Properties

Label 10.11.c.c
Level $10$
Weight $11$
Character orbit 10.c
Analytic conductor $6.354$
Analytic rank $0$
Dimension $6$
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] = [10,11,Mod(3,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.3");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 10.c (of order \(4\), degree \(2\), 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: \(6.35357252674\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 1148x^{3} + 68121x^{2} - 299628x + 658952 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{8}\cdot 5^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 16 \beta_1 - 16) q^{2} + (\beta_{2} - 21 \beta_1 + 21) q^{3} + 512 \beta_1 q^{4} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} - 896 \beta_1 + 910) q^{5} + ( - 16 \beta_{3} - 16 \beta_{2} - 672) q^{6} + (5 \beta_{5} + 5 \beta_{4} + 19 \beta_{3} + 2244 \beta_1 + 2244) q^{7} + ( - 8192 \beta_1 + 8192) q^{8} + (30 \beta_{5} - 61 \beta_{3} + 61 \beta_{2} - 59175 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 16 \beta_1 - 16) q^{2} + (\beta_{2} - 21 \beta_1 + 21) q^{3} + 512 \beta_1 q^{4} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} - 896 \beta_1 + 910) q^{5} + ( - 16 \beta_{3} - 16 \beta_{2} - 672) q^{6} + (5 \beta_{5} + 5 \beta_{4} + 19 \beta_{3} + 2244 \beta_1 + 2244) q^{7} + ( - 8192 \beta_1 + 8192) q^{8} + (30 \beta_{5} - 61 \beta_{3} + 61 \beta_{2} - 59175 \beta_1) q^{9} + (16 \beta_{5} + 16 \beta_{4} - 16 \beta_{3} + 48 \beta_{2} - 224 \beta_1 - 28896) q^{10} + (10 \beta_{4} - 263 \beta_{3} - 263 \beta_{2} + 108144) q^{11} + (512 \beta_{3} + 10752 \beta_1 + 10752) q^{12} + (65 \beta_{5} - 65 \beta_{4} - 988 \beta_{2} + 123466 \beta_1 - 123466) q^{13} + ( - 160 \beta_{5} - 304 \beta_{3} + 304 \beta_{2} - 71808 \beta_1) q^{14} + ( - 27 \beta_{5} - 63 \beta_{4} - 776 \beta_{3} + 2883 \beta_{2} + \cdots + 262272) q^{15}+ \cdots + (2576190 \beta_{5} + 25931375 \beta_{3} + \cdots + 77635512 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 96 q^{2} + 128 q^{3} + 5460 q^{5} - 4096 q^{6} + 13512 q^{7} + 49152 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 96 q^{2} + 128 q^{3} + 5460 q^{5} - 4096 q^{6} + 13512 q^{7} + 49152 q^{8} - 173280 q^{10} + 647832 q^{11} + 65536 q^{12} - 742902 q^{13} + 1577720 q^{15} - 1572864 q^{16} - 755118 q^{17} - 5683744 q^{18} + 2749440 q^{20} + 12277112 q^{21} - 10365312 q^{22} - 15052992 q^{23} + 42644850 q^{25} + 23772864 q^{26} - 47998120 q^{27} - 6918144 q^{28} - 20357760 q^{30} + 153847152 q^{31} + 25165824 q^{32} - 173025784 q^{33} - 208942440 q^{35} + 181879808 q^{36} - 32574498 q^{37} - 17493120 q^{38} + 737280 q^{40} + 226153272 q^{41} - 196433792 q^{42} - 63628752 q^{43} - 174000230 q^{45} + 481695744 q^{46} - 19700448 q^{47} - 33554432 q^{48} - 788800800 q^{50} + 2388638032 q^{51} - 380365824 q^{52} - 950001042 q^{53} - 135145080 q^{55} + 221380608 q^{56} - 338012560 q^{57} - 310652160 q^{58} - 156344320 q^{60} + 1603984392 q^{61} - 2461554432 q^{62} - 5135206432 q^{63} + 1398176070 q^{65} + 5536825088 q^{66} + 2502647712 q^{67} + 386620416 q^{68} + 174645120 q^{70} - 7137533808 q^{71} - 2910076928 q^{72} + 2304462438 q^{73} + 9497972200 q^{75} + 559779840 q^{76} + 597969864 q^{77} + 11288293632 q^{78} - 1431306240 q^{80} - 26068931054 q^{81} - 3618452352 q^{82} - 88180632 q^{83} + 1729841610 q^{85} + 2036120064 q^{86} + 35176248320 q^{87} + 5307039744 q^{88} - 25065537760 q^{90} - 43718253408 q^{91} - 7707131904 q^{92} + 22042053176 q^{93} - 34456200 q^{95} + 1073741824 q^{96} + 33281088582 q^{97} + 12874047264 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 1148x^{3} + 68121x^{2} - 299628x + 658952 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 68121\nu^{5} + 149814\nu^{4} + 329476\nu^{3} - 39101454\nu^{2} + 4554477405\nu - 10394598718 ) / 10016360270 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 60546049 \nu^{5} - 58729384 \nu^{4} + 638377544 \nu^{3} - 84835233476 \nu^{2} + 4158168070345 \nu - 18141291569772 ) / 30049080810 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 60844529 \nu^{5} - 325695636 \nu^{4} - 716280824 \nu^{3} + 85006560996 \nu^{2} - 4158168070345 \nu + 822290385952 ) / 30049080810 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -31840\nu^{5} - 738610\nu^{4} - 8310240\nu^{3} + 18276160\nu^{2} - 20819537159 ) / 10470063 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 517677579 \nu^{5} - 1138493986 \nu^{4} + 16684615276 \nu^{3} + 797964943846 \nu^{2} - 29603038676095 \nu + 67978379651882 ) / 30049080810 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} - 10\beta_{3} + 3\beta _1 + 3 ) / 400 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 25\beta_{3} - 25\beta_{2} + 17383\beta_1 ) / 100 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 261\beta_{5} - 261\beta_{4} + 2610\beta_{2} - 228817\beta _1 + 228817 ) / 400 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13\beta_{4} - 3980\beta_{3} - 3980\beta_{2} - 2268051 ) / 50 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -13165\beta_{5} - 13165\beta_{4} + 159202\beta_{3} + 19926521\beta _1 + 19926521 ) / 80 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
2.29143 2.29143i
10.1043 10.1043i
−12.3957 + 12.3957i
2.29143 + 2.29143i
10.1043 + 10.1043i
−12.3957 12.3957i
−16.0000 + 16.0000i −266.441 266.441i 512.000i 2583.39 + 1758.32i 8526.10 −13021.6 + 13021.6i 8192.00 + 8192.00i 82932.3i −69467.5 + 13201.2i
3.2 −16.0000 + 16.0000i 4.29207 + 4.29207i 512.000i −2978.33 + 946.124i −137.346 21284.7 21284.7i 8192.00 + 8192.00i 59012.2i 32515.4 62791.3i
3.3 −16.0000 + 16.0000i 326.149 + 326.149i 512.000i 3124.94 19.4459i −10436.8 −1507.13 + 1507.13i 8192.00 + 8192.00i 153697.i −49687.9 + 50310.2i
7.1 −16.0000 16.0000i −266.441 + 266.441i 512.000i 2583.39 1758.32i 8526.10 −13021.6 13021.6i 8192.00 8192.00i 82932.3i −69467.5 13201.2i
7.2 −16.0000 16.0000i 4.29207 4.29207i 512.000i −2978.33 946.124i −137.346 21284.7 + 21284.7i 8192.00 8192.00i 59012.2i 32515.4 + 62791.3i
7.3 −16.0000 16.0000i 326.149 326.149i 512.000i 3124.94 + 19.4459i −10436.8 −1507.13 1507.13i 8192.00 8192.00i 153697.i −49687.9 50310.2i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.11.c.c 6
3.b odd 2 1 90.11.g.c 6
4.b odd 2 1 80.11.p.c 6
5.b even 2 1 50.11.c.e 6
5.c odd 4 1 inner 10.11.c.c 6
5.c odd 4 1 50.11.c.e 6
15.e even 4 1 90.11.g.c 6
20.e even 4 1 80.11.p.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.11.c.c 6 1.a even 1 1 trivial
10.11.c.c 6 5.c odd 4 1 inner
50.11.c.e 6 5.b even 2 1
50.11.c.e 6 5.c odd 4 1
80.11.p.c 6 4.b odd 2 1
80.11.p.c 6 20.e even 4 1
90.11.g.c 6 3.b odd 2 1
90.11.g.c 6 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 128T_{3}^{5} + 8192T_{3}^{4} + 20688696T_{3}^{3} + 30028037796T_{3}^{2} - 258527462832T_{3} + 1112900707872 \) acting on \(S_{11}^{\mathrm{new}}(10, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 32 T + 512)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} - 128 T^{5} + \cdots + 1112900707872 \) Copy content Toggle raw display
$5$ \( T^{6} - 5460 T^{5} + \cdots + 93\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{6} - 13512 T^{5} + \cdots + 13\!\cdots\!12 \) Copy content Toggle raw display
$11$ \( (T^{3} - 323916 T^{2} + \cdots + 26\!\cdots\!52)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 742902 T^{5} + \cdots + 77\!\cdots\!52 \) Copy content Toggle raw display
$17$ \( T^{6} + 755118 T^{5} + \cdots + 28\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{6} + 619530339600 T^{4} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{6} + 15052992 T^{5} + \cdots + 71\!\cdots\!92 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{3} - 76923576 T^{2} + \cdots - 10\!\cdots\!88)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 32574498 T^{5} + \cdots + 28\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( (T^{3} - 113076636 T^{2} + \cdots + 19\!\cdots\!72)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 63628752 T^{5} + \cdots + 31\!\cdots\!52 \) Copy content Toggle raw display
$47$ \( T^{6} + 19700448 T^{5} + \cdots + 89\!\cdots\!52 \) Copy content Toggle raw display
$53$ \( T^{6} + 950001042 T^{5} + \cdots + 85\!\cdots\!92 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{3} - 801992196 T^{2} + \cdots + 15\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} - 2502647712 T^{5} + \cdots + 29\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( (T^{3} + 3568766904 T^{2} + \cdots + 58\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 2304462438 T^{5} + \cdots + 16\!\cdots\!12 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} + 88180632 T^{5} + \cdots + 13\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} - 33281088582 T^{5} + \cdots + 11\!\cdots\!32 \) Copy content Toggle raw display
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