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Properties

Label 369.2.n.d
Level $369$
Weight $2$
Character orbit 369.n
Analytic conductor $2.946$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [369,2,Mod(64,369)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(369, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("369.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 369 = 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 369.n (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: \(2.94647983459\)
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: \( x^{16} - 9x^{14} + 51x^{12} - 249x^{10} + 1476x^{8} - 2875x^{6} + 2335x^{4} + 125x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{12} q^{2} + (\beta_{9} - \beta_{7} + \cdots - \beta_{2}) q^{4}+ \cdots + ( - \beta_{15} - 2 \beta_{14} + \cdots - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{12} q^{2} + (\beta_{9} - \beta_{7} + \cdots - \beta_{2}) q^{4}+ \cdots + ( - 5 \beta_{15} + \beta_{14} + \cdots + 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 10 q^{4} + 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 10 q^{4} + 20 q^{7} + 22 q^{10} + 20 q^{13} - 10 q^{16} + 20 q^{19} - 50 q^{22} + 12 q^{25} - 30 q^{28} + 12 q^{31} - 50 q^{34} - 16 q^{37} - 100 q^{40} - 28 q^{43} + 46 q^{46} - 8 q^{49} - 10 q^{52} - 10 q^{58} + 8 q^{61} + 62 q^{64} - 40 q^{67} + 20 q^{70} + 64 q^{73} - 30 q^{76} + 66 q^{82} + 90 q^{88} + 40 q^{91} + 30 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 9x^{14} + 51x^{12} - 249x^{10} + 1476x^{8} - 2875x^{6} + 2335x^{4} + 125x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 16594091 \nu^{14} + 132889182 \nu^{12} - 669375495 \nu^{10} + 3344289682 \nu^{8} + \cdots + 109896768237 ) / 48030072872 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 399459369 \nu^{14} - 3872461014 \nu^{12} + 22604705541 \nu^{10} - 111418189614 \nu^{8} + \cdots - 163815397855 ) / 240150364360 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 641538645 \nu^{14} - 5845087188 \nu^{12} + 33077583727 \nu^{10} - 161239139198 \nu^{8} + \cdots + 220551349875 ) / 240150364360 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 420560697 \nu^{14} - 4175189607 \nu^{12} + 24562062708 \nu^{10} - 121476379662 \nu^{8} + \cdots - 186816081040 ) / 120075182180 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1070034741 \nu^{14} - 9471203848 \nu^{12} + 53263591527 \nu^{10} - 259312649798 \nu^{8} + \cdots + 118775784515 ) / 240150364360 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 724570675 \nu^{14} + 6672080139 \nu^{12} - 37857976976 \nu^{10} + 184639030474 \nu^{8} + \cdots - 251290182000 ) / 120075182180 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 399459369 \nu^{15} - 3872461014 \nu^{13} + 22604705541 \nu^{11} - 111418189614 \nu^{9} + \cdots - 163815397855 \nu ) / 240150364360 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 921163875 \nu^{14} - 8325322917 \nu^{12} + 47647278148 \nu^{10} - 233617980722 \nu^{8} + \cdots - 28361391040 ) / 120075182180 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 827955465 \nu^{15} - 7498577674 \nu^{13} + 42790713341 \nu^{11} - 209491700214 \nu^{9} + \cdots - 25440598855 \nu ) / 240150364360 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1070034741 \nu^{15} - 9471203848 \nu^{13} + 53263591527 \nu^{11} - 259312649798 \nu^{9} + \cdots + 118775784515 \nu ) / 240150364360 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 914980841 \nu^{15} + 8770447689 \nu^{13} - 51196668716 \nu^{11} + 253062739159 \nu^{9} + \cdots + 962789857750 \nu ) / 150093977725 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 12695929441 \nu^{15} + 111086725194 \nu^{13} - 620796945061 \nu^{11} + \cdots - 5798667404425 \nu ) / 1200751821800 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 13576283373 \nu^{15} + 121399649892 \nu^{13} - 682891277823 \nu^{11} + \cdots - 8793919655675 \nu ) / 1200751821800 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 24923473777 \nu^{15} - 228729353598 \nu^{13} + 1310185729437 \nu^{11} + \cdots - 3637970480775 \nu ) / 1200751821800 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} + 3\beta_{6} - 4\beta_{4} + 3\beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{15} - \beta_{14} - 2\beta_{12} - \beta_{11} + 7\beta_{10} - \beta_{8} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} - 6\beta_{7} + 6\beta_{6} + \beta_{5} - 22\beta_{4} + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{15} - 14\beta_{14} + \beta_{13} - 15\beta_{12} - 34\beta_{11} + 43\beta_{10} - 42\beta_{8} - 33\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -9\beta_{9} + 39\beta_{6} + 43\beta_{5} - 39\beta_{4} - 89\beta_{3} + 9\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -9\beta_{15} - 52\beta_{14} + 52\beta_{13} - 43\beta_{12} + 66\beta_{10} - 205\beta_{8} - 14\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -191\beta_{9} + 191\beta_{7} + 501\beta_{6} + 257\beta_{5} - 256\beta_{3} + 257\beta_{2} - 256 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 66\beta_{15} - 66\beta_{14} + 514\beta_{13} + 257\beta_{12} + 1074\beta_{11} + 60\beta_{8} + 60\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 454\beta_{7} + 454\beta_{5} + 1664\beta_{4} - 1664\beta_{3} + 1528\beta_{2} - 4503 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1982 \beta_{15} + 454 \beta_{14} + 1982 \beta_{13} + 3510 \beta_{12} + 3026 \beta_{11} + \cdots - 4079 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 6061\beta_{9} + 3026\beta_{7} - 16171\beta_{6} + 26862\beta_{4} - 16171\beta_{3} + 3026\beta_{2} - 26862 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 18174 \beta_{15} + 12113 \beta_{14} + 3026 \beta_{13} + 24226 \beta_{12} + 19769 \beta_{11} + \cdots + 3026 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 19769\beta_{9} + 34354\beta_{7} - 68004\beta_{6} - 19769\beta_{5} + 160548\beta_{4} - 68004 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 73892 \beta_{15} + 108246 \beta_{14} - 19769 \beta_{13} + 128015 \beta_{12} + 195606 \beta_{11} + \cdots + 175837 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/369\mathbb{Z}\right)^\times\).

\(n\) \(83\) \(334\)
\(\chi(n)\) \(1\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
−1.18970 − 0.386556i
−2.33991 − 0.760284i
2.33991 + 0.760284i
1.18970 + 0.386556i
−0.185814 + 0.255752i
−1.35089 + 1.85934i
1.35089 − 1.85934i
0.185814 − 0.255752i
−0.185814 − 0.255752i
−1.35089 − 1.85934i
1.35089 + 1.85934i
0.185814 + 0.255752i
−1.18970 + 0.386556i
−2.33991 + 0.760284i
2.33991 − 0.760284i
1.18970 − 0.386556i
−0.760284 − 2.33991i 0 −3.27913 + 2.38243i −3.28609 + 2.38749i 0 1.80902 + 0.587785i 4.08684 + 2.96926i 0 8.08487 + 5.87400i
64.2 −0.386556 − 1.18970i 0 0.352078 − 0.255800i 1.13656 − 0.825761i 0 1.80902 + 0.587785i −2.46446 − 1.79053i 0 −1.42175 − 1.03296i
64.3 0.386556 + 1.18970i 0 0.352078 − 0.255800i −1.13656 + 0.825761i 0 1.80902 + 0.587785i 2.46446 + 1.79053i 0 −1.42175 − 1.03296i
64.4 0.760284 + 2.33991i 0 −3.27913 + 2.38243i 3.28609 − 2.38749i 0 1.80902 + 0.587785i −4.08684 − 2.96926i 0 8.08487 + 5.87400i
127.1 −1.85934 + 1.35089i 0 1.01420 − 3.12140i −0.386113 + 1.18833i 0 0.690983 − 0.951057i 0.910502 + 2.80224i 0 −0.887391 − 2.73111i
127.2 −0.255752 + 0.185814i 0 −0.587152 + 1.80707i −0.872208 + 2.68438i 0 0.690983 − 0.951057i −0.380991 − 1.17257i 0 −0.275728 − 0.848603i
127.3 0.255752 − 0.185814i 0 −0.587152 + 1.80707i 0.872208 − 2.68438i 0 0.690983 − 0.951057i 0.380991 + 1.17257i 0 −0.275728 − 0.848603i
127.4 1.85934 − 1.35089i 0 1.01420 − 3.12140i 0.386113 − 1.18833i 0 0.690983 − 0.951057i −0.910502 − 2.80224i 0 −0.887391 − 2.73111i
154.1 −1.85934 − 1.35089i 0 1.01420 + 3.12140i −0.386113 − 1.18833i 0 0.690983 + 0.951057i 0.910502 − 2.80224i 0 −0.887391 + 2.73111i
154.2 −0.255752 − 0.185814i 0 −0.587152 − 1.80707i −0.872208 − 2.68438i 0 0.690983 + 0.951057i −0.380991 + 1.17257i 0 −0.275728 + 0.848603i
154.3 0.255752 + 0.185814i 0 −0.587152 − 1.80707i 0.872208 + 2.68438i 0 0.690983 + 0.951057i 0.380991 − 1.17257i 0 −0.275728 + 0.848603i
154.4 1.85934 + 1.35089i 0 1.01420 + 3.12140i 0.386113 + 1.18833i 0 0.690983 + 0.951057i −0.910502 + 2.80224i 0 −0.887391 + 2.73111i
271.1 −0.760284 + 2.33991i 0 −3.27913 − 2.38243i −3.28609 − 2.38749i 0 1.80902 − 0.587785i 4.08684 − 2.96926i 0 8.08487 − 5.87400i
271.2 −0.386556 + 1.18970i 0 0.352078 + 0.255800i 1.13656 + 0.825761i 0 1.80902 − 0.587785i −2.46446 + 1.79053i 0 −1.42175 + 1.03296i
271.3 0.386556 − 1.18970i 0 0.352078 + 0.255800i −1.13656 − 0.825761i 0 1.80902 − 0.587785i 2.46446 − 1.79053i 0 −1.42175 + 1.03296i
271.4 0.760284 − 2.33991i 0 −3.27913 − 2.38243i 3.28609 + 2.38749i 0 1.80902 − 0.587785i −4.08684 + 2.96926i 0 8.08487 − 5.87400i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
41.f even 10 1 inner
123.l odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 369.2.n.d ✓ 16
3.b odd 2 1 inner 369.2.n.d ✓ 16
41.f even 10 1 inner 369.2.n.d ✓ 16
123.l odd 10 1 inner 369.2.n.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
369.2.n.d ✓ 16 1.a even 1 1 trivial
369.2.n.d ✓ 16 3.b odd 2 1 inner
369.2.n.d ✓ 16 41.f even 10 1 inner
369.2.n.d ✓ 16 123.l odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 9T_{2}^{14} + 51T_{2}^{12} + 249T_{2}^{10} + 1476T_{2}^{8} + 2875T_{2}^{6} + 2335T_{2}^{4} - 125T_{2}^{2} + 25 \) acting on \(S_{2}^{\mathrm{new}}(369, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 9 T^{14} + \cdots + 25 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 4 T^{14} + \cdots + 164025 \) Copy content Toggle raw display
$7$ \( (T^{4} - 5 T^{3} + 10 T^{2} + \cdots + 5)^{4} \) Copy content Toggle raw display
$11$ \( T^{16} - 16 T^{14} + \cdots + 366025 \) Copy content Toggle raw display
$13$ \( (T^{4} - 5 T^{3} + 10 T^{2} + \cdots + 5)^{4} \) Copy content Toggle raw display
$17$ \( T^{16} - 16 T^{14} + \cdots + 366025 \) Copy content Toggle raw display
$19$ \( (T^{4} - 5 T^{3} + 10 T^{2} + \cdots + 5)^{4} \) Copy content Toggle raw display
$23$ \( T^{16} + 76 T^{14} + \cdots + 164025 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 515818422025 \) Copy content Toggle raw display
$31$ \( (T^{8} - 6 T^{7} + \cdots + 164025)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 8 T^{7} + \cdots + 93025)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 7984925229121 \) Copy content Toggle raw display
$43$ \( (T^{8} + 14 T^{7} + \cdots + 32041)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 64 T^{14} + \cdots + 366025 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 189333765625 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 134839512025 \) Copy content Toggle raw display
$61$ \( (T^{8} - 4 T^{7} + \cdots + 60025)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 20 T^{7} + \cdots + 2739025)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 67\!\cdots\!25 \) Copy content Toggle raw display
$73$ \( (T^{4} - 16 T^{3} + \cdots - 2096)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + 500 T^{6} + \cdots + 41990400)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 332 T^{6} + \cdots + 103680)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 122890638216025 \) Copy content Toggle raw display
$97$ \( (T^{8} - 405 T^{6} + \cdots + 912025)^{2} \) Copy content Toggle raw display
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