Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [624,2,Mod(305,624)] 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("624.305"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(624, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 0, 6, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 624.cn (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.98266508613\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: 16.0.9349208943630483456.9
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 305.2
Root \(0.500000 + 1.33108i\) of defining polynomial
Character \(\chi\) \(=\) 624.305
Dual form 624.2.cn.d.401.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.933998 - 1.45865i) q^{3} +(-2.76293 - 2.76293i) q^{5} +(0.657464 - 2.45369i) q^{7} +(-1.25529 + 2.72474i) q^{9} +(0.150860 + 0.563016i) q^{11} +(-1.20856 - 3.39697i) q^{13} +(-1.44957 + 6.61072i) q^{15} +(0.547000 + 0.947432i) q^{17} +(-1.32717 - 0.355613i) q^{19} +(-4.19313 + 1.33273i) q^{21} +(0.876460 - 1.51807i) q^{23} +10.2676i q^{25} +(5.14688 - 0.713876i) q^{27} +(5.12973 + 2.96165i) q^{29} +(-6.49983 + 6.49983i) q^{31} +(0.680339 - 0.745907i) q^{33} +(-8.59591 + 4.96285i) q^{35} +(-2.98942 + 0.801012i) q^{37} +(-3.82618 + 4.93562i) q^{39} +(-5.11781 + 1.37131i) q^{41} +(-3.26299 + 1.88389i) q^{43} +(10.9966 - 4.05999i) q^{45} +(5.51114 - 5.51114i) q^{47} +(0.473846 + 0.273575i) q^{49} +(0.871071 - 1.68278i) q^{51} -3.04435i q^{53} +(1.13876 - 1.97239i) q^{55} +(0.720857 + 2.26801i) q^{57} +(-8.19009 - 2.19453i) q^{59} +(-4.67266 - 8.09329i) q^{61} +(5.86037 + 4.87153i) q^{63} +(-6.04642 + 12.7248i) q^{65} +(-1.70856 - 6.37644i) q^{67} +(-3.03294 + 0.139433i) q^{69} +(-0.220122 + 0.821505i) q^{71} +(-5.18078 - 5.18078i) q^{73} +(14.9768 - 9.58993i) q^{75} +1.48065 q^{77} +13.1089 q^{79} +(-5.84847 - 6.84072i) q^{81} +(5.15394 + 5.15394i) q^{83} +(1.10637 - 4.12902i) q^{85} +(-0.471158 - 10.2486i) q^{87} +(-2.50797 - 9.35988i) q^{89} +(-9.12969 + 0.732051i) q^{91} +(15.5518 + 3.41012i) q^{93} +(2.68434 + 4.64941i) q^{95} +(-0.592450 - 0.158747i) q^{97} +(-1.72345 - 0.295697i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{7} - 24 q^{13} + 16 q^{19} - 24 q^{21} - 16 q^{31} - 24 q^{33} + 16 q^{37} - 48 q^{39} + 24 q^{45} + 24 q^{49} + 24 q^{55} - 24 q^{57} - 24 q^{61} + 24 q^{63} - 32 q^{67} - 48 q^{69} + 56 q^{73}+ \cdots + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{12}\right)\) \(-1\) \(1\)

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
\(3\) −0.933998 − 1.45865i −0.539244 − 0.842150i
\(4\) 0 0
\(5\) −2.76293 − 2.76293i −1.23562 − 1.23562i −0.961773 − 0.273849i \(-0.911703\pi\)
−0.273849 − 0.961773i \(-0.588297\pi\)
\(6\) 0 0
\(7\) 0.657464 − 2.45369i 0.248498 − 0.927407i −0.723095 − 0.690749i \(-0.757281\pi\)
0.971593 − 0.236659i \(-0.0760523\pi\)
\(8\) 0 0
\(9\) −1.25529 + 2.72474i −0.418432 + 0.908248i
\(10\) 0 0
\(11\) 0.150860 + 0.563016i 0.0454859 + 0.169756i 0.984932 − 0.172940i \(-0.0553267\pi\)
−0.939446 + 0.342696i \(0.888660\pi\)
\(12\) 0 0
\(13\) −1.20856 − 3.39697i −0.335195 − 0.942149i
\(14\) 0 0
\(15\) −1.44957 + 6.61072i −0.374277 + 1.70688i
\(16\) 0 0
\(17\) 0.547000 + 0.947432i 0.132667 + 0.229786i 0.924704 − 0.380687i \(-0.124313\pi\)
−0.792037 + 0.610473i \(0.790979\pi\)
\(18\) 0 0
\(19\) −1.32717 − 0.355613i −0.304473 − 0.0815832i 0.103348 − 0.994645i \(-0.467044\pi\)
−0.407821 + 0.913062i \(0.633711\pi\)
\(20\) 0 0
\(21\) −4.19313 + 1.33273i −0.915017 + 0.290826i
\(22\) 0 0
\(23\) 0.876460 − 1.51807i 0.182755 − 0.316540i −0.760063 − 0.649849i \(-0.774832\pi\)
0.942818 + 0.333309i \(0.108165\pi\)
\(24\) 0 0
\(25\) 10.2676i 2.05352i
\(26\) 0 0
\(27\) 5.14688 − 0.713876i 0.990518 − 0.137386i
\(28\) 0 0
\(29\) 5.12973 + 2.96165i 0.952566 + 0.549965i 0.893877 − 0.448312i \(-0.147975\pi\)
0.0586892 + 0.998276i \(0.481308\pi\)
\(30\) 0 0
\(31\) −6.49983 + 6.49983i −1.16740 + 1.16740i −0.184588 + 0.982816i \(0.559095\pi\)
−0.982816 + 0.184588i \(0.940905\pi\)
\(32\) 0 0
\(33\) 0.680339 − 0.745907i 0.118432 − 0.129846i
\(34\) 0 0
\(35\) −8.59591 + 4.96285i −1.45297 + 0.838875i
\(36\) 0 0
\(37\) −2.98942 + 0.801012i −0.491457 + 0.131686i −0.496032 − 0.868304i \(-0.665210\pi\)
0.00457534 + 0.999990i \(0.498544\pi\)
\(38\) 0 0
\(39\) −3.82618 + 4.93562i −0.612679 + 0.790332i
\(40\) 0 0
\(41\) −5.11781 + 1.37131i −0.799268 + 0.214163i −0.635262 − 0.772296i \(-0.719108\pi\)
−0.164005 + 0.986459i \(0.552441\pi\)
\(42\) 0 0
\(43\) −3.26299 + 1.88389i −0.497602 + 0.287290i −0.727723 − 0.685872i \(-0.759421\pi\)
0.230121 + 0.973162i \(0.426088\pi\)
\(44\) 0 0
\(45\) 10.9966 − 4.05999i 1.63927 − 0.605228i
\(46\) 0 0
\(47\) 5.51114 − 5.51114i 0.803883 − 0.803883i −0.179817 − 0.983700i \(-0.557551\pi\)
0.983700 + 0.179817i \(0.0575506\pi\)
\(48\) 0 0
\(49\) 0.473846 + 0.273575i 0.0676922 + 0.0390821i
\(50\) 0 0
\(51\) 0.871071 − 1.68278i 0.121974 − 0.235636i
\(52\) 0 0
\(53\) − 3.04435i − 0.418173i −0.977897 − 0.209087i \(-0.932951\pi\)
0.977897 − 0.209087i \(-0.0670490\pi\)
\(54\) 0 0
\(55\) 1.13876 − 1.97239i 0.153551 − 0.265957i
\(56\) 0 0
\(57\) 0.720857 + 2.26801i 0.0954799 + 0.300405i
\(58\) 0 0
\(59\) −8.19009 − 2.19453i −1.06626 − 0.285703i −0.317304 − 0.948324i \(-0.602777\pi\)
−0.748955 + 0.662621i \(0.769444\pi\)
\(60\) 0 0
\(61\) −4.67266 − 8.09329i −0.598273 − 1.03624i −0.993076 − 0.117474i \(-0.962520\pi\)
0.394803 − 0.918766i \(-0.370813\pi\)
\(62\) 0 0
\(63\) 5.86037 + 4.87153i 0.738337 + 0.613755i
\(64\) 0 0
\(65\) −6.04642 + 12.7248i −0.749966 + 1.57831i
\(66\) 0 0
\(67\) −1.70856 − 6.37644i −0.208734 − 0.779006i −0.988279 − 0.152659i \(-0.951216\pi\)
0.779545 − 0.626346i \(-0.215450\pi\)
\(68\) 0 0
\(69\) −3.03294 + 0.139433i −0.365124 + 0.0167858i
\(70\) 0 0
\(71\) −0.220122 + 0.821505i −0.0261236 + 0.0974947i −0.977757 − 0.209742i \(-0.932738\pi\)
0.951633 + 0.307237i \(0.0994043\pi\)
\(72\) 0 0
\(73\) −5.18078 − 5.18078i −0.606365 − 0.606365i 0.335629 − 0.941994i \(-0.391051\pi\)
−0.941994 + 0.335629i \(0.891051\pi\)
\(74\) 0 0
\(75\) 14.9768 − 9.58993i 1.72937 − 1.10735i
\(76\) 0 0
\(77\) 1.48065 0.168736
\(78\) 0 0
\(79\) 13.1089 1.47486 0.737431 − 0.675422i \(-0.236039\pi\)
0.737431 + 0.675422i \(0.236039\pi\)
\(80\) 0 0
\(81\) −5.84847 − 6.84072i −0.649830 − 0.760080i
\(82\) 0 0
\(83\) 5.15394 + 5.15394i 0.565719 + 0.565719i 0.930926 − 0.365208i \(-0.119002\pi\)
−0.365208 + 0.930926i \(0.619002\pi\)
\(84\) 0 0
\(85\) 1.10637 − 4.12902i 0.120002 − 0.447855i
\(86\) 0 0
\(87\) −0.471158 − 10.2486i −0.0505135 − 1.09877i
\(88\) 0 0
\(89\) −2.50797 − 9.35988i −0.265844 − 0.992145i −0.961731 − 0.273994i \(-0.911655\pi\)
0.695887 − 0.718151i \(-0.255011\pi\)
\(90\) 0 0
\(91\) −9.12969 + 0.732051i −0.957051 + 0.0767398i
\(92\) 0 0
\(93\) 15.5518 + 3.41012i 1.61264 + 0.353613i
\(94\) 0 0
\(95\) 2.68434 + 4.64941i 0.275407 + 0.477019i
\(96\) 0 0
\(97\) −0.592450 − 0.158747i −0.0601542 − 0.0161183i 0.228616 − 0.973517i \(-0.426580\pi\)
−0.288771 + 0.957398i \(0.593247\pi\)
\(98\) 0 0
\(99\) −1.72345 − 0.295697i −0.173213 − 0.0297187i
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 624.2.cn.d.305.2 16
3.2 odd 2 inner 624.2.cn.d.305.4 16
4.3 odd 2 78.2.k.a.71.4 yes 16
12.11 even 2 78.2.k.a.71.1 yes 16
13.11 odd 12 inner 624.2.cn.d.401.4 16
39.11 even 12 inner 624.2.cn.d.401.2 16
52.7 even 12 1014.2.g.c.437.7 16
52.11 even 12 78.2.k.a.11.1 ✓ 16
52.19 even 12 1014.2.g.d.437.3 16
52.35 odd 6 1014.2.g.c.239.3 16
52.43 odd 6 1014.2.g.d.239.7 16
156.11 odd 12 78.2.k.a.11.4 yes 16
156.35 even 6 1014.2.g.c.239.7 16
156.59 odd 12 1014.2.g.c.437.3 16
156.71 odd 12 1014.2.g.d.437.7 16
156.95 even 6 1014.2.g.d.239.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.k.a.11.1 ✓ 16 52.11 even 12
78.2.k.a.11.4 yes 16 156.11 odd 12
78.2.k.a.71.1 yes 16 12.11 even 2
78.2.k.a.71.4 yes 16 4.3 odd 2
624.2.cn.d.305.2 16 1.1 even 1 trivial
624.2.cn.d.305.4 16 3.2 odd 2 inner
624.2.cn.d.401.2 16 39.11 even 12 inner
624.2.cn.d.401.4 16 13.11 odd 12 inner
1014.2.g.c.239.3 16 52.35 odd 6
1014.2.g.c.239.7 16 156.35 even 6
1014.2.g.c.437.3 16 156.59 odd 12
1014.2.g.c.437.7 16 52.7 even 12
1014.2.g.d.239.3 16 156.95 even 6
1014.2.g.d.239.7 16 52.43 odd 6
1014.2.g.d.437.3 16 52.19 even 12
1014.2.g.d.437.7 16 156.71 odd 12