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Properties

Label 550.3.c.b
Level $550$
Weight $3$
Character orbit 550.c
Analytic conductor $14.986$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [550,3,Mod(549,550)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("550.549"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(550, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 550.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,32,0,0,0,0,-80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.9864145398\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.393016351129600000000.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 42 x^{14} - 148 x^{13} + 402 x^{12} - 928 x^{11} + 1834 x^{10} - 2940 x^{9} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{26} \)
Twist minimal: no (minimal twist has level 110)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_{8} + \beta_{7} - \beta_{5}) q^{3} + 2 q^{4} + (\beta_{11} - \beta_{10} + \beta_{9}) q^{6} + (\beta_{15} + \beta_{13} + \cdots - 2 \beta_{2}) q^{7} - 2 \beta_{2} q^{8} + ( - 2 \beta_1 - 5) q^{9}+ \cdots + (5 \beta_{14} - 5 \beta_{11} + \cdots - 88) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4} - 80 q^{9} + 64 q^{14} + 64 q^{16} + 160 q^{26} - 128 q^{31} - 224 q^{34} - 160 q^{36} + 464 q^{49} + 128 q^{56} - 160 q^{59} + 128 q^{64} - 352 q^{66} - 224 q^{69} - 512 q^{71} + 176 q^{81}+ \cdots - 1408 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 42 x^{14} - 148 x^{13} + 402 x^{12} - 928 x^{11} + 1834 x^{10} - 2940 x^{9} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 197934540060 \nu^{15} + 414416439506 \nu^{14} + 2024790634 \nu^{13} + \cdots - 20750347130764 ) / 3490329086195 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 125322462559 \nu^{15} - 1038502435871 \nu^{14} + 5512798058778 \nu^{13} + \cdots - 1248817587602 ) / 608756139145 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 179814 \nu^{15} + 1738538 \nu^{14} - 9750080 \nu^{13} + 37754112 \nu^{12} - 109364886 \nu^{11} + \cdots + 2393844 ) / 658735 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 65084520363244 \nu^{15} + 578670999318598 \nu^{14} + \cdots + 889823157098394 ) / 159059282642315 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 28158435948488 \nu^{15} + 221138963491046 \nu^{14} + \cdots + 89703590216798 ) / 38393619948145 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13574865856 \nu^{15} - 116760997512 \nu^{14} + 628792835280 \nu^{13} - 2301945072568 \nu^{12} + \cdots - 73735392626 ) / 16084465835 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 180305768237 \nu^{15} + 1409916038975 \nu^{14} - 7338791573656 \nu^{13} + \cdots + 573864899468 ) / 176929124185 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 698137876 \nu^{15} - 5312396568 \nu^{14} + 27199514744 \nu^{13} - 92329637004 \nu^{12} + \cdots - 1656473268 ) / 502632977 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 6124283962 \nu^{15} - 46899301818 \nu^{14} + 241162973044 \nu^{13} - 823623947572 \nu^{12} + \cdots - 20741614676 ) / 4051536785 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 25581423136190 \nu^{15} + 196604823448072 \nu^{14} + \cdots + 37043379340952 ) / 14459934785665 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7315091541316 \nu^{15} + 56229690839102 \nu^{14} - 289553633160840 \nu^{13} + \cdots + 12331918069046 ) / 3265146564505 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3701423705735 \nu^{15} + 27771829810177 \nu^{14} - 141431427745807 \nu^{13} + \cdots + 5725695602827 ) / 1219512572285 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 651013779363504 \nu^{15} + \cdots - 15\!\cdots\!08 ) / 159059282642315 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 446435837466419 \nu^{15} + \cdots - 894625320206957 ) / 101219543499655 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 66283780073896 \nu^{15} + 506125578245276 \nu^{14} + \cdots + 160132113792840 ) / 14459934785665 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{15} - 2 \beta_{14} + \beta_{13} + 2 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + \cdots + 8 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} + \beta_{14} + \beta_{13} + 3 \beta_{12} - 3 \beta_{11} + 6 \beta_{9} - 4 \beta_{8} + \cdots - 10 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11 \beta_{15} + 12 \beta_{14} + \beta_{13} + 8 \beta_{12} + 10 \beta_{11} - 14 \beta_{10} + 4 \beta_{9} + \cdots - 52 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4 \beta_{15} + 3 \beta_{14} - 4 \beta_{13} - 3 \beta_{12} + 23 \beta_{11} - 16 \beta_{10} + \cdots + 3 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 29 \beta_{15} - 54 \beta_{14} - 61 \beta_{13} - 80 \beta_{12} + 57 \beta_{11} - 60 \beta_{10} + \cdots + 324 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 181 \beta_{15} - 283 \beta_{14} - 87 \beta_{13} - 121 \beta_{12} - 349 \beta_{11} + 246 \beta_{10} + \cdots + 974 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 79 \beta_{15} - 208 \beta_{14} + 1031 \beta_{13} + 1424 \beta_{12} - 2984 \beta_{11} + 2578 \beta_{10} + \cdots - 3380 ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1060 \beta_{15} + 1464 \beta_{14} + 1060 \beta_{13} + 1464 \beta_{12} - 1491 \beta_{6} - 1164 \beta_{4} + \cdots - 7514 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 10903 \beta_{15} + 15284 \beta_{14} - 535 \beta_{13} - 1136 \beta_{12} + 32726 \beta_{11} + \cdots - 36052 ) / 16 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 9252 \beta_{15} - 13056 \beta_{14} - 22614 \beta_{13} - 32400 \beta_{12} + 41538 \beta_{11} + \cdots + 110680 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 80658 \beta_{15} - 113698 \beta_{14} - 54950 \beta_{13} - 78500 \beta_{12} - 74189 \beta_{11} + \cdots + 467460 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 53090 \beta_{15} - 74459 \beta_{14} + 53090 \beta_{13} + 74459 \beta_{12} - 320565 \beta_{11} + \cdots - 74459 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1213547 \beta_{15} + 1722418 \beta_{14} + 1762257 \beta_{13} + 2489302 \beta_{12} - 1619358 \beta_{11} + \cdots - 10261984 ) / 16 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 2710379 \beta_{15} + 3839531 \beta_{14} + 1070823 \beta_{13} + 1513937 \beta_{12} + 4943999 \beta_{11} + \cdots - 13061198 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 399699 \beta_{15} + 588326 \beta_{14} - 13893519 \beta_{13} - 19640170 \beta_{12} + 42659912 \beta_{11} + \cdots + 46567008 ) / 16 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/550\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(-1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
549.1
0.687711 + 1.66028i
1.26417 − 3.05197i
0.172847 + 0.417290i
−0.831834 − 2.00822i
−0.831834 + 2.00822i
0.172847 − 0.417290i
1.26417 + 3.05197i
0.687711 − 1.66028i
1.83183 − 0.758770i
0.827153 + 0.342618i
−0.264169 + 0.109422i
0.312289 − 0.129354i
0.312289 + 0.129354i
−0.264169 − 0.109422i
0.827153 − 0.342618i
1.83183 + 0.758770i
−1.41421 − 5.11433i 2.00000 0 7.23275i 6.78202 −2.82843 −17.1564 0
549.2 −1.41421 − 4.76766i 2.00000 0 6.74249i −9.73055 −2.82843 −13.7306 0
549.3 −1.41421 − 2.29552i 2.00000 0 3.24636i 4.07370 −2.82843 3.73057 0
549.4 −1.41421 − 1.35781i 2.00000 0 1.92023i −12.4389 −2.82843 7.15636 0
549.5 −1.41421 1.35781i 2.00000 0 − 1.92023i −12.4389 −2.82843 7.15636 0
549.6 −1.41421 2.29552i 2.00000 0 − 3.24636i 4.07370 −2.82843 3.73057 0
549.7 −1.41421 4.76766i 2.00000 0 − 6.74249i −9.73055 −2.82843 −13.7306 0
549.8 −1.41421 5.11433i 2.00000 0 − 7.23275i 6.78202 −2.82843 −17.1564 0
549.9 1.41421 − 5.11433i 2.00000 0 − 7.23275i −6.78202 2.82843 −17.1564 0
549.10 1.41421 − 4.76766i 2.00000 0 − 6.74249i 9.73055 2.82843 −13.7306 0
549.11 1.41421 − 2.29552i 2.00000 0 − 3.24636i −4.07370 2.82843 3.73057 0
549.12 1.41421 − 1.35781i 2.00000 0 − 1.92023i 12.4389 2.82843 7.15636 0
549.13 1.41421 1.35781i 2.00000 0 1.92023i 12.4389 2.82843 7.15636 0
549.14 1.41421 2.29552i 2.00000 0 3.24636i −4.07370 2.82843 3.73057 0
549.15 1.41421 4.76766i 2.00000 0 6.74249i 9.73055 2.82843 −13.7306 0
549.16 1.41421 5.11433i 2.00000 0 7.23275i −6.78202 2.82843 −17.1564 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 549.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.b odd 2 1 inner
55.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 550.3.c.b 16
5.b even 2 1 inner 550.3.c.b 16
5.c odd 4 1 110.3.d.a ✓ 8
5.c odd 4 1 550.3.d.f 8
11.b odd 2 1 inner 550.3.c.b 16
15.e even 4 1 990.3.b.b 8
20.e even 4 1 880.3.j.c 8
55.d odd 2 1 inner 550.3.c.b 16
55.e even 4 1 110.3.d.a ✓ 8
55.e even 4 1 550.3.d.f 8
165.l odd 4 1 990.3.b.b 8
220.i odd 4 1 880.3.j.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.3.d.a ✓ 8 5.c odd 4 1
110.3.d.a ✓ 8 55.e even 4 1
550.3.c.b 16 1.a even 1 1 trivial
550.3.c.b 16 5.b even 2 1 inner
550.3.c.b 16 11.b odd 2 1 inner
550.3.c.b 16 55.d odd 2 1 inner
550.3.d.f 8 5.c odd 4 1
550.3.d.f 8 55.e even 4 1
880.3.j.c 8 20.e even 4 1
880.3.j.c 8 220.i odd 4 1
990.3.b.b 8 15.e even 4 1
990.3.b.b 8 165.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 56T_{3}^{6} + 952T_{3}^{4} + 4704T_{3}^{2} + 5776 \) acting on \(S_{3}^{\mathrm{new}}(550, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{8} \) Copy content Toggle raw display
$3$ \( (T^{8} + 56 T^{6} + \cdots + 5776)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 312 T^{6} + \cdots + 11182336)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 132 T^{6} + \cdots + 214358881)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 896 T^{6} + \cdots + 23001616)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 1056 T^{6} + \cdots + 50069776)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 1664 T^{6} + \cdots + 52070656)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 1462077578896)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 2112 T^{6} + \cdots + 2498400256)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 32 T^{3} + \cdots + 378176)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 976 T^{6} + \cdots + 1014804736)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 32472856662016)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 6808 T^{6} + \cdots + 506465648896)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 13016 T^{6} + \cdots + 12491191696)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 11024 T^{6} + \cdots + 35398164736)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 40 T^{3} + \cdots - 456896)^{4} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 288073767325696)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 16216 T^{6} + \cdots + 326950665616)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 128 T^{3} + \cdots + 7774976)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 50\!\cdots\!56)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 25984 T^{6} + \cdots + 516925696)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 45563364004096)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 184 T^{3} + \cdots - 101033216)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 19\!\cdots\!76)^{2} \) Copy content Toggle raw display
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