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Properties

Label 11.5.d.a
Level $11$
Weight $5$
Character orbit 11.d
Analytic conductor $1.137$
Analytic rank $0$
Dimension $12$
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] = [11,5,Mod(2,11)]
 
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([1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("11.2");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 11.d (of order \(10\), 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: \(1.13706959392\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 115x^{10} + 5030x^{8} + 102975x^{6} + 953170x^{4} + 2910655x^{2} + 73205 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} - \beta_{4} - 2 \beta_{3} + \cdots - 1) q^{2}+ \cdots + ( - 4 \beta_{11} - \beta_{10} + \cdots + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} - \beta_{4} - 2 \beta_{3} + \cdots - 1) q^{2}+ \cdots + ( - 113 \beta_{11} + 364 \beta_{10} + \cdots + 493) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 5 q^{2} - 6 q^{3} + 7 q^{4} - 18 q^{5} + 75 q^{6} - 80 q^{7} - 245 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 5 q^{2} - 6 q^{3} + 7 q^{4} - 18 q^{5} + 75 q^{6} - 80 q^{7} - 245 q^{8} + q^{9} - 43 q^{11} + 594 q^{12} + 250 q^{13} + 610 q^{14} + 1134 q^{15} - 633 q^{16} - 1250 q^{17} - 3150 q^{18} - 1025 q^{19} + 752 q^{20} - 35 q^{22} + 1684 q^{23} + 5345 q^{24} + 197 q^{25} + 3490 q^{26} - 687 q^{27} - 3580 q^{28} - 2690 q^{29} - 6740 q^{30} - 1136 q^{31} + 5939 q^{33} + 2370 q^{34} + 3610 q^{35} - 514 q^{36} - 336 q^{37} + 1900 q^{38} - 6880 q^{39} - 2340 q^{40} - 4550 q^{41} + 1310 q^{42} - 6268 q^{44} + 5136 q^{45} + 4150 q^{46} + 24 q^{47} + 344 q^{48} + 827 q^{49} + 8895 q^{50} + 13155 q^{51} + 14070 q^{52} + 414 q^{53} - 2738 q^{55} - 21340 q^{56} - 26925 q^{57} + 2980 q^{58} - 10011 q^{59} - 6856 q^{60} + 9460 q^{61} - 6200 q^{62} + 9150 q^{63} - 2633 q^{64} - 3210 q^{66} + 12154 q^{67} - 9400 q^{68} - 9022 q^{69} - 9380 q^{70} + 17574 q^{71} + 43045 q^{72} + 27950 q^{73} + 43270 q^{74} - 1761 q^{75} + 4090 q^{77} - 42920 q^{78} - 41540 q^{79} - 2308 q^{80} - 21080 q^{81} - 28175 q^{82} - 18665 q^{83} + 26250 q^{84} - 4230 q^{85} - 10125 q^{86} - 15125 q^{88} + 5554 q^{89} + 18400 q^{90} + 7390 q^{91} + 3904 q^{92} + 36898 q^{93} + 18920 q^{94} + 14110 q^{95} - 21140 q^{96} + 20769 q^{97} - 3269 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 115x^{10} + 5030x^{8} + 102975x^{6} + 953170x^{4} + 2910655x^{2} + 73205 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 13052 \nu^{11} - 117304 \nu^{10} - 1219776 \nu^{9} - 10477907 \nu^{8} - 40135806 \nu^{7} + \cdots - 1427754383 ) / 1339827192 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13052 \nu^{11} - 117304 \nu^{10} + 1219776 \nu^{9} - 10477907 \nu^{8} + 40135806 \nu^{7} + \cdots - 1427754383 ) / 1339827192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 31452 \nu^{11} + 117304 \nu^{10} - 2737673 \nu^{9} + 10477907 \nu^{8} - 79586472 \nu^{7} + \cdots + 757840787 ) / 1339827192 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 45643 \nu^{10} + 4016197 \nu^{8} + 120327945 \nu^{6} + 1386775659 \nu^{4} + 4387794733 \nu^{2} - 676101899 ) / 60901236 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10664 \nu^{11} + 25564 \nu^{10} + 952537 \nu^{9} + 2319614 \nu^{8} + 29205906 \nu^{7} + \cdots + 86861060 ) / 121802472 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10664 \nu^{11} - 25564 \nu^{10} + 952537 \nu^{9} - 2319614 \nu^{8} + 29205906 \nu^{7} + \cdots - 86861060 ) / 121802472 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10664 \nu^{11} - 79937 \nu^{10} + 952537 \nu^{9} - 7147008 \nu^{8} + 29205906 \nu^{7} + \cdots - 209313060 ) / 121802472 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 169940 \nu^{11} - 259930 \nu^{10} - 15878131 \nu^{9} - 23961014 \nu^{8} - 530879748 \nu^{7} + \cdots + 4416548158 ) / 1339827192 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 169940 \nu^{11} - 259930 \nu^{10} + 15878131 \nu^{9} - 23961014 \nu^{8} + 530879748 \nu^{7} + \cdots + 4416548158 ) / 1339827192 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 380623 \nu^{11} + 502073 \nu^{10} - 35561311 \nu^{9} + 44178167 \nu^{8} - 1181014629 \nu^{7} + \cdots - 7437120889 ) / 1339827192 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{9} - 2\beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - \beta _1 - 19 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{9} - 5\beta_{7} - 5\beta_{6} - 24\beta_{4} - 9\beta_{3} - 15\beta_{2} - 25\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 31 \beta_{10} - 31 \beta_{9} + 90 \beta_{8} - 51 \beta_{7} - 39 \beta_{6} + 33 \beta_{5} - 113 \beta_{3} + \cdots + 476 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 32 \beta_{11} - 33 \beta_{10} + 33 \beta_{9} + 271 \beta_{7} + 271 \beta_{6} - 16 \beta_{5} + 1300 \beta_{4} + \cdots + 650 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 954 \beta_{10} + 954 \beta_{9} - 3628 \beta_{8} + 2292 \beta_{7} + 1336 \beta_{6} - 1142 \beta_{5} + \cdots - 12905 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2096 \beta_{11} + 1056 \beta_{10} - 1056 \beta_{9} - 11834 \beta_{7} - 11834 \beta_{6} + 1048 \beta_{5} + \cdots - 28374 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 30477 \beta_{10} - 30477 \beta_{9} + 140790 \beta_{8} - 95925 \beta_{7} - 44865 \beta_{6} + \cdots + 369011 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 99264 \beta_{11} - 36579 \beta_{10} + 36579 \beta_{9} + 478665 \beta_{7} + 478665 \beta_{6} + \cdots + 1150758 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1012445 \beta_{10} + 1012445 \beta_{9} - 5366098 \beta_{8} + 3851627 \beta_{7} + 1514471 \beta_{6} + \cdots - 11069600 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4178328 \beta_{11} + 1343903 \beta_{10} - 1343903 \beta_{9} - 18668485 \beta_{7} - 18668485 \beta_{6} + \cdots - 45055758 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
4.56289i
2.38108i
− 5.04186i
− 4.56289i
− 2.38108i
5.04186i
6.10049i
0.159251i
− 5.08417i
− 6.10049i
− 0.159251i
5.08417i
−5.45760 − 1.77328i −10.4040 + 7.55891i 13.6966 + 9.95117i −9.33639 − 28.7345i 70.1847 − 22.8044i −33.6132 + 46.2646i −3.13669 − 4.31729i 26.0747 − 80.2495i 173.377i
2.2 −3.38257 − 1.09906i 12.7857 − 9.28936i −2.71041 − 1.96923i 5.48545 + 16.8825i −53.4582 + 17.3696i −19.6356 + 27.0262i 40.4526 + 55.6782i 52.1517 − 160.506i − 63.1351i
2.3 3.67706 + 1.19475i −1.64569 + 1.19566i −0.850950 − 0.618251i −1.76709 − 5.43854i −7.47979 + 2.43033i −2.52826 + 3.47985i −38.7511 − 53.3363i −23.7517 + 73.1002i − 22.1090i
6.1 −5.45760 + 1.77328i −10.4040 − 7.55891i 13.6966 − 9.95117i −9.33639 + 28.7345i 70.1847 + 22.8044i −33.6132 − 46.2646i −3.13669 + 4.31729i 26.0747 + 80.2495i − 173.377i
6.2 −3.38257 + 1.09906i 12.7857 + 9.28936i −2.71041 + 1.96923i 5.48545 − 16.8825i −53.4582 − 17.3696i −19.6356 − 27.0262i 40.4526 − 55.6782i 52.1517 + 160.506i 63.1351i
6.3 3.67706 − 1.19475i −1.64569 − 1.19566i −0.850950 + 0.618251i −1.76709 + 5.43854i −7.47979 − 2.43033i −2.52826 − 3.47985i −38.7511 + 53.3363i −23.7517 − 73.1002i 22.1090i
7.1 −2.46775 + 3.39656i −2.26281 + 6.96422i −0.502581 − 1.54679i 14.7329 − 10.7041i −18.0704 − 24.8717i 47.9990 − 15.5958i −57.3923 − 18.6479i 22.1503 + 16.0931i 76.4560i
7.2 1.02443 − 1.41001i 2.68168 − 8.25338i 4.00561 + 12.3280i −8.06057 + 5.85635i −8.89011 − 12.2362i −56.0338 + 18.2065i 48.0070 + 15.5984i 4.60358 + 3.34469i 17.3649i
7.3 4.10644 − 5.65202i −4.15494 + 12.7876i −10.1383 − 31.2024i −10.0543 + 7.30486i 55.2138 + 75.9952i 23.8119 − 7.73696i −111.679 − 36.2869i −80.7285 − 58.6527i 86.8239i
8.1 −2.46775 − 3.39656i −2.26281 − 6.96422i −0.502581 + 1.54679i 14.7329 + 10.7041i −18.0704 + 24.8717i 47.9990 + 15.5958i −57.3923 + 18.6479i 22.1503 − 16.0931i − 76.4560i
8.2 1.02443 + 1.41001i 2.68168 + 8.25338i 4.00561 − 12.3280i −8.06057 − 5.85635i −8.89011 + 12.2362i −56.0338 − 18.2065i 48.0070 − 15.5984i 4.60358 − 3.34469i − 17.3649i
8.3 4.10644 + 5.65202i −4.15494 − 12.7876i −10.1383 + 31.2024i −10.0543 − 7.30486i 55.2138 − 75.9952i 23.8119 + 7.73696i −111.679 + 36.2869i −80.7285 + 58.6527i − 86.8239i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 11.5.d.a ✓ 12
3.b odd 2 1 99.5.k.a 12
4.b odd 2 1 176.5.n.a 12
11.b odd 2 1 121.5.d.d 12
11.c even 5 1 121.5.b.b 12
11.c even 5 1 121.5.d.c 12
11.c even 5 1 121.5.d.d 12
11.c even 5 1 121.5.d.e 12
11.d odd 10 1 inner 11.5.d.a ✓ 12
11.d odd 10 1 121.5.b.b 12
11.d odd 10 1 121.5.d.c 12
11.d odd 10 1 121.5.d.e 12
33.f even 10 1 99.5.k.a 12
44.g even 10 1 176.5.n.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.5.d.a ✓ 12 1.a even 1 1 trivial
11.5.d.a ✓ 12 11.d odd 10 1 inner
99.5.k.a 12 3.b odd 2 1
99.5.k.a 12 33.f even 10 1
121.5.b.b 12 11.c even 5 1
121.5.b.b 12 11.d odd 10 1
121.5.d.c 12 11.c even 5 1
121.5.d.c 12 11.d odd 10 1
121.5.d.d 12 11.b odd 2 1
121.5.d.d 12 11.c even 5 1
121.5.d.e 12 11.c even 5 1
121.5.d.e 12 11.d odd 10 1
176.5.n.a 12 4.b odd 2 1
176.5.n.a 12 44.g even 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(11, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 5 T^{11} + \cdots + 16272080 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 124779204081 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 47826408560896 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 98\!\cdots\!41 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 10\!\cdots\!80 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 16\!\cdots\!05 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 31\!\cdots\!05 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 10\!\cdots\!96)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 61\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 41\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 49\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 72\!\cdots\!05 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 84\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 38\!\cdots\!41 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 15\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 15\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 56\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 14\!\cdots\!05 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 36\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 18\!\cdots\!05 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 66\!\cdots\!44)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 18\!\cdots\!01 \) Copy content Toggle raw display
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