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Properties

Label 1210.3.d.c
Level $1210$
Weight $3$
Character orbit 1210.d
Analytic conductor $32.970$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1210,3,Mod(241,1210)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1210, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1210.241"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1210 = 2 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1210.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.9701119876\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 92 x^{14} - 504 x^{13} + 3078 x^{12} - 12280 x^{11} + 49836 x^{10} - 147672 x^{9} + \cdots + 339856 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
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.

\(f(q)\) \(=\) \( q - \beta_{9} q^{2} + (\beta_{4} - \beta_{3} + \beta_{2} - 1) q^{3} - 2 q^{4} + (2 \beta_{4} + 1) q^{5} + ( - \beta_{14} - \beta_{12} + \cdots + \beta_{8}) q^{6} + (\beta_{13} + \beta_{12} + \cdots + \beta_{8}) q^{7}+ \cdots + ( - 5 \beta_{15} - 2 \beta_{14} + \cdots + 17 \beta_{8}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{3} - 32 q^{4} + 40 q^{9} + 32 q^{12} + 16 q^{14} + 40 q^{15} + 64 q^{16} + 108 q^{23} + 80 q^{25} - 292 q^{27} - 268 q^{31} - 16 q^{34} - 80 q^{36} + 44 q^{37} - 280 q^{38} - 16 q^{42} - 476 q^{47}+ \cdots + 296 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 92 x^{14} - 504 x^{13} + 3078 x^{12} - 12280 x^{11} + 49836 x^{10} - 147672 x^{9} + \cdots + 339856 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 6847 \nu^{14} + 47929 \nu^{13} - 536177 \nu^{12} + 2593985 \nu^{11} - 14744955 \nu^{10} + \cdots + 504479976 ) / 187906840 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3879 \nu^{14} - 27153 \nu^{13} + 320536 \nu^{12} - 1570227 \nu^{11} + 9511204 \nu^{10} + \cdots + 120629080 ) / 93953420 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 31483 \nu^{14} - 220381 \nu^{13} + 2517773 \nu^{12} - 12241685 \nu^{11} + 71798033 \nu^{10} + \cdots + 2438433712 ) / 187906840 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 290 \nu^{14} - 2030 \nu^{13} + 23009 \nu^{12} - 111664 \nu^{11} + 649767 \nu^{10} + \cdots + 14465132 ) / 1540220 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 19183 \nu^{14} - 134281 \nu^{13} + 1495764 \nu^{12} - 7228931 \nu^{11} + 41233831 \nu^{10} + \cdots + 1018240960 ) / 93953420 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 40855 \nu^{14} + 285985 \nu^{13} - 3207332 \nu^{12} + 15526187 \nu^{11} - 90037978 \nu^{10} + \cdots - 3587134496 ) / 187906840 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 41072 \nu^{14} - 287504 \nu^{13} + 3139499 \nu^{12} - 15099442 \nu^{11} + 84477033 \nu^{10} + \cdots - 172033840 ) / 187906840 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 108250 \nu^{15} + 811875 \nu^{14} - 67445001 \nu^{13} + 426079069 \nu^{12} + \cdots + 2203083416072 ) / 1147922885560 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 286766 \nu^{15} + 2150745 \nu^{14} - 19989215 \nu^{13} + 97310265 \nu^{12} + \cdots - 677093564048 ) / 229584577112 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 9074476 \nu^{15} + 68058570 \nu^{14} - 1023489448 \nu^{13} + 5620459767 \nu^{12} + \cdots + 19099979970720 ) / 1147922885560 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 6293502 \nu^{15} + 47201265 \nu^{14} - 656746807 \nu^{13} + 3552968393 \nu^{12} + \cdots + 6234316637296 ) / 573961442780 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 512176 \nu^{15} - 3841320 \nu^{14} + 41411952 \nu^{13} - 210917668 \nu^{12} + \cdots + 12389117708 ) / 9409203980 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 537456 \nu^{15} - 4030920 \nu^{14} + 44156068 \nu^{13} - 225878822 \nu^{12} + \cdots + 22363264616 ) / 9409203980 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 68667902 \nu^{15} + 515009265 \nu^{14} - 5654937033 \nu^{13} + 28946116862 \nu^{12} + \cdots + 4694721525472 ) / 573961442780 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 3031044 \nu^{15} + 22732830 \nu^{14} - 252113280 \nu^{13} + 1293955065 \nu^{12} + \cdots + 100756609096 ) / 18818407960 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{13} - \beta_{12} + 2\beta_{9} + 2\beta_{8} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{13} - \beta_{12} + 2\beta_{9} + 2\beta_{8} + 2\beta_{5} - 4\beta_{4} + 2\beta_{3} + 2\beta_{2} + 2\beta _1 - 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 6 \beta_{14} - 19 \beta_{13} + 6 \beta_{12} - \beta_{11} + 2 \beta_{10} - 35 \beta_{9} - 21 \beta_{8} + \cdots - 26 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 12 \beta_{14} - 39 \beta_{13} + 13 \beta_{12} - 2 \beta_{11} + 4 \beta_{10} - 72 \beta_{9} + \cdots + 181 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 90 \beta_{14} + 265 \beta_{13} - 68 \beta_{12} + 27 \beta_{11} - 34 \beta_{10} + 495 \beta_{9} + \cdots + 496 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 300 \beta_{14} + 893 \beta_{13} - 237 \beta_{12} + 86 \beta_{11} - 112 \beta_{10} + 1666 \beta_{9} + \cdots - 1973 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 10 \beta_{15} - 1080 \beta_{14} - 3273 \beta_{13} + 922 \beta_{12} - 491 \beta_{11} + 492 \beta_{10} + \cdots - 8672 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 40 \beta_{15} - 5748 \beta_{14} - 17351 \beta_{13} + 4825 \beta_{12} - 2370 \beta_{11} + 2500 \beta_{10} + \cdots + 19921 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 380 \beta_{15} + 10598 \beta_{14} + 33593 \beta_{13} - 11168 \beta_{12} + 7019 \beta_{11} + \cdots + 143812 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2200 \beta_{15} + 98260 \beta_{14} + 304545 \beta_{13} - 93753 \beta_{12} + 53482 \beta_{11} + \cdots - 154557 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 8150 \beta_{15} - 64048 \beta_{14} - 215537 \beta_{13} + 94654 \beta_{12} - 71627 \beta_{11} + \cdots - 2269332 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 73760 \beta_{15} - 1566788 \beta_{14} - 4949803 \beta_{13} + 1684253 \beta_{12} - 1059094 \beta_{11} + \cdots + 111729 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 102880 \beta_{15} - 548690 \beta_{14} - 1623695 \beta_{13} + 148132 \beta_{12} + 126627 \beta_{11} + \cdots + 34031232 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1915120 \beta_{15} + 23642652 \beta_{14} + 75200765 \beta_{13} - 28005525 \beta_{12} + \cdots + 31334739 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 175750 \beta_{15} + 31453120 \beta_{14} + 100471855 \beta_{13} - 31368086 \beta_{12} + \cdots - 482904216 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1210\mathbb{Z}\right)^\times\).

\(n\) \(727\) \(1091\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
241.1
0.500000 + 0.322806i
0.500000 + 3.59455i
0.500000 + 3.85038i
0.500000 + 1.09141i
0.500000 + 3.96822i
0.500000 − 2.55400i
0.500000 − 2.43617i
0.500000 − 2.18033i
0.500000 − 0.322806i
0.500000 − 3.59455i
0.500000 − 3.85038i
0.500000 − 1.09141i
0.500000 − 3.96822i
0.500000 + 2.55400i
0.500000 + 2.43617i
0.500000 + 2.18033i
− 1.41421i −5.52516 −2.00000 −2.23607 7.81376i 11.7293i 2.82843i 21.5274 3.16228i
241.2 − 1.41421i −5.33478 −2.00000 2.23607 7.54452i − 8.94509i 2.82843i 19.4599 − 3.16228i
241.3 − 1.41421i −3.25446 −2.00000 −2.23607 4.60250i 6.57525i 2.82843i 1.59149 3.16228i
241.4 − 1.41421i −1.40090 −2.00000 −2.23607 1.98117i − 12.9437i 2.82843i −7.03748 3.16228i
241.5 − 1.41421i 0.468308 −2.00000 2.23607 − 0.662288i 13.4610i 2.82843i −8.78069 − 3.16228i
241.6 − 1.41421i 1.43027 −2.00000 2.23607 − 2.02270i − 7.79353i 2.82843i −6.95434 − 3.16228i
241.7 − 1.41421i 1.70838 −2.00000 −2.23607 − 2.41602i − 2.53238i 2.82843i −6.08143 3.16228i
241.8 − 1.41421i 3.90834 −2.00000 2.23607 − 5.52723i 6.10605i 2.82843i 6.27513 − 3.16228i
241.9 1.41421i −5.52516 −2.00000 −2.23607 − 7.81376i − 11.7293i − 2.82843i 21.5274 − 3.16228i
241.10 1.41421i −5.33478 −2.00000 2.23607 − 7.54452i 8.94509i − 2.82843i 19.4599 3.16228i
241.11 1.41421i −3.25446 −2.00000 −2.23607 − 4.60250i − 6.57525i − 2.82843i 1.59149 − 3.16228i
241.12 1.41421i −1.40090 −2.00000 −2.23607 − 1.98117i 12.9437i − 2.82843i −7.03748 − 3.16228i
241.13 1.41421i 0.468308 −2.00000 2.23607 0.662288i − 13.4610i − 2.82843i −8.78069 3.16228i
241.14 1.41421i 1.43027 −2.00000 2.23607 2.02270i 7.79353i − 2.82843i −6.95434 3.16228i
241.15 1.41421i 1.70838 −2.00000 −2.23607 2.41602i 2.53238i − 2.82843i −6.08143 − 3.16228i
241.16 1.41421i 3.90834 −2.00000 2.23607 5.52723i − 6.10605i − 2.82843i 6.27513 3.16228i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 241.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1210.3.d.c 16
11.b odd 2 1 inner 1210.3.d.c 16
11.c even 5 1 110.3.h.b ✓ 16
11.d odd 10 1 110.3.h.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.3.h.b ✓ 16 11.c even 5 1
110.3.h.b ✓ 16 11.d odd 10 1
1210.3.d.c 16 1.a even 1 1 trivial
1210.3.d.c 16 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 8T_{3}^{7} - 14T_{3}^{6} - 186T_{3}^{5} + 4T_{3}^{4} + 1032T_{3}^{3} - 271T_{3}^{2} - 1374T_{3} + 601 \) acting on \(S_{3}^{\mathrm{new}}(1210, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{8} \) Copy content Toggle raw display
$3$ \( (T^{8} + 8 T^{7} + \cdots + 601)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 209825683300321 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 16\!\cdots\!61 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 27\!\cdots\!81 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 13\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( (T^{8} - 54 T^{7} + \cdots - 1621279)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 87\!\cdots\!41 \) Copy content Toggle raw display
$31$ \( (T^{8} + 134 T^{7} + \cdots + 1276321001)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 2166427529321)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 36\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 40\!\cdots\!61 \) Copy content Toggle raw display
$47$ \( (T^{8} + 238 T^{7} + \cdots - 610336322119)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 5919664588321)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots - 7338146831999)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 26\!\cdots\!61 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 42460000821961)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 23172255911041)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 95\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 42\!\cdots\!01 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 39\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 5123163662521)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 148 T^{7} + \cdots + 187883820521)^{2} \) Copy content Toggle raw display
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