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Properties

Label 175.10.a.c.1.2
Level $175$
Weight $10$
Character 175.1
Self dual yes
Analytic conductor $90.131$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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(1,175)] 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("175.1"); 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([0, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 175.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(90.1312713287\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 175.1

$q$-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)
 
\(f(q)\) \(=\) \(q+14.8284 q^{2} +239.735 q^{3} -292.118 q^{4} +3554.89 q^{6} -2401.00 q^{7} -11923.8 q^{8} +37789.9 q^{9} -16523.6 q^{11} -70030.9 q^{12} -26311.4 q^{13} -35603.1 q^{14} -27246.9 q^{16} +144003. q^{17} +560365. q^{18} -159710. q^{19} -575604. q^{21} -245019. q^{22} -2.07393e6 q^{23} -2.85855e6 q^{24} -390156. q^{26} +4.34086e6 q^{27} +701375. q^{28} -4.94938e6 q^{29} +4.22040e6 q^{31} +5.70096e6 q^{32} -3.96128e6 q^{33} +2.13533e6 q^{34} -1.10391e7 q^{36} +1.29081e7 q^{37} -2.36824e6 q^{38} -6.30776e6 q^{39} -2.87518e7 q^{41} -8.53530e6 q^{42} -3.54825e7 q^{43} +4.82683e6 q^{44} -3.07532e7 q^{46} -5.95633e7 q^{47} -6.53205e6 q^{48} +5.76480e6 q^{49} +3.45225e7 q^{51} +7.68602e6 q^{52} -2.31161e6 q^{53} +6.43681e7 q^{54} +2.86290e7 q^{56} -3.82880e7 q^{57} -7.33915e7 q^{58} -1.68651e8 q^{59} -6.70167e7 q^{61} +6.25819e7 q^{62} -9.07336e7 q^{63} +9.84867e7 q^{64} -5.87395e7 q^{66} +1.56259e8 q^{67} -4.20657e7 q^{68} -4.97195e8 q^{69} +6.95067e7 q^{71} -4.50599e8 q^{72} +7.83438e7 q^{73} +1.91407e8 q^{74} +4.66540e7 q^{76} +3.96731e7 q^{77} -9.35342e7 q^{78} -4.26957e8 q^{79} +2.96838e8 q^{81} -4.26344e8 q^{82} +5.31242e8 q^{83} +1.68144e8 q^{84} -5.26149e8 q^{86} -1.18654e9 q^{87} +1.97024e8 q^{88} -1.14168e8 q^{89} +6.31736e7 q^{91} +6.05833e8 q^{92} +1.01178e9 q^{93} -8.83231e8 q^{94} +1.36672e9 q^{96} +1.46573e9 q^{97} +8.54829e7 q^{98} -6.24424e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 24 q^{2} + 174 q^{3} - 720 q^{4} + 2952 q^{6} - 4802 q^{7} - 20544 q^{8} + 22428 q^{9} + 18566 q^{11} - 41904 q^{12} + 51090 q^{13} - 57624 q^{14} + 112768 q^{16} + 373910 q^{17} + 419472 q^{18} - 143276 q^{19}+ \cdots - 1163466540 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 14.8284 0.655330 0.327665 − 0.944794i \(-0.393738\pi\)
0.327665 + 0.944794i \(0.393738\pi\)
\(3\) 239.735 1.70878 0.854390 − 0.519633i \(-0.173931\pi\)
0.854390 + 0.519633i \(0.173931\pi\)
\(4\) −292.118 −0.570542
\(5\) 0 0
\(6\) 3554.89 1.11981
\(7\) −2401.00 −0.377964
\(8\) −11923.8 −1.02922
\(9\) 37789.9 1.91993
\(10\) 0 0
\(11\) −16523.6 −0.340280 −0.170140 − 0.985420i \(-0.554422\pi\)
−0.170140 + 0.985420i \(0.554422\pi\)
\(12\) −70030.9 −0.974931
\(13\) −26311.4 −0.255505 −0.127752 − 0.991806i \(-0.540776\pi\)
−0.127752 + 0.991806i \(0.540776\pi\)
\(14\) −35603.1 −0.247691
\(15\) 0 0
\(16\) −27246.9 −0.103939
\(17\) 144003. 0.418167 0.209084 − 0.977898i \(-0.432952\pi\)
0.209084 + 0.977898i \(0.432952\pi\)
\(18\) 560365. 1.25819
\(19\) −159710. −0.281151 −0.140576 − 0.990070i \(-0.544895\pi\)
−0.140576 + 0.990070i \(0.544895\pi\)
\(20\) 0 0
\(21\) −575604. −0.645858
\(22\) −245019. −0.222996
\(23\) −2.07393e6 −1.54533 −0.772663 − 0.634817i \(-0.781075\pi\)
−0.772663 + 0.634817i \(0.781075\pi\)
\(24\) −2.85855e6 −1.75872
\(25\) 0 0
\(26\) −390156. −0.167440
\(27\) 4.34086e6 1.57195
\(28\) 701375. 0.215645
\(29\) −4.94938e6 −1.29945 −0.649725 − 0.760169i \(-0.725116\pi\)
−0.649725 + 0.760169i \(0.725116\pi\)
\(30\) 0 0
\(31\) 4.22040e6 0.820779 0.410389 − 0.911910i \(-0.365393\pi\)
0.410389 + 0.911910i \(0.365393\pi\)
\(32\) 5.70096e6 0.961110
\(33\) −3.96128e6 −0.581464
\(34\) 2.13533e6 0.274038
\(35\) 0 0
\(36\) −1.10391e7 −1.09540
\(37\) 1.29081e7 1.13228 0.566142 − 0.824308i \(-0.308435\pi\)
0.566142 + 0.824308i \(0.308435\pi\)
\(38\) −2.36824e6 −0.184247
\(39\) −6.30776e6 −0.436601
\(40\) 0 0
\(41\) −2.87518e7 −1.58905 −0.794525 − 0.607231i \(-0.792280\pi\)
−0.794525 + 0.607231i \(0.792280\pi\)
\(42\) −8.53530e6 −0.423250
\(43\) −3.54825e7 −1.58273 −0.791363 − 0.611347i \(-0.790628\pi\)
−0.791363 + 0.611347i \(0.790628\pi\)
\(44\) 4.82683e6 0.194144
\(45\) 0 0
\(46\) −3.07532e7 −1.01270
\(47\) −5.95633e7 −1.78049 −0.890243 − 0.455485i \(-0.849466\pi\)
−0.890243 + 0.455485i \(0.849466\pi\)
\(48\) −6.53205e6 −0.177608
\(49\) 5.76480e6 0.142857
\(50\) 0 0
\(51\) 3.45225e7 0.714555
\(52\) 7.68602e6 0.145776
\(53\) −2.31161e6 −0.0402414 −0.0201207 − 0.999798i \(-0.506405\pi\)
−0.0201207 + 0.999798i \(0.506405\pi\)
\(54\) 6.43681e7 1.03015
\(55\) 0 0
\(56\) 2.86290e7 0.389010
\(57\) −3.82880e7 −0.480425
\(58\) −7.33915e7 −0.851569
\(59\) −1.68651e8 −1.81198 −0.905992 − 0.423296i \(-0.860873\pi\)
−0.905992 + 0.423296i \(0.860873\pi\)
\(60\) 0 0
\(61\) −6.70167e7 −0.619725 −0.309863 − 0.950781i \(-0.600283\pi\)
−0.309863 + 0.950781i \(0.600283\pi\)
\(62\) 6.25819e7 0.537881
\(63\) −9.07336e7 −0.725664
\(64\) 9.84867e7 0.733783
\(65\) 0 0
\(66\) −5.87395e7 −0.381051
\(67\) 1.56259e8 0.947343 0.473671 − 0.880702i \(-0.342928\pi\)
0.473671 + 0.880702i \(0.342928\pi\)
\(68\) −4.20657e7 −0.238582
\(69\) −4.97195e8 −2.64062
\(70\) 0 0
\(71\) 6.95067e7 0.324612 0.162306 − 0.986740i \(-0.448107\pi\)
0.162306 + 0.986740i \(0.448107\pi\)
\(72\) −4.50599e8 −1.97603
\(73\) 7.83438e7 0.322888 0.161444 − 0.986882i \(-0.448385\pi\)
0.161444 + 0.986882i \(0.448385\pi\)
\(74\) 1.91407e8 0.742020
\(75\) 0 0
\(76\) 4.66540e7 0.160409
\(77\) 3.96731e7 0.128614
\(78\) −9.35342e7 −0.286118
\(79\) −4.26957e8 −1.23328 −0.616641 − 0.787245i \(-0.711507\pi\)
−0.616641 + 0.787245i \(0.711507\pi\)
\(80\) 0 0
\(81\) 2.96838e8 0.766190
\(82\) −4.26344e8 −1.04135
\(83\) 5.31242e8 1.22869 0.614343 − 0.789039i \(-0.289421\pi\)
0.614343 + 0.789039i \(0.289421\pi\)
\(84\) 1.68144e8 0.368489
\(85\) 0 0
\(86\) −5.26149e8 −1.03721
\(87\) −1.18654e9 −2.22047
\(88\) 1.97024e8 0.350225
\(89\) −1.14168e8 −0.192881 −0.0964404 − 0.995339i \(-0.530746\pi\)
−0.0964404 + 0.995339i \(0.530746\pi\)
\(90\) 0 0
\(91\) 6.31736e7 0.0965716
\(92\) 6.05833e8 0.881674
\(93\) 1.01178e9 1.40253
\(94\) −8.83231e8 −1.16681
\(95\) 0 0
\(96\) 1.36672e9 1.64232
\(97\) 1.46573e9 1.68105 0.840524 − 0.541774i \(-0.182247\pi\)
0.840524 + 0.541774i \(0.182247\pi\)
\(98\) 8.54829e7 0.0936186
\(99\) −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.a.c.1.2 2
5.2 odd 4 175.10.b.c.99.4 4
5.3 odd 4 175.10.b.c.99.1 4
5.4 even 2 35.10.a.b.1.1 ✓ 2
15.14 odd 2 315.10.a.b.1.2 2
35.34 odd 2 245.10.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.10.a.b.1.1 ✓ 2 5.4 even 2
175.10.a.c.1.2 2 1.1 even 1 trivial
175.10.b.c.99.1 4 5.3 odd 4
175.10.b.c.99.4 4 5.2 odd 4
245.10.a.c.1.1 2 35.34 odd 2
315.10.a.b.1.2 2 15.14 odd 2