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Properties

Label 375.4.b.d.124.13
Level $375$
Weight $4$
Character 375.124
Analytic conductor $22.126$
Analytic rank $0$
Dimension $16$
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] = [375,4,Mod(124,375)] 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("375.124"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(375, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 375 = 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 375.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,-82] 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: \(22.1257162522\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 99 x^{14} + 3861 x^{12} + 76609 x^{10} + 832051 x^{8} + 5007830 x^{6} + 16593025 x^{4} + \cdots + 19360000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 5^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 124.13
Root \(3.91962i\) of defining polynomial
Character \(\chi\) \(=\) 375.124
Dual form 375.4.b.d.124.4

$q$-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)
 
\(f(q)\) \(=\) \(q+2.91962i q^{2} +3.00000i q^{3} -0.524200 q^{4} -8.75887 q^{6} -12.0754i q^{7} +21.8265i q^{8} -9.00000 q^{9} +45.6229 q^{11} -1.57260i q^{12} +11.8301i q^{13} +35.2557 q^{14} -67.9188 q^{16} +136.463i q^{17} -26.2766i q^{18} -51.4445 q^{19} +36.2263 q^{21} +133.202i q^{22} +23.7841i q^{23} -65.4796 q^{24} -34.5395 q^{26} -27.0000i q^{27} +6.32994i q^{28} -216.013 q^{29} -14.3619 q^{31} -23.6852i q^{32} +136.869i q^{33} -398.419 q^{34} +4.71780 q^{36} +342.589i q^{37} -150.199i q^{38} -35.4904 q^{39} +286.502 q^{41} +105.767i q^{42} +159.491i q^{43} -23.9155 q^{44} -69.4406 q^{46} -314.063i q^{47} -203.756i q^{48} +197.184 q^{49} -409.388 q^{51} -6.20135i q^{52} +138.174i q^{53} +78.8298 q^{54} +263.565 q^{56} -154.333i q^{57} -630.677i q^{58} -760.013 q^{59} -372.123 q^{61} -41.9312i q^{62} +108.679i q^{63} -474.199 q^{64} -399.605 q^{66} -389.961i q^{67} -71.5336i q^{68} -71.3523 q^{69} +1085.49 q^{71} -196.439i q^{72} -257.362i q^{73} -1000.23 q^{74} +26.9672 q^{76} -550.916i q^{77} -103.619i q^{78} -680.342 q^{79} +81.0000 q^{81} +836.478i q^{82} -395.985i q^{83} -18.9898 q^{84} -465.652 q^{86} -648.040i q^{87} +995.788i q^{88} +797.491 q^{89} +142.854 q^{91} -12.4676i q^{92} -43.0856i q^{93} +916.945 q^{94} +71.0555 q^{96} -200.021i q^{97} +575.702i q^{98} -410.606 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 82 q^{4} + 42 q^{6} - 144 q^{9} + 54 q^{11} - 238 q^{14} + 610 q^{16} - 608 q^{19} + 114 q^{21} - 594 q^{24} + 442 q^{26} - 336 q^{29} + 766 q^{31} - 464 q^{34} + 738 q^{36} + 78 q^{39} + 1206 q^{41}+ \cdots - 486 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(251\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.91962i 1.03224i 0.856515 + 0.516121i \(0.172625\pi\)
−0.856515 + 0.516121i \(0.827375\pi\)
\(3\) 3.00000i 0.577350i
\(4\) −0.524200 −0.0655249
\(5\) 0 0
\(6\) −8.75887 −0.595966
\(7\) − 12.0754i − 0.652013i −0.945368 − 0.326006i \(-0.894297\pi\)
0.945368 − 0.326006i \(-0.105703\pi\)
\(8\) 21.8265i 0.964605i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 45.6229 1.25053 0.625264 − 0.780413i \(-0.284991\pi\)
0.625264 + 0.780413i \(0.284991\pi\)
\(12\) − 1.57260i − 0.0378308i
\(13\) 11.8301i 0.252391i 0.992005 + 0.126196i \(0.0402767\pi\)
−0.992005 + 0.126196i \(0.959723\pi\)
\(14\) 35.2557 0.673035
\(15\) 0 0
\(16\) −67.9188 −1.06123
\(17\) 136.463i 1.94688i 0.228932 + 0.973442i \(0.426477\pi\)
−0.228932 + 0.973442i \(0.573523\pi\)
\(18\) − 26.2766i − 0.344081i
\(19\) −51.4445 −0.621167 −0.310584 − 0.950546i \(-0.600524\pi\)
−0.310584 + 0.950546i \(0.600524\pi\)
\(20\) 0 0
\(21\) 36.2263 0.376440
\(22\) 133.202i 1.29085i
\(23\) 23.7841i 0.215623i 0.994171 + 0.107812i \(0.0343843\pi\)
−0.994171 + 0.107812i \(0.965616\pi\)
\(24\) −65.4796 −0.556915
\(25\) 0 0
\(26\) −34.5395 −0.260529
\(27\) − 27.0000i − 0.192450i
\(28\) 6.32994i 0.0427231i
\(29\) −216.013 −1.38320 −0.691598 − 0.722283i \(-0.743093\pi\)
−0.691598 + 0.722283i \(0.743093\pi\)
\(30\) 0 0
\(31\) −14.3619 −0.0832086 −0.0416043 − 0.999134i \(-0.513247\pi\)
−0.0416043 + 0.999134i \(0.513247\pi\)
\(32\) − 23.6852i − 0.130843i
\(33\) 136.869i 0.721993i
\(34\) −398.419 −2.00966
\(35\) 0 0
\(36\) 4.71780 0.0218416
\(37\) 342.589i 1.52220i 0.648636 + 0.761099i \(0.275340\pi\)
−0.648636 + 0.761099i \(0.724660\pi\)
\(38\) − 150.199i − 0.641195i
\(39\) −35.4904 −0.145718
\(40\) 0 0
\(41\) 286.502 1.09132 0.545660 − 0.838007i \(-0.316279\pi\)
0.545660 + 0.838007i \(0.316279\pi\)
\(42\) 105.767i 0.388577i
\(43\) 159.491i 0.565630i 0.959175 + 0.282815i \(0.0912682\pi\)
−0.959175 + 0.282815i \(0.908732\pi\)
\(44\) −23.9155 −0.0819408
\(45\) 0 0
\(46\) −69.4406 −0.222575
\(47\) − 314.063i − 0.974697i −0.873207 − 0.487349i \(-0.837964\pi\)
0.873207 − 0.487349i \(-0.162036\pi\)
\(48\) − 203.756i − 0.612702i
\(49\) 197.184 0.574880
\(50\) 0 0
\(51\) −409.388 −1.12403
\(52\) − 6.20135i − 0.0165379i
\(53\) 138.174i 0.358106i 0.983839 + 0.179053i \(0.0573033\pi\)
−0.983839 + 0.179053i \(0.942697\pi\)
\(54\) 78.8298 0.198655
\(55\) 0 0
\(56\) 263.565 0.628935
\(57\) − 154.333i − 0.358631i
\(58\) − 630.677i − 1.42779i
\(59\) −760.013 −1.67704 −0.838519 − 0.544873i \(-0.816578\pi\)
−0.838519 + 0.544873i \(0.816578\pi\)
\(60\) 0 0
\(61\) −372.123 −0.781074 −0.390537 − 0.920587i \(-0.627711\pi\)
−0.390537 + 0.920587i \(0.627711\pi\)
\(62\) − 41.9312i − 0.0858915i
\(63\) 108.679i 0.217338i
\(64\) −474.199 −0.926169
\(65\) 0 0
\(66\) −399.605 −0.745272
\(67\) − 389.961i − 0.711064i −0.934664 − 0.355532i \(-0.884300\pi\)
0.934664 − 0.355532i \(-0.115700\pi\)
\(68\) − 71.5336i − 0.127570i
\(69\) −71.3523 −0.124490
\(70\) 0 0
\(71\) 1085.49 1.81443 0.907213 − 0.420672i \(-0.138206\pi\)
0.907213 + 0.420672i \(0.138206\pi\)
\(72\) − 196.439i − 0.321535i
\(73\) − 257.362i − 0.412629i −0.978486 − 0.206314i \(-0.933853\pi\)
0.978486 − 0.206314i \(-0.0661470\pi\)
\(74\) −1000.23 −1.57128
\(75\) 0 0
\(76\) 26.9672 0.0407019
\(77\) − 550.916i − 0.815360i
\(78\) − 103.619i − 0.150417i
\(79\) −680.342 −0.968917 −0.484459 − 0.874814i \(-0.660983\pi\)
−0.484459 + 0.874814i \(0.660983\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 836.478i 1.12651i
\(83\) − 395.985i − 0.523675i −0.965112 − 0.261837i \(-0.915672\pi\)
0.965112 − 0.261837i \(-0.0843284\pi\)
\(84\) −18.9898 −0.0246662
\(85\) 0 0
\(86\) −465.652 −0.583867
\(87\) − 648.040i − 0.798588i
\(88\) 995.788i 1.20627i
\(89\) 797.491 0.949818 0.474909 − 0.880035i \(-0.342481\pi\)
0.474909 + 0.880035i \(0.342481\pi\)
\(90\) 0 0
\(91\) 142.854 0.164562
\(92\) − 12.4676i − 0.0141287i
\(93\) − 43.0856i − 0.0480405i
\(94\) 916.945 1.00612
\(95\) 0 0
\(96\) 71.0555 0.0755424
\(97\) − 200.021i − 0.209372i −0.994505 − 0.104686i \(-0.966616\pi\)
0.994505 − 0.104686i \(-0.0333837\pi\)
\(98\) 575.702i 0.593415i
\(99\) −410.606 −0.416843
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 375.4.b.d.124.13 16
5.2 odd 4 375.4.a.h.1.2 yes 8
5.3 odd 4 375.4.a.g.1.7 ✓ 8
5.4 even 2 inner 375.4.b.d.124.4 16
15.2 even 4 1125.4.a.l.1.7 8
15.8 even 4 1125.4.a.p.1.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
375.4.a.g.1.7 ✓ 8 5.3 odd 4
375.4.a.h.1.2 yes 8 5.2 odd 4
375.4.b.d.124.4 16 5.4 even 2 inner
375.4.b.d.124.13 16 1.1 even 1 trivial
1125.4.a.l.1.7 8 15.2 even 4
1125.4.a.p.1.2 8 15.8 even 4