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 = [4,0,0,-4,3,0,0,0,0,9,-14] 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: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 145)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 784.1
Root \(-1.18614 - 1.26217i\) of defining polynomial
Character \(\chi\) \(=\) 1305.784
Dual form 1305.2.c.e.784.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.73205i q^{2} -1.00000 q^{4} +(-0.686141 + 2.12819i) q^{5} +5.04868i q^{7} -1.73205i q^{8} +(3.68614 + 1.18843i) q^{10} -0.627719 q^{11} -4.25639i q^{13} +8.74456 q^{14} -5.00000 q^{16} +1.58457i q^{17} -4.00000 q^{19} +(0.686141 - 2.12819i) q^{20} +1.08724i q^{22} +3.46410i q^{23} +(-4.05842 - 2.92048i) q^{25} -7.37228 q^{26} -5.04868i q^{28} -1.00000 q^{29} -3.37228 q^{31} +5.19615i q^{32} +2.74456 q^{34} +(-10.7446 - 3.46410i) q^{35} +3.16915i q^{37} +6.92820i q^{38} +(3.68614 + 1.18843i) q^{40} +4.74456 q^{41} +10.8896i q^{43} +0.627719 q^{44} +6.00000 q^{46} +10.8896i q^{47} -18.4891 q^{49} +(-5.05842 + 7.02939i) q^{50} +4.25639i q^{52} +4.25639i q^{53} +(0.430703 - 1.33591i) q^{55} +8.74456 q^{56} +1.73205i q^{58} -10.7446 q^{59} +6.00000 q^{61} +5.84096i q^{62} -1.00000 q^{64} +(9.05842 + 2.92048i) q^{65} -1.87953i q^{67} -1.58457i q^{68} +(-6.00000 + 18.6101i) q^{70} -6.74456 q^{71} +6.92820i q^{73} +5.48913 q^{74} +4.00000 q^{76} -3.16915i q^{77} -11.3723 q^{79} +(3.43070 - 10.6410i) q^{80} -8.21782i q^{82} -9.80240i q^{83} +(-3.37228 - 1.08724i) q^{85} +18.8614 q^{86} +1.08724i q^{88} -0.744563 q^{89} +21.4891 q^{91} -3.46410i q^{92} +18.8614 q^{94} +(2.74456 - 8.51278i) q^{95} -6.92820i q^{97} +32.0241i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 3 q^{5} + 9 q^{10} - 14 q^{11} + 12 q^{14} - 20 q^{16} - 16 q^{19} - 3 q^{20} + q^{25} - 18 q^{26} - 4 q^{29} - 2 q^{31} - 12 q^{34} - 20 q^{35} + 9 q^{40} - 4 q^{41} + 14 q^{44} + 24 q^{46}+ \cdots - 12 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.73205i − 1.22474i −0.790569 − 0.612372i \(-0.790215\pi\)
0.790569 − 0.612372i \(-0.209785\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −0.686141 + 2.12819i −0.306851 + 0.951757i
\(6\) 0 0
\(7\) 5.04868i 1.90822i 0.299456 + 0.954110i \(0.403195\pi\)
−0.299456 + 0.954110i \(0.596805\pi\)
\(8\) − 1.73205i − 0.612372i
\(9\) 0 0
\(10\) 3.68614 + 1.18843i 1.16566 + 0.375815i
\(11\) −0.627719 −0.189264 −0.0946322 − 0.995512i \(-0.530167\pi\)
−0.0946322 + 0.995512i \(0.530167\pi\)
\(12\) 0 0
\(13\) − 4.25639i − 1.18051i −0.807217 − 0.590255i \(-0.799027\pi\)
0.807217 − 0.590255i \(-0.200973\pi\)
\(14\) 8.74456 2.33708
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 1.58457i 0.384316i 0.981364 + 0.192158i \(0.0615486\pi\)
−0.981364 + 0.192158i \(0.938451\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0.686141 − 2.12819i 0.153426 − 0.475879i
\(21\) 0 0
\(22\) 1.08724i 0.231800i
\(23\) 3.46410i 0.722315i 0.932505 + 0.361158i \(0.117618\pi\)
−0.932505 + 0.361158i \(0.882382\pi\)
\(24\) 0 0
\(25\) −4.05842 − 2.92048i −0.811684 − 0.584096i
\(26\) −7.37228 −1.44582
\(27\) 0 0
\(28\) − 5.04868i − 0.954110i
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −3.37228 −0.605680 −0.302840 − 0.953041i \(-0.597935\pi\)
−0.302840 + 0.953041i \(0.597935\pi\)
\(32\) 5.19615i 0.918559i
\(33\) 0 0
\(34\) 2.74456 0.470689
\(35\) −10.7446 − 3.46410i −1.81616 − 0.585540i
\(36\) 0 0
\(37\) 3.16915i 0.521005i 0.965473 + 0.260502i \(0.0838882\pi\)
−0.965473 + 0.260502i \(0.916112\pi\)
\(38\) 6.92820i 1.12390i
\(39\) 0 0
\(40\) 3.68614 + 1.18843i 0.582830 + 0.187907i
\(41\) 4.74456 0.740976 0.370488 − 0.928837i \(-0.379190\pi\)
0.370488 + 0.928837i \(0.379190\pi\)
\(42\) 0 0
\(43\) 10.8896i 1.66065i 0.557276 + 0.830327i \(0.311846\pi\)
−0.557276 + 0.830327i \(0.688154\pi\)
\(44\) 0.627719 0.0946322
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 10.8896i 1.58842i 0.607645 + 0.794208i \(0.292114\pi\)
−0.607645 + 0.794208i \(0.707886\pi\)
\(48\) 0 0
\(49\) −18.4891 −2.64130
\(50\) −5.05842 + 7.02939i −0.715369 + 0.994106i
\(51\) 0 0
\(52\) 4.25639i 0.590255i
\(53\) 4.25639i 0.584660i 0.956318 + 0.292330i \(0.0944306\pi\)
−0.956318 + 0.292330i \(0.905569\pi\)
\(54\) 0 0
\(55\) 0.430703 − 1.33591i 0.0580760 − 0.180134i
\(56\) 8.74456 1.16854
\(57\) 0 0
\(58\) 1.73205i 0.227429i
\(59\) −10.7446 −1.39882 −0.699411 − 0.714719i \(-0.746554\pi\)
−0.699411 + 0.714719i \(0.746554\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 5.84096i 0.741803i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 9.05842 + 2.92048i 1.12356 + 0.362241i
\(66\) 0 0
\(67\) − 1.87953i − 0.229621i −0.993387 − 0.114810i \(-0.963374\pi\)
0.993387 − 0.114810i \(-0.0366261\pi\)
\(68\) − 1.58457i − 0.192158i
\(69\) 0 0
\(70\) −6.00000 + 18.6101i −0.717137 + 2.22434i
\(71\) −6.74456 −0.800432 −0.400216 − 0.916421i \(-0.631065\pi\)
−0.400216 + 0.916421i \(0.631065\pi\)
\(72\) 0 0
\(73\) 6.92820i 0.810885i 0.914121 + 0.405442i \(0.132883\pi\)
−0.914121 + 0.405442i \(0.867117\pi\)
\(74\) 5.48913 0.638098
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) − 3.16915i − 0.361158i
\(78\) 0 0
\(79\) −11.3723 −1.27948 −0.639741 − 0.768591i \(-0.720958\pi\)
−0.639741 + 0.768591i \(0.720958\pi\)
\(80\) 3.43070 − 10.6410i 0.383564 − 1.18970i
\(81\) 0 0
\(82\) − 8.21782i − 0.907507i
\(83\) − 9.80240i − 1.07595i −0.842960 − 0.537976i \(-0.819189\pi\)
0.842960 − 0.537976i \(-0.180811\pi\)
\(84\) 0 0
\(85\) −3.37228 − 1.08724i −0.365775 − 0.117928i
\(86\) 18.8614 2.03388
\(87\) 0 0
\(88\) 1.08724i 0.115900i
\(89\) −0.744563 −0.0789235 −0.0394617 − 0.999221i \(-0.512564\pi\)
−0.0394617 + 0.999221i \(0.512564\pi\)
\(90\) 0 0
\(91\) 21.4891 2.25267
\(92\) − 3.46410i − 0.361158i
\(93\) 0 0
\(94\) 18.8614 1.94541
\(95\) 2.74456 − 8.51278i 0.281586 − 0.873393i
\(96\) 0 0
\(97\) − 6.92820i − 0.703452i −0.936103 − 0.351726i \(-0.885595\pi\)
0.936103 − 0.351726i \(-0.114405\pi\)
\(98\) 32.0241i 3.23492i
\(99\) 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.e.784.1 4
3.2 odd 2 145.2.b.a.59.3 yes 4
5.2 odd 4 6525.2.a.bk.1.3 4
5.3 odd 4 6525.2.a.bk.1.2 4
5.4 even 2 inner 1305.2.c.e.784.3 4
12.11 even 2 2320.2.d.c.929.3 4
15.2 even 4 725.2.a.g.1.1 4
15.8 even 4 725.2.a.g.1.4 4
15.14 odd 2 145.2.b.a.59.2 ✓ 4
60.59 even 2 2320.2.d.c.929.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.b.a.59.2 ✓ 4 15.14 odd 2
145.2.b.a.59.3 yes 4 3.2 odd 2
725.2.a.g.1.1 4 15.2 even 4
725.2.a.g.1.4 4 15.8 even 4
1305.2.c.e.784.1 4 1.1 even 1 trivial
1305.2.c.e.784.3 4 5.4 even 2 inner
2320.2.d.c.929.2 4 60.59 even 2
2320.2.d.c.929.3 4 12.11 even 2
6525.2.a.bk.1.2 4 5.3 odd 4
6525.2.a.bk.1.3 4 5.2 odd 4