Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [160,4,Mod(63,160)] 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("160.63"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(160, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 3])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 160.n (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.44030560092\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 63.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 160.63
Dual form 160.4.n.a.127.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.00000 + 2.00000i) q^{3} +(10.0000 - 5.00000i) q^{5} +(-18.0000 - 18.0000i) q^{7} +19.0000i q^{9} -16.0000i q^{11} +(-33.0000 - 33.0000i) q^{13} +(-10.0000 + 30.0000i) q^{15} +(67.0000 - 67.0000i) q^{17} -116.000 q^{19} +72.0000 q^{21} +(110.000 - 110.000i) q^{23} +(75.0000 - 100.000i) q^{25} +(-92.0000 - 92.0000i) q^{27} -132.000i q^{29} -68.0000i q^{31} +(32.0000 + 32.0000i) q^{33} +(-270.000 - 90.0000i) q^{35} +(65.0000 - 65.0000i) q^{37} +132.000 q^{39} -304.000 q^{41} +(-154.000 + 154.000i) q^{43} +(95.0000 + 190.000i) q^{45} +(306.000 + 306.000i) q^{47} +305.000i q^{49} +268.000i q^{51} +(217.000 + 217.000i) q^{53} +(-80.0000 - 160.000i) q^{55} +(232.000 - 232.000i) q^{57} +204.000 q^{59} -748.000 q^{61} +(342.000 - 342.000i) q^{63} +(-495.000 - 165.000i) q^{65} +(-166.000 - 166.000i) q^{67} +440.000i q^{69} +524.000i q^{71} +(277.000 + 277.000i) q^{73} +(50.0000 + 350.000i) q^{75} +(-288.000 + 288.000i) q^{77} +1232.00 q^{79} -145.000 q^{81} +(290.000 - 290.000i) q^{83} +(335.000 - 1005.00i) q^{85} +(264.000 + 264.000i) q^{87} +800.000i q^{89} +1188.00i q^{91} +(136.000 + 136.000i) q^{93} +(-1160.00 + 580.000i) q^{95} +(-651.000 + 651.000i) q^{97} +304.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 20 q^{5} - 36 q^{7} - 66 q^{13} - 20 q^{15} + 134 q^{17} - 232 q^{19} + 144 q^{21} + 220 q^{23} + 150 q^{25} - 184 q^{27} + 64 q^{33} - 540 q^{35} + 130 q^{37} + 264 q^{39} - 608 q^{41}+ \cdots + 608 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(1\)

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
\(3\) −2.00000 + 2.00000i −0.384900 + 0.384900i −0.872864 − 0.487964i \(-0.837740\pi\)
0.487964 + 0.872864i \(0.337740\pi\)
\(4\) 0 0
\(5\) 10.0000 − 5.00000i 0.894427 − 0.447214i
\(6\) 0 0
\(7\) −18.0000 − 18.0000i −0.971909 − 0.971909i 0.0277074 − 0.999616i \(-0.491179\pi\)
−0.999616 + 0.0277074i \(0.991179\pi\)
\(8\) 0 0
\(9\) 19.0000i 0.703704i
\(10\) 0 0
\(11\) − 16.0000i − 0.438562i −0.975662 − 0.219281i \(-0.929629\pi\)
0.975662 − 0.219281i \(-0.0703711\pi\)
\(12\) 0 0
\(13\) −33.0000 − 33.0000i −0.704043 − 0.704043i 0.261233 − 0.965276i \(-0.415871\pi\)
−0.965276 + 0.261233i \(0.915871\pi\)
\(14\) 0 0
\(15\) −10.0000 + 30.0000i −0.172133 + 0.516398i
\(16\) 0 0
\(17\) 67.0000 − 67.0000i 0.955876 − 0.955876i −0.0431911 − 0.999067i \(-0.513752\pi\)
0.999067 + 0.0431911i \(0.0137524\pi\)
\(18\) 0 0
\(19\) −116.000 −1.40064 −0.700322 − 0.713827i \(-0.746960\pi\)
−0.700322 + 0.713827i \(0.746960\pi\)
\(20\) 0 0
\(21\) 72.0000 0.748176
\(22\) 0 0
\(23\) 110.000 − 110.000i 0.997243 − 0.997243i −0.00275336 − 0.999996i \(-0.500876\pi\)
0.999996 + 0.00275336i \(0.000876423\pi\)
\(24\) 0 0
\(25\) 75.0000 − 100.000i 0.600000 − 0.800000i
\(26\) 0 0
\(27\) −92.0000 − 92.0000i −0.655756 − 0.655756i
\(28\) 0 0
\(29\) − 132.000i − 0.845234i −0.906308 − 0.422617i \(-0.861112\pi\)
0.906308 − 0.422617i \(-0.138888\pi\)
\(30\) 0 0
\(31\) − 68.0000i − 0.393973i −0.980406 − 0.196986i \(-0.936884\pi\)
0.980406 − 0.196986i \(-0.0631155\pi\)
\(32\) 0 0
\(33\) 32.0000 + 32.0000i 0.168803 + 0.168803i
\(34\) 0 0
\(35\) −270.000 − 90.0000i −1.30395 − 0.434651i
\(36\) 0 0
\(37\) 65.0000 − 65.0000i 0.288809 − 0.288809i −0.547800 − 0.836609i \(-0.684535\pi\)
0.836609 + 0.547800i \(0.184535\pi\)
\(38\) 0 0
\(39\) 132.000 0.541972
\(40\) 0 0
\(41\) −304.000 −1.15797 −0.578986 − 0.815338i \(-0.696551\pi\)
−0.578986 + 0.815338i \(0.696551\pi\)
\(42\) 0 0
\(43\) −154.000 + 154.000i −0.546158 + 0.546158i −0.925327 − 0.379170i \(-0.876210\pi\)
0.379170 + 0.925327i \(0.376210\pi\)
\(44\) 0 0
\(45\) 95.0000 + 190.000i 0.314706 + 0.629412i
\(46\) 0 0
\(47\) 306.000 + 306.000i 0.949674 + 0.949674i 0.998793 − 0.0491187i \(-0.0156413\pi\)
−0.0491187 + 0.998793i \(0.515641\pi\)
\(48\) 0 0
\(49\) 305.000i 0.889213i
\(50\) 0 0
\(51\) 268.000i 0.735833i
\(52\) 0 0
\(53\) 217.000 + 217.000i 0.562401 + 0.562401i 0.929989 − 0.367588i \(-0.119816\pi\)
−0.367588 + 0.929989i \(0.619816\pi\)
\(54\) 0 0
\(55\) −80.0000 − 160.000i −0.196131 − 0.392262i
\(56\) 0 0
\(57\) 232.000 − 232.000i 0.539108 − 0.539108i
\(58\) 0 0
\(59\) 204.000 0.450145 0.225072 − 0.974342i \(-0.427738\pi\)
0.225072 + 0.974342i \(0.427738\pi\)
\(60\) 0 0
\(61\) −748.000 −1.57003 −0.785013 − 0.619479i \(-0.787344\pi\)
−0.785013 + 0.619479i \(0.787344\pi\)
\(62\) 0 0
\(63\) 342.000 − 342.000i 0.683936 − 0.683936i
\(64\) 0 0
\(65\) −495.000 − 165.000i −0.944572 − 0.314857i
\(66\) 0 0
\(67\) −166.000 − 166.000i −0.302688 − 0.302688i 0.539376 − 0.842065i \(-0.318660\pi\)
−0.842065 + 0.539376i \(0.818660\pi\)
\(68\) 0 0
\(69\) 440.000i 0.767678i
\(70\) 0 0
\(71\) 524.000i 0.875878i 0.899005 + 0.437939i \(0.144291\pi\)
−0.899005 + 0.437939i \(0.855709\pi\)
\(72\) 0 0
\(73\) 277.000 + 277.000i 0.444115 + 0.444115i 0.893392 − 0.449277i \(-0.148318\pi\)
−0.449277 + 0.893392i \(0.648318\pi\)
\(74\) 0 0
\(75\) 50.0000 + 350.000i 0.0769800 + 0.538860i
\(76\) 0 0
\(77\) −288.000 + 288.000i −0.426242 + 0.426242i
\(78\) 0 0
\(79\) 1232.00 1.75457 0.877284 − 0.479972i \(-0.159353\pi\)
0.877284 + 0.479972i \(0.159353\pi\)
\(80\) 0 0
\(81\) −145.000 −0.198903
\(82\) 0 0
\(83\) 290.000 − 290.000i 0.383514 − 0.383514i −0.488853 − 0.872366i \(-0.662584\pi\)
0.872366 + 0.488853i \(0.162584\pi\)
\(84\) 0 0
\(85\) 335.000 − 1005.00i 0.427481 − 1.28244i
\(86\) 0 0
\(87\) 264.000 + 264.000i 0.325331 + 0.325331i
\(88\) 0 0
\(89\) 800.000i 0.952807i 0.879227 + 0.476404i \(0.158060\pi\)
−0.879227 + 0.476404i \(0.841940\pi\)
\(90\) 0 0
\(91\) 1188.00i 1.36853i
\(92\) 0 0
\(93\) 136.000 + 136.000i 0.151640 + 0.151640i
\(94\) 0 0
\(95\) −1160.00 + 580.000i −1.25277 + 0.626387i
\(96\) 0 0
\(97\) −651.000 + 651.000i −0.681433 + 0.681433i −0.960323 − 0.278890i \(-0.910034\pi\)
0.278890 + 0.960323i \(0.410034\pi\)
\(98\) 0 0
\(99\) 304.000 0.308618
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 160.4.n.a.63.1 ✓ 2
4.3 odd 2 160.4.n.b.63.1 yes 2
5.2 odd 4 160.4.n.b.127.1 yes 2
8.3 odd 2 320.4.n.a.63.1 2
8.5 even 2 320.4.n.d.63.1 2
20.7 even 4 inner 160.4.n.a.127.1 yes 2
40.27 even 4 320.4.n.d.127.1 2
40.37 odd 4 320.4.n.a.127.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.4.n.a.63.1 ✓ 2 1.1 even 1 trivial
160.4.n.a.127.1 yes 2 20.7 even 4 inner
160.4.n.b.63.1 yes 2 4.3 odd 2
160.4.n.b.127.1 yes 2 5.2 odd 4
320.4.n.a.63.1 2 8.3 odd 2
320.4.n.a.127.1 2 40.37 odd 4
320.4.n.d.63.1 2 8.5 even 2
320.4.n.d.127.1 2 40.27 even 4