↑

Properties

Label 78.2.k.a.11.1
Level $78$
Weight $2$
Character 78.11
Analytic conductor $0.623$
Analytic rank $0$
Dimension $16$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [78,2,Mod(11,78)] 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("78.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(78, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 78.k (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.622833135766\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 11.1
Root \(0.500000 - 1.74530i\) of defining polynomial
Character \(\chi\) \(=\) 78.11
Dual form 78.2.k.a.71.1

$q$-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+(-0.965926 - 0.258819i) q^{2} +(-1.73022 + 0.0795432i) q^{3} +(0.866025 + 0.500000i) q^{4} +(2.76293 - 2.76293i) q^{5} +(1.69185 + 0.370982i) q^{6} +(-0.657464 - 2.45369i) q^{7} +(-0.707107 - 0.707107i) q^{8} +(2.98735 - 0.275255i) q^{9} +(-3.38389 + 1.95369i) q^{10} +(0.150860 - 0.563016i) q^{11} +(-1.53819 - 0.796225i) q^{12} +(-1.20856 + 3.39697i) q^{13} +2.54025i q^{14} +(-4.56072 + 5.00027i) q^{15} +(0.500000 + 0.866025i) q^{16} +(-0.547000 + 0.947432i) q^{17} +(-2.95680 - 0.507306i) q^{18} +(1.32717 - 0.355613i) q^{19} +(3.77424 - 1.01130i) q^{20} +(1.33273 + 4.19313i) q^{21} +(-0.291439 + 0.504787i) q^{22} +(0.876460 + 1.51807i) q^{23} +(1.27970 + 1.16721i) q^{24} -10.2676i q^{25} +(2.04658 - 2.96842i) q^{26} +(-5.14688 + 0.713876i) q^{27} +(0.657464 - 2.45369i) q^{28} +(-5.12973 + 2.96165i) q^{29} +(5.69948 - 3.64948i) q^{30} +(6.49983 + 6.49983i) q^{31} +(-0.258819 - 0.965926i) q^{32} +(-0.216237 + 0.986144i) q^{33} +(0.773575 - 0.773575i) q^{34} +(-8.59591 - 4.96285i) q^{35} +(2.72474 + 1.25529i) q^{36} +(-2.98942 - 0.801012i) q^{37} -1.37398 q^{38} +(1.82088 - 5.97364i) q^{39} -3.90738 q^{40} +(5.11781 + 1.37131i) q^{41} +(-0.202059 - 4.39519i) q^{42} +(3.26299 + 1.88389i) q^{43} +(0.412157 - 0.412157i) q^{44} +(7.49333 - 9.01435i) q^{45} +(-0.453689 - 1.69319i) q^{46} +(5.51114 + 5.51114i) q^{47} +(-0.933998 - 1.45865i) q^{48} +(0.473846 - 0.273575i) q^{49} +(-2.65745 + 9.91775i) q^{50} +(0.871071 - 1.68278i) q^{51} +(-2.74513 + 2.33758i) q^{52} -3.04435i q^{53} +(5.15627 + 0.642559i) q^{54} +(-1.13876 - 1.97239i) q^{55} +(-1.27012 + 2.19992i) q^{56} +(-2.26801 + 0.720857i) q^{57} +(5.72147 - 1.53306i) q^{58} +(-8.19009 + 2.19453i) q^{59} +(-6.44983 + 2.05000i) q^{60} +(-4.67266 + 8.09329i) q^{61} +(-4.59607 - 7.96063i) q^{62} +(-2.63946 - 7.14905i) q^{63} +1.00000i q^{64} +(6.04642 + 12.7248i) q^{65} +(0.464102 - 0.896575i) q^{66} +(1.70856 - 6.37644i) q^{67} +(-0.947432 + 0.547000i) q^{68} +(-1.63722 - 2.55689i) q^{69} +(7.01853 + 7.01853i) q^{70} +(-0.220122 - 0.821505i) q^{71} +(-2.30701 - 1.91774i) q^{72} +(-5.18078 + 5.18078i) q^{73} +(2.68024 + 1.54744i) q^{74} +(0.816719 + 17.7653i) q^{75} +(1.32717 + 0.355613i) q^{76} -1.48065 q^{77} +(-3.30492 + 5.29882i) q^{78} -13.1089 q^{79} +(3.77424 + 1.01130i) q^{80} +(8.84847 - 1.64456i) q^{81} +(-4.58850 - 2.64917i) q^{82} +(5.15394 - 5.15394i) q^{83} +(-0.942385 + 4.29773i) q^{84} +(1.10637 + 4.12902i) q^{85} +(-2.66422 - 2.66422i) q^{86} +(8.63999 - 5.53235i) q^{87} +(-0.504787 + 0.291439i) q^{88} +(2.50797 - 9.35988i) q^{89} +(-9.57108 + 6.76778i) q^{90} +(9.12969 + 0.732051i) q^{91} +1.75292i q^{92} +(-11.7632 - 10.7291i) q^{93} +(-3.89697 - 6.74974i) q^{94} +(2.68434 - 4.64941i) q^{95} +(0.524648 + 1.65068i) q^{96} +(-0.592450 + 0.158747i) q^{97} +(-0.528506 + 0.141613i) q^{98} +(0.295697 - 1.72345i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{7} - 24 q^{10} - 24 q^{13} + 8 q^{16} - 16 q^{19} - 24 q^{21} - 8 q^{28} + 24 q^{30} + 16 q^{31} - 24 q^{33} + 24 q^{34} + 24 q^{36} + 16 q^{37} + 48 q^{39} + 24 q^{45} + 24 q^{46} + 24 q^{49}+ \cdots + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/78\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(67\)
\(\chi(n)\) \(-1\) \(e\left(\frac{7}{12}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.965926 − 0.258819i −0.683013 − 0.183013i
\(3\) −1.73022 + 0.0795432i −0.998945 + 0.0459243i
\(4\) 0.866025 + 0.500000i 0.433013 + 0.250000i
\(5\) 2.76293 − 2.76293i 1.23562 − 1.23562i 0.273849 − 0.961773i \(-0.411703\pi\)
0.961773 − 0.273849i \(-0.0882968\pi\)
\(6\) 1.69185 + 0.370982i 0.690697 + 0.151453i
\(7\) −0.657464 − 2.45369i −0.248498 − 0.927407i −0.971593 − 0.236659i \(-0.923948\pi\)
0.723095 − 0.690749i \(-0.242719\pi\)
\(8\) −0.707107 − 0.707107i −0.250000 − 0.250000i
\(9\) 2.98735 − 0.275255i 0.995782 − 0.0917517i
\(10\) −3.38389 + 1.95369i −1.07008 + 0.617811i
\(11\) 0.150860 − 0.563016i 0.0454859 − 0.169756i −0.939446 − 0.342696i \(-0.888660\pi\)
0.984932 + 0.172940i \(0.0553267\pi\)
\(12\) −1.53819 − 0.796225i −0.444037 − 0.229850i
\(13\) −1.20856 + 3.39697i −0.335195 + 0.942149i
\(14\) 2.54025i 0.678909i
\(15\) −4.56072 + 5.00027i −1.17757 + 1.29106i
\(16\) 0.500000 + 0.866025i 0.125000 + 0.216506i
\(17\) −0.547000 + 0.947432i −0.132667 + 0.229786i −0.924704 − 0.380687i \(-0.875687\pi\)
0.792037 + 0.610473i \(0.209021\pi\)
\(18\) −2.95680 − 0.507306i −0.696923 − 0.119573i
\(19\) 1.32717 − 0.355613i 0.304473 − 0.0815832i −0.103348 − 0.994645i \(-0.532956\pi\)
0.407821 + 0.913062i \(0.366289\pi\)
\(20\) 3.77424 − 1.01130i 0.843945 − 0.226134i
\(21\) 1.33273 + 4.19313i 0.290826 + 0.915017i
\(22\) −0.291439 + 0.504787i −0.0621349 + 0.107621i
\(23\) 0.876460 + 1.51807i 0.182755 + 0.316540i 0.942818 − 0.333309i \(-0.108165\pi\)
−0.760063 + 0.649849i \(0.774832\pi\)
\(24\) 1.27970 + 1.16721i 0.261217 + 0.238255i
\(25\) − 10.2676i − 2.05352i
\(26\) 2.04658 − 2.96842i 0.401367 − 0.582155i
\(27\) −5.14688 + 0.713876i −0.990518 + 0.137386i
\(28\) 0.657464 − 2.45369i 0.124249 − 0.463704i
\(29\) −5.12973 + 2.96165i −0.952566 + 0.549965i −0.893877 − 0.448312i \(-0.852025\pi\)
−0.0586892 + 0.998276i \(0.518692\pi\)
\(30\) 5.69948 − 3.64948i 1.04058 − 0.666302i
\(31\) 6.49983 + 6.49983i 1.16740 + 1.16740i 0.982816 + 0.184588i \(0.0590950\pi\)
0.184588 + 0.982816i \(0.440905\pi\)
\(32\) −0.258819 − 0.965926i −0.0457532 − 0.170753i
\(33\) −0.216237 + 0.986144i −0.0376420 + 0.171666i
\(34\) 0.773575 − 0.773575i 0.132667 − 0.132667i
\(35\) −8.59591 − 4.96285i −1.45297 − 0.838875i
\(36\) 2.72474 + 1.25529i 0.454124 + 0.209216i
\(37\) −2.98942 − 0.801012i −0.491457 − 0.131686i 0.00457534 − 0.999990i \(-0.498544\pi\)
−0.496032 + 0.868304i \(0.665210\pi\)
\(38\) −1.37398 −0.222890
\(39\) 1.82088 − 5.97364i 0.291573 − 0.956548i
\(40\) −3.90738 −0.617811
\(41\) 5.11781 + 1.37131i 0.799268 + 0.214163i 0.635262 − 0.772296i \(-0.280892\pi\)
0.164005 + 0.986459i \(0.447559\pi\)
\(42\) −0.202059 − 4.39519i −0.0311784 − 0.678193i
\(43\) 3.26299 + 1.88389i 0.497602 + 0.287290i 0.727723 − 0.685872i \(-0.240579\pi\)
−0.230121 + 0.973162i \(0.573912\pi\)
\(44\) 0.412157 − 0.412157i 0.0621349 − 0.0621349i
\(45\) 7.49333 − 9.01435i 1.11704 − 1.34378i
\(46\) −0.453689 − 1.69319i −0.0668928 − 0.249647i
\(47\) 5.51114 + 5.51114i 0.803883 + 0.803883i 0.983700 − 0.179817i \(-0.0575506\pi\)
−0.179817 + 0.983700i \(0.557551\pi\)
\(48\) −0.933998 − 1.45865i −0.134811 − 0.210537i
\(49\) 0.473846 − 0.273575i 0.0676922 − 0.0390821i
\(50\) −2.65745 + 9.91775i −0.375821 + 1.40258i
\(51\) 0.871071 − 1.68278i 0.121974 − 0.235636i
\(52\) −2.74513 + 2.33758i −0.380681 + 0.324164i
\(53\) − 3.04435i − 0.418173i −0.977897 − 0.209087i \(-0.932951\pi\)
0.977897 − 0.209087i \(-0.0670490\pi\)
\(54\) 5.15627 + 0.642559i 0.701679 + 0.0874413i
\(55\) −1.13876 − 1.97239i −0.153551 − 0.265957i
\(56\) −1.27012 + 2.19992i −0.169727 + 0.293976i
\(57\) −2.26801 + 0.720857i −0.300405 + 0.0954799i
\(58\) 5.72147 − 1.53306i 0.751265 − 0.201301i
\(59\) −8.19009 + 2.19453i −1.06626 + 0.285703i −0.748955 − 0.662621i \(-0.769444\pi\)
−0.317304 + 0.948324i \(0.602777\pi\)
\(60\) −6.44983 + 2.05000i −0.832670 + 0.264653i
\(61\) −4.67266 + 8.09329i −0.598273 + 1.03624i 0.394803 + 0.918766i \(0.370813\pi\)
−0.993076 + 0.117474i \(0.962520\pi\)
\(62\) −4.59607 − 7.96063i −0.583702 − 1.01100i
\(63\) −2.63946 − 7.14905i −0.332541 − 0.900695i
\(64\) 1.00000i 0.125000i
\(65\) 6.04642 + 12.7248i 0.749966 + 1.57831i
\(66\) 0.464102 − 0.896575i 0.0571270 − 0.110361i
\(67\) 1.70856 − 6.37644i 0.208734 − 0.779006i −0.779545 − 0.626346i \(-0.784550\pi\)
0.988279 − 0.152659i \(-0.0487838\pi\)
\(68\) −0.947432 + 0.547000i −0.114893 + 0.0663335i
\(69\) −1.63722 − 2.55689i −0.197099 − 0.307813i
\(70\) 7.01853 + 7.01853i 0.838875 + 0.838875i
\(71\) −0.220122 − 0.821505i −0.0261236 − 0.0974947i 0.951633 − 0.307237i \(-0.0994043\pi\)
−0.977757 + 0.209742i \(0.932738\pi\)
\(72\) −2.30701 − 1.91774i −0.271883 − 0.226008i
\(73\) −5.18078 + 5.18078i −0.606365 + 0.606365i −0.941994 − 0.335629i \(-0.891051\pi\)
0.335629 + 0.941994i \(0.391051\pi\)
\(74\) 2.68024 + 1.54744i 0.311571 + 0.179886i
\(75\) 0.816719 + 17.7653i 0.0943066 + 2.05135i
\(76\) 1.32717 + 0.355613i 0.152236 + 0.0407916i
\(77\) −1.48065 −0.168736
\(78\) −3.30492 + 5.29882i −0.374209 + 0.599973i
\(79\) −13.1089 −1.47486 −0.737431 − 0.675422i \(-0.763961\pi\)
−0.737431 + 0.675422i \(0.763961\pi\)
\(80\) 3.77424 + 1.01130i 0.421973 + 0.113067i
\(81\) 8.84847 − 1.64456i 0.983163 − 0.182729i
\(82\) −4.58850 − 2.64917i −0.506715 − 0.292552i
\(83\) 5.15394 − 5.15394i 0.565719 − 0.565719i −0.365208 − 0.930926i \(-0.619002\pi\)
0.930926 + 0.365208i \(0.119002\pi\)
\(84\) −0.942385 + 4.29773i −0.102823 + 0.468920i
\(85\) 1.10637 + 4.12902i 0.120002 + 0.447855i
\(86\) −2.66422 − 2.66422i −0.287290 − 0.287290i
\(87\) 8.63999 − 5.53235i 0.926305 − 0.593130i
\(88\) −0.504787 + 0.291439i −0.0538104 + 0.0310675i
\(89\) 2.50797 − 9.35988i 0.265844 − 0.992145i −0.695887 − 0.718151i \(-0.744989\pi\)
0.961731 − 0.273994i \(-0.0883448\pi\)
\(90\) −9.57108 + 6.76778i −1.00888 + 0.713386i
\(91\) 9.12969 + 0.732051i 0.957051 + 0.0767398i
\(92\) 1.75292i 0.182755i
\(93\) −11.7632 − 10.7291i −1.21978 − 1.11256i
\(94\) −3.89697 − 6.74974i −0.401941 − 0.696183i
\(95\) 2.68434 − 4.64941i 0.275407 − 0.477019i
\(96\) 0.524648 + 1.65068i 0.0535466 + 0.168472i
\(97\) −0.592450 + 0.158747i −0.0601542 + 0.0161183i −0.288771 − 0.957398i \(-0.593247\pi\)
0.228616 + 0.973517i \(0.426580\pi\)
\(98\) −0.528506 + 0.141613i −0.0533872 + 0.0143051i
\(99\) 0.295697 − 1.72345i 0.0297187 − 0.173213i
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 78.2.k.a.11.1 ✓ 16
3.2 odd 2 inner 78.2.k.a.11.4 yes 16
4.3 odd 2 624.2.cn.d.401.4 16
12.11 even 2 624.2.cn.d.401.2 16
13.2 odd 12 1014.2.g.c.239.3 16
13.3 even 3 1014.2.g.c.437.7 16
13.6 odd 12 inner 78.2.k.a.71.4 yes 16
13.10 even 6 1014.2.g.d.437.3 16
13.11 odd 12 1014.2.g.d.239.7 16
39.2 even 12 1014.2.g.c.239.7 16
39.11 even 12 1014.2.g.d.239.3 16
39.23 odd 6 1014.2.g.d.437.7 16
39.29 odd 6 1014.2.g.c.437.3 16
39.32 even 12 inner 78.2.k.a.71.1 yes 16
52.19 even 12 624.2.cn.d.305.2 16
156.71 odd 12 624.2.cn.d.305.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.k.a.11.1 ✓ 16 1.1 even 1 trivial
78.2.k.a.11.4 yes 16 3.2 odd 2 inner
78.2.k.a.71.1 yes 16 39.32 even 12 inner
78.2.k.a.71.4 yes 16 13.6 odd 12 inner
624.2.cn.d.305.2 16 52.19 even 12
624.2.cn.d.305.4 16 156.71 odd 12
624.2.cn.d.401.2 16 12.11 even 2
624.2.cn.d.401.4 16 4.3 odd 2
1014.2.g.c.239.3 16 13.2 odd 12
1014.2.g.c.239.7 16 39.2 even 12
1014.2.g.c.437.3 16 39.29 odd 6
1014.2.g.c.437.7 16 13.3 even 3
1014.2.g.d.239.3 16 39.11 even 12
1014.2.g.d.239.7 16 13.11 odd 12
1014.2.g.d.437.3 16 13.10 even 6
1014.2.g.d.437.7 16 39.23 odd 6