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Properties

Label 623.2.a.f
Level $623$
Weight $2$
Character orbit 623.a
Self dual yes
Analytic conductor $4.975$
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] = [623,2,Mod(1,623)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(623, 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("623.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 623 = 7 \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 623.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: \(4.97468004593\)
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: \( x^{17} - 3 x^{16} - 24 x^{15} + 74 x^{14} + 224 x^{13} - 719 x^{12} - 1025 x^{11} + 3510 x^{10} + \cdots + 21 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + 1) q^{4} - \beta_{11} q^{5} + \beta_{6} q^{6} - q^{7} + ( - \beta_{10} + \beta_{9} + \cdots + \beta_1) q^{8}+ \cdots + ( - \beta_{16} + \beta_{14} + \beta_{5} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + 1) q^{4} - \beta_{11} q^{5} + \beta_{6} q^{6} - q^{7} + ( - \beta_{10} + \beta_{9} + \cdots + \beta_1) q^{8}+ \cdots + ( - \beta_{15} - \beta_{14} - 3 \beta_{13} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 3 q^{2} + 4 q^{3} + 23 q^{4} + 6 q^{5} - 2 q^{6} - 17 q^{7} + 9 q^{8} + 31 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q + 3 q^{2} + 4 q^{3} + 23 q^{4} + 6 q^{5} - 2 q^{6} - 17 q^{7} + 9 q^{8} + 31 q^{9} + 4 q^{10} + 5 q^{11} + 11 q^{12} + 7 q^{13} - 3 q^{14} + 11 q^{15} + 39 q^{16} + 20 q^{17} + 12 q^{18} - 11 q^{19} + 26 q^{20} - 4 q^{21} + 31 q^{22} + 10 q^{23} - 16 q^{24} + 61 q^{25} - 2 q^{26} + 10 q^{27} - 23 q^{28} + 11 q^{29} - 30 q^{30} - 25 q^{31} + 16 q^{32} + 24 q^{33} - 16 q^{34} - 6 q^{35} + q^{36} + 31 q^{37} - 21 q^{38} + 15 q^{39} + 23 q^{40} - 5 q^{41} + 2 q^{42} + 28 q^{43} - 30 q^{44} + 27 q^{45} + 26 q^{46} + 7 q^{47} - 2 q^{48} + 17 q^{49} - 50 q^{50} - 7 q^{51} - 17 q^{52} + 22 q^{53} - 25 q^{54} - 37 q^{55} - 9 q^{56} + 28 q^{57} + 24 q^{58} - 18 q^{59} + 35 q^{60} - 40 q^{62} - 31 q^{63} + 43 q^{64} - 23 q^{65} - 41 q^{66} + 19 q^{67} + 41 q^{68} - 26 q^{69} - 4 q^{70} + 8 q^{71} - 33 q^{72} + 55 q^{73} + 26 q^{74} + 31 q^{75} - 64 q^{76} - 5 q^{77} - 22 q^{78} + 13 q^{79} + 5 q^{80} + 73 q^{81} + 38 q^{82} - 5 q^{83} - 11 q^{84} + 35 q^{85} - 69 q^{86} - 8 q^{87} + 124 q^{88} + 17 q^{89} - 162 q^{90} - 7 q^{91} - 67 q^{92} - 59 q^{94} + 16 q^{95} - 44 q^{96} + 75 q^{97} + 3 q^{98} - 34 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 3 x^{16} - 24 x^{15} + 74 x^{14} + 224 x^{13} - 719 x^{12} - 1025 x^{11} + 3510 x^{10} + \cdots + 21 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 9452729 \nu^{16} + 200496596 \nu^{15} - 225020120 \nu^{14} - 4578778266 \nu^{13} + \cdots + 7709098651 ) / 5561384420 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 9365745 \nu^{16} + 46102944 \nu^{15} + 208114348 \nu^{14} - 1053479970 \nu^{13} + \cdots + 4141978679 ) / 1112276884 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 88247089 \nu^{16} + 504558116 \nu^{15} + 2075324560 \nu^{14} - 12496586066 \nu^{13} + \cdots - 601096769 ) / 5561384420 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 239816849 \nu^{16} - 42605576 \nu^{15} - 5966301480 \nu^{14} + 157121786 \nu^{13} + \cdots + 1853188869 ) / 5561384420 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 29217953 \nu^{16} + 146075128 \nu^{15} + 708232540 \nu^{14} - 3700567990 \nu^{13} + \cdots + 7352112549 ) / 556138442 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 452873067 \nu^{16} + 1202704608 \nu^{15} + 11256622360 \nu^{14} - 29765754818 \nu^{13} + \cdots + 34094341613 ) / 5561384420 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 269383733 \nu^{16} + 683538562 \nu^{15} + 6703035590 \nu^{14} - 16892378882 \nu^{13} + \cdots + 38247766967 ) / 2780692210 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 269383733 \nu^{16} + 683538562 \nu^{15} + 6703035590 \nu^{14} - 16892378882 \nu^{13} + \cdots + 29905690337 ) / 2780692210 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 615166227 \nu^{16} - 1712839288 \nu^{15} - 15101751220 \nu^{14} + 42039574278 \nu^{13} + \cdots - 61641538513 ) / 5561384420 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 79034543 \nu^{16} + 218291472 \nu^{15} + 1941598164 \nu^{14} - 5337878246 \nu^{13} + \cdots + 5593013047 ) / 556138442 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 415547007 \nu^{16} - 988903888 \nu^{15} - 10222302610 \nu^{14} + 24089754868 \nu^{13} + \cdots - 50736995683 ) / 2780692210 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 108426311 \nu^{16} - 297976076 \nu^{15} - 2649079776 \nu^{14} + 7300606344 \nu^{13} + \cdots - 11939534493 ) / 556138442 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 309186459 \nu^{16} + 802185831 \nu^{15} + 7600372065 \nu^{14} - 19769937636 \nu^{13} + \cdots + 38968656716 ) / 1390346105 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 653388483 \nu^{16} - 1569824442 \nu^{15} - 16207194580 \nu^{14} + 38631871782 \nu^{13} + \cdots - 66731596947 ) / 2780692210 \) 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
\(\nu^{3}\)\(=\) \( -\beta_{10} + \beta_{9} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{14} + \beta_{11} - \beta_{10} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{3} + 7\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{16} + 2 \beta_{15} + \beta_{13} + \beta_{11} - 10 \beta_{10} + 9 \beta_{9} + \beta_{8} + \cdots + 30 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{16} + 2 \beta_{15} - 11 \beta_{14} + \beta_{13} + \beta_{12} + 10 \beta_{11} - 11 \beta_{10} + \cdots + 98 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15 \beta_{16} + 27 \beta_{15} - \beta_{14} + 12 \beta_{13} - 2 \beta_{12} + 11 \beta_{11} - 82 \beta_{10} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 33 \beta_{16} + 32 \beta_{15} - 97 \beta_{14} + 12 \beta_{13} + 14 \beta_{12} + 82 \beta_{11} + \cdots + 640 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 155 \beta_{16} + 264 \beta_{15} - 17 \beta_{14} + 110 \beta_{13} - 35 \beta_{12} + 88 \beta_{11} + \cdots - 63 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 373 \beta_{16} + 352 \beta_{15} - 796 \beta_{14} + 105 \beta_{13} + 144 \beta_{12} + 642 \beta_{11} + \cdots + 4328 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1388 \beta_{16} + 2283 \beta_{15} - 192 \beta_{14} + 915 \beta_{13} - 410 \beta_{12} + 625 \beta_{11} + \cdots - 684 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3597 \beta_{16} + 3300 \beta_{15} - 6326 \beta_{14} + 819 \beta_{13} + 1314 \beta_{12} + 4943 \beta_{11} + \cdots + 29893 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 11586 \beta_{16} + 18567 \beta_{15} - 1836 \beta_{14} + 7258 \beta_{13} - 4060 \beta_{12} + 4172 \beta_{11} + \cdots - 6365 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 31868 \beta_{16} + 28372 \beta_{15} - 49404 \beta_{14} + 6047 \beta_{13} + 11263 \beta_{12} + \cdots + 209376 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 92991 \beta_{16} + 145747 \beta_{15} - 16138 \beta_{14} + 56036 \beta_{13} - 36775 \beta_{12} + \cdots - 54539 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 268333 \beta_{16} + 231471 \beta_{15} - 381610 \beta_{14} + 43262 \beta_{13} + 92932 \beta_{12} + \cdots + 1481211 \) 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
−2.71727
−2.54137
−1.96860
−1.82906
−1.38321
−1.22109
−0.451888
0.111515
0.233127
0.424739
0.895426
1.57801
1.73103
2.37485
2.39270
2.64245
2.72865
−2.71727 −0.850513 5.38356 −1.80088 2.31108 −1.00000 −9.19406 −2.27663 4.89349
1.2 −2.54137 3.21298 4.45856 4.36716 −8.16536 −1.00000 −6.24811 7.32322 −11.0986
1.3 −1.96860 −0.0161481 1.87539 −4.23451 0.0317893 −1.00000 0.245308 −2.99974 8.33606
1.4 −1.82906 0.577438 1.34547 1.04992 −1.05617 −1.00000 1.19718 −2.66657 −1.92037
1.5 −1.38321 −3.24014 −0.0867371 3.00833 4.48179 −1.00000 2.88639 7.49853 −4.16114
1.6 −1.22109 1.42812 −0.508951 3.63601 −1.74385 −1.00000 3.06364 −0.960483 −4.43988
1.7 −0.451888 2.77173 −1.79580 −4.34591 −1.25251 −1.00000 1.71528 4.68251 1.96386
1.8 0.111515 −1.76626 −1.98756 2.39924 −0.196964 −1.00000 −0.444673 0.119661 0.267552
1.9 0.233127 −1.50513 −1.94565 −2.87372 −0.350887 −1.00000 −0.919837 −0.734570 −0.669942
1.10 0.424739 3.03850 −1.81960 1.41506 1.29057 −1.00000 −1.62233 6.23249 0.601032
1.11 0.895426 −3.03713 −1.19821 −3.18965 −2.71952 −1.00000 −2.86376 6.22413 −2.85610
1.12 1.57801 0.762849 0.490100 3.81510 1.20378 −1.00000 −2.38263 −2.41806 6.02024
1.13 1.73103 3.17381 0.996454 0.582799 5.49396 −1.00000 −1.73716 7.07309 1.00884
1.14 2.37485 −2.83471 3.63992 −0.0579386 −6.73203 −1.00000 3.89457 5.03561 −0.137596
1.15 2.39270 2.11352 3.72502 0.234219 5.05703 −1.00000 4.12745 1.46698 0.560415
1.16 2.64245 1.38654 4.98252 −2.19326 3.66386 −1.00000 7.88116 −1.07750 −5.79558
1.17 2.72865 −1.21546 5.44551 4.18804 −3.31655 −1.00000 9.40159 −1.52266 11.4277
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)
\(89\) \(-1\)

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 623.2.a.f ✓ 17
3.b odd 2 1 5607.2.a.u 17
4.b odd 2 1 9968.2.a.bn 17
7.b odd 2 1 4361.2.a.k 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
623.2.a.f ✓ 17 1.a even 1 1 trivial
4361.2.a.k 17 7.b odd 2 1
5607.2.a.u 17 3.b odd 2 1
9968.2.a.bn 17 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{17} - 3 T_{2}^{16} - 24 T_{2}^{15} + 74 T_{2}^{14} + 224 T_{2}^{13} - 719 T_{2}^{12} - 1025 T_{2}^{11} + \cdots + 21 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(623))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} - 3 T^{16} + \cdots + 21 \) Copy content Toggle raw display
$3$ \( T^{17} - 4 T^{16} + \cdots - 196 \) Copy content Toggle raw display
$5$ \( T^{17} - 6 T^{16} + \cdots + 14336 \) Copy content Toggle raw display
$7$ \( (T + 1)^{17} \) Copy content Toggle raw display
$11$ \( T^{17} - 5 T^{16} + \cdots - 724512 \) Copy content Toggle raw display
$13$ \( T^{17} - 7 T^{16} + \cdots - 24771400 \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots - 4317313024 \) Copy content Toggle raw display
$19$ \( T^{17} + \cdots + 113000980 \) Copy content Toggle raw display
$23$ \( T^{17} + \cdots + 11214299136 \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots - 385850089472 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots - 122835612688 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots - 1374511185920 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots + 57683399208 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots - 2714949505024 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots + 782285078528 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots - 728034866848 \) Copy content Toggle raw display
$59$ \( T^{17} + 18 T^{16} + \cdots + 67776960 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots - 100039251497000 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots - 153234581218288 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots + 1220779726880 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots - 218387363840 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots - 297416348032 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots + 42250461192128 \) Copy content Toggle raw display
$89$ \( (T - 1)^{17} \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots + 19458314147840 \) Copy content Toggle raw display
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