Properties

Label 350.6.a.c.1.1
Level $350$
Weight $6$
Character 350.1
Self dual yes
Analytic conductor $56.134$
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] = [350,6,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, 6); 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, 6, names="a")
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 350.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(56.1343369345\)
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)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} +16.0000 q^{4} +49.0000 q^{7} -64.0000 q^{8} -243.000 q^{9} +384.000 q^{11} -236.000 q^{13} -196.000 q^{14} +256.000 q^{16} +1172.00 q^{17} +972.000 q^{18} -1100.00 q^{19} -1536.00 q^{22} -1400.00 q^{23} +944.000 q^{26} +784.000 q^{28} -3854.00 q^{29} +88.0000 q^{31} -1024.00 q^{32} -4688.00 q^{34} -3888.00 q^{36} +13240.0 q^{37} +4400.00 q^{38} -13338.0 q^{41} +2504.00 q^{43} +6144.00 q^{44} +5600.00 q^{46} +14728.0 q^{47} +2401.00 q^{49} -3776.00 q^{52} -11232.0 q^{53} -3136.00 q^{56} +15416.0 q^{58} +652.000 q^{59} -1494.00 q^{61} -352.000 q^{62} -11907.0 q^{63} +4096.00 q^{64} -18232.0 q^{67} +18752.0 q^{68} -28356.0 q^{71} +15552.0 q^{72} +70892.0 q^{73} -52960.0 q^{74} -17600.0 q^{76} +18816.0 q^{77} -79828.0 q^{79} +59049.0 q^{81} +53352.0 q^{82} -83712.0 q^{83} -10016.0 q^{86} -24576.0 q^{88} -93290.0 q^{89} -11564.0 q^{91} -22400.0 q^{92} -58912.0 q^{94} -91068.0 q^{97} -9604.00 q^{98} -93312.0 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.707107
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 16.0000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 49.0000 0.377964
\(8\) −64.0000 −0.353553
\(9\) −243.000 −1.00000
\(10\) 0 0
\(11\) 384.000 0.956862 0.478431 − 0.878125i \(-0.341206\pi\)
0.478431 + 0.878125i \(0.341206\pi\)
\(12\) 0 0
\(13\) −236.000 −0.387305 −0.193653 − 0.981070i \(-0.562034\pi\)
−0.193653 + 0.981070i \(0.562034\pi\)
\(14\) −196.000 −0.267261
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) 1172.00 0.983570 0.491785 − 0.870717i \(-0.336345\pi\)
0.491785 + 0.870717i \(0.336345\pi\)
\(18\) 972.000 0.707107
\(19\) −1100.00 −0.699051 −0.349525 − 0.936927i \(-0.613657\pi\)
−0.349525 + 0.936927i \(0.613657\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1536.00 −0.676604
\(23\) −1400.00 −0.551834 −0.275917 − 0.961181i \(-0.588981\pi\)
−0.275917 + 0.961181i \(0.588981\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 944.000 0.273866
\(27\) 0 0
\(28\) 784.000 0.188982
\(29\) −3854.00 −0.850975 −0.425487 − 0.904964i \(-0.639897\pi\)
−0.425487 + 0.904964i \(0.639897\pi\)
\(30\) 0 0
\(31\) 88.0000 0.0164467 0.00822334 − 0.999966i \(-0.497382\pi\)
0.00822334 + 0.999966i \(0.497382\pi\)
\(32\) −1024.00 −0.176777
\(33\) 0 0
\(34\) −4688.00 −0.695489
\(35\) 0 0
\(36\) −3888.00 −0.500000
\(37\) 13240.0 1.58995 0.794975 − 0.606642i \(-0.207484\pi\)
0.794975 + 0.606642i \(0.207484\pi\)
\(38\) 4400.00 0.494303
\(39\) 0 0
\(40\) 0 0
\(41\) −13338.0 −1.23917 −0.619585 − 0.784929i \(-0.712699\pi\)
−0.619585 + 0.784929i \(0.712699\pi\)
\(42\) 0 0
\(43\) 2504.00 0.206521 0.103260 − 0.994654i \(-0.467073\pi\)
0.103260 + 0.994654i \(0.467073\pi\)
\(44\) 6144.00 0.478431
\(45\) 0 0
\(46\) 5600.00 0.390206
\(47\) 14728.0 0.972521 0.486261 − 0.873814i \(-0.338361\pi\)
0.486261 + 0.873814i \(0.338361\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) −3776.00 −0.193653
\(53\) −11232.0 −0.549247 −0.274623 − 0.961552i \(-0.588553\pi\)
−0.274623 + 0.961552i \(0.588553\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3136.00 −0.133631
\(57\) 0 0
\(58\) 15416.0 0.601730
\(59\) 652.000 0.0243847 0.0121924 − 0.999926i \(-0.496119\pi\)
0.0121924 + 0.999926i \(0.496119\pi\)
\(60\) 0 0
\(61\) −1494.00 −0.0514074 −0.0257037 − 0.999670i \(-0.508183\pi\)
−0.0257037 + 0.999670i \(0.508183\pi\)
\(62\) −352.000 −0.0116296
\(63\) −11907.0 −0.377964
\(64\) 4096.00 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −18232.0 −0.496189 −0.248095 − 0.968736i \(-0.579804\pi\)
−0.248095 + 0.968736i \(0.579804\pi\)
\(68\) 18752.0 0.491785
\(69\) 0 0
\(70\) 0 0
\(71\) −28356.0 −0.667574 −0.333787 − 0.942649i \(-0.608327\pi\)
−0.333787 + 0.942649i \(0.608327\pi\)
\(72\) 15552.0 0.353553
\(73\) 70892.0 1.55701 0.778503 − 0.627641i \(-0.215980\pi\)
0.778503 + 0.627641i \(0.215980\pi\)
\(74\) −52960.0 −1.12426
\(75\) 0 0
\(76\) −17600.0 −0.349525
\(77\) 18816.0 0.361660
\(78\) 0 0
\(79\) −79828.0 −1.43909 −0.719544 − 0.694447i \(-0.755649\pi\)
−0.719544 + 0.694447i \(0.755649\pi\)
\(80\) 0 0
\(81\) 59049.0 1.00000
\(82\) 53352.0 0.876226
\(83\) −83712.0 −1.33381 −0.666903 − 0.745145i \(-0.732380\pi\)
−0.666903 + 0.745145i \(0.732380\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −10016.0 −0.146032
\(87\) 0 0
\(88\) −24576.0 −0.338302
\(89\) −93290.0 −1.24842 −0.624209 − 0.781257i \(-0.714579\pi\)
−0.624209 + 0.781257i \(0.714579\pi\)
\(90\) 0 0
\(91\) −11564.0 −0.146388
\(92\) −22400.0 −0.275917
\(93\) 0 0
\(94\) −58912.0 −0.687676
\(95\) 0 0
\(96\) 0 0
\(97\) −91068.0 −0.982735 −0.491368 − 0.870952i \(-0.663503\pi\)
−0.491368 + 0.870952i \(0.663503\pi\)
\(98\) −9604.00 −0.101015
\(99\) −93312.0 −0.956862
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 350.6.a.c.1.1 1
5.2 odd 4 70.6.c.a.29.1 ✓ 2
5.3 odd 4 70.6.c.a.29.2 yes 2
5.4 even 2 350.6.a.j.1.1 1
15.2 even 4 630.6.g.c.379.2 2
15.8 even 4 630.6.g.c.379.1 2
20.3 even 4 560.6.g.a.449.1 2
20.7 even 4 560.6.g.a.449.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.6.c.a.29.1 ✓ 2 5.2 odd 4
70.6.c.a.29.2 yes 2 5.3 odd 4
350.6.a.c.1.1 1 1.1 even 1 trivial
350.6.a.j.1.1 1 5.4 even 2
560.6.g.a.449.1 2 20.3 even 4
560.6.g.a.449.2 2 20.7 even 4
630.6.g.c.379.1 2 15.8 even 4
630.6.g.c.379.2 2 15.2 even 4