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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [272,4,Mod(33,272)] 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("272.33"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(272, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 272 = 2^{4} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 272.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.0485195216\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-37 +3 \sqrt{33}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 74x^{2} + 1072 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 33.4
Root \(7.36435i\) of defining polynomial
Character \(\chi\) \(=\) 272.33
Dual form 272.4.b.d.33.1

$q$-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)
 
\(f(q)\) \(=\) \(q+7.36435i q^{3} -10.1060i q^{5} -17.4703i q^{7} -27.2337 q^{9} +51.5505i q^{11} +75.2119 q^{13} +74.4239 q^{15} +(-12.2119 + 69.0208i) q^{17} +28.0000 q^{19} +128.658 q^{21} +19.1913i q^{23} +22.8695 q^{25} -1.72096i q^{27} +70.7417i q^{29} -41.4445i q^{31} -379.636 q^{33} -176.554 q^{35} +135.460i q^{37} +553.887i q^{39} +288.771i q^{41} -88.2934 q^{43} +275.223i q^{45} -157.576 q^{47} +37.7881 q^{49} +(-508.293 - 89.9330i) q^{51} +120.250 q^{53} +520.967 q^{55} +206.202i q^{57} +696.119 q^{59} -683.544i q^{61} +475.781i q^{63} -760.089i q^{65} -123.826 q^{67} -141.331 q^{69} -225.393i q^{71} +919.423i q^{73} +168.419i q^{75} +900.603 q^{77} +354.830i q^{79} -722.636 q^{81} +955.272 q^{83} +(697.522 + 123.413i) q^{85} -520.967 q^{87} +617.636 q^{89} -1313.98i q^{91} +305.212 q^{93} -282.967i q^{95} -428.533i q^{97} -1403.91i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 40 q^{9} + 140 q^{13} - 24 q^{15} + 112 q^{17} + 112 q^{19} + 124 q^{21} - 460 q^{25} - 1036 q^{33} - 936 q^{35} + 520 q^{43} - 952 q^{47} + 312 q^{49} - 1160 q^{51} - 576 q^{53} - 168 q^{55} + 1176 q^{59}+ \cdots + 1060 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/272\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(239\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 7.36435i 1.41727i 0.705575 + 0.708635i \(0.250689\pi\)
−0.705575 + 0.708635i \(0.749311\pi\)
\(4\) 0 0
\(5\) − 10.1060i − 0.903905i −0.892042 − 0.451952i \(-0.850728\pi\)
0.892042 − 0.451952i \(-0.149272\pi\)
\(6\) 0 0
\(7\) − 17.4703i − 0.943308i −0.881784 − 0.471654i \(-0.843657\pi\)
0.881784 − 0.471654i \(-0.156343\pi\)
\(8\) 0 0
\(9\) −27.2337 −1.00866
\(10\) 0 0
\(11\) 51.5505i 1.41300i 0.707711 + 0.706502i \(0.249728\pi\)
−0.707711 + 0.706502i \(0.750272\pi\)
\(12\) 0 0
\(13\) 75.2119 1.60462 0.802309 − 0.596909i \(-0.203605\pi\)
0.802309 + 0.596909i \(0.203605\pi\)
\(14\) 0 0
\(15\) 74.4239 1.28108
\(16\) 0 0
\(17\) −12.2119 + 69.0208i −0.174225 + 0.984706i
\(18\) 0 0
\(19\) 28.0000 0.338086 0.169043 − 0.985609i \(-0.445932\pi\)
0.169043 + 0.985609i \(0.445932\pi\)
\(20\) 0 0
\(21\) 128.658 1.33692
\(22\) 0 0
\(23\) 19.1913i 0.173985i 0.996209 + 0.0869926i \(0.0277256\pi\)
−0.996209 + 0.0869926i \(0.972274\pi\)
\(24\) 0 0
\(25\) 22.8695 0.182956
\(26\) 0 0
\(27\) − 1.72096i − 0.0122666i
\(28\) 0 0
\(29\) 70.7417i 0.452980i 0.974014 + 0.226490i \(0.0727250\pi\)
−0.974014 + 0.226490i \(0.927275\pi\)
\(30\) 0 0
\(31\) − 41.4445i − 0.240118i −0.992767 − 0.120059i \(-0.961692\pi\)
0.992767 − 0.120059i \(-0.0383083\pi\)
\(32\) 0 0
\(33\) −379.636 −2.00261
\(34\) 0 0
\(35\) −176.554 −0.852661
\(36\) 0 0
\(37\) 135.460i 0.601879i 0.953643 + 0.300939i \(0.0973002\pi\)
−0.953643 + 0.300939i \(0.902700\pi\)
\(38\) 0 0
\(39\) 553.887i 2.27418i
\(40\) 0 0
\(41\) 288.771i 1.09996i 0.835178 + 0.549980i \(0.185365\pi\)
−0.835178 + 0.549980i \(0.814635\pi\)
\(42\) 0 0
\(43\) −88.2934 −0.313131 −0.156565 − 0.987668i \(-0.550042\pi\)
−0.156565 + 0.987668i \(0.550042\pi\)
\(44\) 0 0
\(45\) 275.223i 0.911728i
\(46\) 0 0
\(47\) −157.576 −0.489039 −0.244520 − 0.969644i \(-0.578630\pi\)
−0.244520 + 0.969644i \(0.578630\pi\)
\(48\) 0 0
\(49\) 37.7881 0.110169
\(50\) 0 0
\(51\) −508.293 − 89.9330i −1.39559 − 0.246924i
\(52\) 0 0
\(53\) 120.250 0.311653 0.155826 − 0.987784i \(-0.450196\pi\)
0.155826 + 0.987784i \(0.450196\pi\)
\(54\) 0 0
\(55\) 520.967 1.27722
\(56\) 0 0
\(57\) 206.202i 0.479160i
\(58\) 0 0
\(59\) 696.119 1.53605 0.768026 − 0.640419i \(-0.221239\pi\)
0.768026 + 0.640419i \(0.221239\pi\)
\(60\) 0 0
\(61\) − 683.544i − 1.43473i −0.696695 − 0.717367i \(-0.745347\pi\)
0.696695 − 0.717367i \(-0.254653\pi\)
\(62\) 0 0
\(63\) 475.781i 0.951473i
\(64\) 0 0
\(65\) − 760.089i − 1.45042i
\(66\) 0 0
\(67\) −123.826 −0.225787 −0.112894 − 0.993607i \(-0.536012\pi\)
−0.112894 + 0.993607i \(0.536012\pi\)
\(68\) 0 0
\(69\) −141.331 −0.246584
\(70\) 0 0
\(71\) − 225.393i − 0.376750i −0.982097 − 0.188375i \(-0.939678\pi\)
0.982097 − 0.188375i \(-0.0603220\pi\)
\(72\) 0 0
\(73\) 919.423i 1.47411i 0.675831 + 0.737057i \(0.263785\pi\)
−0.675831 + 0.737057i \(0.736215\pi\)
\(74\) 0 0
\(75\) 168.419i 0.259298i
\(76\) 0 0
\(77\) 900.603 1.33290
\(78\) 0 0
\(79\) 354.830i 0.505335i 0.967553 + 0.252668i \(0.0813079\pi\)
−0.967553 + 0.252668i \(0.918692\pi\)
\(80\) 0 0
\(81\) −722.636 −0.991270
\(82\) 0 0
\(83\) 955.272 1.26331 0.631655 − 0.775250i \(-0.282376\pi\)
0.631655 + 0.775250i \(0.282376\pi\)
\(84\) 0 0
\(85\) 697.522 + 123.413i 0.890080 + 0.157483i
\(86\) 0 0
\(87\) −520.967 −0.641995
\(88\) 0 0
\(89\) 617.636 0.735610 0.367805 − 0.929903i \(-0.380109\pi\)
0.367805 + 0.929903i \(0.380109\pi\)
\(90\) 0 0
\(91\) − 1313.98i − 1.51365i
\(92\) 0 0
\(93\) 305.212 0.340312
\(94\) 0 0
\(95\) − 282.967i − 0.305598i
\(96\) 0 0
\(97\) − 428.533i − 0.448566i −0.974524 − 0.224283i \(-0.927996\pi\)
0.974524 − 0.224283i \(-0.0720041\pi\)
\(98\) 0 0
\(99\) − 1403.91i − 1.42523i
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 272.4.b.d.33.4 4
4.3 odd 2 17.4.b.a.16.1 ✓ 4
12.11 even 2 153.4.d.b.118.4 4
17.16 even 2 inner 272.4.b.d.33.1 4
20.3 even 4 425.4.c.c.424.8 8
20.7 even 4 425.4.c.c.424.1 8
20.19 odd 2 425.4.d.c.101.4 4
68.47 odd 4 289.4.a.e.1.4 4
68.55 odd 4 289.4.a.e.1.3 4
68.67 odd 2 17.4.b.a.16.2 yes 4
204.203 even 2 153.4.d.b.118.3 4
340.67 even 4 425.4.c.c.424.2 8
340.203 even 4 425.4.c.c.424.7 8
340.339 odd 2 425.4.d.c.101.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.b.a.16.1 ✓ 4 4.3 odd 2
17.4.b.a.16.2 yes 4 68.67 odd 2
153.4.d.b.118.3 4 204.203 even 2
153.4.d.b.118.4 4 12.11 even 2
272.4.b.d.33.1 4 17.16 even 2 inner
272.4.b.d.33.4 4 1.1 even 1 trivial
289.4.a.e.1.3 4 68.55 odd 4
289.4.a.e.1.4 4 68.47 odd 4
425.4.c.c.424.1 8 20.7 even 4
425.4.c.c.424.2 8 340.67 even 4
425.4.c.c.424.7 8 340.203 even 4
425.4.c.c.424.8 8 20.3 even 4
425.4.d.c.101.3 4 340.339 odd 2
425.4.d.c.101.4 4 20.19 odd 2