Properties

Label 350.4.a.k.1.1
Level $350$
Weight $4$
Character 350.1
Self dual yes
Analytic conductor $20.651$
Analytic rank $1$
Dimension $1$
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] = [350,4,Mod(1,350)] 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("350.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 350.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,2,-10,4,0,-20,-7,8,73,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(20.6506685020\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 350.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} -10.0000 q^{3} +4.00000 q^{4} -20.0000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +73.0000 q^{9} +9.00000 q^{11} -40.0000 q^{12} +52.0000 q^{13} -14.0000 q^{14} +16.0000 q^{16} -96.0000 q^{17} +146.000 q^{18} -10.0000 q^{19} +70.0000 q^{21} +18.0000 q^{22} -75.0000 q^{23} -80.0000 q^{24} +104.000 q^{26} -460.000 q^{27} -28.0000 q^{28} +189.000 q^{29} -232.000 q^{31} +32.0000 q^{32} -90.0000 q^{33} -192.000 q^{34} +292.000 q^{36} -305.000 q^{37} -20.0000 q^{38} -520.000 q^{39} -438.000 q^{41} +140.000 q^{42} -353.000 q^{43} +36.0000 q^{44} -150.000 q^{46} +486.000 q^{47} -160.000 q^{48} +49.0000 q^{49} +960.000 q^{51} +208.000 q^{52} +354.000 q^{53} -920.000 q^{54} -56.0000 q^{56} +100.000 q^{57} +378.000 q^{58} -672.000 q^{59} +206.000 q^{61} -464.000 q^{62} -511.000 q^{63} +64.0000 q^{64} -180.000 q^{66} -599.000 q^{67} -384.000 q^{68} +750.000 q^{69} -471.000 q^{71} +584.000 q^{72} -614.000 q^{73} -610.000 q^{74} -40.0000 q^{76} -63.0000 q^{77} -1040.00 q^{78} +743.000 q^{79} +2629.00 q^{81} -876.000 q^{82} -996.000 q^{83} +280.000 q^{84} -706.000 q^{86} -1890.00 q^{87} +72.0000 q^{88} +180.000 q^{89} -364.000 q^{91} -300.000 q^{92} +2320.00 q^{93} +972.000 q^{94} -320.000 q^{96} +184.000 q^{97} +98.0000 q^{98} +657.000 q^{99} +O(q^{100})\)

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
\(3\) −10.0000 −1.92450 −0.962250 − 0.272166i \(-0.912260\pi\)
−0.962250 + 0.272166i \(0.912260\pi\)
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) −20.0000 −1.36083
\(7\) −7.00000 −0.377964
\(8\) 8.00000 0.353553
\(9\) 73.0000 2.70370
\(10\) 0 0
\(11\) 9.00000 0.246691 0.123346 − 0.992364i \(-0.460638\pi\)
0.123346 + 0.992364i \(0.460638\pi\)
\(12\) −40.0000 −0.962250
\(13\) 52.0000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) −14.0000 −0.267261
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −96.0000 −1.36961 −0.684806 − 0.728725i \(-0.740113\pi\)
−0.684806 + 0.728725i \(0.740113\pi\)
\(18\) 146.000 1.91181
\(19\) −10.0000 −0.120745 −0.0603726 − 0.998176i \(-0.519229\pi\)
−0.0603726 + 0.998176i \(0.519229\pi\)
\(20\) 0 0
\(21\) 70.0000 0.727393
\(22\) 18.0000 0.174437
\(23\) −75.0000 −0.679938 −0.339969 − 0.940437i \(-0.610417\pi\)
−0.339969 + 0.940437i \(0.610417\pi\)
\(24\) −80.0000 −0.680414
\(25\) 0 0
\(26\) 104.000 0.784465
\(27\) −460.000 −3.27878
\(28\) −28.0000 −0.188982
\(29\) 189.000 1.21022 0.605111 − 0.796141i \(-0.293129\pi\)
0.605111 + 0.796141i \(0.293129\pi\)
\(30\) 0 0
\(31\) −232.000 −1.34414 −0.672071 − 0.740486i \(-0.734595\pi\)
−0.672071 + 0.740486i \(0.734595\pi\)
\(32\) 32.0000 0.176777
\(33\) −90.0000 −0.474757
\(34\) −192.000 −0.968463
\(35\) 0 0
\(36\) 292.000 1.35185
\(37\) −305.000 −1.35518 −0.677590 − 0.735439i \(-0.736976\pi\)
−0.677590 + 0.735439i \(0.736976\pi\)
\(38\) −20.0000 −0.0853797
\(39\) −520.000 −2.13504
\(40\) 0 0
\(41\) −438.000 −1.66839 −0.834196 − 0.551467i \(-0.814068\pi\)
−0.834196 + 0.551467i \(0.814068\pi\)
\(42\) 140.000 0.514344
\(43\) −353.000 −1.25191 −0.625953 − 0.779860i \(-0.715290\pi\)
−0.625953 + 0.779860i \(0.715290\pi\)
\(44\) 36.0000 0.123346
\(45\) 0 0
\(46\) −150.000 −0.480789
\(47\) 486.000 1.50831 0.754153 − 0.656699i \(-0.228048\pi\)
0.754153 + 0.656699i \(0.228048\pi\)
\(48\) −160.000 −0.481125
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 960.000 2.63582
\(52\) 208.000 0.554700
\(53\) 354.000 0.917465 0.458732 − 0.888574i \(-0.348304\pi\)
0.458732 + 0.888574i \(0.348304\pi\)
\(54\) −920.000 −2.31845
\(55\) 0 0
\(56\) −56.0000 −0.133631
\(57\) 100.000 0.232374
\(58\) 378.000 0.855756
\(59\) −672.000 −1.48283 −0.741415 − 0.671047i \(-0.765845\pi\)
−0.741415 + 0.671047i \(0.765845\pi\)
\(60\) 0 0
\(61\) 206.000 0.432387 0.216193 − 0.976351i \(-0.430636\pi\)
0.216193 + 0.976351i \(0.430636\pi\)
\(62\) −464.000 −0.950453
\(63\) −511.000 −1.02190
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) −180.000 −0.335704
\(67\) −599.000 −1.09223 −0.546116 − 0.837710i \(-0.683894\pi\)
−0.546116 + 0.837710i \(0.683894\pi\)
\(68\) −384.000 −0.684806
\(69\) 750.000 1.30854
\(70\) 0 0
\(71\) −471.000 −0.787288 −0.393644 − 0.919263i \(-0.628786\pi\)
−0.393644 + 0.919263i \(0.628786\pi\)
\(72\) 584.000 0.955904
\(73\) −614.000 −0.984428 −0.492214 − 0.870474i \(-0.663812\pi\)
−0.492214 + 0.870474i \(0.663812\pi\)
\(74\) −610.000 −0.958258
\(75\) 0 0
\(76\) −40.0000 −0.0603726
\(77\) −63.0000 −0.0932405
\(78\) −1040.00 −1.50970
\(79\) 743.000 1.05815 0.529076 − 0.848574i \(-0.322539\pi\)
0.529076 + 0.848574i \(0.322539\pi\)
\(80\) 0 0
\(81\) 2629.00 3.60631
\(82\) −876.000 −1.17973
\(83\) −996.000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 280.000 0.363696
\(85\) 0 0
\(86\) −706.000 −0.885232
\(87\) −1890.00 −2.32907
\(88\) 72.0000 0.0872185
\(89\) 180.000 0.214382 0.107191 − 0.994238i \(-0.465814\pi\)
0.107191 + 0.994238i \(0.465814\pi\)
\(90\) 0 0
\(91\) −364.000 −0.419314
\(92\) −300.000 −0.339969
\(93\) 2320.00 2.58680
\(94\) 972.000 1.06653
\(95\) 0 0
\(96\) −320.000 −0.340207
\(97\) 184.000 0.192602 0.0963009 − 0.995352i \(-0.469299\pi\)
0.0963009 + 0.995352i \(0.469299\pi\)
\(98\) 98.0000 0.101015
\(99\) 657.000 0.666980
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 350.4.a.k.1.1 yes 1
5.2 odd 4 350.4.c.a.99.2 2
5.3 odd 4 350.4.c.a.99.1 2
5.4 even 2 350.4.a.j.1.1 ✓ 1
7.6 odd 2 2450.4.a.bp.1.1 1
35.34 odd 2 2450.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
350.4.a.j.1.1 ✓ 1 5.4 even 2
350.4.a.k.1.1 yes 1 1.1 even 1 trivial
350.4.c.a.99.1 2 5.3 odd 4
350.4.c.a.99.2 2 5.2 odd 4
2450.4.a.a.1.1 1 35.34 odd 2
2450.4.a.bp.1.1 1 7.6 odd 2