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Properties

Label 375.2.g.b.151.1
Level $375$
Weight $2$
Character 375.151
Analytic conductor $2.994$
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] = [375,2,Mod(76,375)] 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("375.76"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(375, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 375 = 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 375.g (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,1] 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: no
Analytic conductor: \(2.99439007580\)
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: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 151.1
Root \(-1.21700 - 0.720348i\) of defining polynomial
Character \(\chi\) \(=\) 375.151
Dual form 375.2.g.b.226.1

$q$-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+(-0.655837 - 2.01846i) q^{2} +(0.809017 - 0.587785i) q^{3} +(-2.02602 + 1.47199i) q^{4} +(-1.71700 - 1.24748i) q^{6} -4.35840 q^{7} +(0.865884 + 0.629102i) q^{8} +(0.309017 - 0.951057i) q^{9} +(-0.488218 - 1.50258i) q^{11} +(-0.773871 + 2.38173i) q^{12} +(-0.370184 + 1.13931i) q^{13} +(2.85840 + 8.79726i) q^{14} +(-0.845805 + 2.60312i) q^{16} +(-0.907987 - 0.659691i) q^{17} -2.12233 q^{18} +(-6.21218 - 4.51341i) q^{19} +(-3.52602 + 2.56180i) q^{21} +(-2.71270 + 1.97090i) q^{22} +(0.717004 + 2.20671i) q^{23} +1.07029 q^{24} +2.54243 q^{26} +(-0.309017 - 0.951057i) q^{27} +(8.83021 - 6.41552i) q^{28} +(4.45307 - 3.23535i) q^{29} +(-3.88495 - 2.82258i) q^{31} +7.94959 q^{32} +(-1.27817 - 0.928645i) q^{33} +(-0.736068 + 2.26538i) q^{34} +(0.773871 + 2.38173i) q^{36} +(1.96915 - 6.06043i) q^{37} +(-5.03596 + 15.4991i) q^{38} +(0.370184 + 1.13931i) q^{39} +(-2.30902 + 7.10642i) q^{41} +(7.48339 + 5.43700i) q^{42} +1.24998 q^{43} +(3.20092 + 2.32561i) q^{44} +(3.98392 - 2.89449i) q^{46} +(3.33934 - 2.42617i) q^{47} +(0.845805 + 2.60312i) q^{48} +11.9957 q^{49} -1.12233 q^{51} +(-0.927051 - 2.85317i) q^{52} +(3.03032 - 2.20166i) q^{53} +(-1.71700 + 1.24748i) q^{54} +(-3.77387 - 2.74188i) q^{56} -7.67867 q^{57} +(-9.45090 - 6.86648i) q^{58} +(2.82940 - 8.70799i) q^{59} +(0.431351 + 1.32756i) q^{61} +(-3.14937 + 9.69276i) q^{62} +(-1.34682 + 4.14509i) q^{63} +(-3.52202 - 10.8397i) q^{64} +(-1.03616 + 3.18898i) q^{66} +(-3.12499 - 2.27044i) q^{67} +2.81066 q^{68} +(1.87714 + 1.36382i) q^{69} +(8.57970 - 6.23352i) q^{71} +(0.865884 - 0.629102i) q^{72} +(-1.54407 - 4.75216i) q^{73} -13.5242 q^{74} +19.2297 q^{76} +(2.12785 + 6.54885i) q^{77} +(2.05687 - 1.49440i) q^{78} +(-11.7737 + 8.55407i) q^{79} +(-0.809017 - 0.587785i) q^{81} +15.8584 q^{82} +(-7.06760 - 5.13491i) q^{83} +(3.37284 - 10.3805i) q^{84} +(-0.819784 - 2.52304i) q^{86} +(1.70092 - 5.23490i) q^{87} +(0.522535 - 1.60820i) q^{88} +(-3.10195 - 9.54683i) q^{89} +(1.61341 - 4.96556i) q^{91} +(-4.70092 - 3.41542i) q^{92} -4.80206 q^{93} +(-7.08719 - 5.14914i) q^{94} +(6.43135 - 4.67265i) q^{96} +(-6.06760 + 4.40837i) q^{97} +(-7.86720 - 24.2128i) q^{98} -1.57991 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + 2 q^{3} + q^{4} - q^{6} - 4 q^{7} - 2 q^{9} + 16 q^{11} + 9 q^{12} + 8 q^{13} - 8 q^{14} - 17 q^{16} + q^{17} - 4 q^{18} - 5 q^{19} - 11 q^{21} - 13 q^{22} - 7 q^{23} + 30 q^{24} + 6 q^{26}+ \cdots - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/375\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(251\)
\(\chi(n)\) \(e\left(\frac{2}{5}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.655837 − 2.01846i −0.463747 − 1.42727i −0.860552 − 0.509363i \(-0.829881\pi\)
0.396805 − 0.917903i \(-0.370119\pi\)
\(3\) 0.809017 − 0.587785i 0.467086 − 0.339358i
\(4\) −2.02602 + 1.47199i −1.01301 + 0.735995i
\(5\) 0 0
\(6\) −1.71700 − 1.24748i −0.700964 − 0.509280i
\(7\) −4.35840 −1.64732 −0.823660 − 0.567083i \(-0.808072\pi\)
−0.823660 + 0.567083i \(0.808072\pi\)
\(8\) 0.865884 + 0.629102i 0.306136 + 0.222421i
\(9\) 0.309017 − 0.951057i 0.103006 − 0.317019i
\(10\) 0 0
\(11\) −0.488218 − 1.50258i −0.147203 − 0.453045i 0.850085 − 0.526646i \(-0.176551\pi\)
−0.997288 + 0.0736014i \(0.976551\pi\)
\(12\) −0.773871 + 2.38173i −0.223397 + 0.687546i
\(13\) −0.370184 + 1.13931i −0.102670 + 0.315987i −0.989177 − 0.146729i \(-0.953125\pi\)
0.886506 + 0.462717i \(0.153125\pi\)
\(14\) 2.85840 + 8.79726i 0.763940 + 2.35117i
\(15\) 0 0
\(16\) −0.845805 + 2.60312i −0.211451 + 0.650780i
\(17\) −0.907987 − 0.659691i −0.220219 − 0.159999i 0.472206 − 0.881488i \(-0.343458\pi\)
−0.692425 + 0.721490i \(0.743458\pi\)
\(18\) −2.12233 −0.500239
\(19\) −6.21218 − 4.51341i −1.42517 − 1.03545i −0.990890 − 0.134670i \(-0.957002\pi\)
−0.434281 − 0.900777i \(-0.642998\pi\)
\(20\) 0 0
\(21\) −3.52602 + 2.56180i −0.769441 + 0.559031i
\(22\) −2.71270 + 1.97090i −0.578351 + 0.420196i
\(23\) 0.717004 + 2.20671i 0.149506 + 0.460131i 0.997563 − 0.0697736i \(-0.0222277\pi\)
−0.848057 + 0.529905i \(0.822228\pi\)
\(24\) 1.07029 0.218472
\(25\) 0 0
\(26\) 2.54243 0.498611
\(27\) −0.309017 − 0.951057i −0.0594703 − 0.183031i
\(28\) 8.83021 − 6.41552i 1.66875 − 1.21242i
\(29\) 4.45307 − 3.23535i 0.826915 − 0.600789i −0.0917701 − 0.995780i \(-0.529252\pi\)
0.918685 + 0.394992i \(0.129252\pi\)
\(30\) 0 0
\(31\) −3.88495 − 2.82258i −0.697757 − 0.506950i 0.181444 − 0.983401i \(-0.441923\pi\)
−0.879201 + 0.476451i \(0.841923\pi\)
\(32\) 7.94959 1.40530
\(33\) −1.27817 − 0.928645i −0.222501 − 0.161656i
\(34\) −0.736068 + 2.26538i −0.126235 + 0.388510i
\(35\) 0 0
\(36\) 0.773871 + 2.38173i 0.128979 + 0.396955i
\(37\) 1.96915 − 6.06043i 0.323727 − 0.996329i −0.648285 − 0.761398i \(-0.724514\pi\)
0.972012 − 0.234931i \(-0.0754865\pi\)
\(38\) −5.03596 + 15.4991i −0.816941 + 2.51428i
\(39\) 0.370184 + 1.13931i 0.0592768 + 0.182435i
\(40\) 0 0
\(41\) −2.30902 + 7.10642i −0.360608 + 1.10984i 0.592078 + 0.805881i \(0.298308\pi\)
−0.952686 + 0.303956i \(0.901692\pi\)
\(42\) 7.48339 + 5.43700i 1.15471 + 0.838948i
\(43\) 1.24998 0.190620 0.0953102 − 0.995448i \(-0.469616\pi\)
0.0953102 + 0.995448i \(0.469616\pi\)
\(44\) 3.20092 + 2.32561i 0.482557 + 0.350598i
\(45\) 0 0
\(46\) 3.98392 − 2.89449i 0.587397 − 0.426769i
\(47\) 3.33934 − 2.42617i 0.487092 − 0.353893i −0.316973 − 0.948435i \(-0.602666\pi\)
0.804065 + 0.594541i \(0.202666\pi\)
\(48\) 0.845805 + 2.60312i 0.122081 + 0.375728i
\(49\) 11.9957 1.71367
\(50\) 0 0
\(51\) −1.12233 −0.157158
\(52\) −0.927051 − 2.85317i −0.128559 − 0.395663i
\(53\) 3.03032 − 2.20166i 0.416247 − 0.302421i −0.359879 − 0.932999i \(-0.617182\pi\)
0.776126 + 0.630578i \(0.217182\pi\)
\(54\) −1.71700 + 1.24748i −0.233655 + 0.169760i
\(55\) 0 0
\(56\) −3.77387 − 2.74188i −0.504305 − 0.366399i
\(57\) −7.67867 −1.01707
\(58\) −9.45090 − 6.86648i −1.24096 − 0.901613i
\(59\) 2.82940 − 8.70799i 0.368356 − 1.13368i −0.579496 − 0.814975i \(-0.696751\pi\)
0.947853 − 0.318709i \(-0.103249\pi\)
\(60\) 0 0
\(61\) 0.431351 + 1.32756i 0.0552288 + 0.169977i 0.974866 − 0.222792i \(-0.0715172\pi\)
−0.919637 + 0.392769i \(0.871517\pi\)
\(62\) −3.14937 + 9.69276i −0.399970 + 1.23098i
\(63\) −1.34682 + 4.14509i −0.169683 + 0.522232i
\(64\) −3.52202 − 10.8397i −0.440253 − 1.35496i
\(65\) 0 0
\(66\) −1.03616 + 3.18898i −0.127543 + 0.392536i
\(67\) −3.12499 − 2.27044i −0.381778 − 0.277378i 0.380300 − 0.924863i \(-0.375821\pi\)
−0.762078 + 0.647485i \(0.775821\pi\)
\(68\) 2.81066 0.340843
\(69\) 1.87714 + 1.36382i 0.225981 + 0.164185i
\(70\) 0 0
\(71\) 8.57970 − 6.23352i 1.01822 − 0.739783i 0.0523057 − 0.998631i \(-0.483343\pi\)
0.965918 + 0.258848i \(0.0833430\pi\)
\(72\) 0.865884 − 0.629102i 0.102045 − 0.0741403i
\(73\) −1.54407 − 4.75216i −0.180720 − 0.556198i 0.819129 − 0.573610i \(-0.194457\pi\)
−0.999848 + 0.0174117i \(0.994457\pi\)
\(74\) −13.5242 −1.57215
\(75\) 0 0
\(76\) 19.2297 2.20580
\(77\) 2.12785 + 6.54885i 0.242491 + 0.746310i
\(78\) 2.05687 − 1.49440i 0.232894 − 0.169208i
\(79\) −11.7737 + 8.55407i −1.32464 + 0.962408i −0.324779 + 0.945790i \(0.605290\pi\)
−0.999862 + 0.0166185i \(0.994710\pi\)
\(80\) 0 0
\(81\) −0.809017 − 0.587785i −0.0898908 − 0.0653095i
\(82\) 15.8584 1.75126
\(83\) −7.06760 − 5.13491i −0.775770 − 0.563630i 0.127937 − 0.991782i \(-0.459165\pi\)
−0.903706 + 0.428153i \(0.859165\pi\)
\(84\) 3.37284 − 10.3805i 0.368007 − 1.13261i
\(85\) 0 0
\(86\) −0.819784 − 2.52304i −0.0883996 − 0.272066i
\(87\) 1.70092 − 5.23490i 0.182358 − 0.561240i
\(88\) 0.522535 − 1.60820i 0.0557025 − 0.171435i
\(89\) −3.10195 − 9.54683i −0.328806 − 1.01196i −0.969693 − 0.244326i \(-0.921433\pi\)
0.640887 − 0.767635i \(-0.278567\pi\)
\(90\) 0 0
\(91\) 1.61341 − 4.96556i 0.169131 − 0.520532i
\(92\) −4.70092 − 3.41542i −0.490105 − 0.356082i
\(93\) −4.80206 −0.497950
\(94\) −7.08719 − 5.14914i −0.730988 − 0.531094i
\(95\) 0 0
\(96\) 6.43135 − 4.67265i 0.656397 − 0.476900i
\(97\) −6.06760 + 4.40837i −0.616071 + 0.447602i −0.851547 − 0.524278i \(-0.824335\pi\)
0.235476 + 0.971880i \(0.424335\pi\)
\(98\) −7.86720 − 24.2128i −0.794707 − 2.44586i
\(99\) −1.57991 −0.158787
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 375.2.g.b.151.1 8
5.2 odd 4 375.2.i.b.349.4 16
5.3 odd 4 375.2.i.b.349.1 16
5.4 even 2 75.2.g.b.31.2 ✓ 8
15.14 odd 2 225.2.h.c.181.1 8
25.2 odd 20 1875.2.b.c.1249.2 8
25.3 odd 20 375.2.i.b.274.4 16
25.4 even 10 75.2.g.b.46.2 yes 8
25.11 even 5 1875.2.a.e.1.2 4
25.14 even 10 1875.2.a.h.1.3 4
25.21 even 5 inner 375.2.g.b.226.1 8
25.22 odd 20 375.2.i.b.274.1 16
25.23 odd 20 1875.2.b.c.1249.7 8
75.11 odd 10 5625.2.a.n.1.3 4
75.14 odd 10 5625.2.a.i.1.2 4
75.29 odd 10 225.2.h.c.46.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.b.31.2 ✓ 8 5.4 even 2
75.2.g.b.46.2 yes 8 25.4 even 10
225.2.h.c.46.1 8 75.29 odd 10
225.2.h.c.181.1 8 15.14 odd 2
375.2.g.b.151.1 8 1.1 even 1 trivial
375.2.g.b.226.1 8 25.21 even 5 inner
375.2.i.b.274.1 16 25.22 odd 20
375.2.i.b.274.4 16 25.3 odd 20
375.2.i.b.349.1 16 5.3 odd 4
375.2.i.b.349.4 16 5.2 odd 4
1875.2.a.e.1.2 4 25.11 even 5
1875.2.a.h.1.3 4 25.14 even 10
1875.2.b.c.1249.2 8 25.2 odd 20
1875.2.b.c.1249.7 8 25.23 odd 20
5625.2.a.i.1.2 4 75.14 odd 10
5625.2.a.n.1.3 4 75.11 odd 10