Properties

Label 350.2.o.b
Level 350
Weight 2
Character orbit 350.o
Analytic conductor 2.795
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 350.o (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.79476407074\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(\zeta_{24}^{4}\) \(\zeta_{24}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
0.258819 0.965926i
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
0.965926 0.258819i
−0.965926 + 0.258819i
0.258819 + 0.965926i
−0.258819 0.965926i
−0.965926 0.258819i −0.189469 0.707107i 0.866025 + 0.500000i 0 0.732051i −2.19067 + 1.48356i −0.707107 0.707107i 2.13397 1.23205i 0
143.2 0.965926 + 0.258819i 0.189469 + 0.707107i 0.866025 + 0.500000i 0 0.732051i 2.19067 1.48356i 0.707107 + 0.707107i 2.13397 1.23205i 0
157.1 −0.258819 + 0.965926i 2.63896 0.707107i −0.866025 0.500000i 0 2.73205i −1.48356 2.19067i 0.707107 0.707107i 3.86603 2.23205i 0
157.2 0.258819 0.965926i −2.63896 + 0.707107i −0.866025 0.500000i 0 2.73205i 1.48356 + 2.19067i −0.707107 + 0.707107i 3.86603 2.23205i 0
243.1 −0.258819 0.965926i 2.63896 + 0.707107i −0.866025 + 0.500000i 0 2.73205i −1.48356 + 2.19067i 0.707107 + 0.707107i 3.86603 + 2.23205i 0
243.2 0.258819 + 0.965926i −2.63896 0.707107i −0.866025 + 0.500000i 0 2.73205i 1.48356 2.19067i −0.707107 0.707107i 3.86603 + 2.23205i 0
257.1 −0.965926 + 0.258819i −0.189469 + 0.707107i 0.866025 0.500000i 0 0.732051i −2.19067 1.48356i −0.707107 + 0.707107i 2.13397 + 1.23205i 0
257.2 0.965926 0.258819i 0.189469 0.707107i 0.866025 0.500000i 0 0.732051i 2.19067 + 1.48356i 0.707107 0.707107i 2.13397 + 1.23205i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
35.k even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.o.b yes 8
5.b even 2 1 inner 350.2.o.b yes 8
5.c odd 4 2 350.2.o.a 8
7.d odd 6 1 350.2.o.a 8
35.i odd 6 1 350.2.o.a 8
35.k even 12 2 inner 350.2.o.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.o.a 8 5.c odd 4 2
350.2.o.a 8 7.d odd 6 1
350.2.o.a 8 35.i odd 6 1
350.2.o.b yes 8 1.a even 1 1 trivial
350.2.o.b yes 8 5.b even 2 1 inner
350.2.o.b yes 8 35.k even 12 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 12 T_{3}^{6} + 44 T_{3}^{4} + 48 T_{3}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(350, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{4} + T^{8} \)
$3$ \( 1 - 12 T^{2} + 74 T^{4} - 312 T^{6} + 1027 T^{8} - 2808 T^{10} + 5994 T^{12} - 8748 T^{14} + 6561 T^{16} \)
$5$ \( \)
$7$ \( 1 + 71 T^{4} + 2401 T^{8} \)
$11$ \( ( 1 - 6 T + 8 T^{2} - 36 T^{3} + 267 T^{4} - 396 T^{5} + 968 T^{6} - 7986 T^{7} + 14641 T^{8} )^{2} \)
$13$ \( 1 - 452 T^{4} + 106470 T^{8} - 12909572 T^{12} + 815730721 T^{16} \)
$17$ \( 1 + 72 T^{2} + 2569 T^{4} + 60552 T^{6} + 1123152 T^{8} + 17499528 T^{10} + 214565449 T^{12} + 1737904968 T^{14} + 6975757441 T^{16} \)
$19$ \( ( 1 - 19 T^{2} + 361 T^{4} )^{4} \)
$23$ \( 1 + 697 T^{4} + 205968 T^{8} + 195049177 T^{12} + 78310985281 T^{16} \)
$29$ \( ( 1 - 46 T^{2} + 841 T^{4} )^{4} \)
$31$ \( ( 1 - 4 T + 31 T^{2} )^{4}( 1 + 7 T + 31 T^{2} )^{4} \)
$37$ \( ( 1 - 529 T^{4} + 1874161 T^{8} )( 1 + 2591 T^{4} + 1874161 T^{8} ) \)
$41$ \( ( 1 - 38 T^{2} - 165 T^{4} - 63878 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( 1 - 5444 T^{4} + 14231334 T^{8} - 18611952644 T^{12} + 11688200277601 T^{16} \)
$47$ \( 1 - 144 T^{2} + 11689 T^{4} - 687888 T^{6} + 33208656 T^{8} - 1519544592 T^{10} + 57038591209 T^{12} - 1552207007376 T^{14} + 23811286661761 T^{16} \)
$53$ \( 1 - 108 T^{2} + 5578 T^{4} - 182520 T^{6} + 5887011 T^{8} - 512698680 T^{10} + 44013103018 T^{12} - 2393751001932 T^{14} + 62259690411361 T^{16} \)
$59$ \( ( 1 + 18 T + 152 T^{2} + 972 T^{3} + 6987 T^{4} + 57348 T^{5} + 529112 T^{6} + 3696822 T^{7} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 6 T + 56 T^{2} - 264 T^{3} - 1053 T^{4} - 16104 T^{5} + 208376 T^{6} - 1361886 T^{7} + 13845841 T^{8} )^{2} \)
$67$ \( 1 - 324 T^{2} + 52042 T^{4} - 5524200 T^{6} + 427630467 T^{8} - 24798133800 T^{10} + 1048704639082 T^{12} - 29308515822756 T^{14} + 406067677556641 T^{16} \)
$71$ \( ( 1 - 6 T + 43 T^{2} - 426 T^{3} + 5041 T^{4} )^{4} \)
$73$ \( 1 + 3934 T^{4} - 12921885 T^{8} + 111718680094 T^{12} + 806460091894081 T^{16} \)
$79$ \( ( 1 + 36 T + 697 T^{2} + 9540 T^{3} + 98112 T^{4} + 753660 T^{5} + 4349977 T^{6} + 17749404 T^{7} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 + 3122 T^{4} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 12 T - 43 T^{2} - 108 T^{3} + 14232 T^{4} - 9612 T^{5} - 340603 T^{6} - 8459628 T^{7} + 62742241 T^{8} )^{2} \)
$97$ \( 1 - 22322 T^{4} + 289141683 T^{8} - 1976150610482 T^{12} + 7837433594376961 T^{16} \)
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