↑

Properties

Label 85.2.l.a.76.3
Level $85$
Weight $2$
Character 85.76
Analytic conductor $0.679$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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] = [85,2,Mod(26,85)] 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("85.26"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(85, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 85.l (of order \(8\), 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: \(0.678728417181\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 76.3
Character \(\chi\) \(=\) 85.76
Dual form 85.2.l.a.66.3

$q$-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+(-0.254738 + 0.254738i) q^{2} +(0.0207557 - 0.0501087i) q^{3} +1.87022i q^{4} +(0.923880 + 0.382683i) q^{5} +(0.00747733 + 0.0180519i) q^{6} +(0.275980 - 0.114315i) q^{7} +(-0.985893 - 0.985893i) q^{8} +(2.11924 + 2.11924i) q^{9} +(-0.332832 + 0.137863i) q^{10} +(-1.05900 - 2.55665i) q^{11} +(0.0937141 + 0.0388177i) q^{12} -1.97956i q^{13} +(-0.0411823 + 0.0994229i) q^{14} +(0.0383515 - 0.0383515i) q^{15} -3.23814 q^{16} +(-1.21202 - 3.94094i) q^{17} -1.07970 q^{18} +(-1.99331 + 1.99331i) q^{19} +(-0.715701 + 1.72785i) q^{20} -0.0162017i q^{21} +(0.921046 + 0.381510i) q^{22} +(-2.57919 - 6.22672i) q^{23} +(-0.0698647 + 0.0289389i) q^{24} +(0.707107 + 0.707107i) q^{25} +(0.504269 + 0.504269i) q^{26} +(0.300505 - 0.124473i) q^{27} +(0.213793 + 0.516142i) q^{28} +(4.36632 + 1.80859i) q^{29} +0.0195392i q^{30} +(1.15808 - 2.79584i) q^{31} +(2.79667 - 2.79667i) q^{32} -0.150091 q^{33} +(1.31266 + 0.695159i) q^{34} +0.298718 q^{35} +(-3.96344 + 3.96344i) q^{36} +(-3.60537 + 8.70414i) q^{37} -1.01554i q^{38} +(-0.0991930 - 0.0410871i) q^{39} +(-0.533561 - 1.28813i) q^{40} +(2.87301 - 1.19004i) q^{41} +(0.00412718 + 0.00412718i) q^{42} +(5.78771 + 5.78771i) q^{43} +(4.78150 - 1.98056i) q^{44} +(1.14692 + 2.76892i) q^{45} +(2.24320 + 0.929166i) q^{46} +1.08341i q^{47} +(-0.0672100 + 0.162259i) q^{48} +(-4.88665 + 4.88665i) q^{49} -0.360254 q^{50} +(-0.222632 - 0.0210640i) q^{51} +3.70220 q^{52} +(-1.89858 + 1.89858i) q^{53} +(-0.0448420 + 0.108258i) q^{54} -2.76730i q^{55} +(-0.384788 - 0.159384i) q^{56} +(0.0585096 + 0.141255i) q^{57} +(-1.57299 + 0.651553i) q^{58} +(-6.47310 - 6.47310i) q^{59} +(0.0717257 + 0.0717257i) q^{60} +(10.3418 - 4.28372i) q^{61} +(0.417202 + 1.00722i) q^{62} +(0.827127 + 0.342607i) q^{63} -5.05145i q^{64} +(0.757544 - 1.82887i) q^{65} +(0.0382339 - 0.0382339i) q^{66} -12.5585 q^{67} +(7.37041 - 2.26675i) q^{68} -0.365546 q^{69} +(-0.0760950 + 0.0760950i) q^{70} +(-2.25315 + 5.43960i) q^{71} -4.17869i q^{72} +(-0.200173 - 0.0829144i) q^{73} +(-1.29885 - 3.13570i) q^{74} +(0.0501087 - 0.0207557i) q^{75} +(-3.72792 - 3.72792i) q^{76} +(-0.584525 - 0.584525i) q^{77} +(0.0357347 - 0.0148018i) q^{78} +(-4.07771 - 9.84447i) q^{79} +(-2.99165 - 1.23918i) q^{80} +8.97353i q^{81} +(-0.428717 + 1.03501i) q^{82} +(-11.0129 + 11.0129i) q^{83} +0.0303006 q^{84} +(0.388367 - 4.10477i) q^{85} -2.94870 q^{86} +(0.181252 - 0.181252i) q^{87} +(-1.47653 + 3.56465i) q^{88} -1.55264i q^{89} +(-0.997516 - 0.413185i) q^{90} +(-0.226292 - 0.546318i) q^{91} +(11.6453 - 4.82365i) q^{92} +(-0.116059 - 0.116059i) q^{93} +(-0.275986 - 0.275986i) q^{94} +(-2.60438 + 1.07877i) q^{95} +(-0.0820905 - 0.198184i) q^{96} +(8.28752 + 3.43280i) q^{97} -2.48963i q^{98} +(3.17389 - 7.66244i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 8 q^{6} - 24 q^{9} - 8 q^{11} + 24 q^{12} - 8 q^{15} - 24 q^{16} - 8 q^{17} + 8 q^{18} - 8 q^{19} - 32 q^{22} - 16 q^{23} - 8 q^{24} + 16 q^{26} + 24 q^{27} + 48 q^{28} - 8 q^{29} + 16 q^{34} - 32 q^{35}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/85\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(71\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{8}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.254738 + 0.254738i −0.180127 + 0.180127i −0.791411 − 0.611284i \(-0.790653\pi\)
0.611284 + 0.791411i \(0.290653\pi\)
\(3\) 0.0207557 − 0.0501087i 0.0119833 − 0.0289303i −0.917775 − 0.397102i \(-0.870016\pi\)
0.929758 + 0.368171i \(0.120016\pi\)
\(4\) 1.87022i 0.935108i
\(5\) 0.923880 + 0.382683i 0.413171 + 0.171141i
\(6\) 0.00747733 + 0.0180519i 0.00305261 + 0.00736965i
\(7\) 0.275980 − 0.114315i 0.104310 − 0.0432068i −0.329918 − 0.944010i \(-0.607021\pi\)
0.434228 + 0.900803i \(0.357021\pi\)
\(8\) −0.985893 − 0.985893i −0.348566 − 0.348566i
\(9\) 2.11924 + 2.11924i 0.706413 + 0.706413i
\(10\) −0.332832 + 0.137863i −0.105251 + 0.0435962i
\(11\) −1.05900 − 2.55665i −0.319301 − 0.770860i −0.999291 − 0.0376394i \(-0.988016\pi\)
0.679991 − 0.733221i \(-0.261984\pi\)
\(12\) 0.0937141 + 0.0388177i 0.0270529 + 0.0112057i
\(13\) − 1.97956i − 0.549030i −0.961583 − 0.274515i \(-0.911483\pi\)
0.961583 − 0.274515i \(-0.0885174\pi\)
\(14\) −0.0411823 + 0.0994229i −0.0110064 + 0.0265719i
\(15\) 0.0383515 − 0.0383515i 0.00990232 − 0.00990232i
\(16\) −3.23814 −0.809536
\(17\) −1.21202 − 3.94094i −0.293959 − 0.955818i
\(18\) −1.07970 −0.254489
\(19\) −1.99331 + 1.99331i −0.457296 + 0.457296i −0.897767 − 0.440471i \(-0.854812\pi\)
0.440471 + 0.897767i \(0.354812\pi\)
\(20\) −0.715701 + 1.72785i −0.160036 + 0.386360i
\(21\) − 0.0162017i − 0.00353549i
\(22\) 0.921046 + 0.381510i 0.196368 + 0.0813382i
\(23\) −2.57919 − 6.22672i −0.537799 − 1.29836i −0.926256 − 0.376895i \(-0.876992\pi\)
0.388457 − 0.921467i \(-0.373008\pi\)
\(24\) −0.0698647 + 0.0289389i −0.0142611 + 0.00590713i
\(25\) 0.707107 + 0.707107i 0.141421 + 0.141421i
\(26\) 0.504269 + 0.504269i 0.0988953 + 0.0988953i
\(27\) 0.300505 − 0.124473i 0.0578322 − 0.0239549i
\(28\) 0.213793 + 0.516142i 0.0404031 + 0.0975416i
\(29\) 4.36632 + 1.80859i 0.810806 + 0.335847i 0.749276 − 0.662258i \(-0.230402\pi\)
0.0615305 + 0.998105i \(0.480402\pi\)
\(30\) 0.0195392i 0.00356736i
\(31\) 1.15808 − 2.79584i 0.207997 − 0.502148i −0.785111 − 0.619355i \(-0.787394\pi\)
0.993108 + 0.117207i \(0.0373941\pi\)
\(32\) 2.79667 − 2.79667i 0.494385 − 0.494385i
\(33\) −0.150091 −0.0261275
\(34\) 1.31266 + 0.695159i 0.225119 + 0.119219i
\(35\) 0.298718 0.0504926
\(36\) −3.96344 + 3.96344i −0.660573 + 0.660573i
\(37\) −3.60537 + 8.70414i −0.592719 + 1.43095i 0.288147 + 0.957586i \(0.406961\pi\)
−0.880866 + 0.473365i \(0.843039\pi\)
\(38\) − 1.01554i − 0.164743i
\(39\) −0.0991930 − 0.0410871i −0.0158836 − 0.00657920i
\(40\) −0.533561 − 1.28813i −0.0843635 − 0.203671i
\(41\) 2.87301 − 1.19004i 0.448688 − 0.185853i −0.146885 − 0.989154i \(-0.546925\pi\)
0.595574 + 0.803301i \(0.296925\pi\)
\(42\) 0.00412718 + 0.00412718i 0.000636838 + 0.000636838i
\(43\) 5.78771 + 5.78771i 0.882617 + 0.882617i 0.993800 − 0.111183i \(-0.0354638\pi\)
−0.111183 + 0.993800i \(0.535464\pi\)
\(44\) 4.78150 − 1.98056i 0.720838 − 0.298581i
\(45\) 1.14692 + 2.76892i 0.170973 + 0.412766i
\(46\) 2.24320 + 0.929166i 0.330742 + 0.136998i
\(47\) 1.08341i 0.158032i 0.996873 + 0.0790159i \(0.0251778\pi\)
−0.996873 + 0.0790159i \(0.974822\pi\)
\(48\) −0.0672100 + 0.162259i −0.00970092 + 0.0234201i
\(49\) −4.88665 + 4.88665i −0.698093 + 0.698093i
\(50\) −0.360254 −0.0509477
\(51\) −0.222632 − 0.0210640i −0.0311747 − 0.00294954i
\(52\) 3.70220 0.513403
\(53\) −1.89858 + 1.89858i −0.260790 + 0.260790i −0.825375 − 0.564585i \(-0.809036\pi\)
0.564585 + 0.825375i \(0.309036\pi\)
\(54\) −0.0448420 + 0.108258i −0.00610222 + 0.0147321i
\(55\) − 2.76730i − 0.373143i
\(56\) −0.384788 − 0.159384i −0.0514195 − 0.0212986i
\(57\) 0.0585096 + 0.141255i 0.00774979 + 0.0187096i
\(58\) −1.57299 + 0.651553i −0.206543 + 0.0855531i
\(59\) −6.47310 − 6.47310i −0.842726 − 0.842726i 0.146487 − 0.989213i \(-0.453203\pi\)
−0.989213 + 0.146487i \(0.953203\pi\)
\(60\) 0.0717257 + 0.0717257i 0.00925975 + 0.00925975i
\(61\) 10.3418 − 4.28372i 1.32413 − 0.548474i 0.395158 − 0.918613i \(-0.370690\pi\)
0.928976 + 0.370139i \(0.120690\pi\)
\(62\) 0.417202 + 1.00722i 0.0529847 + 0.127916i
\(63\) 0.827127 + 0.342607i 0.104208 + 0.0431645i
\(64\) − 5.05145i − 0.631432i
\(65\) 0.757544 − 1.82887i 0.0939617 − 0.226844i
\(66\) 0.0382339 − 0.0382339i 0.00470627 − 0.00470627i
\(67\) −12.5585 −1.53427 −0.767133 − 0.641488i \(-0.778317\pi\)
−0.767133 + 0.641488i \(0.778317\pi\)
\(68\) 7.37041 − 2.26675i 0.893793 − 0.274884i
\(69\) −0.365546 −0.0440066
\(70\) −0.0760950 + 0.0760950i −0.00909509 + 0.00909509i
\(71\) −2.25315 + 5.43960i −0.267400 + 0.645561i −0.999359 − 0.0357872i \(-0.988606\pi\)
0.731959 + 0.681348i \(0.238606\pi\)
\(72\) − 4.17869i − 0.492463i
\(73\) −0.200173 − 0.0829144i −0.0234285 − 0.00970440i 0.370938 − 0.928657i \(-0.379036\pi\)
−0.394367 + 0.918953i \(0.629036\pi\)
\(74\) −1.29885 − 3.13570i −0.150988 − 0.364518i
\(75\) 0.0501087 − 0.0207557i 0.00578605 − 0.00239666i
\(76\) −3.72792 − 3.72792i −0.427622 − 0.427622i
\(77\) −0.584525 − 0.584525i −0.0666128 − 0.0666128i
\(78\) 0.0357347 − 0.0148018i 0.00404616 − 0.00167598i
\(79\) −4.07771 − 9.84447i −0.458779 − 1.10759i −0.968892 − 0.247482i \(-0.920397\pi\)
0.510114 − 0.860107i \(-0.329603\pi\)
\(80\) −2.99165 − 1.23918i −0.334477 − 0.138545i
\(81\) 8.97353i 0.997059i
\(82\) −0.428717 + 1.03501i −0.0473438 + 0.114298i
\(83\) −11.0129 + 11.0129i −1.20883 + 1.20883i −0.237421 + 0.971407i \(0.576302\pi\)
−0.971407 + 0.237421i \(0.923698\pi\)
\(84\) 0.0303006 0.00330607
\(85\) 0.388367 − 4.10477i 0.0421243 − 0.445225i
\(86\) −2.94870 −0.317967
\(87\) 0.181252 − 0.181252i 0.0194323 − 0.0194323i
\(88\) −1.47653 + 3.56465i −0.157398 + 0.379993i
\(89\) − 1.55264i − 0.164579i −0.996608 − 0.0822897i \(-0.973777\pi\)
0.996608 − 0.0822897i \(-0.0262233\pi\)
\(90\) −0.997516 − 0.413185i −0.105147 − 0.0435535i
\(91\) −0.226292 − 0.546318i −0.0237219 − 0.0572696i
\(92\) 11.6453 − 4.82365i 1.21411 − 0.502900i
\(93\) −0.116059 − 0.116059i −0.0120348 − 0.0120348i
\(94\) −0.275986 − 0.275986i −0.0284658 − 0.0284658i
\(95\) −2.60438 + 1.07877i −0.267204 + 0.110680i
\(96\) −0.0820905 − 0.198184i −0.00837833 − 0.0202271i
\(97\) 8.28752 + 3.43280i 0.841471 + 0.348549i 0.761433 − 0.648243i \(-0.224496\pi\)
0.0800373 + 0.996792i \(0.474496\pi\)
\(98\) − 2.48963i − 0.251491i
\(99\) 3.17389 − 7.66244i 0.318988 − 0.770104i
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 85.2.l.a.76.3 yes 24
3.2 odd 2 765.2.be.b.586.4 24
5.2 odd 4 425.2.n.f.399.3 24
5.3 odd 4 425.2.n.c.399.4 24
5.4 even 2 425.2.m.b.76.4 24
17.6 odd 16 1445.2.d.j.866.9 24
17.7 odd 16 1445.2.a.q.1.8 12
17.10 odd 16 1445.2.a.p.1.8 12
17.11 odd 16 1445.2.d.j.866.10 24
17.15 even 8 inner 85.2.l.a.66.3 ✓ 24
51.32 odd 8 765.2.be.b.406.4 24
85.24 odd 16 7225.2.a.bq.1.5 12
85.32 odd 8 425.2.n.c.49.4 24
85.44 odd 16 7225.2.a.bs.1.5 12
85.49 even 8 425.2.m.b.151.4 24
85.83 odd 8 425.2.n.f.49.3 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.l.a.66.3 ✓ 24 17.15 even 8 inner
85.2.l.a.76.3 yes 24 1.1 even 1 trivial
425.2.m.b.76.4 24 5.4 even 2
425.2.m.b.151.4 24 85.49 even 8
425.2.n.c.49.4 24 85.32 odd 8
425.2.n.c.399.4 24 5.3 odd 4
425.2.n.f.49.3 24 85.83 odd 8
425.2.n.f.399.3 24 5.2 odd 4
765.2.be.b.406.4 24 51.32 odd 8
765.2.be.b.586.4 24 3.2 odd 2
1445.2.a.p.1.8 12 17.10 odd 16
1445.2.a.q.1.8 12 17.7 odd 16
1445.2.d.j.866.9 24 17.6 odd 16
1445.2.d.j.866.10 24 17.11 odd 16
7225.2.a.bq.1.5 12 85.24 odd 16
7225.2.a.bs.1.5 12 85.44 odd 16