Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [77,2,Mod(4,77)] 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("77.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(77, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([20, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 77.m (of order \(15\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.614848095564\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(5\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label 58.1
Character \(\chi\) \(=\) 77.58
Dual form 77.2.m.b.4.1

$q$-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+(-0.233841 - 2.22485i) q^{2} +(0.893246 - 0.992050i) q^{3} +(-2.93896 + 0.624696i) q^{4} +(-0.791435 + 0.352369i) q^{5} +(-2.41604 - 1.75535i) q^{6} +(2.03186 + 1.69457i) q^{7} +(0.694498 + 2.13745i) q^{8} +(0.127310 + 1.21128i) q^{9} +(0.969038 + 1.67842i) q^{10} +(-3.30459 + 0.282305i) q^{11} +(-2.00549 + 3.47361i) q^{12} +(2.73539 - 1.98738i) q^{13} +(3.29502 - 4.91683i) q^{14} +(-0.357378 + 1.09990i) q^{15} +(-0.896633 + 0.399207i) q^{16} +(0.229511 - 2.18365i) q^{17} +(2.66514 - 0.566492i) q^{18} +(3.51015 + 0.746106i) q^{19} +(2.10587 - 1.53001i) q^{20} +(3.49604 - 0.502041i) q^{21} +(1.40083 + 7.28619i) q^{22} +(-0.834837 + 1.44598i) q^{23} +(2.74081 + 1.22029i) q^{24} +(-2.84345 + 3.15797i) q^{25} +(-5.06126 - 5.62110i) q^{26} +(4.55532 + 3.30963i) q^{27} +(-7.03015 - 3.71097i) q^{28} +(0.473517 - 1.45734i) q^{29} +(2.53067 + 0.537910i) q^{30} +(-8.10945 - 3.61056i) q^{31} +(3.34529 + 5.79421i) q^{32} +(-2.67175 + 3.53048i) q^{33} -4.91197 q^{34} +(-2.20520 - 0.625173i) q^{35} +(-1.13084 - 3.48037i) q^{36} +(2.06447 + 2.29283i) q^{37} +(0.839155 - 7.98402i) q^{38} +(0.471798 - 4.48886i) q^{39} +(-1.30282 - 1.44693i) q^{40} +(0.203209 + 0.625414i) q^{41} +(-1.93448 - 7.66076i) q^{42} -9.76359 q^{43} +(9.53571 - 2.89405i) q^{44} +(-0.527575 - 0.913787i) q^{45} +(3.41230 + 1.51926i) q^{46} +(-12.9072 - 2.74350i) q^{47} +(-0.404881 + 1.24609i) q^{48} +(1.25690 + 6.88623i) q^{49} +(7.69091 + 5.58777i) q^{50} +(-1.96128 - 2.17823i) q^{51} +(-6.79771 + 7.54962i) q^{52} +(5.99629 + 2.66972i) q^{53} +(6.29821 - 10.9088i) q^{54} +(2.51589 - 1.38786i) q^{55} +(-2.21092 + 5.51986i) q^{56} +(3.87560 - 2.81579i) q^{57} +(-3.35308 - 0.712719i) q^{58} +(14.2508 - 3.02910i) q^{59} +(0.363220 - 3.45581i) q^{60} +(5.29525 - 2.35760i) q^{61} +(-6.13662 + 18.8866i) q^{62} +(-1.79391 + 2.67688i) q^{63} +(10.5209 - 7.64386i) q^{64} +(-1.46459 + 2.53675i) q^{65} +(8.47955 + 5.11866i) q^{66} +(-2.64188 - 4.57587i) q^{67} +(0.689595 + 6.56106i) q^{68} +(0.688770 + 2.11982i) q^{69} +(-0.875249 + 5.05241i) q^{70} +(-6.33800 - 4.60483i) q^{71} +(-2.50062 + 1.11335i) q^{72} +(5.78803 - 1.23028i) q^{73} +(4.61843 - 5.12929i) q^{74} +(0.592965 + 5.64168i) q^{75} -10.7823 q^{76} +(-7.19284 - 5.02624i) q^{77} -10.0974 q^{78} +(-1.22867 - 11.6900i) q^{79} +(0.568958 - 0.631892i) q^{80} +(3.77834 - 0.803111i) q^{81} +(1.34393 - 0.598357i) q^{82} +(0.455953 + 0.331269i) q^{83} +(-9.96112 + 3.65945i) q^{84} +(0.587810 + 1.80909i) q^{85} +(2.28312 + 21.7225i) q^{86} +(-1.02278 - 1.77151i) q^{87} +(-2.89844 - 6.86732i) q^{88} +(-1.76101 + 3.05016i) q^{89} +(-1.90967 + 1.38745i) q^{90} +(8.92567 + 0.597229i) q^{91} +(1.55026 - 4.77120i) q^{92} +(-10.8256 + 4.81986i) q^{93} +(-3.08565 + 29.3580i) q^{94} +(-3.04096 + 0.646376i) q^{95} +(8.73630 + 1.85696i) q^{96} +(11.2566 - 8.17838i) q^{97} +(15.0269 - 4.40668i) q^{98} +(-0.762658 - 3.96683i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q - 3 q^{2} - 4 q^{3} - 3 q^{4} + 4 q^{5} - 16 q^{6} - 2 q^{7} - 38 q^{8} + 7 q^{9} + 14 q^{10} - 9 q^{11} - 18 q^{12} + 6 q^{13} - 3 q^{14} - 14 q^{15} - 5 q^{16} - 7 q^{17} + 24 q^{18} - 4 q^{19} - 30 q^{20}+ \cdots - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/77\mathbb{Z}\right)^\times\).

\(n\) \(45\) \(57\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{4}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.233841 − 2.22485i −0.165350 − 1.57320i −0.691224 − 0.722641i \(-0.742928\pi\)
0.525873 − 0.850563i \(-0.323739\pi\)
\(3\) 0.893246 − 0.992050i 0.515716 − 0.572760i −0.427890 − 0.903831i \(-0.640743\pi\)
0.943606 + 0.331070i \(0.107410\pi\)
\(4\) −2.93896 + 0.624696i −1.46948 + 0.312348i
\(5\) −0.791435 + 0.352369i −0.353940 + 0.157584i −0.576002 − 0.817448i \(-0.695388\pi\)
0.222062 + 0.975033i \(0.428721\pi\)
\(6\) −2.41604 − 1.75535i −0.986342 − 0.716620i
\(7\) 2.03186 + 1.69457i 0.767970 + 0.640486i
\(8\) 0.694498 + 2.13745i 0.245542 + 0.755701i
\(9\) 0.127310 + 1.21128i 0.0424368 + 0.403759i
\(10\) 0.969038 + 1.67842i 0.306437 + 0.530764i
\(11\) −3.30459 + 0.282305i −0.996371 + 0.0851181i
\(12\) −2.00549 + 3.47361i −0.578934 + 1.00274i
\(13\) 2.73539 − 1.98738i 0.758661 − 0.551200i −0.139838 − 0.990174i \(-0.544658\pi\)
0.898499 + 0.438975i \(0.144658\pi\)
\(14\) 3.29502 − 4.91683i 0.880630 − 1.31408i
\(15\) −0.357378 + 1.09990i −0.0922745 + 0.283992i
\(16\) −0.896633 + 0.399207i −0.224158 + 0.0998017i
\(17\) 0.229511 − 2.18365i 0.0556647 − 0.529614i −0.930787 − 0.365562i \(-0.880877\pi\)
0.986452 − 0.164052i \(-0.0524565\pi\)
\(18\) 2.66514 − 0.566492i 0.628179 − 0.133523i
\(19\) 3.51015 + 0.746106i 0.805284 + 0.171168i 0.592122 − 0.805848i \(-0.298290\pi\)
0.213162 + 0.977017i \(0.431624\pi\)
\(20\) 2.10587 − 1.53001i 0.470888 − 0.342120i
\(21\) 3.49604 − 0.502041i 0.762899 − 0.109554i
\(22\) 1.40083 + 7.28619i 0.298658 + 1.55342i
\(23\) −0.834837 + 1.44598i −0.174076 + 0.301508i −0.939841 − 0.341612i \(-0.889027\pi\)
0.765765 + 0.643120i \(0.222360\pi\)
\(24\) 2.74081 + 1.22029i 0.559465 + 0.249090i
\(25\) −2.84345 + 3.15797i −0.568690 + 0.631594i
\(26\) −5.06126 − 5.62110i −0.992594 − 1.10239i
\(27\) 4.55532 + 3.30963i 0.876672 + 0.636939i
\(28\) −7.03015 − 3.71097i −1.32857 − 0.701308i
\(29\) 0.473517 − 1.45734i 0.0879300 − 0.270621i −0.897417 − 0.441184i \(-0.854559\pi\)
0.985347 + 0.170563i \(0.0545587\pi\)
\(30\) 2.53067 + 0.537910i 0.462034 + 0.0982085i
\(31\) −8.10945 − 3.61056i −1.45650 − 0.648475i −0.482680 − 0.875797i \(-0.660337\pi\)
−0.973820 + 0.227321i \(0.927003\pi\)
\(32\) 3.34529 + 5.79421i 0.591369 + 1.02428i
\(33\) −2.67175 + 3.53048i −0.465092 + 0.614578i
\(34\) −4.91197 −0.842395
\(35\) −2.20520 − 0.625173i −0.372746 − 0.105674i
\(36\) −1.13084 − 3.48037i −0.188474 − 0.580062i
\(37\) 2.06447 + 2.29283i 0.339397 + 0.376939i 0.888547 − 0.458785i \(-0.151715\pi\)
−0.549150 + 0.835724i \(0.685049\pi\)
\(38\) 0.839155 − 7.98402i 0.136129 − 1.29518i
\(39\) 0.471798 − 4.48886i 0.0755482 − 0.718793i
\(40\) −1.30282 − 1.44693i −0.205994 − 0.228780i
\(41\) 0.203209 + 0.625414i 0.0317360 + 0.0976732i 0.965670 − 0.259773i \(-0.0836476\pi\)
−0.933934 + 0.357446i \(0.883648\pi\)
\(42\) −1.93448 − 7.66076i −0.298497 − 1.18208i
\(43\) −9.76359 −1.48893 −0.744467 − 0.667660i \(-0.767296\pi\)
−0.744467 + 0.667660i \(0.767296\pi\)
\(44\) 9.53571 − 2.89405i 1.43756 − 0.436294i
\(45\) −0.527575 − 0.913787i −0.0786463 − 0.136219i
\(46\) 3.41230 + 1.51926i 0.503117 + 0.224002i
\(47\) −12.9072 − 2.74350i −1.88270 − 0.400181i −0.884862 − 0.465854i \(-0.845747\pi\)
−0.997841 + 0.0656733i \(0.979080\pi\)
\(48\) −0.404881 + 1.24609i −0.0584395 + 0.179858i
\(49\) 1.25690 + 6.88623i 0.179557 + 0.983748i
\(50\) 7.69091 + 5.58777i 1.08766 + 0.790231i
\(51\) −1.96128 − 2.17823i −0.274635 − 0.305013i
\(52\) −6.79771 + 7.54962i −0.942673 + 1.04694i
\(53\) 5.99629 + 2.66972i 0.823654 + 0.366714i 0.774890 − 0.632096i \(-0.217805\pi\)
0.0487639 + 0.998810i \(0.484472\pi\)
\(54\) 6.29821 − 10.9088i 0.857078 − 1.48450i
\(55\) 2.51589 − 1.38786i 0.339243 − 0.187139i
\(56\) −2.21092 + 5.51986i −0.295447 + 0.737622i
\(57\) 3.87560 − 2.81579i 0.513336 − 0.372961i
\(58\) −3.35308 − 0.712719i −0.440281 − 0.0935846i
\(59\) 14.2508 − 3.02910i 1.85530 − 0.394356i 0.861709 − 0.507403i \(-0.169395\pi\)
0.993590 + 0.113047i \(0.0360612\pi\)
\(60\) 0.363220 − 3.45581i 0.0468915 − 0.446143i
\(61\) 5.29525 − 2.35760i 0.677988 − 0.301860i −0.0387029 − 0.999251i \(-0.512323\pi\)
0.716691 + 0.697391i \(0.245656\pi\)
\(62\) −6.13662 + 18.8866i −0.779351 + 2.39860i
\(63\) −1.79391 + 2.67688i −0.226012 + 0.337255i
\(64\) 10.5209 − 7.64386i 1.31511 − 0.955483i
\(65\) −1.46459 + 2.53675i −0.181660 + 0.314645i
\(66\) 8.47955 + 5.11866i 1.04376 + 0.630063i
\(67\) −2.64188 − 4.57587i −0.322757 − 0.559032i 0.658299 − 0.752757i \(-0.271276\pi\)
−0.981056 + 0.193725i \(0.937943\pi\)
\(68\) 0.689595 + 6.56106i 0.0836257 + 0.795645i
\(69\) 0.688770 + 2.11982i 0.0829181 + 0.255196i
\(70\) −0.875249 + 5.05241i −0.104612 + 0.603879i
\(71\) −6.33800 − 4.60483i −0.752183 − 0.546493i 0.144320 − 0.989531i \(-0.453901\pi\)
−0.896503 + 0.443038i \(0.853901\pi\)
\(72\) −2.50062 + 1.11335i −0.294701 + 0.131209i
\(73\) 5.78803 − 1.23028i 0.677437 − 0.143994i 0.143671 − 0.989626i \(-0.454109\pi\)
0.533767 + 0.845632i \(0.320776\pi\)
\(74\) 4.61843 − 5.12929i 0.536882 − 0.596268i
\(75\) 0.592965 + 5.64168i 0.0684697 + 0.651446i
\(76\) −10.7823 −1.23682
\(77\) −7.19284 − 5.02624i −0.819700 − 0.572793i
\(78\) −10.0974 −1.14330
\(79\) −1.22867 − 11.6900i −0.138236 − 1.31523i −0.815186 − 0.579200i \(-0.803365\pi\)
0.676950 − 0.736029i \(-0.263301\pi\)
\(80\) 0.568958 − 0.631892i 0.0636115 − 0.0706477i
\(81\) 3.77834 − 0.803111i 0.419815 − 0.0892345i
\(82\) 1.34393 − 0.598357i 0.148412 − 0.0660774i
\(83\) 0.455953 + 0.331269i 0.0500473 + 0.0363615i 0.612528 − 0.790449i \(-0.290153\pi\)
−0.562480 + 0.826811i \(0.690153\pi\)
\(84\) −9.96112 + 3.65945i −1.08685 + 0.399278i
\(85\) 0.587810 + 1.80909i 0.0637569 + 0.196224i
\(86\) 2.28312 + 21.7225i 0.246196 + 2.34240i
\(87\) −1.02278 − 1.77151i −0.109654 − 0.189926i
\(88\) −2.89844 − 6.86732i −0.308975 − 0.732059i
\(89\) −1.76101 + 3.05016i −0.186667 + 0.323317i −0.944137 − 0.329553i \(-0.893102\pi\)
0.757470 + 0.652870i \(0.226435\pi\)
\(90\) −1.90967 + 1.38745i −0.201297 + 0.146251i
\(91\) 8.92567 + 0.597229i 0.935665 + 0.0626066i
\(92\) 1.55026 − 4.77120i 0.161626 − 0.497432i
\(93\) −10.8256 + 4.81986i −1.12256 + 0.499796i
\(94\) −3.08565 + 29.3580i −0.318260 + 3.02805i
\(95\) −3.04096 + 0.646376i −0.311996 + 0.0663168i
\(96\) 8.73630 + 1.85696i 0.891645 + 0.189525i
\(97\) 11.2566 − 8.17838i 1.14293 − 0.830388i 0.155406 − 0.987851i \(-0.450331\pi\)
0.987525 + 0.157462i \(0.0503313\pi\)
\(98\) 15.0269 − 4.40668i 1.51795 − 0.445142i
\(99\) −0.762658 − 3.96683i −0.0766500 − 0.398682i
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 77.2.m.b.58.1 yes 40
3.2 odd 2 693.2.by.b.289.5 40
7.2 even 3 539.2.f.h.344.1 20
7.3 odd 6 539.2.q.h.410.5 40
7.4 even 3 inner 77.2.m.b.25.5 yes 40
7.5 odd 6 539.2.f.g.344.1 20
7.6 odd 2 539.2.q.h.520.1 40
11.2 odd 10 847.2.e.h.485.9 20
11.3 even 5 847.2.n.i.632.1 40
11.4 even 5 inner 77.2.m.b.37.5 yes 40
11.5 even 5 847.2.n.i.9.5 40
11.6 odd 10 847.2.n.h.9.1 40
11.7 odd 10 847.2.n.j.807.1 40
11.8 odd 10 847.2.n.h.632.5 40
11.9 even 5 847.2.e.i.485.2 20
11.10 odd 2 847.2.n.j.366.5 40
21.11 odd 6 693.2.by.b.487.1 40
33.26 odd 10 693.2.by.b.37.1 40
77.2 odd 30 5929.2.a.by.1.2 10
77.4 even 15 inner 77.2.m.b.4.1 ✓ 40
77.9 even 15 5929.2.a.bw.1.9 10
77.18 odd 30 847.2.n.j.81.5 40
77.25 even 15 847.2.n.i.753.5 40
77.26 odd 30 539.2.f.g.246.1 20
77.32 odd 6 847.2.n.j.487.1 40
77.37 even 15 539.2.f.h.246.1 20
77.39 odd 30 847.2.n.h.130.5 40
77.46 odd 30 847.2.e.h.606.9 20
77.48 odd 10 539.2.q.h.422.5 40
77.53 even 15 847.2.e.i.606.2 20
77.59 odd 30 539.2.q.h.312.1 40
77.60 even 15 847.2.n.i.130.1 40
77.68 even 30 5929.2.a.bz.1.2 10
77.74 odd 30 847.2.n.h.753.1 40
77.75 odd 30 5929.2.a.bx.1.9 10
231.158 odd 30 693.2.by.b.235.5 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.m.b.4.1 ✓ 40 77.4 even 15 inner
77.2.m.b.25.5 yes 40 7.4 even 3 inner
77.2.m.b.37.5 yes 40 11.4 even 5 inner
77.2.m.b.58.1 yes 40 1.1 even 1 trivial
539.2.f.g.246.1 20 77.26 odd 30
539.2.f.g.344.1 20 7.5 odd 6
539.2.f.h.246.1 20 77.37 even 15
539.2.f.h.344.1 20 7.2 even 3
539.2.q.h.312.1 40 77.59 odd 30
539.2.q.h.410.5 40 7.3 odd 6
539.2.q.h.422.5 40 77.48 odd 10
539.2.q.h.520.1 40 7.6 odd 2
693.2.by.b.37.1 40 33.26 odd 10
693.2.by.b.235.5 40 231.158 odd 30
693.2.by.b.289.5 40 3.2 odd 2
693.2.by.b.487.1 40 21.11 odd 6
847.2.e.h.485.9 20 11.2 odd 10
847.2.e.h.606.9 20 77.46 odd 30
847.2.e.i.485.2 20 11.9 even 5
847.2.e.i.606.2 20 77.53 even 15
847.2.n.h.9.1 40 11.6 odd 10
847.2.n.h.130.5 40 77.39 odd 30
847.2.n.h.632.5 40 11.8 odd 10
847.2.n.h.753.1 40 77.74 odd 30
847.2.n.i.9.5 40 11.5 even 5
847.2.n.i.130.1 40 77.60 even 15
847.2.n.i.632.1 40 11.3 even 5
847.2.n.i.753.5 40 77.25 even 15
847.2.n.j.81.5 40 77.18 odd 30
847.2.n.j.366.5 40 11.10 odd 2
847.2.n.j.487.1 40 77.32 odd 6
847.2.n.j.807.1 40 11.7 odd 10
5929.2.a.bw.1.9 10 77.9 even 15
5929.2.a.bx.1.9 10 77.75 odd 30
5929.2.a.by.1.2 10 77.2 odd 30
5929.2.a.bz.1.2 10 77.68 even 30