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Properties

Label 175.10.b.c.99.4
Level $175$
Weight $10$
Character 175.99
Analytic conductor $90.131$
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] = [175,10,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, 10); 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, 10, names="a")
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 175.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,1440] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(90.1312713287\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.4
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 175.99
Dual form 175.10.b.c.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)
 
\(f(q)\) \(=\) \(q+14.8284i q^{2} -239.735i q^{3} +292.118 q^{4} +3554.89 q^{6} -2401.00i q^{7} +11923.8i q^{8} -37789.9 q^{9} -16523.6 q^{11} -70030.9i q^{12} +26311.4i q^{13} +35603.1 q^{14} -27246.9 q^{16} +144003. i q^{17} -560365. i q^{18} +159710. q^{19} -575604. q^{21} -245019. i q^{22} +2.07393e6i q^{23} +2.85855e6 q^{24} -390156. q^{26} +4.34086e6i q^{27} -701375. i q^{28} +4.94938e6 q^{29} +4.22040e6 q^{31} +5.70096e6i q^{32} +3.96128e6i q^{33} -2.13533e6 q^{34} -1.10391e7 q^{36} +1.29081e7i q^{37} +2.36824e6i q^{38} +6.30776e6 q^{39} -2.87518e7 q^{41} -8.53530e6i q^{42} +3.54825e7i q^{43} -4.82683e6 q^{44} -3.07532e7 q^{46} -5.95633e7i q^{47} +6.53205e6i q^{48} -5.76480e6 q^{49} +3.45225e7 q^{51} +7.68602e6i q^{52} +2.31161e6i q^{53} -6.43681e7 q^{54} +2.86290e7 q^{56} -3.82880e7i q^{57} +7.33915e7i q^{58} +1.68651e8 q^{59} -6.70167e7 q^{61} +6.25819e7i q^{62} +9.07336e7i q^{63} -9.84867e7 q^{64} -5.87395e7 q^{66} +1.56259e8i q^{67} +4.20657e7i q^{68} +4.97195e8 q^{69} +6.95067e7 q^{71} -4.50599e8i q^{72} -7.83438e7i q^{73} -1.91407e8 q^{74} +4.66540e7 q^{76} +3.96731e7i q^{77} +9.35342e7i q^{78} +4.26957e8 q^{79} +2.96838e8 q^{81} -4.26344e8i q^{82} -5.31242e8i q^{83} -1.68144e8 q^{84} -5.26149e8 q^{86} -1.18654e9i q^{87} -1.97024e8i q^{88} +1.14168e8 q^{89} +6.31736e7 q^{91} +6.05833e8i q^{92} -1.01178e9i q^{93} +8.83231e8 q^{94} +1.36672e9 q^{96} +1.46573e9i q^{97} -8.54829e7i q^{98} +6.24424e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 1440 q^{4} + 5904 q^{6} - 44856 q^{9} + 37132 q^{11} + 115248 q^{14} + 225536 q^{16} + 286552 q^{19} - 835548 q^{21} + 4583808 q^{24} + 639472 q^{26} + 23155108 q^{29} - 7907520 q^{31} - 8487888 q^{34}+ \cdots + 2326933080 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 14.8284i 0.655330i 0.944794 + 0.327665i \(0.106262\pi\)
−0.944794 + 0.327665i \(0.893738\pi\)
\(3\) − 239.735i − 1.70878i −0.519633 − 0.854390i \(-0.673931\pi\)
0.519633 − 0.854390i \(-0.326069\pi\)
\(4\) 292.118 0.570542
\(5\) 0 0
\(6\) 3554.89 1.11981
\(7\) − 2401.00i − 0.377964i
\(8\) 11923.8i 1.02922i
\(9\) −37789.9 −1.91993
\(10\) 0 0
\(11\) −16523.6 −0.340280 −0.170140 − 0.985420i \(-0.554422\pi\)
−0.170140 + 0.985420i \(0.554422\pi\)
\(12\) − 70030.9i − 0.974931i
\(13\) 26311.4i 0.255505i 0.991806 + 0.127752i \(0.0407762\pi\)
−0.991806 + 0.127752i \(0.959224\pi\)
\(14\) 35603.1 0.247691
\(15\) 0 0
\(16\) −27246.9 −0.103939
\(17\) 144003.i 0.418167i 0.977898 + 0.209084i \(0.0670481\pi\)
−0.977898 + 0.209084i \(0.932952\pi\)
\(18\) − 560365.i − 1.25819i
\(19\) 159710. 0.281151 0.140576 − 0.990070i \(-0.455105\pi\)
0.140576 + 0.990070i \(0.455105\pi\)
\(20\) 0 0
\(21\) −575604. −0.645858
\(22\) − 245019.i − 0.222996i
\(23\) 2.07393e6i 1.54533i 0.634817 + 0.772663i \(0.281075\pi\)
−0.634817 + 0.772663i \(0.718925\pi\)
\(24\) 2.85855e6 1.75872
\(25\) 0 0
\(26\) −390156. −0.167440
\(27\) 4.34086e6i 1.57195i
\(28\) − 701375.i − 0.215645i
\(29\) 4.94938e6 1.29945 0.649725 − 0.760169i \(-0.274884\pi\)
0.649725 + 0.760169i \(0.274884\pi\)
\(30\) 0 0
\(31\) 4.22040e6 0.820779 0.410389 − 0.911910i \(-0.365393\pi\)
0.410389 + 0.911910i \(0.365393\pi\)
\(32\) 5.70096e6i 0.961110i
\(33\) 3.96128e6i 0.581464i
\(34\) −2.13533e6 −0.274038
\(35\) 0 0
\(36\) −1.10391e7 −1.09540
\(37\) 1.29081e7i 1.13228i 0.824308 + 0.566142i \(0.191565\pi\)
−0.824308 + 0.566142i \(0.808435\pi\)
\(38\) 2.36824e6i 0.184247i
\(39\) 6.30776e6 0.436601
\(40\) 0 0
\(41\) −2.87518e7 −1.58905 −0.794525 − 0.607231i \(-0.792280\pi\)
−0.794525 + 0.607231i \(0.792280\pi\)
\(42\) − 8.53530e6i − 0.423250i
\(43\) 3.54825e7i 1.58273i 0.611347 + 0.791363i \(0.290628\pi\)
−0.611347 + 0.791363i \(0.709372\pi\)
\(44\) −4.82683e6 −0.194144
\(45\) 0 0
\(46\) −3.07532e7 −1.01270
\(47\) − 5.95633e7i − 1.78049i −0.455485 − 0.890243i \(-0.650534\pi\)
0.455485 − 0.890243i \(-0.349466\pi\)
\(48\) 6.53205e6i 0.177608i
\(49\) −5.76480e6 −0.142857
\(50\) 0 0
\(51\) 3.45225e7 0.714555
\(52\) 7.68602e6i 0.145776i
\(53\) 2.31161e6i 0.0402414i 0.999798 + 0.0201207i \(0.00640504\pi\)
−0.999798 + 0.0201207i \(0.993595\pi\)
\(54\) −6.43681e7 −1.03015
\(55\) 0 0
\(56\) 2.86290e7 0.389010
\(57\) − 3.82880e7i − 0.480425i
\(58\) 7.33915e7i 0.851569i
\(59\) 1.68651e8 1.81198 0.905992 − 0.423296i \(-0.139127\pi\)
0.905992 + 0.423296i \(0.139127\pi\)
\(60\) 0 0
\(61\) −6.70167e7 −0.619725 −0.309863 − 0.950781i \(-0.600283\pi\)
−0.309863 + 0.950781i \(0.600283\pi\)
\(62\) 6.25819e7i 0.537881i
\(63\) 9.07336e7i 0.725664i
\(64\) −9.84867e7 −0.733783
\(65\) 0 0
\(66\) −5.87395e7 −0.381051
\(67\) 1.56259e8i 0.947343i 0.880702 + 0.473671i \(0.157072\pi\)
−0.880702 + 0.473671i \(0.842928\pi\)
\(68\) 4.20657e7i 0.238582i
\(69\) 4.97195e8 2.64062
\(70\) 0 0
\(71\) 6.95067e7 0.324612 0.162306 − 0.986740i \(-0.448107\pi\)
0.162306 + 0.986740i \(0.448107\pi\)
\(72\) − 4.50599e8i − 1.97603i
\(73\) − 7.83438e7i − 0.322888i −0.986882 − 0.161444i \(-0.948385\pi\)
0.986882 − 0.161444i \(-0.0516151\pi\)
\(74\) −1.91407e8 −0.742020
\(75\) 0 0
\(76\) 4.66540e7 0.160409
\(77\) 3.96731e7i 0.128614i
\(78\) 9.35342e7i 0.286118i
\(79\) 4.26957e8 1.23328 0.616641 − 0.787245i \(-0.288493\pi\)
0.616641 + 0.787245i \(0.288493\pi\)
\(80\) 0 0
\(81\) 2.96838e8 0.766190
\(82\) − 4.26344e8i − 1.04135i
\(83\) − 5.31242e8i − 1.22869i −0.789039 − 0.614343i \(-0.789421\pi\)
0.789039 − 0.614343i \(-0.210579\pi\)
\(84\) −1.68144e8 −0.368489
\(85\) 0 0
\(86\) −5.26149e8 −1.03721
\(87\) − 1.18654e9i − 2.22047i
\(88\) − 1.97024e8i − 0.350225i
\(89\) 1.14168e8 0.192881 0.0964404 − 0.995339i \(-0.469254\pi\)
0.0964404 + 0.995339i \(0.469254\pi\)
\(90\) 0 0
\(91\) 6.31736e7 0.0965716
\(92\) 6.05833e8i 0.881674i
\(93\) − 1.01178e9i − 1.40253i
\(94\) 8.83231e8 1.16681
\(95\) 0 0
\(96\) 1.36672e9 1.64232
\(97\) 1.46573e9i 1.68105i 0.541774 + 0.840524i \(0.317753\pi\)
−0.541774 + 0.840524i \(0.682247\pi\)
\(98\) − 8.54829e7i − 0.0936186i
\(99\) 6.24424e8 0.653313
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 175.10.b.c.99.4 4
5.2 odd 4 35.10.a.b.1.1 ✓ 2
5.3 odd 4 175.10.a.c.1.2 2
5.4 even 2 inner 175.10.b.c.99.1 4
15.2 even 4 315.10.a.b.1.2 2
35.27 even 4 245.10.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.10.a.b.1.1 ✓ 2 5.2 odd 4
175.10.a.c.1.2 2 5.3 odd 4
175.10.b.c.99.1 4 5.4 even 2 inner
175.10.b.c.99.4 4 1.1 even 1 trivial
245.10.a.c.1.1 2 35.27 even 4
315.10.a.b.1.2 2 15.2 even 4