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Properties

Label 2475.2.f.i
Level $2475$
Weight $2$
Character orbit 2475.f
Analytic conductor $19.763$
Analytic rank $0$
Dimension $16$
CM discriminant -55
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2475,2,Mod(2276,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2475.2276");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.f (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 24x^{14} + 300x^{12} - 2112x^{10} + 8919x^{8} - 17520x^{6} + 27500x^{4} - 54000x^{2} + 50625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 495)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + ( - \beta_1 + 2) q^{4} - \beta_{15} q^{7} + ( - 2 \beta_{5} + \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + ( - \beta_1 + 2) q^{4} - \beta_{15} q^{7} + ( - 2 \beta_{5} + \beta_{4}) q^{8} + \beta_{8} q^{11} + \beta_{14} q^{13} + ( - \beta_{11} + \beta_{8}) q^{14} + (\beta_{3} - 2 \beta_1 + 4) q^{16} + \beta_{7} q^{17} + ( - \beta_{15} - \beta_{9}) q^{22} + (\beta_{11} + \beta_{10} - \beta_{6}) q^{26} + ( - \beta_{15} - \beta_{14} + \cdots - \beta_{9}) q^{28}+ \cdots + ( - \beta_{12} + 2 \beta_{7} + \cdots + 3 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{4} + 64 q^{16} - 112 q^{49} + 128 q^{64}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 24x^{14} + 300x^{12} - 2112x^{10} + 8919x^{8} - 17520x^{6} + 27500x^{4} - 54000x^{2} + 50625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 134387 \nu^{14} + 3363298 \nu^{12} - 43947565 \nu^{10} + 324033994 \nu^{8} + \cdots + 1745885250 ) / 2490445125 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 252766 \nu^{14} - 6916514 \nu^{12} + 92087945 \nu^{10} - 692956892 \nu^{8} + 2959207139 \nu^{6} + \cdots - 27519401250 ) / 2490445125 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 176 \nu^{14} - 4467 \nu^{12} + 57012 \nu^{10} - 416532 \nu^{8} + 1784160 \nu^{6} - 3502722 \nu^{4} + \cdots - 7072650 ) / 956025 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4772188 \nu^{15} - 118994471 \nu^{13} + 1516468181 \nu^{11} - 11030317841 \nu^{9} + \cdots - 582586057575 \nu ) / 97127359875 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 25393254 \nu^{15} - 605244631 \nu^{13} + 7399619590 \nu^{11} - 49707720523 \nu^{9} + \cdots - 571642978500 \nu ) / 485636799375 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1232 \nu^{14} - 27569 \nu^{12} + 324884 \nu^{10} - 2076224 \nu^{8} + 7709120 \nu^{6} + \cdots - 37291050 ) / 4571775 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 894992 \nu^{15} - 20843464 \nu^{13} + 249343954 \nu^{11} - 1643388319 \nu^{9} + \cdots - 8226832050 \nu ) / 13875337125 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 335728 \nu^{14} + 7394248 \nu^{12} - 85732672 \nu^{10} + 531901402 \nu^{8} + \cdots + 7659311643 ) / 777018879 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1316449 \nu^{15} - 31514736 \nu^{13} + 395884290 \nu^{11} - 2801304363 \nu^{9} + \cdots - 35433024750 \nu ) / 12452225625 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 32224 \nu^{14} - 714346 \nu^{12} + 8396155 \nu^{10} - 53581288 \nu^{8} + 198868621 \nu^{6} + \cdots - 961719750 ) / 62141625 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 139333 \nu^{14} - 3080782 \nu^{12} + 35927485 \nu^{10} - 224785546 \nu^{8} + 795435382 \nu^{6} + \cdots - 3363275250 ) / 186424875 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 14589717 \nu^{15} + 348509138 \nu^{13} - 4319818070 \nu^{11} + 29809473029 \nu^{9} + \cdots + 776999245500 \nu ) / 69376685625 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 490156 \nu^{15} - 10593627 \nu^{13} + 120964887 \nu^{11} - 728204352 \nu^{9} + \cdots - 12754646325 \nu ) / 1494267075 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 3025552 \nu^{15} - 67317234 \nu^{13} + 788461974 \nu^{11} - 4984102689 \nu^{9} + \cdots - 78694334550 \nu ) / 7471335375 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 18136789 \nu^{15} - 403479621 \nu^{13} + 4740214440 \nu^{11} - 30152425068 \nu^{9} + \cdots - 467919351000 \nu ) / 37356676875 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{12} + \beta_{9} + 2\beta_{5} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{11} + \beta_{10} + 6\beta_{8} + \beta_{2} + 5\beta _1 + 18 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{15} + 2\beta_{14} + 8\beta_{13} + 3\beta_{12} + 12\beta_{9} + 4\beta_{7} - 3\beta_{5} + 4\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 36\beta_{11} - 8\beta_{10} + 54\beta_{8} + 3\beta_{6} + 9\beta_{3} - 8\beta_{2} + 20\beta _1 - 18 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 90 \beta_{15} + 54 \beta_{14} + 45 \beta_{13} - 20 \beta_{12} + 70 \beta_{9} + 51 \beta_{7} + \cdots - 21 \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 156\beta_{11} - 197\beta_{10} + 240\beta_{8} + 324\beta_{6} + 72\beta_{3} - 155\beta_{2} - 82\beta _1 - 1080 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 495 \beta_{15} + 707 \beta_{14} - 133 \beta_{13} - 351 \beta_{12} - 12 \beta_{9} + \cdots - 455 \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 360 \beta_{11} - 2096 \beta_{10} - 567 \beta_{8} + 4116 \beta_{6} - 252 \beta_{3} - 848 \beta_{2} + \cdots - 9261 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 729 \beta_{15} + 4812 \beta_{14} - 3576 \beta_{13} - 2441 \beta_{12} - 3776 \beta_{9} + \cdots - 3912 \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 10005 \beta_{11} - 9134 \beta_{10} - 15834 \beta_{8} + 20520 \beta_{6} - 10440 \beta_{3} + \cdots - 31122 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 20295 \beta_{15} - 127 \beta_{14} - 20146 \beta_{13} - 7986 \beta_{12} - 28017 \beta_{9} + \cdots - 20876 \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 31536 \beta_{11} + 33746 \beta_{10} - 49896 \beta_{8} - 54057 \beta_{6} - 52596 \beta_{3} + \cdots + 51282 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 299520 \beta_{15} - 398175 \beta_{14} + 57855 \beta_{13} + 10114 \beta_{12} - 17342 \beta_{9} + \cdots - 54855 \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 119067 \beta_{11} + 1535014 \beta_{10} - 188292 \beta_{8} - 2874690 \beta_{6} - 402570 \beta_{3} + \cdots + 1344564 ) / 6 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 2656062 \beta_{15} - 4769809 \beta_{14} + 1624310 \beta_{13} + 224688 \beta_{12} + \cdots + 287986 \beta_{4} ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(2026\) \(2377\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2276.1
0.804180 + 1.09134i
0.804180 − 1.09134i
2.48610 − 1.40769i
2.48610 + 1.40769i
−1.34882 + 0.135696i
−1.34882 − 0.135696i
2.71170 − 0.899435i
2.71170 + 0.899435i
−2.71170 − 0.899435i
−2.71170 + 0.899435i
1.34882 + 0.135696i
1.34882 − 0.135696i
−2.48610 − 1.40769i
−2.48610 + 1.40769i
−0.804180 + 1.09134i
−0.804180 − 1.09134i
−2.81538 0 5.92635 0 0 − 2.78952i −11.0541 0 0
2276.2 −2.81538 0 5.92635 0 0 2.78952i −11.0541 0 0
2276.3 −2.18267 0 2.76407 0 0 − 1.20702i −1.66771 0 0
2276.4 −2.18267 0 2.76407 0 0 1.20702i −1.66771 0 0
2276.5 −1.79887 0 1.23593 0 0 − 5.15200i 1.37446 0 0
2276.6 −1.79887 0 1.23593 0 0 5.15200i 1.37446 0 0
2276.7 −0.271391 0 −1.92635 0 0 − 4.49650i 1.06558 0 0
2276.8 −0.271391 0 −1.92635 0 0 4.49650i 1.06558 0 0
2276.9 0.271391 0 −1.92635 0 0 − 4.49650i −1.06558 0 0
2276.10 0.271391 0 −1.92635 0 0 4.49650i −1.06558 0 0
2276.11 1.79887 0 1.23593 0 0 − 5.15200i −1.37446 0 0
2276.12 1.79887 0 1.23593 0 0 5.15200i −1.37446 0 0
2276.13 2.18267 0 2.76407 0 0 − 1.20702i 1.66771 0 0
2276.14 2.18267 0 2.76407 0 0 1.20702i 1.66771 0 0
2276.15 2.81538 0 5.92635 0 0 − 2.78952i 11.0541 0 0
2276.16 2.81538 0 5.92635 0 0 2.78952i 11.0541 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2276.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
55.d odd 2 1 CM by \(\Q(\sqrt{-55}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
11.b odd 2 1 inner
15.d odd 2 1 inner
33.d even 2 1 inner
165.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2475.2.f.i 16
3.b odd 2 1 inner 2475.2.f.i 16
5.b even 2 1 inner 2475.2.f.i 16
5.c odd 4 2 495.2.d.c ✓ 16
11.b odd 2 1 inner 2475.2.f.i 16
15.d odd 2 1 inner 2475.2.f.i 16
15.e even 4 2 495.2.d.c ✓ 16
33.d even 2 1 inner 2475.2.f.i 16
55.d odd 2 1 CM 2475.2.f.i 16
55.e even 4 2 495.2.d.c ✓ 16
165.d even 2 1 inner 2475.2.f.i 16
165.l odd 4 2 495.2.d.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.d.c ✓ 16 5.c odd 4 2
495.2.d.c ✓ 16 15.e even 4 2
495.2.d.c ✓ 16 55.e even 4 2
495.2.d.c ✓ 16 165.l odd 4 2
2475.2.f.i 16 1.a even 1 1 trivial
2475.2.f.i 16 3.b odd 2 1 inner
2475.2.f.i 16 5.b even 2 1 inner
2475.2.f.i 16 11.b odd 2 1 inner
2475.2.f.i 16 15.d odd 2 1 inner
2475.2.f.i 16 33.d even 2 1 inner
2475.2.f.i 16 55.d odd 2 1 CM
2475.2.f.i 16 165.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2475, [\chi])\):

\( T_{2}^{8} - 16T_{2}^{6} + 80T_{2}^{4} - 128T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{29} \) Copy content Toggle raw display
\( T_{37} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 16 T^{6} + 80 T^{4} + \cdots + 9)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + 56 T^{6} + \cdots + 6084)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 11)^{8} \) Copy content Toggle raw display
$13$ \( (T^{8} + 104 T^{6} + \cdots + 26244)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 136 T^{6} + \cdots + 161604)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( (T^{4} - 124 T^{2} + 324)^{4} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( (T^{8} + 344 T^{6} + \cdots + 12404484)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( (T^{4} + 236 T^{2} + 10404)^{4} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( (T^{4} + 284 T^{2} + 6084)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + 584 T^{6} + \cdots + 113167044)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( (T^{8} - 664 T^{6} + \cdots + 185014404)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 176)^{8} \) Copy content Toggle raw display
$97$ \( T^{16} \) Copy content Toggle raw display
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