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Properties

Label 1305.2.c.l.784.8
Level $1305$
Weight $2$
Character 1305.784
Analytic conductor $10.420$
Analytic rank $0$
Dimension $12$
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] = [1305,2,Mod(784,1305)] 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("1305.784"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1305, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,-16,0,0,0,0,0,-10,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.4204774638\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 20x^{10} + 148x^{8} + 502x^{6} + 792x^{4} + 496x^{2} + 45 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 784.8
Root \(1.27263i\) of defining polynomial
Character \(\chi\) \(=\) 1305.784
Dual form 1305.2.c.l.784.5

$q$-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)
 
\(f(q)\) \(=\) \(q+1.27263i q^{2} +0.380419 q^{4} +(-1.10723 - 1.94269i) q^{5} +0.255813i q^{7} +3.02939i q^{8} +(2.47232 - 1.40909i) q^{10} -4.63446 q^{11} -5.02700i q^{13} -0.325555 q^{14} -3.09444 q^{16} -0.336444i q^{17} +2.91437 q^{19} +(-0.421212 - 0.739035i) q^{20} -5.89795i q^{22} -8.65656i q^{23} +(-2.54807 + 4.30201i) q^{25} +6.39750 q^{26} +0.0973162i q^{28} +1.00000 q^{29} -3.26943 q^{31} +2.12070i q^{32} +0.428167 q^{34} +(0.496966 - 0.283245i) q^{35} -3.86954i q^{37} +3.70891i q^{38} +(5.88515 - 3.35424i) q^{40} -5.71649 q^{41} -6.98619i q^{43} -1.76304 q^{44} +11.0166 q^{46} -0.336444i q^{47} +6.93456 q^{49} +(-5.47486 - 3.24275i) q^{50} -1.91237i q^{52} -6.01484i q^{53} +(5.13143 + 9.00332i) q^{55} -0.774958 q^{56} +1.27263i q^{58} -13.2799 q^{59} -7.77792 q^{61} -4.16077i q^{62} -8.88775 q^{64} +(-9.76589 + 5.56606i) q^{65} -11.2605i q^{67} -0.127989i q^{68} +(0.360465 + 0.632452i) q^{70} +13.9668 q^{71} -8.46426i q^{73} +4.92448 q^{74} +1.10868 q^{76} -1.18556i q^{77} +15.3102 q^{79} +(3.42627 + 6.01154i) q^{80} -7.27496i q^{82} -7.60521i q^{83} +(-0.653605 + 0.372521i) q^{85} +8.89082 q^{86} -14.0396i q^{88} -13.0383 q^{89} +1.28597 q^{91} -3.29312i q^{92} +0.428167 q^{94} +(-3.22689 - 5.66171i) q^{95} +11.5445i q^{97} +8.82511i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 16 q^{4} - 10 q^{10} + 12 q^{11} - 16 q^{14} + 16 q^{16} + 20 q^{19} - 14 q^{20} + 8 q^{25} + 56 q^{26} + 12 q^{29} - 16 q^{31} - 4 q^{34} + 16 q^{35} + 16 q^{40} + 32 q^{41} - 68 q^{44} + 20 q^{46}+ \cdots + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times\).

\(n\) \(146\) \(262\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.27263i 0.899884i 0.893058 + 0.449942i \(0.148555\pi\)
−0.893058 + 0.449942i \(0.851445\pi\)
\(3\) 0 0
\(4\) 0.380419 0.190209
\(5\) −1.10723 − 1.94269i −0.495169 − 0.868796i
\(6\) 0 0
\(7\) 0.255813i 0.0966884i 0.998831 + 0.0483442i \(0.0153944\pi\)
−0.998831 + 0.0483442i \(0.984606\pi\)
\(8\) 3.02939i 1.07105i
\(9\) 0 0
\(10\) 2.47232 − 1.40909i 0.781816 − 0.445595i
\(11\) −4.63446 −1.39734 −0.698672 − 0.715442i \(-0.746225\pi\)
−0.698672 + 0.715442i \(0.746225\pi\)
\(12\) 0 0
\(13\) − 5.02700i − 1.39424i −0.716955 − 0.697120i \(-0.754465\pi\)
0.716955 − 0.697120i \(-0.245535\pi\)
\(14\) −0.325555 −0.0870083
\(15\) 0 0
\(16\) −3.09444 −0.773611
\(17\) − 0.336444i − 0.0815995i −0.999167 − 0.0407998i \(-0.987009\pi\)
0.999167 − 0.0407998i \(-0.0129906\pi\)
\(18\) 0 0
\(19\) 2.91437 0.668602 0.334301 − 0.942466i \(-0.391500\pi\)
0.334301 + 0.942466i \(0.391500\pi\)
\(20\) −0.421212 − 0.739035i −0.0941859 − 0.165253i
\(21\) 0 0
\(22\) − 5.89795i − 1.25745i
\(23\) − 8.65656i − 1.80502i −0.430671 − 0.902509i \(-0.641723\pi\)
0.430671 − 0.902509i \(-0.358277\pi\)
\(24\) 0 0
\(25\) −2.54807 + 4.30201i −0.509614 + 0.860403i
\(26\) 6.39750 1.25465
\(27\) 0 0
\(28\) 0.0973162i 0.0183910i
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −3.26943 −0.587207 −0.293603 − 0.955927i \(-0.594854\pi\)
−0.293603 + 0.955927i \(0.594854\pi\)
\(32\) 2.12070i 0.374890i
\(33\) 0 0
\(34\) 0.428167 0.0734301
\(35\) 0.496966 − 0.283245i 0.0840025 − 0.0478771i
\(36\) 0 0
\(37\) − 3.86954i − 0.636148i −0.948066 − 0.318074i \(-0.896964\pi\)
0.948066 − 0.318074i \(-0.103036\pi\)
\(38\) 3.70891i 0.601664i
\(39\) 0 0
\(40\) 5.88515 − 3.35424i 0.930524 − 0.530351i
\(41\) −5.71649 −0.892765 −0.446383 − 0.894842i \(-0.647288\pi\)
−0.446383 + 0.894842i \(0.647288\pi\)
\(42\) 0 0
\(43\) − 6.98619i − 1.06538i −0.846309 − 0.532692i \(-0.821180\pi\)
0.846309 − 0.532692i \(-0.178820\pi\)
\(44\) −1.76304 −0.265788
\(45\) 0 0
\(46\) 11.0166 1.62431
\(47\) − 0.336444i − 0.0490753i −0.999699 − 0.0245377i \(-0.992189\pi\)
0.999699 − 0.0245377i \(-0.00781137\pi\)
\(48\) 0 0
\(49\) 6.93456 0.990651
\(50\) −5.47486 − 3.24275i −0.774263 − 0.458594i
\(51\) 0 0
\(52\) − 1.91237i − 0.265197i
\(53\) − 6.01484i − 0.826202i −0.910685 − 0.413101i \(-0.864446\pi\)
0.910685 − 0.413101i \(-0.135554\pi\)
\(54\) 0 0
\(55\) 5.13143 + 9.00332i 0.691922 + 1.21401i
\(56\) −0.774958 −0.103558
\(57\) 0 0
\(58\) 1.27263i 0.167104i
\(59\) −13.2799 −1.72890 −0.864451 − 0.502718i \(-0.832334\pi\)
−0.864451 + 0.502718i \(0.832334\pi\)
\(60\) 0 0
\(61\) −7.77792 −0.995861 −0.497930 − 0.867217i \(-0.665906\pi\)
−0.497930 + 0.867217i \(0.665906\pi\)
\(62\) − 4.16077i − 0.528418i
\(63\) 0 0
\(64\) −8.88775 −1.11097
\(65\) −9.76589 + 5.56606i −1.21131 + 0.690385i
\(66\) 0 0
\(67\) − 11.2605i − 1.37569i −0.725856 − 0.687847i \(-0.758556\pi\)
0.725856 − 0.687847i \(-0.241444\pi\)
\(68\) − 0.127989i − 0.0155210i
\(69\) 0 0
\(70\) 0.360465 + 0.632452i 0.0430839 + 0.0755925i
\(71\) 13.9668 1.65756 0.828778 − 0.559578i \(-0.189037\pi\)
0.828778 + 0.559578i \(0.189037\pi\)
\(72\) 0 0
\(73\) − 8.46426i − 0.990667i −0.868703 − 0.495333i \(-0.835046\pi\)
0.868703 − 0.495333i \(-0.164954\pi\)
\(74\) 4.92448 0.572459
\(75\) 0 0
\(76\) 1.10868 0.127174
\(77\) − 1.18556i − 0.135107i
\(78\) 0 0
\(79\) 15.3102 1.72253 0.861267 − 0.508153i \(-0.169672\pi\)
0.861267 + 0.508153i \(0.169672\pi\)
\(80\) 3.42627 + 6.01154i 0.383069 + 0.672111i
\(81\) 0 0
\(82\) − 7.27496i − 0.803385i
\(83\) − 7.60521i − 0.834781i −0.908727 − 0.417390i \(-0.862945\pi\)
0.908727 − 0.417390i \(-0.137055\pi\)
\(84\) 0 0
\(85\) −0.653605 + 0.372521i −0.0708934 + 0.0404056i
\(86\) 8.89082 0.958722
\(87\) 0 0
\(88\) − 14.0396i − 1.49662i
\(89\) −13.0383 −1.38206 −0.691029 − 0.722827i \(-0.742843\pi\)
−0.691029 + 0.722827i \(0.742843\pi\)
\(90\) 0 0
\(91\) 1.28597 0.134807
\(92\) − 3.29312i − 0.343331i
\(93\) 0 0
\(94\) 0.428167 0.0441621
\(95\) −3.22689 − 5.66171i −0.331071 − 0.580879i
\(96\) 0 0
\(97\) 11.5445i 1.17216i 0.810253 + 0.586081i \(0.199330\pi\)
−0.810253 + 0.586081i \(0.800670\pi\)
\(98\) 8.82511i 0.891471i
\(99\) 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 1305.2.c.l.784.8 yes 12
3.2 odd 2 1305.2.c.k.784.5 ✓ 12
5.2 odd 4 6525.2.a.cf.1.5 12
5.3 odd 4 6525.2.a.cf.1.8 12
5.4 even 2 inner 1305.2.c.l.784.5 yes 12
15.2 even 4 6525.2.a.ce.1.8 12
15.8 even 4 6525.2.a.ce.1.5 12
15.14 odd 2 1305.2.c.k.784.8 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1305.2.c.k.784.5 ✓ 12 3.2 odd 2
1305.2.c.k.784.8 yes 12 15.14 odd 2
1305.2.c.l.784.5 yes 12 5.4 even 2 inner
1305.2.c.l.784.8 yes 12 1.1 even 1 trivial
6525.2.a.ce.1.5 12 15.8 even 4
6525.2.a.ce.1.8 12 15.2 even 4
6525.2.a.cf.1.5 12 5.2 odd 4
6525.2.a.cf.1.8 12 5.3 odd 4