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Properties

Label 2070.4.a.s.1.2
Level $2070$
Weight $4$
Character 2070.1
Self dual yes
Analytic conductor $122.134$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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] = [2070,4,Mod(1,2070)] 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("2070.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2070, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2070.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,4,0,8,-10,0,-17] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(122.133953712\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{73}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.77200\) of defining polynomial
Character \(\chi\) \(=\) 2070.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} -4.22800 q^{7} +8.00000 q^{8} -10.0000 q^{10} +33.7200 q^{11} -42.9480 q^{13} -8.45600 q^{14} +16.0000 q^{16} -7.49202 q^{17} -25.7200 q^{19} -20.0000 q^{20} +67.4400 q^{22} +23.0000 q^{23} +25.0000 q^{25} -85.8960 q^{26} -16.9120 q^{28} +47.4560 q^{29} +65.0720 q^{31} +32.0000 q^{32} -14.9840 q^{34} +21.1400 q^{35} -215.596 q^{37} -51.4400 q^{38} -40.0000 q^{40} -150.128 q^{41} -83.0160 q^{43} +134.880 q^{44} +46.0000 q^{46} +278.140 q^{47} -325.124 q^{49} +50.0000 q^{50} -171.792 q^{52} +182.404 q^{53} -168.600 q^{55} -33.8240 q^{56} +94.9120 q^{58} +270.060 q^{59} -137.264 q^{61} +130.144 q^{62} +64.0000 q^{64} +214.740 q^{65} -440.652 q^{67} -29.9681 q^{68} +42.2800 q^{70} -1071.29 q^{71} -269.756 q^{73} -431.192 q^{74} -102.880 q^{76} -142.568 q^{77} +195.232 q^{79} -80.0000 q^{80} -300.256 q^{82} -136.676 q^{83} +37.4601 q^{85} -166.032 q^{86} +269.760 q^{88} -629.896 q^{89} +181.584 q^{91} +92.0000 q^{92} +556.280 q^{94} +128.600 q^{95} -419.392 q^{97} -650.248 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 8 q^{4} - 10 q^{5} - 17 q^{7} + 16 q^{8} - 20 q^{10} - 18 q^{11} - 9 q^{13} - 34 q^{14} + 32 q^{16} + 79 q^{17} + 34 q^{19} - 40 q^{20} - 36 q^{22} + 46 q^{23} + 50 q^{25} - 18 q^{26} - 68 q^{28}+ \cdots - 1010 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −4.22800 −0.228290 −0.114145 − 0.993464i \(-0.536413\pi\)
−0.114145 + 0.993464i \(0.536413\pi\)
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) −10.0000 −0.316228
\(11\) 33.7200 0.924270 0.462135 − 0.886810i \(-0.347084\pi\)
0.462135 + 0.886810i \(0.347084\pi\)
\(12\) 0 0
\(13\) −42.9480 −0.916280 −0.458140 − 0.888880i \(-0.651484\pi\)
−0.458140 + 0.888880i \(0.651484\pi\)
\(14\) −8.45600 −0.161426
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −7.49202 −0.106887 −0.0534436 − 0.998571i \(-0.517020\pi\)
−0.0534436 + 0.998571i \(0.517020\pi\)
\(18\) 0 0
\(19\) −25.7200 −0.310557 −0.155278 − 0.987871i \(-0.549627\pi\)
−0.155278 + 0.987871i \(0.549627\pi\)
\(20\) −20.0000 −0.223607
\(21\) 0 0
\(22\) 67.4400 0.653557
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −85.8960 −0.647908
\(27\) 0 0
\(28\) −16.9120 −0.114145
\(29\) 47.4560 0.303874 0.151937 − 0.988390i \(-0.451449\pi\)
0.151937 + 0.988390i \(0.451449\pi\)
\(30\) 0 0
\(31\) 65.0720 0.377009 0.188505 − 0.982072i \(-0.439636\pi\)
0.188505 + 0.982072i \(0.439636\pi\)
\(32\) 32.0000 0.176777
\(33\) 0 0
\(34\) −14.9840 −0.0755806
\(35\) 21.1400 0.102095
\(36\) 0 0
\(37\) −215.596 −0.957940 −0.478970 − 0.877831i \(-0.658990\pi\)
−0.478970 + 0.877831i \(0.658990\pi\)
\(38\) −51.4400 −0.219597
\(39\) 0 0
\(40\) −40.0000 −0.158114
\(41\) −150.128 −0.571856 −0.285928 − 0.958251i \(-0.592302\pi\)
−0.285928 + 0.958251i \(0.592302\pi\)
\(42\) 0 0
\(43\) −83.0160 −0.294414 −0.147207 − 0.989106i \(-0.547028\pi\)
−0.147207 + 0.989106i \(0.547028\pi\)
\(44\) 134.880 0.462135
\(45\) 0 0
\(46\) 46.0000 0.147442
\(47\) 278.140 0.863210 0.431605 − 0.902063i \(-0.357947\pi\)
0.431605 + 0.902063i \(0.357947\pi\)
\(48\) 0 0
\(49\) −325.124 −0.947883
\(50\) 50.0000 0.141421
\(51\) 0 0
\(52\) −171.792 −0.458140
\(53\) 182.404 0.472738 0.236369 − 0.971663i \(-0.424043\pi\)
0.236369 + 0.971663i \(0.424043\pi\)
\(54\) 0 0
\(55\) −168.600 −0.413346
\(56\) −33.8240 −0.0807129
\(57\) 0 0
\(58\) 94.9120 0.214872
\(59\) 270.060 0.595913 0.297956 − 0.954580i \(-0.403695\pi\)
0.297956 + 0.954580i \(0.403695\pi\)
\(60\) 0 0
\(61\) −137.264 −0.288112 −0.144056 − 0.989570i \(-0.546015\pi\)
−0.144056 + 0.989570i \(0.546015\pi\)
\(62\) 130.144 0.266586
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 214.740 0.409773
\(66\) 0 0
\(67\) −440.652 −0.803496 −0.401748 − 0.915750i \(-0.631597\pi\)
−0.401748 + 0.915750i \(0.631597\pi\)
\(68\) −29.9681 −0.0534436
\(69\) 0 0
\(70\) 42.2800 0.0721918
\(71\) −1071.29 −1.79068 −0.895342 − 0.445380i \(-0.853069\pi\)
−0.895342 + 0.445380i \(0.853069\pi\)
\(72\) 0 0
\(73\) −269.756 −0.432501 −0.216250 − 0.976338i \(-0.569383\pi\)
−0.216250 + 0.976338i \(0.569383\pi\)
\(74\) −431.192 −0.677366
\(75\) 0 0
\(76\) −102.880 −0.155278
\(77\) −142.568 −0.211002
\(78\) 0 0
\(79\) 195.232 0.278042 0.139021 − 0.990289i \(-0.455604\pi\)
0.139021 + 0.990289i \(0.455604\pi\)
\(80\) −80.0000 −0.111803
\(81\) 0 0
\(82\) −300.256 −0.404363
\(83\) −136.676 −0.180748 −0.0903742 − 0.995908i \(-0.528806\pi\)
−0.0903742 + 0.995908i \(0.528806\pi\)
\(84\) 0 0
\(85\) 37.4601 0.0478014
\(86\) −166.032 −0.208182
\(87\) 0 0
\(88\) 269.760 0.326779
\(89\) −629.896 −0.750212 −0.375106 − 0.926982i \(-0.622394\pi\)
−0.375106 + 0.926982i \(0.622394\pi\)
\(90\) 0 0
\(91\) 181.584 0.209178
\(92\) 92.0000 0.104257
\(93\) 0 0
\(94\) 556.280 0.610382
\(95\) 128.600 0.138885
\(96\) 0 0
\(97\) −419.392 −0.438998 −0.219499 − 0.975613i \(-0.570442\pi\)
−0.219499 + 0.975613i \(0.570442\pi\)
\(98\) −650.248 −0.670255
\(99\) 0 0
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 2070.4.a.s.1.2 2
3.2 odd 2 230.4.a.f.1.2 ✓ 2
12.11 even 2 1840.4.a.i.1.1 2
15.2 even 4 1150.4.b.k.599.1 4
15.8 even 4 1150.4.b.k.599.4 4
15.14 odd 2 1150.4.a.l.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.f.1.2 ✓ 2 3.2 odd 2
1150.4.a.l.1.1 2 15.14 odd 2
1150.4.b.k.599.1 4 15.2 even 4
1150.4.b.k.599.4 4 15.8 even 4
1840.4.a.i.1.1 2 12.11 even 2
2070.4.a.s.1.2 2 1.1 even 1 trivial