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Properties

Label 1400.2.x.b.993.5
Level $1400$
Weight $2$
Character 1400.993
Analytic conductor $11.179$
Analytic rank $0$
Dimension $24$
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(657,1400)] 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("1400.657"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1400, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.x (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 993.5
Character \(\chi\) \(=\) 1400.993
Dual form 1400.2.x.b.657.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.730185 + 0.730185i) q^{3} +(1.08445 + 2.41329i) q^{7} +1.93366i q^{9} +5.95977 q^{11} +(-0.921623 + 0.921623i) q^{13} +(2.02728 + 2.02728i) q^{17} -3.29475 q^{19} +(-2.55400 - 0.970294i) q^{21} +(-0.0544054 - 0.0544054i) q^{23} +(-3.60248 - 3.60248i) q^{27} -3.78901i q^{29} -4.88150i q^{31} +(-4.35174 + 4.35174i) q^{33} +(3.20809 - 3.20809i) q^{37} -1.34591i q^{39} +10.6672i q^{41} +(1.60483 + 1.60483i) q^{43} +(2.64854 + 2.64854i) q^{47} +(-4.64792 + 5.23420i) q^{49} -2.96059 q^{51} +(9.16786 + 9.16786i) q^{53} +(2.40577 - 2.40577i) q^{57} -13.3231 q^{59} +11.2091i q^{61} +(-4.66648 + 2.09697i) q^{63} +(6.43325 - 6.43325i) q^{67} +0.0794520 q^{69} -8.51221 q^{71} +(-8.66633 + 8.66633i) q^{73} +(6.46310 + 14.3826i) q^{77} -1.76209i q^{79} -0.540019 q^{81} +(-2.36897 + 2.36897i) q^{83} +(2.76668 + 2.76668i) q^{87} +9.73989 q^{89} +(-3.22360 - 1.22468i) q^{91} +(3.56440 + 3.56440i) q^{93} +(-2.05040 - 2.05040i) q^{97} +11.5242i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 4 q^{7} + 8 q^{11} + 16 q^{21} + 32 q^{23} + 8 q^{37} - 16 q^{43} - 24 q^{51} + 16 q^{53} - 20 q^{63} + 32 q^{67} - 32 q^{71} + 40 q^{77} - 72 q^{81} - 64 q^{91} - 72 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) −0.730185 + 0.730185i −0.421573 + 0.421573i −0.885745 − 0.464172i \(-0.846352\pi\)
0.464172 + 0.885745i \(0.346352\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.08445 + 2.41329i 0.409885 + 0.912137i
\(8\) 0 0
\(9\) 1.93366i 0.644553i
\(10\) 0 0
\(11\) 5.95977 1.79694 0.898470 − 0.439036i \(-0.144680\pi\)
0.898470 + 0.439036i \(0.144680\pi\)
\(12\) 0 0
\(13\) −0.921623 + 0.921623i −0.255612 + 0.255612i −0.823267 − 0.567655i \(-0.807851\pi\)
0.567655 + 0.823267i \(0.307851\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.02728 + 2.02728i 0.491689 + 0.491689i 0.908838 − 0.417149i \(-0.136971\pi\)
−0.417149 + 0.908838i \(0.636971\pi\)
\(18\) 0 0
\(19\) −3.29475 −0.755867 −0.377933 − 0.925833i \(-0.623365\pi\)
−0.377933 + 0.925833i \(0.623365\pi\)
\(20\) 0 0
\(21\) −2.55400 − 0.970294i −0.557328 − 0.211736i
\(22\) 0 0
\(23\) −0.0544054 − 0.0544054i −0.0113443 − 0.0113443i 0.701412 − 0.712756i \(-0.252553\pi\)
−0.712756 + 0.701412i \(0.752553\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −3.60248 − 3.60248i −0.693298 − 0.693298i
\(28\) 0 0
\(29\) − 3.78901i − 0.703602i −0.936075 − 0.351801i \(-0.885569\pi\)
0.936075 − 0.351801i \(-0.114431\pi\)
\(30\) 0 0
\(31\) − 4.88150i − 0.876744i −0.898794 − 0.438372i \(-0.855555\pi\)
0.898794 − 0.438372i \(-0.144445\pi\)
\(32\) 0 0
\(33\) −4.35174 + 4.35174i −0.757540 + 0.757540i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.20809 − 3.20809i 0.527406 − 0.527406i −0.392392 − 0.919798i \(-0.628352\pi\)
0.919798 + 0.392392i \(0.128352\pi\)
\(38\) 0 0
\(39\) − 1.34591i − 0.215518i
\(40\) 0 0
\(41\) 10.6672i 1.66593i 0.553323 + 0.832967i \(0.313359\pi\)
−0.553323 + 0.832967i \(0.686641\pi\)
\(42\) 0 0
\(43\) 1.60483 + 1.60483i 0.244734 + 0.244734i 0.818805 − 0.574071i \(-0.194637\pi\)
−0.574071 + 0.818805i \(0.694637\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.64854 + 2.64854i 0.386329 + 0.386329i 0.873376 − 0.487047i \(-0.161926\pi\)
−0.487047 + 0.873376i \(0.661926\pi\)
\(48\) 0 0
\(49\) −4.64792 + 5.23420i −0.663988 + 0.747743i
\(50\) 0 0
\(51\) −2.96059 −0.414565
\(52\) 0 0
\(53\) 9.16786 + 9.16786i 1.25930 + 1.25930i 0.951426 + 0.307876i \(0.0996182\pi\)
0.307876 + 0.951426i \(0.400382\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.40577 − 2.40577i 0.318653 − 0.318653i
\(58\) 0 0
\(59\) −13.3231 −1.73452 −0.867261 − 0.497854i \(-0.834121\pi\)
−0.867261 + 0.497854i \(0.834121\pi\)
\(60\) 0 0
\(61\) 11.2091i 1.43518i 0.696465 + 0.717591i \(0.254755\pi\)
−0.696465 + 0.717591i \(0.745245\pi\)
\(62\) 0 0
\(63\) −4.66648 + 2.09697i −0.587921 + 0.264193i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.43325 − 6.43325i 0.785947 − 0.785947i −0.194880 − 0.980827i \(-0.562432\pi\)
0.980827 + 0.194880i \(0.0624318\pi\)
\(68\) 0 0
\(69\) 0.0794520 0.00956489
\(70\) 0 0
\(71\) −8.51221 −1.01021 −0.505107 − 0.863057i \(-0.668547\pi\)
−0.505107 + 0.863057i \(0.668547\pi\)
\(72\) 0 0
\(73\) −8.66633 + 8.66633i −1.01432 + 1.01432i −0.0144206 + 0.999896i \(0.504590\pi\)
−0.999896 + 0.0144206i \(0.995410\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.46310 + 14.3826i 0.736539 + 1.63905i
\(78\) 0 0
\(79\) − 1.76209i − 0.198250i −0.995075 − 0.0991251i \(-0.968396\pi\)
0.995075 − 0.0991251i \(-0.0316044\pi\)
\(80\) 0 0
\(81\) −0.540019 −0.0600021
\(82\) 0 0
\(83\) −2.36897 + 2.36897i −0.260028 + 0.260028i −0.825065 − 0.565037i \(-0.808862\pi\)
0.565037 + 0.825065i \(0.308862\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.76668 + 2.76668i 0.296619 + 0.296619i
\(88\) 0 0
\(89\) 9.73989 1.03243 0.516213 − 0.856460i \(-0.327341\pi\)
0.516213 + 0.856460i \(0.327341\pi\)
\(90\) 0 0
\(91\) −3.22360 − 1.22468i −0.337925 − 0.128382i
\(92\) 0 0
\(93\) 3.56440 + 3.56440i 0.369611 + 0.369611i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.05040 − 2.05040i −0.208187 − 0.208187i 0.595310 − 0.803496i \(-0.297029\pi\)
−0.803496 + 0.595310i \(0.797029\pi\)
\(98\) 0 0
\(99\) 11.5242i 1.15822i
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.x.b.993.5 24
5.2 odd 4 inner 1400.2.x.b.657.8 24
5.3 odd 4 280.2.x.a.97.5 ✓ 24
5.4 even 2 280.2.x.a.153.8 yes 24
7.6 odd 2 inner 1400.2.x.b.993.8 24
20.3 even 4 560.2.bj.d.97.8 24
20.19 odd 2 560.2.bj.d.433.5 24
35.13 even 4 280.2.x.a.97.8 yes 24
35.27 even 4 inner 1400.2.x.b.657.5 24
35.34 odd 2 280.2.x.a.153.5 yes 24
140.83 odd 4 560.2.bj.d.97.5 24
140.139 even 2 560.2.bj.d.433.8 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.x.a.97.5 ✓ 24 5.3 odd 4
280.2.x.a.97.8 yes 24 35.13 even 4
280.2.x.a.153.5 yes 24 35.34 odd 2
280.2.x.a.153.8 yes 24 5.4 even 2
560.2.bj.d.97.5 24 140.83 odd 4
560.2.bj.d.97.8 24 20.3 even 4
560.2.bj.d.433.5 24 20.19 odd 2
560.2.bj.d.433.8 24 140.139 even 2
1400.2.x.b.657.5 24 35.27 even 4 inner
1400.2.x.b.657.8 24 5.2 odd 4 inner
1400.2.x.b.993.5 24 1.1 even 1 trivial
1400.2.x.b.993.8 24 7.6 odd 2 inner