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Properties

Label 1360.2.e.d.1089.1
Level $1360$
Weight $2$
Character 1360.1089
Analytic conductor $10.860$
Analytic rank $0$
Dimension $8$
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] = [1360,2,Mod(1089,1360)] 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("1360.1089"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1360 = 2^{4} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1360.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,-2,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.8596546749\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.619810816.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1089.1
Root \(-0.252709 - 0.252709i\) of defining polynomial
Character \(\chi\) \(=\) 1360.1089
Dual form 1360.2.e.d.1089.8

$q$-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)
 
\(f(q)\) \(=\) \(q-2.87228i q^{3} +(0.146426 + 2.23127i) q^{5} +1.42058i q^{7} -5.24997 q^{9} -0.0740063 q^{11} -5.70167i q^{13} +(6.40882 - 0.420575i) q^{15} +1.00000i q^{17} -4.90340 q^{19} +4.08029 q^{21} -3.88311i q^{23} +(-4.95712 + 0.653431i) q^{25} +6.46254i q^{27} -5.91424 q^{29} -0.388531 q^{31} +0.212567i q^{33} +(-3.16969 + 0.208009i) q^{35} -9.91424i q^{37} -16.3768 q^{39} -6.61055 q^{41} -6.94628i q^{43} +(-0.768731 - 11.7141i) q^{45} -5.70167i q^{47} +4.98197 q^{49} +2.87228 q^{51} -0.0216729i q^{53} +(-0.0108364 - 0.165128i) q^{55} +14.0839i q^{57} -2.00000 q^{59} -3.47337 q^{61} -7.45798i q^{63} +(12.7220 - 0.834872i) q^{65} -6.71251i q^{67} -11.1534 q^{69} -3.84023 q^{71} +13.5356i q^{73} +(1.87683 + 14.2382i) q^{75} -0.105132i q^{77} -1.06000 q^{79} +2.81228 q^{81} +11.8497i q^{83} +(-2.23127 + 0.146426i) q^{85} +16.9873i q^{87} +1.99452 q^{89} +8.09965 q^{91} +1.11597i q^{93} +(-0.717985 - 10.9408i) q^{95} -5.34109i q^{97} +0.388531 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{5} - 8 q^{9} + 4 q^{11} + 8 q^{19} + 24 q^{21} - 12 q^{25} + 8 q^{29} + 24 q^{31} - 44 q^{39} - 12 q^{41} - 22 q^{45} - 16 q^{49} + 8 q^{51} + 8 q^{55} - 16 q^{59} + 12 q^{61} + 20 q^{65} - 8 q^{69}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1360\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(341\) \(511\) \(817\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 2.87228i − 1.65831i −0.559019 − 0.829155i \(-0.688822\pi\)
0.559019 − 0.829155i \(-0.311178\pi\)
\(4\) 0 0
\(5\) 0.146426 + 2.23127i 0.0654836 + 0.997854i
\(6\) 0 0
\(7\) 1.42058i 0.536927i 0.963290 + 0.268464i \(0.0865159\pi\)
−0.963290 + 0.268464i \(0.913484\pi\)
\(8\) 0 0
\(9\) −5.24997 −1.74999
\(10\) 0 0
\(11\) −0.0740063 −0.0223137 −0.0111569 − 0.999938i \(-0.503551\pi\)
−0.0111569 + 0.999938i \(0.503551\pi\)
\(12\) 0 0
\(13\) − 5.70167i − 1.58136i −0.612230 − 0.790680i \(-0.709727\pi\)
0.612230 − 0.790680i \(-0.290273\pi\)
\(14\) 0 0
\(15\) 6.40882 − 0.420575i 1.65475 − 0.108592i
\(16\) 0 0
\(17\) 1.00000i 0.242536i
\(18\) 0 0
\(19\) −4.90340 −1.12492 −0.562459 − 0.826825i \(-0.690144\pi\)
−0.562459 + 0.826825i \(0.690144\pi\)
\(20\) 0 0
\(21\) 4.08029 0.890391
\(22\) 0 0
\(23\) − 3.88311i − 0.809685i −0.914386 − 0.404842i \(-0.867326\pi\)
0.914386 − 0.404842i \(-0.132674\pi\)
\(24\) 0 0
\(25\) −4.95712 + 0.653431i −0.991424 + 0.130686i
\(26\) 0 0
\(27\) 6.46254i 1.24372i
\(28\) 0 0
\(29\) −5.91424 −1.09825 −0.549123 − 0.835741i \(-0.685038\pi\)
−0.549123 + 0.835741i \(0.685038\pi\)
\(30\) 0 0
\(31\) −0.388531 −0.0697822 −0.0348911 − 0.999391i \(-0.511108\pi\)
−0.0348911 + 0.999391i \(0.511108\pi\)
\(32\) 0 0
\(33\) 0.212567i 0.0370031i
\(34\) 0 0
\(35\) −3.16969 + 0.208009i −0.535775 + 0.0351599i
\(36\) 0 0
\(37\) − 9.91424i − 1.62989i −0.579538 − 0.814945i \(-0.696767\pi\)
0.579538 − 0.814945i \(-0.303233\pi\)
\(38\) 0 0
\(39\) −16.3768 −2.62238
\(40\) 0 0
\(41\) −6.61055 −1.03239 −0.516197 − 0.856470i \(-0.672653\pi\)
−0.516197 + 0.856470i \(0.672653\pi\)
\(42\) 0 0
\(43\) − 6.94628i − 1.05930i −0.848217 − 0.529649i \(-0.822324\pi\)
0.848217 − 0.529649i \(-0.177676\pi\)
\(44\) 0 0
\(45\) −0.768731 − 11.7141i −0.114596 − 1.74623i
\(46\) 0 0
\(47\) − 5.70167i − 0.831674i −0.909439 − 0.415837i \(-0.863489\pi\)
0.909439 − 0.415837i \(-0.136511\pi\)
\(48\) 0 0
\(49\) 4.98197 0.711709
\(50\) 0 0
\(51\) 2.87228 0.402199
\(52\) 0 0
\(53\) − 0.0216729i − 0.00297700i −0.999999 − 0.00148850i \(-0.999526\pi\)
0.999999 − 0.00148850i \(-0.000473804\pi\)
\(54\) 0 0
\(55\) −0.0108364 − 0.165128i −0.00146118 − 0.0222658i
\(56\) 0 0
\(57\) 14.0839i 1.86546i
\(58\) 0 0
\(59\) −2.00000 −0.260378 −0.130189 − 0.991489i \(-0.541558\pi\)
−0.130189 + 0.991489i \(0.541558\pi\)
\(60\) 0 0
\(61\) −3.47337 −0.444720 −0.222360 − 0.974965i \(-0.571376\pi\)
−0.222360 + 0.974965i \(0.571376\pi\)
\(62\) 0 0
\(63\) − 7.45798i − 0.939617i
\(64\) 0 0
\(65\) 12.7220 − 0.834872i 1.57796 − 0.103553i
\(66\) 0 0
\(67\) − 6.71251i − 0.820063i −0.912071 − 0.410032i \(-0.865518\pi\)
0.912071 − 0.410032i \(-0.134482\pi\)
\(68\) 0 0
\(69\) −11.1534 −1.34271
\(70\) 0 0
\(71\) −3.84023 −0.455752 −0.227876 − 0.973690i \(-0.573178\pi\)
−0.227876 + 0.973690i \(0.573178\pi\)
\(72\) 0 0
\(73\) 13.5356i 1.58422i 0.610375 + 0.792112i \(0.291019\pi\)
−0.610375 + 0.792112i \(0.708981\pi\)
\(74\) 0 0
\(75\) 1.87683 + 14.2382i 0.216718 + 1.64409i
\(76\) 0 0
\(77\) − 0.105132i − 0.0119808i
\(78\) 0 0
\(79\) −1.06000 −0.119259 −0.0596295 − 0.998221i \(-0.518992\pi\)
−0.0596295 + 0.998221i \(0.518992\pi\)
\(80\) 0 0
\(81\) 2.81228 0.312476
\(82\) 0 0
\(83\) 11.8497i 1.30067i 0.759647 + 0.650336i \(0.225372\pi\)
−0.759647 + 0.650336i \(0.774628\pi\)
\(84\) 0 0
\(85\) −2.23127 + 0.146426i −0.242015 + 0.0158821i
\(86\) 0 0
\(87\) 16.9873i 1.82123i
\(88\) 0 0
\(89\) 1.99452 0.211419 0.105710 − 0.994397i \(-0.466289\pi\)
0.105710 + 0.994397i \(0.466289\pi\)
\(90\) 0 0
\(91\) 8.09965 0.849075
\(92\) 0 0
\(93\) 1.11597i 0.115720i
\(94\) 0 0
\(95\) −0.717985 − 10.9408i −0.0736637 − 1.12250i
\(96\) 0 0
\(97\) − 5.34109i − 0.542306i −0.962536 − 0.271153i \(-0.912595\pi\)
0.962536 − 0.271153i \(-0.0874049\pi\)
\(98\) 0 0
\(99\) 0.388531 0.0390488
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 1360.2.e.d.1089.1 8
4.3 odd 2 85.2.b.a.69.3 ✓ 8
5.2 odd 4 6800.2.a.bt.1.1 4
5.3 odd 4 6800.2.a.bw.1.4 4
5.4 even 2 inner 1360.2.e.d.1089.8 8
12.11 even 2 765.2.b.c.154.6 8
20.3 even 4 425.2.a.g.1.2 4
20.7 even 4 425.2.a.h.1.3 4
20.19 odd 2 85.2.b.a.69.6 yes 8
60.23 odd 4 3825.2.a.bj.1.3 4
60.47 odd 4 3825.2.a.bh.1.2 4
60.59 even 2 765.2.b.c.154.3 8
68.67 odd 2 1445.2.b.e.579.3 8
340.67 even 4 7225.2.a.w.1.3 4
340.203 even 4 7225.2.a.v.1.2 4
340.339 odd 2 1445.2.b.e.579.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.b.a.69.3 ✓ 8 4.3 odd 2
85.2.b.a.69.6 yes 8 20.19 odd 2
425.2.a.g.1.2 4 20.3 even 4
425.2.a.h.1.3 4 20.7 even 4
765.2.b.c.154.3 8 60.59 even 2
765.2.b.c.154.6 8 12.11 even 2
1360.2.e.d.1089.1 8 1.1 even 1 trivial
1360.2.e.d.1089.8 8 5.4 even 2 inner
1445.2.b.e.579.3 8 68.67 odd 2
1445.2.b.e.579.6 8 340.339 odd 2
3825.2.a.bh.1.2 4 60.47 odd 4
3825.2.a.bj.1.3 4 60.23 odd 4
6800.2.a.bt.1.1 4 5.2 odd 4
6800.2.a.bw.1.4 4 5.3 odd 4
7225.2.a.v.1.2 4 340.203 even 4
7225.2.a.w.1.3 4 340.67 even 4