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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(151,925)] 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("925.151"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 13])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.bb (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,-6] 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: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(12\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 185)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 151.8
Character \(\chi\) \(=\) 925.151
Dual form 925.2.bb.b.876.8

$q$-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+(0.590034 - 0.703175i) q^{2} +(0.605440 - 0.508024i) q^{3} +(0.200981 + 1.13982i) q^{4} -0.725482i q^{6} +(-3.19522 - 1.16297i) q^{7} +(2.50998 + 1.44914i) q^{8} +(-0.412476 + 2.33927i) q^{9} +(-1.45515 + 2.52039i) q^{11} +(0.700739 + 0.587990i) q^{12} +(0.612910 - 0.108073i) q^{13} +(-2.70306 + 1.56061i) q^{14} +(0.324761 - 0.118203i) q^{16} +(5.41828 + 0.955390i) q^{17} +(1.40154 + 1.67029i) q^{18} +(3.15940 + 3.76523i) q^{19} +(-2.52533 + 0.919145i) q^{21} +(0.913687 + 2.51034i) q^{22} +(-3.86235 + 2.22993i) q^{23} +(2.25584 - 0.397765i) q^{24} +(0.285644 - 0.494749i) q^{26} +(2.12419 + 3.67921i) q^{27} +(0.683393 - 3.87572i) q^{28} +(8.86469 + 5.11803i) q^{29} -6.23857i q^{31} +(-1.87404 + 5.14887i) q^{32} +(0.399414 + 2.26519i) q^{33} +(3.86878 - 3.24629i) q^{34} -2.74925 q^{36} +(-4.16454 - 4.43357i) q^{37} +4.51177 q^{38} +(0.316177 - 0.376805i) q^{39} +(-0.0934663 - 0.530074i) q^{41} +(-0.843710 + 2.31807i) q^{42} +9.30569i q^{43} +(-3.16525 - 1.15206i) q^{44} +(-0.710887 + 4.03164i) q^{46} +(0.483688 + 0.837773i) q^{47} +(0.136573 - 0.236551i) q^{48} +(3.49464 + 2.93235i) q^{49} +(3.76581 - 2.17419i) q^{51} +(0.246367 + 0.676887i) q^{52} +(-4.46261 + 1.62426i) q^{53} +(3.84047 + 0.677179i) q^{54} +(-6.33465 - 7.54934i) q^{56} +(3.82566 + 0.674567i) q^{57} +(8.82934 - 3.21362i) q^{58} +(-4.29946 - 11.8127i) q^{59} +(4.37236 - 0.770966i) q^{61} +(-4.38681 - 3.68097i) q^{62} +(4.03844 - 6.99478i) q^{63} +(2.86041 + 4.95438i) q^{64} +(1.82849 + 1.05568i) q^{66} +(-10.9727 - 3.99372i) q^{67} +6.36789i q^{68} +(-1.20556 + 3.31225i) q^{69} +(0.548337 - 0.460110i) q^{71} +(-4.42523 + 5.27378i) q^{72} +5.46858 q^{73} +(-5.57480 + 0.312443i) q^{74} +(-3.65671 + 4.35790i) q^{76} +(7.58064 - 6.36091i) q^{77} +(-0.0784046 - 0.444655i) q^{78} +(-2.49273 + 6.84871i) q^{79} +(-3.54111 - 1.28886i) q^{81} +(-0.427883 - 0.247038i) q^{82} +(-0.0550228 + 0.312050i) q^{83} +(-1.55520 - 2.69369i) q^{84} +(6.54353 + 5.49067i) q^{86} +(7.96712 - 1.40482i) q^{87} +(-7.30477 + 4.21741i) q^{88} +(3.44377 + 9.46169i) q^{89} +(-2.08407 - 0.367477i) q^{91} +(-3.31798 - 3.95422i) q^{92} +(-3.16935 - 3.77708i) q^{93} +(0.874493 + 0.154197i) q^{94} +(1.48114 + 4.06939i) q^{96} +(10.7079 - 6.18220i) q^{97} +(4.12392 - 0.727158i) q^{98} +(-5.29564 - 4.44357i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 6 q^{2} - 6 q^{4} + 6 q^{7} + 18 q^{8} - 30 q^{12} + 30 q^{13} - 18 q^{14} - 18 q^{16} + 30 q^{18} - 18 q^{19} - 30 q^{22} - 30 q^{24} + 24 q^{27} - 30 q^{28} + 18 q^{29} + 36 q^{32} + 12 q^{33}+ \cdots - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(852\)
\(\chi(n)\) \(e\left(\frac{13}{18}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.590034 − 0.703175i 0.417217 − 0.497220i −0.515972 − 0.856605i \(-0.672569\pi\)
0.933189 + 0.359386i \(0.117014\pi\)
\(3\) 0.605440 − 0.508024i 0.349551 − 0.293308i −0.451059 − 0.892494i \(-0.648954\pi\)
0.800610 + 0.599186i \(0.204509\pi\)
\(4\) 0.200981 + 1.13982i 0.100491 + 0.569911i
\(5\) 0 0
\(6\) − 0.725482i − 0.296177i
\(7\) −3.19522 − 1.16297i −1.20768 − 0.439560i −0.341782 − 0.939779i \(-0.611030\pi\)
−0.865899 + 0.500220i \(0.833253\pi\)
\(8\) 2.50998 + 1.44914i 0.887412 + 0.512348i
\(9\) −0.412476 + 2.33927i −0.137492 + 0.779756i
\(10\) 0 0
\(11\) −1.45515 + 2.52039i −0.438743 + 0.759925i −0.997593 − 0.0693438i \(-0.977909\pi\)
0.558850 + 0.829269i \(0.311243\pi\)
\(12\) 0.700739 + 0.587990i 0.202286 + 0.169738i
\(13\) 0.612910 − 0.108073i 0.169991 − 0.0299739i −0.0880049 − 0.996120i \(-0.528049\pi\)
0.257996 + 0.966146i \(0.416938\pi\)
\(14\) −2.70306 + 1.56061i −0.722422 + 0.417091i
\(15\) 0 0
\(16\) 0.324761 − 0.118203i 0.0811902 − 0.0295508i
\(17\) 5.41828 + 0.955390i 1.31413 + 0.231716i 0.786410 − 0.617704i \(-0.211937\pi\)
0.527717 + 0.849420i \(0.323048\pi\)
\(18\) 1.40154 + 1.67029i 0.330346 + 0.393691i
\(19\) 3.15940 + 3.76523i 0.724817 + 0.863803i 0.995089 − 0.0989806i \(-0.0315582\pi\)
−0.270272 + 0.962784i \(0.587114\pi\)
\(20\) 0 0
\(21\) −2.52533 + 0.919145i −0.551072 + 0.200574i
\(22\) 0.913687 + 2.51034i 0.194799 + 0.535205i
\(23\) −3.86235 + 2.22993i −0.805356 + 0.464972i −0.845340 − 0.534228i \(-0.820602\pi\)
0.0399849 + 0.999200i \(0.487269\pi\)
\(24\) 2.25584 − 0.397765i 0.460471 − 0.0811935i
\(25\) 0 0
\(26\) 0.285644 − 0.494749i 0.0560193 − 0.0970283i
\(27\) 2.12419 + 3.67921i 0.408801 + 0.708064i
\(28\) 0.683393 − 3.87572i 0.129149 − 0.732442i
\(29\) 8.86469 + 5.11803i 1.64613 + 0.950395i 0.978589 + 0.205823i \(0.0659871\pi\)
0.667542 + 0.744572i \(0.267346\pi\)
\(30\) 0 0
\(31\) − 6.23857i − 1.12048i −0.828330 − 0.560240i \(-0.810709\pi\)
0.828330 − 0.560240i \(-0.189291\pi\)
\(32\) −1.87404 + 5.14887i −0.331286 + 0.910200i
\(33\) 0.399414 + 2.26519i 0.0695291 + 0.394319i
\(34\) 3.86878 − 3.24629i 0.663490 − 0.556734i
\(35\) 0 0
\(36\) −2.74925 −0.458208
\(37\) −4.16454 − 4.43357i −0.684647 − 0.728875i
\(38\) 4.51177 0.731906
\(39\) 0.316177 − 0.376805i 0.0506288 − 0.0603370i
\(40\) 0 0
\(41\) −0.0934663 − 0.530074i −0.0145970 − 0.0827836i 0.976639 − 0.214887i \(-0.0689382\pi\)
−0.991236 + 0.132103i \(0.957827\pi\)
\(42\) −0.843710 + 2.31807i −0.130187 + 0.357687i
\(43\) 9.30569i 1.41911i 0.704653 + 0.709553i \(0.251103\pi\)
−0.704653 + 0.709553i \(0.748897\pi\)
\(44\) −3.16525 − 1.15206i −0.477179 − 0.173679i
\(45\) 0 0
\(46\) −0.710887 + 4.03164i −0.104815 + 0.594433i
\(47\) 0.483688 + 0.837773i 0.0705532 + 0.122202i 0.899144 − 0.437653i \(-0.144190\pi\)
−0.828591 + 0.559855i \(0.810857\pi\)
\(48\) 0.136573 − 0.236551i 0.0197126 − 0.0341432i
\(49\) 3.49464 + 2.93235i 0.499235 + 0.418908i
\(50\) 0 0
\(51\) 3.76581 − 2.17419i 0.527318 − 0.304447i
\(52\) 0.246367 + 0.676887i 0.0341649 + 0.0938674i
\(53\) −4.46261 + 1.62426i −0.612987 + 0.223109i −0.629810 − 0.776749i \(-0.716867\pi\)
0.0168227 + 0.999858i \(0.494645\pi\)
\(54\) 3.84047 + 0.677179i 0.522622 + 0.0921524i
\(55\) 0 0
\(56\) −6.33465 − 7.54934i −0.846503 − 1.00882i
\(57\) 3.82566 + 0.674567i 0.506721 + 0.0893486i
\(58\) 8.82934 − 3.21362i 1.15935 − 0.421969i
\(59\) −4.29946 − 11.8127i −0.559741 − 1.53788i −0.820015 − 0.572342i \(-0.806035\pi\)
0.260273 − 0.965535i \(-0.416187\pi\)
\(60\) 0 0
\(61\) 4.37236 − 0.770966i 0.559824 − 0.0987120i 0.113426 − 0.993546i \(-0.463818\pi\)
0.446398 + 0.894834i \(0.352707\pi\)
\(62\) −4.38681 − 3.68097i −0.557125 − 0.467484i
\(63\) 4.03844 − 6.99478i 0.508796 − 0.881260i
\(64\) 2.86041 + 4.95438i 0.357552 + 0.619298i
\(65\) 0 0
\(66\) 1.82849 + 1.05568i 0.225072 + 0.129945i
\(67\) −10.9727 − 3.99372i −1.34052 − 0.487911i −0.430547 − 0.902568i \(-0.641679\pi\)
−0.909978 + 0.414657i \(0.863902\pi\)
\(68\) 6.36789i 0.772220i
\(69\) −1.20556 + 3.31225i −0.145133 + 0.398749i
\(70\) 0 0
\(71\) 0.548337 − 0.460110i 0.0650757 − 0.0546050i −0.609670 − 0.792656i \(-0.708698\pi\)
0.674745 + 0.738051i \(0.264253\pi\)
\(72\) −4.42523 + 5.27378i −0.521518 + 0.621521i
\(73\) 5.46858 0.640048 0.320024 − 0.947409i \(-0.396309\pi\)
0.320024 + 0.947409i \(0.396309\pi\)
\(74\) −5.57480 + 0.312443i −0.648057 + 0.0363208i
\(75\) 0 0
\(76\) −3.65671 + 4.35790i −0.419453 + 0.499885i
\(77\) 7.58064 − 6.36091i 0.863894 − 0.724893i
\(78\) −0.0784046 − 0.444655i −0.00887758 − 0.0503472i
\(79\) −2.49273 + 6.84871i −0.280454 + 0.770540i 0.716855 + 0.697222i \(0.245581\pi\)
−0.997309 + 0.0733179i \(0.976641\pi\)
\(80\) 0 0
\(81\) −3.54111 − 1.28886i −0.393456 − 0.143206i
\(82\) −0.427883 − 0.247038i −0.0472518 − 0.0272808i
\(83\) −0.0550228 + 0.312050i −0.00603953 + 0.0342519i −0.987679 − 0.156496i \(-0.949980\pi\)
0.981639 + 0.190748i \(0.0610913\pi\)
\(84\) −1.55520 − 2.69369i −0.169687 − 0.293906i
\(85\) 0 0
\(86\) 6.54353 + 5.49067i 0.705607 + 0.592075i
\(87\) 7.96712 − 1.40482i 0.854165 − 0.150612i
\(88\) −7.30477 + 4.21741i −0.778692 + 0.449578i
\(89\) 3.44377 + 9.46169i 0.365039 + 1.00294i 0.977222 + 0.212221i \(0.0680695\pi\)
−0.612183 + 0.790716i \(0.709708\pi\)
\(90\) 0 0
\(91\) −2.08407 − 0.367477i −0.218470 − 0.0385221i
\(92\) −3.31798 − 3.95422i −0.345923 − 0.412255i
\(93\) −3.16935 − 3.77708i −0.328646 − 0.391665i
\(94\) 0.874493 + 0.154197i 0.0901971 + 0.0159042i
\(95\) 0 0
\(96\) 1.48114 + 4.06939i 0.151168 + 0.415330i
\(97\) 10.7079 − 6.18220i 1.08722 − 0.627708i 0.154386 − 0.988011i \(-0.450660\pi\)
0.932835 + 0.360303i \(0.117327\pi\)
\(98\) 4.12392 − 0.727158i 0.416578 − 0.0734540i
\(99\) −5.29564 − 4.44357i −0.532232 − 0.446596i
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 925.2.bb.b.151.8 72
5.2 odd 4 925.2.ba.b.299.9 72
5.3 odd 4 925.2.ba.c.299.4 72
5.4 even 2 185.2.w.a.151.5 yes 72
37.25 even 18 inner 925.2.bb.b.876.8 72
185.62 odd 36 925.2.ba.c.99.4 72
185.69 odd 36 6845.2.a.t.1.13 36
185.79 odd 36 6845.2.a.u.1.24 36
185.99 even 18 185.2.w.a.136.5 ✓ 72
185.173 odd 36 925.2.ba.b.99.9 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.136.5 ✓ 72 185.99 even 18
185.2.w.a.151.5 yes 72 5.4 even 2
925.2.ba.b.99.9 72 185.173 odd 36
925.2.ba.b.299.9 72 5.2 odd 4
925.2.ba.c.99.4 72 185.62 odd 36
925.2.ba.c.299.4 72 5.3 odd 4
925.2.bb.b.151.8 72 1.1 even 1 trivial
925.2.bb.b.876.8 72 37.25 even 18 inner
6845.2.a.t.1.13 36 185.69 odd 36
6845.2.a.u.1.24 36 185.79 odd 36