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Properties

Label 495.4.d.a
Level $495$
Weight $4$
Character orbit 495.d
Analytic conductor $29.206$
Analytic rank $0$
Dimension $16$
CM discriminant -55
Inner twists $8$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [495,4,Mod(494,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.494");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 495.d (of order \(2\), degree \(1\), 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: \(29.2059454528\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 24x^{14} + 300x^{12} - 2112x^{10} + 8919x^{8} - 17520x^{6} + 27500x^{4} - 54000x^{2} + 50625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{13} q^{2} + ( - \beta_{3} - \beta_{2} - 8) q^{4} + 5 \beta_{5} q^{5} + (2 \beta_{12} + \beta_1) q^{7} + (3 \beta_{15} - 8 \beta_{13} - 2 \beta_{10}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{13} q^{2} + ( - \beta_{3} - \beta_{2} - 8) q^{4} + 5 \beta_{5} q^{5} + (2 \beta_{12} + \beta_1) q^{7} + (3 \beta_{15} - 8 \beta_{13} - 2 \beta_{10}) q^{8} - 5 \beta_{9} q^{10} - 11 \beta_{6} q^{11} + (5 \beta_{9} + 7 \beta_{8}) q^{13} + ( - 3 \beta_{11} + 13 \beta_{7} - 31 \beta_{6}) q^{14} + ( - 7 \beta_{4} + 8 \beta_{3} + \cdots + 64) q^{16}+ \cdots + ( - 193 \beta_{15} + \cdots + 46 \beta_{10}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 128 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 128 q^{4} + 1024 q^{16} - 2000 q^{25} + 5488 q^{49} - 8192 q^{64}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 24x^{14} + 300x^{12} - 2112x^{10} + 8919x^{8} - 17520x^{6} + 27500x^{4} - 54000x^{2} + 50625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 8279417 \nu^{15} - 195553063 \nu^{13} + 2253123820 \nu^{11} - 13290763804 \nu^{9} + \cdots + 860259942000 \nu ) / 485636799375 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 54822 \nu^{14} - 1332658 \nu^{12} + 17299505 \nu^{10} - 126621004 \nu^{8} + 565030403 \nu^{6} + \cdots + 903487950 ) / 498089025 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 129051 \nu^{14} + 3426604 \nu^{12} - 45345170 \nu^{10} + 338996962 \nu^{8} + \cdots + 9755095500 ) / 830148375 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 176 \nu^{14} - 4467 \nu^{12} + 57012 \nu^{10} - 416532 \nu^{8} + 1784160 \nu^{6} - 3502722 \nu^{4} + \cdots - 7072650 ) / 956025 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1232 \nu^{14} - 27569 \nu^{12} + 324884 \nu^{10} - 2076224 \nu^{8} + 7709120 \nu^{6} + \cdots - 37291050 ) / 4571775 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 335728 \nu^{14} + 7394248 \nu^{12} - 85732672 \nu^{10} + 531901402 \nu^{8} + \cdots + 7659311643 ) / 777018879 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 35703 \nu^{14} - 788812 \nu^{12} + 9177110 \nu^{10} - 57068086 \nu^{8} + 198855587 \nu^{6} + \cdots - 800518500 ) / 62141625 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9943988 \nu^{15} - 234203821 \nu^{13} + 2832699631 \nu^{11} - 19015063591 \nu^{9} + \cdots - 270979118325 \nu ) / 97127359875 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 377812 \nu^{15} + 9322529 \nu^{13} - 117720809 \nu^{11} + 844913744 \nu^{9} + \cdots + 39387525975 \nu ) / 2775067425 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 354976 \nu^{15} - 8314242 \nu^{13} + 101534142 \nu^{11} - 690428427 \nu^{9} + \cdots - 9225249750 \nu ) / 1494267075 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 32224 \nu^{14} + 714346 \nu^{12} - 8396155 \nu^{10} + 53581288 \nu^{8} - 198868621 \nu^{6} + \cdots + 961719750 ) / 20713875 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 211980452 \nu^{15} + 5056095553 \nu^{13} - 62090328670 \nu^{11} + 420787957099 \nu^{9} + \cdots + 6865306690500 \nu ) / 485636799375 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 2779034 \nu^{15} - 62142051 \nu^{13} + 733728390 \nu^{11} - 4707675633 \nu^{9} + \cdots - 71907482250 \nu ) / 5336668125 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 5411687 \nu^{15} + 117271218 \nu^{13} - 1341847770 \nu^{11} + 8139143919 \nu^{9} + \cdots + 137561505750 \nu ) / 7471335375 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1825444 \nu^{15} + 40095123 \nu^{13} - 464428803 \nu^{11} + 2875041483 \nu^{9} + \cdots + 47489188725 \nu ) / 2490445125 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} + 2\beta_{13} + \beta_{12} + 7\beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{11} + 9\beta_{7} + 18\beta_{6} - \beta_{3} - 6\beta_{2} + 54 ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -16\beta_{15} + 15\beta_{14} + 3\beta_{13} + 6\beta_{12} - 6\beta_{10} - 4\beta_{9} + 8\beta_{8} + 15\beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{11} + 108\beta_{7} + 162\beta_{6} + 9\beta_{5} + 27\beta_{4} + 28\beta_{3} - 12\beta_{2} - 54 ) / 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 147 \beta_{15} + 100 \beta_{14} - 70 \beta_{13} + 10 \beta_{12} + 9 \beta_{10} + 45 \beta_{9} + \cdots - 200 \beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 145\beta_{11} + 468\beta_{7} + 720\beta_{6} + 972\beta_{5} + 216\beta_{4} + 386\beta_{3} + 237\beta_{2} - 3240 ) / 18 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 721 \beta_{15} + 153 \beta_{14} - 1179 \beta_{13} - 204 \beta_{12} + 840 \beta_{10} + \cdots - 2751 \beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2216 \beta_{11} - 1080 \beta_{7} - 1701 \beta_{6} + 12348 \beta_{5} - 756 \beta_{4} + 1432 \beta_{3} + \cdots - 27783 ) / 18 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 456 \beta_{15} - 3533 \beta_{14} - 9253 \beta_{13} - 3590 \beta_{12} + 8388 \beta_{10} + \cdots - 14789 \beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 12469 \beta_{11} - 30015 \beta_{7} - 47502 \beta_{6} + 61560 \beta_{5} - 31320 \beta_{4} - 15427 \beta_{3} + \cdots - 93366 ) / 18 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 33746 \beta_{15} - 34782 \beta_{14} - 8679 \beta_{13} - 33759 \beta_{12} + 20019 \beta_{10} + \cdots - 4356 \beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 23234 \beta_{11} - 94608 \beta_{7} - 149688 \beta_{6} - 162171 \beta_{5} - 157788 \beta_{4} + \cdots + 153846 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 434475 \beta_{15} - 117182 \beta_{14} + 664196 \beta_{13} - 178334 \beta_{12} - 456030 \beta_{10} + \cdots + 447694 \beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1495325 \beta_{11} - 357201 \beta_{7} - 564876 \beta_{6} - 8624070 \beta_{5} - 1207710 \beta_{4} + \cdots + 4033692 ) / 18 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 3652562 \beta_{15} + 199554 \beta_{14} + 8367294 \beta_{13} + 135780 \beta_{12} - 6394119 \beta_{10} + \cdots + 1750632 \beta_1 ) / 18 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/495\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(397\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
494.1
1.34882 − 0.135696i
−1.34882 − 0.135696i
0.804180 − 1.09134i
−0.804180 − 1.09134i
−2.71170 + 0.899435i
2.71170 + 0.899435i
−2.48610 − 1.40769i
2.48610 − 1.40769i
2.48610 + 1.40769i
−2.48610 + 1.40769i
2.71170 − 0.899435i
−2.71170 − 0.899435i
−0.804180 + 1.09134i
0.804180 + 1.09134i
−1.34882 + 0.135696i
1.34882 + 0.135696i
− 5.42339i 0 −21.4132 − 11.1803i 0 36.8735 72.7449i 0 −60.6354
494.2 − 5.42339i 0 −21.4132 11.1803i 0 −36.8735 72.7449i 0 60.6354
494.3 − 4.97220i 0 −16.7228 − 11.1803i 0 −28.5581 43.3713i 0 −55.5909
494.4 − 4.97220i 0 −16.7228 11.1803i 0 28.5581 43.3713i 0 55.5909
494.5 − 2.69764i 0 0.722764 − 11.1803i 0 −23.5889 − 23.5308i 0 −30.1605
494.6 − 2.69764i 0 0.722764 11.1803i 0 23.5889 − 23.5308i 0 30.1605
494.7 − 1.60836i 0 5.41318 − 11.1803i 0 3.51378 − 21.5732i 0 −17.9820
494.8 − 1.60836i 0 5.41318 11.1803i 0 −3.51378 − 21.5732i 0 17.9820
494.9 1.60836i 0 5.41318 − 11.1803i 0 −3.51378 21.5732i 0 17.9820
494.10 1.60836i 0 5.41318 11.1803i 0 3.51378 21.5732i 0 −17.9820
494.11 2.69764i 0 0.722764 − 11.1803i 0 23.5889 23.5308i 0 30.1605
494.12 2.69764i 0 0.722764 11.1803i 0 −23.5889 23.5308i 0 −30.1605
494.13 4.97220i 0 −16.7228 − 11.1803i 0 28.5581 − 43.3713i 0 55.5909
494.14 4.97220i 0 −16.7228 11.1803i 0 −28.5581 − 43.3713i 0 −55.5909
494.15 5.42339i 0 −21.4132 − 11.1803i 0 −36.8735 − 72.7449i 0 60.6354
494.16 5.42339i 0 −21.4132 11.1803i 0 36.8735 − 72.7449i 0 −60.6354
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 494.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
55.d odd 2 1 CM by \(\Q(\sqrt{-55}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
11.b odd 2 1 inner
15.d odd 2 1 inner
33.d even 2 1 inner
165.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.4.d.a ✓ 16
3.b odd 2 1 inner 495.4.d.a ✓ 16
5.b even 2 1 inner 495.4.d.a ✓ 16
11.b odd 2 1 inner 495.4.d.a ✓ 16
15.d odd 2 1 inner 495.4.d.a ✓ 16
33.d even 2 1 inner 495.4.d.a ✓ 16
55.d odd 2 1 CM 495.4.d.a ✓ 16
165.d even 2 1 inner 495.4.d.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.4.d.a ✓ 16 1.a even 1 1 trivial
495.4.d.a ✓ 16 3.b odd 2 1 inner
495.4.d.a ✓ 16 5.b even 2 1 inner
495.4.d.a ✓ 16 11.b odd 2 1 inner
495.4.d.a ✓ 16 15.d odd 2 1 inner
495.4.d.a ✓ 16 33.d even 2 1 inner
495.4.d.a ✓ 16 55.d odd 2 1 CM
495.4.d.a ✓ 16 165.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 64T_{2}^{6} + 1280T_{2}^{4} + 8192T_{2}^{2} + 13689 \) acting on \(S_{4}^{\mathrm{new}}(495, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 64 T^{6} + \cdots + 13689)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{2} + 125)^{8} \) Copy content Toggle raw display
$7$ \( (T^{8} - 2744 T^{6} + \cdots + 7618147524)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1331)^{8} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 92719913457924)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 12\!\cdots\!24)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( (T^{4} - 119164 T^{2} + 2121707844)^{4} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 57\!\cdots\!24)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( (T^{4} + 821516 T^{2} + 15824484)^{4} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( (T^{4} + 1431644 T^{2} + 497084221764)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 88\!\cdots\!04)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 33\!\cdots\!24)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 1457456)^{8} \) Copy content Toggle raw display
$97$ \( T^{16} \) Copy content Toggle raw display
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