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Properties

Label 1400.2.bh.b.849.1
Level $1400$
Weight $2$
Character 1400.849
Analytic conductor $11.179$
Analytic rank $0$
Dimension $4$
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] = [1400,2,Mod(249,1400)] 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("1400.249"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1400, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.bh (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 849.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1400.849
Dual form 1400.2.bh.b.249.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 0.500000i) q^{3} +(0.866025 - 2.50000i) q^{7} +(-1.00000 + 1.73205i) q^{9} +(-1.00000 - 1.73205i) q^{11} +(3.46410 - 2.00000i) q^{17} +(-1.00000 + 1.73205i) q^{19} +(0.500000 + 2.59808i) q^{21} +(0.866025 + 0.500000i) q^{23} -5.00000i q^{27} -9.00000 q^{29} +(-2.00000 - 3.46410i) q^{31} +(1.73205 + 1.00000i) q^{33} +(-3.46410 - 2.00000i) q^{37} +1.00000 q^{41} -9.00000i q^{43} +(-5.50000 - 4.33013i) q^{49} +(-2.00000 + 3.46410i) q^{51} +(8.66025 - 5.00000i) q^{53} -2.00000i q^{57} +(-5.00000 - 8.66025i) q^{59} +(-4.50000 + 7.79423i) q^{61} +(3.46410 + 4.00000i) q^{63} +(4.33013 - 2.50000i) q^{67} -1.00000 q^{69} +14.0000 q^{71} +(-10.3923 + 6.00000i) q^{73} +(-5.19615 + 1.00000i) q^{77} +(7.00000 - 12.1244i) q^{79} +(-0.500000 - 0.866025i) q^{81} -11.0000i q^{83} +(7.79423 - 4.50000i) q^{87} +(-7.50000 + 12.9904i) q^{89} +(3.46410 + 2.00000i) q^{93} -18.0000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9} - 4 q^{11} - 4 q^{19} + 2 q^{21} - 36 q^{29} - 8 q^{31} + 4 q^{41} - 22 q^{49} - 8 q^{51} - 20 q^{59} - 18 q^{61} - 4 q^{69} + 56 q^{71} + 28 q^{79} - 2 q^{81} - 30 q^{89} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(-1\)

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
\(3\) −0.866025 + 0.500000i −0.500000 + 0.288675i −0.728714 − 0.684819i \(-0.759881\pi\)
0.228714 + 0.973494i \(0.426548\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.866025 − 2.50000i 0.327327 − 0.944911i
\(8\) 0 0
\(9\) −1.00000 + 1.73205i −0.333333 + 0.577350i
\(10\) 0 0
\(11\) −1.00000 − 1.73205i −0.301511 − 0.522233i 0.674967 − 0.737848i \(-0.264158\pi\)
−0.976478 + 0.215615i \(0.930824\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.46410 − 2.00000i 0.840168 − 0.485071i −0.0171533 − 0.999853i \(-0.505460\pi\)
0.857321 + 0.514782i \(0.172127\pi\)
\(18\) 0 0
\(19\) −1.00000 + 1.73205i −0.229416 + 0.397360i −0.957635 − 0.287984i \(-0.907015\pi\)
0.728219 + 0.685344i \(0.240348\pi\)
\(20\) 0 0
\(21\) 0.500000 + 2.59808i 0.109109 + 0.566947i
\(22\) 0 0
\(23\) 0.866025 + 0.500000i 0.180579 + 0.104257i 0.587565 − 0.809177i \(-0.300087\pi\)
−0.406986 + 0.913434i \(0.633420\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 5.00000i − 0.962250i
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 − 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) −2.00000 − 3.46410i −0.359211 − 0.622171i 0.628619 − 0.777714i \(-0.283621\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 1.73205 + 1.00000i 0.301511 + 0.174078i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.46410 − 2.00000i −0.569495 − 0.328798i 0.187453 − 0.982274i \(-0.439977\pi\)
−0.756948 + 0.653476i \(0.773310\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.00000 0.156174 0.0780869 − 0.996947i \(-0.475119\pi\)
0.0780869 + 0.996947i \(0.475119\pi\)
\(42\) 0 0
\(43\) − 9.00000i − 1.37249i −0.727372 − 0.686244i \(-0.759258\pi\)
0.727372 − 0.686244i \(-0.240742\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) 0 0
\(49\) −5.50000 − 4.33013i −0.785714 − 0.618590i
\(50\) 0 0
\(51\) −2.00000 + 3.46410i −0.280056 + 0.485071i
\(52\) 0 0
\(53\) 8.66025 − 5.00000i 1.18958 − 0.686803i 0.231367 − 0.972867i \(-0.425680\pi\)
0.958211 + 0.286064i \(0.0923469\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 2.00000i − 0.264906i
\(58\) 0 0
\(59\) −5.00000 − 8.66025i −0.650945 − 1.12747i −0.982894 − 0.184172i \(-0.941040\pi\)
0.331949 − 0.943297i \(-0.392294\pi\)
\(60\) 0 0
\(61\) −4.50000 + 7.79423i −0.576166 + 0.997949i 0.419748 + 0.907641i \(0.362118\pi\)
−0.995914 + 0.0903080i \(0.971215\pi\)
\(62\) 0 0
\(63\) 3.46410 + 4.00000i 0.436436 + 0.503953i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.33013 − 2.50000i 0.529009 − 0.305424i −0.211604 − 0.977356i \(-0.567869\pi\)
0.740613 + 0.671932i \(0.234535\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 14.0000 1.66149 0.830747 − 0.556650i \(-0.187914\pi\)
0.830747 + 0.556650i \(0.187914\pi\)
\(72\) 0 0
\(73\) −10.3923 + 6.00000i −1.21633 + 0.702247i −0.964130 − 0.265429i \(-0.914486\pi\)
−0.252197 + 0.967676i \(0.581153\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.19615 + 1.00000i −0.592157 + 0.113961i
\(78\) 0 0
\(79\) 7.00000 − 12.1244i 0.787562 − 1.36410i −0.139895 − 0.990166i \(-0.544677\pi\)
0.927457 − 0.373930i \(-0.121990\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) − 11.0000i − 1.20741i −0.797209 − 0.603703i \(-0.793691\pi\)
0.797209 − 0.603703i \(-0.206309\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 7.79423 − 4.50000i 0.835629 − 0.482451i
\(88\) 0 0
\(89\) −7.50000 + 12.9904i −0.794998 + 1.37698i 0.127842 + 0.991795i \(0.459195\pi\)
−0.922840 + 0.385183i \(0.874138\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 3.46410 + 2.00000i 0.359211 + 0.207390i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 18.0000i − 1.82762i −0.406138 − 0.913812i \(-0.633125\pi\)
0.406138 − 0.913812i \(-0.366875\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
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 1400.2.bh.b.849.1 4
5.2 odd 4 1400.2.q.e.401.1 2
5.3 odd 4 280.2.q.a.121.1 yes 2
5.4 even 2 inner 1400.2.bh.b.849.2 4
7.4 even 3 inner 1400.2.bh.b.249.2 4
15.8 even 4 2520.2.bi.a.1801.1 2
20.3 even 4 560.2.q.h.401.1 2
35.2 odd 12 9800.2.a.r.1.1 1
35.3 even 12 1960.2.q.k.361.1 2
35.4 even 6 inner 1400.2.bh.b.249.1 4
35.12 even 12 9800.2.a.bc.1.1 1
35.13 even 4 1960.2.q.k.961.1 2
35.18 odd 12 280.2.q.a.81.1 ✓ 2
35.23 odd 12 1960.2.a.i.1.1 1
35.32 odd 12 1400.2.q.e.1201.1 2
35.33 even 12 1960.2.a.e.1.1 1
105.53 even 12 2520.2.bi.a.361.1 2
140.23 even 12 3920.2.a.m.1.1 1
140.103 odd 12 3920.2.a.y.1.1 1
140.123 even 12 560.2.q.h.81.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.q.a.81.1 ✓ 2 35.18 odd 12
280.2.q.a.121.1 yes 2 5.3 odd 4
560.2.q.h.81.1 2 140.123 even 12
560.2.q.h.401.1 2 20.3 even 4
1400.2.q.e.401.1 2 5.2 odd 4
1400.2.q.e.1201.1 2 35.32 odd 12
1400.2.bh.b.249.1 4 35.4 even 6 inner
1400.2.bh.b.249.2 4 7.4 even 3 inner
1400.2.bh.b.849.1 4 1.1 even 1 trivial
1400.2.bh.b.849.2 4 5.4 even 2 inner
1960.2.a.e.1.1 1 35.33 even 12
1960.2.a.i.1.1 1 35.23 odd 12
1960.2.q.k.361.1 2 35.3 even 12
1960.2.q.k.961.1 2 35.13 even 4
2520.2.bi.a.361.1 2 105.53 even 12
2520.2.bi.a.1801.1 2 15.8 even 4
3920.2.a.m.1.1 1 140.23 even 12
3920.2.a.y.1.1 1 140.103 odd 12
9800.2.a.r.1.1 1 35.2 odd 12
9800.2.a.bc.1.1 1 35.12 even 12