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Properties

Label 85.2.l.a.76.6
Level $85$
Weight $2$
Character 85.76
Analytic conductor $0.679$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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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 76.6
Character \(\chi\) \(=\) 85.76
Dual form 85.2.l.a.66.6

$q$-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+(1.86672 - 1.86672i) q^{2} +(-0.811501 + 1.95914i) q^{3} -4.96928i q^{4} +(0.923880 + 0.382683i) q^{5} +(2.14231 + 5.17200i) q^{6} +(-3.75274 + 1.55444i) q^{7} +(-5.54282 - 5.54282i) q^{8} +(-1.05836 - 1.05836i) q^{9} +(2.43899 - 1.01026i) q^{10} +(-0.669316 - 1.61587i) q^{11} +(9.73550 + 4.03258i) q^{12} +1.67715i q^{13} +(-4.10362 + 9.90701i) q^{14} +(-1.49946 + 1.49946i) q^{15} -10.7552 q^{16} +(4.08018 - 0.593395i) q^{17} -3.95133 q^{18} +(-0.176096 + 0.176096i) q^{19} +(1.90166 - 4.59102i) q^{20} -8.61356i q^{21} +(-4.26580 - 1.76695i) q^{22} +(-0.198886 - 0.480154i) q^{23} +(15.3571 - 6.36113i) q^{24} +(0.707107 + 0.707107i) q^{25} +(3.13077 + 3.13077i) q^{26} +(-2.94507 + 1.21989i) q^{27} +(7.72443 + 18.6484i) q^{28} +(0.449816 + 0.186320i) q^{29} +5.59814i q^{30} +(3.64713 - 8.80495i) q^{31} +(-8.99131 + 8.99131i) q^{32} +3.70886 q^{33} +(6.50885 - 8.72426i) q^{34} -4.06194 q^{35} +(-5.25930 + 5.25930i) q^{36} +(1.63395 - 3.94470i) q^{37} +0.657443i q^{38} +(-3.28577 - 1.36101i) q^{39} +(-2.99975 - 7.24204i) q^{40} +(-4.88624 + 2.02395i) q^{41} +(-16.0791 - 16.0791i) q^{42} +(-2.24825 - 2.24825i) q^{43} +(-8.02972 + 3.32602i) q^{44} +(-0.572782 - 1.38282i) q^{45} +(-1.26758 - 0.525048i) q^{46} +6.26212i q^{47} +(8.72786 - 21.0709i) q^{48} +(6.71704 - 6.71704i) q^{49} +2.63994 q^{50} +(-2.14853 + 8.47517i) q^{51} +8.33425 q^{52} +(-7.24333 + 7.24333i) q^{53} +(-3.22043 + 7.77481i) q^{54} -1.74901i q^{55} +(29.4167 + 12.1848i) q^{56} +(-0.202094 - 0.487898i) q^{57} +(1.18749 - 0.491874i) q^{58} +(8.20143 + 8.20143i) q^{59} +(7.45123 + 7.45123i) q^{60} +(-4.02196 + 1.66595i) q^{61} +(-9.62820 - 23.2445i) q^{62} +(5.61692 + 2.32660i) q^{63} +12.0581i q^{64} +(-0.641819 + 1.54949i) q^{65} +(6.92340 - 6.92340i) q^{66} +9.73489 q^{67} +(-2.94875 - 20.2756i) q^{68} +1.10208 q^{69} +(-7.58250 + 7.58250i) q^{70} +(-0.384985 + 0.929437i) q^{71} +11.7326i q^{72} +(2.69318 + 1.11555i) q^{73} +(-4.31353 - 10.4138i) q^{74} +(-1.95914 + 0.811501i) q^{75} +(0.875070 + 0.875070i) q^{76} +(5.02354 + 5.02354i) q^{77} +(-8.67424 + 3.59299i) q^{78} +(2.60922 + 6.29920i) q^{79} +(-9.93651 - 4.11584i) q^{80} -11.2500i q^{81} +(-5.34310 + 12.8994i) q^{82} +(-2.26046 + 2.26046i) q^{83} -42.8032 q^{84} +(3.99668 + 1.01319i) q^{85} -8.39371 q^{86} +(-0.730053 + 0.730053i) q^{87} +(-5.24658 + 12.6664i) q^{88} -3.30525i q^{89} +(-3.65055 - 1.51211i) q^{90} +(-2.60703 - 6.29392i) q^{91} +(-2.38602 + 0.988323i) q^{92} +(14.2904 + 14.2904i) q^{93} +(11.6896 + 11.6896i) q^{94} +(-0.230080 + 0.0953024i) q^{95} +(-10.3188 - 24.9117i) q^{96} +(1.57147 + 0.650923i) q^{97} -25.0777i q^{98} +(-1.00180 + 2.41856i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.86672 − 1.86672i 1.31997 − 1.31997i 0.406174 − 0.913796i \(-0.366863\pi\)
0.913796 − 0.406174i \(-0.133137\pi\)
\(3\) −0.811501 + 1.95914i −0.468520 + 1.13111i 0.496289 + 0.868157i \(0.334696\pi\)
−0.964809 + 0.262951i \(0.915304\pi\)
\(4\) − 4.96928i − 2.48464i
\(5\) 0.923880 + 0.382683i 0.413171 + 0.171141i
\(6\) 2.14231 + 5.17200i 0.874596 + 2.11146i
\(7\) −3.75274 + 1.55444i −1.41840 + 0.587522i −0.954459 − 0.298343i \(-0.903566\pi\)
−0.463944 + 0.885865i \(0.653566\pi\)
\(8\) −5.54282 − 5.54282i −1.95968 − 1.95968i
\(9\) −1.05836 − 1.05836i −0.352787 − 0.352787i
\(10\) 2.43899 − 1.01026i 0.771275 − 0.319473i
\(11\) −0.669316 − 1.61587i −0.201806 − 0.487203i 0.790282 − 0.612743i \(-0.209934\pi\)
−0.992089 + 0.125539i \(0.959934\pi\)
\(12\) 9.73550 + 4.03258i 2.81040 + 1.16410i
\(13\) 1.67715i 0.465159i 0.972577 + 0.232579i \(0.0747165\pi\)
−0.972577 + 0.232579i \(0.925283\pi\)
\(14\) −4.10362 + 9.90701i −1.09674 + 2.64776i
\(15\) −1.49946 + 1.49946i −0.387158 + 0.387158i
\(16\) −10.7552 −2.68880
\(17\) 4.08018 − 0.593395i 0.989589 − 0.143920i
\(18\) −3.95133 −0.931338
\(19\) −0.176096 + 0.176096i −0.0403992 + 0.0403992i −0.727018 − 0.686619i \(-0.759094\pi\)
0.686619 + 0.727018i \(0.259094\pi\)
\(20\) 1.90166 − 4.59102i 0.425225 − 1.02658i
\(21\) − 8.61356i − 1.87963i
\(22\) −4.26580 − 1.76695i −0.909472 − 0.376716i
\(23\) −0.198886 − 0.480154i −0.0414707 − 0.100119i 0.901787 − 0.432181i \(-0.142256\pi\)
−0.943258 + 0.332062i \(0.892256\pi\)
\(24\) 15.3571 − 6.36113i 3.13476 − 1.29846i
\(25\) 0.707107 + 0.707107i 0.141421 + 0.141421i
\(26\) 3.13077 + 3.13077i 0.613995 + 0.613995i
\(27\) −2.94507 + 1.21989i −0.566779 + 0.234768i
\(28\) 7.72443 + 18.6484i 1.45978 + 3.52422i
\(29\) 0.449816 + 0.186320i 0.0835288 + 0.0345988i 0.424057 − 0.905636i \(-0.360606\pi\)
−0.340528 + 0.940234i \(0.610606\pi\)
\(30\) 5.59814i 1.02208i
\(31\) 3.64713 − 8.80495i 0.655043 − 1.58141i −0.150323 − 0.988637i \(-0.548031\pi\)
0.805366 − 0.592778i \(-0.201969\pi\)
\(32\) −8.99131 + 8.99131i −1.58945 + 1.58945i
\(33\) 3.70886 0.645630
\(34\) 6.50885 − 8.72426i 1.11626 − 1.49620i
\(35\) −4.06194 −0.686593
\(36\) −5.25930 + 5.25930i −0.876550 + 0.876550i
\(37\) 1.63395 − 3.94470i 0.268620 − 0.648505i −0.730799 − 0.682592i \(-0.760852\pi\)
0.999419 + 0.0340875i \(0.0108525\pi\)
\(38\) 0.657443i 0.106651i
\(39\) −3.28577 − 1.36101i −0.526145 − 0.217936i
\(40\) −2.99975 − 7.24204i −0.474302 − 1.14507i
\(41\) −4.88624 + 2.02395i −0.763103 + 0.316087i −0.730075 − 0.683367i \(-0.760515\pi\)
−0.0330276 + 0.999454i \(0.510515\pi\)
\(42\) −16.0791 − 16.0791i −2.48106 − 2.48106i
\(43\) −2.24825 − 2.24825i −0.342855 − 0.342855i 0.514584 − 0.857440i \(-0.327946\pi\)
−0.857440 + 0.514584i \(0.827946\pi\)
\(44\) −8.02972 + 3.32602i −1.21053 + 0.501416i
\(45\) −0.572782 − 1.38282i −0.0853852 − 0.206138i
\(46\) −1.26758 − 0.525048i −0.186894 − 0.0774141i
\(47\) 6.26212i 0.913423i 0.889615 + 0.456712i \(0.150973\pi\)
−0.889615 + 0.456712i \(0.849027\pi\)
\(48\) 8.72786 − 21.0709i 1.25976 − 3.04132i
\(49\) 6.71704 − 6.71704i 0.959578 − 0.959578i
\(50\) 2.63994 0.373344
\(51\) −2.14853 + 8.47517i −0.300854 + 1.18676i
\(52\) 8.33425 1.15575
\(53\) −7.24333 + 7.24333i −0.994948 + 0.994948i −0.999987 − 0.00503923i \(-0.998396\pi\)
0.00503923 + 0.999987i \(0.498396\pi\)
\(54\) −3.22043 + 7.77481i −0.438245 + 1.05802i
\(55\) − 1.74901i − 0.235836i
\(56\) 29.4167 + 12.1848i 3.93097 + 1.62826i
\(57\) −0.202094 − 0.487898i −0.0267680 − 0.0646236i
\(58\) 1.18749 − 0.491874i 0.155925 − 0.0645862i
\(59\) 8.20143 + 8.20143i 1.06773 + 1.06773i 0.997533 + 0.0702021i \(0.0223644\pi\)
0.0702021 + 0.997533i \(0.477636\pi\)
\(60\) 7.45123 + 7.45123i 0.961950 + 0.961950i
\(61\) −4.02196 + 1.66595i −0.514959 + 0.213303i −0.625001 − 0.780624i \(-0.714901\pi\)
0.110042 + 0.993927i \(0.464901\pi\)
\(62\) −9.62820 − 23.2445i −1.22278 − 2.95206i
\(63\) 5.61692 + 2.32660i 0.707665 + 0.293124i
\(64\) 12.0581i 1.50726i
\(65\) −0.641819 + 1.54949i −0.0796078 + 0.192190i
\(66\) 6.92340 − 6.92340i 0.852212 − 0.852212i
\(67\) 9.73489 1.18931 0.594653 − 0.803982i \(-0.297289\pi\)
0.594653 + 0.803982i \(0.297289\pi\)
\(68\) −2.94875 − 20.2756i −0.357588 − 2.45877i
\(69\) 1.10208 0.132675
\(70\) −7.58250 + 7.58250i −0.906282 + 0.906282i
\(71\) −0.384985 + 0.929437i −0.0456894 + 0.110304i −0.945077 − 0.326849i \(-0.894013\pi\)
0.899387 + 0.437153i \(0.144013\pi\)
\(72\) 11.7326i 1.38270i
\(73\) 2.69318 + 1.11555i 0.315213 + 0.130565i 0.534680 − 0.845054i \(-0.320432\pi\)
−0.219467 + 0.975620i \(0.570432\pi\)
\(74\) −4.31353 − 10.4138i −0.501437 − 1.21058i
\(75\) −1.95914 + 0.811501i −0.226222 + 0.0937041i
\(76\) 0.875070 + 0.875070i 0.100377 + 0.100377i
\(77\) 5.02354 + 5.02354i 0.572485 + 0.572485i
\(78\) −8.67424 + 3.59299i −0.982164 + 0.406826i
\(79\) 2.60922 + 6.29920i 0.293560 + 0.708716i 1.00000 0.000942651i \(0.000300055\pi\)
−0.706440 + 0.707773i \(0.749700\pi\)
\(80\) −9.93651 − 4.11584i −1.11094 − 0.460165i
\(81\) − 11.2500i − 1.25000i
\(82\) −5.34310 + 12.8994i −0.590047 + 1.42450i
\(83\) −2.26046 + 2.26046i −0.248118 + 0.248118i −0.820198 − 0.572080i \(-0.806137\pi\)
0.572080 + 0.820198i \(0.306137\pi\)
\(84\) −42.8032 −4.67021
\(85\) 3.99668 + 1.01319i 0.433501 + 0.109896i
\(86\) −8.39371 −0.905118
\(87\) −0.730053 + 0.730053i −0.0782699 + 0.0782699i
\(88\) −5.24658 + 12.6664i −0.559288 + 1.35024i
\(89\) − 3.30525i − 0.350356i −0.984537 − 0.175178i \(-0.943950\pi\)
0.984537 − 0.175178i \(-0.0560501\pi\)
\(90\) −3.65055 − 1.51211i −0.384802 − 0.159390i
\(91\) −2.60703 − 6.29392i −0.273291 − 0.659782i
\(92\) −2.38602 + 0.988323i −0.248760 + 0.103040i
\(93\) 14.2904 + 14.2904i 1.48185 + 1.48185i
\(94\) 11.6896 + 11.6896i 1.20569 + 1.20569i
\(95\) −0.230080 + 0.0953024i −0.0236057 + 0.00977782i
\(96\) −10.3188 − 24.9117i −1.05315 − 2.54254i
\(97\) 1.57147 + 0.650923i 0.159558 + 0.0660912i 0.461033 − 0.887383i \(-0.347479\pi\)
−0.301475 + 0.953474i \(0.597479\pi\)
\(98\) − 25.0777i − 2.53323i
\(99\) −1.00180 + 2.41856i −0.100685 + 0.243074i
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.76.6 yes 24
3.2 odd 2 765.2.be.b.586.1 24
5.2 odd 4 425.2.n.f.399.6 24
5.3 odd 4 425.2.n.c.399.1 24
5.4 even 2 425.2.m.b.76.1 24
17.6 odd 16 1445.2.d.j.866.22 24
17.7 odd 16 1445.2.a.q.1.2 12
17.10 odd 16 1445.2.a.p.1.2 12
17.11 odd 16 1445.2.d.j.866.21 24
17.15 even 8 inner 85.2.l.a.66.6 ✓ 24
51.32 odd 8 765.2.be.b.406.1 24
85.24 odd 16 7225.2.a.bq.1.11 12
85.32 odd 8 425.2.n.c.49.1 24
85.44 odd 16 7225.2.a.bs.1.11 12
85.49 even 8 425.2.m.b.151.1 24
85.83 odd 8 425.2.n.f.49.6 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.l.a.66.6 ✓ 24 17.15 even 8 inner
85.2.l.a.76.6 yes 24 1.1 even 1 trivial
425.2.m.b.76.1 24 5.4 even 2
425.2.m.b.151.1 24 85.49 even 8
425.2.n.c.49.1 24 85.32 odd 8
425.2.n.c.399.1 24 5.3 odd 4
425.2.n.f.49.6 24 85.83 odd 8
425.2.n.f.399.6 24 5.2 odd 4
765.2.be.b.406.1 24 51.32 odd 8
765.2.be.b.586.1 24 3.2 odd 2
1445.2.a.p.1.2 12 17.10 odd 16
1445.2.a.q.1.2 12 17.7 odd 16
1445.2.d.j.866.21 24 17.11 odd 16
1445.2.d.j.866.22 24 17.6 odd 16
7225.2.a.bq.1.11 12 85.24 odd 16
7225.2.a.bs.1.11 12 85.44 odd 16