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Properties

Label 465.2.n.e
Level $465$
Weight $2$
Character orbit 465.n
Analytic conductor $3.713$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [465,2,Mod(16,465)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(465, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("465.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.n (of order \(5\), 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: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} + 14 x^{18} - 34 x^{17} + 102 x^{16} - 176 x^{15} + 468 x^{14} - 806 x^{13} + 2268 x^{12} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + \beta_{13} q^{3} + ( - \beta_{9} - \beta_{5}) q^{4} - q^{5} + \beta_{6} q^{6} + (\beta_{16} + \beta_{6} - \beta_{4} + \cdots - \beta_1) q^{7} + (\beta_{19} - \beta_{17} - \beta_{13} + \cdots + 1) q^{8}+ \cdots + ( - \beta_{17} + \beta_{15} - \beta_{13} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{2} + 5 q^{3} - 7 q^{4} - 20 q^{5} + 6 q^{6} + 3 q^{7} + 7 q^{8} - 5 q^{9} + q^{10} - 6 q^{11} + 2 q^{12} + 2 q^{13} - 21 q^{14} - 5 q^{15} + 15 q^{16} - 8 q^{17} - q^{18} - 15 q^{19} + 7 q^{20}+ \cdots + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} + 14 x^{18} - 34 x^{17} + 102 x^{16} - 176 x^{15} + 468 x^{14} - 806 x^{13} + 2268 x^{12} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 50\!\cdots\!91 \nu^{19} + \cdots + 19\!\cdots\!75 ) / 66\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 19\!\cdots\!19 \nu^{19} + \cdots - 12\!\cdots\!11 ) / 66\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 15\!\cdots\!55 \nu^{19} + \cdots - 51\!\cdots\!56 ) / 16\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 60\!\cdots\!66 \nu^{19} + \cdots - 28\!\cdots\!07 ) / 33\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10\!\cdots\!35 \nu^{19} + \cdots - 60\!\cdots\!66 ) / 33\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 51\!\cdots\!56 \nu^{19} + \cdots + 10\!\cdots\!06 ) / 16\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12\!\cdots\!21 \nu^{19} + \cdots + 92\!\cdots\!39 ) / 33\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 17\!\cdots\!54 \nu^{19} + \cdots + 85\!\cdots\!11 ) / 33\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 31\!\cdots\!25 \nu^{19} + \cdots + 27\!\cdots\!11 ) / 33\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 66\!\cdots\!15 \nu^{19} + \cdots - 32\!\cdots\!41 ) / 66\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 27\!\cdots\!83 \nu^{19} + \cdots + 16\!\cdots\!14 ) / 16\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11\!\cdots\!19 \nu^{19} + \cdots - 22\!\cdots\!89 ) / 66\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 82\!\cdots\!21 \nu^{19} + \cdots + 11\!\cdots\!03 ) / 33\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 84\!\cdots\!10 \nu^{19} + \cdots + 51\!\cdots\!13 ) / 33\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 12\!\cdots\!68 \nu^{19} + \cdots + 82\!\cdots\!73 ) / 33\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 13\!\cdots\!77 \nu^{19} + \cdots - 91\!\cdots\!68 ) / 33\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 31\!\cdots\!25 \nu^{19} + \cdots + 37\!\cdots\!83 ) / 66\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 37\!\cdots\!97 \nu^{19} + \cdots - 25\!\cdots\!11 ) / 66\!\cdots\!48 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{19} + \beta_{18} - 3\beta_{7} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{18} + \beta_{17} - \beta_{15} - \beta_{14} + \beta_{13} + \beta_{10} + \beta_{8} + \beta_{7} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{18} + \beta_{17} - \beta_{16} - 2 \beta_{15} + 4 \beta_{13} + 2 \beta_{12} + \beta_{10} + \cdots - 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{19} + 2 \beta_{17} - 8 \beta_{16} - 2 \beta_{15} + 8 \beta_{14} + 11 \beta_{13} + 8 \beta_{12} + \cdots - 23 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 49 \beta_{19} - 12 \beta_{18} + 11 \beta_{17} - 9 \beta_{16} + 11 \beta_{14} + 85 \beta_{13} + \cdots - 97 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 96 \beta_{19} - 96 \beta_{18} + 26 \beta_{15} + 26 \beta_{14} - 11 \beta_{13} - 11 \beta_{12} + \cdots + 46 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 114 \beta_{19} - 350 \beta_{18} - 94 \beta_{17} + 71 \beta_{16} + 191 \beta_{15} + 94 \beta_{14} + \cdots + 288 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 289 \beta_{18} - 251 \beta_{17} + 421 \beta_{16} + 612 \beta_{15} - 1000 \beta_{13} - 515 \beta_{12} + \cdots + 1251 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1000 \beta_{19} - 807 \beta_{17} + 936 \beta_{16} + 807 \beta_{15} - 741 \beta_{14} - 2780 \beta_{13} + \cdots + 5950 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 6202 \beta_{19} + 2654 \beta_{18} - 2473 \beta_{17} + 1380 \beta_{16} - 2187 \beta_{14} - 6764 \beta_{13} + \cdots + 9157 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 19021 \beta_{19} + 19021 \beta_{18} - 6573 \beta_{15} - 6573 \beta_{14} + 2187 \beta_{13} + 2187 \beta_{12} + \cdots + 1031 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 22984 \beta_{19} + 48631 \beta_{18} + 17438 \beta_{17} - 11638 \beta_{16} - 35649 \beta_{15} + \cdots - 53860 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 69689 \beta_{18} + 52964 \beta_{17} - 60269 \beta_{16} - 95918 \beta_{15} + 204857 \beta_{13} + \cdots - 257821 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 192432 \beta_{19} + 148169 \beta_{17} - 178356 \beta_{16} - 148169 \beta_{15} + 126105 \beta_{14} + \cdots - 856900 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 1095557 \beta_{19} - 566914 \beta_{18} + 325220 \beta_{17} - 275583 \beta_{16} + 423752 \beta_{14} + \cdots - 1758540 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 2956130 \beta_{19} - 2956130 \beta_{18} + 1189781 \beta_{15} + 1189781 \beta_{14} - 423752 \beta_{13} + \cdots - 282391 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 4564894 \beta_{19} - 8420086 \beta_{18} - 2469903 \beta_{17} + 2182587 \beta_{16} + 5842271 \beta_{15} + \cdots + 9214639 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 12753259 \beta_{18} - 9476834 \beta_{17} + 10602673 \beta_{16} + 16444944 \beta_{15} + \cdots + 45515203 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{5} + \beta_{7} + \beta_{13}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
0.637536 + 1.96213i
0.409100 + 1.25908i
−0.0381448 − 0.117398i
−0.378783 − 1.16577i
−0.747742 − 2.30131i
2.26261 − 1.64389i
1.33101 − 0.967034i
0.438498 − 0.318588i
−0.148654 + 0.108004i
−1.76543 + 1.28266i
2.26261 + 1.64389i
1.33101 + 0.967034i
0.438498 + 0.318588i
−0.148654 − 0.108004i
−1.76543 − 1.28266i
0.637536 − 1.96213i
0.409100 − 1.25908i
−0.0381448 + 0.117398i
−0.378783 + 1.16577i
−0.747742 + 2.30131i
−1.66909 + 1.21267i 0.809017 + 0.587785i 0.697273 − 2.14598i −1.00000 −2.06311 0.615965 − 1.89574i 0.163478 + 0.503134i 0.309017 + 0.951057i 1.66909 − 1.21267i
16.2 −1.07104 + 0.778154i 0.809017 + 0.587785i −0.0764373 + 0.235250i −1.00000 −1.32387 −0.947997 + 2.91763i −0.919393 − 2.82960i 0.309017 + 0.951057i 1.07104 − 0.778154i
16.3 0.0998644 − 0.0725558i 0.809017 + 0.587785i −0.613325 + 1.88762i −1.00000 0.123439 0.472867 − 1.45534i 0.151998 + 0.467802i 0.309017 + 0.951057i −0.0998644 + 0.0725558i
16.4 0.991667 − 0.720489i 0.809017 + 0.587785i −0.153733 + 0.473143i −1.00000 1.22577 −1.37543 + 4.23313i 0.946008 + 2.91151i 0.309017 + 0.951057i −0.991667 + 0.720489i
16.5 1.95761 − 1.42229i 0.809017 + 0.587785i 1.19131 − 3.66647i −1.00000 2.41974 0.307542 − 0.946518i −1.38718 − 4.26929i 0.309017 + 0.951057i −1.95761 + 1.42229i
256.1 −0.864242 + 2.65986i −0.309017 − 0.951057i −4.70992 − 3.42196i −1.00000 2.79674 3.88423 + 2.82206i 8.64721 − 6.28257i −0.809017 + 0.587785i 0.864242 − 2.65986i
256.2 −0.508400 + 1.56469i −0.309017 − 0.951057i −0.571764 − 0.415411i −1.00000 1.64522 0.192639 + 0.139961i −1.72134 + 1.25063i −0.809017 + 0.587785i 0.508400 − 1.56469i
256.3 −0.167492 + 0.515486i −0.309017 − 0.951057i 1.38036 + 1.00289i −1.00000 0.542014 −2.90722 − 2.11222i −1.62517 + 1.18076i −0.809017 + 0.587785i 0.167492 − 0.515486i
256.4 0.0567809 − 0.174754i −0.309017 − 0.951057i 1.59072 + 1.15573i −1.00000 −0.183747 2.27111 + 1.65006i 0.589599 − 0.428368i −0.809017 + 0.587785i −0.0567809 + 0.174754i
256.5 0.674335 − 2.07539i −0.309017 − 0.951057i −2.23448 − 1.62345i −1.00000 −2.18219 −1.01371 − 0.736504i −1.34521 + 0.977354i −0.809017 + 0.587785i −0.674335 + 2.07539i
376.1 −0.864242 − 2.65986i −0.309017 + 0.951057i −4.70992 + 3.42196i −1.00000 2.79674 3.88423 − 2.82206i 8.64721 + 6.28257i −0.809017 − 0.587785i 0.864242 + 2.65986i
376.2 −0.508400 − 1.56469i −0.309017 + 0.951057i −0.571764 + 0.415411i −1.00000 1.64522 0.192639 − 0.139961i −1.72134 − 1.25063i −0.809017 − 0.587785i 0.508400 + 1.56469i
376.3 −0.167492 − 0.515486i −0.309017 + 0.951057i 1.38036 − 1.00289i −1.00000 0.542014 −2.90722 + 2.11222i −1.62517 − 1.18076i −0.809017 − 0.587785i 0.167492 + 0.515486i
376.4 0.0567809 + 0.174754i −0.309017 + 0.951057i 1.59072 − 1.15573i −1.00000 −0.183747 2.27111 − 1.65006i 0.589599 + 0.428368i −0.809017 − 0.587785i −0.0567809 − 0.174754i
376.5 0.674335 + 2.07539i −0.309017 + 0.951057i −2.23448 + 1.62345i −1.00000 −2.18219 −1.01371 + 0.736504i −1.34521 − 0.977354i −0.809017 − 0.587785i −0.674335 − 2.07539i
436.1 −1.66909 − 1.21267i 0.809017 − 0.587785i 0.697273 + 2.14598i −1.00000 −2.06311 0.615965 + 1.89574i 0.163478 − 0.503134i 0.309017 − 0.951057i 1.66909 + 1.21267i
436.2 −1.07104 − 0.778154i 0.809017 − 0.587785i −0.0764373 − 0.235250i −1.00000 −1.32387 −0.947997 − 2.91763i −0.919393 + 2.82960i 0.309017 − 0.951057i 1.07104 + 0.778154i
436.3 0.0998644 + 0.0725558i 0.809017 − 0.587785i −0.613325 − 1.88762i −1.00000 0.123439 0.472867 + 1.45534i 0.151998 − 0.467802i 0.309017 − 0.951057i −0.0998644 − 0.0725558i
436.4 0.991667 + 0.720489i 0.809017 − 0.587785i −0.153733 − 0.473143i −1.00000 1.22577 −1.37543 − 4.23313i 0.946008 − 2.91151i 0.309017 − 0.951057i −0.991667 − 0.720489i
436.5 1.95761 + 1.42229i 0.809017 − 0.587785i 1.19131 + 3.66647i −1.00000 2.41974 0.307542 + 0.946518i −1.38718 + 4.26929i 0.309017 − 0.951057i −1.95761 − 1.42229i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 465.2.n.e ✓ 20
31.d even 5 1 inner 465.2.n.e ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.n.e ✓ 20 1.a even 1 1 trivial
465.2.n.e ✓ 20 31.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + T_{2}^{19} + 9 T_{2}^{18} + T_{2}^{17} + 42 T_{2}^{16} + 49 T_{2}^{15} + 253 T_{2}^{14} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(465, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + T^{19} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{5} \) Copy content Toggle raw display
$5$ \( (T + 1)^{20} \) Copy content Toggle raw display
$7$ \( T^{20} - 3 T^{19} + \cdots + 358801 \) Copy content Toggle raw display
$11$ \( T^{20} + 6 T^{19} + \cdots + 19927296 \) Copy content Toggle raw display
$13$ \( T^{20} - 2 T^{19} + \cdots + 34117281 \) Copy content Toggle raw display
$17$ \( T^{20} + 8 T^{19} + \cdots + 614656 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 11319044881 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 451562496256 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 582694062336 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 819628286980801 \) Copy content Toggle raw display
$37$ \( (T^{10} - 5 T^{9} + \cdots + 2238541)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} - 18 T^{19} + \cdots + 33362176 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 17593670859361 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 390495010816 \) Copy content Toggle raw display
$53$ \( T^{20} + 14 T^{19} + \cdots + 3936256 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 28209865199616 \) Copy content Toggle raw display
$61$ \( (T^{10} + 2 T^{9} + \cdots + 12905251)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + 4 T^{9} + \cdots - 685379)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 18046160896 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 75581723750625 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 45309268838656 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 745979429765376 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 1095222947841 \) Copy content Toggle raw display
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