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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 = [4,0,-12,0,12,0,-44] 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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{29})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 15x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 5 \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 257.2
Root \(2.19258i\) of defining polynomial
Character \(\chi\) \(=\) 320.257
Dual form 320.5.p.k.193.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-3.00000 + 3.00000i) q^{3} +(24.5407 - 4.77033i) q^{5} +(42.8516 + 42.8516i) q^{7} +63.0000i q^{9} -181.703 q^{11} +(-25.1484 + 25.1484i) q^{13} +(-59.3110 + 87.9330i) q^{15} +(160.407 + 160.407i) q^{17} +80.2967i q^{19} -257.110 q^{21} +(-324.555 + 324.555i) q^{23} +(579.488 - 234.134i) q^{25} +(-432.000 - 432.000i) q^{27} -1374.52i q^{29} +151.484 q^{31} +(545.110 - 545.110i) q^{33} +(1256.02 + 847.191i) q^{35} +(1428.48 + 1428.48i) q^{37} -150.890i q^{39} -3016.66 q^{41} +(-1615.74 + 1615.74i) q^{43} +(300.531 + 1546.06i) q^{45} +(397.522 + 397.522i) q^{47} +1271.53i q^{49} -962.440 q^{51} +(-923.225 + 923.225i) q^{53} +(-4459.12 + 866.785i) q^{55} +(-240.890 - 240.890i) q^{57} +4549.02i q^{59} -631.912 q^{61} +(-2699.65 + 2699.65i) q^{63} +(-497.191 + 737.123i) q^{65} +(-2687.74 - 2687.74i) q^{67} -1947.33i q^{69} -4543.02 q^{71} +(5853.14 - 5853.14i) q^{73} +(-1036.06 + 2440.87i) q^{75} +(-7786.29 - 7786.29i) q^{77} +9423.38i q^{79} -2511.00 q^{81} +(-8391.13 + 8391.13i) q^{83} +(4701.68 + 3171.29i) q^{85} +(4123.55 + 4123.55i) q^{87} +8365.60i q^{89} -2155.30 q^{91} +(-454.451 + 454.451i) q^{93} +(383.042 + 1970.53i) q^{95} +(1014.34 + 1014.34i) q^{97} -11447.3i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} + 12 q^{5} - 44 q^{7} - 296 q^{11} - 316 q^{13} - 108 q^{15} - 220 q^{17} + 264 q^{21} - 652 q^{23} + 1284 q^{25} - 1728 q^{27} + 2760 q^{31} + 888 q^{33} + 7092 q^{35} + 2052 q^{37} - 4312 q^{41}+ \cdots + 11812 q^{97}+O(q^{100}) \) 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
\(3\) −3.00000 + 3.00000i −0.333333 + 0.333333i −0.853851 − 0.520518i \(-0.825739\pi\)
0.520518 + 0.853851i \(0.325739\pi\)
\(4\) 0 0
\(5\) 24.5407 − 4.77033i 0.981626 − 0.190813i
\(6\) 0 0
\(7\) 42.8516 + 42.8516i 0.874523 + 0.874523i 0.992961 − 0.118438i \(-0.0377887\pi\)
−0.118438 + 0.992961i \(0.537789\pi\)
\(8\) 0 0
\(9\) 63.0000i 0.777778i
\(10\) 0 0
\(11\) −181.703 −1.50168 −0.750840 − 0.660484i \(-0.770351\pi\)
−0.750840 + 0.660484i \(0.770351\pi\)
\(12\) 0 0
\(13\) −25.1484 + 25.1484i −0.148807 + 0.148807i −0.777585 − 0.628778i \(-0.783555\pi\)
0.628778 + 0.777585i \(0.283555\pi\)
\(14\) 0 0
\(15\) −59.3110 + 87.9330i −0.263604 + 0.390813i
\(16\) 0 0
\(17\) 160.407 + 160.407i 0.555040 + 0.555040i 0.927891 − 0.372851i \(-0.121620\pi\)
−0.372851 + 0.927891i \(0.621620\pi\)
\(18\) 0 0
\(19\) 80.2967i 0.222429i 0.993796 + 0.111214i \(0.0354740\pi\)
−0.993796 + 0.111214i \(0.964526\pi\)
\(20\) 0 0
\(21\) −257.110 −0.583016
\(22\) 0 0
\(23\) −324.555 + 324.555i −0.613525 + 0.613525i −0.943863 − 0.330337i \(-0.892837\pi\)
0.330337 + 0.943863i \(0.392837\pi\)
\(24\) 0 0
\(25\) 579.488 − 234.134i 0.927181 − 0.374615i
\(26\) 0 0
\(27\) −432.000 − 432.000i −0.592593 − 0.592593i
\(28\) 0 0
\(29\) − 1374.52i − 1.63438i −0.576366 − 0.817192i \(-0.695530\pi\)
0.576366 − 0.817192i \(-0.304470\pi\)
\(30\) 0 0
\(31\) 151.484 0.157631 0.0788156 − 0.996889i \(-0.474886\pi\)
0.0788156 + 0.996889i \(0.474886\pi\)
\(32\) 0 0
\(33\) 545.110 − 545.110i 0.500560 − 0.500560i
\(34\) 0 0
\(35\) 1256.02 + 847.191i 1.02533 + 0.691585i
\(36\) 0 0
\(37\) 1428.48 + 1428.48i 1.04345 + 1.04345i 0.999012 + 0.0444340i \(0.0141484\pi\)
0.0444340 + 0.999012i \(0.485852\pi\)
\(38\) 0 0
\(39\) − 150.890i − 0.0992045i
\(40\) 0 0
\(41\) −3016.66 −1.79456 −0.897281 − 0.441459i \(-0.854461\pi\)
−0.897281 + 0.441459i \(0.854461\pi\)
\(42\) 0 0
\(43\) −1615.74 + 1615.74i −0.873843 + 0.873843i −0.992889 − 0.119046i \(-0.962017\pi\)
0.119046 + 0.992889i \(0.462017\pi\)
\(44\) 0 0
\(45\) 300.531 + 1546.06i 0.148410 + 0.763487i
\(46\) 0 0
\(47\) 397.522 + 397.522i 0.179956 + 0.179956i 0.791336 − 0.611381i \(-0.209386\pi\)
−0.611381 + 0.791336i \(0.709386\pi\)
\(48\) 0 0
\(49\) 1271.53i 0.529582i
\(50\) 0 0
\(51\) −962.440 −0.370027
\(52\) 0 0
\(53\) −923.225 + 923.225i −0.328667 + 0.328667i −0.852079 − 0.523413i \(-0.824659\pi\)
0.523413 + 0.852079i \(0.324659\pi\)
\(54\) 0 0
\(55\) −4459.12 + 866.785i −1.47409 + 0.286540i
\(56\) 0 0
\(57\) −240.890 − 240.890i −0.0741428 − 0.0741428i
\(58\) 0 0
\(59\) 4549.02i 1.30681i 0.757006 + 0.653407i \(0.226661\pi\)
−0.757006 + 0.653407i \(0.773339\pi\)
\(60\) 0 0
\(61\) −631.912 −0.169823 −0.0849116 − 0.996388i \(-0.527061\pi\)
−0.0849116 + 0.996388i \(0.527061\pi\)
\(62\) 0 0
\(63\) −2699.65 + 2699.65i −0.680185 + 0.680185i
\(64\) 0 0
\(65\) −497.191 + 737.123i −0.117678 + 0.174467i
\(66\) 0 0
\(67\) −2687.74 − 2687.74i −0.598738 − 0.598738i 0.341238 − 0.939977i \(-0.389154\pi\)
−0.939977 + 0.341238i \(0.889154\pi\)
\(68\) 0 0
\(69\) − 1947.33i − 0.409017i
\(70\) 0 0
\(71\) −4543.02 −0.901214 −0.450607 − 0.892722i \(-0.648792\pi\)
−0.450607 + 0.892722i \(0.648792\pi\)
\(72\) 0 0
\(73\) 5853.14 − 5853.14i 1.09836 − 1.09836i 0.103754 − 0.994603i \(-0.466915\pi\)
0.994603 − 0.103754i \(-0.0330854\pi\)
\(74\) 0 0
\(75\) −1036.06 + 2440.87i −0.184189 + 0.433932i
\(76\) 0 0
\(77\) −7786.29 − 7786.29i −1.31325 − 1.31325i
\(78\) 0 0
\(79\) 9423.38i 1.50992i 0.655773 + 0.754958i \(0.272343\pi\)
−0.655773 + 0.754958i \(0.727657\pi\)
\(80\) 0 0
\(81\) −2511.00 −0.382716
\(82\) 0 0
\(83\) −8391.13 + 8391.13i −1.21805 + 1.21805i −0.249733 + 0.968315i \(0.580343\pi\)
−0.968315 + 0.249733i \(0.919657\pi\)
\(84\) 0 0
\(85\) 4701.68 + 3171.29i 0.650751 + 0.438933i
\(86\) 0 0
\(87\) 4123.55 + 4123.55i 0.544794 + 0.544794i
\(88\) 0 0
\(89\) 8365.60i 1.05613i 0.849204 + 0.528065i \(0.177082\pi\)
−0.849204 + 0.528065i \(0.822918\pi\)
\(90\) 0 0
\(91\) −2155.30 −0.260270
\(92\) 0 0
\(93\) −454.451 + 454.451i −0.0525437 + 0.0525437i
\(94\) 0 0
\(95\) 383.042 + 1970.53i 0.0424423 + 0.218342i
\(96\) 0 0
\(97\) 1014.34 + 1014.34i 0.107805 + 0.107805i 0.758952 − 0.651147i \(-0.225712\pi\)
−0.651147 + 0.758952i \(0.725712\pi\)
\(98\) 0 0
\(99\) − 11447.3i − 1.16797i
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.k.257.2 4
4.3 odd 2 320.5.p.n.257.2 4
5.3 odd 4 inner 320.5.p.k.193.2 4
8.3 odd 2 40.5.l.b.17.1 ✓ 4
8.5 even 2 80.5.p.f.17.1 4
20.3 even 4 320.5.p.n.193.2 4
24.11 even 2 360.5.v.b.217.2 4
40.3 even 4 40.5.l.b.33.1 yes 4
40.13 odd 4 80.5.p.f.33.1 4
40.19 odd 2 200.5.l.c.57.2 4
40.27 even 4 200.5.l.c.193.2 4
40.29 even 2 400.5.p.g.257.1 4
40.37 odd 4 400.5.p.g.193.1 4
120.83 odd 4 360.5.v.b.73.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.5.l.b.17.1 ✓ 4 8.3 odd 2
40.5.l.b.33.1 yes 4 40.3 even 4
80.5.p.f.17.1 4 8.5 even 2
80.5.p.f.33.1 4 40.13 odd 4
200.5.l.c.57.2 4 40.19 odd 2
200.5.l.c.193.2 4 40.27 even 4
320.5.p.k.193.2 4 5.3 odd 4 inner
320.5.p.k.257.2 4 1.1 even 1 trivial
320.5.p.n.193.2 4 20.3 even 4
320.5.p.n.257.2 4 4.3 odd 2
360.5.v.b.73.2 4 120.83 odd 4
360.5.v.b.217.2 4 24.11 even 2
400.5.p.g.193.1 4 40.37 odd 4
400.5.p.g.257.1 4 40.29 even 2