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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 15.14
Character \(\chi\) \(=\) 448.15
Dual form 448.5.k.a.239.14

$q$-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)
 
\(f(q)\) \(=\) \(q+(-6.54673 - 6.54673i) q^{3} +(-10.9039 - 10.9039i) q^{5} +18.5203 q^{7} +4.71923i q^{9} +(4.98403 - 4.98403i) q^{11} +(-55.7937 + 55.7937i) q^{13} +142.770i q^{15} -492.666 q^{17} +(337.985 + 337.985i) q^{19} +(-121.247 - 121.247i) q^{21} +290.895 q^{23} -387.208i q^{25} +(-499.389 + 499.389i) q^{27} +(-1132.19 + 1132.19i) q^{29} -998.354i q^{31} -65.2582 q^{33} +(-201.944 - 201.944i) q^{35} +(-319.046 - 319.046i) q^{37} +730.532 q^{39} -433.353i q^{41} +(-499.804 + 499.804i) q^{43} +(51.4582 - 51.4582i) q^{45} +1763.91i q^{47} +343.000 q^{49} +(3225.35 + 3225.35i) q^{51} +(3086.00 + 3086.00i) q^{53} -108.691 q^{55} -4425.39i q^{57} +(-1177.76 + 1177.76i) q^{59} +(3815.32 - 3815.32i) q^{61} +87.4013i q^{63} +1216.74 q^{65} +(722.719 + 722.719i) q^{67} +(-1904.41 - 1904.41i) q^{69} +6982.08 q^{71} +2567.95i q^{73} +(-2534.94 + 2534.94i) q^{75} +(92.3055 - 92.3055i) q^{77} -5338.40i q^{79} +6920.99 q^{81} +(7409.85 + 7409.85i) q^{83} +(5372.00 + 5372.00i) q^{85} +14824.2 q^{87} +1263.43i q^{89} +(-1033.31 + 1033.31i) q^{91} +(-6535.95 + 6535.95i) q^{93} -7370.74i q^{95} +1211.89 q^{97} +(23.5208 + 23.5208i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 96 q^{11} - 1408 q^{19} + 1152 q^{23} - 3648 q^{27} - 864 q^{29} + 1824 q^{37} + 5376 q^{39} + 6176 q^{43} + 32928 q^{49} + 8064 q^{51} + 480 q^{53} - 23552 q^{55} + 7488 q^{59} + 7552 q^{61} + 4032 q^{65}+ \cdots + 59552 q^{99}+O(q^{100}) \) 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
\(3\) −6.54673 − 6.54673i −0.727414 − 0.727414i 0.242690 − 0.970104i \(-0.421970\pi\)
−0.970104 + 0.242690i \(0.921970\pi\)
\(4\) 0 0
\(5\) −10.9039 − 10.9039i −0.436158 − 0.436158i 0.454559 − 0.890717i \(-0.349797\pi\)
−0.890717 + 0.454559i \(0.849797\pi\)
\(6\) 0 0
\(7\) 18.5203 0.377964
\(8\) 0 0
\(9\) 4.71923i 0.0582621i
\(10\) 0 0
\(11\) 4.98403 − 4.98403i 0.0411903 − 0.0411903i −0.686212 − 0.727402i \(-0.740728\pi\)
0.727402 + 0.686212i \(0.240728\pi\)
\(12\) 0 0
\(13\) −55.7937 + 55.7937i −0.330140 + 0.330140i −0.852640 − 0.522500i \(-0.825000\pi\)
0.522500 + 0.852640i \(0.325000\pi\)
\(14\) 0 0
\(15\) 142.770i 0.634535i
\(16\) 0 0
\(17\) −492.666 −1.70473 −0.852363 − 0.522950i \(-0.824831\pi\)
−0.852363 + 0.522950i \(0.824831\pi\)
\(18\) 0 0
\(19\) 337.985 + 337.985i 0.936246 + 0.936246i 0.998086 − 0.0618397i \(-0.0196967\pi\)
−0.0618397 + 0.998086i \(0.519697\pi\)
\(20\) 0 0
\(21\) −121.247 − 121.247i −0.274937 − 0.274937i
\(22\) 0 0
\(23\) 290.895 0.549896 0.274948 − 0.961459i \(-0.411339\pi\)
0.274948 + 0.961459i \(0.411339\pi\)
\(24\) 0 0
\(25\) − 387.208i − 0.619533i
\(26\) 0 0
\(27\) −499.389 + 499.389i −0.685033 + 0.685033i
\(28\) 0 0
\(29\) −1132.19 + 1132.19i −1.34624 + 1.34624i −0.456533 + 0.889706i \(0.650909\pi\)
−0.889706 + 0.456533i \(0.849091\pi\)
\(30\) 0 0
\(31\) − 998.354i − 1.03887i −0.854510 − 0.519435i \(-0.826143\pi\)
0.854510 − 0.519435i \(-0.173857\pi\)
\(32\) 0 0
\(33\) −65.2582 −0.0599248
\(34\) 0 0
\(35\) −201.944 − 201.944i −0.164852 − 0.164852i
\(36\) 0 0
\(37\) −319.046 − 319.046i −0.233051 − 0.233051i 0.580914 − 0.813965i \(-0.302695\pi\)
−0.813965 + 0.580914i \(0.802695\pi\)
\(38\) 0 0
\(39\) 730.532 0.480297
\(40\) 0 0
\(41\) − 433.353i − 0.257795i −0.991658 − 0.128897i \(-0.958856\pi\)
0.991658 − 0.128897i \(-0.0411438\pi\)
\(42\) 0 0
\(43\) −499.804 + 499.804i −0.270310 + 0.270310i −0.829225 − 0.558915i \(-0.811218\pi\)
0.558915 + 0.829225i \(0.311218\pi\)
\(44\) 0 0
\(45\) 51.4582 − 51.4582i 0.0254115 − 0.0254115i
\(46\) 0 0
\(47\) 1763.91i 0.798510i 0.916840 + 0.399255i \(0.130731\pi\)
−0.916840 + 0.399255i \(0.869269\pi\)
\(48\) 0 0
\(49\) 343.000 0.142857
\(50\) 0 0
\(51\) 3225.35 + 3225.35i 1.24004 + 1.24004i
\(52\) 0 0
\(53\) 3086.00 + 3086.00i 1.09861 + 1.09861i 0.994573 + 0.104039i \(0.0331767\pi\)
0.104039 + 0.994573i \(0.466823\pi\)
\(54\) 0 0
\(55\) −108.691 −0.0359310
\(56\) 0 0
\(57\) − 4425.39i − 1.36208i
\(58\) 0 0
\(59\) −1177.76 + 1177.76i −0.338339 + 0.338339i −0.855742 − 0.517403i \(-0.826899\pi\)
0.517403 + 0.855742i \(0.326899\pi\)
\(60\) 0 0
\(61\) 3815.32 − 3815.32i 1.02535 − 1.02535i 0.0256768 − 0.999670i \(-0.491826\pi\)
0.999670 − 0.0256768i \(-0.00817409\pi\)
\(62\) 0 0
\(63\) 87.4013i 0.0220210i
\(64\) 0 0
\(65\) 1216.74 0.287986
\(66\) 0 0
\(67\) 722.719 + 722.719i 0.160998 + 0.160998i 0.783009 − 0.622011i \(-0.213684\pi\)
−0.622011 + 0.783009i \(0.713684\pi\)
\(68\) 0 0
\(69\) −1904.41 − 1904.41i −0.400002 − 0.400002i
\(70\) 0 0
\(71\) 6982.08 1.38506 0.692529 − 0.721390i \(-0.256497\pi\)
0.692529 + 0.721390i \(0.256497\pi\)
\(72\) 0 0
\(73\) 2567.95i 0.481883i 0.970540 + 0.240941i \(0.0774562\pi\)
−0.970540 + 0.240941i \(0.922544\pi\)
\(74\) 0 0
\(75\) −2534.94 + 2534.94i −0.450657 + 0.450657i
\(76\) 0 0
\(77\) 92.3055 − 92.3055i 0.0155685 − 0.0155685i
\(78\) 0 0
\(79\) − 5338.40i − 0.855375i −0.903927 − 0.427688i \(-0.859328\pi\)
0.903927 − 0.427688i \(-0.140672\pi\)
\(80\) 0 0
\(81\) 6920.99 1.05487
\(82\) 0 0
\(83\) 7409.85 + 7409.85i 1.07561 + 1.07561i 0.996898 + 0.0787086i \(0.0250797\pi\)
0.0787086 + 0.996898i \(0.474920\pi\)
\(84\) 0 0
\(85\) 5372.00 + 5372.00i 0.743530 + 0.743530i
\(86\) 0 0
\(87\) 14824.2 1.95855
\(88\) 0 0
\(89\) 1263.43i 0.159503i 0.996815 + 0.0797517i \(0.0254127\pi\)
−0.996815 + 0.0797517i \(0.974587\pi\)
\(90\) 0 0
\(91\) −1033.31 + 1033.31i −0.124781 + 0.124781i
\(92\) 0 0
\(93\) −6535.95 + 6535.95i −0.755688 + 0.755688i
\(94\) 0 0
\(95\) − 7370.74i − 0.816703i
\(96\) 0 0
\(97\) 1211.89 0.128801 0.0644007 − 0.997924i \(-0.479486\pi\)
0.0644007 + 0.997924i \(0.479486\pi\)
\(98\) 0 0
\(99\) 23.5208 + 23.5208i 0.00239983 + 0.00239983i
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.15.14 96
4.3 odd 2 112.5.k.a.43.17 ✓ 96
16.3 odd 4 inner 448.5.k.a.239.14 96
16.13 even 4 112.5.k.a.99.17 yes 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.17 ✓ 96 4.3 odd 2
112.5.k.a.99.17 yes 96 16.13 even 4
448.5.k.a.15.14 96 1.1 even 1 trivial
448.5.k.a.239.14 96 16.3 odd 4 inner