LMFDB - Embedded newform 3024.2.b.j.1567.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3024,2,Mod(1567,3024)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3024.1567");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3024 = 2^{4} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3024.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$24.1467615712$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1567.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3024.1567
Dual form			3024.2.b.j.1567.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{5} +(0.500000 + 2.59808i) q^{7} +O(q^{10})$	$q-1.73205i q^{5} +(0.500000 + 2.59808i) q^{7} +5.19615i q^{11} -3.46410i q^{13} -3.46410i q^{17} -8.00000 q^{19} -6.92820i q^{23} +2.00000 q^{25} +6.00000 q^{29} +5.00000 q^{31} +(4.50000 - 0.866025i) q^{35} +4.00000 q^{37} -6.92820i q^{41} +3.46410i q^{43} +6.00000 q^{47} +(-6.50000 + 2.59808i) q^{49} +3.00000 q^{53} +9.00000 q^{55} +12.0000 q^{59} +13.8564i q^{61} -6.00000 q^{65} -3.46410i q^{67} +3.46410i q^{71} +1.73205i q^{73} +(-13.5000 + 2.59808i) q^{77} +3.46410i q^{79} +15.0000 q^{83} -6.00000 q^{85} -10.3923i q^{89} +(9.00000 - 1.73205i) q^{91} +13.8564i q^{95} -8.66025i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{7}+O(q^{10}) $ 2 * q + q^7	$ 2 q + q^{7} - 16 q^{19} + 4 q^{25} + 12 q^{29} + 10 q^{31} + 9 q^{35} + 8 q^{37} + 12 q^{47} - 13 q^{49} + 6 q^{53} + 18 q^{55} + 24 q^{59} - 12 q^{65} - 27 q^{77} + 30 q^{83} - 12 q^{85} + 18 q^{91}+O(q^{100}) $ 2 * q + q^7 - 16 * q^19 + 4 * q^25 + 12 * q^29 + 10 * q^31 + 9 * q^35 + 8 * q^37 + 12 * q^47 - 13 * q^49 + 6 * q^53 + 18 * q^55 + 24 * q^59 - 12 * q^65 - 27 * q^77 + 30 * q^83 - 12 * q^85 + 18 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1135$	$2593$
$\chi(n)$	$1$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	1.73205i		−	0.774597i	−0.921954	−	0.387298i	$-0.873408\pi$
$5$		−	1.73205i		−	0.774597i	0.921954	−	0.387298i	$-0.126592\pi$
$6$		0			0
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			5.19615i			1.56670i	0.621582	+	0.783349i	$0.286490\pi$
$11$			5.19615i			1.56670i	−0.621582	+	0.783349i	$0.713510\pi$
$12$		0			0
$13$		−	3.46410i		−	0.960769i	−0.877058	−	0.480384i	$-0.840497\pi$
$13$		−	3.46410i		−	0.960769i	0.877058	−	0.480384i	$-0.159503\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	3.46410i		−	0.840168i	−0.907485	−	0.420084i	$-0.862001\pi$
$17$		−	3.46410i		−	0.840168i	0.907485	−	0.420084i	$-0.137999\pi$
$18$		0			0
$19$	−8.00000			−1.83533			−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000			−1.83533			−0.917663	+	0.397360i	$0.869927\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	6.92820i		−	1.44463i	−0.691564	−	0.722315i	$-0.743078\pi$
$23$		−	6.92820i		−	1.44463i	0.691564	−	0.722315i	$-0.256922\pi$
$24$		0			0
$25$	2.00000			0.400000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	5.00000			0.898027			0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000			0.898027			0.449013	+	0.893525i	$0.351776\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.50000	−	0.866025i	0.760639	−	0.146385i
$36$		0			0
$37$	4.00000			0.657596			0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000			0.657596			0.328798	+	0.944400i	$0.393356\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	6.92820i		−	1.08200i	−0.841021	−	0.541002i	$-0.818045\pi$
$41$		−	6.92820i		−	1.08200i	0.841021	−	0.541002i	$-0.181955\pi$
$42$		0			0
$43$			3.46410i			0.528271i	0.964486	+	0.264135i	$0.0850865\pi$
$43$			3.46410i			0.528271i	−0.964486	+	0.264135i	$0.914913\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.00000			0.875190			0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000			0.875190			0.437595	+	0.899172i	$0.355830\pi$
$48$		0			0
$49$	−6.50000	+	2.59808i	−0.928571	+	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.00000			0.412082			0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000			0.412082			0.206041	+	0.978543i	$0.433942\pi$
$54$		0			0
$55$	9.00000			1.21356
$56$		0			0
$57$		0			0
$58$		0			0
$59$	12.0000			1.56227			0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000			1.56227			0.781133	+	0.624364i	$0.214642\pi$
$60$		0			0
$61$			13.8564i			1.77413i	0.461644	+	0.887066i	$0.347260\pi$
$61$			13.8564i			1.77413i	−0.461644	+	0.887066i	$0.652740\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.00000			−0.744208
$66$		0			0
$67$		−	3.46410i		−	0.423207i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$		−	3.46410i		−	0.423207i	0.977356	−	0.211604i	$-0.0678686\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			3.46410i			0.411113i	0.978645	+	0.205557i	$0.0659005\pi$
$71$			3.46410i			0.411113i	−0.978645	+	0.205557i	$0.934100\pi$
$72$		0			0
$73$			1.73205i			0.202721i	0.994850	+	0.101361i	$0.0323196\pi$
$73$			1.73205i			0.202721i	−0.994850	+	0.101361i	$0.967680\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−13.5000	+	2.59808i	−1.53847	+	0.296078i
$78$		0			0
$79$			3.46410i			0.389742i	0.980829	+	0.194871i	$0.0624288\pi$
$79$			3.46410i			0.389742i	−0.980829	+	0.194871i	$0.937571\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	15.0000			1.64646			0.823232	−	0.567705i	$-0.192169\pi$
$83$	15.0000			1.64646			0.823232	+	0.567705i	$0.192169\pi$
$84$		0			0
$85$	−6.00000			−0.650791
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	10.3923i		−	1.10158i	−0.834643	−	0.550791i	$-0.814326\pi$
$89$		−	10.3923i		−	1.10158i	0.834643	−	0.550791i	$-0.185674\pi$
$90$		0			0
$91$	9.00000	−	1.73205i	0.943456	−	0.181568i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			13.8564i			1.42164i
$96$		0			0
$97$		−	8.66025i		−	0.879316i	−0.898165	−	0.439658i	$-0.855100\pi$
$97$		−	8.66025i		−	0.879316i	0.898165	−	0.439658i	$-0.144900\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			8.66025i			0.861727i	0.902417	+	0.430864i	$0.141791\pi$
$101$			8.66025i			0.861727i	−0.902417	+	0.430864i	$0.858209\pi$
$102$		0			0
$103$	−16.0000			−1.57653			−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000			−1.57653			−0.788263	+	0.615338i	$0.789020\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			8.66025i			0.837218i	0.908166	+	0.418609i	$0.137482\pi$
$107$			8.66025i			0.837218i	−0.908166	+	0.418609i	$0.862518\pi$
$108$		0			0
$109$	20.0000			1.91565			0.957826	−	0.287348i	$-0.0927736\pi$
$109$	20.0000			1.91565			0.957826	+	0.287348i	$0.0927736\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	12.0000			1.12887			0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000			1.12887			0.564433	+	0.825479i	$0.309095\pi$
$114$		0			0
$115$	−12.0000			−1.11901
$116$		0			0
$117$		0			0
$118$		0			0
$119$	9.00000	−	1.73205i	0.825029	−	0.158777i
$120$		0			0
$121$	−16.0000			−1.45455
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	12.1244i		−	1.08444i
$126$		0			0
$127$		−	12.1244i		−	1.07586i	−0.842989	−	0.537931i	$-0.819206\pi$
$127$		−	12.1244i		−	1.07586i	0.842989	−	0.537931i	$-0.180794\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	15.0000			1.31056			0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000			1.31056			0.655278	+	0.755388i	$0.272551\pi$
$132$		0			0
$133$	−4.00000	−	20.7846i	−0.346844	−	1.80225i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.00000			−0.512615			−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000			−0.512615			−0.256307	+	0.966595i	$0.582506\pi$
$138$		0			0
$139$	8.00000			0.678551			0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000			0.678551			0.339276	+	0.940687i	$0.389818\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	18.0000			1.50524
$144$		0			0
$145$		−	10.3923i		−	0.863034i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	21.0000			1.72039			0.860194	−	0.509968i	$-0.170343\pi$
$149$	21.0000			1.72039			0.860194	+	0.509968i	$0.170343\pi$
$150$		0			0
$151$		−	12.1244i		−	0.986666i	−0.869841	−	0.493333i	$-0.835778\pi$
$151$		−	12.1244i		−	0.986666i	0.869841	−	0.493333i	$-0.164222\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	8.66025i		−	0.695608i
$156$		0			0
$157$		−	6.92820i		−	0.552931i	−0.961024	−	0.276465i	$-0.910837\pi$
$157$		−	6.92820i		−	0.552931i	0.961024	−	0.276465i	$-0.0891631\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	18.0000	−	3.46410i	1.41860	−	0.273009i
$162$		0			0
$163$		−	10.3923i		−	0.813988i	−0.913431	−	0.406994i	$-0.866577\pi$
$163$		−	10.3923i		−	0.813988i	0.913431	−	0.406994i	$-0.133423\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.00000			−0.464294			−0.232147	−	0.972681i	$-0.574575\pi$
$167$	−6.00000			−0.464294			−0.232147	+	0.972681i	$0.574575\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	1.73205i		−	0.131685i	−0.997830	−	0.0658427i	$-0.979026\pi$
$173$		−	1.73205i		−	0.131685i	0.997830	−	0.0658427i	$-0.0209736\pi$
$174$		0			0
$175$	1.00000	+	5.19615i	0.0755929	+	0.392792i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	19.0526i		−	1.42406i	−0.702152	−	0.712028i	$-0.747777\pi$
$179$		−	19.0526i		−	1.42406i	0.702152	−	0.712028i	$-0.252223\pi$
$180$		0			0
$181$			20.7846i			1.54491i	0.635071	+	0.772454i	$0.280971\pi$
$181$			20.7846i			1.54491i	−0.635071	+	0.772454i	$0.719029\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	6.92820i		−	0.509372i
$186$		0			0
$187$	18.0000			1.31629
$188$		0			0
$189$		0			0
$190$		0			0
$191$			10.3923i			0.751961i	0.926628	+	0.375980i	$0.122694\pi$
$191$			10.3923i			0.751961i	−0.926628	+	0.375980i	$0.877306\pi$
$192$		0			0
$193$	−5.00000			−0.359908			−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000			−0.359908			−0.179954	+	0.983675i	$0.557595\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−27.0000			−1.92367			−0.961835	−	0.273629i	$-0.911776\pi$
$197$	−27.0000			−1.92367			−0.961835	+	0.273629i	$0.911776\pi$
$198$		0			0
$199$	−11.0000			−0.779769			−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000			−0.779769			−0.389885	+	0.920864i	$0.627485\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.00000	+	15.5885i	0.210559	+	1.09410i
$204$		0			0
$205$	−12.0000			−0.838116
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	41.5692i		−	2.87540i
$210$		0			0
$211$			13.8564i			0.953914i	0.878927	+	0.476957i	$0.158260\pi$
$211$			13.8564i			0.953914i	−0.878927	+	0.476957i	$0.841740\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	6.00000			0.409197
$216$		0			0
$217$	2.50000	+	12.9904i	0.169711	+	0.881845i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−12.0000			−0.807207
$222$		0			0
$223$	−8.00000			−0.535720			−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000			−0.535720			−0.267860	+	0.963458i	$0.586316\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	12.0000			0.796468			0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000			0.796468			0.398234	+	0.917284i	$0.369623\pi$
$228$		0			0
$229$			3.46410i			0.228914i	0.993428	+	0.114457i	$0.0365129\pi$
$229$			3.46410i			0.228914i	−0.993428	+	0.114457i	$0.963487\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$		0			0
$235$		−	10.3923i		−	0.677919i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	6.92820i		−	0.448148i	−0.974572	−	0.224074i	$-0.928064\pi$
$239$		−	6.92820i		−	0.448148i	0.974572	−	0.224074i	$-0.0719358\pi$
$240$		0			0
$241$			20.7846i			1.33885i	0.742878	+	0.669427i	$0.233460\pi$
$241$			20.7846i			1.33885i	−0.742878	+	0.669427i	$0.766540\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.50000	+	11.2583i	0.287494	+	0.719268i
$246$		0			0
$247$			27.7128i			1.76332i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$	36.0000			2.26330
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	24.2487i		−	1.51259i	−0.654229	−	0.756297i	$-0.727007\pi$
$257$		−	24.2487i		−	1.51259i	0.654229	−	0.756297i	$-0.272993\pi$
$258$		0			0
$259$	2.00000	+	10.3923i	0.124274	+	0.645746i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	13.8564i		−	0.854423i	−0.904152	−	0.427211i	$-0.859496\pi$
$263$		−	13.8564i		−	0.854423i	0.904152	−	0.427211i	$-0.140504\pi$
$264$		0			0
$265$		−	5.19615i		−	0.319197i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			13.8564i			0.844840i	0.906400	+	0.422420i	$0.138819\pi$
$269$			13.8564i			0.844840i	−0.906400	+	0.422420i	$0.861181\pi$
$270$		0			0
$271$	17.0000			1.03268			0.516338	−	0.856385i	$-0.327295\pi$
$271$	17.0000			1.03268			0.516338	+	0.856385i	$0.327295\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			10.3923i			0.626680i
$276$		0			0
$277$	−8.00000			−0.480673			−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000			−0.480673			−0.240337	+	0.970690i	$0.577258\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$	4.00000			0.237775			0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000			0.237775			0.118888	+	0.992908i	$0.462067\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	18.0000	−	3.46410i	1.06251	−	0.204479i
$288$		0			0
$289$	5.00000			0.294118
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		−	20.7846i		−	1.21013i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−24.0000			−1.38796
$300$		0			0
$301$	−9.00000	+	1.73205i	−0.518751	+	0.0998337i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	24.0000			1.37424
$306$		0			0
$307$	−14.0000			−0.799022			−0.399511	−	0.916728i	$-0.630820\pi$
$307$	−14.0000			−0.799022			−0.399511	+	0.916728i	$0.630820\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		−	15.5885i		−	0.881112i	−0.897725	−	0.440556i	$-0.854781\pi$
$313$		−	15.5885i		−	0.881112i	0.897725	−	0.440556i	$-0.145219\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.00000			0.168497			0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000			0.168497			0.0842484	+	0.996445i	$0.473151\pi$
$318$		0			0
$319$			31.1769i			1.74557i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			27.7128i			1.54198i
$324$		0			0
$325$		−	6.92820i		−	0.384308i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.00000	+	15.5885i	0.165395	+	0.859419i
$330$		0			0
$331$			24.2487i			1.33283i	0.745581	+	0.666415i	$0.232172\pi$
$331$			24.2487i			1.33283i	−0.745581	+	0.666415i	$0.767828\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6.00000			−0.327815
$336$		0			0
$337$	−10.0000			−0.544735			−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000			−0.544735			−0.272367	+	0.962193i	$0.587807\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			25.9808i			1.40694i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	25.9808i		−	1.39472i	−0.716721	−	0.697360i	$-0.754358\pi$
$347$		−	25.9808i		−	1.39472i	0.716721	−	0.697360i	$-0.245642\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			6.92820i			0.368751i	0.982856	+	0.184376i	$0.0590263\pi$
$353$			6.92820i			0.368751i	−0.982856	+	0.184376i	$0.940974\pi$
$354$		0			0
$355$	6.00000			0.318447
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	3.46410i		−	0.182828i	−0.995813	−	0.0914141i	$-0.970861\pi$
$359$		−	3.46410i		−	0.182828i	0.995813	−	0.0914141i	$-0.0291387\pi$
$360$		0			0
$361$	45.0000			2.36842
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3.00000			0.157027
$366$		0			0
$367$	13.0000			0.678594			0.339297	−	0.940679i	$-0.389811\pi$
$367$	13.0000			0.678594			0.339297	+	0.940679i	$0.389811\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.50000	+	7.79423i	0.0778761	+	0.404656i
$372$		0			0
$373$	−22.0000			−1.13912			−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000			−1.13912			−0.569558	+	0.821951i	$0.692886\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	20.7846i		−	1.07046i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	18.0000			0.919757			0.459879	−	0.887982i	$-0.347893\pi$
$383$	18.0000			0.919757			0.459879	+	0.887982i	$0.347893\pi$
$384$		0			0
$385$	4.50000	+	23.3827i	0.229341	+	1.19169i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.00000			0.152106			0.0760530	−	0.997104i	$-0.475768\pi$
$389$	3.00000			0.152106			0.0760530	+	0.997104i	$0.475768\pi$
$390$		0			0
$391$	−24.0000			−1.21373
$392$		0			0
$393$		0			0
$394$		0			0
$395$	6.00000			0.301893
$396$		0			0
$397$		−	3.46410i		−	0.173858i	−0.996214	−	0.0869291i	$-0.972295\pi$
$397$		−	3.46410i		−	0.173858i	0.996214	−	0.0869291i	$-0.0277054\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−24.0000			−1.19850			−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000			−1.19850			−0.599251	+	0.800561i	$0.704535\pi$
$402$		0			0
$403$		−	17.3205i		−	0.862796i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			20.7846i			1.03025i
$408$		0			0
$409$		−	15.5885i		−	0.770800i	−0.922750	−	0.385400i	$-0.874064\pi$
$409$		−	15.5885i		−	0.770800i	0.922750	−	0.385400i	$-0.125936\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	6.00000	+	31.1769i	0.295241	+	1.53412i
$414$		0			0
$415$		−	25.9808i		−	1.27535i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−36.0000			−1.75872			−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000			−1.75872			−0.879358	+	0.476162i	$0.842028\pi$
$420$		0			0
$421$	−20.0000			−0.974740			−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000			−0.974740			−0.487370	+	0.873195i	$0.662044\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	6.92820i		−	0.336067i
$426$		0			0
$427$	−36.0000	+	6.92820i	−1.74216	+	0.335279i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			17.3205i			0.834300i	0.908838	+	0.417150i	$0.136971\pi$
$431$			17.3205i			0.834300i	−0.908838	+	0.417150i	$0.863029\pi$
$432$		0			0
$433$		−	22.5167i		−	1.08208i	−0.840996	−	0.541041i	$-0.818030\pi$
$433$		−	22.5167i		−	1.08208i	0.840996	−	0.541041i	$-0.181970\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			55.4256i			2.65137i
$438$		0			0
$439$	7.00000			0.334092			0.167046	−	0.985949i	$-0.446577\pi$
$439$	7.00000			0.334092			0.167046	+	0.985949i	$0.446577\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	17.3205i		−	0.822922i	−0.911427	−	0.411461i	$-0.865019\pi$
$443$		−	17.3205i		−	0.822922i	0.911427	−	0.411461i	$-0.134981\pi$
$444$		0			0
$445$	−18.0000			−0.853282
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−24.0000			−1.13263			−0.566315	−	0.824189i	$-0.691631\pi$
$449$	−24.0000			−1.13263			−0.566315	+	0.824189i	$0.691631\pi$
$450$		0			0
$451$	36.0000			1.69517
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.00000	−	15.5885i	−0.140642	−	0.730798i
$456$		0			0
$457$	17.0000			0.795226			0.397613	−	0.917553i	$-0.369839\pi$
$457$	17.0000			0.795226			0.397613	+	0.917553i	$0.369839\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	5.19615i		−	0.242009i	−0.992652	−	0.121004i	$-0.961388\pi$
$461$		−	5.19615i		−	0.242009i	0.992652	−	0.121004i	$-0.0386115\pi$
$462$		0			0
$463$			12.1244i			0.563467i	0.959493	+	0.281733i	$0.0909093\pi$
$463$			12.1244i			0.563467i	−0.959493	+	0.281733i	$0.909091\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−27.0000			−1.24941			−0.624705	−	0.780860i	$-0.714781\pi$
$467$	−27.0000			−1.24941			−0.624705	+	0.780860i	$0.714781\pi$
$468$		0			0
$469$	9.00000	−	1.73205i	0.415581	−	0.0799787i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−18.0000			−0.827641
$474$		0			0
$475$	−16.0000			−0.734130
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−18.0000			−0.822441			−0.411220	−	0.911536i	$-0.634897\pi$
$479$	−18.0000			−0.822441			−0.411220	+	0.911536i	$0.634897\pi$
$480$		0			0
$481$		−	13.8564i		−	0.631798i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−15.0000			−0.681115
$486$		0			0
$487$			3.46410i			0.156973i	0.996915	+	0.0784867i	$0.0250088\pi$
$487$			3.46410i			0.156973i	−0.996915	+	0.0784867i	$0.974991\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	22.5167i		−	1.01616i	−0.861309	−	0.508081i	$-0.830355\pi$
$491$		−	22.5167i		−	1.01616i	0.861309	−	0.508081i	$-0.169645\pi$
$492$		0			0
$493$		−	20.7846i		−	0.936092i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9.00000	+	1.73205i	−0.403705	+	0.0776931i
$498$		0			0
$499$		−	31.1769i		−	1.39567i	−0.716258	−	0.697835i	$-0.754147\pi$
$499$		−	31.1769i		−	1.39567i	0.716258	−	0.697835i	$-0.245853\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−12.0000			−0.535054			−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000			−0.535054			−0.267527	+	0.963550i	$0.586206\pi$
$504$		0			0
$505$	15.0000			0.667491
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	8.66025i		−	0.383859i	−0.981409	−	0.191930i	$-0.938526\pi$
$509$		−	8.66025i		−	0.383859i	0.981409	−	0.191930i	$-0.0614745\pi$
$510$		0			0
$511$	−4.50000	+	0.866025i	−0.199068	+	0.0383107i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			27.7128i			1.22117i
$516$		0			0
$517$			31.1769i			1.37116i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			31.1769i			1.36589i	0.730472	+	0.682943i	$0.239300\pi$
$521$			31.1769i			1.36589i	−0.730472	+	0.682943i	$0.760700\pi$
$522$		0			0
$523$	2.00000			0.0874539			0.0437269	−	0.999044i	$-0.486077\pi$
$523$	2.00000			0.0874539			0.0437269	+	0.999044i	$0.486077\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	17.3205i		−	0.754493i
$528$		0			0
$529$	−25.0000			−1.08696
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−24.0000			−1.03956
$534$		0			0
$535$	15.0000			0.648507
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−13.5000	−	33.7750i	−0.581486	−	1.45479i
$540$		0			0
$541$	16.0000			0.687894			0.343947	−	0.938989i	$-0.388236\pi$
$541$	16.0000			0.687894			0.343947	+	0.938989i	$0.388236\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	34.6410i		−	1.48386i
$546$		0			0
$547$			24.2487i			1.03680i	0.855138	+	0.518400i	$0.173472\pi$
$547$			24.2487i			1.03680i	−0.855138	+	0.518400i	$0.826528\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−48.0000			−2.04487
$552$		0			0
$553$	−9.00000	+	1.73205i	−0.382719	+	0.0736543i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	21.0000			0.889799			0.444899	−	0.895581i	$-0.353239\pi$
$557$	21.0000			0.889799			0.444899	+	0.895581i	$0.353239\pi$
$558$		0			0
$559$	12.0000			0.507546
$560$		0			0
$561$		0			0
$562$		0			0
$563$	21.0000			0.885044			0.442522	−	0.896758i	$-0.354084\pi$
$563$	21.0000			0.885044			0.442522	+	0.896758i	$0.354084\pi$
$564$		0			0
$565$		−	20.7846i		−	0.874415i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	36.0000			1.50920			0.754599	−	0.656186i	$-0.227831\pi$
$569$	36.0000			1.50920			0.754599	+	0.656186i	$0.227831\pi$
$570$		0			0
$571$		−	10.3923i		−	0.434904i	−0.976071	−	0.217452i	$-0.930225\pi$
$571$		−	10.3923i		−	0.434904i	0.976071	−	0.217452i	$-0.0697746\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	13.8564i		−	0.577852i
$576$		0			0
$577$			34.6410i			1.44212i	0.692870	+	0.721062i	$0.256346\pi$
$577$			34.6410i			1.44212i	−0.692870	+	0.721062i	$0.743654\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	7.50000	+	38.9711i	0.311152	+	1.61680i
$582$		0			0
$583$			15.5885i			0.645608i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−39.0000			−1.60970			−0.804851	−	0.593477i	$-0.797755\pi$
$587$	−39.0000			−1.60970			−0.804851	+	0.593477i	$0.797755\pi$
$588$		0			0
$589$	−40.0000			−1.64817
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	31.1769i		−	1.28028i	−0.768257	−	0.640141i	$-0.778876\pi$
$593$		−	31.1769i		−	1.28028i	0.768257	−	0.640141i	$-0.221124\pi$
$594$		0			0
$595$	−3.00000	−	15.5885i	−0.122988	−	0.639064i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			38.1051i			1.55693i	0.627686	+	0.778466i	$0.284002\pi$
$599$			38.1051i			1.55693i	−0.627686	+	0.778466i	$0.715998\pi$
$600$		0			0
$601$			19.0526i			0.777170i	0.921413	+	0.388585i	$0.127036\pi$
$601$			19.0526i			0.777170i	−0.921413	+	0.388585i	$0.872964\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			27.7128i			1.12669i
$606$		0			0
$607$	28.0000			1.13648			0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000			1.13648			0.568242	+	0.822861i	$0.307624\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	20.7846i		−	0.840855i
$612$		0			0
$613$	2.00000			0.0807792			0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000			0.0807792			0.0403896	+	0.999184i	$0.487140\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	36.0000			1.44931			0.724653	−	0.689114i	$-0.242000\pi$
$617$	36.0000			1.44931			0.724653	+	0.689114i	$0.242000\pi$
$618$		0			0
$619$	−10.0000			−0.401934			−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000			−0.401934			−0.200967	+	0.979598i	$0.564408\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	27.0000	−	5.19615i	1.08173	−	0.208179i
$624$		0			0
$625$	−11.0000			−0.440000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	13.8564i		−	0.552491i
$630$		0			0
$631$		−	5.19615i		−	0.206856i	−0.994637	−	0.103428i	$-0.967019\pi$
$631$		−	5.19615i		−	0.206856i	0.994637	−	0.103428i	$-0.0329811\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−21.0000			−0.833360
$636$		0			0
$637$	9.00000	+	22.5167i	0.356593	+	0.892143i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	24.0000			0.947943			0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000			0.947943			0.473972	+	0.880540i	$0.342820\pi$
$642$		0			0
$643$	28.0000			1.10421			0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000			1.10421			0.552106	+	0.833774i	$0.313824\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−36.0000			−1.41531			−0.707653	−	0.706560i	$-0.750246\pi$
$647$	−36.0000			−1.41531			−0.707653	+	0.706560i	$0.750246\pi$
$648$		0			0
$649$			62.3538i			2.44760i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−27.0000			−1.05659			−0.528296	−	0.849060i	$-0.677169\pi$
$653$	−27.0000			−1.05659			−0.528296	+	0.849060i	$0.677169\pi$
$654$		0			0
$655$		−	25.9808i		−	1.01515i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	19.0526i		−	0.742182i	−0.928596	−	0.371091i	$-0.878984\pi$
$659$		−	19.0526i		−	0.742182i	0.928596	−	0.371091i	$-0.121016\pi$
$660$		0			0
$661$		−	13.8564i		−	0.538952i	−0.963007	−	0.269476i	$-0.913150\pi$
$661$		−	13.8564i		−	0.538952i	0.963007	−	0.269476i	$-0.0868504\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−36.0000	+	6.92820i	−1.39602	+	0.268664i
$666$		0			0
$667$		−	41.5692i		−	1.60957i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−72.0000			−2.77953
$672$		0			0
$673$	1.00000			0.0385472			0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000			0.0385472			0.0192736	+	0.999814i	$0.493865\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			41.5692i			1.59763i	0.601574	+	0.798817i	$0.294541\pi$
$677$			41.5692i			1.59763i	−0.601574	+	0.798817i	$0.705459\pi$
$678$		0			0
$679$	22.5000	−	4.33013i	0.863471	−	0.166175i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			17.3205i			0.662751i	0.943499	+	0.331375i	$0.107513\pi$
$683$			17.3205i			0.662751i	−0.943499	+	0.331375i	$0.892487\pi$
$684$		0			0
$685$			10.3923i			0.397070i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	10.3923i		−	0.395915i
$690$		0			0
$691$	−40.0000			−1.52167			−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000			−1.52167			−0.760836	+	0.648944i	$0.775211\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	13.8564i		−	0.525603i
$696$		0			0
$697$	−24.0000			−0.909065
$698$		0			0
$699$		0			0
$700$		0			0
$701$	39.0000			1.47301			0.736505	−	0.676432i	$-0.236475\pi$
$701$	39.0000			1.47301			0.736505	+	0.676432i	$0.236475\pi$
$702$		0			0
$703$	−32.0000			−1.20690
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−22.5000	+	4.33013i	−0.846200	+	0.162851i
$708$		0			0
$709$	32.0000			1.20179			0.600893	−	0.799330i	$-0.294812\pi$
$709$	32.0000			1.20179			0.600893	+	0.799330i	$0.294812\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	34.6410i		−	1.29732i
$714$		0			0
$715$		−	31.1769i		−	1.16595i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−12.0000			−0.447524			−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000			−0.447524			−0.223762	+	0.974644i	$0.571834\pi$
$720$		0			0
$721$	−8.00000	−	41.5692i	−0.297936	−	1.54812i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	12.0000			0.445669
$726$		0			0
$727$	29.0000			1.07555			0.537775	−	0.843088i	$-0.319265\pi$
$727$	29.0000			1.07555			0.537775	+	0.843088i	$0.319265\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	12.0000			0.443836
$732$		0			0
$733$		−	6.92820i		−	0.255899i	−0.991781	−	0.127950i	$-0.959160\pi$
$733$		−	6.92820i		−	0.255899i	0.991781	−	0.127950i	$-0.0408395\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	18.0000			0.663039
$738$		0			0
$739$			45.0333i			1.65658i	0.560301	+	0.828289i	$0.310685\pi$
$739$			45.0333i			1.65658i	−0.560301	+	0.828289i	$0.689315\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			10.3923i			0.381257i	0.981662	+	0.190628i	$0.0610525\pi$
$743$			10.3923i			0.381257i	−0.981662	+	0.190628i	$0.938947\pi$
$744$		0			0
$745$		−	36.3731i		−	1.33261i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−22.5000	+	4.33013i	−0.822132	+	0.158219i
$750$		0			0
$751$			25.9808i			0.948051i	0.880511	+	0.474026i	$0.157200\pi$
$751$			25.9808i			0.948051i	−0.880511	+	0.474026i	$0.842800\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−21.0000			−0.764268
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			13.8564i			0.502294i	0.967949	+	0.251147i	$0.0808078\pi$
$761$			13.8564i			0.502294i	−0.967949	+	0.251147i	$0.919192\pi$
$762$		0			0
$763$	10.0000	+	51.9615i	0.362024	+	1.88113i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	41.5692i		−	1.50098i
$768$		0			0
$769$		−	39.8372i		−	1.43657i	−0.695752	−	0.718283i	$-0.744929\pi$
$769$		−	39.8372i		−	1.43657i	0.695752	−	0.718283i	$-0.255071\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			48.4974i			1.74433i	0.489211	+	0.872166i	$0.337285\pi$
$773$			48.4974i			1.74433i	−0.489211	+	0.872166i	$0.662715\pi$
$774$		0			0
$775$	10.0000			0.359211
$776$		0			0
$777$		0			0
$778$		0			0
$779$			55.4256i			1.98583i
$780$		0			0
$781$	−18.0000			−0.644091
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−12.0000			−0.428298
$786$		0			0
$787$	−40.0000			−1.42585			−0.712923	−	0.701242i	$-0.752629\pi$
$787$	−40.0000			−1.42585			−0.712923	+	0.701242i	$0.752629\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	6.00000	+	31.1769i	0.213335	+	1.10852i
$792$		0			0
$793$	48.0000			1.70453
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	19.0526i		−	0.674876i	−0.941348	−	0.337438i	$-0.890440\pi$
$797$		−	19.0526i		−	0.674876i	0.941348	−	0.337438i	$-0.109560\pi$
$798$		0			0
$799$		−	20.7846i		−	0.735307i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−9.00000			−0.317603
$804$		0			0
$805$	−6.00000	−	31.1769i	−0.211472	−	1.09884i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−54.0000			−1.89854			−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000			−1.89854			−0.949269	+	0.314464i	$0.898175\pi$
$810$		0			0
$811$	−28.0000			−0.983213			−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000			−0.983213			−0.491606	+	0.870817i	$0.663590\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−18.0000			−0.630512
$816$		0			0
$817$		−	27.7128i		−	0.969549i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−42.0000			−1.46581			−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000			−1.46581			−0.732905	+	0.680331i	$0.761836\pi$
$822$		0			0
$823$			29.4449i			1.02638i	0.858274	+	0.513192i	$0.171537\pi$
$823$			29.4449i			1.02638i	−0.858274	+	0.513192i	$0.828463\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	10.3923i		−	0.361376i	−0.983540	−	0.180688i	$-0.942168\pi$
$827$		−	10.3923i		−	0.361376i	0.983540	−	0.180688i	$-0.0578324\pi$
$828$		0			0
$829$		−	3.46410i		−	0.120313i	−0.998189	−	0.0601566i	$-0.980840\pi$
$829$		−	3.46410i		−	0.120313i	0.998189	−	0.0601566i	$-0.0191600\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	9.00000	+	22.5167i	0.311832	+	0.780156i
$834$		0			0
$835$			10.3923i			0.359641i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−30.0000			−1.03572			−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000			−1.03572			−0.517858	+	0.855467i	$0.673270\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	1.73205i		−	0.0595844i
$846$		0			0
$847$	−8.00000	−	41.5692i	−0.274883	−	1.42834i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	27.7128i		−	0.949983i
$852$		0			0
$853$			17.3205i			0.593043i	0.955026	+	0.296521i	$0.0958266\pi$
$853$			17.3205i			0.593043i	−0.955026	+	0.296521i	$0.904173\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			45.0333i			1.53831i	0.639063	+	0.769154i	$0.279322\pi$
$857$			45.0333i			1.53831i	−0.639063	+	0.769154i	$0.720678\pi$
$858$		0			0
$859$	−8.00000			−0.272956			−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000			−0.272956			−0.136478	+	0.990643i	$0.543578\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$	−3.00000			−0.102003
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−18.0000			−0.610608
$870$		0			0
$871$	−12.0000			−0.406604
$872$		0			0
$873$		0			0
$874$		0			0
$875$	31.5000	−	6.06218i	1.06489	−	0.204939i
$876$		0			0
$877$	26.0000			0.877958			0.438979	−	0.898497i	$-0.355340\pi$
$877$	26.0000			0.877958			0.438979	+	0.898497i	$0.355340\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	45.0333i		−	1.51721i	−0.651550	−	0.758606i	$-0.725881\pi$
$881$		−	45.0333i		−	1.51721i	0.651550	−	0.758606i	$-0.274119\pi$
$882$		0			0
$883$			34.6410i			1.16576i	0.812557	+	0.582882i	$0.198075\pi$
$883$			34.6410i			1.16576i	−0.812557	+	0.582882i	$0.801925\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	18.0000			0.604381			0.302190	−	0.953248i	$-0.402282\pi$
$887$	18.0000			0.604381			0.302190	+	0.953248i	$0.402282\pi$
$888$		0			0
$889$	31.5000	−	6.06218i	1.05648	−	0.203319i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−48.0000			−1.60626
$894$		0			0
$895$	−33.0000			−1.10307
$896$		0			0
$897$		0			0
$898$		0			0
$899$	30.0000			1.00056
$900$		0			0
$901$		−	10.3923i		−	0.346218i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	36.0000			1.19668
$906$		0			0
$907$			6.92820i			0.230047i	0.993363	+	0.115024i	$0.0366944\pi$
$907$			6.92820i			0.230047i	−0.993363	+	0.115024i	$0.963306\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	10.3923i		−	0.344312i	−0.985070	−	0.172156i	$-0.944927\pi$
$911$		−	10.3923i		−	0.344312i	0.985070	−	0.172156i	$-0.0550734\pi$
$912$		0			0
$913$			77.9423i			2.57951i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	7.50000	+	38.9711i	0.247672	+	1.28694i
$918$		0			0
$919$			5.19615i			0.171405i	0.996321	+	0.0857026i	$0.0273135\pi$
$919$			5.19615i			0.171405i	−0.996321	+	0.0857026i	$0.972687\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	12.0000			0.394985
$924$		0			0
$925$	8.00000			0.263038
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	6.92820i		−	0.227307i	−0.993520	−	0.113653i	$-0.963745\pi$
$929$		−	6.92820i		−	0.227307i	0.993520	−	0.113653i	$-0.0362554\pi$
$930$		0			0
$931$	52.0000	−	20.7846i	1.70423	−	0.681188i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	31.1769i		−	1.01959i
$936$		0			0
$937$		−	25.9808i		−	0.848755i	−0.905485	−	0.424377i	$-0.860493\pi$
$937$		−	25.9808i		−	0.848755i	0.905485	−	0.424377i	$-0.139507\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			25.9808i			0.846949i	0.905908	+	0.423474i	$0.139190\pi$
$941$			25.9808i			0.846949i	−0.905908	+	0.423474i	$0.860810\pi$
$942$		0			0
$943$	−48.0000			−1.56310
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	19.0526i		−	0.619125i	−0.950879	−	0.309562i	$-0.899817\pi$
$947$		−	19.0526i		−	0.619125i	0.950879	−	0.309562i	$-0.100183\pi$
$948$		0			0
$949$	6.00000			0.194768
$950$		0			0
$951$		0			0
$952$		0			0
$953$	36.0000			1.16615			0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000			1.16615			0.583077	+	0.812417i	$0.301849\pi$
$954$		0			0
$955$	18.0000			0.582466
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3.00000	−	15.5885i	−0.0968751	−	0.503378i
$960$		0			0
$961$	−6.00000			−0.193548
$962$		0			0
$963$		0			0
$964$		0			0
$965$			8.66025i			0.278783i
$966$		0			0
$967$			53.6936i			1.72667i	0.504632	+	0.863334i	$0.331628\pi$
$967$			53.6936i			1.72667i	−0.504632	+	0.863334i	$0.668372\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−27.0000			−0.866471			−0.433236	−	0.901281i	$-0.642628\pi$
$971$	−27.0000			−0.866471			−0.433236	+	0.901281i	$0.642628\pi$
$972$		0			0
$973$	4.00000	+	20.7846i	0.128234	+	0.666324i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	18.0000			0.575871			0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000			0.575871			0.287936	+	0.957650i	$0.407031\pi$
$978$		0			0
$979$	54.0000			1.72585
$980$		0			0
$981$		0			0
$982$		0			0
$983$	18.0000			0.574111			0.287055	−	0.957914i	$-0.407324\pi$
$983$	18.0000			0.574111			0.287055	+	0.957914i	$0.407324\pi$
$984$		0			0
$985$			46.7654i			1.49007i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	24.0000			0.763156
$990$		0			0
$991$			43.3013i			1.37551i	0.725943	+	0.687755i	$0.241404\pi$
$991$			43.3013i			1.37551i	−0.725943	+	0.687755i	$0.758596\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			19.0526i			0.604007i
$996$		0			0
$997$			51.9615i			1.64564i	0.568304	+	0.822819i	$0.307600\pi$
$997$			51.9615i			1.64564i	−0.568304	+	0.822819i	$0.692400\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3024.2.b.j.1567.1	yes	2
3.2	odd	2		3024.2.b.i.1567.2	yes	2
4.3	odd	2		3024.2.b.h.1567.1	yes	2
7.6	odd	2		3024.2.b.h.1567.2	yes	2
12.11	even	2		3024.2.b.g.1567.2	yes	2
21.20	even	2		3024.2.b.g.1567.1	✓	2
28.27	even	2	inner	3024.2.b.j.1567.2	yes	2
84.83	odd	2		3024.2.b.i.1567.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3024.2.b.g.1567.1	✓	2	21.20	even	2
3024.2.b.g.1567.2	yes	2	12.11	even	2
3024.2.b.h.1567.1	yes	2	4.3	odd	2
3024.2.b.h.1567.2	yes	2	7.6	odd	2
3024.2.b.i.1567.1	yes	2	84.83	odd	2
3024.2.b.i.1567.2	yes	2	3.2	odd	2
3024.2.b.j.1567.1	yes	2	1.1	even	1	trivial
3024.2.b.j.1567.2	yes	2	28.27	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3024.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{5} +(0.500000 + 2.59808i) q^{7} +O(q^{10})\)	\(q-1.73205i q^{5} +(0.500000 + 2.59808i) q^{7} +5.19615i q^{11} -3.46410i q^{13} -3.46410i q^{17} -8.00000 q^{19} -6.92820i q^{23} +2.00000 q^{25} +6.00000 q^{29} +5.00000 q^{31} +(4.50000 - 0.866025i) q^{35} +4.00000 q^{37} -6.92820i q^{41} +3.46410i q^{43} +6.00000 q^{47} +(-6.50000 + 2.59808i) q^{49} +3.00000 q^{53} +9.00000 q^{55} +12.0000 q^{59} +13.8564i q^{61} -6.00000 q^{65} -3.46410i q^{67} +3.46410i q^{71} +1.73205i q^{73} +(-13.5000 + 2.59808i) q^{77} +3.46410i q^{79} +15.0000 q^{83} -6.00000 q^{85} -10.3923i q^{89} +(9.00000 - 1.73205i) q^{91} +13.8564i q^{95} -8.66025i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{7}+O(q^{10}) \) 2 * q + q^7	\( 2 q + q^{7} - 16 q^{19} + 4 q^{25} + 12 q^{29} + 10 q^{31} + 9 q^{35} + 8 q^{37} + 12 q^{47} - 13 q^{49} + 6 q^{53} + 18 q^{55} + 24 q^{59} - 12 q^{65} - 27 q^{77} + 30 q^{83} - 12 q^{85} + 18 q^{91}+O(q^{100}) \) 2 * q + q^7 - 16 * q^19 + 4 * q^25 + 12 * q^29 + 10 * q^31 + 9 * q^35 + 8 * q^37 + 12 * q^47 - 13 * q^49 + 6 * q^53 + 18 * q^55 + 24 * q^59 - 12 * q^65 - 27 * q^77 + 30 * q^83 - 12 * q^85 + 18 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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