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Properties

Label 810.4.c.e
Level $810$
Weight $4$
Character orbit 810.c
Analytic conductor $47.792$
Analytic rank $0$
Dimension $18$
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] = [810,4,Mod(649,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("810.649");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 810.c (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: \(47.7915471046\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 452 x^{16} + 85462 x^{14} + 8762268 x^{12} + 527229657 x^{10} + 18762394824 x^{8} + \cdots + 3061100160000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{12}\cdot 5 \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 4 q^{4} - \beta_{7} q^{5} + ( - \beta_{3} - \beta_1) q^{7} + 4 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 4 q^{4} - \beta_{7} q^{5} + ( - \beta_{3} - \beta_1) q^{7} + 4 \beta_1 q^{8} + \beta_{5} q^{10} + ( - \beta_{15} + \beta_{14} + \beta_{7} + \cdots + 3) q^{11}+ \cdots + ( - \beta_{17} + 9 \beta_{16} + \cdots + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 72 q^{4} + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 72 q^{4} + 8 q^{5} + 52 q^{11} - 68 q^{14} + 288 q^{16} - 32 q^{20} + 144 q^{25} + 312 q^{26} - 750 q^{29} - 72 q^{31} + 286 q^{35} + 706 q^{41} - 208 q^{44} + 252 q^{46} - 1080 q^{49} + 488 q^{50} - 90 q^{55} + 272 q^{56} - 2000 q^{59} - 18 q^{61} - 1152 q^{64} + 126 q^{65} + 36 q^{70} + 2796 q^{71} - 1784 q^{74} - 684 q^{79} + 128 q^{80} - 828 q^{85} + 1976 q^{86} - 4642 q^{89} + 504 q^{91} + 612 q^{94} - 2772 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 452 x^{16} + 85462 x^{14} + 8762268 x^{12} + 527229657 x^{10} + 18762394824 x^{8} + \cdots + 3061100160000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2915 \nu^{17} - 1199446 \nu^{15} - 197957846 \nu^{13} - 16388690856 \nu^{11} + \cdots + 31\!\cdots\!00 \nu ) / 76\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1921774 \nu^{17} + 5075049 \nu^{16} + 780124976 \nu^{15} + 4219834794 \nu^{14} + \cdots + 96\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 60508393 \nu^{17} + 20391917330 \nu^{15} + 2575078369090 \nu^{13} + \cdots - 57\!\cdots\!00 \nu ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19217740 \nu^{17} + 154405427 \nu^{16} + 7801249760 \nu^{15} + 73408440862 \nu^{14} + \cdots - 24\!\cdots\!00 ) / 74\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 361271 \nu^{17} + 25619451 \nu^{16} + 12708401 \nu^{15} + 10564973346 \nu^{14} + \cdots - 24\!\cdots\!00 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10207439 \nu^{16} - 3214844434 \nu^{14} - 357990171614 \nu^{12} - 14130437943168 \nu^{10} + \cdots - 86\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3843548 \nu^{17} + 273249927 \nu^{16} + 1560249952 \nu^{15} + 108571002462 \nu^{14} + \cdots + 72\!\cdots\!00 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 491699021 \nu^{17} - 173357065450 \nu^{15} - 22901386845770 \nu^{13} + \cdots + 21\!\cdots\!00 \nu ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 8767 \nu^{17} + 3108206 \nu^{15} + 414578446 \nu^{13} + 23948661288 \nu^{11} + \cdots - 47\!\cdots\!00 \nu ) / 770868161280000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 282736105 \nu^{17} + 100000272002 \nu^{15} + 13607425409602 \nu^{13} + \cdots - 49\!\cdots\!00 \nu ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 190676 \nu^{17} + 95641107 \nu^{16} - 332383432 \nu^{15} + 38618157846 \nu^{14} + \cdots + 10\!\cdots\!00 ) / 89\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 26443160 \nu^{17} + 1768593573 \nu^{16} - 7547081740 \nu^{15} + 713095712838 \nu^{14} + \cdots + 24\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 19217740 \nu^{17} - 3736995633 \nu^{16} + 7801249760 \nu^{15} - 1550281423698 \nu^{14} + \cdots - 17\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 861648809 \nu^{17} + 2619105198 \nu^{16} + 349618374658 \nu^{15} + 1014702205548 \nu^{14} + \cdots - 25\!\cdots\!00 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 312839923 \nu^{17} - 2981774366 \nu^{16} + 126941124566 \nu^{15} - 1158381140956 \nu^{14} + \cdots + 20\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 861648809 \nu^{17} + 8945323098 \nu^{16} + 349618374658 \nu^{15} + 3475143422868 \nu^{14} + \cdots - 61\!\cdots\!00 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 1377181133 \nu^{17} - 227647730 \nu^{16} - 563130092650 \nu^{15} - 80170718980 \nu^{14} + \cdots - 30\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} - \beta_{8} - \beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{16} + 3 \beta_{15} - 4 \beta_{14} + 2 \beta_{13} + 2 \beta_{12} + 3 \beta_{11} + 5 \beta_{7} + \cdots - 451 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{17} - 6 \beta_{16} - 6 \beta_{15} - 15 \beta_{12} + 6 \beta_{11} + 24 \beta_{10} + 61 \beta_{9} + \cdots + 30 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 119 \beta_{16} - 228 \beta_{15} + 109 \beta_{14} - 233 \beta_{13} - 221 \beta_{12} - 345 \beta_{11} + \cdots + 33193 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 543 \beta_{17} + 1320 \beta_{16} + 1320 \beta_{15} + 2301 \beta_{12} - 222 \beta_{11} - 2904 \beta_{10} + \cdots - 4602 ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 24083 \beta_{16} + 20136 \beta_{15} + 3947 \beta_{14} + 24749 \beta_{13} + 23051 \beta_{12} + \cdots - 2719375 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 70797 \beta_{17} - 183858 \beta_{16} - 183858 \beta_{15} - 266757 \beta_{12} - 16194 \beta_{11} + \cdots + 533514 ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3042395 \beta_{16} - 1906410 \beta_{15} - 1135985 \beta_{14} - 2592443 \beta_{13} - 2336519 \beta_{12} + \cdots + 236833021 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 8300739 \beta_{17} + 22103916 \beta_{16} + 22103916 \beta_{15} + 28232493 \beta_{12} + 4198458 \beta_{11} + \cdots - 56464986 ) / 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 335890055 \beta_{16} + 185649960 \beta_{15} + 150240095 \beta_{14} + 268680041 \beta_{13} + \cdots - 21456581599 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 922987389 \beta_{17} - 2475755358 \beta_{16} - 2475755358 \beta_{15} - 2879261889 \beta_{12} + \cdots + 5758523778 ) / 9 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 35002094327 \beta_{16} - 18271151094 \beta_{15} - 16730943233 \beta_{14} - 27565177139 \beta_{13} + \cdots + 1995631558477 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 99423606171 \beta_{17} + 266383770432 \beta_{16} + 266383770432 \beta_{15} + 288875879385 \beta_{12} + \cdots - 577751758770 ) / 9 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 3543535238363 \beta_{16} + 1804971791316 \beta_{15} + 1738563447047 \beta_{14} + \cdots - 188907870881647 ) / 9 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 10492746312837 \beta_{17} - 27961751289066 \beta_{16} - 27961751289066 \beta_{15} + \cdots + 57540145801962 ) / 9 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 353178010651763 \beta_{16} - 178555013963706 \beta_{15} - 174622996688057 \beta_{14} + \cdots + 18\!\cdots\!33 ) / 9 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 10\!\cdots\!47 \beta_{17} + \cdots - 57\!\cdots\!34 ) / 9 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).

\(n\) \(487\) \(731\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
8.53195i
0.514526i
− 4.56169i
− 7.33141i
9.85525i
6.36774i
− 9.96721i
− 7.83944i
2.43029i
− 8.53195i
− 0.514526i
4.56169i
7.33141i
− 9.85525i
− 6.36774i
9.96721i
7.83944i
− 2.43029i
− 2.00000i 0 −4.00000 −11.0633 + 1.61374i 0 − 10.9736i 8.00000i 0 3.22747 + 22.1265i
649.2 − 2.00000i 0 −4.00000 −10.7621 − 3.02956i 0 17.1190i 8.00000i 0 −6.05911 + 21.5241i
649.3 − 2.00000i 0 −4.00000 −7.31132 − 8.45840i 0 − 4.99739i 8.00000i 0 −16.9168 + 14.6226i
649.4 − 2.00000i 0 −4.00000 −3.04411 + 10.7579i 0 − 14.4641i 8.00000i 0 21.5159 + 6.08823i
649.5 − 2.00000i 0 −4.00000 4.11732 + 10.3946i 0 5.81829i 8.00000i 0 20.7892 − 8.23464i
649.6 − 2.00000i 0 −4.00000 5.06568 − 9.96689i 0 − 22.2870i 8.00000i 0 −19.9338 − 10.1314i
649.7 − 2.00000i 0 −4.00000 6.19931 − 9.30422i 0 17.7903i 8.00000i 0 −18.6084 − 12.3986i
649.8 − 2.00000i 0 −4.00000 10.0618 + 4.87452i 0 − 35.0584i 8.00000i 0 9.74903 − 20.1235i
649.9 − 2.00000i 0 −4.00000 10.7367 + 3.11828i 0 30.0529i 8.00000i 0 6.23655 − 21.4734i
649.10 2.00000i 0 −4.00000 −11.0633 − 1.61374i 0 10.9736i − 8.00000i 0 3.22747 − 22.1265i
649.11 2.00000i 0 −4.00000 −10.7621 + 3.02956i 0 − 17.1190i − 8.00000i 0 −6.05911 − 21.5241i
649.12 2.00000i 0 −4.00000 −7.31132 + 8.45840i 0 4.99739i − 8.00000i 0 −16.9168 − 14.6226i
649.13 2.00000i 0 −4.00000 −3.04411 − 10.7579i 0 14.4641i − 8.00000i 0 21.5159 − 6.08823i
649.14 2.00000i 0 −4.00000 4.11732 − 10.3946i 0 − 5.81829i − 8.00000i 0 20.7892 + 8.23464i
649.15 2.00000i 0 −4.00000 5.06568 + 9.96689i 0 22.2870i − 8.00000i 0 −19.9338 + 10.1314i
649.16 2.00000i 0 −4.00000 6.19931 + 9.30422i 0 − 17.7903i − 8.00000i 0 −18.6084 + 12.3986i
649.17 2.00000i 0 −4.00000 10.0618 − 4.87452i 0 35.0584i − 8.00000i 0 9.74903 + 20.1235i
649.18 2.00000i 0 −4.00000 10.7367 − 3.11828i 0 − 30.0529i − 8.00000i 0 6.23655 + 21.4734i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.4.c.e 18
3.b odd 2 1 810.4.c.d 18
5.b even 2 1 inner 810.4.c.e 18
9.c even 3 2 90.4.i.a ✓ 36
9.d odd 6 2 270.4.i.a 36
15.d odd 2 1 810.4.c.d 18
45.h odd 6 2 270.4.i.a 36
45.j even 6 2 90.4.i.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.4.i.a ✓ 36 9.c even 3 2
90.4.i.a ✓ 36 45.j even 6 2
270.4.i.a 36 9.d odd 6 2
270.4.i.a 36 45.h odd 6 2
810.4.c.d 18 3.b odd 2 1
810.4.c.d 18 15.d odd 2 1
810.4.c.e 18 1.a even 1 1 trivial
810.4.c.e 18 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(810, [\chi])\):

\( T_{7}^{18} + 3627 T_{7}^{16} + 5167953 T_{7}^{14} + 3767579175 T_{7}^{12} + 1541154044655 T_{7}^{10} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
\( T_{11}^{9} - 26 T_{11}^{8} - 6628 T_{11}^{7} + 231596 T_{11}^{6} + 9949203 T_{11}^{5} + \cdots - 617832495080 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{9} \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 74\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$11$ \( (T^{9} - 26 T^{8} + \cdots - 617832495080)^{2} \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{9} + \cdots + 47\!\cdots\!80)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{9} + \cdots + 37\!\cdots\!52)^{2} \) Copy content Toggle raw display
$31$ \( (T^{9} + \cdots + 54\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{9} + \cdots - 21\!\cdots\!89)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 27\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{9} + \cdots + 85\!\cdots\!08)^{2} \) Copy content Toggle raw display
$61$ \( (T^{9} + \cdots - 48\!\cdots\!50)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 83\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{9} + \cdots + 55\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 42\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{9} + \cdots - 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{9} + \cdots + 22\!\cdots\!40)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
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