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Properties

Label 1210.3.d.f
Level $1210$
Weight $3$
Character orbit 1210.d
Analytic conductor $32.970$
Analytic rank $0$
Dimension $16$
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] = [1210,3,Mod(241,1210)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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

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: \(32.9701119876\)
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: \( x^{16} - 8 x^{15} - 28 x^{14} + 336 x^{13} + 362 x^{12} - 6904 x^{11} - 3132 x^{10} + 87908 x^{9} + \cdots + 24267881 \) 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

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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + (\beta_{9} + 1) q^{3} - 2 q^{4} + (2 \beta_1 - 1) q^{5} + (\beta_{11} + \beta_{7} - \beta_{6}) q^{6} + (\beta_{12} + \beta_{11} - \beta_{8}) q^{7} - 2 \beta_{7} q^{8} + ( - \beta_{15} + 3 \beta_{9} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} + (\beta_{9} + 1) q^{3} - 2 q^{4} + (2 \beta_1 - 1) q^{5} + (\beta_{11} + \beta_{7} - \beta_{6}) q^{6} + (\beta_{12} + \beta_{11} - \beta_{8}) q^{7} - 2 \beta_{7} q^{8} + ( - \beta_{15} + 3 \beta_{9} + \cdots + 1) q^{9}+ \cdots + (8 \beta_{13} + 3 \beta_{12} + \cdots - 5 \beta_{6}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 24 q^{3} - 32 q^{4} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 24 q^{3} - 32 q^{4} + 40 q^{9} - 48 q^{12} + 16 q^{14} + 64 q^{16} - 12 q^{23} + 80 q^{25} + 228 q^{27} + 212 q^{31} - 176 q^{34} - 80 q^{36} + 164 q^{37} + 120 q^{38} - 16 q^{42} + 164 q^{47} + 96 q^{48} + 396 q^{49} + 244 q^{53} - 32 q^{56} - 288 q^{58} - 180 q^{59} - 128 q^{64} + 148 q^{67} + 784 q^{69} + 228 q^{71} + 120 q^{75} - 392 q^{78} + 56 q^{81} + 176 q^{82} - 496 q^{86} - 16 q^{89} - 160 q^{91} + 24 q^{92} - 324 q^{93} + 1056 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} - 28 x^{14} + 336 x^{13} + 362 x^{12} - 6904 x^{11} - 3132 x^{10} + 87908 x^{9} + \cdots + 24267881 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 43168 \nu^{14} - 302176 \nu^{13} - 3235452 \nu^{12} + 23341000 \nu^{11} + 72222984 \nu^{10} + \cdots - 677992820483 ) / 35174628995 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 26885531 \nu^{14} + 188198717 \nu^{13} + 776237449 \nu^{12} - 7104008015 \nu^{11} + \cdots + 100263437182601 ) / 703492579900 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 63908283 \nu^{14} + 447357981 \nu^{13} + 1737732052 \nu^{12} - 16242046065 \nu^{11} + \cdots + 170196388533138 ) / 703492579900 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 718104862 \nu^{15} - 486520465809 \nu^{14} + 4266488946046 \nu^{13} + 7273407353184 \nu^{12} + \cdots + 98\!\cdots\!58 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15016612927 \nu^{15} + 504370784967 \nu^{14} - 2707835078035 \nu^{13} + \cdots - 44\!\cdots\!47 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 22646585502 \nu^{15} + 293337983118 \nu^{14} - 196171108889 \nu^{13} + \cdots - 16\!\cdots\!17 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 192564 \nu^{15} - 1444230 \nu^{14} - 5682124 \nu^{13} + 58837961 \nu^{12} + \cdots + 804134838023 ) / 75965869100 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 52412624098 \nu^{15} + 1060621445519 \nu^{14} - 3056484645650 \nu^{13} + \cdots - 48\!\cdots\!25 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 53247910785 \nu^{15} + 412243909071 \nu^{14} - 6955077986834 \nu^{13} + \cdots - 20\!\cdots\!30 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 75894496287 \nu^{15} + 365883109659 \nu^{14} - 8487747163887 \nu^{13} + \cdots - 36\!\cdots\!94 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6087788336 \nu^{15} - 45658412520 \nu^{14} - 190771610314 \nu^{13} + 1932501390261 \nu^{12} + \cdots + 21\!\cdots\!46 ) / 893400401844005 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 7505920960 \nu^{15} + 56294407200 \nu^{14} + 228445174192 \nu^{13} + \cdots - 18\!\cdots\!96 ) / 893400401844005 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 151788992574 \nu^{15} - 1138417444305 \nu^{14} - 3884208682413 \nu^{13} + \cdots + 35\!\cdots\!55 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 711228254349 \nu^{15} - 5675644708552 \nu^{14} - 15689067031628 \nu^{13} + \cdots + 24\!\cdots\!99 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1473088060717 \nu^{15} - 11859763695336 \nu^{14} - 28205307545489 \nu^{13} + \cdots + 39\!\cdots\!42 ) / 17\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{13} - 2\beta_{9} - \beta_{7} - \beta_{3} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - \beta_{9} - \beta_{7} - \beta_{6} + 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{15} + 23 \beta_{13} + 3 \beta_{12} + 3 \beta_{10} - 12 \beta_{9} - 27 \beta_{7} - 3 \beta_{6} + \cdots + 28 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} + 16 \beta_{13} + 4 \beta_{12} + 16 \beta_{10} - 11 \beta_{9} - 2 \beta_{8} - 32 \beta_{7} + \cdots + 36 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 13 \beta_{15} - 4 \beta_{14} + 285 \beta_{13} + 85 \beta_{12} + 75 \beta_{10} - 8 \beta_{9} + \cdots + 120 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 17 \beta_{15} - 6 \beta_{14} + 332 \beta_{13} + 136 \beta_{12} + \beta_{11} + 182 \beta_{10} + \cdots - 273 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 84 \beta_{15} - 155 \beta_{14} + 2724 \beta_{13} + 1407 \beta_{12} - 119 \beta_{11} + 1015 \beta_{10} + \cdots - 2857 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 91 \beta_{15} - 282 \beta_{14} + 3700 \beta_{13} + 2336 \beta_{12} - 180 \beta_{11} + 1564 \beta_{10} + \cdots - 10099 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 433 \beta_{15} - 3458 \beta_{14} + 17488 \beta_{13} + 15645 \beta_{12} - 5238 \beta_{11} + 8295 \beta_{10} + \cdots - 78856 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1651 \beta_{15} - 6572 \beta_{14} + 18358 \beta_{13} + 22446 \beta_{12} - 10243 \beta_{11} + \cdots - 162223 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 22608 \beta_{15} - 54770 \beta_{14} - 27290 \beta_{13} + 77242 \beta_{12} - 140800 \beta_{11} + \cdots - 1114930 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 48519 \beta_{15} - 96539 \beta_{14} - 222848 \beta_{13} + 5048 \beta_{12} - 292168 \beta_{11} + \cdots - 1717536 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 371201 \beta_{15} - 584271 \beta_{14} - 3127825 \beta_{13} - 1340976 \beta_{12} - 2872545 \beta_{11} + \cdots - 9397995 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 623301 \beta_{15} - 780497 \beta_{14} - 7204103 \beta_{13} - 4335779 \beta_{12} - 5783060 \beta_{11} + \cdots - 8521734 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 3206483 \beta_{15} - 1784771 \beta_{14} - 58004286 \beta_{13} - 43688122 \beta_{12} - 46657209 \beta_{11} + \cdots - 1549227 ) / 2 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
241.1
3.46412 − 0.437016i
3.56435 + 1.14412i
3.27989 + 1.14412i
2.96901 − 0.437016i
−2.46412 − 0.437016i
−2.56435 + 1.14412i
−2.27989 + 1.14412i
−1.96901 − 0.437016i
3.46412 + 0.437016i
3.56435 − 1.14412i
3.27989 − 1.14412i
2.96901 + 0.437016i
−2.46412 + 0.437016i
−2.56435 − 1.14412i
−2.27989 − 1.14412i
−1.96901 + 0.437016i
− 1.41421i −2.80911 −2.00000 2.23607 3.97268i − 1.46309i 2.82843i −1.10889 − 3.16228i
241.2 − 1.41421i −2.39561 −2.00000 −2.23607 3.38790i 8.66248i 2.82843i −3.26106 3.16228i
241.3 − 1.41421i −0.448632 −2.00000 −2.23607 0.634461i − 6.70718i 2.82843i −8.79873 3.16228i
241.4 − 1.41421i 0.375992 −2.00000 2.23607 − 0.531733i 2.29527i 2.82843i −8.85863 − 3.16228i
241.5 − 1.41421i 3.11912 −2.00000 2.23607 − 4.41110i 5.50596i 2.82843i 0.728899 − 3.16228i
241.6 − 1.41421i 3.73310 −2.00000 −2.23607 − 5.27940i − 2.99502i 2.82843i 4.93604 3.16228i
241.7 − 1.41421i 5.11114 −2.00000 −2.23607 − 7.22824i 3.86814i 2.82843i 17.1237 3.16228i
241.8 − 1.41421i 5.31400 −2.00000 2.23607 − 7.51513i − 3.50971i 2.82843i 19.2386 − 3.16228i
241.9 1.41421i −2.80911 −2.00000 2.23607 − 3.97268i 1.46309i − 2.82843i −1.10889 3.16228i
241.10 1.41421i −2.39561 −2.00000 −2.23607 − 3.38790i − 8.66248i − 2.82843i −3.26106 − 3.16228i
241.11 1.41421i −0.448632 −2.00000 −2.23607 − 0.634461i 6.70718i − 2.82843i −8.79873 − 3.16228i
241.12 1.41421i 0.375992 −2.00000 2.23607 0.531733i − 2.29527i − 2.82843i −8.85863 3.16228i
241.13 1.41421i 3.11912 −2.00000 2.23607 4.41110i − 5.50596i − 2.82843i 0.728899 3.16228i
241.14 1.41421i 3.73310 −2.00000 −2.23607 5.27940i 2.99502i − 2.82843i 4.93604 − 3.16228i
241.15 1.41421i 5.11114 −2.00000 −2.23607 7.22824i − 3.86814i − 2.82843i 17.1237 − 3.16228i
241.16 1.41421i 5.31400 −2.00000 2.23607 7.51513i 3.50971i − 2.82843i 19.2386 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.f 16
11.b odd 2 1 inner 1210.3.d.f 16
11.c even 5 1 110.3.h.a ✓ 16
11.d odd 10 1 110.3.h.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.3.h.a ✓ 16 11.c even 5 1
110.3.h.a ✓ 16 11.d odd 10 1
1210.3.d.f 16 1.a even 1 1 trivial
1210.3.d.f 16 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 12T_{3}^{7} + 26T_{3}^{6} + 154T_{3}^{5} - 536T_{3}^{4} - 488T_{3}^{3} + 2189T_{3}^{2} + 226T_{3} - 359 \) 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
$3$ \( (T^{8} - 12 T^{7} + \cdots - 359)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 1908029761 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 13797948564481 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 234979436663521 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 16983967987921 \) Copy content Toggle raw display
$23$ \( (T^{8} + 6 T^{7} + \cdots - 31128714199)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 67\!\cdots\!01 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 1065752908841)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 82 T^{7} + \cdots - 593028592199)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 29\!\cdots\!41 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 18\!\cdots\!61 \) Copy content Toggle raw display
$47$ \( (T^{8} - 82 T^{7} + \cdots - 47738600399)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 122 T^{7} + \cdots - 359139393239)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 108730158935761)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots - 168814808814199)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 85604120483761)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 90\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 52\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 44\!\cdots\!21 \) Copy content Toggle raw display
$89$ \( (T^{8} + 8 T^{7} + \cdots - 2521899959)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 17\!\cdots\!21)^{2} \) Copy content Toggle raw display
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