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Properties

Label 2370.2.d.f.949.3
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 949.3
Character \(\chi\) \(=\) 2370.949
Dual form 2370.2.d.f.949.15

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(-1.83569 + 1.27681i) q^{5} -1.00000 q^{6} +2.44522i q^{7} +1.00000i q^{8} -1.00000 q^{9} +(1.27681 + 1.83569i) q^{10} -4.69150 q^{11} +1.00000i q^{12} +6.20818i q^{13} +2.44522 q^{14} +(1.27681 + 1.83569i) q^{15} +1.00000 q^{16} -6.36099i q^{17} +1.00000i q^{18} +5.23811 q^{19} +(1.83569 - 1.27681i) q^{20} +2.44522 q^{21} +4.69150i q^{22} -0.416090i q^{23} +1.00000 q^{24} +(1.73949 - 4.68766i) q^{25} +6.20818 q^{26} +1.00000i q^{27} -2.44522i q^{28} -5.88979 q^{29} +(1.83569 - 1.27681i) q^{30} -1.12694 q^{31} -1.00000i q^{32} +4.69150i q^{33} -6.36099 q^{34} +(-3.12209 - 4.48865i) q^{35} +1.00000 q^{36} +4.34009i q^{37} -5.23811i q^{38} +6.20818 q^{39} +(-1.27681 - 1.83569i) q^{40} +5.92283 q^{41} -2.44522i q^{42} -2.90068i q^{43} +4.69150 q^{44} +(1.83569 - 1.27681i) q^{45} -0.416090 q^{46} -5.81022i q^{47} -1.00000i q^{48} +1.02091 q^{49} +(-4.68766 - 1.73949i) q^{50} -6.36099 q^{51} -6.20818i q^{52} -10.1435i q^{53} +1.00000 q^{54} +(8.61212 - 5.99017i) q^{55} -2.44522 q^{56} -5.23811i q^{57} +5.88979i q^{58} -5.32331 q^{59} +(-1.27681 - 1.83569i) q^{60} -13.9191 q^{61} +1.12694i q^{62} -2.44522i q^{63} -1.00000 q^{64} +(-7.92670 - 11.3963i) q^{65} +4.69150 q^{66} -9.36998i q^{67} +6.36099i q^{68} -0.416090 q^{69} +(-4.48865 + 3.12209i) q^{70} +7.61769 q^{71} -1.00000i q^{72} -7.70055i q^{73} +4.34009 q^{74} +(-4.68766 - 1.73949i) q^{75} -5.23811 q^{76} -11.4717i q^{77} -6.20818i q^{78} -1.00000 q^{79} +(-1.83569 + 1.27681i) q^{80} +1.00000 q^{81} -5.92283i q^{82} -5.48498i q^{83} -2.44522 q^{84} +(8.12181 + 11.6768i) q^{85} -2.90068 q^{86} +5.88979i q^{87} -4.69150i q^{88} -3.98493 q^{89} +(-1.27681 - 1.83569i) q^{90} -15.1804 q^{91} +0.416090i q^{92} +1.12694i q^{93} -5.81022 q^{94} +(-9.61552 + 6.68809i) q^{95} -1.00000 q^{96} -14.9399i q^{97} -1.02091i q^{98} +4.69150 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 24 q^{4} - 24 q^{6} - 24 q^{9} - 2 q^{10} + 16 q^{11} - 8 q^{14} - 2 q^{15} + 24 q^{16} - 2 q^{19} - 8 q^{21} + 24 q^{24} - 6 q^{25} + 18 q^{26} - 40 q^{29} - 6 q^{35} + 24 q^{36} + 18 q^{39} + 2 q^{40}+ \cdots - 16 q^{99}+O(q^{100}) \) 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
\(5\) −1.83569 + 1.27681i −0.820944 + 0.571009i
\(6\) −1.00000 −0.408248
\(7\) 2.44522i 0.924206i 0.886826 + 0.462103i \(0.152905\pi\)
−0.886826 + 0.462103i \(0.847095\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 1.27681 + 1.83569i 0.403764 + 0.580495i
\(11\) −4.69150 −1.41454 −0.707270 − 0.706944i \(-0.750073\pi\)
−0.707270 + 0.706944i \(0.750073\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 6.20818i 1.72184i 0.508740 + 0.860920i \(0.330112\pi\)
−0.508740 + 0.860920i \(0.669888\pi\)
\(14\) 2.44522 0.653512
\(15\) 1.27681 + 1.83569i 0.329672 + 0.473972i
\(16\) 1.00000 0.250000
\(17\) − 6.36099i − 1.54277i −0.636370 − 0.771384i \(-0.719565\pi\)
0.636370 − 0.771384i \(-0.280435\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 5.23811 1.20170 0.600852 − 0.799360i \(-0.294828\pi\)
0.600852 + 0.799360i \(0.294828\pi\)
\(20\) 1.83569 − 1.27681i 0.410472 − 0.285504i
\(21\) 2.44522 0.533590
\(22\) 4.69150i 1.00023i
\(23\) − 0.416090i − 0.0867608i −0.999059 − 0.0433804i \(-0.986187\pi\)
0.999059 − 0.0433804i \(-0.0138127\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.73949 − 4.68766i 0.347898 − 0.937532i
\(26\) 6.20818 1.21753
\(27\) 1.00000i 0.192450i
\(28\) − 2.44522i − 0.462103i
\(29\) −5.88979 −1.09371 −0.546853 − 0.837228i \(-0.684174\pi\)
−0.546853 + 0.837228i \(0.684174\pi\)
\(30\) 1.83569 − 1.27681i 0.335149 − 0.233113i
\(31\) −1.12694 −0.202404 −0.101202 − 0.994866i \(-0.532269\pi\)
−0.101202 + 0.994866i \(0.532269\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 4.69150i 0.816685i
\(34\) −6.36099 −1.09090
\(35\) −3.12209 − 4.48865i −0.527730 − 0.758721i
\(36\) 1.00000 0.166667
\(37\) 4.34009i 0.713506i 0.934199 + 0.356753i \(0.116116\pi\)
−0.934199 + 0.356753i \(0.883884\pi\)
\(38\) − 5.23811i − 0.849733i
\(39\) 6.20818 0.994105
\(40\) −1.27681 − 1.83569i −0.201882 − 0.290248i
\(41\) 5.92283 0.924990 0.462495 − 0.886622i \(-0.346954\pi\)
0.462495 + 0.886622i \(0.346954\pi\)
\(42\) − 2.44522i − 0.377305i
\(43\) − 2.90068i − 0.442349i −0.975234 − 0.221174i \(-0.929011\pi\)
0.975234 − 0.221174i \(-0.0709890\pi\)
\(44\) 4.69150 0.707270
\(45\) 1.83569 − 1.27681i 0.273648 − 0.190336i
\(46\) −0.416090 −0.0613492
\(47\) − 5.81022i − 0.847508i −0.905777 − 0.423754i \(-0.860712\pi\)
0.905777 − 0.423754i \(-0.139288\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 1.02091 0.145844
\(50\) −4.68766 − 1.73949i −0.662936 − 0.246001i
\(51\) −6.36099 −0.890717
\(52\) − 6.20818i − 0.860920i
\(53\) − 10.1435i − 1.39332i −0.717400 − 0.696661i \(-0.754668\pi\)
0.717400 − 0.696661i \(-0.245332\pi\)
\(54\) 1.00000 0.136083
\(55\) 8.61212 − 5.99017i 1.16126 − 0.807714i
\(56\) −2.44522 −0.326756
\(57\) − 5.23811i − 0.693804i
\(58\) 5.88979i 0.773367i
\(59\) −5.32331 −0.693036 −0.346518 − 0.938043i \(-0.612636\pi\)
−0.346518 + 0.938043i \(0.612636\pi\)
\(60\) −1.27681 − 1.83569i −0.164836 − 0.236986i
\(61\) −13.9191 −1.78216 −0.891078 − 0.453851i \(-0.850050\pi\)
−0.891078 + 0.453851i \(0.850050\pi\)
\(62\) 1.12694i 0.143122i
\(63\) − 2.44522i − 0.308069i
\(64\) −1.00000 −0.125000
\(65\) −7.92670 − 11.3963i −0.983186 − 1.41353i
\(66\) 4.69150 0.577483
\(67\) − 9.36998i − 1.14472i −0.820001 − 0.572362i \(-0.806027\pi\)
0.820001 − 0.572362i \(-0.193973\pi\)
\(68\) 6.36099i 0.771384i
\(69\) −0.416090 −0.0500914
\(70\) −4.48865 + 3.12209i −0.536497 + 0.373161i
\(71\) 7.61769 0.904053 0.452026 − 0.892005i \(-0.350701\pi\)
0.452026 + 0.892005i \(0.350701\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 7.70055i − 0.901281i −0.892705 − 0.450641i \(-0.851196\pi\)
0.892705 − 0.450641i \(-0.148804\pi\)
\(74\) 4.34009 0.504525
\(75\) −4.68766 − 1.73949i −0.541285 − 0.200859i
\(76\) −5.23811 −0.600852
\(77\) − 11.4717i − 1.30733i
\(78\) − 6.20818i − 0.702938i
\(79\) −1.00000 −0.112509
\(80\) −1.83569 + 1.27681i −0.205236 + 0.142752i
\(81\) 1.00000 0.111111
\(82\) − 5.92283i − 0.654067i
\(83\) − 5.48498i − 0.602055i −0.953616 − 0.301027i \(-0.902670\pi\)
0.953616 − 0.301027i \(-0.0973295\pi\)
\(84\) −2.44522 −0.266795
\(85\) 8.12181 + 11.6768i 0.880934 + 1.26653i
\(86\) −2.90068 −0.312788
\(87\) 5.88979i 0.631452i
\(88\) − 4.69150i − 0.500115i
\(89\) −3.98493 −0.422402 −0.211201 − 0.977443i \(-0.567737\pi\)
−0.211201 + 0.977443i \(0.567737\pi\)
\(90\) −1.27681 − 1.83569i −0.134588 − 0.193498i
\(91\) −15.1804 −1.59134
\(92\) 0.416090i 0.0433804i
\(93\) 1.12694i 0.116858i
\(94\) −5.81022 −0.599279
\(95\) −9.61552 + 6.68809i −0.986531 + 0.686183i
\(96\) −1.00000 −0.102062
\(97\) − 14.9399i − 1.51692i −0.651722 − 0.758458i \(-0.725953\pi\)
0.651722 − 0.758458i \(-0.274047\pi\)
\(98\) − 1.02091i − 0.103127i
\(99\) 4.69150 0.471513
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.949.3 ✓ 24
5.4 even 2 inner 2370.2.d.f.949.15 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2370.2.d.f.949.3 ✓ 24 1.1 even 1 trivial
2370.2.d.f.949.15 yes 24 5.4 even 2 inner