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Properties

Label 75.2.g.b.16.1
Level $75$
Weight $2$
Character 75.16
Analytic conductor $0.599$
Analytic rank $0$
Dimension $8$
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] = [75,2,Mod(16,75)] 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("75.16"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 75.g (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] 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: \(0.598878015160\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.26265625.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 2x^{6} + x^{4} + 8x^{2} - 24x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 16.1
Root \(1.33631 + 0.462894i\) of defining polynomial
Character \(\chi\) \(=\) 75.16
Dual form 75.2.g.b.61.1

$q$-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+(-2.18949 - 1.59076i) q^{2} +(0.309017 + 0.951057i) q^{3} +(1.64533 + 5.06380i) q^{4} +(0.336312 + 2.21063i) q^{5} +(0.836312 - 2.57390i) q^{6} +0.470294 q^{7} +(2.78023 - 8.55667i) q^{8} +(-0.809017 + 0.587785i) q^{9} +(2.78023 - 5.37515i) q^{10} +(2.57387 + 1.87003i) q^{11} +(-4.30753 + 3.12960i) q^{12} +(-0.455836 + 0.331184i) q^{13} +(-1.02971 - 0.748125i) q^{14} +(-1.99851 + 1.00297i) q^{15} +(-11.0839 + 8.05289i) q^{16} +(-0.527295 + 1.62285i) q^{17} +2.70636 q^{18} +(1.15575 - 3.55705i) q^{19} +(-10.6409 + 5.34023i) q^{20} +(0.145329 + 0.447276i) q^{21} +(-2.66071 - 8.18882i) q^{22} +(1.83631 + 1.33416i) q^{23} +8.99702 q^{24} +(-4.77379 + 1.48692i) q^{25} +1.52488 q^{26} +(-0.809017 - 0.587785i) q^{27} +(0.773789 + 2.38148i) q^{28} +(-2.57238 - 7.91697i) q^{29} +(5.97122 + 0.983144i) q^{30} +(1.67999 - 5.17047i) q^{31} +19.0842 q^{32} +(-0.983131 + 3.02577i) q^{33} +(3.73607 - 2.71441i) q^{34} +(0.158166 + 1.03965i) q^{35} +(-4.30753 - 3.12960i) q^{36} +(-0.825886 + 0.600041i) q^{37} +(-8.18892 + 5.94960i) q^{38} +(-0.455836 - 0.331184i) q^{39} +(19.8507 + 3.26836i) q^{40} +(-1.19098 + 0.865300i) q^{41} +(0.393313 - 1.21049i) q^{42} +6.72721 q^{43} +(-5.23458 + 16.1104i) q^{44} +(-1.57146 - 1.59076i) q^{45} +(-1.89827 - 5.84226i) q^{46} +(-1.37005 - 4.21658i) q^{47} +(-11.0839 - 8.05289i) q^{48} -6.77882 q^{49} +(12.8175 + 4.33834i) q^{50} -1.70636 q^{51} +(-2.42705 - 1.76336i) q^{52} +(-2.17907 - 6.70648i) q^{53} +(0.836312 + 2.57390i) q^{54} +(-3.26832 + 6.31879i) q^{55} +(1.30753 - 4.02415i) q^{56} +3.74010 q^{57} +(-6.96179 + 21.4262i) q^{58} +(10.4136 - 7.56596i) q^{59} +(-8.36707 - 8.46984i) q^{60} +(-0.102655 - 0.0745831i) q^{61} +(-11.9033 + 8.64825i) q^{62} +(-0.380476 + 0.276432i) q^{63} +(-19.6170 - 14.2526i) q^{64} +(-0.885429 - 0.896304i) q^{65} +(6.96582 - 5.06097i) q^{66} +(-0.863607 + 2.65791i) q^{67} -9.08535 q^{68} +(-0.701409 + 2.15871i) q^{69} +(1.30753 - 2.52790i) q^{70} +(4.95838 + 15.2603i) q^{71} +(2.78023 + 8.55667i) q^{72} +(7.91925 + 5.75367i) q^{73} +2.76279 q^{74} +(-2.88933 - 4.08066i) q^{75} +19.9138 q^{76} +(1.21048 + 0.879462i) q^{77} +(0.471215 + 1.45025i) q^{78} +(-1.46937 - 4.52227i) q^{79} +(-21.5296 - 21.7940i) q^{80} +(0.309017 - 0.951057i) q^{81} +3.98413 q^{82} +(3.61256 - 11.1183i) q^{83} +(-2.02580 + 1.47183i) q^{84} +(-3.76485 - 0.619872i) q^{85} +(-14.7292 - 10.7014i) q^{86} +(6.73458 - 4.89296i) q^{87} +(23.1572 - 16.8247i) q^{88} +(5.01630 + 3.64456i) q^{89} +(0.910182 + 5.98277i) q^{90} +(-0.214377 + 0.155754i) q^{91} +(-3.73458 + 11.4938i) q^{92} +5.43656 q^{93} +(-3.70785 + 11.4116i) q^{94} +(8.25202 + 1.35867i) q^{95} +(5.89735 + 18.1502i) q^{96} +(2.61256 + 8.04064i) q^{97} +(14.8422 + 10.7835i) q^{98} -3.18148 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} - 2 q^{3} + q^{4} - 5 q^{5} - q^{6} + 4 q^{7} - 2 q^{9} + 16 q^{11} - 9 q^{12} - 8 q^{13} - 8 q^{14} + 5 q^{15} - 17 q^{16} - q^{17} + 4 q^{18} - 5 q^{19} - 10 q^{20} - 11 q^{21} + 13 q^{22}+ \cdots - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.18949 − 1.59076i −1.54821 − 1.12484i −0.944915 − 0.327315i \(-0.893856\pi\)
−0.603290 − 0.797522i \(-0.706144\pi\)
\(3\) 0.309017 + 0.951057i 0.178411 + 0.549093i
\(4\) 1.64533 + 5.06380i 0.822664 + 2.53190i
\(5\) 0.336312 + 2.21063i 0.150403 + 0.988625i
\(6\) 0.836312 − 2.57390i 0.341423 − 1.05079i
\(7\) 0.470294 0.177754 0.0888772 − 0.996043i \(-0.471672\pi\)
0.0888772 + 0.996043i \(0.471672\pi\)
\(8\) 2.78023 − 8.55667i 0.982960 − 3.02524i
\(9\) −0.809017 + 0.587785i −0.269672 + 0.195928i
\(10\) 2.78023 − 5.37515i 0.879187 − 1.69977i
\(11\) 2.57387 + 1.87003i 0.776051 + 0.563834i 0.903791 − 0.427974i \(-0.140772\pi\)
−0.127740 + 0.991808i \(0.540772\pi\)
\(12\) −4.30753 + 3.12960i −1.24348 + 0.903438i
\(13\) −0.455836 + 0.331184i −0.126426 + 0.0918540i −0.649201 − 0.760617i \(-0.724897\pi\)
0.522775 + 0.852471i \(0.324897\pi\)
\(14\) −1.02971 − 0.748125i −0.275200 − 0.199945i
\(15\) −1.99851 + 1.00297i −0.516013 + 0.258967i
\(16\) −11.0839 + 8.05289i −2.77096 + 2.01322i
\(17\) −0.527295 + 1.62285i −0.127888 + 0.393598i −0.994416 − 0.105530i \(-0.966346\pi\)
0.866528 + 0.499128i \(0.166346\pi\)
\(18\) 2.70636 0.637896
\(19\) 1.15575 − 3.55705i 0.265148 − 0.816043i −0.726511 − 0.687155i \(-0.758859\pi\)
0.991659 − 0.128888i \(-0.0411406\pi\)
\(20\) −10.6409 + 5.34023i −2.37937 + 1.19411i
\(21\) 0.145329 + 0.447276i 0.0317134 + 0.0976037i
\(22\) −2.66071 − 8.18882i −0.567265 − 1.74586i
\(23\) 1.83631 + 1.33416i 0.382898 + 0.278191i 0.762539 − 0.646942i \(-0.223953\pi\)
−0.379641 + 0.925134i \(0.623953\pi\)
\(24\) 8.99702 1.83651
\(25\) −4.77379 + 1.48692i −0.954758 + 0.297385i
\(26\) 1.52488 0.299054
\(27\) −0.809017 − 0.587785i −0.155695 − 0.113119i
\(28\) 0.773789 + 2.38148i 0.146232 + 0.450057i
\(29\) −2.57238 − 7.91697i −0.477679 − 1.47014i −0.842310 − 0.538993i \(-0.818805\pi\)
0.364631 − 0.931152i \(-0.381195\pi\)
\(30\) 5.97122 + 0.983144i 1.09019 + 0.179497i
\(31\) 1.67999 − 5.17047i 0.301735 − 0.928644i −0.679141 − 0.734008i \(-0.737647\pi\)
0.980876 − 0.194636i \(-0.0623526\pi\)
\(32\) 19.0842 3.37364
\(33\) −0.983131 + 3.02577i −0.171141 + 0.526718i
\(34\) 3.73607 − 2.71441i 0.640730 − 0.465518i
\(35\) 0.158166 + 1.03965i 0.0267349 + 0.175732i
\(36\) −4.30753 − 3.12960i −0.717921 − 0.521600i
\(37\) −0.825886 + 0.600041i −0.135775 + 0.0986462i −0.653600 − 0.756841i \(-0.726742\pi\)
0.517825 + 0.855487i \(0.326742\pi\)
\(38\) −8.18892 + 5.94960i −1.32842 + 0.965153i
\(39\) −0.455836 − 0.331184i −0.0729922 − 0.0530319i
\(40\) 19.8507 + 3.26836i 3.13867 + 0.516773i
\(41\) −1.19098 + 0.865300i −0.186000 + 0.135137i −0.676889 − 0.736085i \(-0.736672\pi\)
0.490889 + 0.871222i \(0.336672\pi\)
\(42\) 0.393313 − 1.21049i 0.0606895 − 0.186783i
\(43\) 6.72721 1.02589 0.512945 − 0.858421i \(-0.328554\pi\)
0.512945 + 0.858421i \(0.328554\pi\)
\(44\) −5.23458 + 16.1104i −0.789142 + 2.42873i
\(45\) −1.57146 − 1.59076i −0.234259 − 0.237136i
\(46\) −1.89827 − 5.84226i −0.279884 − 0.861395i
\(47\) −1.37005 − 4.21658i −0.199842 − 0.615052i −0.999886 − 0.0151095i \(-0.995190\pi\)
0.800043 − 0.599942i \(-0.204810\pi\)
\(48\) −11.0839 − 8.05289i −1.59982 − 1.16234i
\(49\) −6.77882 −0.968403
\(50\) 12.8175 + 4.33834i 1.81267 + 0.613534i
\(51\) −1.70636 −0.238938
\(52\) −2.42705 − 1.76336i −0.336571 − 0.244533i
\(53\) −2.17907 − 6.70648i −0.299318 − 0.921206i −0.981737 − 0.190244i \(-0.939072\pi\)
0.682419 − 0.730961i \(-0.260928\pi\)
\(54\) 0.836312 + 2.57390i 0.113808 + 0.350264i
\(55\) −3.26832 + 6.31879i −0.440700 + 0.852026i
\(56\) 1.30753 − 4.02415i 0.174726 − 0.537750i
\(57\) 3.74010 0.495388
\(58\) −6.96179 + 21.4262i −0.914128 + 2.81340i
\(59\) 10.4136 − 7.56596i 1.35574 − 0.985004i 0.357038 − 0.934090i \(-0.383787\pi\)
0.998703 − 0.0509138i \(-0.0162134\pi\)
\(60\) −8.36707 − 8.46984i −1.08018 − 1.09345i
\(61\) −0.102655 − 0.0745831i −0.0131436 − 0.00954939i 0.581194 − 0.813765i \(-0.302586\pi\)
−0.594338 + 0.804216i \(0.702586\pi\)
\(62\) −11.9033 + 8.64825i −1.51172 + 1.09833i
\(63\) −0.380476 + 0.276432i −0.0479355 + 0.0348272i
\(64\) −19.6170 − 14.2526i −2.45213 − 1.78158i
\(65\) −0.885429 − 0.896304i −0.109824 − 0.111173i
\(66\) 6.96582 − 5.06097i 0.857434 − 0.622962i
\(67\) −0.863607 + 2.65791i −0.105506 + 0.324715i −0.989849 − 0.142123i \(-0.954607\pi\)
0.884343 + 0.466838i \(0.154607\pi\)
\(68\) −9.08535 −1.10176
\(69\) −0.701409 + 2.15871i −0.0844397 + 0.259879i
\(70\) 1.30753 − 2.52790i 0.156279 − 0.302142i
\(71\) 4.95838 + 15.2603i 0.588451 + 1.81107i 0.584945 + 0.811073i \(0.301116\pi\)
0.00350617 + 0.999994i \(0.498884\pi\)
\(72\) 2.78023 + 8.55667i 0.327653 + 1.00841i
\(73\) 7.91925 + 5.75367i 0.926878 + 0.673416i 0.945226 − 0.326415i \(-0.105841\pi\)
−0.0183484 + 0.999832i \(0.505841\pi\)
\(74\) 2.76279 0.321168
\(75\) −2.88933 − 4.08066i −0.333631 − 0.471194i
\(76\) 19.9138 2.28427
\(77\) 1.21048 + 0.879462i 0.137947 + 0.100224i
\(78\) 0.471215 + 1.45025i 0.0533546 + 0.164209i
\(79\) −1.46937 − 4.52227i −0.165317 − 0.508795i 0.833742 − 0.552154i \(-0.186194\pi\)
−0.999060 + 0.0433593i \(0.986194\pi\)
\(80\) −21.5296 − 21.7940i −2.40708 − 2.43665i
\(81\) 0.309017 − 0.951057i 0.0343352 − 0.105673i
\(82\) 3.98413 0.439974
\(83\) 3.61256 − 11.1183i 0.396530 − 1.22039i −0.531233 − 0.847226i \(-0.678271\pi\)
0.927763 − 0.373169i \(-0.121729\pi\)
\(84\) −2.02580 + 1.47183i −0.221033 + 0.160590i
\(85\) −3.76485 − 0.619872i −0.408356 − 0.0672346i
\(86\) −14.7292 − 10.7014i −1.58829 − 1.15396i
\(87\) 6.73458 − 4.89296i 0.722023 − 0.524580i
\(88\) 23.1572 − 16.8247i 2.46856 − 1.79351i
\(89\) 5.01630 + 3.64456i 0.531727 + 0.386322i 0.821003 − 0.570923i \(-0.193415\pi\)
−0.289277 + 0.957246i \(0.593415\pi\)
\(90\) 0.910182 + 5.98277i 0.0959416 + 0.630639i
\(91\) −0.214377 + 0.155754i −0.0224728 + 0.0163275i
\(92\) −3.73458 + 11.4938i −0.389357 + 1.19832i
\(93\) 5.43656 0.563745
\(94\) −3.70785 + 11.4116i −0.382436 + 1.17702i
\(95\) 8.25202 + 1.35867i 0.846639 + 0.139397i
\(96\) 5.89735 + 18.1502i 0.601895 + 1.85244i
\(97\) 2.61256 + 8.04064i 0.265265 + 0.816403i 0.991632 + 0.129096i \(0.0412074\pi\)
−0.726367 + 0.687307i \(0.758793\pi\)
\(98\) 14.8422 + 10.7835i 1.49929 + 1.08930i
\(99\) −3.18148 −0.319751
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 75.2.g.b.16.1 ✓ 8
3.2 odd 2 225.2.h.c.91.2 8
5.2 odd 4 375.2.i.b.49.4 16
5.3 odd 4 375.2.i.b.49.1 16
5.4 even 2 375.2.g.b.76.2 8
25.2 odd 20 375.2.i.b.199.1 16
25.6 even 5 1875.2.a.h.1.4 4
25.8 odd 20 1875.2.b.c.1249.1 8
25.11 even 5 inner 75.2.g.b.61.1 yes 8
25.14 even 10 375.2.g.b.301.2 8
25.17 odd 20 1875.2.b.c.1249.8 8
25.19 even 10 1875.2.a.e.1.1 4
25.23 odd 20 375.2.i.b.199.4 16
75.11 odd 10 225.2.h.c.136.2 8
75.44 odd 10 5625.2.a.n.1.4 4
75.56 odd 10 5625.2.a.i.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.b.16.1 ✓ 8 1.1 even 1 trivial
75.2.g.b.61.1 yes 8 25.11 even 5 inner
225.2.h.c.91.2 8 3.2 odd 2
225.2.h.c.136.2 8 75.11 odd 10
375.2.g.b.76.2 8 5.4 even 2
375.2.g.b.301.2 8 25.14 even 10
375.2.i.b.49.1 16 5.3 odd 4
375.2.i.b.49.4 16 5.2 odd 4
375.2.i.b.199.1 16 25.2 odd 20
375.2.i.b.199.4 16 25.23 odd 20
1875.2.a.e.1.1 4 25.19 even 10
1875.2.a.h.1.4 4 25.6 even 5
1875.2.b.c.1249.1 8 25.8 odd 20
1875.2.b.c.1249.8 8 25.17 odd 20
5625.2.a.i.1.1 4 75.56 odd 10
5625.2.a.n.1.4 4 75.44 odd 10