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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [153,4,Mod(118,153)] 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("153.118"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(153, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 153.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.02729223088\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 118.3
Root \(-7.36435i\) of defining polynomial
Character \(\chi\) \(=\) 153.118
Dual form 153.4.d.b.118.4

$q$-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)
 
\(f(q)\) \(=\) \(q+3.37228 q^{2} +3.37228 q^{4} -10.1060i q^{5} -17.4703i q^{7} -15.6060 q^{8} -34.0802i q^{10} -51.5505i q^{11} +75.2119 q^{13} -58.9148i q^{14} -79.6060 q^{16} +(12.2119 + 69.0208i) q^{17} -28.0000 q^{19} -34.0802i q^{20} -173.843i q^{22} -19.1913i q^{23} +22.8695 q^{25} +253.636 q^{26} -58.9148i q^{28} +70.7417i q^{29} -41.4445i q^{31} -143.606 q^{32} +(41.1821 + 232.757i) q^{34} -176.554 q^{35} -135.460i q^{37} -94.4239 q^{38} +157.713i q^{40} +288.771i q^{41} +88.2934 q^{43} -173.843i q^{44} -64.7184i q^{46} -157.576 q^{47} +37.7881 q^{49} +77.1224 q^{50} +253.636 q^{52} -120.250 q^{53} -520.967 q^{55} +272.641i q^{56} +238.561i q^{58} +696.119 q^{59} +683.544i q^{61} -139.763i q^{62} +152.568 q^{64} -760.089i q^{65} +123.826 q^{67} +(41.1821 + 232.757i) q^{68} -595.391 q^{70} +225.393i q^{71} -919.423i q^{73} -456.810i q^{74} -94.4239 q^{76} -900.603 q^{77} +354.830i q^{79} +804.495i q^{80} +973.815i q^{82} +955.272 q^{83} +(697.522 - 123.413i) q^{85} +297.750 q^{86} +804.495i q^{88} -617.636 q^{89} -1313.98i q^{91} -64.7184i q^{92} -531.391 q^{94} +282.967i q^{95} +428.533i q^{97} +127.432 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 2 q^{4} + 18 q^{8} + 140 q^{13} - 238 q^{16} - 112 q^{17} - 112 q^{19} - 460 q^{25} + 532 q^{26} - 494 q^{32} + 406 q^{34} - 936 q^{35} - 56 q^{38} - 520 q^{43} - 952 q^{47} + 312 q^{49}+ \cdots - 306 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/153\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(137\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.37228 1.19228 0.596141 − 0.802880i \(-0.296700\pi\)
0.596141 + 0.802880i \(0.296700\pi\)
\(3\) 0 0
\(4\) 3.37228 0.421535
\(5\) − 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 \(-0.156343\pi\)
\(8\) −15.6060 −0.689693
\(9\) 0 0
\(10\) − 34.0802i − 1.07771i
\(11\) − 51.5505i − 1.41300i −0.707711 − 0.706502i \(-0.750272\pi\)
0.707711 − 0.706502i \(-0.249728\pi\)
\(12\) 0 0
\(13\) 75.2119 1.60462 0.802309 − 0.596909i \(-0.203605\pi\)
0.802309 + 0.596909i \(0.203605\pi\)
\(14\) − 58.9148i − 1.12469i
\(15\) 0 0
\(16\) −79.6060 −1.24384
\(17\) 12.2119 + 69.0208i 0.174225 + 0.984706i
\(18\) 0 0
\(19\) −28.0000 −0.338086 −0.169043 − 0.985609i \(-0.554068\pi\)
−0.169043 + 0.985609i \(0.554068\pi\)
\(20\) − 34.0802i − 0.381028i
\(21\) 0 0
\(22\) − 173.843i − 1.68470i
\(23\) − 19.1913i − 0.173985i −0.996209 − 0.0869926i \(-0.972274\pi\)
0.996209 − 0.0869926i \(-0.0277256\pi\)
\(24\) 0 0
\(25\) 22.8695 0.182956
\(26\) 253.636 1.91316
\(27\) 0 0
\(28\) − 58.9148i − 0.397638i
\(29\) 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 \(-0.0383083\pi\)
\(32\) −143.606 −0.793318
\(33\) 0 0
\(34\) 41.1821 + 232.757i 0.207726 + 1.17405i
\(35\) −176.554 −0.852661
\(36\) 0 0
\(37\) − 135.460i − 0.601879i −0.953643 − 0.300939i \(-0.902700\pi\)
0.953643 − 0.300939i \(-0.0973002\pi\)
\(38\) −94.4239 −0.403094
\(39\) 0 0
\(40\) 157.713i 0.623417i
\(41\) 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 \(-0.449958\pi\)
0.156565 + 0.987668i \(0.449958\pi\)
\(44\) − 173.843i − 0.595631i
\(45\) 0 0
\(46\) − 64.7184i − 0.207439i
\(47\) −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
\(50\) 77.1224 0.218135
\(51\) 0 0
\(52\) 253.636 0.676403
\(53\) −120.250 −0.311653 −0.155826 − 0.987784i \(-0.549804\pi\)
−0.155826 + 0.987784i \(0.549804\pi\)
\(54\) 0 0
\(55\) −520.967 −1.27722
\(56\) 272.641i 0.650593i
\(57\) 0 0
\(58\) 238.561i 0.540079i
\(59\) 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 \(0.254653\pi\)
−0.696695 + 0.717367i \(0.745347\pi\)
\(62\) − 139.763i − 0.286288i
\(63\) 0 0
\(64\) 152.568 0.297984
\(65\) − 760.089i − 1.45042i
\(66\) 0 0
\(67\) 123.826 0.225787 0.112894 − 0.993607i \(-0.463988\pi\)
0.112894 + 0.993607i \(0.463988\pi\)
\(68\) 41.1821 + 232.757i 0.0734421 + 0.415088i
\(69\) 0 0
\(70\) −595.391 −1.01661
\(71\) 225.393i 0.376750i 0.982097 + 0.188375i \(0.0603220\pi\)
−0.982097 + 0.188375i \(0.939678\pi\)
\(72\) 0 0
\(73\) − 919.423i − 1.47411i −0.675831 − 0.737057i \(-0.736215\pi\)
0.675831 − 0.737057i \(-0.263785\pi\)
\(74\) − 456.810i − 0.717609i
\(75\) 0 0
\(76\) −94.4239 −0.142515
\(77\) −900.603 −1.33290
\(78\) 0 0
\(79\) 354.830i 0.505335i 0.967553 + 0.252668i \(0.0813079\pi\)
−0.967553 + 0.252668i \(0.918692\pi\)
\(80\) 804.495i 1.12432i
\(81\) 0 0
\(82\) 973.815i 1.31146i
\(83\) 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
\(86\) 297.750 0.373340
\(87\) 0 0
\(88\) 804.495i 0.974539i
\(89\) −617.636 −0.735610 −0.367805 − 0.929903i \(-0.619891\pi\)
−0.367805 + 0.929903i \(0.619891\pi\)
\(90\) 0 0
\(91\) − 1313.98i − 1.51365i
\(92\) − 64.7184i − 0.0733408i
\(93\) 0 0
\(94\) −531.391 −0.583072
\(95\) 282.967i 0.305598i
\(96\) 0 0
\(97\) 428.533i 0.448566i 0.974524 + 0.224283i \(0.0720041\pi\)
−0.974524 + 0.224283i \(0.927996\pi\)
\(98\) 127.432 0.131353
\(99\) 0 0
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 153.4.d.b.118.3 4
3.2 odd 2 17.4.b.a.16.2 yes 4
12.11 even 2 272.4.b.d.33.1 4
15.2 even 4 425.4.c.c.424.2 8
15.8 even 4 425.4.c.c.424.7 8
15.14 odd 2 425.4.d.c.101.3 4
17.16 even 2 inner 153.4.d.b.118.4 4
51.38 odd 4 289.4.a.e.1.4 4
51.47 odd 4 289.4.a.e.1.3 4
51.50 odd 2 17.4.b.a.16.1 ✓ 4
204.203 even 2 272.4.b.d.33.4 4
255.152 even 4 425.4.c.c.424.1 8
255.203 even 4 425.4.c.c.424.8 8
255.254 odd 2 425.4.d.c.101.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.b.a.16.1 ✓ 4 51.50 odd 2
17.4.b.a.16.2 yes 4 3.2 odd 2
153.4.d.b.118.3 4 1.1 even 1 trivial
153.4.d.b.118.4 4 17.16 even 2 inner
272.4.b.d.33.1 4 12.11 even 2
272.4.b.d.33.4 4 204.203 even 2
289.4.a.e.1.3 4 51.47 odd 4
289.4.a.e.1.4 4 51.38 odd 4
425.4.c.c.424.1 8 255.152 even 4
425.4.c.c.424.2 8 15.2 even 4
425.4.c.c.424.7 8 15.8 even 4
425.4.c.c.424.8 8 255.203 even 4
425.4.d.c.101.3 4 15.14 odd 2
425.4.d.c.101.4 4 255.254 odd 2