LMFDB - Embedded newform 675.2.b.a.649.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,2,Mod(649,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("675.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	675.b (of order $2$, degree $1$, not 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:	$5.38990213644$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	675.649
Dual form			675.2.b.a.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -2.00000 q^{4} -3.00000i q^{7} +O(q^{10})$	$q-2.00000i q^{2} -2.00000 q^{4} -3.00000i q^{7} -2.00000 q^{11} +5.00000i q^{13} -6.00000 q^{14} -4.00000 q^{16} -8.00000i q^{17} -1.00000 q^{19} +4.00000i q^{22} -6.00000i q^{23} +10.0000 q^{26} +6.00000i q^{28} -2.00000 q^{29} +8.00000i q^{32} -16.0000 q^{34} +5.00000i q^{37} +2.00000i q^{38} -10.0000 q^{41} -4.00000i q^{43} +4.00000 q^{44} -12.0000 q^{46} +4.00000i q^{47} -2.00000 q^{49} -10.0000i q^{52} +2.00000i q^{53} +4.00000i q^{58} +8.00000 q^{59} +7.00000 q^{61} +8.00000 q^{64} -9.00000i q^{67} +16.0000i q^{68} +2.00000 q^{71} +5.00000i q^{73} +10.0000 q^{74} +2.00000 q^{76} +6.00000i q^{77} +3.00000 q^{79} +20.0000i q^{82} -6.00000i q^{83} -8.00000 q^{86} +12.0000 q^{89} +15.0000 q^{91} +12.0000i q^{92} +8.00000 q^{94} -13.0000i q^{97} +4.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4}+O(q^{10}) $ 2 * q - 4 * q^4	$ 2 q - 4 q^{4} - 4 q^{11} - 12 q^{14} - 8 q^{16} - 2 q^{19} + 20 q^{26} - 4 q^{29} - 32 q^{34} - 20 q^{41} + 8 q^{44} - 24 q^{46} - 4 q^{49} + 16 q^{59} + 14 q^{61} + 16 q^{64} + 4 q^{71} + 20 q^{74} + 4 q^{76} + 6 q^{79} - 16 q^{86} + 24 q^{89} + 30 q^{91} + 16 q^{94}+O(q^{100}) $ 2 * q - 4 * q^4 - 4 * q^11 - 12 * q^14 - 8 * q^16 - 2 * q^19 + 20 * q^26 - 4 * q^29 - 32 * q^34 - 20 * q^41 + 8 * q^44 - 24 * q^46 - 4 * q^49 + 16 * q^59 + 14 * q^61 + 16 * q^64 + 4 * q^71 + 20 * q^74 + 4 * q^76 + 6 * q^79 - 16 * q^86 + 24 * q^89 + 30 * q^91 + 16 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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 $

$2$	− 2.00000i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	− 2.00000i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.00000i	− 1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$	− 3.00000i	− 1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	5.00000i	1.38675i	0.720577	+	0.693375i	$0.243877\pi$
$13$	5.00000i	1.38675i	−0.720577	+	0.693375i	$0.756123\pi$
$14$	−6.00000	−1.60357
$15$	0	0
$16$	−4.00000	−1.00000
$17$	− 8.00000i	− 1.94029i	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	− 8.00000i	− 1.94029i	0.242536	−	0.970143i	$-0.422021\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	0	0
$22$	4.00000i	0.852803i
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	0	0
$25$	0	0
$26$	10.0000	1.96116
$27$	0	0
$28$	6.00000i	1.13389i
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	8.00000i	1.41421i
$33$	0	0
$34$	−16.0000	−2.74398
$35$	0	0
$36$	0	0
$37$	5.00000i	0.821995i	0.911636	+	0.410997i	$0.134819\pi$
$37$	5.00000i	0.821995i	−0.911636	+	0.410997i	$0.865181\pi$
$38$	2.00000i	0.324443i
$39$	0	0
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−12.0000	−1.76930
$47$	4.00000i	0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$	4.00000i	0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	0	0
$52$	− 10.0000i	− 1.38675i
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	4.00000i	0.525226i
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	0	0
$63$	0	0
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	− 9.00000i	− 1.09952i	−0.835321	−	0.549762i	$-0.814718\pi$
$67$	− 9.00000i	− 1.09952i	0.835321	−	0.549762i	$-0.185282\pi$
$68$	16.0000i	1.94029i
$69$	0	0
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	0	0
$73$	5.00000i	0.585206i	0.956234	+	0.292603i	$0.0945214\pi$
$73$	5.00000i	0.585206i	−0.956234	+	0.292603i	$0.905479\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	2.00000	0.229416
$77$	6.00000i	0.683763i
$78$	0	0
$79$	3.00000	0.337526	0.168763	−	0.985657i	$-0.446023\pi$
$79$	3.00000	0.337526	0.168763	+	0.985657i	$0.446023\pi$
$80$	0	0
$81$	0	0
$82$	20.0000i	2.20863i
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	0	0
$85$	0	0
$86$	−8.00000	−0.862662
$87$	0	0
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	15.0000	1.57243
$92$	12.0000i	1.25109i
$93$	0	0
$94$	8.00000	0.825137
$95$	0	0
$96$	0	0
$97$	− 13.0000i	− 1.31995i	−0.751288	−	0.659975i	$-0.770567\pi$
$97$	− 13.0000i	− 1.31995i	0.751288	−	0.659975i	$-0.229433\pi$
$98$	4.00000i	0.404061i
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	− 17.0000i	− 1.67506i	−0.546392	−	0.837530i	$-0.683999\pi$
$103$	− 17.0000i	− 1.67506i	0.546392	−	0.837530i	$-0.316001\pi$
$104$	0	0
$105$	0	0
$106$	4.00000	0.388514
$107$	6.00000i	0.580042i	0.957020	+	0.290021i	$0.0936623\pi$
$107$	6.00000i	0.580042i	−0.957020	+	0.290021i	$0.906338\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	12.0000i	1.13389i
$113$	− 10.0000i	− 0.940721i	−0.882474	−	0.470360i	$-0.844124\pi$
$113$	− 10.0000i	− 0.940721i	0.882474	−	0.470360i	$-0.155876\pi$
$114$	0	0
$115$	0	0
$116$	4.00000	0.371391
$117$	0	0
$118$	− 16.0000i	− 1.47292i
$119$	−24.0000	−2.20008
$120$	0	0
$121$	−7.00000	−0.636364
$122$	− 14.0000i	− 1.26750i
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	3.00000i	0.260133i
$134$	−18.0000	−1.55496
$135$	0	0
$136$	0	0
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	0	0
$139$	13.0000	1.10265	0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000	1.10265	0.551323	+	0.834292i	$0.314123\pi$
$140$	0	0
$141$	0	0
$142$	− 4.00000i	− 0.335673i
$143$	− 10.0000i	− 0.836242i
$144$	0	0
$145$	0	0
$146$	10.0000	0.827606
$147$	0	0
$148$	− 10.0000i	− 0.821995i
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	0	0
$151$	1.00000	0.0813788	0.0406894	−	0.999172i	$-0.487045\pi$
$151$	1.00000	0.0813788	0.0406894	+	0.999172i	$0.487045\pi$
$152$	0	0
$153$	0	0
$154$	12.0000	0.966988
$155$	0	0
$156$	0	0
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	− 6.00000i	− 0.477334i
$159$	0	0
$160$	0	0
$161$	−18.0000	−1.41860
$162$	0	0
$163$	19.0000i	1.48819i	0.668071	+	0.744097i	$0.267120\pi$
$163$	19.0000i	1.48819i	−0.668071	+	0.744097i	$0.732880\pi$
$164$	20.0000	1.56174
$165$	0	0
$166$	−12.0000	−0.931381
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	8.00000i	0.609994i
$173$	12.0000i	0.912343i	0.889892	+	0.456172i	$0.150780\pi$
$173$	12.0000i	0.912343i	−0.889892	+	0.456172i	$0.849220\pi$
$174$	0	0
$175$	0	0
$176$	8.00000	0.603023
$177$	0	0
$178$	− 24.0000i	− 1.79888i
$179$	−22.0000	−1.64436	−0.822179	−	0.569230i	$-0.807242\pi$
$179$	−22.0000	−1.64436	−0.822179	+	0.569230i	$0.807242\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	− 30.0000i	− 2.22375i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	16.0000i	1.17004i
$188$	− 8.00000i	− 0.583460i
$189$	0	0
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	− 5.00000i	− 0.359908i	−0.983675	−	0.179954i	$-0.942405\pi$
$193$	− 5.00000i	− 0.359908i	0.983675	−	0.179954i	$-0.0575949\pi$
$194$	−26.0000	−1.86669
$195$	0	0
$196$	4.00000	0.285714
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	17.0000	1.20510	0.602549	−	0.798082i	$-0.294152\pi$
$199$	17.0000	1.20510	0.602549	+	0.798082i	$0.294152\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.00000i	0.421117i
$204$	0	0
$205$	0	0
$206$	−34.0000	−2.36889
$207$	0	0
$208$	− 20.0000i	− 1.38675i
$209$	2.00000	0.138343
$210$	0	0
$211$	23.0000	1.58339	0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000	1.58339	0.791693	+	0.610920i	$0.209200\pi$
$212$	− 4.00000i	− 0.274721i
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	0	0
$217$	0	0
$218$	− 20.0000i	− 1.35457i
$219$	0	0
$220$	0	0
$221$	40.0000	2.69069
$222$	0	0
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	24.0000	1.60357
$225$	0	0
$226$	−20.0000	−1.33038
$227$	− 10.0000i	− 0.663723i	−0.943328	−	0.331862i	$-0.892323\pi$
$227$	− 10.0000i	− 0.663723i	0.943328	−	0.331862i	$-0.107677\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	24.0000i	1.57229i	0.618041	+	0.786146i	$0.287927\pi$
$233$	24.0000i	1.57229i	−0.618041	+	0.786146i	$0.712073\pi$
$234$	0	0
$235$	0	0
$236$	−16.0000	−1.04151
$237$	0	0
$238$	48.0000i	3.11138i
$239$	26.0000	1.68180	0.840900	−	0.541190i	$-0.182026\pi$
$239$	26.0000	1.68180	0.840900	+	0.541190i	$0.182026\pi$
$240$	0	0
$241$	1.00000	0.0644157	0.0322078	−	0.999481i	$-0.489746\pi$
$241$	1.00000	0.0644157	0.0322078	+	0.999481i	$0.489746\pi$
$242$	14.0000i	0.899954i
$243$	0	0
$244$	−14.0000	−0.896258
$245$	0	0
$246$	0	0
$247$	− 5.00000i	− 0.318142i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	0	0
$253$	12.0000i	0.754434i
$254$	−16.0000	−1.00393
$255$	0	0
$256$	16.0000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	15.0000	0.932055
$260$	0	0
$261$	0	0
$262$	− 24.0000i	− 1.48272i
$263$	28.0000i	1.72655i	0.504730	+	0.863277i	$0.331592\pi$
$263$	28.0000i	1.72655i	−0.504730	+	0.863277i	$0.668408\pi$
$264$	0	0
$265$	0	0
$266$	6.00000	0.367884
$267$	0	0
$268$	18.0000i	1.09952i
$269$	−16.0000	−0.975537	−0.487769	−	0.872973i	$-0.662189\pi$
$269$	−16.0000	−0.975537	−0.487769	+	0.872973i	$0.662189\pi$
$270$	0	0
$271$	13.0000	0.789694	0.394847	−	0.918747i	$-0.370798\pi$
$271$	13.0000	0.789694	0.394847	+	0.918747i	$0.370798\pi$
$272$	32.0000i	1.94029i
$273$	0	0
$274$	−12.0000	−0.724947
$275$	0	0
$276$	0	0
$277$	− 30.0000i	− 1.80253i	−0.433273	−	0.901263i	$-0.642641\pi$
$277$	− 30.0000i	− 1.80253i	0.433273	−	0.901263i	$-0.357359\pi$
$278$	− 26.0000i	− 1.55938i
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	− 12.0000i	− 0.713326i	−0.934233	−	0.356663i	$-0.883914\pi$
$283$	− 12.0000i	− 0.713326i	0.934233	−	0.356663i	$-0.116086\pi$
$284$	−4.00000	−0.237356
$285$	0	0
$286$	−20.0000	−1.18262
$287$	30.0000i	1.77084i
$288$	0	0
$289$	−47.0000	−2.76471
$290$	0	0
$291$	0	0
$292$	− 10.0000i	− 0.585206i
$293$	− 18.0000i	− 1.05157i	−0.850617	−	0.525786i	$-0.823771\pi$
$293$	− 18.0000i	− 1.05157i	0.850617	−	0.525786i	$-0.176229\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	− 8.00000i	− 0.463428i
$299$	30.0000	1.73494
$300$	0	0
$301$	−12.0000	−0.691669
$302$	− 2.00000i	− 0.115087i
$303$	0	0
$304$	4.00000	0.229416
$305$	0	0
$306$	0	0
$307$	16.0000i	0.913168i	0.889680	+	0.456584i	$0.150927\pi$
$307$	16.0000i	0.913168i	−0.889680	+	0.456584i	$0.849073\pi$
$308$	− 12.0000i	− 0.683763i
$309$	0	0
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	7.00000i	0.395663i	0.980236	+	0.197832i	$0.0633900\pi$
$313$	7.00000i	0.395663i	−0.980236	+	0.197832i	$0.936610\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	−6.00000	−0.337526
$317$	20.0000i	1.12331i	0.827371	+	0.561656i	$0.189836\pi$
$317$	20.0000i	1.12331i	−0.827371	+	0.561656i	$0.810164\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	0	0
$321$	0	0
$322$	36.0000i	2.00620i
$323$	8.00000i	0.445132i
$324$	0	0
$325$	0	0
$326$	38.0000	2.10463
$327$	0	0
$328$	0	0
$329$	12.0000	0.661581
$330$	0	0
$331$	−21.0000	−1.15426	−0.577132	−	0.816651i	$-0.695828\pi$
$331$	−21.0000	−1.15426	−0.577132	+	0.816651i	$0.695828\pi$
$332$	12.0000i	0.658586i
$333$	0	0
$334$	−24.0000	−1.31322
$335$	0	0
$336$	0	0
$337$	7.00000i	0.381314i	0.981657	+	0.190657i	$0.0610619\pi$
$337$	7.00000i	0.381314i	−0.981657	+	0.190657i	$0.938938\pi$
$338$	24.0000i	1.30543i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 15.0000i	− 0.809924i
$344$	0	0
$345$	0	0
$346$	24.0000	1.29025
$347$	10.0000i	0.536828i	0.963304	+	0.268414i	$0.0864995\pi$
$347$	10.0000i	0.536828i	−0.963304	+	0.268414i	$0.913500\pi$
$348$	0	0
$349$	−19.0000	−1.01705	−0.508523	−	0.861048i	$-0.669808\pi$
$349$	−19.0000	−1.01705	−0.508523	+	0.861048i	$0.669808\pi$
$350$	0	0
$351$	0	0
$352$	− 16.0000i	− 0.852803i
$353$	12.0000i	0.638696i	0.947638	+	0.319348i	$0.103464\pi$
$353$	12.0000i	0.638696i	−0.947638	+	0.319348i	$0.896536\pi$
$354$	0	0
$355$	0	0
$356$	−24.0000	−1.27200
$357$	0	0
$358$	44.0000i	2.32547i
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	− 10.0000i	− 0.525588i
$363$	0	0
$364$	−30.0000	−1.57243
$365$	0	0
$366$	0	0
$367$	21.0000i	1.09619i	0.836416	+	0.548096i	$0.184647\pi$
$367$	21.0000i	1.09619i	−0.836416	+	0.548096i	$0.815353\pi$
$368$	24.0000i	1.25109i
$369$	0	0
$370$	0	0
$371$	6.00000	0.311504
$372$	0	0
$373$	11.0000i	0.569558i	0.958593	+	0.284779i	$0.0919203\pi$
$373$	11.0000i	0.569558i	−0.958593	+	0.284779i	$0.908080\pi$
$374$	32.0000	1.65468
$375$	0	0
$376$	0	0
$377$	− 10.0000i	− 0.515026i
$378$	0	0
$379$	11.0000	0.565032	0.282516	−	0.959263i	$-0.408831\pi$
$379$	11.0000	0.565032	0.282516	+	0.959263i	$0.408831\pi$
$380$	0	0
$381$	0	0
$382$	8.00000i	0.409316i
$383$	24.0000i	1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$	24.0000i	1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$	0	0
$385$	0	0
$386$	−10.0000	−0.508987
$387$	0	0
$388$	26.0000i	1.31995i
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	−48.0000	−2.42746
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	− 34.0000i	− 1.70427i
$399$	0	0
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	12.0000	0.595550
$407$	− 10.0000i	− 0.495682i
$408$	0	0
$409$	7.00000	0.346128	0.173064	−	0.984911i	$-0.444633\pi$
$409$	7.00000	0.346128	0.173064	+	0.984911i	$0.444633\pi$
$410$	0	0
$411$	0	0
$412$	34.0000i	1.67506i
$413$	− 24.0000i	− 1.18096i
$414$	0	0
$415$	0	0
$416$	−40.0000	−1.96116
$417$	0	0
$418$	− 4.00000i	− 0.195646i
$419$	−32.0000	−1.56330	−0.781651	−	0.623716i	$-0.785622\pi$
$419$	−32.0000	−1.56330	−0.781651	+	0.623716i	$0.785622\pi$
$420$	0	0
$421$	1.00000	0.0487370	0.0243685	−	0.999703i	$-0.492242\pi$
$421$	1.00000	0.0487370	0.0243685	+	0.999703i	$0.492242\pi$
$422$	− 46.0000i	− 2.23924i
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 21.0000i	− 1.01626i
$428$	− 12.0000i	− 0.580042i
$429$	0	0
$430$	0	0
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	0	0
$433$	38.0000i	1.82616i	0.407777	+	0.913082i	$0.366304\pi$
$433$	38.0000i	1.82616i	−0.407777	+	0.913082i	$0.633696\pi$
$434$	0	0
$435$	0	0
$436$	−20.0000	−0.957826
$437$	6.00000i	0.287019i
$438$	0	0
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	0	0
$441$	0	0
$442$	− 80.0000i	− 3.80521i
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	0	0
$445$	0	0
$446$	−16.0000	−0.757622
$447$	0	0
$448$	− 24.0000i	− 1.13389i
$449$	−4.00000	−0.188772	−0.0943858	−	0.995536i	$-0.530089\pi$
$449$	−4.00000	−0.188772	−0.0943858	+	0.995536i	$0.530089\pi$
$450$	0	0
$451$	20.0000	0.941763
$452$	20.0000i	0.940721i
$453$	0	0
$454$	−20.0000	−0.938647
$455$	0	0
$456$	0	0
$457$	22.0000i	1.02912i	0.857455	+	0.514558i	$0.172044\pi$
$457$	22.0000i	1.02912i	−0.857455	+	0.514558i	$0.827956\pi$
$458$	12.0000i	0.560723i
$459$	0	0
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	− 33.0000i	− 1.53364i	−0.641862	−	0.766820i	$-0.721838\pi$
$463$	− 33.0000i	− 1.53364i	0.641862	−	0.766820i	$-0.278162\pi$
$464$	8.00000	0.371391
$465$	0	0
$466$	48.0000	2.22356
$467$	− 28.0000i	− 1.29569i	−0.761774	−	0.647843i	$-0.775671\pi$
$467$	− 28.0000i	− 1.29569i	0.761774	−	0.647843i	$-0.224329\pi$
$468$	0	0
$469$	−27.0000	−1.24674
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.00000i	0.367840i
$474$	0	0
$475$	0	0
$476$	48.0000	2.20008
$477$	0	0
$478$	− 52.0000i	− 2.37842i
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	0	0
$481$	−25.0000	−1.13990
$482$	− 2.00000i	− 0.0910975i
$483$	0	0
$484$	14.0000	0.636364
$485$	0	0
$486$	0	0
$487$	5.00000i	0.226572i	0.993562	+	0.113286i	$0.0361376\pi$
$487$	5.00000i	0.226572i	−0.993562	+	0.113286i	$0.963862\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−22.0000	−0.992846	−0.496423	−	0.868081i	$-0.665354\pi$
$491$	−22.0000	−0.992846	−0.496423	+	0.868081i	$0.665354\pi$
$492$	0	0
$493$	16.0000i	0.720604i
$494$	−10.0000	−0.449921
$495$	0	0
$496$	0	0
$497$	− 6.00000i	− 0.269137i
$498$	0	0
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	0	0
$501$	0	0
$502$	− 12.0000i	− 0.535586i
$503$	− 26.0000i	− 1.15928i	−0.814872	−	0.579641i	$-0.803193\pi$
$503$	− 26.0000i	− 1.15928i	0.814872	−	0.579641i	$-0.196807\pi$
$504$	0	0
$505$	0	0
$506$	24.0000	1.06693
$507$	0	0
$508$	16.0000i	0.709885i
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	0	0
$511$	15.0000	0.663561
$512$	− 32.0000i	− 1.41421i
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 8.00000i	− 0.351840i
$518$	− 30.0000i	− 1.31812i
$519$	0	0
$520$	0	0
$521$	38.0000	1.66481	0.832405	−	0.554168i	$-0.186963\pi$
$521$	38.0000	1.66481	0.832405	+	0.554168i	$0.186963\pi$
$522$	0	0
$523$	− 19.0000i	− 0.830812i	−0.909636	−	0.415406i	$-0.863640\pi$
$523$	− 19.0000i	− 0.830812i	0.909636	−	0.415406i	$-0.136360\pi$
$524$	−24.0000	−1.04844
$525$	0	0
$526$	56.0000	2.44172
$527$	0	0
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	− 6.00000i	− 0.260133i
$533$	− 50.0000i	− 2.16574i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	32.0000i	1.37962i
$539$	4.00000	0.172292
$540$	0	0
$541$	33.0000	1.41878	0.709390	−	0.704816i	$-0.248970\pi$
$541$	33.0000	1.41878	0.709390	+	0.704816i	$0.248970\pi$
$542$	− 26.0000i	− 1.11680i
$543$	0	0
$544$	64.0000	2.74398
$545$	0	0
$546$	0	0
$547$	37.0000i	1.58201i	0.611812	+	0.791003i	$0.290441\pi$
$547$	37.0000i	1.58201i	−0.611812	+	0.791003i	$0.709559\pi$
$548$	12.0000i	0.512615i
$549$	0	0
$550$	0	0
$551$	2.00000	0.0852029
$552$	0	0
$553$	− 9.00000i	− 0.382719i
$554$	−60.0000	−2.54916
$555$	0	0
$556$	−26.0000	−1.10265
$557$	12.0000i	0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$	12.0000i	0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$	0	0
$559$	20.0000	0.845910
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 36.0000i	− 1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$	− 36.0000i	− 1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$	0	0
$565$	0	0
$566$	−24.0000	−1.00880
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	5.00000	0.209243	0.104622	−	0.994512i	$-0.466637\pi$
$571$	5.00000	0.209243	0.104622	+	0.994512i	$0.466637\pi$
$572$	20.0000i	0.836242i
$573$	0	0
$574$	60.0000	2.50435
$575$	0	0
$576$	0	0
$577$	17.0000i	0.707719i	0.935299	+	0.353860i	$0.115131\pi$
$577$	17.0000i	0.707719i	−0.935299	+	0.353860i	$0.884869\pi$
$578$	94.0000i	3.90988i
$579$	0	0
$580$	0	0
$581$	−18.0000	−0.746766
$582$	0	0
$583$	− 4.00000i	− 0.165663i
$584$	0	0
$585$	0	0
$586$	−36.0000	−1.48715
$587$	− 18.0000i	− 0.742940i	−0.928445	−	0.371470i	$-0.878854\pi$
$587$	− 18.0000i	− 0.742940i	0.928445	−	0.371470i	$-0.121146\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	− 20.0000i	− 0.821995i
$593$	22.0000i	0.903432i	0.892162	+	0.451716i	$0.149188\pi$
$593$	22.0000i	0.903432i	−0.892162	+	0.451716i	$0.850812\pi$
$594$	0	0
$595$	0	0
$596$	−8.00000	−0.327693
$597$	0	0
$598$	− 60.0000i	− 2.45358i
$599$	10.0000	0.408589	0.204294	−	0.978909i	$-0.434510\pi$
$599$	10.0000	0.408589	0.204294	+	0.978909i	$0.434510\pi$
$600$	0	0
$601$	−34.0000	−1.38689	−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000	−1.38689	−0.693444	+	0.720510i	$0.743908\pi$
$602$	24.0000i	0.978167i
$603$	0	0
$604$	−2.00000	−0.0813788
$605$	0	0
$606$	0	0
$607$	1.00000i	0.0405887i	0.999794	+	0.0202944i	$0.00646034\pi$
$607$	1.00000i	0.0405887i	−0.999794	+	0.0202944i	$0.993540\pi$
$608$	− 8.00000i	− 0.324443i
$609$	0	0
$610$	0	0
$611$	−20.0000	−0.809113
$612$	0	0
$613$	− 1.00000i	− 0.0403896i	−0.999796	−	0.0201948i	$-0.993571\pi$
$613$	− 1.00000i	− 0.0403896i	0.999796	−	0.0201948i	$-0.00642864\pi$
$614$	32.0000	1.29141
$615$	0	0
$616$	0	0
$617$	18.0000i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$	18.0000i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$	0	0
$619$	19.0000	0.763674	0.381837	−	0.924230i	$-0.375291\pi$
$619$	19.0000	0.763674	0.381837	+	0.924230i	$0.375291\pi$
$620$	0	0
$621$	0	0
$622$	48.0000i	1.92462i
$623$	− 36.0000i	− 1.44231i
$624$	0	0
$625$	0	0
$626$	14.0000	0.559553
$627$	0	0
$628$	− 4.00000i	− 0.159617i
$629$	40.0000	1.59490
$630$	0	0
$631$	−11.0000	−0.437903	−0.218952	−	0.975736i	$-0.570264\pi$
$631$	−11.0000	−0.437903	−0.218952	+	0.975736i	$0.570264\pi$
$632$	0	0
$633$	0	0
$634$	40.0000	1.58860
$635$	0	0
$636$	0	0
$637$	− 10.0000i	− 0.396214i
$638$	− 8.00000i	− 0.316723i
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	12.0000i	0.473234i	0.971603	+	0.236617i	$0.0760386\pi$
$643$	12.0000i	0.473234i	−0.971603	+	0.236617i	$0.923961\pi$
$644$	36.0000	1.41860
$645$	0	0
$646$	16.0000	0.629512
$647$	− 38.0000i	− 1.49393i	−0.664861	−	0.746967i	$-0.731509\pi$
$647$	− 38.0000i	− 1.49393i	0.664861	−	0.746967i	$-0.268491\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	0	0
$652$	− 38.0000i	− 1.48819i
$653$	4.00000i	0.156532i	0.996933	+	0.0782660i	$0.0249384\pi$
$653$	4.00000i	0.156532i	−0.996933	+	0.0782660i	$0.975062\pi$
$654$	0	0
$655$	0	0
$656$	40.0000	1.56174
$657$	0	0
$658$	− 24.0000i	− 0.935617i
$659$	32.0000	1.24654	0.623272	−	0.782006i	$-0.285803\pi$
$659$	32.0000	1.24654	0.623272	+	0.782006i	$0.285803\pi$
$660$	0	0
$661$	7.00000	0.272268	0.136134	−	0.990690i	$-0.456532\pi$
$661$	7.00000	0.272268	0.136134	+	0.990690i	$0.456532\pi$
$662$	42.0000i	1.63238i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	12.0000i	0.464642i
$668$	24.0000i	0.928588i
$669$	0	0
$670$	0	0
$671$	−14.0000	−0.540464
$672$	0	0
$673$	− 33.0000i	− 1.27206i	−0.771666	−	0.636028i	$-0.780576\pi$
$673$	− 33.0000i	− 1.27206i	0.771666	−	0.636028i	$-0.219424\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	24.0000	0.923077
$677$	12.0000i	0.461197i	0.973049	+	0.230599i	$0.0740685\pi$
$677$	12.0000i	0.461197i	−0.973049	+	0.230599i	$0.925932\pi$
$678$	0	0
$679$	−39.0000	−1.49668
$680$	0	0
$681$	0	0
$682$	0	0
$683$	18.0000i	0.688751i	0.938832	+	0.344375i	$0.111909\pi$
$683$	18.0000i	0.688751i	−0.938832	+	0.344375i	$0.888091\pi$
$684$	0	0
$685$	0	0
$686$	−30.0000	−1.14541
$687$	0	0
$688$	16.0000i	0.609994i
$689$	−10.0000	−0.380970
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	− 24.0000i	− 0.912343i
$693$	0	0
$694$	20.0000	0.759190
$695$	0	0
$696$	0	0
$697$	80.0000i	3.03022i
$698$	38.0000i	1.43832i
$699$	0	0
$700$	0	0
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	0	0
$703$	− 5.00000i	− 0.188579i
$704$	−16.0000	−0.603023
$705$	0	0
$706$	24.0000	0.903252
$707$	0	0
$708$	0	0
$709$	−25.0000	−0.938895	−0.469447	−	0.882960i	$-0.655547\pi$
$709$	−25.0000	−0.938895	−0.469447	+	0.882960i	$0.655547\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	44.0000	1.64436
$717$	0	0
$718$	− 36.0000i	− 1.34351i
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	0	0
$721$	−51.0000	−1.89934
$722$	36.0000i	1.33978i
$723$	0	0
$724$	−10.0000	−0.371647
$725$	0	0
$726$	0	0
$727$	− 16.0000i	− 0.593407i	−0.954970	−	0.296704i	$-0.904113\pi$
$727$	− 16.0000i	− 0.593407i	0.954970	−	0.296704i	$-0.0958873\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−32.0000	−1.18356
$732$	0	0
$733$	22.0000i	0.812589i	0.913742	+	0.406294i	$0.133179\pi$
$733$	22.0000i	0.812589i	−0.913742	+	0.406294i	$0.866821\pi$
$734$	42.0000	1.55025
$735$	0	0
$736$	48.0000	1.76930
$737$	18.0000i	0.663039i
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	0	0
$742$	− 12.0000i	− 0.440534i
$743$	32.0000i	1.17397i	0.809599	+	0.586983i	$0.199684\pi$
$743$	32.0000i	1.17397i	−0.809599	+	0.586983i	$0.800316\pi$
$744$	0	0
$745$	0	0
$746$	22.0000	0.805477
$747$	0	0
$748$	− 32.0000i	− 1.17004i
$749$	18.0000	0.657706
$750$	0	0
$751$	25.0000	0.912263	0.456131	−	0.889912i	$-0.349235\pi$
$751$	25.0000	0.912263	0.456131	+	0.889912i	$0.349235\pi$
$752$	− 16.0000i	− 0.583460i
$753$	0	0
$754$	−20.0000	−0.728357
$755$	0	0
$756$	0	0
$757$	− 53.0000i	− 1.92632i	−0.268933	−	0.963159i	$-0.586671\pi$
$757$	− 53.0000i	− 1.92632i	0.268933	−	0.963159i	$-0.413329\pi$
$758$	− 22.0000i	− 0.799076i
$759$	0	0
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	0	0
$763$	− 30.0000i	− 1.08607i
$764$	8.00000	0.289430
$765$	0	0
$766$	48.0000	1.73431
$767$	40.0000i	1.44432i
$768$	0	0
$769$	37.0000	1.33425	0.667127	−	0.744944i	$-0.267524\pi$
$769$	37.0000	1.33425	0.667127	+	0.744944i	$0.267524\pi$
$770$	0	0
$771$	0	0
$772$	10.0000i	0.359908i
$773$	− 18.0000i	− 0.647415i	−0.946157	−	0.323708i	$-0.895071\pi$
$773$	− 18.0000i	− 0.647415i	0.946157	−	0.323708i	$-0.104929\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	48.0000i	1.72088i
$779$	10.0000	0.358287
$780$	0	0
$781$	−4.00000	−0.143131
$782$	96.0000i	3.43295i
$783$	0	0
$784$	8.00000	0.285714
$785$	0	0
$786$	0	0
$787$	− 41.0000i	− 1.46149i	−0.682649	−	0.730746i	$-0.739172\pi$
$787$	− 41.0000i	− 1.46149i	0.682649	−	0.730746i	$-0.260828\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−30.0000	−1.06668
$792$	0	0
$793$	35.0000i	1.24289i
$794$	−4.00000	−0.141955
$795$	0	0
$796$	−34.0000	−1.20510
$797$	− 16.0000i	− 0.566749i	−0.959009	−	0.283375i	$-0.908546\pi$
$797$	− 16.0000i	− 0.566749i	0.959009	−	0.283375i	$-0.0914540\pi$
$798$	0	0
$799$	32.0000	1.13208
$800$	0	0
$801$	0	0
$802$	− 12.0000i	− 0.423735i
$803$	− 10.0000i	− 0.352892i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	16.0000	0.562530	0.281265	−	0.959630i	$-0.409246\pi$
$809$	16.0000	0.562530	0.281265	+	0.959630i	$0.409246\pi$
$810$	0	0
$811$	−12.0000	−0.421377	−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000	−0.421377	−0.210688	+	0.977553i	$0.567571\pi$
$812$	− 12.0000i	− 0.421117i
$813$	0	0
$814$	−20.0000	−0.701000
$815$	0	0
$816$	0	0
$817$	4.00000i	0.139942i
$818$	− 14.0000i	− 0.489499i
$819$	0	0
$820$	0	0
$821$	48.0000	1.67521	0.837606	−	0.546275i	$-0.183955\pi$
$821$	48.0000	1.67521	0.837606	+	0.546275i	$0.183955\pi$
$822$	0	0
$823$	− 11.0000i	− 0.383436i	−0.981450	−	0.191718i	$-0.938594\pi$
$823$	− 11.0000i	− 0.383436i	0.981450	−	0.191718i	$-0.0614059\pi$
$824$	0	0
$825$	0	0
$826$	−48.0000	−1.67013
$827$	28.0000i	0.973655i	0.873498	+	0.486828i	$0.161846\pi$
$827$	28.0000i	0.973655i	−0.873498	+	0.486828i	$0.838154\pi$
$828$	0	0
$829$	51.0000	1.77130	0.885652	−	0.464350i	$-0.153712\pi$
$829$	51.0000	1.77130	0.885652	+	0.464350i	$0.153712\pi$
$830$	0	0
$831$	0	0
$832$	40.0000i	1.38675i
$833$	16.0000i	0.554367i
$834$	0	0
$835$	0	0
$836$	−4.00000	−0.138343
$837$	0	0
$838$	64.0000i	2.21084i
$839$	4.00000	0.138095	0.0690477	−	0.997613i	$-0.478004\pi$
$839$	4.00000	0.138095	0.0690477	+	0.997613i	$0.478004\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	− 2.00000i	− 0.0689246i
$843$	0	0
$844$	−46.0000	−1.58339
$845$	0	0
$846$	0	0
$847$	21.0000i	0.721569i
$848$	− 8.00000i	− 0.274721i
$849$	0	0
$850$	0	0
$851$	30.0000	1.02839
$852$	0	0
$853$	− 21.0000i	− 0.719026i	−0.933140	−	0.359513i	$-0.882943\pi$
$853$	− 21.0000i	− 0.719026i	0.933140	−	0.359513i	$-0.117057\pi$
$854$	−42.0000	−1.43721
$855$	0	0
$856$	0	0
$857$	− 26.0000i	− 0.888143i	−0.895991	−	0.444072i	$-0.853534\pi$
$857$	− 26.0000i	− 0.888143i	0.895991	−	0.444072i	$-0.146466\pi$
$858$	0	0
$859$	−7.00000	−0.238837	−0.119418	−	0.992844i	$-0.538103\pi$
$859$	−7.00000	−0.238837	−0.119418	+	0.992844i	$0.538103\pi$
$860$	0	0
$861$	0	0
$862$	− 60.0000i	− 2.04361i
$863$	4.00000i	0.136162i	0.997680	+	0.0680808i	$0.0216876\pi$
$863$	4.00000i	0.136162i	−0.997680	+	0.0680808i	$0.978312\pi$
$864$	0	0
$865$	0	0
$866$	76.0000	2.58259
$867$	0	0
$868$	0	0
$869$	−6.00000	−0.203536
$870$	0	0
$871$	45.0000	1.52477
$872$	0	0
$873$	0	0
$874$	12.0000	0.405906
$875$	0	0
$876$	0	0
$877$	9.00000i	0.303908i	0.988388	+	0.151954i	$0.0485566\pi$
$877$	9.00000i	0.303908i	−0.988388	+	0.151954i	$0.951443\pi$
$878$	− 64.0000i	− 2.15990i
$879$	0	0
$880$	0	0
$881$	−38.0000	−1.28025	−0.640126	−	0.768270i	$-0.721118\pi$
$881$	−38.0000	−1.28025	−0.640126	+	0.768270i	$0.721118\pi$
$882$	0	0
$883$	19.0000i	0.639401i	0.947519	+	0.319700i	$0.103582\pi$
$883$	19.0000i	0.639401i	−0.947519	+	0.319700i	$0.896418\pi$
$884$	−80.0000	−2.69069
$885$	0	0
$886$	24.0000	0.806296
$887$	− 48.0000i	− 1.61168i	−0.592132	−	0.805841i	$-0.701714\pi$
$887$	− 48.0000i	− 1.61168i	0.592132	−	0.805841i	$-0.298286\pi$
$888$	0	0
$889$	−24.0000	−0.804934
$890$	0	0
$891$	0	0
$892$	16.0000i	0.535720i
$893$	− 4.00000i	− 0.133855i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	8.00000i	0.266963i
$899$	0	0
$900$	0	0
$901$	16.0000	0.533037
$902$	− 40.0000i	− 1.33185i
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 21.0000i	− 0.697294i	−0.937254	−	0.348647i	$-0.886641\pi$
$907$	− 21.0000i	− 0.697294i	0.937254	−	0.348647i	$-0.113359\pi$
$908$	20.0000i	0.663723i
$909$	0	0
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	12.0000i	0.397142i
$914$	44.0000	1.45539
$915$	0	0
$916$	12.0000	0.396491
$917$	− 36.0000i	− 1.18882i
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	0	0
$922$	24.0000i	0.790398i
$923$	10.0000i	0.329154i
$924$	0	0
$925$	0	0
$926$	−66.0000	−2.16889
$927$	0	0
$928$	− 16.0000i	− 0.525226i
$929$	34.0000	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	− 48.0000i	− 1.57229i
$933$	0	0
$934$	−56.0000	−1.83238
$935$	0	0
$936$	0	0
$937$	33.0000i	1.07806i	0.842286	+	0.539032i	$0.181210\pi$
$937$	33.0000i	1.07806i	−0.842286	+	0.539032i	$0.818790\pi$
$938$	54.0000i	1.76316i
$939$	0	0
$940$	0	0
$941$	−22.0000	−0.717180	−0.358590	−	0.933495i	$-0.616742\pi$
$941$	−22.0000	−0.717180	−0.358590	+	0.933495i	$0.616742\pi$
$942$	0	0
$943$	60.0000i	1.95387i
$944$	−32.0000	−1.04151
$945$	0	0
$946$	16.0000	0.520205
$947$	48.0000i	1.55979i	0.625910	+	0.779895i	$0.284728\pi$
$947$	48.0000i	1.55979i	−0.625910	+	0.779895i	$0.715272\pi$
$948$	0	0
$949$	−25.0000	−0.811534
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 40.0000i	− 1.29573i	−0.761756	−	0.647864i	$-0.775663\pi$
$953$	− 40.0000i	− 1.29573i	0.761756	−	0.647864i	$-0.224337\pi$
$954$	0	0
$955$	0	0
$956$	−52.0000	−1.68180
$957$	0	0
$958$	12.0000i	0.387702i
$959$	−18.0000	−0.581250
$960$	0	0
$961$	−31.0000	−1.00000
$962$	50.0000i	1.61206i
$963$	0	0
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	0	0
$967$	− 13.0000i	− 0.418052i	−0.977910	−	0.209026i	$-0.932971\pi$
$967$	− 13.0000i	− 0.418052i	0.977910	−	0.209026i	$-0.0670293\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	− 39.0000i	− 1.25028i
$974$	10.0000	0.320421
$975$	0	0
$976$	−28.0000	−0.896258
$977$	− 62.0000i	− 1.98356i	−0.127971	−	0.991778i	$-0.540847\pi$
$977$	− 62.0000i	− 1.98356i	0.127971	−	0.991778i	$-0.459153\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	0	0
$981$	0	0
$982$	44.0000i	1.40410i
$983$	− 54.0000i	− 1.72233i	−0.508323	−	0.861166i	$-0.669735\pi$
$983$	− 54.0000i	− 1.72233i	0.508323	−	0.861166i	$-0.330265\pi$
$984$	0	0
$985$	0	0
$986$	32.0000	1.01909
$987$	0	0
$988$	10.0000i	0.318142i
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−25.0000	−0.794151	−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000	−0.794151	−0.397076	+	0.917786i	$0.629975\pi$
$992$	0	0
$993$	0	0
$994$	−12.0000	−0.380617
$995$	0	0
$996$	0	0
$997$	− 42.0000i	− 1.33015i	−0.746775	−	0.665077i	$-0.768399\pi$
$997$	− 42.0000i	− 1.33015i	0.746775	−	0.665077i	$-0.231601\pi$
$998$	32.0000i	1.01294i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.2.b.a.649.1		2
3.2	odd	2		675.2.b.b.649.2		2
5.2	odd	4		675.2.a.i.1.1		1
5.3	odd	4		135.2.a.a.1.1	✓	1
5.4	even	2	inner	675.2.b.a.649.2		2
15.2	even	4		675.2.a.a.1.1		1
15.8	even	4		135.2.a.b.1.1	yes	1
15.14	odd	2		675.2.b.b.649.1		2
20.3	even	4		2160.2.a.j.1.1		1
35.13	even	4		6615.2.a.a.1.1		1
40.3	even	4		8640.2.a.ce.1.1		1
40.13	odd	4		8640.2.a.bh.1.1		1
45.13	odd	12		405.2.e.h.136.1		2
45.23	even	12		405.2.e.b.136.1		2
45.38	even	12		405.2.e.b.271.1		2
45.43	odd	12		405.2.e.h.271.1		2
60.23	odd	4		2160.2.a.v.1.1		1
105.83	odd	4		6615.2.a.j.1.1		1
120.53	even	4		8640.2.a.c.1.1		1
120.83	odd	4		8640.2.a.bb.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.2.a.a.1.1	✓	1	5.3	odd	4
135.2.a.b.1.1	yes	1	15.8	even	4
405.2.e.b.136.1		2	45.23	even	12
405.2.e.b.271.1		2	45.38	even	12
405.2.e.h.136.1		2	45.13	odd	12
405.2.e.h.271.1		2	45.43	odd	12
675.2.a.a.1.1		1	15.2	even	4
675.2.a.i.1.1		1	5.2	odd	4
675.2.b.a.649.1		2	1.1	even	1	trivial
675.2.b.a.649.2		2	5.4	even	2	inner
675.2.b.b.649.1		2	15.14	odd	2
675.2.b.b.649.2		2	3.2	odd	2
2160.2.a.j.1.1		1	20.3	even	4
2160.2.a.v.1.1		1	60.23	odd	4
6615.2.a.a.1.1		1	35.13	even	4
6615.2.a.j.1.1		1	105.83	odd	4
8640.2.a.c.1.1		1	120.53	even	4
8640.2.a.bb.1.1		1	120.83	odd	4
8640.2.a.bh.1.1		1	40.13	odd	4
8640.2.a.ce.1.1		1	40.3	even	4

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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -2.00000 q^{4} -3.00000i q^{7} +O(q^{10})\)	\(q-2.00000i q^{2} -2.00000 q^{4} -3.00000i q^{7} -2.00000 q^{11} +5.00000i q^{13} -6.00000 q^{14} -4.00000 q^{16} -8.00000i q^{17} -1.00000 q^{19} +4.00000i q^{22} -6.00000i q^{23} +10.0000 q^{26} +6.00000i q^{28} -2.00000 q^{29} +8.00000i q^{32} -16.0000 q^{34} +5.00000i q^{37} +2.00000i q^{38} -10.0000 q^{41} -4.00000i q^{43} +4.00000 q^{44} -12.0000 q^{46} +4.00000i q^{47} -2.00000 q^{49} -10.0000i q^{52} +2.00000i q^{53} +4.00000i q^{58} +8.00000 q^{59} +7.00000 q^{61} +8.00000 q^{64} -9.00000i q^{67} +16.0000i q^{68} +2.00000 q^{71} +5.00000i q^{73} +10.0000 q^{74} +2.00000 q^{76} +6.00000i q^{77} +3.00000 q^{79} +20.0000i q^{82} -6.00000i q^{83} -8.00000 q^{86} +12.0000 q^{89} +15.0000 q^{91} +12.0000i q^{92} +8.00000 q^{94} -13.0000i q^{97} +4.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4}+O(q^{10}) \) 2 * q - 4 * q^4	\( 2 q - 4 q^{4} - 4 q^{11} - 12 q^{14} - 8 q^{16} - 2 q^{19} + 20 q^{26} - 4 q^{29} - 32 q^{34} - 20 q^{41} + 8 q^{44} - 24 q^{46} - 4 q^{49} + 16 q^{59} + 14 q^{61} + 16 q^{64} + 4 q^{71} + 20 q^{74} + 4 q^{76} + 6 q^{79} - 16 q^{86} + 24 q^{89} + 30 q^{91} + 16 q^{94}+O(q^{100}) \) 2 * q - 4 * q^4 - 4 * q^11 - 12 * q^14 - 8 * q^16 - 2 * q^19 + 20 * q^26 - 4 * q^29 - 32 * q^34 - 20 * q^41 + 8 * q^44 - 24 * q^46 - 4 * q^49 + 16 * q^59 + 14 * q^61 + 16 * q^64 + 4 * q^71 + 20 * q^74 + 4 * q^76 + 6 * q^79 - 16 * q^86 + 24 * q^89 + 30 * q^91 + 16 * q^94

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