Properties

Label 175.2.a.d.1.1
Level $175$
Weight $2$
Character 175.1
Self dual yes
Analytic conductor $1.397$
Analytic rank $0$
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] = [175,2,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, 2); 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, 2, names="a")
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 175.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.39738203537\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) 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)
 
\(f(q)\) \(=\) \(q-1.61803 q^{2} -1.23607 q^{3} +0.618034 q^{4} +2.00000 q^{6} -1.00000 q^{7} +2.23607 q^{8} -1.47214 q^{9} +4.23607 q^{11} -0.763932 q^{12} +3.23607 q^{13} +1.61803 q^{14} -4.85410 q^{16} +6.47214 q^{17} +2.38197 q^{18} +4.47214 q^{19} +1.23607 q^{21} -6.85410 q^{22} -1.76393 q^{23} -2.76393 q^{24} -5.23607 q^{26} +5.52786 q^{27} -0.618034 q^{28} +5.00000 q^{29} -9.70820 q^{31} +3.38197 q^{32} -5.23607 q^{33} -10.4721 q^{34} -0.909830 q^{36} -3.00000 q^{37} -7.23607 q^{38} -4.00000 q^{39} +9.23607 q^{41} -2.00000 q^{42} -6.23607 q^{43} +2.61803 q^{44} +2.85410 q^{46} +2.00000 q^{47} +6.00000 q^{48} +1.00000 q^{49} -8.00000 q^{51} +2.00000 q^{52} +0.472136 q^{53} -8.94427 q^{54} -2.23607 q^{56} -5.52786 q^{57} -8.09017 q^{58} -1.70820 q^{59} +3.70820 q^{61} +15.7082 q^{62} +1.47214 q^{63} +4.23607 q^{64} +8.47214 q^{66} -0.236068 q^{67} +4.00000 q^{68} +2.18034 q^{69} -4.70820 q^{71} -3.29180 q^{72} +13.2361 q^{73} +4.85410 q^{74} +2.76393 q^{76} -4.23607 q^{77} +6.47214 q^{78} +11.1803 q^{79} -2.41641 q^{81} -14.9443 q^{82} -5.70820 q^{83} +0.763932 q^{84} +10.0902 q^{86} -6.18034 q^{87} +9.47214 q^{88} +12.7639 q^{89} -3.23607 q^{91} -1.09017 q^{92} +12.0000 q^{93} -3.23607 q^{94} -4.18034 q^{96} -0.763932 q^{97} -1.61803 q^{98} -6.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 2 q^{3} - q^{4} + 4 q^{6} - 2 q^{7} + 6 q^{9} + 4 q^{11} - 6 q^{12} + 2 q^{13} + q^{14} - 3 q^{16} + 4 q^{17} + 7 q^{18} - 2 q^{21} - 7 q^{22} - 8 q^{23} - 10 q^{24} - 6 q^{26} + 20 q^{27}+ \cdots - 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.61803 −1.14412 −0.572061 − 0.820211i \(-0.693856\pi\)
−0.572061 + 0.820211i \(0.693856\pi\)
\(3\) −1.23607 −0.713644 −0.356822 − 0.934172i \(-0.616140\pi\)
−0.356822 + 0.934172i \(0.616140\pi\)
\(4\) 0.618034 0.309017
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) −1.00000 −0.377964
\(8\) 2.23607 0.790569
\(9\) −1.47214 −0.490712
\(10\) 0 0
\(11\) 4.23607 1.27722 0.638611 − 0.769529i \(-0.279509\pi\)
0.638611 + 0.769529i \(0.279509\pi\)
\(12\) −0.763932 −0.220528
\(13\) 3.23607 0.897524 0.448762 − 0.893651i \(-0.351865\pi\)
0.448762 + 0.893651i \(0.351865\pi\)
\(14\) 1.61803 0.432438
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) 6.47214 1.56972 0.784862 − 0.619671i \(-0.212734\pi\)
0.784862 + 0.619671i \(0.212734\pi\)
\(18\) 2.38197 0.561435
\(19\) 4.47214 1.02598 0.512989 − 0.858395i \(-0.328538\pi\)
0.512989 + 0.858395i \(0.328538\pi\)
\(20\) 0 0
\(21\) 1.23607 0.269732
\(22\) −6.85410 −1.46130
\(23\) −1.76393 −0.367805 −0.183903 − 0.982944i \(-0.558873\pi\)
−0.183903 + 0.982944i \(0.558873\pi\)
\(24\) −2.76393 −0.564185
\(25\) 0 0
\(26\) −5.23607 −1.02688
\(27\) 5.52786 1.06384
\(28\) −0.618034 −0.116797
\(29\) 5.00000 0.928477 0.464238 − 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) −9.70820 −1.74364 −0.871822 − 0.489822i \(-0.837062\pi\)
−0.871822 + 0.489822i \(0.837062\pi\)
\(32\) 3.38197 0.597853
\(33\) −5.23607 −0.911482
\(34\) −10.4721 −1.79596
\(35\) 0 0
\(36\) −0.909830 −0.151638
\(37\) −3.00000 −0.493197 −0.246598 − 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) −7.23607 −1.17385
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 9.23607 1.44243 0.721216 − 0.692711i \(-0.243584\pi\)
0.721216 + 0.692711i \(0.243584\pi\)
\(42\) −2.00000 −0.308607
\(43\) −6.23607 −0.950991 −0.475496 − 0.879718i \(-0.657731\pi\)
−0.475496 + 0.879718i \(0.657731\pi\)
\(44\) 2.61803 0.394683
\(45\) 0 0
\(46\) 2.85410 0.420814
\(47\) 2.00000 0.291730 0.145865 − 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 6.00000 0.866025
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −8.00000 −1.12022
\(52\) 2.00000 0.277350
\(53\) 0.472136 0.0648529 0.0324264 − 0.999474i \(-0.489677\pi\)
0.0324264 + 0.999474i \(0.489677\pi\)
\(54\) −8.94427 −1.21716
\(55\) 0 0
\(56\) −2.23607 −0.298807
\(57\) −5.52786 −0.732183
\(58\) −8.09017 −1.06229
\(59\) −1.70820 −0.222389 −0.111195 − 0.993799i \(-0.535468\pi\)
−0.111195 + 0.993799i \(0.535468\pi\)
\(60\) 0 0
\(61\) 3.70820 0.474787 0.237393 − 0.971414i \(-0.423707\pi\)
0.237393 + 0.971414i \(0.423707\pi\)
\(62\) 15.7082 1.99494
\(63\) 1.47214 0.185472
\(64\) 4.23607 0.529508
\(65\) 0 0
\(66\) 8.47214 1.04285
\(67\) −0.236068 −0.0288403 −0.0144201 − 0.999896i \(-0.504590\pi\)
−0.0144201 + 0.999896i \(0.504590\pi\)
\(68\) 4.00000 0.485071
\(69\) 2.18034 0.262482
\(70\) 0 0
\(71\) −4.70820 −0.558761 −0.279381 − 0.960180i \(-0.590129\pi\)
−0.279381 + 0.960180i \(0.590129\pi\)
\(72\) −3.29180 −0.387942
\(73\) 13.2361 1.54916 0.774582 − 0.632473i \(-0.217960\pi\)
0.774582 + 0.632473i \(0.217960\pi\)
\(74\) 4.85410 0.564278
\(75\) 0 0
\(76\) 2.76393 0.317045
\(77\) −4.23607 −0.482745
\(78\) 6.47214 0.732825
\(79\) 11.1803 1.25789 0.628943 − 0.777451i \(-0.283488\pi\)
0.628943 + 0.777451i \(0.283488\pi\)
\(80\) 0 0
\(81\) −2.41641 −0.268490
\(82\) −14.9443 −1.65032
\(83\) −5.70820 −0.626557 −0.313278 − 0.949661i \(-0.601427\pi\)
−0.313278 + 0.949661i \(0.601427\pi\)
\(84\) 0.763932 0.0833518
\(85\) 0 0
\(86\) 10.0902 1.08805
\(87\) −6.18034 −0.662602
\(88\) 9.47214 1.00973
\(89\) 12.7639 1.35297 0.676487 − 0.736455i \(-0.263501\pi\)
0.676487 + 0.736455i \(0.263501\pi\)
\(90\) 0 0
\(91\) −3.23607 −0.339232
\(92\) −1.09017 −0.113658
\(93\) 12.0000 1.24434
\(94\) −3.23607 −0.333775
\(95\) 0 0
\(96\) −4.18034 −0.426654
\(97\) −0.763932 −0.0775655 −0.0387828 − 0.999248i \(-0.512348\pi\)
−0.0387828 + 0.999248i \(0.512348\pi\)
\(98\) −1.61803 −0.163446
\(99\) −6.23607 −0.626748
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 175.2.a.d.1.1 ✓ 2
3.2 odd 2 1575.2.a.s.1.2 2
4.3 odd 2 2800.2.a.bh.1.2 2
5.2 odd 4 175.2.b.c.99.1 4
5.3 odd 4 175.2.b.c.99.4 4
5.4 even 2 175.2.a.e.1.2 yes 2
7.6 odd 2 1225.2.a.n.1.1 2
15.2 even 4 1575.2.d.k.1324.4 4
15.8 even 4 1575.2.d.k.1324.1 4
15.14 odd 2 1575.2.a.n.1.1 2
20.3 even 4 2800.2.g.s.449.3 4
20.7 even 4 2800.2.g.s.449.2 4
20.19 odd 2 2800.2.a.bp.1.1 2
35.13 even 4 1225.2.b.k.99.4 4
35.27 even 4 1225.2.b.k.99.1 4
35.34 odd 2 1225.2.a.u.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.a.d.1.1 ✓ 2 1.1 even 1 trivial
175.2.a.e.1.2 yes 2 5.4 even 2
175.2.b.c.99.1 4 5.2 odd 4
175.2.b.c.99.4 4 5.3 odd 4
1225.2.a.n.1.1 2 7.6 odd 2
1225.2.a.u.1.2 2 35.34 odd 2
1225.2.b.k.99.1 4 35.27 even 4
1225.2.b.k.99.4 4 35.13 even 4
1575.2.a.n.1.1 2 15.14 odd 2
1575.2.a.s.1.2 2 3.2 odd 2
1575.2.d.k.1324.1 4 15.8 even 4
1575.2.d.k.1324.4 4 15.2 even 4
2800.2.a.bh.1.2 2 4.3 odd 2
2800.2.a.bp.1.1 2 20.19 odd 2
2800.2.g.s.449.2 4 20.7 even 4
2800.2.g.s.449.3 4 20.3 even 4