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Properties

Label 11.6.c.a
Level $11$
Weight $6$
Character orbit 11.c
Analytic conductor $1.764$
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] = [11,6,Mod(3,11)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(11, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("11.3");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 11.c (of order \(5\), degree \(4\), 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: \(1.76422201794\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 5 x^{15} + 86 x^{14} - 146 x^{13} + 7205 x^{12} - 23732 x^{11} + 774165 x^{10} + \cdots + 393784336 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{4} + \beta_{3} + \cdots + 1) q^{2}+ \cdots + ( - 4 \beta_{15} + \beta_{13} + \cdots - 42) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_{4} + \beta_{3} + \cdots + 1) q^{2}+ \cdots + ( - 244 \beta_{15} + 107 \beta_{14} + \cdots + 54518) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{2} - 24 q^{3} - 73 q^{4} - 10 q^{5} + 121 q^{6} + 196 q^{7} + 527 q^{8} - 530 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - q^{2} - 24 q^{3} - 73 q^{4} - 10 q^{5} + 121 q^{6} + 196 q^{7} + 527 q^{8} - 530 q^{9} - 672 q^{10} - 692 q^{11} - 1562 q^{12} + 1162 q^{13} + 560 q^{14} + 1796 q^{15} + 6399 q^{16} - 22 q^{17} + 834 q^{18} - 3236 q^{19} - 5514 q^{20} - 7772 q^{21} - 13059 q^{22} - 10848 q^{23} + 13233 q^{24} + 15686 q^{25} + 16216 q^{26} + 22500 q^{27} + 8838 q^{28} + 13070 q^{29} - 22830 q^{30} - 14764 q^{31} - 58812 q^{32} - 48544 q^{33} - 26966 q^{34} + 43368 q^{35} + 63696 q^{36} + 4638 q^{37} + 68144 q^{38} + 20300 q^{39} + 52284 q^{40} - 14806 q^{41} - 69922 q^{42} - 24376 q^{43} - 104960 q^{44} - 97480 q^{45} + 50452 q^{46} + 40364 q^{47} + 30236 q^{48} + 32246 q^{49} + 10839 q^{50} + 75564 q^{51} - 80654 q^{52} - 11654 q^{53} + 43796 q^{54} + 7052 q^{55} + 70632 q^{56} - 40020 q^{57} + 10276 q^{58} + 70804 q^{59} + 46116 q^{60} - 31446 q^{61} - 153388 q^{62} - 7064 q^{63} + 17695 q^{64} - 41284 q^{65} + 48006 q^{66} - 64200 q^{67} - 94114 q^{68} - 62412 q^{69} - 103768 q^{70} - 184380 q^{71} - 14729 q^{72} - 1750 q^{73} + 306048 q^{74} + 246596 q^{75} + 174806 q^{76} + 384646 q^{77} + 360916 q^{78} + 24324 q^{79} - 386444 q^{80} - 259412 q^{81} - 316255 q^{82} - 46028 q^{83} - 271956 q^{84} + 63914 q^{85} + 22931 q^{86} - 47592 q^{87} + 211871 q^{88} + 148364 q^{89} - 347186 q^{90} - 258448 q^{91} - 58658 q^{92} - 387478 q^{93} - 55370 q^{94} - 4716 q^{95} + 328396 q^{96} + 484296 q^{97} + 743692 q^{98} + 724964 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 5 x^{15} + 86 x^{14} - 146 x^{13} + 7205 x^{12} - 23732 x^{11} + 774165 x^{10} + \cdots + 393784336 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 25\!\cdots\!53 \nu^{15} + \cdots + 31\!\cdots\!52 ) / 22\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 19\!\cdots\!81 \nu^{15} + \cdots + 38\!\cdots\!44 ) / 12\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 21\!\cdots\!19 \nu^{15} + \cdots - 37\!\cdots\!20 ) / 12\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 35\!\cdots\!61 \nu^{15} + \cdots + 32\!\cdots\!44 ) / 12\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 48\!\cdots\!69 \nu^{15} + \cdots + 18\!\cdots\!36 ) / 99\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 78\!\cdots\!15 \nu^{15} + \cdots + 19\!\cdots\!24 ) / 99\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 39\!\cdots\!39 \nu^{15} + \cdots + 65\!\cdots\!40 ) / 49\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 47\!\cdots\!26 \nu^{15} + \cdots - 20\!\cdots\!72 ) / 12\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 28\!\cdots\!59 \nu^{15} + \cdots + 15\!\cdots\!04 ) / 49\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 64\!\cdots\!06 \nu^{15} + \cdots + 17\!\cdots\!80 ) / 49\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 76\!\cdots\!69 \nu^{15} + \cdots + 11\!\cdots\!76 ) / 49\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 10\!\cdots\!27 \nu^{15} + \cdots - 17\!\cdots\!96 ) / 49\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 26\!\cdots\!26 \nu^{15} + \cdots - 64\!\cdots\!38 ) / 12\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 27\!\cdots\!65 \nu^{15} + \cdots + 68\!\cdots\!96 ) / 81\!\cdots\!36 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - 43\beta_{8} + \beta_{7} - 2\beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{14} - 4 \beta_{12} + 4 \beta_{11} - 4 \beta_{10} - 12 \beta_{8} + 34 \beta_{7} - 28 \beta_{6} + \cdots - 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{15} - 87 \beta_{14} + 2 \beta_{13} + 4 \beta_{11} - 12 \beta_{10} - 4 \beta_{9} + 135 \beta_{8} + \cdots - 833 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 14 \beta_{14} + 14 \beta_{13} + 336 \beta_{12} - 336 \beta_{11} + 16 \beta_{9} + 9007 \beta_{8} + \cdots - 3705 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 724 \beta_{15} - 724 \beta_{14} + 7321 \beta_{13} + 1008 \beta_{12} - 1608 \beta_{11} + 1608 \beta_{10} + \cdots - 272598 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 4252 \beta_{15} + 25030 \beta_{13} + 3904 \beta_{12} + 25788 \beta_{10} + 25788 \beta_{9} + \cdots - 164691 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 622115 \beta_{15} + 105294 \beta_{14} - 57324 \beta_{12} + 161348 \beta_{11} - 57324 \beta_{10} + \cdots + 1467091 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2363932 \beta_{15} + 710642 \beta_{14} - 2363932 \beta_{13} + 2009960 \beta_{11} - 1484760 \beta_{10} + \cdots + 72285022 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 53543269 \beta_{14} - 53543269 \beta_{13} + 4603200 \beta_{12} - 4603200 \beta_{11} - 9980928 \beta_{9} + \cdots + 1965217305 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 217581914 \beta_{15} + 217581914 \beta_{14} - 94872624 \beta_{13} - 58722752 \beta_{12} + \cdots + 2995735314 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4657355935 \beta_{15} - 1286346842 \beta_{13} - 929050408 \beta_{12} + 330009108 \beta_{10} + \cdots + 75162756573 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 11293502550 \beta_{15} - 19835222816 \beta_{14} - 13154027264 \beta_{12} + 7079589488 \beta_{11} + \cdots - 165692995286 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 129838068780 \beta_{15} - 408574340305 \beta_{14} + 129838068780 \beta_{13} + 21012853800 \beta_{11} + \cdots - 8094359895355 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1256105108580 \beta_{14} + 1256105108580 \beta_{13} + 1093932232300 \beta_{12} + \cdots - 60157769831531 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
2.73971 + 8.43195i
1.27432 + 3.92196i
−1.19154 − 3.66717i
−2.13151 − 6.56011i
2.73971 − 8.43195i
1.27432 − 3.92196i
−1.19154 + 3.66717i
−2.13151 + 6.56011i
7.81150 + 5.67539i
1.13051 + 0.821361i
0.136182 + 0.0989419i
−7.26917 − 5.28136i
7.81150 − 5.67539i
1.13051 − 0.821361i
0.136182 − 0.0989419i
−7.26917 + 5.28136i
−6.36363 + 4.62345i −5.98507 − 18.4202i 9.23097 − 28.4100i −59.3249 − 43.1020i 123.251 + 89.5474i −51.6143 + 158.852i −5.17243 − 15.9191i −106.890 + 77.6601i 576.801
3.2 −2.52720 + 1.83612i 4.03595 + 12.4214i −6.87313 + 21.1533i 23.3906 + 16.9942i −33.0068 − 23.9808i 27.3683 − 84.2310i −52.3600 − 161.147i 58.5894 − 42.5677i −90.3160
3.3 3.92850 − 2.85422i −7.68717 − 23.6587i −2.60202 + 8.00819i 68.6567 + 49.8820i −97.7261 − 71.0022i 19.5777 − 60.2541i 60.6528 + 186.670i −304.049 + 220.905i 412.092
3.4 6.38938 − 4.64216i 3.63629 + 11.1914i 9.38602 − 28.8872i −51.9930 − 37.7751i 75.1856 + 54.6256i −33.5385 + 103.221i 3.96879 + 12.2147i 84.5674 − 61.4418i −507.561
4.1 −6.36363 − 4.62345i −5.98507 + 18.4202i 9.23097 + 28.4100i −59.3249 + 43.1020i 123.251 − 89.5474i −51.6143 − 158.852i −5.17243 + 15.9191i −106.890 − 77.6601i 576.801
4.2 −2.52720 − 1.83612i 4.03595 − 12.4214i −6.87313 − 21.1533i 23.3906 − 16.9942i −33.0068 + 23.9808i 27.3683 + 84.2310i −52.3600 + 161.147i 58.5894 + 42.5677i −90.3160
4.3 3.92850 + 2.85422i −7.68717 + 23.6587i −2.60202 − 8.00819i 68.6567 − 49.8820i −97.7261 + 71.0022i 19.5777 + 60.2541i 60.6528 − 186.670i −304.049 − 220.905i 412.092
4.4 6.38938 + 4.64216i 3.63629 − 11.1914i 9.38602 + 28.8872i −51.9930 + 37.7751i 75.1856 − 54.6256i −33.5385 − 103.221i 3.96879 − 12.2147i 84.5674 + 61.4418i −507.561
5.1 −3.29275 − 10.1340i 2.26410 + 1.64496i −65.9678 + 47.9284i 20.4388 − 62.9043i 9.21501 − 28.3609i 58.3278 − 42.3777i 427.066 + 310.282i −72.6709 − 223.658i −704.774
5.2 −0.740832 − 2.28005i −21.7558 − 15.8065i 21.2388 − 15.4309i −5.23817 + 16.1214i −19.9222 + 61.3143i 87.7560 − 63.7585i −112.982 − 82.0864i 148.378 + 456.662i 40.6382
5.3 −0.361034 − 1.11115i 12.7716 + 9.27912i 24.7842 − 18.0068i 1.31139 − 4.03604i 5.69949 − 17.5412i −146.551 + 106.476i −59.2025 − 43.0132i 1.92087 + 5.91183i −4.95810
5.4 2.46756 + 7.59437i 0.720120 + 0.523198i −25.6970 + 18.6700i −2.24156 + 6.89880i −2.19642 + 6.75988i 136.674 − 99.2995i 1.52919 + 1.11102i −74.8463 − 230.353i −57.9232
9.1 −3.29275 + 10.1340i 2.26410 − 1.64496i −65.9678 − 47.9284i 20.4388 + 62.9043i 9.21501 + 28.3609i 58.3278 + 42.3777i 427.066 − 310.282i −72.6709 + 223.658i −704.774
9.2 −0.740832 + 2.28005i −21.7558 + 15.8065i 21.2388 + 15.4309i −5.23817 − 16.1214i −19.9222 − 61.3143i 87.7560 + 63.7585i −112.982 + 82.0864i 148.378 − 456.662i 40.6382
9.3 −0.361034 + 1.11115i 12.7716 − 9.27912i 24.7842 + 18.0068i 1.31139 + 4.03604i 5.69949 + 17.5412i −146.551 − 106.476i −59.2025 + 43.0132i 1.92087 − 5.91183i −4.95810
9.4 2.46756 − 7.59437i 0.720120 − 0.523198i −25.6970 − 18.6700i −2.24156 − 6.89880i −2.19642 − 6.75988i 136.674 + 99.2995i 1.52919 − 1.11102i −74.8463 + 230.353i −57.9232
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 11.6.c.a ✓ 16
3.b odd 2 1 99.6.f.a 16
11.c even 5 1 inner 11.6.c.a ✓ 16
11.c even 5 1 121.6.a.g 8
11.d odd 10 1 121.6.a.i 8
33.f even 10 1 1089.6.a.bb 8
33.h odd 10 1 99.6.f.a 16
33.h odd 10 1 1089.6.a.bg 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.6.c.a ✓ 16 1.a even 1 1 trivial
11.6.c.a ✓ 16 11.c even 5 1 inner
99.6.f.a 16 3.b odd 2 1
99.6.f.a 16 33.h odd 10 1
121.6.a.g 8 11.c even 5 1
121.6.a.i 8 11.d odd 10 1
1089.6.a.bb 8 33.f even 10 1
1089.6.a.bg 8 33.h odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(11, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 50434379776 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 61\!\cdots\!25 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 45\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 64\!\cdots\!61 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots - 13\!\cdots\!36)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 80\!\cdots\!41 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 31\!\cdots\!84)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 32\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 46\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 97\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots - 81\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 39\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 35\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 72\!\cdots\!25 \) Copy content Toggle raw display
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