Properties

Label 1850.2.c.k
Level $1850$
Weight $2$
Character orbit 1850.c
Analytic conductor $14.772$
Analytic rank $0$
Dimension $12$
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] = [1850,2,Mod(1849,1850)]
 
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([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1850.1849");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.c (of order \(2\), degree \(1\), not 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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 30x^{10} + 347x^{8} + 1968x^{6} + 5635x^{4} + 7246x^{2} + 2601 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_1 q^{6} + ( - \beta_{4} - \beta_{3}) q^{7} - q^{8} + ( - \beta_{7} + \beta_{6} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_1 q^{6} + ( - \beta_{4} - \beta_{3}) q^{7} - q^{8} + ( - \beta_{7} + \beta_{6} - 2) q^{9} + ( - \beta_{9} - 1) q^{11} + \beta_1 q^{12} + (\beta_{9} + \beta_{8} - \beta_{6}) q^{13} + (\beta_{4} + \beta_{3}) q^{14} + q^{16} + (\beta_{9} + \beta_{7} - \beta_{6} - 1) q^{17} + (\beta_{7} - \beta_{6} + 2) q^{18} + (\beta_{5} + \beta_{4}) q^{19} + ( - \beta_{11} + \beta_{10} - \beta_{8} + \beta_{7} - \beta_{6} + 1) q^{21} + (\beta_{9} + 1) q^{22} + (\beta_{9} - 2 \beta_{7} + \beta_{6} - 1) q^{23} - \beta_1 q^{24} + ( - \beta_{9} - \beta_{8} + \beta_{6}) q^{26} + ( - \beta_{5} - \beta_{4} - 3 \beta_{3} - 2 \beta_{2} - \beta_1) q^{27} + ( - \beta_{4} - \beta_{3}) q^{28} + (\beta_{11} + \beta_{10} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{29} + ( - \beta_{5} + 2 \beta_{3}) q^{31} - q^{32} + (2 \beta_{5} - 3 \beta_{3} - 2 \beta_1) q^{33} + ( - \beta_{9} - \beta_{7} + \beta_{6} + 1) q^{34} + ( - \beta_{7} + \beta_{6} - 2) q^{36} + (\beta_{11} + \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} + 2 \beta_{3} + 1) q^{37} + ( - \beta_{5} - \beta_{4}) q^{38} + (\beta_{11} + \beta_{10} - \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{39} + ( - \beta_{11} + \beta_{10} - 2 \beta_{9} - \beta_{8} - \beta_{7} + 2 \beta_{6} - 1) q^{41} + (\beta_{11} - \beta_{10} + \beta_{8} - \beta_{7} + \beta_{6} - 1) q^{42} + ( - 2 \beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{6} + 2) q^{43} + ( - \beta_{9} - 1) q^{44} + ( - \beta_{9} + 2 \beta_{7} - \beta_{6} + 1) q^{46} + (\beta_{11} + \beta_{10} - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_1) q^{47} + \beta_1 q^{48} + (2 \beta_{9} - \beta_{7} + \beta_{6} - 5) q^{49} + ( - \beta_{5} + \beta_{4} + 6 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{51} + (\beta_{9} + \beta_{8} - \beta_{6}) q^{52} + ( - 2 \beta_{5} + 2 \beta_1) q^{53} + (\beta_{5} + \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + \beta_1) q^{54} + (\beta_{4} + \beta_{3}) q^{56} + (\beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - 4) q^{57} + ( - \beta_{11} - \beta_{10} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{58} + (\beta_{11} + \beta_{10} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1) q^{59} + (2 \beta_{5} + 3 \beta_{3} + 2 \beta_{2} + \beta_1) q^{61} + (\beta_{5} - 2 \beta_{3}) q^{62} + (\beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{63} + q^{64} + ( - 2 \beta_{5} + 3 \beta_{3} + 2 \beta_1) q^{66} + \beta_{4} q^{67} + (\beta_{9} + \beta_{7} - \beta_{6} - 1) q^{68} + ( - 3 \beta_{5} - \beta_{4} - 5 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{69} + (\beta_{9} - \beta_{7} + 3 \beta_{6}) q^{71} + (\beta_{7} - \beta_{6} + 2) q^{72} + (2 \beta_{5} + 3 \beta_{3} + 3 \beta_{2} - \beta_1) q^{73} + ( - \beta_{11} - \beta_{9} - \beta_{8} + \beta_{6} + \beta_{5} - 2 \beta_{3} - 1) q^{74} + (\beta_{5} + \beta_{4}) q^{76} + (\beta_{11} + \beta_{10} + \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_1) q^{77} + ( - \beta_{11} - \beta_{10} + \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{78} + ( - \beta_{11} - \beta_{10} - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{79} + ( - \beta_{11} + \beta_{10} + \beta_{8} + 4 \beta_{7} - \beta_{6} - 1) q^{81} + (\beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} + 1) q^{82} + ( - \beta_{5} + 5 \beta_{3} - 2 \beta_{2}) q^{83} + ( - \beta_{11} + \beta_{10} - \beta_{8} + \beta_{7} - \beta_{6} + 1) q^{84} + (2 \beta_{9} - \beta_{8} - 2 \beta_{7} - \beta_{6} - 2) q^{86} + ( - \beta_{9} - 4 \beta_{8} + 2 \beta_{7} + \beta_{6} - 7) q^{87} + (\beta_{9} + 1) q^{88} + (\beta_{11} + \beta_{10} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2}) q^{89} + (\beta_{4} - 8 \beta_{3} - 3 \beta_{2} - 4 \beta_1) q^{91} + (\beta_{9} - 2 \beta_{7} + \beta_{6} - 1) q^{92} + ( - 2 \beta_{9} - \beta_{7} + 3) q^{93} + ( - \beta_{11} - \beta_{10} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_1) q^{94} - \beta_1 q^{96} + (\beta_{8} + \beta_{7} + 2 \beta_{6}) q^{97} + ( - 2 \beta_{9} + \beta_{7} - \beta_{6} + 5) q^{98} + (\beta_{9} + 3 \beta_{7} - 2 \beta_{6} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 12 q^{4} - 12 q^{8} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} + 12 q^{4} - 12 q^{8} - 24 q^{9} - 16 q^{11} + 12 q^{16} - 8 q^{17} + 24 q^{18} + 12 q^{21} + 16 q^{22} - 8 q^{23} - 12 q^{32} + 8 q^{34} - 24 q^{36} + 14 q^{37} - 20 q^{41} - 12 q^{42} + 12 q^{43} - 16 q^{44} + 8 q^{46} - 52 q^{49} - 40 q^{57} + 12 q^{64} - 8 q^{68} + 4 q^{71} + 24 q^{72} - 14 q^{74} - 20 q^{81} + 20 q^{82} + 12 q^{84} - 12 q^{86} - 72 q^{87} + 16 q^{88} - 8 q^{92} + 28 q^{93} - 4 q^{97} + 52 q^{98} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 30x^{10} + 347x^{8} + 1968x^{6} + 5635x^{4} + 7246x^{2} + 2601 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{11} - 51\nu^{9} - 472\nu^{7} - 1980\nu^{5} - 3650\nu^{3} - 2207\nu ) / 102 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{11} - 123\nu^{9} - 1069\nu^{7} - 4023\nu^{5} - 6080\nu^{3} - 2180\nu ) / 102 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{11} + 120\nu^{9} + 1012\nu^{7} + 3711\nu^{5} + 5597\nu^{3} + 2216\nu ) / 51 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{11} + 77\nu^{9} + 709\nu^{7} + 2869\nu^{5} + 4748\nu^{3} + 1936\nu ) / 34 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -9\nu^{10} - 222\nu^{8} - 1939\nu^{6} - 7365\nu^{4} - 11316\nu^{2} - 4165 ) / 34 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -9\nu^{10} - 222\nu^{8} - 1939\nu^{6} - 7365\nu^{4} - 11350\nu^{2} - 4335 ) / 34 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 8\nu^{10} + 203\nu^{8} + 1835\nu^{6} + 7221\nu^{4} + 11515\nu^{2} + 4522 ) / 34 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -11\nu^{10} - 277\nu^{8} - 2487\nu^{6} - 9761\nu^{4} - 15576\nu^{2} - 6018 ) / 34 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 41 \nu^{11} + 57 \nu^{10} + 1023 \nu^{9} + 1389 \nu^{8} + 9070 \nu^{7} + 11946 \nu^{6} + 35088 \nu^{5} + 44724 \nu^{4} + 55469 \nu^{3} + 68625 \nu^{2} + 22385 \nu + 26979 ) / 204 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 41 \nu^{11} - 57 \nu^{10} + 1023 \nu^{9} - 1389 \nu^{8} + 9070 \nu^{7} - 11946 \nu^{6} + 35088 \nu^{5} - 44724 \nu^{4} + 55469 \nu^{3} - 68625 \nu^{2} + 22385 \nu - 26979 ) / 204 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{6} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} - \beta_{4} - 3\beta_{3} - 2\beta_{2} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{8} + 13\beta_{7} - 10\beta_{6} + 35 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{11} + 2\beta_{10} + 10\beta_{5} + 13\beta_{4} + 50\beta_{3} + 26\beta_{2} + 58\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 15\beta_{11} - 15\beta_{10} - 6\beta_{9} - 21\beta_{8} - 144\beta_{7} + 101\beta_{6} - 289 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -36\beta_{11} - 36\beta_{10} - 89\beta_{5} - 140\beta_{4} - 617\beta_{3} - 296\beta_{2} - 540\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -176\beta_{11} + 176\beta_{10} + 118\beta_{9} + 300\beta_{8} + 1542\beta_{7} - 1048\beta_{6} + 2659 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 476\beta_{11} + 476\beta_{10} + 812\beta_{5} + 1452\beta_{4} + 6972\beta_{3} + 3242\beta_{2} + 5367\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1928\beta_{11} - 1928\beta_{10} - 1618\beta_{9} - 3694\beta_{8} - 16393\beta_{7} + 11013\beta_{6} - 26143 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -5622\beta_{11} - 5622\beta_{10} - 7777\beta_{5} - 15031\beta_{4} - 76199\beta_{3} - 34956\beta_{2} - 55167\beta_1 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1849.1
3.26032i
2.40433i
2.30444i
2.15864i
1.74941i
0.747618i
0.747618i
1.74941i
2.15864i
2.30444i
2.40433i
3.26032i
−1.00000 3.26032i 1.00000 0 3.26032i 3.90506i −1.00000 −7.62966 0
1849.2 −1.00000 2.40433i 1.00000 0 2.40433i 3.96576i −1.00000 −2.78082 0
1849.3 −1.00000 2.30444i 1.00000 0 2.30444i 3.13780i −1.00000 −2.31045 0
1849.4 −1.00000 2.15864i 1.00000 0 2.15864i 4.71880i −1.00000 −1.65972 0
1849.5 −1.00000 1.74941i 1.00000 0 1.74941i 0.281746i −1.00000 −0.0604218 0
1849.6 −1.00000 0.747618i 1.00000 0 0.747618i 2.19796i −1.00000 2.44107 0
1849.7 −1.00000 0.747618i 1.00000 0 0.747618i 2.19796i −1.00000 2.44107 0
1849.8 −1.00000 1.74941i 1.00000 0 1.74941i 0.281746i −1.00000 −0.0604218 0
1849.9 −1.00000 2.15864i 1.00000 0 2.15864i 4.71880i −1.00000 −1.65972 0
1849.10 −1.00000 2.30444i 1.00000 0 2.30444i 3.13780i −1.00000 −2.31045 0
1849.11 −1.00000 2.40433i 1.00000 0 2.40433i 3.96576i −1.00000 −2.78082 0
1849.12 −1.00000 3.26032i 1.00000 0 3.26032i 3.90506i −1.00000 −7.62966 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1849.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.c.k 12
5.b even 2 1 1850.2.c.l 12
5.c odd 4 1 1850.2.d.g 12
5.c odd 4 1 1850.2.d.h yes 12
37.b even 2 1 1850.2.c.l 12
185.d even 2 1 inner 1850.2.c.k 12
185.h odd 4 1 1850.2.d.g 12
185.h odd 4 1 1850.2.d.h yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1850.2.c.k 12 1.a even 1 1 trivial
1850.2.c.k 12 185.d even 2 1 inner
1850.2.c.l 12 5.b even 2 1
1850.2.c.l 12 37.b even 2 1
1850.2.d.g 12 5.c odd 4 1
1850.2.d.g 12 185.h odd 4 1
1850.2.d.h yes 12 5.c odd 4 1
1850.2.d.h yes 12 185.h odd 4 1

Hecke kernels

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

\( T_{3}^{12} + 30T_{3}^{10} + 347T_{3}^{8} + 1968T_{3}^{6} + 5635T_{3}^{4} + 7246T_{3}^{2} + 2601 \) Copy content Toggle raw display
\( T_{7}^{12} + 68T_{7}^{10} + 1764T_{7}^{8} + 21656T_{7}^{6} + 124304T_{7}^{4} + 263748T_{7}^{2} + 20164 \) Copy content Toggle raw display
\( T_{11}^{6} + 8T_{11}^{5} - 3T_{11}^{4} - 138T_{11}^{3} - 311T_{11}^{2} - 120T_{11} + 141 \) Copy content Toggle raw display
\( T_{13}^{6} - 50T_{13}^{4} - 56T_{13}^{3} + 560T_{13}^{2} + 978T_{13} - 258 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 30 T^{10} + 347 T^{8} + \cdots + 2601 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 68 T^{10} + 1764 T^{8} + \cdots + 20164 \) Copy content Toggle raw display
$11$ \( (T^{6} + 8 T^{5} - 3 T^{4} - 138 T^{3} + \cdots + 141)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 50 T^{4} - 56 T^{3} + 560 T^{2} + \cdots - 258)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 4 T^{5} - 43 T^{4} - 140 T^{3} + \cdots - 297)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 94 T^{10} + 3191 T^{8} + \cdots + 1766241 \) Copy content Toggle raw display
$23$ \( (T^{6} + 4 T^{5} - 78 T^{4} - 264 T^{3} + \cdots + 2088)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 308 T^{10} + \cdots + 2815788096 \) Copy content Toggle raw display
$31$ \( T^{12} + 76 T^{10} + 1352 T^{8} + \cdots + 900 \) Copy content Toggle raw display
$37$ \( T^{12} - 14 T^{11} + \cdots + 2565726409 \) Copy content Toggle raw display
$41$ \( (T^{6} + 10 T^{5} - 133 T^{4} + \cdots + 78357)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 6 T^{5} - 188 T^{4} + 1372 T^{3} + \cdots + 93684)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 404 T^{10} + \cdots + 2368963584 \) Copy content Toggle raw display
$53$ \( T^{12} + 248 T^{10} + 13296 T^{8} + \cdots + 2985984 \) Copy content Toggle raw display
$59$ \( T^{12} + 484 T^{10} + \cdots + 111260304 \) Copy content Toggle raw display
$61$ \( T^{12} + 380 T^{10} + \cdots + 2784672900 \) Copy content Toggle raw display
$67$ \( T^{12} + 66 T^{10} + 1595 T^{8} + \cdots + 36481 \) Copy content Toggle raw display
$71$ \( (T^{6} - 2 T^{5} - 284 T^{4} + \cdots - 395406)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 550 T^{10} + 110799 T^{8} + \cdots + 88209 \) Copy content Toggle raw display
$79$ \( T^{12} + 692 T^{10} + \cdots + 14841086976 \) Copy content Toggle raw display
$83$ \( T^{12} + 302 T^{10} + 23523 T^{8} + \cdots + 729 \) Copy content Toggle raw display
$89$ \( T^{12} + 578 T^{10} + \cdots + 3557287449 \) Copy content Toggle raw display
$97$ \( (T^{6} + 2 T^{5} - 176 T^{4} - 112 T^{3} + \cdots - 2256)^{2} \) Copy content Toggle raw display
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