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Properties

Label 637.2.e.n
Level $637$
Weight $2$
Character orbit 637.e
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(79,637)]
 
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([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.79");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.e (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: \( x^{12} - 2 x^{11} + 9 x^{10} - 6 x^{9} + 34 x^{8} - 18 x^{7} + 85 x^{6} - 2 x^{5} + 92 x^{4} - 26 x^{3} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{11} - \beta_{3}) q^{2} + ( - \beta_{8} + \beta_{2} + \beta_1) q^{3} + (\beta_{9} - \beta_{8} - \beta_{7} + \cdots - \beta_1) q^{4}+ \cdots + ( - \beta_{11} + \beta_{3} - 3 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{11} - \beta_{3}) q^{2} + ( - \beta_{8} + \beta_{2} + \beta_1) q^{3} + (\beta_{9} - \beta_{8} - \beta_{7} + \cdots - \beta_1) q^{4}+ \cdots + (\beta_{6} + \beta_{5} + \cdots - 5 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{3} - 4 q^{4} - 6 q^{5} + 8 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{3} - 4 q^{4} - 6 q^{5} + 8 q^{6} - 6 q^{9} - 4 q^{10} - 4 q^{11} + 4 q^{12} + 12 q^{13} + 24 q^{15} - 16 q^{17} + 4 q^{18} - 2 q^{19} + 32 q^{20} - 24 q^{22} + 6 q^{23} - 12 q^{24} + 4 q^{25} + 40 q^{27} - 12 q^{29} - 6 q^{31} + 20 q^{32} - 4 q^{33} - 48 q^{36} - 8 q^{38} - 8 q^{39} - 4 q^{40} - 16 q^{41} + 4 q^{43} + 4 q^{44} - 14 q^{45} - 8 q^{46} - 30 q^{47} - 16 q^{48} + 16 q^{50} + 4 q^{51} - 4 q^{52} + 14 q^{53} + 48 q^{54} - 16 q^{55} + 8 q^{57} + 8 q^{58} - 24 q^{59} - 12 q^{60} + 56 q^{62} - 40 q^{64} - 6 q^{65} + 4 q^{66} - 16 q^{67} - 28 q^{68} - 40 q^{69} + 16 q^{71} - 28 q^{72} + 6 q^{73} + 12 q^{74} - 12 q^{75} - 32 q^{76} + 8 q^{78} + 22 q^{79} + 28 q^{80} - 46 q^{81} + 40 q^{82} + 100 q^{83} - 16 q^{85} + 16 q^{86} + 16 q^{87} + 44 q^{88} - 26 q^{89} - 80 q^{90} + 40 q^{92} - 16 q^{93} + 32 q^{94} + 6 q^{95} + 20 q^{96} - 28 q^{97} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 9 x^{10} - 6 x^{9} + 34 x^{8} - 18 x^{7} + 85 x^{6} - 2 x^{5} + 92 x^{4} - 26 x^{3} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 512378 \nu^{11} + 5336 \nu^{10} + 3065721 \nu^{9} + 4369060 \nu^{8} + 17395862 \nu^{7} + \cdots + 9713364 ) / 62842357 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 517714 \nu^{11} - 1551017 \nu^{10} + 4377607 \nu^{9} - 4394050 \nu^{8} + 8967106 \nu^{7} + \cdots + 115458972 ) / 62842357 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2297118 \nu^{11} - 4668361 \nu^{10} + 18068728 \nu^{9} - 6968746 \nu^{8} + 52032662 \nu^{7} + \cdots + 103693971 ) / 62842357 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1878 \nu^{11} + 1119 \nu^{10} - 12173 \nu^{9} - 9570 \nu^{8} - 57462 \nu^{7} - 38216 \nu^{6} + \cdots - 66913 ) / 45971 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6705029 \nu^{11} - 7815976 \nu^{10} + 48698198 \nu^{9} + 12543658 \nu^{8} + 184020371 \nu^{7} + \cdots + 159636402 ) / 62842357 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 9197775 \nu^{11} + 20220925 \nu^{10} - 86127174 \nu^{9} + 69981075 \nu^{8} - 318884784 \nu^{7} + \cdots + 10820061 ) / 62842357 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9713364 \nu^{11} - 19939106 \nu^{10} + 87414940 \nu^{9} - 61345905 \nu^{8} + 325885316 \nu^{7} + \cdots + 52540010 ) / 62842357 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 14791857 \nu^{11} + 31867988 \nu^{10} - 138901006 \nu^{9} + 113931690 \nu^{8} + \cdots + 17525090 ) / 62842357 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15232235 \nu^{11} - 36300822 \nu^{10} + 151870530 \nu^{9} - 148661200 \nu^{8} + 577528481 \nu^{7} + \cdots - 20852300 ) / 62842357 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 18914350 \nu^{11} - 39883548 \nu^{10} + 171764159 \nu^{9} - 127060870 \nu^{8} + 634374770 \nu^{7} + \cdots + 95366656 ) / 62842357 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - 2\beta_{8} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{3} + 4\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{11} - \beta_{9} + 7\beta_{8} - \beta_{7} + 5\beta_{3} - 6\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 8 \beta_{11} - \beta_{10} - 2 \beta_{9} + 8 \beta_{8} - 6 \beta_{7} - \beta_{6} - 6 \beta_{5} + \cdots - 18 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{6} - 9\beta_{5} + 9\beta_{4} - 26\beta_{3} - 33\beta_{2} + 30 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 51\beta_{11} + 9\beta_{10} + 20\beta_{9} - 49\beta_{8} + 33\beta_{7} - 51\beta_{3} + 87\beta _1 + 49 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 140 \beta_{11} + 20 \beta_{10} + 62 \beta_{9} - 143 \beta_{8} + 62 \beta_{7} + 20 \beta_{6} + \cdots + 178 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 62\beta_{6} + 182\beta_{5} - 144\beta_{4} + 302\beta_{3} + 441\beta_{2} - 278 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -767\beta_{11} - 144\beta_{10} - 388\beta_{9} + 724\beta_{8} - 384\beta_{7} + 767\beta_{3} - 959\beta _1 - 724 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1731 \beta_{11} - 388 \beta_{10} - 916 \beta_{9} + 1539 \beta_{8} - 1011 \beta_{7} + \cdots - 2306 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(197\) \(248\)
\(\chi(n)\) \(1\) \(-1 + \beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.379209 − 0.656810i
−0.0731214 + 0.126650i
0.954516 − 1.65327i
−0.602377 + 1.04335i
1.17550 − 2.03602i
−0.833726 + 1.44406i
0.379209 + 0.656810i
−0.0731214 − 0.126650i
0.954516 + 1.65327i
−0.602377 − 1.04335i
1.17550 + 2.03602i
−0.833726 − 1.44406i
−1.09161 + 1.89072i −0.879209 − 1.52284i −1.38322 − 2.39581i −1.05533 + 1.82788i 3.83901 0 1.67333 −0.0460183 + 0.0797060i −2.30401 − 3.99066i
79.2 −0.916185 + 1.58688i −0.426879 − 0.739375i −0.678791 − 1.17570i −1.31278 + 2.27379i 1.56440 0 −1.17715 1.13555 − 1.96683i −2.40549 − 4.16643i
79.3 −0.132313 + 0.229173i −1.45452 − 2.51930i 0.964986 + 1.67141i 0.717577 − 1.24288i 0.769807 0 −1.03998 −2.73124 + 4.73064i 0.189890 + 0.328899i
79.4 0.328092 − 0.568272i 0.102377 + 0.177322i 0.784711 + 1.35916i 0.679981 − 1.17776i 0.134356 0 2.34220 1.47904 − 2.56177i −0.446193 − 0.772828i
79.5 0.588093 − 1.01861i −1.67550 − 2.90205i 0.308293 + 0.533979i −1.57431 + 2.72679i −3.94140 0 3.07759 −4.11459 + 7.12667i 1.85168 + 3.20721i
79.6 1.22392 − 2.11990i 0.333726 + 0.578030i −1.99598 − 3.45713i −0.455143 + 0.788331i 1.63382 0 −4.87599 1.27725 − 2.21227i 1.11412 + 1.92971i
508.1 −1.09161 − 1.89072i −0.879209 + 1.52284i −1.38322 + 2.39581i −1.05533 − 1.82788i 3.83901 0 1.67333 −0.0460183 − 0.0797060i −2.30401 + 3.99066i
508.2 −0.916185 − 1.58688i −0.426879 + 0.739375i −0.678791 + 1.17570i −1.31278 − 2.27379i 1.56440 0 −1.17715 1.13555 + 1.96683i −2.40549 + 4.16643i
508.3 −0.132313 − 0.229173i −1.45452 + 2.51930i 0.964986 − 1.67141i 0.717577 + 1.24288i 0.769807 0 −1.03998 −2.73124 − 4.73064i 0.189890 − 0.328899i
508.4 0.328092 + 0.568272i 0.102377 − 0.177322i 0.784711 − 1.35916i 0.679981 + 1.17776i 0.134356 0 2.34220 1.47904 + 2.56177i −0.446193 + 0.772828i
508.5 0.588093 + 1.01861i −1.67550 + 2.90205i 0.308293 − 0.533979i −1.57431 − 2.72679i −3.94140 0 3.07759 −4.11459 − 7.12667i 1.85168 − 3.20721i
508.6 1.22392 + 2.11990i 0.333726 − 0.578030i −1.99598 + 3.45713i −0.455143 − 0.788331i 1.63382 0 −4.87599 1.27725 + 2.21227i 1.11412 − 1.92971i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c 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.e.n 12
7.b odd 2 1 637.2.e.o 12
7.c even 3 1 637.2.a.n yes 6
7.c even 3 1 inner 637.2.e.n 12
7.d odd 6 1 637.2.a.m ✓ 6
7.d odd 6 1 637.2.e.o 12
21.g even 6 1 5733.2.a.bu 6
21.h odd 6 1 5733.2.a.br 6
91.r even 6 1 8281.2.a.cd 6
91.s odd 6 1 8281.2.a.cc 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.a.m ✓ 6 7.d odd 6 1
637.2.a.n yes 6 7.c even 3 1
637.2.e.n 12 1.a even 1 1 trivial
637.2.e.n 12 7.c even 3 1 inner
637.2.e.o 12 7.b odd 2 1
637.2.e.o 12 7.d odd 6 1
5733.2.a.br 6 21.h odd 6 1
5733.2.a.bu 6 21.g even 6 1
8281.2.a.cc 6 91.s odd 6 1
8281.2.a.cd 6 91.r even 6 1

Hecke kernels

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

\( T_{2}^{12} + 8T_{2}^{10} + 50T_{2}^{8} - 4T_{2}^{7} + 108T_{2}^{6} - 64T_{2}^{5} + 180T_{2}^{4} - 56T_{2}^{3} + 44T_{2}^{2} + 8T_{2} + 4 \) Copy content Toggle raw display
\( T_{3}^{12} + 8 T_{3}^{11} + 44 T_{3}^{10} + 136 T_{3}^{9} + 316 T_{3}^{8} + 424 T_{3}^{7} + 452 T_{3}^{6} + \cdots + 4 \) Copy content Toggle raw display
\( T_{5}^{12} + 6 T_{5}^{11} + 31 T_{5}^{10} + 78 T_{5}^{9} + 200 T_{5}^{8} + 278 T_{5}^{7} + 637 T_{5}^{6} + \cdots + 961 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 8 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{12} + 8 T^{11} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{12} + 6 T^{11} + \cdots + 961 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 4 T^{11} + \cdots + 315844 \) Copy content Toggle raw display
$13$ \( (T - 1)^{12} \) Copy content Toggle raw display
$17$ \( T^{12} + 16 T^{11} + \cdots + 28026436 \) Copy content Toggle raw display
$19$ \( T^{12} + 2 T^{11} + \cdots + 5329 \) Copy content Toggle raw display
$23$ \( T^{12} - 6 T^{11} + \cdots + 279841 \) Copy content Toggle raw display
$29$ \( (T^{6} + 6 T^{5} + \cdots + 529)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 1957974001 \) Copy content Toggle raw display
$37$ \( T^{12} + 82 T^{10} + \cdots + 64516 \) Copy content Toggle raw display
$41$ \( (T^{6} + 8 T^{5} + \cdots + 28784)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 2 T^{5} + \cdots + 35153)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 18391970689 \) Copy content Toggle raw display
$53$ \( T^{12} - 14 T^{11} + \cdots + 1739761 \) Copy content Toggle raw display
$59$ \( T^{12} + 24 T^{11} + \cdots + 2347024 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 46908629056 \) Copy content Toggle raw display
$67$ \( T^{12} + 16 T^{11} + \cdots + 37356544 \) Copy content Toggle raw display
$71$ \( (T^{6} - 8 T^{5} + \cdots - 1206162)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 20351019649 \) Copy content Toggle raw display
$79$ \( T^{12} - 22 T^{11} + \cdots + 62615569 \) Copy content Toggle raw display
$83$ \( (T^{6} - 50 T^{5} + \cdots - 167041)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 26 T^{11} + \cdots + 99181681 \) Copy content Toggle raw display
$97$ \( (T^{6} + 14 T^{5} + \cdots + 217287)^{2} \) Copy content Toggle raw display
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