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Properties

Label 879.2.a.c
Level $879$
Weight $2$
Character orbit 879.a
Self dual yes
Analytic conductor $7.019$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [879,2,Mod(1,879)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(879, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("879.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 879 = 3 \cdot 293 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 879.a (trivial)

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: yes
Analytic conductor: \(7.01885033767\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 6 x^{16} - 9 x^{15} + 112 x^{14} - 52 x^{13} - 800 x^{12} + 920 x^{11} + 2743 x^{10} - 4265 x^{9} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{6} + 1) q^{5} + \beta_1 q^{6} - \beta_{8} q^{7} + (\beta_{3} + \beta_1 + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{6} + 1) q^{5} + \beta_1 q^{6} - \beta_{8} q^{7} + (\beta_{3} + \beta_1 + 1) q^{8} + q^{9} + (\beta_{15} - \beta_{12} - \beta_{10} + \cdots - 1) q^{10}+ \cdots + (\beta_{11} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 6 q^{2} + 17 q^{3} + 20 q^{4} + 12 q^{5} + 6 q^{6} - q^{7} + 18 q^{8} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q + 6 q^{2} + 17 q^{3} + 20 q^{4} + 12 q^{5} + 6 q^{6} - q^{7} + 18 q^{8} + 17 q^{9} + 3 q^{10} + 19 q^{11} + 20 q^{12} + q^{13} + 9 q^{14} + 12 q^{15} + 18 q^{16} + 12 q^{17} + 6 q^{18} + 13 q^{20} - q^{21} - 9 q^{22} + 18 q^{23} + 18 q^{24} + 25 q^{25} + 8 q^{26} + 17 q^{27} - 23 q^{28} + 10 q^{29} + 3 q^{30} + 14 q^{31} + 22 q^{32} + 19 q^{33} - 12 q^{34} + 15 q^{35} + 20 q^{36} - 11 q^{37} - 17 q^{38} + q^{39} - 16 q^{40} + 44 q^{41} + 9 q^{42} - 27 q^{43} + 15 q^{44} + 12 q^{45} - 26 q^{46} + 36 q^{47} + 18 q^{48} + 20 q^{49} - 3 q^{50} + 12 q^{51} - 40 q^{52} + 13 q^{53} + 6 q^{54} - 19 q^{55} - 10 q^{56} - 47 q^{58} + 39 q^{59} + 13 q^{60} - 6 q^{61} + 6 q^{62} - q^{63} - 14 q^{64} + 16 q^{65} - 9 q^{66} - 32 q^{67} - 16 q^{68} + 18 q^{69} - 69 q^{70} + 25 q^{71} + 18 q^{72} + 8 q^{73} - 14 q^{74} + 25 q^{75} - 34 q^{76} - 4 q^{77} + 8 q^{78} - 14 q^{79} - 3 q^{80} + 17 q^{81} + 5 q^{82} + 27 q^{83} - 23 q^{84} - 19 q^{85} - 21 q^{86} + 10 q^{87} - 58 q^{88} + 55 q^{89} + 3 q^{90} - 18 q^{91} + 45 q^{92} + 14 q^{93} - 12 q^{94} - 7 q^{95} + 22 q^{96} + 37 q^{97} - 21 q^{98} + 19 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 6 x^{16} - 9 x^{15} + 112 x^{14} - 52 x^{13} - 800 x^{12} + 920 x^{11} + 2743 x^{10} - 4265 x^{9} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 22845 \nu^{16} + 174198 \nu^{15} - 81912 \nu^{14} - 2762162 \nu^{13} + 6853836 \nu^{12} + \cdots + 9019288 ) / 1243241 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 758005 \nu^{16} + 4723644 \nu^{15} + 6493380 \nu^{14} - 89430214 \nu^{13} + 44247250 \nu^{12} + \cdots + 36832035 ) / 26108061 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 797140 \nu^{16} + 7128138 \nu^{15} - 2852475 \nu^{14} - 126510736 \nu^{13} + 235000336 \nu^{12} + \cdots + 145566564 ) / 26108061 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1762423 \nu^{16} - 11980431 \nu^{15} - 9755850 \nu^{14} + 217625404 \nu^{13} - 203685544 \nu^{12} + \cdots + 55398006 ) / 26108061 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1896427 \nu^{16} - 7360881 \nu^{15} - 33192618 \nu^{14} + 145364746 \nu^{13} + 215795849 \nu^{12} + \cdots + 31740174 ) / 26108061 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2287738 \nu^{16} - 8991921 \nu^{15} - 36500316 \nu^{14} + 166102870 \nu^{13} + 191061431 \nu^{12} + \cdots - 26989257 ) / 26108061 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2462536 \nu^{16} - 4272993 \nu^{15} - 63394173 \nu^{14} + 98469103 \nu^{13} + 667046027 \nu^{12} + \cdots + 191438817 ) / 26108061 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2781616 \nu^{16} + 11479014 \nu^{15} + 45364710 \nu^{14} - 219953191 \nu^{13} + \cdots + 73538109 ) / 26108061 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3489568 \nu^{16} - 18378984 \nu^{15} - 47276508 \nu^{14} + 364139401 \nu^{13} + 128431787 \nu^{12} + \cdots - 122306325 ) / 26108061 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 3875258 \nu^{16} + 19570599 \nu^{15} + 50939211 \nu^{14} - 372448031 \nu^{13} - 122005141 \nu^{12} + \cdots + 3358662 ) / 26108061 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1536240 \nu^{16} + 5133062 \nu^{15} + 29753895 \nu^{14} - 101041316 \nu^{13} - 231492685 \nu^{12} + \cdots - 53091181 ) / 8702687 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1614852 \nu^{16} - 7002348 \nu^{15} - 26070951 \nu^{14} + 138831688 \nu^{13} + 137504244 \nu^{12} + \cdots - 54759617 ) / 8702687 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 9265318 \nu^{16} + 39836262 \nu^{15} + 147199167 \nu^{14} - 774246604 \nu^{13} + \cdots + 125723385 ) / 26108061 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{16} + \beta_{12} + \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{6} - 3 \beta_{5} + \cdots + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{16} + \beta_{15} + 2 \beta_{14} - 2 \beta_{13} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + \cdots + 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11 \beta_{16} + \beta_{15} + \beta_{14} - 2 \beta_{13} + 10 \beta_{12} + 10 \beta_{10} + 12 \beta_{9} + \cdots + 85 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13 \beta_{16} + 14 \beta_{15} + 24 \beta_{14} - 24 \beta_{13} - \beta_{12} - 13 \beta_{11} + 12 \beta_{10} + \cdots + 43 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 95 \beta_{16} + 16 \beta_{15} + 16 \beta_{14} - 29 \beta_{13} + 78 \beta_{12} - 3 \beta_{11} + \cdots + 512 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 126 \beta_{16} + 139 \beta_{15} + 213 \beta_{14} - 214 \beta_{13} - 14 \beta_{12} - 122 \beta_{11} + \cdots + 267 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 750 \beta_{16} + 180 \beta_{15} + 180 \beta_{14} - 296 \beta_{13} + 553 \beta_{12} - 61 \beta_{11} + \cdots + 3195 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1089 \beta_{16} + 1207 \beta_{15} + 1702 \beta_{14} - 1716 \beta_{13} - 138 \beta_{12} - 1019 \beta_{11} + \cdots + 1741 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 5665 \beta_{16} + 1739 \beta_{15} + 1751 \beta_{14} - 2630 \beta_{13} + 3732 \beta_{12} - 793 \beta_{11} + \cdots + 20403 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 8867 \beta_{16} + 9793 \beta_{15} + 13008 \beta_{14} - 13099 \beta_{13} - 1187 \beta_{12} - 8090 \beta_{11} + \cdots + 11964 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 41747 \beta_{16} + 15429 \beta_{15} + 15752 \beta_{14} - 21767 \beta_{13} + 24445 \beta_{12} + \cdots + 132483 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 69653 \beta_{16} + 76400 \beta_{15} + 97462 \beta_{14} - 97418 \beta_{13} - 9579 \beta_{12} + \cdots + 85730 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 303158 \beta_{16} + 129752 \beta_{15} + 135087 \beta_{14} - 172849 \beta_{13} + 156825 \beta_{12} + \cdots + 871566 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.52316
−2.11476
−1.85943
−1.57486
−1.35929
−0.515578
−0.342023
0.0669141
0.223365
1.01260
1.06087
1.96377
2.05242
2.19462
2.37779
2.60253
2.73421
−2.52316 1.00000 4.36632 2.75995 −2.52316 −0.718103 −5.97060 1.00000 −6.96380
1.2 −2.11476 1.00000 2.47223 −2.79285 −2.11476 −5.13552 −0.998650 1.00000 5.90622
1.3 −1.85943 1.00000 1.45747 3.18519 −1.85943 0.989995 1.00879 1.00000 −5.92263
1.4 −1.57486 1.00000 0.480168 0.803286 −1.57486 2.46086 2.39351 1.00000 −1.26506
1.5 −1.35929 1.00000 −0.152323 −0.294299 −1.35929 −2.60136 2.92564 1.00000 0.400038
1.6 −0.515578 1.00000 −1.73418 −1.69773 −0.515578 2.71275 1.92526 1.00000 0.875311
1.7 −0.342023 1.00000 −1.88302 3.68628 −0.342023 4.34086 1.32808 1.00000 −1.26079
1.8 0.0669141 1.00000 −1.99552 −3.60766 0.0669141 −2.76371 −0.267357 1.00000 −0.241403
1.9 0.223365 1.00000 −1.95011 3.80344 0.223365 −3.94047 −0.882315 1.00000 0.849555
1.10 1.01260 1.00000 −0.974637 1.27618 1.01260 2.71183 −3.01212 1.00000 1.29226
1.11 1.06087 1.00000 −0.874548 0.0171292 1.06087 0.756371 −3.04953 1.00000 0.0181719
1.12 1.96377 1.00000 1.85638 4.33637 1.96377 −1.72011 −0.282037 1.00000 8.51561
1.13 2.05242 1.00000 2.21245 2.20576 2.05242 0.133348 0.436030 1.00000 4.52717
1.14 2.19462 1.00000 2.81637 0.619808 2.19462 4.89185 1.79163 1.00000 1.36025
1.15 2.37779 1.00000 3.65390 −2.82870 2.37779 0.523685 3.93263 1.00000 −6.72606
1.16 2.60253 1.00000 4.77316 −1.45796 2.60253 0.0762303 7.21723 1.00000 −3.79437
1.17 2.73421 1.00000 5.47589 1.98578 2.73421 −3.71851 9.50380 1.00000 5.42953
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(293\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 879.2.a.c ✓ 17
3.b odd 2 1 2637.2.a.e 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
879.2.a.c ✓ 17 1.a even 1 1 trivial
2637.2.a.e 17 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{17} - 6 T_{2}^{16} - 9 T_{2}^{15} + 112 T_{2}^{14} - 52 T_{2}^{13} - 800 T_{2}^{12} + 920 T_{2}^{11} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(879))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} - 6 T^{16} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( (T - 1)^{17} \) Copy content Toggle raw display
$5$ \( T^{17} - 12 T^{16} + \cdots - 529 \) Copy content Toggle raw display
$7$ \( T^{17} + T^{16} + \cdots + 1024 \) Copy content Toggle raw display
$11$ \( T^{17} - 19 T^{16} + \cdots + 10431769 \) Copy content Toggle raw display
$13$ \( T^{17} - T^{16} + \cdots + 78848 \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots - 5523731456 \) Copy content Toggle raw display
$19$ \( T^{17} - 193 T^{15} + \cdots + 4115456 \) Copy content Toggle raw display
$23$ \( T^{17} + \cdots - 327761089 \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots - 1060701282593 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots - 16637132133 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots - 604532421487 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots - 199185388694803 \) Copy content Toggle raw display
$43$ \( T^{17} + 27 T^{16} + \cdots - 74971719 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots + 10777782310571 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots + 40750263296 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots - 781700096 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots - 140668271 \) Copy content Toggle raw display
$67$ \( T^{17} + 32 T^{16} + \cdots + 212087 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots + 32904383562752 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots + 11155956046911 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots + 29698177143808 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots + 152479460424704 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots - 682552498599 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots + 11537448183359 \) Copy content Toggle raw display
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