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Properties

Label 225.2.h.c.46.1
Level $225$
Weight $2$
Character 225.46
Analytic conductor $1.797$
Analytic rank $0$
Dimension $8$
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] = [225,2,Mod(46,225)] 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("225.46"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 225.h (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,1] 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: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.26265625.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 2x^{6} + x^{4} + 8x^{2} - 24x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 46.1
Root \(-1.21700 + 0.720348i\) of defining polynomial
Character \(\chi\) \(=\) 225.46
Dual form 225.2.h.c.181.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.655837 + 2.01846i) q^{2} +(-2.02602 - 1.47199i) q^{4} +(2.21700 - 0.291365i) q^{5} +4.35840 q^{7} +(0.865884 - 0.629102i) q^{8} +(-0.865884 + 4.66602i) q^{10} +(0.488218 - 1.50258i) q^{11} +(0.370184 + 1.13931i) q^{13} +(-2.85840 + 8.79726i) q^{14} +(-0.845805 - 2.60312i) q^{16} +(-0.907987 + 0.659691i) q^{17} +(-6.21218 + 4.51341i) q^{19} +(-4.92058 - 2.67310i) q^{20} +(2.71270 + 1.97090i) q^{22} +(0.717004 - 2.20671i) q^{23} +(4.83021 - 1.29192i) q^{25} -2.54243 q^{26} +(-8.83021 - 6.41552i) q^{28} +(-4.45307 - 3.23535i) q^{29} +(-3.88495 + 2.82258i) q^{31} +7.94959 q^{32} +(-0.736068 - 2.26538i) q^{34} +(9.66259 - 1.26989i) q^{35} +(-1.96915 - 6.06043i) q^{37} +(-5.03596 - 15.4991i) q^{38} +(1.73637 - 1.64701i) q^{40} +(2.30902 + 7.10642i) q^{41} -1.24998 q^{43} +(-3.20092 + 2.32561i) q^{44} +(3.98392 + 2.89449i) q^{46} +(3.33934 + 2.42617i) q^{47} +11.9957 q^{49} +(-0.560152 + 10.5969i) q^{50} +(0.927051 - 2.85317i) q^{52} +(3.03032 + 2.20166i) q^{53} +(0.644581 - 3.47347i) q^{55} +(3.77387 - 2.74188i) q^{56} +(9.45090 - 6.86648i) q^{58} +(-2.82940 - 8.70799i) q^{59} +(0.431351 - 1.32756i) q^{61} +(-3.14937 - 9.69276i) q^{62} +(-3.52202 + 10.8397i) q^{64} +(1.15265 + 2.41799i) q^{65} +(3.12499 - 2.27044i) q^{67} +2.81066 q^{68} +(-3.77387 + 20.3364i) q^{70} +(-8.57970 - 6.23352i) q^{71} +(1.54407 - 4.75216i) q^{73} +13.5242 q^{74} +19.2297 q^{76} +(2.12785 - 6.54885i) q^{77} +(-11.7737 - 8.55407i) q^{79} +(-2.63361 - 5.52469i) q^{80} -15.8584 q^{82} +(-7.06760 + 5.13491i) q^{83} +(-1.82080 + 1.72709i) q^{85} +(0.819784 - 2.52304i) q^{86} +(-0.522535 - 1.60820i) q^{88} +(3.10195 - 9.54683i) q^{89} +(1.61341 + 4.96556i) q^{91} +(-4.70092 + 3.41542i) q^{92} +(-7.08719 + 5.14914i) q^{94} +(-12.4574 + 11.8163i) q^{95} +(6.06760 + 4.40837i) q^{97} +(-7.86720 + 24.2128i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + q^{4} + 5 q^{5} + 4 q^{7} - 16 q^{11} - 8 q^{13} + 8 q^{14} - 17 q^{16} + q^{17} - 5 q^{19} + 10 q^{20} + 13 q^{22} - 7 q^{23} - 15 q^{25} - 6 q^{26} - 17 q^{28} - 5 q^{29} - 19 q^{31}+ \cdots - 73 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.655837 + 2.01846i −0.463747 + 1.42727i 0.396805 + 0.917903i \(0.370119\pi\)
−0.860552 + 0.509363i \(0.829881\pi\)
\(3\) 0 0
\(4\) −2.02602 − 1.47199i −1.01301 − 0.735995i
\(5\) 2.21700 − 0.291365i 0.991474 − 0.130303i
\(6\) 0 0
\(7\) 4.35840 1.64732 0.823660 − 0.567083i \(-0.191928\pi\)
0.823660 + 0.567083i \(0.191928\pi\)
\(8\) 0.865884 − 0.629102i 0.306136 − 0.222421i
\(9\) 0 0
\(10\) −0.865884 + 4.66602i −0.273817 + 1.47553i
\(11\) 0.488218 − 1.50258i 0.147203 − 0.453045i −0.850085 − 0.526646i \(-0.823449\pi\)
0.997288 + 0.0736014i \(0.0234493\pi\)
\(12\) 0 0
\(13\) 0.370184 + 1.13931i 0.102670 + 0.315987i 0.989177 − 0.146729i \(-0.0468747\pi\)
−0.886506 + 0.462717i \(0.846875\pi\)
\(14\) −2.85840 + 8.79726i −0.763940 + 2.35117i
\(15\) 0 0
\(16\) −0.845805 − 2.60312i −0.211451 − 0.650780i
\(17\) −0.907987 + 0.659691i −0.220219 + 0.159999i −0.692425 − 0.721490i \(-0.743458\pi\)
0.472206 + 0.881488i \(0.343458\pi\)
\(18\) 0 0
\(19\) −6.21218 + 4.51341i −1.42517 + 1.03545i −0.434281 + 0.900777i \(0.642998\pi\)
−0.990890 + 0.134670i \(0.957002\pi\)
\(20\) −4.92058 − 2.67310i −1.10028 − 0.597722i
\(21\) 0 0
\(22\) 2.71270 + 1.97090i 0.578351 + 0.420196i
\(23\) 0.717004 − 2.20671i 0.149506 − 0.460131i −0.848057 − 0.529905i \(-0.822228\pi\)
0.997563 + 0.0697736i \(0.0222277\pi\)
\(24\) 0 0
\(25\) 4.83021 − 1.29192i 0.966042 − 0.258383i
\(26\) −2.54243 −0.498611
\(27\) 0 0
\(28\) −8.83021 − 6.41552i −1.66875 − 1.21242i
\(29\) −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 \(0.441923\pi\)
\(32\) 7.94959 1.40530
\(33\) 0 0
\(34\) −0.736068 − 2.26538i −0.126235 − 0.388510i
\(35\) 9.66259 − 1.26989i 1.63328 − 0.214650i
\(36\) 0 0
\(37\) −1.96915 − 6.06043i −0.323727 − 0.996329i −0.972012 − 0.234931i \(-0.924514\pi\)
0.648285 − 0.761398i \(-0.275486\pi\)
\(38\) −5.03596 − 15.4991i −0.816941 − 2.51428i
\(39\) 0 0
\(40\) 1.73637 − 1.64701i 0.274544 − 0.260415i
\(41\) 2.30902 + 7.10642i 0.360608 + 1.10984i 0.952686 + 0.303956i \(0.0983077\pi\)
−0.592078 + 0.805881i \(0.701692\pi\)
\(42\) 0 0
\(43\) −1.24998 −0.190620 −0.0953102 − 0.995448i \(-0.530384\pi\)
−0.0953102 + 0.995448i \(0.530384\pi\)
\(44\) −3.20092 + 2.32561i −0.482557 + 0.350598i
\(45\) 0 0
\(46\) 3.98392 + 2.89449i 0.587397 + 0.426769i
\(47\) 3.33934 + 2.42617i 0.487092 + 0.353893i 0.804065 − 0.594541i \(-0.202666\pi\)
−0.316973 + 0.948435i \(0.602666\pi\)
\(48\) 0 0
\(49\) 11.9957 1.71367
\(50\) −0.560152 + 10.5969i −0.0792174 + 1.49862i
\(51\) 0 0
\(52\) 0.927051 − 2.85317i 0.128559 − 0.395663i
\(53\) 3.03032 + 2.20166i 0.416247 + 0.302421i 0.776126 − 0.630578i \(-0.217182\pi\)
−0.359879 + 0.932999i \(0.617182\pi\)
\(54\) 0 0
\(55\) 0.644581 − 3.47347i 0.0869153 − 0.468363i
\(56\) 3.77387 − 2.74188i 0.504305 − 0.366399i
\(57\) 0 0
\(58\) 9.45090 − 6.86648i 1.24096 − 0.901613i
\(59\) −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 \(0.0715172\pi\)
\(62\) −3.14937 − 9.69276i −0.399970 − 1.23098i
\(63\) 0 0
\(64\) −3.52202 + 10.8397i −0.440253 + 1.35496i
\(65\) 1.15265 + 2.41799i 0.142969 + 0.299915i
\(66\) 0 0
\(67\) 3.12499 − 2.27044i 0.381778 − 0.277378i −0.380300 − 0.924863i \(-0.624179\pi\)
0.762078 + 0.647485i \(0.224179\pi\)
\(68\) 2.81066 0.340843
\(69\) 0 0
\(70\) −3.77387 + 20.3364i −0.451064 + 2.43066i
\(71\) −8.57970 − 6.23352i −1.01822 − 0.739783i −0.0523057 − 0.998631i \(-0.516657\pi\)
−0.965918 + 0.258848i \(0.916657\pi\)
\(72\) 0 0
\(73\) 1.54407 − 4.75216i 0.180720 − 0.556198i −0.819129 − 0.573610i \(-0.805543\pi\)
0.999848 + 0.0174117i \(0.00554259\pi\)
\(74\) 13.5242 1.57215
\(75\) 0 0
\(76\) 19.2297 2.20580
\(77\) 2.12785 − 6.54885i 0.242491 − 0.746310i
\(78\) 0 0
\(79\) −11.7737 − 8.55407i −1.32464 − 0.962408i −0.999862 − 0.0166185i \(-0.994710\pi\)
−0.324779 − 0.945790i \(-0.605290\pi\)
\(80\) −2.63361 − 5.52469i −0.294447 − 0.617679i
\(81\) 0 0
\(82\) −15.8584 −1.75126
\(83\) −7.06760 + 5.13491i −0.775770 + 0.563630i −0.903706 − 0.428153i \(-0.859165\pi\)
0.127937 + 0.991782i \(0.459165\pi\)
\(84\) 0 0
\(85\) −1.82080 + 1.72709i −0.197493 + 0.187330i
\(86\) 0.819784 − 2.52304i 0.0883996 − 0.272066i
\(87\) 0 0
\(88\) −0.522535 − 1.60820i −0.0557025 − 0.171435i
\(89\) 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
\(92\) −4.70092 + 3.41542i −0.490105 + 0.356082i
\(93\) 0 0
\(94\) −7.08719 + 5.14914i −0.730988 + 0.531094i
\(95\) −12.4574 + 11.8163i −1.27810 + 1.21232i
\(96\) 0 0
\(97\) 6.06760 + 4.40837i 0.616071 + 0.447602i 0.851547 − 0.524278i \(-0.175665\pi\)
−0.235476 + 0.971880i \(0.575665\pi\)
\(98\) −7.86720 + 24.2128i −0.794707 + 2.44586i
\(99\) 0 0
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 225.2.h.c.46.1 8
3.2 odd 2 75.2.g.b.46.2 yes 8
15.2 even 4 375.2.i.b.274.4 16
15.8 even 4 375.2.i.b.274.1 16
15.14 odd 2 375.2.g.b.226.1 8
25.6 even 5 inner 225.2.h.c.181.1 8
25.9 even 10 5625.2.a.n.1.3 4
25.16 even 5 5625.2.a.i.1.2 4
75.8 even 20 375.2.i.b.349.4 16
75.17 even 20 375.2.i.b.349.1 16
75.38 even 20 1875.2.b.c.1249.2 8
75.41 odd 10 1875.2.a.h.1.3 4
75.44 odd 10 375.2.g.b.151.1 8
75.56 odd 10 75.2.g.b.31.2 ✓ 8
75.59 odd 10 1875.2.a.e.1.2 4
75.62 even 20 1875.2.b.c.1249.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.b.31.2 ✓ 8 75.56 odd 10
75.2.g.b.46.2 yes 8 3.2 odd 2
225.2.h.c.46.1 8 1.1 even 1 trivial
225.2.h.c.181.1 8 25.6 even 5 inner
375.2.g.b.151.1 8 75.44 odd 10
375.2.g.b.226.1 8 15.14 odd 2
375.2.i.b.274.1 16 15.8 even 4
375.2.i.b.274.4 16 15.2 even 4
375.2.i.b.349.1 16 75.17 even 20
375.2.i.b.349.4 16 75.8 even 20
1875.2.a.e.1.2 4 75.59 odd 10
1875.2.a.h.1.3 4 75.41 odd 10
1875.2.b.c.1249.2 8 75.38 even 20
1875.2.b.c.1249.7 8 75.62 even 20
5625.2.a.i.1.2 4 25.16 even 5
5625.2.a.n.1.3 4 25.9 even 10