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Properties

Label 2370.2.d.f.949.13
Level $2370$
Weight $2$
Character 2370.949
Analytic conductor $18.925$
Analytic rank $0$
Dimension $24$
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] = [2370,2,Mod(949,2370)] 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("2370.949"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2370, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2370 = 2 \cdot 3 \cdot 5 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2370.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,-24,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.9245452790\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 949.13
Character \(\chi\) \(=\) 2370.949
Dual form 2370.2.d.f.949.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +(-2.23590 - 0.0277372i) q^{5} -1.00000 q^{6} -3.93532i q^{7} -1.00000i q^{8} -1.00000 q^{9} +(0.0277372 - 2.23590i) q^{10} +0.564136 q^{11} -1.00000i q^{12} +3.10833i q^{13} +3.93532 q^{14} +(0.0277372 - 2.23590i) q^{15} +1.00000 q^{16} -4.81132i q^{17} -1.00000i q^{18} +2.38257 q^{19} +(2.23590 + 0.0277372i) q^{20} +3.93532 q^{21} +0.564136i q^{22} +6.17431i q^{23} +1.00000 q^{24} +(4.99846 + 0.124035i) q^{25} -3.10833 q^{26} -1.00000i q^{27} +3.93532i q^{28} -0.938404 q^{29} +(2.23590 + 0.0277372i) q^{30} -2.56096 q^{31} +1.00000i q^{32} +0.564136i q^{33} +4.81132 q^{34} +(-0.109155 + 8.79897i) q^{35} +1.00000 q^{36} +8.12270i q^{37} +2.38257i q^{38} -3.10833 q^{39} +(-0.0277372 + 2.23590i) q^{40} -5.68588 q^{41} +3.93532i q^{42} -1.05897i q^{43} -0.564136 q^{44} +(2.23590 + 0.0277372i) q^{45} -6.17431 q^{46} -2.20261i q^{47} +1.00000i q^{48} -8.48675 q^{49} +(-0.124035 + 4.99846i) q^{50} +4.81132 q^{51} -3.10833i q^{52} +5.13224i q^{53} +1.00000 q^{54} +(-1.26135 - 0.0156476i) q^{55} -3.93532 q^{56} +2.38257i q^{57} -0.938404i q^{58} -11.0109 q^{59} +(-0.0277372 + 2.23590i) q^{60} +7.25881 q^{61} -2.56096i q^{62} +3.93532i q^{63} -1.00000 q^{64} +(0.0862165 - 6.94990i) q^{65} -0.564136 q^{66} -2.58380i q^{67} +4.81132i q^{68} -6.17431 q^{69} +(-8.79897 - 0.109155i) q^{70} -15.3978 q^{71} +1.00000i q^{72} +0.0979291i q^{73} -8.12270 q^{74} +(-0.124035 + 4.99846i) q^{75} -2.38257 q^{76} -2.22006i q^{77} -3.10833i q^{78} -1.00000 q^{79} +(-2.23590 - 0.0277372i) q^{80} +1.00000 q^{81} -5.68588i q^{82} +17.0366i q^{83} -3.93532 q^{84} +(-0.133453 + 10.7576i) q^{85} +1.05897 q^{86} -0.938404i q^{87} -0.564136i q^{88} -12.1335 q^{89} +(-0.0277372 + 2.23590i) q^{90} +12.2323 q^{91} -6.17431i q^{92} -2.56096i q^{93} +2.20261 q^{94} +(-5.32717 - 0.0660858i) q^{95} -1.00000 q^{96} -0.877030i q^{97} -8.48675i q^{98} -0.564136 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 24 q^{4} - 24 q^{6} - 24 q^{9} - 2 q^{10} + 16 q^{11} - 8 q^{14} - 2 q^{15} + 24 q^{16} - 2 q^{19} - 8 q^{21} + 24 q^{24} - 6 q^{25} + 18 q^{26} - 40 q^{29} - 6 q^{35} + 24 q^{36} + 18 q^{39} + 2 q^{40}+ \cdots - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2370\mathbb{Z}\right)^\times\).

\(n\) \(791\) \(1741\) \(1897\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 1.00000i 0.577350i
\(4\) −1.00000 −0.500000
\(5\) −2.23590 − 0.0277372i −0.999923 − 0.0124045i
\(6\) −1.00000 −0.408248
\(7\) − 3.93532i − 1.48741i −0.668507 − 0.743706i \(-0.733066\pi\)
0.668507 − 0.743706i \(-0.266934\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0.0277372 − 2.23590i 0.00877128 − 0.707052i
\(11\) 0.564136 0.170093 0.0850467 − 0.996377i \(-0.472896\pi\)
0.0850467 + 0.996377i \(0.472896\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) 3.10833i 0.862095i 0.902329 + 0.431048i \(0.141856\pi\)
−0.902329 + 0.431048i \(0.858144\pi\)
\(14\) 3.93532 1.05176
\(15\) 0.0277372 − 2.23590i 0.00716172 − 0.577306i
\(16\) 1.00000 0.250000
\(17\) − 4.81132i − 1.16692i −0.812143 − 0.583459i \(-0.801699\pi\)
0.812143 − 0.583459i \(-0.198301\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 2.38257 0.546598 0.273299 − 0.961929i \(-0.411885\pi\)
0.273299 + 0.961929i \(0.411885\pi\)
\(20\) 2.23590 + 0.0277372i 0.499962 + 0.00620223i
\(21\) 3.93532 0.858757
\(22\) 0.564136i 0.120274i
\(23\) 6.17431i 1.28743i 0.765264 + 0.643717i \(0.222609\pi\)
−0.765264 + 0.643717i \(0.777391\pi\)
\(24\) 1.00000 0.204124
\(25\) 4.99846 + 0.124035i 0.999692 + 0.0248070i
\(26\) −3.10833 −0.609594
\(27\) − 1.00000i − 0.192450i
\(28\) 3.93532i 0.743706i
\(29\) −0.938404 −0.174257 −0.0871286 − 0.996197i \(-0.527769\pi\)
−0.0871286 + 0.996197i \(0.527769\pi\)
\(30\) 2.23590 + 0.0277372i 0.408217 + 0.00506410i
\(31\) −2.56096 −0.459963 −0.229981 − 0.973195i \(-0.573867\pi\)
−0.229981 + 0.973195i \(0.573867\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0.564136i 0.0982035i
\(34\) 4.81132 0.825135
\(35\) −0.109155 + 8.79897i −0.0184506 + 1.48730i
\(36\) 1.00000 0.166667
\(37\) 8.12270i 1.33536i 0.744447 + 0.667681i \(0.232713\pi\)
−0.744447 + 0.667681i \(0.767287\pi\)
\(38\) 2.38257i 0.386503i
\(39\) −3.10833 −0.497731
\(40\) −0.0277372 + 2.23590i −0.00438564 + 0.353526i
\(41\) −5.68588 −0.887985 −0.443992 − 0.896031i \(-0.646438\pi\)
−0.443992 + 0.896031i \(0.646438\pi\)
\(42\) 3.93532i 0.607233i
\(43\) − 1.05897i − 0.161492i −0.996735 − 0.0807461i \(-0.974270\pi\)
0.996735 − 0.0807461i \(-0.0257303\pi\)
\(44\) −0.564136 −0.0850467
\(45\) 2.23590 + 0.0277372i 0.333308 + 0.00413482i
\(46\) −6.17431 −0.910353
\(47\) − 2.20261i − 0.321283i −0.987013 − 0.160642i \(-0.948644\pi\)
0.987013 − 0.160642i \(-0.0513563\pi\)
\(48\) 1.00000i 0.144338i
\(49\) −8.48675 −1.21239
\(50\) −0.124035 + 4.99846i −0.0175412 + 0.706889i
\(51\) 4.81132 0.673720
\(52\) − 3.10833i − 0.431048i
\(53\) 5.13224i 0.704968i 0.935818 + 0.352484i \(0.114663\pi\)
−0.935818 + 0.352484i \(0.885337\pi\)
\(54\) 1.00000 0.136083
\(55\) −1.26135 − 0.0156476i −0.170080 − 0.00210992i
\(56\) −3.93532 −0.525879
\(57\) 2.38257i 0.315579i
\(58\) − 0.938404i − 0.123218i
\(59\) −11.0109 −1.43350 −0.716751 − 0.697329i \(-0.754372\pi\)
−0.716751 + 0.697329i \(0.754372\pi\)
\(60\) −0.0277372 + 2.23590i −0.00358086 + 0.288653i
\(61\) 7.25881 0.929395 0.464698 − 0.885469i \(-0.346163\pi\)
0.464698 + 0.885469i \(0.346163\pi\)
\(62\) − 2.56096i − 0.325243i
\(63\) 3.93532i 0.495804i
\(64\) −1.00000 −0.125000
\(65\) 0.0862165 − 6.94990i 0.0106938 − 0.862029i
\(66\) −0.564136 −0.0694404
\(67\) − 2.58380i − 0.315661i −0.987466 − 0.157831i \(-0.949550\pi\)
0.987466 − 0.157831i \(-0.0504500\pi\)
\(68\) 4.81132i 0.583459i
\(69\) −6.17431 −0.743300
\(70\) −8.79897 − 0.109155i −1.05168 − 0.0130465i
\(71\) −15.3978 −1.82738 −0.913689 − 0.406415i \(-0.866779\pi\)
−0.913689 + 0.406415i \(0.866779\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 0.0979291i 0.0114617i 0.999984 + 0.00573087i \(0.00182420\pi\)
−0.999984 + 0.00573087i \(0.998176\pi\)
\(74\) −8.12270 −0.944244
\(75\) −0.124035 + 4.99846i −0.0143223 + 0.577173i
\(76\) −2.38257 −0.273299
\(77\) − 2.22006i − 0.252999i
\(78\) − 3.10833i − 0.351949i
\(79\) −1.00000 −0.112509
\(80\) −2.23590 − 0.0277372i −0.249981 − 0.00310112i
\(81\) 1.00000 0.111111
\(82\) − 5.68588i − 0.627900i
\(83\) 17.0366i 1.87001i 0.354639 + 0.935003i \(0.384604\pi\)
−0.354639 + 0.935003i \(0.615396\pi\)
\(84\) −3.93532 −0.429379
\(85\) −0.133453 + 10.7576i −0.0144750 + 1.16683i
\(86\) 1.05897 0.114192
\(87\) − 0.938404i − 0.100607i
\(88\) − 0.564136i − 0.0601371i
\(89\) −12.1335 −1.28615 −0.643077 − 0.765802i \(-0.722342\pi\)
−0.643077 + 0.765802i \(0.722342\pi\)
\(90\) −0.0277372 + 2.23590i −0.00292376 + 0.235684i
\(91\) 12.2323 1.28229
\(92\) − 6.17431i − 0.643717i
\(93\) − 2.56096i − 0.265560i
\(94\) 2.20261 0.227181
\(95\) −5.32717 − 0.0660858i −0.546556 − 0.00678026i
\(96\) −1.00000 −0.102062
\(97\) − 0.877030i − 0.0890489i −0.999008 − 0.0445244i \(-0.985823\pi\)
0.999008 − 0.0445244i \(-0.0141773\pi\)
\(98\) − 8.48675i − 0.857291i
\(99\) −0.564136 −0.0566978
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 2370.2.d.f.949.13 yes 24
5.4 even 2 inner 2370.2.d.f.949.1 ✓ 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2370.2.d.f.949.1 ✓ 24 5.4 even 2 inner
2370.2.d.f.949.13 yes 24 1.1 even 1 trivial