Properties

Label 770.2.m.c
Level $770$
Weight $2$
Character orbit 770.m
Analytic conductor $6.148$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.14848095564\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{8} q^{2} + ( 2 + 2 \zeta_{8}^{2} ) q^{3} + \zeta_{8}^{2} q^{4} + ( -1 + 2 \zeta_{8}^{2} ) q^{5} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{6} -\zeta_{8} q^{7} + \zeta_{8}^{3} q^{8} + 5 \zeta_{8}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{8} q^{2} + ( 2 + 2 \zeta_{8}^{2} ) q^{3} + \zeta_{8}^{2} q^{4} + ( -1 + 2 \zeta_{8}^{2} ) q^{5} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{6} -\zeta_{8} q^{7} + \zeta_{8}^{3} q^{8} + 5 \zeta_{8}^{2} q^{9} + ( -\zeta_{8} + 2 \zeta_{8}^{3} ) q^{10} + ( 3 - \zeta_{8} - \zeta_{8}^{3} ) q^{11} + ( -2 + 2 \zeta_{8}^{2} ) q^{12} + 2 \zeta_{8}^{3} q^{13} -\zeta_{8}^{2} q^{14} + ( -6 + 2 \zeta_{8}^{2} ) q^{15} - q^{16} -2 \zeta_{8} q^{17} + 5 \zeta_{8}^{3} q^{18} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{19} + ( -2 - \zeta_{8}^{2} ) q^{20} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{21} + ( 1 + 3 \zeta_{8} - \zeta_{8}^{2} ) q^{22} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{24} + ( -3 - 4 \zeta_{8}^{2} ) q^{25} -2 q^{26} + ( -4 + 4 \zeta_{8}^{2} ) q^{27} -\zeta_{8}^{3} q^{28} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{29} + ( -6 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{30} + 2 q^{31} -\zeta_{8} q^{32} + ( 6 + 6 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{33} -2 \zeta_{8}^{2} q^{34} + ( \zeta_{8} - 2 \zeta_{8}^{3} ) q^{35} -5 q^{36} + ( -4 + 4 \zeta_{8}^{2} ) q^{37} + ( 3 + 3 \zeta_{8}^{2} ) q^{38} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{39} + ( -2 \zeta_{8} - \zeta_{8}^{3} ) q^{40} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{41} + ( 2 - 2 \zeta_{8}^{2} ) q^{42} -8 \zeta_{8}^{3} q^{43} + ( \zeta_{8} + 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{44} + ( -10 - 5 \zeta_{8}^{2} ) q^{45} + ( 5 - 5 \zeta_{8}^{2} ) q^{47} + ( -2 - 2 \zeta_{8}^{2} ) q^{48} + \zeta_{8}^{2} q^{49} + ( -3 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{50} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{51} -2 \zeta_{8} q^{52} + ( -6 - 6 \zeta_{8}^{2} ) q^{53} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{54} + ( -3 + 3 \zeta_{8} + 6 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{55} + q^{56} + 12 \zeta_{8} q^{57} + ( -1 - \zeta_{8}^{2} ) q^{58} + 8 \zeta_{8}^{2} q^{59} + ( -2 - 6 \zeta_{8}^{2} ) q^{60} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{61} + 2 \zeta_{8} q^{62} -5 \zeta_{8}^{3} q^{63} -\zeta_{8}^{2} q^{64} + ( -4 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{65} + ( 4 + 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{66} + ( 3 - 3 \zeta_{8}^{2} ) q^{67} -2 \zeta_{8}^{3} q^{68} + ( 2 + \zeta_{8}^{2} ) q^{70} + 8 q^{71} -5 \zeta_{8} q^{72} -16 \zeta_{8}^{3} q^{73} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{74} + ( 2 - 14 \zeta_{8}^{2} ) q^{75} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{76} + ( -1 - 3 \zeta_{8} + \zeta_{8}^{2} ) q^{77} + ( -4 - 4 \zeta_{8}^{2} ) q^{78} + ( 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{79} + ( 1 - 2 \zeta_{8}^{2} ) q^{80} - q^{81} + ( -3 + 3 \zeta_{8}^{2} ) q^{82} + 10 \zeta_{8}^{3} q^{83} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{84} + ( 2 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{85} + 8 q^{86} -4 \zeta_{8} q^{87} + ( 1 + \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{88} -6 \zeta_{8}^{2} q^{89} + ( -10 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{90} + 2 q^{91} + ( 4 + 4 \zeta_{8}^{2} ) q^{93} + ( 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{94} + ( 3 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{95} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{96} + ( 2 - 2 \zeta_{8}^{2} ) q^{97} + \zeta_{8}^{3} q^{98} + ( 5 \zeta_{8} + 15 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{3} - 4q^{5} + O(q^{10}) \) \( 4q + 8q^{3} - 4q^{5} + 12q^{11} - 8q^{12} - 24q^{15} - 4q^{16} - 8q^{20} + 4q^{22} - 12q^{25} - 8q^{26} - 16q^{27} + 8q^{31} + 24q^{33} - 20q^{36} - 16q^{37} + 12q^{38} + 8q^{42} - 40q^{45} + 20q^{47} - 8q^{48} - 24q^{53} - 12q^{55} + 4q^{56} - 4q^{58} - 8q^{60} + 16q^{66} + 12q^{67} + 8q^{70} + 32q^{71} + 8q^{75} - 4q^{77} - 16q^{78} + 4q^{80} - 4q^{81} - 12q^{82} + 32q^{86} + 4q^{88} + 8q^{91} + 16q^{93} + 8q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/770\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(617\) \(661\)
\(\chi(n)\) \(-1\) \(-\zeta_{8}^{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i 2.00000 + 2.00000i 1.00000i −1.00000 + 2.00000i 2.82843i 0.707107 + 0.707107i 0.707107 0.707107i 5.00000i 2.12132 0.707107i
43.2 0.707107 + 0.707107i 2.00000 + 2.00000i 1.00000i −1.00000 + 2.00000i 2.82843i −0.707107 0.707107i −0.707107 + 0.707107i 5.00000i −2.12132 + 0.707107i
197.1 −0.707107 + 0.707107i 2.00000 2.00000i 1.00000i −1.00000 2.00000i 2.82843i 0.707107 0.707107i 0.707107 + 0.707107i 5.00000i 2.12132 + 0.707107i
197.2 0.707107 0.707107i 2.00000 2.00000i 1.00000i −1.00000 2.00000i 2.82843i −0.707107 + 0.707107i −0.707107 0.707107i 5.00000i −2.12132 0.707107i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.b odd 2 1 inner
55.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.m.c 4
5.c odd 4 1 inner 770.2.m.c 4
11.b odd 2 1 inner 770.2.m.c 4
55.e even 4 1 inner 770.2.m.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.m.c 4 1.a even 1 1 trivial
770.2.m.c 4 5.c odd 4 1 inner
770.2.m.c 4 11.b odd 2 1 inner
770.2.m.c 4 55.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\):

\( T_{3}^{2} - 4 T_{3} + 8 \)
\( T_{17}^{4} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{4} \)
$3$ \( ( 8 - 4 T + T^{2} )^{2} \)
$5$ \( ( 5 + 2 T + T^{2} )^{2} \)
$7$ \( 1 + T^{4} \)
$11$ \( ( 11 - 6 T + T^{2} )^{2} \)
$13$ \( 16 + T^{4} \)
$17$ \( 16 + T^{4} \)
$19$ \( ( -18 + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( ( -2 + T^{2} )^{2} \)
$31$ \( ( -2 + T )^{4} \)
$37$ \( ( 32 + 8 T + T^{2} )^{2} \)
$41$ \( ( 18 + T^{2} )^{2} \)
$43$ \( 4096 + T^{4} \)
$47$ \( ( 50 - 10 T + T^{2} )^{2} \)
$53$ \( ( 72 + 12 T + T^{2} )^{2} \)
$59$ \( ( 64 + T^{2} )^{2} \)
$61$ \( ( 32 + T^{2} )^{2} \)
$67$ \( ( 18 - 6 T + T^{2} )^{2} \)
$71$ \( ( -8 + T )^{4} \)
$73$ \( 65536 + T^{4} \)
$79$ \( ( -50 + T^{2} )^{2} \)
$83$ \( 10000 + T^{4} \)
$89$ \( ( 36 + T^{2} )^{2} \)
$97$ \( ( 8 - 4 T + T^{2} )^{2} \)
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