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Properties

Label 1400.2.bh.i.849.6
Level $1400$
Weight $2$
Character 1400.849
Analytic conductor $11.179$
Analytic rank $0$
Dimension $12$
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 = [12,0,0,0,0,0,0,0,18,0,-6] 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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.32905425960566784.37
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 36x^{10} + 432x^{8} + 2040x^{6} + 3780x^{4} + 2592x^{2} + 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 849.6
Root \(-2.22358i\) of defining polynomial
Character \(\chi\) \(=\) 1400.849
Dual form 1400.2.bh.i.249.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.84918 - 1.64497i) q^{3} +(-0.0641892 + 2.64497i) q^{7} +(3.91187 - 6.77556i) q^{9} +(-2.91187 - 5.04351i) q^{11} -2.75615i q^{13} +(1.73205 - 1.00000i) q^{17} +(-0.378076 + 0.654846i) q^{19} +(4.16802 + 7.64158i) q^{21} +(-0.462279 - 0.266897i) q^{23} -15.8698i q^{27} +0.823739 q^{29} +(1.28995 + 2.23425i) q^{31} +(-16.5929 - 9.57989i) q^{33} +(-4.11895 - 2.37808i) q^{37} +(-4.53379 - 7.85276i) q^{39} +6.06759 q^{41} -0.710055i q^{43} +(11.1642 + 6.44566i) q^{47} +(-6.99176 - 0.339557i) q^{49} +(3.28995 - 5.69835i) q^{51} +(7.27776 - 4.20181i) q^{53} +2.48770i q^{57} +(4.00000 + 6.92820i) q^{59} +(-4.70181 + 8.14378i) q^{61} +(17.6701 + 10.7817i) q^{63} +(10.2796 - 5.93492i) q^{67} -1.75615 q^{69} +(-3.04174 + 1.75615i) q^{73} +(13.5268 - 7.37808i) q^{77} +(-4.75615 + 8.23790i) q^{79} +(-14.3698 - 24.8893i) q^{81} +6.71005i q^{83} +(2.34698 - 1.35503i) q^{87} +(0.878076 - 1.52087i) q^{89} +(7.28995 + 0.176915i) q^{91} +(7.35056 + 4.24385i) q^{93} -2.00000i q^{97} -45.5634 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 18 q^{9} - 6 q^{11} + 6 q^{19} - 48 q^{29} - 24 q^{31} - 36 q^{39} + 36 q^{41} + 24 q^{49} + 48 q^{59} + 12 q^{61} - 36 q^{79} - 54 q^{81} + 48 q^{91} - 252 q^{99}+O(q^{100}) \) 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
\(3\) 2.84918 − 1.64497i 1.64497 − 0.949725i 0.665944 − 0.746002i \(-0.268029\pi\)
0.979028 − 0.203724i \(-0.0653044\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −0.0641892 + 2.64497i −0.0242612 + 0.999706i
\(8\) 0 0
\(9\) 3.91187 − 6.77556i 1.30396 − 2.25852i
\(10\) 0 0
\(11\) −2.91187 − 5.04351i −0.877962 − 1.52067i −0.853574 − 0.520972i \(-0.825570\pi\)
−0.0243876 − 0.999703i \(-0.507764\pi\)
\(12\) 0 0
\(13\) − 2.75615i − 0.764419i −0.924076 − 0.382209i \(-0.875163\pi\)
0.924076 − 0.382209i \(-0.124837\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.73205 − 1.00000i 0.420084 − 0.242536i −0.275029 − 0.961436i \(-0.588688\pi\)
0.695113 + 0.718900i \(0.255354\pi\)
\(18\) 0 0
\(19\) −0.378076 + 0.654846i −0.0867365 + 0.150232i −0.906130 − 0.423000i \(-0.860977\pi\)
0.819393 + 0.573232i \(0.194310\pi\)
\(20\) 0 0
\(21\) 4.16802 + 7.64158i 0.909537 + 1.66753i
\(22\) 0 0
\(23\) −0.462279 − 0.266897i −0.0963918 − 0.0556518i 0.451029 − 0.892509i \(-0.351057\pi\)
−0.547421 + 0.836857i \(0.684390\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 15.8698i − 3.05415i
\(28\) 0 0
\(29\) 0.823739 0.152964 0.0764822 − 0.997071i \(-0.475631\pi\)
0.0764822 + 0.997071i \(0.475631\pi\)
\(30\) 0 0
\(31\) 1.28995 + 2.23425i 0.231681 + 0.401283i 0.958303 − 0.285754i \(-0.0922441\pi\)
−0.726622 + 0.687038i \(0.758911\pi\)
\(32\) 0 0
\(33\) −16.5929 − 9.57989i −2.88845 − 1.66764i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.11895 − 2.37808i −0.677151 − 0.390953i 0.121630 − 0.992576i \(-0.461188\pi\)
−0.798781 + 0.601622i \(0.794521\pi\)
\(38\) 0 0
\(39\) −4.53379 − 7.85276i −0.725988 − 1.25745i
\(40\) 0 0
\(41\) 6.06759 0.947598 0.473799 − 0.880633i \(-0.342882\pi\)
0.473799 + 0.880633i \(0.342882\pi\)
\(42\) 0 0
\(43\) − 0.710055i − 0.108282i −0.998533 − 0.0541412i \(-0.982758\pi\)
0.998533 − 0.0541412i \(-0.0172421\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.1642 + 6.44566i 1.62847 + 0.940197i 0.984550 + 0.175102i \(0.0560255\pi\)
0.643918 + 0.765095i \(0.277308\pi\)
\(48\) 0 0
\(49\) −6.99176 − 0.339557i −0.998823 − 0.0485082i
\(50\) 0 0
\(51\) 3.28995 − 5.69835i 0.460684 − 0.797929i
\(52\) 0 0
\(53\) 7.27776 − 4.20181i 0.999677 − 0.577164i 0.0915241 − 0.995803i \(-0.470826\pi\)
0.908153 + 0.418639i \(0.137493\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.48770i 0.329504i
\(58\) 0 0
\(59\) 4.00000 + 6.92820i 0.520756 + 0.901975i 0.999709 + 0.0241347i \(0.00768307\pi\)
−0.478953 + 0.877841i \(0.658984\pi\)
\(60\) 0 0
\(61\) −4.70181 + 8.14378i −0.602006 + 1.04270i 0.390511 + 0.920598i \(0.372298\pi\)
−0.992517 + 0.122106i \(0.961035\pi\)
\(62\) 0 0
\(63\) 17.6701 + 10.7817i 2.22622 + 1.35837i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 10.2796 − 5.93492i 1.25585 − 0.725066i 0.283585 − 0.958947i \(-0.408476\pi\)
0.972265 + 0.233881i \(0.0751428\pi\)
\(68\) 0 0
\(69\) −1.75615 −0.211416
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −3.04174 + 1.75615i −0.356009 + 0.205542i −0.667329 − 0.744763i \(-0.732562\pi\)
0.311320 + 0.950305i \(0.399229\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 13.5268 − 7.37808i 1.54153 − 0.840810i
\(78\) 0 0
\(79\) −4.75615 + 8.23790i −0.535109 + 0.926836i 0.464049 + 0.885809i \(0.346396\pi\)
−0.999158 + 0.0410263i \(0.986937\pi\)
\(80\) 0 0
\(81\) −14.3698 − 24.8893i −1.59665 − 2.76548i
\(82\) 0 0
\(83\) 6.71005i 0.736524i 0.929722 + 0.368262i \(0.120047\pi\)
−0.929722 + 0.368262i \(0.879953\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.34698 − 1.35503i 0.251622 − 0.145274i
\(88\) 0 0
\(89\) 0.878076 − 1.52087i 0.0930758 − 0.161212i −0.815728 − 0.578436i \(-0.803663\pi\)
0.908804 + 0.417223i \(0.136997\pi\)
\(90\) 0 0
\(91\) 7.28995 + 0.176915i 0.764194 + 0.0185458i
\(92\) 0 0
\(93\) 7.35056 + 4.24385i 0.762218 + 0.440067i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 2.00000i − 0.203069i −0.994832 − 0.101535i \(-0.967625\pi\)
0.994832 − 0.101535i \(-0.0323753\pi\)
\(98\) 0 0
\(99\) −45.5634 −4.57929
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.i.849.6 12
5.2 odd 4 1400.2.q.j.401.1 6
5.3 odd 4 280.2.q.e.121.3 yes 6
5.4 even 2 inner 1400.2.bh.i.849.1 12
7.4 even 3 inner 1400.2.bh.i.249.1 12
15.8 even 4 2520.2.bi.q.1801.3 6
20.3 even 4 560.2.q.l.401.1 6
35.2 odd 12 9800.2.a.ce.1.3 3
35.3 even 12 1960.2.q.w.361.1 6
35.4 even 6 inner 1400.2.bh.i.249.6 12
35.12 even 12 9800.2.a.cf.1.1 3
35.13 even 4 1960.2.q.w.961.1 6
35.18 odd 12 280.2.q.e.81.3 ✓ 6
35.23 odd 12 1960.2.a.w.1.1 3
35.32 odd 12 1400.2.q.j.1201.1 6
35.33 even 12 1960.2.a.v.1.3 3
105.53 even 12 2520.2.bi.q.361.3 6
140.23 even 12 3920.2.a.cc.1.3 3
140.103 odd 12 3920.2.a.cb.1.1 3
140.123 even 12 560.2.q.l.81.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.q.e.81.3 ✓ 6 35.18 odd 12
280.2.q.e.121.3 yes 6 5.3 odd 4
560.2.q.l.81.1 6 140.123 even 12
560.2.q.l.401.1 6 20.3 even 4
1400.2.q.j.401.1 6 5.2 odd 4
1400.2.q.j.1201.1 6 35.32 odd 12
1400.2.bh.i.249.1 12 7.4 even 3 inner
1400.2.bh.i.249.6 12 35.4 even 6 inner
1400.2.bh.i.849.1 12 5.4 even 2 inner
1400.2.bh.i.849.6 12 1.1 even 1 trivial
1960.2.a.v.1.3 3 35.33 even 12
1960.2.a.w.1.1 3 35.23 odd 12
1960.2.q.w.361.1 6 35.3 even 12
1960.2.q.w.961.1 6 35.13 even 4
2520.2.bi.q.361.3 6 105.53 even 12
2520.2.bi.q.1801.3 6 15.8 even 4
3920.2.a.cb.1.1 3 140.103 odd 12
3920.2.a.cc.1.3 3 140.23 even 12
9800.2.a.ce.1.3 3 35.2 odd 12
9800.2.a.cf.1.1 3 35.12 even 12