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Properties

Label 450.3.d.a.251.1
Level $450$
Weight $3$
Character 450.251
Analytic conductor $12.262$
Analytic rank $0$
Dimension $2$
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] = [450,3,Mod(251,450)] 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("450.251"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 450.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-4,0,0,-22,0,0,0,0,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.2616118962\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 251.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 450.251
Dual form 450.3.d.a.251.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.41421i q^{2} -2.00000 q^{4} -11.0000 q^{7} +2.82843i q^{8} -4.24264i q^{11} +7.00000 q^{13} +15.5563i q^{14} +4.00000 q^{16} +12.7279i q^{17} +29.0000 q^{19} -6.00000 q^{22} +38.1838i q^{23} -9.89949i q^{26} +22.0000 q^{28} +46.6690i q^{29} +29.0000 q^{31} -5.65685i q^{32} +18.0000 q^{34} -56.0000 q^{37} -41.0122i q^{38} +67.8823i q^{41} -5.00000 q^{43} +8.48528i q^{44} +54.0000 q^{46} -63.6396i q^{47} +72.0000 q^{49} -14.0000 q^{52} +67.8823i q^{53} -31.1127i q^{56} +66.0000 q^{58} -29.6985i q^{59} -55.0000 q^{61} -41.0122i q^{62} -8.00000 q^{64} +37.0000 q^{67} -25.4558i q^{68} +33.9411i q^{71} +16.0000 q^{73} +79.1960i q^{74} -58.0000 q^{76} +46.6690i q^{77} +104.000 q^{79} +96.0000 q^{82} +29.6985i q^{83} +7.07107i q^{86} +12.0000 q^{88} -135.765i q^{89} -77.0000 q^{91} -76.3675i q^{92} -90.0000 q^{94} -41.0000 q^{97} -101.823i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 22 q^{7} + 14 q^{13} + 8 q^{16} + 58 q^{19} - 12 q^{22} + 44 q^{28} + 58 q^{31} + 36 q^{34} - 112 q^{37} - 10 q^{43} + 108 q^{46} + 144 q^{49} - 28 q^{52} + 132 q^{58} - 110 q^{61} - 16 q^{64}+ \cdots - 82 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.41421i − 0.707107i
\(3\) 0 0
\(4\) −2.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −11.0000 −1.57143 −0.785714 − 0.618590i \(-0.787704\pi\)
−0.785714 + 0.618590i \(0.787704\pi\)
\(8\) 2.82843i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) − 4.24264i − 0.385695i −0.981229 − 0.192847i \(-0.938228\pi\)
0.981229 − 0.192847i \(-0.0617722\pi\)
\(12\) 0 0
\(13\) 7.00000 0.538462 0.269231 − 0.963076i \(-0.413231\pi\)
0.269231 + 0.963076i \(0.413231\pi\)
\(14\) 15.5563i 1.11117i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 12.7279i 0.748701i 0.927287 + 0.374351i \(0.122134\pi\)
−0.927287 + 0.374351i \(0.877866\pi\)
\(18\) 0 0
\(19\) 29.0000 1.52632 0.763158 − 0.646212i \(-0.223648\pi\)
0.763158 + 0.646212i \(0.223648\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −6.00000 −0.272727
\(23\) 38.1838i 1.66016i 0.557642 + 0.830082i \(0.311706\pi\)
−0.557642 + 0.830082i \(0.688294\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) − 9.89949i − 0.380750i
\(27\) 0 0
\(28\) 22.0000 0.785714
\(29\) 46.6690i 1.60928i 0.593765 + 0.804639i \(0.297641\pi\)
−0.593765 + 0.804639i \(0.702359\pi\)
\(30\) 0 0
\(31\) 29.0000 0.935484 0.467742 − 0.883865i \(-0.345068\pi\)
0.467742 + 0.883865i \(0.345068\pi\)
\(32\) − 5.65685i − 0.176777i
\(33\) 0 0
\(34\) 18.0000 0.529412
\(35\) 0 0
\(36\) 0 0
\(37\) −56.0000 −1.51351 −0.756757 − 0.653697i \(-0.773217\pi\)
−0.756757 + 0.653697i \(0.773217\pi\)
\(38\) − 41.0122i − 1.07927i
\(39\) 0 0
\(40\) 0 0
\(41\) 67.8823i 1.65566i 0.560976 + 0.827832i \(0.310426\pi\)
−0.560976 + 0.827832i \(0.689574\pi\)
\(42\) 0 0
\(43\) −5.00000 −0.116279 −0.0581395 − 0.998308i \(-0.518517\pi\)
−0.0581395 + 0.998308i \(0.518517\pi\)
\(44\) 8.48528i 0.192847i
\(45\) 0 0
\(46\) 54.0000 1.17391
\(47\) − 63.6396i − 1.35403i −0.735967 − 0.677017i \(-0.763272\pi\)
0.735967 − 0.677017i \(-0.236728\pi\)
\(48\) 0 0
\(49\) 72.0000 1.46939
\(50\) 0 0
\(51\) 0 0
\(52\) −14.0000 −0.269231
\(53\) 67.8823i 1.28080i 0.768043 + 0.640399i \(0.221231\pi\)
−0.768043 + 0.640399i \(0.778769\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) − 31.1127i − 0.555584i
\(57\) 0 0
\(58\) 66.0000 1.13793
\(59\) − 29.6985i − 0.503364i −0.967810 − 0.251682i \(-0.919016\pi\)
0.967810 − 0.251682i \(-0.0809837\pi\)
\(60\) 0 0
\(61\) −55.0000 −0.901639 −0.450820 − 0.892615i \(-0.648868\pi\)
−0.450820 + 0.892615i \(0.648868\pi\)
\(62\) − 41.0122i − 0.661487i
\(63\) 0 0
\(64\) −8.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 37.0000 0.552239 0.276119 − 0.961123i \(-0.410951\pi\)
0.276119 + 0.961123i \(0.410951\pi\)
\(68\) − 25.4558i − 0.374351i
\(69\) 0 0
\(70\) 0 0
\(71\) 33.9411i 0.478044i 0.971014 + 0.239022i \(0.0768268\pi\)
−0.971014 + 0.239022i \(0.923173\pi\)
\(72\) 0 0
\(73\) 16.0000 0.219178 0.109589 − 0.993977i \(-0.465047\pi\)
0.109589 + 0.993977i \(0.465047\pi\)
\(74\) 79.1960i 1.07022i
\(75\) 0 0
\(76\) −58.0000 −0.763158
\(77\) 46.6690i 0.606092i
\(78\) 0 0
\(79\) 104.000 1.31646 0.658228 − 0.752819i \(-0.271306\pi\)
0.658228 + 0.752819i \(0.271306\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 96.0000 1.17073
\(83\) 29.6985i 0.357813i 0.983866 + 0.178907i \(0.0572560\pi\)
−0.983866 + 0.178907i \(0.942744\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 7.07107i 0.0822217i
\(87\) 0 0
\(88\) 12.0000 0.136364
\(89\) − 135.765i − 1.52544i −0.646727 − 0.762722i \(-0.723863\pi\)
0.646727 − 0.762722i \(-0.276137\pi\)
\(90\) 0 0
\(91\) −77.0000 −0.846154
\(92\) − 76.3675i − 0.830082i
\(93\) 0 0
\(94\) −90.0000 −0.957447
\(95\) 0 0
\(96\) 0 0
\(97\) −41.0000 −0.422680 −0.211340 − 0.977413i \(-0.567783\pi\)
−0.211340 + 0.977413i \(0.567783\pi\)
\(98\) − 101.823i − 1.03901i
\(99\) 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 450.3.d.a.251.1 ✓ 2
3.2 odd 2 inner 450.3.d.a.251.2 yes 2
4.3 odd 2 3600.3.l.k.1601.2 2
5.2 odd 4 450.3.b.a.449.3 4
5.3 odd 4 450.3.b.a.449.2 4
5.4 even 2 450.3.d.g.251.2 yes 2
12.11 even 2 3600.3.l.k.1601.1 2
15.2 even 4 450.3.b.a.449.1 4
15.8 even 4 450.3.b.a.449.4 4
15.14 odd 2 450.3.d.g.251.1 yes 2
20.3 even 4 3600.3.c.f.449.2 4
20.7 even 4 3600.3.c.f.449.4 4
20.19 odd 2 3600.3.l.a.1601.2 2
60.23 odd 4 3600.3.c.f.449.1 4
60.47 odd 4 3600.3.c.f.449.3 4
60.59 even 2 3600.3.l.a.1601.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
450.3.b.a.449.1 4 15.2 even 4
450.3.b.a.449.2 4 5.3 odd 4
450.3.b.a.449.3 4 5.2 odd 4
450.3.b.a.449.4 4 15.8 even 4
450.3.d.a.251.1 ✓ 2 1.1 even 1 trivial
450.3.d.a.251.2 yes 2 3.2 odd 2 inner
450.3.d.g.251.1 yes 2 15.14 odd 2
450.3.d.g.251.2 yes 2 5.4 even 2
3600.3.c.f.449.1 4 60.23 odd 4
3600.3.c.f.449.2 4 20.3 even 4
3600.3.c.f.449.3 4 60.47 odd 4
3600.3.c.f.449.4 4 20.7 even 4
3600.3.l.a.1601.1 2 60.59 even 2
3600.3.l.a.1601.2 2 20.19 odd 2
3600.3.l.k.1601.1 2 12.11 even 2
3600.3.l.k.1601.2 2 4.3 odd 2