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Properties

Label 55.5.c.a
Level $55$
Weight $5$
Character orbit 55.c
Analytic conductor $5.685$
Analytic rank $0$
Dimension $16$
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] = [55,5,Mod(21,55)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(55, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("55.21");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 55.c (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: \(5.68534796961\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 198 x^{14} + 15656 x^{12} + 635290 x^{10} + 14206695 x^{8} + 175523080 x^{6} + \cdots + 4034278000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 5^{5} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{5} - 1) q^{3} + (\beta_{2} - 9) q^{4} + \beta_{4} q^{5} + (\beta_{10} - 2 \beta_1) q^{6} + (\beta_{13} + 3 \beta_1) q^{7} + (\beta_{3} - 8 \beta_1) q^{8} + ( - \beta_{8} - 2 \beta_{6} + \cdots + 52) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{5} - 1) q^{3} + (\beta_{2} - 9) q^{4} + \beta_{4} q^{5} + (\beta_{10} - 2 \beta_1) q^{6} + (\beta_{13} + 3 \beta_1) q^{7} + (\beta_{3} - 8 \beta_1) q^{8} + ( - \beta_{8} - 2 \beta_{6} + \cdots + 52) q^{9}+ \cdots + (3 \beta_{15} + 14 \beta_{14} + \cdots - 201) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{3} - 140 q^{4} + 836 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{3} - 140 q^{4} + 836 q^{9} - 312 q^{11} + 384 q^{12} - 1056 q^{14} - 100 q^{15} + 872 q^{16} + 300 q^{20} - 1280 q^{22} - 2736 q^{23} + 2000 q^{25} + 4476 q^{26} - 2032 q^{27} + 1476 q^{31} - 1376 q^{33} - 6176 q^{34} - 9528 q^{36} + 3544 q^{37} + 10080 q^{38} + 9160 q^{42} + 2412 q^{44} - 2976 q^{47} - 21016 q^{48} - 7420 q^{49} - 5016 q^{53} - 1300 q^{55} + 19632 q^{56} + 29120 q^{58} - 1440 q^{59} + 16500 q^{60} - 17196 q^{64} + 1140 q^{66} + 25744 q^{67} + 152 q^{69} - 18900 q^{70} - 44268 q^{71} - 2000 q^{75} + 25320 q^{77} - 59360 q^{78} + 10800 q^{80} + 43288 q^{81} - 3320 q^{82} - 43404 q^{86} + 5560 q^{88} + 28716 q^{89} - 21944 q^{91} + 42144 q^{92} + 63208 q^{93} - 47976 q^{97} - 3256 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 198 x^{14} + 15656 x^{12} + 635290 x^{10} + 14206695 x^{8} + 175523080 x^{6} + \cdots + 4034278000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 40\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13037 \nu^{14} - 2444141 \nu^{12} - 178939302 \nu^{10} - 6511155010 \nu^{8} + \cdots - 4415101764100 ) / 153272708160 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 6840173 \nu^{14} - 1402552868 \nu^{12} - 115499723200 \nu^{10} - 4876452668754 \nu^{8} + \cdots - 12\!\cdots\!60 ) / 71118536586240 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 41806991 \nu^{14} - 7841472748 \nu^{12} - 577038294336 \nu^{10} - 21241062834950 \nu^{8} + \cdots - 35\!\cdots\!60 ) / 71118536586240 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 177075185 \nu^{14} - 33319558468 \nu^{12} - 2443628585504 \nu^{10} + \cdots - 45\!\cdots\!40 ) / 71118536586240 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1295365 \nu^{14} + 247824372 \nu^{12} + 18589488736 \nu^{10} + 694499552962 \nu^{8} + \cdots + 988215893365400 ) / 497332423680 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13037 \nu^{15} + 2444141 \nu^{13} + 178939302 \nu^{11} + 6511155010 \nu^{9} + \cdots + 4261829055940 \nu ) / 153272708160 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 6840173 \nu^{15} - 1402552868 \nu^{13} - 115499723200 \nu^{11} + \cdots - 12\!\cdots\!20 \nu ) / 71118536586240 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 240403981 \nu^{15} + 51503827972 \nu^{13} + 4346737753408 \nu^{11} + \cdots + 19\!\cdots\!00 \nu ) / 22\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 239208323 \nu^{15} + 74930906 \nu^{14} + 46325111212 \nu^{13} + 18417554088 \nu^{12} + \cdots + 43\!\cdots\!40 ) / 400849933486080 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 768210351 \nu^{15} + 147294741580 \nu^{13} + 11117462438912 \nu^{11} + \cdots + 65\!\cdots\!00 \nu ) / 11\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3112099515 \nu^{15} + 824239966 \nu^{14} - 612583879276 \nu^{13} + 202593094968 \nu^{12} + \cdots + 48\!\cdots\!40 ) / 44\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 81670871 \nu^{15} - 16450843340 \nu^{13} - 1326814836352 \nu^{11} + \cdots - 14\!\cdots\!00 \nu ) / 71118536586240 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 25 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 40\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{14} - \beta_{12} - \beta_{11} + \beta_{8} + \beta_{7} - 8\beta_{5} + 6\beta_{4} - 57\beta_{2} + 1000 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3 \beta_{15} - 5 \beta_{14} - 7 \beta_{13} + 5 \beta_{12} - \beta_{11} - 12 \beta_{10} + \cdots + 1878 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 90 \beta_{14} + 90 \beta_{12} + 90 \beta_{11} - 82 \beta_{8} - 86 \beta_{7} - 24 \beta_{6} + \cdots - 46771 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 262 \beta_{15} + 422 \beta_{14} + 622 \beta_{13} - 422 \beta_{12} + 78 \beta_{11} + \cdots - 94074 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 5957 \beta_{14} - 5957 \beta_{12} - 5957 \beta_{11} + 5273 \beta_{8} + 5717 \beta_{7} + \cdots + 2334516 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 17187 \beta_{15} - 27697 \beta_{14} - 42383 \beta_{13} + 27697 \beta_{12} - 4685 \beta_{11} + \cdots + 4864810 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 353438 \beta_{14} + 353438 \beta_{12} + 353438 \beta_{11} - 316858 \beta_{8} - 347754 \beta_{7} + \cdots - 120417567 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1023734 \beta_{15} + 1676978 \beta_{14} + 2634262 \beta_{13} - 1676978 \beta_{12} + \cdots - 255948194 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 19944117 \beta_{14} - 19944117 \beta_{12} - 19944117 \beta_{11} + 18597101 \beta_{8} + \cdots + 6325328260 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 58485335 \beta_{15} - 98089897 \beta_{14} - 157031963 \beta_{13} + 98089897 \beta_{12} + \cdots + 13599179162 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1096593098 \beta_{14} + 1096593098 \beta_{12} + 1096593098 \beta_{11} - 1080742730 \beta_{8} + \cdots - 335776291283 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3273928926 \beta_{15} + 5639401078 \beta_{14} + 9161660342 \beta_{13} - 5639401078 \beta_{12} + \cdots - 726777317922 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/55\mathbb{Z}\right)^\times\).

\(n\) \(12\) \(46\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1
− 7.40793i
− 7.24168i
− 5.74801i
− 5.16959i
− 3.78312i
− 3.12956i
− 1.98078i
− 1.69904i
1.69904i
1.98078i
3.12956i
3.78312i
5.16959i
5.74801i
7.24168i
7.40793i
− 7.40793i 1.13523 −38.8775 −11.1803 − 8.40974i 23.6201i 169.475i −79.7112 82.8232i
21.2 − 7.24168i −16.1202 −36.4420 11.1803 116.737i − 72.0665i 148.034i 178.861 − 80.9645i
21.3 − 5.74801i 7.97807 −17.0396 11.1803 − 45.8580i − 49.5190i 5.97573i −17.3504 − 64.2647i
21.4 − 5.16959i 16.5915 −10.7246 −11.1803 − 85.7711i − 10.9903i − 27.2715i 194.277 57.7977i
21.5 − 3.78312i −15.5112 1.68801 −11.1803 58.6808i 62.7441i − 66.9158i 159.598 42.2966i
21.6 − 3.12956i −3.97942 6.20586 −11.1803 12.4538i − 62.9202i − 69.4945i −65.1642 34.9895i
21.7 − 1.98078i 11.1430 12.0765 11.1803 − 22.0719i 78.9677i − 55.6134i 43.1675 − 22.1458i
21.8 − 1.69904i −9.23697 13.1133 11.1803 15.6940i − 21.4921i − 49.4647i 4.32166 − 18.9959i
21.9 1.69904i −9.23697 13.1133 11.1803 − 15.6940i 21.4921i 49.4647i 4.32166 18.9959i
21.10 1.98078i 11.1430 12.0765 11.1803 22.0719i − 78.9677i 55.6134i 43.1675 22.1458i
21.11 3.12956i −3.97942 6.20586 −11.1803 − 12.4538i 62.9202i 69.4945i −65.1642 − 34.9895i
21.12 3.78312i −15.5112 1.68801 −11.1803 − 58.6808i − 62.7441i 66.9158i 159.598 − 42.2966i
21.13 5.16959i 16.5915 −10.7246 −11.1803 85.7711i 10.9903i 27.2715i 194.277 − 57.7977i
21.14 5.74801i 7.97807 −17.0396 11.1803 45.8580i 49.5190i − 5.97573i −17.3504 64.2647i
21.15 7.24168i −16.1202 −36.4420 11.1803 − 116.737i 72.0665i − 148.034i 178.861 80.9645i
21.16 7.40793i 1.13523 −38.8775 −11.1803 8.40974i − 23.6201i − 169.475i −79.7112 − 82.8232i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 55.5.c.a ✓ 16
3.b odd 2 1 495.5.b.a 16
4.b odd 2 1 880.5.j.c 16
5.b even 2 1 275.5.c.i 16
5.c odd 4 2 275.5.d.d 32
11.b odd 2 1 inner 55.5.c.a ✓ 16
33.d even 2 1 495.5.b.a 16
44.c even 2 1 880.5.j.c 16
55.d odd 2 1 275.5.c.i 16
55.e even 4 2 275.5.d.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.5.c.a ✓ 16 1.a even 1 1 trivial
55.5.c.a ✓ 16 11.b odd 2 1 inner
275.5.c.i 16 5.b even 2 1
275.5.c.i 16 55.d odd 2 1
275.5.d.d 32 5.c odd 4 2
275.5.d.d 32 55.e even 4 2
495.5.b.a 16 3.b odd 2 1
495.5.b.a 16 33.d even 2 1
880.5.j.c 16 4.b odd 2 1
880.5.j.c 16 44.c even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(55, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 4034278000 \) Copy content Toggle raw display
$3$ \( (T^{8} + 8 T^{7} + \cdots + 15390000)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 125)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 21\!\cdots\!21 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 75\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots - 32\!\cdots\!24)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots - 12\!\cdots\!64)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 11\!\cdots\!36)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 37\!\cdots\!76)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 54\!\cdots\!00)^{2} \) Copy content Toggle raw display
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