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Properties

Label 1400.2.x.b.657.5
Level $1400$
Weight $2$
Character 1400.657
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 657.5
Character \(\chi\) \(=\) 1400.657
Dual form 1400.2.x.b.993.5

$q$-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 \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).

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

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 0.464172 − 0.885745i \(-0.346352\pi\)
−0.885745 + 0.464172i \(0.846352\pi\)
\(4\) 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 0.567655 − 0.823267i \(-0.307851\pi\)
−0.823267 + 0.567655i \(0.807851\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.02728 − 2.02728i 0.491689 − 0.491689i −0.417149 − 0.908838i \(-0.636971\pi\)
0.908838 + 0.417149i \(0.136971\pi\)
\(18\) 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 −0.712756 − 0.701412i \(-0.752553\pi\)
0.701412 + 0.712756i \(0.252553\pi\)
\(24\) 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 \(0.114431\pi\)
−0.936075 + 0.351801i \(0.885569\pi\)
\(30\) 0 0
\(31\) 4.88150i 0.876744i 0.898794 + 0.438372i \(0.144445\pi\)
−0.898794 + 0.438372i \(0.855555\pi\)
\(32\) 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 0.919798 − 0.392392i \(-0.128352\pi\)
−0.392392 + 0.919798i \(0.628352\pi\)
\(38\) 0 0
\(39\) 1.34591i 0.215518i
\(40\) 0 0
\(41\) − 10.6672i − 1.66593i −0.553323 − 0.832967i \(-0.686641\pi\)
0.553323 − 0.832967i \(-0.313359\pi\)
\(42\) 0 0
\(43\) 1.60483 − 1.60483i 0.244734 − 0.244734i −0.574071 − 0.818805i \(-0.694637\pi\)
0.818805 + 0.574071i \(0.194637\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.64854 − 2.64854i 0.386329 − 0.386329i −0.487047 − 0.873376i \(-0.661926\pi\)
0.873376 + 0.487047i \(0.161926\pi\)
\(48\) 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 0.307876 − 0.951426i \(-0.400382\pi\)
0.951426 − 0.307876i \(-0.0996182\pi\)
\(54\) 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 \(-0.745245\pi\)
0.696465 − 0.717591i \(-0.254755\pi\)
\(62\) 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 0.980827 − 0.194880i \(-0.0624318\pi\)
−0.194880 + 0.980827i \(0.562432\pi\)
\(68\) 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 −0.999896 − 0.0144206i \(-0.995410\pi\)
−0.0144206 − 0.999896i \(-0.504590\pi\)
\(74\) 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 \(0.0316044\pi\)
−0.995075 + 0.0991251i \(0.968396\pi\)
\(80\) 0 0
\(81\) −0.540019 −0.0600021
\(82\) 0 0
\(83\) −2.36897 − 2.36897i −0.260028 − 0.260028i 0.565037 − 0.825065i \(-0.308862\pi\)
−0.825065 + 0.565037i \(0.808862\pi\)
\(84\) 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 −0.803496 − 0.595310i \(-0.797029\pi\)
0.595310 + 0.803496i \(0.297029\pi\)
\(98\) 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.657.5 24
5.2 odd 4 280.2.x.a.153.5 yes 24
5.3 odd 4 inner 1400.2.x.b.993.8 24
5.4 even 2 280.2.x.a.97.8 yes 24
7.6 odd 2 inner 1400.2.x.b.657.8 24
20.7 even 4 560.2.bj.d.433.8 24
20.19 odd 2 560.2.bj.d.97.5 24
35.13 even 4 inner 1400.2.x.b.993.5 24
35.27 even 4 280.2.x.a.153.8 yes 24
35.34 odd 2 280.2.x.a.97.5 ✓ 24
140.27 odd 4 560.2.bj.d.433.5 24
140.139 even 2 560.2.bj.d.97.8 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.x.a.97.5 ✓ 24 35.34 odd 2
280.2.x.a.97.8 yes 24 5.4 even 2
280.2.x.a.153.5 yes 24 5.2 odd 4
280.2.x.a.153.8 yes 24 35.27 even 4
560.2.bj.d.97.5 24 20.19 odd 2
560.2.bj.d.97.8 24 140.139 even 2
560.2.bj.d.433.5 24 140.27 odd 4
560.2.bj.d.433.8 24 20.7 even 4
1400.2.x.b.657.5 24 1.1 even 1 trivial
1400.2.x.b.657.8 24 7.6 odd 2 inner
1400.2.x.b.993.5 24 35.13 even 4 inner
1400.2.x.b.993.8 24 5.3 odd 4 inner