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Properties

Label 1352.2.o.e.361.4
Level $1352$
Weight $2$
Character 1352.361
Analytic conductor $10.796$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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] = [1352,2,Mod(361,1352)] 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("1352.361"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1352, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1352 = 2^{3} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1352.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,2,0,0,0,0,0,-6,0,0,0,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.7957743533\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.1731891456.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 104)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 361.4
Root \(-2.21837 + 1.28078i\) of defining polynomial
Character \(\chi\) \(=\) 1352.361
Dual form 1352.2.o.e.1161.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.28078 + 2.21837i) q^{3} +2.56155i q^{5} +(-3.95042 - 2.28078i) q^{7} +(-1.78078 + 3.08440i) q^{9} +(-2.70469 + 1.56155i) q^{11} +(-5.68247 + 3.28078i) q^{15} +(-0.280776 + 0.486319i) q^{17} +(-1.73205 - 1.00000i) q^{19} -11.6847i q^{21} +(-2.56155 - 4.43674i) q^{23} -1.56155 q^{25} -1.43845 q^{27} +(-1.56155 - 2.70469i) q^{29} +3.12311i q^{31} +(-6.92820 - 4.00000i) q^{33} +(5.84233 - 10.1192i) q^{35} +(4.70983 - 2.71922i) q^{37} +(-6.92820 + 4.00000i) q^{41} +(-2.71922 + 4.70983i) q^{43} +(-7.90084 - 4.56155i) q^{45} -3.43845i q^{47} +(6.90388 + 11.9579i) q^{49} -1.43845 q^{51} -4.24621 q^{53} +(-4.00000 - 6.92820i) q^{55} -5.12311i q^{57} +(-8.66025 - 5.00000i) q^{59} +(-5.56155 + 9.63289i) q^{61} +(14.0696 - 8.12311i) q^{63} +(-0.759413 + 0.438447i) q^{67} +(6.56155 - 11.3649i) q^{69} +(4.92306 + 2.84233i) q^{71} +15.3693i q^{73} +(-2.00000 - 3.46410i) q^{75} +14.2462 q^{77} -2.87689 q^{79} +(3.50000 + 6.06218i) q^{81} -8.24621i q^{83} +(-1.24573 - 0.719224i) q^{85} +(4.00000 - 6.92820i) q^{87} +(4.43674 - 2.56155i) q^{89} +(-6.92820 + 4.00000i) q^{93} +(2.56155 - 4.43674i) q^{95} +(8.87348 + 5.12311i) q^{97} -11.1231i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} - 6 q^{9} + 6 q^{17} - 4 q^{23} + 4 q^{25} - 28 q^{27} + 4 q^{29} + 22 q^{35} - 30 q^{43} + 14 q^{49} - 28 q^{51} + 32 q^{53} - 32 q^{55} - 28 q^{61} + 36 q^{69} - 16 q^{75} + 48 q^{77} - 56 q^{79}+ \cdots + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1352\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1015\) \(1185\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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 0
\(3\) 1.28078 + 2.21837i 0.739457 + 1.28078i 0.952740 + 0.303786i \(0.0982508\pi\)
−0.213284 + 0.976990i \(0.568416\pi\)
\(4\) 0 0
\(5\) 2.56155i 1.14556i 0.819709 + 0.572781i \(0.194135\pi\)
−0.819709 + 0.572781i \(0.805865\pi\)
\(6\) 0 0
\(7\) −3.95042 − 2.28078i −1.49312 − 0.862052i −0.493150 − 0.869944i \(-0.664155\pi\)
−0.999969 + 0.00789196i \(0.997488\pi\)
\(8\) 0 0
\(9\) −1.78078 + 3.08440i −0.593592 + 1.02813i
\(10\) 0 0
\(11\) −2.70469 + 1.56155i −0.815494 + 0.470826i −0.848860 − 0.528617i \(-0.822711\pi\)
0.0333659 + 0.999443i \(0.489377\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −5.68247 + 3.28078i −1.46721 + 0.847093i
\(16\) 0 0
\(17\) −0.280776 + 0.486319i −0.0680983 + 0.117950i −0.898064 − 0.439864i \(-0.855026\pi\)
0.829966 + 0.557814i \(0.188360\pi\)
\(18\) 0 0
\(19\) −1.73205 − 1.00000i −0.397360 − 0.229416i 0.287984 − 0.957635i \(-0.407015\pi\)
−0.685344 + 0.728219i \(0.740348\pi\)
\(20\) 0 0
\(21\) − 11.6847i − 2.54980i
\(22\) 0 0
\(23\) −2.56155 − 4.43674i −0.534121 − 0.925124i −0.999205 − 0.0398580i \(-0.987309\pi\)
0.465085 − 0.885266i \(-0.346024\pi\)
\(24\) 0 0
\(25\) −1.56155 −0.312311
\(26\) 0 0
\(27\) −1.43845 −0.276829
\(28\) 0 0
\(29\) −1.56155 − 2.70469i −0.289973 − 0.502248i 0.683830 − 0.729641i \(-0.260313\pi\)
−0.973803 + 0.227393i \(0.926980\pi\)
\(30\) 0 0
\(31\) 3.12311i 0.560926i 0.959865 + 0.280463i \(0.0904881\pi\)
−0.959865 + 0.280463i \(0.909512\pi\)
\(32\) 0 0
\(33\) −6.92820 − 4.00000i −1.20605 − 0.696311i
\(34\) 0 0
\(35\) 5.84233 − 10.1192i 0.987534 − 1.71046i
\(36\) 0 0
\(37\) 4.70983 − 2.71922i 0.774292 − 0.447038i −0.0601117 − 0.998192i \(-0.519146\pi\)
0.834404 + 0.551154i \(0.185812\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.92820 + 4.00000i −1.08200 + 0.624695i −0.931436 − 0.363905i \(-0.881443\pi\)
−0.150567 + 0.988600i \(0.548110\pi\)
\(42\) 0 0
\(43\) −2.71922 + 4.70983i −0.414678 + 0.718243i −0.995395 − 0.0958627i \(-0.969439\pi\)
0.580717 + 0.814106i \(0.302772\pi\)
\(44\) 0 0
\(45\) −7.90084 − 4.56155i −1.17779 − 0.679996i
\(46\) 0 0
\(47\) − 3.43845i − 0.501549i −0.968046 − 0.250775i \(-0.919315\pi\)
0.968046 − 0.250775i \(-0.0806853\pi\)
\(48\) 0 0
\(49\) 6.90388 + 11.9579i 0.986269 + 1.70827i
\(50\) 0 0
\(51\) −1.43845 −0.201423
\(52\) 0 0
\(53\) −4.24621 −0.583262 −0.291631 − 0.956531i \(-0.594198\pi\)
−0.291631 + 0.956531i \(0.594198\pi\)
\(54\) 0 0
\(55\) −4.00000 − 6.92820i −0.539360 − 0.934199i
\(56\) 0 0
\(57\) − 5.12311i − 0.678572i
\(58\) 0 0
\(59\) −8.66025 − 5.00000i −1.12747 − 0.650945i −0.184172 − 0.982894i \(-0.558960\pi\)
−0.943297 + 0.331949i \(0.892294\pi\)
\(60\) 0 0
\(61\) −5.56155 + 9.63289i −0.712084 + 1.23337i 0.251990 + 0.967730i \(0.418915\pi\)
−0.964074 + 0.265636i \(0.914418\pi\)
\(62\) 0 0
\(63\) 14.0696 − 8.12311i 1.77261 − 1.02342i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −0.759413 + 0.438447i −0.0927770 + 0.0535648i −0.545671 − 0.838000i \(-0.683725\pi\)
0.452894 + 0.891565i \(0.350392\pi\)
\(68\) 0 0
\(69\) 6.56155 − 11.3649i 0.789918 − 1.36818i
\(70\) 0 0
\(71\) 4.92306 + 2.84233i 0.584260 + 0.337322i 0.762824 − 0.646606i \(-0.223812\pi\)
−0.178565 + 0.983928i \(0.557145\pi\)
\(72\) 0 0
\(73\) 15.3693i 1.79884i 0.437083 + 0.899421i \(0.356012\pi\)
−0.437083 + 0.899421i \(0.643988\pi\)
\(74\) 0 0
\(75\) −2.00000 − 3.46410i −0.230940 − 0.400000i
\(76\) 0 0
\(77\) 14.2462 1.62351
\(78\) 0 0
\(79\) −2.87689 −0.323676 −0.161838 − 0.986817i \(-0.551742\pi\)
−0.161838 + 0.986817i \(0.551742\pi\)
\(80\) 0 0
\(81\) 3.50000 + 6.06218i 0.388889 + 0.673575i
\(82\) 0 0
\(83\) − 8.24621i − 0.905139i −0.891729 − 0.452570i \(-0.850507\pi\)
0.891729 − 0.452570i \(-0.149493\pi\)
\(84\) 0 0
\(85\) −1.24573 − 0.719224i −0.135119 − 0.0780108i
\(86\) 0 0
\(87\) 4.00000 − 6.92820i 0.428845 − 0.742781i
\(88\) 0 0
\(89\) 4.43674 − 2.56155i 0.470293 − 0.271524i −0.246069 − 0.969252i \(-0.579139\pi\)
0.716363 + 0.697728i \(0.245806\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −6.92820 + 4.00000i −0.718421 + 0.414781i
\(94\) 0 0
\(95\) 2.56155 − 4.43674i 0.262810 − 0.455200i
\(96\) 0 0
\(97\) 8.87348 + 5.12311i 0.900965 + 0.520173i 0.877513 − 0.479552i \(-0.159201\pi\)
0.0234521 + 0.999725i \(0.492534\pi\)
\(98\) 0 0
\(99\) − 11.1231i − 1.11791i
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 1352.2.o.e.361.4 8
13.2 odd 12 1352.2.a.e.1.1 2
13.3 even 3 104.2.f.a.25.2 yes 4
13.4 even 6 inner 1352.2.o.e.1161.3 8
13.5 odd 4 1352.2.i.h.1329.2 4
13.6 odd 12 1352.2.i.h.529.2 4
13.7 odd 12 1352.2.i.g.529.2 4
13.8 odd 4 1352.2.i.g.1329.2 4
13.9 even 3 inner 1352.2.o.e.1161.4 8
13.10 even 6 104.2.f.a.25.1 ✓ 4
13.11 odd 12 1352.2.a.d.1.1 2
13.12 even 2 inner 1352.2.o.e.361.3 8
39.23 odd 6 936.2.c.d.649.4 4
39.29 odd 6 936.2.c.d.649.1 4
52.3 odd 6 208.2.f.b.129.4 4
52.11 even 12 2704.2.a.s.1.2 2
52.15 even 12 2704.2.a.t.1.2 2
52.23 odd 6 208.2.f.b.129.3 4
65.3 odd 12 2600.2.f.b.649.1 4
65.23 odd 12 2600.2.f.a.649.1 4
65.29 even 6 2600.2.k.a.2001.3 4
65.42 odd 12 2600.2.f.a.649.4 4
65.49 even 6 2600.2.k.a.2001.4 4
65.62 odd 12 2600.2.f.b.649.4 4
104.3 odd 6 832.2.f.f.129.1 4
104.29 even 6 832.2.f.i.129.3 4
104.75 odd 6 832.2.f.f.129.2 4
104.101 even 6 832.2.f.i.129.4 4
156.23 even 6 1872.2.c.j.1585.4 4
156.107 even 6 1872.2.c.j.1585.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
104.2.f.a.25.1 ✓ 4 13.10 even 6
104.2.f.a.25.2 yes 4 13.3 even 3
208.2.f.b.129.3 4 52.23 odd 6
208.2.f.b.129.4 4 52.3 odd 6
832.2.f.f.129.1 4 104.3 odd 6
832.2.f.f.129.2 4 104.75 odd 6
832.2.f.i.129.3 4 104.29 even 6
832.2.f.i.129.4 4 104.101 even 6
936.2.c.d.649.1 4 39.29 odd 6
936.2.c.d.649.4 4 39.23 odd 6
1352.2.a.d.1.1 2 13.11 odd 12
1352.2.a.e.1.1 2 13.2 odd 12
1352.2.i.g.529.2 4 13.7 odd 12
1352.2.i.g.1329.2 4 13.8 odd 4
1352.2.i.h.529.2 4 13.6 odd 12
1352.2.i.h.1329.2 4 13.5 odd 4
1352.2.o.e.361.3 8 13.12 even 2 inner
1352.2.o.e.361.4 8 1.1 even 1 trivial
1352.2.o.e.1161.3 8 13.4 even 6 inner
1352.2.o.e.1161.4 8 13.9 even 3 inner
1872.2.c.j.1585.1 4 156.107 even 6
1872.2.c.j.1585.4 4 156.23 even 6
2600.2.f.a.649.1 4 65.23 odd 12
2600.2.f.a.649.4 4 65.42 odd 12
2600.2.f.b.649.1 4 65.3 odd 12
2600.2.f.b.649.4 4 65.62 odd 12
2600.2.k.a.2001.3 4 65.29 even 6
2600.2.k.a.2001.4 4 65.49 even 6
2704.2.a.s.1.2 2 52.11 even 12
2704.2.a.t.1.2 2 52.15 even 12