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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [405,2,Mod(244,405)] 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("405.244"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-6,-3,0,0,0,0,-1,-6] 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: \(3.23394128186\)
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: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 244.1
Root \(-1.18614 + 1.26217i\) of defining polynomial
Character \(\chi\) \(=\) 405.244
Dual form 405.2.b.a.244.4

$q$-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)
 
\(f(q)\) \(=\) \(q-2.52434i q^{2} -4.37228 q^{4} +(-2.18614 + 0.469882i) q^{5} +3.46410i q^{7} +5.98844i q^{8} +(1.18614 + 5.51856i) q^{10} +1.37228 q^{11} +4.10891i q^{13} +8.74456 q^{14} +6.37228 q^{16} -2.52434i q^{17} -5.37228 q^{19} +(9.55842 - 2.05446i) q^{20} -3.46410i q^{22} +5.04868i q^{23} +(4.55842 - 2.05446i) q^{25} +10.3723 q^{26} -15.1460i q^{28} -5.74456 q^{29} +0.627719 q^{31} -4.10891i q^{32} -6.37228 q^{34} +(-1.62772 - 7.57301i) q^{35} +7.57301i q^{37} +13.5615i q^{38} +(-2.81386 - 13.0916i) q^{40} -1.37228 q^{41} +3.46410i q^{43} -6.00000 q^{44} +12.7446 q^{46} +8.51278i q^{47} -5.00000 q^{49} +(-5.18614 - 11.5070i) q^{50} -17.9653i q^{52} -5.34363i q^{53} +(-3.00000 + 0.644810i) q^{55} -20.7446 q^{56} +14.5012i q^{58} -7.37228 q^{59} +3.62772 q^{61} -1.58457i q^{62} +2.37228 q^{64} +(-1.93070 - 8.98266i) q^{65} -8.21782i q^{67} +11.0371i q^{68} +(-19.1168 + 4.10891i) q^{70} -4.11684 q^{71} -7.57301i q^{73} +19.1168 q^{74} +23.4891 q^{76} +4.75372i q^{77} +4.74456 q^{79} +(-13.9307 + 2.99422i) q^{80} +3.46410i q^{82} -5.34363i q^{83} +(1.18614 + 5.51856i) q^{85} +8.74456 q^{86} +8.21782i q^{88} +3.00000 q^{89} -14.2337 q^{91} -22.0742i q^{92} +21.4891 q^{94} +(11.7446 - 2.52434i) q^{95} +18.6101i q^{97} +12.6217i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{4} - 3 q^{5} - q^{10} - 6 q^{11} + 12 q^{14} + 14 q^{16} - 10 q^{19} + 21 q^{20} + q^{25} + 30 q^{26} + 14 q^{31} - 14 q^{34} - 18 q^{35} - 17 q^{40} + 6 q^{41} - 24 q^{44} + 28 q^{46} - 20 q^{49}+ \cdots + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\) \(82\) \(326\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.52434i − 1.78498i −0.451071 − 0.892488i \(-0.648958\pi\)
0.451071 − 0.892488i \(-0.351042\pi\)
\(3\) 0 0
\(4\) −4.37228 −2.18614
\(5\) −2.18614 + 0.469882i −0.977672 + 0.210138i
\(6\) 0 0
\(7\) 3.46410i 1.30931i 0.755929 + 0.654654i \(0.227186\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 5.98844i 2.11723i
\(9\) 0 0
\(10\) 1.18614 + 5.51856i 0.375091 + 1.74512i
\(11\) 1.37228 0.413758 0.206879 − 0.978366i \(-0.433669\pi\)
0.206879 + 0.978366i \(0.433669\pi\)
\(12\) 0 0
\(13\) 4.10891i 1.13961i 0.821781 + 0.569804i \(0.192981\pi\)
−0.821781 + 0.569804i \(0.807019\pi\)
\(14\) 8.74456 2.33708
\(15\) 0 0
\(16\) 6.37228 1.59307
\(17\) − 2.52434i − 0.612242i −0.951993 − 0.306121i \(-0.900969\pi\)
0.951993 − 0.306121i \(-0.0990312\pi\)
\(18\) 0 0
\(19\) −5.37228 −1.23249 −0.616243 − 0.787556i \(-0.711346\pi\)
−0.616243 + 0.787556i \(0.711346\pi\)
\(20\) 9.55842 − 2.05446i 2.13733 − 0.459390i
\(21\) 0 0
\(22\) − 3.46410i − 0.738549i
\(23\) 5.04868i 1.05272i 0.850261 + 0.526361i \(0.176444\pi\)
−0.850261 + 0.526361i \(0.823556\pi\)
\(24\) 0 0
\(25\) 4.55842 − 2.05446i 0.911684 − 0.410891i
\(26\) 10.3723 2.03417
\(27\) 0 0
\(28\) − 15.1460i − 2.86233i
\(29\) −5.74456 −1.06674 −0.533369 − 0.845883i \(-0.679074\pi\)
−0.533369 + 0.845883i \(0.679074\pi\)
\(30\) 0 0
\(31\) 0.627719 0.112742 0.0563708 − 0.998410i \(-0.482047\pi\)
0.0563708 + 0.998410i \(0.482047\pi\)
\(32\) − 4.10891i − 0.726360i
\(33\) 0 0
\(34\) −6.37228 −1.09284
\(35\) −1.62772 − 7.57301i −0.275135 − 1.28007i
\(36\) 0 0
\(37\) 7.57301i 1.24500i 0.782621 + 0.622498i \(0.213882\pi\)
−0.782621 + 0.622498i \(0.786118\pi\)
\(38\) 13.5615i 2.19996i
\(39\) 0 0
\(40\) −2.81386 − 13.0916i −0.444910 − 2.06996i
\(41\) −1.37228 −0.214314 −0.107157 − 0.994242i \(-0.534175\pi\)
−0.107157 + 0.994242i \(0.534175\pi\)
\(42\) 0 0
\(43\) 3.46410i 0.528271i 0.964486 + 0.264135i \(0.0850865\pi\)
−0.964486 + 0.264135i \(0.914913\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) 12.7446 1.87908
\(47\) 8.51278i 1.24172i 0.783923 + 0.620858i \(0.213216\pi\)
−0.783923 + 0.620858i \(0.786784\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) −5.18614 − 11.5070i −0.733431 − 1.62734i
\(51\) 0 0
\(52\) − 17.9653i − 2.49134i
\(53\) − 5.34363i − 0.734004i −0.930220 − 0.367002i \(-0.880384\pi\)
0.930220 − 0.367002i \(-0.119616\pi\)
\(54\) 0 0
\(55\) −3.00000 + 0.644810i −0.404520 + 0.0869462i
\(56\) −20.7446 −2.77211
\(57\) 0 0
\(58\) 14.5012i 1.90410i
\(59\) −7.37228 −0.959789 −0.479895 − 0.877326i \(-0.659325\pi\)
−0.479895 + 0.877326i \(0.659325\pi\)
\(60\) 0 0
\(61\) 3.62772 0.464482 0.232241 − 0.972658i \(-0.425394\pi\)
0.232241 + 0.972658i \(0.425394\pi\)
\(62\) − 1.58457i − 0.201241i
\(63\) 0 0
\(64\) 2.37228 0.296535
\(65\) −1.93070 − 8.98266i −0.239474 − 1.11416i
\(66\) 0 0
\(67\) − 8.21782i − 1.00397i −0.864877 − 0.501983i \(-0.832604\pi\)
0.864877 − 0.501983i \(-0.167396\pi\)
\(68\) 11.0371i 1.33845i
\(69\) 0 0
\(70\) −19.1168 + 4.10891i −2.28490 + 0.491109i
\(71\) −4.11684 −0.488579 −0.244290 − 0.969702i \(-0.578555\pi\)
−0.244290 + 0.969702i \(0.578555\pi\)
\(72\) 0 0
\(73\) − 7.57301i − 0.886354i −0.896434 − 0.443177i \(-0.853851\pi\)
0.896434 − 0.443177i \(-0.146149\pi\)
\(74\) 19.1168 2.22229
\(75\) 0 0
\(76\) 23.4891 2.69439
\(77\) 4.75372i 0.541737i
\(78\) 0 0
\(79\) 4.74456 0.533805 0.266903 − 0.963724i \(-0.414000\pi\)
0.266903 + 0.963724i \(0.414000\pi\)
\(80\) −13.9307 + 2.99422i −1.55750 + 0.334764i
\(81\) 0 0
\(82\) 3.46410i 0.382546i
\(83\) − 5.34363i − 0.586540i −0.956030 − 0.293270i \(-0.905257\pi\)
0.956030 − 0.293270i \(-0.0947434\pi\)
\(84\) 0 0
\(85\) 1.18614 + 5.51856i 0.128655 + 0.598572i
\(86\) 8.74456 0.942950
\(87\) 0 0
\(88\) 8.21782i 0.876023i
\(89\) 3.00000 0.317999 0.159000 − 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) −14.2337 −1.49210
\(92\) − 22.0742i − 2.30140i
\(93\) 0 0
\(94\) 21.4891 2.21643
\(95\) 11.7446 − 2.52434i 1.20497 − 0.258992i
\(96\) 0 0
\(97\) 18.6101i 1.88957i 0.327688 + 0.944786i \(0.393731\pi\)
−0.327688 + 0.944786i \(0.606269\pi\)
\(98\) 12.6217i 1.27498i
\(99\) 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 405.2.b.a.244.1 ✓ 4
3.2 odd 2 405.2.b.b.244.4 yes 4
5.2 odd 4 2025.2.a.w.1.4 4
5.3 odd 4 2025.2.a.w.1.1 4
5.4 even 2 inner 405.2.b.a.244.4 yes 4
9.2 odd 6 405.2.j.d.109.2 4
9.4 even 3 405.2.j.e.379.2 4
9.5 odd 6 405.2.j.a.379.1 4
9.7 even 3 405.2.j.b.109.1 4
15.2 even 4 2025.2.a.x.1.1 4
15.8 even 4 2025.2.a.x.1.4 4
15.14 odd 2 405.2.b.b.244.1 yes 4
45.4 even 6 405.2.j.b.379.1 4
45.14 odd 6 405.2.j.d.379.2 4
45.29 odd 6 405.2.j.a.109.1 4
45.34 even 6 405.2.j.e.109.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
405.2.b.a.244.1 ✓ 4 1.1 even 1 trivial
405.2.b.a.244.4 yes 4 5.4 even 2 inner
405.2.b.b.244.1 yes 4 15.14 odd 2
405.2.b.b.244.4 yes 4 3.2 odd 2
405.2.j.a.109.1 4 45.29 odd 6
405.2.j.a.379.1 4 9.5 odd 6
405.2.j.b.109.1 4 9.7 even 3
405.2.j.b.379.1 4 45.4 even 6
405.2.j.d.109.2 4 9.2 odd 6
405.2.j.d.379.2 4 45.14 odd 6
405.2.j.e.109.2 4 45.34 even 6
405.2.j.e.379.2 4 9.4 even 3
2025.2.a.w.1.1 4 5.3 odd 4
2025.2.a.w.1.4 4 5.2 odd 4
2025.2.a.x.1.1 4 15.2 even 4
2025.2.a.x.1.4 4 15.8 even 4