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Properties

Label 2370.2.d.f.949.4
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 949.4
Character \(\chi\) \(=\) 2370.949
Dual form 2370.2.d.f.949.16

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(-1.15652 + 1.91376i) q^{5} -1.00000 q^{6} -0.907474i q^{7} +1.00000i q^{8} -1.00000 q^{9} +(1.91376 + 1.15652i) q^{10} +2.72699 q^{11} +1.00000i q^{12} -0.430124i q^{13} -0.907474 q^{14} +(1.91376 + 1.15652i) q^{15} +1.00000 q^{16} +3.00349i q^{17} +1.00000i q^{18} -6.32655 q^{19} +(1.15652 - 1.91376i) q^{20} -0.907474 q^{21} -2.72699i q^{22} -1.55113i q^{23} +1.00000 q^{24} +(-2.32493 - 4.42659i) q^{25} -0.430124 q^{26} +1.00000i q^{27} +0.907474i q^{28} +1.14837 q^{29} +(1.15652 - 1.91376i) q^{30} +5.44238 q^{31} -1.00000i q^{32} -2.72699i q^{33} +3.00349 q^{34} +(1.73669 + 1.04951i) q^{35} +1.00000 q^{36} -3.09537i q^{37} +6.32655i q^{38} -0.430124 q^{39} +(-1.91376 - 1.15652i) q^{40} +8.78465 q^{41} +0.907474i q^{42} +0.114235i q^{43} -2.72699 q^{44} +(1.15652 - 1.91376i) q^{45} -1.55113 q^{46} -11.7675i q^{47} -1.00000i q^{48} +6.17649 q^{49} +(-4.42659 + 2.32493i) q^{50} +3.00349 q^{51} +0.430124i q^{52} -10.3019i q^{53} +1.00000 q^{54} +(-3.15381 + 5.21879i) q^{55} +0.907474 q^{56} +6.32655i q^{57} -1.14837i q^{58} +6.83951 q^{59} +(-1.91376 - 1.15652i) q^{60} -0.608520 q^{61} -5.44238i q^{62} +0.907474i q^{63} -1.00000 q^{64} +(0.823153 + 0.497446i) q^{65} -2.72699 q^{66} +6.44538i q^{67} -3.00349i q^{68} -1.55113 q^{69} +(1.04951 - 1.73669i) q^{70} +11.7881 q^{71} -1.00000i q^{72} -6.00998i q^{73} -3.09537 q^{74} +(-4.42659 + 2.32493i) q^{75} +6.32655 q^{76} -2.47467i q^{77} +0.430124i q^{78} -1.00000 q^{79} +(-1.15652 + 1.91376i) q^{80} +1.00000 q^{81} -8.78465i q^{82} -7.81468i q^{83} +0.907474 q^{84} +(-5.74795 - 3.47359i) q^{85} +0.114235 q^{86} -1.14837i q^{87} +2.72699i q^{88} -15.2155 q^{89} +(-1.91376 - 1.15652i) q^{90} -0.390327 q^{91} +1.55113i q^{92} -5.44238i q^{93} -11.7675 q^{94} +(7.31677 - 12.1075i) q^{95} -1.00000 q^{96} -13.5686i q^{97} -6.17649i q^{98} -2.72699 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 24 q^{4} - 24 q^{6} - 24 q^{9} - 2 q^{10} + 16 q^{11} - 8 q^{14} - 2 q^{15} + 24 q^{16} - 2 q^{19} - 8 q^{21} + 24 q^{24} - 6 q^{25} + 18 q^{26} - 40 q^{29} - 6 q^{35} + 24 q^{36} + 18 q^{39} + 2 q^{40}+ \cdots - 16 q^{99}+O(q^{100}) \) 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
\(5\) −1.15652 + 1.91376i −0.517211 + 0.855858i
\(6\) −1.00000 −0.408248
\(7\) − 0.907474i − 0.342993i −0.985185 − 0.171497i \(-0.945140\pi\)
0.985185 − 0.171497i \(-0.0548602\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 1.91376 + 1.15652i 0.605183 + 0.365723i
\(11\) 2.72699 0.822218 0.411109 − 0.911586i \(-0.365142\pi\)
0.411109 + 0.911586i \(0.365142\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 0.430124i − 0.119295i −0.998220 − 0.0596475i \(-0.981002\pi\)
0.998220 − 0.0596475i \(-0.0189977\pi\)
\(14\) −0.907474 −0.242533
\(15\) 1.91376 + 1.15652i 0.494130 + 0.298612i
\(16\) 1.00000 0.250000
\(17\) 3.00349i 0.728454i 0.931310 + 0.364227i \(0.118667\pi\)
−0.931310 + 0.364227i \(0.881333\pi\)
\(18\) 1.00000i 0.235702i
\(19\) −6.32655 −1.45141 −0.725705 − 0.688006i \(-0.758486\pi\)
−0.725705 + 0.688006i \(0.758486\pi\)
\(20\) 1.15652 − 1.91376i 0.258605 − 0.427929i
\(21\) −0.907474 −0.198027
\(22\) − 2.72699i − 0.581396i
\(23\) − 1.55113i − 0.323433i −0.986837 − 0.161716i \(-0.948297\pi\)
0.986837 − 0.161716i \(-0.0517030\pi\)
\(24\) 1.00000 0.204124
\(25\) −2.32493 − 4.42659i −0.464986 − 0.885318i
\(26\) −0.430124 −0.0843543
\(27\) 1.00000i 0.192450i
\(28\) 0.907474i 0.171497i
\(29\) 1.14837 0.213247 0.106623 − 0.994299i \(-0.465996\pi\)
0.106623 + 0.994299i \(0.465996\pi\)
\(30\) 1.15652 − 1.91376i 0.211150 − 0.349403i
\(31\) 5.44238 0.977480 0.488740 − 0.872429i \(-0.337457\pi\)
0.488740 + 0.872429i \(0.337457\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 2.72699i − 0.474708i
\(34\) 3.00349 0.515095
\(35\) 1.73669 + 1.04951i 0.293553 + 0.177400i
\(36\) 1.00000 0.166667
\(37\) − 3.09537i − 0.508875i −0.967089 − 0.254438i \(-0.918110\pi\)
0.967089 − 0.254438i \(-0.0818903\pi\)
\(38\) 6.32655i 1.02630i
\(39\) −0.430124 −0.0688750
\(40\) −1.91376 − 1.15652i −0.302592 − 0.182862i
\(41\) 8.78465 1.37193 0.685966 − 0.727634i \(-0.259380\pi\)
0.685966 + 0.727634i \(0.259380\pi\)
\(42\) 0.907474i 0.140026i
\(43\) 0.114235i 0.0174206i 0.999962 + 0.00871031i \(0.00277261\pi\)
−0.999962 + 0.00871031i \(0.997227\pi\)
\(44\) −2.72699 −0.411109
\(45\) 1.15652 − 1.91376i 0.172404 − 0.285286i
\(46\) −1.55113 −0.228701
\(47\) − 11.7675i − 1.71647i −0.513256 − 0.858235i \(-0.671561\pi\)
0.513256 − 0.858235i \(-0.328439\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 6.17649 0.882356
\(50\) −4.42659 + 2.32493i −0.626014 + 0.328795i
\(51\) 3.00349 0.420573
\(52\) 0.430124i 0.0596475i
\(53\) − 10.3019i − 1.41508i −0.706674 − 0.707540i \(-0.749805\pi\)
0.706674 − 0.707540i \(-0.250195\pi\)
\(54\) 1.00000 0.136083
\(55\) −3.15381 + 5.21879i −0.425260 + 0.703702i
\(56\) 0.907474 0.121266
\(57\) 6.32655i 0.837972i
\(58\) − 1.14837i − 0.150788i
\(59\) 6.83951 0.890428 0.445214 − 0.895424i \(-0.353128\pi\)
0.445214 + 0.895424i \(0.353128\pi\)
\(60\) −1.91376 − 1.15652i −0.247065 − 0.149306i
\(61\) −0.608520 −0.0779131 −0.0389565 − 0.999241i \(-0.512403\pi\)
−0.0389565 + 0.999241i \(0.512403\pi\)
\(62\) − 5.44238i − 0.691183i
\(63\) 0.907474i 0.114331i
\(64\) −1.00000 −0.125000
\(65\) 0.823153 + 0.497446i 0.102100 + 0.0617006i
\(66\) −2.72699 −0.335669
\(67\) 6.44538i 0.787429i 0.919233 + 0.393715i \(0.128810\pi\)
−0.919233 + 0.393715i \(0.871190\pi\)
\(68\) − 3.00349i − 0.364227i
\(69\) −1.55113 −0.186734
\(70\) 1.04951 − 1.73669i 0.125440 − 0.207574i
\(71\) 11.7881 1.39899 0.699494 − 0.714638i \(-0.253409\pi\)
0.699494 + 0.714638i \(0.253409\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 6.00998i − 0.703415i −0.936110 − 0.351707i \(-0.885601\pi\)
0.936110 − 0.351707i \(-0.114399\pi\)
\(74\) −3.09537 −0.359829
\(75\) −4.42659 + 2.32493i −0.511138 + 0.268460i
\(76\) 6.32655 0.725705
\(77\) − 2.47467i − 0.282015i
\(78\) 0.430124i 0.0487020i
\(79\) −1.00000 −0.112509
\(80\) −1.15652 + 1.91376i −0.129303 + 0.213965i
\(81\) 1.00000 0.111111
\(82\) − 8.78465i − 0.970102i
\(83\) − 7.81468i − 0.857772i −0.903359 − 0.428886i \(-0.858906\pi\)
0.903359 − 0.428886i \(-0.141094\pi\)
\(84\) 0.907474 0.0990136
\(85\) −5.74795 − 3.47359i −0.623453 − 0.376764i
\(86\) 0.114235 0.0123182
\(87\) − 1.14837i − 0.123118i
\(88\) 2.72699i 0.290698i
\(89\) −15.2155 −1.61283 −0.806417 − 0.591347i \(-0.798596\pi\)
−0.806417 + 0.591347i \(0.798596\pi\)
\(90\) −1.91376 − 1.15652i −0.201728 − 0.121908i
\(91\) −0.390327 −0.0409174
\(92\) 1.55113i 0.161716i
\(93\) − 5.44238i − 0.564348i
\(94\) −11.7675 −1.21373
\(95\) 7.31677 − 12.1075i 0.750684 − 1.24220i
\(96\) −1.00000 −0.102062
\(97\) − 13.5686i − 1.37768i −0.724914 − 0.688840i \(-0.758120\pi\)
0.724914 − 0.688840i \(-0.241880\pi\)
\(98\) − 6.17649i − 0.623920i
\(99\) −2.72699 −0.274073
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.949.4 ✓ 24
5.4 even 2 inner 2370.2.d.f.949.16 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2370.2.d.f.949.4 ✓ 24 1.1 even 1 trivial
2370.2.d.f.949.16 yes 24 5.4 even 2 inner