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Properties

Label 637.2.h.l
Level $637$
Weight $2$
Character orbit 637.h
Analytic conductor $5.086$
Analytic rank $0$
Dimension $12$
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] = [637,2,Mod(165,637)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("637.165");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.h (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: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 7x^{10} - 2x^{9} + 33x^{8} - 11x^{7} + 55x^{6} + 17x^{5} + 47x^{4} + x^{3} + 8x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{11} + \beta_{5} + \beta_{3}) q^{2} - \beta_{11} q^{3} + ( - \beta_{10} - \beta_{6} + 1) q^{4} + \beta_{9} q^{5} + ( - \beta_{10} - \beta_{8} - 2 \beta_{7} + \cdots + 2) q^{6}+ \cdots + ( - \beta_{8} + \beta_{6}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{11} + \beta_{5} + \beta_{3}) q^{2} - \beta_{11} q^{3} + ( - \beta_{10} - \beta_{6} + 1) q^{4} + \beta_{9} q^{5} + ( - \beta_{10} - \beta_{8} - 2 \beta_{7} + \cdots + 2) q^{6}+ \cdots + (2 \beta_{11} - 4 \beta_{5} + 2 \beta_{4} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{2} - q^{3} + 8 q^{4} - q^{5} + 9 q^{6} - 6 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{2} - q^{3} + 8 q^{4} - q^{5} + 9 q^{6} - 6 q^{8} + 3 q^{9} - 4 q^{10} + 4 q^{11} - 5 q^{12} + 2 q^{13} - 2 q^{15} - 16 q^{16} + 10 q^{17} + 3 q^{18} + q^{19} + q^{20} - 5 q^{22} + 2 q^{23} + 11 q^{24} + 7 q^{25} + 16 q^{26} + 8 q^{27} + 3 q^{29} - 5 q^{30} - 16 q^{31} - 16 q^{32} - 16 q^{33} - 32 q^{34} - 21 q^{36} + 26 q^{37} + 17 q^{38} - 20 q^{39} + 5 q^{40} + 8 q^{41} - 11 q^{43} + 21 q^{44} - 14 q^{45} - 32 q^{46} + q^{47} - 21 q^{48} + 6 q^{50} - 20 q^{51} - 41 q^{52} - 2 q^{53} - 36 q^{54} - 9 q^{55} + 42 q^{57} - 8 q^{58} + 26 q^{59} + 20 q^{60} + 5 q^{61} - 5 q^{62} - 30 q^{64} - 5 q^{65} - 18 q^{66} - 11 q^{67} + 58 q^{68} - 23 q^{69} + 6 q^{71} + 25 q^{72} + 30 q^{73} + 6 q^{74} - 6 q^{75} + 9 q^{76} + 16 q^{78} + 7 q^{79} + 7 q^{80} - 6 q^{81} - q^{82} + 54 q^{83} - q^{85} - 7 q^{86} + 32 q^{87} + 8 q^{89} + 16 q^{90} + 54 q^{92} + 14 q^{93} - 45 q^{94} + 12 q^{95} - 19 q^{96} + 35 q^{97} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 7x^{10} - 2x^{9} + 33x^{8} - 11x^{7} + 55x^{6} + 17x^{5} + 47x^{4} + x^{3} + 8x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 29696 \nu^{11} - 478424 \nu^{10} + 682506 \nu^{9} - 3846008 \nu^{8} + 2684563 \nu^{7} + \cdots - 2119374 ) / 3318773 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 73788 \nu^{11} - 498559 \nu^{10} + 495146 \nu^{9} - 4188508 \nu^{8} + 1631143 \nu^{7} + \cdots - 2229034 ) / 3318773 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 109660 \nu^{11} + 153752 \nu^{10} - 747485 \nu^{9} + 406680 \nu^{8} - 3276280 \nu^{7} + \cdots - 6198231 ) / 3318773 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 439315 \nu^{11} - 329655 \nu^{10} + 2921453 \nu^{9} - 131145 \nu^{8} + 14090715 \nu^{7} + \cdots + 572347 ) / 3318773 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 566698 \nu^{11} - 1732988 \nu^{10} + 5617249 \nu^{9} - 9944902 \nu^{8} + 24340355 \nu^{7} + \cdots - 6707921 ) / 3318773 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 572347 \nu^{11} + 1011662 \nu^{10} - 4336084 \nu^{9} + 4066147 \nu^{8} - 19018596 \nu^{7} + \cdots + 3392561 ) / 3318773 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1035034 \nu^{11} + 1869572 \nu^{10} - 7924683 \nu^{9} + 7725614 \nu^{8} - 34760912 \nu^{7} + \cdots + 6345807 ) / 3318773 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1166290 \nu^{11} - 1650363 \nu^{10} + 8811506 \nu^{9} - 5639321 \nu^{8} + 40119354 \nu^{7} + \cdots + 566698 ) / 3318773 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2686072 \nu^{11} + 3882058 \nu^{10} - 19974443 \nu^{9} + 13501144 \nu^{8} - 90433689 \nu^{7} + \cdots - 1035561 ) / 3318773 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( \nu^{11} - \nu^{10} + 7\nu^{9} - 2\nu^{8} + 33\nu^{7} - 11\nu^{6} + 55\nu^{5} + 17\nu^{4} + 47\nu^{3} + \nu^{2} + 8\nu + 1 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{8} + 2\beta_{7} - \beta_{4} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{9} + 5\beta_{5} + \beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{8} - 8\beta_{7} + \beta_{6} - \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5\beta_{11} - \beta_{10} - 7\beta_{9} + \beta_{8} - \beta_{7} - 24\beta_{5} + \beta_{4} - 24\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{11} - 7\beta_{10} - 9\beta_{9} - 7\beta_{6} - 11\beta_{5} + 24\beta_{4} - \beta_{3} + 9\beta_{2} + 36 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -11\beta_{8} + 12\beta_{7} - 9\beta_{6} - 24\beta_{3} + 40\beta_{2} + 117\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 11 \beta_{11} + 40 \beta_{10} + 60 \beta_{9} - 117 \beta_{8} + 170 \beta_{7} + 85 \beta_{5} + \cdots - 170 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 117 \beta_{11} + 60 \beta_{10} + 217 \beta_{9} + 60 \beta_{6} + 581 \beta_{5} - 85 \beta_{4} + \cdots - 99 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 581\beta_{8} - 828\beta_{7} + 217\beta_{6} + 85\beta_{3} - 362\beta_{2} - 571\beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 581 \beta_{11} - 362 \beta_{10} - 1160 \beta_{9} + 571 \beta_{8} - 695 \beta_{7} - 2933 \beta_{5} + \cdots + 695 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(248\)
\(\chi(n)\) \(-1 + \beta_{7}\) \(-1 + \beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
−0.181721 + 0.314749i
1.16700 − 2.02131i
0.756174 − 1.30973i
−0.437442 + 0.757672i
−1.02197 + 1.77010i
0.217953 − 0.377506i
−0.181721 − 0.314749i
1.16700 + 2.02131i
0.756174 + 1.30973i
−0.437442 − 0.757672i
−1.02197 − 1.77010i
0.217953 + 0.377506i
−2.38804 −1.37574 − 2.38285i 3.70272 0.491140 + 0.850679i 3.28532 + 5.69033i 0 −4.06616 −2.28532 + 3.95828i −1.17286 − 2.03145i
165.2 −1.90556 0.214224 + 0.371047i 1.63116 −0.736565 − 1.27577i −0.408216 − 0.707051i 0 0.702849 1.40822 − 2.43910i 1.40357 + 2.43105i
165.3 −0.851125 0.330612 + 0.572636i −1.27559 1.72074 + 2.98041i −0.281392 − 0.487385i 0 2.78793 1.28139 − 2.21944i −1.46456 − 2.53670i
165.4 −0.268125 −0.571504 − 0.989875i −1.92811 −1.28088 − 2.21854i 0.153235 + 0.265410i 0 1.05323 0.846765 − 1.46664i 0.343436 + 0.594848i
165.5 1.55469 −0.244626 − 0.423704i 0.417051 −0.595756 − 1.03188i −0.380316 − 0.658727i 0 −2.46099 1.38032 − 2.39078i −0.926214 − 1.60425i
165.6 1.85816 1.14703 + 1.98672i 1.45276 −0.0986811 − 0.170921i 2.13137 + 3.69165i 0 −1.01686 −1.13137 + 1.95960i −0.183365 − 0.317598i
471.1 −2.38804 −1.37574 + 2.38285i 3.70272 0.491140 − 0.850679i 3.28532 − 5.69033i 0 −4.06616 −2.28532 − 3.95828i −1.17286 + 2.03145i
471.2 −1.90556 0.214224 − 0.371047i 1.63116 −0.736565 + 1.27577i −0.408216 + 0.707051i 0 0.702849 1.40822 + 2.43910i 1.40357 − 2.43105i
471.3 −0.851125 0.330612 − 0.572636i −1.27559 1.72074 − 2.98041i −0.281392 + 0.487385i 0 2.78793 1.28139 + 2.21944i −1.46456 + 2.53670i
471.4 −0.268125 −0.571504 + 0.989875i −1.92811 −1.28088 + 2.21854i 0.153235 − 0.265410i 0 1.05323 0.846765 + 1.46664i 0.343436 − 0.594848i
471.5 1.55469 −0.244626 + 0.423704i 0.417051 −0.595756 + 1.03188i −0.380316 + 0.658727i 0 −2.46099 1.38032 + 2.39078i −0.926214 + 1.60425i
471.6 1.85816 1.14703 − 1.98672i 1.45276 −0.0986811 + 0.170921i 2.13137 − 3.69165i 0 −1.01686 −1.13137 − 1.95960i −0.183365 + 0.317598i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 165.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.h.l 12
7.b odd 2 1 91.2.h.b yes 12
7.c even 3 1 637.2.f.j 12
7.c even 3 1 637.2.g.l 12
7.d odd 6 1 91.2.g.b ✓ 12
7.d odd 6 1 637.2.f.k 12
13.c even 3 1 637.2.g.l 12
21.c even 2 1 819.2.s.d 12
21.g even 6 1 819.2.n.d 12
91.g even 3 1 637.2.f.j 12
91.h even 3 1 inner 637.2.h.l 12
91.h even 3 1 8281.2.a.ca 6
91.k even 6 1 8281.2.a.cf 6
91.l odd 6 1 8281.2.a.ce 6
91.m odd 6 1 637.2.f.k 12
91.m odd 6 1 1183.2.e.h 12
91.n odd 6 1 91.2.g.b ✓ 12
91.n odd 6 1 1183.2.e.h 12
91.p odd 6 1 1183.2.e.g 12
91.t odd 6 1 1183.2.e.g 12
91.v odd 6 1 91.2.h.b yes 12
91.v odd 6 1 8281.2.a.bz 6
273.r even 6 1 819.2.s.d 12
273.bn even 6 1 819.2.n.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.g.b ✓ 12 7.d odd 6 1
91.2.g.b ✓ 12 91.n odd 6 1
91.2.h.b yes 12 7.b odd 2 1
91.2.h.b yes 12 91.v odd 6 1
637.2.f.j 12 7.c even 3 1
637.2.f.j 12 91.g even 3 1
637.2.f.k 12 7.d odd 6 1
637.2.f.k 12 91.m odd 6 1
637.2.g.l 12 7.c even 3 1
637.2.g.l 12 13.c even 3 1
637.2.h.l 12 1.a even 1 1 trivial
637.2.h.l 12 91.h even 3 1 inner
819.2.n.d 12 21.g even 6 1
819.2.n.d 12 273.bn even 6 1
819.2.s.d 12 21.c even 2 1
819.2.s.d 12 273.r even 6 1
1183.2.e.g 12 91.p odd 6 1
1183.2.e.g 12 91.t odd 6 1
1183.2.e.h 12 91.m odd 6 1
1183.2.e.h 12 91.n odd 6 1
8281.2.a.bz 6 91.v odd 6 1
8281.2.a.ca 6 91.h even 3 1
8281.2.a.ce 6 91.l odd 6 1
8281.2.a.cf 6 91.k even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\):

\( T_{2}^{6} + 2T_{2}^{5} - 6T_{2}^{4} - 11T_{2}^{3} + 8T_{2}^{2} + 14T_{2} + 3 \) Copy content Toggle raw display
\( T_{3}^{12} + T_{3}^{11} + 8 T_{3}^{10} + T_{3}^{9} + 47 T_{3}^{8} + 17 T_{3}^{7} + 55 T_{3}^{6} + \cdots + 1 \) Copy content Toggle raw display
\( T_{5}^{12} + T_{5}^{11} + 12 T_{5}^{10} + 25 T_{5}^{9} + 133 T_{5}^{8} + 203 T_{5}^{7} + 379 T_{5}^{6} + \cdots + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 2 T^{5} - 6 T^{4} + \cdots + 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{12} + T^{11} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - 4 T^{11} + \cdots + 6561 \) Copy content Toggle raw display
$13$ \( T^{12} - 2 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( (T^{6} - 5 T^{5} - 12 T^{4} + \cdots - 9)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - T^{11} + \cdots + 762129 \) Copy content Toggle raw display
$23$ \( (T^{6} - T^{5} + \cdots - 24387)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} - 3 T^{11} + \cdots + 40401 \) Copy content Toggle raw display
$31$ \( T^{12} + 16 T^{11} + \cdots + 6135529 \) Copy content Toggle raw display
$37$ \( (T^{6} - 13 T^{5} + \cdots - 13477)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 8 T^{11} + \cdots + 4173849 \) Copy content Toggle raw display
$43$ \( T^{12} + 11 T^{11} + \cdots + 1369 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 318515409 \) Copy content Toggle raw display
$53$ \( T^{12} + 2 T^{11} + \cdots + 4761 \) Copy content Toggle raw display
$59$ \( (T^{6} - 13 T^{5} + \cdots + 9123)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 1055015361 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 276324129 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 530979849 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 196812841 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 110859841 \) Copy content Toggle raw display
$83$ \( (T^{6} - 27 T^{5} + \cdots + 2673)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 4 T^{5} + \cdots - 304479)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} - 35 T^{11} + \cdots + 15202201 \) Copy content Toggle raw display
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