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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1035,4,Mod(1,1035)] 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("1035.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1035, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1035 = 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1035.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [13,0,0,64,65,0,44] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(61.0669768559\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 84 x^{11} + 2633 x^{9} - 44 x^{8} - 38474 x^{7} + 1204 x^{6} + 266796 x^{5} - 1960 x^{4} + \cdots + 89472 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 5) q^{4} + 5 q^{5} + ( - \beta_{4} + \beta_1 + 3) q^{7} + ( - \beta_{3} - 5 \beta_1) q^{8} - 5 \beta_1 q^{10} + (\beta_{8} + \beta_{2} + 7) q^{11} + (\beta_{10} + \beta_{2} + 11) q^{13}+ \cdots + ( - 10 \beta_{12} + 18 \beta_{11} + \cdots - 168) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 64 q^{4} + 65 q^{5} + 44 q^{7} + 90 q^{11} + 142 q^{13} - 94 q^{14} + 380 q^{16} + 12 q^{17} + 178 q^{19} + 320 q^{20} - 50 q^{22} + 299 q^{23} + 325 q^{25} - 4 q^{26} + 502 q^{28} + 436 q^{29} + 248 q^{31}+ \cdots - 2360 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 84 x^{11} + 2633 x^{9} - 44 x^{8} - 38474 x^{7} + 1204 x^{6} + 266796 x^{5} - 1960 x^{4} + \cdots + 89472 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 21\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 212952149 \nu^{12} - 1535377624 \nu^{11} + 13456729612 \nu^{10} + 109870720760 \nu^{9} + \cdots + 32919010363824 ) / 1544349547632 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 155851217 \nu^{12} + 1581374839 \nu^{11} - 12652344694 \nu^{10} - 122378803634 \nu^{9} + \cdots - 402505520424 ) / 772174773816 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 705127289 \nu^{12} - 2637749536 \nu^{11} + 48365931268 \nu^{10} + 188763892124 \nu^{9} + \cdots + 166203153471168 ) / 1544349547632 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 129656145 \nu^{12} + 281848875 \nu^{11} - 9492960532 \nu^{10} - 19448967052 \nu^{9} + \cdots + 8315982288992 ) / 257391591272 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 164104508 \nu^{12} - 213237887 \nu^{11} + 12489079252 \nu^{10} + 14771273598 \nu^{9} + \cdots - 9451701291120 ) / 257391591272 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1433695477 \nu^{12} - 2965206602 \nu^{11} + 108939475340 \nu^{10} + 211543351756 \nu^{9} + \cdots + 67016091914352 ) / 1544349547632 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 955818236 \nu^{12} + 3531843235 \nu^{11} - 67641944332 \nu^{10} - 256943807168 \nu^{9} + \cdots - 87550502686656 ) / 772174773816 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1984446053 \nu^{12} - 6117030184 \nu^{11} + 143875720588 \nu^{10} + \cdots + 133513457050608 ) / 1544349547632 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1029079933 \nu^{12} - 3922191402 \nu^{11} + 74067394492 \nu^{10} + 284020630036 \nu^{9} + \cdots + 12016767409680 ) / 514783182544 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 21\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{10} + \beta_{7} + \beta_{6} + 30\beta_{2} + \beta _1 + 277 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{12} - \beta_{9} + 3 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 5 \beta_{4} + \cdots + 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 47 \beta_{11} + 45 \beta_{10} - 6 \beta_{9} + 10 \beta_{8} + 49 \beta_{7} + 39 \beta_{6} + 10 \beta_{4} + \cdots + 6905 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 110 \beta_{12} + 32 \beta_{11} + 18 \beta_{10} - 69 \beta_{9} + 161 \beta_{8} - 110 \beta_{7} + \cdots + 1514 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 48 \beta_{12} + 1771 \beta_{11} + 1607 \beta_{10} - 342 \beta_{9} + 592 \beta_{8} + 1829 \beta_{7} + \cdots + 185695 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4510 \beta_{12} + 2196 \beta_{11} + 1088 \beta_{10} - 3269 \beta_{9} + 6511 \beta_{8} - 4382 \beta_{7} + \cdots + 73128 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3416 \beta_{12} + 61523 \beta_{11} + 53653 \beta_{10} - 14006 \beta_{9} + 25130 \beta_{8} + \cdots + 5218329 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 164782 \beta_{12} + 102624 \beta_{11} + 46662 \beta_{10} - 131225 \beta_{9} + 236865 \beta_{8} + \cdots + 2958318 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 164288 \beta_{12} + 2049843 \beta_{11} + 1741235 \beta_{10} - 507958 \beta_{9} + 936972 \beta_{8} + \cdots + 150798283 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
5.59409
4.75972
3.46027
3.41679
2.16267
1.32823
−0.121296
−1.28321
−1.97327
−2.99840
−4.21952
−4.64871
−5.47736
−5.59409 0 23.2939 5.00000 0 −2.30830 −85.5554 0 −27.9705
1.2 −4.75972 0 14.6550 5.00000 0 34.9481 −31.6757 0 −23.7986
1.3 −3.46027 0 3.97346 5.00000 0 17.6754 13.9329 0 −17.3013
1.4 −3.41679 0 3.67448 5.00000 0 −14.0420 14.7794 0 −17.0840
1.5 −2.16267 0 −3.32287 5.00000 0 −27.2331 24.4876 0 −10.8133
1.6 −1.32823 0 −6.23582 5.00000 0 21.1602 18.9084 0 −6.64113
1.7 0.121296 0 −7.98529 5.00000 0 −10.1226 −1.93895 0 0.606479
1.8 1.28321 0 −6.35337 5.00000 0 27.0439 −18.4184 0 6.41605
1.9 1.97327 0 −4.10622 5.00000 0 −0.0675157 −23.8888 0 9.86633
1.10 2.99840 0 0.990411 5.00000 0 −3.75884 −21.0176 0 14.9920
1.11 4.21952 0 9.80435 5.00000 0 −28.8160 7.61348 0 21.0976
1.12 4.64871 0 13.6105 5.00000 0 24.5642 26.0818 0 23.2436
1.13 5.47736 0 22.0015 5.00000 0 4.95663 76.6913 0 27.3868
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( -1 \)
\(23\) \( -1 \)

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 1035.4.a.v yes 13
3.b odd 2 1 1035.4.a.u ✓ 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1035.4.a.u ✓ 13 3.b odd 2 1
1035.4.a.v yes 13 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1035))\):

\( T_{2}^{13} - 84 T_{2}^{11} + 2633 T_{2}^{9} + 44 T_{2}^{8} - 38474 T_{2}^{7} - 1204 T_{2}^{6} + \cdots - 89472 \) Copy content Toggle raw display
\( T_{7}^{13} - 44 T_{7}^{12} - 1648 T_{7}^{11} + 86104 T_{7}^{10} + 702282 T_{7}^{9} + \cdots + 2812410246528 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} - 84 T^{11} + \cdots - 89472 \) Copy content Toggle raw display
$3$ \( T^{13} \) Copy content Toggle raw display
$5$ \( (T - 5)^{13} \) Copy content Toggle raw display
$7$ \( T^{13} + \cdots + 2812410246528 \) Copy content Toggle raw display
$11$ \( T^{13} + \cdots + 69\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{13} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{13} + \cdots - 41\!\cdots\!52 \) Copy content Toggle raw display
$19$ \( T^{13} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( (T - 23)^{13} \) Copy content Toggle raw display
$29$ \( T^{13} + \cdots + 76\!\cdots\!72 \) Copy content Toggle raw display
$31$ \( T^{13} + \cdots - 48\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{13} + \cdots + 12\!\cdots\!92 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots + 32\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots + 11\!\cdots\!92 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots + 54\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots - 10\!\cdots\!08 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots + 27\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots - 15\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots - 90\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots + 13\!\cdots\!48 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots + 45\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots - 88\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots - 31\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots - 57\!\cdots\!00 \) Copy content Toggle raw display
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