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Properties

Label 375.2.i.b.274.1
Level $375$
Weight $2$
Character 375.274
Analytic conductor $2.994$
Analytic rank $0$
Dimension $16$
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] = [375,2,Mod(49,375)] 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("375.49"); 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, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 375 = 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 375.i (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.99439007580\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: 16.0.45212176000000000000.9
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 5x^{14} + 6x^{12} - 20x^{10} - 79x^{8} - 80x^{6} + 96x^{4} + 320x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 274.1
Root \(-0.132563 - 1.40799i\) of defining polynomial
Character \(\chi\) \(=\) 375.274
Dual form 375.2.i.b.349.1

$q$-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+(-2.01846 - 0.655837i) q^{2} +(0.587785 - 0.809017i) q^{3} +(2.02602 + 1.47199i) q^{4} +(-1.71700 + 1.24748i) q^{6} -4.35840i q^{7} +(-0.629102 - 0.865884i) q^{8} +(-0.309017 - 0.951057i) q^{9} +(-0.488218 + 1.50258i) q^{11} +(2.38173 - 0.773871i) q^{12} +(-1.13931 + 0.370184i) q^{13} +(-2.85840 + 8.79726i) q^{14} +(-0.845805 - 2.60312i) q^{16} +(-0.659691 - 0.907987i) q^{17} +2.12233i q^{18} +(6.21218 - 4.51341i) q^{19} +(-3.52602 - 2.56180i) q^{21} +(1.97090 - 2.71270i) q^{22} +(-2.20671 - 0.717004i) q^{23} -1.07029 q^{24} +2.54243 q^{26} +(-0.951057 - 0.309017i) q^{27} +(6.41552 - 8.83021i) q^{28} +(-4.45307 - 3.23535i) q^{29} +(-3.88495 + 2.82258i) q^{31} +7.94959i q^{32} +(0.928645 + 1.27817i) q^{33} +(0.736068 + 2.26538i) q^{34} +(0.773871 - 2.38173i) q^{36} +(-6.06043 + 1.96915i) q^{37} +(-15.4991 + 5.03596i) q^{38} +(-0.370184 + 1.13931i) q^{39} +(-2.30902 - 7.10642i) q^{41} +(5.43700 + 7.48339i) q^{42} -1.24998i q^{43} +(-3.20092 + 2.32561i) q^{44} +(3.98392 + 2.89449i) q^{46} +(-2.42617 + 3.33934i) q^{47} +(-2.60312 - 0.845805i) q^{48} -11.9957 q^{49} -1.12233 q^{51} +(-2.85317 - 0.927051i) q^{52} +(2.20166 - 3.03032i) q^{53} +(1.71700 + 1.24748i) q^{54} +(-3.77387 + 2.74188i) q^{56} -7.67867i q^{57} +(6.86648 + 9.45090i) q^{58} +(-2.82940 - 8.70799i) q^{59} +(0.431351 - 1.32756i) q^{61} +(9.69276 - 3.14937i) q^{62} +(-4.14509 + 1.34682i) q^{63} +(3.52202 - 10.8397i) q^{64} +(-1.03616 - 3.18898i) q^{66} +(-2.27044 - 3.12499i) q^{67} -2.81066i q^{68} +(-1.87714 + 1.36382i) q^{69} +(8.57970 + 6.23352i) q^{71} +(-0.629102 + 0.865884i) q^{72} +(4.75216 + 1.54407i) q^{73} +13.5242 q^{74} +19.2297 q^{76} +(6.54885 + 2.12785i) q^{77} +(1.49440 - 2.05687i) q^{78} +(11.7737 + 8.55407i) q^{79} +(-0.809017 + 0.587785i) q^{81} +15.8584i q^{82} +(5.13491 + 7.06760i) q^{83} +(-3.37284 - 10.3805i) q^{84} +(-0.819784 + 2.52304i) q^{86} +(-5.23490 + 1.70092i) q^{87} +(1.60820 - 0.522535i) q^{88} +(3.10195 - 9.54683i) q^{89} +(1.61341 + 4.96556i) q^{91} +(-3.41542 - 4.70092i) q^{92} +4.80206i q^{93} +(7.08719 - 5.14914i) q^{94} +(6.43135 + 4.67265i) q^{96} +(4.40837 - 6.06760i) q^{97} +(24.2128 + 7.86720i) q^{98} +1.57991 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{4} - 2 q^{6} + 4 q^{9} + 32 q^{11} + 16 q^{14} - 34 q^{16} + 10 q^{19} - 22 q^{21} - 60 q^{24} + 12 q^{26} - 10 q^{29} - 38 q^{31} - 24 q^{34} - 18 q^{36} + 16 q^{39} - 28 q^{41} + 6 q^{44}+ \cdots + 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(251\)
\(\chi(n)\) \(e\left(\frac{1}{10}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.01846 − 0.655837i −1.42727 − 0.463747i −0.509363 − 0.860552i \(-0.670119\pi\)
−0.917903 + 0.396805i \(0.870119\pi\)
\(3\) 0.587785 − 0.809017i 0.339358 − 0.467086i
\(4\) 2.02602 + 1.47199i 1.01301 + 0.735995i
\(5\) 0 0
\(6\) −1.71700 + 1.24748i −0.700964 + 0.509280i
\(7\) − 4.35840i − 1.64732i −0.567083 − 0.823660i \(-0.691928\pi\)
0.567083 − 0.823660i \(-0.308072\pi\)
\(8\) −0.629102 − 0.865884i −0.222421 − 0.306136i
\(9\) −0.309017 − 0.951057i −0.103006 − 0.317019i
\(10\) 0 0
\(11\) −0.488218 + 1.50258i −0.147203 + 0.453045i −0.997288 − 0.0736014i \(-0.976551\pi\)
0.850085 + 0.526646i \(0.176551\pi\)
\(12\) 2.38173 − 0.773871i 0.687546 − 0.223397i
\(13\) −1.13931 + 0.370184i −0.315987 + 0.102670i −0.462717 − 0.886506i \(-0.653125\pi\)
0.146729 + 0.989177i \(0.453125\pi\)
\(14\) −2.85840 + 8.79726i −0.763940 + 2.35117i
\(15\) 0 0
\(16\) −0.845805 − 2.60312i −0.211451 − 0.650780i
\(17\) −0.659691 − 0.907987i −0.159999 − 0.220219i 0.721490 − 0.692425i \(-0.243458\pi\)
−0.881488 + 0.472206i \(0.843458\pi\)
\(18\) 2.12233i 0.500239i
\(19\) 6.21218 − 4.51341i 1.42517 − 1.03545i 0.434281 − 0.900777i \(-0.357002\pi\)
0.990890 − 0.134670i \(-0.0429975\pi\)
\(20\) 0 0
\(21\) −3.52602 − 2.56180i −0.769441 − 0.559031i
\(22\) 1.97090 − 2.71270i 0.420196 − 0.578351i
\(23\) −2.20671 − 0.717004i −0.460131 − 0.149506i 0.0697736 − 0.997563i \(-0.477772\pi\)
−0.529905 + 0.848057i \(0.677772\pi\)
\(24\) −1.07029 −0.218472
\(25\) 0 0
\(26\) 2.54243 0.498611
\(27\) −0.951057 − 0.309017i −0.183031 − 0.0594703i
\(28\) 6.41552 − 8.83021i 1.21242 − 1.66875i
\(29\) −4.45307 − 3.23535i −0.826915 − 0.600789i 0.0917701 − 0.995780i \(-0.470748\pi\)
−0.918685 + 0.394992i \(0.870748\pi\)
\(30\) 0 0
\(31\) −3.88495 + 2.82258i −0.697757 + 0.506950i −0.879201 − 0.476451i \(-0.841923\pi\)
0.181444 + 0.983401i \(0.441923\pi\)
\(32\) 7.94959i 1.40530i
\(33\) 0.928645 + 1.27817i 0.161656 + 0.222501i
\(34\) 0.736068 + 2.26538i 0.126235 + 0.388510i
\(35\) 0 0
\(36\) 0.773871 − 2.38173i 0.128979 − 0.396955i
\(37\) −6.06043 + 1.96915i −0.996329 + 0.323727i −0.761398 − 0.648285i \(-0.775486\pi\)
−0.234931 + 0.972012i \(0.575486\pi\)
\(38\) −15.4991 + 5.03596i −2.51428 + 0.816941i
\(39\) −0.370184 + 1.13931i −0.0592768 + 0.182435i
\(40\) 0 0
\(41\) −2.30902 − 7.10642i −0.360608 − 1.10984i −0.952686 − 0.303956i \(-0.901692\pi\)
0.592078 − 0.805881i \(-0.298308\pi\)
\(42\) 5.43700 + 7.48339i 0.838948 + 1.15471i
\(43\) − 1.24998i − 0.190620i −0.995448 − 0.0953102i \(-0.969616\pi\)
0.995448 − 0.0953102i \(-0.0303843\pi\)
\(44\) −3.20092 + 2.32561i −0.482557 + 0.350598i
\(45\) 0 0
\(46\) 3.98392 + 2.89449i 0.587397 + 0.426769i
\(47\) −2.42617 + 3.33934i −0.353893 + 0.487092i −0.948435 − 0.316973i \(-0.897334\pi\)
0.594541 + 0.804065i \(0.297334\pi\)
\(48\) −2.60312 − 0.845805i −0.375728 − 0.122081i
\(49\) −11.9957 −1.71367
\(50\) 0 0
\(51\) −1.12233 −0.157158
\(52\) −2.85317 − 0.927051i −0.395663 − 0.128559i
\(53\) 2.20166 − 3.03032i 0.302421 − 0.416247i −0.630578 − 0.776126i \(-0.717182\pi\)
0.932999 + 0.359879i \(0.117182\pi\)
\(54\) 1.71700 + 1.24748i 0.233655 + 0.169760i
\(55\) 0 0
\(56\) −3.77387 + 2.74188i −0.504305 + 0.366399i
\(57\) − 7.67867i − 1.01707i
\(58\) 6.86648 + 9.45090i 0.901613 + 1.24096i
\(59\) −2.82940 − 8.70799i −0.368356 − 1.13368i −0.947853 − 0.318709i \(-0.896751\pi\)
0.579496 − 0.814975i \(-0.303249\pi\)
\(60\) 0 0
\(61\) 0.431351 − 1.32756i 0.0552288 − 0.169977i −0.919637 − 0.392769i \(-0.871517\pi\)
0.974866 + 0.222792i \(0.0715172\pi\)
\(62\) 9.69276 − 3.14937i 1.23098 − 0.399970i
\(63\) −4.14509 + 1.34682i −0.522232 + 0.169683i
\(64\) 3.52202 − 10.8397i 0.440253 − 1.35496i
\(65\) 0 0
\(66\) −1.03616 − 3.18898i −0.127543 − 0.392536i
\(67\) −2.27044 − 3.12499i −0.277378 − 0.381778i 0.647485 − 0.762078i \(-0.275821\pi\)
−0.924863 + 0.380300i \(0.875821\pi\)
\(68\) − 2.81066i − 0.340843i
\(69\) −1.87714 + 1.36382i −0.225981 + 0.164185i
\(70\) 0 0
\(71\) 8.57970 + 6.23352i 1.01822 + 0.739783i 0.965918 − 0.258848i \(-0.0833430\pi\)
0.0523057 + 0.998631i \(0.483343\pi\)
\(72\) −0.629102 + 0.865884i −0.0741403 + 0.102045i
\(73\) 4.75216 + 1.54407i 0.556198 + 0.180720i 0.573610 − 0.819129i \(-0.305543\pi\)
−0.0174117 + 0.999848i \(0.505543\pi\)
\(74\) 13.5242 1.57215
\(75\) 0 0
\(76\) 19.2297 2.20580
\(77\) 6.54885 + 2.12785i 0.746310 + 0.242491i
\(78\) 1.49440 − 2.05687i 0.169208 − 0.232894i
\(79\) 11.7737 + 8.55407i 1.32464 + 0.962408i 0.999862 + 0.0166185i \(0.00529009\pi\)
0.324779 + 0.945790i \(0.394710\pi\)
\(80\) 0 0
\(81\) −0.809017 + 0.587785i −0.0898908 + 0.0653095i
\(82\) 15.8584i 1.75126i
\(83\) 5.13491 + 7.06760i 0.563630 + 0.775770i 0.991782 − 0.127937i \(-0.0408355\pi\)
−0.428153 + 0.903706i \(0.640835\pi\)
\(84\) −3.37284 − 10.3805i −0.368007 − 1.13261i
\(85\) 0 0
\(86\) −0.819784 + 2.52304i −0.0883996 + 0.272066i
\(87\) −5.23490 + 1.70092i −0.561240 + 0.182358i
\(88\) 1.60820 − 0.522535i 0.171435 − 0.0557025i
\(89\) 3.10195 − 9.54683i 0.328806 − 1.01196i −0.640887 − 0.767635i \(-0.721433\pi\)
0.969693 − 0.244326i \(-0.0785668\pi\)
\(90\) 0 0
\(91\) 1.61341 + 4.96556i 0.169131 + 0.520532i
\(92\) −3.41542 − 4.70092i −0.356082 − 0.490105i
\(93\) 4.80206i 0.497950i
\(94\) 7.08719 − 5.14914i 0.730988 − 0.531094i
\(95\) 0 0
\(96\) 6.43135 + 4.67265i 0.656397 + 0.476900i
\(97\) 4.40837 − 6.06760i 0.447602 − 0.616071i −0.524278 − 0.851547i \(-0.675665\pi\)
0.971880 + 0.235476i \(0.0756648\pi\)
\(98\) 24.2128 + 7.86720i 2.44586 + 0.794707i
\(99\) 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.i.b.274.1 16
5.2 odd 4 75.2.g.b.46.2 yes 8
5.3 odd 4 375.2.g.b.226.1 8
5.4 even 2 inner 375.2.i.b.274.4 16
15.2 even 4 225.2.h.c.46.1 8
25.6 even 5 inner 375.2.i.b.349.4 16
25.8 odd 20 375.2.g.b.151.1 8
25.9 even 10 1875.2.b.c.1249.7 8
25.12 odd 20 1875.2.a.h.1.3 4
25.13 odd 20 1875.2.a.e.1.2 4
25.16 even 5 1875.2.b.c.1249.2 8
25.17 odd 20 75.2.g.b.31.2 ✓ 8
25.19 even 10 inner 375.2.i.b.349.1 16
75.17 even 20 225.2.h.c.181.1 8
75.38 even 20 5625.2.a.n.1.3 4
75.62 even 20 5625.2.a.i.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.b.31.2 ✓ 8 25.17 odd 20
75.2.g.b.46.2 yes 8 5.2 odd 4
225.2.h.c.46.1 8 15.2 even 4
225.2.h.c.181.1 8 75.17 even 20
375.2.g.b.151.1 8 25.8 odd 20
375.2.g.b.226.1 8 5.3 odd 4
375.2.i.b.274.1 16 1.1 even 1 trivial
375.2.i.b.274.4 16 5.4 even 2 inner
375.2.i.b.349.1 16 25.19 even 10 inner
375.2.i.b.349.4 16 25.6 even 5 inner
1875.2.a.e.1.2 4 25.13 odd 20
1875.2.a.h.1.3 4 25.12 odd 20
1875.2.b.c.1249.2 8 25.16 even 5
1875.2.b.c.1249.7 8 25.9 even 10
5625.2.a.i.1.2 4 75.62 even 20
5625.2.a.n.1.3 4 75.38 even 20