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Properties

Label 121.10.a.i
Level $121$
Weight $10$
Character orbit 121.a
Self dual yes
Analytic conductor $62.319$
Analytic rank $0$
Dimension $16$
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] = [121,10,Mod(1,121)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(121, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("121.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 121.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: \(62.3193361758\)
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: \( x^{16} - 5809 x^{14} - 4272 x^{13} + 13066060 x^{12} + 16497800 x^{11} - 14560551456 x^{10} + \cdots + 24\!\cdots\!44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{4}\cdot 11^{9} \)
Twist minimal: no (minimal twist has level 11)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 3) q^{2} + ( - \beta_{5} - \beta_{2} - \beta_1 + 14) q^{3} + ( - \beta_{3} - 2 \beta_{2} + \cdots + 230) q^{4}+ \cdots + ( - \beta_{13} + \beta_{10} + \cdots + 5061) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 3) q^{2} + ( - \beta_{5} - \beta_{2} - \beta_1 + 14) q^{3} + ( - \beta_{3} - 2 \beta_{2} + \cdots + 230) q^{4}+ \cdots + (7558 \beta_{15} - 10745 \beta_{14} + \cdots + 208272698) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 48 q^{2} + 222 q^{3} + 3670 q^{4} - 1875 q^{5} + 6191 q^{6} + 12575 q^{7} + 41046 q^{8} + 78904 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 48 q^{2} + 222 q^{3} + 3670 q^{4} - 1875 q^{5} + 6191 q^{6} + 12575 q^{7} + 41046 q^{8} + 78904 q^{9} + 8136 q^{10} + 169811 q^{12} + 119783 q^{13} - 150396 q^{14} + 128978 q^{15} + 1706770 q^{16} + 570930 q^{17} + 1377185 q^{18} + 708062 q^{19} - 2134062 q^{20} + 1713601 q^{21} - 2022726 q^{23} + 5319405 q^{24} - 222677 q^{25} - 4839084 q^{26} + 6603192 q^{27} + 4933520 q^{28} + 12196593 q^{29} + 20118540 q^{30} - 1920467 q^{31} + 28274430 q^{32} + 4257577 q^{34} + 17645076 q^{35} + 10442417 q^{36} + 8193077 q^{37} + 17969289 q^{38} + 26233656 q^{39} - 78857762 q^{40} + 27052578 q^{41} + 126693168 q^{42} + 2191717 q^{43} - 82573010 q^{45} - 82764586 q^{46} + 61772571 q^{47} + 26253395 q^{48} - 35102109 q^{49} + 162127428 q^{50} + 27137348 q^{51} + 174822050 q^{52} + 230906967 q^{53} + 168706262 q^{54} - 163593804 q^{56} + 68054133 q^{57} - 373835684 q^{58} - 92927886 q^{59} + 190895818 q^{60} + 463219765 q^{61} + 573749574 q^{62} + 784007783 q^{63} + 1128452290 q^{64} + 434503197 q^{65} - 473101677 q^{67} + 1073658963 q^{68} - 975630076 q^{69} + 524415506 q^{70} + 374085717 q^{71} + 3278349663 q^{72} + 1061898004 q^{73} + 1887418362 q^{74} + 1708808113 q^{75} + 2116654933 q^{76} - 2149007830 q^{78} + 1193641695 q^{79} - 2800403982 q^{80} - 160274720 q^{81} + 406045707 q^{82} + 3036904890 q^{83} + 3741117150 q^{84} + 2067806293 q^{85} + 4258123329 q^{86} + 2016430701 q^{87} - 779711079 q^{89} + 2671912588 q^{90} - 2737301136 q^{91} + 211938438 q^{92} - 2375160030 q^{93} + 8271211980 q^{94} + 3152016348 q^{95} + 7750699293 q^{96} + 6028874386 q^{97} + 3685530690 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 5809 x^{14} - 4272 x^{13} + 13066060 x^{12} + 16497800 x^{11} - 14560551456 x^{10} + \cdots + 24\!\cdots\!44 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 16\!\cdots\!43 \nu^{15} + \cdots - 45\!\cdots\!72 ) / 89\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16\!\cdots\!43 \nu^{15} + \cdots + 45\!\cdots\!32 ) / 16\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10\!\cdots\!55 \nu^{15} + \cdots - 17\!\cdots\!32 ) / 22\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\!\cdots\!15 \nu^{15} + \cdots + 45\!\cdots\!64 ) / 17\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 26\!\cdots\!61 \nu^{15} + \cdots + 59\!\cdots\!60 ) / 25\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 30\!\cdots\!09 \nu^{15} + \cdots - 24\!\cdots\!92 ) / 25\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 86\!\cdots\!15 \nu^{15} + \cdots - 21\!\cdots\!04 ) / 76\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11\!\cdots\!81 \nu^{15} + \cdots - 30\!\cdots\!36 ) / 76\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 50\!\cdots\!13 \nu^{15} + \cdots - 13\!\cdots\!28 ) / 15\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 62\!\cdots\!61 \nu^{15} + \cdots + 14\!\cdots\!20 ) / 15\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 24\!\cdots\!84 \nu^{15} + \cdots + 63\!\cdots\!60 ) / 38\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 37\!\cdots\!21 \nu^{15} + \cdots + 10\!\cdots\!16 ) / 51\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 13\!\cdots\!39 \nu^{15} + \cdots + 47\!\cdots\!96 ) / 15\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 18\!\cdots\!47 \nu^{15} + \cdots - 55\!\cdots\!12 ) / 15\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 10\!\cdots\!27 \nu^{15} + \cdots + 23\!\cdots\!76 ) / 76\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{2} + 11\beta _1 - 1 ) / 11 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -4\beta_{4} - 11\beta_{3} - 22\beta_{2} + 9\beta _1 + 7994 ) / 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 6 \beta_{12} - 11 \beta_{9} + 22 \beta_{8} - 11 \beta_{6} + 119 \beta_{5} + 38 \beta_{4} + \cdots + 6461 ) / 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 134 \beta_{15} - 66 \beta_{14} + 16 \beta_{13} + 74 \beta_{12} - 120 \beta_{11} - 77 \beta_{9} + \cdots + 10489225 ) / 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 812 \beta_{15} + 176 \beta_{14} - 948 \beta_{13} - 17130 \beta_{12} + 1456 \beta_{11} + \cdots + 23548189 ) / 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 326842 \beta_{15} - 201366 \beta_{14} + 65224 \beta_{13} + 204518 \beta_{12} - 324312 \beta_{11} + \cdots + 16594787665 ) / 11 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2537604 \beta_{15} + 843656 \beta_{14} - 2478196 \beta_{13} - 39452698 \beta_{12} + \cdots + 64938220561 ) / 11 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 652688178 \beta_{15} - 478287678 \beta_{14} + 159913736 \beta_{13} + 462726078 \beta_{12} + \cdots + 28351797214157 ) / 11 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 5996934820 \beta_{15} + 2202558776 \beta_{14} - 5199942788 \beta_{13} - 84173325234 \beta_{12} + \cdots + 148245695216493 ) / 11 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1237818613306 \beta_{15} - 1041698295638 \beta_{14} + 336150572168 \beta_{13} + \cdots + 50\!\cdots\!01 ) / 11 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 12940008323396 \beta_{15} + 4521762083432 \beta_{14} - 10249237782036 \beta_{13} + \cdots + 30\!\cdots\!13 ) / 11 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 23\!\cdots\!70 \beta_{15} + \cdots + 90\!\cdots\!61 ) / 11 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 26\!\cdots\!92 \beta_{15} + \cdots + 59\!\cdots\!17 ) / 11 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 43\!\cdots\!62 \beta_{15} + \cdots + 16\!\cdots\!89 ) / 11 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 53\!\cdots\!44 \beta_{15} + \cdots + 11\!\cdots\!93 ) / 11 \) Copy content Toggle raw display

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

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
42.0736
43.5025
27.2199
25.8013
18.6674
19.4858
10.5435
1.85437
−8.86670
−8.64074
−14.2614
−16.9861
−29.6361
−29.8620
−37.3912
−43.5041
−41.3097 −59.8414 1194.49 388.500 2472.03 4825.97 −28193.4 −16102.0 −16048.8
1.2 −38.2664 −21.6619 952.317 −893.315 828.923 52.1192 −16849.4 −19213.8 34184.0
1.3 −26.4559 248.603 187.917 −1841.70 −6577.01 2332.33 8573.93 42120.2 48724.0
1.4 −20.5653 141.054 −89.0699 1332.53 −2900.82 2657.03 12361.2 213.371 −27403.7
1.5 −17.9035 −13.5236 −191.465 730.872 242.119 −6805.88 12594.5 −19500.1 −13085.2
1.6 −14.2497 −209.100 −308.945 204.402 2979.62 −1958.12 11698.2 24039.9 −2912.67
1.7 −9.77960 −249.697 −416.359 −853.278 2441.93 8440.94 9078.98 42665.5 8344.72
1.8 3.38170 167.072 −500.564 −2122.94 564.987 −8366.46 −3424.19 8230.03 −7179.16
1.9 9.63063 166.602 −419.251 2454.19 1604.49 7441.36 −8968.54 8073.32 23635.4
1.10 13.8768 −68.3315 −319.434 −768.757 −948.222 11279.4 −11537.7 −15013.8 −10667.9
1.11 15.0253 43.4121 −286.239 −1775.96 652.281 −2400.82 −11993.8 −17798.4 −26684.4
1.12 22.2222 −52.8463 −18.1743 1935.00 −1174.36 −9526.26 −11781.6 −16890.3 43000.0
1.13 30.4001 −148.103 412.164 743.956 −4502.34 −1393.39 −3035.03 2251.51 22616.3
1.14 35.0980 224.882 719.872 1007.13 7892.93 8984.27 7295.88 30889.1 35348.4
1.15 42.6273 −150.567 1305.08 −1950.19 −6418.28 −4420.29 33807.0 2987.56 −83131.1
1.16 44.2681 204.046 1447.66 −465.433 9032.73 1432.84 41419.9 21951.9 −20603.8
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)

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 121.10.a.i 16
11.b odd 2 1 121.10.a.h 16
11.d odd 10 2 11.10.c.a ✓ 32
33.f even 10 2 99.10.f.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.10.c.a ✓ 32 11.d odd 10 2
99.10.f.a 32 33.f even 10 2
121.10.a.h 16 11.b odd 2 1
121.10.a.i 16 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 48 T_{2}^{15} - 4779 T_{2}^{14} + 236190 T_{2}^{13} + 8487096 T_{2}^{12} + \cdots - 65\!\cdots\!04 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(121))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots - 65\!\cdots\!04 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots - 14\!\cdots\!95 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots - 22\!\cdots\!80 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots - 85\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots - 19\!\cdots\!91 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots - 52\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots - 21\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots - 42\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 91\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 37\!\cdots\!41 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots - 56\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 79\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots - 25\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 24\!\cdots\!20 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots - 12\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots - 28\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 24\!\cdots\!61 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots - 91\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots - 10\!\cdots\!49 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
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