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Properties

Label 1089.3.b.i
Level $1089$
Weight $3$
Character orbit 1089.b
Analytic conductor $29.673$
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] = [1089,3,Mod(485,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1089.485");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1089.b (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: \(29.6731007888\)
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: \( x^{16} + 48x^{14} + 921x^{12} + 8986x^{10} + 46812x^{8} + 125072x^{6} + 152129x^{4} + 65614x^{2} + 5041 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 99)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} + \beta_{13} q^{5} + (\beta_{9} - \beta_{5} + \beta_{3} - 1) q^{7} + ( - \beta_{13} - \beta_{12} + \cdots - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} + \beta_{13} q^{5} + (\beta_{9} - \beta_{5} + \beta_{3} - 1) q^{7} + ( - \beta_{13} - \beta_{12} + \cdots - 2 \beta_1) q^{8}+ \cdots + (7 \beta_{15} - 4 \beta_{14} + \cdots + 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{4} - 8 q^{7} + 24 q^{10} + 4 q^{13} + 28 q^{16} - 20 q^{19} - 44 q^{25} + 16 q^{28} + 28 q^{31} + 148 q^{34} - 148 q^{37} + 224 q^{40} + 272 q^{43} + 208 q^{46} + 348 q^{49} + 520 q^{52} - 44 q^{58} + 224 q^{61} + 436 q^{64} + 24 q^{67} - 664 q^{70} - 4 q^{73} + 1052 q^{76} + 216 q^{79} + 348 q^{82} + 416 q^{85} - 168 q^{91} + 1140 q^{94} - 44 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 48x^{14} + 921x^{12} + 8986x^{10} + 46812x^{8} + 125072x^{6} + 152129x^{4} + 65614x^{2} + 5041 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{14} - 38\nu^{12} - 532\nu^{10} - 3324\nu^{8} - 8784\nu^{6} - 7244\nu^{4} + 1095\nu^{2} + 1448 ) / 792 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 115 \nu^{15} + 373 \nu^{13} + 131438 \nu^{11} + 2591160 \nu^{9} + 20689524 \nu^{7} + \cdots + 48462527 \nu ) / 1237104 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{14} + 161\nu^{12} + 2470\nu^{10} + 18084\nu^{8} + 65124\nu^{6} + 109760\nu^{4} + 72840\nu^{2} + 8977 ) / 792 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 127 \nu^{14} - 5015 \nu^{12} - 74458 \nu^{10} - 513480 \nu^{8} - 1642716 \nu^{6} - 2151404 \nu^{4} + \cdots + 72971 ) / 17424 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 103 \nu^{14} + 4139 \nu^{12} + 63418 \nu^{10} + 465168 \nu^{8} + 1696932 \nu^{6} + 2964164 \nu^{4} + \cdots + 145345 ) / 17424 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 41 \nu^{14} - 1612 \nu^{12} - 23810 \nu^{10} - 163086 \nu^{8} - 515538 \nu^{6} - 653278 \nu^{4} + \cdots - 6692 ) / 4356 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 161 \nu^{14} + 6289 \nu^{12} + 92078 \nu^{10} + 624624 \nu^{8} + 1979532 \nu^{6} + 2767276 \nu^{4} + \cdots + 339227 ) / 17424 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 127 \nu^{15} + 5741 \nu^{13} + 102838 \nu^{11} + 926376 \nu^{9} + 4389372 \nu^{7} + \cdots + 6703315 \nu ) / 412368 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 83 \nu^{15} + 3629 \nu^{13} + 62314 \nu^{11} + 534116 \nu^{9} + 2407744 \nu^{7} + 5555816 \nu^{5} + \cdots + 2563575 \nu ) / 68728 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -\nu^{15} - 38\nu^{13} - 532\nu^{11} - 3324\nu^{9} - 8784\nu^{7} - 7244\nu^{5} + 1095\nu^{3} + 656\nu ) / 792 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1528 \nu^{15} + 62339 \nu^{13} + 976318 \nu^{11} + 7403088 \nu^{9} + 28530000 \nu^{7} + \cdots + 16374319 \nu ) / 618552 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 592 \nu^{15} + 21245 \nu^{13} + 265918 \nu^{11} + 1252548 \nu^{9} + 449556 \nu^{7} + \cdots - 2699819 \nu ) / 206184 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 607 \nu^{15} + 25373 \nu^{13} + 411154 \nu^{11} + 3268980 \nu^{9} + 13372824 \nu^{7} + \cdots + 6633991 \nu ) / 206184 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{13} - \beta_{12} + \beta_{11} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - \beta_{8} + \beta_{6} - \beta_{5} + 5\beta_{3} - 13\beta_{2} + 55 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{15} + 2\beta_{14} + 13\beta_{13} + 20\beta_{12} - 16\beta_{11} + 3\beta_{10} + \beta_{4} + 110\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -19\beta_{9} + 23\beta_{8} - 4\beta_{7} - 21\beta_{6} + 26\beta_{5} - 104\beta_{3} + 163\beta_{2} - 556 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 58 \beta_{15} - 49 \beta_{14} - 144 \beta_{13} - 323 \beta_{12} + 226 \beta_{11} - 52 \beta_{10} + \cdots - 1262 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 300\beta_{9} - 385\beta_{8} + 107\beta_{7} + 337\beta_{6} - 479\beta_{5} + 1705\beta_{3} - 2050\beta_{2} + 5876 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1113 \beta_{15} + 886 \beta_{14} + 1561 \beta_{13} + 4853 \beta_{12} - 3072 \beta_{11} + \cdots + 14948 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 4446 \beta_{9} + 5741 \beta_{8} - 1999 \beta_{7} - 4857 \beta_{6} + 7738 \beta_{5} - 25702 \beta_{3} + \cdots - 64004 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 18183 \beta_{15} - 14183 \beta_{14} - 17131 \beta_{13} - 70235 \beta_{12} + 40991 \beta_{11} + \cdots - 181597 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 63680 \beta_{9} - 80998 \beta_{8} + 32366 \beta_{7} + 66278 \beta_{6} - 116784 \beta_{5} + 371616 \beta_{3} + \cdots + 714871 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 274142 \beta_{15} + 212830 \beta_{14} + 192314 \beta_{13} + 992430 \beta_{12} - 541350 \beta_{11} + \cdots + 2250309 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 892116 \beta_{9} + 1108664 \beta_{8} - 486972 \beta_{7} - 877608 \beta_{6} + 1692232 \beta_{5} + \cdots - 8166432 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3948668 \beta_{15} - 3071320 \beta_{14} - 2213375 \beta_{13} - 13788227 \beta_{12} + \cdots - 28311016 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
485.1
− 3.62039i
− 3.25431i
− 2.99491i
− 2.98036i
− 1.96467i
− 1.35141i
− 0.816689i
− 0.311356i
0.311356i
0.816689i
1.35141i
1.96467i
2.98036i
2.99491i
3.25431i
3.62039i
− 3.62039i 0 −9.10725 0.165198i 0 10.2989 18.4903i 0 0.598083
485.2 − 3.25431i 0 −6.59053 1.59288i 0 −10.8798 8.43039i 0 5.18373
485.3 − 2.99491i 0 −4.96950 1.61173i 0 −3.32981 2.90358i 0 4.82698
485.4 − 2.98036i 0 −4.88257 − 7.85680i 0 2.80793 2.63038i 0 −23.4161
485.5 − 1.96467i 0 0.140085 9.62261i 0 −5.23056 − 8.13389i 0 18.9052
485.6 − 1.35141i 0 2.17369 5.10469i 0 −13.1679 − 8.34318i 0 6.89852
485.7 − 0.816689i 0 3.33302 1.04696i 0 6.83027 − 5.98880i 0 0.855043
485.8 − 0.311356i 0 3.90306 − 5.94641i 0 8.67101 − 2.46067i 0 −1.85145
485.9 0.311356i 0 3.90306 5.94641i 0 8.67101 2.46067i 0 −1.85145
485.10 0.816689i 0 3.33302 − 1.04696i 0 6.83027 5.98880i 0 0.855043
485.11 1.35141i 0 2.17369 − 5.10469i 0 −13.1679 8.34318i 0 6.89852
485.12 1.96467i 0 0.140085 − 9.62261i 0 −5.23056 8.13389i 0 18.9052
485.13 2.98036i 0 −4.88257 7.85680i 0 2.80793 − 2.63038i 0 −23.4161
485.14 2.99491i 0 −4.96950 − 1.61173i 0 −3.32981 − 2.90358i 0 4.82698
485.15 3.25431i 0 −6.59053 − 1.59288i 0 −10.8798 − 8.43039i 0 5.18373
485.16 3.62039i 0 −9.10725 − 0.165198i 0 10.2989 − 18.4903i 0 0.598083
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 485.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.3.b.i 16
3.b odd 2 1 inner 1089.3.b.i 16
11.b odd 2 1 1089.3.b.j 16
11.c even 5 2 99.3.l.a ✓ 32
33.d even 2 1 1089.3.b.j 16
33.h odd 10 2 99.3.l.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.3.l.a ✓ 32 11.c even 5 2
99.3.l.a ✓ 32 33.h odd 10 2
1089.3.b.i 16 1.a even 1 1 trivial
1089.3.b.i 16 3.b odd 2 1 inner
1089.3.b.j 16 11.b odd 2 1
1089.3.b.j 16 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1089, [\chi])\):

\( T_{2}^{16} + 48 T_{2}^{14} + 921 T_{2}^{12} + 8986 T_{2}^{10} + 46812 T_{2}^{8} + 125072 T_{2}^{6} + \cdots + 5041 \) Copy content Toggle raw display
\( T_{7}^{8} + 4T_{7}^{7} - 275T_{7}^{6} - 566T_{7}^{5} + 23990T_{7}^{4} + 14910T_{7}^{3} - 659963T_{7}^{2} - 237648T_{7} + 4273551 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 48 T^{14} + \cdots + 5041 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 222 T^{14} + \cdots + 1038361 \) Copy content Toggle raw display
$7$ \( (T^{8} + 4 T^{7} + \cdots + 4273551)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( (T^{8} - 2 T^{7} + \cdots - 14305556)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 98\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{8} + 10 T^{7} + \cdots + 918027484)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 47\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( (T^{8} - 14 T^{7} + \cdots + 37407528581)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 74 T^{7} + \cdots - 36000957284)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 32\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{8} - 136 T^{7} + \cdots - 2337129036)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 97\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 20\!\cdots\!61 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$61$ \( (T^{8} - 112 T^{7} + \cdots - 99578003184)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 40427575967196)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 4747380152636)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots - 8796984423319)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 69\!\cdots\!01 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 29900401075089)^{2} \) Copy content Toggle raw display
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