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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(99,925)] 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("925.99"); 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([9, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.ba (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,0,0,6,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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 299.9
Character \(\chi\) \(=\) 925.299
Dual form 925.2.ba.b.99.9

$q$-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+(0.703175 + 0.590034i) q^{2} +(-0.508024 - 0.605440i) q^{3} +(-0.200981 - 1.13982i) q^{4} -0.725482i q^{6} +(1.16297 - 3.19522i) q^{7} +(1.44914 - 2.50998i) q^{8} +(0.412476 - 2.33927i) q^{9} +(-1.45515 + 2.52039i) q^{11} +(-0.587990 + 0.700739i) q^{12} +(-0.108073 - 0.612910i) q^{13} +(2.70306 - 1.56061i) q^{14} +(0.324761 - 0.118203i) q^{16} +(-0.955390 + 5.41828i) q^{17} +(1.67029 - 1.40154i) q^{18} +(-3.15940 - 3.76523i) q^{19} +(-2.52533 + 0.919145i) q^{21} +(-2.51034 + 0.913687i) q^{22} +(2.22993 + 3.86235i) q^{23} +(-2.25584 + 0.397765i) q^{24} +(0.285644 - 0.494749i) q^{26} +(-3.67921 + 2.12419i) q^{27} +(-3.87572 - 0.683393i) q^{28} +(-8.86469 - 5.11803i) q^{29} -6.23857i q^{31} +(-5.14887 - 1.87404i) q^{32} +(2.26519 - 0.399414i) q^{33} +(-3.86878 + 3.24629i) q^{34} -2.74925 q^{36} +(4.43357 - 4.16454i) q^{37} -4.51177i q^{38} +(-0.316177 + 0.376805i) q^{39} +(-0.0934663 - 0.530074i) q^{41} +(-2.31807 - 0.843710i) q^{42} +9.30569 q^{43} +(3.16525 + 1.15206i) q^{44} +(-0.710887 + 4.03164i) q^{46} +(-0.837773 + 0.483688i) q^{47} +(-0.236551 - 0.136573i) q^{48} +(-3.49464 - 2.93235i) q^{49} +(3.76581 - 2.17419i) q^{51} +(-0.676887 + 0.246367i) q^{52} +(1.62426 + 4.46261i) q^{53} +(-3.84047 - 0.677179i) q^{54} +(-6.33465 - 7.54934i) q^{56} +(-0.674567 + 3.82566i) q^{57} +(-3.21362 - 8.82934i) q^{58} +(4.29946 + 11.8127i) q^{59} +(4.37236 - 0.770966i) q^{61} +(3.68097 - 4.38681i) q^{62} +(-6.99478 - 4.03844i) q^{63} +(-2.86041 - 4.95438i) q^{64} +(1.82849 + 1.05568i) q^{66} +(3.99372 - 10.9727i) q^{67} +6.36789 q^{68} +(1.20556 - 3.31225i) q^{69} +(0.548337 - 0.460110i) q^{71} +(-5.27378 - 4.42523i) q^{72} -5.46858i q^{73} +(5.57480 - 0.312443i) q^{74} +(-3.65671 + 4.35790i) q^{76} +(6.36091 + 7.58064i) q^{77} +(-0.444655 + 0.0784046i) q^{78} +(2.49273 - 6.84871i) q^{79} +(-3.54111 - 1.28886i) q^{81} +(0.247038 - 0.427883i) q^{82} +(0.312050 + 0.0550228i) q^{83} +(1.55520 + 2.69369i) q^{84} +(6.54353 + 5.49067i) q^{86} +(1.40482 + 7.96712i) q^{87} +(4.21741 + 7.30477i) q^{88} +(-3.44377 - 9.46169i) q^{89} +(-2.08407 - 0.367477i) q^{91} +(3.95422 - 3.31798i) q^{92} +(-3.77708 + 3.16935i) q^{93} +(-0.874493 - 0.154197i) q^{94} +(1.48114 + 4.06939i) q^{96} +(6.18220 + 10.7079i) q^{97} +(-0.727158 - 4.12392i) q^{98} +(5.29564 + 4.44357i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 6 q^{4} - 6 q^{7} - 18 q^{8} - 42 q^{12} - 18 q^{13} + 18 q^{14} - 18 q^{16} + 12 q^{17} + 24 q^{18} + 18 q^{19} + 24 q^{22} - 24 q^{23} + 30 q^{24} - 48 q^{28} - 18 q^{29} + 24 q^{33} - 66 q^{34}+ \cdots + 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.703175 + 0.590034i 0.497220 + 0.417217i 0.856605 − 0.515972i \(-0.172569\pi\)
−0.359386 + 0.933189i \(0.617014\pi\)
\(3\) −0.508024 − 0.605440i −0.293308 − 0.349551i 0.599186 − 0.800610i \(-0.295491\pi\)
−0.892494 + 0.451059i \(0.851046\pi\)
\(4\) −0.200981 − 1.13982i −0.100491 − 0.569911i
\(5\) 0 0
\(6\) − 0.725482i − 0.296177i
\(7\) 1.16297 − 3.19522i 0.439560 − 1.20768i −0.500220 − 0.865899i \(-0.666747\pi\)
0.939779 − 0.341782i \(-0.111030\pi\)
\(8\) 1.44914 − 2.50998i 0.512348 − 0.887412i
\(9\) 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 \(0.311243\pi\)
\(12\) −0.587990 + 0.700739i −0.169738 + 0.202286i
\(13\) −0.108073 − 0.612910i −0.0299739 − 0.169991i 0.966146 − 0.257996i \(-0.0830620\pi\)
−0.996120 + 0.0880049i \(0.971951\pi\)
\(14\) 2.70306 − 1.56061i 0.722422 − 0.417091i
\(15\) 0 0
\(16\) 0.324761 − 0.118203i 0.0811902 − 0.0295508i
\(17\) −0.955390 + 5.41828i −0.231716 + 1.31413i 0.617704 + 0.786410i \(0.288063\pi\)
−0.849420 + 0.527717i \(0.823048\pi\)
\(18\) 1.67029 − 1.40154i 0.393691 − 0.330346i
\(19\) −3.15940 − 3.76523i −0.724817 − 0.863803i 0.270272 − 0.962784i \(-0.412886\pi\)
−0.995089 + 0.0989806i \(0.968442\pi\)
\(20\) 0 0
\(21\) −2.52533 + 0.919145i −0.551072 + 0.200574i
\(22\) −2.51034 + 0.913687i −0.535205 + 0.194799i
\(23\) 2.22993 + 3.86235i 0.464972 + 0.805356i 0.999200 − 0.0399849i \(-0.0127310\pi\)
−0.534228 + 0.845340i \(0.679398\pi\)
\(24\) −2.25584 + 0.397765i −0.460471 + 0.0811935i
\(25\) 0 0
\(26\) 0.285644 − 0.494749i 0.0560193 − 0.0970283i
\(27\) −3.67921 + 2.12419i −0.708064 + 0.408801i
\(28\) −3.87572 − 0.683393i −0.732442 − 0.129149i
\(29\) −8.86469 − 5.11803i −1.64613 − 0.950395i −0.978589 − 0.205823i \(-0.934013\pi\)
−0.667542 − 0.744572i \(-0.732654\pi\)
\(30\) 0 0
\(31\) − 6.23857i − 1.12048i −0.828330 − 0.560240i \(-0.810709\pi\)
0.828330 − 0.560240i \(-0.189291\pi\)
\(32\) −5.14887 − 1.87404i −0.910200 − 0.331286i
\(33\) 2.26519 − 0.399414i 0.394319 − 0.0695291i
\(34\) −3.86878 + 3.24629i −0.663490 + 0.556734i
\(35\) 0 0
\(36\) −2.74925 −0.458208
\(37\) 4.43357 − 4.16454i 0.728875 − 0.684647i
\(38\) − 4.51177i − 0.731906i
\(39\) −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 \(0.957827\pi\)
\(42\) −2.31807 − 0.843710i −0.357687 − 0.130187i
\(43\) 9.30569 1.41911 0.709553 − 0.704653i \(-0.248897\pi\)
0.709553 + 0.704653i \(0.248897\pi\)
\(44\) 3.16525 + 1.15206i 0.477179 + 0.173679i
\(45\) 0 0
\(46\) −0.710887 + 4.03164i −0.104815 + 0.594433i
\(47\) −0.837773 + 0.483688i −0.122202 + 0.0705532i −0.559855 − 0.828591i \(-0.689143\pi\)
0.437653 + 0.899144i \(0.355810\pi\)
\(48\) −0.236551 − 0.136573i −0.0341432 − 0.0197126i
\(49\) −3.49464 − 2.93235i −0.499235 − 0.418908i
\(50\) 0 0
\(51\) 3.76581 − 2.17419i 0.527318 − 0.304447i
\(52\) −0.676887 + 0.246367i −0.0938674 + 0.0341649i
\(53\) 1.62426 + 4.46261i 0.223109 + 0.612987i 0.999858 − 0.0168227i \(-0.00535507\pi\)
−0.776749 + 0.629810i \(0.783133\pi\)
\(54\) −3.84047 − 0.677179i −0.522622 − 0.0921524i
\(55\) 0 0
\(56\) −6.33465 − 7.54934i −0.846503 − 1.00882i
\(57\) −0.674567 + 3.82566i −0.0893486 + 0.506721i
\(58\) −3.21362 − 8.82934i −0.421969 − 1.15935i
\(59\) 4.29946 + 11.8127i 0.559741 + 1.53788i 0.820015 + 0.572342i \(0.193965\pi\)
−0.260273 + 0.965535i \(0.583813\pi\)
\(60\) 0 0
\(61\) 4.37236 − 0.770966i 0.559824 − 0.0987120i 0.113426 − 0.993546i \(-0.463818\pi\)
0.446398 + 0.894834i \(0.352707\pi\)
\(62\) 3.68097 − 4.38681i 0.467484 − 0.557125i
\(63\) −6.99478 − 4.03844i −0.881260 − 0.508796i
\(64\) −2.86041 − 4.95438i −0.357552 − 0.619298i
\(65\) 0 0
\(66\) 1.82849 + 1.05568i 0.225072 + 0.129945i
\(67\) 3.99372 − 10.9727i 0.487911 − 1.34052i −0.414657 − 0.909978i \(-0.636098\pi\)
0.902568 − 0.430547i \(-0.141679\pi\)
\(68\) 6.36789 0.772220
\(69\) 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 \(0.264253\pi\)
\(72\) −5.27378 − 4.42523i −0.621521 − 0.521518i
\(73\) − 5.46858i − 0.640048i −0.947409 − 0.320024i \(-0.896309\pi\)
0.947409 − 0.320024i \(-0.103691\pi\)
\(74\) 5.57480 − 0.312443i 0.648057 − 0.0363208i
\(75\) 0 0
\(76\) −3.65671 + 4.35790i −0.419453 + 0.499885i
\(77\) 6.36091 + 7.58064i 0.724893 + 0.863894i
\(78\) −0.444655 + 0.0784046i −0.0503472 + 0.00887758i
\(79\) 2.49273 − 6.84871i 0.280454 − 0.770540i −0.716855 − 0.697222i \(-0.754419\pi\)
0.997309 − 0.0733179i \(-0.0233588\pi\)
\(80\) 0 0
\(81\) −3.54111 − 1.28886i −0.393456 − 0.143206i
\(82\) 0.247038 − 0.427883i 0.0272808 − 0.0472518i
\(83\) 0.312050 + 0.0550228i 0.0342519 + 0.00603953i 0.190748 − 0.981639i \(-0.438909\pi\)
−0.156496 + 0.987679i \(0.550020\pi\)
\(84\) 1.55520 + 2.69369i 0.169687 + 0.293906i
\(85\) 0 0
\(86\) 6.54353 + 5.49067i 0.705607 + 0.592075i
\(87\) 1.40482 + 7.96712i 0.150612 + 0.854165i
\(88\) 4.21741 + 7.30477i 0.449578 + 0.778692i
\(89\) −3.44377 − 9.46169i −0.365039 − 1.00294i −0.977222 − 0.212221i \(-0.931930\pi\)
0.612183 − 0.790716i \(-0.290292\pi\)
\(90\) 0 0
\(91\) −2.08407 − 0.367477i −0.218470 − 0.0385221i
\(92\) 3.95422 − 3.31798i 0.412255 − 0.345923i
\(93\) −3.77708 + 3.16935i −0.391665 + 0.328646i
\(94\) −0.874493 − 0.154197i −0.0901971 − 0.0159042i
\(95\) 0 0
\(96\) 1.48114 + 4.06939i 0.151168 + 0.415330i
\(97\) 6.18220 + 10.7079i 0.627708 + 1.08722i 0.988011 + 0.154386i \(0.0493400\pi\)
−0.360303 + 0.932835i \(0.617327\pi\)
\(98\) −0.727158 − 4.12392i −0.0734540 − 0.416578i
\(99\) 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.ba.b.299.9 72
5.2 odd 4 185.2.w.a.151.5 yes 72
5.3 odd 4 925.2.bb.b.151.8 72
5.4 even 2 925.2.ba.c.299.4 72
37.25 even 18 925.2.ba.c.99.4 72
185.32 even 36 6845.2.a.t.1.13 36
185.42 even 36 6845.2.a.u.1.24 36
185.62 odd 36 185.2.w.a.136.5 ✓ 72
185.99 even 18 inner 925.2.ba.b.99.9 72
185.173 odd 36 925.2.bb.b.876.8 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.136.5 ✓ 72 185.62 odd 36
185.2.w.a.151.5 yes 72 5.2 odd 4
925.2.ba.b.99.9 72 185.99 even 18 inner
925.2.ba.b.299.9 72 1.1 even 1 trivial
925.2.ba.c.99.4 72 37.25 even 18
925.2.ba.c.299.4 72 5.4 even 2
925.2.bb.b.151.8 72 5.3 odd 4
925.2.bb.b.876.8 72 185.173 odd 36
6845.2.a.t.1.13 36 185.32 even 36
6845.2.a.u.1.24 36 185.42 even 36