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Properties

Label 175.5.c.d.174.10
Level $175$
Weight $5$
Character 175.174
Analytic conductor $18.090$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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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] = [175,5,Mod(174,175)] 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("175.174"); S:= CuspForms(chi, 5); 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, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 175.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.0897435397\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 174.10
Character \(\chi\) \(=\) 175.174
Dual form 175.5.c.d.174.9

$q$-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)
 
\(f(q)\) \(=\) \(q+1.17355i q^{2} -9.10378 q^{3} +14.6228 q^{4} -10.6837i q^{6} +(27.5814 - 40.5002i) q^{7} +35.9373i q^{8} +1.87877 q^{9} +106.806 q^{11} -133.123 q^{12} -243.104 q^{13} +(47.5290 + 32.3681i) q^{14} +191.790 q^{16} -148.252 q^{17} +2.20483i q^{18} -244.752i q^{19} +(-251.095 + 368.705i) q^{21} +125.342i q^{22} -621.143i q^{23} -327.166i q^{24} -285.294i q^{26} +720.302 q^{27} +(403.316 - 592.226i) q^{28} +774.981 q^{29} -403.109i q^{31} +800.073i q^{32} -972.334 q^{33} -173.981i q^{34} +27.4729 q^{36} -1290.58i q^{37} +287.229 q^{38} +2213.16 q^{39} -2536.73i q^{41} +(-432.694 - 294.672i) q^{42} -2242.99i q^{43} +1561.79 q^{44} +728.942 q^{46} +2494.61 q^{47} -1746.02 q^{48} +(-879.536 - 2234.10i) q^{49} +1349.65 q^{51} -3554.85 q^{52} +281.688i q^{53} +845.310i q^{54} +(1455.47 + 991.201i) q^{56} +2228.17i q^{57} +909.479i q^{58} -5822.15i q^{59} +6268.19i q^{61} +473.068 q^{62} +(51.8192 - 76.0908i) q^{63} +2129.72 q^{64} -1141.08i q^{66} +4128.43i q^{67} -2167.85 q^{68} +5654.74i q^{69} -793.446 q^{71} +67.5182i q^{72} +1600.72 q^{73} +1514.56 q^{74} -3578.96i q^{76} +(2945.84 - 4325.65i) q^{77} +2597.25i q^{78} -9856.44 q^{79} -6709.65 q^{81} +2976.98 q^{82} +617.012 q^{83} +(-3671.70 + 5391.49i) q^{84} +2632.26 q^{86} -7055.26 q^{87} +3838.31i q^{88} +7345.17i q^{89} +(-6705.13 + 9845.75i) q^{91} -9082.83i q^{92} +3669.81i q^{93} +2927.55i q^{94} -7283.69i q^{96} -67.6288 q^{97} +(2621.83 - 1032.18i) q^{98} +200.664 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 244 q^{4} + 868 q^{9} + 252 q^{11} - 156 q^{14} + 1156 q^{16} - 1284 q^{21} + 4380 q^{29} - 7164 q^{36} - 17268 q^{39} + 22392 q^{44} + 15688 q^{46} + 17592 q^{49} + 14412 q^{51} - 13212 q^{56} - 40292 q^{64}+ \cdots - 46168 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.17355i 0.293387i 0.989182 + 0.146694i \(0.0468632\pi\)
−0.989182 + 0.146694i \(0.953137\pi\)
\(3\) −9.10378 −1.01153 −0.505765 − 0.862671i \(-0.668790\pi\)
−0.505765 + 0.862671i \(0.668790\pi\)
\(4\) 14.6228 0.913924
\(5\) 0 0
\(6\) − 10.6837i − 0.296770i
\(7\) 27.5814 − 40.5002i 0.562885 − 0.826535i
\(8\) 35.9373i 0.561521i
\(9\) 1.87877 0.0231947
\(10\) 0 0
\(11\) 106.806 0.882690 0.441345 − 0.897337i \(-0.354501\pi\)
0.441345 + 0.897337i \(0.354501\pi\)
\(12\) −133.123 −0.924462
\(13\) −243.104 −1.43848 −0.719242 − 0.694760i \(-0.755511\pi\)
−0.719242 + 0.694760i \(0.755511\pi\)
\(14\) 47.5290 + 32.3681i 0.242495 + 0.165143i
\(15\) 0 0
\(16\) 191.790 0.749181
\(17\) −148.252 −0.512982 −0.256491 − 0.966547i \(-0.582566\pi\)
−0.256491 + 0.966547i \(0.582566\pi\)
\(18\) 2.20483i 0.00680505i
\(19\) − 244.752i − 0.677984i −0.940789 − 0.338992i \(-0.889914\pi\)
0.940789 − 0.338992i \(-0.110086\pi\)
\(20\) 0 0
\(21\) −251.095 + 368.705i −0.569376 + 0.836066i
\(22\) 125.342i 0.258970i
\(23\) − 621.143i − 1.17418i −0.809521 − 0.587091i \(-0.800273\pi\)
0.809521 − 0.587091i \(-0.199727\pi\)
\(24\) − 327.166i − 0.567996i
\(25\) 0 0
\(26\) − 285.294i − 0.422033i
\(27\) 720.302 0.988069
\(28\) 403.316 − 592.226i 0.514434 − 0.755390i
\(29\) 774.981 0.921500 0.460750 − 0.887530i \(-0.347581\pi\)
0.460750 + 0.887530i \(0.347581\pi\)
\(30\) 0 0
\(31\) − 403.109i − 0.419468i −0.977759 − 0.209734i \(-0.932740\pi\)
0.977759 − 0.209734i \(-0.0672598\pi\)
\(32\) 800.073i 0.781321i
\(33\) −972.334 −0.892869
\(34\) − 173.981i − 0.150502i
\(35\) 0 0
\(36\) 27.4729 0.0211982
\(37\) − 1290.58i − 0.942716i −0.881942 − 0.471358i \(-0.843764\pi\)
0.881942 − 0.471358i \(-0.156236\pi\)
\(38\) 287.229 0.198912
\(39\) 2213.16 1.45507
\(40\) 0 0
\(41\) − 2536.73i − 1.50906i −0.656264 − 0.754531i \(-0.727864\pi\)
0.656264 − 0.754531i \(-0.272136\pi\)
\(42\) −432.694 − 294.672i −0.245291 − 0.167048i
\(43\) − 2242.99i − 1.21308i −0.795052 − 0.606542i \(-0.792556\pi\)
0.795052 − 0.606542i \(-0.207444\pi\)
\(44\) 1561.79 0.806712
\(45\) 0 0
\(46\) 728.942 0.344490
\(47\) 2494.61 1.12930 0.564648 − 0.825332i \(-0.309012\pi\)
0.564648 + 0.825332i \(0.309012\pi\)
\(48\) −1746.02 −0.757819
\(49\) −879.536 − 2234.10i −0.366321 − 0.930489i
\(50\) 0 0
\(51\) 1349.65 0.518897
\(52\) −3554.85 −1.31466
\(53\) 281.688i 0.100280i 0.998742 + 0.0501402i \(0.0159668\pi\)
−0.998742 + 0.0501402i \(0.984033\pi\)
\(54\) 845.310i 0.289887i
\(55\) 0 0
\(56\) 1455.47 + 991.201i 0.464117 + 0.316072i
\(57\) 2228.17i 0.685802i
\(58\) 909.479i 0.270356i
\(59\) − 5822.15i − 1.67255i −0.548309 − 0.836276i \(-0.684728\pi\)
0.548309 − 0.836276i \(-0.315272\pi\)
\(60\) 0 0
\(61\) 6268.19i 1.68454i 0.539053 + 0.842272i \(0.318782\pi\)
−0.539053 + 0.842272i \(0.681218\pi\)
\(62\) 473.068 0.123067
\(63\) 51.8192 − 76.0908i 0.0130560 − 0.0191713i
\(64\) 2129.72 0.519951
\(65\) 0 0
\(66\) − 1141.08i − 0.261956i
\(67\) 4128.43i 0.919678i 0.888002 + 0.459839i \(0.152093\pi\)
−0.888002 + 0.459839i \(0.847907\pi\)
\(68\) −2167.85 −0.468826
\(69\) 5654.74i 1.18772i
\(70\) 0 0
\(71\) −793.446 −0.157399 −0.0786993 − 0.996898i \(-0.525077\pi\)
−0.0786993 + 0.996898i \(0.525077\pi\)
\(72\) 67.5182i 0.0130243i
\(73\) 1600.72 0.300380 0.150190 − 0.988657i \(-0.452011\pi\)
0.150190 + 0.988657i \(0.452011\pi\)
\(74\) 1514.56 0.276581
\(75\) 0 0
\(76\) − 3578.96i − 0.619626i
\(77\) 2945.84 − 4325.65i 0.496853 − 0.729575i
\(78\) 2597.25i 0.426899i
\(79\) −9856.44 −1.57931 −0.789653 − 0.613554i \(-0.789739\pi\)
−0.789653 + 0.613554i \(0.789739\pi\)
\(80\) 0 0
\(81\) −6709.65 −1.02266
\(82\) 2976.98 0.442740
\(83\) 617.012 0.0895648 0.0447824 − 0.998997i \(-0.485741\pi\)
0.0447824 + 0.998997i \(0.485741\pi\)
\(84\) −3671.70 + 5391.49i −0.520366 + 0.764101i
\(85\) 0 0
\(86\) 2632.26 0.355903
\(87\) −7055.26 −0.932125
\(88\) 3838.31i 0.495649i
\(89\) 7345.17i 0.927304i 0.886018 + 0.463652i \(0.153461\pi\)
−0.886018 + 0.463652i \(0.846539\pi\)
\(90\) 0 0
\(91\) −6705.13 + 9845.75i −0.809701 + 1.18896i
\(92\) − 9082.83i − 1.07311i
\(93\) 3669.81i 0.424305i
\(94\) 2927.55i 0.331321i
\(95\) 0 0
\(96\) − 7283.69i − 0.790330i
\(97\) −67.6288 −0.00718767 −0.00359383 − 0.999994i \(-0.501144\pi\)
−0.00359383 + 0.999994i \(0.501144\pi\)
\(98\) 2621.83 − 1032.18i 0.272994 − 0.107474i
\(99\) 200.664 0.0204738
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 175.5.c.d.174.10 24
5.2 odd 4 175.5.d.i.76.6 12
5.3 odd 4 35.5.d.a.6.7 ✓ 12
5.4 even 2 inner 175.5.c.d.174.15 24
7.6 odd 2 inner 175.5.c.d.174.16 24
15.8 even 4 315.5.h.a.181.6 12
20.3 even 4 560.5.f.b.321.9 12
35.13 even 4 35.5.d.a.6.8 yes 12
35.27 even 4 175.5.d.i.76.5 12
35.34 odd 2 inner 175.5.c.d.174.9 24
105.83 odd 4 315.5.h.a.181.5 12
140.83 odd 4 560.5.f.b.321.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.d.a.6.7 ✓ 12 5.3 odd 4
35.5.d.a.6.8 yes 12 35.13 even 4
175.5.c.d.174.9 24 35.34 odd 2 inner
175.5.c.d.174.10 24 1.1 even 1 trivial
175.5.c.d.174.15 24 5.4 even 2 inner
175.5.c.d.174.16 24 7.6 odd 2 inner
175.5.d.i.76.5 12 35.27 even 4
175.5.d.i.76.6 12 5.2 odd 4
315.5.h.a.181.5 12 105.83 odd 4
315.5.h.a.181.6 12 15.8 even 4
560.5.f.b.321.4 12 140.83 odd 4
560.5.f.b.321.9 12 20.3 even 4