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Properties

Label 1850.2.d.i
Level $1850$
Weight $2$
Character orbit 1850.d
Analytic conductor $14.772$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1850,2,Mod(1701,1850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1850, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1850.1701");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.d (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: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 48 x^{18} + 878 x^{16} + 8102 x^{14} + 41081 x^{12} + 115688 x^{10} + 175041 x^{8} + 134990 x^{6} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 370)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - \beta_1 q^{3} - q^{4} - \beta_{2} q^{6} + \beta_{9} q^{7} + \beta_{3} q^{8} + ( - \beta_{7} - \beta_{6} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - \beta_1 q^{3} - q^{4} - \beta_{2} q^{6} + \beta_{9} q^{7} + \beta_{3} q^{8} + ( - \beta_{7} - \beta_{6} + 1) q^{9} - \beta_{7} q^{11} + \beta_1 q^{12} + \beta_{17} q^{13} + \beta_{4} q^{14} + q^{16} + ( - \beta_{16} + 2 \beta_{3}) q^{17} + ( - \beta_{18} + \beta_{16} - \beta_{3}) q^{18} + ( - \beta_{19} - \beta_{12} - \beta_{4}) q^{19} + (\beta_{14} + \beta_{7} + \beta_{6} - 1) q^{21} - \beta_{18} q^{22} + ( - \beta_{18} - \beta_{17} + \cdots - \beta_{3}) q^{23}+ \cdots + (\beta_{14} + \beta_{13} - 2 \beta_{7} + \cdots + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 20 q^{4} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 20 q^{4} + 16 q^{9} + 20 q^{16} - 24 q^{21} + 4 q^{26} + 36 q^{34} - 16 q^{36} - 8 q^{41} - 20 q^{46} + 16 q^{49} - 20 q^{64} - 40 q^{71} - 16 q^{74} + 116 q^{81} + 24 q^{84} + 20 q^{86} + 164 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 48 x^{18} + 878 x^{16} + 8102 x^{14} + 41081 x^{12} + 115688 x^{10} + 175041 x^{8} + 134990 x^{6} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 54063874 \nu^{18} + 2574618933 \nu^{16} + 46502302940 \nu^{14} + 420755397517 \nu^{12} + \cdots + 23257566513 ) / 23403378928 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 145812415 \nu^{19} - 7053059794 \nu^{17} - 130597919303 \nu^{15} - 1227874489270 \nu^{13} + \cdots - 1925816411076 \nu ) / 23403378928 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 145812415 \nu^{19} + 7053059794 \nu^{17} + 130597919303 \nu^{15} + 1227874489270 \nu^{13} + \cdots + 1949219790004 \nu ) / 23403378928 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 613303985 \nu^{19} + 32361753535 \nu^{17} + 677080261599 \nu^{15} + 7453904939383 \nu^{13} + \cdots + 37340664846289 \nu ) / 93613515712 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 390695550 \nu^{18} - 18488174953 \nu^{16} - 330462274588 \nu^{14} - 2940177837725 \nu^{12} + \cdots - 13582288129 ) / 23403378928 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 477639065 \nu^{18} + 22592949918 \nu^{16} + 403550267529 \nu^{14} + 3586552255918 \nu^{12} + \cdots - 68796237176 ) / 23403378928 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 585766813 \nu^{18} - 27742187784 \nu^{16} - 496554873409 \nu^{14} - 4428063050952 \nu^{12} + \cdots + 139297998790 ) / 23403378928 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1757120435 \nu^{18} + 83402607063 \nu^{16} + 1498220961617 \nu^{14} + 13437365531935 \nu^{12} + \cdots - 67439214595 ) / 46806757856 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 4057864633 \nu^{18} - 192353961175 \nu^{16} - 3448085901127 \nu^{14} - 30824311286719 \nu^{12} + \cdots - 455501597545 ) / 93613515712 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2241687127 \nu^{19} + 106747249363 \nu^{17} + 1927702590245 \nu^{15} + \cdots + 5898561406361 \nu ) / 46806757856 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1206828866 \nu^{19} - 56678790431 \nu^{17} - 1000449551428 \nu^{15} + \cdots + 5276190286737 \nu ) / 23403378928 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1208955653 \nu^{19} + 57331395806 \nu^{17} + 1028299056857 \nu^{15} + 9198988235086 \nu^{13} + \cdots - 1089782663136 \nu ) / 23403378928 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 652713292 \nu^{18} - 30975328581 \nu^{16} - 556220975934 \nu^{14} - 4985087030133 \nu^{12} + \cdots + 39702760795 ) / 11701689464 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 926692235 \nu^{18} + 43943784426 \nu^{16} + 788127547195 \nu^{14} + 7050093005546 \nu^{12} + \cdots + 33311845352 ) / 11701689464 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 8135398299 \nu^{18} + 385482610181 \nu^{16} + 6905411302437 \nu^{14} + 61664032838653 \nu^{12} + \cdots - 277209785317 ) / 93613515712 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 4287569963 \nu^{19} - 204436380315 \nu^{17} - 3699591846113 \nu^{15} + \cdots - 11559854198745 \nu ) / 46806757856 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 2291028611 \nu^{19} + 109181047801 \nu^{17} + 1974157614537 \nu^{15} + \cdots + 6814000675891 \nu ) / 23403378928 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 5270572201 \nu^{19} - 251903976945 \nu^{17} - 4576183967811 \nu^{15} + \cdots - 23348494829651 \nu ) / 46806757856 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 13437406989 \nu^{19} + 638839429843 \nu^{17} + 11506290522083 \nu^{15} + \cdots + 24205664106461 \nu ) / 93613515712 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + 2\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{19} - 3\beta_{18} + 3\beta_{16} + \beta_{12} + \beta_{10} - \beta_{4} - 13\beta_{3} - 11\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 4 \beta_{15} + \beta_{14} + 3 \beta_{13} - 4 \beta_{9} + 4 \beta_{8} - 19 \beta_{7} - 17 \beta_{6} + \cdots + 58 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 24 \beta_{19} + 75 \beta_{18} + 15 \beta_{17} - 65 \beta_{16} - 24 \beta_{12} + 5 \beta_{11} + \cdots + 170 \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 104 \beta_{15} - 31 \beta_{14} - 90 \beta_{13} + 92 \beta_{9} - 122 \beta_{8} + 363 \beta_{7} + \cdots - 902 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 498 \beta_{19} - 1547 \beta_{18} - 420 \beta_{17} + 1281 \beta_{16} + 499 \beta_{12} + \cdots - 3035 \beta_{2} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2192 \beta_{15} + 719 \beta_{14} + 2024 \beta_{13} - 1824 \beta_{9} + 2728 \beta_{8} - 7000 \beta_{7} + \cdots + 16010 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 9911 \beta_{19} + 30528 \beta_{18} + 9144 \beta_{17} - 24882 \beta_{16} - 9962 \beta_{12} + \cdots + 57149 \beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 43718 \beta_{15} - 14996 \beta_{14} - 41531 \beta_{13} + 35354 \beta_{9} - 55684 \beta_{8} + \cdots - 300055 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 194242 \beta_{19} - 595144 \beta_{18} - 184921 \beta_{17} + 482240 \beta_{16} + \cdots - 1096608 \beta_{2} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 856464 \beta_{15} + 299676 \beta_{14} + 823548 \beta_{13} - 683952 \beta_{9} + 1100888 \beta_{8} + \cdots + 5743281 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 3783420 \beta_{19} + 11560432 \beta_{18} + 3643640 \beta_{17} - 9346116 \beta_{16} + \cdots + 21183641 \beta_{2} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 16672504 \beta_{15} - 5883936 \beta_{14} - 16113116 \beta_{13} + 13243336 \beta_{9} - 21507272 \beta_{8} + \cdots - 110816833 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 73503269 \beta_{19} - 224302179 \beta_{18} - 71094828 \beta_{17} + 181176015 \beta_{16} + \cdots - 410202023 \beta_{2} \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 323791056 \beta_{15} + 114685917 \beta_{14} + 313577999 \beta_{13} - 256622800 \beta_{9} + \cdots + 2144779450 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 1426455392 \beta_{19} + 4350427347 \beta_{18} + 1381985471 \beta_{17} - 3512712533 \beta_{16} + \cdots + 7950232518 \beta_{2} \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 6282527968 \beta_{15} - 2228640711 \beta_{14} - 6089482710 \beta_{13} + 4974697420 \beta_{9} + \cdots - 41559602026 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 27670205198 \beta_{19} - 84367618639 \beta_{18} - 26824434004 \beta_{17} + \cdots - 154136502515 \beta_{2} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\)
\(\chi(n)\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1701.1
− 2.40359i
− 0.786469i
− 0.762160i
0.0120170i
0.622139i
1.37786i
1.98798i
2.76216i
2.78647i
4.40359i
2.40359i
0.786469i
0.762160i
− 0.0120170i
− 0.622139i
− 1.37786i
− 1.98798i
− 2.76216i
− 2.78647i
− 4.40359i
− 1.00000i −3.40359 −1.00000 0 3.40359i 2.06225 1.00000i 8.58443 0
1701.2 − 1.00000i −1.78647 −1.00000 0 1.78647i 3.14934 1.00000i 0.191472 0
1701.3 − 1.00000i −1.76216 −1.00000 0 1.76216i −1.22131 1.00000i 0.105209 0
1701.4 − 1.00000i −0.987983 −1.00000 0 0.987983i −4.78937 1.00000i −2.02389 0
1701.5 − 1.00000i −0.377861 −1.00000 0 0.377861i 0.631751 1.00000i −2.85722 0
1701.6 − 1.00000i 0.377861 −1.00000 0 − 0.377861i −0.631751 1.00000i −2.85722 0
1701.7 − 1.00000i 0.987983 −1.00000 0 − 0.987983i 4.78937 1.00000i −2.02389 0
1701.8 − 1.00000i 1.76216 −1.00000 0 − 1.76216i 1.22131 1.00000i 0.105209 0
1701.9 − 1.00000i 1.78647 −1.00000 0 − 1.78647i −3.14934 1.00000i 0.191472 0
1701.10 − 1.00000i 3.40359 −1.00000 0 − 3.40359i −2.06225 1.00000i 8.58443 0
1701.11 1.00000i −3.40359 −1.00000 0 − 3.40359i 2.06225 − 1.00000i 8.58443 0
1701.12 1.00000i −1.78647 −1.00000 0 − 1.78647i 3.14934 − 1.00000i 0.191472 0
1701.13 1.00000i −1.76216 −1.00000 0 − 1.76216i −1.22131 − 1.00000i 0.105209 0
1701.14 1.00000i −0.987983 −1.00000 0 − 0.987983i −4.78937 − 1.00000i −2.02389 0
1701.15 1.00000i −0.377861 −1.00000 0 − 0.377861i 0.631751 − 1.00000i −2.85722 0
1701.16 1.00000i 0.377861 −1.00000 0 0.377861i −0.631751 − 1.00000i −2.85722 0
1701.17 1.00000i 0.987983 −1.00000 0 0.987983i 4.78937 − 1.00000i −2.02389 0
1701.18 1.00000i 1.76216 −1.00000 0 1.76216i 1.22131 − 1.00000i 0.105209 0
1701.19 1.00000i 1.78647 −1.00000 0 1.78647i −3.14934 − 1.00000i 0.191472 0
1701.20 1.00000i 3.40359 −1.00000 0 3.40359i −2.06225 − 1.00000i 8.58443 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1701.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1850.2.d.i 20
5.b even 2 1 inner 1850.2.d.i 20
5.c odd 4 1 370.2.c.a ✓ 10
5.c odd 4 1 370.2.c.b yes 10
15.e even 4 1 3330.2.e.c 10
15.e even 4 1 3330.2.e.d 10
37.b even 2 1 inner 1850.2.d.i 20
185.d even 2 1 inner 1850.2.d.i 20
185.h odd 4 1 370.2.c.a ✓ 10
185.h odd 4 1 370.2.c.b yes 10
555.n even 4 1 3330.2.e.c 10
555.n even 4 1 3330.2.e.d 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.c.a ✓ 10 5.c odd 4 1
370.2.c.a ✓ 10 185.h odd 4 1
370.2.c.b yes 10 5.c odd 4 1
370.2.c.b yes 10 185.h odd 4 1
1850.2.d.i 20 1.a even 1 1 trivial
1850.2.d.i 20 5.b even 2 1 inner
1850.2.d.i 20 37.b even 2 1 inner
1850.2.d.i 20 185.d even 2 1 inner
3330.2.e.c 10 15.e even 4 1
3330.2.e.c 10 555.n even 4 1
3330.2.e.d 10 15.e even 4 1
3330.2.e.d 10 555.n even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1850, [\chi])\):

\( T_{3}^{10} - 19T_{3}^{8} + 103T_{3}^{6} - 210T_{3}^{4} + 140T_{3}^{2} - 16 \) Copy content Toggle raw display
\( T_{7}^{10} - 39T_{7}^{8} + 438T_{7}^{6} - 1684T_{7}^{4} + 2048T_{7}^{2} - 576 \) Copy content Toggle raw display
\( T_{11}^{5} - 28T_{11}^{3} + 51T_{11}^{2} + 16T_{11} - 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{10} \) Copy content Toggle raw display
$3$ \( (T^{10} - 19 T^{8} + \cdots - 16)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( (T^{10} - 39 T^{8} + \cdots - 576)^{2} \) Copy content Toggle raw display
$11$ \( (T^{5} - 28 T^{3} + \cdots - 48)^{4} \) Copy content Toggle raw display
$13$ \( (T^{10} + 79 T^{8} + \cdots + 238144)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + 73 T^{8} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + 118 T^{8} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} + 151 T^{8} + \cdots + 589824)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + 182 T^{8} + \cdots + 2262016)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + 116 T^{8} + \cdots + 60516)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 48\!\cdots\!49 \) Copy content Toggle raw display
$41$ \( (T^{5} + 2 T^{4} - 44 T^{3} + \cdots - 36)^{4} \) Copy content Toggle raw display
$43$ \( (T^{10} + 201 T^{8} + \cdots + 6885376)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} - 310 T^{8} + \cdots - 4596736)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} - 257 T^{8} + \cdots - 39337984)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + 238 T^{8} + \cdots + 21827584)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + 150 T^{8} + \cdots + 82944)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} - 367 T^{8} + \cdots - 559417104)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + 10 T^{4} + \cdots + 4608)^{4} \) Copy content Toggle raw display
$73$ \( (T^{10} - 189 T^{8} + \cdots - 589824)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + 291 T^{8} + \cdots + 186486336)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} - 466 T^{8} + \cdots - 1024)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + 412 T^{8} + \cdots + 1230045184)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + 357 T^{8} + \cdots + 1364224)^{2} \) Copy content Toggle raw display
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