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Properties

Label 370.2.g.b.327.1
Level $370$
Weight $2$
Character 370.327
Analytic conductor $2.954$
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] = [370,2,Mod(43,370)] 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("370.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(370, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 327.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 370.327
Dual form 370.2.g.b.43.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} +(-2.00000 - 2.00000i) q^{7} -1.00000i q^{8} -3.00000i q^{9} +(-1.00000 - 2.00000i) q^{10} -4.00000i q^{13} +(2.00000 - 2.00000i) q^{14} +1.00000 q^{16} -2.00000 q^{17} +3.00000 q^{18} +(2.00000 + 2.00000i) q^{19} +(2.00000 - 1.00000i) q^{20} -4.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +4.00000 q^{26} +(2.00000 + 2.00000i) q^{28} +(-7.00000 + 7.00000i) q^{29} +(-4.00000 - 4.00000i) q^{31} +1.00000i q^{32} -2.00000i q^{34} +(6.00000 + 2.00000i) q^{35} +3.00000i q^{36} +(-6.00000 - 1.00000i) q^{37} +(-2.00000 + 2.00000i) q^{38} +(1.00000 + 2.00000i) q^{40} -4.00000i q^{43} +(3.00000 + 6.00000i) q^{45} +4.00000 q^{46} +(-2.00000 - 2.00000i) q^{47} +1.00000i q^{49} +(4.00000 + 3.00000i) q^{50} +4.00000i q^{52} +(-1.00000 + 1.00000i) q^{53} +(-2.00000 + 2.00000i) q^{56} +(-7.00000 - 7.00000i) q^{58} +(2.00000 + 2.00000i) q^{59} +(1.00000 + 1.00000i) q^{61} +(4.00000 - 4.00000i) q^{62} +(-6.00000 + 6.00000i) q^{63} -1.00000 q^{64} +(4.00000 + 8.00000i) q^{65} +2.00000 q^{68} +(-2.00000 + 6.00000i) q^{70} +12.0000 q^{71} -3.00000 q^{72} +(-5.00000 - 5.00000i) q^{73} +(1.00000 - 6.00000i) q^{74} +(-2.00000 - 2.00000i) q^{76} +(12.0000 + 12.0000i) q^{79} +(-2.00000 + 1.00000i) q^{80} -9.00000 q^{81} +(4.00000 - 4.00000i) q^{83} +(4.00000 - 2.00000i) q^{85} +4.00000 q^{86} +(-7.00000 + 7.00000i) q^{89} +(-6.00000 + 3.00000i) q^{90} +(-8.00000 + 8.00000i) q^{91} +4.00000i q^{92} +(2.00000 - 2.00000i) q^{94} +(-6.00000 - 2.00000i) q^{95} -12.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 4 q^{5} - 4 q^{7} - 2 q^{10} + 4 q^{14} + 2 q^{16} - 4 q^{17} + 6 q^{18} + 4 q^{19} + 4 q^{20} + 6 q^{25} + 8 q^{26} + 4 q^{28} - 14 q^{29} - 8 q^{31} + 12 q^{35} - 12 q^{37} - 4 q^{38}+ \cdots - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(4\) −1.00000 −0.500000
\(5\) −2.00000 + 1.00000i −0.894427 + 0.447214i
\(6\) 0 0
\(7\) −2.00000 − 2.00000i −0.755929 − 0.755929i 0.219650 − 0.975579i \(-0.429509\pi\)
−0.975579 + 0.219650i \(0.929509\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) − 3.00000i − 1.00000i
\(10\) −1.00000 − 2.00000i −0.316228 − 0.632456i
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) − 4.00000i − 1.10940i −0.832050 − 0.554700i \(-0.812833\pi\)
0.832050 − 0.554700i \(-0.187167\pi\)
\(14\) 2.00000 − 2.00000i 0.534522 − 0.534522i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 3.00000 0.707107
\(19\) 2.00000 + 2.00000i 0.458831 + 0.458831i 0.898272 − 0.439440i \(-0.144823\pi\)
−0.439440 + 0.898272i \(0.644823\pi\)
\(20\) 2.00000 − 1.00000i 0.447214 − 0.223607i
\(21\) 0 0
\(22\) 0 0
\(23\) − 4.00000i − 0.834058i −0.908893 − 0.417029i \(-0.863071\pi\)
0.908893 − 0.417029i \(-0.136929\pi\)
\(24\) 0 0
\(25\) 3.00000 − 4.00000i 0.600000 − 0.800000i
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) 2.00000 + 2.00000i 0.377964 + 0.377964i
\(29\) −7.00000 + 7.00000i −1.29987 + 1.29987i −0.371391 + 0.928477i \(0.621119\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) −4.00000 − 4.00000i −0.718421 − 0.718421i 0.249861 − 0.968282i \(-0.419615\pi\)
−0.968282 + 0.249861i \(0.919615\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) − 2.00000i − 0.342997i
\(35\) 6.00000 + 2.00000i 1.01419 + 0.338062i
\(36\) 3.00000i 0.500000i
\(37\) −6.00000 − 1.00000i −0.986394 − 0.164399i
\(38\) −2.00000 + 2.00000i −0.324443 + 0.324443i
\(39\) 0 0
\(40\) 1.00000 + 2.00000i 0.158114 + 0.316228i
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 − 0.304997i \(-0.901344\pi\)
0.952353 − 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 3.00000 + 6.00000i 0.447214 + 0.894427i
\(46\) 4.00000 0.589768
\(47\) −2.00000 − 2.00000i −0.291730 − 0.291730i 0.546033 − 0.837763i \(-0.316137\pi\)
−0.837763 + 0.546033i \(0.816137\pi\)
\(48\) 0 0
\(49\) 1.00000i 0.142857i
\(50\) 4.00000 + 3.00000i 0.565685 + 0.424264i
\(51\) 0 0
\(52\) 4.00000i 0.554700i
\(53\) −1.00000 + 1.00000i −0.137361 + 0.137361i −0.772444 − 0.635083i \(-0.780966\pi\)
0.635083 + 0.772444i \(0.280966\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.00000 + 2.00000i −0.267261 + 0.267261i
\(57\) 0 0
\(58\) −7.00000 − 7.00000i −0.919145 − 0.919145i
\(59\) 2.00000 + 2.00000i 0.260378 + 0.260378i 0.825208 − 0.564830i \(-0.191058\pi\)
−0.564830 + 0.825208i \(0.691058\pi\)
\(60\) 0 0
\(61\) 1.00000 + 1.00000i 0.128037 + 0.128037i 0.768221 − 0.640184i \(-0.221142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 4.00000 − 4.00000i 0.508001 − 0.508001i
\(63\) −6.00000 + 6.00000i −0.755929 + 0.755929i
\(64\) −1.00000 −0.125000
\(65\) 4.00000 + 8.00000i 0.496139 + 0.992278i
\(66\) 0 0
\(67\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) −2.00000 + 6.00000i −0.239046 + 0.717137i
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) −3.00000 −0.353553
\(73\) −5.00000 − 5.00000i −0.585206 − 0.585206i 0.351123 − 0.936329i \(-0.385800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 1.00000 − 6.00000i 0.116248 − 0.697486i
\(75\) 0 0
\(76\) −2.00000 − 2.00000i −0.229416 − 0.229416i
\(77\) 0 0
\(78\) 0 0
\(79\) 12.0000 + 12.0000i 1.35011 + 1.35011i 0.885537 + 0.464568i \(0.153790\pi\)
0.464568 + 0.885537i \(0.346210\pi\)
\(80\) −2.00000 + 1.00000i −0.223607 + 0.111803i
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 4.00000 − 4.00000i 0.439057 − 0.439057i −0.452638 − 0.891695i \(-0.649517\pi\)
0.891695 + 0.452638i \(0.149517\pi\)
\(84\) 0 0
\(85\) 4.00000 − 2.00000i 0.433861 − 0.216930i
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) −7.00000 + 7.00000i −0.741999 + 0.741999i −0.972962 − 0.230964i \(-0.925812\pi\)
0.230964 + 0.972962i \(0.425812\pi\)
\(90\) −6.00000 + 3.00000i −0.632456 + 0.316228i
\(91\) −8.00000 + 8.00000i −0.838628 + 0.838628i
\(92\) 4.00000i 0.417029i
\(93\) 0 0
\(94\) 2.00000 − 2.00000i 0.206284 − 0.206284i
\(95\) −6.00000 − 2.00000i −0.615587 − 0.205196i
\(96\) 0 0
\(97\) −12.0000 −1.21842 −0.609208 − 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) −1.00000 −0.101015
\(99\) 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 370.2.g.b.327.1 yes 2
5.3 odd 4 370.2.h.a.253.1 yes 2
37.6 odd 4 370.2.h.a.117.1 yes 2
185.43 even 4 inner 370.2.g.b.43.1 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.g.b.43.1 ✓ 2 185.43 even 4 inner
370.2.g.b.327.1 yes 2 1.1 even 1 trivial
370.2.h.a.117.1 yes 2 37.6 odd 4
370.2.h.a.253.1 yes 2 5.3 odd 4