 \beta_{19} - 50581146 \beta_{18} - 28971841 \beta_{16} + 27801478 \beta_{15} + \cdots + 29086355 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/276\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(139\) \(185\)
\(\chi(n)\) \(-1 + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} + \beta_{18}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
1.84381 + 0.541390i
−0.962045 − 0.282482i
−1.54238 + 1.78001i
1.74521 − 2.01408i
−0.0616736 + 0.0396352i
2.31834 − 1.48991i
0.302381 + 2.10310i
−0.404188 − 2.81119i
1.84381 − 0.541390i
−0.962045 + 0.282482i
0.302381 − 2.10310i
−0.404188 + 2.81119i
−0.262998 − 0.575885i
1.02355 + 2.24127i
−0.0616736 − 0.0396352i
2.31834 + 1.48991i
−0.262998 + 0.575885i
1.02355 − 2.24127i
−1.54238 − 1.78001i
1.74521 + 2.01408i
0 0.841254 + 0.540641i 0 −1.21471 + 2.65985i 0 0.960219 + 0.281946i 0 0.415415 + 0.909632i 0
13.2 0 0.841254 + 0.540641i 0 1.11647 − 2.44474i 0 0.161591 + 0.0474474i 0 0.415415 + 0.909632i 0
25.1 0 −0.142315 − 0.989821i 0 −2.38954 − 0.701632i 0 2.64891 − 3.05701i 0 −0.959493 + 0.281733i 0
25.2 0 −0.142315 − 0.989821i 0 3.91931 + 1.15081i 0 −0.0825209 + 0.0952342i 0 −0.959493 + 0.281733i 0
49.1 0 0.415415 − 0.909632i 0 −2.38152 + 2.74842i 0 −3.67577 + 2.36227i 0 −0.654861 − 0.755750i 0
49.2 0 0.415415 − 0.909632i 0 0.735636 − 0.848969i 0 0.891451 − 0.572901i 0 −0.654861 − 0.755750i 0
73.1 0 −0.959493 + 0.281733i 0 −1.52130 − 0.977682i 0 0.485296 + 3.37531i 0 0.841254 − 0.540641i 0
73.2 0 −0.959493 + 0.281733i 0 −0.332496 − 0.213682i 0 −0.440112 − 3.06105i 0 0.841254 − 0.540641i 0
85.1 0 0.841254 − 0.540641i 0 −1.21471 − 2.65985i 0 0.960219 − 0.281946i 0 0.415415 − 0.909632i 0
85.2 0 0.841254 − 0.540641i 0 1.11647 + 2.44474i 0 0.161591 − 0.0474474i 0 0.415415 − 0.909632i 0
121.1 0 −0.959493 − 0.281733i 0 −1.52130 + 0.977682i 0 0.485296 − 3.37531i 0 0.841254 + 0.540641i 0
121.2 0 −0.959493 − 0.281733i 0 −0.332496 + 0.213682i 0 −0.440112 + 3.06105i 0 0.841254 + 0.540641i 0
133.1 0 −0.654861 + 0.755750i 0 −0.149019 + 1.03645i 0 0.607780 + 1.33085i 0 −0.142315 − 0.989821i 0
133.2 0 −0.654861 + 0.755750i 0 0.217172 − 1.51046i 0 −1.55685 − 3.40903i 0 −0.142315 − 0.989821i 0
169.1 0 0.415415 + 0.909632i 0 −2.38152 − 2.74842i 0 −3.67577 − 2.36227i 0 −0.654861 + 0.755750i 0
169.2 0 0.415415 + 0.909632i 0 0.735636 + 0.848969i 0 0.891451 + 0.572901i 0 −0.654861 + 0.755750i 0
193.1 0 −0.654861 − 0.755750i 0 −0.149019 − 1.03645i 0 0.607780 − 1.33085i 0 −0.142315 + 0.989821i 0
193.2 0 −0.654861 − 0.755750i 0 0.217172 + 1.51046i 0 −1.55685 + 3.40903i 0 −0.142315 + 0.989821i 0
265.1 0 −0.142315 + 0.989821i 0 −2.38954 + 0.701632i 0 2.64891 + 3.05701i 0 −0.959493 − 0.281733i 0
265.2 0 −0.142315 + 0.989821i 0 3.91931 − 1.15081i 0 −0.0825209 − 0.0952342i 0 −0.959493 − 0.281733i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 276.2.i.a ✓ 20
3.b odd 2 1 828.2.q.c 20
23.c even 11 1 inner 276.2.i.a ✓ 20
23.c even 11 1 6348.2.a.s 10
23.d odd 22 1 6348.2.a.t 10
69.h odd 22 1 828.2.q.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.2.i.a ✓ 20 1.a even 1 1 trivial
276.2.i.a ✓ 20 23.c even 11 1 inner
828.2.q.c 20 3.b odd 2 1
828.2.q.c 20 69.h odd 22 1
6348.2.a.s 10 23.c even 11 1
6348.2.a.t 10 23.d odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} + 4 T_{5}^{19} + 3 T_{5}^{18} - 62 T_{5}^{17} - 199 T_{5}^{16} - 188 T_{5}^{15} + \cdots + 139129 \) acting on \(S_{2}^{\mathrm{new}}(276, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{10} + T^{9} + T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} + 4 T^{19} + \cdots + 139129 \) Copy content Toggle raw display
$7$ \( T^{20} + 24 T^{18} + \cdots + 529 \) Copy content Toggle raw display
$11$ \( T^{20} + 10 T^{18} + \cdots + 36590401 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 746546329 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 181252369 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 512524321 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 41426511213649 \) Copy content Toggle raw display
$29$ \( T^{20} - 32 T^{19} + \cdots + 94249 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 1366115521 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 41\!\cdots\!41 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 389752131263881 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 577670697717961 \) Copy content Toggle raw display
$47$ \( (T^{10} + 9 T^{9} + \cdots - 337853)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 11\!\cdots\!49 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 1812553308721 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 16910706981169 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 24475292457001 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 14637426640321 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 41\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 684944324016481 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 12\!\cdots\!21 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 68\!\cdots\!09 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 58\!\cdots\!61 \) Copy content Toggle raw display
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