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Properties

Label 1400.2.q.j.1201.1
Level $1400$
Weight $2$
Character 1400.1201
Analytic conductor $11.179$
Analytic rank $0$
Dimension $6$
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] = [1400,2,Mod(401,1400)] 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("1400.401"); 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, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.11337408.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 18x^{4} + 81x^{2} + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1201.1
Root \(2.78499i\) of defining polynomial
Character \(\chi\) \(=\) 1400.1201
Dual form 1400.2.q.j.401.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.64497 + 2.84918i) q^{3} +(-2.64497 + 0.0641892i) q^{7} +(-3.91187 - 6.77556i) q^{9} +(-2.91187 + 5.04351i) q^{11} -2.75615 q^{13} +(1.00000 - 1.73205i) q^{17} +(0.378076 + 0.654846i) q^{19} +(4.16802 - 7.64158i) q^{21} +(-0.266897 - 0.462279i) q^{23} +15.8698 q^{27} -0.823739 q^{29} +(1.28995 - 2.23425i) q^{31} +(-9.57989 - 16.5929i) q^{33} +(2.37808 + 4.11895i) q^{37} +(4.53379 - 7.85276i) q^{39} +6.06759 q^{41} -0.710055 q^{43} +(-6.44566 - 11.1642i) q^{47} +(6.99176 - 0.339557i) q^{49} +(3.28995 + 5.69835i) q^{51} +(-4.20181 + 7.27776i) q^{53} -2.48770 q^{57} +(-4.00000 + 6.92820i) q^{59} +(-4.70181 - 8.14378i) q^{61} +(10.7817 + 17.6701i) q^{63} +(5.93492 - 10.2796i) q^{67} +1.75615 q^{69} +(1.75615 - 3.04174i) q^{73} +(7.37808 - 13.5268i) q^{77} +(4.75615 + 8.23790i) q^{79} +(-14.3698 + 24.8893i) q^{81} +6.71005 q^{83} +(1.35503 - 2.34698i) q^{87} +(-0.878076 - 1.52087i) q^{89} +(7.28995 - 0.176915i) q^{91} +(4.24385 + 7.35056i) q^{93} +2.00000 q^{97} +45.5634 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{7} - 9 q^{9} - 3 q^{11} - 6 q^{13} + 6 q^{17} - 3 q^{19} + 3 q^{23} + 36 q^{27} + 24 q^{29} - 12 q^{31} - 18 q^{33} + 9 q^{37} + 18 q^{39} + 18 q^{41} - 24 q^{43} - 15 q^{47} - 12 q^{49} + 9 q^{53}+ \cdots + 126 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) −1.64497 + 2.84918i −0.949725 + 1.64497i −0.203724 + 0.979028i \(0.565304\pi\)
−0.746002 + 0.665944i \(0.768029\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.64497 + 0.0641892i −0.999706 + 0.0242612i
\(8\) 0 0
\(9\) −3.91187 − 6.77556i −1.30396 − 2.25852i
\(10\) 0 0
\(11\) −2.91187 + 5.04351i −0.877962 + 1.52067i −0.0243876 + 0.999703i \(0.507764\pi\)
−0.853574 + 0.520972i \(0.825570\pi\)
\(12\) 0 0
\(13\) −2.75615 −0.764419 −0.382209 − 0.924076i \(-0.624837\pi\)
−0.382209 + 0.924076i \(0.624837\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.00000 − 1.73205i 0.242536 − 0.420084i −0.718900 − 0.695113i \(-0.755354\pi\)
0.961436 + 0.275029i \(0.0886875\pi\)
\(18\) 0 0
\(19\) 0.378076 + 0.654846i 0.0867365 + 0.150232i 0.906130 − 0.423000i \(-0.139023\pi\)
−0.819393 + 0.573232i \(0.805690\pi\)
\(20\) 0 0
\(21\) 4.16802 − 7.64158i 0.909537 − 1.66753i
\(22\) 0 0
\(23\) −0.266897 − 0.462279i −0.0556518 − 0.0963918i 0.836857 − 0.547421i \(-0.184390\pi\)
−0.892509 + 0.451029i \(0.851057\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 15.8698 3.05415
\(28\) 0 0
\(29\) −0.823739 −0.152964 −0.0764822 − 0.997071i \(-0.524369\pi\)
−0.0764822 + 0.997071i \(0.524369\pi\)
\(30\) 0 0
\(31\) 1.28995 − 2.23425i 0.231681 − 0.401283i −0.726622 − 0.687038i \(-0.758911\pi\)
0.958303 + 0.285754i \(0.0922441\pi\)
\(32\) 0 0
\(33\) −9.57989 − 16.5929i −1.66764 − 2.88845i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.37808 + 4.11895i 0.390953 + 0.677151i 0.992576 − 0.121630i \(-0.0388121\pi\)
−0.601622 + 0.798781i \(0.705479\pi\)
\(38\) 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
\(43\) −0.710055 −0.108282 −0.0541412 − 0.998533i \(-0.517242\pi\)
−0.0541412 + 0.998533i \(0.517242\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.44566 − 11.1642i −0.940197 − 1.62847i −0.765095 − 0.643918i \(-0.777308\pi\)
−0.175102 − 0.984550i \(-0.556026\pi\)
\(48\) 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
\(53\) −4.20181 + 7.27776i −0.577164 + 0.999677i 0.418639 + 0.908153i \(0.362507\pi\)
−0.995803 + 0.0915241i \(0.970826\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.48770 −0.329504
\(58\) 0 0
\(59\) −4.00000 + 6.92820i −0.520756 + 0.901975i 0.478953 + 0.877841i \(0.341016\pi\)
−0.999709 + 0.0241347i \(0.992317\pi\)
\(60\) 0 0
\(61\) −4.70181 − 8.14378i −0.602006 − 1.04270i −0.992517 − 0.122106i \(-0.961035\pi\)
0.390511 − 0.920598i \(-0.372298\pi\)
\(62\) 0 0
\(63\) 10.7817 + 17.6701i 1.35837 + 2.22622i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.93492 − 10.2796i 0.725066 − 1.25585i −0.233881 − 0.972265i \(-0.575143\pi\)
0.958947 − 0.283585i \(-0.0915239\pi\)
\(68\) 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
\(73\) 1.75615 − 3.04174i 0.205542 − 0.356009i −0.744763 − 0.667329i \(-0.767438\pi\)
0.950305 + 0.311320i \(0.100771\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7.37808 − 13.5268i 0.840810 − 1.54153i
\(78\) 0 0
\(79\) 4.75615 + 8.23790i 0.535109 + 0.926836i 0.999158 + 0.0410263i \(0.0130627\pi\)
−0.464049 + 0.885809i \(0.653604\pi\)
\(80\) 0 0
\(81\) −14.3698 + 24.8893i −1.59665 + 2.76548i
\(82\) 0 0
\(83\) 6.71005 0.736524 0.368262 − 0.929722i \(-0.379953\pi\)
0.368262 + 0.929722i \(0.379953\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.35503 − 2.34698i 0.145274 − 0.251622i
\(88\) 0 0
\(89\) −0.878076 − 1.52087i −0.0930758 − 0.161212i 0.815728 − 0.578436i \(-0.196337\pi\)
−0.908804 + 0.417223i \(0.863003\pi\)
\(90\) 0 0
\(91\) 7.28995 − 0.176915i 0.764194 − 0.0185458i
\(92\) 0 0
\(93\) 4.24385 + 7.35056i 0.440067 + 0.762218i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 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.q.j.1201.1 6
5.2 odd 4 1400.2.bh.i.249.6 12
5.3 odd 4 1400.2.bh.i.249.1 12
5.4 even 2 280.2.q.e.81.3 ✓ 6
7.2 even 3 inner 1400.2.q.j.401.1 6
7.3 odd 6 9800.2.a.cf.1.1 3
7.4 even 3 9800.2.a.ce.1.3 3
15.14 odd 2 2520.2.bi.q.361.3 6
20.19 odd 2 560.2.q.l.81.1 6
35.2 odd 12 1400.2.bh.i.849.1 12
35.4 even 6 1960.2.a.w.1.1 3
35.9 even 6 280.2.q.e.121.3 yes 6
35.19 odd 6 1960.2.q.w.961.1 6
35.23 odd 12 1400.2.bh.i.849.6 12
35.24 odd 6 1960.2.a.v.1.3 3
35.34 odd 2 1960.2.q.w.361.1 6
105.44 odd 6 2520.2.bi.q.1801.3 6
140.39 odd 6 3920.2.a.cc.1.3 3
140.59 even 6 3920.2.a.cb.1.1 3
140.79 odd 6 560.2.q.l.401.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.q.e.81.3 ✓ 6 5.4 even 2
280.2.q.e.121.3 yes 6 35.9 even 6
560.2.q.l.81.1 6 20.19 odd 2
560.2.q.l.401.1 6 140.79 odd 6
1400.2.q.j.401.1 6 7.2 even 3 inner
1400.2.q.j.1201.1 6 1.1 even 1 trivial
1400.2.bh.i.249.1 12 5.3 odd 4
1400.2.bh.i.249.6 12 5.2 odd 4
1400.2.bh.i.849.1 12 35.2 odd 12
1400.2.bh.i.849.6 12 35.23 odd 12
1960.2.a.v.1.3 3 35.24 odd 6
1960.2.a.w.1.1 3 35.4 even 6
1960.2.q.w.361.1 6 35.34 odd 2
1960.2.q.w.961.1 6 35.19 odd 6
2520.2.bi.q.361.3 6 15.14 odd 2
2520.2.bi.q.1801.3 6 105.44 odd 6
3920.2.a.cb.1.1 3 140.59 even 6
3920.2.a.cc.1.3 3 140.39 odd 6
9800.2.a.ce.1.3 3 7.4 even 3
9800.2.a.cf.1.1 3 7.3 odd 6