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Properties

Label 266.2.d.a
Level $266$
Weight $2$
Character orbit 266.d
Analytic conductor $2.124$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [266,2,Mod(265,266)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(266, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("266.265");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 266 = 2 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 266.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: \(2.12402069377\)
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: \( x^{16} + 44x^{14} + 708x^{12} + 5378x^{10} + 20592x^{8} + 38856x^{6} + 33265x^{4} + 10216x^{2} + 900 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{11} q^{2} + \beta_{4} q^{3} - q^{4} + \beta_{7} q^{5} - \beta_{10} q^{6} + \beta_{12} q^{7} - \beta_{11} q^{8} + ( - \beta_{6} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{11} q^{2} + \beta_{4} q^{3} - q^{4} + \beta_{7} q^{5} - \beta_{10} q^{6} + \beta_{12} q^{7} - \beta_{11} q^{8} + ( - \beta_{6} + 2) q^{9} - \beta_{2} q^{10} + ( - \beta_1 - 1) q^{11} - \beta_{4} q^{12} + ( - \beta_{9} + \beta_{8} + \cdots - \beta_{3}) q^{13}+ \cdots + (\beta_{13} - \beta_{12} + \beta_{6} - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} + 6 q^{7} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 6 q^{7} + 24 q^{9} - 12 q^{11} + 16 q^{16} + 20 q^{23} - 28 q^{25} - 6 q^{28} - 16 q^{30} + 8 q^{35} - 24 q^{36} - 44 q^{39} + 18 q^{42} - 4 q^{43} + 12 q^{44} + 10 q^{49} - 12 q^{57} - 16 q^{58} + 20 q^{63} - 16 q^{64} + 12 q^{74} - 4 q^{77} + 88 q^{81} + 16 q^{85} - 20 q^{92} - 144 q^{93} + 12 q^{95} - 84 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 44x^{14} + 708x^{12} + 5378x^{10} + 20592x^{8} + 38856x^{6} + 33265x^{4} + 10216x^{2} + 900 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4684 \nu^{14} - 62369 \nu^{12} - 7419046 \nu^{10} - 122870856 \nu^{8} - 791532402 \nu^{6} + \cdots - 581108283 ) / 44810181 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 40712 \nu^{14} + 1832436 \nu^{12} + 30245083 \nu^{10} + 233190572 \nu^{8} + 869437060 \nu^{6} + \cdots + 61394868 ) / 29873454 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 245873 \nu^{15} + 1038150 \nu^{14} - 10399862 \nu^{13} + 44598900 \nu^{12} + \cdots + 5245163640 ) / 1792407240 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11500 \nu^{14} - 479398 \nu^{12} - 7072472 \nu^{10} - 46978521 \nu^{8} - 146700948 \nu^{6} + \cdots - 18087156 ) / 5271786 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 245873 \nu^{15} + 1038150 \nu^{14} + 10399862 \nu^{13} + 44598900 \nu^{12} + 157844884 \nu^{11} + \cdots + 5245163640 ) / 1792407240 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11500 \nu^{14} - 479398 \nu^{12} - 7072472 \nu^{10} - 46978521 \nu^{8} - 146700948 \nu^{6} + \cdots - 2271798 ) / 2635893 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 216968 \nu^{15} - 9869032 \nu^{13} - 167295244 \nu^{11} - 1372581589 \nu^{9} + \cdots - 1736652438 \nu ) / 149367270 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2118938 \nu^{15} + 1093745 \nu^{14} - 94420697 \nu^{13} + 47832545 \nu^{12} + \cdots + 9715121460 ) / 896203620 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2118938 \nu^{15} - 1093745 \nu^{14} - 94420697 \nu^{13} - 47832545 \nu^{12} + \cdots - 9715121460 ) / 896203620 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 142393 \nu^{15} + 6150292 \nu^{13} + 96020264 \nu^{11} + 695064834 \nu^{9} + 2462371446 \nu^{7} + \cdots + 514402458 \nu ) / 52717860 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 142393 \nu^{15} + 6150292 \nu^{13} + 96020264 \nu^{11} + 695064834 \nu^{9} + 2462371446 \nu^{7} + \cdots + 461684598 \nu ) / 52717860 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 4980447 \nu^{15} + 6972320 \nu^{14} - 221025498 \nu^{13} + 301307180 \nu^{12} + \cdots + 14857203060 ) / 1792407240 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 4980447 \nu^{15} - 6972320 \nu^{14} - 221025498 \nu^{13} - 301307180 \nu^{12} + \cdots - 14857203060 ) / 1792407240 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 3212683 \nu^{15} + 142253242 \nu^{13} + 2316632594 \nu^{11} + 17996184174 \nu^{9} + \cdots + 29377335738 \nu ) / 896203620 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 75762 \nu^{15} - 3301198 \nu^{13} - 52226056 \nu^{11} - 385079021 \nu^{9} + \cdots - 268858162 \nu ) / 8786310 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{11} + \beta_{10} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 2\beta_{4} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{15} - 2\beta_{13} - 2\beta_{12} + 16\beta_{11} - 10\beta_{10} + \beta_{7} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{13} - 3\beta_{12} - 18\beta_{6} + 8\beta_{5} + 32\beta_{4} + 8\beta_{3} + 4\beta_{2} - \beta _1 + 76 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 70 \beta_{15} - 5 \beta_{14} + 47 \beta_{13} + 47 \beta_{12} - 276 \beta_{11} + 137 \beta_{10} + \cdots - 15 \beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 91 \beta_{13} + 91 \beta_{12} - 6 \beta_{9} + 6 \beta_{8} + 330 \beta_{6} - 202 \beta_{5} + \cdots - 1207 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1386 \beta_{15} + 182 \beta_{14} - 953 \beta_{13} - 953 \beta_{12} + 4873 \beta_{11} + \cdots + 427 \beta_{3} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2062 \beta_{13} - 2062 \beta_{12} + 224 \beta_{9} - 224 \beta_{8} - 6141 \beta_{6} + 4152 \beta_{5} + \cdots + 20884 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 26325 \beta_{15} - 4500 \beta_{14} + 18496 \beta_{13} + 18496 \beta_{12} - 87580 \beta_{11} + \cdots - 9318 \beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 42131 \beta_{13} + 42131 \beta_{12} - 5672 \beta_{9} + 5672 \beta_{8} + 114504 \beta_{6} + \cdots - 374548 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 493174 \beta_{15} + 96283 \beta_{14} - 351603 \beta_{13} - 351603 \beta_{12} + 1593780 \beta_{11} + \cdots + 186373 \beta_{3} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 821681 \beta_{13} - 821681 \beta_{12} + 123116 \beta_{9} - 123116 \beta_{8} - 2132182 \beta_{6} + \cdots + 6828087 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 9189934 \beta_{15} - 1925794 \beta_{14} + 6610369 \beta_{13} + 6610369 \beta_{12} + \cdots - 3591913 \beta_{3} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 15657388 \beta_{13} + 15657388 \beta_{12} - 2482564 \beta_{9} + 2482564 \beta_{8} + 39644409 \beta_{6} + \cdots - 125478022 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 170845427 \beta_{15} + 37214228 \beta_{14} - 123528356 \beta_{13} - 123528356 \beta_{12} + \cdots + 67994198 \beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/266\mathbb{Z}\right)^\times\).

\(n\) \(115\) \(211\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
265.1
− 2.30643i
− 1.23100i
− 0.384810i
0.584675i
1.41533i
2.38481i
3.23100i
4.30643i
2.30643i
1.23100i
0.384810i
− 0.584675i
− 1.41533i
− 2.38481i
− 3.23100i
− 4.30643i
− 1.00000i −3.30643 −1.00000 2.68862i 3.30643i −0.592978 − 2.57844i 1.00000i 7.93246 2.68862
265.2 − 1.00000i −2.23100 −1.00000 − 3.34318i 2.23100i 2.62955 + 0.292339i 1.00000i 1.97734 −3.34318
265.3 − 1.00000i −1.38481 −1.00000 1.03210i 1.38481i 1.71160 + 2.01753i 1.00000i −1.08230 1.03210
265.4 − 1.00000i −0.415325 −1.00000 2.74394i 0.415325i −2.24817 + 1.39489i 1.00000i −2.82751 2.74394
265.5 − 1.00000i 0.415325 −1.00000 − 2.74394i − 0.415325i −2.24817 − 1.39489i 1.00000i −2.82751 −2.74394
265.6 − 1.00000i 1.38481 −1.00000 − 1.03210i − 1.38481i 1.71160 − 2.01753i 1.00000i −1.08230 −1.03210
265.7 − 1.00000i 2.23100 −1.00000 3.34318i − 2.23100i 2.62955 − 0.292339i 1.00000i 1.97734 3.34318
265.8 − 1.00000i 3.30643 −1.00000 − 2.68862i − 3.30643i −0.592978 + 2.57844i 1.00000i 7.93246 −2.68862
265.9 1.00000i −3.30643 −1.00000 − 2.68862i − 3.30643i −0.592978 + 2.57844i − 1.00000i 7.93246 2.68862
265.10 1.00000i −2.23100 −1.00000 3.34318i − 2.23100i 2.62955 − 0.292339i − 1.00000i 1.97734 −3.34318
265.11 1.00000i −1.38481 −1.00000 − 1.03210i − 1.38481i 1.71160 − 2.01753i − 1.00000i −1.08230 1.03210
265.12 1.00000i −0.415325 −1.00000 − 2.74394i − 0.415325i −2.24817 − 1.39489i − 1.00000i −2.82751 2.74394
265.13 1.00000i 0.415325 −1.00000 2.74394i 0.415325i −2.24817 + 1.39489i − 1.00000i −2.82751 −2.74394
265.14 1.00000i 1.38481 −1.00000 1.03210i 1.38481i 1.71160 + 2.01753i − 1.00000i −1.08230 −1.03210
265.15 1.00000i 2.23100 −1.00000 − 3.34318i 2.23100i 2.62955 + 0.292339i − 1.00000i 1.97734 3.34318
265.16 1.00000i 3.30643 −1.00000 2.68862i 3.30643i −0.592978 − 2.57844i − 1.00000i 7.93246 −2.68862
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 265.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
19.b odd 2 1 inner
133.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 266.2.d.a ✓ 16
3.b odd 2 1 2394.2.e.c 16
4.b odd 2 1 2128.2.m.f 16
7.b odd 2 1 inner 266.2.d.a ✓ 16
19.b odd 2 1 inner 266.2.d.a ✓ 16
21.c even 2 1 2394.2.e.c 16
28.d even 2 1 2128.2.m.f 16
57.d even 2 1 2394.2.e.c 16
76.d even 2 1 2128.2.m.f 16
133.c even 2 1 inner 266.2.d.a ✓ 16
399.h odd 2 1 2394.2.e.c 16
532.b odd 2 1 2128.2.m.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
266.2.d.a ✓ 16 1.a even 1 1 trivial
266.2.d.a ✓ 16 7.b odd 2 1 inner
266.2.d.a ✓ 16 19.b odd 2 1 inner
266.2.d.a ✓ 16 133.c even 2 1 inner
2128.2.m.f 16 4.b odd 2 1
2128.2.m.f 16 28.d even 2 1
2128.2.m.f 16 76.d even 2 1
2128.2.m.f 16 532.b odd 2 1
2394.2.e.c 16 3.b odd 2 1
2394.2.e.c 16 21.c even 2 1
2394.2.e.c 16 57.d even 2 1
2394.2.e.c 16 399.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(266, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( (T^{8} - 18 T^{6} + \cdots + 18)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + 27 T^{6} + \cdots + 648)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 3 T^{7} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 3 T^{3} - 23 T^{2} + \cdots - 48)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} - 93 T^{6} + \cdots + 276768)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 51 T^{6} + \cdots + 1152)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 16983563041 \) Copy content Toggle raw display
$23$ \( (T^{4} - 5 T^{3} - 20 T^{2} + \cdots + 96)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} + 216 T^{6} + \cdots + 5062500)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 72 T^{6} + \cdots + 4608)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 139 T^{6} + \cdots + 11664)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 185 T^{6} + \cdots + 41472)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + T^{3} - 57 T^{2} + \cdots + 128)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + 253 T^{6} + \cdots + 5971968)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 176 T^{6} + \cdots + 26244)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 70 T^{6} + \cdots + 36450)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 311 T^{6} + \cdots + 2880000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 269 T^{6} + \cdots + 82944)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 299 T^{6} + \cdots + 2073600)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 395 T^{6} + \cdots + 10616832)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 247 T^{6} + \cdots + 5308416)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 238 T^{6} + \cdots + 93312)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 365 T^{6} + \cdots + 6480000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 361 T^{6} + \cdots + 7558272)^{2} \) Copy content Toggle raw display
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