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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [252,6,Mod(37,252)] 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("252.37"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(252, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 252.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(40.4167225929\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 703x^{6} + 2770x^{5} + 427565x^{4} + 718170x^{3} + 42175732x^{2} - 40929504x + 3559792896 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{3}\cdot 7 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 109.3
Root \(-5.49618 + 9.51967i\) of defining polynomial
Character \(\chi\) \(=\) 252.109
Dual form 252.6.k.f.37.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(23.0577 + 39.9371i) q^{5} +(112.271 + 64.8240i) q^{7} +(-315.582 + 546.605i) q^{11} -1079.22 q^{13} +(80.5778 - 139.565i) q^{17} +(-588.428 - 1019.19i) q^{19} +(1081.73 + 1873.61i) q^{23} +(499.186 - 864.615i) q^{25} +4492.01 q^{29} +(-159.130 + 275.621i) q^{31} +(-0.166415 + 5978.48i) q^{35} +(-7593.41 - 13152.2i) q^{37} -20587.2 q^{41} -455.118 q^{43} +(-10381.4 - 17981.1i) q^{47} +(8402.69 + 14555.8i) q^{49} +(-9650.03 + 16714.3i) q^{53} -29106.4 q^{55} +(3184.15 - 5515.11i) q^{59} +(24572.6 + 42560.9i) q^{61} +(-24884.4 - 43101.1i) q^{65} +(-17027.0 + 29491.7i) q^{67} -62962.4 q^{71} +(4433.88 - 7679.70i) q^{73} +(-70864.0 + 40910.7i) q^{77} +(-17206.6 - 29802.7i) q^{79} +7041.42 q^{83} +7431.75 q^{85} +(-10121.4 - 17530.7i) q^{89} +(-121166. - 69959.7i) q^{91} +(27135.6 - 47000.2i) q^{95} +54066.6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 42 q^{7} + 462 q^{11} - 1204 q^{13} - 228 q^{17} + 358 q^{19} + 2148 q^{23} - 5454 q^{25} + 11064 q^{29} + 830 q^{31} - 7692 q^{35} - 3914 q^{37} + 16632 q^{41} - 29036 q^{43} - 41700 q^{47} + 41876 q^{49}+ \cdots - 433356 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 23.0577 + 39.9371i 0.412469 + 0.714416i 0.995159 − 0.0982777i \(-0.0313333\pi\)
−0.582691 + 0.812694i \(0.698000\pi\)
\(6\) 0 0
\(7\) 112.271 + 64.8240i 0.866011 + 0.500024i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −315.582 + 546.605i −0.786378 + 1.36205i 0.141795 + 0.989896i \(0.454713\pi\)
−0.928173 + 0.372150i \(0.878621\pi\)
\(12\) 0 0
\(13\) −1079.22 −1.77114 −0.885571 − 0.464503i \(-0.846233\pi\)
−0.885571 + 0.464503i \(0.846233\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 80.5778 − 139.565i 0.0676228 − 0.117126i −0.830232 − 0.557419i \(-0.811792\pi\)
0.897854 + 0.440292i \(0.145125\pi\)
\(18\) 0 0
\(19\) −588.428 − 1019.19i −0.373946 − 0.647694i 0.616222 − 0.787572i \(-0.288662\pi\)
−0.990169 + 0.139878i \(0.955329\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1081.73 + 1873.61i 0.426382 + 0.738516i 0.996548 − 0.0830136i \(-0.0264545\pi\)
−0.570166 + 0.821529i \(0.693121\pi\)
\(24\) 0 0
\(25\) 499.186 − 864.615i 0.159739 − 0.276677i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4492.01 0.991850 0.495925 − 0.868365i \(-0.334829\pi\)
0.495925 + 0.868365i \(0.334829\pi\)
\(30\) 0 0
\(31\) −159.130 + 275.621i −0.0297405 + 0.0515120i −0.880513 − 0.474023i \(-0.842801\pi\)
0.850772 + 0.525535i \(0.176135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.166415 + 5978.48i −2.29627e−5 + 0.824937i
\(36\) 0 0
\(37\) −7593.41 − 13152.2i −0.911869 − 1.57940i −0.811422 − 0.584461i \(-0.801306\pi\)
−0.100447 − 0.994942i \(-0.532027\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −20587.2 −1.91266 −0.956330 − 0.292289i \(-0.905583\pi\)
−0.956330 + 0.292289i \(0.905583\pi\)
\(42\) 0 0
\(43\) −455.118 −0.0375364 −0.0187682 − 0.999824i \(-0.505974\pi\)
−0.0187682 + 0.999824i \(0.505974\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −10381.4 − 17981.1i −0.685504 − 1.18733i −0.973278 − 0.229630i \(-0.926248\pi\)
0.287774 − 0.957698i \(-0.407085\pi\)
\(48\) 0 0
\(49\) 8402.69 + 14555.8i 0.499952 + 0.866053i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9650.03 + 16714.3i −0.471888 + 0.817334i −0.999483 − 0.0321622i \(-0.989761\pi\)
0.527595 + 0.849496i \(0.323094\pi\)
\(54\) 0 0
\(55\) −29106.4 −1.29742
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3184.15 − 5515.11i 0.119087 − 0.206264i −0.800319 − 0.599574i \(-0.795337\pi\)
0.919406 + 0.393310i \(0.128670\pi\)
\(60\) 0 0
\(61\) 24572.6 + 42560.9i 0.845524 + 1.46449i 0.885165 + 0.465276i \(0.154045\pi\)
−0.0396416 + 0.999214i \(0.512622\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −24884.4 − 43101.1i −0.730541 − 1.26533i
\(66\) 0 0
\(67\) −17027.0 + 29491.7i −0.463395 + 0.802624i −0.999128 − 0.0417639i \(-0.986702\pi\)
0.535732 + 0.844388i \(0.320036\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −62962.4 −1.48230 −0.741149 − 0.671341i \(-0.765719\pi\)
−0.741149 + 0.671341i \(0.765719\pi\)
\(72\) 0 0
\(73\) 4433.88 − 7679.70i 0.0973815 − 0.168670i −0.813219 − 0.581958i \(-0.802287\pi\)
0.910600 + 0.413289i \(0.135620\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −70864.0 + 40910.7i −1.36207 + 0.786340i
\(78\) 0 0
\(79\) −17206.6 − 29802.7i −0.310190 − 0.537265i 0.668213 − 0.743970i \(-0.267059\pi\)
−0.978403 + 0.206705i \(0.933726\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7041.42 0.112193 0.0560964 − 0.998425i \(-0.482135\pi\)
0.0560964 + 0.998425i \(0.482135\pi\)
\(84\) 0 0
\(85\) 7431.75 0.111569
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10121.4 − 17530.7i −0.135445 − 0.234598i 0.790322 − 0.612692i \(-0.209913\pi\)
−0.925768 + 0.378093i \(0.876580\pi\)
\(90\) 0 0
\(91\) −121166. − 69959.7i −1.53383 − 0.885614i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 27135.6 − 47000.2i 0.308482 − 0.534307i
\(96\) 0 0
\(97\) 54066.6 0.583444 0.291722 − 0.956503i \(-0.405772\pi\)
0.291722 + 0.956503i \(0.405772\pi\)
\(98\) 0 0
\(99\) 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 252.6.k.f.109.3 8
3.2 odd 2 84.6.i.c.25.2 ✓ 8
7.2 even 3 inner 252.6.k.f.37.3 8
12.11 even 2 336.6.q.i.193.2 8
21.2 odd 6 84.6.i.c.37.2 yes 8
21.5 even 6 588.6.i.o.373.3 8
21.11 odd 6 588.6.a.n.1.3 4
21.17 even 6 588.6.a.p.1.2 4
21.20 even 2 588.6.i.o.361.3 8
84.23 even 6 336.6.q.i.289.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.6.i.c.25.2 ✓ 8 3.2 odd 2
84.6.i.c.37.2 yes 8 21.2 odd 6
252.6.k.f.37.3 8 7.2 even 3 inner
252.6.k.f.109.3 8 1.1 even 1 trivial
336.6.q.i.193.2 8 12.11 even 2
336.6.q.i.289.2 8 84.23 even 6
588.6.a.n.1.3 4 21.11 odd 6
588.6.a.p.1.2 4 21.17 even 6
588.6.i.o.361.3 8 21.20 even 2
588.6.i.o.373.3 8 21.5 even 6