LMFDB - Embedded newform 2700.2.s.c.2449.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2700,2,Mod(1549,2700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2700, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 3]))

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

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

chi := DirichletCharacter("2700.1549");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2700.s (of order $6$, degree $2$, 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:	$21.5596085457$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 $ x^12 - 16x^8 - 24x^7 + 96x^5 + 304x^4 + 384x^3 + 288x^2 + 144*x + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2449.6
Root			$-1.50511 - 0.403293i$ of defining polynomial
Character	$\chi$	$=$	2700.2449
Dual form			2700.2.s.c.1549.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.55142 + 2.05042i) q^{7} +(-1.90841 + 3.30545i) q^{11} +(5.03751 - 2.90841i) q^{13} +3.81681i q^{17} +1.81681 q^{19} +(-1.81937 + 1.05042i) q^{23} +(3.60083 - 6.23682i) q^{29} +(0.908405 + 1.57340i) q^{31} +6.01847i q^{37} +(5.50924 + 9.54228i) q^{41} +(-5.03751 - 2.90841i) q^{43} +(-10.3210 - 5.95882i) q^{47} +(4.90841 + 8.50161i) q^{49} -4.20166i q^{53} +(2.10083 + 3.63875i) q^{59} +(1.50924 - 2.61407i) q^{61} +(-3.21813 + 1.85799i) q^{67} +2.01847 q^{71} +8.00000i q^{73} +(-13.5551 + 7.82605i) q^{77} +(1.00000 - 1.73205i) q^{79} +(-3.37678 - 1.94958i) q^{83} -3.00000 q^{89} +23.8538 q^{91} +(10.5669 + 6.10083i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 24 q^{19} + 6 q^{29} - 12 q^{31} + 6 q^{41} + 36 q^{49} - 12 q^{59} - 42 q^{61} - 96 q^{71} + 12 q^{79} - 36 q^{89}+O(q^{100}) $ 12 * q - 24 * q^19 + 6 * q^29 - 12 * q^31 + 6 * q^41 + 36 * q^49 - 12 * q^59 - 42 * q^61 - 96 * q^71 + 12 * q^79 - 36 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$-1$

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
$5$		0			0
$5$		0			0
$6$		0			0
$7$	3.55142	+	2.05042i	1.34231	+	0.774984i	0.987146	−	0.159819i	$-0.0510912\pi$
$7$	3.55142	+	2.05042i	1.34231	+	0.774984i	0.355166	+	0.934803i	$0.384424\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.90841	+	3.30545i	−0.575406	+	0.996632i	0.420592	+	0.907250i	$0.361823\pi$
$11$	−1.90841	+	3.30545i	−0.575406	+	0.996632i	−0.995997	+	0.0893820i	$0.971511\pi$
$12$		0			0
$13$	5.03751	−	2.90841i	1.39715	−	0.806646i	0.403059	−	0.915174i	$-0.367947\pi$
$13$	5.03751	−	2.90841i	1.39715	−	0.806646i	0.994093	+	0.108527i	$0.0346135\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			3.81681i			0.925712i	0.886433	+	0.462856i	$0.153175\pi$
$17$			3.81681i			0.925712i	−0.886433	+	0.462856i	$0.846825\pi$
$18$		0			0
$19$	1.81681			0.416805			0.208402	−	0.978043i	$-0.433174\pi$
$19$	1.81681			0.416805			0.208402	+	0.978043i	$0.433174\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.81937	+	1.05042i	−0.379365	+	0.219027i	−0.677542	−	0.735484i	$-0.736955\pi$
$23$	−1.81937	+	1.05042i	−0.379365	+	0.219027i	0.298177	+	0.954511i	$0.403622\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.60083	−	6.23682i	0.668657	−	1.15815i	−0.309622	−	0.950860i	$-0.600203\pi$
$29$	3.60083	−	6.23682i	0.668657	−	1.15815i	0.978280	−	0.207289i	$-0.0664640\pi$
$30$		0			0
$31$	0.908405	+	1.57340i	0.163154	+	0.282592i	0.935998	−	0.352004i	$-0.114500\pi$
$31$	0.908405	+	1.57340i	0.163154	+	0.282592i	−0.772844	+	0.634596i	$0.781166\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$			6.01847i			0.989431i	0.869055	+	0.494715i	$0.164728\pi$
$37$			6.01847i			0.989431i	−0.869055	+	0.494715i	$0.835272\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.50924	+	9.54228i	0.860398	+	1.49025i	0.871545	+	0.490315i	$0.163118\pi$
$41$	5.50924	+	9.54228i	0.860398	+	1.49025i	−0.0111471	+	0.999938i	$0.503548\pi$
$42$		0			0
$43$	−5.03751	−	2.90841i	−0.768212	−	0.443528i	0.0640242	−	0.997948i	$-0.479607\pi$
$43$	−5.03751	−	2.90841i	−0.768212	−	0.443528i	−0.832237	+	0.554421i	$0.812940\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−10.3210	−	5.95882i	−1.50547	−	0.869183i	−0.999980	−	0.00635068i	$-0.997979\pi$
$47$	−10.3210	−	5.95882i	−1.50547	−	0.869183i	−0.505490	−	0.862833i	$-0.668688\pi$
$48$		0			0
$49$	4.90841	+	8.50161i	0.701201	+	1.21452i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	4.20166i		−	0.577143i	−0.957458	−	0.288571i	$-0.906820\pi$
$53$		−	4.20166i		−	0.577143i	0.957458	−	0.288571i	$-0.0931802\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.10083	+	3.63875i	0.273505	+	0.473724i	0.969757	−	0.244073i	$-0.0784837\pi$
$59$	2.10083	+	3.63875i	0.273505	+	0.473724i	−0.696252	+	0.717797i	$0.745150\pi$
$60$		0			0
$61$	1.50924	−	2.61407i	0.193238	−	0.334698i	−0.753084	−	0.657925i	$-0.771434\pi$
$61$	1.50924	−	2.61407i	0.193238	−	0.334698i	0.946321	+	0.323227i	$0.104768\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−3.21813	+	1.85799i	−0.393158	+	0.226990i	−0.683527	−	0.729925i	$-0.739555\pi$
$67$	−3.21813	+	1.85799i	−0.393158	+	0.226990i	0.290370	+	0.956914i	$0.406222\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.01847			0.239548			0.119774	−	0.992801i	$-0.461783\pi$
$71$	2.01847			0.239548			0.119774	+	0.992801i	$0.461783\pi$
$72$		0			0
$73$			8.00000i			0.936329i	0.883641	+	0.468165i	$0.155085\pi$
$73$			8.00000i			0.936329i	−0.883641	+	0.468165i	$0.844915\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−13.5551	+	7.82605i	−1.54475	+	0.891861i
$78$		0			0
$79$	1.00000	−	1.73205i	0.112509	−	0.194871i	−0.804272	−	0.594261i	$-0.797445\pi$
$79$	1.00000	−	1.73205i	0.112509	−	0.194871i	0.916781	+	0.399390i	$0.130778\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.37678	−	1.94958i	−0.370650	−	0.213995i	0.303093	−	0.952961i	$-0.401981\pi$
$83$	−3.37678	−	1.94958i	−0.370650	−	0.213995i	−0.673742	+	0.738966i	$0.735314\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.00000			−0.317999			−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000			−0.317999			−0.159000	+	0.987279i	$0.550827\pi$
$90$		0			0
$91$	23.8538			2.50055
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	10.5669	+	6.10083i	1.07291	+	0.619445i	0.928975	−	0.370142i	$-0.120691\pi$
$97$	10.5669	+	6.10083i	1.07291	+	0.619445i	0.143936	+	0.989587i	$0.454024\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	8.10083	−	14.0310i	0.806063	−	1.39614i	−0.109508	−	0.993986i	$-0.534928\pi$
$101$	8.10083	−	14.0310i	0.806063	−	1.39614i	0.915571	−	0.402156i	$-0.131739\pi$
$102$		0			0
$103$	−8.18431	+	4.72522i	−0.806424	+	0.465589i	−0.845713	−	0.533639i	$-0.820824\pi$
$103$	−8.18431	+	4.72522i	−0.806424	+	0.465589i	0.0392883	+	0.999228i	$0.487491\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			4.66887i			0.451357i	0.974202	+	0.225678i	$0.0724599\pi$
$107$			4.66887i			0.451357i	−0.974202	+	0.225678i	$0.927540\pi$
$108$		0			0
$109$	−8.81681			−0.844497			−0.422249	−	0.906480i	$-0.638759\pi$
$109$	−8.81681			−0.844497			−0.422249	+	0.906480i	$0.638759\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	8.83490	−	5.10083i	0.831117	−	0.479846i	−0.0231178	−	0.999733i	$-0.507359\pi$
$113$	8.83490	−	5.10083i	0.831117	−	0.479846i	0.854235	+	0.519887i	$0.174026\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−7.82605	+	13.5551i	−0.717412	+	1.24259i
$120$		0			0
$121$	−1.78402	−	3.09001i	−0.162184	−	0.280910i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	0.284020i		−	0.0252027i	−0.999921	−	0.0126014i	$-0.995989\pi$
$127$		−	0.284020i		−	0.0252027i	0.999921	−	0.0126014i	$-0.00401124\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	10.6336	+	18.4180i	0.929064	+	1.60919i	0.784892	+	0.619633i	$0.212719\pi$
$131$	10.6336	+	18.4180i	0.929064	+	1.60919i	0.144172	+	0.989553i	$0.453948\pi$
$132$		0			0
$133$	6.45226	+	3.72522i	0.559482	+	0.323017i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.22399	+	1.28402i	0.190008	+	0.109701i	0.591986	−	0.805948i	$-0.298344\pi$
$137$	2.22399	+	1.28402i	0.190008	+	0.109701i	−0.401978	+	0.915649i	$0.631677\pi$
$138$		0			0
$139$	−2.00000	−	3.46410i	−0.169638	−	0.293821i	0.768655	−	0.639664i	$-0.220926\pi$
$139$	−2.00000	−	3.46410i	−0.169638	−	0.293821i	−0.938293	+	0.345843i	$0.887593\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			22.2017i			1.85660i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	7.22522	+	12.5144i	0.591913	+	1.02522i	0.993975	+	0.109610i	$0.0349603\pi$
$149$	7.22522	+	12.5144i	0.591913	+	1.02522i	−0.402062	+	0.915612i	$0.631706\pi$
$150$		0			0
$151$	4.29326	−	7.43614i	0.349380	−	0.605144i	−0.636759	−	0.771063i	$-0.719726\pi$
$151$	4.29326	−	7.43614i	0.349380	−	0.605144i	0.986139	+	0.165918i	$0.0530588\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−17.1779	+	9.91764i	−1.37094	+	0.791514i	−0.991047	−	0.133515i	$-0.957373\pi$
$157$	−17.1779	+	9.91764i	−1.37094	+	0.791514i	−0.379896	+	0.925029i	$0.624040\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−8.61515			−0.678969
$162$		0			0
$163$		−	13.8168i		−	1.08222i	−0.840953	−	0.541108i	$-0.818005\pi$
$163$		−	13.8168i		−	1.08222i	0.840953	−	0.541108i	$-0.181995\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	5.26748	−	3.04118i	0.407610	−	0.235334i	−0.282153	−	0.959370i	$-0.591048\pi$
$167$	5.26748	−	3.04118i	0.407610	−	0.235334i	0.689762	+	0.724036i	$0.257715\pi$
$168$		0			0
$169$	10.4176	−	18.0439i	0.801357	−	1.38799i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.78140	−	2.18319i	−0.287494	−	0.165985i	0.349317	−	0.937005i	$-0.386414\pi$
$173$	−3.78140	−	2.18319i	−0.287494	−	0.165985i	−0.636811	+	0.771020i	$0.719747\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	6.38485			0.477226			0.238613	−	0.971115i	$-0.423307\pi$
$179$	6.38485			0.477226			0.238613	+	0.971115i	$0.423307\pi$
$180$		0			0
$181$	10.0656			0.748169			0.374084	−	0.927395i	$-0.377957\pi$
$181$	10.0656			0.748169			0.374084	+	0.927395i	$0.377957\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−12.6163	−	7.28402i	−0.922595	−	0.532660i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−7.82605	+	13.5551i	−0.566273	+	0.980813i	0.430657	+	0.902516i	$0.358282\pi$
$191$	−7.82605	+	13.5551i	−0.566273	+	0.980813i	−0.996930	+	0.0782977i	$0.975052\pi$
$192$		0			0
$193$	11.6484	−	6.72522i	0.838471	−	0.484092i	−0.0182730	−	0.999833i	$-0.505817\pi$
$193$	11.6484	−	6.72522i	0.838471	−	0.484092i	0.856744	+	0.515741i	$0.172483\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	1.79834i		−	0.128126i	−0.997946	−	0.0640632i	$-0.979594\pi$
$197$		−	1.79834i		−	0.128126i	0.997946	−	0.0640632i	$-0.0204059\pi$
$198$		0			0
$199$	22.2201			1.57514			0.787572	−	0.616223i	$-0.211338\pi$
$199$	22.2201			1.57514			0.787572	+	0.616223i	$0.211338\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	25.5761	−	14.7664i	1.79509	−	1.03640i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−3.46721	+	6.00538i	−0.239832	+	0.415401i
$210$		0			0
$211$	14.1101	+	24.4394i	0.971377	+	1.68247i	0.691407	+	0.722466i	$0.256991\pi$
$211$	14.1101	+	24.4394i	0.971377	+	1.68247i	0.279970	+	0.960009i	$0.409675\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$			7.45043i			0.505768i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	11.1008	+	19.2272i	0.746723	+	1.29336i
$222$		0			0
$223$	−1.66073	−	0.958820i	−0.111210	−	0.0642074i	0.443363	−	0.896342i	$-0.353785\pi$
$223$	−1.66073	−	0.958820i	−0.111210	−	0.0642074i	−0.554573	+	0.832135i	$0.687119\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.94420	+	4.00924i	0.460903	+	0.266102i	0.712424	−	0.701750i	$-0.247597\pi$
$227$	6.94420	+	4.00924i	0.460903	+	0.266102i	−0.251521	+	0.967852i	$0.580931\pi$
$228$		0			0
$229$	−9.41764	−	16.3118i	−0.622335	−	1.07792i	−0.989050	−	0.147583i	$-0.952851\pi$
$229$	−9.41764	−	16.3118i	−0.622335	−	1.07792i	0.366715	−	0.930334i	$-0.380483\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			14.0185i			0.918381i	0.888338	+	0.459190i	$0.151860\pi$
$233$			14.0185i			0.918381i	−0.888338	+	0.459190i	$0.848140\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−5.10083	−	8.83490i	−0.329945	−	0.571482i	0.652555	−	0.757741i	$-0.273697\pi$
$239$	−5.10083	−	8.83490i	−0.329945	−	0.571482i	−0.982501	+	0.186259i	$0.940364\pi$
$240$		0			0
$241$	−8.61007	+	14.9131i	−0.554623	+	0.960635i	0.443310	+	0.896369i	$0.353804\pi$
$241$	−8.61007	+	14.9131i	−0.554623	+	0.960635i	−0.997933	+	0.0642669i	$0.979529\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	9.15219	−	5.28402i	0.582340	−	0.336214i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	0.384851			0.0242916			0.0121458	−	0.999926i	$-0.496134\pi$
$251$	0.384851			0.0242916			0.0121458	+	0.999926i	$0.496134\pi$
$252$		0			0
$253$		−	8.01847i		−	0.504117i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	8.02567	−	4.63362i	0.500627	−	0.289037i	−0.228345	−	0.973580i	$-0.573332\pi$
$257$	8.02567	−	4.63362i	0.500627	−	0.289037i	0.728973	+	0.684543i	$0.239998\pi$
$258$		0			0
$259$	−12.3404	+	21.3741i	−0.766793	+	1.32812i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−22.3900	−	12.9269i	−1.38063	−	0.797105i	−0.388393	−	0.921494i	$-0.626970\pi$
$263$	−22.3900	−	12.9269i	−1.38063	−	0.797105i	−0.992234	+	0.124388i	$0.960303\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	28.0841			1.71231			0.856157	−	0.516715i	$-0.172845\pi$
$269$	28.0841			1.71231			0.856157	+	0.516715i	$0.172845\pi$
$270$		0			0
$271$	−21.4504			−1.30302			−0.651510	−	0.758640i	$-0.725864\pi$
$271$	−21.4504			−1.30302			−0.651510	+	0.758640i	$0.725864\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−6.76956	−	3.90841i	−0.406743	−	0.234833i	0.282646	−	0.959224i	$-0.408788\pi$
$277$	−6.76956	−	3.90841i	−0.406743	−	0.234833i	−0.689389	+	0.724391i	$0.742121\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.59159	−	4.48877i	0.154602	−	0.267778i	−0.778312	−	0.627877i	$-0.783924\pi$
$281$	2.59159	−	4.48877i	0.154602	−	0.267778i	0.932914	+	0.360100i	$0.117257\pi$
$282$		0			0
$283$	−15.6758	+	9.05042i	−0.931828	+	0.537991i	−0.887389	−	0.461021i	$-0.847483\pi$
$283$	−15.6758	+	9.05042i	−0.931828	+	0.537991i	−0.0444390	+	0.999012i	$0.514150\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			45.1849i			2.66718i
$288$		0			0
$289$	2.43196			0.143056
$290$		0			0
$291$		0			0
$292$		0			0
$293$	14.3643	−	8.29326i	0.839174	−	0.484497i	−0.0178096	−	0.999841i	$-0.505669\pi$
$293$	14.3643	−	8.29326i	0.839174	−	0.484497i	0.856983	+	0.515344i	$0.172336\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.11007	+	10.5829i	−0.353354	+	0.612027i
$300$		0			0
$301$	−11.9269	−	20.6580i	−0.687454	−	1.19070i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			22.3025i			1.27287i	0.771330	+	0.636435i	$0.219592\pi$
$307$			22.3025i			1.27287i	−0.771330	+	0.636435i	$0.780408\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−13.8260	−	23.9474i	−0.784003	−	1.35793i	−0.929593	−	0.368588i	$-0.879841\pi$
$311$	−13.8260	−	23.9474i	−0.784003	−	1.35793i	0.145590	−	0.989345i	$-0.453492\pi$
$312$		0			0
$313$	−10.0910	−	5.82605i	−0.570377	−	0.329308i	0.186923	−	0.982375i	$-0.440149\pi$
$313$	−10.0910	−	5.82605i	−0.570377	−	0.329308i	−0.757300	+	0.653067i	$0.773482\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−10.0590	−	5.80757i	−0.564971	−	0.326186i	0.190168	−	0.981752i	$-0.439097\pi$
$317$	−10.0590	−	5.80757i	−0.564971	−	0.326186i	−0.755138	+	0.655566i	$0.772430\pi$
$318$		0			0
$319$	13.7437	+	23.8048i	0.769499	+	1.33281i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			6.93442i			0.385841i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−24.4361	−	42.3246i	−1.34721	−	2.33343i
$330$		0			0
$331$	9.71598	−	16.8286i	0.534039	−	0.924982i	−0.465171	−	0.885221i	$-0.654007\pi$
$331$	9.71598	−	16.8286i	0.534039	−	0.924982i	0.999209	−	0.0397609i	$-0.0126596\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	8.51760	−	4.91764i	0.463983	−	0.267881i	−0.249734	−	0.968314i	$-0.580343\pi$
$337$	8.51760	−	4.91764i	0.463983	−	0.267881i	0.713718	+	0.700433i	$0.247010\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6.93442			−0.375520
$342$		0			0
$343$			11.5513i			0.623709i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	19.5605	−	11.2933i	1.05006	−	0.606254i	0.127395	−	0.991852i	$-0.459338\pi$
$347$	19.5605	−	11.2933i	1.05006	−	0.606254i	0.922667	+	0.385598i	$0.126005\pi$
$348$		0			0
$349$	2.93196	−	5.07830i	0.156944	−	0.271835i	−0.776821	−	0.629721i	$-0.783169\pi$
$349$	2.93196	−	5.07830i	0.156944	−	0.271835i	0.933765	+	0.357886i	$0.116502\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−20.7846	−	12.0000i	−1.10625	−	0.638696i	−0.168397	−	0.985719i	$-0.553859\pi$
$353$	−20.7846	−	12.0000i	−1.10625	−	0.638696i	−0.937856	+	0.347024i	$0.887192\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−19.2488			−1.01591			−0.507956	−	0.861383i	$-0.669599\pi$
$359$	−19.2488			−1.01591			−0.507956	+	0.861383i	$0.669599\pi$
$360$		0			0
$361$	−15.6992			−0.826274
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	16.0964	+	9.29326i	0.840225	+	0.485104i	0.857341	−	0.514750i	$-0.172115\pi$
$367$	16.0964	+	9.29326i	0.840225	+	0.485104i	−0.0171158	+	0.999854i	$0.505448\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	8.61515	−	14.9219i	0.447276	−	0.774705i
$372$		0			0
$373$	0.174643	−	0.100830i	0.00904269	−	0.00522080i	−0.495472	−	0.868624i	$-0.665005\pi$
$373$	0.174643	−	0.100830i	0.00904269	−	0.00522080i	0.504515	+	0.863403i	$0.331671\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	41.8907i		−	2.15748i
$378$		0			0
$379$	2.36638			0.121553			0.0607764	−	0.998151i	$-0.480642\pi$
$379$	2.36638			0.121553			0.0607764	+	0.998151i	$0.480642\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−22.3900	+	12.9269i	−1.14408	+	0.660533i	−0.947437	−	0.319943i	$-0.896336\pi$
$383$	−22.3900	+	12.9269i	−1.14408	+	0.660533i	−0.196639	+	0.980476i	$0.563003\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	8.89409	−	15.4050i	0.450948	−	0.781065i	−0.547497	−	0.836808i	$-0.684419\pi$
$389$	8.89409	−	15.4050i	0.450948	−	0.781065i	0.998445	+	0.0557426i	$0.0177526\pi$
$390$		0			0
$391$	−4.00924	−	6.94420i	−0.202756	−	0.351183i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	13.6706i		−	0.686106i	−0.939316	−	0.343053i	$-0.888539\pi$
$397$		−	13.6706i		−	0.686106i	0.939316	−	0.343053i	$-0.111461\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.73445	−	6.46826i	−0.186490	−	0.323009i	0.757588	−	0.652733i	$-0.226378\pi$
$401$	−3.73445	−	6.46826i	−0.186490	−	0.323009i	−0.944077	+	0.329724i	$0.893044\pi$
$402$		0			0
$403$	9.15219	+	5.28402i	0.455903	+	0.263216i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−19.8938	−	11.4857i	−0.986098	−	0.569324i
$408$		0			0
$409$	−14.6521	−	25.3782i	−0.724499	−	1.25487i	−0.959180	−	0.282797i	$-0.908738\pi$
$409$	−14.6521	−	25.3782i	−0.724499	−	1.25487i	0.234680	−	0.972073i	$-0.424596\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			17.2303i			0.847848i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	18.1101	+	31.3676i	0.884735	+	1.53241i	0.846018	+	0.533155i	$0.178994\pi$
$419$	18.1101	+	31.3676i	0.884735	+	1.53241i	0.0387171	+	0.999250i	$0.487673\pi$
$420$		0			0
$421$	13.0185	−	22.5487i	0.634481	−	1.09895i	−0.352143	−	0.935946i	$-0.614547\pi$
$421$	13.0185	−	22.5487i	0.634481	−	1.09895i	0.986625	−	0.163008i	$-0.0521196\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	10.7199	−	6.18912i	0.518771	−	0.299512i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−22.0369			−1.06148			−0.530741	−	0.847534i	$-0.678086\pi$
$431$	−22.0369			−1.06148			−0.530741	+	0.847534i	$0.678086\pi$
$432$		0			0
$433$		−	0.183190i		−	0.00880354i	−0.999990	−	0.00440177i	$-0.998599\pi$
$433$		−	0.183190i		−	0.00880354i	0.999990	−	0.00440177i	$-0.00140113\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.30545	+	1.90841i	−0.158121	+	0.0912914i
$438$		0			0
$439$	−11.3025	+	19.5765i	−0.539438	+	0.934335i	0.459496	+	0.888180i	$0.348030\pi$
$439$	−11.3025	+	19.5765i	−0.539438	+	0.934335i	−0.998934	+	0.0461549i	$0.985303\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−25.9094	−	14.9588i	−1.23099	−	0.710715i	−0.263757	−	0.964589i	$-0.584962\pi$
$443$	−25.9094	−	14.9588i	−1.23099	−	0.710715i	−0.967237	+	0.253874i	$0.918295\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	12.1647			0.574089			0.287044	−	0.957917i	$-0.407327\pi$
$449$	12.1647			0.574089			0.287044	+	0.957917i	$0.407327\pi$
$450$		0			0
$451$	−42.0554			−1.98031
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−29.4929	−	17.0277i	−1.37962	−	0.796523i	−0.387504	−	0.921868i	$-0.626663\pi$
$457$	−29.4929	−	17.0277i	−1.37962	−	0.796523i	−0.992113	+	0.125345i	$0.959996\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	19.2529	−	33.3470i	0.896698	−	1.55313i	0.0650090	−	0.997885i	$-0.479292\pi$
$461$	19.2529	−	33.3470i	0.896698	−	1.55313i	0.831689	−	0.555242i	$-0.187374\pi$
$462$		0			0
$463$	−0.0159964	+	0.00923555i	−0.000743418	+	0.000429212i	−0.500372	−	0.865811i	$-0.666803\pi$
$463$	−0.0159964	+	0.00923555i	−0.000743418	+	0.000429212i	0.499628	+	0.866240i	$0.333470\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	10.5865i		−	0.489885i	−0.969538	−	0.244943i	$-0.921231\pi$
$467$		−	10.5865i		−	0.489885i	0.969538	−	0.244943i	$-0.0787691\pi$
$468$		0			0
$469$	−15.2386			−0.703653
$470$		0			0
$471$		0			0
$472$		0			0
$473$	19.2272	−	11.1008i	0.884068	−	0.510417i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	3.38485	−	5.86273i	0.154658	−	0.267875i	−0.778277	−	0.627922i	$-0.783906\pi$
$479$	3.38485	−	5.86273i	0.154658	−	0.267875i	0.932934	+	0.360046i	$0.117239\pi$
$480$		0			0
$481$	17.5042	+	30.3181i	0.798121	+	1.38239i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	31.2857i		−	1.41769i	−0.705364	−	0.708845i	$-0.749216\pi$
$487$		−	31.2857i		−	1.41769i	0.705364	−	0.708845i	$-0.250784\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	9.08236	+	15.7311i	0.409881	+	0.709935i	0.994876	−	0.101102i	$-0.0322368\pi$
$491$	9.08236	+	15.7311i	0.409881	+	0.709935i	−0.584995	+	0.811037i	$0.698904\pi$
$492$		0			0
$493$	23.8048	+	13.7437i	1.07211	+	0.618985i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	7.16845	+	4.13870i	0.321549	+	0.185646i
$498$		0			0
$499$	−1.26555	−	2.19200i	−0.0566538	−	0.0981272i	0.836308	−	0.548261i	$-0.184710\pi$
$499$	−1.26555	−	2.19200i	−0.0566538	−	0.0981272i	−0.892961	+	0.450133i	$0.851376\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	18.6873i		−	0.833227i	−0.909084	−	0.416614i	$-0.863217\pi$
$503$		−	18.6873i		−	0.833227i	0.909084	−	0.416614i	$-0.136783\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−4.93196	−	8.54240i	−0.218605	−	0.378635i	0.735777	−	0.677224i	$-0.236817\pi$
$509$	−4.93196	−	8.54240i	−0.218605	−	0.378635i	−0.954382	+	0.298589i	$0.903484\pi$
$510$		0			0
$511$	−16.4033	+	28.4114i	−0.725640	+	1.25685i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	39.3932	−	22.7437i	1.73251	−	1.00027i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−9.60498			−0.420802			−0.210401	−	0.977615i	$-0.567477\pi$
$521$	−9.60498			−0.420802			−0.210401	+	0.977615i	$0.567477\pi$
$522$		0			0
$523$			4.32096i			0.188942i	0.995528	+	0.0944712i	$0.0301160\pi$
$523$			4.32096i			0.188942i	−0.995528	+	0.0944712i	$0.969884\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.00538	+	3.46721i	−0.261599	+	0.151034i
$528$		0			0
$529$	−9.29326	+	16.0964i	−0.404055	+	0.699843i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	55.5056	+	32.0462i	2.40421	+	1.38807i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−37.4689			−1.61390
$540$		0			0
$541$	4.23030			0.181875			0.0909374	−	0.995857i	$-0.471014\pi$
$541$	4.23030			0.181875			0.0909374	+	0.995857i	$0.471014\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−8.92222	−	5.15125i	−0.381487	−	0.220251i	0.296978	−	0.954884i	$-0.404021\pi$
$547$	−8.92222	−	5.15125i	−0.381487	−	0.220251i	−0.678465	+	0.734633i	$0.737354\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6.54203	−	11.3311i	0.278700	−	0.482722i
$552$		0			0
$553$	7.10285	−	4.10083i	0.302044	−	0.174385i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			6.00000i			0.254228i	0.991888	+	0.127114i	$0.0405714\pi$
$557$			6.00000i			0.254228i	−0.991888	+	0.127114i	$0.959429\pi$
$558$		0			0
$559$	−33.8353			−1.43108
$560$		0			0
$561$		0			0
$562$		0			0
$563$	8.23964	−	4.75716i	0.347260	−	0.200490i	−0.316218	−	0.948687i	$-0.602413\pi$
$563$	8.23964	−	4.75716i	0.347260	−	0.200490i	0.663478	+	0.748196i	$0.269080\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	10.6336	−	18.4180i	0.445785	−	0.772122i	−0.552322	−	0.833631i	$-0.686258\pi$
$569$	10.6336	−	18.4180i	0.445785	−	0.772122i	0.998107	+	0.0615094i	$0.0195914\pi$
$570$		0			0
$571$	−8.20166	−	14.2057i	−0.343229	−	0.594490i	0.641802	−	0.766871i	$-0.278187\pi$
$571$	−8.20166	−	14.2057i	−0.343229	−	0.594490i	−0.985030	+	0.172381i	$0.944854\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	34.3328i		−	1.42929i	−0.699485	−	0.714647i	$-0.746587\pi$
$577$		−	34.3328i		−	1.42929i	0.699485	−	0.714647i	$-0.253413\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−7.99492	−	13.8476i	−0.331685	−	0.574495i
$582$		0			0
$583$	13.8884	+	8.01847i	0.575199	+	0.332091i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−15.3265	−	8.84875i	−0.632592	−	0.365227i	0.149163	−	0.988813i	$-0.452342\pi$
$587$	−15.3265	−	8.84875i	−0.632592	−	0.365227i	−0.781755	+	0.623585i	$0.785675\pi$
$588$		0			0
$589$	1.65040	+	2.85858i	0.0680035	+	0.117786i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			45.3227i			1.86118i	0.366065	+	0.930589i	$0.380705\pi$
$593$			45.3227i			1.86118i	−0.366065	+	0.930589i	$0.619295\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	5.01847	+	8.69225i	0.205049	+	0.355156i	0.950148	−	0.311798i	$-0.100931\pi$
$599$	5.01847	+	8.69225i	0.205049	+	0.355156i	−0.745099	+	0.666954i	$0.767598\pi$
$600$		0			0
$601$	−16.6521	+	28.8423i	−0.679253	+	1.17650i	0.295953	+	0.955202i	$0.404363\pi$
$601$	−16.6521	+	28.8423i	−0.679253	+	1.17650i	−0.975206	+	0.221298i	$0.928971\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	3.25013	−	1.87646i	0.131919	−	0.0761632i	−0.432588	−	0.901592i	$-0.642400\pi$
$607$	3.25013	−	1.87646i	0.131919	−	0.0761632i	0.564507	+	0.825428i	$0.309066\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−69.3227			−2.80449
$612$		0			0
$613$		−	3.45043i		−	0.139362i	−0.997569	−	0.0696808i	$-0.977802\pi$
$613$		−	3.45043i		−	0.139362i	0.997569	−	0.0696808i	$-0.0221981\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	11.6644	−	6.73445i	0.469592	−	0.271119i	−0.246477	−	0.969149i	$-0.579273\pi$
$617$	11.6644	−	6.73445i	0.469592	−	0.271119i	0.716069	+	0.698030i	$0.245940\pi$
$618$		0			0
$619$	1.27478	−	2.20799i	0.0512379	−	0.0887467i	−0.839269	−	0.543717i	$-0.817017\pi$
$619$	1.27478	−	2.20799i	0.0512379	−	0.0887467i	0.890507	+	0.454970i	$0.150350\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−10.6543	−	6.15125i	−0.426854	−	0.246444i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−22.9714			−0.915928
$630$		0			0
$631$	30.2017			1.20231			0.601155	−	0.799133i	$-0.294708\pi$
$631$	30.2017			1.20231			0.601155	+	0.799133i	$0.294708\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	49.4522	+	28.5513i	1.95937	+	1.13124i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	8.20674	−	14.2145i	0.324147	−	0.561439i	−0.657192	−	0.753723i	$-0.728256\pi$
$641$	8.20674	−	14.2145i	0.324147	−	0.561439i	0.981339	+	0.192284i	$0.0615895\pi$
$642$		0			0
$643$	−1.97802	+	1.14201i	−0.0780055	+	0.0450365i	−0.538495	−	0.842628i	$-0.681007\pi$
$643$	−1.97802	+	1.14201i	−0.0780055	+	0.0450365i	0.460490	+	0.887665i	$0.347674\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			6.08236i			0.239122i	0.992827	+	0.119561i	$0.0381487\pi$
$647$			6.08236i			0.239122i	−0.992827	+	0.119561i	$0.961851\pi$
$648$		0			0
$649$	−16.0369			−0.629505
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.97742	+	5.76047i	−0.390447	+	0.225424i	−0.682354	−	0.731022i	$-0.739044\pi$
$653$	−9.97742	+	5.76047i	−0.390447	+	0.225424i	0.291907	+	0.956447i	$0.405710\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	14.5328	−	25.1715i	0.566117	−	0.980544i	−0.430828	−	0.902434i	$-0.641778\pi$
$659$	14.5328	−	25.1715i	0.566117	−	0.980544i	0.996945	−	0.0781094i	$-0.0248883\pi$
$660$		0			0
$661$	−20.4689	−	35.4532i	−0.796148	−	1.37897i	−0.922108	−	0.386933i	$-0.873534\pi$
$661$	−20.4689	−	35.4532i	−0.796148	−	1.37897i	0.125960	−	0.992035i	$-0.459799\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			15.1295i			0.585815i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5.76047	+	9.97742i	0.222380	+	0.385174i
$672$		0			0
$673$	−24.2647	−	14.0092i	−0.935336	−	0.540016i	−0.0468406	−	0.998902i	$-0.514915\pi$
$673$	−24.2647	−	14.0092i	−0.935336	−	0.540016i	−0.888495	+	0.458886i	$0.848249\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	44.7320	+	25.8260i	1.71919	+	0.992576i	0.920410	+	0.390954i	$0.127855\pi$
$677$	44.7320	+	25.8260i	1.71919	+	0.992576i	0.798781	+	0.601622i	$0.205479\pi$
$678$		0			0
$679$	25.0185	+	43.3333i	0.960121	+	1.66298i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	13.8538i		−	0.530099i	−0.964235	−	0.265050i	$-0.914612\pi$
$683$		−	13.8538i		−	0.530099i	0.964235	−	0.265050i	$-0.0853883\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−12.2201	−	21.1659i	−0.465550	−	0.806356i
$690$		0			0
$691$	−6.45043	+	11.1725i	−0.245386	+	0.425021i	−0.962240	−	0.272202i	$-0.912248\pi$
$691$	−6.45043	+	11.1725i	−0.245386	+	0.425021i	0.716854	+	0.697223i	$0.245581\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−36.4211	+	21.0277i	−1.37955	+	0.796481i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.75123			−0.0661430			−0.0330715	−	0.999453i	$-0.510529\pi$
$701$	−1.75123			−0.0661430			−0.0330715	+	0.999453i	$0.510529\pi$
$702$		0			0
$703$			10.9344i			0.412399i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	57.5390	−	33.2201i	2.16398	−	1.24937i
$708$		0			0
$709$	7.16887	−	12.4168i	0.269233	−	0.466325i	−0.699431	−	0.714700i	$-0.746563\pi$
$709$	7.16887	−	12.4168i	0.269233	−	0.466325i	0.968664	+	0.248375i	$0.0798966\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3.30545	−	1.90841i	−0.123790	−	0.0714703i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−12.6050			−0.470087			−0.235043	−	0.971985i	$-0.575523\pi$
$719$	−12.6050			−0.470087			−0.235043	+	0.971985i	$0.575523\pi$
$720$		0			0
$721$	−38.7546			−1.44330
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−37.9231	−	21.8949i	−1.40649	−	0.812038i	−0.411443	−	0.911435i	$-0.634975\pi$
$727$	−37.9231	−	21.8949i	−1.40649	−	0.812038i	−0.995048	+	0.0993972i	$0.968309\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	11.1008	−	19.2272i	0.410579	−	0.711144i
$732$		0			0
$733$	6.18003	−	3.56804i	0.228265	−	0.131789i	−0.381507	−	0.924366i	$-0.624595\pi$
$733$	6.18003	−	3.56804i	0.228265	−	0.131789i	0.609771	+	0.792578i	$0.291261\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	14.1832i		−	0.522445i
$738$		0			0
$739$	44.6419			1.64218			0.821090	−	0.570799i	$-0.193366\pi$
$739$	44.6419			1.64218			0.821090	+	0.570799i	$0.193366\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	4.18601	−	2.41679i	0.153570	−	0.0886636i	−0.421246	−	0.906946i	$-0.638407\pi$
$743$	4.18601	−	2.41679i	0.153570	−	0.0886636i	0.574816	+	0.818283i	$0.305074\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−9.57312	+	16.5811i	−0.349794	+	0.605862i
$750$		0			0
$751$	−14.4412	−	25.0129i	−0.526967	−	0.912733i	−0.999506	−	0.0314236i	$-0.989996\pi$
$751$	−14.4412	−	25.0129i	−0.526967	−	0.912733i	0.472539	−	0.881310i	$-0.343337\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	30.5865i		−	1.11169i	−0.831287	−	0.555843i	$-0.812396\pi$
$757$		−	30.5865i		−	1.11169i	0.831287	−	0.555843i	$-0.187604\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−17.6193	−	30.5175i	−0.638699	−	1.10626i	−0.985718	−	0.168401i	$-0.946140\pi$
$761$	−17.6193	−	30.5175i	−0.638699	−	1.10626i	0.347019	−	0.937858i	$-0.387194\pi$
$762$		0			0
$763$	−31.3122	−	18.0781i	−1.13358	−	0.654472i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	21.1659	+	12.2201i	0.764256	+	0.441243i
$768$		0			0
$769$	−22.6193	−	39.1778i	−0.815673	−	1.41279i	−0.908844	−	0.417137i	$-0.863034\pi$
$769$	−22.6193	−	39.1778i	−0.815673	−	1.41279i	0.0931709	−	0.995650i	$-0.470300\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	27.2672i		−	0.980734i	−0.871516	−	0.490367i	$-0.836863\pi$
$773$		−	27.2672i		−	0.980734i	0.871516	−	0.490367i	$-0.163137\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	10.0092	+	17.3365i	0.358618	+	0.621145i
$780$		0			0
$781$	−3.85206	+	6.67196i	−0.137838	+	0.238742i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−11.6484	+	6.72522i	−0.415221	+	0.239728i	−0.693031	−	0.720908i	$-0.743725\pi$
$787$	−11.6484	+	6.72522i	−0.415221	+	0.239728i	0.277809	+	0.960636i	$0.410392\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	41.8353			1.48749
$792$		0			0
$793$		−	17.5579i		−	0.623498i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2.78152	+	1.60591i	−0.0985266	+	0.0568844i	−0.548454	−	0.836181i	$-0.684783\pi$
$797$	−2.78152	+	1.60591i	−0.0985266	+	0.0568844i	0.449927	+	0.893065i	$0.351450\pi$
$798$		0			0
$799$	22.7437	−	39.3932i	0.804614	−	1.39363i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−26.4436	−	15.2672i	−0.933176	−	0.538769i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	43.4689			1.52829			0.764143	−	0.645047i	$-0.223162\pi$
$809$	43.4689			1.52829			0.764143	+	0.645047i	$0.223162\pi$
$810$		0			0
$811$	−18.4033			−0.646228			−0.323114	−	0.946360i	$-0.604730\pi$
$811$	−18.4033			−0.646228			−0.323114	+	0.946360i	$0.604730\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−9.15219	−	5.28402i	−0.320195	−	0.184864i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−7.69243	+	13.3237i	−0.268467	+	0.464999i	−0.968466	−	0.249145i	$-0.919851\pi$
$821$	−7.69243	+	13.3237i	−0.268467	+	0.464999i	0.699999	+	0.714144i	$0.253184\pi$
$822$		0			0
$823$	−17.2812	+	9.97729i	−0.602384	+	0.347787i	−0.769979	−	0.638069i	$-0.779733\pi$
$823$	−17.2812	+	9.97729i	−0.602384	+	0.347787i	0.167595	+	0.985856i	$0.446400\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	37.0571i		−	1.28860i	−0.764772	−	0.644301i	$-0.777148\pi$
$827$		−	37.0571i		−	1.28860i	0.764772	−	0.644301i	$-0.222852\pi$
$828$		0			0
$829$	−14.9815			−0.520330			−0.260165	−	0.965564i	$-0.583777\pi$
$829$	−14.9815			−0.520330			−0.260165	+	0.965564i	$0.583777\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−32.4490	+	18.7345i	−1.12429	+	0.649110i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	3.27478	−	5.67209i	0.113058	−	0.195822i	−0.803944	−	0.594705i	$-0.797269\pi$
$839$	3.27478	−	5.67209i	0.113058	−	0.195822i	0.917002	+	0.398883i	$0.130602\pi$
$840$		0			0
$841$	−11.4320	−	19.8007i	−0.394206	−	0.682784i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	14.6319i		−	0.502759i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6.32189	−	10.9498i	−0.216712	−	0.375356i
$852$		0			0
$853$	2.46423	+	1.42272i	0.0843736	+	0.0487131i	0.541593	−	0.840641i	$-0.317821\pi$
$853$	2.46423	+	1.42272i	0.0843736	+	0.0487131i	−0.457220	+	0.889354i	$0.651155\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−15.7791	−	9.11007i	−0.539004	−	0.311194i	0.205671	−	0.978621i	$-0.434062\pi$
$857$	−15.7791	−	9.11007i	−0.539004	−	0.311194i	−0.744675	+	0.667427i	$0.767396\pi$
$858$		0			0
$859$	−23.0277	−	39.8852i	−0.785695	−	1.36086i	−0.928583	−	0.371125i	$-0.878972\pi$
$859$	−23.0277	−	39.8852i	−0.785695	−	1.36086i	0.142888	−	0.989739i	$-0.454361\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	36.4521i		−	1.24084i	−0.784268	−	0.620422i	$-0.786961\pi$
$863$		−	36.4521i		−	1.24084i	0.784268	−	0.620422i	$-0.213039\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	3.81681	+	6.61091i	0.129476	+	0.224260i
$870$		0			0
$871$	−10.8076	+	18.7193i	−0.366201	+	0.634278i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−45.6548	+	26.3588i	−1.54165	+	0.890075i	−0.542920	+	0.839785i	$0.682681\pi$
$877$	−45.6548	+	26.3588i	−1.54165	+	0.890075i	−0.998735	+	0.0502901i	$0.983985\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−17.8824			−0.602473			−0.301237	−	0.953549i	$-0.597399\pi$
$881$	−17.8824			−0.602473			−0.301237	+	0.953549i	$0.597399\pi$
$882$		0			0
$883$			29.3496i			0.987693i	0.869549	+	0.493846i	$0.164409\pi$
$883$			29.3496i			0.987693i	−0.869549	+	0.493846i	$0.835591\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	14.4459	−	8.34036i	0.485047	−	0.280042i	−0.237470	−	0.971395i	$-0.576318\pi$
$887$	14.4459	−	8.34036i	0.485047	−	0.280042i	0.722517	+	0.691353i	$0.242985\pi$
$888$		0			0
$889$	0.582359	−	1.00868i	0.0195317	−	0.0338299i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−18.7513	−	10.8260i	−0.627487	−	0.362280i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	13.0841			0.436378
$900$		0			0
$901$	16.0369			0.534268
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.94603	+	1.12354i	0.0646168	+	0.0373065i	0.531960	−	0.846769i	$-0.321456\pi$
$907$	1.94603	+	1.12354i	0.0646168	+	0.0373065i	−0.467344	+	0.884076i	$0.654789\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−5.14794	+	8.91649i	−0.170559	+	0.295417i	−0.938615	−	0.344965i	$-0.887891\pi$
$911$	−5.14794	+	8.91649i	−0.170559	+	0.295417i	0.768057	+	0.640382i	$0.221224\pi$
$912$		0			0
$913$	12.8885	−	7.44120i	0.426548	−	0.246268i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			87.2133i			2.88004i
$918$		0			0
$919$	14.0369			0.463036			0.231518	−	0.972831i	$-0.425631\pi$
$919$	14.0369			0.463036			0.231518	+	0.972831i	$0.425631\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	10.1681	−	5.87053i	0.334686	−	0.193231i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−27.2201	+	47.1467i	−0.893064	+	1.54683i	−0.0568805	+	0.998381i	$0.518115\pi$
$929$	−27.2201	+	47.1467i	−0.893064	+	1.54683i	−0.836183	+	0.548450i	$0.815218\pi$
$930$		0			0
$931$	8.91764	+	15.4458i	0.292264	+	0.506216i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			33.0656i			1.08021i	0.841599	+	0.540103i	$0.181615\pi$
$937$			33.0656i			1.08021i	−0.841599	+	0.540103i	$0.818385\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−6.60083	−	11.4330i	−0.215181	−	0.372704i	0.738148	−	0.674639i	$-0.235701\pi$
$941$	−6.60083	−	11.4330i	−0.215181	−	0.372704i	−0.953329	+	0.301935i	$0.902367\pi$
$942$		0			0
$943$	−20.0467	−	11.5740i	−0.652810	−	0.376900i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.595256	+	0.343671i	0.0193432	+	0.0111678i	0.509640	−	0.860387i	$-0.329778\pi$
$947$	0.595256	+	0.343671i	0.0193432	+	0.0111678i	−0.490297	+	0.871555i	$0.663112\pi$
$948$		0			0
$949$	23.2672	+	40.3000i	0.755287	+	1.30819i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			32.7882i			1.06211i	0.847336	+	0.531057i	$0.178205\pi$
$953$			32.7882i			1.06211i	−0.847336	+	0.531057i	$0.821795\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	5.26555	+	9.12020i	0.170033	+	0.294507i
$960$		0			0
$961$	13.8496	−	23.9882i	0.446761	−	0.773813i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	4.99817	−	2.88570i	0.160730	−	0.0927977i	−0.417477	−	0.908687i	$-0.637086\pi$
$967$	4.99817	−	2.88570i	0.160730	−	0.0927977i	0.578208	+	0.815890i	$0.303752\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	41.1210			1.31964			0.659818	−	0.751426i	$-0.270633\pi$
$971$	41.1210			1.31964			0.659818	+	0.751426i	$0.270633\pi$
$972$		0			0
$973$		−	16.4033i		−	0.525866i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−11.3922	+	6.57728i	−0.364468	+	0.210426i	−0.671039	−	0.741422i	$-0.734152\pi$
$977$	−11.3922	+	6.57728i	−0.364468	+	0.210426i	0.306571	+	0.951848i	$0.400818\pi$
$978$		0			0
$979$	5.72522	−	9.91636i	0.182979	−	0.316928i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	15.7078	+	9.06889i	0.501000	+	0.289253i	0.729126	−	0.684379i	$-0.239927\pi$
$983$	15.7078	+	9.06889i	0.501000	+	0.289253i	−0.228126	+	0.973632i	$0.573260\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	12.2201			0.388578
$990$		0			0
$991$	11.4320			0.363148			0.181574	−	0.983377i	$-0.441881\pi$
$991$	11.4320			0.363148			0.181574	+	0.983377i	$0.441881\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−7.38815	−	4.26555i	−0.233985	−	0.135091i	0.378424	−	0.925632i	$-0.376466\pi$
$997$	−7.38815	−	4.26555i	−0.233985	−	0.135091i	−0.612409	+	0.790541i	$0.709799\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.2.s.c.2449.6		12
3.2	odd	2		900.2.s.c.349.2		12
5.2	odd	4		540.2.i.b.181.1		6
5.3	odd	4		2700.2.i.c.1801.3		6
5.4	even	2	inner	2700.2.s.c.2449.1		12
9.2	odd	6		8100.2.d.p.649.1		6
9.4	even	3	inner	2700.2.s.c.1549.1		12
9.5	odd	6		900.2.s.c.49.5		12
9.7	even	3		8100.2.d.o.649.1		6
15.2	even	4		180.2.i.b.61.3	✓	6
15.8	even	4		900.2.i.c.601.1		6
15.14	odd	2		900.2.s.c.349.5		12
20.7	even	4		2160.2.q.i.721.3		6
45.2	even	12		1620.2.a.i.1.3		3
45.4	even	6	inner	2700.2.s.c.1549.6		12
45.7	odd	12		1620.2.a.j.1.3		3
45.13	odd	12		2700.2.i.c.901.3		6
45.14	odd	6		900.2.s.c.49.2		12
45.22	odd	12		540.2.i.b.361.1		6
45.23	even	12		900.2.i.c.301.1		6
45.29	odd	6		8100.2.d.p.649.6		6
45.32	even	12		180.2.i.b.121.3	yes	6
45.34	even	6		8100.2.d.o.649.6		6
45.38	even	12		8100.2.a.v.1.1		3
45.43	odd	12		8100.2.a.u.1.1		3
60.47	odd	4		720.2.q.k.241.1		6
180.7	even	12		6480.2.a.bw.1.1		3
180.47	odd	12		6480.2.a.bt.1.1		3
180.67	even	12		2160.2.q.i.1441.3		6
180.167	odd	12		720.2.q.k.481.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.i.b.61.3	✓	6	15.2	even	4
180.2.i.b.121.3	yes	6	45.32	even	12
540.2.i.b.181.1		6	5.2	odd	4
540.2.i.b.361.1		6	45.22	odd	12
720.2.q.k.241.1		6	60.47	odd	4
720.2.q.k.481.1		6	180.167	odd	12
900.2.i.c.301.1		6	45.23	even	12
900.2.i.c.601.1		6	15.8	even	4
900.2.s.c.49.2		12	45.14	odd	6
900.2.s.c.49.5		12	9.5	odd	6
900.2.s.c.349.2		12	3.2	odd	2
900.2.s.c.349.5		12	15.14	odd	2
1620.2.a.i.1.3		3	45.2	even	12
1620.2.a.j.1.3		3	45.7	odd	12
2160.2.q.i.721.3		6	20.7	even	4
2160.2.q.i.1441.3		6	180.67	even	12
2700.2.i.c.901.3		6	45.13	odd	12
2700.2.i.c.1801.3		6	5.3	odd	4
2700.2.s.c.1549.1		12	9.4	even	3	inner
2700.2.s.c.1549.6		12	45.4	even	6	inner
2700.2.s.c.2449.1		12	5.4	even	2	inner
2700.2.s.c.2449.6		12	1.1	even	1	trivial
6480.2.a.bt.1.1		3	180.47	odd	12
6480.2.a.bw.1.1		3	180.7	even	12
8100.2.a.u.1.1		3	45.43	odd	12
8100.2.a.v.1.1		3	45.38	even	12
8100.2.d.o.649.1		6	9.7	even	3
8100.2.d.o.649.6		6	45.34	even	6
8100.2.d.p.649.1		6	9.2	odd	6
8100.2.d.p.649.6		6	45.29	odd	6

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:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2700.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(3.55142 + 2.05042i) q^{7} +(-1.90841 + 3.30545i) q^{11} +(5.03751 - 2.90841i) q^{13} +3.81681i q^{17} +1.81681 q^{19} +(-1.81937 + 1.05042i) q^{23} +(3.60083 - 6.23682i) q^{29} +(0.908405 + 1.57340i) q^{31} +6.01847i q^{37} +(5.50924 + 9.54228i) q^{41} +(-5.03751 - 2.90841i) q^{43} +(-10.3210 - 5.95882i) q^{47} +(4.90841 + 8.50161i) q^{49} -4.20166i q^{53} +(2.10083 + 3.63875i) q^{59} +(1.50924 - 2.61407i) q^{61} +(-3.21813 + 1.85799i) q^{67} +2.01847 q^{71} +8.00000i q^{73} +(-13.5551 + 7.82605i) q^{77} +(1.00000 - 1.73205i) q^{79} +(-3.37678 - 1.94958i) q^{83} -3.00000 q^{89} +23.8538 q^{91} +(10.5669 + 6.10083i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 24 q^{19} + 6 q^{29} - 12 q^{31} + 6 q^{41} + 36 q^{49} - 12 q^{59} - 42 q^{61} - 96 q^{71} + 12 q^{79} - 36 q^{89}+O(q^{100}) \) 12 * q - 24 * q^19 + 6 * q^29 - 12 * q^31 + 6 * q^41 + 36 * q^49 - 12 * q^59 - 42 * q^61 - 96 * q^71 + 12 * q^79 - 36 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

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