Properties

Label 1850.2.b.n
Level $1850$
Weight $2$
Character orbit 1850.b
Analytic conductor $14.772$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{2} + 5\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9 \) 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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
0.403032 0.403032i
−0.854638 + 0.854638i
1.45161 1.45161i
1.45161 + 1.45161i
−0.854638 0.854638i
0.403032 + 0.403032i
1.00000i 2.67513i −1.00000 0 −2.67513 3.28726i 1.00000i −4.15633 0
149.2 1.00000i 1.53919i −1.00000 0 −1.53919 2.87936i 1.00000i 0.630898 0
149.3 1.00000i 1.21432i −1.00000 0 1.21432 3.59210i 1.00000i 1.52543 0
149.4 1.00000i 1.21432i −1.00000 0 1.21432 3.59210i 1.00000i 1.52543 0
149.5 1.00000i 1.53919i −1.00000 0 −1.53919 2.87936i 1.00000i 0.630898 0
149.6 1.00000i 2.67513i −1.00000 0 −2.67513 3.28726i 1.00000i −4.15633 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b 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.b.n 6
5.b even 2 1 inner 1850.2.b.n 6
5.c odd 4 1 1850.2.a.ba 3
5.c odd 4 1 1850.2.a.bb yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1850.2.a.ba 3 5.c odd 4 1
1850.2.a.bb yes 3 5.c odd 4 1
1850.2.b.n 6 1.a even 1 1 trivial
1850.2.b.n 6 5.b 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 \(S_{2}^{\mathrm{new}}(1850, [\chi])\):

\( T_{3}^{6} + 11T_{3}^{4} + 31T_{3}^{2} + 25 \) Copy content Toggle raw display
\( T_{7}^{6} + 32T_{7}^{4} + 336T_{7}^{2} + 1156 \) Copy content Toggle raw display
\( T_{13}^{6} + 56T_{13}^{4} + 832T_{13}^{2} + 2116 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + 11 T^{4} + 31 T^{2} + 25 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 32 T^{4} + 336 T^{2} + \cdots + 1156 \) Copy content Toggle raw display
$11$ \( (T^{3} + T^{2} - 23 T - 25)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 56 T^{4} + 832 T^{2} + \cdots + 2116 \) Copy content Toggle raw display
$17$ \( T^{6} + 43 T^{4} + 383 T^{2} + \cdots + 841 \) Copy content Toggle raw display
$19$ \( (T^{3} + 3 T^{2} - 25 T - 79)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 64 T^{4} + 512 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$29$ \( (T^{3} - 2 T^{2} - 52 T + 40)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 6 T^{2} - 18 T - 54)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$41$ \( (T^{3} + 21 T^{2} + 131 T + 215)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 320 T^{4} + 33600 T^{2} + \cdots + 1157776 \) Copy content Toggle raw display
$47$ \( T^{6} + 112 T^{4} + 2816 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( T^{6} + 236 T^{4} + 17584 T^{2} + \cdots + 399424 \) Copy content Toggle raw display
$59$ \( (T^{3} + 8 T^{2} - 128 T - 1076)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 8 T^{2} - 44 T - 290)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 187 T^{4} + 1823 T^{2} + \cdots + 4489 \) Copy content Toggle raw display
$71$ \( (T^{3} + 8 T^{2} - 44 T - 290)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 331 T^{4} + 34187 T^{2} + \cdots + 1100401 \) Copy content Toggle raw display
$79$ \( (T^{3} + 22 T^{2} + 140 T + 232)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 503 T^{4} + 76995 T^{2} + \cdots + 3308761 \) Copy content Toggle raw display
$89$ \( (T^{3} - 3 T^{2} - 27 T + 27)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 44 T^{4} + 496 T^{2} + \cdots + 1600 \) Copy content Toggle raw display
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