Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [320,5,Mod(193,320)] 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("320.193"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 320.p (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,8,0,4,0,-40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(33.0783881868\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.313431616.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 27x^{4} + 145x^{2} + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 257.1
Root \(4.49029i\) of defining polynomial
Character \(\chi\) \(=\) 320.257
Dual form 320.5.p.p.193.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-8.28874 + 8.28874i) q^{3} +(10.2887 + 22.7847i) q^{5} +(-3.20721 - 3.20721i) q^{7} -56.4063i q^{9} +19.5694 q^{11} +(-219.840 + 219.840i) q^{13} +(-274.137 - 103.576i) q^{15} +(281.352 + 281.352i) q^{17} -305.648i q^{19} +53.1675 q^{21} +(-633.243 + 633.243i) q^{23} +(-413.284 + 468.851i) q^{25} +(-203.851 - 203.851i) q^{27} +885.210i q^{29} +1383.32 q^{31} +(-162.205 + 162.205i) q^{33} +(40.0772 - 106.073i) q^{35} +(-453.987 - 453.987i) q^{37} -3644.39i q^{39} -193.003 q^{41} +(1016.66 - 1016.66i) q^{43} +(1285.20 - 580.350i) q^{45} +(-1203.53 - 1203.53i) q^{47} -2380.43i q^{49} -4664.11 q^{51} +(530.174 - 530.174i) q^{53} +(201.344 + 445.882i) q^{55} +(2533.43 + 2533.43i) q^{57} +934.801i q^{59} +1122.32 q^{61} +(-180.907 + 180.907i) q^{63} +(-7270.86 - 2747.11i) q^{65} +(-4760.40 - 4760.40i) q^{67} -10497.6i q^{69} -8831.10 q^{71} +(3008.63 - 3008.63i) q^{73} +(-460.584 - 7311.79i) q^{75} +(-62.7631 - 62.7631i) q^{77} -3349.09i q^{79} +7948.24 q^{81} +(3166.08 - 3166.08i) q^{83} +(-3515.76 + 9305.28i) q^{85} +(-7337.27 - 7337.27i) q^{87} -1732.33i q^{89} +1410.15 q^{91} +(-11466.0 + 11466.0i) q^{93} +(6964.09 - 3144.73i) q^{95} +(1659.43 + 1659.43i) q^{97} -1103.84i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{3} + 4 q^{5} - 40 q^{7} - 72 q^{11} + 62 q^{13} - 280 q^{15} + 418 q^{17} - 1736 q^{21} - 1760 q^{23} - 2974 q^{25} - 2248 q^{27} + 2864 q^{31} - 8 q^{33} + 4224 q^{35} - 5366 q^{37} + 1656 q^{41}+ \cdots - 9418 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\) \(1\)

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
\(3\) −8.28874 + 8.28874i −0.920971 + 0.920971i −0.997098 − 0.0761273i \(-0.975744\pi\)
0.0761273 + 0.997098i \(0.475744\pi\)
\(4\) 0 0
\(5\) 10.2887 + 22.7847i 0.411549 + 0.911387i
\(6\) 0 0
\(7\) −3.20721 − 3.20721i −0.0654533 − 0.0654533i 0.673622 − 0.739076i \(-0.264737\pi\)
−0.739076 + 0.673622i \(0.764737\pi\)
\(8\) 0 0
\(9\) − 56.4063i − 0.696374i
\(10\) 0 0
\(11\) 19.5694 0.161730 0.0808652 − 0.996725i \(-0.474232\pi\)
0.0808652 + 0.996725i \(0.474232\pi\)
\(12\) 0 0
\(13\) −219.840 + 219.840i −1.30083 + 1.30083i −0.372996 + 0.927833i \(0.621669\pi\)
−0.927833 + 0.372996i \(0.878331\pi\)
\(14\) 0 0
\(15\) −274.137 − 103.576i −1.21839 − 0.460336i
\(16\) 0 0
\(17\) 281.352 + 281.352i 0.973537 + 0.973537i 0.999659 − 0.0261217i \(-0.00831574\pi\)
−0.0261217 + 0.999659i \(0.508316\pi\)
\(18\) 0 0
\(19\) − 305.648i − 0.846670i −0.905973 − 0.423335i \(-0.860859\pi\)
0.905973 − 0.423335i \(-0.139141\pi\)
\(20\) 0 0
\(21\) 53.1675 0.120561
\(22\) 0 0
\(23\) −633.243 + 633.243i −1.19706 + 1.19706i −0.222014 + 0.975044i \(0.571263\pi\)
−0.975044 + 0.222014i \(0.928737\pi\)
\(24\) 0 0
\(25\) −413.284 + 468.851i −0.661254 + 0.750162i
\(26\) 0 0
\(27\) −203.851 − 203.851i −0.279630 − 0.279630i
\(28\) 0 0
\(29\) 885.210i 1.05257i 0.850309 + 0.526284i \(0.176415\pi\)
−0.850309 + 0.526284i \(0.823585\pi\)
\(30\) 0 0
\(31\) 1383.32 1.43946 0.719730 − 0.694254i \(-0.244266\pi\)
0.719730 + 0.694254i \(0.244266\pi\)
\(32\) 0 0
\(33\) −162.205 + 162.205i −0.148949 + 0.148949i
\(34\) 0 0
\(35\) 40.0772 − 106.073i 0.0327160 − 0.0865906i
\(36\) 0 0
\(37\) −453.987 − 453.987i −0.331619 − 0.331619i 0.521582 − 0.853201i \(-0.325342\pi\)
−0.853201 + 0.521582i \(0.825342\pi\)
\(38\) 0 0
\(39\) − 3644.39i − 2.39605i
\(40\) 0 0
\(41\) −193.003 −0.114814 −0.0574071 − 0.998351i \(-0.518283\pi\)
−0.0574071 + 0.998351i \(0.518283\pi\)
\(42\) 0 0
\(43\) 1016.66 − 1016.66i 0.549846 − 0.549846i −0.376550 − 0.926396i \(-0.622890\pi\)
0.926396 + 0.376550i \(0.122890\pi\)
\(44\) 0 0
\(45\) 1285.20 − 580.350i 0.634667 − 0.286592i
\(46\) 0 0
\(47\) −1203.53 − 1203.53i −0.544828 − 0.544828i 0.380112 − 0.924940i \(-0.375885\pi\)
−0.924940 + 0.380112i \(0.875885\pi\)
\(48\) 0 0
\(49\) − 2380.43i − 0.991432i
\(50\) 0 0
\(51\) −4664.11 −1.79320
\(52\) 0 0
\(53\) 530.174 − 530.174i 0.188741 − 0.188741i −0.606410 − 0.795152i \(-0.707391\pi\)
0.795152 + 0.606410i \(0.207391\pi\)
\(54\) 0 0
\(55\) 201.344 + 445.882i 0.0665600 + 0.147399i
\(56\) 0 0
\(57\) 2533.43 + 2533.43i 0.779758 + 0.779758i
\(58\) 0 0
\(59\) 934.801i 0.268544i 0.990945 + 0.134272i \(0.0428695\pi\)
−0.990945 + 0.134272i \(0.957130\pi\)
\(60\) 0 0
\(61\) 1122.32 0.301617 0.150809 − 0.988563i \(-0.451812\pi\)
0.150809 + 0.988563i \(0.451812\pi\)
\(62\) 0 0
\(63\) −180.907 + 180.907i −0.0455800 + 0.0455800i
\(64\) 0 0
\(65\) −7270.86 − 2747.11i −1.72091 − 0.650204i
\(66\) 0 0
\(67\) −4760.40 − 4760.40i −1.06046 − 1.06046i −0.998051 − 0.0624093i \(-0.980122\pi\)
−0.0624093 − 0.998051i \(-0.519878\pi\)
\(68\) 0 0
\(69\) − 10497.6i − 2.20491i
\(70\) 0 0
\(71\) −8831.10 −1.75185 −0.875927 − 0.482443i \(-0.839750\pi\)
−0.875927 + 0.482443i \(0.839750\pi\)
\(72\) 0 0
\(73\) 3008.63 − 3008.63i 0.564578 − 0.564578i −0.366027 − 0.930604i \(-0.619282\pi\)
0.930604 + 0.366027i \(0.119282\pi\)
\(74\) 0 0
\(75\) −460.584 − 7311.79i −0.0818816 − 1.29987i
\(76\) 0 0
\(77\) −62.7631 − 62.7631i −0.0105858 − 0.0105858i
\(78\) 0 0
\(79\) − 3349.09i − 0.536627i −0.963332 − 0.268313i \(-0.913534\pi\)
0.963332 − 0.268313i \(-0.0864663\pi\)
\(80\) 0 0
\(81\) 7948.24 1.21144
\(82\) 0 0
\(83\) 3166.08 − 3166.08i 0.459586 − 0.459586i −0.438934 − 0.898519i \(-0.644644\pi\)
0.898519 + 0.438934i \(0.144644\pi\)
\(84\) 0 0
\(85\) −3515.76 + 9305.28i −0.486611 + 1.28793i
\(86\) 0 0
\(87\) −7337.27 − 7337.27i −0.969384 − 0.969384i
\(88\) 0 0
\(89\) − 1732.33i − 0.218700i −0.994003 − 0.109350i \(-0.965123\pi\)
0.994003 − 0.109350i \(-0.0348770\pi\)
\(90\) 0 0
\(91\) 1410.15 0.170287
\(92\) 0 0
\(93\) −11466.0 + 11466.0i −1.32570 + 1.32570i
\(94\) 0 0
\(95\) 6964.09 − 3144.73i 0.771644 − 0.348446i
\(96\) 0 0
\(97\) 1659.43 + 1659.43i 0.176366 + 0.176366i 0.789770 − 0.613404i \(-0.210200\pi\)
−0.613404 + 0.789770i \(0.710200\pi\)
\(98\) 0 0
\(99\) − 1103.84i − 0.112625i
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 320.5.p.p.257.1 6
4.3 odd 2 320.5.p.o.257.3 6
5.3 odd 4 inner 320.5.p.p.193.1 6
8.3 odd 2 80.5.p.g.17.1 6
8.5 even 2 40.5.l.c.17.3 ✓ 6
20.3 even 4 320.5.p.o.193.3 6
24.5 odd 2 360.5.v.c.217.3 6
40.3 even 4 80.5.p.g.33.1 6
40.13 odd 4 40.5.l.c.33.3 yes 6
40.19 odd 2 400.5.p.o.257.3 6
40.27 even 4 400.5.p.o.193.3 6
40.29 even 2 200.5.l.e.57.1 6
40.37 odd 4 200.5.l.e.193.1 6
120.53 even 4 360.5.v.c.73.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.5.l.c.17.3 ✓ 6 8.5 even 2
40.5.l.c.33.3 yes 6 40.13 odd 4
80.5.p.g.17.1 6 8.3 odd 2
80.5.p.g.33.1 6 40.3 even 4
200.5.l.e.57.1 6 40.29 even 2
200.5.l.e.193.1 6 40.37 odd 4
320.5.p.o.193.3 6 20.3 even 4
320.5.p.o.257.3 6 4.3 odd 2
320.5.p.p.193.1 6 5.3 odd 4 inner
320.5.p.p.257.1 6 1.1 even 1 trivial
360.5.v.c.73.3 6 120.53 even 4
360.5.v.c.217.3 6 24.5 odd 2
400.5.p.o.193.3 6 40.27 even 4
400.5.p.o.257.3 6 40.19 odd 2