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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [85,4,Mod(16,85)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("85.16"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(85, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 85.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.01516235049\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 113 x^{14} + 5162 x^{12} + 121550 x^{10} + 1552797 x^{8} + 10318089 x^{6} + 30998912 x^{4} + \cdots + 2663424 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{4} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + \beta_{12} q^{3} + ( - \beta_{2} + 6) q^{4} - \beta_{7} q^{5} + (\beta_{13} + \beta_{12} + \cdots + \beta_1) q^{6} + (\beta_{12} - \beta_{10} - \beta_{7}) q^{7} + (\beta_{6} - 6 \beta_{3} + \beta_{2} + 4) q^{8}+ \cdots + (9 \beta_{15} + 37 \beta_{14} + \cdots - 38 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{2} + 98 q^{4} + 78 q^{8} - 116 q^{9} + 48 q^{13} + 180 q^{15} + 490 q^{16} - 132 q^{17} - 470 q^{18} - 44 q^{19} - 136 q^{21} - 400 q^{25} + 720 q^{26} - 40 q^{30} - 1614 q^{32} + 2304 q^{33}+ \cdots - 4470 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 113 x^{14} + 5162 x^{12} + 121550 x^{10} + 1552797 x^{8} + 10318089 x^{6} + 30998912 x^{4} + \cdots + 2663424 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 5\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 943 \nu^{14} + 91938 \nu^{12} + 3428718 \nu^{10} + 60471672 \nu^{8} + 498880455 \nu^{6} + \cdots - 53921280 ) / 538811200 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 178 \nu^{14} - 67707 \nu^{12} - 5023972 \nu^{10} - 154297406 \nu^{8} - 2233601606 \nu^{6} + \cdots - 1879009344 ) / 107762240 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1595 \nu^{14} - 400484 \nu^{12} - 27188974 \nu^{10} - 802792684 \nu^{8} - 11342296411 \nu^{6} + \cdots - 27078114624 ) / 538811200 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6125 \nu^{14} + 583588 \nu^{12} + 21281818 \nu^{10} + 364969668 \nu^{8} + 2842979037 \nu^{6} + \cdots - 13396478592 ) / 538811200 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2065 \nu^{15} + 329531 \nu^{13} + 20037206 \nu^{11} + 600729986 \nu^{9} + \cdots + 208204410496 \nu ) / 10991748480 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 17699 \nu^{14} - 1928765 \nu^{12} - 84117030 \nu^{10} - 1856218462 \nu^{8} + \cdots - 13051070496 ) / 269405600 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 57275 \nu^{14} - 5472263 \nu^{12} - 204361838 \nu^{10} - 3762854418 \nu^{8} + \cdots - 85488796928 ) / 538811200 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 40465 \nu^{15} + 3130571 \nu^{13} + 86021806 \nu^{11} + 1015034946 \nu^{9} + \cdots + 1798405926016 \nu ) / 18319580800 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 123031 \nu^{15} + 13661579 \nu^{13} + 610212914 \nu^{11} + 13977600362 \nu^{9} + \cdots + 2756116108288 \nu ) / 54958742400 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 502871 \nu^{15} + 52818271 \nu^{13} + 2215147126 \nu^{11} + 46990171954 \nu^{9} + \cdots + 1461472995200 \nu ) / 109917484800 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 518533 \nu^{15} - 63335189 \nu^{13} - 3039665354 \nu^{11} - 72459542918 \nu^{9} + \cdots - 70515753856 \nu ) / 109917484800 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1210435 \nu^{15} + 126042023 \nu^{13} + 5232772238 \nu^{11} + 109213322258 \nu^{9} + \cdots - 11231359020032 \nu ) / 109917484800 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 924935 \nu^{15} - 105932191 \nu^{13} - 4842488086 \nu^{11} - 111606936946 \nu^{9} + \cdots - 2597262082176 \nu ) / 36639161600 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{13} - 5\beta_{12} - 5\beta_{11} + 5\beta_{10} - 3\beta_{7} - 24\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{6} - 2\beta_{5} + 2\beta_{4} + 5\beta_{3} - 28\beta_{2} + 302 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 20 \beta_{15} - 10 \beta_{14} + 210 \beta_{13} + 120 \beta_{12} + 180 \beta_{11} - 160 \beta_{10} + \cdots + 609 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{9} + 6\beta_{8} + 50\beta_{6} + 80\beta_{5} - 92\beta_{4} - 294\beta_{3} + 743\beta_{2} - 7116 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 910 \beta_{15} + 280 \beta_{14} - 6825 \beta_{13} - 2055 \beta_{12} - 5985 \beta_{11} + \cdots - 15622 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 208 \beta_{9} - 322 \beta_{8} - 1987 \beta_{6} - 2494 \beta_{5} + 3146 \beta_{4} + 12203 \beta_{3} + \cdots + 174960 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 30710 \beta_{15} - 3930 \beta_{14} + 202710 \beta_{13} + 18790 \beta_{12} + 194320 \beta_{11} + \cdots + 403663 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 7896 \beta_{9} + 12404 \beta_{8} + 70568 \beta_{6} + 71376 \beta_{5} - 96784 \beta_{4} + \cdots - 4428218 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 933700 \beta_{15} - 37000 \beta_{14} - 5788145 \beta_{13} + 481955 \beta_{12} - 6163945 \beta_{11} + \cdots - 10511088 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 266064 \beta_{9} - 419396 \beta_{8} - 2342961 \beta_{6} - 1972162 \beta_{5} + 2838266 \beta_{4} + \cdots + 114512018 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 27139880 \beta_{15} + 5241310 \beta_{14} + 162310150 \beta_{13} - 38542360 \beta_{12} + \cdots + 275910569 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 8452620 \beta_{9} + 13263210 \beta_{8} + 74507034 \beta_{6} + 53753448 \beta_{5} - 81274092 \beta_{4} + \cdots - 3010102032 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 772902930 \beta_{15} - 255641760 \beta_{14} - 4516072845 \beta_{13} + 1691397465 \beta_{12} + \cdots - 7300984966 \beta_1 ) / 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/85\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(71\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
5.33303i
− 5.33303i
− 4.11126i
4.11126i
1.74738i
− 1.74738i
1.63349i
− 1.63349i
− 0.289794i
0.289794i
3.79130i
− 3.79130i
4.83075i
− 4.83075i
− 4.91331i
4.91331i
−5.33303 − 5.58969i 20.4412 5.00000i 29.8100i 7.17613i −66.3491 −4.24466 − 26.6651i
16.2 −5.33303 5.58969i 20.4412 − 5.00000i − 29.8100i − 7.17613i −66.3491 −4.24466 26.6651i
16.3 −4.11126 − 3.82687i 8.90247 − 5.00000i 15.7333i − 15.9229i −3.71030 12.3551 20.5563i
16.4 −4.11126 3.82687i 8.90247 5.00000i − 15.7333i 15.9229i −3.71030 12.3551 − 20.5563i
16.5 −1.74738 − 0.322521i −4.94666 5.00000i 0.563566i − 7.55169i 22.6227 26.8960 − 8.73690i
16.6 −1.74738 0.322521i −4.94666 − 5.00000i − 0.563566i 7.55169i 22.6227 26.8960 8.73690i
16.7 −1.63349 − 9.29997i −5.33171 5.00000i 15.1914i − 29.4708i 21.7772 −59.4894 − 8.16745i
16.8 −1.63349 9.29997i −5.33171 − 5.00000i − 15.1914i 29.4708i 21.7772 −59.4894 8.16745i
16.9 0.289794 − 2.72824i −7.91602 5.00000i − 0.790629i 29.7500i −4.61237 19.5567 1.44897i
16.10 0.289794 2.72824i −7.91602 − 5.00000i 0.790629i − 29.7500i −4.61237 19.5567 − 1.44897i
16.11 3.79130 − 5.21292i 6.37398 − 5.00000i − 19.7638i − 18.4051i −6.16475 −0.174523 − 18.9565i
16.12 3.79130 5.21292i 6.37398 5.00000i 19.7638i 18.4051i −6.16475 −0.174523 18.9565i
16.13 4.83075 − 1.17308i 15.3362 − 5.00000i − 5.66685i 17.9463i 35.4393 25.6239 − 24.1538i
16.14 4.83075 1.17308i 15.3362 5.00000i 5.66685i − 17.9463i 35.4393 25.6239 24.1538i
16.15 4.91331 − 10.2724i 16.1406 5.00000i − 50.4717i 21.7147i 39.9972 −78.5231 24.5665i
16.16 4.91331 10.2724i 16.1406 − 5.00000i 50.4717i − 21.7147i 39.9972 −78.5231 − 24.5665i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 16.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 85.4.d.b ✓ 16
3.b odd 2 1 765.4.g.c 16
5.b even 2 1 425.4.d.g 16
5.c odd 4 1 425.4.c.e 16
5.c odd 4 1 425.4.c.f 16
17.b even 2 1 inner 85.4.d.b ✓ 16
17.c even 4 1 1445.4.a.o 8
17.c even 4 1 1445.4.a.p 8
51.c odd 2 1 765.4.g.c 16
85.c even 2 1 425.4.d.g 16
85.g odd 4 1 425.4.c.e 16
85.g odd 4 1 425.4.c.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.4.d.b ✓ 16 1.a even 1 1 trivial
85.4.d.b ✓ 16 17.b even 2 1 inner
425.4.c.e 16 5.c odd 4 1
425.4.c.e 16 85.g odd 4 1
425.4.c.f 16 5.c odd 4 1
425.4.c.f 16 85.g odd 4 1
425.4.d.g 16 5.b even 2 1
425.4.d.g 16 85.c even 2 1
765.4.g.c 16 3.b odd 2 1
765.4.g.c 16 51.c odd 2 1
1445.4.a.o 8 17.c even 4 1
1445.4.a.p 8 17.c even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - T_{2}^{7} - 56T_{2}^{6} + 38T_{2}^{5} + 975T_{2}^{4} - 101T_{2}^{3} - 5352T_{2}^{2} - 4096T_{2} + 1632 \) acting on \(S_{4}^{\mathrm{new}}(85, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{7} - 56 T^{6} + \cdots + 1632)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 120912016 \) Copy content Toggle raw display
$5$ \( (T^{2} + 25)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots - 1309766957360)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 33\!\cdots\!21 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots - 747381242730240)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 12\!\cdots\!12)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots - 20\!\cdots\!76)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 79\!\cdots\!84)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 18\!\cdots\!40)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 25\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 39\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 97\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 24\!\cdots\!80)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 33\!\cdots\!56 \) Copy content Toggle raw display
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