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Properties

Label 165.2.p.a
Level $165$
Weight $2$
Character orbit 165.p
Analytic conductor $1.318$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [165,2,Mod(41,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 165.p (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: \(1.31753163335\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{14} + 15x^{12} - 59x^{10} + 104x^{8} - 59x^{6} + 15x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{3}) q^{2} + ( - \beta_{10} - \beta_{8} + \cdots + \beta_{2}) q^{3}+ \cdots + ( - 2 \beta_{14} - 2 \beta_{10} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{3}) q^{2} + ( - \beta_{10} - \beta_{8} + \cdots + \beta_{2}) q^{3}+ \cdots + ( - 2 \beta_{15} - 4 \beta_{14} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{3} - 12 q^{4} + 10 q^{6} - 10 q^{7} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{3} - 12 q^{4} + 10 q^{6} - 10 q^{7} - 20 q^{9} + 4 q^{12} - 12 q^{15} - 16 q^{16} + 20 q^{18} + 40 q^{19} + 30 q^{22} - 70 q^{24} + 4 q^{25} - 4 q^{27} - 110 q^{28} - 10 q^{30} + 10 q^{31} + 12 q^{33} + 100 q^{34} + 40 q^{36} - 2 q^{37} - 10 q^{39} + 50 q^{40} - 10 q^{42} - 32 q^{45} - 40 q^{46} + 22 q^{48} + 42 q^{49} - 40 q^{52} + 6 q^{55} + 40 q^{57} - 20 q^{58} + 14 q^{60} - 50 q^{61} + 70 q^{63} + 42 q^{64} + 30 q^{66} - 108 q^{67} - 12 q^{69} + 40 q^{70} - 40 q^{72} - 50 q^{73} + 12 q^{75} + 40 q^{78} - 40 q^{79} - 4 q^{81} + 50 q^{82} - 150 q^{84} - 20 q^{85} + 70 q^{88} - 20 q^{90} + 10 q^{91} - 50 q^{94} + 40 q^{96} - 58 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{14} + 15x^{12} - 59x^{10} + 104x^{8} - 59x^{6} + 15x^{4} - x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 27\nu^{14} + 50\nu^{12} + 355\nu^{10} - 484\nu^{8} - 1348\nu^{6} + 4523\nu^{4} - 750\nu^{2} - 365 ) / 384 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 21\nu^{14} - 22\nu^{12} + 329\nu^{10} - 1260\nu^{8} + 2436\nu^{6} - 1995\nu^{4} + 1498\nu^{2} - 119 ) / 384 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} - 35\nu^{13} + 40\nu^{11} - 560\nu^{9} + 1972\nu^{7} - 3075\nu^{5} + 1041\nu^{3} - 12\nu ) / 96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\nu^{15} + \nu^{13} + 181\nu^{11} - 524\nu^{9} + 688\nu^{7} - 128\nu^{5} + 717\nu^{3} - 231\nu ) / 96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9\nu^{15} - 23\nu^{13} + 148\nu^{11} - 736\nu^{9} + 1748\nu^{7} - 1867\nu^{5} + 685\nu^{3} + 16\nu ) / 96 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9\nu^{14} - 17\nu^{12} + 138\nu^{10} - 648\nu^{8} + 1332\nu^{6} - 1107\nu^{4} + 155\nu^{2} + 30 ) / 96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -14\nu^{14} + 7\nu^{12} - 195\nu^{10} + 724\nu^{8} - 920\nu^{6} - 210\nu^{4} + 555\nu^{2} - 23 ) / 96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 63\nu^{15} - 42\nu^{13} + 947\nu^{11} - 3428\nu^{9} + 5644\nu^{7} - 2945\nu^{5} + 1990\nu^{3} - 685\nu ) / 384 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -17\nu^{14} + 5\nu^{12} - 246\nu^{10} + 824\nu^{8} - 1108\nu^{6} - 101\nu^{4} + 201\nu^{2} - 58 ) / 96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -77\nu^{15} + 134\nu^{13} - 1185\nu^{11} + 5388\nu^{9} - 10980\nu^{7} + 9107\nu^{5} - 2666\nu^{3} + 543\nu ) / 384 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 7\nu^{14} - 17\nu^{12} + 112\nu^{10} - 560\nu^{8} + 1276\nu^{6} - 1269\nu^{4} + 411\nu^{2} - 12 ) / 48 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -15\nu^{14} + 9\nu^{12} - 224\nu^{10} + 800\nu^{8} - 1276\nu^{6} + 557\nu^{4} - 307\nu^{2} + 28 ) / 48 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -155\nu^{15} + 166\nu^{13} - 2315\nu^{11} + 9284\nu^{9} - 16444\nu^{7} + 9013\nu^{5} - 410\nu^{3} - 1499\nu ) / 384 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -91\nu^{15} + 70\nu^{13} - 1355\nu^{11} + 5060\nu^{9} - 8380\nu^{7} + 3765\nu^{5} - 890\nu^{3} - 155\nu ) / 192 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 375\nu^{15} - 270\nu^{13} + 5559\nu^{11} - 20564\nu^{9} + 33388\nu^{7} - 13145\nu^{5} + 2290\nu^{3} + 391\nu ) / 384 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} + 3\beta_{14} - 4\beta_{13} + 3\beta_{8} + 2\beta_{5} + \beta_{4} - 2\beta_{3} ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{12} - 2\beta_{11} - 2\beta_{7} + \beta_{6} + 11\beta_{2} - \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} - 3\beta_{8} - 3\beta_{5} + 2\beta_{4} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -12\beta_{12} + 7\beta_{11} + 5\beta_{9} - 13\beta_{7} - 41\beta_{6} - 11\beta_{2} - 19\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -24\beta_{15} - 73\beta_{14} + 24\beta_{13} - 20\beta_{10} - 38\beta_{8} + 18\beta_{5} - 6\beta_{4} - 18\beta_{3} ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 5\beta_{12} + 11\beta_{11} - 6\beta_{9} + 11\beta_{7} + 6\beta_{6} - 23\beta_{2} + 17\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -36\beta_{15} + 153\beta_{14} - 109\beta_{13} + 333\beta_{8} + 262\beta_{5} - 109\beta_{4} - 182\beta_{3} ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 237\beta_{12} - 237\beta_{11} - 110\beta_{9} + 98\beta_{7} + 631\beta_{6} + 631\beta_{2} + 139\beta _1 - 110 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 124 \beta_{15} + 226 \beta_{14} - 35 \beta_{13} + 67 \beta_{10} - 62 \beta_{8} - 195 \beta_{5} + \cdots + 159 \beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1032 \beta_{12} - 408 \beta_{11} + 660 \beta_{9} - 1248 \beta_{7} - 2316 \beta_{6} + 624 \beta_{2} + \cdots - 445 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 799 \beta_{15} - 5763 \beta_{14} + 2684 \beta_{13} - 840 \beta_{10} - 5763 \beta_{8} + \cdots + 1342 \beta_{3} ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -350\beta_{12} + 1106\beta_{11} + 350\beta_{7} - 1141\beta_{6} - 2771\beta_{2} + 553\beta _1 + 588 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 9421 \beta_{15} - 5857 \beta_{14} - 3564 \beta_{13} - 3780 \beta_{10} + 21543 \beta_{8} + \cdots - 18842 \beta_{3} ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 24372 \beta_{12} - 7607 \beta_{11} - 12985 \beta_{9} + 19793 \beta_{7} + 60721 \beta_{6} + \cdots + 31979 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 8280 \beta_{15} + 25393 \beta_{14} - 8280 \beta_{13} + 5480 \beta_{10} + 10610 \beta_{8} + \cdots + 5130 \beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/165\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(67\)
\(\chi(n)\) \(\beta_{2}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
1.23158 + 1.69513i
0.280526 + 0.386111i
−0.280526 − 0.386111i
−1.23158 − 1.69513i
−1.28932 − 0.418926i
−0.701538 − 0.227943i
0.701538 + 0.227943i
1.28932 + 0.418926i
−1.28932 + 0.418926i
−0.701538 + 0.227943i
0.701538 − 0.227943i
1.28932 − 0.418926i
1.23158 − 1.69513i
0.280526 − 0.386111i
−0.280526 + 0.386111i
−1.23158 + 1.69513i
−1.99274 + 1.44781i −0.0877853 + 1.72982i 1.25683 − 3.86812i 0.587785 − 0.809017i −2.32953 − 3.57419i −4.90846 − 1.59485i 1.57346 + 4.84261i −2.98459 − 0.303706i 2.46317i
41.2 −0.453901 + 0.329779i 1.08779 + 1.34786i −0.520762 + 1.60274i −0.587785 + 0.809017i −0.938242 − 0.253067i −0.254663 − 0.0827449i −0.638924 − 1.96641i −0.633446 + 2.93236i − 0.561053i
41.3 0.453901 − 0.329779i −0.0877853 + 1.72982i −0.520762 + 1.60274i 0.587785 − 0.809017i 0.530613 + 0.814119i −0.254663 − 0.0827449i 0.638924 + 1.96641i −2.98459 − 0.303706i − 0.561053i
41.4 1.99274 − 1.44781i 1.08779 + 1.34786i 1.25683 − 3.86812i −0.587785 + 0.809017i 4.11912 + 1.11103i −4.90846 − 1.59485i −1.57346 − 4.84261i −0.633446 + 2.93236i 2.46317i
101.1 −0.796845 + 2.45244i −0.451057 − 1.67229i −3.76145 − 2.73286i 0.951057 − 0.309017i 4.46061 + 0.226367i 2.05478 − 2.82816i 5.52712 − 4.01569i −2.59310 + 1.50859i 2.57865i
101.2 −0.433574 + 1.33440i 1.45106 + 0.945746i 0.0253869 + 0.0184446i −0.951057 + 0.309017i −1.89115 + 1.52624i 0.608337 − 0.837304i −2.30584 + 1.67529i 1.21113 + 2.74466i − 1.40308i
101.3 0.433574 − 1.33440i −0.451057 − 1.67229i 0.0253869 + 0.0184446i 0.951057 − 0.309017i −2.42707 − 0.123169i 0.608337 − 0.837304i 2.30584 − 1.67529i −2.59310 + 1.50859i − 1.40308i
101.4 0.796845 − 2.45244i 1.45106 + 0.945746i −3.76145 − 2.73286i −0.951057 + 0.309017i 3.47565 − 2.80501i 2.05478 − 2.82816i −5.52712 + 4.01569i 1.21113 + 2.74466i 2.57865i
116.1 −0.796845 − 2.45244i −0.451057 + 1.67229i −3.76145 + 2.73286i 0.951057 + 0.309017i 4.46061 − 0.226367i 2.05478 + 2.82816i 5.52712 + 4.01569i −2.59310 − 1.50859i − 2.57865i
116.2 −0.433574 − 1.33440i 1.45106 − 0.945746i 0.0253869 − 0.0184446i −0.951057 − 0.309017i −1.89115 − 1.52624i 0.608337 + 0.837304i −2.30584 − 1.67529i 1.21113 − 2.74466i 1.40308i
116.3 0.433574 + 1.33440i −0.451057 + 1.67229i 0.0253869 − 0.0184446i 0.951057 + 0.309017i −2.42707 + 0.123169i 0.608337 + 0.837304i 2.30584 + 1.67529i −2.59310 − 1.50859i 1.40308i
116.4 0.796845 + 2.45244i 1.45106 − 0.945746i −3.76145 + 2.73286i −0.951057 − 0.309017i 3.47565 + 2.80501i 2.05478 + 2.82816i −5.52712 − 4.01569i 1.21113 − 2.74466i − 2.57865i
161.1 −1.99274 − 1.44781i −0.0877853 − 1.72982i 1.25683 + 3.86812i 0.587785 + 0.809017i −2.32953 + 3.57419i −4.90846 + 1.59485i 1.57346 − 4.84261i −2.98459 + 0.303706i − 2.46317i
161.2 −0.453901 − 0.329779i 1.08779 − 1.34786i −0.520762 − 1.60274i −0.587785 − 0.809017i −0.938242 + 0.253067i −0.254663 + 0.0827449i −0.638924 + 1.96641i −0.633446 − 2.93236i 0.561053i
161.3 0.453901 + 0.329779i −0.0877853 − 1.72982i −0.520762 − 1.60274i 0.587785 + 0.809017i 0.530613 − 0.814119i −0.254663 + 0.0827449i 0.638924 − 1.96641i −2.98459 + 0.303706i 0.561053i
161.4 1.99274 + 1.44781i 1.08779 − 1.34786i 1.25683 + 3.86812i −0.587785 − 0.809017i 4.11912 − 1.11103i −4.90846 + 1.59485i −1.57346 + 4.84261i −0.633446 − 2.93236i − 2.46317i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
33.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.2.p.a ✓ 16
3.b odd 2 1 inner 165.2.p.a ✓ 16
5.b even 2 1 825.2.bi.d 16
5.c odd 4 1 825.2.bs.e 16
5.c odd 4 1 825.2.bs.f 16
11.d odd 10 1 inner 165.2.p.a ✓ 16
15.d odd 2 1 825.2.bi.d 16
15.e even 4 1 825.2.bs.e 16
15.e even 4 1 825.2.bs.f 16
33.f even 10 1 inner 165.2.p.a ✓ 16
55.h odd 10 1 825.2.bi.d 16
55.l even 20 1 825.2.bs.e 16
55.l even 20 1 825.2.bs.f 16
165.r even 10 1 825.2.bi.d 16
165.u odd 20 1 825.2.bs.e 16
165.u odd 20 1 825.2.bs.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.p.a ✓ 16 1.a even 1 1 trivial
165.2.p.a ✓ 16 3.b odd 2 1 inner
165.2.p.a ✓ 16 11.d odd 10 1 inner
165.2.p.a ✓ 16 33.f even 10 1 inner
825.2.bi.d 16 5.b even 2 1
825.2.bi.d 16 15.d odd 2 1
825.2.bi.d 16 55.h odd 10 1
825.2.bi.d 16 165.r even 10 1
825.2.bs.e 16 5.c odd 4 1
825.2.bs.e 16 15.e even 4 1
825.2.bs.e 16 55.l even 20 1
825.2.bs.e 16 165.u odd 20 1
825.2.bs.f 16 5.c odd 4 1
825.2.bs.f 16 15.e even 4 1
825.2.bs.f 16 55.l even 20 1
825.2.bs.f 16 165.u odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 10T_{2}^{14} + 65T_{2}^{12} + 375T_{2}^{10} + 2450T_{2}^{8} + 5625T_{2}^{6} + 5375T_{2}^{4} - 625T_{2}^{2} + 625 \) acting on \(S_{2}^{\mathrm{new}}(165, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 10 T^{14} + \cdots + 625 \) Copy content Toggle raw display
$3$ \( (T^{8} - 4 T^{7} + 13 T^{6} + \cdots + 81)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 5 T^{7} - 5 T^{6} + \cdots + 25)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( (T^{8} - 5 T^{6} + 15 T^{5} + \cdots + 25)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 130 T^{14} + \cdots + 625 \) Copy content Toggle raw display
$19$ \( (T^{8} - 20 T^{7} + \cdots + 3025)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 59 T^{6} + 741 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + 50 T^{14} + \cdots + 81450625 \) Copy content Toggle raw display
$31$ \( (T^{8} - 5 T^{7} + \cdots + 600625)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + T^{7} + \cdots + 22201)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 184062450625 \) Copy content Toggle raw display
$43$ \( (T^{8} + 40 T^{6} + \cdots + 25)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 1908029761 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 492884401 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 5465500541281 \) Copy content Toggle raw display
$61$ \( (T^{8} + 25 T^{7} + \cdots + 15625)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 27 T^{3} + \cdots + 281)^{4} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 360750390625 \) Copy content Toggle raw display
$73$ \( (T^{8} + 25 T^{7} + \cdots + 8673025)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 20 T^{7} + \cdots + 483025)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 133974300625 \) Copy content Toggle raw display
$89$ \( (T^{8} + 490 T^{6} + \cdots + 161925625)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 29 T^{7} + \cdots + 5958481)^{2} \) Copy content Toggle raw display
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