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Properties

Label 2268.2.i.n
Level $2268$
Weight $2$
Character orbit 2268.i
Analytic conductor $18.110$
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] = [2268,2,Mod(865,2268)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2268, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2268.865");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.i (of order \(3\), degree \(2\), not 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: \(18.1100711784\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 9x^{14} + 31x^{12} - 282x^{10} + 1695x^{8} - 3318x^{6} + 4606x^{4} - 4116x^{2} + 2401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{7} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{13} q^{5} + \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{13} q^{5} + \beta_{3} q^{7} - \beta_{12} q^{11} + (\beta_{9} - \beta_{8} - 2 \beta_{4} + 2) q^{13} + ( - \beta_{13} - \beta_{11} - \beta_{6}) q^{17} + (\beta_{9} - \beta_{8} + \beta_{7} + \cdots + 2) q^{19}+ \cdots + (\beta_{8} + 3 \beta_{4} - 2 \beta_{3} + \cdots + 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{7} + 10 q^{13} + 8 q^{19} + 16 q^{31} - 4 q^{37} - 10 q^{43} + 10 q^{49} - 32 q^{55} - 56 q^{61} - 36 q^{67} + 40 q^{79} - 38 q^{85} - 2 q^{91} + 42 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 9x^{14} + 31x^{12} - 282x^{10} + 1695x^{8} - 3318x^{6} + 4606x^{4} - 4116x^{2} + 2401 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 92249 \nu^{14} - 22952 \nu^{12} - 2800495 \nu^{10} - 21569239 \nu^{8} - 19347710 \nu^{6} + \cdots + 170475459 ) / 982062165 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 244351 \nu^{14} - 3103286 \nu^{12} + 15156266 \nu^{10} - 92194120 \nu^{8} + 665006344 \nu^{6} + \cdots - 4446117606 ) / 982062165 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 50804 \nu^{14} - 431490 \nu^{12} + 910001 \nu^{10} - 11273818 \nu^{8} + 75102634 \nu^{6} + \cdots - 188652058 ) / 140294595 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 95779 \nu^{14} - 763661 \nu^{12} + 2336349 \nu^{10} - 25351497 \nu^{8} + 138865326 \nu^{6} + \cdots - 129671150 ) / 196412433 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 784170 \nu^{15} + 214456 \nu^{13} + 30604253 \nu^{11} + 93415788 \nu^{9} + \cdots - 6997483661 \nu ) / 6874435155 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 202439 \nu^{15} - 3779851 \nu^{13} + 20969988 \nu^{11} - 90097992 \nu^{9} + \cdots - 2189789518 \nu ) / 1374887031 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1700901 \nu^{14} + 13906205 \nu^{12} - 39713699 \nu^{10} + 439930867 \nu^{8} + \cdots + 401714397 ) / 982062165 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 53324 \nu^{14} + 366096 \nu^{12} - 722548 \nu^{10} + 12291821 \nu^{8} - 61209692 \nu^{6} + \cdots + 6024452 ) / 22838655 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 160457 \nu^{14} - 1409701 \nu^{12} + 4498055 \nu^{10} - 43172212 \nu^{8} + 259237135 \nu^{6} + \cdots - 305714528 ) / 42698355 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 635345 \nu^{15} + 5091886 \nu^{13} - 13365107 \nu^{11} + 152655738 \nu^{9} + \cdots - 2003509256 \nu ) / 982062165 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 751964 \nu^{15} - 2837787 \nu^{13} - 5745335 \nu^{11} - 140264304 \nu^{9} + \cdots + 3200801814 \nu ) / 982062165 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 7006527 \nu^{15} - 54909609 \nu^{13} + 140871131 \nu^{11} - 1753147101 \nu^{9} + \cdots - 6552594825 \nu ) / 6874435155 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 7841464 \nu^{15} - 71750653 \nu^{13} + 246634027 \nu^{11} - 2192423802 \nu^{9} + \cdots - 8080133320 \nu ) / 6874435155 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1635500 \nu^{15} - 12087451 \nu^{13} + 27574530 \nu^{11} - 390459270 \nu^{9} + \cdots - 504509782 \nu ) / 1374887031 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 12267456 \nu^{15} + 107332095 \nu^{13} - 343863724 \nu^{11} + 3308249367 \nu^{9} + \cdots + 38302053342 \nu ) / 6874435155 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + \beta_{13} + \beta_{12} - \beta_{6} + 2\beta_{5} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{9} + \beta_{8} - 3\beta_{7} + 7\beta_{4} + 3\beta_{3} + \beta_{2} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{15} + 9\beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} + 6\beta_{10} + 9\beta_{6} + \beta_{5} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -6\beta_{9} - 6\beta_{8} - 13\beta_{7} - 31\beta_{4} - 8\beta_{3} + 20\beta_{2} - 10\beta _1 + 25 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 10 \beta_{15} - 16 \beta_{14} - 21 \beta_{13} + 44 \beta_{12} + 17 \beta_{11} + 2 \beta_{10} + \cdots + 22 \beta_{5} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 11\beta_{9} - 22\beta_{8} + 71\beta_{7} + 10\beta_{4} + 31\beta_{3} + 49\beta_{2} + 21\beta _1 + 322 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 70 \beta_{15} - 12 \beta_{14} + 23 \beta_{13} + 178 \beta_{12} + 14 \beta_{11} + 33 \beta_{10} + \cdots + 356 \beta_{5} ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -234\beta_{9} + 117\beta_{8} - 169\beta_{7} + 1409\beta_{4} + 538\beta_{3} + 26\beta_{2} - 304\beta _1 + 187 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 804 \beta_{15} + 1814 \beta_{14} - 25 \beta_{13} - 467 \beta_{12} + 221 \beta_{11} + \cdots + 467 \beta_{5} ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1148 \beta_{9} - 1148 \beta_{8} - 2191 \beta_{7} - 3861 \beta_{4} - 2165 \beta_{3} + 3234 \beta_{2} + \cdots + 2713 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1322 \beta_{15} - 2061 \beta_{14} - 5174 \beta_{13} + 5600 \beta_{12} + 4591 \beta_{11} + \cdots + 2800 \beta_{5} ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1284\beta_{9} - 2568\beta_{8} + 5956\beta_{7} - 1481\beta_{4} - 1678\beta_{3} + 3388\beta_{2} - 197\beta _1 + 19573 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 3218 \beta_{15} - 21675 \beta_{14} - 5929 \beta_{13} + 34676 \beta_{12} + 13164 \beta_{11} + \cdots + 69352 \beta_{5} ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 24730 \beta_{9} + 12365 \beta_{8} + 31071 \beta_{7} + 328079 \beta_{4} + 80961 \beta_{3} + \cdots + 43866 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 108952 \beta_{15} + 307191 \beta_{14} - 7798 \beta_{13} - 141578 \beta_{12} + 66890 \beta_{11} + \cdots + 141578 \beta_{5} ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(1\) \(-1 + \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
865.1
0.817131 − 0.735533i
−2.40332 + 0.123797i
1.04556 − 0.339889i
1.30887 − 2.01944i
−1.30887 + 2.01944i
−1.04556 + 0.339889i
2.40332 − 0.123797i
−0.817131 + 0.735533i
0.817131 + 0.735533i
−2.40332 − 0.123797i
1.04556 + 0.339889i
1.30887 + 2.01944i
−1.30887 − 2.01944i
−1.04556 − 0.339889i
2.40332 + 0.123797i
−0.817131 − 0.735533i
0 0 0 −1.83843 + 3.18426i 0 −1.55575 + 2.14001i 0 0 0
865.2 0 0 0 −1.15101 + 1.99360i 0 2.41508 + 1.08045i 0 0 0
865.3 0 0 0 −0.515559 + 0.892975i 0 −2.63118 + 0.277320i 0 0 0
865.4 0 0 0 −0.171869 + 0.297685i 0 0.271847 − 2.63175i 0 0 0
865.5 0 0 0 0.171869 − 0.297685i 0 0.271847 − 2.63175i 0 0 0
865.6 0 0 0 0.515559 − 0.892975i 0 −2.63118 + 0.277320i 0 0 0
865.7 0 0 0 1.15101 − 1.99360i 0 2.41508 + 1.08045i 0 0 0
865.8 0 0 0 1.83843 − 3.18426i 0 −1.55575 + 2.14001i 0 0 0
2053.1 0 0 0 −1.83843 − 3.18426i 0 −1.55575 − 2.14001i 0 0 0
2053.2 0 0 0 −1.15101 − 1.99360i 0 2.41508 − 1.08045i 0 0 0
2053.3 0 0 0 −0.515559 − 0.892975i 0 −2.63118 − 0.277320i 0 0 0
2053.4 0 0 0 −0.171869 − 0.297685i 0 0.271847 + 2.63175i 0 0 0
2053.5 0 0 0 0.171869 + 0.297685i 0 0.271847 + 2.63175i 0 0 0
2053.6 0 0 0 0.515559 + 0.892975i 0 −2.63118 − 0.277320i 0 0 0
2053.7 0 0 0 1.15101 + 1.99360i 0 2.41508 − 1.08045i 0 0 0
2053.8 0 0 0 1.83843 + 3.18426i 0 −1.55575 − 2.14001i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 865.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
63.h even 3 1 inner
63.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.i.n 16
3.b odd 2 1 inner 2268.2.i.n 16
7.c even 3 1 2268.2.l.n 16
9.c even 3 1 2268.2.k.g ✓ 16
9.c even 3 1 2268.2.l.n 16
9.d odd 6 1 2268.2.k.g ✓ 16
9.d odd 6 1 2268.2.l.n 16
21.h odd 6 1 2268.2.l.n 16
63.g even 3 1 2268.2.k.g ✓ 16
63.h even 3 1 inner 2268.2.i.n 16
63.j odd 6 1 inner 2268.2.i.n 16
63.n odd 6 1 2268.2.k.g ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2268.2.i.n 16 1.a even 1 1 trivial
2268.2.i.n 16 3.b odd 2 1 inner
2268.2.i.n 16 63.h even 3 1 inner
2268.2.i.n 16 63.j odd 6 1 inner
2268.2.k.g ✓ 16 9.c even 3 1
2268.2.k.g ✓ 16 9.d odd 6 1
2268.2.k.g ✓ 16 63.g even 3 1
2268.2.k.g ✓ 16 63.n odd 6 1
2268.2.l.n 16 7.c even 3 1
2268.2.l.n 16 9.c even 3 1
2268.2.l.n 16 9.d odd 6 1
2268.2.l.n 16 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2268, [\chi])\):

\( T_{5}^{16} + 20T_{5}^{14} + 306T_{5}^{12} + 1706T_{5}^{10} + 7087T_{5}^{8} + 7818T_{5}^{6} + 6723T_{5}^{4} + 783T_{5}^{2} + 81 \) Copy content Toggle raw display
\( T_{13}^{8} - 5T_{13}^{7} + 45T_{13}^{6} - 152T_{13}^{5} + 1177T_{13}^{4} - 3990T_{13}^{3} + 12936T_{13}^{2} - 18522T_{13} + 21609 \) Copy content Toggle raw display
\( T_{19}^{8} - 4T_{19}^{7} + 40T_{19}^{6} + 110T_{19}^{5} + 541T_{19}^{4} + 224T_{19}^{3} + 217T_{19}^{2} - 49T_{19} + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 20 T^{14} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( (T^{8} + 3 T^{7} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 47 T^{14} + \cdots + 194481 \) Copy content Toggle raw display
$13$ \( (T^{8} - 5 T^{7} + \cdots + 21609)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 62 T^{14} + \cdots + 194481 \) Copy content Toggle raw display
$19$ \( (T^{8} - 4 T^{7} + 40 T^{6} + \cdots + 49)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 44 T^{14} + \cdots + 50625 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 10016218555281 \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{3} - 17 T^{2} + \cdots + 63)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 2 T^{7} + 64 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 9845600625 \) Copy content Toggle raw display
$43$ \( (T^{8} + 5 T^{7} + \cdots + 3200521)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 429 T^{6} + \cdots + 62583921)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 1315703055681 \) Copy content Toggle raw display
$59$ \( (T^{8} - 200 T^{6} + \cdots + 3272481)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 14 T^{3} + \cdots - 2649)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 9 T^{3} + \cdots - 917)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} - 305 T^{6} + \cdots + 505521)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 128 T^{6} + \cdots + 7458361)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 10 T^{3} + \cdots - 1497)^{4} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 60\!\cdots\!21 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 4631487063921 \) Copy content Toggle raw display
$97$ \( (T^{8} - 21 T^{7} + \cdots + 8614225)^{2} \) Copy content Toggle raw display
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