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Properties

Label 1690.2.b.g
Level $1690$
Weight $2$
Character orbit 1690.b
Analytic conductor $13.495$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1690,2,Mod(339,1690)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1690, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1690.339"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1690.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,0,0,-18,0,14,0,0,-16,2,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.4947179416\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 29x^{16} + 336x^{14} + 1977x^{12} + 6147x^{10} + 9369x^{8} + 5559x^{6} + 1342x^{4} + 116x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + ( - \beta_{15} + \beta_{4}) q^{3} - q^{4} + ( - \beta_{13} + \beta_{8} - \beta_{7} + \cdots + 1) q^{5} + ( - \beta_{5} + 1) q^{6} + (\beta_{17} + \beta_{16} + \cdots - \beta_{2}) q^{7}+ \cdots + (3 \beta_{17} - 3 \beta_{16} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 18 q^{4} + 14 q^{6} - 16 q^{9} + 2 q^{10} + 8 q^{11} - 2 q^{14} + 8 q^{15} + 18 q^{16} - 12 q^{19} + 16 q^{21} - 14 q^{24} + 22 q^{25} - 30 q^{29} - 14 q^{30} - 12 q^{31} - 24 q^{34} - 4 q^{35} + 16 q^{36}+ \cdots - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 29x^{16} + 336x^{14} + 1977x^{12} + 6147x^{10} + 9369x^{8} + 5559x^{6} + 1342x^{4} + 116x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 432 \nu^{16} - 12547 \nu^{14} - 144249 \nu^{12} - 827089 \nu^{10} - 2409607 \nu^{8} - 3075069 \nu^{6} + \cdots + 3368 ) / 30658 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 856 \nu^{16} + 26281 \nu^{14} + 322446 \nu^{12} + 2001364 \nu^{10} + 6492859 \nu^{8} + 10011455 \nu^{6} + \cdots + 707 ) / 15329 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2515 \nu^{17} - 73294 \nu^{15} - 847341 \nu^{13} - 4897329 \nu^{11} - 14407793 \nu^{9} + \cdots + 406523 \nu ) / 30658 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3368 \nu^{17} + 98104 \nu^{15} + 1144195 \nu^{13} + 6802785 \nu^{11} + 21530185 \nu^{9} + \cdots + 361509 \nu ) / 30658 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4171 \nu^{16} + 118836 \nu^{14} + 1346644 \nu^{12} + 7697386 \nu^{10} + 22952260 \nu^{8} + \cdots - 46428 ) / 30658 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3029 \nu^{17} - 85171 \nu^{15} - 950203 \nu^{13} - 5331200 \nu^{11} - 15531886 \nu^{9} + \cdots - 46501 \nu ) / 15329 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3029 \nu^{16} - 85171 \nu^{14} - 950203 \nu^{12} - 5331200 \nu^{10} - 15531886 \nu^{8} + \cdots - 31172 ) / 15329 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6131 \nu^{16} - 180588 \nu^{14} - 2119481 \nu^{12} - 12541401 \nu^{10} - 38534249 \nu^{8} + \cdots - 92149 ) / 30658 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13766 \nu^{17} - 4110 \nu^{16} - 398613 \nu^{15} - 114474 \nu^{14} - 4601498 \nu^{13} + \cdots + 79733 ) / 61316 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 13766 \nu^{17} + 4110 \nu^{16} - 398613 \nu^{15} + 114474 \nu^{14} - 4601498 \nu^{13} + \cdots - 79733 ) / 61316 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 13766 \nu^{17} + 4110 \nu^{16} - 398613 \nu^{15} + 114474 \nu^{14} - 4601498 \nu^{13} + \cdots - 79733 ) / 61316 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10150 \nu^{17} - 291319 \nu^{15} - 3329358 \nu^{13} - 19216808 \nu^{11} - 57967748 \nu^{9} + \cdots - 202467 \nu ) / 30658 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 32545 \nu^{17} + 15404 \nu^{16} + 940588 \nu^{15} + 435329 \nu^{14} + 10824192 \nu^{13} + \cdots + 54770 ) / 61316 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 32545 \nu^{17} + 15404 \nu^{16} - 940588 \nu^{15} + 435329 \nu^{14} - 10824192 \nu^{13} + \cdots + 54770 ) / 61316 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 21536 \nu^{17} - 621517 \nu^{15} - 7146796 \nu^{13} - 41526008 \nu^{11} - 126143694 \nu^{9} + \cdots - 795555 \nu ) / 30658 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 44912 \nu^{17} - 4711 \nu^{16} - 1291933 \nu^{15} - 138352 \nu^{14} - 14809199 \nu^{13} + \cdots + 127283 ) / 61316 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 44912 \nu^{17} + 8135 \nu^{16} - 1291933 \nu^{15} + 243476 \nu^{14} - 14809199 \nu^{13} + \cdots - 124455 ) / 61316 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{11} + \beta_{10} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - \beta_{17} + \beta_{16} + \beta_{14} + \beta_{13} + \beta_{11} - \beta_{9} - 2 \beta_{8} + 3 \beta_{7} + \cdots - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} + 7\beta_{11} - 8\beta_{10} - \beta_{9} - \beta_{6} + \beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6 \beta_{17} - 6 \beta_{16} - 8 \beta_{14} - 8 \beta_{13} - 7 \beta_{11} + 7 \beta_{9} + 13 \beta_{8} + \cdots + 31 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{17} - 3 \beta_{16} - 22 \beta_{15} + 13 \beta_{14} - 13 \beta_{13} - 6 \beta_{12} + \cdots + 3 \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 37 \beta_{17} + 37 \beta_{16} + 64 \beta_{14} + 64 \beta_{13} + 50 \beta_{11} - 50 \beta_{9} + \cdots - 206 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 39 \beta_{17} + 39 \beta_{16} + 212 \beta_{15} - 134 \beta_{14} + 134 \beta_{13} + 40 \beta_{12} + \cdots - 39 \beta_{2} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 244 \beta_{17} - 244 \beta_{16} - 523 \beta_{14} - 523 \beta_{13} - 378 \beta_{11} + 378 \beta_{9} + \cdots + 1454 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 384 \beta_{17} - 384 \beta_{16} - 1933 \beta_{15} + 1259 \beta_{14} - 1259 \beta_{13} + \cdots + 384 \beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1717 \beta_{17} + 1717 \beta_{16} + 4334 \beta_{14} + 4334 \beta_{13} + 2976 \beta_{11} + \cdots - 10802 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3441 \beta_{17} + 3441 \beta_{16} + 17095 \beta_{15} - 11293 \beta_{14} + 11293 \beta_{13} + \cdots - 3441 \beta_{2} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 12753 \beta_{17} - 12753 \beta_{16} - 36206 \beta_{14} - 36206 \beta_{13} - 24046 \beta_{11} + \cdots + 83499 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 29641 \beta_{17} - 29641 \beta_{16} - 148458 \beta_{15} + 98734 \beta_{14} - 98734 \beta_{13} + \cdots + 29641 \beta_{2} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 98668 \beta_{17} + 98668 \beta_{16} + 303776 \beta_{14} + 303776 \beta_{13} + 197402 \beta_{11} + \cdots - 664159 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 250928 \beta_{17} + 250928 \beta_{16} + 1274792 \beta_{15} - 850436 \beta_{14} + 850436 \beta_{13} + \cdots - 250928 \beta_{2} \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 785637 \beta_{17} - 785637 \beta_{16} - 2554209 \beta_{14} - 2554209 \beta_{13} - 1636073 \beta_{11} + \cdots + 5387045 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 2108660 \beta_{17} - 2108660 \beta_{16} - 10868358 \beta_{15} + 7259729 \beta_{14} + \cdots + 2108660 \beta_{2} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1690\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
339.1
0.671265i
2.08395i
− 2.90009i
− 0.506152i
2.20982i
2.17115i
− 0.430845i
0.0982708i
− 2.39737i
2.39737i
− 0.0982708i
0.430845i
− 2.17115i
− 2.20982i
0.506152i
2.90009i
− 2.08395i
− 0.671265i
− 1.00000i − 1.92714i −1.00000 2.03261 + 0.931932i −1.92714 − 1.36403i 1.00000i −0.713864 0.931932 − 2.03261i
339.2 − 1.00000i − 1.49535i −1.00000 −1.69382 + 1.45979i −1.49535 − 1.01867i 1.00000i 0.763924 1.45979 + 1.69382i
339.3 − 1.00000i − 0.303663i −1.00000 1.92689 − 1.13450i −0.303663 1.08593i 1.00000i 2.90779 −1.13450 − 1.92689i
339.4 − 1.00000i 0.0670024i −1.00000 −1.65112 + 1.50791i 0.0670024 5.03568i 1.00000i 2.99551 1.50791 + 1.65112i
339.5 − 1.00000i 0.666723i −1.00000 2.18940 + 0.454429i 0.666723 − 2.41461i 1.00000i 2.55548 0.454429 − 2.18940i
339.6 − 1.00000i 0.956446i −1.00000 −0.426774 − 2.19496i 0.956446 2.15124i 1.00000i 2.08521 −2.19496 + 0.426774i
339.7 − 1.00000i 2.55203i −1.00000 −1.58997 − 1.57226i 2.55203 − 1.86919i 1.00000i −3.51288 −1.57226 + 1.58997i
339.8 − 1.00000i 3.06821i −1.00000 −2.23031 − 0.160405i 3.06821 − 3.06611i 1.00000i −6.41393 −0.160405 + 2.23031i
339.9 − 1.00000i 3.41573i −1.00000 1.44308 + 1.70807i 3.41573 0.459770i 1.00000i −8.66724 1.70807 − 1.44308i
339.10 1.00000i − 3.41573i −1.00000 1.44308 − 1.70807i 3.41573 − 0.459770i − 1.00000i −8.66724 1.70807 + 1.44308i
339.11 1.00000i − 3.06821i −1.00000 −2.23031 + 0.160405i 3.06821 3.06611i − 1.00000i −6.41393 −0.160405 − 2.23031i
339.12 1.00000i − 2.55203i −1.00000 −1.58997 + 1.57226i 2.55203 1.86919i − 1.00000i −3.51288 −1.57226 − 1.58997i
339.13 1.00000i − 0.956446i −1.00000 −0.426774 + 2.19496i 0.956446 − 2.15124i − 1.00000i 2.08521 −2.19496 − 0.426774i
339.14 1.00000i − 0.666723i −1.00000 2.18940 − 0.454429i 0.666723 2.41461i − 1.00000i 2.55548 0.454429 + 2.18940i
339.15 1.00000i − 0.0670024i −1.00000 −1.65112 − 1.50791i 0.0670024 − 5.03568i − 1.00000i 2.99551 1.50791 − 1.65112i
339.16 1.00000i 0.303663i −1.00000 1.92689 + 1.13450i −0.303663 − 1.08593i − 1.00000i 2.90779 −1.13450 + 1.92689i
339.17 1.00000i 1.49535i −1.00000 −1.69382 − 1.45979i −1.49535 1.01867i − 1.00000i 0.763924 1.45979 − 1.69382i
339.18 1.00000i 1.92714i −1.00000 2.03261 − 0.931932i −1.92714 1.36403i − 1.00000i −0.713864 0.931932 + 2.03261i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 339.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1690.2.b.g yes 18
5.b even 2 1 inner 1690.2.b.g yes 18
5.c odd 4 1 8450.2.a.ct 9
5.c odd 4 1 8450.2.a.da 9
13.b even 2 1 1690.2.b.f ✓ 18
13.d odd 4 1 1690.2.c.g 18
13.d odd 4 1 1690.2.c.h 18
65.d even 2 1 1690.2.b.f ✓ 18
65.g odd 4 1 1690.2.c.g 18
65.g odd 4 1 1690.2.c.h 18
65.h odd 4 1 8450.2.a.cw 9
65.h odd 4 1 8450.2.a.cx 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1690.2.b.f ✓ 18 13.b even 2 1
1690.2.b.f ✓ 18 65.d even 2 1
1690.2.b.g yes 18 1.a even 1 1 trivial
1690.2.b.g yes 18 5.b even 2 1 inner
1690.2.c.g 18 13.d odd 4 1
1690.2.c.g 18 65.g odd 4 1
1690.2.c.h 18 13.d odd 4 1
1690.2.c.h 18 65.g odd 4 1
8450.2.a.ct 9 5.c odd 4 1
8450.2.a.cw 9 65.h odd 4 1
8450.2.a.cx 9 65.h odd 4 1
8450.2.a.da 9 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1690, [\chi])\):

\( T_{3}^{18} + 35 T_{3}^{16} + 469 T_{3}^{14} + 3044 T_{3}^{12} + 10052 T_{3}^{10} + 16443 T_{3}^{8} + \cdots + 1 \) Copy content Toggle raw display
\( T_{11}^{9} - 4 T_{11}^{8} - 45 T_{11}^{7} + 126 T_{11}^{6} + 672 T_{11}^{5} - 938 T_{11}^{4} + \cdots - 1688 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$3$ \( T^{18} + 35 T^{16} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{18} - 11 T^{16} + \cdots + 1953125 \) Copy content Toggle raw display
$7$ \( T^{18} + 53 T^{16} + \cdots + 10816 \) Copy content Toggle raw display
$11$ \( (T^{9} - 4 T^{8} + \cdots - 1688)^{2} \) Copy content Toggle raw display
$13$ \( T^{18} \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 821624896 \) Copy content Toggle raw display
$19$ \( (T^{9} + 6 T^{8} + \cdots - 1912)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + 225 T^{16} + \cdots + 322624 \) Copy content Toggle raw display
$29$ \( (T^{9} + 15 T^{8} + \cdots + 369496)^{2} \) Copy content Toggle raw display
$31$ \( (T^{9} + 6 T^{8} + \cdots + 3017216)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 52583993344 \) Copy content Toggle raw display
$41$ \( (T^{9} - 27 T^{8} + \cdots + 138557)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 7526073470161 \) Copy content Toggle raw display
$47$ \( T^{18} + 373 T^{16} + \cdots + 1236544 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 782191661056 \) Copy content Toggle raw display
$59$ \( (T^{9} + 20 T^{8} + \cdots - 1232008)^{2} \) Copy content Toggle raw display
$61$ \( (T^{9} + 19 T^{8} + \cdots - 17534728)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 13\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( (T^{9} + 10 T^{8} + \cdots + 5913536)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 48110204567104 \) Copy content Toggle raw display
$79$ \( (T^{9} - 14 T^{8} + \cdots - 487936)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 46485110364001 \) Copy content Toggle raw display
$89$ \( (T^{9} + 43 T^{8} + \cdots - 263242657)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 115021163443264 \) Copy content Toggle raw display
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