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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [448,5,Mod(15,448)] 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("448.15"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 448.k (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(46.3097434616\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 15.16
Character \(\chi\) \(=\) 448.15
Dual form 448.5.k.a.239.16

$q$-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)
 
\(f(q)\) \(=\) \(q+(-4.91576 - 4.91576i) q^{3} +(-5.29630 - 5.29630i) q^{5} +18.5203 q^{7} -32.6706i q^{9} +(-65.1032 + 65.1032i) q^{11} +(2.97917 - 2.97917i) q^{13} +52.0707i q^{15} +212.501 q^{17} +(-301.655 - 301.655i) q^{19} +(-91.0412 - 91.0412i) q^{21} -474.668 q^{23} -568.898i q^{25} +(-558.777 + 558.777i) q^{27} +(355.127 - 355.127i) q^{29} +801.728i q^{31} +640.063 q^{33} +(-98.0888 - 98.0888i) q^{35} +(151.471 + 151.471i) q^{37} -29.2898 q^{39} -788.369i q^{41} +(-853.638 + 853.638i) q^{43} +(-173.033 + 173.033i) q^{45} -2384.15i q^{47} +343.000 q^{49} +(-1044.60 - 1044.60i) q^{51} +(1527.65 + 1527.65i) q^{53} +689.612 q^{55} +2965.73i q^{57} +(-3615.57 + 3615.57i) q^{59} +(-4237.89 + 4237.89i) q^{61} -605.068i q^{63} -31.5572 q^{65} +(2926.57 + 2926.57i) q^{67} +(2333.35 + 2333.35i) q^{69} +2711.36 q^{71} +803.219i q^{73} +(-2796.57 + 2796.57i) q^{75} +(-1205.73 + 1205.73i) q^{77} +10318.3i q^{79} +2847.31 q^{81} +(-322.720 - 322.720i) q^{83} +(-1125.47 - 1125.47i) q^{85} -3491.43 q^{87} +8475.90i q^{89} +(55.1751 - 55.1751i) q^{91} +(3941.10 - 3941.10i) q^{93} +3195.31i q^{95} +3166.40 q^{97} +(2126.96 + 2126.96i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 96 q^{11} - 1408 q^{19} + 1152 q^{23} - 3648 q^{27} - 864 q^{29} + 1824 q^{37} + 5376 q^{39} + 6176 q^{43} + 32928 q^{49} + 8064 q^{51} + 480 q^{53} - 23552 q^{55} + 7488 q^{59} + 7552 q^{61} + 4032 q^{65}+ \cdots + 59552 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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
\(3\) −4.91576 − 4.91576i −0.546196 − 0.546196i 0.379143 − 0.925338i \(-0.376219\pi\)
−0.925338 + 0.379143i \(0.876219\pi\)
\(4\) 0 0
\(5\) −5.29630 − 5.29630i −0.211852 − 0.211852i 0.593202 − 0.805054i \(-0.297864\pi\)
−0.805054 + 0.593202i \(0.797864\pi\)
\(6\) 0 0
\(7\) 18.5203 0.377964
\(8\) 0 0
\(9\) − 32.6706i − 0.403341i
\(10\) 0 0
\(11\) −65.1032 + 65.1032i −0.538043 + 0.538043i −0.922954 − 0.384911i \(-0.874232\pi\)
0.384911 + 0.922954i \(0.374232\pi\)
\(12\) 0 0
\(13\) 2.97917 − 2.97917i 0.0176282 − 0.0176282i −0.698238 − 0.715866i \(-0.746032\pi\)
0.715866 + 0.698238i \(0.246032\pi\)
\(14\) 0 0
\(15\) 52.0707i 0.231425i
\(16\) 0 0
\(17\) 212.501 0.735297 0.367648 − 0.929965i \(-0.380163\pi\)
0.367648 + 0.929965i \(0.380163\pi\)
\(18\) 0 0
\(19\) −301.655 − 301.655i −0.835609 − 0.835609i 0.152668 − 0.988277i \(-0.451213\pi\)
−0.988277 + 0.152668i \(0.951213\pi\)
\(20\) 0 0
\(21\) −91.0412 − 91.0412i −0.206443 − 0.206443i
\(22\) 0 0
\(23\) −474.668 −0.897293 −0.448646 − 0.893709i \(-0.648094\pi\)
−0.448646 + 0.893709i \(0.648094\pi\)
\(24\) 0 0
\(25\) − 568.898i − 0.910237i
\(26\) 0 0
\(27\) −558.777 + 558.777i −0.766499 + 0.766499i
\(28\) 0 0
\(29\) 355.127 − 355.127i 0.422267 − 0.422267i −0.463717 − 0.885984i \(-0.653484\pi\)
0.885984 + 0.463717i \(0.153484\pi\)
\(30\) 0 0
\(31\) 801.728i 0.834264i 0.908846 + 0.417132i \(0.136965\pi\)
−0.908846 + 0.417132i \(0.863035\pi\)
\(32\) 0 0
\(33\) 640.063 0.587753
\(34\) 0 0
\(35\) −98.0888 − 98.0888i −0.0800725 − 0.0800725i
\(36\) 0 0
\(37\) 151.471 + 151.471i 0.110644 + 0.110644i 0.760261 − 0.649618i \(-0.225071\pi\)
−0.649618 + 0.760261i \(0.725071\pi\)
\(38\) 0 0
\(39\) −29.2898 −0.0192569
\(40\) 0 0
\(41\) − 788.369i − 0.468988i −0.972118 − 0.234494i \(-0.924657\pi\)
0.972118 − 0.234494i \(-0.0753433\pi\)
\(42\) 0 0
\(43\) −853.638 + 853.638i −0.461675 + 0.461675i −0.899204 − 0.437529i \(-0.855854\pi\)
0.437529 + 0.899204i \(0.355854\pi\)
\(44\) 0 0
\(45\) −173.033 + 173.033i −0.0854485 + 0.0854485i
\(46\) 0 0
\(47\) − 2384.15i − 1.07929i −0.841893 − 0.539645i \(-0.818558\pi\)
0.841893 − 0.539645i \(-0.181442\pi\)
\(48\) 0 0
\(49\) 343.000 0.142857
\(50\) 0 0
\(51\) −1044.60 − 1044.60i −0.401616 − 0.401616i
\(52\) 0 0
\(53\) 1527.65 + 1527.65i 0.543841 + 0.543841i 0.924653 − 0.380812i \(-0.124355\pi\)
−0.380812 + 0.924653i \(0.624355\pi\)
\(54\) 0 0
\(55\) 689.612 0.227971
\(56\) 0 0
\(57\) 2965.73i 0.912812i
\(58\) 0 0
\(59\) −3615.57 + 3615.57i −1.03866 + 1.03866i −0.0394362 + 0.999222i \(0.512556\pi\)
−0.999222 + 0.0394362i \(0.987444\pi\)
\(60\) 0 0
\(61\) −4237.89 + 4237.89i −1.13891 + 1.13891i −0.150267 + 0.988645i \(0.548013\pi\)
−0.988645 + 0.150267i \(0.951987\pi\)
\(62\) 0 0
\(63\) − 605.068i − 0.152448i
\(64\) 0 0
\(65\) −31.5572 −0.00746916
\(66\) 0 0
\(67\) 2926.57 + 2926.57i 0.651943 + 0.651943i 0.953461 − 0.301518i \(-0.0974933\pi\)
−0.301518 + 0.953461i \(0.597493\pi\)
\(68\) 0 0
\(69\) 2333.35 + 2333.35i 0.490097 + 0.490097i
\(70\) 0 0
\(71\) 2711.36 0.537861 0.268930 − 0.963160i \(-0.413330\pi\)
0.268930 + 0.963160i \(0.413330\pi\)
\(72\) 0 0
\(73\) 803.219i 0.150726i 0.997156 + 0.0753630i \(0.0240115\pi\)
−0.997156 + 0.0753630i \(0.975988\pi\)
\(74\) 0 0
\(75\) −2796.57 + 2796.57i −0.497168 + 0.497168i
\(76\) 0 0
\(77\) −1205.73 + 1205.73i −0.203361 + 0.203361i
\(78\) 0 0
\(79\) 10318.3i 1.65331i 0.562709 + 0.826655i \(0.309759\pi\)
−0.562709 + 0.826655i \(0.690241\pi\)
\(80\) 0 0
\(81\) 2847.31 0.433976
\(82\) 0 0
\(83\) −322.720 − 322.720i −0.0468458 − 0.0468458i 0.683296 − 0.730142i \(-0.260546\pi\)
−0.730142 + 0.683296i \(0.760546\pi\)
\(84\) 0 0
\(85\) −1125.47 − 1125.47i −0.155774 − 0.155774i
\(86\) 0 0
\(87\) −3491.43 −0.461281
\(88\) 0 0
\(89\) 8475.90i 1.07005i 0.844835 + 0.535027i \(0.179699\pi\)
−0.844835 + 0.535027i \(0.820301\pi\)
\(90\) 0 0
\(91\) 55.1751 − 55.1751i 0.00666285 − 0.00666285i
\(92\) 0 0
\(93\) 3941.10 − 3941.10i 0.455671 − 0.455671i
\(94\) 0 0
\(95\) 3195.31i 0.354051i
\(96\) 0 0
\(97\) 3166.40 0.336528 0.168264 − 0.985742i \(-0.446184\pi\)
0.168264 + 0.985742i \(0.446184\pi\)
\(98\) 0 0
\(99\) 2126.96 + 2126.96i 0.217015 + 0.217015i
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 448.5.k.a.15.16 96
4.3 odd 2 112.5.k.a.43.9 ✓ 96
16.3 odd 4 inner 448.5.k.a.239.16 96
16.13 even 4 112.5.k.a.99.9 yes 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.9 ✓ 96 4.3 odd 2
112.5.k.a.99.9 yes 96 16.13 even 4
448.5.k.a.15.16 96 1.1 even 1 trivial
448.5.k.a.239.16 96 16.3 odd 4 inner