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(99,350)] 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("350.99"); 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([1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 350.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-8,0,4,0,0,52,0,-130] 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: no
Analytic conductor: \(20.6506685020\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 350.99
Dual form 350.4.c.h.99.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000i q^{2} -1.00000i q^{3} -4.00000 q^{4} +2.00000 q^{6} -7.00000i q^{7} -8.00000i q^{8} +26.0000 q^{9} -65.0000 q^{11} +4.00000i q^{12} +13.0000i q^{13} +14.0000 q^{14} +16.0000 q^{16} +73.0000i q^{17} +52.0000i q^{18} +142.000 q^{19} -7.00000 q^{21} -130.000i q^{22} +130.000i q^{23} -8.00000 q^{24} -26.0000 q^{26} -53.0000i q^{27} +28.0000i q^{28} -111.000 q^{29} +256.000 q^{31} +32.0000i q^{32} +65.0000i q^{33} -146.000 q^{34} -104.000 q^{36} +266.000i q^{37} +284.000i q^{38} +13.0000 q^{39} -424.000 q^{41} -14.0000i q^{42} +534.000i q^{43} +260.000 q^{44} -260.000 q^{46} +269.000i q^{47} -16.0000i q^{48} -49.0000 q^{49} +73.0000 q^{51} -52.0000i q^{52} -132.000i q^{53} +106.000 q^{54} -56.0000 q^{56} -142.000i q^{57} -222.000i q^{58} +224.000 q^{59} -572.000 q^{61} +512.000i q^{62} -182.000i q^{63} -64.0000 q^{64} -130.000 q^{66} +108.000i q^{67} -292.000i q^{68} +130.000 q^{69} +560.000 q^{71} -208.000i q^{72} +586.000i q^{73} -532.000 q^{74} -568.000 q^{76} +455.000i q^{77} +26.0000i q^{78} -57.0000 q^{79} +649.000 q^{81} -848.000i q^{82} +252.000i q^{83} +28.0000 q^{84} -1068.00 q^{86} +111.000i q^{87} +520.000i q^{88} +184.000 q^{89} +91.0000 q^{91} -520.000i q^{92} -256.000i q^{93} -538.000 q^{94} +32.0000 q^{96} +605.000i q^{97} -98.0000i q^{98} -1690.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} + 4 q^{6} + 52 q^{9} - 130 q^{11} + 28 q^{14} + 32 q^{16} + 284 q^{19} - 14 q^{21} - 16 q^{24} - 52 q^{26} - 222 q^{29} + 512 q^{31} - 292 q^{34} - 208 q^{36} + 26 q^{39} - 848 q^{41} + 520 q^{44}+ \cdots - 3380 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 0.707107i
\(3\) − 1.00000i − 0.192450i −0.995360 − 0.0962250i \(-0.969323\pi\)
0.995360 − 0.0962250i \(-0.0306768\pi\)
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) 2.00000 0.136083
\(7\) − 7.00000i − 0.377964i
\(8\) − 8.00000i − 0.353553i
\(9\) 26.0000 0.962963
\(10\) 0 0
\(11\) −65.0000 −1.78166 −0.890829 − 0.454339i \(-0.849876\pi\)
−0.890829 + 0.454339i \(0.849876\pi\)
\(12\) 4.00000i 0.0962250i
\(13\) 13.0000i 0.277350i 0.990338 + 0.138675i \(0.0442844\pi\)
−0.990338 + 0.138675i \(0.955716\pi\)
\(14\) 14.0000 0.267261
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 73.0000i 1.04148i 0.853716 + 0.520738i \(0.174343\pi\)
−0.853716 + 0.520738i \(0.825657\pi\)
\(18\) 52.0000i 0.680918i
\(19\) 142.000 1.71458 0.857290 − 0.514833i \(-0.172146\pi\)
0.857290 + 0.514833i \(0.172146\pi\)
\(20\) 0 0
\(21\) −7.00000 −0.0727393
\(22\) − 130.000i − 1.25982i
\(23\) 130.000i 1.17856i 0.807929 + 0.589280i \(0.200588\pi\)
−0.807929 + 0.589280i \(0.799412\pi\)
\(24\) −8.00000 −0.0680414
\(25\) 0 0
\(26\) −26.0000 −0.196116
\(27\) − 53.0000i − 0.377772i
\(28\) 28.0000i 0.188982i
\(29\) −111.000 −0.710765 −0.355382 − 0.934721i \(-0.615649\pi\)
−0.355382 + 0.934721i \(0.615649\pi\)
\(30\) 0 0
\(31\) 256.000 1.48319 0.741596 − 0.670847i \(-0.234069\pi\)
0.741596 + 0.670847i \(0.234069\pi\)
\(32\) 32.0000i 0.176777i
\(33\) 65.0000i 0.342880i
\(34\) −146.000 −0.736435
\(35\) 0 0
\(36\) −104.000 −0.481481
\(37\) 266.000i 1.18190i 0.806710 + 0.590948i \(0.201246\pi\)
−0.806710 + 0.590948i \(0.798754\pi\)
\(38\) 284.000i 1.21239i
\(39\) 13.0000 0.0533761
\(40\) 0 0
\(41\) −424.000 −1.61507 −0.807533 − 0.589823i \(-0.799198\pi\)
−0.807533 + 0.589823i \(0.799198\pi\)
\(42\) − 14.0000i − 0.0514344i
\(43\) 534.000i 1.89382i 0.321500 + 0.946910i \(0.395813\pi\)
−0.321500 + 0.946910i \(0.604187\pi\)
\(44\) 260.000 0.890829
\(45\) 0 0
\(46\) −260.000 −0.833368
\(47\) 269.000i 0.834844i 0.908713 + 0.417422i \(0.137066\pi\)
−0.908713 + 0.417422i \(0.862934\pi\)
\(48\) − 16.0000i − 0.0481125i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) 73.0000 0.200432
\(52\) − 52.0000i − 0.138675i
\(53\) − 132.000i − 0.342106i −0.985262 − 0.171053i \(-0.945283\pi\)
0.985262 − 0.171053i \(-0.0547169\pi\)
\(54\) 106.000 0.267125
\(55\) 0 0
\(56\) −56.0000 −0.133631
\(57\) − 142.000i − 0.329971i
\(58\) − 222.000i − 0.502587i
\(59\) 224.000 0.494277 0.247138 − 0.968980i \(-0.420510\pi\)
0.247138 + 0.968980i \(0.420510\pi\)
\(60\) 0 0
\(61\) −572.000 −1.20061 −0.600304 − 0.799772i \(-0.704954\pi\)
−0.600304 + 0.799772i \(0.704954\pi\)
\(62\) 512.000i 1.04878i
\(63\) − 182.000i − 0.363966i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) −130.000 −0.242453
\(67\) 108.000i 0.196930i 0.995141 + 0.0984649i \(0.0313932\pi\)
−0.995141 + 0.0984649i \(0.968607\pi\)
\(68\) − 292.000i − 0.520738i
\(69\) 130.000 0.226814
\(70\) 0 0
\(71\) 560.000 0.936053 0.468027 − 0.883714i \(-0.344965\pi\)
0.468027 + 0.883714i \(0.344965\pi\)
\(72\) − 208.000i − 0.340459i
\(73\) 586.000i 0.939536i 0.882790 + 0.469768i \(0.155662\pi\)
−0.882790 + 0.469768i \(0.844338\pi\)
\(74\) −532.000 −0.835726
\(75\) 0 0
\(76\) −568.000 −0.857290
\(77\) 455.000i 0.673403i
\(78\) 26.0000i 0.0377426i
\(79\) −57.0000 −0.0811772 −0.0405886 − 0.999176i \(-0.512923\pi\)
−0.0405886 + 0.999176i \(0.512923\pi\)
\(80\) 0 0
\(81\) 649.000 0.890261
\(82\) − 848.000i − 1.14202i
\(83\) 252.000i 0.333260i 0.986019 + 0.166630i \(0.0532886\pi\)
−0.986019 + 0.166630i \(0.946711\pi\)
\(84\) 28.0000 0.0363696
\(85\) 0 0
\(86\) −1068.00 −1.33913
\(87\) 111.000i 0.136787i
\(88\) 520.000i 0.629911i
\(89\) 184.000 0.219146 0.109573 − 0.993979i \(-0.465052\pi\)
0.109573 + 0.993979i \(0.465052\pi\)
\(90\) 0 0
\(91\) 91.0000 0.104828
\(92\) − 520.000i − 0.589280i
\(93\) − 256.000i − 0.285440i
\(94\) −538.000 −0.590324
\(95\) 0 0
\(96\) 32.0000 0.0340207
\(97\) 605.000i 0.633283i 0.948545 + 0.316641i \(0.102555\pi\)
−0.948545 + 0.316641i \(0.897445\pi\)
\(98\) − 98.0000i − 0.101015i
\(99\) −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.c.h.99.2 2
5.2 odd 4 70.4.a.c.1.1 ✓ 1
5.3 odd 4 350.4.a.r.1.1 1
5.4 even 2 inner 350.4.c.h.99.1 2
15.2 even 4 630.4.a.x.1.1 1
20.7 even 4 560.4.a.i.1.1 1
35.2 odd 12 490.4.e.o.361.1 2
35.12 even 12 490.4.e.n.361.1 2
35.13 even 4 2450.4.a.bc.1.1 1
35.17 even 12 490.4.e.n.471.1 2
35.27 even 4 490.4.a.d.1.1 1
35.32 odd 12 490.4.e.o.471.1 2
40.27 even 4 2240.4.a.r.1.1 1
40.37 odd 4 2240.4.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.4.a.c.1.1 ✓ 1 5.2 odd 4
350.4.a.r.1.1 1 5.3 odd 4
350.4.c.h.99.1 2 5.4 even 2 inner
350.4.c.h.99.2 2 1.1 even 1 trivial
490.4.a.d.1.1 1 35.27 even 4
490.4.e.n.361.1 2 35.12 even 12
490.4.e.n.471.1 2 35.17 even 12
490.4.e.o.361.1 2 35.2 odd 12
490.4.e.o.471.1 2 35.32 odd 12
560.4.a.i.1.1 1 20.7 even 4
630.4.a.x.1.1 1 15.2 even 4
2240.4.a.r.1.1 1 40.27 even 4
2240.4.a.v.1.1 1 40.37 odd 4
2450.4.a.bc.1.1 1 35.13 even 4