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Properties

Label 175.4.b.d.99.2
Level $175$
Weight $4$
Character 175.99
Analytic conductor $10.325$
Analytic rank $0$
Dimension $4$
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 = [4,0,0,-10,0,-46] 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: \(4\)
Coefficient field: \(\Q(i, \sqrt{41})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 21x^{2} + 100 \) 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 99.2
Root \(-2.70156i\) of defining polynomial
Character \(\chi\) \(=\) 175.99
Dual form 175.4.b.d.99.3

$q$-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)
 
\(f(q)\) \(=\) \(q-2.70156i q^{2} -0.701562i q^{3} +0.701562 q^{4} -1.89531 q^{6} +7.00000i q^{7} -23.5078i q^{8} +26.5078 q^{9} +4.01562 q^{11} -0.492189i q^{12} -51.6125i q^{13} +18.9109 q^{14} -57.8953 q^{16} -67.5078i q^{17} -71.6125i q^{18} +50.9109 q^{19} +4.91093 q^{21} -10.8485i q^{22} -0.507811i q^{23} -16.4922 q^{24} -139.434 q^{26} -37.5391i q^{27} +4.91093i q^{28} +120.058 q^{29} -292.303 q^{31} -31.6547i q^{32} -2.81721i q^{33} -182.377 q^{34} +18.5969 q^{36} -144.989i q^{37} -137.539i q^{38} -36.2094 q^{39} -57.2047 q^{41} -13.2672i q^{42} +283.020i q^{43} +2.81721 q^{44} -1.37188 q^{46} -233.769i q^{47} +40.6172i q^{48} -49.0000 q^{49} -47.3609 q^{51} -36.2094i q^{52} +406.334i q^{53} -101.414 q^{54} +164.555 q^{56} -35.7172i q^{57} -324.344i q^{58} +577.328 q^{59} +322.116 q^{61} +789.675i q^{62} +185.555i q^{63} -548.680 q^{64} -7.61086 q^{66} +985.459i q^{67} -47.3609i q^{68} -0.356261 q^{69} +1033.57 q^{71} -623.141i q^{72} -692.720i q^{73} -391.697 q^{74} +35.7172 q^{76} +28.1093i q^{77} +97.8219i q^{78} +428.236 q^{79} +689.375 q^{81} +154.542i q^{82} +537.592i q^{83} +3.44533 q^{84} +764.597 q^{86} -84.2280i q^{87} -94.3985i q^{88} +802.073 q^{89} +361.287 q^{91} -0.356261i q^{92} +205.069i q^{93} -631.541 q^{94} -22.2077 q^{96} +1752.82i q^{97} +132.377i q^{98} +106.445 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{4} - 46 q^{6} + 42 q^{9} - 112 q^{11} - 14 q^{14} - 270 q^{16} + 114 q^{19} - 70 q^{21} - 130 q^{24} - 276 q^{26} + 826 q^{29} - 324 q^{31} - 102 q^{34} + 100 q^{36} - 68 q^{39} - 1010 q^{41}+ \cdots + 874 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.70156i − 0.955146i −0.878592 − 0.477573i \(-0.841517\pi\)
0.878592 − 0.477573i \(-0.158483\pi\)
\(3\) − 0.701562i − 0.135016i −0.997719 − 0.0675078i \(-0.978495\pi\)
0.997719 − 0.0675078i \(-0.0215048\pi\)
\(4\) 0.701562 0.0876953
\(5\) 0 0
\(6\) −1.89531 −0.128960
\(7\) 7.00000i 0.377964i
\(8\) − 23.5078i − 1.03891i
\(9\) 26.5078 0.981771
\(10\) 0 0
\(11\) 4.01562 0.110069 0.0550343 − 0.998484i \(-0.482473\pi\)
0.0550343 + 0.998484i \(0.482473\pi\)
\(12\) − 0.492189i − 0.0118402i
\(13\) − 51.6125i − 1.10113i −0.834791 − 0.550567i \(-0.814412\pi\)
0.834791 − 0.550567i \(-0.185588\pi\)
\(14\) 18.9109 0.361011
\(15\) 0 0
\(16\) −57.8953 −0.904614
\(17\) − 67.5078i − 0.963121i −0.876413 − 0.481560i \(-0.840070\pi\)
0.876413 − 0.481560i \(-0.159930\pi\)
\(18\) − 71.6125i − 0.937735i
\(19\) 50.9109 0.614725 0.307362 − 0.951593i \(-0.400554\pi\)
0.307362 + 0.951593i \(0.400554\pi\)
\(20\) 0 0
\(21\) 4.91093 0.0510311
\(22\) − 10.8485i − 0.105132i
\(23\) − 0.507811i − 0.00460373i −0.999997 − 0.00230187i \(-0.999267\pi\)
0.999997 − 0.00230187i \(-0.000732707\pi\)
\(24\) −16.4922 −0.140269
\(25\) 0 0
\(26\) −139.434 −1.05174
\(27\) − 37.5391i − 0.267570i
\(28\) 4.91093i 0.0331457i
\(29\) 120.058 0.768765 0.384382 − 0.923174i \(-0.374414\pi\)
0.384382 + 0.923174i \(0.374414\pi\)
\(30\) 0 0
\(31\) −292.303 −1.69352 −0.846761 − 0.531973i \(-0.821451\pi\)
−0.846761 + 0.531973i \(0.821451\pi\)
\(32\) − 31.6547i − 0.174869i
\(33\) − 2.81721i − 0.0148610i
\(34\) −182.377 −0.919921
\(35\) 0 0
\(36\) 18.5969 0.0860966
\(37\) − 144.989i − 0.644218i −0.946703 − 0.322109i \(-0.895608\pi\)
0.946703 − 0.322109i \(-0.104392\pi\)
\(38\) − 137.539i − 0.587152i
\(39\) −36.2094 −0.148670
\(40\) 0 0
\(41\) −57.2047 −0.217899 −0.108950 − 0.994047i \(-0.534749\pi\)
−0.108950 + 0.994047i \(0.534749\pi\)
\(42\) − 13.2672i − 0.0487422i
\(43\) 283.020i 1.00373i 0.864947 + 0.501863i \(0.167352\pi\)
−0.864947 + 0.501863i \(0.832648\pi\)
\(44\) 2.81721 0.00965250
\(45\) 0 0
\(46\) −1.37188 −0.00439724
\(47\) − 233.769i − 0.725504i −0.931886 − 0.362752i \(-0.881837\pi\)
0.931886 − 0.362752i \(-0.118163\pi\)
\(48\) 40.6172i 0.122137i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) −47.3609 −0.130036
\(52\) − 36.2094i − 0.0965642i
\(53\) 406.334i 1.05310i 0.850144 + 0.526550i \(0.176515\pi\)
−0.850144 + 0.526550i \(0.823485\pi\)
\(54\) −101.414 −0.255569
\(55\) 0 0
\(56\) 164.555 0.392670
\(57\) − 35.7172i − 0.0829975i
\(58\) − 324.344i − 0.734283i
\(59\) 577.328 1.27393 0.636964 − 0.770894i \(-0.280190\pi\)
0.636964 + 0.770894i \(0.280190\pi\)
\(60\) 0 0
\(61\) 322.116 0.676110 0.338055 − 0.941126i \(-0.390231\pi\)
0.338055 + 0.941126i \(0.390231\pi\)
\(62\) 789.675i 1.61756i
\(63\) 185.555i 0.371074i
\(64\) −548.680 −1.07164
\(65\) 0 0
\(66\) −7.61086 −0.0141944
\(67\) 985.459i 1.79691i 0.439065 + 0.898455i \(0.355310\pi\)
−0.439065 + 0.898455i \(0.644690\pi\)
\(68\) − 47.3609i − 0.0844611i
\(69\) −0.356261 −0.000621576 0
\(70\) 0 0
\(71\) 1033.57 1.72764 0.863821 − 0.503799i \(-0.168065\pi\)
0.863821 + 0.503799i \(0.168065\pi\)
\(72\) − 623.141i − 1.01997i
\(73\) − 692.720i − 1.11064i −0.831637 − 0.555320i \(-0.812596\pi\)
0.831637 − 0.555320i \(-0.187404\pi\)
\(74\) −391.697 −0.615322
\(75\) 0 0
\(76\) 35.7172 0.0539084
\(77\) 28.1093i 0.0416020i
\(78\) 97.8219i 0.142002i
\(79\) 428.236 0.609877 0.304939 − 0.952372i \(-0.401364\pi\)
0.304939 + 0.952372i \(0.401364\pi\)
\(80\) 0 0
\(81\) 689.375 0.945645
\(82\) 154.542i 0.208126i
\(83\) 537.592i 0.710945i 0.934687 + 0.355472i \(0.115680\pi\)
−0.934687 + 0.355472i \(0.884320\pi\)
\(84\) 3.44533 0.00447519
\(85\) 0 0
\(86\) 764.597 0.958705
\(87\) − 84.2280i − 0.103795i
\(88\) − 94.3985i − 0.114351i
\(89\) 802.073 0.955277 0.477638 − 0.878557i \(-0.341493\pi\)
0.477638 + 0.878557i \(0.341493\pi\)
\(90\) 0 0
\(91\) 361.287 0.416189
\(92\) − 0.356261i 0 0.000403725i
\(93\) 205.069i 0.228652i
\(94\) −631.541 −0.692962
\(95\) 0 0
\(96\) −22.2077 −0.0236101
\(97\) 1752.82i 1.83477i 0.398004 + 0.917384i \(0.369703\pi\)
−0.398004 + 0.917384i \(0.630297\pi\)
\(98\) 132.377i 0.136449i
\(99\) 106.445 0.108062
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.d.99.2 4
5.2 odd 4 175.4.a.d.1.2 ✓ 2
5.3 odd 4 175.4.a.e.1.1 yes 2
5.4 even 2 inner 175.4.b.d.99.3 4
15.2 even 4 1575.4.a.v.1.1 2
15.8 even 4 1575.4.a.s.1.2 2
35.13 even 4 1225.4.a.t.1.1 2
35.27 even 4 1225.4.a.r.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.4.a.d.1.2 ✓ 2 5.2 odd 4
175.4.a.e.1.1 yes 2 5.3 odd 4
175.4.b.d.99.2 4 1.1 even 1 trivial
175.4.b.d.99.3 4 5.4 even 2 inner
1225.4.a.r.1.2 2 35.27 even 4
1225.4.a.t.1.1 2 35.13 even 4
1575.4.a.s.1.2 2 15.8 even 4
1575.4.a.v.1.1 2 15.2 even 4