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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,10,Mod(49,75)] 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("75.49"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-3584] 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: \(38.6276877123\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 1470x^{5} + 317749x^{4} + 221032x^{3} + 2888x^{2} + 10631640x + 19569212100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2}\cdot 5^{4}\cdot 23^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.5
Root \(-13.7982 - 13.7982i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.10.b.h.49.4

$q$-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)
 
\(f(q)\) \(=\) \(q+11.8931i q^{2} +81.0000i q^{3} +370.553 q^{4} -963.344 q^{6} -10946.0i q^{7} +10496.3i q^{8} -6561.00 q^{9} -39453.3 q^{11} +30014.8i q^{12} +50257.6i q^{13} +130183. q^{14} +64888.9 q^{16} +461329. i q^{17} -78030.9i q^{18} +370082. q^{19} +886628. q^{21} -469223. i q^{22} +2.29014e6i q^{23} -850203. q^{24} -597721. q^{26} -531441. i q^{27} -4.05608e6i q^{28} +1.28309e6 q^{29} +6.51912e6 q^{31} +6.14585e6i q^{32} -3.19571e6i q^{33} -5.48665e6 q^{34} -2.43120e6 q^{36} +1.45530e7i q^{37} +4.40144e6i q^{38} -4.07086e6 q^{39} +1.34566e7 q^{41} +1.05448e7i q^{42} +2.24211e7i q^{43} -1.46195e7 q^{44} -2.72369e7 q^{46} -1.45871e7i q^{47} +5.25600e6i q^{48} -7.94618e7 q^{49} -3.73677e7 q^{51} +1.86231e7i q^{52} +6.49845e7i q^{53} +6.32050e6 q^{54} +1.14893e8 q^{56} +2.99766e7i q^{57} +1.52599e7i q^{58} -1.04242e8 q^{59} +1.46085e8 q^{61} +7.75328e7i q^{62} +7.18168e7i q^{63} -3.98704e7 q^{64} +3.80071e7 q^{66} -9.72944e7i q^{67} +1.70947e8i q^{68} -1.85501e8 q^{69} -2.89431e8 q^{71} -6.88664e7i q^{72} +6.21359e7i q^{73} -1.73081e8 q^{74} +1.37135e8 q^{76} +4.31856e8i q^{77} -4.84154e7i q^{78} -3.55247e8 q^{79} +4.30467e7 q^{81} +1.60042e8i q^{82} -2.13970e8i q^{83} +3.28543e8 q^{84} -2.66658e8 q^{86} +1.03930e8i q^{87} -4.14114e8i q^{88} +8.61695e8 q^{89} +5.50121e8 q^{91} +8.48617e8i q^{92} +5.28049e8i q^{93} +1.73486e8 q^{94} -4.97814e8 q^{96} -1.00809e9i q^{97} -9.45050e8i q^{98} +2.58853e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 3584 q^{4} - 324 q^{6} - 52488 q^{9} + 209392 q^{11} + 362124 q^{14} + 2639600 q^{16} - 786616 q^{19} + 2111832 q^{21} + 3991032 q^{24} - 12955004 q^{26} - 9853232 q^{29} - 195032 q^{31} - 33657288 q^{34}+ \cdots - 1373820912 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 11.8931i 0.525608i 0.964849 + 0.262804i \(0.0846472\pi\)
−0.964849 + 0.262804i \(0.915353\pi\)
\(3\) 81.0000i 0.577350i
\(4\) 370.553 0.723737
\(5\) 0 0
\(6\) −963.344 −0.303460
\(7\) − 10946.0i − 1.72312i −0.507657 − 0.861559i \(-0.669488\pi\)
0.507657 − 0.861559i \(-0.330512\pi\)
\(8\) 10496.3i 0.906009i
\(9\) −6561.00 −0.333333
\(10\) 0 0
\(11\) −39453.3 −0.812486 −0.406243 − 0.913765i \(-0.633161\pi\)
−0.406243 + 0.913765i \(0.633161\pi\)
\(12\) 30014.8i 0.417850i
\(13\) 50257.6i 0.488041i 0.969770 + 0.244021i \(0.0784665\pi\)
−0.969770 + 0.244021i \(0.921534\pi\)
\(14\) 130183. 0.905684
\(15\) 0 0
\(16\) 64888.9 0.247532
\(17\) 461329.i 1.33965i 0.742520 + 0.669824i \(0.233631\pi\)
−0.742520 + 0.669824i \(0.766369\pi\)
\(18\) − 78030.9i − 0.175203i
\(19\) 370082. 0.651489 0.325744 − 0.945458i \(-0.394385\pi\)
0.325744 + 0.945458i \(0.394385\pi\)
\(20\) 0 0
\(21\) 886628. 0.994843
\(22\) − 469223.i − 0.427049i
\(23\) 2.29014e6i 1.70642i 0.521567 + 0.853210i \(0.325348\pi\)
−0.521567 + 0.853210i \(0.674652\pi\)
\(24\) −850203. −0.523085
\(25\) 0 0
\(26\) −597721. −0.256518
\(27\) − 531441.i − 0.192450i
\(28\) − 4.05608e6i − 1.24708i
\(29\) 1.28309e6 0.336872 0.168436 − 0.985713i \(-0.446128\pi\)
0.168436 + 0.985713i \(0.446128\pi\)
\(30\) 0 0
\(31\) 6.51912e6 1.26783 0.633916 − 0.773402i \(-0.281447\pi\)
0.633916 + 0.773402i \(0.281447\pi\)
\(32\) 6.14585e6i 1.03611i
\(33\) − 3.19571e6i − 0.469089i
\(34\) −5.48665e6 −0.704129
\(35\) 0 0
\(36\) −2.43120e6 −0.241246
\(37\) 1.45530e7i 1.27657i 0.769800 + 0.638286i \(0.220356\pi\)
−0.769800 + 0.638286i \(0.779644\pi\)
\(38\) 4.40144e6i 0.342427i
\(39\) −4.07086e6 −0.281771
\(40\) 0 0
\(41\) 1.34566e7 0.743720 0.371860 − 0.928289i \(-0.378720\pi\)
0.371860 + 0.928289i \(0.378720\pi\)
\(42\) 1.05448e7i 0.522897i
\(43\) 2.24211e7i 1.00011i 0.865992 + 0.500057i \(0.166688\pi\)
−0.865992 + 0.500057i \(0.833312\pi\)
\(44\) −1.46195e7 −0.588026
\(45\) 0 0
\(46\) −2.72369e7 −0.896907
\(47\) − 1.45871e7i − 0.436042i −0.975944 − 0.218021i \(-0.930040\pi\)
0.975944 − 0.218021i \(-0.0699601\pi\)
\(48\) 5.25600e6i 0.142912i
\(49\) −7.94618e7 −1.96914
\(50\) 0 0
\(51\) −3.73677e7 −0.773447
\(52\) 1.86231e7i 0.353213i
\(53\) 6.49845e7i 1.13128i 0.824654 + 0.565638i \(0.191370\pi\)
−0.824654 + 0.565638i \(0.808630\pi\)
\(54\) 6.32050e6 0.101153
\(55\) 0 0
\(56\) 1.14893e8 1.56116
\(57\) 2.99766e7i 0.376137i
\(58\) 1.52599e7i 0.177063i
\(59\) −1.04242e8 −1.11997 −0.559986 − 0.828502i \(-0.689193\pi\)
−0.559986 + 0.828502i \(0.689193\pi\)
\(60\) 0 0
\(61\) 1.46085e8 1.35089 0.675445 − 0.737410i \(-0.263952\pi\)
0.675445 + 0.737410i \(0.263952\pi\)
\(62\) 7.75328e7i 0.666382i
\(63\) 7.18168e7i 0.574373i
\(64\) −3.98704e7 −0.297058
\(65\) 0 0
\(66\) 3.80071e7 0.246557
\(67\) − 9.72944e7i − 0.589863i −0.955518 − 0.294932i \(-0.904703\pi\)
0.955518 − 0.294932i \(-0.0952969\pi\)
\(68\) 1.70947e8i 0.969553i
\(69\) −1.85501e8 −0.985202
\(70\) 0 0
\(71\) −2.89431e8 −1.35171 −0.675854 − 0.737036i \(-0.736225\pi\)
−0.675854 + 0.737036i \(0.736225\pi\)
\(72\) − 6.88664e7i − 0.302003i
\(73\) 6.21359e7i 0.256088i 0.991768 + 0.128044i \(0.0408699\pi\)
−0.991768 + 0.128044i \(0.959130\pi\)
\(74\) −1.73081e8 −0.670975
\(75\) 0 0
\(76\) 1.37135e8 0.471506
\(77\) 4.31856e8i 1.40001i
\(78\) − 4.84154e7i − 0.148101i
\(79\) −3.55247e8 −1.02614 −0.513072 − 0.858346i \(-0.671493\pi\)
−0.513072 + 0.858346i \(0.671493\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) 1.60042e8i 0.390905i
\(83\) − 2.13970e8i − 0.494881i −0.968903 − 0.247440i \(-0.920411\pi\)
0.968903 − 0.247440i \(-0.0795894\pi\)
\(84\) 3.28543e8 0.720004
\(85\) 0 0
\(86\) −2.66658e8 −0.525668
\(87\) 1.03930e8i 0.194493i
\(88\) − 4.14114e8i − 0.736120i
\(89\) 8.61695e8 1.45579 0.727895 − 0.685689i \(-0.240499\pi\)
0.727895 + 0.685689i \(0.240499\pi\)
\(90\) 0 0
\(91\) 5.50121e8 0.840953
\(92\) 8.48617e8i 1.23500i
\(93\) 5.28049e8i 0.731983i
\(94\) 1.73486e8 0.229187
\(95\) 0 0
\(96\) −4.97814e8 −0.598200
\(97\) − 1.00809e9i − 1.15618i −0.815974 − 0.578089i \(-0.803799\pi\)
0.815974 − 0.578089i \(-0.196201\pi\)
\(98\) − 9.45050e8i − 1.03499i
\(99\) 2.58853e8 0.270829
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 75.10.b.h.49.5 8
3.2 odd 2 225.10.b.n.199.4 8
5.2 odd 4 75.10.a.j.1.2 ✓ 4
5.3 odd 4 75.10.a.k.1.3 yes 4
5.4 even 2 inner 75.10.b.h.49.4 8
15.2 even 4 225.10.a.t.1.3 4
15.8 even 4 225.10.a.r.1.2 4
15.14 odd 2 225.10.b.n.199.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.10.a.j.1.2 ✓ 4 5.2 odd 4
75.10.a.k.1.3 yes 4 5.3 odd 4
75.10.b.h.49.4 8 5.4 even 2 inner
75.10.b.h.49.5 8 1.1 even 1 trivial
225.10.a.r.1.2 4 15.8 even 4
225.10.a.t.1.3 4 15.2 even 4
225.10.b.n.199.4 8 3.2 odd 2
225.10.b.n.199.5 8 15.14 odd 2