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Properties

Label 1344.3.f.c.769.1
Level $1344$
Weight $3$
Character 1344.769
Analytic conductor $36.621$
Analytic rank $0$
Dimension $2$
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] = [1344,3,Mod(769,1344)] 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("1344.769"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1344, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1344.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,2,0,-6,0,-20,0,0,0,24,0,0,0,0,0,24,0,-28,0,-46] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(36.6213475300\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 769.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1344.769
Dual form 1344.3.f.c.769.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.73205i q^{3} +6.92820i q^{5} +(1.00000 + 6.92820i) q^{7} -3.00000 q^{9} -10.0000 q^{11} -6.92820i q^{13} +12.0000 q^{15} +20.7846i q^{19} +(12.0000 - 1.73205i) q^{21} -14.0000 q^{23} -23.0000 q^{25} +5.19615i q^{27} +38.0000 q^{29} +27.7128i q^{31} +17.3205i q^{33} +(-48.0000 + 6.92820i) q^{35} -26.0000 q^{37} -12.0000 q^{39} -69.2820i q^{41} -26.0000 q^{43} -20.7846i q^{45} +27.7128i q^{47} +(-47.0000 + 13.8564i) q^{49} -10.0000 q^{53} -69.2820i q^{55} +36.0000 q^{57} -76.2102i q^{59} -34.6410i q^{61} +(-3.00000 - 20.7846i) q^{63} +48.0000 q^{65} -74.0000 q^{67} +24.2487i q^{69} -62.0000 q^{71} -41.5692i q^{73} +39.8372i q^{75} +(-10.0000 - 69.2820i) q^{77} -46.0000 q^{79} +9.00000 q^{81} -90.0666i q^{83} -65.8179i q^{87} -41.5692i q^{89} +(48.0000 - 6.92820i) q^{91} +48.0000 q^{93} -144.000 q^{95} +55.4256i q^{97} +30.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{7} - 6 q^{9} - 20 q^{11} + 24 q^{15} + 24 q^{21} - 28 q^{23} - 46 q^{25} + 76 q^{29} - 96 q^{35} - 52 q^{37} - 24 q^{39} - 52 q^{43} - 94 q^{49} - 20 q^{53} + 72 q^{57} - 6 q^{63} + 96 q^{65}+ \cdots + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.73205i − 0.577350i
\(4\) 0 0
\(5\) 6.92820i 1.38564i 0.721110 + 0.692820i \(0.243632\pi\)
−0.721110 + 0.692820i \(0.756368\pi\)
\(6\) 0 0
\(7\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) −10.0000 −0.909091 −0.454545 − 0.890724i \(-0.650198\pi\)
−0.454545 + 0.890724i \(0.650198\pi\)
\(12\) 0 0
\(13\) − 6.92820i − 0.532939i −0.963843 − 0.266469i \(-0.914143\pi\)
0.963843 − 0.266469i \(-0.0858571\pi\)
\(14\) 0 0
\(15\) 12.0000 0.800000
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 20.7846i 1.09393i 0.837157 + 0.546963i \(0.184216\pi\)
−0.837157 + 0.546963i \(0.815784\pi\)
\(20\) 0 0
\(21\) 12.0000 − 1.73205i 0.571429 − 0.0824786i
\(22\) 0 0
\(23\) −14.0000 −0.608696 −0.304348 − 0.952561i \(-0.598439\pi\)
−0.304348 + 0.952561i \(0.598439\pi\)
\(24\) 0 0
\(25\) −23.0000 −0.920000
\(26\) 0 0
\(27\) 5.19615i 0.192450i
\(28\) 0 0
\(29\) 38.0000 1.31034 0.655172 − 0.755479i \(-0.272596\pi\)
0.655172 + 0.755479i \(0.272596\pi\)
\(30\) 0 0
\(31\) 27.7128i 0.893962i 0.894544 + 0.446981i \(0.147501\pi\)
−0.894544 + 0.446981i \(0.852499\pi\)
\(32\) 0 0
\(33\) 17.3205i 0.524864i
\(34\) 0 0
\(35\) −48.0000 + 6.92820i −1.37143 + 0.197949i
\(36\) 0 0
\(37\) −26.0000 −0.702703 −0.351351 − 0.936244i \(-0.614278\pi\)
−0.351351 + 0.936244i \(0.614278\pi\)
\(38\) 0 0
\(39\) −12.0000 −0.307692
\(40\) 0 0
\(41\) − 69.2820i − 1.68981i −0.534920 − 0.844903i \(-0.679658\pi\)
0.534920 − 0.844903i \(-0.320342\pi\)
\(42\) 0 0
\(43\) −26.0000 −0.604651 −0.302326 − 0.953205i \(-0.597763\pi\)
−0.302326 + 0.953205i \(0.597763\pi\)
\(44\) 0 0
\(45\) − 20.7846i − 0.461880i
\(46\) 0 0
\(47\) 27.7128i 0.589634i 0.955554 + 0.294817i \(0.0952587\pi\)
−0.955554 + 0.294817i \(0.904741\pi\)
\(48\) 0 0
\(49\) −47.0000 + 13.8564i −0.959184 + 0.282784i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −10.0000 −0.188679 −0.0943396 − 0.995540i \(-0.530074\pi\)
−0.0943396 + 0.995540i \(0.530074\pi\)
\(54\) 0 0
\(55\) − 69.2820i − 1.25967i
\(56\) 0 0
\(57\) 36.0000 0.631579
\(58\) 0 0
\(59\) − 76.2102i − 1.29170i −0.763465 − 0.645849i \(-0.776503\pi\)
0.763465 − 0.645849i \(-0.223497\pi\)
\(60\) 0 0
\(61\) − 34.6410i − 0.567886i −0.958841 − 0.283943i \(-0.908357\pi\)
0.958841 − 0.283943i \(-0.0916426\pi\)
\(62\) 0 0
\(63\) −3.00000 − 20.7846i −0.0476190 − 0.329914i
\(64\) 0 0
\(65\) 48.0000 0.738462
\(66\) 0 0
\(67\) −74.0000 −1.10448 −0.552239 − 0.833686i \(-0.686226\pi\)
−0.552239 + 0.833686i \(0.686226\pi\)
\(68\) 0 0
\(69\) 24.2487i 0.351431i
\(70\) 0 0
\(71\) −62.0000 −0.873239 −0.436620 − 0.899646i \(-0.643824\pi\)
−0.436620 + 0.899646i \(0.643824\pi\)
\(72\) 0 0
\(73\) − 41.5692i − 0.569441i −0.958611 − 0.284721i \(-0.908099\pi\)
0.958611 − 0.284721i \(-0.0919009\pi\)
\(74\) 0 0
\(75\) 39.8372i 0.531162i
\(76\) 0 0
\(77\) −10.0000 − 69.2820i −0.129870 − 0.899767i
\(78\) 0 0
\(79\) −46.0000 −0.582278 −0.291139 − 0.956681i \(-0.594034\pi\)
−0.291139 + 0.956681i \(0.594034\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) − 90.0666i − 1.08514i −0.840011 − 0.542570i \(-0.817451\pi\)
0.840011 − 0.542570i \(-0.182549\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 65.8179i − 0.756528i
\(88\) 0 0
\(89\) − 41.5692i − 0.467070i −0.972348 − 0.233535i \(-0.924971\pi\)
0.972348 − 0.233535i \(-0.0750293\pi\)
\(90\) 0 0
\(91\) 48.0000 − 6.92820i 0.527473 − 0.0761341i
\(92\) 0 0
\(93\) 48.0000 0.516129
\(94\) 0 0
\(95\) −144.000 −1.51579
\(96\) 0 0
\(97\) 55.4256i 0.571398i 0.958319 + 0.285699i \(0.0922258\pi\)
−0.958319 + 0.285699i \(0.907774\pi\)
\(98\) 0 0
\(99\) 30.0000 0.303030
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 1344.3.f.c.769.1 2
4.3 odd 2 1344.3.f.b.769.2 2
7.6 odd 2 inner 1344.3.f.c.769.2 2
8.3 odd 2 336.3.f.a.97.1 2
8.5 even 2 21.3.d.a.13.2 yes 2
24.5 odd 2 63.3.d.b.55.2 2
24.11 even 2 1008.3.f.d.433.2 2
28.27 even 2 1344.3.f.b.769.1 2
40.13 odd 4 525.3.e.a.349.1 4
40.29 even 2 525.3.h.a.76.1 2
40.37 odd 4 525.3.e.a.349.4 4
56.5 odd 6 147.3.f.d.31.1 2
56.13 odd 2 21.3.d.a.13.1 ✓ 2
56.27 even 2 336.3.f.a.97.2 2
56.37 even 6 147.3.f.b.31.1 2
56.45 odd 6 147.3.f.b.19.1 2
56.53 even 6 147.3.f.d.19.1 2
168.5 even 6 441.3.m.d.325.1 2
168.53 odd 6 441.3.m.d.19.1 2
168.83 odd 2 1008.3.f.d.433.1 2
168.101 even 6 441.3.m.f.19.1 2
168.125 even 2 63.3.d.b.55.1 2
168.149 odd 6 441.3.m.f.325.1 2
280.13 even 4 525.3.e.a.349.2 4
280.69 odd 2 525.3.h.a.76.2 2
280.237 even 4 525.3.e.a.349.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.3.d.a.13.1 ✓ 2 56.13 odd 2
21.3.d.a.13.2 yes 2 8.5 even 2
63.3.d.b.55.1 2 168.125 even 2
63.3.d.b.55.2 2 24.5 odd 2
147.3.f.b.19.1 2 56.45 odd 6
147.3.f.b.31.1 2 56.37 even 6
147.3.f.d.19.1 2 56.53 even 6
147.3.f.d.31.1 2 56.5 odd 6
336.3.f.a.97.1 2 8.3 odd 2
336.3.f.a.97.2 2 56.27 even 2
441.3.m.d.19.1 2 168.53 odd 6
441.3.m.d.325.1 2 168.5 even 6
441.3.m.f.19.1 2 168.101 even 6
441.3.m.f.325.1 2 168.149 odd 6
525.3.e.a.349.1 4 40.13 odd 4
525.3.e.a.349.2 4 280.13 even 4
525.3.e.a.349.3 4 280.237 even 4
525.3.e.a.349.4 4 40.37 odd 4
525.3.h.a.76.1 2 40.29 even 2
525.3.h.a.76.2 2 280.69 odd 2
1008.3.f.d.433.1 2 168.83 odd 2
1008.3.f.d.433.2 2 24.11 even 2
1344.3.f.b.769.1 2 28.27 even 2
1344.3.f.b.769.2 2 4.3 odd 2
1344.3.f.c.769.1 2 1.1 even 1 trivial
1344.3.f.c.769.2 2 7.6 odd 2 inner