Properties

Label 770.2.i.l
Level $770$
Weight $2$
Character orbit 770.i
Analytic conductor $6.148$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.14848095564\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - x^{7} + 8 x^{6} + 3 x^{5} + 50 x^{4} - 12 x^{3} + 11 x^{2} + 2 x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{5} ) q^{2} + \beta_{1} q^{3} + \beta_{5} q^{4} + ( 1 + \beta_{5} ) q^{5} + \beta_{3} q^{6} + ( -\beta_{1} - \beta_{3} + \beta_{4} ) q^{7} + q^{8} + ( -1 + \beta_{1} + \beta_{3} - \beta_{5} + \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{5} ) q^{2} + \beta_{1} q^{3} + \beta_{5} q^{4} + ( 1 + \beta_{5} ) q^{5} + \beta_{3} q^{6} + ( -\beta_{1} - \beta_{3} + \beta_{4} ) q^{7} + q^{8} + ( -1 + \beta_{1} + \beta_{3} - \beta_{5} + \beta_{7} ) q^{9} -\beta_{5} q^{10} -\beta_{5} q^{11} + ( -\beta_{1} - \beta_{3} ) q^{12} + ( -1 + \beta_{2} + \beta_{3} - \beta_{6} ) q^{13} + ( \beta_{1} + \beta_{6} ) q^{14} -\beta_{3} q^{15} + ( -1 - \beta_{5} ) q^{16} + \beta_{4} q^{17} + ( 1 - \beta_{1} - \beta_{2} + \beta_{5} - \beta_{7} ) q^{18} + ( -4 - 2 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{19} - q^{20} + ( 2 + \beta_{1} + \beta_{2} - \beta_{5} ) q^{21} - q^{22} + ( -2 + 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{23} + \beta_{1} q^{24} + \beta_{5} q^{25} + ( -\beta_{1} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{26} + ( -2 - \beta_{2} + 2 \beta_{3} - \beta_{6} ) q^{27} + ( \beta_{3} - \beta_{4} - \beta_{6} ) q^{28} + ( 2 + \beta_{3} - \beta_{6} ) q^{29} + ( \beta_{1} + \beta_{3} ) q^{30} + ( -1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{5} + \beta_{7} ) q^{31} + \beta_{5} q^{32} + ( \beta_{1} + \beta_{3} ) q^{33} + \beta_{6} q^{34} + ( -\beta_{1} - \beta_{6} ) q^{35} + ( \beta_{2} - \beta_{3} ) q^{36} + ( 2 + \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{37} + ( 1 - \beta_{2} + 2 \beta_{4} + 4 \beta_{5} - \beta_{7} ) q^{38} + ( 1 + 3 \beta_{1} - \beta_{2} - \beta_{4} + 4 \beta_{5} - \beta_{7} ) q^{39} + ( 1 + \beta_{5} ) q^{40} -2 \beta_{6} q^{41} + ( -4 + \beta_{3} - 3 \beta_{5} + \beta_{7} ) q^{42} + ( 6 + 2 \beta_{3} - 2 \beta_{6} ) q^{43} + ( 1 + \beta_{5} ) q^{44} + ( -1 + \beta_{1} + \beta_{2} - \beta_{5} + \beta_{7} ) q^{45} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} ) q^{46} + ( 4 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} ) q^{47} + \beta_{3} q^{48} + ( -1 - \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{7} ) q^{49} + q^{50} + ( -1 + \beta_{1} + \beta_{3} - \beta_{5} ) q^{51} + ( 1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{7} ) q^{52} + ( 1 + 2 \beta_{1} - \beta_{2} + 3 \beta_{4} + 6 \beta_{5} - \beta_{7} ) q^{53} + ( 3 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{54} + q^{55} + ( -\beta_{1} - \beta_{3} + \beta_{4} ) q^{56} + ( 3 + 5 \beta_{3} - \beta_{6} ) q^{57} + ( -2 - \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{58} + ( -1 - 3 \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{59} -\beta_{1} q^{60} + ( -3 \beta_{1} - 3 \beta_{3} - \beta_{7} ) q^{61} + ( -1 - \beta_{2} + 2 \beta_{3} ) q^{62} + ( -4 + 5 \beta_{1} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{63} + q^{64} + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{65} -\beta_{1} q^{66} + ( -2 \beta_{1} + 7 \beta_{5} ) q^{67} + ( -\beta_{4} - \beta_{6} ) q^{68} + ( -4 - 2 \beta_{2} + 2 \beta_{3} ) q^{69} + ( -\beta_{3} + \beta_{4} + \beta_{6} ) q^{70} + ( 7 - \beta_{2} + 4 \beta_{3} + 2 \beta_{6} ) q^{71} + ( -1 + \beta_{1} + \beta_{3} - \beta_{5} + \beta_{7} ) q^{72} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} + 5 \beta_{5} - \beta_{7} ) q^{73} + ( -2 + 2 \beta_{2} - \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{74} + ( -\beta_{1} - \beta_{3} ) q^{75} + ( 3 + \beta_{2} + 2 \beta_{6} ) q^{76} + ( -\beta_{3} + \beta_{4} + \beta_{6} ) q^{77} + ( 3 + \beta_{2} + 3 \beta_{3} - \beta_{6} ) q^{78} + ( 6 - 3 \beta_{1} - 3 \beta_{3} + 5 \beta_{4} + 6 \beta_{5} + 5 \beta_{6} - \beta_{7} ) q^{79} -\beta_{5} q^{80} + ( -1 - 4 \beta_{1} + \beta_{2} + \beta_{4} + 3 \beta_{5} + \beta_{7} ) q^{81} + ( 2 \beta_{4} + 2 \beta_{6} ) q^{82} + ( -2 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} ) q^{83} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} + 4 \beta_{5} - \beta_{7} ) q^{84} -\beta_{6} q^{85} + ( -6 - 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 6 \beta_{5} + 2 \beta_{6} ) q^{86} + ( 1 + 2 \beta_{1} - \beta_{2} + 3 \beta_{5} - \beta_{7} ) q^{87} -\beta_{5} q^{88} + ( 2 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{89} + ( -\beta_{2} + \beta_{3} ) q^{90} + ( -4 - 3 \beta_{1} - 3 \beta_{3} - 3 \beta_{4} - 10 \beta_{5} + \beta_{6} + \beta_{7} ) q^{91} + ( 2 - 2 \beta_{3} + 2 \beta_{6} ) q^{92} + ( -7 + 7 \beta_{1} + 7 \beta_{3} - \beta_{4} - 7 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{93} + ( -2 \beta_{1} - 2 \beta_{4} - 4 \beta_{5} ) q^{94} + ( -1 + \beta_{2} - 2 \beta_{4} - 4 \beta_{5} + \beta_{7} ) q^{95} + ( -\beta_{1} - \beta_{3} ) q^{96} + ( 1 - \beta_{2} + 3 \beta_{3} + 3 \beta_{6} ) q^{97} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{98} + ( -\beta_{2} + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{2} + q^{3} - 4q^{4} + 4q^{5} - 2q^{6} + 3q^{7} + 8q^{8} - 3q^{9} + O(q^{10}) \) \( 8q - 4q^{2} + q^{3} - 4q^{4} + 4q^{5} - 2q^{6} + 3q^{7} + 8q^{8} - 3q^{9} + 4q^{10} + 4q^{11} + q^{12} - 2q^{13} - 3q^{14} + 2q^{15} - 4q^{16} + 2q^{17} - 3q^{18} - 10q^{19} - 8q^{20} + 25q^{21} - 8q^{22} - 6q^{23} + q^{24} - 4q^{25} + q^{26} - 20q^{27} + 18q^{29} - q^{30} + 8q^{31} - 4q^{32} - q^{33} - 4q^{34} + 3q^{35} + 6q^{36} + 2q^{37} - 10q^{38} - 13q^{39} + 4q^{40} + 8q^{41} - 20q^{42} + 52q^{43} + 4q^{44} + 3q^{45} - 6q^{46} + 10q^{47} - 2q^{48} - 13q^{49} + 8q^{50} - 5q^{51} + q^{52} - 14q^{53} + 10q^{54} + 8q^{55} + 3q^{56} + 18q^{57} - 9q^{58} + q^{59} - q^{60} + q^{61} - 16q^{62} - 7q^{63} + 8q^{64} - q^{65} - q^{66} - 30q^{67} + 2q^{68} - 44q^{69} + 36q^{71} - 3q^{72} - 17q^{73} + 2q^{74} + q^{75} + 20q^{76} + 26q^{78} + 15q^{79} + 4q^{80} - 16q^{81} - 4q^{82} + 4q^{83} - 5q^{84} + 4q^{85} - 26q^{86} - 8q^{87} + 4q^{88} + 6q^{89} - 6q^{90} + 3q^{91} + 12q^{92} - 29q^{93} + 10q^{94} + 10q^{95} + q^{96} - 14q^{97} + 2q^{98} - 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} + 8 x^{6} + 3 x^{5} + 50 x^{4} - 12 x^{3} + 11 x^{2} + 2 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 84 \nu^{7} - 31 \nu^{6} + 636 \nu^{5} + 456 \nu^{4} + 4685 \nu^{3} + 156 \nu^{2} + 36 \nu - 3755 \)\()/1381\)
\(\beta_{3}\)\(=\)\((\)\( 371 \nu^{7} - 252 \nu^{6} + 2809 \nu^{5} + 2014 \nu^{4} + 19196 \nu^{3} + 689 \nu^{2} + 159 \nu + 793 \)\()/4143\)
\(\beta_{4}\)\(=\)\((\)\( 634 \nu^{7} - 1056 \nu^{6} + 5984 \nu^{5} - 1885 \nu^{4} + 34144 \nu^{3} - 26048 \nu^{2} + 36375 \nu - 2464 \)\()/4143\)
\(\beta_{5}\)\(=\)\((\)\( 793 \nu^{7} - 1164 \nu^{6} + 6596 \nu^{5} - 430 \nu^{4} + 37636 \nu^{3} - 28712 \nu^{2} + 8034 \nu - 2716 \)\()/4143\)
\(\beta_{6}\)\(=\)\((\)\( 2716 \nu^{7} - 1923 \nu^{6} + 20564 \nu^{5} + 14744 \nu^{4} + 135370 \nu^{3} + 5044 \nu^{2} + 1164 \nu + 9323 \)\()/4143\)
\(\beta_{7}\)\(=\)\((\)\( 2801 \nu^{7} - 4404 \nu^{6} + 23575 \nu^{5} - 3734 \nu^{4} + 131348 \nu^{3} - 111394 \nu^{2} + 27834 \nu + 4915 \)\()/4143\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} - 4 \beta_{5} + \beta_{3} + \beta_{1} - 4\)
\(\nu^{3}\)\(=\)\(-\beta_{6} + 8 \beta_{3} - \beta_{2} - 2\)
\(\nu^{4}\)\(=\)\(-8 \beta_{7} + 30 \beta_{5} + \beta_{4} - 8 \beta_{2} - 13 \beta_{1} + 8\)
\(\nu^{5}\)\(=\)\(-13 \beta_{7} + 8 \beta_{6} + 43 \beta_{5} + 8 \beta_{4} - 66 \beta_{3} - 66 \beta_{1} + 43\)
\(\nu^{6}\)\(=\)\(13 \beta_{6} - 140 \beta_{3} + 66 \beta_{2} + 177\)
\(\nu^{7}\)\(=\)\(140 \beta_{7} - 481 \beta_{5} - 66 \beta_{4} + 140 \beta_{2} + 568 \beta_{1} - 140\)

Character values

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

\(n\) \(211\) \(617\) \(661\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1
−1.15777 + 2.00531i
−0.132681 + 0.229810i
0.267083 0.462601i
1.52336 2.63854i
−1.15777 2.00531i
−0.132681 0.229810i
0.267083 + 0.462601i
1.52336 + 2.63854i
−0.500000 0.866025i −1.15777 + 2.00531i −0.500000 + 0.866025i 0.500000 + 0.866025i 2.31553 −0.873699 2.49733i 1.00000 −1.18085 2.04528i 0.500000 0.866025i
221.2 −0.500000 0.866025i −0.132681 + 0.229810i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.265362 −1.51690 + 2.16772i 1.00000 1.46479 + 2.53709i 0.500000 0.866025i
221.3 −0.500000 0.866025i 0.267083 0.462601i −0.500000 + 0.866025i 0.500000 + 0.866025i −0.534166 1.70312 2.02469i 1.00000 1.35733 + 2.35097i 0.500000 0.866025i
221.4 −0.500000 0.866025i 1.52336 2.63854i −0.500000 + 0.866025i 0.500000 + 0.866025i −3.04673 2.18747 + 1.48827i 1.00000 −3.14128 5.44086i 0.500000 0.866025i
331.1 −0.500000 + 0.866025i −1.15777 2.00531i −0.500000 0.866025i 0.500000 0.866025i 2.31553 −0.873699 + 2.49733i 1.00000 −1.18085 + 2.04528i 0.500000 + 0.866025i
331.2 −0.500000 + 0.866025i −0.132681 0.229810i −0.500000 0.866025i 0.500000 0.866025i 0.265362 −1.51690 2.16772i 1.00000 1.46479 2.53709i 0.500000 + 0.866025i
331.3 −0.500000 + 0.866025i 0.267083 + 0.462601i −0.500000 0.866025i 0.500000 0.866025i −0.534166 1.70312 + 2.02469i 1.00000 1.35733 2.35097i 0.500000 + 0.866025i
331.4 −0.500000 + 0.866025i 1.52336 + 2.63854i −0.500000 0.866025i 0.500000 0.866025i −3.04673 2.18747 1.48827i 1.00000 −3.14128 + 5.44086i 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 331.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.i.l 8
7.c even 3 1 inner 770.2.i.l 8
7.c even 3 1 5390.2.a.ce 4
7.d odd 6 1 5390.2.a.cf 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.i.l 8 1.a even 1 1 trivial
770.2.i.l 8 7.c even 3 1 inner
5390.2.a.ce 4 7.c even 3 1
5390.2.a.cf 4 7.d odd 6 1

Hecke kernels

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

\(T_{3}^{8} - \cdots\)
\( T_{13}^{4} + T_{13}^{3} - 42 T_{13}^{2} - 56 T_{13} + 40 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{4} \)
$3$ \( 1 + 2 T + 11 T^{2} - 12 T^{3} + 50 T^{4} + 3 T^{5} + 8 T^{6} - T^{7} + T^{8} \)
$5$ \( ( 1 - T + T^{2} )^{4} \)
$7$ \( 2401 - 1029 T + 539 T^{2} - 231 T^{3} + 135 T^{4} - 33 T^{5} + 11 T^{6} - 3 T^{7} + T^{8} \)
$11$ \( ( 1 - T + T^{2} )^{4} \)
$13$ \( ( 40 - 56 T - 42 T^{2} + T^{3} + T^{4} )^{2} \)
$17$ \( 36 - 90 T + 183 T^{2} - 129 T^{3} + 85 T^{4} - 16 T^{5} + 11 T^{6} - 2 T^{7} + T^{8} \)
$19$ \( 678976 + 295816 T + 118169 T^{2} + 21147 T^{3} + 4583 T^{4} + 588 T^{5} + 113 T^{6} + 10 T^{7} + T^{8} \)
$23$ \( 147456 - 46080 T + 26688 T^{2} - 768 T^{3} + 1360 T^{4} + 48 T^{5} + 68 T^{6} + 6 T^{7} + T^{8} \)
$29$ \( ( -3 + 6 T + 19 T^{2} - 9 T^{3} + T^{4} )^{2} \)
$31$ \( 784 + 196 T + 749 T^{2} + 273 T^{3} + 653 T^{4} + 186 T^{5} + 89 T^{6} - 8 T^{7} + T^{8} \)
$37$ \( 1175056 + 222220 T + 147173 T^{2} - 15549 T^{3} + 8735 T^{4} - 216 T^{5} + 101 T^{6} - 2 T^{7} + T^{8} \)
$41$ \( ( -96 + 120 T - 28 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$43$ \( ( 320 - 504 T + 208 T^{2} - 26 T^{3} + T^{4} )^{2} \)
$47$ \( 1016064 + 338688 T + 157248 T^{2} + 5376 T^{3} + 4288 T^{4} - 232 T^{5} + 144 T^{6} - 10 T^{7} + T^{8} \)
$53$ \( 4235364 - 2117682 T + 1217307 T^{2} + 21609 T^{3} + 18277 T^{4} + 980 T^{5} + 273 T^{6} + 14 T^{7} + T^{8} \)
$59$ \( 921600 - 276480 T + 194304 T^{2} + 35328 T^{3} + 12208 T^{4} + 692 T^{5} + 117 T^{6} - T^{7} + T^{8} \)
$61$ \( 192721 + 158040 T + 93163 T^{2} + 29002 T^{3} + 6968 T^{4} + 803 T^{5} + 84 T^{6} - T^{7} + T^{8} \)
$67$ \( 2621161 + 2036702 T + 1083912 T^{2} + 290324 T^{3} + 55505 T^{4} + 6724 T^{5} + 592 T^{6} + 30 T^{7} + T^{8} \)
$71$ \( ( -4458 + 2181 T - 89 T^{2} - 18 T^{3} + T^{4} )^{2} \)
$73$ \( 99856 - 8532 T + 25693 T^{2} + 12877 T^{3} + 6098 T^{4} + 1289 T^{5} + 210 T^{6} + 17 T^{7} + T^{8} \)
$79$ \( 40000 + 360800 T + 3279216 T^{2} - 217696 T^{3} + 42236 T^{4} - 1748 T^{5} + 349 T^{6} - 15 T^{7} + T^{8} \)
$83$ \( ( 1152 + 384 T - 184 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$89$ \( 146458404 + 8023626 T + 3477171 T^{2} - 21189 T^{3} + 54877 T^{4} + 180 T^{5} + 287 T^{6} - 6 T^{7} + T^{8} \)
$97$ \( ( 1832 - 888 T - 194 T^{2} + 7 T^{3} + T^{4} )^{2} \)
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