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Properties

Label 1870.2.c.e
Level $1870$
Weight $2$
Character orbit 1870.c
Analytic conductor $14.932$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1870,2,Mod(441,1870)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1870, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1870.441");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1870 = 2 \cdot 5 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1870.c (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: \(14.9320251780\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 39 x^{16} + 629 x^{14} + 5475 x^{12} + 28167 x^{10} + 87917 x^{8} + 163243 x^{6} + 166385 x^{4} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_{5} q^{5} - \beta_1 q^{6} - \beta_{7} q^{7} - q^{8} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_{5} q^{5} - \beta_1 q^{6} - \beta_{7} q^{7} - q^{8} + (\beta_{2} - 1) q^{9} + \beta_{5} q^{10} + \beta_{5} q^{11} + \beta_1 q^{12} + (\beta_{9} + \beta_{2} + 1) q^{13} + \beta_{7} q^{14} - \beta_{4} q^{15} + q^{16} + ( - \beta_{17} + \beta_{10} - \beta_{7}) q^{17} + ( - \beta_{2} + 1) q^{18} + ( - \beta_{11} - 1) q^{19} - \beta_{5} q^{20} + (\beta_{14} - \beta_{13} - \beta_{4} + \cdots + 2) q^{21}+ \cdots + (\beta_{6} - \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 18 q^{2} + 18 q^{4} - 18 q^{8} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 18 q^{2} + 18 q^{4} - 18 q^{8} - 24 q^{9} + 10 q^{13} - 2 q^{15} + 18 q^{16} + 2 q^{17} + 24 q^{18} - 26 q^{19} + 24 q^{21} - 18 q^{25} - 10 q^{26} + 2 q^{30} - 18 q^{32} + 2 q^{33} - 2 q^{34} - 24 q^{36} + 26 q^{38} - 24 q^{42} - 4 q^{43} - 2 q^{47} - 30 q^{49} + 18 q^{50} + 14 q^{51} + 10 q^{52} - 42 q^{53} + 18 q^{55} - 26 q^{59} - 2 q^{60} + 18 q^{64} - 2 q^{66} + 20 q^{67} + 2 q^{68} + 20 q^{69} + 24 q^{72} - 26 q^{76} - 14 q^{81} - 4 q^{83} + 24 q^{84} + 6 q^{85} + 4 q^{86} - 78 q^{87} + 50 q^{89} - 58 q^{93} + 2 q^{94} + 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 39 x^{16} + 629 x^{14} + 5475 x^{12} + 28167 x^{10} + 87917 x^{8} + 163243 x^{6} + 166385 x^{4} + \cdots + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2005 \nu^{16} + 65976 \nu^{14} + 839751 \nu^{12} + 5211994 \nu^{10} + 16225927 \nu^{8} + \cdots - 1751616 ) / 625376 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 732 \nu^{16} - 26631 \nu^{14} - 391246 \nu^{12} - 3000545 \nu^{10} - 12964740 \nu^{8} + \cdots - 1139008 ) / 156344 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17797 \nu^{17} + 647235 \nu^{15} + 9489929 \nu^{13} + 72398831 \nu^{11} + 309253219 \nu^{9} + \cdots - 17387520 \nu ) / 10006016 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6085 \nu^{17} + 221139 \nu^{15} + 3229993 \nu^{13} + 24390111 \nu^{11} + 101817379 \nu^{9} + \cdots - 35611648 \nu ) / 2501504 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14725 \nu^{17} + 517531 \nu^{15} + 7289241 \nu^{13} + 53262999 \nu^{11} + 219299907 \nu^{9} + \cdots - 9168000 \nu ) / 5003008 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 38745 \nu^{17} - 1541663 \nu^{15} - 25224013 \nu^{13} - 219552987 \nu^{11} + \cdots - 526054784 \nu ) / 10006016 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1862 \nu^{16} + 66300 \nu^{14} + 945667 \nu^{12} + 6966203 \nu^{10} + 28547800 \nu^{8} + \cdots + 2136524 ) / 78172 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 21667 \nu^{17} - 215648 \nu^{16} + 732765 \nu^{15} - 7735312 \nu^{14} + 9747951 \nu^{13} + \cdots - 332381696 ) / 10006016 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 16358 \nu^{16} + 595283 \nu^{14} + 8740716 \nu^{12} + 66918017 \nu^{10} + 288404302 \nu^{8} + \cdots + 21666976 ) / 625376 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 21667 \nu^{17} + 215648 \nu^{16} + 732765 \nu^{15} + 7735312 \nu^{14} + 9747951 \nu^{13} + \cdots + 332381696 ) / 10006016 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 117853 \nu^{17} - 33320 \nu^{16} + 4232923 \nu^{15} - 1120592 \nu^{14} + 61093921 \nu^{13} + \cdots - 33871360 ) / 10006016 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 117853 \nu^{17} + 33320 \nu^{16} + 4232923 \nu^{15} + 1120592 \nu^{14} + 61093921 \nu^{13} + \cdots + 33871360 ) / 10006016 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 48535 \nu^{16} - 1761514 \nu^{14} - 25772001 \nu^{12} - 196355844 \nu^{10} - 840824037 \nu^{8} + \cdots - 80492064 ) / 625376 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 204973 \nu^{17} + 7234571 \nu^{15} + 101894385 \nu^{13} + 736852279 \nu^{11} + \cdots + 407180288 \nu ) / 10006016 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 263959 \nu^{17} + 9567169 \nu^{15} + 139680323 \nu^{13} + 1060664229 \nu^{11} + \cdots + 385565696 \nu ) / 10006016 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} - \beta_{14} - \beta_{13} + \beta_{12} + \beta_{10} - \beta_{7} - \beta_{6} - \beta_{5} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{15} - 2\beta_{11} - \beta_{9} + \beta_{4} + \beta_{3} - 12\beta_{2} + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 13 \beta_{17} - \beta_{16} + 14 \beta_{14} + 14 \beta_{13} - 11 \beta_{12} - 11 \beta_{10} + \cdots + 45 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 15\beta_{15} + \beta_{14} - \beta_{13} + 28\beta_{11} + 17\beta_{9} - 18\beta_{4} - 20\beta_{3} + 127\beta_{2} - 206 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 138 \beta_{17} + 17 \beta_{16} - 155 \beta_{14} - 155 \beta_{13} + 106 \beta_{12} + 106 \beta_{10} + \cdots - 382 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 172 \beta_{15} - 21 \beta_{14} + 21 \beta_{13} + 8 \beta_{12} - 310 \beta_{11} - 8 \beta_{10} + \cdots + 1804 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1396 \beta_{17} - 198 \beta_{16} + 1594 \beta_{14} + 1594 \beta_{13} - 1010 \beta_{12} + \cdots + 3473 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1792 \beta_{15} + 332 \beta_{14} - 332 \beta_{13} - 226 \beta_{12} + 3188 \beta_{11} + 226 \beta_{10} + \cdots - 16594 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 13953 \beta_{17} + 1978 \beta_{16} - 15915 \beta_{14} - 15915 \beta_{13} + 9691 \beta_{12} + \cdots - 32798 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 17877 \beta_{15} - 4612 \beta_{14} + 4612 \beta_{13} + 4118 \beta_{12} - 31830 \beta_{11} + \cdots + 156544 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 139205 \beta_{17} - 18183 \beta_{16} + 156888 \beta_{14} + 156888 \beta_{13} - 93701 \beta_{12} + \cdots + 316449 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 174571 \beta_{15} + 59125 \beta_{14} - 59125 \beta_{13} - 61670 \beta_{12} + 313776 \beta_{11} + \cdots - 1497784 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1390274 \beta_{17} + 158003 \beta_{16} - 1538541 \beta_{14} - 1538541 \beta_{13} + 911328 \beta_{12} + \cdots - 3091754 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 1686808 \beta_{15} - 718013 \beta_{14} + 718013 \beta_{13} + 827306 \beta_{12} - 3077082 \beta_{11} + \cdots + 14455622 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 13909056 \beta_{17} - 1308584 \beta_{16} + 15065092 \beta_{14} + 15065092 \beta_{13} + \cdots + 30441073 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1870\mathbb{Z}\right)^\times\).

\(n\) \(1431\) \(1497\) \(1531\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
441.1
− 3.19168i
− 3.02501i
− 2.35473i
− 2.11699i
− 1.77959i
− 1.69204i
− 1.43334i
− 1.21981i
− 0.252582i
0.252582i
1.21981i
1.43334i
1.69204i
1.77959i
2.11699i
2.35473i
3.02501i
3.19168i
−1.00000 − 3.19168i 1.00000 − 1.00000i 3.19168i 1.31415i −1.00000 −7.18680 1.00000i
441.2 −1.00000 − 3.02501i 1.00000 1.00000i 3.02501i 3.23390i −1.00000 −6.15066 − 1.00000i
441.3 −1.00000 − 2.35473i 1.00000 1.00000i 2.35473i 2.21743i −1.00000 −2.54474 − 1.00000i
441.4 −1.00000 − 2.11699i 1.00000 − 1.00000i 2.11699i − 5.00934i −1.00000 −1.48164 1.00000i
441.5 −1.00000 − 1.77959i 1.00000 − 1.00000i 1.77959i 3.32458i −1.00000 −0.166944 1.00000i
441.6 −1.00000 − 1.69204i 1.00000 − 1.00000i 1.69204i 2.36158i −1.00000 0.136984 1.00000i
441.7 −1.00000 − 1.43334i 1.00000 1.00000i 1.43334i − 0.837735i −1.00000 0.945545 − 1.00000i
441.8 −1.00000 − 1.21981i 1.00000 1.00000i 1.21981i − 4.05299i −1.00000 1.51206 − 1.00000i
441.9 −1.00000 − 0.252582i 1.00000 − 1.00000i 0.252582i − 1.43037i −1.00000 2.93620 1.00000i
441.10 −1.00000 0.252582i 1.00000 1.00000i − 0.252582i 1.43037i −1.00000 2.93620 − 1.00000i
441.11 −1.00000 1.21981i 1.00000 − 1.00000i − 1.21981i 4.05299i −1.00000 1.51206 1.00000i
441.12 −1.00000 1.43334i 1.00000 − 1.00000i − 1.43334i 0.837735i −1.00000 0.945545 1.00000i
441.13 −1.00000 1.69204i 1.00000 1.00000i − 1.69204i − 2.36158i −1.00000 0.136984 − 1.00000i
441.14 −1.00000 1.77959i 1.00000 1.00000i − 1.77959i − 3.32458i −1.00000 −0.166944 − 1.00000i
441.15 −1.00000 2.11699i 1.00000 1.00000i − 2.11699i 5.00934i −1.00000 −1.48164 − 1.00000i
441.16 −1.00000 2.35473i 1.00000 − 1.00000i − 2.35473i − 2.21743i −1.00000 −2.54474 1.00000i
441.17 −1.00000 3.02501i 1.00000 − 1.00000i − 3.02501i − 3.23390i −1.00000 −6.15066 1.00000i
441.18 −1.00000 3.19168i 1.00000 1.00000i − 3.19168i − 1.31415i −1.00000 −7.18680 − 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 441.18
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 1870.2.c.e ✓ 18
17.b even 2 1 inner 1870.2.c.e ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1870.2.c.e ✓ 18 1.a even 1 1 trivial
1870.2.c.e ✓ 18 17.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} + 39 T_{3}^{16} + 629 T_{3}^{14} + 5475 T_{3}^{12} + 28167 T_{3}^{10} + 87917 T_{3}^{8} + \cdots + 4096 \) acting on \(S_{2}^{\mathrm{new}}(1870, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{18} \) Copy content Toggle raw display
$3$ \( T^{18} + 39 T^{16} + \cdots + 4096 \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{18} + 78 T^{16} + \cdots + 3240000 \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$13$ \( (T^{9} - 5 T^{8} + \cdots - 1072)^{2} \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 118587876497 \) Copy content Toggle raw display
$19$ \( (T^{9} + 13 T^{8} + \cdots + 6588)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 65029080064 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 1119000077584 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 361456144 \) Copy content Toggle raw display
$37$ \( T^{18} + 110 T^{16} + \cdots + 45589504 \) Copy content Toggle raw display
$41$ \( T^{18} + 424 T^{16} + \cdots + 14017536 \) Copy content Toggle raw display
$43$ \( (T^{9} + 2 T^{8} + \cdots + 7824384)^{2} \) Copy content Toggle raw display
$47$ \( (T^{9} + T^{8} + \cdots - 474688)^{2} \) Copy content Toggle raw display
$53$ \( (T^{9} + 21 T^{8} + \cdots - 13605044)^{2} \) Copy content Toggle raw display
$59$ \( (T^{9} + 13 T^{8} + \cdots + 14400)^{2} \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 805333813587216 \) Copy content Toggle raw display
$67$ \( (T^{9} - 10 T^{8} + \cdots - 5552928)^{2} \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 869306256 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 32\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 43\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( (T^{9} + 2 T^{8} + \cdots - 33691264)^{2} \) Copy content Toggle raw display
$89$ \( (T^{9} - 25 T^{8} + \cdots + 3864844)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 9521322320896 \) Copy content Toggle raw display
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