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Properties

Label 300.2.j.b.7.1
Level $300$
Weight $2$
Character 300.7
Analytic conductor $2.396$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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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] = [300,2,Mod(7,300)] 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("300.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(300, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 300.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.39551206064\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{2}, \sqrt{7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 7.1
Root \(-1.28897 - 0.581861i\) of defining polynomial
Character \(\chi\) \(=\) 300.7
Dual form 300.2.j.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)
 
\(f(q)\) \(=\) \(q+(-1.28897 - 0.581861i) q^{2} +(-0.707107 - 0.707107i) q^{3} +(1.32288 + 1.50000i) q^{4} +(0.500000 + 1.32288i) q^{6} +(-1.41421 + 1.41421i) q^{7} +(-0.832353 - 2.70318i) q^{8} +1.00000i q^{9} -5.29150i q^{11} +(0.125246 - 1.99607i) q^{12} +(3.74166 - 3.74166i) q^{13} +(2.64575 - 1.00000i) q^{14} +(-0.500000 + 3.96863i) q^{16} +(0.581861 - 1.28897i) q^{18} -5.29150 q^{19} +2.00000 q^{21} +(-3.07892 + 6.82058i) q^{22} +(-2.82843 - 2.82843i) q^{23} +(-1.32288 + 2.50000i) q^{24} +(-7.00000 + 2.64575i) q^{26} +(0.707107 - 0.707107i) q^{27} +(-3.99215 - 0.250492i) q^{28} -8.00000i q^{29} -5.29150i q^{31} +(2.95367 - 4.82450i) q^{32} +(-3.74166 + 3.74166i) q^{33} +(-1.50000 + 1.32288i) q^{36} +(-3.74166 - 3.74166i) q^{37} +(6.82058 + 3.07892i) q^{38} -5.29150 q^{39} -2.00000 q^{41} +(-2.57794 - 1.16372i) q^{42} +(5.65685 + 5.65685i) q^{43} +(7.93725 - 7.00000i) q^{44} +(2.00000 + 5.29150i) q^{46} +(3.15980 - 2.45269i) q^{48} +3.00000i q^{49} +(10.5622 + 0.662739i) q^{52} +(-7.48331 + 7.48331i) q^{53} +(-1.32288 + 0.500000i) q^{54} +(5.00000 + 2.64575i) q^{56} +(3.74166 + 3.74166i) q^{57} +(-4.65489 + 10.3117i) q^{58} +5.29150 q^{59} +6.00000 q^{61} +(-3.07892 + 6.82058i) q^{62} +(-1.41421 - 1.41421i) q^{63} +(-6.61438 + 4.50000i) q^{64} +(7.00000 - 2.64575i) q^{66} +(8.48528 - 8.48528i) q^{67} +4.00000i q^{69} +(2.70318 - 0.832353i) q^{72} +(7.48331 - 7.48331i) q^{73} +(2.64575 + 7.00000i) q^{74} +(-7.00000 - 7.93725i) q^{76} +(7.48331 + 7.48331i) q^{77} +(6.82058 + 3.07892i) q^{78} +5.29150 q^{79} -1.00000 q^{81} +(2.57794 + 1.16372i) q^{82} +(8.48528 + 8.48528i) q^{83} +(2.64575 + 3.00000i) q^{84} +(-4.00000 - 10.5830i) q^{86} +(-5.65685 + 5.65685i) q^{87} +(-14.3039 + 4.40440i) q^{88} +6.00000i q^{89} +10.5830i q^{91} +(0.500983 - 7.98430i) q^{92} +(-3.74166 + 3.74166i) q^{93} +(-5.50000 + 1.32288i) q^{96} +(-7.48331 - 7.48331i) q^{97} +(1.74558 - 3.86690i) q^{98} +5.29150 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{6} - 4 q^{16} + 16 q^{21} - 56 q^{26} - 12 q^{36} - 16 q^{41} + 16 q^{46} + 40 q^{56} + 48 q^{61} + 56 q^{66} - 56 q^{76} - 8 q^{81} - 32 q^{86} - 44 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.28897 − 0.581861i −0.911438 − 0.411438i
\(3\) −0.707107 − 0.707107i −0.408248 − 0.408248i
\(4\) 1.32288 + 1.50000i 0.661438 + 0.750000i
\(5\) 0 0
\(6\) 0.500000 + 1.32288i 0.204124 + 0.540062i
\(7\) −1.41421 + 1.41421i −0.534522 + 0.534522i −0.921915 − 0.387392i \(-0.873376\pi\)
0.387392 + 0.921915i \(0.373376\pi\)
\(8\) −0.832353 − 2.70318i −0.294281 − 0.955719i
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) − 5.29150i − 1.59545i −0.603023 − 0.797724i \(-0.706037\pi\)
0.603023 − 0.797724i \(-0.293963\pi\)
\(12\) 0.125246 − 1.99607i 0.0361554 − 0.576217i
\(13\) 3.74166 − 3.74166i 1.03775 − 1.03775i 0.0384901 − 0.999259i \(-0.487745\pi\)
0.999259 − 0.0384901i \(-0.0122548\pi\)
\(14\) 2.64575 − 1.00000i 0.707107 − 0.267261i
\(15\) 0 0
\(16\) −0.500000 + 3.96863i −0.125000 + 0.992157i
\(17\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(18\) 0.581861 − 1.28897i 0.137146 − 0.303813i
\(19\) −5.29150 −1.21395 −0.606977 − 0.794719i \(-0.707618\pi\)
−0.606977 + 0.794719i \(0.707618\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) −3.07892 + 6.82058i −0.656428 + 1.45415i
\(23\) −2.82843 − 2.82843i −0.589768 − 0.589768i 0.347801 − 0.937568i \(-0.386929\pi\)
−0.937568 + 0.347801i \(0.886929\pi\)
\(24\) −1.32288 + 2.50000i −0.270031 + 0.510310i
\(25\) 0 0
\(26\) −7.00000 + 2.64575i −1.37281 + 0.518875i
\(27\) 0.707107 − 0.707107i 0.136083 − 0.136083i
\(28\) −3.99215 − 0.250492i −0.754445 − 0.0473385i
\(29\) − 8.00000i − 1.48556i −0.669534 − 0.742781i \(-0.733506\pi\)
0.669534 − 0.742781i \(-0.266494\pi\)
\(30\) 0 0
\(31\) − 5.29150i − 0.950382i −0.879883 − 0.475191i \(-0.842379\pi\)
0.879883 − 0.475191i \(-0.157621\pi\)
\(32\) 2.95367 − 4.82450i 0.522141 − 0.852859i
\(33\) −3.74166 + 3.74166i −0.651339 + 0.651339i
\(34\) 0 0
\(35\) 0 0
\(36\) −1.50000 + 1.32288i −0.250000 + 0.220479i
\(37\) −3.74166 − 3.74166i −0.615125 − 0.615125i 0.329152 − 0.944277i \(-0.393237\pi\)
−0.944277 + 0.329152i \(0.893237\pi\)
\(38\) 6.82058 + 3.07892i 1.10644 + 0.499467i
\(39\) −5.29150 −0.847319
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) −2.57794 − 1.16372i −0.397784 − 0.179566i
\(43\) 5.65685 + 5.65685i 0.862662 + 0.862662i 0.991647 − 0.128984i \(-0.0411717\pi\)
−0.128984 + 0.991647i \(0.541172\pi\)
\(44\) 7.93725 − 7.00000i 1.19659 − 1.05529i
\(45\) 0 0
\(46\) 2.00000 + 5.29150i 0.294884 + 0.780189i
\(47\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(48\) 3.15980 − 2.45269i 0.456077 − 0.354015i
\(49\) 3.00000i 0.428571i
\(50\) 0 0
\(51\) 0 0
\(52\) 10.5622 + 0.662739i 1.46472 + 0.0919053i
\(53\) −7.48331 + 7.48331i −1.02791 + 1.02791i −0.0283132 + 0.999599i \(0.509014\pi\)
−0.999599 + 0.0283132i \(0.990986\pi\)
\(54\) −1.32288 + 0.500000i −0.180021 + 0.0680414i
\(55\) 0 0
\(56\) 5.00000 + 2.64575i 0.668153 + 0.353553i
\(57\) 3.74166 + 3.74166i 0.495595 + 0.495595i
\(58\) −4.65489 + 10.3117i −0.611217 + 1.35400i
\(59\) 5.29150 0.688895 0.344447 − 0.938806i \(-0.388066\pi\)
0.344447 + 0.938806i \(0.388066\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −3.07892 + 6.82058i −0.391023 + 0.866214i
\(63\) −1.41421 − 1.41421i −0.178174 − 0.178174i
\(64\) −6.61438 + 4.50000i −0.826797 + 0.562500i
\(65\) 0 0
\(66\) 7.00000 − 2.64575i 0.861640 − 0.325669i
\(67\) 8.48528 − 8.48528i 1.03664 − 1.03664i 0.0373395 − 0.999303i \(-0.488112\pi\)
0.999303 − 0.0373395i \(-0.0118883\pi\)
\(68\) 0 0
\(69\) 4.00000i 0.481543i
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 2.70318 − 0.832353i 0.318573 − 0.0980937i
\(73\) 7.48331 − 7.48331i 0.875856 − 0.875856i −0.117247 − 0.993103i \(-0.537407\pi\)
0.993103 + 0.117247i \(0.0374069\pi\)
\(74\) 2.64575 + 7.00000i 0.307562 + 0.813733i
\(75\) 0 0
\(76\) −7.00000 − 7.93725i −0.802955 − 0.910465i
\(77\) 7.48331 + 7.48331i 0.852803 + 0.852803i
\(78\) 6.82058 + 3.07892i 0.772278 + 0.348619i
\(79\) 5.29150 0.595341 0.297670 − 0.954669i \(-0.403790\pi\)
0.297670 + 0.954669i \(0.403790\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 2.57794 + 1.16372i 0.284685 + 0.128512i
\(83\) 8.48528 + 8.48528i 0.931381 + 0.931381i 0.997792 − 0.0664117i \(-0.0211551\pi\)
−0.0664117 + 0.997792i \(0.521155\pi\)
\(84\) 2.64575 + 3.00000i 0.288675 + 0.327327i
\(85\) 0 0
\(86\) −4.00000 − 10.5830i −0.431331 − 1.14119i
\(87\) −5.65685 + 5.65685i −0.606478 + 0.606478i
\(88\) −14.3039 + 4.40440i −1.52480 + 0.469510i
\(89\) 6.00000i 0.635999i 0.948091 + 0.317999i \(0.103011\pi\)
−0.948091 + 0.317999i \(0.896989\pi\)
\(90\) 0 0
\(91\) 10.5830i 1.10940i
\(92\) 0.500983 − 7.98430i 0.0522311 − 0.832421i
\(93\) −3.74166 + 3.74166i −0.387992 + 0.387992i
\(94\) 0 0
\(95\) 0 0
\(96\) −5.50000 + 1.32288i −0.561341 + 0.135015i
\(97\) −7.48331 − 7.48331i −0.759815 − 0.759815i 0.216473 − 0.976289i \(-0.430545\pi\)
−0.976289 + 0.216473i \(0.930545\pi\)
\(98\) 1.74558 − 3.86690i 0.176330 − 0.390616i
\(99\) 5.29150 0.531816
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 300.2.j.b.7.1 ✓ 8
3.2 odd 2 900.2.k.h.307.4 8
4.3 odd 2 inner 300.2.j.b.7.2 yes 8
5.2 odd 4 inner 300.2.j.b.43.3 yes 8
5.3 odd 4 inner 300.2.j.b.43.2 yes 8
5.4 even 2 inner 300.2.j.b.7.4 yes 8
12.11 even 2 900.2.k.h.307.3 8
15.2 even 4 900.2.k.h.343.2 8
15.8 even 4 900.2.k.h.343.3 8
15.14 odd 2 900.2.k.h.307.1 8
20.3 even 4 inner 300.2.j.b.43.1 yes 8
20.7 even 4 inner 300.2.j.b.43.4 yes 8
20.19 odd 2 inner 300.2.j.b.7.3 yes 8
60.23 odd 4 900.2.k.h.343.4 8
60.47 odd 4 900.2.k.h.343.1 8
60.59 even 2 900.2.k.h.307.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.2.j.b.7.1 ✓ 8 1.1 even 1 trivial
300.2.j.b.7.2 yes 8 4.3 odd 2 inner
300.2.j.b.7.3 yes 8 20.19 odd 2 inner
300.2.j.b.7.4 yes 8 5.4 even 2 inner
300.2.j.b.43.1 yes 8 20.3 even 4 inner
300.2.j.b.43.2 yes 8 5.3 odd 4 inner
300.2.j.b.43.3 yes 8 5.2 odd 4 inner
300.2.j.b.43.4 yes 8 20.7 even 4 inner
900.2.k.h.307.1 8 15.14 odd 2
900.2.k.h.307.2 8 60.59 even 2
900.2.k.h.307.3 8 12.11 even 2
900.2.k.h.307.4 8 3.2 odd 2
900.2.k.h.343.1 8 60.47 odd 4
900.2.k.h.343.2 8 15.2 even 4
900.2.k.h.343.3 8 15.8 even 4
900.2.k.h.343.4 8 60.23 odd 4