LMFDB - Embedded newform 2700.2.i.e.901.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2700,2,Mod(901,2700)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2700, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2700.901"); 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.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,1,0,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$21.5596085457$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.142635249.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 3x^{6} + 3x^{5} - 11x^{4} + 6x^{3} + 12x^{2} - 24x + 16 $ x^8 - 3x^7 + 3x^6 + 3x^5 - 11x^4 + 6x^3 + 12x^2 - 24*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{5} $
Twist minimal:	no (minimal twist has level 900)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			901.1
Root			$0.620769 - 1.27069i$ of defining polynomial
Character	$\chi$	$=$	2700.901
Dual form			2700.2.i.e.1801.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.70089 - 2.94604i) q^{7} +(-2.20089 - 3.81206i) q^{11} +(1.03160 - 1.78679i) q^{13} -1.40179 q^{17} -6.35717 q^{19} +(-0.0539129 + 0.0933799i) q^{23} +(4.54089 + 7.86505i) q^{29} +(-1.53160 + 2.65281i) q^{31} -1.95538 q^{37} +(-4.34788 + 7.53074i) q^{41} +(-3.56320 - 6.17165i) q^{43} +(3.74179 + 6.48096i) q^{47} +(-2.28608 + 3.95961i) q^{49} +13.0975 q^{53} +(-6.58966 + 11.4136i) q^{59} +(0.862308 + 1.49356i) q^{61} +(-6.18787 + 10.7177i) q^{67} +7.50961 q^{71} +5.42037 q^{73} +(-7.48698 + 12.9678i) q^{77} +(-4.71806 - 8.17193i) q^{79} +(-4.48698 - 7.77167i) q^{83} -4.01576 q^{89} -7.01858 q^{91} +(1.08551 + 1.88017i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{7} - 3 q^{11} - 2 q^{13} + 18 q^{17} - 8 q^{19} + 3 q^{23} + 9 q^{29} - 2 q^{31} - 2 q^{37} - 9 q^{41} - 8 q^{43} - 12 q^{47} - 9 q^{49} + 24 q^{53} + 15 q^{59} + q^{61} - 11 q^{67} + 24 q^{71}+ \cdots - 5 q^{97}+O(q^{100}) $ 8 * q + q^7 - 3 * q^11 - 2 * q^13 + 18 * q^17 - 8 * q^19 + 3 * q^23 + 9 * q^29 - 2 * q^31 - 2 * q^37 - 9 * q^41 - 8 * q^43 - 12 * q^47 - 9 * q^49 + 24 * q^53 + 15 * q^59 + q^61 - 11 * q^67 + 24 * q^71 - 20 * q^73 - 36 * q^77 + 7 * q^79 - 12 * q^83 - 6 * q^89 - 22 * q^91 - 5 * q^97

Character values

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

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

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
$7$	−1.70089	−	2.94604i	−0.642878	−	1.11350i	−0.984787	−	0.173764i	$-0.944407\pi$
$7$	−1.70089	−	2.94604i	−0.642878	−	1.11350i	0.341910	−	0.939733i	$-0.388926\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.20089	−	3.81206i	−0.663595	−	1.14938i	−0.979664	−	0.200644i	$-0.935697\pi$
$11$	−2.20089	−	3.81206i	−0.663595	−	1.14938i	0.316070	−	0.948736i	$-0.397637\pi$
$12$		0			0
$13$	1.03160	−	1.78679i	0.286115	−	0.495565i	−0.686764	−	0.726880i	$-0.740969\pi$
$13$	1.03160	−	1.78679i	0.286115	−	0.495565i	0.972879	+	0.231315i	$0.0743028\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.40179			−0.339984			−0.169992	−	0.985445i	$-0.554374\pi$
$17$	−1.40179			−0.339984			−0.169992	+	0.985445i	$0.554374\pi$
$18$		0			0
$19$	−6.35717			−1.45843			−0.729217	−	0.684283i	$-0.760115\pi$
$19$	−6.35717			−1.45843			−0.729217	+	0.684283i	$0.760115\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.0539129	+	0.0933799i	−0.0112416	+	0.0194711i	−0.871591	−	0.490233i	$-0.836912\pi$
$23$	−0.0539129	+	0.0933799i	−0.0112416	+	0.0194711i	0.860350	+	0.509704i	$0.170245\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.54089	+	7.86505i	0.843222	+	1.46050i	0.887156	+	0.461469i	$0.152678\pi$
$29$	4.54089	+	7.86505i	0.843222	+	1.46050i	−0.0439339	+	0.999034i	$0.513989\pi$
$30$		0			0
$31$	−1.53160	+	2.65281i	−0.275084	+	0.476459i	−0.970156	−	0.242481i	$-0.922039\pi$
$31$	−1.53160	+	2.65281i	−0.275084	+	0.476459i	0.695073	+	0.718940i	$0.255372\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.95538			−0.321462			−0.160731	−	0.986998i	$-0.551385\pi$
$37$	−1.95538			−0.321462			−0.160731	+	0.986998i	$0.551385\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.34788	+	7.53074i	−0.679024	+	1.17610i	0.296251	+	0.955110i	$0.404264\pi$
$41$	−4.34788	+	7.53074i	−0.679024	+	1.17610i	−0.975275	+	0.220994i	$0.929070\pi$
$42$		0			0
$43$	−3.56320	−	6.17165i	−0.543383	−	0.941167i	−0.998707	−	0.0508414i	$-0.983810\pi$
$43$	−3.56320	−	6.17165i	−0.543383	−	0.941167i	0.455323	−	0.890326i	$-0.349524\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.74179	+	6.48096i	0.545795	+	0.945345i	0.998556	+	0.0537135i	$0.0171058\pi$
$47$	3.74179	+	6.48096i	0.545795	+	0.945345i	−0.452761	+	0.891632i	$0.649561\pi$
$48$		0			0
$49$	−2.28608	+	3.95961i	−0.326583	+	0.565659i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	13.0975			1.79909			0.899543	−	0.436833i	$-0.143900\pi$
$53$	13.0975			1.79909			0.899543	+	0.436833i	$0.143900\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.58966	+	11.4136i	−0.857901	+	1.48593i	0.0160267	+	0.999872i	$0.494898\pi$
$59$	−6.58966	+	11.4136i	−0.857901	+	1.48593i	−0.873928	+	0.486056i	$0.838435\pi$
$60$		0			0
$61$	0.862308	+	1.49356i	0.110407	+	0.191231i	0.915935	−	0.401328i	$-0.131451\pi$
$61$	0.862308	+	1.49356i	0.110407	+	0.191231i	−0.805527	+	0.592559i	$0.798118\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−6.18787	+	10.7177i	−0.755969	+	1.30938i	0.188922	+	0.981992i	$0.439501\pi$
$67$	−6.18787	+	10.7177i	−0.755969	+	1.30938i	−0.944891	+	0.327385i	$0.893833\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	7.50961			0.891227			0.445614	−	0.895225i	$-0.352986\pi$
$71$	7.50961			0.891227			0.445614	+	0.895225i	$0.352986\pi$
$72$		0			0
$73$	5.42037			0.634406			0.317203	−	0.948358i	$-0.397256\pi$
$73$	5.42037			0.634406			0.317203	+	0.948358i	$0.397256\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−7.48698	+	12.9678i	−0.853220	+	1.47782i
$78$		0			0
$79$	−4.71806	−	8.17193i	−0.530824	−	0.919413i	−0.999353	−	0.0359656i	$-0.988549\pi$
$79$	−4.71806	−	8.17193i	−0.530824	−	0.919413i	0.468529	−	0.883448i	$-0.344784\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.48698	−	7.77167i	−0.492510	−	0.853052i	0.507453	−	0.861679i	$-0.330587\pi$
$83$	−4.48698	−	7.77167i	−0.492510	−	0.853052i	−0.999963	+	0.00862744i	$0.997254\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.01576			−0.425670			−0.212835	−	0.977088i	$-0.568270\pi$
$89$	−4.01576			−0.425670			−0.212835	+	0.977088i	$0.568270\pi$
$90$		0			0
$91$	−7.01858			−0.735747
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.08551	+	1.88017i	0.110217	+	0.190902i	0.915858	−	0.401503i	$-0.131512\pi$
$97$	1.08551	+	1.88017i	0.110217	+	0.190902i	−0.805641	+	0.592405i	$0.798179\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.48698	+	2.57552i	0.147960	+	0.256274i	0.930473	−	0.366360i	$-0.119396\pi$
$101$	1.48698	+	2.57552i	0.147960	+	0.256274i	−0.782513	+	0.622634i	$0.786063\pi$
$102$		0			0
$103$	3.31768	−	5.74640i	0.326901	−	0.566209i	−0.654994	−	0.755634i	$-0.727329\pi$
$103$	3.31768	−	5.74640i	0.326901	−	0.566209i	0.981895	+	0.189425i	$0.0606622\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	13.7878			1.33292			0.666459	−	0.745541i	$-0.267809\pi$
$107$	13.7878			1.33292			0.666459	+	0.745541i	$0.267809\pi$
$108$		0			0
$109$	−18.1347			−1.73699			−0.868495	−	0.495699i	$-0.834912\pi$
$109$	−18.1347			−1.73699			−0.868495	+	0.495699i	$0.834912\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.0408909	−	0.0708250i	0.00384669	−	0.00666266i	−0.864096	−	0.503328i	$-0.832109\pi$
$113$	0.0408909	−	0.0708250i	0.00384669	−	0.00666266i	0.867942	+	0.496665i	$0.165442\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.38429	+	4.12972i	0.218568	+	0.378571i
$120$		0			0
$121$	−4.18787	+	7.25361i	−0.380716	+	0.659419i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−21.5811			−1.91501			−0.957507	−	0.288410i	$-0.906873\pi$
$127$	−21.5811			−1.91501			−0.957507	+	0.288410i	$0.906873\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.39391	+	9.34253i	−0.471268	+	0.816260i	−0.999460	−	0.0328649i	$-0.989537\pi$
$131$	−5.39391	+	9.34253i	−0.471268	+	0.816260i	0.528192	+	0.849125i	$0.322870\pi$
$132$		0			0
$133$	10.8129	+	18.7284i	0.937594	+	1.62396i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.75481	−	11.6997i	−0.577102	−	0.999570i	−0.995810	−	0.0914491i	$-0.970850\pi$
$137$	−6.75481	−	11.6997i	−0.577102	−	0.999570i	0.418708	−	0.908121i	$-0.362483\pi$
$138$		0			0
$139$	−7.30499	+	12.6526i	−0.619601	+	1.07318i	0.369958	+	0.929049i	$0.379372\pi$
$139$	−7.30499	+	12.6526i	−0.619601	+	1.07318i	−0.989559	+	0.144132i	$0.953961\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−9.08178			−0.759457
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.0669350	−	0.115935i	0.00548353	−	0.00949775i	−0.863271	−	0.504741i	$-0.831588\pi$
$149$	0.0669350	−	0.115935i	0.00548353	−	0.00949775i	0.868754	+	0.495244i	$0.164921\pi$
$150$		0			0
$151$	−3.07249	−	5.32171i	−0.250036	−	0.433075i	0.713500	−	0.700656i	$-0.247109\pi$
$151$	−3.07249	−	5.32171i	−0.250036	−	0.433075i	−0.963535	+	0.267581i	$0.913776\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	7.38736	−	12.7953i	0.589575	−	1.02117i	−0.404713	−	0.914444i	$-0.632628\pi$
$157$	7.38736	−	12.7953i	0.589575	−	1.02117i	0.994288	−	0.106730i	$-0.0340382\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.366801			0.0289079
$162$		0			0
$163$	15.9451			1.24892			0.624458	−	0.781058i	$-0.285320\pi$
$163$	15.9451			1.24892			0.624458	+	0.781058i	$0.285320\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−10.2497	+	17.7529i	−0.793143	+	1.37376i	0.130869	+	0.991400i	$0.458223\pi$
$167$	−10.2497	+	17.7529i	−0.793143	+	1.37376i	−0.924012	+	0.382364i	$0.875110\pi$
$168$		0			0
$169$	4.37160	+	7.57183i	0.336277	+	0.582448i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−11.2957	−	19.5647i	−0.858796	−	1.48748i	−0.873078	−	0.487580i	$-0.837880\pi$
$173$	−11.2957	−	19.5647i	−0.858796	−	1.48748i	0.0142821	−	0.999898i	$-0.495454\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−20.5879			−1.53881			−0.769407	−	0.638759i	$-0.779448\pi$
$179$	−20.5879			−1.53881			−0.769407	+	0.638759i	$0.779448\pi$
$180$		0			0
$181$	−8.32830			−0.619038			−0.309519	−	0.950893i	$-0.600168\pi$
$181$	−8.32830			−0.619038			−0.309519	+	0.950893i	$0.600168\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	3.08519	+	5.34370i	0.225611	+	0.390770i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.39217	−	7.60747i	−0.317807	−	0.550457i	0.662224	−	0.749306i	$-0.269613\pi$
$191$	−4.39217	−	7.60747i	−0.317807	−	0.550457i	−0.980030	+	0.198849i	$0.936280\pi$
$192$		0			0
$193$	3.61538	−	6.26202i	0.260241	−	0.450750i	−0.706065	−	0.708147i	$-0.749531\pi$
$193$	3.61538	−	6.26202i	0.260241	−	0.450750i	0.966306	+	0.257397i	$0.0828647\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4.17932			−0.297764			−0.148882	−	0.988855i	$-0.547568\pi$
$197$	−4.17932			−0.297764			−0.148882	+	0.988855i	$0.547568\pi$
$198$		0			0
$199$	15.6518			1.10953			0.554763	−	0.832009i	$-0.312809\pi$
$199$	15.6518			1.10953			0.554763	+	0.832009i	$0.312809\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	15.4472	−	26.7553i	1.08418	−	1.87785i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	13.9914	+	24.2339i	0.967809	+	1.67629i
$210$		0			0
$211$	12.0770	−	20.9179i	0.831412	−	1.44005i	−0.0655057	−	0.997852i	$-0.520866\pi$
$211$	12.0770	−	20.9179i	0.831412	−	1.44005i	0.896918	−	0.442196i	$-0.145801\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	10.4204			0.707381
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.44609	+	2.50470i	−0.0972743	+	0.168484i
$222$		0			0
$223$	0.723206	+	1.25263i	0.0484295	+	0.0838823i	0.889224	−	0.457472i	$-0.151245\pi$
$223$	0.723206	+	1.25263i	0.0484295	+	0.0838823i	−0.840794	+	0.541354i	$0.817912\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.2627	−	17.7755i	−0.681158	−	1.17980i	−0.974628	−	0.223832i	$-0.928143\pi$
$227$	−10.2627	−	17.7755i	−0.681158	−	1.17980i	0.293469	−	0.955969i	$-0.405190\pi$
$228$		0			0
$229$	−10.4743	+	18.1420i	−0.692160	+	1.19886i	0.278969	+	0.960300i	$0.410007\pi$
$229$	−10.4743	+	18.1420i	−0.692160	+	1.19886i	−0.971129	+	0.238556i	$0.923326\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−9.90112			−0.648644			−0.324322	−	0.945947i	$-0.605136\pi$
$233$	−9.90112			−0.648644			−0.324322	+	0.945947i	$0.605136\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3.05391	−	5.28953i	0.197541	−	0.342151i	−0.750189	−	0.661223i	$-0.770038\pi$
$239$	3.05391	−	5.28953i	0.197541	−	0.342151i	0.947731	+	0.319072i	$0.103371\pi$
$240$		0			0
$241$	0.148392	+	0.257022i	0.00955875	+	0.0165562i	0.870765	−	0.491699i	$-0.163624\pi$
$241$	0.148392	+	0.257022i	0.00955875	+	0.0165562i	−0.861206	+	0.508255i	$0.830291\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−6.55806	+	11.3589i	−0.417279	+	0.722749i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4.51643			0.285075			0.142537	−	0.989789i	$-0.454474\pi$
$251$	4.51643			0.285075			0.142537	+	0.989789i	$0.454474\pi$
$252$		0			0
$253$	0.474626			0.0298395
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.79911	+	11.7764i	−0.424117	+	0.734591i	−0.996337	−	0.0855081i	$-0.972749\pi$
$257$	−6.79911	+	11.7764i	−0.424117	+	0.734591i	0.572221	+	0.820100i	$0.306082\pi$
$258$		0			0
$259$	3.32589	+	5.76061i	0.206661	+	0.357947i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.56179	+	6.16921i	0.219630	+	0.380409i	0.954695	−	0.297587i	$-0.0961818\pi$
$263$	3.56179	+	6.16921i	0.219630	+	0.380409i	−0.735065	+	0.677996i	$0.762848\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−8.80358			−0.536764			−0.268382	−	0.963313i	$-0.586489\pi$
$269$	−8.80358			−0.536764			−0.268382	+	0.963313i	$0.586489\pi$
$270$		0			0
$271$	−9.95885			−0.604957			−0.302478	−	0.953156i	$-0.597814\pi$
$271$	−9.95885			−0.604957			−0.302478	+	0.953156i	$0.597814\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−4.46467	−	7.73303i	−0.268256	−	0.464633i	0.700156	−	0.713990i	$-0.253114\pi$
$277$	−4.46467	−	7.73303i	−0.268256	−	0.464633i	−0.968411	+	0.249358i	$0.919781\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	16.0402	+	27.7825i	0.956879	+	1.65736i	0.730008	+	0.683439i	$0.239516\pi$
$281$	16.0402	+	27.7825i	0.956879	+	1.65736i	0.226872	+	0.973925i	$0.427150\pi$
$282$		0			0
$283$	−3.18373	+	5.51437i	−0.189253	+	0.327796i	−0.945001	−	0.327066i	$-0.893940\pi$
$283$	−3.18373	+	5.51437i	−0.189253	+	0.327796i	0.755749	+	0.654862i	$0.227273\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	29.5811			1.74612
$288$		0			0
$289$	−15.0350			−0.884411
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−13.9506	+	24.1631i	−0.815000	+	1.41162i	0.0943273	+	0.995541i	$0.469930\pi$
$293$	−13.9506	+	24.1631i	−0.815000	+	1.41162i	−0.909328	+	0.416081i	$0.863403\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.111233	+	0.192662i	0.00643278	+	0.0111419i
$300$		0			0
$301$	−12.1213	+	20.9946i	−0.698658	+	1.21011i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	12.2054			0.696597			0.348299	−	0.937384i	$-0.386760\pi$
$307$	12.2054			0.696597			0.348299	+	0.937384i	$0.386760\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2.45123	+	4.24565i	−0.138996	+	0.240749i	−0.927117	−	0.374772i	$-0.877721\pi$
$311$	−2.45123	+	4.24565i	−0.138996	+	0.240749i	0.788121	+	0.615521i	$0.211054\pi$
$312$		0			0
$313$	−8.79429	−	15.2322i	−0.497083	−	0.860972i	0.502912	−	0.864338i	$-0.332262\pi$
$313$	−8.79429	−	15.2322i	−0.497083	−	0.860972i	−0.999994	+	0.00336551i	$0.998929\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.78608	−	16.9500i	−0.549641	−	0.952007i	−0.998299	−	0.0583028i	$-0.981431\pi$
$317$	−9.78608	−	16.9500i	−0.549641	−	0.952007i	0.448658	−	0.893704i	$-0.351902\pi$
$318$		0			0
$319$	19.9880	−	34.6203i	1.11912	−	1.93836i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	8.91140			0.495844
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	12.7288	−	22.0469i	0.701759	−	1.21548i
$330$		0			0
$331$	−7.29537	−	12.6360i	−0.400990	−	0.694535i	0.592856	−	0.805309i	$-0.298000\pi$
$331$	−7.29537	−	12.6360i	−0.400990	−	0.694535i	−0.993846	+	0.110774i	$0.964667\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−4.95429	+	8.58108i	−0.269877	+	0.467441i	−0.968830	−	0.247727i	$-0.920317\pi$
$337$	−4.95429	+	8.58108i	−0.269877	+	0.467441i	0.698953	+	0.715168i	$0.253650\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	13.4836			0.730176
$342$		0			0
$343$	−8.25897			−0.445943
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−5.16000	+	8.93739i	−0.277004	+	0.479784i	−0.970639	−	0.240542i	$-0.922675\pi$
$347$	−5.16000	+	8.93739i	−0.277004	+	0.479784i	0.693635	+	0.720327i	$0.256008\pi$
$348$		0			0
$349$	−8.56767	−	14.8396i	−0.458617	−	0.794348i	0.540271	−	0.841491i	$-0.318322\pi$
$349$	−8.56767	−	14.8396i	−0.458617	−	0.794348i	−0.998888	+	0.0471429i	$0.984988\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6.63055	+	11.4845i	0.352909	+	0.611256i	0.986758	−	0.162201i	$-0.0518593\pi$
$353$	6.63055	+	11.4845i	0.352909	+	0.611256i	−0.633849	+	0.773457i	$0.718526\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−14.2817			−0.753758			−0.376879	−	0.926263i	$-0.623003\pi$
$359$	−14.2817			−0.753758			−0.376879	+	0.926263i	$0.623003\pi$
$360$		0			0
$361$	21.4136			1.12703
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	4.79056	+	8.29749i	0.250065	+	0.433125i	0.963543	−	0.267552i	$-0.0862147\pi$
$367$	4.79056	+	8.29749i	0.250065	+	0.433125i	−0.713478	+	0.700677i	$0.752881\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−22.2775	−	38.5858i	−1.15659	−	2.00328i
$372$		0			0
$373$	−12.0529	+	20.8763i	−0.624076	+	1.08093i	0.364642	+	0.931148i	$0.381191\pi$
$373$	−12.0529	+	20.8763i	−0.624076	+	1.08093i	−0.988719	+	0.149784i	$0.952142\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	18.7376			0.965033
$378$		0			0
$379$	2.73973			0.140730			0.0703651	−	0.997521i	$-0.477584\pi$
$379$	2.73973			0.140730			0.0703651	+	0.997521i	$0.477584\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−10.6766	+	18.4924i	−0.545548	+	0.944917i	0.453024	+	0.891498i	$0.350345\pi$
$383$	−10.6766	+	18.4924i	−0.545548	+	0.944917i	−0.998572	+	0.0534187i	$0.982988\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7.33485	−	12.7043i	−0.371892	−	0.644136i	0.617965	−	0.786206i	$-0.287957\pi$
$389$	−7.33485	−	12.7043i	−0.371892	−	0.644136i	−0.989857	+	0.142070i	$0.954624\pi$
$390$		0			0
$391$	0.0755745	−	0.130899i	0.00382197	−	0.00661984i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−6.43065			−0.322745			−0.161373	−	0.986894i	$-0.551592\pi$
$397$	−6.43065			−0.322745			−0.161373	+	0.986894i	$0.551592\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	9.03748	−	15.6534i	0.451310	−	0.781693i	−0.547157	−	0.837030i	$-0.684290\pi$
$401$	9.03748	−	15.6534i	0.451310	−	0.781693i	0.998468	+	0.0553373i	$0.0176234\pi$
$402$		0			0
$403$	3.16000	+	5.47329i	0.157411	+	0.272644i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4.30358	+	7.45402i	0.213320	+	0.369482i
$408$		0			0
$409$	9.39532	−	16.2732i	0.464569	−	0.804657i	−0.534613	−	0.845097i	$-0.679543\pi$
$409$	9.39532	−	16.2732i	0.464569	−	0.804657i	0.999182	+	0.0404403i	$0.0128761\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	44.8333			2.20610
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	19.8297	−	34.3461i	0.968745	−	1.67792i	0.269547	−	0.962987i	$-0.413126\pi$
$419$	19.8297	−	34.3461i	0.968745	−	1.67792i	0.699198	−	0.714928i	$-0.253540\pi$
$420$		0			0
$421$	18.7682	+	32.5076i	0.914708	+	1.58432i	0.807328	+	0.590103i	$0.200913\pi$
$421$	18.7682	+	32.5076i	0.914708	+	1.58432i	0.107380	+	0.994218i	$0.465754\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	2.93339	−	5.08078i	0.141957	−	0.245876i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−16.7357			−0.806132			−0.403066	−	0.915171i	$-0.632055\pi$
$431$	−16.7357			−0.806132			−0.403066	+	0.915171i	$0.632055\pi$
$432$		0			0
$433$	−21.0008			−1.00924			−0.504618	−	0.863343i	$-0.668367\pi$
$433$	−21.0008			−1.00924			−0.504618	+	0.863343i	$0.668367\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.342733	−	0.593631i	0.0163952	−	0.0283972i
$438$		0			0
$439$	3.01576	+	5.22345i	0.143934	+	0.249302i	0.928975	−	0.370143i	$-0.120691\pi$
$439$	3.01576	+	5.22345i	0.143934	+	0.249302i	−0.785041	+	0.619444i	$0.787358\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−3.16274	−	5.47803i	−0.150266	−	0.260269i	0.781059	−	0.624457i	$-0.214680\pi$
$443$	−3.16274	−	5.47803i	−0.150266	−	0.260269i	−0.931325	+	0.364188i	$0.881346\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	18.6594			0.880593			0.440296	−	0.897853i	$-0.354873\pi$
$449$	18.6594			0.880593			0.440296	+	0.897853i	$0.354873\pi$
$450$		0			0
$451$	38.2769			1.80239
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−13.7923	−	23.8889i	−0.645176	−	1.11748i	−0.984261	−	0.176722i	$-0.943451\pi$
$457$	−13.7923	−	23.8889i	−0.645176	−	1.11748i	0.339085	−	0.940756i	$-0.389883\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−6.76442	−	11.7163i	−0.315051	−	0.545684i	0.664398	−	0.747379i	$-0.268688\pi$
$461$	−6.76442	−	11.7163i	−0.315051	−	0.545684i	−0.979448	+	0.201696i	$0.935355\pi$
$462$		0			0
$463$	13.7421	−	23.8020i	0.638650	−	1.10617i	−0.347079	−	0.937836i	$-0.612827\pi$
$463$	13.7421	−	23.8020i	0.638650	−	1.10617i	0.985729	−	0.168338i	$-0.0538402\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	32.7515			1.51556			0.757779	−	0.652511i	$-0.226284\pi$
$467$	32.7515			1.51556			0.757779	+	0.652511i	$0.226284\pi$
$468$		0			0
$469$	42.0997			1.94398
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−15.6845	+	27.1663i	−0.721172	+	1.24911i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2.92783	+	5.07116i	0.133776	+	0.231707i	0.925129	−	0.379652i	$-0.123956\pi$
$479$	2.92783	+	5.07116i	0.133776	+	0.231707i	−0.791353	+	0.611359i	$0.790623\pi$
$480$		0			0
$481$	−2.01717	+	3.49384i	−0.0919750	+	0.159305i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	16.4864			0.747070			0.373535	−	0.927616i	$-0.378146\pi$
$487$	16.4864			0.747070			0.373535	+	0.927616i	$0.378146\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−6.85302	+	11.8698i	−0.309272	+	0.535676i	−0.978203	−	0.207649i	$-0.933419\pi$
$491$	−6.85302	+	11.8698i	−0.309272	+	0.535676i	0.668931	+	0.743324i	$0.266752\pi$
$492$		0			0
$493$	−6.36537	−	11.0251i	−0.286682	−	0.496548i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−12.7731	−	22.1236i	−0.572950	−	0.992379i
$498$		0			0
$499$	−2.23482	+	3.87082i	−0.100044	+	0.173282i	−0.911703	−	0.410851i	$-0.865232\pi$
$499$	−2.23482	+	3.87082i	−0.100044	+	0.173282i	0.811658	+	0.584132i	$0.198565\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−18.1010			−0.807084			−0.403542	−	0.914961i	$-0.632221\pi$
$503$	−18.1010			−0.807084			−0.403542	+	0.914961i	$0.632221\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−10.6044	+	18.3674i	−0.470033	+	0.814120i	−0.999413	−	0.0342644i	$-0.989091\pi$
$509$	−10.6044	+	18.3674i	−0.470033	+	0.814120i	0.529380	+	0.848385i	$0.322425\pi$
$510$		0			0
$511$	−9.21947	−	15.9686i	−0.407846	−	0.706409i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	16.4705	−	28.5278i	0.724374	−	1.25465i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	28.4507			1.24645			0.623224	−	0.782043i	$-0.285822\pi$
$521$	28.4507			1.24645			0.623224	+	0.782043i	$0.285822\pi$
$522$		0			0
$523$	−26.5111			−1.15925			−0.579625	−	0.814884i	$-0.696801\pi$
$523$	−26.5111			−1.15925			−0.579625	+	0.814884i	$0.696801\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.14698	−	3.71868i	0.0935240	−	0.161988i
$528$		0			0
$529$	11.4942	+	19.9085i	0.499747	+	0.865588i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	8.97055	+	15.5374i	0.388558	+	0.673001i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	20.1257			0.866876
$540$		0			0
$541$	−14.3132			−0.615372			−0.307686	−	0.951488i	$-0.599555\pi$
$541$	−14.3132			−0.615372			−0.307686	+	0.951488i	$0.599555\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	6.82556	+	11.8222i	0.291840	+	0.505482i	0.974245	−	0.225493i	$-0.0723992\pi$
$547$	6.82556	+	11.8222i	0.291840	+	0.505482i	−0.682405	+	0.730974i	$0.739066\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−28.8672	−	49.9994i	−1.22978	−	2.13005i
$552$		0			0
$553$	−16.0499	+	27.7992i	−0.682509	+	1.18214i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−23.4665			−0.994306			−0.497153	−	0.867663i	$-0.665621\pi$
$557$	−23.4665			−0.994306			−0.497153	+	0.867663i	$0.665621\pi$
$558$		0			0
$559$	−14.7032			−0.621880
$560$		0			0
$561$		0			0
$562$		0			0
$563$	0.291226	−	0.504418i	0.0122737	−	0.0212587i	−0.859823	−	0.510592i	$-0.829426\pi$
$563$	0.291226	−	0.504418i	0.0122737	−	0.0212587i	0.872097	+	0.489333i	$0.162760\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2.77994	−	4.81500i	−0.116541	−	0.201855i	0.801854	−	0.597521i	$-0.203847\pi$
$569$	−2.77994	−	4.81500i	−0.116541	−	0.201855i	−0.918395	+	0.395665i	$0.870514\pi$
$570$		0			0
$571$	11.3706	−	19.6945i	0.475845	−	0.824188i	−0.523772	−	0.851858i	$-0.675476\pi$
$571$	11.3706	−	19.6945i	0.475845	−	0.824188i	0.999617	+	0.0276708i	$0.00880903\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	30.5522			1.27191			0.635953	−	0.771728i	$-0.280607\pi$
$577$	30.5522			1.27191			0.635953	+	0.771728i	$0.280607\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−15.2638	+	26.4376i	−0.633247	+	1.09682i
$582$		0			0
$583$	−28.8263	−	49.9286i	−1.19386	−	2.06783i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−10.1817	−	17.6353i	−0.420245	−	0.727886i	0.575718	−	0.817648i	$-0.304722\pi$
$587$	−10.1817	−	17.6353i	−0.420245	−	0.727886i	−0.995963	+	0.0897626i	$0.971389\pi$
$588$		0			0
$589$	9.73664	−	16.8644i	0.401191	−	0.694884i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	15.5199			0.637326			0.318663	−	0.947868i	$-0.396766\pi$
$593$	15.5199			0.637326			0.318663	+	0.947868i	$0.396766\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	14.6288	−	25.3379i	0.597717	−	1.03528i	−0.395440	−	0.918492i	$-0.629408\pi$
$599$	14.6288	−	25.3379i	0.597717	−	1.03528i	0.993157	−	0.116785i	$-0.0372588\pi$
$600$		0			0
$601$	9.15254	+	15.8527i	0.373340	+	0.646644i	0.990077	−	0.140526i	$-0.0448792\pi$
$601$	9.15254	+	15.8527i	0.373340	+	0.646644i	−0.616737	+	0.787169i	$0.711546\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	8.42037	−	14.5845i	0.341772	−	0.591967i	−0.642990	−	0.765875i	$-0.722306\pi$
$607$	8.42037	−	14.5845i	0.341772	−	0.591967i	0.984762	+	0.173908i	$0.0556395\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	15.4401			0.624640
$612$		0			0
$613$	−13.5954			−0.549113			−0.274556	−	0.961571i	$-0.588531\pi$
$613$	−13.5954			−0.549113			−0.274556	+	0.961571i	$0.588531\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	14.6567	−	25.3861i	0.590056	−	1.02201i	−0.404168	−	0.914685i	$-0.632439\pi$
$617$	14.6567	−	25.3861i	0.590056	−	1.02201i	0.994224	−	0.107322i	$-0.0342277\pi$
$618$		0			0
$619$	−6.08211	−	10.5345i	−0.244461	−	0.423418i	0.717519	−	0.696539i	$-0.245278\pi$
$619$	−6.08211	−	10.5345i	−0.244461	−	0.423418i	−0.961980	+	0.273121i	$0.911944\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	6.83038	+	11.8306i	0.273653	+	0.473982i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.74103			0.109292
$630$		0			0
$631$	5.45471			0.217148			0.108574	−	0.994088i	$-0.465371\pi$
$631$	5.45471			0.217148			0.108574	+	0.994088i	$0.465371\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	4.71665	+	8.16948i	0.186881	+	0.323687i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.839996	−	1.45492i	−0.0331779	−	0.0574657i	0.848960	−	0.528458i	$-0.177229\pi$
$641$	−0.839996	−	1.45492i	−0.0331779	−	0.0574657i	−0.882138	+	0.470992i	$0.843896\pi$
$642$		0			0
$643$	−2.84173	+	4.92202i	−0.112067	+	0.194106i	−0.916603	−	0.399797i	$-0.869080\pi$
$643$	−2.84173	+	4.92202i	−0.112067	+	0.194106i	0.804537	+	0.593903i	$0.202414\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−34.3064			−1.34872			−0.674361	−	0.738401i	$-0.735581\pi$
$647$	−34.3064			−1.34872			−0.674361	+	0.738401i	$0.735581\pi$
$648$		0			0
$649$	58.0126			2.27719
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−6.31486	+	10.9377i	−0.247120	+	0.428024i	−0.962725	−	0.270480i	$-0.912817\pi$
$653$	−6.31486	+	10.9377i	−0.247120	+	0.428024i	0.715606	+	0.698504i	$0.246151\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−3.16697	−	5.48535i	−0.123368	−	0.213679i	0.797726	−	0.603020i	$-0.206036\pi$
$659$	−3.16697	−	5.48535i	−0.123368	−	0.213679i	−0.921094	+	0.389341i	$0.872703\pi$
$660$		0			0
$661$	18.8448	−	32.6401i	0.732978	−	1.26955i	−0.222628	−	0.974904i	$-0.571463\pi$
$661$	18.8448	−	32.6401i	0.732978	−	1.26955i	0.955605	−	0.294651i	$-0.0952033\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−0.979251			−0.0379167
$668$		0			0
$669$		0			0
$670$		0			0
$671$	3.79570	−	6.57434i	0.146531	−	0.253800i
$672$		0			0
$673$	15.3273	+	26.5477i	0.590824	+	1.02334i	0.994122	+	0.108268i	$0.0345306\pi$
$673$	15.3273	+	26.5477i	0.590824	+	1.02334i	−0.403298	+	0.915069i	$0.632136\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−14.4358	−	25.0035i	−0.554813	−	0.960964i	−0.997918	−	0.0644945i	$-0.979456\pi$
$677$	−14.4358	−	25.0035i	−0.554813	−	0.960964i	0.443105	−	0.896470i	$-0.353877\pi$
$678$		0			0
$679$	3.69269	−	6.39593i	0.141712	−	0.245453i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−47.0352			−1.79975			−0.899875	−	0.436147i	$-0.856343\pi$
$683$	−47.0352			−1.79975			−0.899875	+	0.436147i	$0.856343\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	13.5114	−	23.4025i	0.514745	−	0.891564i
$690$		0			0
$691$	−0.425186	−	0.736443i	−0.0161748	−	0.0280156i	0.857825	−	0.513942i	$-0.171815\pi$
$691$	−0.425186	−	0.736443i	−0.0161748	−	0.0280156i	−0.874000	+	0.485927i	$0.838482\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	6.09480	−	10.5565i	0.230857	−	0.399856i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	35.6821			1.34770			0.673848	−	0.738870i	$-0.264640\pi$
$701$	35.6821			1.34770			0.673848	+	0.738870i	$0.264640\pi$
$702$		0			0
$703$	12.4307			0.468831
$704$		0			0
$705$		0			0
$706$		0			0
$707$	5.05838	−	8.76138i	0.190240	−	0.329506i
$708$		0			0
$709$	−7.69543	−	13.3289i	−0.289008	−	0.500576i	0.684565	−	0.728951i	$-0.259992\pi$
$709$	−7.69543	−	13.3289i	−0.289008	−	0.500576i	−0.973573	+	0.228375i	$0.926659\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.165146	−	0.286042i	−0.00618477	−	0.0107123i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−25.5846			−0.954144			−0.477072	−	0.878864i	$-0.658302\pi$
$719$	−25.5846			−0.954144			−0.477072	+	0.878864i	$0.658302\pi$
$720$		0			0
$721$	−22.5721			−0.840630
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	3.06867	+	5.31509i	0.113811	+	0.197126i	0.917304	−	0.398188i	$-0.130361\pi$
$727$	3.06867	+	5.31509i	0.113811	+	0.197126i	−0.803493	+	0.595314i	$0.797028\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.99486	+	8.65135i	0.184741	+	0.319982i
$732$		0			0
$733$	2.56520	−	4.44306i	0.0947478	−	0.164108i	−0.814756	−	0.579805i	$-0.803129\pi$
$733$	2.56520	−	4.44306i	0.0947478	−	0.164108i	0.909503	+	0.415697i	$0.136462\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	54.4754			2.00663
$738$		0			0
$739$	18.8784			0.694454			0.347227	−	0.937781i	$-0.387123\pi$
$739$	18.8784			0.694454			0.347227	+	0.937781i	$0.387123\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−10.9784	+	19.0152i	−0.402759	+	0.697600i	−0.994058	−	0.108853i	$-0.965282\pi$
$743$	−10.9784	+	19.0152i	−0.402759	+	0.697600i	0.591298	+	0.806453i	$0.298616\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−23.4516	−	40.6194i	−0.856904	−	1.48420i
$750$		0			0
$751$	1.58037	−	2.73728i	0.0576686	−	0.0998849i	−0.835750	−	0.549110i	$-0.814967\pi$
$751$	1.58037	−	2.73728i	0.0576686	−	0.0998849i	0.893418	+	0.449225i	$0.148300\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−39.1753			−1.42385			−0.711926	−	0.702255i	$-0.752177\pi$
$757$	−39.1753			−1.42385			−0.711926	+	0.702255i	$0.752177\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	5.03916	−	8.72807i	0.182669	−	0.316392i	−0.760119	−	0.649783i	$-0.774860\pi$
$761$	5.03916	−	8.72807i	0.182669	−	0.316392i	0.942789	+	0.333391i	$0.108193\pi$
$762$		0			0
$763$	30.8452	+	53.4255i	1.11667	+	1.93413i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	13.5958	+	23.5486i	0.490916	+	0.850292i
$768$		0			0
$769$	−10.5790	+	18.3233i	−0.381487	+	0.660755i	−0.991275	−	0.131810i	$-0.957921\pi$
$769$	−10.5790	+	18.3233i	−0.381487	+	0.660755i	0.609788	+	0.792565i	$0.291255\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	43.2543			1.55575			0.777874	−	0.628420i	$-0.216298\pi$
$773$	43.2543			1.55575			0.777874	+	0.628420i	$0.216298\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	27.6402	−	47.8742i	0.990312	−	1.71527i
$780$		0			0
$781$	−16.5279	−	28.6271i	−0.591414	−	1.02436i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−0.589986	+	1.02189i	−0.0210307	+	0.0364263i	−0.876349	−	0.481676i	$-0.840028\pi$
$787$	−0.589986	+	1.02189i	−0.0210307	+	0.0364263i	0.855318	+	0.518103i	$0.173361\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−0.278204			−0.00989180
$792$		0			0
$793$	3.55823			0.126357
$794$		0			0
$795$		0			0
$796$		0			0
$797$	0.571407	−	0.989706i	0.0202403	−	0.0350572i	−0.855728	−	0.517426i	$-0.826890\pi$
$797$	0.571407	−	0.989706i	0.0202403	−	0.0350572i	0.875968	+	0.482369i	$0.160224\pi$
$798$		0			0
$799$	−5.24519	−	9.08494i	−0.185562	−	0.321402i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−11.9297	−	20.6628i	−0.420988	−	0.729173i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	11.3723			0.399828			0.199914	−	0.979813i	$-0.435934\pi$
$809$	11.3723			0.399828			0.199914	+	0.979813i	$0.435934\pi$
$810$		0			0
$811$	2.08590			0.0732459			0.0366230	−	0.999329i	$-0.488340\pi$
$811$	2.08590			0.0732459			0.0366230	+	0.999329i	$0.488340\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	22.6519	+	39.2342i	0.792489	+	1.37263i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	7.89564	+	13.6757i	0.275560	+	0.477284i	0.970276	−	0.242000i	$-0.0778034\pi$
$821$	7.89564	+	13.6757i	0.275560	+	0.477284i	−0.694716	+	0.719284i	$0.744470\pi$
$822$		0			0
$823$	3.66042	−	6.34003i	0.127594	−	0.221000i	−0.795150	−	0.606413i	$-0.792608\pi$
$823$	3.66042	−	6.34003i	0.127594	−	0.221000i	0.922744	+	0.385413i	$0.125941\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−14.8401			−0.516040			−0.258020	−	0.966140i	$-0.583070\pi$
$827$	−14.8401			−0.516040			−0.258020	+	0.966140i	$0.583070\pi$
$828$		0			0
$829$	−23.3346			−0.810444			−0.405222	−	0.914218i	$-0.632806\pi$
$829$	−23.3346			−0.810444			−0.405222	+	0.914218i	$0.632806\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.20461	−	5.55054i	0.111033	−	0.192315i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−7.33826	−	12.7102i	−0.253345	−	0.438806i	0.711100	−	0.703091i	$-0.248197\pi$
$839$	−7.33826	−	12.7102i	−0.253345	−	0.438806i	−0.964445	+	0.264285i	$0.914864\pi$
$840$		0			0
$841$	−26.7394	+	46.3140i	−0.922048	+	1.59703i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	28.4925			0.979014
$848$		0			0
$849$		0			0
$850$		0			0
$851$	0.105420	−	0.182593i	0.00361375	−	0.00625920i
$852$		0			0
$853$	12.2611	+	21.2369i	0.419812	+	0.727136i	0.995920	−	0.0902375i	$-0.0287626\pi$
$853$	12.2611	+	21.2369i	0.419812	+	0.727136i	−0.576108	+	0.817374i	$0.695429\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−2.79396	−	4.83929i	−0.0954400	−	0.165307i	0.814352	−	0.580371i	$-0.197092\pi$
$857$	−2.79396	−	4.83929i	−0.0954400	−	0.165307i	−0.909792	+	0.415064i	$0.863759\pi$
$858$		0			0
$859$	20.0309	−	34.6946i	0.683447	−	1.18376i	−0.290476	−	0.956882i	$-0.593814\pi$
$859$	20.0309	−	34.6946i	0.683447	−	1.18376i	0.973922	−	0.226882i	$-0.0728531\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−27.2157			−0.926432			−0.463216	−	0.886246i	$-0.653305\pi$
$863$	−27.2157			−0.926432			−0.463216	+	0.886246i	$0.653305\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−20.7679	+	35.9711i	−0.704503	+	1.22024i
$870$		0			0
$871$	12.7668	+	22.1128i	0.432588	+	0.749264i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−11.9399	+	20.6806i	−0.403183	+	0.698334i	−0.994108	−	0.108393i	$-0.965430\pi$
$877$	−11.9399	+	20.6806i	−0.403183	+	0.698334i	0.590925	+	0.806727i	$0.298763\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−28.3585			−0.955421			−0.477710	−	0.878517i	$-0.658533\pi$
$881$	−28.3585			−0.955421			−0.477710	+	0.878517i	$0.658533\pi$
$882$		0			0
$883$	−28.3449			−0.953881			−0.476941	−	0.878936i	$-0.658254\pi$
$883$	−28.3449			−0.953881			−0.476941	+	0.878936i	$0.658254\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−0.257544	+	0.446080i	−0.00864749	+	0.0149779i	−0.870317	−	0.492492i	$-0.836086\pi$
$887$	−0.257544	+	0.446080i	−0.00864749	+	0.0149779i	0.861669	+	0.507470i	$0.169419\pi$
$888$		0			0
$889$	36.7072	+	63.5787i	1.23112	+	2.13236i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−23.7871	−	41.2005i	−0.796007	−	1.37872i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−27.8193			−0.927827
$900$		0			0
$901$	−18.3600			−0.611660
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.08211	+	1.87426i	0.0359308	+	0.0622339i	0.883432	−	0.468560i	$-0.155227\pi$
$907$	1.08211	+	1.87426i	0.0359308	+	0.0622339i	−0.847501	+	0.530794i	$0.821894\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	19.2332	+	33.3129i	0.637226	+	1.10371i	0.986039	+	0.166515i	$0.0532513\pi$
$911$	19.2332	+	33.3129i	0.637226	+	1.10371i	−0.348813	+	0.937192i	$0.613415\pi$
$912$		0			0
$913$	−19.7507	+	34.2093i	−0.653654	+	1.13216i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	36.6979			1.21187
$918$		0			0
$919$	18.6992			0.616830			0.308415	−	0.951252i	$-0.400201\pi$
$919$	18.6992			0.616830			0.308415	+	0.951252i	$0.400201\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	7.74693	−	13.4181i	0.254993	−	0.441661i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.10335	−	1.91106i	−0.0361999	−	0.0627000i	0.847358	−	0.531022i	$-0.178192\pi$
$929$	−1.10335	−	1.91106i	−0.0361999	−	0.0627000i	−0.883558	+	0.468322i	$0.844859\pi$
$930$		0			0
$931$	14.5330	−	25.1719i	0.476300	−	0.824976i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	21.3429			0.697242			0.348621	−	0.937264i	$-0.386650\pi$
$937$	21.3429			0.697242			0.348621	+	0.937264i	$0.386650\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.589661	+	1.02132i	−0.0192224	+	0.0332942i	−0.875477	−	0.483261i	$-0.839452\pi$
$941$	−0.589661	+	1.02132i	−0.0192224	+	0.0332942i	0.856254	+	0.516555i	$0.172786\pi$
$942$		0			0
$943$	−0.468813	−	0.812008i	−0.0152667	−	0.0264426i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	11.8445	+	20.5152i	0.384894	+	0.666655i	0.991754	−	0.128153i	$-0.0409048\pi$
$947$	11.8445	+	20.5152i	0.384894	+	0.666655i	−0.606861	+	0.794808i	$0.707572\pi$
$948$		0			0
$949$	5.59166	−	9.68504i	0.181513	−	0.314390i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−12.8125			−0.415038			−0.207519	−	0.978231i	$-0.566539\pi$
$953$	−12.8125			−0.415038			−0.207519	+	0.978231i	$0.566539\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−22.9784	+	39.7998i	−0.742012	+	1.28520i
$960$		0			0
$961$	10.8084	+	18.7207i	0.348658	+	0.603893i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−5.46326	+	9.46264i	−0.175686	+	0.304298i	−0.940399	−	0.340074i	$-0.889548\pi$
$967$	−5.46326	+	9.46264i	−0.175686	+	0.304298i	0.764712	+	0.644372i	$0.222881\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−15.7933			−0.506831			−0.253415	−	0.967358i	$-0.581554\pi$
$971$	−15.7933			−0.506831			−0.253415	+	0.967358i	$0.581554\pi$
$972$		0			0
$973$	49.7001			1.59331
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−18.7490	+	32.4742i	−0.599833	+	1.03894i	0.393012	+	0.919533i	$0.371433\pi$
$977$	−18.7490	+	32.4742i	−0.599833	+	1.03894i	−0.992845	+	0.119409i	$0.961900\pi$
$978$		0			0
$979$	8.83826	+	15.3083i	0.282472	+	0.489256i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	11.7661	+	20.3795i	0.375280	+	0.650004i	0.990369	−	0.138454i	$-0.0442132\pi$
$983$	11.7661	+	20.3795i	0.375280	+	0.650004i	−0.615089	+	0.788458i	$0.710880\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	0.768410			0.0244340
$990$		0			0
$991$	46.7019			1.48353			0.741767	−	0.670658i	$-0.233988\pi$
$991$	46.7019			1.48353			0.741767	+	0.670658i	$0.233988\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−21.4290	−	37.1161i	−0.678663	−	1.17548i	−0.975384	−	0.220514i	$-0.929226\pi$
$997$	−21.4290	−	37.1161i	−0.678663	−	1.17548i	0.296721	−	0.954964i	$-0.404107\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.2.i.e.901.1		8
3.2	odd	2		900.2.i.d.301.1	✓	8
5.2	odd	4		2700.2.s.d.1549.7		16
5.3	odd	4		2700.2.s.d.1549.2		16
5.4	even	2		2700.2.i.d.901.4		8
9.2	odd	6		900.2.i.d.601.1	yes	8
9.4	even	3		8100.2.a.y.1.4		4
9.5	odd	6		8100.2.a.x.1.4		4
9.7	even	3	inner	2700.2.i.e.1801.1		8
15.2	even	4		900.2.s.d.49.5		16
15.8	even	4		900.2.s.d.49.4		16
15.14	odd	2		900.2.i.e.301.4	yes	8
45.2	even	12		900.2.s.d.349.4		16
45.4	even	6		8100.2.a.ba.1.1		4
45.7	odd	12		2700.2.s.d.2449.2		16
45.13	odd	12		8100.2.d.s.649.2		8
45.14	odd	6		8100.2.a.z.1.1		4
45.22	odd	12		8100.2.d.s.649.7		8
45.23	even	12		8100.2.d.q.649.2		8
45.29	odd	6		900.2.i.e.601.4	yes	8
45.32	even	12		8100.2.d.q.649.7		8
45.34	even	6		2700.2.i.d.1801.4		8
45.38	even	12		900.2.s.d.349.5		16
45.43	odd	12		2700.2.s.d.2449.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.2.i.d.301.1	✓	8	3.2	odd	2
900.2.i.d.601.1	yes	8	9.2	odd	6
900.2.i.e.301.4	yes	8	15.14	odd	2
900.2.i.e.601.4	yes	8	45.29	odd	6
900.2.s.d.49.4		16	15.8	even	4
900.2.s.d.49.5		16	15.2	even	4
900.2.s.d.349.4		16	45.2	even	12
900.2.s.d.349.5		16	45.38	even	12
2700.2.i.d.901.4		8	5.4	even	2
2700.2.i.d.1801.4		8	45.34	even	6
2700.2.i.e.901.1		8	1.1	even	1	trivial
2700.2.i.e.1801.1		8	9.7	even	3	inner
2700.2.s.d.1549.2		16	5.3	odd	4
2700.2.s.d.1549.7		16	5.2	odd	4
2700.2.s.d.2449.2		16	45.7	odd	12
2700.2.s.d.2449.7		16	45.43	odd	12
8100.2.a.x.1.4		4	9.5	odd	6
8100.2.a.y.1.4		4	9.4	even	3
8100.2.a.z.1.1		4	45.14	odd	6
8100.2.a.ba.1.1		4	45.4	even	6
8100.2.d.q.649.2		8	45.23	even	12
8100.2.d.q.649.7		8	45.32	even	12
8100.2.d.s.649.2		8	45.13	odd	12
8100.2.d.s.649.7		8	45.22	odd	12

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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(-1.70089 - 2.94604i) q^{7} +(-2.20089 - 3.81206i) q^{11} +(1.03160 - 1.78679i) q^{13} -1.40179 q^{17} -6.35717 q^{19} +(-0.0539129 + 0.0933799i) q^{23} +(4.54089 + 7.86505i) q^{29} +(-1.53160 + 2.65281i) q^{31} -1.95538 q^{37} +(-4.34788 + 7.53074i) q^{41} +(-3.56320 - 6.17165i) q^{43} +(3.74179 + 6.48096i) q^{47} +(-2.28608 + 3.95961i) q^{49} +13.0975 q^{53} +(-6.58966 + 11.4136i) q^{59} +(0.862308 + 1.49356i) q^{61} +(-6.18787 + 10.7177i) q^{67} +7.50961 q^{71} +5.42037 q^{73} +(-7.48698 + 12.9678i) q^{77} +(-4.71806 - 8.17193i) q^{79} +(-4.48698 - 7.77167i) q^{83} -4.01576 q^{89} -7.01858 q^{91} +(1.08551 + 1.88017i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{7} - 3 q^{11} - 2 q^{13} + 18 q^{17} - 8 q^{19} + 3 q^{23} + 9 q^{29} - 2 q^{31} - 2 q^{37} - 9 q^{41} - 8 q^{43} - 12 q^{47} - 9 q^{49} + 24 q^{53} + 15 q^{59} + q^{61} - 11 q^{67} + 24 q^{71}+ \cdots - 5 q^{97}+O(q^{100}) \) 8 * q + q^7 - 3 * q^11 - 2 * q^13 + 18 * q^17 - 8 * q^19 + 3 * q^23 + 9 * q^29 - 2 * q^31 - 2 * q^37 - 9 * q^41 - 8 * q^43 - 12 * q^47 - 9 * q^49 + 24 * q^53 + 15 * q^59 + q^61 - 11 * q^67 + 24 * q^71 - 20 * q^73 - 36 * q^77 + 7 * q^79 - 12 * q^83 - 6 * q^89 - 22 * q^91 - 5 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more