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Properties

Label 350.4.a.r.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 = [1,2,1,4,0,2,-7,8,-26,0,-65] 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: no (minimal twist has level 70)
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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +1.00000 q^{3} +4.00000 q^{4} +2.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} -26.0000 q^{9} -65.0000 q^{11} +4.00000 q^{12} -13.0000 q^{13} -14.0000 q^{14} +16.0000 q^{16} +73.0000 q^{17} -52.0000 q^{18} -142.000 q^{19} -7.00000 q^{21} -130.000 q^{22} -130.000 q^{23} +8.00000 q^{24} -26.0000 q^{26} -53.0000 q^{27} -28.0000 q^{28} +111.000 q^{29} +256.000 q^{31} +32.0000 q^{32} -65.0000 q^{33} +146.000 q^{34} -104.000 q^{36} +266.000 q^{37} -284.000 q^{38} -13.0000 q^{39} -424.000 q^{41} -14.0000 q^{42} -534.000 q^{43} -260.000 q^{44} -260.000 q^{46} +269.000 q^{47} +16.0000 q^{48} +49.0000 q^{49} +73.0000 q^{51} -52.0000 q^{52} +132.000 q^{53} -106.000 q^{54} -56.0000 q^{56} -142.000 q^{57} +222.000 q^{58} -224.000 q^{59} -572.000 q^{61} +512.000 q^{62} +182.000 q^{63} +64.0000 q^{64} -130.000 q^{66} +108.000 q^{67} +292.000 q^{68} -130.000 q^{69} +560.000 q^{71} -208.000 q^{72} -586.000 q^{73} +532.000 q^{74} -568.000 q^{76} +455.000 q^{77} -26.0000 q^{78} +57.0000 q^{79} +649.000 q^{81} -848.000 q^{82} -252.000 q^{83} -28.0000 q^{84} -1068.00 q^{86} +111.000 q^{87} -520.000 q^{88} -184.000 q^{89} +91.0000 q^{91} -520.000 q^{92} +256.000 q^{93} +538.000 q^{94} +32.0000 q^{96} +605.000 q^{97} +98.0000 q^{98} +1690.00 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.192450 0.0962250 − 0.995360i \(-0.469323\pi\)
0.0962250 + 0.995360i \(0.469323\pi\)
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) 2.00000 0.136083
\(7\) −7.00000 −0.377964
\(8\) 8.00000 0.353553
\(9\) −26.0000 −0.962963
\(10\) 0 0
\(11\) −65.0000 −1.78166 −0.890829 − 0.454339i \(-0.849876\pi\)
−0.890829 + 0.454339i \(0.849876\pi\)
\(12\) 4.00000 0.0962250
\(13\) −13.0000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) −14.0000 −0.267261
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 73.0000 1.04148 0.520738 − 0.853716i \(-0.325657\pi\)
0.520738 + 0.853716i \(0.325657\pi\)
\(18\) −52.0000 −0.680918
\(19\) −142.000 −1.71458 −0.857290 − 0.514833i \(-0.827854\pi\)
−0.857290 + 0.514833i \(0.827854\pi\)
\(20\) 0 0
\(21\) −7.00000 −0.0727393
\(22\) −130.000 −1.25982
\(23\) −130.000 −1.17856 −0.589280 − 0.807929i \(-0.700588\pi\)
−0.589280 + 0.807929i \(0.700588\pi\)
\(24\) 8.00000 0.0680414
\(25\) 0 0
\(26\) −26.0000 −0.196116
\(27\) −53.0000 −0.377772
\(28\) −28.0000 −0.188982
\(29\) 111.000 0.710765 0.355382 − 0.934721i \(-0.384351\pi\)
0.355382 + 0.934721i \(0.384351\pi\)
\(30\) 0 0
\(31\) 256.000 1.48319 0.741596 − 0.670847i \(-0.234069\pi\)
0.741596 + 0.670847i \(0.234069\pi\)
\(32\) 32.0000 0.176777
\(33\) −65.0000 −0.342880
\(34\) 146.000 0.736435
\(35\) 0 0
\(36\) −104.000 −0.481481
\(37\) 266.000 1.18190 0.590948 − 0.806710i \(-0.298754\pi\)
0.590948 + 0.806710i \(0.298754\pi\)
\(38\) −284.000 −1.21239
\(39\) −13.0000 −0.0533761
\(40\) 0 0
\(41\) −424.000 −1.61507 −0.807533 − 0.589823i \(-0.799198\pi\)
−0.807533 + 0.589823i \(0.799198\pi\)
\(42\) −14.0000 −0.0514344
\(43\) −534.000 −1.89382 −0.946910 − 0.321500i \(-0.895813\pi\)
−0.946910 + 0.321500i \(0.895813\pi\)
\(44\) −260.000 −0.890829
\(45\) 0 0
\(46\) −260.000 −0.833368
\(47\) 269.000 0.834844 0.417422 − 0.908713i \(-0.362934\pi\)
0.417422 + 0.908713i \(0.362934\pi\)
\(48\) 16.0000 0.0481125
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 73.0000 0.200432
\(52\) −52.0000 −0.138675
\(53\) 132.000 0.342106 0.171053 − 0.985262i \(-0.445283\pi\)
0.171053 + 0.985262i \(0.445283\pi\)
\(54\) −106.000 −0.267125
\(55\) 0 0
\(56\) −56.0000 −0.133631
\(57\) −142.000 −0.329971
\(58\) 222.000 0.502587
\(59\) −224.000 −0.494277 −0.247138 − 0.968980i \(-0.579490\pi\)
−0.247138 + 0.968980i \(0.579490\pi\)
\(60\) 0 0
\(61\) −572.000 −1.20061 −0.600304 − 0.799772i \(-0.704954\pi\)
−0.600304 + 0.799772i \(0.704954\pi\)
\(62\) 512.000 1.04878
\(63\) 182.000 0.363966
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) −130.000 −0.242453
\(67\) 108.000 0.196930 0.0984649 − 0.995141i \(-0.468607\pi\)
0.0984649 + 0.995141i \(0.468607\pi\)
\(68\) 292.000 0.520738
\(69\) −130.000 −0.226814
\(70\) 0 0
\(71\) 560.000 0.936053 0.468027 − 0.883714i \(-0.344965\pi\)
0.468027 + 0.883714i \(0.344965\pi\)
\(72\) −208.000 −0.340459
\(73\) −586.000 −0.939536 −0.469768 − 0.882790i \(-0.655662\pi\)
−0.469768 + 0.882790i \(0.655662\pi\)
\(74\) 532.000 0.835726
\(75\) 0 0
\(76\) −568.000 −0.857290
\(77\) 455.000 0.673403
\(78\) −26.0000 −0.0377426
\(79\) 57.0000 0.0811772 0.0405886 − 0.999176i \(-0.487077\pi\)
0.0405886 + 0.999176i \(0.487077\pi\)
\(80\) 0 0
\(81\) 649.000 0.890261
\(82\) −848.000 −1.14202
\(83\) −252.000 −0.333260 −0.166630 − 0.986019i \(-0.553289\pi\)
−0.166630 + 0.986019i \(0.553289\pi\)
\(84\) −28.0000 −0.0363696
\(85\) 0 0
\(86\) −1068.00 −1.33913
\(87\) 111.000 0.136787
\(88\) −520.000 −0.629911
\(89\) −184.000 −0.219146 −0.109573 − 0.993979i \(-0.534948\pi\)
−0.109573 + 0.993979i \(0.534948\pi\)
\(90\) 0 0
\(91\) 91.0000 0.104828
\(92\) −520.000 −0.589280
\(93\) 256.000 0.285440
\(94\) 538.000 0.590324
\(95\) 0 0
\(96\) 32.0000 0.0340207
\(97\) 605.000 0.633283 0.316641 − 0.948545i \(-0.397445\pi\)
0.316641 + 0.948545i \(0.397445\pi\)
\(98\) 98.0000 0.101015
\(99\) 1690.00 1.71567
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.r.1.1 1
5.2 odd 4 350.4.c.h.99.2 2
5.3 odd 4 350.4.c.h.99.1 2
5.4 even 2 70.4.a.c.1.1 ✓ 1
7.6 odd 2 2450.4.a.bc.1.1 1
15.14 odd 2 630.4.a.x.1.1 1
20.19 odd 2 560.4.a.i.1.1 1
35.4 even 6 490.4.e.o.471.1 2
35.9 even 6 490.4.e.o.361.1 2
35.19 odd 6 490.4.e.n.361.1 2
35.24 odd 6 490.4.e.n.471.1 2
35.34 odd 2 490.4.a.d.1.1 1
40.19 odd 2 2240.4.a.r.1.1 1
40.29 even 2 2240.4.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.4.a.c.1.1 ✓ 1 5.4 even 2
350.4.a.r.1.1 1 1.1 even 1 trivial
350.4.c.h.99.1 2 5.3 odd 4
350.4.c.h.99.2 2 5.2 odd 4
490.4.a.d.1.1 1 35.34 odd 2
490.4.e.n.361.1 2 35.19 odd 6
490.4.e.n.471.1 2 35.24 odd 6
490.4.e.o.361.1 2 35.9 even 6
490.4.e.o.471.1 2 35.4 even 6
560.4.a.i.1.1 1 20.19 odd 2
630.4.a.x.1.1 1 15.14 odd 2
2240.4.a.r.1.1 1 40.19 odd 2
2240.4.a.v.1.1 1 40.29 even 2
2450.4.a.bc.1.1 1 7.6 odd 2