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Properties

Label 175.4.b.e.99.2
Level $175$
Weight $4$
Character 175.99
Analytic conductor $10.325$
Analytic rank $0$
Dimension $6$
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] = [175,4,Mod(99,175)] 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("175.99"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 175.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,-26,0,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.3253342510\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.3299353600.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 34x^{4} + 289x^{2} + 196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.2
Root \(-4.48565i\) of defining polynomial
Character \(\chi\) \(=\) 175.99
Dual form 175.4.b.e.99.5

$q$-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)
 
\(f(q)\) \(=\) \(q-3.48565i q^{2} +0.850238i q^{3} -4.14976 q^{4} +2.96363 q^{6} -7.00000i q^{7} -13.4206i q^{8} +26.2771 q^{9} -6.90764 q^{11} -3.52829i q^{12} -22.1364i q^{13} -24.3996 q^{14} -79.9776 q^{16} -88.3030i q^{17} -91.5928i q^{18} -36.9560 q^{19} +5.95167 q^{21} +24.0776i q^{22} -95.5283i q^{23} +11.4107 q^{24} -77.1598 q^{26} +45.2982i q^{27} +29.0483i q^{28} -269.029 q^{29} +197.114 q^{31} +171.409i q^{32} -5.87314i q^{33} -307.793 q^{34} -109.044 q^{36} -2.14546i q^{37} +128.816i q^{38} +18.8212 q^{39} +174.127 q^{41} -20.7454i q^{42} -17.0345i q^{43} +28.6650 q^{44} -332.978 q^{46} +528.029i q^{47} -68.0000i q^{48} -49.0000 q^{49} +75.0786 q^{51} +91.8608i q^{52} -641.114i q^{53} +157.894 q^{54} -93.9441 q^{56} -31.4214i q^{57} +937.742i q^{58} +642.975 q^{59} +142.967 q^{61} -687.070i q^{62} -183.940i q^{63} -42.3480 q^{64} -20.4717 q^{66} -478.797i q^{67} +366.436i q^{68} +81.2218 q^{69} +105.550 q^{71} -352.654i q^{72} +986.512i q^{73} -7.47834 q^{74} +153.358 q^{76} +48.3534i q^{77} -65.6042i q^{78} +1099.86 q^{79} +670.967 q^{81} -606.947i q^{82} -1236.62i q^{83} -24.6980 q^{84} -59.3763 q^{86} -228.739i q^{87} +92.7045i q^{88} +711.698 q^{89} -154.955 q^{91} +396.420i q^{92} +167.594i q^{93} +1840.52 q^{94} -145.739 q^{96} +636.553i q^{97} +170.797i q^{98} -181.513 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 26 q^{4} + 48 q^{6} - 162 q^{9} - 148 q^{11} + 42 q^{14} - 158 q^{16} - 336 q^{19} + 28 q^{21} - 840 q^{24} - 892 q^{26} - 664 q^{29} + 640 q^{31} - 1164 q^{34} + 362 q^{36} + 1964 q^{39} + 724 q^{41}+ \cdots + 6976 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.48565i − 1.23236i −0.787604 − 0.616182i \(-0.788679\pi\)
0.787604 − 0.616182i \(-0.211321\pi\)
\(3\) 0.850238i 0.163628i 0.996648 + 0.0818142i \(0.0260714\pi\)
−0.996648 + 0.0818142i \(0.973929\pi\)
\(4\) −4.14976 −0.518720
\(5\) 0 0
\(6\) 2.96363 0.201650
\(7\) − 7.00000i − 0.377964i
\(8\) − 13.4206i − 0.593112i
\(9\) 26.2771 0.973226
\(10\) 0 0
\(11\) −6.90764 −0.189339 −0.0946696 − 0.995509i \(-0.530179\pi\)
−0.0946696 + 0.995509i \(0.530179\pi\)
\(12\) − 3.52829i − 0.0848774i
\(13\) − 22.1364i − 0.472272i −0.971720 − 0.236136i \(-0.924119\pi\)
0.971720 − 0.236136i \(-0.0758810\pi\)
\(14\) −24.3996 −0.465790
\(15\) 0 0
\(16\) −79.9776 −1.24965
\(17\) − 88.3030i − 1.25980i −0.776676 − 0.629901i \(-0.783096\pi\)
0.776676 − 0.629901i \(-0.216904\pi\)
\(18\) − 91.5928i − 1.19937i
\(19\) −36.9560 −0.446225 −0.223113 − 0.974793i \(-0.571622\pi\)
−0.223113 + 0.974793i \(0.571622\pi\)
\(20\) 0 0
\(21\) 5.95167 0.0618457
\(22\) 24.0776i 0.233335i
\(23\) − 95.5283i − 0.866045i −0.901383 − 0.433022i \(-0.857447\pi\)
0.901383 − 0.433022i \(-0.142553\pi\)
\(24\) 11.4107 0.0970500
\(25\) 0 0
\(26\) −77.1598 −0.582010
\(27\) 45.2982i 0.322876i
\(28\) 29.0483i 0.196058i
\(29\) −269.029 −1.72267 −0.861336 − 0.508035i \(-0.830372\pi\)
−0.861336 + 0.508035i \(0.830372\pi\)
\(30\) 0 0
\(31\) 197.114 1.14202 0.571012 − 0.820942i \(-0.306551\pi\)
0.571012 + 0.820942i \(0.306551\pi\)
\(32\) 171.409i 0.946911i
\(33\) − 5.87314i − 0.0309813i
\(34\) −307.793 −1.55253
\(35\) 0 0
\(36\) −109.044 −0.504832
\(37\) − 2.14546i − 0.00953276i −0.999989 − 0.00476638i \(-0.998483\pi\)
0.999989 − 0.00476638i \(-0.00151719\pi\)
\(38\) 128.816i 0.549912i
\(39\) 18.8212 0.0772771
\(40\) 0 0
\(41\) 174.127 0.663271 0.331636 − 0.943408i \(-0.392400\pi\)
0.331636 + 0.943408i \(0.392400\pi\)
\(42\) − 20.7454i − 0.0762164i
\(43\) − 17.0345i − 0.0604125i −0.999544 − 0.0302062i \(-0.990384\pi\)
0.999544 − 0.0302062i \(-0.00961641\pi\)
\(44\) 28.6650 0.0982140
\(45\) 0 0
\(46\) −332.978 −1.06728
\(47\) 528.029i 1.63874i 0.573264 + 0.819371i \(0.305677\pi\)
−0.573264 + 0.819371i \(0.694323\pi\)
\(48\) − 68.0000i − 0.204478i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) 75.0786 0.206139
\(52\) 91.8608i 0.244977i
\(53\) − 641.114i − 1.66158i −0.556586 − 0.830790i \(-0.687889\pi\)
0.556586 − 0.830790i \(-0.312111\pi\)
\(54\) 157.894 0.397900
\(55\) 0 0
\(56\) −93.9441 −0.224175
\(57\) − 31.4214i − 0.0730151i
\(58\) 937.742i 2.12296i
\(59\) 642.975 1.41878 0.709391 − 0.704815i \(-0.248970\pi\)
0.709391 + 0.704815i \(0.248970\pi\)
\(60\) 0 0
\(61\) 142.967 0.300083 0.150042 − 0.988680i \(-0.452059\pi\)
0.150042 + 0.988680i \(0.452059\pi\)
\(62\) − 687.070i − 1.40739i
\(63\) − 183.940i − 0.367845i
\(64\) −42.3480 −0.0827109
\(65\) 0 0
\(66\) −20.4717 −0.0381802
\(67\) − 478.797i − 0.873050i −0.899692 − 0.436525i \(-0.856209\pi\)
0.899692 − 0.436525i \(-0.143791\pi\)
\(68\) 366.436i 0.653484i
\(69\) 81.2218 0.141710
\(70\) 0 0
\(71\) 105.550 0.176430 0.0882150 − 0.996101i \(-0.471884\pi\)
0.0882150 + 0.996101i \(0.471884\pi\)
\(72\) − 352.654i − 0.577232i
\(73\) 986.512i 1.58168i 0.612024 + 0.790839i \(0.290356\pi\)
−0.612024 + 0.790839i \(0.709644\pi\)
\(74\) −7.47834 −0.0117478
\(75\) 0 0
\(76\) 153.358 0.231466
\(77\) 48.3534i 0.0715635i
\(78\) − 65.6042i − 0.0952335i
\(79\) 1099.86 1.56638 0.783190 − 0.621783i \(-0.213591\pi\)
0.783190 + 0.621783i \(0.213591\pi\)
\(80\) 0 0
\(81\) 670.967 0.920394
\(82\) − 606.947i − 0.817391i
\(83\) − 1236.62i − 1.63538i −0.575657 − 0.817691i \(-0.695254\pi\)
0.575657 − 0.817691i \(-0.304746\pi\)
\(84\) −24.6980 −0.0320806
\(85\) 0 0
\(86\) −59.3763 −0.0744501
\(87\) − 228.739i − 0.281878i
\(88\) 92.7045i 0.112299i
\(89\) 711.698 0.847638 0.423819 − 0.905747i \(-0.360689\pi\)
0.423819 + 0.905747i \(0.360689\pi\)
\(90\) 0 0
\(91\) −154.955 −0.178502
\(92\) 396.420i 0.449235i
\(93\) 167.594i 0.186867i
\(94\) 1840.52 2.01953
\(95\) 0 0
\(96\) −145.739 −0.154942
\(97\) 636.553i 0.666311i 0.942872 + 0.333156i \(0.108113\pi\)
−0.942872 + 0.333156i \(0.891887\pi\)
\(98\) 170.797i 0.176052i
\(99\) −181.513 −0.184270
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 175.4.b.e.99.2 6
5.2 odd 4 35.4.a.c.1.3 ✓ 3
5.3 odd 4 175.4.a.f.1.1 3
5.4 even 2 inner 175.4.b.e.99.5 6
15.2 even 4 315.4.a.p.1.1 3
15.8 even 4 1575.4.a.ba.1.3 3
20.7 even 4 560.4.a.u.1.2 3
35.2 odd 12 245.4.e.m.116.1 6
35.12 even 12 245.4.e.n.116.1 6
35.13 even 4 1225.4.a.y.1.1 3
35.17 even 12 245.4.e.n.226.1 6
35.27 even 4 245.4.a.l.1.3 3
35.32 odd 12 245.4.e.m.226.1 6
40.27 even 4 2240.4.a.bv.1.2 3
40.37 odd 4 2240.4.a.bt.1.2 3
105.62 odd 4 2205.4.a.bm.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.c.1.3 ✓ 3 5.2 odd 4
175.4.a.f.1.1 3 5.3 odd 4
175.4.b.e.99.2 6 1.1 even 1 trivial
175.4.b.e.99.5 6 5.4 even 2 inner
245.4.a.l.1.3 3 35.27 even 4
245.4.e.m.116.1 6 35.2 odd 12
245.4.e.m.226.1 6 35.32 odd 12
245.4.e.n.116.1 6 35.12 even 12
245.4.e.n.226.1 6 35.17 even 12
315.4.a.p.1.1 3 15.2 even 4
560.4.a.u.1.2 3 20.7 even 4
1225.4.a.y.1.1 3 35.13 even 4
1575.4.a.ba.1.3 3 15.8 even 4
2205.4.a.bm.1.1 3 105.62 odd 4
2240.4.a.bt.1.2 3 40.37 odd 4
2240.4.a.bv.1.2 3 40.27 even 4