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Properties

Label 774.2.d.b.773.1
Level $774$
Weight $2$
Character 774.773
Analytic conductor $6.180$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [774,2,Mod(773,774)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("774.773"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(774, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 774 = 2 \cdot 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 774.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.18042111645\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.30233088.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 6x^{3} + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 773.1
Root \(1.64497 - 0.542278i\) of defining polynomial
Character \(\chi\) \(=\) 774.773
Dual form 774.2.d.b.773.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -3.28995 q^{5} -1.08456i q^{7} +1.00000 q^{8} -3.28995 q^{10} -2.49877i q^{11} +5.28995 q^{13} -1.08456i q^{14} +1.00000 q^{16} -1.41421i q^{17} -5.73724i q^{19} -3.28995 q^{20} -2.49877i q^{22} -6.06690i q^{23} +5.82374 q^{25} +5.28995 q^{26} -1.08456i q^{28} +0.222358 q^{29} -3.04610 q^{31} +1.00000 q^{32} -1.41421i q^{34} +3.56813i q^{35} -9.30537i q^{37} -5.73724i q^{38} -3.28995 q^{40} -3.89779i q^{41} +(-3.82374 + 5.32720i) q^{43} -2.49877i q^{44} -6.06690i q^{46} +2.15392i q^{47} +5.82374 q^{49} +5.82374 q^{50} +5.28995 q^{52} +5.40758i q^{53} +8.22081i q^{55} -1.08456i q^{56} +0.222358 q^{58} +5.40758i q^{59} +13.1380i q^{61} -3.04610 q^{62} +1.00000 q^{64} -17.4036 q^{65} +3.53379 q^{67} -1.41421i q^{68} +3.56813i q^{70} -1.53379 q^{71} -2.48357i q^{73} -9.30537i q^{74} -5.73724i q^{76} -2.71005 q^{77} +11.6475 q^{79} -3.28995 q^{80} -3.89779i q^{82} +6.80660i q^{83} +4.65268i q^{85} +(-3.82374 + 5.32720i) q^{86} -2.49877i q^{88} -15.6475 q^{89} -5.73724i q^{91} -6.06690i q^{92} +2.15392i q^{94} +18.8752i q^{95} +7.40363 q^{97} +5.82374 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 12 q^{13} + 6 q^{16} + 6 q^{25} + 12 q^{26} + 12 q^{31} + 6 q^{32} + 6 q^{43} + 6 q^{49} + 6 q^{50} + 12 q^{52} + 12 q^{62} + 6 q^{64} - 36 q^{65} + 12 q^{67} - 36 q^{77}+ \cdots + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/774\mathbb{Z}\right)^\times\).

\(n\) \(173\) \(433\)
\(\chi(n)\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −3.28995 −1.47131 −0.735654 − 0.677357i \(-0.763125\pi\)
−0.735654 + 0.677357i \(0.763125\pi\)
\(6\) 0 0
\(7\) − 1.08456i − 0.409924i −0.978770 − 0.204962i \(-0.934293\pi\)
0.978770 − 0.204962i \(-0.0657070\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −3.28995 −1.04037
\(11\) − 2.49877i − 0.753407i −0.926334 − 0.376704i \(-0.877058\pi\)
0.926334 − 0.376704i \(-0.122942\pi\)
\(12\) 0 0
\(13\) 5.28995 1.46717 0.733583 − 0.679599i \(-0.237846\pi\)
0.733583 + 0.679599i \(0.237846\pi\)
\(14\) − 1.08456i − 0.289860i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 1.41421i − 0.342997i −0.985184 − 0.171499i \(-0.945139\pi\)
0.985184 − 0.171499i \(-0.0548609\pi\)
\(18\) 0 0
\(19\) − 5.73724i − 1.31621i −0.752925 − 0.658107i \(-0.771358\pi\)
0.752925 − 0.658107i \(-0.228642\pi\)
\(20\) −3.28995 −0.735654
\(21\) 0 0
\(22\) − 2.49877i − 0.532739i
\(23\) − 6.06690i − 1.26504i −0.774546 − 0.632518i \(-0.782021\pi\)
0.774546 − 0.632518i \(-0.217979\pi\)
\(24\) 0 0
\(25\) 5.82374 1.16475
\(26\) 5.28995 1.03744
\(27\) 0 0
\(28\) − 1.08456i − 0.204962i
\(29\) 0.222358 0.0412908 0.0206454 − 0.999787i \(-0.493428\pi\)
0.0206454 + 0.999787i \(0.493428\pi\)
\(30\) 0 0
\(31\) −3.04610 −0.547095 −0.273548 − 0.961858i \(-0.588197\pi\)
−0.273548 + 0.961858i \(0.588197\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) − 1.41421i − 0.242536i
\(35\) 3.56813i 0.603124i
\(36\) 0 0
\(37\) − 9.30537i − 1.52979i −0.644153 − 0.764897i \(-0.722790\pi\)
0.644153 − 0.764897i \(-0.277210\pi\)
\(38\) − 5.73724i − 0.930703i
\(39\) 0 0
\(40\) −3.28995 −0.520186
\(41\) − 3.89779i − 0.608732i −0.952555 − 0.304366i \(-0.901555\pi\)
0.952555 − 0.304366i \(-0.0984446\pi\)
\(42\) 0 0
\(43\) −3.82374 + 5.32720i −0.583115 + 0.812390i
\(44\) − 2.49877i − 0.376704i
\(45\) 0 0
\(46\) − 6.06690i − 0.894515i
\(47\) 2.15392i 0.314181i 0.987584 + 0.157090i \(0.0502114\pi\)
−0.987584 + 0.157090i \(0.949789\pi\)
\(48\) 0 0
\(49\) 5.82374 0.831963
\(50\) 5.82374 0.823601
\(51\) 0 0
\(52\) 5.28995 0.733583
\(53\) 5.40758i 0.742789i 0.928475 + 0.371394i \(0.121120\pi\)
−0.928475 + 0.371394i \(0.878880\pi\)
\(54\) 0 0
\(55\) 8.22081i 1.10849i
\(56\) − 1.08456i − 0.144930i
\(57\) 0 0
\(58\) 0.222358 0.0291970
\(59\) 5.40758i 0.704007i 0.935999 + 0.352004i \(0.114500\pi\)
−0.935999 + 0.352004i \(0.885500\pi\)
\(60\) 0 0
\(61\) 13.1380i 1.68214i 0.540923 + 0.841072i \(0.318075\pi\)
−0.540923 + 0.841072i \(0.681925\pi\)
\(62\) −3.04610 −0.386855
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −17.4036 −2.15865
\(66\) 0 0
\(67\) 3.53379 0.431722 0.215861 − 0.976424i \(-0.430744\pi\)
0.215861 + 0.976424i \(0.430744\pi\)
\(68\) − 1.41421i − 0.171499i
\(69\) 0 0
\(70\) 3.56813i 0.426473i
\(71\) −1.53379 −0.182028 −0.0910139 − 0.995850i \(-0.529011\pi\)
−0.0910139 + 0.995850i \(0.529011\pi\)
\(72\) 0 0
\(73\) − 2.48357i − 0.290680i −0.989382 − 0.145340i \(-0.953572\pi\)
0.989382 − 0.145340i \(-0.0464276\pi\)
\(74\) − 9.30537i − 1.08173i
\(75\) 0 0
\(76\) − 5.73724i − 0.658107i
\(77\) −2.71005 −0.308839
\(78\) 0 0
\(79\) 11.6475 1.31044 0.655222 − 0.755437i \(-0.272575\pi\)
0.655222 + 0.755437i \(0.272575\pi\)
\(80\) −3.28995 −0.367827
\(81\) 0 0
\(82\) − 3.89779i − 0.430439i
\(83\) 6.80660i 0.747121i 0.927606 + 0.373561i \(0.121863\pi\)
−0.927606 + 0.373561i \(0.878137\pi\)
\(84\) 0 0
\(85\) 4.65268i 0.504655i
\(86\) −3.82374 + 5.32720i −0.412324 + 0.574446i
\(87\) 0 0
\(88\) − 2.49877i − 0.266370i
\(89\) −15.6475 −1.65863 −0.829315 − 0.558782i \(-0.811269\pi\)
−0.829315 + 0.558782i \(0.811269\pi\)
\(90\) 0 0
\(91\) − 5.73724i − 0.601426i
\(92\) − 6.06690i − 0.632518i
\(93\) 0 0
\(94\) 2.15392i 0.222159i
\(95\) 18.8752i 1.93656i
\(96\) 0 0
\(97\) 7.40363 0.751725 0.375862 − 0.926676i \(-0.377347\pi\)
0.375862 + 0.926676i \(0.377347\pi\)
\(98\) 5.82374 0.588286
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 774.2.d.b.773.1 yes 6
3.2 odd 2 774.2.d.a.773.5 ✓ 6
4.3 odd 2 6192.2.l.f.2321.2 6
12.11 even 2 6192.2.l.g.2321.6 6
43.42 odd 2 774.2.d.a.773.6 yes 6
129.128 even 2 inner 774.2.d.b.773.2 yes 6
172.171 even 2 6192.2.l.g.2321.5 6
516.515 odd 2 6192.2.l.f.2321.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
774.2.d.a.773.5 ✓ 6 3.2 odd 2
774.2.d.a.773.6 yes 6 43.42 odd 2
774.2.d.b.773.1 yes 6 1.1 even 1 trivial
774.2.d.b.773.2 yes 6 129.128 even 2 inner
6192.2.l.f.2321.1 6 516.515 odd 2
6192.2.l.f.2321.2 6 4.3 odd 2
6192.2.l.g.2321.5 6 172.171 even 2
6192.2.l.g.2321.6 6 12.11 even 2