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Properties

Label 297.3.c.b
Level $297$
Weight $3$
Character orbit 297.c
Analytic conductor $8.093$
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] = [297,3,Mod(109,297)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(297, 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("297.109");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 297.c (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: \(8.09266385150\)
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: \( x^{16} + 24x^{14} + 420x^{12} + 1908x^{10} + 20196x^{8} - 91800x^{6} + 597348x^{4} - 4428432x^{2} + 8714304 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{10} \)
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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} + (\beta_{2} - 2) q^{4} - \beta_{10} q^{5} + \beta_{6} q^{7} + (\beta_{12} - 3 \beta_{8}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{2} + (\beta_{2} - 2) q^{4} - \beta_{10} q^{5} + \beta_{6} q^{7} + (\beta_{12} - 3 \beta_{8}) q^{8} + \beta_{3} q^{10} + (\beta_{13} - \beta_{8}) q^{11} + \beta_{7} q^{13} + (\beta_{14} + \beta_{13} + \cdots + \beta_{9}) q^{14}+ \cdots + ( - 2 \beta_{15} - 2 \beta_{14} + \cdots - 54 \beta_{8}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{4} + 160 q^{16} + 102 q^{22} + 12 q^{25} + 68 q^{31} + 156 q^{34} - 124 q^{37} - 272 q^{49} + 182 q^{55} + 492 q^{58} - 680 q^{64} - 400 q^{67} + 324 q^{70} - 444 q^{82} - 510 q^{88} + 120 q^{91} + 152 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 24x^{14} + 420x^{12} + 1908x^{10} + 20196x^{8} - 91800x^{6} + 597348x^{4} - 4428432x^{2} + 8714304 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 3463743 \nu^{14} + 774131840 \nu^{12} + 24962448882 \nu^{10} + 435264116454 \nu^{8} + \cdots - 165797395825680 ) / 63784681752744 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6639806 \nu^{14} - 134103918 \nu^{12} - 2346672834 \nu^{10} - 4452205761 \nu^{8} + \cdots + 96239493784818 ) / 15946170438186 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 43302579 \nu^{14} + 30491668 \nu^{12} - 10882411878 \nu^{10} - 408550881888 \nu^{8} + \cdots - 124608498995880 ) / 63784681752744 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 269730203 \nu^{14} + 5971191264 \nu^{12} + 134409572244 \nu^{10} + 1163775668436 \nu^{8} + \cdots - 20\!\cdots\!24 ) / 127569363505488 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 19205843 \nu^{14} + 412414608 \nu^{12} + 5581163412 \nu^{10} - 15545599092 \nu^{8} + \cdots - 38030050672992 ) / 5546494065456 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 211559797 \nu^{14} - 5896585505 \nu^{12} - 110238514680 \nu^{10} + \cdots + 539814893161428 ) / 31892340876372 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 517384676 \nu^{14} - 13764995848 \nu^{12} - 254500596285 \nu^{10} + \cdots + 14\!\cdots\!64 ) / 31892340876372 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1779435417 \nu^{15} + 40528593640 \nu^{13} + 703376790036 \nu^{11} + \cdots - 36\!\cdots\!00 \nu ) / 10\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1779435417 \nu^{15} - 40528593640 \nu^{13} - 703376790036 \nu^{11} + \cdots + 14\!\cdots\!16 \nu ) / 34\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 25488582509 \nu^{15} - 746877698796 \nu^{13} - 13716179035740 \nu^{11} + \cdots + 57\!\cdots\!52 \nu ) / 31\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13837062385 \nu^{15} + 320771257656 \nu^{13} + 5560682420772 \nu^{11} + \cdots - 79\!\cdots\!16 \nu ) / 10\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 9189263367 \nu^{15} + 217721027824 \nu^{13} + 3769074495924 \nu^{11} + \cdots - 17\!\cdots\!04 \nu ) / 52\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 75996048679 \nu^{15} + 2292928258584 \nu^{13} + 45712410778620 \nu^{11} + \cdots - 17\!\cdots\!80 \nu ) / 31\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 68196624391 \nu^{15} - 1821951301062 \nu^{13} - 34187260905612 \nu^{11} + \cdots + 23\!\cdots\!64 \nu ) / 15\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 39850044393 \nu^{15} + 1066853023837 \nu^{13} + 19522141729098 \nu^{11} + \cdots - 95\!\cdots\!48 \nu ) / 26\!\cdots\!04 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + 3\beta_{8} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} + 3\beta_{2} + 2\beta _1 - 9 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} - 3\beta_{11} - 5\beta_{9} - 2\beta_{8} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{7} + 8\beta_{6} - \beta_{5} - \beta_{4} - 8\beta_{3} + 3\beta_{2} - 8\beta _1 - 33 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} + 16\beta_{14} + 14\beta_{13} + 11\beta_{12} + 45\beta_{11} + 15\beta_{10} + 43\beta_{9} - 143\beta_{8} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 50\beta_{7} - 116\beta_{6} - 26\beta_{5} - 22\beta_{4} + 66\beta_{3} - 324\beta_{2} + 28\beta _1 + 1338 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 40\beta_{15} - 88\beta_{14} - 164\beta_{13} - 582\beta_{12} - 126\beta_{11} + 72\beta_{9} + 3710\beta_{8} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 408\beta_{7} - 696\beta_{6} + 666\beta_{5} + 906\beta_{4} + 672\beta_{3} + 6336\beta_{2} + 696\beta _1 - 22500 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 954 \beta_{15} - 4200 \beta_{14} - 1956 \beta_{13} + 8616 \beta_{12} - 10368 \beta_{11} + \cdots - 47022 \beta_{8} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 23136 \beta_{7} + 49848 \beta_{6} - 5754 \beta_{5} - 9474 \beta_{4} - 38556 \beta_{3} - 48474 \beta_{2} + \cdots + 160362 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3318 \beta_{15} + 109596 \beta_{14} + 100680 \beta_{13} - 9666 \beta_{12} + 276606 \beta_{11} + \cdots - 6 \beta_{8} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 378396 \beta_{7} - 879300 \beta_{6} - 76860 \beta_{5} - 98532 \beta_{4} + 693144 \beta_{3} + \cdots + 2990520 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 232236 \beta_{15} - 1040616 \beta_{14} - 1570824 \beta_{13} - 2427948 \beta_{12} + \cdots + 14449176 \beta_{8} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1584360 \beta_{7} + 4630608 \beta_{6} + 3476736 \beta_{5} + 4985280 \beta_{4} - 4102416 \beta_{3} + \cdots - 120592476 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 5851224 \beta_{15} - 9843120 \beta_{14} + 3684528 \beta_{13} + 58128516 \beta_{12} + \cdots - 338759064 \beta_{8} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/297\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(244\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−1.73205 − 3.81469i
1.73205 − 3.81469i
1.73205 − 2.58810i
−1.73205 − 2.58810i
−1.73205 − 1.64800i
1.73205 − 1.64800i
1.73205 − 0.184385i
−1.73205 − 0.184385i
1.73205 + 0.184385i
−1.73205 + 0.184385i
−1.73205 + 1.64800i
1.73205 + 1.64800i
1.73205 + 2.58810i
−1.73205 + 2.58810i
−1.73205 + 3.81469i
1.73205 + 3.81469i
− 3.81469i 0 −10.5518 −2.90548 0 − 12.6815i 24.9932i 0 11.0835i
109.2 − 3.81469i 0 −10.5518 2.90548 0 12.6815i 24.9932i 0 − 11.0835i
109.3 − 2.58810i 0 −2.69827 −2.26558 0 − 2.77010i − 3.36900i 0 5.86356i
109.4 − 2.58810i 0 −2.69827 2.26558 0 2.77010i − 3.36900i 0 − 5.86356i
109.5 − 1.64800i 0 1.28411 −8.56484 0 4.92814i − 8.70819i 0 14.1148i
109.6 − 1.64800i 0 1.28411 8.56484 0 − 4.92814i − 8.70819i 0 − 14.1148i
109.7 − 0.184385i 0 3.96600 −4.00859 0 8.43922i − 1.46881i 0 0.739122i
109.8 − 0.184385i 0 3.96600 4.00859 0 − 8.43922i − 1.46881i 0 − 0.739122i
109.9 0.184385i 0 3.96600 −4.00859 0 − 8.43922i 1.46881i 0 − 0.739122i
109.10 0.184385i 0 3.96600 4.00859 0 8.43922i 1.46881i 0 0.739122i
109.11 1.64800i 0 1.28411 −8.56484 0 − 4.92814i 8.70819i 0 − 14.1148i
109.12 1.64800i 0 1.28411 8.56484 0 4.92814i 8.70819i 0 14.1148i
109.13 2.58810i 0 −2.69827 −2.26558 0 2.77010i 3.36900i 0 − 5.86356i
109.14 2.58810i 0 −2.69827 2.26558 0 − 2.77010i 3.36900i 0 5.86356i
109.15 3.81469i 0 −10.5518 −2.90548 0 12.6815i − 24.9932i 0 − 11.0835i
109.16 3.81469i 0 −10.5518 2.90548 0 − 12.6815i − 24.9932i 0 11.0835i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 297.3.c.b ✓ 16
3.b odd 2 1 inner 297.3.c.b ✓ 16
11.b odd 2 1 inner 297.3.c.b ✓ 16
33.d even 2 1 inner 297.3.c.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.3.c.b ✓ 16 1.a even 1 1 trivial
297.3.c.b ✓ 16 3.b odd 2 1 inner
297.3.c.b ✓ 16 11.b odd 2 1 inner
297.3.c.b ✓ 16 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 24T_{2}^{6} + 156T_{2}^{4} + 270T_{2}^{2} + 9 \) acting on \(S_{3}^{\mathrm{new}}(297, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 24 T^{6} + 156 T^{4} + \cdots + 9)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 103 T^{6} + \cdots + 51076)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 264 T^{6} + \cdots + 2134521)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 45\!\cdots\!61 \) Copy content Toggle raw display
$13$ \( (T^{8} + 606 T^{6} + \cdots + 74442384)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 1809 T^{6} + \cdots + 10693628100)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 1926 T^{6} + \cdots + 12230590464)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 1069 T^{6} + \cdots + 38837824)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 1683 T^{6} + \cdots + 5069724804)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 17 T^{3} + \cdots + 519082)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 31 T^{3} + \cdots + 384952)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + 5559 T^{6} + \cdots + 17055315216)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 2607 T^{6} + \cdots + 24679781604)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 1831946008036)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 13399 T^{6} + \cdots + 99131781904)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 5545440975376)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 159547819738176)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 100 T^{3} + \cdots + 598228)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 125406491838016)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 25567706828304)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 751514895267984)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 2470071149316)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 2082202824256)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 38 T^{3} + \cdots + 159460717)^{4} \) Copy content Toggle raw display
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