Properties

Label 850.2.c.b
Level $850$
Weight $2$
Character orbit 850.c
Analytic conductor $6.787$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 850.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.78728417181\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 34)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} + 2 i q^{3} - q^{4} -2 q^{6} -4 i q^{7} -i q^{8} - q^{9} +O(q^{10})\) \( q + i q^{2} + 2 i q^{3} - q^{4} -2 q^{6} -4 i q^{7} -i q^{8} - q^{9} + 6 q^{11} -2 i q^{12} -2 i q^{13} + 4 q^{14} + q^{16} -i q^{17} -i q^{18} + 4 q^{19} + 8 q^{21} + 6 i q^{22} + 2 q^{24} + 2 q^{26} + 4 i q^{27} + 4 i q^{28} -4 q^{31} + i q^{32} + 12 i q^{33} + q^{34} + q^{36} -4 i q^{37} + 4 i q^{38} + 4 q^{39} + 6 q^{41} + 8 i q^{42} -8 i q^{43} -6 q^{44} + 2 i q^{48} -9 q^{49} + 2 q^{51} + 2 i q^{52} + 6 i q^{53} -4 q^{54} -4 q^{56} + 8 i q^{57} -4 q^{61} -4 i q^{62} + 4 i q^{63} - q^{64} -12 q^{66} + 8 i q^{67} + i q^{68} + i q^{72} -2 i q^{73} + 4 q^{74} -4 q^{76} -24 i q^{77} + 4 i q^{78} -8 q^{79} -11 q^{81} + 6 i q^{82} -8 q^{84} + 8 q^{86} -6 i q^{88} + 6 q^{89} -8 q^{91} -8 i q^{93} -2 q^{96} + 14 i q^{97} -9 i q^{98} -6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 4q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 4q^{6} - 2q^{9} + 12q^{11} + 8q^{14} + 2q^{16} + 8q^{19} + 16q^{21} + 4q^{24} + 4q^{26} - 8q^{31} + 2q^{34} + 2q^{36} + 8q^{39} + 12q^{41} - 12q^{44} - 18q^{49} + 4q^{51} - 8q^{54} - 8q^{56} - 8q^{61} - 2q^{64} - 24q^{66} + 8q^{74} - 8q^{76} - 16q^{79} - 22q^{81} - 16q^{84} + 16q^{86} + 12q^{89} - 16q^{91} - 4q^{96} - 12q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/850\mathbb{Z}\right)^\times\).

\(n\) \(477\) \(751\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
749.1
1.00000i
1.00000i
1.00000i 2.00000i −1.00000 0 −2.00000 4.00000i 1.00000i −1.00000 0
749.2 1.00000i 2.00000i −1.00000 0 −2.00000 4.00000i 1.00000i −1.00000 0
\(n\): e.g. 2-40 or 990-1000
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 850.2.c.b 2
5.b even 2 1 inner 850.2.c.b 2
5.c odd 4 1 34.2.a.a 1
5.c odd 4 1 850.2.a.e 1
15.e even 4 1 306.2.a.a 1
15.e even 4 1 7650.2.a.ci 1
20.e even 4 1 272.2.a.d 1
20.e even 4 1 6800.2.a.b 1
35.f even 4 1 1666.2.a.m 1
40.i odd 4 1 1088.2.a.l 1
40.k even 4 1 1088.2.a.d 1
55.e even 4 1 4114.2.a.a 1
60.l odd 4 1 2448.2.a.k 1
65.h odd 4 1 5746.2.a.b 1
85.f odd 4 1 578.2.b.a 2
85.g odd 4 1 578.2.a.a 1
85.i odd 4 1 578.2.b.a 2
85.k odd 8 2 578.2.c.e 4
85.n odd 8 2 578.2.c.e 4
85.o even 16 4 578.2.d.e 8
85.r even 16 4 578.2.d.e 8
120.q odd 4 1 9792.2.a.bj 1
120.w even 4 1 9792.2.a.y 1
255.o even 4 1 5202.2.a.d 1
340.r even 4 1 4624.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
34.2.a.a 1 5.c odd 4 1
272.2.a.d 1 20.e even 4 1
306.2.a.a 1 15.e even 4 1
578.2.a.a 1 85.g odd 4 1
578.2.b.a 2 85.f odd 4 1
578.2.b.a 2 85.i odd 4 1
578.2.c.e 4 85.k odd 8 2
578.2.c.e 4 85.n odd 8 2
578.2.d.e 8 85.o even 16 4
578.2.d.e 8 85.r even 16 4
850.2.a.e 1 5.c odd 4 1
850.2.c.b 2 1.a even 1 1 trivial
850.2.c.b 2 5.b even 2 1 inner
1088.2.a.d 1 40.k even 4 1
1088.2.a.l 1 40.i odd 4 1
1666.2.a.m 1 35.f even 4 1
2448.2.a.k 1 60.l odd 4 1
4114.2.a.a 1 55.e even 4 1
4624.2.a.a 1 340.r even 4 1
5202.2.a.d 1 255.o even 4 1
5746.2.a.b 1 65.h odd 4 1
6800.2.a.b 1 20.e even 4 1
7650.2.a.ci 1 15.e even 4 1
9792.2.a.y 1 120.w even 4 1
9792.2.a.bj 1 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(850, [\chi])\):

\( T_{3}^{2} + 4 \)
\( T_{7}^{2} + 16 \)
\( T_{11} - 6 \)
\( T_{13}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 4 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 16 + T^{2} \)
$11$ \( ( -6 + T )^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 1 + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( 16 + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 4 + T )^{2} \)
$67$ \( 64 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 4 + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( 196 + T^{2} \)
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