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Properties

Label 350.4.c.f.99.1
Level $350$
Weight $4$
Character 350.99
Analytic conductor $20.651$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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(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,-8,0,0,46,0,-54] 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} -2.00000i q^{3} -4.00000 q^{4} -4.00000 q^{6} -7.00000i q^{7} +8.00000i q^{8} +23.0000 q^{9} -27.0000 q^{11} +8.00000i q^{12} -64.0000i q^{13} -14.0000 q^{14} +16.0000 q^{16} +24.0000i q^{17} -46.0000i q^{18} -62.0000 q^{19} -14.0000 q^{21} +54.0000i q^{22} -105.000i q^{23} +16.0000 q^{24} -128.000 q^{26} -100.000i q^{27} +28.0000i q^{28} -141.000 q^{29} -124.000 q^{31} -32.0000i q^{32} +54.0000i q^{33} +48.0000 q^{34} -92.0000 q^{36} +439.000i q^{37} +124.000i q^{38} -128.000 q^{39} -354.000 q^{41} +28.0000i q^{42} -211.000i q^{43} +108.000 q^{44} -210.000 q^{46} +102.000i q^{47} -32.0000i q^{48} -49.0000 q^{49} +48.0000 q^{51} +256.000i q^{52} -306.000i q^{53} -200.000 q^{54} +56.0000 q^{56} +124.000i q^{57} +282.000i q^{58} -348.000 q^{59} +410.000 q^{61} +248.000i q^{62} -161.000i q^{63} -64.0000 q^{64} +108.000 q^{66} +349.000i q^{67} -96.0000i q^{68} -210.000 q^{69} -339.000 q^{71} +184.000i q^{72} -70.0000i q^{73} +878.000 q^{74} +248.000 q^{76} +189.000i q^{77} +256.000i q^{78} -731.000 q^{79} +421.000 q^{81} +708.000i q^{82} +528.000i q^{83} +56.0000 q^{84} -422.000 q^{86} +282.000i q^{87} -216.000i q^{88} -960.000 q^{89} -448.000 q^{91} +420.000i q^{92} +248.000i q^{93} +204.000 q^{94} -64.0000 q^{96} -1340.00i q^{97} +98.0000i q^{98} -621.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} - 8 q^{6} + 46 q^{9} - 54 q^{11} - 28 q^{14} + 32 q^{16} - 124 q^{19} - 28 q^{21} + 32 q^{24} - 256 q^{26} - 282 q^{29} - 248 q^{31} + 96 q^{34} - 184 q^{36} - 256 q^{39} - 708 q^{41} + 216 q^{44}+ \cdots - 1242 q^{99}+O(q^{100}) \) 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
\(3\) − 2.00000i − 0.384900i −0.981307 − 0.192450i \(-0.938357\pi\)
0.981307 − 0.192450i \(-0.0616434\pi\)
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) −4.00000 −0.272166
\(7\) − 7.00000i − 0.377964i
\(8\) 8.00000i 0.353553i
\(9\) 23.0000 0.851852
\(10\) 0 0
\(11\) −27.0000 −0.740073 −0.370037 − 0.929017i \(-0.620655\pi\)
−0.370037 + 0.929017i \(0.620655\pi\)
\(12\) 8.00000i 0.192450i
\(13\) − 64.0000i − 1.36542i −0.730691 − 0.682708i \(-0.760802\pi\)
0.730691 − 0.682708i \(-0.239198\pi\)
\(14\) −14.0000 −0.267261
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 24.0000i 0.342403i 0.985236 + 0.171202i \(0.0547649\pi\)
−0.985236 + 0.171202i \(0.945235\pi\)
\(18\) − 46.0000i − 0.602350i
\(19\) −62.0000 −0.748620 −0.374310 − 0.927304i \(-0.622120\pi\)
−0.374310 + 0.927304i \(0.622120\pi\)
\(20\) 0 0
\(21\) −14.0000 −0.145479
\(22\) 54.0000i 0.523311i
\(23\) − 105.000i − 0.951914i −0.879469 − 0.475957i \(-0.842102\pi\)
0.879469 − 0.475957i \(-0.157898\pi\)
\(24\) 16.0000 0.136083
\(25\) 0 0
\(26\) −128.000 −0.965495
\(27\) − 100.000i − 0.712778i
\(28\) 28.0000i 0.188982i
\(29\) −141.000 −0.902864 −0.451432 − 0.892306i \(-0.649087\pi\)
−0.451432 + 0.892306i \(0.649087\pi\)
\(30\) 0 0
\(31\) −124.000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) − 32.0000i − 0.176777i
\(33\) 54.0000i 0.284854i
\(34\) 48.0000 0.242116
\(35\) 0 0
\(36\) −92.0000 −0.425926
\(37\) 439.000i 1.95057i 0.220947 + 0.975286i \(0.429085\pi\)
−0.220947 + 0.975286i \(0.570915\pi\)
\(38\) 124.000i 0.529354i
\(39\) −128.000 −0.525549
\(40\) 0 0
\(41\) −354.000 −1.34843 −0.674214 − 0.738536i \(-0.735517\pi\)
−0.674214 + 0.738536i \(0.735517\pi\)
\(42\) 28.0000i 0.102869i
\(43\) − 211.000i − 0.748307i −0.927367 − 0.374153i \(-0.877933\pi\)
0.927367 − 0.374153i \(-0.122067\pi\)
\(44\) 108.000 0.370037
\(45\) 0 0
\(46\) −210.000 −0.673105
\(47\) 102.000i 0.316558i 0.987394 + 0.158279i \(0.0505946\pi\)
−0.987394 + 0.158279i \(0.949405\pi\)
\(48\) − 32.0000i − 0.0962250i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) 48.0000 0.131791
\(52\) 256.000i 0.682708i
\(53\) − 306.000i − 0.793063i −0.918021 − 0.396531i \(-0.870214\pi\)
0.918021 − 0.396531i \(-0.129786\pi\)
\(54\) −200.000 −0.504010
\(55\) 0 0
\(56\) 56.0000 0.133631
\(57\) 124.000i 0.288144i
\(58\) 282.000i 0.638421i
\(59\) −348.000 −0.767894 −0.383947 − 0.923355i \(-0.625435\pi\)
−0.383947 + 0.923355i \(0.625435\pi\)
\(60\) 0 0
\(61\) 410.000 0.860576 0.430288 − 0.902692i \(-0.358412\pi\)
0.430288 + 0.902692i \(0.358412\pi\)
\(62\) 248.000i 0.508001i
\(63\) − 161.000i − 0.321970i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 108.000 0.201422
\(67\) 349.000i 0.636375i 0.948028 + 0.318188i \(0.103074\pi\)
−0.948028 + 0.318188i \(0.896926\pi\)
\(68\) − 96.0000i − 0.171202i
\(69\) −210.000 −0.366392
\(70\) 0 0
\(71\) −339.000 −0.566646 −0.283323 − 0.959024i \(-0.591437\pi\)
−0.283323 + 0.959024i \(0.591437\pi\)
\(72\) 184.000i 0.301175i
\(73\) − 70.0000i − 0.112231i −0.998424 − 0.0561156i \(-0.982128\pi\)
0.998424 − 0.0561156i \(-0.0178715\pi\)
\(74\) 878.000 1.37926
\(75\) 0 0
\(76\) 248.000 0.374310
\(77\) 189.000i 0.279721i
\(78\) 256.000i 0.371619i
\(79\) −731.000 −1.04106 −0.520531 − 0.853843i \(-0.674266\pi\)
−0.520531 + 0.853843i \(0.674266\pi\)
\(80\) 0 0
\(81\) 421.000 0.577503
\(82\) 708.000i 0.953482i
\(83\) 528.000i 0.698259i 0.937074 + 0.349130i \(0.113523\pi\)
−0.937074 + 0.349130i \(0.886477\pi\)
\(84\) 56.0000 0.0727393
\(85\) 0 0
\(86\) −422.000 −0.529133
\(87\) 282.000i 0.347512i
\(88\) − 216.000i − 0.261655i
\(89\) −960.000 −1.14337 −0.571684 − 0.820474i \(-0.693710\pi\)
−0.571684 + 0.820474i \(0.693710\pi\)
\(90\) 0 0
\(91\) −448.000 −0.516079
\(92\) 420.000i 0.475957i
\(93\) 248.000i 0.276520i
\(94\) 204.000 0.223840
\(95\) 0 0
\(96\) −64.0000 −0.0680414
\(97\) − 1340.00i − 1.40264i −0.712845 − 0.701322i \(-0.752594\pi\)
0.712845 − 0.701322i \(-0.247406\pi\)
\(98\) 98.0000i 0.101015i
\(99\) −621.000 −0.630433
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.f.99.1 2
5.2 odd 4 350.4.a.q.1.1 yes 1
5.3 odd 4 350.4.a.e.1.1 ✓ 1
5.4 even 2 inner 350.4.c.f.99.2 2
35.13 even 4 2450.4.a.h.1.1 1
35.27 even 4 2450.4.a.bi.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
350.4.a.e.1.1 ✓ 1 5.3 odd 4
350.4.a.q.1.1 yes 1 5.2 odd 4
350.4.c.f.99.1 2 1.1 even 1 trivial
350.4.c.f.99.2 2 5.4 even 2 inner
2450.4.a.h.1.1 1 35.13 even 4
2450.4.a.bi.1.1 1 35.27 even 4